doctor of philosophy - shodhgangashodhganga.inflibnet.ac.in/bitstream/10603/26741/1/final...bhabi...

“Some Relativistic Phenomenain Astrophysics”

THE THESIS SUBMITTED TO

KUMAUN UNIVERSITY, NAINITALFOR THE AWARD OF DEGREE OF

DOCTOR OF PHILOSOPHY

IN

(PHYSICS)BY

PRATIBHA FULORIAUNDER THE SUPERVISION

OF

Supervisor Co-supervisor

Prof. B. C. Joshi Dr. B. C. TewariDepartment OF PHYSICS Department of Mathematics

Kumaun University Kumaun UniversityS. S. J. Campus, Almora S. S. J. Campus, Almora

Uttarakhand Uttarakhand

2012

Acknowledgements

It is a great honour to express my deep sense of gratitude and indebtedness to my

supervisor Prof. B. C. Joshi, Department of Physics, S. S. J. Campus Almora , Kumaun

University , Nainital who gave me an opportunity to work under his kind guidance-ship. I

sincerely thank Dr. B. C. Tiwari, Department of Mathematics, S. S. J. Campus Almora for his

valuable help and direct involvement in the study. His guidance helped me in all the time of

research and encouraged me to complete my thesis work.

I am profoundly indebted to Prof. M. C. Durgapal , Head of Physics Department, S.

S. J. campus Almora, Kumaun University, Nainital for his continuous support and

precious guidance during the progress of my work. I take this opportunity to express my

sincere gratitude and appreciation to Prof. K. L. Sah (Retd.), Prof. O. P. S. Negi, Dr. P. S.

Bisht for their time to time cooperation, invaluable advices and help rendered to me during

the entire period of this work. I now avail this opportunity to extend my grateful thanks to

Pushpa, Gaurav, Bhupesh, Pawan, chauhan, vishal for all possible support rendered to me

during this period.

I also acknowledge time to time cooperation from the members of non - teaching staff

of Physics Department, Soban Singh Jeena Campus, Almora Bhuwan Negi ji, R. S. Rayal ji,

J. C. Upadhayay ji, Heera Singh Kharayat ji, Pramod Nailwal ji, Rajendra Singh Rana ji ,

Bheem Singh ji and Dan Singh ji. In particular, I am grateful to Dr. Santosh, Dr. Ashok for

providing me the research papers and books whenever I was in the urgent need of them

I wish to express my profound gratitude to Late Prof. Mahesh Chandra Durgapal for

enlightening me the first glance of research. His motivation, enthusiasm, and immense

knowledge paved the way for the completion of this difficult task.

iv

I also take this opportunity to express my gratitude to Prof. Kavita Pandey, formerly

Head of Physics Department, Kumaun University, D.S. B. Campus, Nainital for her

inspiration and moral support given to me in many ways. My sincere thanks also go to Dr

Neeraj Pant (Associate Professor) Maths Department, N.D.A., Khadakwasla, Pune for his

stimulating discussions and constructive suggestions during the progress of this work. His

time to time discussions of research problems motivated me to do more and more work. I am

also thankful to Dr. P. S. Negi, Physics Department, D. S. B. Campus, Nainital, Kumaun

University for his guidance and support given to me during this period.

I am deeply indebted to my husband Dr. C. P. Fuloria for his indistinctly help

persistent encouragement and affectionate cooperation, which made it possible to

complete this difficult task. I can not forget the affection and love of my child Dear

Chetan which always enforced me to complete my work without any tension.

It is a pleasure to convey my deep sense of gratitude to my brother Sri Rajesh Joshi,

Bhabi Smt. Anandi Joshi for perpetual inspiration, motivation and invaluable help

given to me whenever I was in the difficult moments. I now avail this opportunity to

express my thanks to my elder sisters Pushpa didi, Geeta didi for their great affection,

invaluable advices and every possible support given to me during the entire course of

this study. My heartfelt thanks are to my Jijaji Mr B.C. Sharma and Dr. Suresh

Mathpal for providing me unflinching encouragement and support in various ways. I

am thankful to Dear Ashu, Gaurav , Udita, Prachi , Kanha , Alok, Abhishek, Yogesh,

Anuj, Champa, Prashant and all my nieces for always creating peaceful and

conducive atmosphere while doing my research work. In the midst of all their activity,

I always felt more full of strength and hope.

I am really very thankful and deeply indebted to my dear mother Mrs. Radha

joshi who always inspired me to do something very special in the academic field and

did’nt involve me too much in the household activities leaving a lot of time for the

enhancement of my knowledge in the related field.

v

My warmest thanks are to my mother in law Mrs. Hansa Devi for the continuous

support and affection given to me during this period. I gratefully acknowledge the constant

inspiration and loving support of my all in laws, which always enforced me to complete my

work with great zeal. Last but not the least the whole credit for my venturing into higher

education goes to my father Late Sri J. C. Joshi who had always inspired me to think high in

the life and had motivated me in very difficult moments for not losing patience .

Above all, I owe a great debt of gratitude to Great Almighty, whose presence and

power was felt in every moment during the completion of this arduous task. The thought of

his presence always enlightened a ray of hope in me and gave me more strength during this

period.

Pratibha Fuloria

vi

Preface

The present thesis entitled “Some Relativistic Phenomena in Astrophysics”

comprises the investigations carried by myself over the period of three and half years

under the supervision of Prof. B. C. Joshi, Department of Physics, Kumaun university

S. S. J. Campus, Almora and co supervision of Dr. B. C. Tiwari, Department of

Mathematics, Kumaun University S. S. J. Campus, Almora. The present manuscript

embodies the investigations towards the study of various astrophysical objects

and their characteristics.

The study of massive fluid spheres can be most successfully done within the

framework of general relativity. The general theory of relativity was propounded by

Einstein in 1915-1920, establishing a landmark by opening the doors for the

theoreticians to get deep insight into some untouched problems. Einstein invoked the

principle of Equivalence for examing the physical significance of his General theory

of relativity. The practical importance of the principle of Equivalence lies in the fact

that it enables us to apply the results of development of physical events in an

accelerated system of reference to phenomena taking place in a homogeneous

gravitational field.

Some interesting astronomical exotica viz. neutron star, pulsar, Quark star,

Quasar, Black Hole can be studied with in the framework of general relativity and

provide a best working ground for the testing of Einstein’s General theory of

relativity. The existence of neutron star and black Holes was suggested in the 1930’s

on purely theoretical grounds chiefly through the work of J. Robert Oppenheimer and

his collaborators. The discovery of pulsars confirmed the existence of neutron stars

and advances have been made to reveal the internal structures of these objects. The

neutron star consists not only of neutron fluid as envisaged originally by Landau but

also a large spectrum of densities and various regions comprising different

elementary particles.

vii

To get a deep insight into the internal structure of various compacts objects the

exact solutions of Einstein’s field equations play very important role. We have made

an attempt to find some new solutions of Einstein’s field equations and Einstein’s

Maxwell field equations in this monograph. These solutions have been used for

constructing the approximate models of immense gravity objects. The problems dealt

with in the present work are the models of compact objects like neutron star, pulsar,

white Dwarf based on exact solutions of Einstein’s field equations for the perfect

fluid spheres.

Neutral interior solutions of Einstein field equations are normally found very

useful for modeling of stellar objects. We have also tried to present the models of

stellar objects, which are close to reality and relevant in nature. Further, the problem

is augmented with charge matter which is more close to the reality of astrophysical

scenario. The compact objects like neutron star, white Dwarf, Quark Star can be

better understood by including charge in them. Despite a large amount of work done

on the neutron star and its equation of state a new venture for the deep understanding

of its internal structure is always desired.

The quasars with very high red shifts in their spectrum and with total

luminosity hundred times greater than that of giant galaxies are extremely unusual in

their properties. Though a plethora of models exist to explain these astrophysical

objects yet the various phenomena associated with them are still far from properly

understood. Radiating fluid spheres have been found to be useful for modeling of

Quasars. Radiating fluid spheres also under gravitational collapse while emitting

radiation in the form of neutrinos and photons. Gravitational collapse is an important

phenomena associated with most of the stellar objects. It is responsible for all the

structure formation in the universe. During Gravitational Collapse the physical

conditions within the star get changed and needs to be investigated deeply.

viii

Any collapsing stellar object may end either into a black hole or into a naked

singularity. Various scenarios of gravitational collapse have been considered which

admit the possibility of naked singularity. Although Cosmic Censorship Conjecture

says that a naked singularity cannot arise in our universe from realistic initial

conditions. Various models of radiating stellar objects have been discovered in which

horizon is never encountered.

The non-static solutions of Einstein’s field equations are found to be useful for

the study of radiating fluid distributions. Radiating fluid distribution may be used for

constructing the approximate models of Quasars. All these problems involve the

solutions of Einstein’s field equations in different coordinate systems.

The whole work is divided into six chapters.

Chapter I : This chapter deals with the general introduction of overall work

undertaken in the present study. It introduces the problems associated with the work

done in the thesis. The mathematical formulation pertaining to achieve the objectives

carried out in the thesis has also been discussed. This chapter also contains the brief

summary of the entire work done in the thesis.

Chapter II: This chapter introduces with some new exact solutions of Einstein’s

field equations. The solutions have been examined to be physically realizable and

their various properties have been discussed. Based on these new solutions we have

done the mathematical modeling of stellar objects, some of which hold close to

reality. By assuming the surface density 314102 cmg the models of neutron star

have been constructed.

Chapter III: In this chapter a charging concept of Durgapal’s fifth solution has

been developed with the suitable choice of electric intensity function.

ix

We have obtained a variety of new classes of exact solutions of Einstein-

Maxwell field equations which are well behaved and regular. Keeping in view of

well behaved nature of these solutions the models of super massive stars with

charged and perfect fluid matter have been constructed. The properties of

charged fluid spheres have been also discussed extensively.

Chapter IV: This chapter includes the study of a known non-static solution of

Einstein’s field equations. We have studied the BCT solution II in great detail

exposing its importance for constructing the radiating fluid ball models. We have

constructed the approximate models of Quasars for different combinations of the

constants X, Y and Z appearing in the solution. The variation of different physical

parameters within the radiating fluid sphere has been discussed. One of the most

important parameter of all these models is the mass –radius gradient that determines

whether the collapse will be horizon free or horizon will be formed during the

collapse. In horizon free collapse, collapse will keep on going and left over core will

be a black hole of point dimension (naked singularity).

Chapter V: The fifth chapter describes the adiabatic collapse of uniform density

sphere with pressure. Adiabatic Collapse solution of uniform density spheres have

been known for about three decades. An analysis of these solutions has been done by

considering the baryonic conservation law and the no heat transfer condition. It has

been shown that if the fluid is Isentropic or the surface temperature remains constant

during the collapse the pressure can not remain finite (it vanishes). We can say that

when the exterior geometry is defined by Schwarz schild vaccum solution then the

solution given by Oppenheimer is the only valid solution.

Chapter VI: In this chapter we have obtained a new time dependent solution of

Einstein’s field equations and have discussed its properties. We have also

investigated the physical viability of a known non-static solution in conformally flat

space-time metric.

x

Keeping in view the well behaved conditions we have shown that the

solution can be used for modeling of radiating astrophysical objects.While

undergoing collapse the stellar object also emits radiation and no horizon is formed.

We have used geometrical units popular in general relativity only in section A

of chapter I, while in rest part of this manuscript we have used conventional units.

This makes it easier to understand the physics of various astrophysical objects,

although it becomes little difficult to formulate the problem mathematically.

xi

Contents

CERTIFICATE i-ii

Declaration iii

ACKNOWLEDGEMENT iv

PREFACE vii

CHAPTER I………………………………………… 1- 47

General Introduction

1.1 Introduction

1.2 Stellar evolution

1.3 Some Astrophysical objects & their properties:

1.3.1 White Dwarf

1.3.2 Neutron star

1.3.1 Quasar

1.4 How a Neutron star is formed

1.5 The maximum mass limit for neutron star

1.6 Neutron star as pulsar

1.7 The equation of state for neutron star

1.8 The Coordinate Systems used in the present investigations:

1.9 Einstein’s Field Equations and their importance

1.10 Field Equations in isotropic coordinates

1.11 Local maxima and local minima at the centre

1.12 Darmois Conditions (Junction conditions in isotropic co-ordinateSystem):

1.13 Exact Solutions of Einstein’s field equations

1.14 Charged fluid spheres in General Relativity

1.15 Einstein’s –Maxwell equation for charged fluid distribution

1.16 Mathematical formulation of red shift

1.17 Radiating Fluid Distribution & gravitational Collapse

1.18 Hydrodynamics of the radiating fluid sphere

1.19 Video Metric and its derivation

1.20 Junction conditions

1.21 Gravitational Collapse

1.22 Naked Singularity & Cosmic Censor Hypothesis

1.23 Objective of the thesis

1.24 References

CHAPTER II……………..…………………………….. 48 -81

New solutions of Einstein’s field Equations for static perfectfluid matter.

Section A

Solution I: A non-singular solution with infinite central density

2.1 Introduction

2.2 Einstein’s Field Equations and their solutions

2.3 Boundary Conditions

2.4 Results and Discussions

Section B

Solution II : A NEW WELL BEHAVED EXACT SOLUTION IN ISOTROPICCOORDINATE SYSTEM FOR PERFECT FLUID.

2.5 Introduction

2.6 Conditions for well behaved solution

2.7 Field equations in isotropic coordinates

2.8 New class of solution

2.9 Properties of the new solution

2.10 Boundary conditions

2.11 Slowly rotating structures (Crab and the Vela Pulsars)


2.13 References

CHAPTER III ……………………………………...82 -122

A Parametric class of Regular and well behaved relativisticcharged fluid spheres:

3.1. Introduction

3.2. The solutions that are used as seed solutions for making charged fluidModel

3.3. Assumptions that must be satisfied in order for the solution to be wellBehaved

3.4. Einstein – Maxwell equations for charged fluid distribution.

3.5. A new Generalised solution of Einstein - Maxwell field equations

3.6. Properties of the new generalised solution

3.7 Variety Of classes of solutions

3.7.1 Case 1 (n = 0) The solution for n = 0

3.7.2 Case 2 (n = 1) The solution for n = 1

3.8 Properties of the new solution for n = 1


3.10 New well behaved solution (n = 2)



3.13 New well behaved solution (n = 3)



3.15 a. Modeling of superdense star for n = 1

3.15 b. Modeling of superdense star for n = 2

3.15 c. Modeling of super dense star for n = 3

3.16 References

CHAPTER IV ……………………………………..123-154

Radiating fluid ball models with horizon free

Gravitational collapse

4.1. Introduction

4.2. Conditions for solution to be physically realizable

4.3. Junction conditions and solution of the field equations

4.4. Different cases of BCT solution II for Quasar Model

4.4.1. Case I (B = 1, C = 1, D = 2)

4.4.2. Case II (B = 1, C = 1. D = 3)

4.4.3. Case III (B = 1, C = 1, D = 4)

4.4.4 Case IV (B = 1, C = 1, D = 5)

4.4.5 Case V (B = 1, C = 1, D = 6)

4.4.6 Case VI (B = 1, C = 1.5, D = 2)

4.4.7 Case VII (B = 1, C = 2, D = 2)


4.6 References

CHAPTER V ………………….………………………….. 155-165

Adiabatic Collapse of Uniform Density Sphere with Pressure

5.1 Introduction

5.2 The metric and uniform density sphere

5.3 The boundary condition and thermodynamic relation.

5.4 Collapse of uniform density sphere

5.4 (a) using NHT condition (eq. 5.6a)

5.4 (b) Isentropic case:

5.4 (c) Non-isentropic case with constant surface temperatures

5.4 (d) General case

5.5 Explanation of inconsistency

5.5 (a) using NHT condition ( eq. 5.6b)

5.5 (b) using NHT condition (eq. 5.6c)

5.6. Results & Discussions

5.7 References

CHAPTER VI………………………………………………............. 166-183

A new time dependent solution of Einstein’s field equations andradiating fluid spheres in conformally flat space-time.

6.1 Introduction

6.2 The metric and Field Equations

6.3 New solution of the field equations

6.4 Properties of the New solution :

6.5 Field Equations of a radiating fluid ball in conformably flat space-time

6.6 Boundary conditions for radiating fluid ball in conformably flat space-time

6.7 Different models of radiating fluid spheres

6.8 The variation of pressure and density with time

6.9 Results and discussions

6.10 References

List of Publication……………………………………................. 184

CHAPTER IGENERAL INTRODUCTION

1

Chapter IGENERAL INTRODUCTION

This chapter deals with a brief introduction of the overall work undertaken in the

present study . It contains a general review of the exact solutions of Einstein field equations

obtained so far . A detailed formulation of static fluid ball problems in general relativity has

been presented in canonical and isotropic coordinates. The laws governing the geometry of

radiating fluid spheres have been formulated. A brief introduction of some stellar objects viz.

White Dwarf, Neutron Star, Quasar has been also given. This chapter also contains the brief

summary of the entire work done in the thesis.

2

1.1 Introduction

General theory of relativity is a theory of Gravitation where one

recognises the power of geometry in describing the physics. The pivotal point

in the theory is that a gravitational field implies in the background of space-

time and conversely, a curve space time satisfying the laws of general relativity

indicates the possible existence of an intrinsically associated real gravitational

field . General theory of relativity and Newton's gravitational theory make

essentially identical predictions as long as the strength of the gravitational field

is weak. However for the case of strong gravitational field both differ in

their predictions. Further Einstein’s General Theory of relativity successfully

explains the phenomena taking place in the presence of strong gravitational

field. Thus Einstein’s theory of relativity has important astrophysical

implications. For the mathematical formulation of this theory Einstein’ field

equations play very important role. The explanation of the anomalous

precession of the perihelion of Mercury emerged naturally from the Einstein’s

theory of General relativity. The deflection of light rays in the gravitational

field of massive stars, the gravitational red shift, gravitational time delay etc.

can be most successfully explained by General theory of relativity.

The predictions of Einstein’s theory of relativity have been confirmed in

various observations and experiments done so far.General relativity is the

relativistic theory of gravity that is consistent with experimental data.Under the

normal conditions the general relativistic effects are very small and extremely

difficult to detect.In the neighbourhood of an object of mass M and radius R

general relativistic effects are of the order of ,2cR

GMG being the Gravitational

constant , c the speed of light. The ratio is equal to ~ 10-6 in the case of sun,

hence it is very difficult to detect these effects. For the massive and compact

objects for which 1~2cR

GMthe general relativistic effects can be easily detected.

Neutron star, White Dwarf, Black Hole are very compact objects in which the

relativistic effects come into existence and can not be ignored. The dominant

role of gravitation and general relativity became very much evident with the

3

discovery of pulsars and their identification as fast rotating neutron stars. The

identification of pulsars as neutron star, the large red shift of quasars, the high

energy generation in quasars have rendered the study of relativistic structures

in astrophysics important. The existence of all these astronomical exotica and

their properties may be successfully explained within the realm of General

Theory of Relativity. Einstein proposed that matter produces curvature in

space-time, and that free-falling objects move along locally straight paths in

curved space-time called geodesics. The space time curvature is expressed in

terms of metric tensor which is linked with the source mass or the stress energy

tensor. Einstein’s field equations of general relativity, which relate the presence

of matter and the curvature of space- time are of vital importance in the

present context. The astrophysical objects in which the amount of radiation

emitted is very large may be also studied within the frame work of General

relativity by using Vaidya type solutions of Einstein’s field equations. Non

static solutions of Einstein’s field equations are very important while

discussing gravitational collapse , very high energy events like quasars,

and supernova bursts .

1.2 Stellar Evolution:

As we are studying the massive fluid spheres in the relativistic range, a

brief introduction of stellar evolution is important in this context. Various

compact states of a star will be formed at different stages of the stellar evolution. At

every layer within a stable star, there must be balance between the inward pull of

gravitation and the gas pressure. Any stellar structure is in equilibrium under the

influence of two forces:

(a) The gravitational force,

(b) The pressure of gas and radiation

The equation of hydrostatic equilibrium is given by

2

)()(

r

rrGM

dr

dp (1.1)

Where )(rM is the mass interior to radius r and )(r is the density at r. prepresents

the total pressure due to both gas and radiation.

4

For a sphere of constant density 3

3

4)( rrM (1.2)

Thus with in any given layer of a star there must be hydrostatic equilibrium

between the outward pressure due to both gas and radiation from below and the

weight of the material above pressing inward. Whenever one force dominates the

other due to some reasons the equilibrium of the star gets disturbed. A star needs a

source of energy at the centre to compensate for the radiation loss from its surface

and to maintain a high temperature necessary to provide a pressure which balances

the gravitational force.

The formation of a star begins with gravitational instability within a molecular

cloud . As the cloud collapses the gravitational energy is converted into heat and its

central temperature rises. When the temperature rises sufficiently hydrogen burning

sets in at the core of the star generating sufficient radiation pressure to stop the

contraction. Gradually hydrogen is converted into helium at the core and star again

begins to contract. Hydrogen burning becomes restricted in a shell-layer

surrounding the core. Eventually the core is compressed enough to start helium

fusion . Depletion of helium at the core gives rise to a carbon core. The core

contracts until the temperature and pressure are sufficient to fuse carbon. This

process continues, with the successive stages being fueled by neon, oxygen and

silicon . Near the end of the star's life, fusion can occur along a series of onion-layer

shells within the star. Each shell fuses a different element, with the outermost shell

fusing hydrogen, the next shell fusing helium, and so forth.

The final stage is reached when the star begins producing iron. The iron is the

element having highest binding energy and no energy will be released due to its

ignition and the star will continue to collapse until some pressure is generated at the

centre to counterbalance the collapse. Fig 1.1 shows the position of different

elements during stellar evolution.

5

Fig 1.1: The onion like layers of a massive evolved star just before

core collapse.

( Image taken from http://en.wikipedia.org/stellar_evolution.)

Thus with the exhaustion of all nuclear fuel the star meets the fate of death and

becomes according to its initial mass white dwarf, neutron star, and black hole. Due

to gradual contraction of the star the density at the centre of the star will go on

increasing till a state of density is reached when electron degeneracy pressure

generates within the stellar object. This electron degeneracy pressure is what

supports a white dwarf against gravitational collapse

If the mass of the stellar object is less than the Chandrasekhar limit (1.44 MΘ )

[1] the dead state of the star will be white star. If the mass of the star exceeds the

Chandrasekhar limit the gravitational contraction can not be counter balanced by

electron degeneracy pressure and consequently the star continues to contract until

some new pressure develops at the centre of the stellar object to counter balance the

gravitational contraction. The neutron degeneracy pressure will bring the stellar

object again into a new equilibrium state known as neutron star. If this pressure also

fails in preventing the gravitation contraction then the contraction will continue

forever and no force in the universe can prevent the collapse to a point singularity

and the concept of black hole comes into picture.

5


core collapse.


















5


core collapse.


















6

Table 1.1 : The possible progenitor mass , Remnant mass, size, density and

means of support for different dead states of a star.

1.3 Some Astrophysical objects & Their properties:

1.3.1 White Dwarf

A white dwarf, is a small star composed mostly of electron-degenerate

matter. A white dwarf ’s mass is comparable to that of the Sun and its radius is

comparable to that of the Earth hence they are very dense.The average density of

White Dwarf lies in the range 105- 106 gm cm-3[2]. Its faint luminosity comes from

the emission of stored thermal energy. For stellar masses less than about 1.44 solar

masses, the energy from the gravitational collapse is not sufficient to produce the

neutrons of a neutron star, so the collapse is halted by electron degeneracy to form

white dwarfs. If electron degeneracy pressure is not able to balance the force of

gravity, then it would collapse into a denser object such as neutron star. This

maximum mass for a white dwarf is called the Chandrasekhar limit. As the star

contracts, all the lowest electron energy levels are filled and the electrons are forced

into higher and higher energy levels, filling the lowest unoccupied energy levels.

This creates an effective pressure called electron degeneracy pressure. This electron

End points of stellar evolution

Deadstates ofstars

Progenitormass

RemnantMass

Size Density Means ofsupport

FinalStage

WhiteDwarf

M* ≈ 8MΘ MW.D.<1.4MΘ

RW.D.~Rearth

105gm/cm3e- degeneracy

Planetarynebula

Neutronstar

8 MΘ <M*>20MΘ MN.S.<3MΘ RN.S.~10k

m1014gm/cm3

n degeneracySupernova

BlackHole

M* > 20MΘ MB.H.>3MΘ Rgrav =2GM/c2

∞ None ?

7

degeneracy pressure is what supports a white dwarf against gravitational collapse.

When a medium sized star nears the end of its life and has used up all of its

available hydrogen, it will slowly expand into a red giant which fuses helium into

carbon and oxygen. Once this process has completed, the star will throw off its outer

layers to form a planetary nebula. The core that remains will be a white dwarf

composed of carbon and oxygen nuclei compressed by gravity and stripped of their

electrons. This extremely dense matter makes up a stellar remnant (white dwarf).

Due to its very high density the classical equation of state for a perfect gas do not

apply for white Dwarf , instead the pressure is given by the equation of state for

degenerate matter that is dense , cold matter [3].

1.3.2 Neutron star:

A neutron star is about 20 km in diameter and has mass of about 1.4 times that

of our sun. Because of its small size and high density, a neutron star possesses a

surface gravitational field about 2 × 1011 times that of the earth. A NS is made up of

cold catalysed matter i.e. matter which has reached the end point of stellar evolution.

The magnetic field of the neutron star is of the order of ~ 1012 Gauss.

The temperature inside a newly formed neutron star is from around 1011 to 1012

Kelvin. However, the huge number of neutrinos it emits carry away so much

energy that the temperature falls within a few years to around 1 million Kelvin.

Neutron stars are known to have rotation periods between about 1.4 ms to 30

seconds. A neutron star's structure is very simple, and it has three main layers: A

solid core, a liquid mantle, and a thin, solid crust. Neutron stars also have a very tiny

(a few centimeters - about an inch) atmosphere, but this is not very important in the

functioning of the star. An approximation of neutron star dimension [4,5] is that it

has thick metallic surface layer, below which there is about 1 km thick solid layer of

material of density 105-1014 gm cm-3. Under this solid crust comes the main part of

the star which consists of a nuclear superfluid with a density of about nuclear

density. This superfluid core of NS comprises most of the mass of the star and

extends upto several km and holds itself up by neutron degeneracy pressure against

gravitational collapse. In the very centre of the star there is the possibility of the

existence of some

8

exotic matter. There may be pion condensate and probably a neutron solid and

hyperons.

1.3.3 Quasar (Quasi Stellar Object) :

Quasars are the most luminous, powerful, and energetic objects known in the

universe. They are the most distant known objects in the universe. The quasars ,

with total luminosity hundred times greater than that of giant galaxies, are

extremely unusual in their properties. The most luminous quasars radiate at a rate

that can exceed the output of average galaxies. Quasars have large red shifts

relative to normal stars and galaxies [6,7]. The accretion of material into super

massive black holes in the nuclei of distant galaxies is believed to be one of the

main cause of energy content of Quasar. All observed quasar spectra have redshifts

between 0.065 and 5.46 [8] .

If the lines in the spectrum of the light from a star or galaxy appear at a lower

frequency , the object exhibits positive red shift. The accepted explanation for this

effect is that the object must be moving away from us. Cosmological red shift is

seen due to the expansion of the universe. Gravitational red shift is a relativistic

effect observed in electromagnetic radiation of very compact objects. Quasars are

very high red shift objects indicating that they are very far away from us. The

brightest quasar is 3C273, two billion light years away from us . The red shift of

3C 273 is z = 0.158[9], meaning that the wavelengths of its spectral lines are

stretched by 15.8%. Models of Quasars have been proposed by Durgapal and

Gehlot and it has been shown that the maximum surface red-shift can be as large as

4.828[10] . We have also tried to construct the approximate models of Quasars in

chapter IV. The red shift has been also obtained for these models.

1.4 HOW A NEUTRON STAR IS FORMED :

Neutron stars are the product of supernovae, gravitational collapse events in

which the core of a massive star reaches nuclear densities and stabilizes against

further collapse. Neutron stars are one of the possible dead states for a star. They

result from massive stars which have mass greater than 4 to 8 times that of our Sun.

After these stars have finished burning their nuclear fuel, they undergo a supernova

9

explosion. This explosion blows off the outer layers of a star into a supernova

remnant. The central region of the star collapses under gravity. It collapses so much

that protons and electrons combine to form neutrons through the reaction .

enep (1.3)

The neutrinos escape the star. Enough electrons and protons must remain so that the

Pauli principle prevents neutron beta decay,

eepn (1.4)

The condition for the neutrons to be stable against beta decay is that the electron

Fermi sea should be filled up to a momentum greater than the maximum momentum

kmax of the electron emitted in neutron beta decay[11].

max, kk eF (1.5)

The neutron degeneracy pressure within the star counter balances the

gravitational collapse paving the way for the formation of neutron star. Neutron star

contains matter in one of the densest forms found in the universe. The pressure in the

star’s core is so high that most of the charged particles, electrons and protons, merge

resulting in a star composed mostly of uncharged particles called neutrons. The

central density of neutron star ranges from a few times the density of normal

nuclear matter to about one order of magnitude higher, depending on the star’s mass

and the equation of state. Neutron stars therefore provide us with a powerful tool for

exploring the properties of such dense matter. The discovery of pulsars by Hewish et

al. [12] have confirmed that they can be neutron stars only. Oppenheimer and

Volkoff [13] were the first to obtain the mass of neutron star within the framework

of general relativity. Oppenheimer and Volkoff concluded that the maximum mass

for neutron stars is 0.7 MΘ. Oppenheimer and Volkoff realized that for a neutron star

the modification of Newtonian gravity due to general relativity would have to be

taken into account. Although Oppenheimer and Volkoff had properly taken into

account the gravitational effects, they had assumed that the neutrons could be

regarded as an ideal fermi gas. The assumption of an ideal gas is quite valid for

electrons, but is a very poor assumption for neutrons. A gas may be regarded as an

ideal if the energy of interaction between the particles can be neglected. For more

realistic modeling of neutron star nucleon-nucleon interaction must be taken into

account.

10

1.5 The maximum mass limit for Neutron star:

The maximum mass of neutron star is a quantity of great importance to study

the final stages of stellar evolution. The upper mass limit is needed to distinguish a

neutron star from a black hole. The Tolman–Oppenheimer –Volkoff limit is an

upper bound to the mass of stars composed of neutron-degenerate matter i.e. neutron

star. The TOV limit is analogous to the Chandrasekhar limit for white dwarf

stars. This limit was obtained by J. Robert Oppenheimer and George Volkoff

in 1939, using the work of Tolman. Oppenheimer and Volkoff assumed that the

neutrons in a neutron star formed a degenerate cold Fermi gas. They obtained a

mass of approximately 0.7 solar masses at a radius of 9.6 km[11]. A black hole

formed by the collapse of an individual star must have mass exceeding the Tolman–

Oppenheimer–Volkoff limit.

The maximum mass limit for neutron star has been discussed by many authors

drawing different conclusions. Arnett and Bowers [14] have estimated the upper

mass limit as 1-3 MΘ. Rhodes and Ruffini [15] have estimated the upper mass limit

as 3.2 MΘ. Brecher and Caporasso [16] suggest this limit as 4.8 MΘ. Kamfer finds a

limit 3.75 MΘ [17]. Buitrago and mediavilla [18] found an upper limit of about 2 MΘ

for gravitational collapse in stars of uniform density in the neutron phase. The

maximum mass predicted by the various stiff equations of state is larger than the

measured masses of radio pulsars. Durgapal and Rawat [19] estimated the mass of

neutron star as 3.34 MΘ, when everywhere the speed of sound remains less than the

speed of light. Durgapal et al. [20] have obtained exact solution for a massive fluid

sphere under the extreme causality condition 1

d

dp. They obtained the maximum

mass of neutron star as 4.8 MΘ and size as 20.1 km. Pant et. al [21] have estimated

the maximum mass of neutron star as 6.33 MΘ with linear dimension 48.08 km . We

have also estimated the maximum mass of neutron star by obtaining exact solutions

of Einstein’s field equations in Chapter II.

11

1.6 Neutron star as pulsar:

Pulsars are spinning neutron stars that emit sharp pulses at exactly spaced

intervals of time [12]. Pulsar is a highly magnetized, rapidly rotating neutron

star[22]. Neutron stars are very dense, and have short, regular rotational periods. This

produces a very precise interval, between pulses that range from roughly

milliseconds to seconds for an individual pulsar. Extremely short periods of pulsars ,

suggests that pulsars are rotating neutron stars possessing a superhigh magnetic field

[23] and [24,25]. Neutron stars have very intense magnetic fields, as compared to

Earth's magnetic field. However, the axis of the magnetic field is not aligned with the

neutron star's rotation axis. The magnetic axis of the pulsar determines the direction

of the electromagnetic beam . Millisecond pulsars have provided us with best

working ground for testing of general relativity. The suggestion that pulsars were

rotating neutron stars was put forth independently by Thomas Gold and Franco Paciii

in 1968, and was soon proven by the discovery of a pulsar with a very short ( 33-

millisecond ) pulse period in the Crab nebula. The discovery of a pulsar at the centre

of crab-nebula, where the astronomers had predicted NS, established the oneness of

NS and pulsars.

The close observation of the pulsars can help to elucidate the interior

properties of neutron stars and can provide us with windows into the interiors of

neutron stars. The discovery of pulsars allowed astronomers to be acquainted with

the conditions of an intense gravitational field. We have also constructed the

approximate models of pulsars (Crab pulsar and vela pulsar) in chapter II.

1.7 The equation of state for Neutron star:

The equation of state for a neutron star is still not known exactly. Being both

very compact and extremely dense, neutron stars are unique laboratories for

probing the equation of state of neutron- rich matter . To have a complete picture of

NS a deep physical insight into the equation of state for dense 14105E gm cm-3

strong interacting hadronic matter is necessary. Despite a large amount of work

that has been done in literature on neutron star and equation of state of matter at

nuclear densities, the final picture regarding the properties of matter at super nuclear

densities is yet to be clearly understood .The structure of neutron stars is sensitive to

12

the equation of state of cold, fully catalysed, neutron-rich matter over an enormous

range of densities [26-28]. It is assumed that EOS for a neutron star differs

significantly from that of a white dwarf, whose EOS is that of a degenerate gas

which can be described in close agreement with special relativity. However, with a

neutron star the increased effects of general relativity can no longer be ignored.

Several EOS have been proposed for neutron star and current research is still

attempting to make predictions of neutron star matter. An understanding of an

equation of state is needed to estimate the parameters i.e. mass, size etc. of neutron

stars.

The mass of neutron star is an important parameter because after knowing the

mass of a neutron star only we can have an idea of the mass of black hole. A neutron

star requires many equations of state to completely describe the internal structure of

star.

The important density regions inside neutron star are as follows :

(i) 15 g cm-3 ≤ ρ ≤ 104 g cm-3; The Fermi- Thomas statistical model gives the

equation of state in this region [29].

(ii) 104 g cm-3 ≤ ρ ≤ 107 cm-3; In this region matter is so compressed that all

atoms are fully ionized, and we have a regular Coulomb lattice of nuclei,

neutralized by a gas of degenerate electrons; this is the outer crust [30]. The

electrons become relativistic at the upper end of the density range of this region.

(iii) 107 g cm-3 ≤ ρ ≤ 1011 g cm-3; Protons are converted into neutrons through

inverse beta decay process. The nuclei become more and more neutron rich.

(iv) 1011 g cm-3 ≤ ρ ≤ 4.5×1012 g cm-3 ; at ρ = 1011 g cm-3, the nuclei become very

neutron rich and neutron begins to drip out of the nuclei. Neutron degeneracy

pressure increases with increase in density. The material consists of nuclei,

degenerate electrons and neutrons.

(v) 4.5×1012 g cm-3 ≤ ρ ≤ 1014 g cm-3; in this region there is a giant nucleus

composed of three degenerate gases, viz. electrons, protons and neutrons. The

number density of protons and electrons is much lesser as compared to that of

neutrons.

(vi) 1014 g cm-3 ≤ ρ ≤ 1016 g cm-3; in this region along with neutrons , electrons

and protons, other elementary particles such as muons, pions and baryons may

also appear. At still higher densities the composition remains somewhat

13

uncertain, although hyperons, meson condensates or even deconfined quarks

might appear [31, 32].

1.8 The Coordinate Systems used in the present investigations:

The infinitesimal distance between two adjacent points in four dimensional

Riemannian space-time is given by

jiij dxdxgds 2

( 1.6)

Where ijg is the metric tensor in coordinates ( x1, x2 , x3 , x4 ) .

Here ( x1, x2 , x3 ) are space- like coordinates and x4 is time- like coordinate.

If we assume the spherical polar coordinates (r, θ, φ, ct ) , the coordinate r

increases as we move outwards from centre of the system. A gravitational system

is said to be spherically symmetric with origin at O, if the system is invariant

under spatial rotations about O. By rotational symmetry the metric properties on a

given sphere will be independent of the choice of θ and φ. The metric having this

property is given by

d Ω2 = (dθ2 + sin2θ dφ2 ) (1.7)

Thus, the spherically symmetric static space- time metric in Canonical or

curvature co-ordinates is given by

22222222 sin drdrdredteds (1.8)

Here λ and ν are functions of r.

The static, spherically symmetric space- time metric in isotropic coordinate

system is given by

22222222 sin dtceddrdreds (1.9)

Here coordinates (r, θ, φ, ct ) , are referred to as isotropic Coordinates and α,

β are functions of r. For non static case α and β will be functions of both r and t.

The another form of metric that is useful in the present study is spherically

symmetric non static metric conformal to the flat space time metric and will be

given by

14

222222222 sin),( ddrdrdtctrAds (1.10)

Where A is function of both r and t.

1.9 Einstein’s Field Equations and their importance :The line element of a static spherically symmetrical system is given by

22222222 sin drdrdredteds (1.11)

Where ν and λ are functions of r alone.

For the spherically symmetric mass distribution described in curvature coordinates

by the metric (1.11 ) the field equation is given as [33, 34]:

Tc

GRgR

4

8

2

1 (1.12)

Where R is Ricci tensor, R is the curvature invariant and T is the energy

momentum tensor.

The energy momentum tensor T is defined as

PguucPT )( 2 (1.13)

where P denotes the pressure distribution , the density distribution andu the velocity vector.

For a static case

)0,0,0,( 2

eu (1.14)

The components of the energy momentum tensor of a perfect fluid are given by

200

33

22

11 , cTPTTT (1.15)

)(0 T (1.16)

By evaluating the values of R , Rand using eq. (1.13) the resulting field

equations for the metric given by eq (1.11) are as follows:

2224

00 1188

rrre

c

G

c

TG (1.17)

2244

11 1188

rrre

c

PG

c

TG (1.18)

15

4242

888 2

44

33

4

22

re

c

PG

c

TG

c

TG

(1.19)

Here we have a system of three differential equations with four unknowns P, ,

λ, ν. When we want to study static massive spheres in general relativity we seek

to obtain the four variables pressure (P), density ( ), red shift parameter (ν) and

volume correction factor (λ) as a function of radial distance (r) measured from the

centre of the sphere. Now as there are only three independent field equations

hence we need one more equation to get all the parameters. The fourth equation

may be taken in the following form.

(i) as a function of r (e.g. Wyman , Kuchowich , Tolman,s III, VI, and VII

solutions)[35-37].

(ii) ν as a function of r ( Durgapal and Pandey, Durgapal, Pant )[38-40].

(iii) λ as a function of r ( Kuchowich, Durgapal and fuloria )[41-42].

(iv) P as a function of (Shapiro et al., pandey et al.,)[43-44].

From equations (1.18) and (1.19) ,Tolman [37] obtained following differential

equation

r

e

dr

de

r

e

dr

d

r

e

dr

d

22

12

(1.20)

By assuming various possible relations among ν and λ , Tolman has solved the above

equation and has found eight solutions of Einstein’s field equations.

For the space-time outside the mass distributions the solution is as follows:

arforarforP 0,0 (1.21)

arforr

mee

21 (1.22)

Eqs.(1.21) and (1.22) express exterior Schwarzschild solution which depends only

upon the configuration mass and not at all upon the details of mass distribution as long

as the distribution is spherically symmetric [45].

16

In geometrical units popular in General relativity, the unit of length is metre: but that

of mass and time are so chosen that the Gravitational Constant G and the speed of light

c are equal to unity i.e. c = G =1.

In geometrical units equations (1.17) , (1.18) and (1.19) will reduce to the following

form

224

4

1188

rrreT

(1.23)

221

1

1188

rrrePT

(1.24)

4242888

23

32

2

rePTT (1.25)

The equations (1.23) to (1.25) along with a fourth equation of the following form

P = P(ρ) , )(r , )(r , )(r (1.26)

can be computed to obtain various physical parameters of the stellar structure viz.

p, ρ, ν and λ under consideration.

The study of the relativistic stellar structures involves solutions of Einstein’s field

equations. In order to visualise a clear picture of the interior, one should obtain

exact solution of the Einstein’s field equations and use the results to construct

stellar models.

1.10 Field Equations in isotropic coordinates:

The space-time metric in isotropic coordinates is given by

22222222 sin ddrdredtceds (1.27)

For the metric (1.27) the field equation (1.12) reduces to the following equations

rrep

c

G

24

8 2

4

(1.28)

17

rr

pc

G

22422

8 2

4

(1.29)

re

c

G

2

4

8 2

2

(1.30)

where prime ( ' ) denotes the differentiation with respect to r .

