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International Journal of Reciprocal Symmetry and Theoretical Physics, Volume 1, No 1 (2014)

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International Journal of Reciprocal Symmetry and Theoretical Physics

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Established: 2014

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EDITORIAL BOARD

Editor-in-Chief

Dr. Mushfiq Ahmad Department of Physics, Rajshahi University, Rajshahi, Bangladesh

Senior Editor

Dr. M. Abdus Sobhan Department of Applied Physics, Rajshahi University, Rajshahi, Bangladesh

Dr. Osman Goni Talukdar Vice Chancellor, Varendra University, Rajshahi, Bangladesh

Associate Editor Suresh B. Rana

ProCure Proton Therapy Center, Medical Physics, Oklahoma City, USA

Dr. Hayder Jabbar Abood Al Dabbagh College of Education for Pure Sciences, University of Babylon, Babylon, Iraq

Dr. Ahmed Hashim Mohaisen Al-Yasari Department of Physics, University of Babylon, Babylon, Iraq

Dr. Suat Pat Department of Physics, Eskişehir Osmangazi Üniversity, Turky

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International Journal of Reciprocal Symmetry and Theoretical Physics

Blind Peer-Reviewed Journal Volume 1, Number 1/2014

Contents

1. Non-Associativity of Lorentz Transformation and Associative Reciprocal Symmetric Transformation Mushfiq Ahmad, & Muhammad Shah Alam

9-