From (1.28) and (1.29) we obtain following differential equation in α and β

0

22

22

rr

(1.31)

The new solution of Eq. (1.31) can be explored by considering various possible

relations among the unknown variables. However, the solution must satisfy all

the necessary conditions to be physically realizable.

1.11 Local maxima and local minima at the centre:

In order to study the trend of physical variables , following theorem may be

useful:

Theorem-

If xlrk ;0

xdx

dland

02

2

xdx

ldare nonzero finite, where 2rx ,

Then (i) maxima of k(r) will exist at r = 0 if0

xdx

dlis finitely negative.

(ii) minima of k( r) will exist at r = 0 if0

xdx

dlis finitely nonzero positive.

Proof : For maxima and minima we have

0,002)(0

xdx

dlasr

dx

dlr

dr

dx

dx

dlrk

dr

d

(1.32)

18

00

2

22

002

2

2422

rrrrdx

dl

dx

ldr

dx

dl

dx

dlr

dr

drk

dr

d, (1.33)

Provided0

2

2

xdx

ldis finite.

For the maximum at the centre (r = 0)

0200

2

2

xrdx

dlrk

dr

d

(1.34)

For the minima at the centre (r = 0)

0200

2

2

xrdx

dlrk

dr

d

(1.35)

This theorem is useful for showing the monotonically decreasing or increasing

nature of various physical parameters for well behaved nature of the solution.

1.12 Darmois Conditions (Junction conditions in isotropiccoordinates):

A solution of the field equation (1.20) for spherical matter distribution will be

valid in the space - time region occupied by the matter. If no matter exists

outside this spherical distribution then the laws governing the space time geometry

in the exterior region will be given by

0R (1.36)

The space- time must be continuous at the junction of interior space time region

and the exterior space time region. The geometry of the junction hyper surface

should satisfy eqs.(1.12) and (1.36) simultaneously. Three different sets of

boundary conditions have been given by Darmois , Lichernowicz , Brian and

Syange. The conditions due to Darmois and Lichernowicz are equivalent.

The Darmois set of condition is most convenient and reliable [46].

The exterior space–time metric to the static fluid ball in isotropic coordinate is

given by.

19

222222

1

222

22 sin

21

21 dRdRdR

Rc

GMdtc

Rc

GMds

(1.37)

Where t and R are the time and the radial coordinates respectively of the

exterior region. M is a Schwarzschild mass of the ball. The time coordinate t is

same for both the interior and exterior region, since the fluid ball is static so

time will be same for exterior also.

According to Darmois conditions the metric coefficients ijg and their first

derivatives kjig , in interior solution as well as in exterior solution should be

continuous upto and on the boundary B. The continuity of metric coefficients ijg of

interior and exterior space-time metric on the boundary is known first

fundamental form. The continuity of derivatives of metric coefficients gij of

internal and external solutions on the boundary is known second fundamental

form.

Since Schwarz schild’s metric (1.37) is considered as the exterior solution, the

following conditions are obtained by matching first and second fundamental

forms with canonical coordinate metric (1.8).

Rb = rb (1.38)

bRc

GMe b

2

21

(1.39)

bRc

GMe b

2

21

(1.40)

2

1

222 2

12

1

bb Rc

GM

Rc

GMe

b

(1.41)

Equations (1.38) to (1.41) are four conditions, known as boundary conditions in

canonical coordinates. Equations (1.38) and (1.41) are equivalent to zero pressure of

interior solution on the boundary. Similarly by matching the Schwarzschild’s metric

(1.37) with the isotropic coordinate metric (1.27) we arrive at the following

conclusions:

bRc

GMe b

2

21

(1.42)

20

2b

erR bb

(1.43)

2

1

2

21

2

2

1

bb

b Rc

GMr

r

(1.44)

2

1

22

21

2

1

bbb Rc

GM

Rc

GMr (1.45)

Equations (1.42) to (1.45) are four conditions, known as boundary conditions in

isotropic coordinates. Equations (1.43) and (1.45) are equivalent to zero

pressure of interior solution on the boundary.

1.13 Exact Solutions of Einstein’s Field equation:

Stellar Relativistic models have been studied ever since the first solution of

Einsein's field equation was obtained by Schwarzschild for the interior of a

compact object in hydrostatic equilibrium. The search for the exact solutions is

of continuous interest to physicists because a well behaved solution of Einstein’s

field equation can give us a deep insight into the interiors of massive fluid spheres.

The Einstein field equations describe the fundamental interaction of gravitation

as a result of space time being curved by matter and energy. The Einstein field

equations are complicated in nature. They are coupled, nonlinear partial differential

equations, Hence, it is very hard to solve them. The nonlinearity of the Einstein’s

field Equations distinguishes general relativity from many other fundamental

physical theories where we come across the linear equations. Despite the non linear

character of Einstein’s field equations, various exact solutions for static and

spherically symmetric metric are available in the literature.

The first two exact solutions of Einstein’s field equations were obtained by

Schwarzschild [47]. The first solution corresponds to the geometry of the space-time

exterior to a prefect fluid sphere in hydrostatic equilibrium. While the other solution

describes the interior geometry of a fluid sphere of constant energy-density E and

known as interior Schwarzschild solution. Tolman [37] obtained five different types

21

of exact solutions for static cases. The III solution corresponds to the constant

density solution obtained earlier by Schwarzschild [47]. The V and VI solutions

correspond to infinite density and infinite pressure at the centre, hence not considered

physically viable. Thus only the IV and VII solutions of Tolman are of physical

relevance. The VII solution has been studied extensively by Durgapal and Rawat[19].

The various other solutions of Einstein’s field equations have been obtained

by Adler [48], Adams and Cohen [49], and Kuchowicz [50], Buchdahl’s solution

[51] for vanishing surface density. The solution obtained by Vaidya and Tikekar

[52], has also been obtained by Durgapal and Bannerji [36]. The class of exact

solutions has been obtained by Durgapal [38]. Durgapal and Fuloria [42] solution is

also physically realizable. The most general exact solution for isentropic superdense

star was obtained by Gupta and Jasim [53]. Durgapal et al. [54] have tested the

suitability of the exact solutions for application to the stellar models . Pant, N.[ 55]

has presented three new categories of exact and spherically symmetric solutions of

Einstein’s field equations and obtained the mass of neutron star as 3.369 MΘ with

linear dimension 37.77 km. The various solutions of Einstein’s Field Equations that

are available in literature can be categorized as follows:

Category I

If the solutions are well behaved and regular (Delgaty and Lake [56]; Pant et al.

[57] ), these solutions completely describe interior of the neutron Star and other

compact stellar objects. The latest account of these solutions has been furnished by

Delgaty and Lake [56] and they found that only nine of them are regular and well

behaved. Out of which only six are well behaved in curvature coordinates and rest

three solutions are in isotropic coordinates.

Category 2-

If the solutions are not regular and well behaved but with finite central

parameters, such type of solutions may be used as seed solutions of super dense star

with charge matter since at the centre the charge distribution is zero. Many of the

authors electrified the well known exact solutions which are not well behaved e.g.

Kuchowich solution [58] by Nduka [59], Adler solution [60] by singh and

Yadav[61] ; by Pant and Tewari [62], Tolman solution [37] by Cataldo and

22

mitskievic [63], Heintzmann solution [64] by Pant. N, et al. [65]. These solutions are

useful for describing the interior of superdense astrophysical objects with charge

matter.

1.14 Charged fluid spheres in General Relativity:

The neutral solutions of Einstein’s field equations have very important

astrophysical implications. The Various compact stellar objects like neutron star,

white Dwarf, pulsar can be explained theoretically by studying the physically

realizable solutions of Einstein’s field equations. But many solutions of Einstein’s

field equations are not well behaved , hence can not be used for modeling of

astrophysical objects. The solutions of Einstein’s field equations which are not well

behaved in neutral arena can be made well behaved after including charge in them .

The charged interior solutions of Einstein field equations are normally found very

useful to predict or explain the various properties of massive compact objects. It is

observed that in the presence of charge, the gravitational catastrophic collapse of a

spherically symmetric material ball to a point singularity can be avoided by virtue of

the Columbian repulsive force along with the thermal pressure gradient. Exact

solutions of Einstein-Maxwell field equations are important in the modeling of

relativistic astrophysical objects. Such models successfully explain the characteristics

of massive objects like Neutron stars, Pulsars, Quark stars, or other super-dense

objects.

23

1.15 Einstein’s –Maxwell equations for charged fluid distribution:

Let us consider a spherical symmetric metric in curvature coordinates

22222222 )sin( dtedrdrdreds (1.46)

where the functions )(r and )(r satisfy the Einstein-Maxwell equations

mn

mnijjm

imij

jiij

ij

ij FFFFpvvpc

c

GRRT

c

G

4

1

4

1)(

8

2

18 244

(1.47)

where , p, iv , Fij denote energy density, fluid pressure, velocity vector and skew-

symmetric electromagnetic field tensor respectively.

In view of the metric (1.46), the field equation (1.47) gives

4

2

42

81

r

qp

c

G

r

ee

r

(1.48)

4

2

4

2 8

2442 r

qp

c

Ge

r

(1.49)

4

2

22

81

r

q

c

G

r

ee

r

(1.50)

where prime ( ' ) denotes the differentiation with respect to r and q(r) represents the

total charge contained with in the sphere of radius r.

24

1.16 Mathematical formulation of red shift:Gravitational red shift is the process by which electromagnetic radiation

originating from a high gravity star is reduced in frequency or red shifted, when

observed in a region of a weaker gravitational field. The red shift of the spectral

lines of the light originating from the dense stars is an important effect of the

gravitational field. This is the manifestation of slowing down of the time in the

gravitational field. The trajectory of light is null and is represented by ds = 0.The

speed of light originating at any position is given in terms of coordinate distance r

and coordinate time t by [66]

r

me

dt

dr 212

for Schwarzchild geometry (1.51)

a

m21 (at the surface of the star) (1.52)

For a fixed direction θ and may be taken as constants. From Eq.(1.51) we see

thatdt

dris independent of time t. Hence it may be concluded that the successive

pulses of light separated by the time t would always be separated by the

coordinate period of the observer. To the same interval of world time t , there

correspond at different points of space different intervals of proper time . The

relation between the proper period and coordinate period is given by [66]

2

1

21

r

mts

(1.53)

2

1

00gt (1.54)

2

et (1.55)

Let 0f = The frequency emitted from star = proper frequency

= number of oscillations per unit proper time

=

stellard

dN

(1.56)

25

ef = Observed frequency on earth

earthd

dN

Thus2

1

00

000

)(

)(

earthg

stellarg

f

f

e

(1.57)

If instead of earth the observer is at rest at a spatial infinity, the frequency

observed by the observer will be given by:

dt

d

d

dN

dt

dNf

(1.58)

= 2

1

000 gf (1.59)

Equations (1.57) and (1.59) show that instead of observing a frequency 0f , we

observe a reduced frequency. Hence the observed wavelength will be larger and

the red shift can be expressed as

2

1

000

1 gZ g

(1.60)

= 2

)(r

e

(1.61)

Where Zg is called the gravitational red shift.

1.17 Radiating Fluid Distribution & gravitational Collapse:

In a normal star the stellar radiation is a very slow process and any change

in the interior and exterior gravitational field is generally insignificant. However,

the situation is different for high energy astrophysical objects such as quasars and

supernova burst, where this radiation process is very strong. Therefore it is

desirable to study the solution of general relativistic field equations in terms of the

out flowing radiation. For the consideration of the above astrophysical problem

in the frame work of general relativity a proper mathematical formulation is

26

desirable. For a general relativistic treatment of strong gravity objects like Quasar

a radiating fluid ball is a close model.

It is already established fact that gravitational collapse is highly dissipating

energy process which plays a dominant role in the formation and evolution of

stars. However, the dissipation of energy from collapsing fluid distribution is

described in two limiting cases. The first case, the free streaming approximation

applies whenever the mean free path of particles responsible for the propagation of

energy in the stellar interior is larger (or equal to) than the typical length of the

object. In this case dissipation is modeled by means of an out flowing null fluid.

The second case, the diffusion approximation applies when the mean free path of

particles responsible for the propagation of energy in stellar interior is very much

small as compared with the typical length of the object . In this case dissipation is

modeled by means of a heat-flow type vector. The models of radiating fluid

spheres have been constructed both in free streaming case and in diffusion case by

many authors.

Following Tolman’s approach [37] , Vaidya [67-68] initiated the problem

for physically meaningful models of radiating fluid spheres in free streaming

limiting case. Bayin [69] has obtained exact solutions describing radiating perfect

fluid spheres. Some of them are physically reasonable but some are not

physically sound. Herrera et al. [70] have proposed a method to construct

radiating fluid ball models from the known static solutions of Einstein’s field

equations. Solutions for the radiating fluid ball problem corresponding to isotropic

coordinates form and in general metric form have been discussed by Tiwari [71-

72]. The radiating fluid sphere in conformally flat metric form has been discussed

by Pant and Tiwari [73]. Santos [74] has extensively studied the model proposed

by Glass [75] and has discussed the boundary conditions at the junction of the

interior and exterior metrics. The interior space time metric is matched with

vaidya’s exterior space-time metric.

27

1.18 Hydrodynamics of the Radiating Fluid spheres:

The most general space-time metric in spherical polar coordinates (ct, r, θ,

) which describes the geometry of a dynamic spherical distribution of matter

energy is expressed as

22222222 sin dderdredtceds (1.62)

Where ),(),,(,),( trtrtr ,

We have used the metric (1.62) because it is consistent with the system of

coordinates co moving with the matter particles of the distribution.

The Einstein’s field equations for space-time region occupied by matter energy

are expressed as:

ijjiji Tc

GgRR

4

8

2

1

(1.63)

i , j takes the values 0,1,2,3.

Rij is contracted curvature tensor, R is the scalar curvature, G is the Newtonian

constant of gravitation.

For a radiating fluid ball we can divide Tij into two parts. One part represents the

matter content of the distribution and the other part represents the radiation:

radiation

ijmatter

ij

ij TTT

(1.64)

To simplify the problem we assume that the fluid ball is composed of perfect fluid

through which energy is flowing out in the radial direction. For a perfect fluid we

have

ij

ijmatter

ij gpvvcpT 2 (1.65)

Here p and ε respectively denote the isotropic pressure and density of a perfect

fluid particle measured in its local rest frame and vi its unit time –like four-

velocity.

1iivv (1.66)

28

The energy momentum tensor for the radiation is given by

jiradiation

ij ww

c

qT (1.67)

where q is the rate of radiation or the rate of flow of energy.

w i its four velocity which is null:

0ii ww (1.68)

If the fluid distribution and its motion is spherically symmetric , we have

032 vv (1.69)

We assume that the radial coordinate r is co moving with fluid particles, so that

01 v (1.70)

From ( 1.62), (1.66) and ( 1.70) we have

20

ev (1.71)

The outward flow of radiation is in the radial direction only so that

032 ww (1.72)

From (1.68) and (1.72) we get

120 wew

(1.73)

if 1w is known we can determine iw completely .

The components of ijT in co moving coordinates are given by

11

11 ww

c

qpT

(1.74)

PTT 33

22 (1.75)

29

00

200 ww

c

qcT

(1.76)

01

01 ww

c

qT

(1.77)

1.19 Vaidya Metric and its derivation:

The geometry outside a spherically symmetric star when the exterior is

taken to be non–empty due to radiation from the star is given by the Vaidya metric

[67, 76]. The Vaidya metric describes exterior gravitational field due to a radiating

star. A spherically symmetric body that emits a continuous stream of photons with

each photon travelling radially outwards will be described by a metric that will

have energy- momentum tensor of the radially outgoing null rays. The Vaidya

metric is capable of describing this situation and provides an interesting model for

a time dependent spherically symmetric metric.

The space-time metric outside a radiating stellar object can not be defined by

Schwarzschild metric as it corresponds to an empty exterior given by Tij = 0. In

the case of a normal star, the effect of radiation on the overall exterior space-time

could be negligible. However the radiation effects would be important during the

late stages of gravitational collapse when the star could be throwing away

considerable mass as radiation or when abundant neutrinos are radiated away

from a collapsing supernova core. Such a radiating stellar object would then be

surrounded by an ever expanding zone of radiation. Vaidya metric is a simple and

interesting generalization of the Schwarzschild metric, which can be interpreted as

a space time with an outgoing spherically symmetric radiation of mass less

particles.

The energy- momentum tensor in the region permeated by radial energy flux with

null four- velocity wi and density q is given by

jiij wwc

qT (1.78)

With 032 ww and

30

011

00 wwww (1.79)

The field equations to be solved are

ijijij Tc

GgRR

4

8

2

1 (1.80)

In the canonical coordinates ( ct, R, θ, ) the space-time metric will be of the form

22222,22,2 sin ddRdRedtceds TRTR (1.81)

In radiation coordinates ( u, R, θ, ) eq. (1.81) transforms into

dRedTcefdu 221

(1.82)

Here TRf , is an integrating factor. The metric (1.81) then takes the form [77]

22222222 sin2

ddRdRduefdufds (1.83)

In this form Vaidya obtained a solution of (1.81) which is given below:

222222

2 sin2)(2

1 ddRdRduduRc

uMGds

(1.84)

At spatial infinity (R = ∞) the space-time is flat. Also when M is a constant, the

metric reduces to that due to Schwarzschild. We call M(u) Vaidya mass and u the

retarded time coordinate. From (1.82) we find that in vaidya field ijg is given as

22

2

1

2

sin

1000

01

00

002

11

0010

R

R

Rc

uMG

g ij (1.85)

31

To obtain the expression for luminosity in Vaidya field as measuredby an observer

at R = ∞, we define q to be the energy flux density as measured locally by an

observer having four velocity vi :

jiji Tvv

c

q (1.86)

Where 1ii vv (1.87)

We assume vi to be radial and d to be the proper time of the observer.

032 vv (1.88)

We define

d

dRvU 1 (1.89)

From equations (1.84) and (1.87) we have

URc

MGU

vd

du

1

21

2

0 (1.90)

Where2

1

22 2

1

Rc

GMU (1.91)

Using (1.90) , the equation (1.89) can be written as

du

dRUU 1 (1.92)

The only non zero component of Ricci tensor for the metric (1.84) is

expressed as

du

dM

Rc

GR

2200

2 (1.93)

From (1.80), (1.86), and (1.93) we get

2

225

28 Udu

dM

Rc

GRvvq

c

Gji

ji

(1.94)

32

Since 0q we conclude

0ud

dM(1.95)

That is, M(u) is a decreasing function of u.

The luminosity or the rate of energy flux through the hypersurface of coordinate

radius R as measured locally by an observer with four velocity vi is given by

qRL 24 (1.96)

Which, in view of eq.(1.94), can be written as

23 Udu

dMcL (1.97)

Also using (1.90), the equation (1.97) can be written as

12 ULd

Mdc

(1.98)

For an observer at rest at infinity RU ,0 , the apparent luminosity is

obtained by taking limiting case of (1.97):

du

dMcLL UR

30,lim (1.99)

(1.20) Junction conditions:

The space –time region surrounding the isolated radiating fluid ball is permeated

by pure radiation. We assume that the outward neutrino flux in the exterior region is

radial which implies that the energy –momentum distribution and consequently the

gravitational field is spherically symmetric. A metric for the spherically symmetric

radiation filled region has been obtained by vaidya using radiation coordinates [76,

78].

In this section we shall discuss the junction of the interior space time metric with the

exterior Vaidya metric over the hyperspace

33

r = rs (1.100)

where rs is a constant, or equivalently R = Rs(u) (1.101)

Suffix s denotes the value of the quantities over the hyper surface (1.100). For

smooth junction of the two space-time regions, Darmois conditions (1927) are to

be satisfied by the geometries of the two regions over the common boundary.

These conditions state that across the boundary the first and the second

fundamental forms should be continuous.

These boundary conditions have been obtained for the radiating fluid sphere by

Misner [79].

The continuity of the first fundamental forms across the hyper surface (1.100)

implies that the interior metric and the exterior metric must match over the hyper

surface.

This leads to the following equations

)(

,2

1

uRer s

tr

s

s

(1.102)

dudu

dR

Rc

GMdtce s

s

trs

2

1

2, 2

21

(1.103)

dt

dReU str

ss 2/,

(1.104)

In view of eq.(1.104), eq.(1.103) can be rewritten as

2

1

2

22/, 21

sss

tr

Rc

GMUU

du

dtce s

(1.105)

From (1.92), (1.104), (1.105) the following relation is obtained.

ss UU (1.106)

The continuity of the second fundamental form across (1.100) leads to the following

equations

34

0, trp s (1.107)

uMtrm s , (1.108)

The expression for the luminosity as observed at infinity is given by

du

dMcL 3 (1.109)

All these relations are useful for the study of radiating fluid spheres.

(1.21) Gravitational Collapse:

Gravitational collapse is the inward fall of a body due to the influence of its

own gravity. In any stable body, this gravitational force is counterbalanced by the

internal pressure of the body, in the opposite direction to the force of gravity.

Gravitational collapse is usually associated with very massive bodies, such as

neutron stars, quasars and massive collections of stars such as globular clusters

and galaxies .

Gravitational collapse is at the heart of structure formation in the universe. For

example, a star is born through the gradual gravitational collapse of a cloud of

interstellar matter. The compression caused by the collapse raises the temperature

until nuclear fuel reignites in the centre of the star and the collapse comes to a

halt. The thermal pressure gradient compensates the gravity and a star is in

dynamical equilibrium between these two forces. After a star has exhausted its

nuclear fuel, it can no longer remain in equilibrium and must ultimately undergo

gravitational collapse.

The importance of gravitational collapse processes in relativistic astrophysics

was realized when Datt [80] and oppenheimer and Synder [81] used general

relativity to study the dynamical collapse of a homogeneous Spherical dust cloud

under its own gravity. This model gave rise to concept of a black hole. A black

hole is a region of space time from which no light or matter can escape away to

far away external observers. In order to create a black hole as the final state of

35

gravitational collapse of the star, an event horizon must develop in the space time

earlier than the time when the final space time singularity forms.

If the event horizon is developed prior to the formation of the singularity , neither

the singularity nor the collapsing matter that has fallen within it would be observable

to an external observer, and a black hole is said to have formed as the final end state

of the collapsing star. The event horizon is where light loses the ability to escape

from the black hole.

1.22 Naked Singularity &Cosmic Censor Hypothesis:

In General Relativity , a naked singularity is a gravitational singularity , without

an event horizon . In a black hole, there is a region around the singularity, the event

horizon, where the gravitational force of the singularity is strong enough so that light

cannot escape. Hence, the singularity cannot be directly observed. A naked

singularity, by contrast, is observable from the outside. Radiating stellar models have

been proposed by Bannerjee et al. [82] in which the horizon is never encountered and

naked singularity comes into existence. Pant, N. and Tewari, B. C. [83] have also

discussed the horizon –free gravitational collapse of radiating fluid spheres .

The theoretical existence of naked singularities is important because their

existence would mean that it would be possible to observe the collapse of an

object to infinite density. A naked singularity could allow scientists to observe an

infinitely dense material, which would under normal circumstances be impossible

by the cosmic censorship hypothesis. The cosmic censorship hypothesis says that a

naked singularity cannot arise in our universe from realistic initial conditions.

According to cosmic censorship hypothesis conceived by Roger Penrose in 1969

no naked singularities other than the Big Bang singularity exist in the universe.

Cosmic Censorship Hypothesis states that the universe contains no singularities

that can be visible to a very far away observer.

We have investigated the models of radiating stellar objects (Quasars)

undergoing gravitational collapse in which the horizon is not formed i.e. the

collapse process will go on forever ultimately reaching to a point singularity called

36

a naked singularity. While constructing the models of radiating stellar objects in

chapter IV we have shown that the mass radius gradient is less than one, or

collapse is horizon free . Theoretically we can say that there is possibility of the

existence of naked singularity, the singularity that will be observable by external

observer.

1.23 Objective of the Thesis:

The exact solutions of Einstein’s field equations and Einstein - Maxwell field

equations provide us with a powerful tool for exploring the interior properties of

relativistic astrophysical objects. Due to nonlinearity of Einstein’s field equations

there is relative scarcity of exact solutions , however their significance in

understanding the interiors of stellar objects can not be ignored. Keeping in view the

relevancy of solutions of Einstein’s field equations and their application to various

astrophysical objects the present thesis embodies the exploration of some new

solutions of Einstein’s field equations , Einstein-Maxwell field equations and their

use in modeling of immense gravity objects. The most important tool for such

modeling is static, spherically symmetric exact solution with perfect fluid matter and

finite central parameters. The characteristics of massive objects like , neutron star,

pulsar or other super-dense objects can be successfully explained by these models.

The non static solutions of Einstein’s Field equations also have manifold importance

in understanding certain aspects of the astrophysical objects emitting a large amount

of energy in the form of radiation. The stellar objects emitting radiation also undergo

gravitational collapse and may end into either black hole or naked singularity.The

time dependent solutions of Einstein’s field equations are useful for understanding

such problems.

Consequently chapter II [84,85] deals with an attempt to find some new

solutions of Einstein’s field equations. One solution in curvature coordinates and

another one in isotropic coordinates have been found. The characteristics of these

solutions have been studied extensively. Keeping in view the well behaved nature

we have ventured into the modeling of compact stellar objects like neutron star.

37

Chapter III [86, 87] covers the study of well behaved , static charged fluidspheres. The presence of charge in massive fluid spheres has considerable impacton the overall properties of stellar objects. The gravitational collapse of the stellarobject to a point singularity can be avoided due to the repulsive effect of chargepresent in the fluid sphere. We have obtained a new generalized solution ofEinstein -Maxwell field equations which can provide us a series of well behavedsolutions. These solutions have been used for the modeling of immese gravityastrophysical objects. For this class of solutions the outmarch of pressure, density,pressure-density ratio and the adiabatic sound speed is monotonically decreasing.

Chapter IV is devoted to the extensive study of BCT solution II [71] for themodeling of radiating stellar objects. We have made an attempt to construct theapproximate models of Quasars for different combinations of the constants X, Yand Z appearing in the solution . The variation of pressure, density, pressure-density ratio and luminosity has been studied with in the radiating fluid sphere(Quasar) . It has been observed that the final outcome of the gravitational collapseof radiating fluid sphere will be naked singularity .

In Chapter V [ 88] we have studied adiabatic collapse of a uniform densitysphere [with Schwarzschild geometry in the exterior] using baryon conservationlaw and no-heat-transfer condition . It is concluded that a uniform density sphere[with Schwarzschild geometry in the exterior] always collapses adiabatically withvanishing pressure.In Chapter VI we have made an attempt to find a new time dependent solution ofEinstein’s field equations . Although the solution is singular at the centre but itmay give some insight in understanding the various phenomena concerned withradiating fluid spheres. We have also made an extensive study of non staticsolution obtained by Tewari [73] in conformally flat space time metric. It has beenshown that the solution is well behaved and may be useful for modeling ofradiating stellar objects.

38

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Contraction”, Phys. Rev. 56, 455 (1939).

[82] Banerjee, A., Chatterjee, S., Dadhich, N. : “Spherical Collapse with

Heat Flow and without Horizon”, Mod. Phy. Lett. A 17, 2335(2002).

47

[83] Pant, N., Tewari, B.C. : “Horizon-free gravitational collapse of

radiating fluid sphere”, Astrophysics &Space Sci. 331 (2), 645 ( 2011) .

[84] Fuloria, P., Durgapal M. C. : “A non singular solution for spherical

configuration with infinite central density”, Astrophys. Space Sci.,

314, 249 (2008).

[85] Pant, N., Fuloria, P., Tewari, B. C. : “A new well behaved exact

solution in general relativity for perfect fluid” , Astrophys. Space

Sci., 340, 407 (2012).

[86] Fuloria, P., Tewari, B. C., Joshi, B. C.: “Well Behaved Class of Charge

Analogue of Durgapal's Relativistic Exact Solution”, J. Modern

Physics 2, No.10, 1156 (2011).

[87] Fuloria, P., Tewari, B. C.: “A family of charge analogue of Durgapal

solution”, Astrophys. Space Sci. 341, 469 (2012).

[88] Durgapal, M. C. , Fuloria, P. : “On collapse of uniform density sphere

with pressure”, J. Modern Physics 1, 143 (2010).

Chapter IINew solutions of Einstein’s fieldequations for static perfect fluid

matter

48

Chapter IINew solutions of Einstein’s field

equations for static perfect fluid matter2

Chapter II has been divided into two sections. In section A a non singular exact

solution with an infinite central density has been obtained for the interior of static

and spherically symmetric matter distribution . Both the energy density and the

pressure are infinite at the centre but we have 1)0( e and 0)0( e . The

solution admits the possibility of receiving signals from the region of infinite

pressure. In section B we have presented a new spherically symmetric solution of the

general relativistic field equations in isotropic coordinates . The solution is well

behaved having positive finite central pressure and positive finite central density.

The ratio of pressure and density is less than one and casualty condition is obeyed at

the centre. The out march of pressure, density , pressure –density ratio and square of

adiabatic speed of sound is monotonically decreasing. The solution is well behaved

for all the values of u lying in the range 0 < u ≤ .186. Further, we have constructed

a neutron star model with all degree of suitability and by assuming the surface

density 314102 cmg . The maximum mass of the Neutron star comes out to

be M = 1.591MΘ with radius Rb =12.685 km.

2A part of this chapter (section A) has been published in Astrophys. Space Sci.,314, 249 (2008). Another part of this chapter (section B) has been published inAstrophys. Space Sci. 340 407 (2012).

49

Section A

Solution 1: A non singular solution with infinite central

Density.

2.1 Introduction:

One of the main obstacle to the better understanding of interior of stellar objects

is the lack of exact solutions of Einstein’s field equations. The non linearity of field

equations restricts the number of exact solutions to be very less. Although a large

number of exact solutions are available in literature but not all of them are physically

relevant hence not suitable for modeling of super massive structures. The well

behaved exact solutions might provide useful models for the internal structures of

super massive astrophysical configurations. Exact solutions have played a crucial

role in the development of many areas of astrophysics. The solutions are obtained

either by solving Einstein’s field equations analytically or by choosing some equation

of state for the matter within the configuration and then using numerical

computation. Exact solutions with well behaved nature of Einstein’s field equations

are of vital importance in relativistic astrophysics because the distribution of matter

in the interior of stellar object can be easily understood in terms of simple algebraic

relations. A new solution is always welcome which may give us a deep insight into

the interior of compact objects formed during the late stages of stellar evolution. The

first exact solution of Einstein’s field equations for a perfect fluid sphere of constant

density was obtained by Schwarzschild [1]. Tolman [2] gave five new exact solutions

for the fluid spheres. The III solution corresponds to a sphere of constant density.

The V and VI solutions correspond to infinite central density. IV and VII solutions

are physically relevant hence suitable for modeling of star. The stability analysis of

Tolman’s VII solution with vanishing surface density has been undertaken in detail

by Negi and Durgapal [3,4] and they have shown that this solution also corresponds

to stable Ultra-Compact Objects (UCOs) which are entities of physical interest.

Although all the solutions obtained so far may not be physically relevant in every

respect but their importance in understanding the stellar interior can not be denied.

Search for new analytic solutions remains valuable due to the fact that once such a

solutions are found one can immediately study all of it's physical properties because

of the non-linear nature of the equations.

50

Durgapal [5] obtained a class of new exact solutions. Durgapal and Fuloria [6]

obtained an analytic relativistic model for a superdense star which stood all the tests

of physical reality. The solution obtained by Adler [7], Adams and Cohen [8], and

Kuchowicz [9] were important with deep relevancy for describing the interior

parameters of neutron star. Buchdahl’s solution [10] for vanishing surface density

also gives deep insight in understanding the interiors of stars. The solution obtained

by Vaidya and Tikekar [11], which is also obtained by Durgapal and Bannerji [12] is

also useful for the study of immese gravity objects . The class of exact solutions

discussed by Durgapal & Rawat [13], Buchdahl [14], kuchowich [15] , Leibovitz

[16], Mehra[17], Pant and Pant [18], Pant [19], and Pant Neeraj[20] are some of the

solutions which have been used for modeling of stellar objects.

In general the metric chosen for obtaining these interior solutions is theSchwarzschild metric given by

)sin( 2222222 ddrdredteds (2.1)

Where ν and λ are functions of r alone.

There are many solutions with finite mass and infinite central density (Tolman[2],

Zeldovich[21], Misner Zapolsky [22] and Gehlot[23] ). Misner and Zaplosky

discussed neutron star models with extreme density distribution ( central density, ρc

= ∞). But in most cases of the above mentioned solutions the value of 1)0( e and

the curvature at the centre is infinite, the solution become singular at the centre. The

spherical symmetry demands that e at the centre should be 1. Also, in all these

solutions 0)0( e which makes it impossible to have any information out of this

region. We have reinvestigated these solutions to obtain a non singular interior

solution with infinite central density. Such solution can throw much light on the

structures at very late stages of their evolution.

2.2 Einstein’s Field Equations and their solutions:

p

c

G

r

ee

r 42

81

(2.2)

pc

Ge

r 4

2 8

2442

(2.3)

51

22

81

c

G

r

ee

r

(2.4)

We can write down the Einstein’s field equations in the following form [6] :

( Here we have used relativistic units in which G = 1, c = 1 )

0)1(2 22 zzxyzxyzx (2.5)

x

z

y

yz

C

P

14

8(2.6)

z

x

z

C

2

18 (2.7)

Where ez ; 2

eBy ; 2rCx

Where P = pressure; = energy density

B and C are constants and the prime represents differentiation with respect to x.

There are three equations and four variables. Hence we require one more equation to

obtain a solution of the field equations.

We assume that

3

1

1 xz

(2.8)

Similar type of general solutions are available in the widely available literature

(Kramer et al. [24] and references given there in); but this particular solution is

obtained in order to discuss non-singular nature at the centre of a structure when both

the pressure and energy density tend to infinity and still 0)0( e .

From equation (2.7) and (2.8) we obtain

3

2

3

58 x

C

(2.9)

52

It can be seen that at x = 0, we get

0 and 1)0( e (2.10)

A particular solution of eq (2.5) for the value of z given by eq. (2.8) is [2, 23]

xyi (2.11)

A more general solution of equation (2.5) is obtained by using the following relation

[25]

zy

dxconstyy

i

i 21 (2.12)

Using equations (2.8), (2.11) and (2.12) we obtain,

xxxxxxxAxy log

16

511log

8

15

8

15

4

511 3

1

3

2

3

1

3

1

(2.13)

Where A is a constant. This is a special case of the generalized solution obtained by

Wymann [26] in the form of hyper geometric functions. It can be shown easily that at

x = 0

Ayy 0 (2.14)

Equations (2.6) and (2.13) give us the expression for pressure as:

2

1

3

1

3

1

15418

xy

Ax

xC

P( 2.15)

2.3 Boundary Conditions :

The values of the unknown constats A, B, C can be obtained from the boundary

conditions at the surface of the structure. The solutions are required to be

continuous with the Schwarzschild solutions at r = a or x = X (= Ca2)

Thus

53

uee aa 21)()( (2.16)

Where u = M/a; M = Total mass of the configuration and a = radius of the

configuration.

And pressure at the boundary must be zero i.e. P(X) = 0 (2.17)

Using boundary conditions (2.16) and (2.17) we obtain

1

3230

2

121log155.75.2121

5.21

218

u

uuuuu

u

uuyA

(2.18)

0

5.21

y

uB

and C =

2

38

a

u

(2.19)

It is seen that the central pressure is infinite but the central red shift is given by

2

1

0

2

1

)5.21(1

10

uyB

eZc

(2.20)

The central red shift remains finite for all the values of u < 0.4

The solution thus remains valid for u < 0.4, at u = 0.4, 0)0( e

Assuming that these structures have densities greater than the nuclear densities, that

is, > 314102 cmg [27].

The size and mass of these neutron stars are given by:

R in km = 2

1

9.29 u and M( in solar mass ) = 2

3

8.20 u (2.21)

54

Table 2.1 :

The variation of radius, mass , central red shift and surface red shift with u.

u Radius (in km) Mass(in solar mass) central red shift surface red

shift

0.02 4.22 0.058 0.0526 0.0206

0.04 5.98 0.1664 0.1111 0.0425

0.06 7.32 0.3056 0.1764 0.0661

0.08 8.45 0.4706 0.2500 0.0910

0.1 9.45 0.6577 0.3333 0.1180

0.15 11.58 1.208 0.6000 0.1952

0.2 13.37 1.860 1.0000 0.2900

0.25 14.95 2.600 1.6666 0.4142

0.3 16..37 3.417 3.000 0.5811

0.35 17.68 4.366 7.000 0.8257

Fig 2.1 : The variation of mass, radius, central red shift, surface red shift with u.

0

5

10

15

20

0 0.1 0.2 0.3 0.4

u

radius ( in km)

Mass (in solarmass)

Central red shift

Surface red shift

55

2.4 Results and Discussions :

The solution obtained here is important because it is non-singular and

corresponds to infinite central density and pressure. Although the pressure and

density are infinite at the centre, but the red shift is finite. This solution is

important to study those stages of stellar evolution when the stellar object tends to

attain infinite pressure and infinite energy density. It would be possible to get the

information of these extreme stages of stellar evolution . Table 2.1 shows the

values of different parameters of Neutron star corresponding to different values of

u.The variation of mass, radius, central red shift and surface red shift with u has

been shown in the Fig 2.1. It has been observed that radius, mass, central red shift,

surface red shift increase with u. Central red shift increases more rapidly as

compared to the surface red shift. Eqs. (2.9) and (2.15) show that pressure and

density become infinite at the centre. The maximum mass of the neutron star based

on this solution comes out to be 4.366 MΘ with radius 17.68 km.

56

Section B

Solution II : A NEW WELL BEHAVED EXACT SOLUTION IN

ISOTROPIC COORDINATE SYSTEM FOR PERFECT FLUID.

2.5 Introduction:Neutron stars and quark stars are assumed to be the state of perfect fluid

balls in a static equilibrium. Hence in order to study such astrophysical problems

in the framework of General Relativity, the static solutions of Einstein’s field

equations for perfect fluid balls are desired.

Few successful attempts have been made to obtain exact static solutions of

Einstein’s field equations for perfect fluid balls in isotropic coordinates [28-30].

The desirability to obtain new exact solutions in isotropic coordinates lies in the

fact that those solutions which have very complicated form in canonical

coordinates may assume simple form in isotropic coordinates. Thus one can

consider Einstein’s gravitational field equations for perfect fluid and get their

physically significant solutions. In this chapter we present a new physically

relevant exact solution of Einstein’s field equations in isotropic coordinates.

The properties and characteristics of enormous gravity objects like neutron

star, white Dwarf , Quark star yet not solved fully in terms of maximization of

mass, can easily and significantly be understood with the help of Einstein’s field

equations of General Relativity. A considerable number of known solutions [31]

of Einstein’s field equations are of finite central pressure and finite central density

which may be useful for the modeling of some stellar objects. These solutions are

of paramount importance with pioneer relevancy for describing interior of the

Neutron star, as the central parameters are completely defined. The solutions

obtained by Tolman IV [2], Adler[7], Heintz[32], Finch and Skea[33] etc. do not

satisfy the one of the well behaved condition i.e. adiabatic sound speed is not

monotonically decreasing. Delgaty- Lake also pointed out that only nine solutions

so far are regular and well behaved; out of which seven in curvature coordinates

(Tolman vii [2], Patvardhav –Vaidya[34], Mehra[17], Kuchowicz[35] , Matese-

Whitman[36] , Durgapal’s two solutions[37] ) and only two solutions (Nariai

[38],Goldman[39]) in isotropic coordinates. The two new well behaved solutions

in curvature coordinates have been explored by N. Pant[40] , Gupta-Maurya [41]

57

and a new well behaved solution in isotropic coordinates has been obtained by

Pant et al. [42]. In this chapter we also present a new solution in isotropic

coordinates which is not only well behaved but also simple in terms of

expressions of field and physical variables. keeping in view the well behaved

nature of the solution we have constructed the models for supermassive

astrophysical configurations.

2.6 Conditions for well behaved solution:

From the physical point of view, the mathematical solutions must satisfy certain

physical requirements to render them physically meaningful. The following

conditions or requirements have been accepted [31, 42] .

(i) The solution should be free from physical and geometrical singularities i.e. finite and

positive values of central pressure, central density and non zero positive values of e

and e .

(ii) The solution should have positive and monotonically decreasing expressions for

pressure and density ( andp ) with the increase of r. The solution should have

positive value of ratio of pressure-density and less than 1(weak energy condition) ,

less than 1/3 (strong energy condition) throughout within the star.

(iii) The solution should have positive and monotonically decreasing expression for fluid

parameter 2c

p

with the increase of r.

(iv) The solution should have positive and monotonically decreasing expression for

velocity of sound

d

dpwith the increase of r and causality condition should be

obeyed at the centre i.e. 12

dc

dp

(v) d

PdP , everywhere within the ball.

(vi) Adiabatic index

P

d

dP

d

dP

Pd

Pd

e

e log

log

For realistic matter 1

58

(vii) The red shift Z should be positive, finite and monotonically decreasing in nature

with the increase of r.

(viii) The central red shift Z0 and surface red shift Zb should be positive and finite i.e.

0)1( 02

0

reZ

0121 5.0 uZb

Under these conditions, we have to assume the one of the gravitational potential

component in such a way that the field equation (2.30) can be integrated and solution

should be well behaved. Further, the mass of the such modeled super dense object

can be maximized by assuming surface density, 2 × 1014 g / cm3.

2.7 Field equations in isotropic coordinates

Let us consider a spherical symmetric metric in curvature coordinates

)sin( 222222 ddrdredteds (2.22)

Where α and β are functions of r. Einstein’s field equations of gravitation for anon empty space-time are

jijiji RgRTc

G

2

184

(2.23)

Where jiR is a Ricci tensor , jiT is energy- momentum tensor and R is the scalar

curvature. The energy - momentum tensor jiT is defined as

Tij = ( P + ρ c2 ) vi vj − P gij (2.24)

Where P denotes the pressure distribution , ρ the density distribution and i the

velocity vector, satisfying the relation

1jijig

(2.25)

Since we are dealing with static field, therefore,

(2.26)44

4321 10

gvandvvv

59

For the metric (2.22) the field equation (2.23) reduces to the following equations

rrep

c

G

24

8 2

4

(2.27)

rr

pc

G

22422

8 2

4

(2.28)

re

c

G

2

4

8 2

2

(2.29)

Where prime ( ' ) denotes the differentiation with respect to r .

From (2.27) and (2.28) we obtain following differential equation in α and β

0

22

22

rr

(2.30)

We have obtained a new solution of equation (2.30) and have studied all its

Properties.

2.8 New class of solution:

The equation (2.30) is solved by assuming

7

122/ 1

Cre B (2.31)

We get

2

7

227

12

2

111

B

CrCrA

e

(2.32)

Where A , B , and C are arbitrary constants.

The expressions for density and pressure are obtained as :

60

22

7

1222

2621

149

48rCC

CrBc

G

(2.33)

7

12

227

12

22

7

1222

4

)1(1

)75()1()47(

)1(49

48

CrA

CrCCrArCC

CrBc

pG

(2.34)

2.9 Properties of the new solution:

The central values of pressure and density are given by

A

ACC

Bc

pG

1

77

49

482

04

(2.35)

20

2 7

128

B

C

c

G

(2.36)

The central values of pressure and density will be non zero positive definite, if the

following conditions will be satisfied.

A > - 1/2, C > 0 (2.37)

Subjecting the condition that positive value of ratio of pressure-density and less than

1 at the centre i.e. 12

0

0 c

p

which leads to the following inequality,

11

13

1

A

A(2.38)

All the values of A which satisfy equation (2.37) , will also lead to the condition

.12

0

0 c

p

Differentiating (2.34) with respect to r,

61

332

3327

12

3327

222

7

192

2

7

122

7

4016

7

120441

7

9030)1(

11149

4

rCrC

rCrCCrA

rCrCCrA

CrCrABdr

dp

(2.39)

Thus extrema of p occur at the centre if

00 rp (2.40)

22222204

1644301

1

49

48CACCA

ABp

c

Gr

(2.41)

= -ve if A > -1/2, C > 0 (2.42)

Thus the expression of right hand side of equation (2.42) is negative showing thereby

that the pressure p is maximum at the centre and monotonically decreasing.

Now differentiating equation (2.33) with respect to r.

32

7

192 7

6060

149

4rCCr

Cr

C

dr

d(2.43)

Thus the extrema of ρ occur at the centre if

00 r (2.44)

2

2

02 49

2408

B

C

c

Gr

(2.45)

The right hand side of this equation will be –ve for positive as well negative values

of B and C. The expressions of right hand side of (2.43) and (2.45) are negative

showing thereby that the density ρ is maximum at the centre and monotonically

decreasing.

In view of Eqs. (2.35) and (2.36), we observe that pressure and density are

maximum at the centre and monotonically decreasing with the increase of radial

coordinate r.

62

The square of adiabatic sound speed at the centre,

02

1

rd

dp

c , is given by

30

82215

1

11 2

20

2

AA

Ad

dp

cr

(2.46)

)(1 veand (2.47)

If following condition will be satisfied

A > -1/2, C > 0 (2.48)

The causality condition is obeyed at the centre for all values of constants

satisfying (2.48).

Further, it is mentioned here that the boundary of the super dense star is established

only when -1/2 < A < 0 .

In view of (2.33) and (2.34 ) the ratio of pressure-density is given by

222

7

12

22222

2 621

11

75147

rCC

CrA

CrCCrArCC

c

p

(2.49)

Differentiating (2.49) with respect to r, we get ;

dr

d

c

p2

(2.51)

222

c

p where (2.52)

Where

7

12

22222

11

75147

CrA

CrCCrArCC

(2.53)

63

= 222 621 rCC (2.54)

A

ACCr 1

770 (2.55)

20 21Cr (2.56)

2

7

127

52

222

22

7

627

12

2

1117

752

8480

1117

8

CrACr

CrCCrA

CrC

CrCrA

ACrrC

dr

d

(2.57)

212 Crdr

d

(2.58)

222

2

7

127

52

52332523

7

627

12

33

2 )621(

111

3014714772

760

168168

111

84

rCC

CrACr

rCrCrCArCrCr

CrCrA

ACrC

c

p

(2.59)

Thus extrema of2c

p

occur at the centre if

002

r

c

p

(2.60)

Differentiating (2.59) w. r. t. r, we get

(2.61) 4

22

2

c

p

64

Where

dr

dand

22 (2.62)

212C (2.63)

98670756

11149

5887201260

11149

8

422

7

122

2

7

12

22

422

7

1327

12

22

rCCr

CrCrA

CA

rCrC

CrCrA

ACC

rCr

CrCrA

rCA75

11149

8 3

7

112

3

7

12

33

(2.64)

2222

20

1

2

1

128

A

CA

A

ACCr

(2.65)

20 12Cr (2.66)

vec

pr

02

provided A > -1/2 , C > 0 (2.67)

2.10 Boundary conditions:

The solutions so obtained are to be matched over the boundary with Schwarzschild’s

exterior solution;

222222

1

222

22 sin

21

21 dRdRdR

Rc

GMdtc

Rc

GMds

(2.68)

Where M is the mass of the ball as determined by the external observer and R is the

radial coordinate of the exterior region. The boundary conditions are that the first and

second fundamental forms are continuous over the boundary r = rb or equivalently R =

Rb.

65

Applying the boundary conditions we get the values of the arbitrary constants in terms of

Schwarzschild’s parametersbRc

GMu

2 and bR

In view of Eqs. (1.43) and (1.45) we obtain two values of constants A as follows:

7

1

22

1

7

1

2

1

2

1

22

1

1

121

2175

12171

121

C

ku

u

u

kuA

(2.69)

7

822

1

7

122

2

122

)1(7)21()1(6

)21(4)1(7

bbb

bb

CruuCrCr

uCrCruA

(2.70)

From equation (2.69) and (2.70) we obtain the value of k as

u

Cr

CruuCrCr

uCrCru

k

b

bbb

bb

21

11

)1(7)21()1(6

)21(4)1(7 2

7

822

1

7

122

2

122

(2.71)

whereb

b

r

Rk

66

7

1

2

1

2

1

2175

12171

u

ukB

(2.72)

244.00

2175

1217

22

1

2

1

ufor

ru

uC

b

(2.73)

Surface density is given by

uu

Rc

G

b

b 10821838

2

1

22

(2.74)

Surface density will be positive

Provided, u < .3

Central red shift is given by

1

1

2

0 A

BZ

(2.75)

The surface red shift is given by

121 5.0 uZ b (2.76)

67

Table 2.2 : The central values of pressure, density, pressure- density ratio, square of

sound speed, red shift for different values of u.

S. No. u 02

4)

8( rbrp

c

G0

2

4)

8( rbrc

G

02

)( rc

p

02))(

1( rd

dp

c (Z)0

1 0.05 0.0175 0.6638 0.0263 0.1107 0.0859

2 0.10 0.0860 1.5288 0.0563 0.1234 0.2033

3 0.15 0.2621 2.8583 0.0917 0.1392 0.6282

4 0.175 0.4472 3.9605 0.1129 0.1489 0.5154

5 0.20 0.8169 5.9011 0.1384 0.1611 0.7204

6 0.24 6.7529 31.3999 0.2150 0.1999 2.0226

68

Table 2.3 : By assuming the surface density 314 /102 cmgb , the variation of

maximum Neutron star mass, radius Rb, central red shift Z0 and surface red shift

121 5.0 uZb with u . ( gmsunofMassM 33102 )

S. No. u 2

2

8bb r

c

G

M

MRb in km Z0 Zb

1 .01 0.1161 0.026 3.973 0.0153 0.0102

2 .02 0.2243 0.075 5.560 0.0315 0.0206

3 .04 0.4175 0.207 7.686 0.0667 0.0426

4 .06 0.5793 0.371 9.183 0.1063 0.066

5 .08 0.7100 0.556 10.318 0.1513 0.0911

6 .1 0.8094 0.754 11.190 0.2033 0.1180

7 .15 0.9229 1.262 12.479 0.3807 0.1953

8 .175 0.9084 1.498 12.688 0.5150 0.2401

9 .186 0.8874 1.591 12.685 0.5933 0.2618

Fig 2.2: The variation of radius (in km) with u .

02468

1012141618

0 0.05 0.1 0.15 0.2u

radius

69

Fig 2.3: The variation of mass , density, central red shift, surface red shift with u.

Table 2.4 : The march of pressure, density, pressure-density ratio and square of adiabatic

sound speed within the ball corresponding to u = .1

r/rb2

4

8brp

c

G 2

4

8brc

G

2c

p

)(

12 d

dp

c red shift

0.0 0.086 1.528 0.05629 0.12347 0.2033

0.1 0.084 1.516 0.05573 0.12337 0.2021

0.2 0.0799 1.479 0.05404 0.12308 0.1986

0.3 0.0727 1.420 0.05123 0.12260 0.1930

0.4 0.063 1.346 0.04728 0.12193 0.1855

0.5 0.0531 1.260 0.04220 0.12107 0.1765

0.6 0.0420 1.167 0.03600 0.12003 0.1661

0.7 0.0307 1.073 0.02866 0.11881 0.1548

0.8 0.0198 0.980 0.02020 0.11740 0.1428

0.9 0.0094 0.892 0.01064 0.11580 0.1305

1.0 0.0000 0.809 0.00000 0.11400 0.1180

00.20.40.60.8

11.21.41.6

0 0.05 0.1 0.15 0.2

u

M/Mo

central redshift

surface red shift

density

70

Fig 2.4: The variation of p ,2c

p

, )(

12 d

dp

c, Z from centre to surface (u = 0.1)

Fig 2.5 : The variation of density from centre to surface (u = 0.1).

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5

r/rb

pressure

pressure/density

d(pressure)/d(density)

red shift

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 1.2

r/rb

density

71

Table 2.5: The march of pressure, density, pressure-density ratio and square of adiabatic

sound speed within the ball corresponding to u = .06

r/rb2

4

8brp

c

G 2

4

8brc

G

2c

p

)(

12 d

dp

c red shift

0 0.026 0.816 0.0320 0.1131 0.1063

0.1 0.025 0.812 0.0317 0.1130 0.1058

0.2 0.024 0.803 0.0307 0.1128 0.1044

0.3 0.023 0.788 0.0291 0.1125 0.1021

0.4 0.020 0.767 0.0269 0.1121 0.0989

0.5 0.017 0.742 0.0240 0.1116 0.0949

0.6 0.014 0.713 0.0205 0.1109 0.0902

0.7 0.011 0.681 0.0163 0.1101 0.0848

0.8 0.007 0.648 0.0115 0.1093 0.0789

0.9 0.003 0.614 0.0060 0.1083 0.0726

1.0 0.000 0.579 0.0000 0.1072 0.0660

72

Fig 2.6: The variation of p ,2c

p

, z

d

dp

c,)(

12

from centre to surface for u = 0.06

Fig 2.7: The variation of density from centre to surface for u = 0.06

0

0.02

0.04

0.06

0.08

0.1

0.12

0 0.5 1 1.5

r/rb

pressure

pressure/density


red shift

00.10.20.30.40.50.60.70.80.9

0 0.2 0.4 0.6 0.8 1 1.2

r/rb

Density

73

Table 2.6: The march of pressure, density, pressure-density ratio, red shift and square

of adiabatic sound speed within the ball corresponding to u = .186

r/rb2

4

8brp

c

G 2

4

8brc

G

2c

p

)(

12 d

dp

c red shift

0 0.5745 4.654 0.1234 0.15394 0.5933

0.1 0.5482 4.483 0.1222 0.15393 0.5852

0.2 0.4790 4.033 0.1187 0.15388 0.5623

0.3 0.3882 3.443 0.1127 0.15377 0.5286

0.4 0.2962 2.844 0.1041 0.15353 0.4887

0.5 0.2147 2.313 0.0928 0.15307 0.4463

0.6 0.1479 1.876 0.0788 0. 15522 0.4044

0.7 0.0954 1.530 0.0623 0.15107 0.3644

0.8 0.0549 1.260 0.0435 0.14930 0.3271

0.9 0.0238 1.050 0.0226 0.14700 0.2929

1.0 0.0000 0.8874 0.0000 0.14410 0.2618

74

Fig 2.8: The variation of p , ,2c

p

, )(

12 d

dp

cfrom centre to surface for

u = 0.186.

2.11: Slowly rotating structures (Crab and the Vela Pulsars):

For slowly rotating structures like the Vela pulsars (rotation velocity

about 70 rad /sec), and the Crab pulsars (rotation velocity about 188 rad/sec)

one can calculate the moment of inertia in the first-order approximation which

appears in the Lense-Thrirring frame dragging effect. However, for the present

case of an exact solution, it is very useful to apply an approximate, but very

precise, empirical formula which is based on the numerical results obtained for

a large number of theoretical equations of state (EOS) of dense nuclear matter.

For the type of solution considered in the present study, the formula yields in

the following form [43-44].

I =(2/5)(1+x) MR2 ; (2.78)

where x is the dimensionless compactness parameter measured in units of

M (in km)/km, i.e.

x = (M/R)/MΘ (in km) km-1 (2.79)

Equation (2.78) is used to calculate the moment of inertia for the models

presented in Table 2.3 .

00.5

11.5

22.5

33.5

44.5

5

0 0.2 0.4 0.6 0.8 1

r/rb

pressure

density

pressure/density


red shift

75

Table 2.7: The moment of Inertia for different values of mass and radius of the

Stellar object.

S. No. M in units of MΘ Rb in km Moment of Inertia in gm cm2

1 0.026 3.973 0.003 × 1045

2 0.075 5.560 0.0187× 1045

3 0.207 7.686 0.1005× 1045

4 0.371 9.183 0.2603× 1045

5 0.556 10.318 0.499× 1045

6 0.754 11.190 0.807× 1045

7 1.262 12.479 1.73× 1045

8 1.498 12.688 2.15× 1045

9 1.591 12.685 2.30× 1045

Table 2.7 shows the value of moment of inertia for different

configurations of super massive structures. Corresponding to M = 0.7549 MΘ

and R = 11.190 km we obtain the moment of inertia for the configuration as

0.807 × 1045 gm cm2. For the mass 1.262 MΘ and R = 12.479 km equation

(2.78) yields the moment of inertia as 1.73 ×1045 gm cm2. Corresponding to

1.591 MΘ and 12.685 km the moment of inertia for the model comes about to

be 2.30 × 1045 gm cm2.These values of masses and moment of inertia agree

quite well with those of the masses and the moment of inertia of the Vela

pulsars.

76

2.12 Results and Discussions :

The density of the massive fluid sphere is positive upto the value of u = .3, beyond

this value of u the density becomes negative. Hence we are restricted to the values of u less

than .3. From Table 2.3 we observe that linear dimension, mass and density of the stellar

object firstly increases with u upto u = .175 and then decreases. Central red shift and

surface red shift are positive within the stellar object and increases with increasing values of

u. The behaviour of all the parameters is well behaved and realistic only upto the value of

u = .186 . Beyond this value of u although the behaviour of all the parameters is well behaved

but the behaviour ofdc

dp2

is not normal,dc

dp2 firstly increases and then decreases as we go

from centre to surface or we can say that the trend of adiabatic sound speed is erratic. Thus

the solution is well behaved for all values of u satisfying the inequality 0 < u <0.186.

From the Table 2.4 and Table 2.5, we observe that the pressure and density are

positive and monotonically decreasing with the increase of radial coordinate r. The fluid

parameters2c

p

, )(

12 d

dp

care positive and monotonically decreasing with the increase of r.

Since )(1

2 d

dp

cis positive and less than 1 hence the causality condition is obeyed everywhere

within the ball . We now here present a model of super dense star based on the particular

solution discussed above by assuming surface density ; 2 × 1014 g / cm3 . Corresponding to u

= 0.186, the resulting well behaved model has maximum mass M = 1.591MΘ with radius Rb =

12.685 km. The model has mass within the range of neutron star. The good matching of our

results for Vela pulsars shows the robustness of our model.

77

2.13 References :

[1] Schwarzschild K.: “ On the Gravitational Field of a Mass Point

According to Einstein's Theory”, Sitzer. Preuss. Akad. Wiss. Berlin , pp.

189 (1916).

[2] Tolman, R.C.: “Static solutions of Einstein’s field equations for

spheres of fluid”, Phys. Rev. 55, 364(1939).

[3] Negi P. S. and Durgapal, M. C.: “Motion of test particles in

parabolic density distributions”, Astrophys. & Space Science 245, 97

(1996).

[4] Negi P. S. and Durgapal,M.C.: “Stable ultracompact objects”, Gen.

Rel. Grav. , 31, 13(1999).

[5] Durgapal, M. C.: “A class of new exact solutions in relativity”, J.

Phys. A. Math. Gen. 15, 2637 (1982).

[6] Durgapal, M. C. and Fuloria, R. S.: “Analytic stellar model for super-

dense star”, Gen. Rel. Grav. 17, 671(1985).

[7] Adlar, R. J. : “A fluid sphere in general relativity ”, J. Math. Phys.

15, 727(1974) .

[8] Adams, R. C. and Cohen, J. M.: “Analytic stellar models in general

relativity”,Astrophys & Space Sci. 198, 507(1975).

[9] Kuchowicz, B.: “A Physically Realistic Sphere of Perfect Fluid to

Serve as a Model of Neutron Stars”, Astrophys. & Space Sci. 131,

33 (1975).

[10] Buchdahl, H. A. : “General-Relativistic Fluid Spheres. III. a Static

Gaseous Model”, Astrophys. J. 147, 310 (1967).

78

.

[11] Vaidya, P. C. and Tikekar, R.: “Exact relativistic model for a

superdense star” J. Astrophys. & Astron. 3, 325(1982).

[12] Durgapal, M. C. and Bannerji, R.: “A new analytical stellar model

in general relativity”, Phys. Rev. D 27, 328(1983) . Erratum D28,

2695.

[13] Durgapal, M. C. and Rawat, P. S. : “Non-rigid spheres in general

relativity”, Mon. Not. R. Astr. Soc. 192, 659(1980).

[14] Buchdahl , H. A. : “A relativistic fluid sphere resembling the

Polytrope of index 5”, Ap. J. 140, 1512 (1964).

[15] Kuchowicz, B.: “General Relativistic fluid spheres. II. Solutions of

the equation for e ”,Acta Physica Polonica 34, 131(1968c).

[16] Leibovitz, C.: “Spherically Symmetric Static Solutions of Einstein's

Equations”, Phy. Rev.D 85,1664 (1969).

[17] Mehra A. L. : “Radially symmetric distribution of matter” , J.Aust.

Math. Soc. 6, 153 (1966).

[18] Pant, D. N. and Pant, N.: “A new class of exact solutions in general

relativity representing perfect fluid balls”, Journal of Mathematical

Physics 34, 2440(1993).

[19] Pant, D. N. : “Varieties of new classes of interior solutions in general

relativity ”, Astrophysics and Space Science, 215( 1), 97(1994).

[20] Pant, N.: “Uniform radial motion of sound in a relativistic fluid

ball”, Astrophysics and Space Science 240, 187 (1996).

79

[21] Zeldovich, Ya. B. : The equation of state at ultrahigh densities and its

relativistic limitations”, Soviet Phys. JEPT 14, 1143(1962).

[22] Misner , C. W. and Zapolsky, H. S.: “High-Density Behavior and

Dynamical Stability of Neutron Star Models”, Phys. Rev. Lett. 12,

635 (1964).

[23] Durgapal, M. C. and Gehlot, G. L. : “Spheres with two density

distribution”, Phys. Rev. 183, 1102 (1971).

[24] Kramer, D., Stephani, H., Mac Callum, M and Herlt, E. : “Exact

solutions of Einstein’s field Equations”,(Cambridge University

Press, Cambridge, England)(1981).

[25] Durgapal , M. C. , Phuloria , R. S. and Pandey, A. K.: “Isothermal

neutron star cores”, Astrophys. & Space Sci. 102, 49(1984).

[26] Wyman , M. : “Radially Symmetric Distributions of Matter”, Phy.

Rev. 75, 1930 (1949).

[27] Brecher , K. and Caporasso, G. : “Obese 'neutron' stars”, Nature

259, 377 (1976).

[28] Buchdahl, H. A.: “A Relativistic Fluid Sphere Resembling the Emden

Polytrope of Index 5”, Ap. J. 140 1512 (1964).

[29] Pant, D. N. and Sah, A. : “Massive fluid spheres in general

relativity”, Physical. Review D 32, 1358 (1985).

[30] Walter Simon.: “Static perfect fluids with Pant-Sah equations of

state”, General Relativity and Gravitation. 40,2591 (2008).

[31] Delgaty, M. S. R., Lake, K. : “Physical Acceptability of Isolated, Static,

Spherically Symmetric, Perfect Fluid Solutions of Einstein’s Equations”,

80

Comput. Phys. Commun.115, 395 (1998).

[32] Heintzmann, H.: “New exact static solutions of Einsteins field

equations”, Z. Phys. 228, 489(1969).

[33] Finch , Skea: “A realistic stellar model based on an ansatz of Duorah

and Ray” , Class. Quan. Gravity 4, 467(1989).

[34] Patvardhav –Vaidya: J. Univ. Bombay 12, PartIII, 23(1943).

[35] Kuchowicz, B.: “General relativistic fluid spheres. II. Solutions of the

equation for e ”, Acta Phys. Pol 34 ,131(1968b).

[36] Matese-Whitman: “New method for extracting static equilibrium

configurations in general relativity”, Phys. Rev. D 22, 1270 (1980).

[37] Durgapal, M. C.: “A class of new exact solutions in relativity”, J.

Phys. A. Math. Gen. 15, 2637 (1980).

[38] Nariai H. : “On some static solutions of Einstein's gravitational field

equations in a spherically symmetric case”, Sci. Rep Tohoku Univ Serl

34,160 (1950).

[39] Goldman, S. P.: “Physical solutions to general-relativistic fluid

spheres”, Astrophys J. 226 1079(1978).

[40] Pant, N.: “Some new exact solutions with finite central parameters and

uniform radial motion of sound ”, Astrophys. Space Sci. 331, 633

(2011).

[41] Maurya, S.K., Gupta, Y.K.: “On a family of well behaved perfect fluid

balls as astrophysical objects in general relativity”, Astrophys Space

Sci.334(1),145 (2011).

81

[42] Pant, N. et al.: “New class of regular and well behaved exact solution

in general relativity”, Astrophys. Space Sci. 330, 353(2010).

[43] Bejger, M., Haensal, P. : “Moments of inertia for neutron and strange

stars Limits derived for the Crab pulsar”, Astron. Astrophys. 396 917

(2002).

[44] Pant N. , Negi P.S.: “Variety of well behaved exact solutions of

Einstein– Maxwell field equations, an application to Strange Quark

stars, Neutron stars and Pulsars”, Astrophys Space Sci ,338 (1), 163

(2012).

Chapter IIIA Parametric class of Regularand well behaved relativistic

Charged fluid spheres

82

Chapter III

A Parametric class of Regular and well behaved

relativistic Charged fluid spheres3

In this chapter we have studied well behaved parametric class of charge

Analogue of Durgapal’s V solution [1]. We have obtained a generalized solution of

Einstein-Maxwell field equations of general relativity for a charged , static,

spherically symmetric fluid ball. We present charged super-dense star models after

prescribing particular forms of the metric potential and electric intensity. The metric

describing the super dense stars joins smoothly with the Reissner-Nordstrom metric

at the pressure free boundary. The electric density assumed is

3

2

2

21

1

2

)61()1(2

xxKx

x

qc

c

E n where n may take the values 0 ,1, 2, 3, 4 and so on

and K is a positive parameter. For n = 0 we rediscovered the Gupta, Maurya

solution [2]. For n = 1 we get the Fuloria, et al. solution [3]. The solution for n =

1, n = 2 and n = 3 have been discussed extensively keeping in view of well behaved

nature of the charged solution of Einstein–Maxwell field equations.

3 Some part of this chapter has been published in J. Modern physics, 2,(10)1156 (2011) & Another part has been published in Astrophysics & SpaceScience, 341,469 (2012).

83

3.1 Introduction:

Exact solutions with well-behaved nature of Einstein-Maxwell field equations

are of great interest in relativistic astrophysics. Such solutions may be used to make a

suitable model of super dense object with charge matter like Neutron star, quark star,

pulsar and analogous stars. It is interesting to observe that, in the presence of charge,

the gravitational collapse of a spherical symmetric distribution of the matter to a

point singularity may be avoided because the presence of some charge in a spherical

material distribution provides an additional resistance against the gravitational

contraction by means of electric repulsion. The inclusion of charge plays very

important role in the stability of massive fluid spheres. A spherical fluid

distribution of uniform density with a net charge on the surface is more stable than a

surface without charge[4,5] . According to De Felice et al.[6] the inclusion of charge

inhibits the growth of space time curvature and which therefore plays a key role in

avoiding singularities. Thus a stable massive charged configuration can be produced

in which repulsive force from the charge counter balances the gravitational attraction.

The singularities which appear in the Schwarzschild solution corresponding to

spherically symmetric static perfect fluid distribution can be avoided to a great extent

by including charge in them. The negative pressure of Bayin’s solution[7] makes it

physically unreasonable, however the inclusion of charge in Bayin’ solution makes it

physically reasonable i.e. the pressure is a decreasing function of radius from centre

to surface (Ray et al.) [8] . Bonner [9] has shown that dust cloud of arbitrarily large

mass and small radius can remain in equilibrium if it has an electric charge density

related to the mass density by . All these advantages of charged spherical

models motivated us to consider the charge analogue of Durgapal’ solution[1].

To include the electric charge a number of authors make additional assumptions

such as an equation of state or a relationship between metric variables [10-13].

Bonnor and Wickramasuriya [14] have studied electrically charged matter, with

electrostatic repulsion balancing the gravitational attraction. Most works have been

done under static conditions, including the ones by Ivanov [15], who exhaustively

surveyed static charged perfect fluid spheres in general relativity, and Ray et al. [16],

84

who studied the effect of electric charge in compact stars and its consequences on the

gravitational collapse. Bekenstein [17] found that for highly relativistic stars, whose

radius is on the verge of forming an event horizon, the large gravitational pull can be

balanced by large amounts of net charge.

The Reissner-Nordstrom solution for the external field of a charged fluid

sphere has two distinct singularities at finite radial positions unlike Schwarzschild

solution which has the singularity at the centre also. Thus the Reissner-Nordstrom

solution describes the bridge between two asymptotically flat spaces and an electric

flux flowing across the bridge. Graves and Brill [18] pointed out that the region of

minimum radius pulsates periodically between these two surfaces due to Maxwell

pressure of the electric field. Thus the study of charged fluid spheres is important

due to having many important implications. Many Exact solutions of Einstein’s field

equations which are not well behaved and regular become well behaved after

including charge in them. A classical model of an electron may be represented by

charged dust if most of its characteristics remain finite and non- trivial while the

junction radius shrinks to zero. Ray and Das[19] , Bonner and Cooperstock [20]

have assumed that charged fluid balls with electromagnetic mass may be used to

depict the model of an electron.

On account of the non linearity of Einstein Maxwell field equations , not many

realistic well behaved, analytic solutions are known for the description of charged

fluid spheres. Astrophysicists have been using exact solutions of Einstein’s field

equations as seed solutions for the modeling of charged fluid spheres. Bonnor [21],

Efinger [22], Kyle and Martin [23], Krori and Barua [24] and Nduka [25] have

obtained internal solutions for static spherically symmetric charged fluid spheres

under different conditions.

Our aim is also to find a new parametric class of exact solutions of Einstein–

Maxwell field equations for a static, spherically symmetric distribution of the

charged and perfect fluid with well behaved nature and to construct suitable models

of super dense objects with charge matter. Many of the authors electrified the well

known uncharged fluid spheres e.g. Durgapal-Fuloria solution [26] by Gupta and

Maurya [27], Schwarzschild solution [28] by Bijalwan and Gupta [29] and by Gupta

and Kumar [30], Kuchowicz solution [31] by Nduka[32] , Tolman solution [33] by

Cataldo and Mitskievic [34], Heintzmann’s solution [35] by Pant et al. [36] , Adler’s

solution by Pant et al.[37] and M. J. Pant andTewari [38] and so on. These coupled

85

solutions are well behaved with some positive values of charge parameter K and

completely describe interior of the super-dense stellar objects with charge matter.

We have tried to charge the Durgapal’s V solution [1] after prescribing

particular forms of metric potential g44 and electric intensity .Some new solutions of

Einstein’s Maxwell field equations have been obtained keeping in view their well

behaved nature. Models of super massive charged fluid spheres have been

constructed by working out new solutions of Einstein’s Maxwell Field Equations.

3.2 The solutions that are used as seed solutions for making

charged fluid model:

The solutions that may be used as seed solutions for the construction of charged

fluid models may be categorized as follows:[39]

Type I : If the solutions are well behaved; such solutions may be used to describe

the interior of super massive stellar objects and may be used as seed solutions for

constructing relativistic charged fluid ball models. Such charged solutions and their

neutral counterparts both show well behaved nature and regularity .

Type II : If the solutions are not well behaved, but with finite parameters; such

solutions may be taken as seed solutions of super dense star with charge matter,

Schwarzschild’s interior solution is not well behaved as causality condition is not

obeyed throughout within the fluid sphere. However, charge analogues of the

solution are well behaved for wide range of constants (Gupta and kumar [30];

Bijalwan, N. and Gupta [29]; Florides [40] ). Adler solution[41] and Durgapal and

Fuloria solution [26] are also not well behaved as the speed of sound is

monotonically increasing from centre to boundary. However, charge Analogues of

the solution is well behaved for wide range of constants [42]. Heintzmann’s neutral

solution [35] is not well behaved as the speed of sound is monotonically increasing

from centre to boundary . However the charge analogue of solution is well behaved

for wide range of constant [36].

86

3.3 Assumptions that must be satisfied in order for the solution to

be well behaved [43-44]:

(i) The solution should be free from physical and geometrical singularities i.e. finite and

positive values of central pressure , central density i.e. p0 > 0 and ρ0 > 0 and non

zero positive values of e and e . For such solutions the tangent 3-space at the centre

is flat and it is an essential condition. For curvature coordinates, mathematically it is

expressed as 10

re and 0re positive constant[45-46].

(ii) The solution should have positive and monotonically decreasing expressions for

pressure and density( andp )with the increase of r. i.e.

(a) pandpandrp r 000 0 is negative valued function for r > 0.

(b) andandr r 000 0 is negative valued function for r > 0.

(iii) The solution should have positive value of ratio of pressure-density and less than 1

with in the ball i. e. 102

c

p

.

(iv) The solution should have positive and monotonically decreasing expression for

fluid parameter 2c

p

with the increase of r .

(v) The solution should have positive and monotonically decreasing expression for

velocity of sound

d

dpwith the increase of r and causality condition should be


dc

dp.

(vi) The central red shift Z0 and surface red shift Zb should be positive and finite i.e.

010

20

r

eZ

and 012

b

eZb

and both should be bounded and

monotonically decreasing in nature with the increase of r i.e. 00

rdr

dz

(vii) Electric intensity E is positive and monotonically increasing from centre to

boundary and at the centre the electric intensity is zero.

(viii) The pressure at the boundary should be zero.

87

3.4 Einstein – Maxwell equations for charged fluidDistribution:

We consider a spherical symmetric metric in curvature coordinates

22222222 )sin( dtedrdrdreds (3.1)

where the functions )(r and )(r satisfy the Einstein-Maxwell equations

mn

mnijjm

imij

jiij

ij

ij FFFFpvvpc

c

GRRT

c

G

4

1

4

1)(

8

2

18 244

(3.2)

where , p, iv , Fij denote energy density, fluid pressure, velocity vector and skew-

symmetric electromagnetic field tensor respectively.

In view of the metric (3.1), the field equation (3.2) gives [47]

4

2

42

81

r

qp

c

G

r

ee

r

(3.3)

4

2

4

2 8

2442 r

qp

c

Ge

r

(3.4)

4

2

22

81

r

q

c

G

r

ee

r

(3.5)

where prime ( ' ) denotes the differentiation with respect to r and q(r)

represents the total charge contained with in the sphere of radius r.

By using the transformation

ZeandrcxxBe 21

5 ,)1( . (3.6)

where B being the positive constant. Now putting (3.6) into (3.3)-(3.5), we have

88

41

2

21 81)1(

)1(

10

c

PG

cx

qc

x

Z

x

Z

(3.7)

41

2

21 81

2)1(

c

G

cx

qc

dX

dZ

x

Z

(3.8)

and Z satisfying the equation

)61(

)1(1)/2(

)61(1

1214 122

xx

xxcqZ

xxxx

xx

dx

dZ

(3.9)

3.5 A New Generalised solution of Einstein-Maxwell Field

Equations:

In order to solve the differential equation (3.9) we consider the electric intensity E

of the following form

3

2

2

21

1

2

)61()1(2

xxKx

x

qc

c

E n (3.10)

where K is a positive constant. The electric density is so assumed that the model is

physically significant and well behaved i.e. E remains regular and positive

throughout the sphere.

In view of (3.10) differential equation (3.9) yields the following solution:

3

3

1

2

33/1

2

)1(

)61(112

)854309(1

)1(

1

)61(

)1(

)5(x

x

Axxxx

xx

xx

n

KZ

n

(3.11)

where A is an arbitrary constant of integration.

89

3.6 Properties of the new generalised solution:

Using (3.11) into (3.7) and (3.8), we get the following expressions for pressure and

energy density,

3

1)61)(5(2

))652()29(2()1(

)61(

)111(

112

)20010504125475(

1

181

2

3

1

32

441

xn

xnxnxK

x

xAxxx

xp

c

G

c

n

(3.12)

3

4

32

3

4

232

421

)61)(5(2

})60256()40178()553(6{)1(

)61(

)22113(

112

)120450151935(

1

181

xn

xnxnxnxK

x

xxA

xxx

xc

G

c

n

(3.13)

The expressions for central pressure and central density are given by

)5(112

4758104

1 n

KAp

c

G

c

(3.14)

)5(2

63

112

19358102

1 n

KA

c

G

c

(3.15)

Differentiating (3.12) and (3.13) w. r. t. x, we get:

3

4)61)(5(2

])36372520(

)12302740(

)58245()325([)1(

)61()1(

)4431(5

)1(112

)860411241(2581 32

22

21

3

45

2

5

32

41 xn

xnn

xnn

xnnnxK

xx

xxA

x

xxx

dx

dp

c

G

c

n

(3.16)

90

)8866395(

)61()1(

5)83657515(

)1(112

1581 32

3

75

3254

1

xxx

xx

Axxx

xdx

d

c

G

c

42

32

222

3

7

1

)36021362560(

)30022644040(

)709061730()5164255()115(

)61)(5(2

)1(

xnn

xnn

xnnxnnn

xn

xK n

(3.17)

The expression for the square of velocity of sound is given by

)5)((10)5()61)((112/30)()1(

)]5)((10)5()61()(112/50)()1([)61(13/74

3/44

2 nxATnxxHxMxK

nxQAnxxPxNxKx

d

dp

c n

n

(3.18)

Where

32 83657515)( xxxxH 32 8866395)( xxxxT

42

32222

)36021362560(

)30022644040()709061730()5164255()115()(

xnn

xnnxnnxnnnxM

32222 )36372520()12320740()58245()325()( xnnxnnxnnnxN

32 860411241)( xxxxP 24431)( xxxQ

)5(50)5(56

772527

)5(10)5(56

602531

1

02

nAnK

nAnK

d

dp

c

(3.19)

91

3.7 Variety Of classes of solutions:

3.7.1 Case 1 (n =0):-- The solution for n = 0 :

When n = 0, we get Gupta, Maurya Solution [2].

The resulting Solution is

3

3

1

2

33/1

2

)1(

)61(112

)854309(1

)1(

1

)61(

)1(

5

x

x

Axxxx

xx

xxKeZ

(3.20)


In view of Eqs. (3.12) and (3.13) the expressions for pressure and energy density aregiven by

3

1)61(

)52292(

10)61(

)111(

112

)20010504125475(

1

181 2

3

1

32

441 x

xxK

x

xAxxx

xp

c

G

c

(3.21)

3

4

32

3

4

232

21 )61(

)256178536(

10)61(

)22113(

112

)120450151935(

41

181

x

xxxK

x

xxA

xxx

xc

G

c

(3.22)

The solution gives wide range of constant K ( 0 < K 50) for which the solution

is well behaved and therefore suitable for modeling of superdense star.

3.7.2 Case 2( n = 1): The solution for n = 1

For n = 1 we get Fuloria et al. Solution [3] .

In order to solve the differential equation (3.9) we consider the electric intensity E of

the following form (n = 1)

92

3

2

2

21

1

2

)61()1(2

xxKx

x

qc

c

E (3.23)

where K is a positive constant. In view of equation (3.11) we get the following solution

3

3

1

2

33/1

3

)1(

)61(112

)854309(1

)1(

1

)61(

)1(

6x

x

Axxxx

xx

xxKe

(3.24)


3.8 Properties of the new solution for n = 1:

In view of Equations (3.12) and (3.13), we get the following expressions for pressure and

energy density

3

1)61(

)29151()1(

6)61(

)111(

112

)20010504125475(

1

181 2

3

1

32

441 x

xxxK

x

xAxxx

xp

c

G

c

(3.25)

3

4

32

3

4

232

21 )61(

)158109293()1(

6)61(

)22113(

112

)120450151935(

41

181

x

xxxxk

x

xxA

xxx

xc

G

c

(3.26)

The expressions for central pressure and central density are given by the

following equations:

6112

4758104

1

KAp

c

G

c

(3.27)

23

112

19358102

1

KA

c

G

c

(3.28)

93

In view of equations ( 3.16) and (3.17) we get:

3

4)61(6

)46452715214(

)61()1(

)4431(5

)1(112

)860411241(2581 32

3

45

2

5

32

41 x

xxxK

xx

xxA

x

xxx

dx

dp

c

G

c

(3.29)

432

3

7

32

3

75

3254

1

2528330213532128

)61(6

)8866395(

)61()1(

5)83657515(

)1(112

1581

xxxx

x

K

xxx

xx

Axxx

xdx

d

c

G

c

(3.30)

3

75

112

6025)

81( 04

1

KA

dx

dp

c

G

c

(3.31)

3

425

112

7725)

81( 04

1

KA

dx

d

c

G

c

(3.32)

The velocity of sound is given by the following expression:

53

7432

3

73232

53

432

3

4232

2

)1()61)(2528330213532128(30

112

)61)(8866395(112)83657515(3

)1()61)(46452715214(30

112

)61)(4431(112)860411241(5

1

xxxxxxK

xxxxAxxx

xxxxxK

xxxAxxx

d

dp

c

(3.33)

94

The velocity of sound at the centre of the massive fluid sphere is given by

KA

KA

d

dp

c x

15

4485601545

15

7841121205

)1

( 02

(3.34)

The expression for gravitational red-shift z is given by

1)1( 2

5

B

xz (3.35)

The central value of gravitational red shift to be non zero positive finite, we have

01 B (3.36a)

Differentiating (3.35) w.r.t. x, we get,

02

5

0

Bdx

dz

x

(3.36b)

The expression of right hand side of (3.36b) is negative, thus the gravitational redshift is

maximum at the centre and monotonically decreasing.\

3.9 Boundary Conditions :

The solutions so obtained are to be matched over the boundary with Reissner-Nordstrom metric:

22

222222

1

2

22 2

1)sin(2

1 dtr

e

r

GMddrdr

r

e

r

GMds

(3.37)

which requires the continuity of ee , and q across the boundary r = r b

2

2

2)( 2

1bb

rb

r

e

rc

GMe (3.38)

95

2

2

2)( 2

1bb

rb

r

e

rc

GMe (3.39)

q(rb) = e (3.40)

p(rb) = 0 (3.41)

The condition (3.41) can be utilized to compute the values of arbitrary constantsA as follows:

Pressure at p ( r = rb) = 0 gives

)29151()111(

)1(

6112

20010504125475

)111(

)61( 25323

1

XXX

XkXXX

X

XA

(3.42)

In view of (3.38) and (3.39) we get,

3

13

2

3

3

1

3

5

)61()1(112

854309(1

)1(

1

)61(

)1(

6)1(

1

XX

AXXXX

XX

XXK

xB

(3.43)

The expression for mass can be written as:

53

22

2)1()61()1(

21

2XBXXX

Kr

c

GM b (3.44)

The expression for surface density is given by

96

3

4

32

3

4

2322

2

)61(

)158109293()1(

6

)61(

)22113(

112

)120450151935(

41

18

X

XXXXK

X

XXA

XXX

Xr

c

Gbb

(3.45)

We have obtained the expressions for pressure, density, pressure-density ratio and

square of adiabatic sound speed within the charged fluid sphere for the solution n =1.

Now we study the variation of different physical quantities from centre to surface .

If pressure, density, pressure-density ratio , red shift would be positive within the

charged fluid sphere and monotonically decreasing from centre to surface ,then the

solution will be suitable for modeling of super massive charged fluid balls.

Table 3.1 : The variation of various physical parameters at the centre, surface density,electric field intensity on the boundary, mass and radius of stars with different values of Kand X = .2

K 041

81p

c

G

c

02

1

81

c

G

c 0

02

1

p

c 02)(

1xd

dp

c 0zbr

c

E

1

22

2

8bb r

c

G

M

Mbr

km

1 5.167 14.496 0.3565 0.9298 1.63 0.202 1.3728 4.41 19.20

2 4.664 16.0037 0.2914 0.7017 1.68 0.404 1.277 4.60 18.52

4 3.660 19.0156 0.1925 0.4806 1.814 0.808 1.060 4.83 16.87

5 3.158 20.5217 0.1538 0.4186 1.885 1.01 0.955 4.88 16.02

6 2.656 22.0275 0.1205 0.3728 1.962 1.212 0.851 4.89 15.12

8 1.652 25.0394 0.0659 0.3411 2.134 1.616 0.643 4.75 13.14

10 0.648 28.0513 0.0231 0.2655 2.342 2.02 0.434 4.31 10.80

97

Table 3.2 : The variation of various physical parameters at the centre, surface density,

electric field intensity on the boundary, mass and radius of stars with different values of X

and K = 2

X 041

81p

c

G

c

02

1

81

c

G

c 0

02

1

p

c 02)(

1xd

dp

c 0zbr

c

E

1

22

2

8bb r

c

G

M

Mbr

In km

.10 3.6036 19.1889 0.1889 0.5628 0.754 0.1504 1.181 2.95 17.81

.15 4.2883 17.1349 0.2502 0.6416 1.196 0.2646 1.282 3.89 18.55

.20 4.6643 16.0037 0.2914 0.7017 1.688 0.404 1.277 4.60 18.52

.25 4.8138 15.5583 0.3094 0.7300 2.276 0.5756 1.168 5.13 17.71

.30 4.764 15.7075 0.3032 0.7202 3.000 0.7747 0.989 5.50 16.29

.35 4.5018 16.4343 0.2751 0.6738 3.985 1.0045 0.7276 5.52 13.98

98

Table 3.3 : The variation of various physical parameters at the center, surface density, electric fieldintensity on the boundary, mass and radius of stars with different values of K and X = .1

_________________________________________________________________________________

K 041

81p

c

G

c

02

1

81

c

G

c 0

02

1

p

c 02)(

1xd

dp

c 0zbr

c

E

1

22

2

8bb r

c

G

M

Mbr in km

________________________________________________________________________________

1 3.79 18.61 0.2037 0.6366 0.760 0.075 1.200 2.95 17.95

2 3.60 19.18 0.1877 0.5628 0.766 0.150 1.181 2.99 17.81

3 3.22 20.32 0.1585 0.4542 0.780 0.300 1.142 3.08 17.51

4 3.03 20.89 0.1451 0.4131 0.787 0.376 1.122 3.12 17.36

5 2.84 21.46 0.1324 0.3781 0.794 0.451 1.103 3.16 17.21

6 2.46 22.60 0.1089 0.3210 0.808 0.601 1.064 3.23 16.90

8 2.08 23.74 0.0877 0.278 0.823 0.752 1.024 3.30 16.59

10 1.89 24.31 0.0778 0.260 0.830 0.827 1.005 3.33 16.43

12 1.70 24.88 0.0684 0.244 0.838 0.902 0.985 3.36 16.27

14 1.32 26.02 0.0594 0.216 0.853 1.053 0.946 3.42 15.94

16 0.94 27.16 0.0347 0.193 0.869 1.283 0.907 3.47 15.61

18 0.86 28.30 0.0199 0.173 0.885 1.354 0.868 3.51 15.27

20 0.184 29.44 0.0062 0.157 0.902 1.504 0.829 3.55 14.92

99

Table 3.4 : The variation of various physical parameters at the center, surface density, electricld intensity on the boundary, mass and radius of stars with different values of K and X =.01

K 041

81p

c

G

c

02

1

81

c

G

c 0

02

1

p

c 02)(

1xd

dp

c 0zbr

c

E

1

22

2

8bb r

c

G

M

Mbr

In km

2 0.653 28.03 0.0233 0.405 0.07479 0.010 0.2640 0.2565 8.422

4 0.623 28.13 0.0221 0.363 0.07485 0.021 0.2638 0.2569 8.418

6 0.592 28.22 0.0210 0.326 0.07491 0.031 0.2636 0.2574 8.415

8 0.562 28.31 0.0198 0.293 0.07496 0.042 0.2634 0.2579 8.411

10 0.531 28.40 0.0187 0.264 0.07502 0.052 0.2631 0.2583 8.407

12 0.501 28.49 0.0175 0.238 0.07510 0.063 0.2629 0.2588 8.404

14 0.470 28.58 0.0164 0.214 0.07514 0.073 0.2627 0.2592 8.400

16 0.440 28.67 0.0153 0.193 0.07516 0.084 0.2624 0.2596 8.396

20 0.379 28.86 0.0131 0.156 0.07533 0.105 0.2620 0.2606 8.389

30 0.226 29.32 0.0077 0.086 0.07550 0.157 0.2609 0.2629 8.371

40 0.073 29.78 0.0024 0.037 0.07581 0.210 0.2597 0.2651 8.353

44 0.012 29.96 0.0004 0.021 0.07598 0.231 0.2593 0.2660 8.345

100

Table 3.5 : The march of pressure, density, pressure-density ratio, square of adiabatic

sound speed , red shift, electric intensity within the perfect fluid spherecorresponding to K = 1 and X = .1

_________________________________________________________________________

r/rb2

4

8brp

c

G 2

2

8brc

G

2c

p

)(

12 d

dp

cz

1

2

c

E-

__________________________________________________________________________________________

0.0 0.3603 1.9189 0.1877 0.5628 0.7668 0

0 .1 0.3549 1.9093 0.1859 0.5601 0.7623 0.0010

0.2 0.3391 1.8810 0.1802 0.5519 0.7492 0.0040

0.3 0.3135 1.8349 0.1708 0.5381 0.7276 0.0094

0.4 0.2795 1.7726 0.1576 0.5188 0.6980 0.0172

0.5 0.2386 1.6961 0.1407 0.4938 0.6610 0.0281

0.6 0.1928 1.6077 0.1199 0.4631 0.6172 0.0424

0.7 0.1440 1.5097 0.0950 0.4269 0.5676 0.0610

0.8 0.0942 1.404 0.0671 0.3853 0.5129 0.0845

0.9 0.0456 1.294 0.0352 0.3388 0.4541 0.1140

1.0 0.0000 1.181 0.0000 0.2882 0.3922 0.1504

101

Fig 3.1 : The march of various physical parameters from centre to surface. ( K1, X = .1)

3.10 New well behaved solution ( n = 2)

Case III- When n = 2

We have the expression for electric density as

3

22

1

2

)61()1(2

1xxxK

c

E

(3.46)

201

2 K

c

E

dx

d

x

(3.47)

The electric intensity is minimum at the centre and monotonically increasing for all

values K > 0. Also at the centre it is zero. Thus we have, 0,0 1 cK .

In view of equation (3.11) the resulting class of solution is

3

3

1

2

33/1

4

)1(

)61(112

)854309(1

)1(

1

)61(

)1(

7

x

x

Axxxx

xx

xxKe

(3.48)

In view of Equations (3.12) and (3.13) we get,

0

0.5

1

1.5

2

2.5

0 0.5 1

fractional radius r/rb

pressure

density

pressure/density


Red shift

Electric density

102

3

1)61(

)64312()1(

14

)61(

)111(

112

)20010504125475(

1

181

22

3

1

32

441

x

xxxK

x

xAxxx

xp

c

G

c

(3.49)

3

4

322

3

4

232

21

)61(

)376258636()1(

14

)61(

)22113(

112

)120450151935(

41

181

x

xxxxK

x

xxA

xxx

xc

G

c

(3.50)

3.11 Properties of the new class of solution:

Central values of pressure and density are given by

7112

4758104

1

KAp

c

G

c

(3.51)

7

33

112

19358102

1

KA

c

G

c

(3.52)

In view of equations (3.16) and (3.17) we get,

3

4)61(14

)1408139236531()1(

)61()1(

)4431(5

)1(112

)860411241(2581

32

3

45

2

5

32

41

x

xxxxK

xx

xxA

x

xxx

dx

dp

c

G

c

(3.53)

43

2

3

7

3

45

32

5

32

41

82729768

382260327

)61(14

)1(

)61()1(

)8866395(5

)1(112

)83657515(1581

xx

xx

x

xK

xx

xxxA

x

xxx

dx

d

c

G

c

(3.54)

103

AK

dx

dp

c

G

c5

112

6025

14

3181

0

41

(3.55)

AK

dx

d

c

G

c25

112

7725

14

2781

0

41

(3.56)

)(70)61)((16

30)()1(

)](70)61()(16

50)()1([

)61(1

3/76

3/46

2

xATxxHxMxK

xQAxxPxNxKx

d

dp

c

(3.57)

where

32 83657515)( xxxxH 432 82749768382260327)( xxxxxM

32 8866395)( xxxxT 32 1408139236531)( xxxxN

32 860411241)( xxxxP

24431)( xxxQ

AK

AK

d

dp

c750

8

772527

708

602531

1

02

(3.58)

3.12 Boundary Conditions:The solutions so obtained are to be matched over the boundary with Reissner-

Nordstrom metric:

22

222222

1

2

22 2

1)sin(2

1 dtr

e

r

GMddrdr

r

e

r

GMds

(3.59)

which requires the continuity of ee , and q across the boundary r = rb

The condition (3.41) can be utilized to compute the values of arbitrary constantsA as follows:

Pressure at p( r = rb) = 0 gives

104

)24312()111(

)1(

14112

20010504125475

)111(

)61( 26323

1

XXX

XkXXX

X

XA

(3.60)

In view of (3.38 ) and (3.39) we get

3

13

2

3

3

1

3

5

)61()1(112

854309(1

)1(

1

)61(

)1(

6)1(

1

XX

AXXXX

XX

XXK

xB

(3.61)

The expression for mass can be written as

53

222

2)1()61()1(

21

2XBXXX

Kr

c

GM b (3.62)


3

4

32

3

4

2

32

2

2

)61(

)158109293()1(

6

)61(

)22113(112

)120450151935(

41

18

X

XXXXK

X

XXA

XXX

Xr

c

Gbb

(3.63)

105

Table 3.6 : The variation of various physical parameters at the centre, surface density,electric field intensity on the boundary, mass and radius of stars with different

values of K and X = .1 .

K 041

81p

c

G

c

02

1

81

c

G

c 0

02

1

p

c 02)(

1xd

dp

c 0zbr

c

E

1

22

2

8bb r

c

G

M

Mbr in

km

2 3.5774 19.2676 0.1856 0.5603 0.31 0.150 1.177 0.4917.78

4 3.17103 20.4857 0.1548 0.4514 0.34 0.300 1.343 0.8417.45

6 2.7653 21.7039 0.12741 0.3757 0.38 0.451 1.091 1.1717.12

8 2.3593 22.922 0.1029 0.3199 0.42 0.601 1.048 1.5016.78

10 1.9538 24.1401 0.08091 0.2772 0.45 0.752 1.005 1.8016.43

12 1.5472 25.3582 0.0610 0.2434 0.50 0.902 0.962 2.1216.07

14 1.1412 26.5763 0.04294 0.2161 0.55 1.053 0.919 2.3715.71

16 0.7332 27.7944 0.02645 0.1934 0.60 1.203 0.876 2.6315.34

106

Table 3.7: The march of pressure, density, pressure-density ratio , square of adiabatic soundspeed ,red shift, electric intensity within the perfect fluid sphere corresponding to K = 1 andX = .2 .

K r/rb 2

4

8brp

c

G 2

2

8brc

G

2c

p

)(

12 d

dp

cz

1

2

c

E

0.0 1.019 2.9429 0.3462 0.9748 1.6410 0.0000

0.1 0.999 2.9217 0.3422 0.9688 1.6279 0.0010

0.2 0.944 2.8588 0.3302 0.9516 1.5889 0.0041

0.3 0.855 2.7562 0.3105 0.9245 1.5258 0.0099

0.4 0.7420 2.6176 0.2834 0.8890 1.4410 0.01915

0.5 0.6106 2.4480 0.2494 0.8462 1.3378 0.03283

0.6 0.4709 2.2535 0.2089 0.7960 1.2197 0.05256

0.7 0.3320 2.0408 0.1627 0.7375 1.0906 0.08040

0.8 0.2025 1.8162 0.1115 0.6680 0.9543 0.11906

0.9 0.0898 1.5853 0.0566 0.5829 0.8145 0.17198

1.0 0.0000 1.3521 0.0000 0.4712 0.6742 0.24358

In view of Table 3.6 we observe that pressure, density, pressure density ratio, red shift,

electric intensity all are positive at the centre. Assuming the surface density

314 /102 cmgb the mass and radius has been estimated for different values of K

corresponding to X =.1. In view of the Table 3.7( K=1, X= .2) we observe that pressure,

density, pressure-density ratio and square of adiabatic sound speed and red-shift decrease

monotonically with the increase of radial coordinate and electric intensity increases

monotonically with the increase of radial coordinate within the perfect fluid sphere .

107

Fig 3.2: The Behaviour of pressure, density, pressure-density ratio , square of

adiabatic sound speed, red shift Z, electric intensity1

2

c

Eversus radius for

K = 1, X = .2.

3.13: The solution for n = 3

Case IV- When n = 3

We have the expression for electric intensity as

3

23

1

2

)61()1(2

1xxxK

c

E

(3.64)

201

2 K

c

E

dx

d

x

(3.65)

The electric intensity is minimum at the centre and monotonically increasing for all values

K > 0. Also at the centre it is zero. Thus we have, 0,0 1 cK .

In view of equations (3.11), (3.12), (3.13) we get ,

0

0.5

1

1.5

2

2.5

3

3.5

0 0.2 0.4 0.6 0.8 1


pressure

density

pressure/density


electric field

red shift

108

3

3

1

2

33/1

5

)1(

)61(112

)854309(1

)1(

1

)61(

)1(

8

x

x

Axxxx

xx

xxKe

(3.66)

3

1)61(

)70322()1(

16)61(

)111(

112

)20010504125475(

1

181 23

3

1

32

441 x

xxxK

x

xAxxx

xp

c

G

c

(3.67)

3

4

323

3

4

232

21 )61(

)436298686()1(

16)61(

)22113(

112

)120450151935(

41

181

x

xxxxK

x

xxA

xxx

xc

G

c

(3.68)

3.14: Properties of the new class of solution (n = 3)


8112

4758104

1

KAp

c

G

c

(3.69)

8

33

112

19358102

1

KA

c

G

c

(3.70)

In view of equations (3.16) and (3.17) we get,

34

)61(16

)1960175437034()1(

)61()1(

)4431(5

)1(112

)860411241(2581 322

3

45

2

5

32

41 x

xxxxK

xx

xxA

x

xxx

dx

dp

c

G

c

(3.71)

109

432

3

7

2

3

45

32

5

32

41

1220813532507879238

)61(16

)1(

)61()1(

)8866395(5

)1(112

)83657515(1581

xxxx

x

xK

xx

xxxA

x

xxx

dx

d

c

G

c

(3.72)

AK

dx

dp

c

G

c5

112

6025

16

3481

04

1

(3.73)

AK

dx

d

c

G

c25

112

7725

16

3881

04

1

(3.74)

In view of equation (3.18) we get

)(80)61)((14

30)()1(

)](80)61()(14

50)()1([)61(1

3/77

3/47

2

xATxxHxMxK

xQAxxPxNxKx

d

dp

c

(3.75)

where

32 83657515)( xxxxH 432 1220813532507879238)( xxxxxM

32 8866395)( xxxxT 32 860411241)( xxxxP

24431)( xxxQ

AK

AK

d

dp

c400

7

772538

807

602534

1

02

(3.76)

Pressure at )( brrp = 0 gives

110

)70322()111(

)1(

16112

20010504125475

)111(

)61( 27323

1

XXX

XKXXX

X

XA

(3.77)

In view of (3.38) and (3.39) we get

3

13

2

3

3

1

5

5

)61()1(112

854309(1

)1(

1

)61(

)1(

6)1(

1

XX

AXXXX

XX

XXK

xB

(3.78)

3

18

2

8

3

1

)61()1(112

854309(1

)1(

1

)61(6

XX

AXXXX

XX

XKB

(3.79)


53

232

2)1()61()1(

21

2XBXXX

Kr

c

GM b (3.80)


3

4

323

3

4

232

2

2

)61(

)436298686()1(

16

)61(

)22113(

112

)120450151935(

41

1

8

X

XXXXK

X

XXA

XXX

XXr

c

Gbb

(3.81)

111

Table 3.8: The variation of various physical parameters at the centre, surface density,

electric field intensity on the boundary, mass and radius of stars with different

values of K and X = .1

K K04

1

81p

c

G

c

02

1

81

c

G

c 0

02

1

p

c02

)(1

xd

dp

c 0z

brc

E

1

2

2

2

8bb r

c

G

M

Mbr in

in kmk

0.02 3.98 18.06 0.220 0.728 0.7536 0.0018 1.219 2.909 18. 10

0.08 3.96 18.09 0.219 0.721 0.7540 0.0072 1.218 2.912 18.09

0.2 3.94 18.16 0.217 0.709 0.7550 0.0182 1.215 2.919 18.06

0.4 3.90 18.28 0.213 0.689 0.7566 0.0364 1.210 2.930 18.03

0.6 3.86 18.40 0.209 0.670 0.7582 0.0546 1.206 2.941 17.99

0.8 3..82 18.52 0.206 0.652 0.7598 0.072 1.201 2.952 17.96

4.0 3.18 20.44 0.155 0.452 0.7862 0.364 1.125 3.118 17.38

8.0 2.38 22.83 0.104 0.320 0.8210 0.728 1.031 3.296 16.64

12.0 1.59 25.22 0. 063 0.243 0.8578 1.092 0.936 3.438 15.86

19.0 0.19 29.41 0.006 0.165 0.9280 1.729 0.771 3.59 14.39

112

Table 3.9: The values of various physical parameters at the center, surface density,electric field intensity on the boundary, mass and radius of star with different values of Xand K = 2

K X04

1

81p

c

G

c

02

1

81

c

G

c 0

02

1

p

c 02)(

1xd

dp

c 0z

brc

E

1

2

2

2

8bb r

c

G

M

Mbr in km

in k

0.001 0.10 28.67 0.003 0.389 0.007 0.001 0.029 0.009 2.81

0.004 0.31 29.06 0.010 0.395 0.029 0.004 0.113 0.071 5.53

0.01 0.68 27.93 0.024 0.406 0.074 0.017 0.264 0.256 8.42

0.02 1.23 26.30 0.046 0.424 0.149 0.022 0.470 0.624 11.23

0.04 2.08 23.74 0.087 0.459 0.299 0.051 0.765 1.360 14.33

0.06 2.72 21.83 0.124 0.495 0.451 0.087 0.959 2.003 16.05

0.08 3.20 20.37 0.157 0.528 0.607 0.130 1.088 2.549 17.09

0.1 3.58 19.24 0.186 0.561 0.769 0.182 1.172 3.017 17.75

0.2 4.38 16.83 0. 260 0.656 1.741 0.584 1.177 4.712 17.78

0.3 3.78 18.64 0.203 0.580 3.497 1.309 0.588 5.128 12.57

From the Table 3.9 we observe that central pressure, the ratio of pressure anddensity, surface density, radius increases with the increasing values of X upto X =.2 and then decreases corresponding to K = 2. Beyond the value of X= .3 the solution is not well behaved

113

Table 3.10 : The march of pressure, density, pressure-density ratio, square ofadiabatic sound speed , red shift, electric intensity within the perfect fluidsphere corresponding to K = 1 and X = .1

r/rb2

4

8brp

c

G 2

2

8brc

G

2c

p

)(

12 d

dp

cz

1

2

c

E

0.0 0.376 1.870 0.2013 0.6331 0.761 0 .0000

0.1 0.371 1.861 0.1994 0.6319 0.757 0.0005

0.2 0.355 1.835 0.1934 0.6281 0.744 0.0020

0.3 0.329 1.794 0.1834 0.6216 0.722 0.0047

0.4 0.294 1.734 0.1693 0.6120 0.692 0.0089

0.5 0.252 1.668 0.1512 0.5990 0.656 0.0147

0.6 0.205 1.588 0.1290 0.5821 0.612 0.0228

0.7 0.154 1.498 0.1028 0.5606 0.562 0.0335

0.8 0.101 1.402 0.0724 0.5340 0.508 0.0478

0.9 0.049 1.301 0.0380 0.5014 0.449 0.0666

1.0 0.000 1.196 0.0000 0.4620 0.388 0.0910

114

Fig 3.3: The Behaviour of pressure , density, pressure-density ratio , redshift Z, electric

intensity1

2

c

E,

d

dp versus radius for K = 1, X = .1 .

3.15 Results and Disscusions :

We have electrified the uncharged fluid sphere e.g. Durgapal V solution

[1]. The charged solution is well behaved with positive values of charge parameter

K and completely describes interior of the super-dense astrophysical objects with

charge matter. The electric field intensity assumed is

3

2

2

21

1

2

)61()1(2

xxKx

x

qc

c

E n . The electric intensity is so assumed that the

model is physically significant and well behaved. E vanishes at the centre of the star

and increases as we move towards the surface and is positive throughout the star.

Thus by assigning different positive integral values to parameter n we get a variety of

classes of exact solutions. We have obtained a generalized solution of Einstein-

Maxwell field equations of general relativity for a static, spherically symmetric

distribution of the charged fluid with well behaved nature.

For n = 0 we get the Gupta, Maurya solution [2], which is well behaved for a

wide range of constant K. For n = 1, we get the Fuloria et al. solution [3] which is

00.20.40.60.8

11.21.41.61.8

2

0 0.2 0.4 0.6 0.8 1 1.2


pressure

density

ratio of pressure and density

red shift

electric density


115

also well behaved for a wide range of constant K, hence suitable for modeling of

super dense star.

In view of Table 3.1 we observe that all the physical parameters (p, , ,2c

p

Eandzd

dp,

) are positive at the centre and within the limit of realistic equation

of state and well behaved conditions for all values of K satisfying the inequality

100 K corresponding to X = .2 . However, for any value of K > 10,

corresponding to X= .2 surface density is negative. Table 3.2 shows the values of

different physical parameters for different values of K corresponding to X = .1 .

From Table 3.2 we observe that the solution is well behaved for the values of K up

to 20 corresponding to X = .1. Beyond this value of K the solution is not well

behaved. The mass and red shift increases with the increasing values of K but the

radius decreases with K. Table 3.3 shows the values of different parameters for

different values of K corresponding to X = .01. From Table 3.4 we observe that the

solution is well behaved for the values of K up to 44 . Hence we may conclude that

for the solution (n =1), as the value of X is decreased the solution becomes well

behaved for larger values of K. The solution for n = 2 is also well behaved for all

the values of K satisfying the inequality 160 K corresponding to X = .2.

Similarly the solution for n = 3 is well behaved for the values of K satisfying the

inequality 19K corresponding to X=.1 .

3.15 a. Modeling of superdense star for the solution n = 1

We present here a model of super dense star based on the particular solution

(n = 1)discussed above corresponding to K = 0.35 with 2.X ,by assuming surface

density;314 /102 cmgb .The resulting well behaved model has the heaviest star

occupying a mass 5.523 MΘ with its radius 13.98 km. Corresponding to K = 1 and

X = .1 ,the maximum mass of the star comes out to be 2.95 MΘ with linear dimension

17.95 km.Corresponding to K = 20 and X = .1 the maximum mass of the star comes

out to be 3.55 MΘ with linear dimension 14.92 km. Corresponding to K = 44 and X =

116

.01 the maximum mass of the star comes out to be 0.2660 MΘ and radius 8.345 km.

Thus we get the mass of the stellar object with in the range of neutron star and white

Dwarf. From Tables 3.1, 3.2, 3.3 it is clear that as the value of X is decreased the

solution becomes well behaved for larger values of K. In absence of the charge we

are left behind with the regular and well behaved fifth model of Durgapal [1].

3.15 b. Modeling of superdense star for n = 2 :

Table 3.6 shows the variation of different physical parameters for different

values of K corresponding to X = .1. The solution is well behaved for the value of K

up to 16. For the solution n = 2 the mass of a star is maximized with all degrees of

suitability and by assuming the surface density 314 /102 cmgb . Corresponding

to K = 16 and X = .1 the maximum mass of the star comes about to be 2.63 MΘ with

linear dimension 15.34 km. Corresponding to K = 6 and X = .1 the maximum mass

of the star comes about to be 1.17 MΘ with linear dimension 17.12 km. Table 3.7

shows that pressure, density, pressure-density ratio , red shift decrease as we move

from centre to the surface of superdense star. However the electric intensity increases

as the surface of the star approaches. Thus for the solution corresponding to n = 2 we

may construct the models for super massive objects. we arrive at the conclusion that

under well behaved conditions this class of solutions gives us the mass of super

dense object within the range of neutron star and quark star.

3.15 c. Modeling of super dense star for n = 3.

Table 3.8 shows the values of different physical quantities for different values of K

corresponding to X = .1. The solution is well behaved for the values of .001< K < 19.

As the value of K increases the mass and red shift increase but radius decreases.For the

solution n = 3 the mass of a star is maximized with all degrees of suitability and by

assuming the surface density 314 /102 cmgb . Corresponding to K = 0.02 and X = .1

the maximum mass of the star comes about to be 2.0909 MΘ with linear dimension

18.10 km. Corresponding to K = 19 and X = .1 the maximum mass of the star comes about

to be 3.59 MΘ with linear dimension 14.39 km. Table 3.9 also shows the values

117

of different physical quantities for different values of X corresponding to K = 2.

Corresponding to K = 2 and X = 0.08 the maximum mass of the neutron star comes out to

be 2.549 MΘ with linear dimension 17.09 km. Corresponding to K = 2 and X = 0.1 the

maximum mass of the star comes out to be 3.017 MΘ with linear dimension 17.75 km.

We observe that with the increase of the value of n, although the solutions

become more complicated in terms of expressions, but they show the well behaved

nature and satisfy all the necessary conditions to be physically realizable. For n = 1, n

= 2 and n = 3 we have obtained new solutions of Einstein- Maxwell field equations

, which have been studied extensively exposing their well behaved nature.The solutions

for n = 4, n = 5, n = 6 and so on can be also studied likewise. In the absence of the

charge we are left behind with the regular and well behaved fifth model of Durgapal [1].

118

3.16 References:

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relativity” , Journal of Phy. A Math Gen, 15 , 2637 ( 1982).

[2] Gupta,Y.K., Maurya S.K., : “A class of regular and well behaved

relativistic super dense star models” , Astrophys. Space Sci.,

334(1),155 (2011) .

[3] Fuloria, P., Tiwari, B.C., Joshi, B. C., : “Well behaved class of

charge Analogue of Durgapal’s relativistic exact solution”, Journal Of

Modern Physics, 2,1156 (2011).

[4] R. Stettner, “On the stability of homogeneous, spherically symmetric,

charged fluids in relativity”, Ann. Phys. (NY) 80 212 (1973).

[5] P. G. Whitman, R.C. Burch: “Charged spheres in general relativity”,

Phys. Rev. D 24 (1981) 2049.

[6] de Felice F., Y. Yu and J. Fang J., Mon. Not. R. Astron. Soc. 277, L17

(1995).

[7] Bayin, S. S.: “Solutions of Einstein's field equations for static fluid

spheres” , Phys. Rev. D18, 2745 (1978).

[8] Ray, S. et al.: “Dark Energy Models with a Time-Dependent

Gravitational Constant”, Int. J. Mod. Phys. D16, 1745, (2007).

[9] Bonnor, W. B.: “The equilibrium of a charged sphere”, Mon. Not. R.

Astron. Soc. 129, 443 (1965).

[10] Herrera, L. and Santos, N. O.: “Local anisotropy in self-gravitating

systems”, Phys. Rep. 286 53(1997).

119

[11] Bondi, H. : “Gravitational Bounce in General Relativity”, Mon. Not.

R. Astr. Soc. 142 333 (1969).

[12] Tikekar, R.: “Spherical charged fluid distributions in general

relativity”, J. Math. Phys. 25, 1481(1984).

[13] Humi, M. and Mansour, J.: “Interior solutions to Einstein-Maxwell

equations in spherical and plane symmetry when p=nρ”, Phys. Rev. D,

29 1076 (1984).

[14] Bonnor, W. B. and Wickramasuriya, S .B. P.: “Are very large

gravitational redshifts possible”, Mon. Not. R. Astr. Soc. 170, 643

(1975).

[15] Ivanov, B. V. “Static charged perfect fluid spheres in

general relativity” , Phys. Rev. D, 65 104001(2002).

[16] Ray, S., et al. : “Electrically charged compact stars and formation of

charged black holes”, Phys. Rev. D 68, id. 084004(2003).

[17] Bekenstein, J.: Phys. Rev 4 2185(1975).

[18] Graves, J. C. and Brill, D.R. : “Oscillatory Character of Reissner

Nordstorm Metric for an Ideal Charged Worm-hole”, Physical

Review, 120(4),1507(1960).

[19] Ray, S., Das, B.: “Tolman-Bayin type static charged fluid spheres in

general relativity” , Mon. Not. R. Astron. Soc. 349, 1331(2004).

[20] Bonner, W.B., Cooperstock, F.I.: “Does the electron contain negative

mass?”, Phys. Lett. A 139, 442(1989).

[21] Bonnor, W. B, “The mass of a static charged sphere”, Z. Phys. 160,

59 (1960).

120

[22] Efinger, H. J.: Z. Phys. 188 31(1965).

[23] Kyle, C. F., Martin, A. W.: “Self-energy considerations in general

relativity and the exact fields of charge and mass distributions”,

Nuovo Cimento 50, 583 (1967).

[24] Krori, K. D., Barua, J.: “A singularity-free solution for a charged fluid

sphere in general relativity”, J. Phys. A Mathematical and general 8,

508(1975).

[25] Nduka, A.: “Charged fluid sphere in general relativity”, Gen. Relativ.

Gravitation 7, 493 (1976).

[26] Durgapal, M.C., Fuloria, R.S.: “Analytic stellar model for super-dense

star”, Gen. Relativ. Gravit. 17, 671(1985).

[27] Gupta, Y.K., Maurya, S. K.: “A class of charged analogues of

Durgapal and Fuloria superdense star”, Astrophys. Space Sci.

331(1),135 (2010).

[28] Schwarzschild, K.: “On the Gravitational Field of a Mass Point

According to Einstein's Theory”, Sitzer. Preuss. Akad. Wiss. Berlin ,

424,189 (1916).

[29] Bijalwan, Naveen, Gupta, Y. K.: “Nonsingular charged analogues of

Schwarzschild's interior solution”, Astrophysics and Space Science,

317,251(2008).

[30] Gupta,Y. K., Kumar, M. : “A superdense star model as charged

analogue of Schwarzschild's interior solution”, General Relativity and

Gravitation 37(3), 575 (2005).

121

[31] Kuchowich, B.: “General Relativistic fluid spheres. I. New solutions

for spherically symmetric matter distribution”, Acta Phys. Pol.33 ,

541(1968b).

[32] Nduka, A.: “Some exact solutions of charged general relativistic fluid

spheres”, Acta Phys. Pol. B8, 75 (1977).

[33] Tolman, R. C.: “Static solutions of Einstein’s field equations for

spheres of Fluid ”, Phys. Rev. 55, 367 (1939).

[34] Cataldo, M., Mitskievic, N. V. : “Static charged fluid surrounded by a

black antihole: an enlarged Klein solutio”, Class. Quantum Gravity

9,545 (1992).

[35] Heintzmann, H.: “New exact static solutions of Einstein field

equations”, Z. Phys. 228, 489 (1969).

[36] Pant, N. et al.: “Well behaved class of charge analogue of

Heintzmann’s relativistic exact solution”, Astrophys. Space Sci.,

332(2), 473 (2011a).

[37] Pant, N. , Tewari, B. C., Fuloria, P.: “Well Behaved Parametric Class of

Exact Solutions of Einstein-Maxwell Field Equations in General

Relativity”, Journal of Modern Physics, 2 (12) 1538 ( 2011).

[38] Pant, M. J. and Tewari, B. C., “Well Behaved Class of Charge Analogue

of Adler’s Relativistic Exact Solution” , Journal of Modern Physics, 2 (

6) 481( 2011).

[39] Pant, N., Mehta, R. N., (Joshi) Pant, M.: “Well behaved class of charge

analogue of Heintzmann’s relativistic exact solution”, Astrophys. Space

Sci. 332(2), 473(2011).

122

[40] Florides, P. S.: “The complete field of charged perfect fluid spheres and

of other static spherically symmetric charged distributions”, J. Phys. A,

Math. Gen. 16, 1419 (1983).

[41] Adler, R.: “A fluid sphere in general relativity”, J. Math. Phys. 15, 727

(1974).

[42] Singh, T., Yadav, R. B. S.: “Some exact solutions of charged fluid

spheres in General relativity”, Acta Phys. Pol. B 9, 475 (1978).

[43] Delgaty, M.S.R., Lake, K.: “Physical Acceptability of Isolated, Static,

Spherically Symmetric, Perfect Fluid Solutions of Einstein’s

Equations”, Comput. Phys. Commun. 115, 395 (1998) .

[44] Pant, N. et al. : “New class of regular and well behaved exact solutions in

general relativity”, Astrophys. Space Sci. 330(2) 353 (2010).

[45] Leibovitz, C.: “Spherically symmetric static solutions of Einstein’s field

equations”, Phys. Rev. D 185, 1664 (1969).

[46] Pant, N.: “Some new exact solutions with finite central parameters and

uniform radial motion of sound ”, Astroph. Space Sci. 331, 633 (2011).

[47] Dionysiou, D.D. :“Equilibrium of a Static Charged Perfect Fluid

Sphere”,Astrophys. Space Sci. 85(1-2), 331 (1982).

Chapter IVRadiating fluid ball models with

horizon free gravitational collapse

123

Chapter IV

Radiating fluid ball models withhorizon free gravitational collapse5

In this chapter we have revisited a solution of the radiating fluid ball problem

proposed by Tewari[1], (hereafter referred to as the BCT solution II ) in which the

horizon is never encountered. By assuming the life time of quasars 107 years we

have constructed different approximate models of quasars for different

combinations of the constants X , Y and Z appearing in the solution. The interesting

feature of all the models is that the rate of decrease of mass is balanced by the rate

of contraction of the boundary and the collapsing process continuous without the

reach of horizon of the black hole. There is an only singularity i.e. the naked

singularity. We have also studied the variation of pressure, density, pressure density

ratio and luminosity within the radiating fluid ball with the increase of r at any

particular time. At any particular point within the sphere the pressure and density

increases with time and attains the infinite value as t tends to 107 years. The

expressions for mass and radius of the collapsing ball are linear functions of time.

5A part of this chapter has been communicated for publication.

124

4.1 Introduction:

One of the most important and outstanding problem of gravitational collapse

is whether it ends in a black hole or a naked singularity. According to general theory of

relativity the collapsing massive star must terminate into a space-time singularity, where the

matter energy densities , space-time curvature and other physical quantities blow up. It then

becomes important to know whether such ultra dense regions , formed during stellar collapse

will be visible to an external observer in the universe, or whether they will be always hidden

within an event horizon of gravity that could form as the star collapses. This is one of the

most important issue in the physics of gravitational collapse today. According to cosmic

censorship hypothesis the space time singularity forming in collapse always hides within the

event horizon , never to be seen by external observers. Or there is no possibility of the

existence of naked singularity. However, many models of radiating fluid spheres have been

discovered which admit the existence of naked singularity . The collapse of radiating fluid

spheres may be studied to ascertain the final outcome of the collapse. We know that

gravitational collapse is a highly dissipating energy process which plays important role in

the structure formation in the universe. The dissipation of energy from collapsing fluid

distribution is described in two limiting cases. The first case , the free streaming

approximation is modeled by means of an outflowing null fluid and the second case ,the

diffusion approximation is modeled by heat flow type vector. Various scenarios of

gravitational collapse have been considered within the realm of relativistic astrophysics both

in the free streaming approximation and in the diffusion approximation .There exist a

number of solutions in free streaming limiting case in which the horizon is never

encountered. These solutions may be useful for constructing the approximate models of

quasar in which the radiation emitted by the collapsing object is significant.

We also know that during some stages of evolution star may radiate away a

large chunk of energy in the form of photons or neutrinos or both. In a normal star the

stellar radiation is a very slow process and does not have any significant effect in the

interior and exterior gravitational field of a star. However there are some high energy

astrophysical events such as Quasars and Supernova burst, where this radiation process

is strong and cannot be ignored . So to make the real assessment of such events the

general relativistic field equations must be studied taking into account the out flowing

125

radiation. Our aim is also to investigate the solution of Einstein field equations for the

radiating fluid ball problem in which the horizon is never encountered. Vaidya initiated the

problem of solving the field equations of general relativity for physically realizable radiating

fluid distributions. Solutions for the radiating fluid structures corresponding to isotropic

coordinate form, general metric form, and conformally metric form have been obtained [1-3]

by solving the modified field equations proposed by Misner[4], Lindquist et al.[5] for an

adiabatic distribution of matter. Bayin [6] has also obtained some solutions of Einstein’s

field equations for radiating fluid distribution. Herrera et al.[7] have constructed the radiating

fluid models from known static solutions of Einstein’s field equations. On the similar

grounds a number of models have been proposed by De’Oliveira et al. [8-10], Bonnor et al.

[11], Kramer [12], Maharaj et al. [13-14] and others. A number of studies have been

proposed by Herrera et al. [15-20], Herrera and Santos [21], Mitra [22], Naidu and Govinder

[23] and references there in for the radiating fluid ball models. All these studies describe a

collapsing fluid dissipating energy.

Realistic models of radiating fluid spheres in general relativity are few on

account of associated mathematical complexities. Vaidya, P. C. [24-25] solved the field

equations of general relativity for physical meaningful models of radiating fluid spheres.

The interior space-time metric is matched with Vaidya exterior space-time metric [26] at

zero pressure boundary. Baneerji et al.[27] and Pant and Tewari[28] have proposed the

radiating stellar models in which the horizon is never encountered. We have revisited the

solution of radiating fluid ball problem proposed by Tewari[1]. We have shown that the

collapse of radiating fluid ball will result into the formation of naked singularity i.e.

horizon will never form.

We have also constructed the approximate models of Quasar by considering

the BCT solution II in detail. To investigate the final outcome of collapse we

have worked out the mass-radius gradient for all the proposed models of Quasars.

If 1)(/)(20 2 uRcuMG s the model will be horizon free favouring the

existence of naked singularity.

126

4.2 Conditions for solution to be physically realizable:

For modeling of a massive radiating object like quasar the following conditions

must be satisfied by the solution:

1. The Einstein’s field equations should be time dependent .

2. The solution should be free from physical and geometrical singularities.

3. The solution should have positive values for pressure, density, radiation flux

density and luminosity.

4. The solution should have a negative value of the rate of contraction U, as the

contraction of a radiating ball results as a natural consequence of energy loss.

5. The solution should have monotonically decreasing expressions for p and2c

p

6. The solution should verify the Zeldovich condition i.e. 12

0

0 c

p

4.3 Junction conditions and solution of the field equations:

The field equations of general relativity for a distribution of mixture of a

perfect fluid and radiation are

ji

ji

ji T

c

GgRR

4

8

2

1 (4.1)

where

jij

ij

ij

i wwc

qgpvvcpT )( 2 (4.2)

where p and respectively denote the isotropic pressure and density of the

matter within the distribution and vi its four velocity:

1ii vv (4.3)

q denotes the radiation flux density and wi its four velocity which is null:

0ii ww (4.4)

The interior gravitational field of the radiating sphere is given by the line element

)]sin([),(),( 2222222222 ddrdrtrBdtctrAds (4.5)

127

We choose r as a co moving coordinate so that ).0,0,0,( 1 Av i

We choose iw such that [4]

,1 11

rB

tcA

xw

ii

(4.6)

which implies that q is the energy density of the radiation in the rest frame of the

fluid [4,5].

jiji wwqvvq (4.7)

From Equation (4.1) we obtain four independent equations, which in view of Eq.

(4.2) can be written as

22Tp (4.8)

,22

00

11

2 TTTc (4.9)

,)( 01

213

TwB

A

c

q (4.10)

01

22

11 T

B

ATT (4.11)

The luminosity or neutrino flux is given by

qBrL 224 (4.12)

The space time external to a radiating fluid ball of mass M(u) is filled with pure

radiation , for which a suitable metric is Vaidya’s radiating metric (Vaidya,

1953)[24]

)sin(22

1 222222

2 ddRdRduduRc

GMds

(4.13)

where R is the radial coordinate. The condition describing the junction of (4.5) and

(4.13) over the hyper surface r = rs or equivalently )(uRR s are ( Misner,1965)[4]

),,()( trBruR sss (4.14)

,1),(s

s B

Br

A

BrdudtctrA

(4.15)

,0),( trp s (4.16)

128

s

s B

Brr

A

BBr

G

ctrmuM

232

2

232 )(2

2),()(

(4.17)

The expression for L is given by

2

3 1s

B

Br

A

BrL

du

dMcL

(4.18)

A prime and a dot here after denote differentiation with respect to r and ct, respectively

In order to solve the field equations Tewari [1]used the following method,

Tewari wrote

),()(),( tlrftrA (4.19)

),()(),( tnrhtrB (4.20)

The equation (4.11) transform into

022

22

2

f

fh

k

n

rf

f

f

f

hr

h

hf

hf

h

h

h

h (4.21)

Tewari assumed that

lsn2

1 (4.22)

where s is an arbitrary constant. Equation (4.21) then reduces to the r-dependent

equation

02

22

2

f

fhs

rf

f

f

f

hr

h

hf

hf

h

h

h

h(4.23)

This is an ordinary differential equation in two unknown functions h and f.

Tewari assumed that

02

2

2

fr

f

f

f

f

f(4.24)

The solution obtained by Tewari [1] is

)/( 22 rXWf (4.25)

1

75234322222

7

2

5

4

3

2)()(

rrXrXsrXYZrXWh (4.26)

t is assumed as the proper time of the observer on the hyper surface r = rs,

This requires that

129

W2(rs,t)=1 (4.27)

From equation (4.19) and (4.25) we get,

tconsWrXtl s tan/)()( 22 (4.28)

From equation (4.22) and (4.28) the expression for n(t) is obtained as,

tWrXscKtn s 2/)( 22 , (4.29)

Where K is an arbitrary non-negative constant.

The expression for pressure, density, and radiation flux density are given by[1]

3222

22

22

2

22222

22224)(

4)73(

)3(434

)3)((4)(

18

rXs

rXsr

rXY

h

W

h

WsrYrXr

rXnWp

c

G

s

(4.30)

3222

22

22

2

22222

22222 )(4

3)237(

)113(6224

)3)((12)(

18

rXs

rXsr

rXY

h

W

h

WsrYrXr

rXnWc

G

s

(4.31)

hnW

srq

c

G25

28

(4.32)

As radiation is being emitted from the collapsing fluid sphere , we are

restricted to s < 0. From equation (4.12), (4.20), (4.25), (4.26) and (4.32) the

luminosity or neutrino flux is given by

)7

2

3

4

3

2()(

)()(

75234322

22235

rrXrXsrXYZ

rXsr

G

cL (4.33)

From equation (4.14), (4.20),(4.26) and (4.29) the radius of the sphere at any time

is given by

)

7

2

3

4

3

2()(

)(2

)()(

75234322

22222

rrXrXsrXYZ

trXW

scKrXWr

uRss

s (4.34)

130

The total energy inside the surface is given by [1]

375234322

275234322

2275234322

22322222224

622

2232

)7

2

3

4

3

2()(

)7

2

3

4

3

2()(

38)}7

2

3

4

3

2()({26

)5()(226)(4

22)(

rrXrXsrXYZ

rrXrXsrXYZ

rXrrXrXsrXYZsrY

rXrXsrYrrXs

rX

trXW

scKr

G

cWuM

ss

ssssss

ss

(4.35)

From Eq.(4.34) and (4.35) we observe that radius and mass are linear functions of

time .

The luminosity observed on the surface is given by

)7

2

3

4

3

2()(

)()(

75234322

22235

ssss

ss

rrXrXsrXYZ

rXsr

G

cL (4.36)

With the aid of equations (4.12), (4.16), (4.17), (4.25), (4.26) and (4.27) the

following equation is obtained .

2

75234322

232222

22

2

)7

2

3

4

3

2()(2

312)(

)(

41

ssss

ssss

s

s

rrXrXsrXYZ

rsXsrCrrX

rX

rdudtc

(4.37)

The total luminosity for an observer at rest at infinity is given by

131

)7

2

3

4

3

2()(2

312)(

)(

41

75234322

232222

22

2

ssss

ssss

s

ss

rrXrXsrXYZ

rsXsrYrrX

rX

rLL ( 4.38)

which is constant for all time.

The effective surface temperature measured by an external observer is

expressed as [29] :

L

tnrhrT

s

s 24

)()(

1

(4.39)

Where for photons the constant is given by

3

42

15

k (4.40)

Where k and denote Boltzmann and Plank constant respectively.

The red shift is given as

11

dt

du

cZ r (4.41)

Now to investigate the nature of variation of pressure and density with respect to

time at any particular point within the radiating fluid sphere we differentiate

Equations (4.30) and (4.31) with respect to time. Consequently we get

3222

22

22

2

22222

334)(

4)73(

)3(434

)3)((48

rXs

rXsr

rXY

h

W

h

WsrYrXr

nW

scp

c

G

(4.42 )

3222

2222

2

22222

332 )(4

3

)237()113(6224

)3)((128

rXs

rXsrrXY

h

W

h

WsrYrXr

nW

cs

c

G

(4.43)

132

Where dot represents the differentiation with respect to time.

From Equations (4.42) and (4.43) it is clear that pressure and density increase with the

passage of time . On the other hand we can say that collapsing radiating fluid sphere

will approach towards space -time singularity, that will be visible by an external

observer .

To investigate the trend of variation of surface temperature with time, we may write

equation (4.39) as follows

4

1

2

14

1

)()()(

1

L

tnrhr

T

s

s

(4.44)

Differentiating Equation (4.44) with repect to time we get

4

1

2

3

2

14

1

22

)()(

)(

4

1

L

nrhr

rXcs

WT

s

ss

(4.45)

From Equation (4.45) we observe that surface temperature will also increase as time

evolves towards the infinity.

4.4. Different cases of BCT solution II for Quasar Model:

4.4.1 Case I

When X = 1, Y = 1, Z = 2, s = -2×10-10

And 2541.0sr

We have calculated the various parameters: like mass, radius, luminosity etc.

at any instant of time for the radiating fluid sphere. An approximate model of Quasar have

been constructed for the above mentioned values of constants. Equations (4.31) and (4.32)

give the march of p, ρ,2c

p

at any time t within the radiating fluid ball and

Eq.(4.33) gives the march of luminosity. We observe that pressure, density, luminosity

are positive within the radiating fluid sphere and3

12

c

p

everywhere within the ball.

Pressure and pressure-density ratio are monotonically decreasing however, luminosity

133

sis monotonically increasing with the increase of r. This combination of constants pertaining

to case I may be used for constructing the radiating fluid ball models . An approximate model

for Quasar has been constructed for the above said combination of constants.

Table 4.1:

The march of pressure , density, pressure-density ratio and Luminosity at anyinstant of time within the sphere (0 ≤ r ≤ rs).

(p in dynes cm-2, ε in g cm-3, and L in ergs s-1)

r/ rs pnWc

G)(

8 224

)(8 22

2nW

c

G2c

p

L

0.0 13.5000 108.0000 0.125 0

0.1 13.3821 108.3957 0.123 3.98×1044

0.2 13.0268 109.5898 0.118 3.18×1045

0.3 12.4288 111.6030 0.111 1.07×1046

0.4 11.5775 114.4712 0.101 2.52×1046

0.5 10.4600 118.2460 0.088 4.90×1046

0.6 9.05779 122.9974 0.073 8.40×1046

0.7 7.34590 128.8158 0.057 1.32×1047

0.8 5.29352 135.8162 0.038 1.95×1047

0.9 2.86151 144.1430 0.019 2.75×1047

1.0 0.0000 153.9762 0.000 3.71×1047

134

Fig 4.1: The variation of pressure with radius for the case I.

Fig 4 .2: The variation ofp

with radius for the case I.

02468

10121416

0 0.2 0.4 0.6 0.8 1 1.2

r/ rs

pressure

00.020.040.060.08

0.10.120.14

0 0.5 1 1.5

r/ rs

ratio of pressure anddensity

135

Fig 4.3: The variation of Luminosity from centre to surface.

Model of Quasar for case I:

In view of Equation (4.29 ) , we have

tWKtn 1806.2)( (4.46)

Consequently Equations (4.34), (4.35), (4.37 ) give us

tWKuRs 806.20788.0)( (4.47)

tWKuM 806.210525.1)( 26 (4.48)

dtcdu 1847.1 (4.49)

In view of Equations (4.41) and (4.49) the gravitational red shift Zr = .1847

For a life time of 107 years ,

our model has an initial mass of 6.65 × 107 MΘ

and an initial linear dimension 6.8 × 1013 cm.

Our model is radiating energy at a constant rate i.e. L∞ = 3.359 × 1047 ergs/ sec.

0

5E+46

1E+47

1.5E+47

2E+47

2.5E+47

0 0.2 0.4 0.6 0.8 1

r/ rs

Luminosity inergs/second

136

4.4.2 Case II:

X= 1, Y= 1, Z= 3, s = -2×10-10

rs = 0.29741

we have studied the march of pressure, density, pressure-density ratio and

luminosity with in the radiating fluid sphere for case II. We have calculated the various

parameters: like mass, radius, luminosity, etc. at any instant of time and have

constructed an approximate model of Quasar. Equations (4.30) and (4.31) give

the march of p, ρ,2c

p

at any time t within the radiating fluid ball and in view of

Eq.(4.33) we get the march of luminosity .We observe that pressure, density,

luminosity are positive within the radiating fluid sphere and3

12

c

p

everywhere

with in the ball. Pressure and pressure-density ratio are monotonically decreasing ,

however , luminosity is monotonically increasing with the increase of r. We have

also found out an important parameter)(

)(22 uRc

uGM

s

that will tell us about

the nature of the gravitational collapse.

137

Table 4.2:

The march of p, ε, p/εc2, and L at any instant of time within the sphere (0 ≤ r ≤ rs)

(p in dynes cm-2, ε in g cm-3, and L in ergs s-1)

r/rs pnWc

G)(

8 224

)(

8 222

nWc

G2c

p

L

0.0 44.44 240.00 0.1857 0

0.1 44.036 241.06 0.1828 4.79×1044

0.2 42.807 244.28 0.1758 3.82×1045

0.3 40.747 249.73 0.1644 1.28×1046

0.4 37.836 257.55 0.1490 3.01×1046

0.5 34.046 267.95 0.2993 5.82×1046

0.6 29.337 281.20 0.1077 9.93×1046

0.7 23.657 297.67 0.0830 1.55×1047

0.8 16.938 317.86 0.0564 2.27×1047

0.9 9.0915 342.39 0.0286 3.17×1047

1.0 0.0000 372.07 0.0000 4.24×1047

________________________________________________________________________________

138

Fig 4.4: The variation of pressure from centre to surface .

Fig 4.5 : The variation of pressure-density ratio from centre to surface.

From Table 4.2 we observe that pressure, density, luminosity are positive and

p/εc2 < 1, everywhere with in the radiating fluid sphere. Pressure and pressure-

density ratio are monotonically decreasing, however, luminosity is monotonically

increasing with the increase of r.

In view of equation (4.29) ,we have

05

101520253035404550

0 0.2 0.4 0.6 0.8 1 1.2

Pressure

r/ rs

00.020.040.060.08

0.10.120.140.160.18

0.2

0 0.2 0.4 0.6 0.8 1 1.2

fractional radius r/ rs

139

tWKtn 1734.2)( (4.50)

Consequently we have

tWKuRs 734.206577.0)( (4.51)

tWKuM 734.210075.2)( 26 (4.52)

dtcdu 370.1 (4.53)

Assuming the life time of quasar 107 years ,

Our model has an initial mass of 8.82 ×107 MΘ and

An initial linear dimension 5.51 ×1013 cm.

In view of equations (4.41) and (4.53) we get the gravitational red shift as .370.

Our model is radiating energy at a constant rate i.e. L∞= 2.902 × 1047 ergs /sec.

4.4.3. Case III

X = 1, Y = 1, Z = 4, s = -2×10-10

In view of Eq. (4.16) we get

rs = 0.317143

We have studied the march of pressure, density, pressure-density ratio and

luminosity with in the radiating fluid sphere. For the model to be realistic the pressure

, pressure-density ratio should be monotonically decreasing as we go from centre to

surface.

140

Table 4.3:

The march of Pressure, density, pressure-density ratio and Luminosity at any instantwithin the ball (p in dynes cm-2, ε in g cm-3, and L in ergs s-1)

r/rs pnWc

G)(

8 224

)(

8 224

nWc

G2c

p

L

0 92.1875 420.0000 0.2194 0

0.1 91.4414 421.9962 0.2166 4.64×1044

0.2 89.1858 428.0419 0.2083 3.70×1045

0.3 85.3675 438.3115 0.1947 1.24×1046

0.4 79.8946 453.1075 0.1763 2.91×1046

0.5 72.6303 472.8782 0.1535 5.60×1046

0.6 63.3846 498.2465 0.1272 9.53×1046

0.7 51.9007 530.0504 0.0979 1.48×1047

0.8 37.8369 569.4025 0.0664 2.16×1047

0.9 20.7389 617.7722 0.0335 2.99×1047

1.0 0.00000 677.1053 0.0000 3.98×1047

From Table 4.3 it is clear that Pressure, Pressure- density ratio are decreasing with the

increase of r. Luminosity is monotonically increasing with the increase of r.

141

Model of quasar for the case III :


tWKtn 16982.2)( (4.54)


tWKuRs 6982.205426.0)( (4.55)

(4.56)

dtcdu 5245.1 (4.57)

The gravitational red shift obtained is .52450.

For a life time of 107 years,

Our model has an initial mass 8.74 ×107 MΘ


Our model is radiating energy at a constant rate i.e. L∞ = 2.611 × 1047 ergs /sec.

4.4.4 Case IV

X = 1, Y = 1 , Z = 5 , s = -2×10-10

For the above mentioned values of constants the value of rs is obtained as

rs = 0.328141

tWKuM 6982.2100838.2)( 26

142

Table 4.4:

The march of Pressure, density, pressure-density ratio and luminosity at any instantwithin the ball (p in dynes cm-2, ε in g cm-3, and L in ergs s-1) .

r/rs pnWc

G)(

8 224

)(

8 222

nWc

G2c

p

L

0.0 156.096 648.000 0.24088 0

0.1 154.686 651.186 0.23780 4.28×1044

0.2 150.441 660.845 0.22863 3.41×1045

0.3 143.317 677.279 0.21367 1.14×1046

0.4 133.235 701.012 0.19339 2.67×1046

0.5 120.076 732.828 0.16839 5.15×1046

0.6 103.686 773.819 0.13939 8.73×1046

0.7 83.8332 825.470 0.10721 1.35×1047

0.8 60.2232 889.767 0.07215 1.97×1047

0.9 32.457 969.363 0.03668 2.71×1047

1.0 0.0000 1067.81 0.00000 3.59×1047

From Table (4.4) we observe that pressure, density , luminosity are positive and p/εc2<1,

everywhere within the fluid sphere. Pressure and pressure-density ratio are monotonically

decreasing, however luminosity is monotonically increasing with the increase of r.

143

Model of Quasar for the case IV :


tWKtn 16769.2)( (4.58)


tKWuRs 6769.20457.0)( (4.59)

tWKuM 6769.2109460.1)( 26 (4.60)

dtcdu 6463.1 (4.61)

Zr = .6463 (4.62)

For a life time of 107 years , our model has an initial mass of 8 ×107 MΘ and an

initial linear dimension 3.8 ×1013 cm. The gravitational red-shift obtained is .6463.

Our model is radiating energy at a constant rate i.e. L∞ = 2.417 × 1047 ergs /sec.

4.4.5. Case V:

X = 1, Y = 1, Z = 6 ,

rs = 0.3351

s = -2×10-10

For these values of constants also we may construct the radiating fluid ball model.

We have discussed the march of pressure, density, pressure-density ratio and

luminosity form centre to boundary. From Table 4.5 it is clear that pressure and

pressure –density ratio are decreasing from centre to boundary and luminosity is

increasing as we move from centre to boundary. The luminosity is zero at the centre.

144

Table 4.5 : The march of Pressure, density, pressure-density ratio and Luminosity atany instant within the ball (p in dynes cm-2, ε in g cm-3, and L in ergs s-1)

r/rs pnWc

G)(

8 224

)(

8 224

nWc

G2c

p

L

0.0 236.051 924.000 0.2554 0

0.1 234.195 928.635 0.2525 3.91×1044

0.2 228.568 942.694 0.2424 3.11×1045

0.3 219.025 966.639 0.2265 1.04×1046

0.4 205.310 1001.276 0.2050 2.44×1046

0.5 187.039 1047.809 0.1785 4.68×1046

0.6 163.671 1107.93 0.1477 7.93×1046

0.7 134.470 1183.942 0.1135 1.23×1047

0.8 98.4391 1278.951 0.0769 1.78×1047

0.9 54.2280 1397.136 0.0388 2.45×1047

1.0 00.0000 1544.143 0.0000 3.23×1047

Model of Quasar for case V:


tWKtn 1663.2)( (4.63)


tWKuRs 663.20394.0)( (4.64)

tWKuM 663.210688.1)( 26 (4.65)

145

For a life time of 107 years , our model has an initial mass of 6.99 ×107 MΘ

and an initial linear dimension 3.26 ×1013 cm. The gravitational red shift Zr =

.74459. Our model is radiating energy at a constant rate i.e. 47102816.2 L

ergs/sec.

Table 4.6 : The variation of initial mass, initial linear dimension and red

shift of Quasar with Z . (X = 1, Y =1)

Z M/MΘ R in cm Zr)(

)(22 uRc

uGM

s

2 6.65 × 107 6.8× 1013 .184 .3449

3 8.82× 107 5.5× 1013 .370 .5873

4 8.74× 107 4.5× 1013 .524 .5691

5 8.00× 107 3.8× 1013 .646 .6304

6 6.99× 107 3.2× 1013 .744 .6350

7 6.18× 107 2.8× 1013 1.84 .6440

10 5.20× 107 2.0× 1013 2.98 .7499

12 3.63× 107 1.7× 1013 3.33 .6183

16 3.54× 107 1.3× 1013 3.82 .7919

20 2.90× 107 1.0× 1013 4.16 .8080

30 1.96× 107 .72× 1013 4.68 .8052

4.4.6. Case VI : (X = 1, Y = 1.5, Z = 2)

Now we consider the case in which the value of Y is different from 1 and it is equal to

1.5. We discuss the various characterstics of radiating fluid distribution for this combination

of constants . we also check the physical validity of the solution to represent Quasar model.

X = 1, Y = 1.5, Z = 2 ,

146

rs = 0.3972

Table 4.7: The march of pressure, density, pressure-density ratio and luminosityat any instant within the ball (p in dynes cm-2, ε in g cm-3, and L in ergs s-1).

r/rs pnWc

G)(

8 224

)(

8 222

nWc

G2c

p

L

0.0 61.25 105.0 0.58 0

0.1 60.70 106.1 0.57 1.3×1045

0.2 59.06 109.7 0.54 1.0×1046

0.3 56.30 115.7 0.49 3.4×1046

0.4 52.39 124.6 0.43 8.1×1046

0.5 47.29 136.6 0.36 1.5×1047

0.6 40.91 152.3 0.28 2.6×1047

0.7 33.17 172.7 0.20 4.2×1047

0.8 23.92 198.7 0.13 6.1×1047

0.9 12.96 231.4 0.06 8.4×1047

1.0 00.00 274.8 0.00 1.1×1048

From Table 4.7 it is clear that the pressure and pressure density ratio are

monotonically decreasing with the increase of r within the radiating fluid sphere.

Luminosity is monotonically increasing with the increase of r.

147

Model of Quasar for case VII


tWKtn 1526.2)( (4.66)


tKWuRs 526.209728.0)( (4.67)

tWKuM 526.2102.4)( 26 (4.68)

For a life time of 107 years ,

Our model has an initial mass of 1.65 ×109 MΘ,


The gravitational red shift Zr = . 6708

Our model is radiating energy at a constant rate i.e. 4710382.2 L ergs/sec.

148

4.4.7. Case VII : X = 1, Y = 2, Z = 2 rs = 0.4595

Table 4.8: The march of pressure, density, pressure-density ratio andluminosity at any instant within the ball.

(p in dynes cm-2, ε in g cm-3, and L in ergs s-1).

______________________________________________________________________

r/rs pnWc

G)(

8 224

)(

8 224

nWc

G2c

p

L

_____________________________________________________________________________

0.0 131.0 96.00 1.364 0

0.1 129.8 97.83 1.329 1.7×1045

0.2 126.3 103.42 1.231 1.4×1046

0.3 120.4 113.08 1.085 4.7×1046

0.4 112.1 127.35 0.911 1.1×1047

0.5 101.2 147.07 0.726 2.1×1047

0.6 87.6 173.51 0.546 3.6×1047

0.7 71.85 208.55 0.380 5.6×1047

0.8 51.46 254.99 0.233 8.2×1047

0.9 27.99 317.04 0.106 1.1×1048

1.0 00.00 401.29 0.000 1.4×1048

____________________________________________________________________________

From the Table 4.8 we observe that pressure is decreasing as we go from centre to surface ,

2c

p

is also monotonically decreasing in nature with the increase of r but it is not less than

one throughout the radiating fluid sphere. Hence it is not suitable for modeling of radiatingfluid sphere.

149

Table 4.9: Physical validity of the solution for different

combinations of X , Y and Z.

X Y Z Physically realizable

Yes /No

1 1 2, 3, 4, 5, 6, 7, Yes

1 1 10,12, 16, 20,30 Yes

2 any value any value No

1 1.5 2 Yes

1 2 2 No

1 3 2 No

1 3 3 No

4.5 Results and Discussions:

We have done an extensive study of BCT solution II [1] and have constructed

the approximate models for quasars for different combinations of constants X, Y and Z

appearing in the solution. Tables 4.1, 4.2, 4.3, 4.4, 4.5 show the variation of pressure,

density, pressure-density ratio and luminosity within the radiating fluid sphere. From these

Tables we observe that pressure, density, luminosity are positive and2c

p

< 1/3 everywhere

within the radiating balls. The pressure and pressure – density ratio are monotonically

decreasing , however luminosity is monotonically increasing with the increase of r. We have

considered the different fluid ball models of Quasar for different combinations of constants

X, Y and Z ( X =1, Y = 1, Z = 2, 3, 4, 5, 6).

The similarity of all these models is that constants X and Y are same but Z is

different for each model. Assuming the life time of quasar 107 years our models have

150

initial mass in the range of 107 MΘ with initial linear dimension in the range of 1013 cm .

From the Tables 4.1, 4.2, 4.3, 4.4, 4.5 we observe that as the value of Z increases the

pressure and density inside the radiating fluid sphere increases. In Table 4.6 the different

models of quasar have been presented for the value of Z up to 30. From Table 4.6 we

observe that the initial mass of Quasar firstly increases with Z up to Z = 4 and then

decreases. The initial linear dimension Of Quasar decreases as the value of Z increases,

however the red shift increases with the increase in the value of Z. By assigning the values

of both X and Y as 1 if we increase the value of Z from 2 onwards the solutions are

physically realizable. These solutions can be used for modeling of radiating fluid

distributions. For all these cases)(

)(22 uRc

uGM

s

< 1, hence for all the models of radiating fluid

spheres discussed above the collapse will be horizon free i.e. collapse process keeps on

going without horizon being formed in the space time. The state of naked singularity is

attained.

We have also considered the case in which Y is different from 1, i.e. case VI

( X = 1, Y = 1.5, Z = 2) . This case also gives us the physically relevant solution . Quasar

model has been constructed for the above said combination of constants. Assuming the life

time of quasar as 107 years , our model has an initial mass of 1.65 ×109 MΘ, an initial linear

dimension 7.64 ×1013 cm. The gravitational red shift Zr = . 6708 and our model is radiating

energy at a constant rate i.e. 4710382.2 L ergs/sec.Table 4.7 shows the variation of

pressure, density , pressure-density ratio and luminosity within the radiating fluid ball.

151

From the Table 4.8, case VII (X = 1, Y= 2, Z = 2) we observe that pressure, energy

density, pressure –energy density ratio all are positive within the radiating fluid sphere.

Pressure, pressure-density ratio are monotonically decreasing and luminosity is increasing as

we move from centre to surface. But the pressure-density ratio is not less than 1 everywhere

within the radiating fluid sphere . Hence the combination of the constants (case VIII) is not

suitable for modeling of radiating fluid sphere. By considering the various possible

combinations among the constants X, Y, Z the physical validity of the solutions has been

shown in Table 4.9 and it may be concluded that constants X, Y, Z play a dominant role in

determining the physical validity of the radiating fluid ball models.

One interesting feature of all the models considered above is that they are

horizon free strongly favouring the existence of naked singularity. The theoretical existence

of naked singularities is important because their existence would mean that it would be

possible to observe the collapse of an object to infinite density. A naked singularity could

allow scientists to observe an infinitely dense material, which would under normal

circumstances be impossible by the cosmic censorship hypothesis. Theoretically we can say

that there is possibility of existence of naked singularity .

152

4.6 References:

[1] Tewari, B. C. : “Radiating fluid spheres in general relativity”, Astrophys.

Space Sci. 149, 233 (1988).

[2] Tewari, B. C. : “Radiating fluid distributions”, Indian J. Pure Appl. Phys.

32, 504 (1994).

[3] Pant, D. N., Tewari, B. C. : “Conformally-flat metric representing a

radiating fluid ball”, Astrophysics. Space Sci. 163, 273 (1990).

[4] Misner, C. W. : “Relativistic Equations for Spherical Gravitational

Collapse with Escaping Neutrinos”, Phys. Rev. B137, 1350 (1965).

[5] Lindquist, R. W., Schwartz, R. A. , Misner, C. W. : “Vaidya's Radiating

Schwarzschild metric”, Phys. Rev. 137, B1364 (1965).

[6] Bayin, S. S. “Radiating fluid spheres in general relativity”, Phys. Rev. D

19, 2858 (1979).

[7] Herrera, L., Jimenez, J., Ruggezi, G. L. : “Evolution of radiating fluid

spheres in general relativity”, Phys. Rev. D 22, 2305 (1980).

[8] De Oliveira, A. K. G. , Santos, N. O., Kolassis, C. A. : “More about

collapse of a radiating star”, MNRAS 216, 1001 (1985).

[9] De Oliveira, A. K. G. , Santos, N. O., “Nonadiabatic gravitational

collapse”, Astrophys. J. 312, 640 (1987).

[10] De Oliveira, A. K. G. , Kolassis, C. A., Santos, N. O., “Collapse of a

radiating star revisited”, MNRAS 231, 1011 (1988).

[11] Bonnor, W. B., de Oliveira, A.K. G., Santos, N. O.: “Radiating

153

spherical collapse”, Phys.Rev. 181, 269 (1989).

[12] Kramer, D.: “Spherically symmetric radiating solution with heat flow in

general relativity”, J. Math. Phys. 33, 1458 (1992).

[13] Maharaj, S. D., Govender, M.: “Behaviour of the Kramer radiating

star”, Aust. J. Phys. 50, 959 (1997).

[14] Maharaj, S. D. ,Govender, M. : “Radiating Collapse with Vanishing

Weyl Stresses”, Int. J. mod. Phys. D14, 667 (2005).

[15] Herrera, L., Di Prisco, A. , Hernandez, Pastora, A., Santos, N. O. : “

On the role of density inhomogeneity and local anisotropy in the fate

of spherical collapse”, Phys. Lett. A 237,113 (1998).

[16] Herrera , L., LeDenmat, G., Santos, N. O., Wang, A.: “Shear-Free

Radiating Collapse and Conformal Flatness”, Int. J. Mod. Phys. D

13, 583 (2004 a).

[17] Herrera, L., Di Prisco, A., Martin, J., Ospino, J., Santosh, N.O.,

Triconis U.: “Spherically symmetric dissipative anisotropic fluids: A

general study”, Phys. Rev. D 69, 084026 (2004b).

[18] Herrera, L., Di Prisco, A., Ospino, J.: “Some analytical models of

radiating collapsing spheres”, Phys. Rev. D, 74, 044001 (2006).

[19] Herrera, L., Di Prisco, A., Carot, J.: “Frame dragging and

superenergy”, Phys. Rev. D, 76, 044012 (2007).

[20] Herrera, L., Ospino, j., Di Prisco, A., Fuenmayor, E., Troconis, O.:

“Structure and evolution of self-gravitating objects and the orthogonal

splitting of the Riemann tensor” , Phys. Rev. D 79, 064025 (2009).

[21] Herrera, L., Santosh, N.O.: “Dynamics of dissipative gravitational

154

collapse”, Phys. Rev. D 70, 084004(2004).

[22] Mitra, A.: “Why gravitational contraction must be accompanied by

emission of radiation in both Newtonian and Einstein gravity”, Phys.

Rev. D 74, 024010 (2006).

[23] Naidu, N. F., Govinder, M. : “Causal temperature profiles in horizon-

free collapse”, J. Astrophys. Astron. 28,167 (2007).

[24] Vaidya, P. C. : “Newtonian' Time in General Relativity”, Nature 171,

260 (1953).

[25] Vaidya , P. C. “Nonstatic Solutions of Einstein's Field Equations for

Spheres of fluids radiating Energy”, Phys. Rev. 83, 10 (1951).

[26] Vaidya, P. C.: “An Analytical Solution for Gravitational Collapse with

Radiation”, Astrophys. J. 144, 343 (1966).

[27] Banerjee, A., Chatterjee, S., Dadhich, N. : “Spherical Collapse with Heat

Flow and without Horizon”, Mod. Phy. Lett. A, 35,2335(2002).

[28] Pant N. , Tewari, B.C. : “Horizon-free gravitational

collapse of radiating fluid Sphere”,

Astrophysics &Space Sci. 331 (2), 645 ( 2011).

[29] Schwarzschid, M. : “Structure and Evolution of stars”,

Dover. New York (1958).

Chapter VAdiabatic Collapse of Uniform

Density Sphere with Pressure

155

Chapter VAdiabatic Collapse of Uniform Density

Sphere with Pressure5

After studying adiabatic collapse of a uniform density sphere [with Schwarzschildgeometry in the exterior] using baryon conservation law and no-heat-transfercondition it is concluded that a uniform density sphere [with Schwarzschild geometryin the exterior] always collapses adiabatically with vanishing pressure. Alternatively,we can say that when the exterior geometry is defined by Schwarzschild vacuumsolution then the solution given by Oppenheimer and Snyder (1939)[1] is the onlyvalid solution for the collapse of a uniform density sphere.

5A part of this chapter has been published in Journal of Modern Physics

1 , 143 (2010).

156

5.1 Introduction

After a star has exhausted its nuclear fuel, it can no longer remain in

equilibrium and must undergo gravitational collapse. During gravitational collapse the

physical conditions within the star does not remain same. There are some drastic changes

that takes place within the star. We have examined the various conditions under which the

pressure within the collapsing sphere will vanish.

Radial adiabatic motion of perfect fluid spheres of uniform density, E = E(t),

but non-uniform pressure were discussed by Bonnor and Faulkes [2], Thompson and

Whitrow [3-4] and Bondi [5] under various assumed relationships between central pressure

and density. These authors discussed the problem of collapse and bounce under two

assumptions: first, that the motion is isotropic or shear-free; and second, that the density is

uniform. But Mishra and Shrivastava [6] showed that the condition of uniform density and

regularity at the centre necessarily lead to the isotropic motion. The problem of collapse of a

dust ball has been studied in detail by Durgapal and Pandey [7 ] and they have shown that

there will be no material pressure within the ball as observed by a comoving observer,

although for an external observer there will be material pressure in the ball. It has been also

discussed by Mitra[8] that matter within a collapsing homogeneous sphere is bound to be a

dust (p = 0). We have also shown that the matter with in a collapsing homogeneous sphere

will be dust or pressure will vanish witnin the sphere.

The theme of our work is rather different from that of the other authors. We have

examined whether the pressure can remain finite or not. We have considered the no-heat

transfer (NHT) conditions and baryon conservation law during the collapse. It is shown that

if the fluid is isentropic or (and) the surface temperature remains constant during the collapse

the pressure cannot remain finite (it vanishes). On the other hand if the fluid is neither

isentropic nor the surface temperature remains constant during the collapse, then the results

obtained by earlier authors (Bondi [5]) are found to be inconsistent with the baryonic

conservation and NHT condition.

157

5.2 The metric and uniform density sphere:

Vanishing shear implies that we can simultaneously introduce isotropic and co-moving

coordinates

ds2 y 2dt 2 R2(dr 2 r2d 2) (5.1)

2222 sin),,(),,( dddtrRRtryy

It is assumed that the fluid’s viscosity vanishes, and the adiabatic flow condition makes T10

component of energy momentum tensor vanish in the co-moving coordinates. The energy

momentum tensor can thus be written as

)( PgUUEPT (5.2)

Where E and P are energy density and pressure, respectively;

and the four-velocity will be given by

U (y,0,0,0) (5.3)

The hydrodynamic equations,

T; 0 and

UT; 0 , and the equation of baryon

conservation,

(nU ); 0 (Where n = number density) give us (Misner and Sharp[9],

Demianski [10])

( y /y) P /(P E) (5.4)

And

Us, 0 or s 0 and s 0 (5.5)

.t.ation w.r.differentipartial)(;.tr..ation wdifferentipartial)( tr

5.3 The boundary condition and thermodynamic relation.

For the exterior solution some authors have chosen Schwarzschild vacuum

solution while others have chosen Vaidya’s radiative solutions in the exterior.

In the later case the heat flow is given by Kramer [11].

158

))(/( 2 TyyRKq

Here, K is thermal conductivity. But in the cases where the exterior solution is chosen as

Schwarzschild solution we get NHT conditions (q = 0) given by either

)(where

,is that,0)(

bb

bb

rrTT

yTTyTy

5.6 (a)

Or T = 0 (for cold stars) 5.6 (b)

Or K = 0 5.6(c)

The basic law of thermodynamic change is

Tds dU Pd(1/n) 5.7(a)

nTds dE hdn and (E /s)n nT 5.7(b)

.1and,,0 that,sochosenareofunitsTheenthalpy.specific

/)(andentropyspecificenergy,internalspecificwhere,

hnEPn

nEPhsU

Writing Bondi’s results [5] in the present notations, we get

nR3 B(r) (5.8)

)(and)1/( 32 rBRnrR (5.9)

y FR /R (5.10)

)1/()/(/ 22 rrRR (5.11)

yb / y (P E) /E E

3ERR

(5.12)

And

P /E (rb2 r2)

(1rb2)[ r2( )]

(5.13)

159

(t), (t), F F(t), yb y(r rb ), rb r at the boundary.

Since

E E (t) or E 0 , we write [using equation 5.7(b)]

(5.15)(5.12)]equation[using)/(

(5.14))/()/( 2

sEyTy

EPsTnn

b

5.4 Collapse of uniform density sphere

The collapse of uniform density sphere is discussed under various physical conditions. [We

have assumed that ttA offunctionarbitraryany)( and B(r)

any arbitrary function of r]

5.4(a) using NHT condition given by eq. (5.6a):

Using equation 5.6 (a) in (5.15) we get ,

(5.16))(1)/(

or)/(/ 2

bb

b

ssTnE

sETnn

)( bb rrss .

It is obvious from equation (5.16) that the entropy of an adiabatic uniform density sphere is

minimum at the boundary.

5.4(b) : Isentropic case: Let the entropy be constant throughout the sphere, that is, s

= constant = sb. Equation (5.16) gives

E = n (5.17)

[Using equation (5.8)] )(3 rBER

)()( rBtAR (5.18)

)(/ tARR

[From equation (5.10)] y = )(tA , or

160

0y (5.19)

[From equation (5.4)] 0P , or

)(tPP (5.20)

Since, )(0)( tPrrP b , the pressure vanishes within the sphere. Hence, an isentropic

uniform sphere undergoes a collapse with vanishing pressure only.

5.4 (c) Non-isentropic case with constant surface temperatures: We

assume that the surface temperature remains constant during the collapse. This is very likely

because there is no energy loss to the surrounding from the surface of the sphere. With Tb =

constant during the collapse one gets

n E /[1 Tb (s sb )] A(t)B(r) (5.21)

[From equation (5.8)]

R A ( t ) B (r ) (5.22)

Arguments similar to those in 5.4(b) show that the pressure vanishes inside thesphere.

Hence, an adiabatic uniform density sphere with constant surface temperature

collapses with vanishing pressure.

5.4(d) General case: Neither the fluid is isentropic nor the temperature of the

surface remains constant. In this case

E n[1 Tb (s sb )] (5.23)

On differentiating with respect to time we obtain

E

En

nTb(s sb )

1 Tb (s sb )(5.24)

For an adiabatic motion the total mass energy is a constant of motion,

that is,

M (4 /3)ERb3 constant

Or

E /E 3Rb /R (5.25)

161

Using equations (5.8), (5.11), (5.24), and (5.25) we get

22222

22

22

22

2

2

2

2

)(1

)(3

)1()1(

)(3

1

3

1

3

)(1

)(

bb

b

b

b

b

b

bb

bb

rrrr

rr

rr

rr

r

r

r

r

ssT

ssT

(5.26)

No choice of functions

s s(r), (t) and Tb Tb (t) can satisfy this equation. The

solutions obtained by various authors for collapsing/expanding uniform density

[with Schwarzschild exterior solutions] are inconsistent with the conservation law

and NHT .

5.5 Explanation of inconsistency:

Equation 5.7(b) shows that

(E /s)n nT

but from equation (5.16) we see that

(E /s)n nTb . Therefore,

nT nTb or T Tb . Since,

Ty Tbyb [from equation (5.6 a)] we get

y yb A(t).

Hence,

0)(

0(5.4)]equation[fromor0

brrPP

Py

The pressure vanishes throughout the sphere.

5.5(a) using NHT condition given by eq. 5.6(b):

When T = 0 equation (5.14) gives )(or0 tnnn

or,

R A ( t ) B (r ) [from equation (5.8)]. As shown in 5.4(b) the pressure

vanishes inside the sphere.

5.5 (b) using NHT condition given by eq. 5.6(c):

When thermal conductivity K = 0, it seems that all the relations of Bondi’s paper are

inconsistent . However, let us analyse this condition in some details. From equation (5.7 b)

we can see that

n /n E /(P E) and n /n E /(P E) (T /h) s (5.27)

162

And for

E E ( t ) ,

(T /h) s n /n (5.28)

When K = 0, no heat enters or leaves any layer within the structure during the

collapse that is we can consider temperature of each layer to be independent of time

or

T = T (r).

Eliminating n from the twin equations (5.27) we obtain [11]

Th

s E P P EP E

0

E PP E

for E E(t)(5.29), or

nT (T /h)P E ( P / s ) (5.30)

It can be seen from equation (5.13), that the right hand side of equation (5.30) cannot be

made zero in any case.

Now, we consider a hypothetical case that during the collapse, though K = 0,

somehow the temperature of each layer changes with time making

T T ( r, t ) , but at the

surface the temperature will not change with time, that is,

Tb 0 . It can be seen that

brrb TPsPEET )/( (5.31)

The right hand side of equation (5.31) cannot be made zero.

5.6. Results & Discussions

After studying adiabatic collapse of a uniform density sphere [with

Schwarzschild geometry in the exterior] using baryon conservation law and NHT

condition it is concluded that,

(i) If the fluid (with

K 0 ) is isentropic or (and) the sphere’s surface temperature

remains constant during the collapse, then the pressure vanishes inside the sphere.

163

(ii) If neither the fluid (with

K 0 ) is isentropic nor the surface temperature of the

sphere remains constant during the collapse the solution obtained by Bondi (1969)

and other authors are found to be inconsistent with the baryonic conservation and

NHT condition. Moreover, it is again seen that the pressure vanishes.

(iii) If the temperature is zero (cold stars) the pressure vanishes inside the

configuration.

(iv) If

K 0 , we can show that the pressure vanishes throughout the sphere.

Thus, we conclude that a uniform density sphere [with Schwarzschild geometry

in the exterior] always collapses adiabatically with vanishing pressure. The matter

with in a collapsing homogeneous sphere is bound to be dust (p = 0). Or we can say

that when the exterior geometry is defined by Schwarzschild vacuum solution then

the solution given by Oppenheimer and Snyder [1] is the only valid solution for the

collapse of a uniform density sphere.

164

5.7 References:

[1] Oppenheimer, J. R. and Snyder, H.: “On Continued Gravitational

Contraction” , Phy. Rev. 56, 455(1939).

[2] Bonnor , W. B. and Faulkes, M. C. : “Exact Solutions for Oscillating

Spheres in General Relativity”, Mon. Not. R. astr. Soc. 137 239

(1967).

[3] Thompson, I. H. and Whitrow, G. J.: “Time-Dependent Internal

Solutions for Spherically Symmetrical Bodies in General

Relativity-I. Adiabatic collapse”, Mon. Not. R.astr.Soc. 136 208(1967).

[4] Thompson, I. H. and Whitrow, G. J.: “Time-dependent internal

solutions for spherically symmetrical bodies in general relativity-II.

Adiabatic radial motions of uniformly dense spheres”, Mon. Not. R.

Astr. Soc. 139,499(1968).

[5] Bondi, H. : “Gravitational Bounce in General Relativity”, Mon. Not.

R. Astr. Soc. 142 333(1969).

[6] Misra, R. M. and Srivastava, D. C.: “Relativity-Bounce of Fluid

Spheres”, Nature 238, 116(1972).

[7] Durgapal M. C. and Pande A. K. : “Gravitational Collapse of a dust

ball- The external point of view” ,Astrophysics &Space Science

116, 349 1985.

[8] Mitra, A. : “The matter in the Big-Bang model is dust and not any

arbitrary perfect fluid”, Astrophysics & Space Science DOI

10.1007/s10509-011- 0635-8(2011).

165

[9] Misner, C.W. and Sharp, D. H. : “Relativistic Equations for

Adiabatic, Spherically Symmetric Gravitational Collapse” , Phys.

Rev. B, 136, 571(1964).

[10] Demianski, M.: “Relativistic Astrophysics”, (Pergamon Press)

New York 188(1985).

[11] Kramer, D.: “Spherically Symmetric Radiating Solution with Heat

Flow in General Relativity” , J. Math. Phys. 33 (4), 1458(1992).

[12] Nariai, H.: “A Simple Model for Gravitational Collapse with

Pressure Gradient”, Progress of Theoretical physics 38

,92 (1967).

Chapter VIA New Time Dependent solution of

Einstein’s field equations

&

Radiating fluid balls in conformallyflat space-time.

166

CHAPTER VI

A new time dependent solution of Einstein’s field equations&Radiating fluid balls in conformally flat space-time.

In this chapter we have obtained a new time dependent solution of Einstein field

equations . we have discussed various conditions under which the B.C.T. solution [1] in

conformally flat space time region will be suitable for the modeling of radiating stellar

objects . The variation of pressure, density, pressure-density ratio , luminosity from centre to

surface have been studied. With suitable choice of constants appearing in the solution we

have shown that pressure, density, luminosity, pressure-density ratio all are positive with in

the radiating fluid ball. It has been also shown that as the time will evolve the pressure and

density will also tend to increase indicating that the astrophysical object will move towards

more and more compact stage.

167

6.1 Introduction:

To study the astrophysical objects emitting a large amount of energy in the form of

neutrinos and photons the non static solution of Einstein’s field equations with outflowing

radiation becomes significant. The problem of such massive stellar objects was initiated by

Vaidya [2-3] with extension of Tolman [4] taking account of out flowing radiation. We will

study the radiating fluid balls in conformally flat space time metric. In conformally flat

space-time metric there is only one variable that is to be determined by solving the field

equations.

In case of an empty space-time, a space-time conformal to the flat space time

is itself flat [5]. But in case of nonempty space-time it may not be true. The study of

nonempty space-time conformal to the flat space-time may have significance on the

radiating astrophysical objects due to the mathematical simplicity involved. Realistic

models of radiating fluid spheres in general relativity are few on account of

associated mathematical complexities. P.C. Vaidya [2-3] solved the field equation of

general relativity for physically meaningful models of radiating fluid balls. Bayin,

S. S. [6] suggested a generalization of Vaidya's metric which, designed to depict the

external field of radiating mass. Tewari, B. C. et al.[1] have considered the

spherically symmetric space time in conformally flat form which had simplified the

field equation to a great extent. Jasim et al. [7] reconsidered the spherically

symmetric metric in conformally flat form and obtained a class of radiating fluid

sphere. A relativistic model of fluid sphere filled with matter and radiation has been

constructed by Jasim et al. [8]. The immense gravity objects radiate a large amount

of energy while undergoing gravitational collapse. Oppenheimer and Synder [9]

first studied gravitational collapse for spherically symmetric fluid distribution

neglecting radiation , pressure and rotation. But Vaidya [2-3] [10] gave the idea that

the gravitational collapse is a highly dissipating energy process and took into

account the outflowing radiation also while solving the Einstein’s field equations.

Thus the radiating fluid ball models may be studied by solving the Einstein’s field

equations either by using the general metric of space-time having three variable or

by using the conformally flat space-time metric having only one variable. Herrera et al.

proposed a general method for obtaining collapsing radiating models from exact

168

static solutions of Einstein’s field equations for a spherically symmetric fluid distribution.

Some of the authors studied the radiating fluid spheres with physically significant solutions

in the free streaming case (Tewari [11-13], Pant et al. [14] , Pant and Tewari [1]) by solving

the modified field equations proposed by Misner [15], Lindquist et al. [16]. We have also

studied the solution obtained by Tewari in conformally flat metric form and checked its

validity for representing the radiating fluid spheres. We have also obtained a new solution of

Einstein’s field equaions by using the general metric of space-time having three variables.

6.2 The metric and Field Equations:

For the gravitational field within the radiating fluid sphere the space time metric in three

variables is given by

22222222 sin dderdredtceds (6.1)

Where α , β , are functions of r and t.

The field equations of general relativity for a distribution of mixture of a

perfect fluid and radiation are

ji

ji

ji T

c

GgRR

4

8

2

1 (6.2)

where

jij

ij

ij

i wwc

qgpvvcpT )( 2 (6.3)

where p and respectively denote the isotropic pressure and density of the matter

within the distribution and vi its four velocity, q denotes the radiation flux density and iw its

four velocity which is null.

0ii ww (6.4)

We choose iw such that[15]

,1 11

rB

tcA

xw

ii

(6.5)

which implies that q is the energy density of the radiation in the rest frame of the fluid

[15,16]

169

jiji wwqvvq (6.6)

From Equation (6.2) we obtain four independent equations, which in view of eq. (6.3) can

be written as

22Tp (6.7)

,22

00

11

2 TTTc (6.8)

,)( 0

1212

3

Twec

q

(6.9)

0

122

21

1 TeTT

(6.10)


qerL 24 (6.11)

6.3 New solution of the field equations:

In order to solve the field equations we assume the following separable forms for the

gravitational field variables α, β, [12]

tgrfe tr 22, (6.12)

tnrhe tr 22, (6.13)

tnrke tr 22, (6.14)

The field equation given in Eq. (6.10) with the metric given in Eq. (6.1) reduces then to the

following form

021

22

2

22

2

f

fh

g

n

rk

h

rrf

f

hr

h

hk

kh

hf

hf

kf

kf

f

f

k

k

k

k (6.15)

we consider

lg

n2aconstant (6.16)

Now Eq. (6.15) becomes only r dependent equation in three function h, f and k

To explore the solution of Eq.(6.15) we make certain assumptions

rf

f 1

and22 2 kh (6.17)

We get rAf and 0f (6.18)

Now Eq. (6.15) may be written as

170

0232

22

2

rA

kl

kr

k

k

k

k

k

(6.19)

or

023222

rA

l

k

k

rk

k

(6.20)

The solution of Eq.(6.20) is written as follows :

14

4log

4

2

C

Brr

A

lk

(6.21)

14

4log

4

22

C

Brr

A

lh

(6.22)

Where A, B, C are integration constants.

Assuming t as the proper time of the distant observer at rest on the hyper surface

rs we have

1)()( 22 tgrf s (6.23)

From Eq. (6.18) and Eq. (6.23) we obtain

srA

tg1

(6.24)

The expression for n(t) is obtained as

trA

clKtn

s2)(

(6.25)

Where K is a non negative constant.

171

6.4 Properties of the New solution :

The expressions for pressure, density, radiation flux density, luminosity are

given by

2

2

22

4482

2

2

224

2

log2

8

log

8

log2727

2164

3

2

18

CA

rCl

A

rl

A

rrlB

A

BlBC

rrB

A

l

nrc

PG

(6.26)

2

2

2

22

284

44

2

2

224

42

3

2

log2

8

log

4

7

2

2

2

21

8

log221

216

27

8

7

2

2log

18

A

l

A

lC

A

rCl

A

lrBr

A

lBBCr

A

rlBr

A

B

A

rl

A

l

A

Clr

nrc

EG

(6.27)

The expression for q is given as

ghfn

fnq

c

G225

28

(6.28)

In view of Eq.(6.28) we get

C

Brr

A

l

rAn

lq

c

G

4log

4

2

2

8 4

225

(6.29)

For the radiating fluid spheres we are restricted to 0l

The luminosity will be given by the expression

qnkrL 2224 (6.30)

or

145

4log

4

2

22

C

Brr

A

l

A

l

G

cL

(6.31)

From eq. (6.31) we observe that luminosity is constant with respect to time.

172

We have obtained a new time dependent solution of Einstein’s field equations.

Although the solution is singular at the centre, it may be useful for the study of late

stages of the stellar evolution when some kind of radiation may be emitted from the

stellar object and pressure, density approach towards infinity.

6.5 Field Equations of a radiating fluid ball in conformally flatspace time metric :

The line element for a spherically symmetric distribution of collapsing fluid inconformally flat space –time metric is given by

)sin(),( 222222222 ddrdrdtctrBds (6.32)

For metric (6.32) the expressions for the physical parameters viz. pressure,matter –density and the radiation flux density are given as

2

22

24

22218

B

B

B

B

rB

B

B

B

B

B

Bp

c

G

(6.33)

2

22

22

363

18

B

B

Br

B

B

B

Bc

G

(6.34)

225

4218

B

BB

B

B

Bq

c

G

(6.35)


qBrL 224 (6.36)

The rate of contraction of boundary is given as

B

BrU

(6.37)

6.6 Boundary conditions for radiating fluid ball in conformally flatspace time:

uRtrBr sss , (6.38)

173

s

s B

Br

B

BrdudtctrB

1, (6.39)

0, trp s (6.40)

s

s B

BrBr

B

Br

G

ctrmuM

22,

232

232

(6.41)

The expression for L∞ is given as

2

1s

s B

Br

B

BrLL

(6.42)

Now we study the solution obtained by Tewari [1] in conformally flat space time.

The general solution of the field equation as obtained by Tewari[1] is

1)(

drctrfrtGB(6.43)

The particular solution obtained is [1]

)( ctrbectrf and )()( ctrb sehtG (6.44)

where b is a constant and h is a nonnegative constant satisfying the followingrelation:

0342 3224 srhbhbhb (6.45)

From which the following relation is obtained

hb

hbhbrs 3

224

4

32

(6.46)

The expression for pressure , matter density, and radiation flux density are obtainedas[1]

ss rbrrbbrbct ehbebrheb

epc

G 222)(22

24

)21(238

(6.47)

ss rbrrbbrbct ehbebrhe

be

c

G 222)(22

22

3638

(6.48)

174

)(

2)()(

5)1(

12

8 ctrbctrbctrb ebrb

ehebrqc

G

(6.49)

The luminosity is given by

rbrb

br

ebrehb

erb

G

cL

s

1

22

335

(6.50)

brU (6.51)

The pressure and matter density will be positive at the centre if following inequalitywill be satisfied

31 2 sabrheb (6.52)

The condition 02

0 cp will be satisfied at the centre if

sbrheb 2

2

3

(6.53)

From (6.35), (6.41), (6.43), the following relation is obtained [1]

(6.54)

The Vaidya mass as obtained by Tewari is expressed as

tcrb sehbhb

hbhbb

G

cuM

2224

32242

12108

32)( (6.55)

Also, 128

23)(2224

2242

2

hahb

hbhb

Rc

uMG

s

(6.56)

The luminosity as observed at rest at infinity is given as [1]

12432

23224

32425

hbhb

hbhb

G

cL (6.57)

If)(

)(22 uRc

uMG

s

< 1 the collapse will be horizon free. For suitable choice of

constants , we can get the collapse in which the horizon is never encountered.

ctrbs

sehbhb

hbhbbR

224

242

23

23

175

6.7 Different cases of radiating fluid spheres:

Case I: b = -1.5 , h = 1, rs = 0.180556

The value of srbheb 2 comes out to be 1.716173 for case I, hence the inequalities (6.52)

and ( 6.53) are satisfied indicating that pressure and matter density will be positive

inside the radiating fluid ball. Table 1 shows the march of pressure, matter density,

ratio of pressure-density, and luminosity from centre to surface.

Table 6.1: Variation of pressure, density, pressure-density ratio and luminosity at anyinstant within the ball (p in dyne-cm-2, ε in g-cm-3 and L in erg-s-1).

r/rstcbep

c

G 24

8 tcbec

G 22

8

2c

p

L

0.0 1.5498 2.5936 0.5975 0

0.1 1.3583 2.7557 0.4929 1.04× 1045

0.2 1.1759 2.8666 0.4102 8.08× 1046

0.3 1.0022 2.9448 0.3403 2.66× 1046

0.4 0.8368 3.0027 0.2787 6.13× 1047

0.5 0.6794 3.0482 0.2228 1.17× 1047

0.6 0.5296 3.0865 0.1715 1.97×1047

0.7 0.3870 3.1206 0.1240 3.04× 1047

0.8 0.2515 3.1524 0.0797 4.43× 1047

0.9 0.1225 3.1830 0.0385 6.15× 1047

1.0 0.0000 3.2130 0.0000 8.22× 047

For the above mentioned values of constants we get from equation (6.56)

176

sRc

uMG2

)(2= 0.03125 < 1 (6.58)

From equation (6.58) we can say that the collapse process will keep on going without

any horizon being formed. The left over core will be a black hole of point dimension.

We have the case of naked singularity.

Case ii : b = -1.4 , h =1.5, rs= 0.014359

Table 6.2: The march of pressure, density, pressure-density ratio,luminosity (p in dyne-cm-2, ε in g-cm-3 and L in erg-s-1) at any

instant within the radiating fluid ball.

r/rstcbep

c

G 24

8 tbcec

G 22

8

2c

p

L

0.0 0.2340 11.1780 0.02099 0

0.1 0.2108 11.1975 0.01883 3.04 × 1041

0.2 0.1870 11.2105 0.01668 2.43× 1042

0.3 0.1634 11.2193 0.01456 8.18× 1042

0.4 0.1398 11.2256 0.01245 1.94× 1043

0.5 0.1163 11.2303 0.01035 3.77× 1043

0.6 0.0928 11.2343 0.00826 6.51× 1043

0.7 0.0695 11.2380 0.00618 1.03× 1044

0.8 0.0462 11.2415 0.00411 1.54× 1044

0.9 0.0230 11.2452 0.00205 2.18× 1044

1.0 0.0000 11.2491 0.00000 2.99× 1044

The value ofsRc

uMG2

)(2= 0.0002 < 1 (6.59)

177

For case ii , (sRc

uMG2

)(2 < 1) hence, we get the horizon free gravitational collapse. A

naked singularity will be formed. A naked singularity is a gravitational singularity,

without an event horizon. In a black hole, there is a region around the singularity, the

event horizon, where the gravitational force of the singularity is strong enough so that

light cannot escape. Hence, the singularity cannot be directly observed. A naked

singularity, by contrast, is observable from the outside.

Case iii:

b = -1.2 , h = 1.5, rs = 0.256019

Table 6.3: The march of pressure, density, pressure-density ratio, luminosity at anyinstant within the radiating fluid ball. (p in dyne-cm-2, ε in g-cm-3 and L in erg-s-1)

r/rstcbep

c

G 24

8 tbcec

G 22

8

2c

p

L

0.0 2.5371 3.1746 0.7991 0

0.1 2.2147 3.4587 0.6403 1.58 × 1045

0.2 1.9097 3.6672 0.5207 1.23× 1046

0.3 1.6212 3.8222 0.4241 4.03× 1046

0.4 1.3484 3.9400 0.3422 9.26× 1046

0.5 1.0905 4.0319 0.2704 1.76 1047

0.6 0.8468 4.1060 0.2062 2.95× 1047

0.7 0.6165 4.1681 0.1479 4.55× 1047

0.8 0.3990 4.2221 0.0945 6.6× 1047

0.9 0.1937 4.2706 0.0453 9.14× 1047

1.0 0.0000 4.3154 0.0000 1.22× 1048

178

For the case (iii) we get

0196.0)(2

2

sRc

uMG< 1 (case of naked simgularity) (6.60)

Case iv: b = -1 , h = 2.5, rs = 0.175

Table 6.4: The march of pressure, density, pressure-density ratio, luminosity atany instant with in the radiating fluid ball. (p in dyne-cm-2, ε in g-cm-3 and L inerg-s-1)

r/rstbcep

c

G 24

8 tbcec

G 22

8

2c

p

L

0.0 2.7929 10.2129 0.2734 0

0.1 2.4726 10.4965 0.2555 2.48 × 1044

0.2 2.1620 10.7111 0.2018 1.95× 1045

0.3 1.8611 10.8747 0.1711 6.46 × 1045

0.4 1.5694 11.0008 0.1426 1.51× 1046

0.5 1.2867 11.0998 0.1159 2.89× 1046

0.6 1.0127 11.1793 0.0905 4.91× 1046

0.7 0.7474 11.2451 0.0664 7.67× 1046

0.8 0.4902 11.3014 0.0433 1.1 × 1047

0.9 0.2412 11.3513 0.0212 1.58× 1047

1.0 0.0000 11.3970 0.0000 2.13× 1047

For the case iv we get

sRc

uMG2

)(2= 0.01388 < 1 (6.61)

179

Case v: b = -1 , h = 2.0, rs = 0.375

Table 6.5: The march of pressure, density, pressure-density ratio, luminosity atany instant with in the radiating fluid ball. (p in dyne-cm-2, ε in g-cm-3 and L inerg-s-1)

r/rstbcep

c

G 24

8 tbcec

G 22

8

2c

p

L

0.0 3.8596 2.6683 1.4464 0

0.1 3.3431 3.1383 1.0652 3.12 × 1045

0.2 2.8605 3.5009 0.8171 2.40× 1046

0.3 2.4099 3.7820 0.6372 7.83× 1046

0.4 1.9893 4.0018 0.4971 1.79× 1047

0.5 1.5968 4.1754 0.3824 3.38 × 1047

0.6 1.2308 4.3144 0.2852 5.63× 1047

0.7 0.8895 4.4276 0.2009 8.64× 1047

0.8 0.5716 4.5215 0.1264 1.25× 1048

0.9 0.2755 4.6010 0.0598 1.72× 1048

1.0 0.0000 4.6698 0.0000 2.28× 1048

For the case v we get

sRc

uMG2

)(2= 0.02778 < 1 (6.62)

For the case v although the pressure and pressure-density ratio are

decreasing from centre to surface but pressure-density ratio is not less than 1

every where within the radiating fluid sphere. Hence the combination of

constants ( b = -1 , h = 2.0, rs = 0.375) is not suitable for the modeling of

180

radiating fluid spheres. Similarly for other combinations of constants b and h

the solution is not well behaved , hence not suitable for modeling of radiating

fluid spheres.

6.8 The behaviour of pressure and density w. r. t. time

Differentiating equation (6.47) we get


bcepc

G 222)(22

24

)21(23

28

(6.63)

Differentiating equation (6.63) again we get


ecbpc

G 222)(22

2224

)21(23

48

(6.64)

From Eq. (6.64) it is clear that pressure will increase with time.

Similarly if we differentiate equation (6.48) we get


bbce

c

G 222)(22

22

363

28

(6.65)

Differentiating equation (6.65) again we get


becb

c

G 222)(22

2222

363

48

(6.66)

In view of equation (6.65) and ( 6.66) we can say that the density will increasewith time. As the time will evolve, the pressure and density will tend towardsinfinity.

181

6.9 Results and Discussions:

We have obtained a new time dependent solution of Einstein’s field equations in

general metric form using three variables. The expressions for pressure, energy density,

luminosity and radiation flux density have been obtained. Although the solution is

singular at the centre it may give some insight in understanding certain stages of

radiating stars. We have also studied a known non static solution of Einstein’s field

equations in conformally flat space time which was obtained by Tewari [1]. The

different cases of radiating fluid ball have been considered for the different

combination of constants. The constants have been chosen in such a way , so that the

inequalities (6.52) and ( 6.53) may get satisfied. From the Tables 6.1, 6.2, 6.3 , 6.4 and

6.5 we observe that the pressure , pressure density ratio decrease as we go from centre to

surface and pressure becomes zero at the boundary. Luminosity increases as we go

from centre to surface. Tables 6.1, 6.2, 6.3, 6.4 show that the value of pressure-density

ratio is less than one within the radiating fluid sphere. But in the Table 6.5

corresponding to case v we observe that although pressure and pressure-density ratio is

decreasing from centre to surface, but pressure-density ratio is not less than 1

everywhere within the radiating fluid sphere. Hence case v can not be considered a

realistic case for radiating fluid ball problem. One important parameter that we have

investigated for all these models issRc

uMG2

)(2. We observe that for all the four cases

sRc

uMG2

)(2< 1, which indicates that collapse will be horizon free and naked singularity

may come into existence. If the horizon is formed during the collapse we will be

deprived of the events happening inside the horizon.

From equations (6.47) and (6.48) it is evident that with the passage of time both

the pressure and density will tend towards infinity and we will come towards more

and more compact stage. In this way the solution obtained in conformally flat metric

case [1] may be applied for constructing the radiating fluid ball models.

182

6.10 References :

[1] Pant, D. N., Tewari, B. C. : "conformally flat metric representing a

radiating fluid ball", Astrophysics. Space Sci. 163, 223 (1990).

[2] Vaidya, P.C.: “Non-static solutions of Einstein's field equations for

spheres of fluid radiating energy”, Phys. Rev. 83, 10 (1951).

[3] Vaidya, P. C.: “Newtonian Time in General Relativity”, Nature

171, 260 (1953).

[4] Tolman, R. C.: “Static solutions of Einstein’s field equations for

spheres of fluid”, Phys. Rev. 55 ,364 (1939).

[5] Singh, K. P. and Roy, S. R. : Proc. Nat. Inst. Sci. India, 32, 223

(1966).

[6] Bayin, S. S.: “Radiating fluid spheres in general relativity”,

Phys. Rev. D 19, 2858 (1979).

[7] Jasim, M. K. et al., On conformally-flat radiating fluid

spheres", Proceeding of GR15, Dec. 16-21, IUCAA, Pune, India,

(1997).

[8] Jasim, M. K. et al. : “A New Generating Solution of a Relativistic

Radiating Fluid Spheres Model”, Applied Mathematical Sciences,

5(80),4005 (2011).

[9] Oppenheimer, J. R . and Snyder, H . : “On Continued Gravitational

Contraction” , Phy. Rev. 56, 455(1939).

[10] Vaidya, P.C.: “An analytical solution to gravitational collapse with

radiation” , Astrophys. J. 144, 343 (1966).

183

[11] Tewari, B.C. : “Radiating fluid spheres in general relativity”,

. Space sci. 149, 233 (1988).

[12] Tewari, B. C.: “Relativistic radiating fluid distribution”, Indian J.

Appl. Phys. 32,504 (1994).

[13] Tewari, B. C.: “Relativistic model for radiating star”,

Astrophys.Space Sci. 306, 273 (2006).

[14] Pant, N. , Tewari, B.C.: “Horizon-free gravitational collapse of

radiating fluid sphere”, 331(2), 645 (2011).

[15] Misner, C. W.: “Relativistic Equations for Spherical

Gravitational Collapse with Escaping Neutrinos”, Phys. Rev.

B137, 1350 (1965).

[16] Lindquist, R .W., Schwartz, R. A. , Misner, C. W. :“Vaidya's

Radiating Schwarzschild Metric”, Phys. Rev. 137,

B1364 (1965).

184

List of Publication

1. Fuloria Pratibha , Durgapal M. C. : “A non singular solution for sphericalconfiguration with infinite central density”, Astrophys. Space Sci., 314, 249(2008).

2. Durgapal M. C., Fuloria Pratibha : “On Collapse of uniform Density Sphere withPressure”, J. Modern Physics, 1, 143 (2010).

3. Fuloria Pratibha , Tewari B. C., Joshi B. C .: “Well behaved class of chargeAnalogue of Durgapal’s relativistic exact solution”, J. Modern physics, 2, No.101156 (2011).

4. Pant N., Tewari B. C., Fuloria Pratibha , “Well Behaved parametric class ofExact solutions of Einstein-Maxwell Field equations in general relativity”, J.Modern Physics 2, 1538 (2011).

5. Pant N., Fuloria Pratibha, Tewari B.C.: “A new well behaved exact solution ingeneral relativity for perfect fluid”, Astrophys & Space Sci 340, 407(2012).

6. Fuloria Pratibha , Tewari B. C.: “A Family of charge Analogue of DurgapalSolution”, Astrophys. Space Sci. 341,469(2012).

7. “Radiating fluid ball models with horizon free gravitational collapse”, Paperpresented in “National Seminar on Recent Trends in Micro and Macro Physics”(NSRTMMP-2011) at Deptt. Of Physics , Gov. Post Graduate college, Gopeshwar,Chamoli.

8. “A New Relativistic Exact solution for Static fluid sphere” , paper communicated in“ 5th Uttarakhand State Science And Technology congress”.

9. “A parametric Class of well Behaved Relativistic Charged Fluid Distribution” ,paper presented in “6th Uttarakhand State Science And Technology congress” atKumaun University, S.S.J. Campus, Almora.

Astrophys Space Sci (2008) 314: 249–250DOI 10.1007/s10509-007-9730-2

O R I G I NA L A RT I C L E

A non singular solution for spherical configuration with infinitecentral density

Pratibha Fuloria · M.C. Durgapal

Received: 3 October 2007 / Accepted: 14 December 2007 / Published online: 22 March 2008© Springer Science+Business Media B.V. 2007

Abstract A non-singular exact solution with an infinitecentral density is obtained for the interior of sphericallysymmetric and static structures. Both the energy density andthe pressure are infinite at the center but we have eλ(0) = 1and eν(0) �= 0. The solution admits the possibility of receiv-ing signals from the region of infinite pressure.

Keywords General relativity · Exact solution ·Astrophysics

1 Introduction

Many interior solutions for spherically symmetric and sta-tic cases are available for relativistic structures in the vastlyavailable literature. These solutions are obtained by eithersolving Einstein field equations analytically or by choosingsome equation of state for the matter within the configura-tion and then using numerical computation. In general, themetric chosen for obtaining these interior solutions is theSchwarzschild metric given by

ds2 = eνdt2 − eλdr2 − r2 sin2 θdφ2 (1)

where ν and λ are functions of ‘r’ alone.There are many solutions with finite mass and infinite

central density (Tolman 1939; Zeldovich 1962; Misner andZapolsky 1964; Durgapal and Gehlot 1968). Misner andZapolsky discussed neutron star models with such extremedensity distribution (Central density, Ec = ∞). But in most

P. Fuloria · M.C. Durgapal (�)Department of Physics, Kumaun University Campus, 692/3Dugalkhola, Almora, Uttaranchal 263601, Indiae-mail: [email protected]

of the above mentioned solutions the value of eλ(0) �= 1 andthe curvature at the center is infinite; the solutions becomesingular at the center. The spherical symmetry demands thateλ at the center should be 1. Also, in all these solutionseν(0) = 0 which makes it impossible to have any informa-tion out of this region.

In this paper we have reinvestigated these solutions toobtain a non-singular interior solution with infinite centraldensity. Such solutions can throw much light on the structureat very late stages of their evolution.

(Because at r = 0, Ec = ∞, eλ(0) = 1 and eν(0) �= 0.)

2 Field equations and their solution

Using the metric in (1) and the relations

T 11 = T 2

2 = T 33 = −P, T 0

0 = E (2)

We can write down the Einstein’s field equations in a simpleform (Durgapal and Fuloria 1985)

x2zy′′ + 2x2z′y′ + (xz′ − z + 1)y = 0 (3)

8πP

C= 4z

(y′

y

)− (1 − z)

x(4)

8πE

C= (1 − z)

x− 2z′ (5)

where z = e−λ; By = eν2 ; x = Cr2; P = pressure; E =

energy density; B and C are constants and the prime rep-resents differentiation with respect to x.

There are three equations and four variables. Hence werequire one more equation to obtain a solution of the fieldequations.

http://dx.doi.org/10.1007/s10509-007-9730-2

mailto:[email protected]

250 Astrophys Space Sci (2008) 314: 249–250

Let us assume that

z = 1 − x13 (6)

[Similar type of general solutions are available in the widelyavailable literature (see Kramer et al. 1981 and referencesgiven there in); but this particular solution is obtained in or-der to discuss non-singular nature at the center of a structurewhen both the pressure and energy density tend to infinityand still eν(0) �= 0.]

From (5) and (6) we obtain

8πE

C= 5

3x

−23 (7)

It can be seen that at x = 0, we get

E0 = ∞ and eλ(0) = 1 (8)

A particular solution of (3) for the value of Z given by (5) is[Tolman V solution (1939); Durgapal and Gehlot (1968)]

yi = x (9)

A more general solution of (3) is obtained by using the rela-tion (Durgapal and Fuloria 1985)

y = yi

[1 + const

∫dx

y2i

√z

](10)

Using (6), (9) and (10) we obtain

y = x + A

{√1 − x

13

(1 + 5

4x

13 + 15

8x

23

)

+ 15

8x log

(√1 − x

13 + 1

)− 5

16x logx

}(11)

where A is a constant [this is a special case of the gener-alized solution obtained by Wymann (1949) in the form ofhyper geometric functions, making the interpretation some-what obscure].

It can be shown easily that at x = 0

y = y0 = A

Equations (4) and (11) give us the expression for pressureas:

8πP

C= 1

x

{(4 − 5x

13

)− A

y

(1 − x

13

) 12}

(12)

The values of the unknown constants A, B , C can be ob-tained from the boundary conditions at the surface of thestructure. The solutions are required to be continuous withthe Schwarzschild solutions at r = a or x = X(≡ Ca2).Thus

eν(a) = e−λ(a) = 1 − 2u (13)

where u = Ma

; M = total mass of the configuration and a =radius of the configuration, and

P(X) = 0 (14)

Using boundary conditions (13) and (14) we obtain

A = y0 = 8u3

{√1 − 2u

1 − 2.5u

−[√

1 − 2u(1 + 2.5u + 7.5u2)

+ 15u3 log

(√1 − 2u + 1√

2u

)]}−1

B = (1 − 2.5u)

y0and C = 8u3

a2

It is seen that the central pressure is infinite but the centralredshift is given by

1 + Zc = e−ν(0)/2 = 1/By0 = (1 − 2.5u)−1/2

The central redshift remains finite for all the values ofu < 0.4. The solution thus remains valid for u < 0.4, atu = 0.4, eν(0) = 0. The maximum surface redshift that canbe obtained is (

√5 − 1).

3 Discussion

The solution obtained here is important because it is non-singular and corresponds to infinite central density and pres-sure. The important feature of this solution may be writtenas:

eλ(0) = 1, eν(0) = finite

when E(r = 0) = infinite and P(r = 0) = infinite.Since the central redshift, Zc, is finite, the solution pro-

vides a possibility of obtaining information from the regionstending to attain infinite energy density and infinite pressure.[In other generally known solutions the central redshift andpressure become infinite simultaneously.]

References

Durgapal, M.C., Fuloria, R.S.: Gen. Relativ. Gravit. 17, 671 (1985)Durgapal, M.C., Gehlot, G.L.: Phys. Rev. 172, 1308 (1968)Kramer, D., Stephani, H., MacCallam, M., Herlt, E.: Exact Solutions

of Einstein’s Field Equations. CUP, Cambridge (1981)Misner, C.W., Zapolsky, H.S.: Phys. Rev. Lett. 12, 49 (1964)Tolman, R.C.: Phys. Rev. 55, 364 (1939)Wymann, M.: Phys. Rev. 75, 1930 (1949)Zeldovich, Y.B.: Sov. Phys. JETP 14, 1143 (1962)

J. Modern Physics, 2010, 1, 143-146 doi:10.4236/jmp.2010.12020 Published Online June 2010 (http://www.SciRP.org/journal/jmp)

Copyright © 2010 SciRes. JMP

On Collapse of Uniform Density Sphere with Pressure

Mahesh Chandra Durgapal1, Pratibha Fuloria2

1Retired Professor of Physics, Kumaun University, Naintal, India; 2Department of Physics, SSJ Campus, Kumaun University, Al-mora, India. Email: [email protected]

Received March 1st, 2010; revised April 18th, 2010; accepted May 10th, 2010.

ABSTRACT

Adiabatic collapse solutions of uniform density sphere have been discussed by so many authors. An analysis of these solutions has been done by considering the baryonic conservation law and the no heat transfer condition. We have ex-amined whether the pressure can remain finite or not during the collapse.

Keywords: Genral Relativity, Astrophysics, Collapse

1. Introduction

Radial adiabatic motion of perfect fluid spheres of uni-form density, E = E(t), but non-uniform pressure were discussed by Bonnor and Faulkes [1], Thompson and Whitrow [2,3] and Bondi [4] under various assumed re-lationships between central pressure and density. These authors discussed the problem of collapse and bounce under two assumptions: first, that the motion is isotropic or shear-free; and second, that the density is uniform. But Mishra and Shrivastava [5] showed that the condition of uniform density and regularity at the centre necessarily lead to the isotropic motion.

The theme of this paper is rather different from that of the other authors. We have examined whether the pres-sure can remain finite or not. We have considered the no-heat transfers (NHT) conditions (explained in the text) and baryon conservation law during the collapse. It is shown that if the fluid is isentropic or (and) the surface temperature remains constant during the collapse the pressure can not remain finite (it vanishes). On the other hand if the fluid is neither isentropic nor the surface temperature remains constant during the collapse, then the results obtained by earlier authors (Bondi, 1969) are found to be inconsistent with the baryonic conservation and NHT condition.

2. The Metric and Uniform Density Sphere

Vanishing shear implies that one can simultaneously

introduce isotropic and co-moving coordinates

2 2 2 2 2 2( + )ds y dt R dr r d (1)

2 2 2 2( , ), ( , ), siny y r t R R r t d d d

It is assumed that the fluid’s viscosity vanishes, and the adiabatic flow condition makes T10 component of energy momentum tensor vanish in the co-moving coor-dinates. The energy momentum tensor can thus be writ-ten as

( )T P E U U Pg (2)

where E and P are energy density and pressure, respec-tively and the four-velocity,

( ,0,0,0)U y (3)

The hydrodynamic equations,

; 0T and ; 0U T

, and the equation of baryon

conservation, ;( ) 0nU (where n = number density)

give us (Misner and Sharp) [6], (Demianski) [7]

( / ) / ( )y y P P E (4)

and

, 0U s or 0s and 0s (5)

( ) partial differentiation w.r.t. r; ; ( )r partial dif-

ferentiation w. r. t. t.



144

3. The Boundary Condition and Thermodynamic Relation

For the exterior solution some authors have chosen Schwarzschild vacuum solution while others have cho-sen Vaidya’s radiative solutions in the exterior. In the later case the heat flow is given by Kramer [8]

2( / )( )q K yR Ty

Here, K is thermal conductivity. But in the cases where the exterior solution is chosen as Schwarzschild solution we get NHT conditions (q = 0) given by either

( ) 0,Ty that is, b bTy T y where ( )b bT T r r (6)

Or 0T (for cold stars) (7)

Or 0K (8)

The basic law of thermodynamic change is

(1/ )Tds dU Pd n (9)

nTds dE hdn and ( / )nE s nT (10)

where, U specific internal energy, s specific en-tropy and ( ) /h P E n specific enthalpy. The units

of n are chosen so that, 0, ,P E n and 1h .

Writing Bondi’s results (1969) in the present notations, one gets

3 ( )nR B r (11)

2/ (1 )R r and 3 ( )nR B r (12)

/y FR R (13)

2 2/ ( / ) / (1 )R R r r (14)

/ ( ) /3b

E Ry y P E E

E R

(15)

And 2 2

2 2

( )/

(1 )[ ( )]b

b

r rP E

r r

(16)

( ), ( ), ( ), ( ),b b bt t F F t y y r r r r at the

boundary. Since = ( )E E t or 0E , we write [using Equation

(10)] 2( / ) / ( )n n Ts P E (17)

( / ) [using Equation (15)]bTy Ey s (18)

4. Collapse of Uniform Density Sphere

The collapse of uniform density sphere is discussed un-der various physical conditions. [We have assumed that

( ) any arbitrary function of A t t and B(r) any arbi-

trary function of r]

4(a) using NHT condition (6a): Using Equation (6) in (18) one gets,

2/ ( / )bn n T E s or

( / ) 1 ( )b bE n T s s (19)

( )b bs s r r .

It is obvious from Equation (19) that the entropy of an adiabatic uniform density sphere is minimum at the boundary.

4(a) (i): Isentropic case: Let the entropy be constant throughout the sphere, that is, s = constant = sb. Equation (19) gives

E n (20)

[Using Equation (11)]

3 ( )ER B r ( ) ( )R A t B r (21)

/ ( )R R A t

[From Equation (13)] y =A(t) or 0y (22)

[From Equation (4)]

0,P or P=P(t) (23)

Since, ( ) 0 ( )bP r r P t , the pressure vanishes

within the sphere. Hence, an isentropic uniform sphere undergoes a collapse with vanishing pressure only.

4(a) (ii) Non-isentropic case with constant surface temperatures: We assume that the surface temperature remains constant during the collapse. This is very likely because there is no energy loss to the surrounding from the surface of the sphere. With Tb = constant during the collapse one gets

/ [1 ( )] ( ) ( )b bn E T s s A t B r (24)

[From Equation (11)]

( ) ( )R A r B r (25)

Arguments similar to those in 4(i) show that the pres-sure vanishes inside the sphere.

Hence, an adiabatic uniform density sphere with con-stant surface temperature collapses with vanishing pres-sure.

4(a) (iii) General case: Neither the fluid is isentropic nor the temperature of the surface remains constant. In this case

[1 ( )]b bE n T s s

On differentiating with respect to time we obtain



145

( )

1 ( )b b

b b

T s sE n

E n T s s

(26)

For an adiabatic motion the total mass energy is a con-stant of motion, that is,

3(4 / 3) constantbM ER or / 3 /bE E R R (27)

Using Equations (11), (14), (26) and (27) we get 0 22

2 2

2 2

2 2

2 2

2 2 2 2 2

( ) 33

1 ( ) 1 1

3 ( )

(1 )(1 )

3 ( )

1 ( )

b b b

b b b

b

b

b

b b

T s s rr

T s s r r

r r

r r

r r

r r r r

(28)

No choice of functions ( ),s s r ( )t and Tb =

( )bT t can satisfy this equation. The solutions obtained

by various authors for collapsing/expanding uniform density [with Schwarzschild exterior solutions] are in-consistent with the conservation law and NHT.

4(a) (iv) Explanation of inconsistency: Equation (10) shows that ( / ) ,nE s nT but from Equation (19) we

see that ( / )n bE s nT . Therefore, bnT nT or T =

bT . Since, b bTy T y [from Equation (6)] we get y =

( )by A t .

Hence, 0y or [from Equation(4)] 0P P

( ) 0bP r r

The pressure vanishes throughout the sphere.

4(b) using NHT condition (7): When T = 0 Equation (14) gives 0n or ( )n n t or,

(t) ( )R A B r [from Equation (9)]. As shown in 4(a) (i)

the pressure vanishes inside the sphere.

4(c) using NHT condition (8): When thermal conductivity K = 0, it seems that all the relations of Bondi’s paper are consistent. However, let us analyse this condition in some details. From Equation (10) we can see that

/ / ( )n n E P E and / / ( ) ( / )n n E P E T h s

(29)

And for ( )E E t , ( / ) /T h s n n (30)

When K = 0, no heat enters or leaves any layer within the structure during the collapse that is we can consider temperature of each layer to be independent of time or T = T (r).

Eliminating n from the twin Equations (29) we obtain (Nariai ) [9]

0T E P P E

sh P E

for ( )E P

E E TP E

(31)

or

( / ) ( / )nT T h P E P s (32)

It can be seen from Equation (16), that the right hand side of Equation (32) can not be made zero in any case.

Now, we consider a hypothetical case that during the collapse, though K = 0, somehow the temperature of each layer changes with time making T = T (r, t), but at the surface the temperature will not change with time,

that is, 0bT . It can be seen that

[ ( / ) ] br rbET E P s P T (33)

The right hand side of equation can not be made zero.

5. Conclusions

After studying adiabatic collapse of a uniform density sphere using baryon conservation law and NHT condi-tion it is concluded that, a uniform density sphere [with Schwarzschild geometry in the exterior] always collapses adiabatically with vanishing pressure. Collapse with pressure will involve violation of either the baryonic con-servation law or the no-heat flow condition. Or we can say that when the exterior geometry is defined by Sch- warzschild vacuum solution then the solution given by Oppenheimer and Snyder [10] is the only valid solution for the collapse of a uniform density sphere.

REFERENCES

[1] W. B. Bonnor and M. C. Faulkes, “Exact Solutions for Oscillating Spheres in General Relativity,” Monthly No-tices of the Royal Astronomical Society, Vol. 137, 1967, pp. 239-251.

[2] I. H. Thompson and G. J. Whitrow, “Time-Dependent Internal Solutions for Spherically Symmetrical Bodies in General Relativity-I. Adiabatic collapse,” Monthly No-tices of the Royal Astronomical Society, Vol. 136, 1967, pp. 207-217.

[3] I. H. Thompson and G. J. Whitrow, “Time-dependent internal solutions for spherically symmetrical bodies in general relativity-II. Adiabatic radial motions of uni-formly dense spheres,” Monthly Notices of the Royal As-tronomical Society, Vol. 139, 1968, pp. 499-513.

[4] H. Bondi, “Gravitational Bounce in General Relativity,” Monthly Notices of the Royal Astronomical Society, Vol. 142, 1969, pp. 333-353.

[5] R. M. Misra and D. C. Srivastava, “Relativity-Bounce of Fluid Spheres,” Nature Physical Science, Vol. 238, 1972, p. 116.

[6] C. W. Misner and D. H. Sharp,“Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Col-lapse,” Physical Review B, Vol. 136, No. 2B, 1964, pp.



146

B571-576.

[7] M. Demianski, “Relativistic Astrophysics,” Pergamon Press, New York, 1985.

[8] D. Kramer, “Spherically Symmetric Radiating Solution with Heat Flow in General Relativity,” Journal of Mathematical Physics, Vol. 33, No. 4, 1992, pp. 1458- 1462.

[9] H. Nariai, “A Simple Model for Gravitational Collapse with Pressure Gradient,” Progress of Theoretical physics, Vol. 38, No. 1, 1967, pp. 92-106.

[10] J. R. Oppenheimer and H. Snyder, “On Continued Gravi-tational Contraction,” Physical Review, Vol. 56, No. 5, 1939, pp. 455-459.

Journal of Modern Physics, 2011, 2, 1156-1160 doi:10.4236/jmp.2011.210143 Published Online October 2011 (http://www.SciRP.org/journal/jmp)


Well Behaved Class of Charge Analogue of Durgapal’s Relativistic Exact Solution

Pratibha Fuloria1, B. C. Tewari2, B. C. Joshi3 1Department or Physics, Kumaun University, S.S.J.Campus, Almora, India

2Department of Mathematics, Kumaun University, S.S.J.Campus, Almora, India 3Department or Physics, Kumaun University, S.S.J.Campus, Almora, India

E-mail: [email protected], [email protected] Received July 12, 2011; revised August 26, 2011; accepted September 6, 2011

Abstract We obtain a new class of charged super-dense star models after prescribing particular Forms of the metric potential and electric intensity. The metric describing the superdense stars joins smoothly with the Reiss-ner-Nordstrom metric at the pressure free boundary. The interior of the stars possess their energy density, pressure, pressure density ratio and velocity of sound to be monotonically decreasing towards the pressure free interface. In view of the surface density 14 32 10 g cmb , the heaviest star occupies a mass 5.523 Mwith its radius 13.98 km. In absence of the charge we are left behind with the regular and well behaved fifth model of Durgapal [1]. Keywords: Charged Fluids, Reissener-Nordstom Metric, General Relativity

1. Introduction Exact interior solutions of the Einstein-Maxwell field equations joining smoothly to the Nordstrom solution at the pressure free interface are gathering big applause due to some of the following reasons: 1) Gravitational col-lapse of a charged fluid sphere to a point singularity may be avoided. 2) Solutions of Einstein-Maxwell equations are useful in the study of cosmic sensership. 3) Charged- dust models and electromagnetic mass models are ex-pected to provide some clue about structure of an elec-tron . In the present paper our aim is to obtain a class of regular and well behaved charged fluid models whose neutral analogues are also regular and well behaved. Here we consider a spherically symmetric metric which shares its metric potential with one of the Durgapal’s interior metric [1].

On account of the nonlinearity of Einstein-Maxwell field equations, not many realistic well behaved, analytic solutions are known for the description of relativistic charge fluid spheres. For well behaved model of relativ-istic star with charged and perfect fluid matter, following conditions should be satisfied (Pant et al. [2]):

1) The solution should be free from physical and geo-metrical singularities i.e. finite and positive values of central pressure, central density and non zero positive

values of e and e . 2) The solution should have positive and monotoni-

cally decreasing expressions for pressure and density ( andp ) with the increase of r. The solution should have positive value of ratio of pressure-density and less than 1(weak energy condition) and less than 1/3(strong energy condition) throughout within the star.

3) The solution should have positive and monotonically decreasing expression for fluid parameter 2P c with the increase of r.

4) The solution should have positive and monotoni-

cally decreasing expression for velocity of sound d

( )d

p

with the increase of r and causality condition should be

obeyed at the centre, i.e., 2

d1

d

p

c ..

5) The red shift Z should be positive, finite and monotonically decreasing in nature with the increase of r.

Electric intensity E is positive and monotonically in-creasing from centre to boundary and at the centre the Electric intensity is zero.

Under these well behaved conditions, one has to as-sume the gravitational potential and electric field inten-sity in such a way that the field equation can be inte-grated and solution should be well behaved. Keeping in

P. FULORIA ET AL.

1157

view of this aspect ,several authors obtained the para-metric class of exact solutions Pant et al. [3,4], Gupta- Maurya [5-7], Pant [8,9], N. Bijalwan [10] etc. These coupled solutions are well behaved with some positive values of charge parameter K and completely describe interior of the super-dense astrophysical object with charge matter. Further, The mass of the such modeled super dense object can be maximized by assuming sur-face density is 14 32 10 g cmb . In the present paper we have obtained yet another new parametric class of well behaved exact solutions of Einstein-Maxwell field equations, which is compatible within the range of Neu-tron star and quark star. 2. Einstein’s–Maxwell Equation for Charged

Fluid Distribution Let us consider a spherical symmetric metric in curvature coordinates

2 2 2 2 2 2 2d d d sin d 2ds e r r r e t (1)

where the functions r and satisfy the Ein-stein-Maxwell equations

r

4

24

8 1

28

1 1

4 4

i i ij j j

i j ij

im i mnjm j mn

GT R R

cG

c p v v pc

F F F F

(2)

where , p, , Fij denote energy density, fluid pres-sure, velocity vector and skew-symmetric electromag-netic field tensor respectively.

iv

In view of the metric (1), the field equation (2) gives Dionysiou [11]

2

2 4

1 8e G qe p

r r c

4r (3)

2

4

8

2 4 4 2

G qe p

r c r

2

4 (4)

2

2 2

1 8e G qe

r r c

4r (5)

where prime ( ' ) denotes the differentiation with respect to r and represents the total charge contained with in the sphere of radius r.

q r


5 211 , ande B x x c r e Z (6)

where B being the positive constants. Now putting (6)

into (3)-(5), we have

2

12

1

110 1 8

1

Z c q4

Z GP

x x cx c

(7)

21

21

1 d 1 82

d

Z c q4

Z G

x X cx c

(8)


22

12 1 1d 14 2 1

d 1 1 6 1 6

q c x xZ x xZ

x x x x x x x

(9)

where 21 ,x c r e Z .

3. New Class of Solutions In order to solve the differential equation (9) let us con-sider the electric intensity E of the following form

22 2

1 32

1

1 1 62

c qE Kxx x

c x (10)

where K is a positive constant. The electric density is so assumed that the model is physically significant and well behaved, i.e., E remains regular and positive throughout the sphere.

In view of (10) differential equation (9) yields the fol-lowing solution

3

1/3

2

3

3

1

3

1

6 1 6

309 54 811

1121

11 6

x xKe

x

x x x

x

Axx

x

(11)

where A is an arbitrary constant of integration. 4. Properties of the New Class of Solutions Using (11), into (7) and (8), we get the following expres-sions for pressure and energy density

41

2 3

4 1

3

2

1 8

475 4125 1050 200 1 111

1121 1 6

1 1 15 29

16 1 63

Gp

c c

x x x A x

x x

x x xK

x

(12)


P. FULORIA ET AL.


1158

41

2 3

2

4

3

2 3

4

3

1 8

1935 15 450 1201

1 4 112

3 11 22

1 6

1 3 29 109 158

6 1 6

G

c c

x x x

x

x xA

x

x x x xk

x

(13)

41 0

1 8 d 6025 75

d 112

G p KA

c xc

3 (18)

41 0

1 8 d 7725 425

d 112 3

G KA

c xc

(19)

The velocity of sound is given by the following ex-pression (20)-(21)

20

7841205 1121 d 15

448d 1545 56015

x

A Kp

c A K

(21)

The expression for gravitational red-shift (z) is given by

041

1 8 475

112 6

Gp A

c c

K (14)

5

211

xz

B

(22)

041

1 8 19353

112 2

G KA

c c

(15)

The central value of gravitational red shift to be non zero positive finite, we have Differenting (12) and (13) w.r.t. x, we get:

41

2 3

5

2 2

45

3

1 8 d

d

25 241 411 60 8

112 1

5 1 3 44 14 152 527 464

46 1 61 1 6

3

G p

c xc

x x x

x

3A x x K x x x

xx x

1 B 0 (22a)

Differenting (22) w.r.t. x, we get,

0

d 50

d 2x

z

x B

(22b)

the expression of right hand side of (22b) is negative, thus the gravitational red-shift is maximum at the center and monotonically decreasing. (16)

41

2 35

2 37

53

2 37

3

1 8 d

d

15515 57 36 8

112 1

55 39 66 88

1 1 6

8 212 1353 3302 2528

6(1 6 )

G

c xc

x x xx

Ax x x

x x

K

5. Boundary Conditions

4x x x

x

x

The solutions so obtained are to be matched over the boundary with Reissner-Nordstrom metric:

122 2

2

22 2 2 2

2

2d 1 d

2d sin d 1 d

GM es r

r r

GM er t

r r

2

(23)

which requires the continuity of ,e e and q across the boundary r = rb (17)

2 d

d

pv

4 452 3 2 2 33 3

7 7252 3 2 3 2 3 43 3

1125 241 411 60 8 112 1 3 44 1 6 14 152 527 464 1 6 11 d 30

112d 3 515 57 36 8 112 5 39 66 88 1 6 (8 212 1353 3302 2528 ) 1 6 130

x x x A x x x K x x x x xp

c x x x A x x x x K x x x x x x

(20)

2( )

2

21rb

b b

GM ee

c r r

2 (24)

2( )

2

21rb

b b

GM ee

c r r

2 (25)

bq r e (26)

0bp r (27)

The condition (27) can be utilized to compute the val-ues of arbitrary constants A as follows:

Pressure at p(r = rb) = 0 gives

12 33

5

2

1 6 475 4125 1050 200

1 11 112

11 15 29

6 1 11

X X X XA

X

XkX X

X

(28)

In view of (24) and (25) we get

3

5 1 3

3

2

13

3

1 (1 ) 1

61 11 6

309 54 81

112 1 1 6

K X XB

x XX

X X X AX

X X

(29)


2

52 32

1 1 1 6 12 2brGM K

X X X B Xc

(30)


2 3

24

2

4

3

2 3

4

3

1935 15 450 1208 1

1 4 112

3 11 22

1 6

1 3 29 109 158

6 1 6

b b

X X XGr

Xc

X XA

X

X X X XK

X

(31) 6. Discussion In view of Table 1 it has been observed that all the

physical parameters (p, ,2

,p

cd

,d

pz

and E) are

positive at the centre and within the limit of realistic equation of state and well behaved conditions for all values of K satisfying the inequalities . How-ever, corresponding to any value of K > 10, there exist no value of X for which surface density is positive .From Table 2 we observe that for K = 2 we obtain increasing mass with increasing values of X and central redshift also increases. We have taken a different expression for elec-

0 K 10

Table 1. The variation of various physical parameters at the center,surface density, electric field intensity on the boundary, mass and radius of stars with different values of K and X = 0.2.

K 041

1 8 Gp

c c

02

1

1 8 G

c c

02

0

1 p

c

20

1 d

d x

p

c

0z 2

1br

E

c

22

8b b

Gr

c

M

M

br in km

1 5.167 14.496 0.3565 0.9298 1.63 0.202 1.3728 4.41 19.20 2 4.664 16.0037 0.2914 0.7017 1.68 0.404 1.277 4.60 18.52 4 3.660 19.0156 0.1925 0.4806 1.814 0.808 1.060 4.83 16.87 5 3.158 20.5217 0.1538 0.4186 1.885 1.01 0.955 4.88 16.02 6 2.656 22.0275 0.1205 0.3728 1.962 1.212 0.851 4.89 15.12 8 1.652 25.0394 0.0659 0.3411 2.134 1.616 0.643 4.75 13.14

10 0.648 28.0513 0.0231 0.2655 2.342 2.02 0.434 4.31 10.80 Table 2. The variation of various physical parameters at the center,surface density, electric field intensity on the boundary, mass and radius of stars with different values of X and K = 2.

21 bX c r 04

1

1 8 Gp

c c

02

1

1 8 G

c c

02

0

1 p

c

20

1 d

d x

p

c

0z 2

1br

E

c

22

8b b

Gr

c

M

M

br in km

0.10 3.6036 19.1889 0.1889 0.5628 0.754 0.1504 1.1811 2.955 17.81 0.15 4.2883 17.1349 0.2502 0.64168 1.196 0.2646 1.2822 3.89 18.55 0.20 4.6643 16.0037 0.2914 0.7017 1.688 0.404 1.277 4.60 18.52 0.25 4.8138 15.5583 0.3094 0.7300 2.276 0.5756 1.168 5.13 17.71 0.30 4.764 15.7075 0.3032 0.7202 3.000 0.7747 0.9890 5.500 16.29 0.35 4.5018 16.4343 0.2751 0.6738 3.985 1.0045 0.7276 5.523 13.98


1160 P. FULORIA ET AL.

tric intensity as compared to that of Gupta and Mau-raya’s solution [5]. Our solution satisfies all the neces-sary physical conditions giving us a possibitity for dif-ferent charge variations within the fluid sphere. Owing to the various conditions that we obtain here we arrive at the conclusion that under well behaved conditions this class of solutions gives us the mass of super dense object within the range of neutron star and quark star.

We now present here a model of super dense star based on the particular solution discussed above corre-sponding to K = 0.35 with , by assuming sur-face density;

0.2X 14 32 10 g cmb .The resulting well

behaved model has the heaviest star occupying a mass 5.523 M　with its radius 13.98 km. In absence of the charge we are left behind with the regular and well be-haved fifth model of Durgapal [1]. 7. References [1] M. C. Durgapal, “A Class of New Exact Solutions in

General Relativity,” Journal of Physics A: Mathematical and General, Vol. 15, August 1982, pp. 2637-2644.

[2] N. Pant, “Some New Exact Solutions with Finite Central Parameters and Uniform Radial Motion of Sound,” As-trophysics and Space Science, Vol. 331, No. 2, 2011, pp. 633-644. doi:10.1007/s10509-010-0453-4

[3] N. Pant, et al., “Well Behaved Class of Charge Analogue of Heintzmann’s Relativistic Exact Solution,” Astrophys-ics and Space Science, Vol. 332, No. 2, 2011, pp. 473- 479. doi:10.1007/s10509-010-0509-5

[4] N. Pant, et al., “Variety of Well Behaved Parametric Classes of Relativistic Charged Fluid Spheres in General Relativity,” Astrophysics and Space Science, Vol. 333, No. 1, 2011, pp. 161-168.

[5] Y. K. Gupta and S. K. Maurya, “A Class of Regular and Well Behaved Relativistic Super Dense Star Models,” Astrophysics and Space Science, Vol. 334, No. 1, 2011, pp. 155-162. doi:10.1007/s10509-010-0503-y

[6] S. K. Maurya and Y. K. Gupta, “A Family of Well Be-haved Charge Analogue of a Well Behaved Neutral Solu-tion in Genetral Relativity,” Astrophysics and Space Sci-ence, Vol. 332, No. 2, 2011, pp. 481-490. doi:10.1007/s10509-010-0541-5

[7] S. K. Maurya and Y. K. Gupta, “Charged Analogue of Vlasenko-Pronin Super Dense Star in General Relativ-ity,” Astrophysics and Space Science, Vol. 333, No. 1, 2011, pp. 149-160. doi:10.1007/s10509-011-0616-y

[8] N. Pant, “Well Behaved Parametric Class of Relativistic in Charged Fluid Ball General Relativity,” Astrophysics and Space Science, Vol. 332, No.2, 2011, pp.403-408. doi:10.1007/s10509-010-0521-9

[9] N. Pant, “New Class of Well Behaved Exact Solutions of Relativistic Charged White-Dwarf Star with Perfect Fluid,” Astrophysics and Space Science, Vol. 334, No. 2, 2011, pp. 267-271. doi:10.1007/s10509-011-0720-z

[10] N. Bijalwan, “Static Electrically Charged Fluids in Terms Pressure: General Relativity,” Astrophysics and Space Science, Vol. 334, No. 1, 2011, pp.139-143. doi:10.1007/s10509-011-0691-0

[11] D. D. Dionysiou, “Equilibrium of a Static Charged Per-fect Fluid Sphere,” Astrophysics and Space Science, Vol. 85, No. 1-2, 1982, pp. 331-343. doi:10.1007/BF00653455


http://dx.doi.org/10.1007/s10509-010-0453-4

http://dx.doi.org/10.1007/s10509-010-0509-5

http://dx.doi.org/10.1007/s10509-010-0503-y

http://dx.doi.org/10.1007/s10509-010-0541-5

http://dx.doi.org/10.1007/s10509-011-0616-y

http://dx.doi.org/10.1007/s10509-010-0521-9

http://dx.doi.org/10.1007/s10509-011-0720-z

http://dx.doi.org/10.1007/s10509-011-0691-0

http://dx.doi.org/10.1007/BF00653455

Journal of Modern Physics, 2011, 2, 1538-1543 doi:10.4236/jmp.2011.212186 Published Online December 2011 (http://www.SciRP.org/journal/jmp)


Well Behaved Parametric Class of Exact Solutions of Einstein-Maxwell Field Equations in General Relativity

Neeraj Pant1, B. C. Tewari2, Pratibha Fuloria3 1Department of Mathematics, National Defence Academy Khadakwasla, Pune, India

2Department of Mathematics, Kumaun University, S.S.J. Campus, Almora, India 3Department of Physics Kumaun University, S.S.J. Campus, Almora, India

E-mail: {neeraj.pant, p.fuloria}@yahoo.com Received September 15, 2011; revised October 27, 2011; accepted November 22, 2011

Abstract We present a new well behaved class of exact solutions of Einstein-Maxwell field equations. This solution describes charge fluid balls with positively finite central pressure, positively finite central density; their ratio is less than one and causality condition is obeyed at the centre. The gravitational red shift is positive throughout positive within the ball. Outmarch of pressure, density, pressure-density ratio, the adiabatic speed of sound and gravitational red shift is monotonically decreasing, however, the electric intensity is monotoni-cally increasing in nature. The solution gives us wide range of parameter K (0.72 ≤ K ≤ 2.41) for which the solution is well behaved hence, suitable for modeling of super dense star. For this solution the mass of a star is maximized with all degree of suitability and by assuming the surface density ρb = 2 × 1014 g/cm3. Corre-sponding to K = 0.72 with X = 0.15, the resulting well behaved model has the mass M = 1.94 MΘ with radius rb 15.2 km and for K = 2.41 with X = 0.15, the resulting well behaved model has the mass M = 2.26 MΘ with radius rb 14.65 km. Keywords: Charge Fluid, Reissner-Nordstrom, General Relativity, Exact Solution

1. Introduction Ever since the formulation of Einstein-Maxwell field equations, the relativists have been proposing different models of immense gravity astrophysical objects by con-sidering the distinct nature of matter or radiation (en-ergy-momentum tensor) present in them. Such models successfully explain the characteristics of massive ob-jects like quasar, neutron star, pulsar, quark star, blackhole or other super-dense object. These stars are speci-fied in terms of their masses as white dwarfs (Mass < 1.44 solar mass), Quark star (2 solar mass - 3 solar mass) and Neutron star (1.35 solar mass - 2.1 solar mass).

It is well known that the Reissener-Nordstrom solution for the external field of a ball of charged mass has two distinct singularities at finite radial positions other than at the centre. Thus the solution describes a bridge (worm hole) between two asymptotically flat spaces and an electric flux flowing across the bridge. Graves and Bill [1] pointed out that the region of minimum radius or the throat of worm hole pulsates periodically between these two surfaces due to Maxwell pressure of the electric field.

Consequently, unlike Schwarzschild’s exterior solution of chargeless matter,in Reissener-Nordstrom solution has no surface which can catastrophically hit the geometric singularity at r = 0.All these aspects show that the presence of some charge in a spherical material distribution provides an additional resistance against the gravitational contraction by mean of electric repulsion and hence ,the catastrophic collapse of the entire mass to a point singularity can be avoided.

The above result has been supported by a physically reasonable charge spherical model of Bonnor [2], that a dust distribution of arbitrarily large mass and small ra-dius can remain in equilibrium against the pull of gravity by a repulsive force produced by a small amount of charge. Thus it is desirable to study the implications of Einstein-Maxwell field equations with reference to the general relativistic prediction of gravitational collapse. For this purpose charged fluid ball models are required. The external field of such ball is to be matched with Reissener-Nordstrom solution.

For obtaining significant charged fluid ball models of Einstein-Maxwell field equations, the Astrophysicists

N. PANT ET AL. 1539 have been using exact solutions with finite central pa-rameters of Einstein field equations, as seed solutions. There are two type of exact solutions of this category.

Type 1. If the solutions are well behaved (Delgaty- Lake [3], Pant [4]). These solutions their self completely describe interior of the Neutron star or analogous super dense astrophysical objects with chargeless matter. Del- gaty-Lake [3] studied most of the exact solutions so far obtained and pointed out that only nine solutions are regular and well behaved. Out of which only six of them are well behaved in curvature coordinates and rest three solutions are in isotropic coordinates. In previous papers (Pant et al. [5], Pant [4], we obtained a new well behaved solution in isotropic coordinates and two new well be-haved solutions in curvature coordinates respectively.

Type 2. If the solutions are not well behaved but with finite central parameters, such solutions are taken as seed solutions of astrophysical objects with charge matter since at centre the charge distribution is zero.

For well behaved nature of the solution in curvature coordinates, the following conditions should be satisfied (augmentation of (Delgaty-Lake [3] and Pant [4]) condi-tions).

1) The solution should be free from physical and geo-metrical singularities i.e. finite and positive values of central pressure , central density and non zero positive values of e and e i.e. 0 an 00p 0 . For well behaved solution in curvature coordinates, it should have

0r, i.e. the tangent-3space at the centre in flat

but converse is not true. e 1

2) The solution should have positive and monotoni-cally decreasing expressions for pressure and density ( and p ) with the increase of r. The solution should have positive value of ratio of pressure-density and less than 1 (weak energy condition) and less than 1/3 (strong energy condition) throughout within the star.

3) The solution should have positive and monotoni-

cally decreasing expression for fluid parameter 2

p

c

with the increase of r, i.e.

20

pr

c

0 and

20

0r

p

c

and 2

p

c

0r is negative valued function for .

d d d d1

d d

d log 1 1

de

e

pp

p

P

og

where d log

de

e

P

og

is adiabatic index and for realistic

matter 1 . Thus we have, 2

d0

d

pp

c

,

decreases with the increase of r. 4) The solution should have positive and monotoni-

cally decreasing expression for velocity of sound d

d

p

with the increase of r and causality condition should be


d1

d

p

c .

5) The red shift Z should be positive, finite and mono-tonically decreasing in nature with the increase of r.

6) Electric intensity E is positive and monotonically increasing from centre to boundary and at the centre the Electric intensity is zero.

Under these well behaved conditions, one has to as-sume the gravitational potential and electric field in-tensity in such a way that the field equation can be inte-grated and solution should be well behaved. Keeping in view of this aspect ,several authors obtained the para-metric class of exact solutions Pant et al. [6,7], Gupta and Maury [8,9], Pant [10], M. J. Pant and Tewari [11] etc. These coupled solutions are well behaved with some positive values of charge parameter K and completely describe interior of the super-dense astrophysical object with charge matter. Further, The mass of the such mod-eled super dense object can be maximized by assuming surface density is ρb = 2 × 1014 g/cm3. In the present pa-per we have obtained yet another new parametric class of well behaved exact solutions of Einstein –Maxwell field equations, which is compatible within the range of Quark star and neutron star . 2. Einstein-Maxwell Equation for Charged

Fluid Distribution Let us consider a spherical symmetric metric in curvature coordinates

2 2 2 2 2 2d e d d sin d e d 2s r r t (1)

where the functions r and satisfy the Ein-stein-Maxwell equations

v r

24 4

8π 1 8π

2

1 1

4π 4

i i i i ij j j j j

im i mnjm j mn

G GT R R c p v v p

c c

F F F F

(2)

where denote energy density, fluid pressure, velocity vector and skew-symmetric electromagnetic field tensor respectively.

, , ,iijp v F

In view of the metric (1), the field Equation (2) gives


N. PANT ET AL.


1540

2

2 4

1 e 8πe

vp

r r c

Our task is to explore the solutions of Equation (9) and obtain the fluid parameters and p from Equation (7) and Equation (8).

4

G q

r (3)

2 2

4

8πe

2 4 4 2

v v v v G qp

r c r

4

(4) 3. New Class of Solutions

2

2 2

1 e 8πe

G q

r r c

In order to solve the differential Equation (9), In this paper we consider the electric intensity E of the follow-ing form

4r

(5)

where, prime (/)denotes the differentiation with respect to r and q(r) represents the total charge contained within the sphere of radius r.

22 1

1 32

1

1 3 12

c qE K x xc x

x (10)

Now let us set where K is a positive constant. The electric intensity is so assumed that the model is physically significant and well behaved i.e. E remains regular and positive throughout the sphere. In addition, E vanishes at the centre of the star.

2211ve B c r (6)

which is the same as that of the metric obtained by Adler [12].

Putting (6) into (3) - (5) , we have In view of Equation (10) differential Equation (9) yields the following solution 2

12 4

1

14 π

1

Z c q 1 8Z Gp

x x cx c

(7)

3

2

3 3

(1 )e 1

3 1 3 1 3

2

K x x AxZ

x x (11a)

21

2 21

1 d 1 8π2

d

Z c qZ G

x x cx c

(8)


and Z satisfying the equation 21ve B x (11b)

21

1 2d 11

d 1 3 1 3

x c qZ xZ

x x x x x x

(9) Using (11a), (11b) into Equations (7) and (8), we get the following expressions for pressure and energy den-sity where 2

1 , ex c r Z .

2

4 21 3 3

19 15 2 1 1 51 8π 4

6 11 3 1 1 3

2

x x x A xG Kp

c xc x x x (12)

3 2

5

2 5

1 3 3

1 73 92 37 6 3 51 8π

6 1 3 1 3

x x x x A xG K

c c x x (13)

4. Properties of the New Class of Solutions

For 0 andp 0 must be positive and 0

0

1p

, we

The central values of pressure and density are given by have

041

1 8π4

3

G Kp A

c c (14)

4 13 3

K KA , and (16) 0, 0K A

021

1 8π3

GK A

c c (15) Differentiating (12) and (13) w. r. t. x, we get;

3 2 2

4 521 3 3

63 123 65 13 1 51 8π d 42

d 6 11 3 1 1 3

x x x xG p KA

c xc 5 2xx x x

(17)

N. PANT ET AL. 1541

4 3 2

2 81 3 3

1241 2602 1914 602 73 11 8π d. 1

d 6 1 3(1 3 )

x x x x xG KA

c xc 80

xx

(18)

41 0

41 0

1 8π d 132 4

d 6

1 8π d0

d

x

x

G p KA

c xc

G pve

c xc

(19)

The expression of right hand side of (19) is negative, thus the pressure p is maximum at the centre and mono-tonically decreasing.

21 0

1 8π d 7310

d 6

x

G KA

c xc (20)

21 0

1 8π d0

d

x

G

c xc (21)

The expression of right hand side of (20) is negative, the density is maximum at the centre and monotoni-cally decreasing. and hence the velocity of sound v is given by the fol-lowing expression

2 d d d

d dd

p p pv

x x

3 2 2

5 52

3 3

24 3 2

8 8

3 3

63 123 65 13 1 5 42

6 11 3 1 1 31 d

d 1241 2602 1914 602 73 110

6 1 3 1 3

x x x xKA

xx x xp

c x x x x xKA

x x

2

(22)

20

1 d 13 12 241

d 73 60r

p K A

K Ac

, for all values of K

and A satisfied by(16). The expression for gravitational red-shift(z) is given

by

11

1x

zB

(23)

The central value of gravitational red shift to be non zero positive finite, we have

1 B 0 (24)

Differentiating equation (24) w.r.t. x, we get,

0

d 10

d x

z

x B

(25)

The expression of right hand side of (25) is negative, thus the gravitational red-shift is maximum at the centre

and monotonically decreasing. Differentiating Equation (10) w.r.t. x, we get,

22

2 31

1 6 7d

d 2 1 3

x xE K

x c x

(26)

2

1 0

d ve

d 2x

E K

x c

(27)

The expression of right hand side of (27) is positive, thus the electric intensity is minimum at the centre and monotonically increasing for all values of K > 0. Also at the centre it is zero. 5. Boundary Conditions The solutions so obtained are to be matched over the boundary with Reissner-Nordstrom metric.;

12 2

2 2 2 2 2 22 2 2 2

2 2d 1 d d sin d 1

b b b b

GM e GM e 2ds r r tc r r c r r

(28)

which requires the continuity of across the boundary .

, ande e q

br r2

( )2

21br

b b

GM ee

c r r

2( )

2

21br

b b

GM ee

c r r

2 (30)

2 (29) bq r e (31)


N. PANT ET AL. 1542

0bp r (32)

The condition (30) can be utilized to compute the val-ues of arbitrary constants A as follows:

On setting br r

21 ,bx X c r ( being the radius of

the charged sphere) br

Pressure at gives ( ) 0br rp

2 2 2 31 19 15 2 1 34

6 1 5 1 5

X X X XKA

X X (33)


22

1 7 22 1 3 2 3 6brGM X K

X X X AXc

(34)

In view of (29) and (30) we get,

3

2 3 2 3

2

11

3 1 3 1 3

1

XKX XA

X XB

X

Centre red shift is given by 1/2

0 1z B (35)

In view of and TableI and Table-II We observe that pressure, density, pressure-density ratio, square of adiabatic sound speed and gravitational red shift decrease monotonically with the increase of radial coordinate

however, the charge distribution is increasing in nature. 6. Discussion In view of and Tables 1 and 2, it has been observed

, 2

,p

c ,

dpz

d that all the physical parameters (p,

and E) are positive at the centre and within the limit of realistic equation of state and well behaved conditions for all values of K satisfying the inequalities 0.72 ≤ K ≤ 2.41. However, corresponding to any value of K < 0.72, there exist no value of X for which the nature of adiabatic sound speed is monotonically decreasing from centre to pressure free interface and for K > 2.41 , the pressure is negative some where within the ball for all values of X. It has been observed that under well behaved conditions this class of solutions gives us the mass of super dense object within the range of quark star and neutron star.

We now present here two models of super dense star based on the particular solution discussed above by as-suming surface density; ρb = 2 × 1014 g/cm3. Corre-sponding to K = 0.72 with X = 0.15, the resulting well behaved model has the mass M = 1.94 MΘ

.with radius rb 15.2 km and for K = 2.41 with X = 0.15, the resulting well behaved model has the mass M = 2.26 MΘwith ra-dius rb 14.65 km.

Table 1. The march of pressure, density, pressure-density ratio, square of adiabatic sound speed, gravitational red shift and electric field intensity within the ball corresponding to K = 0.72 with X = 0.15.

br r 2

4

8πb

Gp r

c 2

2

8πb

Gr

c 2

p

c

2

1 d

d

p

c

z 2

bE r

0 0.133143 1.400572 0.095063 0.216271 0.426164 0

0.1 0.131095 1.392716 0.094129 0.216101 0.424028 0.00008

0.2 0.125065 1.369595 0.091315 0.215525 0.417658 0.000332

0.3 0.115385 1.332484 0.086594 0.214356 0.407167 0.000769

0.4 0.10257 1.283297 0.079927 0.212291 0.392738 0.001422

0.5 0.087277 1.224339 0.071285 0.208939 0.374616 0.002334

0.6 0.070247 1.158033 0.060661 0.203855 0.353097 0.003562

0.7 0.052258 1.086683 0.048089 0.196582 0.328518 0.005177

0.8 0.034079 1.01231 0.033664 0.186713 0.301245 0.007265

0.9 0.016437 0.936551 0.017551 0.173954 0.271658 0.00993

1.0 0 0.860627 0 0.158177 0.240143 0.013294


N. PANT ET AL. 1543 Table 2. The march of pressure, density, pressure-density ratio, square of adiabatic sound speed, gravitational red shift and electric field intensity within the ball corresponding to K = 2.41 with X = 0.15.

br r 2

4

8πb

Gp r

c 2

2

8πb

Gr

c 2

p

c

2

1 d

d

p

c

z 2

bE r

0 0.068295 1.595114 0.042815 0.102667 0.444341 0

0.1 0.066626 1.584203 0.042057 0.101874 0.442177 0.000273

0.2 0.061773 1.552015 0.039802 0.099459 0.435726 0.001111

0.3 0.054182 1.500104 0.036119 0.095324 0.425102 0.002573

0.4 0.044551 1.430801 0.031137 0.089335 0.410489 0.004759

0.5 0.033775 1.346889 0.025076 0.081351 0.392136 0.007813

0.6 0.02288 1.251268 0.018286 0.071278 0.370342 0.011924

0.7 0.012961 1.146656 0.011304 0.059104 0.34545 0.017328

0.8 0.005127 1.035372 0.004952 0.044947 0.317829 0.024317

0.9 0.000462 0.919216 0.000502 0.02907 0.287865 0.033237

1.0 0 0.79943 0 0.011874 0.255948 0.044497

7. Acknowledgements 1) First author (NP) acknowledges his gratitude to Lt. Gen. Jatinder Singh AVSM, SM, Comdt, NDA for his motivation and encouragement.

2) First author (NP) also acknowledges his gratitude to Prof A. N. Srivastava HOD Mathematics NDA for his invaluable suggestions. 8. References [1] J. C. Graves and D. R. Brill, “Oscillatory Character of

Reissner Nordstorm Metric for an Ideal Charged Worm-hole,” Physical Review, Vol. 120, No. 4, 1960, pp. 1507- 1513. doi:10.1103/PhysRev.120.1507

[2] W. B. Bonnor, “The Equilibrium of Charged Sphere,” Monthly Notices of the Royal Astronomical Society, Vol. 137, No. 3, 1965, pp. 239-251.

[3] M. S. R. Delgaty and K. Lake, “Physical Acceptability of Isolated, Static, Spherically Symmetric, Perfect Fluid So- lutions of Einstein’s Equations,” Computer Physics Com- munications, Vol. 115, No. 2-3, 1998, pp. 395-399. doi:10.1016/S0010-4655(98)00130-1

[4] N. Pant, “Some new Exact Solutions with Finite Central Parameters and Uniform Radial Motion of Sound,” As- trophysics and Space Science, Vol. 331, No. 2, 2011, pp. 633-644.

[5] N. Pant, et al., “New Class of Regular and Well Behaved Exact Solutions in General Relativity,” Astrophysics and Space Science, Vol. 330, No. 2, 2010, pp. 353-370.

doi:10.1007/s10509-010-0383-1

[6] N. Pant, et al., “Well Behaved Class of Charge Analogue of Heintzmann’s Relativistic Exact Solution,” Astrophys- ics and Space Science, Vol. 332, No. 2, 2011, pp. 473- 479. doi:10.1007/s10509-010-0509-5

[7] N. Pant, et al., “Variety of Well Behaved Parametric Classes of Relativistic Charged Fluid Spheres in General Relativity,” Astrophysics and Space Science, Vol. 333, No. 1, 2011, pp. 161-168. doi:10.1007/s10509-011-0607-z

[8] S. K. Maurya and Y. K. Gupta, “A Family of Well Be-haved Charge Analogue of a Well Behaved Neutral Solu- tion in Genetral Relativity,” Astrophysics and Space Sci- ence, Vol. 332, No. 2, 2011, pp. 481-490. doi:10.1007/s10509-010-0541-5

[9] Y. K. Gupta and S. K. Maurya, “A Class of Regular and Well Behaved Relativistic Super Dense Star Models,” Astrophysics and Space Science, Vol. 334, No. 1, 2011, pp. 155-162. doi:10.1007/s10509-010-0503-y

[10] N. Pant, “Well Behaved Parametric Class of Relativistic in Charged Fluid Ball General Relativity,” Astrophysics and Space Science, Vol. 332, No. 2, 2011, pp. 403-408. doi:10.1007/s10509-010-0521-9

[11] M. J. Pant and B. C. Tewari, “Well Behaved Class of Charge Analogue of Adler’s Relativistic Exact Solution,” Journal of Modern Physics, Vol. 2, No. 6, 2011, pp. 481- 487. doi:10.4236/jmp.2011.26058

[12] R. J. Adler, “A Fluid Sphere in General Relativity,” Jour- nal of Mathematical Physics, Vol. 15, No. 6, 1974, pp. 727-729. doi:10.1063/1.1666717


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http://dx.doi.org/10.4236/jmp.2011.26058

http://dx.doi.org/10.1063/1.1666717

1 23

Astrophysics and Space ScienceAn International Journal of Astronomy,Astrophysics and Space Science ISSN 0004-640XVolume 340Number 2 Astrophys Space Sci (2012) 340:407-412DOI 10.1007/s10509-012-1068-8

A new well behaved exact solution ingeneral relativity for perfect fluid

Neeraj Pant, Pratibha Fuloria &B. C. Tewari

1 23

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Astrophys Space Sci (2012) 340:407–412DOI 10.1007/s10509-012-1068-8


A new well behaved exact solution in general relativity for perfectfluid

Neeraj Pant · Pratibha Fuloria · B.C. Tewari

Received: 14 February 2012 / Accepted: 31 March 2012 / Published online: 20 April 2012© Springer Science+Business Media B.V. 2012

Abstract We present a new spherically symmetric solutionof the general relativistic field equations in isotropic coordi-nates. The solution is having positive finite central pressureand positive finite central density. The ratio of pressure anddensity is less than one and casualty condition is obeyed atthe centre. Further, the outmarch of pressure, density andpressure-density ratio, and the ratio of sound speed to lightis monotonically decreasing. The solution is well behavedfor all the values of u lying in the range 0 < u ≤ .186. Thecentral red shift and surface red shift are positive and mono-tonically decreasing. Further, we have constructed a neutronstar model with all degree of suitability and by assuming thesurface density ρb = 2×1014 g/cm3. The maximum mass ofthe Neutron star comes out to be M = 1.591 M� with radiusRb ≈ 12.685 km. The most striking feature of the solution isthat the solution not only well behaved but also having oneof the simplest expressions so far known well behaved solu-tions. Moreover, the good matching of our results for Velapulsars show the robustness of our model.

Keywords Isotropic coordinates · General relativity ·Einstein’s field equations · Fluid ball

N. Pant (�)Maths Dept., National Defence Academy Khadakwasla, Pune,Indiae-mail: [email protected]

P. FuloriaPhysics Dept., S.S.J.Campus Almora, Almora, Indiae-mail: [email protected]

B.C. TewariMaths Dept., S.S.J.Campus Almora, Almora, Indiae-mail: [email protected]

1 Introduction

A Stellar object is formed when a sufficiently massive gascloud condenses and contracts under the central pull of itsown gravity, thereby raising the temperature. As long as thegravitational binding energy remains greater than the ther-mal energy of the cloud, it goes on contracting thereby, con-tinually raising the temperature. For low masses (less thanthe solar mass), a stage may come when a resulting ther-mal radiation pressure together with normal hydrodynamicpressure balances the gravity and eventually a quasi staticequilibrium state is arrived which ends up into a planet.For very large masses the radiation pressure may rise toofast resulting into instability. However, in some cases thestar goes on contracting till a core temperature ∼108 °Cis attained so that the hydrogen fusion reactions trigger upand the resulting released energy generates sufficient pres-sure to sustain the star against further contraction. When theentire nuclear fuel has been used, the star gradually coolsdown consequently, loss of radiation pressure causes con-traction under the central pull of gravity. Eventually thestate of high density ∼106 gm/cc is realized in which pres-sure is dominated by the electron Fermi pressure which sup-ports the star against the any further collapse, thus forminga White Dwarf. No stable White Dwarf can be more mas-sive than Chandrasekhar limit. A Star with a mass exceedingthe Chandrasekhar limit contracts further, and with the riseof density, electrons are gradually absorbed in the nuclei,where with proton they form neutrons and neutrinos. Theneutrinos have negligible density remain unaffected thus bythe gravity. These neutrinos escape out and develop the out-ward pressure known as neutrinos pressure. This resultingneutrinos pressure if becomes large, may drive out the outerstellar layer causing a supernova burst. The neutron stars areformed in the leftover core. The leftover core stellar fluid

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408 Astrophys Space Sci (2012) 340:407–412

is rich in degenerated neutrons and neutron Fermi pressureleads in providing the balancing resistance against the enor-mous gravity. The stability as well as the fluid of such stel-lar object is merely associated with neutrons hence knownas Neutron stars. The properties and characteristics of suchenormous gravity objects, yet not solved fully in terms ofmaximization of mass, can easily and significantly be under-stood with the help of Einstein’s field equations of GeneralRelativity.

A considerable number of known solutions (Delgaty andLake 1998) of Einstein’s field equations are of finite cen-tral pressure and finite central density which describe thecausal model with realistic equation of state of stellar ob-jects. These solutions are of paramount importance with pi-oneer relevancy for describing interior of the Neutron star,as the central parameters are completely defined. Moreover,the relevancy is enhanced in manifolds if the solutions areregular, well behaved and simple in algebraic expressions interms of field variables and physical variables. Even somewell known solutions like Tolman (1939, iv), Adler (1974),Heintzmann (1969), Finch and Skea (1989) etc. do not sat-isfy the one of the well behaved condition i.e. adiabaticsound speed is not monotonically deceasing. Delgaty andLake (1998) also pointed out that only nine solutions so farare regular and well behaved; out of which seven in curva-ture coordinates (Tolman 1939, vii; Patvardhav and Vaidya1943; Mehra 1966; Kuchowicz 1968; Matese and Whitman1980; Durgapal (two solutions) 1982) and only two solu-tions (Nariai 1950; Goldman 1978) in isotropic coordinates.In recent past two new well behaved solutions in curvaturecoordinates have been explored (Pant 2011), Maurya andGupta (2011) and one successful attempt has been made inisotropic coordinates (Pant et al. 2010). In this paper wepresent yet another new solution in isotropic coordinateswhich is not only well behaved but also simple in terms ofexpressions of field and physical variables.

2 Field equations in isotropic coordinates

We consider the static and spherically symmetric metric inisotropic co-ordinates

ds2 = −eω[dr2 + r2(dθ2 + sin2 θdφ2)] + c2eυdt2 (1)

where ω and υ are functions of r . Einstein’s field equationsof gravitation for a non empty space-time are

Rij − 1

2Rgij = −8πG

c4Tij (2)

where Rij is a Ricci tensor, Tij is energy-momentum tensorand R the scalar curvature. The energy-momentum tensorTij is defined as

Tij = (p + ρc2)vivj − pgij (3)

where p denotes the pressure distribution, ρ the density dis-tribution and vi the velocity vector, satisfying the relation

gij vivj = 1 (4)

Since the field is static, therefore

v1 = v2 = v3 = 0 and v4 = 1√g44

(5)

Thus we find that for the metric (1) under these conditionsand for matter distributions with isotropic pressure the fieldequation (2) reduces the following:

8πG

c4p = e−ω

((ω′)2

4+ ω′

r+ ω′υ ′

2+ υ ′

r

)(6)

8πG

c4p = e−ω

[ω′′

2+ υ ′′

2+ (υ ′)2

4+ ω′

2r+ υ ′

2r

](7)

8πG

c2ρ = −e−ω

[ω′′ + (ω′)2

4+ 2ω′

r

](8)

where, prime (′) denotes differentiation with respect to r .From (6) and (7) we obtain following differential equationin ω and υ .

ω′′ + ν′′ + (υ ′)2

2− (ω′)2

2− ω′υ ′ −

(ω′

r+ υ ′

r

)= 0 (9)

Our task is to explore the solutions of (9) and obtain the fluidparameters p and ρ from (6) and (8).

3 Conditions for well behaved solution

For well behaved nature of the solution in isotropic coordi-nates, the following conditions should be satisfied:

(i) The solution should be free from physical and geomet-rical singularities i.e. finite and positive values of cen-tral pressure , central density and non zero positive val-ues of eω and eυ , i.e. p0 > 0 an ρ0 > 0.

(ii) The solution should have positive and monotoni-cally decreasing expressions for pressure and density(p and ρ) with the increase of r . The solution shouldhave positive value of ratio of pressure-density andless than 1 (weak energy condition) and less than1/3 (strong energy condition) throughout within thestar, monotonically decreasing as well (Pant and Negi2012).

(iii) The casualty condition (dp/c2dρ)1/2 i.e. velocity ofsound should be less than that of light throughout themodel. In addition to the above the velocity of soundshould be decreasing towards the surface i.e. d

dr(dpdρ

) <

0 or (d2p

dρ2 ) > 0 for 0 ≤ r ≤ rb i.e. the velocity of soundis increasing with the increase of density. In this con-text it is worth mentioning that the equation of state atultra-high distribution, has the property that the soundspeed is decreasing outwards, Canuto and Lodenquai(1975).


Astrophys Space Sci (2012) 340:407–412 409

(iv) pρ

≤ dpdρ

, everywhere within the ball.

γ = d loge P

dogeρ= ρ

p

dp

dρ⇒ dp

dρ= γ

p

ρ,

for realistic matter γ ≥ 1 (Pant and Maurya 2012)(v) The red shift z should be positive, finite and monotoni-

cally decreasing in nature with the increase of r .

Under these conditions , we have to assume the one ofthe gravitational potential component in such a way that thefield equation (9) can be integrated and solution should bewell behaved. Further, the mass of the such modeled superdense object can be maximized by assuming surface density,ρb = 2 × 1014 g/cm3.

4 New class of solution

Equation (9) is solved by assuming

eω/2 = B(1 + Cr2)− 1

7 (10)

We get

eν2 = {1 + A(1 + Cr2)

17 }(1 + Cr2)

27

B2(11)

where A, B , and C are arbitrary constants.The expressions for pressure and density are given by

8πGρ

c2= 4

49B2(1 + Cr2)127

(21C + 6C2r2), (12)

8πGp

c4= 4

49B2(1 + Cr2)127

×[(7C + 4C2r2) + A(1 + Cr2)

17 (5C2r2 + 7C)

(1 + A(1 + Cr2)17 )

]

(13)

5 Properties of the new solution

The central values of pressure and density are given by(

8πGp

c4

)

0= 4

49B2

[7C + 7AC

(1 + A)

], (14)

(8πGρ

c2

)

0= 12C

7B2(15)

The central values of pressure and density will be nonzero positive definite, if the following conditions will be sat-isfied.

A > −1/2, C > 0 (16)

Subjecting the condition that positive value of ratio ofpressure-density and less than 1 at the centre i.e. p0

ρ0c2 ≤ 1

which leads to the following inequality,

1

3

[2A + 1

1 + A

]< 1 (17)

All the values of A which satisfy (16), will also lead tothe condition p0

ρ0c2 ≤ 1. Differentiating (13) with respect to r ,

dp

dr= 4

49B2{1 + A(1 + Cr2)17 }2(1 + Cr2)

197

⎡

⎢⎢⎣

A2(1 + Cr2) 27

(−30C2r − 90

7C3r3

)

+A(1 + Cr2) 1

7

(−44C2r − 120

7C3r3

)− 16C2r − 40

7C3r3

⎤

⎥⎥⎦ (18)

Thus extrema of p occur at the centre if

p′ = 0 ⇒ r = 0, (19)

8πG

c4

(p′′)

r=0 = 4

49B2

1

(1 + A)2

× [−30A2C2 − 44AC2 − 16C2] = −ve

(20)

Thus the expression of right hand side of (20) is nega-tive for all values of A satisfying (16), showing thereby thatthe pressure p is maximum at the centre and monotonicallydecreasing.

Now differentiating (12) with respect to r .

dρ

dr= 4C

49(1 + Cr2)197

[−60Cr − 60

7C2r3

](21)

Thus the extrema of ρ occur at the centre if

ρ′ = 0 ⇒ r = 0, (22)

8πG

c2

(ρ′′)

r=0 = −240C2

49B2(23)

Thus, the expressions of right hand side of (21) and (23)are negative showing thereby that the density ρ is maximumat the centre and monotonically decreasing.

The square of adiabatic sound speed at the centre,1c2 (

dpdρ

)r=0, is given by



1

c2

(dp

dρ

)

r=0= 1

(1 + A)2

(15A2 + 22A + 8)

30

< 1 and positive (24)

The causality condition is obeyed at the centre for all val-ues of constant satisfying (16).

Further, it is mentioned here that the boundary of the su-per dense star is established only when −1/2 < A < 0.

6 Boundary conditions

The solutions so obtained are to be matched over the bound-ary with Schwarzschild exterior solution

ds2 =(

1 − 2GM

c2R

)c2dt2 −

(1 − 2GM

c2R

)−1

dR2

− R2dθ2 − R2 sin2 θdφ2 (25)

where M is the mass of the ball as determined by the ex-ternal observer and R is the radial coordinate of the exteriorregion. The usual boundary conditions are that the first andsecond fundamental forms are continuous over the boundaryr = rb or equivalently R = Rb . Thus we get,

eυb = 1 − 2GM

c2Rb

, (26)

Rb = rbeωb /2, (27)

1

2

(ω′ + 2

r

)

b

rb =(

1 − 2GM

c2Rb

)1/2

, (28)

1

2

(ν′)

brb = GM

c2Rb

(1 − 2

GM

c2Rb

)−1/2

(29)

Applying the boundary conditions we get the values ofthe arbitrary constants in terms of Schwarzschild parametersu = GM

c2Rband radius of the star Rb

A = 7u(1 + Cr2b ) − 4Cr2

b (1 − 2u)12

6Cr2b (1 + Cr2

b )17 (1 − 2u)

12 − 7u(1 + Cr2

b )87

, (30)

k =

√√√√√√{ 7u(1+Cr2

b )−4Cr2b (1−2u)

12

6Cr2b (1+Cr2

b )17 (1−2u)

12 −7u(1+Cr2

b )87}(1 + Cr2

b )17 + 1

√1 − 2u

(31)

where k = Rb

rb

B = k

[1 + 7{(1 − 2u)

12 − 1}

{5 − 7(1 − 2u)12 }

] 17

(32)

C = 7{(1 − 2u)12 − 1}

{5 − 7(1 − 2u)12 }r2

b

> 0 for u ≤ 0.244 (33)

Fig. 1 The variation of mass and radius of star with u

Fig. 2 The variation of p, ρ, p

ρc2 , 1c2 (

dpdρ

) from centre to surface foru = 0.186

Surface density is given by

8πG

c2ρb = 3

R2b

{8(1 − 2u)

12 − 8 + 10u

}> 0 (34)

Central red shift is given by

Z0 =[

B2

1 + A− 1

](35)

The surface red shift is given by

Zb = [(1 − 2u)−0.5 − 1

](36)

7 Application: slowly rotating structures and theirapplication to the Vela pulsars

For slowly rotating structure like the Vela pulsars (rotationvelocity about 70 rad/sec), one can calculate the momentof inertia in the first-order approximation which appears inthe Lense-Thrirring frame dragging effect. However, for thepresent case of an exact solution, it is very useful to applyan approximate, but very precise, empirical formula whichis based on the numerical results obtained for a large num-ber of theoretical equations of state (EOS) of dense nuclear



Table 1 By assuming thesurface densityρb = 2 × 1014 g/cm3, thevariation of maximum Neutronstar mass, radius Rb , central redshift Z0 and surface red shiftZb = [(1 − 2u)−0.5 − 1] with u

u 8πGc2 ρbr

2b

MM�

Rb in km Z0 Zb I (g cm2)

.01 0.1161 0.026 3.973 0.0153 0.0102 0.003 × 1045

.02 0.2243 0.075 5.560 0.0315 0.0206 0.0187 × 1045

.04 0.4175 0.2072 7.686 0.0667 0.0426 0.1005 × 1045

.06 0.5793 0.3710 9.183 0.1063 0.066 0.2603 × 1045

.08 0.7100 0.5564 10.318 0.1513 0.0911 0.499 × 1045

.1 0.8094 0.7549 11.190 0.2033 0.1180 0.807 × 1045

.15 0.9229 1.262 12.479 0.3807 0.1953 1.73 × 1045

.175 0.9084 1.498 12.688 0.5150 0.2401 2.15 × 1045

.186 0.8874 1.591 12.685 0.5933 0.2618 2.30 × 1045

Table 2 The march of pressure,density, pressure-density ratio,red shift and square of adiabaticsound speed and adiabatic indexwithin the ball corresponding tou = 0.186

r/rb8πG

c4 pr2b

8πG

c2 ρr2b

p

ρc21c2 (

dpdρ

) γ = dpdρ

/pρ

Z

0 0.5745 4.654 0.1234 0.15394 1.247 0.5933

0.1 0.5482 4.483 0.1222 0.15393 1.258 0.5852

0.2 0.4790 4.033 0.1187 0.15388 1.295 0.5623

0.3 0.3882 3.443 0.1127 0.15377 1.363 0.5286

0.4 0.2962 2.84 0.1041 0.15353 1.474 0.4887

0.5 0.2147 2.313 0.0928 0.15307 1.649 0.4463

0.6 0.1479 1.876 0.0788 0.15228 1.930 0.4044

0.7 0.0954 1.530 0.0623 0.15107 2.421 0.3644

0.8 0.0549 1.260 0.0435 0.14935 3.428 0.3271

0.9 0.0238 1.050 0.0226 0.14706 6.486 0.2929

1.0 0.000 0.8874 0.0000 0.14419 ∞ 0.2618

matter. For the type of solution considered in the presentstudy, the formula yields in the following form (Bejger andHaensel 2002; Pant and Negi 2012).

I = (2/5)(1 + y)MR2; (37)

where y is the dimensionless compactness parameter mea-sured in units of M� (in km)/km, i.e.

y = (M/R)/M� (in km) km−1 (38)

With the help of (37), we can calculate the moment of in-ertia, for various super dense objects as shown in Tables 1and 2. These values of masses and moment of inertia agreequite well with those of the masses and the moment of iner-tia calculated for the Vela pulsars.

8 Discussions and conclusions

It has been observed that the physical parameters(p,ρ,

p

ρc2 , z) are positive at the centre and within the limitof realistic state equation and monotonically decreasingfor 0.244 > u > 0. However, corresponding to any valueof 0.244 > u > 0.186, though the causality condition isobeyed throughout within the ball but the trend of adiabatic

sound speed is erratic. Thus, the solution is well behavedfor all values of u satisfying the inequality 0. < u ≤ 0.186.We now here present a model of super dense star basedon the particular solution discussed above by assumingsurface density; ρb = 2 × 1014 g/cm3. Corresponding tou = 0.186, the resulting well behaved model has maximummass M = 1.591 M� with radius Rb ≈ 12.685 km (seeFig. 1 and Fig. 2). The model has mass within the rangeof neutron star. The good matching of our results for Velapulsars show the robustness of our model.

Acknowledgements One of us (Neeraj Pant) acknowledges his grat-itude to Lt. Gen. Jatinder Singh AVSM**, SM, Comdt, NDA for hismotivation and encouragement. Neeraj Pant also extends his gratitudeto Prof. O.P. Shukla, Principal NDA, for his encouragement.

Authors are grateful to the anonymous referee for his relevant sug-gestions.

References

Adler, R.J.: J. Math. Phys. 15, 727 (1974)Bejger, M., Haensel, P.: Astron. Astrophys. 396, 917 (2002)Canuto, V., Lodenquai, J.: Phys. Rev. C 12, 2033 (1975)Delgaty, M.S.R., Lake, K.: Comput. Phys. Commun. 115, 395 (1998)Durgapal, M.C.: J. Phys. A, Math. Gen. 15, 2637 (1982)Finch, M., Skea, J.E.F.: Class. Quantum Gravity 4, 467 (1989)



Goldman, S.P.: Astrophys. J. 226, 1079 (1978)Heintzmann, H.: Z. Phys. 228, 489 (1969)Kuchowicz, B.: Acta Phys. Pol. 34, 131 (1968)Matese, J.J., Whitman, P.G.: Phys. Rev. D, Part. Fields 22, 1270 (1980)Maurya, S.K., Gupta, Y.K.: Astrophys. Space Sci. 334, 145 (2011).

doi:10.1007s10509-011-0705-yMehra, A.L.: J. Aust. Math. Soc. 6, 153 (1966)Nariai: Sci. Rep. Tohoku Univ., Ser. 1 34, 160 (1950)

Pant, N.: Astrophys. Space Sci. 331, 633 (2011)Pant, N., Negi, P.S.: Astrophys. Space Sci. 338, 163 (2012)Pant, N., et al.: Astrophys. Space Sci. 330, 353 (2010)Pant, N., Maurya, S.K.: Appl. Math. Comput. 218, 8260 (2012).

doi:10.1016/j.amc.2012.01.044Patvardhav, Vaidya: J. Univ. Bombay 12(Part III), 23 (1943)Tolman, R.C.: Phys. Rev. 55, 364 (1939)


http://dx.doi.org/10.1007s10509-011-0705-y

http://dx.doi.org/10.1016/j.amc.2012.01.044

1 23

Astrophysics and Space ScienceAn International Journal of Astronomy,Astrophysics and Space Science ISSN 0004-640XVolume 341Number 2 Astrophys Space Sci (2012) 341:469-475DOI 10.1007/s10509-012-1105-7

A family of charge analogue of Durgapalsolution

Pratibha Fuloria & B. C. Tewari

1 23

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Astrophys Space Sci (2012) 341:469–475DOI 10.1007/s10509-012-1105-7


A family of charge analogue of Durgapal solution

Pratibha Fuloria · B.C. Tewari

Received: 26 March 2012 / Accepted: 1 May 2012 / Published online: 16 May 2012© Springer Science+Business Media B.V. 2012

Abstract We obtain a new parametric class of exact solu-tions of Einstein–Maxwell field equations which are wellbehaved. We present a charged super-dense star model af-ter prescribing particular forms of the metric potential andelectric intensity. The metric describing the super densestars joins smoothly with the Reissner–Nordstrom metricat the pressure free boundary. The electric density assumed

is E2

c1= Kx

2 (1 + x)n(1 + 6x)23 where n may take the val-

ues 0,1,2,3,4 and so on and K is a positive constant. Forn = 0,1 we rediscover the solutions by Gupta and Maurya(Astrophys. Space Sci. 334(1):155, 2011) and Fuloria et al.(J. Math. 2:1156, 2011) respectively. The solution for n = 2have been discussed extensively keeping in view of well be-haved nature of the charged solution of Einstein–Maxwellfield equations. The solution for n = 3 and n = 4 can be alsostudied likewise. In absence of the charge we are left behindwith the regular and well behaved fifth model of Durgapal (J.Phys. A 15:2637, 1982). The outmarch of pressure, density,pressure-density ratio and the velocity of sound is monoton-ically decreasing, however, the electric intensity is mono-tonically increasing in nature. For this class of solutions themass of a star is maximized with all degree of suitability,compatible with Neutron stars and Pulsars.

By assuming the surface density ρb = 2 × 1014 g/cm3

(Brecher and Caporaso in Nature 259:377, 1976), corre-sponding to K = 12 with X = 0.1, the resulting well be-haved model has the mass M = 2.12M�, radius Rb ≈

P. Fuloria (�)Physics Deptt., S.S.J. Campus, Almora, Indiae-mail: [email protected]

B.C. TewariMath Deptt., S.S.J. Campus, Almora, Indiae-mail: [email protected]

16.07 km and moment of inertia I = 4.95 × 1045 g cm2; forK = 8 with X = 0.1, the resulting well behaved model hasthe mass M = 1.50M�, radius Rb ≈ 16.78 km and momentof inertia I = 3.68 × 1045 g cm2. These values of massesand moment of inertia are found to be consistent with othermodels of Neutron stars and Pulsars available in the litera-ture and are applicable for the Crab Pulsars.

Keywords General relativity · Reissner–Nordstrom ·Einstein–Maxwell · Charged fluid spheres

1 Introduction

Exact solutions with well-behaved nature of Einstein–Maxwell field equations are of vital importance in rela-tivistic astrophysics. Such solutions may be used to makea suitable model of super dense object with charge mat-ter like Neutron star, quark star, pulsar and analogous stars.It is interesting to observe that, in the presence of charge,the gravitational collapse of a spherical symmetric distri-bution of the matter to a point singularity may be avoidedbecause the presence of some charge in a spherical ma-terial distribution provides an additional resistance againstthe gravitational contraction by means of electric repulsion.The inclusion of charge plays very important role in thestability of massive fluid spheres. A spherical fluid distri-bution of uniform density with a net charge on the sur-face is more stable than a surface without charge (de Fe-lice et al. 1995). Thus a stable massive charged configu-ration can be produced in which repulsive force from thecharge counter balances the gravitational attraction. Thesingularities which appear in the Schwarzschild solutioncorresponding to spherically symmetric static perfect fluiddistribution can be avoided to a great extent by including





charge in them. The negative pressure of Bayin’s solution(1978) makes it physically unreasonable, however the in-clusion of charge in Bayin’s solution makes it physicallyreasonable i.e. the pressure is a decreasing function of ra-dius from centre to surface (Ray et al. 2007). Bonnor (1965)has shown that dust cloud of arbitrarily Large mass andsmall radius can remain in equilibrium if it has an elec-tric charge density related to the mass density by σ = ±ρ.Usmani et al. (2011) have shown that the de Sitter void ofa charged gravastar, represents the same thing as electro-magnetic mass model which is generating the gravitationalmass and provides stability to the astrophysical configura-tion. All these advantages of charged spherical models mo-tivated us to consider the charge analogue of Durgapal’ so-lution (1982). We have found out a new parametric classof exact solutions of Einstein–Maxwell field equations fora static, spherically symmetric distribution of the chargedand perfect fluid with well behaved nature and to constructa suitable model of super dense object with charge matter.Many of the authors electrified the well known unchargedfluid spheres e.g. Durgapal and Fuloria (1985) solution byGupta and Maurya (2010), Schwarzschild solution by Guptaand Gupta (1986) and by Gupta and Kumar (2005), Ku-chowich (1968) solution by Nduka (1977), Tolman (1939)solution by Cataldo and Mitskievic (1992) and Pant andNegi (2012), Heintzmann’s solution (1969) by Pant et al.(2011), Pant and Maurya (2012), Adler’s (1974) solutionby Pant and Tewari (2011), Pant et al. (2011) and so on.We have tried to charge the Durgapal (1982) solution byassuming a parametric expression for electric density. Forthe values of n = 0,1,2 the solutions are well behaved. Thesolutions for the n = 3, n = 4 may be obtained and theirwell behaved nature may be verified by applying the variousconditions.

For well behaved nature of the solution the followingconditions should be satisfied: Delgaty and Lake (1998) andPant et al. (2010).

(i) The solution should be free from physical and geo-metrical singularities i.e. finite and positive values of centralpressure, central density and non zero positive values of eλ

and eυ .(ii) The solution should have positive and monotonically

decreasing expressions for pressure and density (p and ρ)with the increase of r . The solution should have positivevalue of ratio of pressure-density and less than 1 (weak en-ergy condition) and less than 1/3 (strong energy condition)throughout within the star, monotonically decreasing as well(Pant and Negi 2012).

(iii) The casualty condition (dp/c2dρ)1/2 i.e. velocity ofsound should be less than that of light throughout the model.In addition to the above the velocity of sound should be.

Decreasing towards the surface i.e. ddr

(dpdρ

) < 0 or

(d2p

dρ2 ) > 0 for 0 ≤ r ≤ rb i.e. the velocity of sound is in-

creasing with the increase of density. In this context it isworth mentioning that the equation of state at ultra-high dis-tribution, has the property that the sound speed is decreasingoutwards.

(iv) pρ

≤ dpdρ

, everywhere within the ball. γ = d loge P

loge ρ=

ρp

dpdρ

⇒ dpdρ

= γpρ

, for realistic matter γ ≥ 1 (Pant and Mau-rya 2012).

(v) The red shift z should be positive, finite and mono-tonically decreasing in nature with the increase of r .

(vi) Electric intensity E is positive and monotonically in-creasing from centre to boundary and at the centre the Elec-tric intensity is zero.

2 Einstein–Maxwell equation for charged fluiddistribution

Let us consider a spherical symmetric metric in curvaturecoordinates

ds2 = −eλdr2 − r2(dθ2 + r2 sin2 θdφ2) + eνdt2 (1)

where the functions λ(r) and ν(r) satisfy the Einstein–Maxwell equations

−8πG

c4T i

j = Rij − 1

2Rδi

j

= −8πG

c4

[(c2ρ + p

)vivj − pδi

j

+ 1

4π

(−F imFjm + 1

4δijFmnF

mn

)](2)

where ρ, p, vi , Fij denote energy density, fluid pressure, ve-locity vector and skew-symmetric electromagnetic field ten-sor respectively.

In view of the metric (1), the field equation (2) givesDionysiou (1982)

ν′

re−λ − (1 − e−λ)

r2= 8πG

c4p − q2

r4(3)

(ν′′

2− λ′ν′

4+ ν′2

4+ ν′ − λ′

2r

)e−λ = 8πG

c4p + q2

r4(4)

λ′

re−λ + (1 − e−λ)

r2= 8πG

c2ρ + q2

r4(5)

where prime (′) denotes the differentiation with respect to r

and q(r) represents the total charge contained with in thesphere of radius r .


eν = B(1 + x)5, x = c1r2 and e−λ = Z. (6)

where B being the positive constant. Now putting (6) into(3)–(5), we have



10Z

(1 + x)− (1 − Z)

x+ c1q

2

x2= 1

c1

8πGP

c4(7)

(1 − Z)

x− 2

dZ

dX+ c1q

2

x2= 1

c1

8πGρ

c4(8)


dZ

dx+ 14x2 − 2x − 1

x(1 + x)(1 + 6x)xZ = {(2q2c1/x) − 1}(1 + x)

x(1 + 6x)

(9)

where x = c1r2, e−λ = Z.

3 New class of solutions

In order to solve the differential equation (9) let us considerthe electric intensity E of the following form

E2

c1= c1q

2

x2= Kx

2(1 + x)n(1 + 6x)

23 (10)

where K is a positive constant. The electric density is soassumed that the model is physically significant and well

behaved i.e. E remains regular and positive throughout thesphere.

In view of (10) differential equation (9) yields the follow-ing solution

e−λ = K

(5 + n)

x(1 + x)2+n

(1 + 6x)1/3

+ 1

(1 + x)3

[1 − x(309 + 54x + 8x2)

112

]

+ Ax

(1 + 6x)13

(1 + x)3 (11)


4 Properties of the new class of solutions

Using (11), into (7) and (8), we get the following expres-sions for pressure and energy density

1

c1

8πG

c4p = 1

(1 + x)4

[(475 − 4125x − 1050x2 − 200x3)

112+ A(1 + 11x)

(1 + 6x)13

]+ K (1 + x)n(2 + (n + 29)x + (52 + 6n)x2)

2(5 + n)(1 + 6x) 13

(12)1

c1

8πG

c4p0 =

(475

112+ A + K

(5 + n)

)(13)

1

c1

8πG

c2ρ = 1

(1 + x)4

[(1935 + 15x + 450x2 + 120x3)

112− A

(3 + 11x − 22x2)

(1 + 6x)43

]

− K(1 + x)n{6 + (53 + 5n)x + (178 + 40n)x2 + (256 + 60n)x3}2(5 + n)(1 + 6x)

43

(14)

1

c1

8πG

c2ρ0 =

(1935

112− 3A − 6K

2(5 + n)

)(15)

Differentiating (12) and (14) w.r.t. x, we get:

1

c1

8πG

c4

dp

dx= − 25(241 − 411x − 60x2 − 8x3)

112(1 + x)5+ 5A(1 − 3x − 44x2)

(1 + x)5(1 + 6x)43

+

(K(1 + x)n−1[(25 + 3n) + (245 + 58n + n2)x+(740 + 302n + 12n2)x2 + (520 + 372n + 36n2)x3]

)

2(5 + n)(1 + 6x) 43

(16)

1

c1

8πG

c2

dρ

dx= −15

112(1 + x)5(515 − 57x + 36x2 + 8x3) + 5A

(1 + x)5(1 + 6x)73

(5 + 39x + 66x2 − 88x3)

− K(1 + x)n−1

2(5 + n)(1 + 6x)73

((5 + 11n) + (255 + 164n + 5n2)x + (1730 + 906n + 70n2)x2

+(4040 + 2264n + 300n2)x3 + (2560 + 2136n + 360n2)x4

)(17)

1

c2

dp

dρ= − (1 + 6x) [K(1 + x)n+4N(x) − 50/112P(x)(1 + 6x)4/3(5 + n) + 10AQ(x)(5 + n)]

[K(1 + x)n+4M(x) − 30/112H(x)(1 + 6x)7/3(5 + n) + 10AT (x)(5 + n)] (18)



where

H(x) = 515 − 57x + 36x2 + 8x3 T (x) = 5 + 39x + 66x2 − 88x3

M(x) = (5 + 11n) + (255 + 164n + 5n2)x + (1730 + 906n + 70n2)x2 + (4040 + 2264n + 300n2)x3

+ (2560 + 2136n + 360n2)x4

N(x) = (25 + 3n) + (245 + 58n + n2)x + (

740 + 320n + 12n2)x2 + (520 + 372n + 36n2)x3

P(x) = 241 − 411x − 60x2 − 8x3 Q(x) = 1 − 3x − 44x2

(1

c2

dp

dρ

)

0= −

[31K − 6025

56 (5 + n) + 10A(5 + n)]

[27K − 7725

56 (5 + n) + 50A(5 + n)] (19)

5 Variety of classes of solutions

Case 1 When n = 0, Gupta and Maurya Solution (2011)The resulting Solution is

e−λ = K

5

x(1 + x)2

(1 + 6x)1/3

+ 1

(1 + x)3

[1 − x(309 + 54x + 8x2)

112

]

+ Ax

(1 + 6x)13

(1 + x)−3 (20)

where A is an arbitrary constant of integrationThe expressions for pressure and energy density are given

by

1

c1

8πG

c4p = 1

(1 + x)4

[(475 − 4125x − 1050x2 − 200x3)

112

+ A(1 + 11x)

(1 + 6x)13

]+ K

10

(2 + 29x + 52x2)

(1 + 6x) 13

(21)

1

c1

8πG

c2ρ = 1

(1 + x)4

[(1935 + 15x + 450x2 + 120x3)

112

− A(3 + 11x − 22x2)

(1 + 6x)43

]− K

10

× (6 + 53x + 178x2 + 256x3)

(1 + 6x)43

(22)

The solution gives wide range of constant K (0 < K ≤ 50)for which the solution is well behaved and therefore suitablefor modelling of superdense star.Case 2 When n = 1 Fuloria et al. (2011) solution

The resulting Solution is

e−λ = K

6

x(1 + x)3

(1 + 6x)1/3

+ 1

(1 + x)3

[1 − x(309 + 54x + 8x2)

112

]

+ Ax

(1 + 6x)13

(1 + x)3 (23)


The expression for pressure and density are expressed as,

1

c1

8πG

c4p = 1

(1 + x)4

[(475 − 4125x − 1050x2 − 200x3)

112

+ A(1 + 11x)

(1 + 6x)13

]

+ K

6

(1 + x)(1 + 15x + 29x2)

(1 + 6x) 13

(24)

1

c1

8πG

c2ρ = 1

(1 + x)4

[(1935 + 15x + 450x2 + 120x3)

112

− A(3 + 11x − 22x2)

(1 + 6x)43

]

− K

6

(1 + x)(3 + 29x + 109x2 + 158x3)

(1 + 6x)43

(25)

The solution is well behaved for all the values of K satisfy-ing the inequality 0 < K < 10 and X = 0.2.

6 New well behaved solution

Case 3 When n = 2We have the expression for electric density as

E2

c1= 1

2Kx(1 + x)2(1 + 6x)

23 (26)

(d

dx

E2

c1

)

x=0= K

2(27)

The electric intensity is minimum at the centre and mono-tonically increasing for all values K > 0. Also at the centreit is zero. Thus we have, K ≥ 0, c1 > 0.

In view of (26) differential equation (9) yields the follow-ing solution

e−λ = K

7

x(1 + x)4

(1 + 6x)1/3

+ 1

(1 + x)3

[1 − x(309 + 54x + 8x2)

112

]

+ Ax

(1 + 6x)13

(1 + x)−3 (28)



Where A is an arbitrary constant of integration.

1

c1

8πG

c4p = 1

(1 + x)4

[(475 − 4125x − 1050x2 − 200x3)

112

+ A(1 + 11x)

(1 + 6x)13

]

+ K

14

(1 + x)2(2 + 31x + 64x2)

(1 + 6x) 13

(29)

1

c1

8πG

c2ρ = 1

(1 + x)4

[(1935 + 15x + 450x2 + 120x3)

112

− A(3 + 11x − 22x2)

(1 + 6x)43

]

− K

14

(1 + x)2(6 + 63x + 258x2 + 376x3)

(1 + 6x)43

(30)

7 Properties of the new solution (n = 2)


1

c1

8πG

c4p0 =

(475

112+ A + K

7

)(31)

1

c1

8πG

c2ρ0 =

(1935

112− 3A − 3K

7

)(32)

Differentiating (29) and (30) with respect to x we get,

1

c1

8πG

c4

dp

dx

= −25(241 − 411x − 60x2 − 8x3)

112(1 + x)5

+ 5A(1 − 3x − 44x2)

(1 + x)5(1 + 6x)43

+ K(1 + x)(31 + 365x + 1392x2 + 1408x3)

14(1 + 6x) 43

(33)

1

c1

8πG

c4

dρ

dx

= −15(515 − 57x + 36x2 + 8x3)

112(1 + x)5

+ 5A(5 + 39x + 66x2 − 88x3)

(1 + x)5(1 + 6x)43

− K(1 + x)

14(1 + 6x)73

× (27 + 603x + 3822x2 + 9768x3 + 8272x4) (34)

(1

c1

8πG

c4

dp

dx

)

0= 31K

14− 6025

112+ 5A (35)

(1

c1

8πG

c4

dρ

dx

)

0= −27K

14− 7725

112+ 25A (36)

1

c2

dp

dρ= − (1 + 6x)

×[K(1 + x)6N(x) − 50

16P(x)(1 + 6x)4/3 + 70AQ(x)]

[K(1 + x)6M(x) − 30

16H(x)(1 + 6x)7/3 + 70AT (x)]

(37)

where

H(x) = 515 − 57x + 36x2 + 8x3

T (x) = 5 + 39x + 66x2 − 88x3

P(x) = 241 − 411x − 60x2 − 8x3

M(x) = 27 + 603x + 3822x2 + 9768x3 + 8274x4

N(x) = 31 + 365x + 1392x2 + 1408x3

Q(x) = 1 − 3x − 44x2

(1

c2

dp

dρ

)

0= −

[31K − 6025

8 + 70A]

[27K − 7725

8 + 750A] (38)

8 Boundary conditions

The solutions so obtained are to be matched over the bound-ary with Reissner–Nordstrom metric:

ds2 = −(

1 − 2GM

r+ e2

r2

)−1

dr2 − r2(dθ2 + sin2 θdφ2)

+(

1 − 2GM

r+ e2

r2

)dt2 (39)

which requires the continuity of eλ, eν and q across theboundary r = rb

e−λ(rb) = 1 − 2GM

c2rb+ e2

r2b

(40)

eν(rb) = 1 − 2GM

c2rb+ e2

r2b

(41)

q (rb) = e (42)

p (rb) = 0 (43)

The condition (43) can be utilized to compute the values ofarbitrary constants A as follows:

Pressure at p (r = rb) = 0 gives

A = (1 + 6X)13

(1 + 11X)

(−475 + 4125X + 1050X2 + 200X3

112

)

− k

14

(1 + X)6

(1 + 11X)

(2 + 31X + 24X2) (44)

In view of (40) and (41) we get

B = 1

(1 + x)5

[K

6

X(1 + X)3

(1 + 6X)13

+ 1

(1 + X)3

(1 − X(309 + 54X + 8X2)

112

)

+ AX

(1 + X)3(1 + 6X)13

](45)



Table 1 The variation of various physical parameters at the centre, surface density, electric field intensity on the boundary, mass and radius ofstars with different values of K and X = 0.1

K 1c1

8πG

c4 p01c1

8πG

c2 ρ01c2

p0ρ0

1c2 (

dpdρ

)x=0 z0 ( E2

c1)rb

8πG

c2 ρbr2b

MM� ≈ rb in km

2 3.5774 19.2676 0.1856 0.5603 0.31 0.150 1.177 0.49 17.78

4 3.17103 20.4857 0.1548 0.4514 0.34 0.300 1.343 0.84 17.45

6 2.7653 21.7039 0.12741 0.3757 0.38 0.451 1.091 1.17 17.12

8 2.3593 22.922 0.1029 0.3199 0.42 0.601 1.048 1.50 16.78

10 1.9538 24.1401 0.08091 0.2772 0.45 0.752 1.005 1.80 16.43

12 1.5472 25.3582 0.0610 0.2434 0.50 0.902 0.9624 2.12 16.07

14 1.1412 26.5763 0.04294 0.2161 0.55 1.053 0.919 2.37 15.71

16 0.7332 27.7944 0.02645 0.1934 0.60 1.203 0.876 2.63 15.34

Table 2 The march of pressure, density, pressure-density ratio, square of adiabatic sound speed, red shift, electric intensity within the perfect fluidsphere corresponding to K = 1 and X = 0.2

r/rb8πG

c4 pr2b

8πG

c2 ρr2b

p

ρc21c2 (

dpdρ

) z (E2

c1)

0.0 1.019 2.9429 0.3462 0.9748 1.6410 0.0000

0.1 0.999 2.9217 0.3422 0.9688 1.6279 0.0010

0.3 0.855 2.7562 0.3105 0.9245 1.5258 0.0099

0.4 0.7420 2.6176 0.2834 0.8890 1.4410 0.01915

0.5 0.6106 2.4480 0.2494 0.8462 1.3378 0.03283

0.6 0.4709 2.2535 0.2089 0.7960 1.2197 0.05256

0.7 0.3320 2.0408 0.1627 0.7375 1.0906 0.08040

0.8 0.2025 1.8162 0.1115 0.6680 0.9543 0.11906

0.9 0.0898 1.5853 0.0566 0.5829 0.8145 0.17198

1.0 0.0000 1.3521 0.0000 0.4712 0.6742 0.24358


GM

c2= rb

2

[1 + K

2X2(1 + X)2(1 + 6X)

23 − B(1 + X)5

]

(46)


8πG

c2ρbr

2b = 1

(1 + X)4

[(1935 + 15X + 450X2 − 120X3)

112

− A(3 + 11X − 22X2)

(1 + 6X)43

]

− K

6

(1 + X)(3 + 29X + 109X2 + 158X3)

(1 + 6X)43

(47)

In view of Table 1 we observe that pressure, density,pressure density ratio, red shift, electric density all is pos-itive at the centre. Assuming the surface density ρb = 2 ×1014 g/cm3 the mass and radius has been calculated for dif-ferent values of K and X = 0.1.

In view of the Table 2 we observe that pressure, density,pressure-density ratio and square of adiabatic sound speed

and red-shift decrease monotonically with the increase ofradial coordinate and electric intensity increases monotoni-cally with the increase of radial coordinate within the per-fect fluid sphere. At the centre of the star the matter is sodense that electrons are forced to combine with protons re-sulting into neutrons and neutrinos. This process favours themaximum matter at the centre to be neutral i.e. the electricintensity will be zero at the centre. As we move towardssurface the density goes on decreasing approaching zero atthe surface. The possibility of combination of electrons andprotons becomes less and less as we move towards surface,hence the electric intensity will be maximum towards thesurface.

9 Slowly rotating structures and their application toCrab Pulsars

For slowly rotating structures like the Crab pulsars (rotationvelocity about 188 rad/sec) we can calculate the momentof inertia in the first-order approximation which appears inthe Lense–Thrirring frame dragging effect. However, for the



exact solution that we have considered here, it is very use-ful to apply an approximate, but very precise, empirical for-mula which is based on the numerical results obtained for alarge number of theoretical equations of state of dense nu-clear matter.

For the type of solution considered in the present study,the formula yields in the following form (Bejger andHaensel 2002):

I = (2.5)(1 + u)MR2; (48)

where u is the dimensionless compactness parameter mea-sured in units of MΘ (in km)/km, i.e.

u = (M/R)/M� (in km) km−1 (49)

Equation (48) is used to calculate the moment of iner-tia for the models presented in Table 1. Corresponding toM = 2.12 M� and R = 16.07 km we obtain the moment ofinertia for the configuration as 4.95 × 1045 gm cm2. For themass 1.50 M� and R = 16.78 km equation (48) yields themoment of inertia as 3.68 × 1045 gm cm2.

These values of masses and moment of inertia agree quitewell with those of the masses and the moment of inertia cal-culated for the Crab pulsars on the basis of newly estimatedvalue of Crab nebula mass M = 4.6M� (Bejger and Haensel2002; and Pant and Negi 2012).

10 Results and discussions

We have electrified the uncharged fluid sphere e.g. Durga-pal (1982) Vth solution. The charged solution is well be-haved with positive values of charge parameter K and com-pletely describes interior of the super-dense astrophysicalobjects with charge matter. The electric density assumed isE2

c1= c1q

2

x2 = Kx2 (1 + x)n(1 + 6x)

23 . The electric density is

so assumed that the model is physically significant and wellbehaved. E vanishes at the centre of the star and increasesas we move towards the surface and is positive throughoutthe star. In view of Table 1 it has been observed that all thephysical parameters (p,ρ,

p

ρc2 , Z and E2

c1) are positive at the

centre and within the limit of realistic equation of state.We observe that with the increase of the value of n, al-

though the solutions become more complicated in terms ofexpressions, but they show the well behaved nature and sat-isfy all the necessary conditions to be physically realizable.Thus by assigning different positive integral value to param-eter n we get a variety of classes of exact solutions. We haveobtained a generalized solution of Einstein–Maxwell fieldequations of general relativity for a static, spherically sym-metric distribution of the charged fluid with well behavednature.

For n = 0 we get the Gupta and Maurya (2011) solution,which is well behaved for a wide range of constant K .

For n = 1, we get the Fuloria et al. (2011) solution whichis also well behaved for K < 10 hence, suitable for modelingof super dense star.

For n = 2, we have obtained a new solution which hasbeen studied in detail exposing its well behaved nature. Thesolution is well behaved for a wide range of constant K .

For this class of solution (n = 2) the mass of a star ismaximized with all degree of suitability compatible withquark stars, neutron stars and pulsars. By assuming the sur-face density ρb = 2 × 1014 g/cm3 corresponding to K = 12and X = 0.1 the maximum mass of the star comes outto be 2.12M� with its radius 16.07 km. Correspondingto this model we obtain the moment of inertia as 4.95 ×1045 gm cm2. The values of mass, radius and moment of in-ertia obtained in the present study strongly favour our modelto represent the crab pulsar. In absence of the charge we areleft behind with the regular and well behaved fifth model ofDurgapal (1982).

Acknowledgements Authors acknowledge their gratitude to Dr.Neeraj Pant (Associate Professor) Maths Deptt., National DefenseAcademy Khadakwasla, Pune for his stimulating discussions and mo-tivation. Authors are also grateful to the referee for pointing out theerrors in original manuscript and making constructive suggestions.

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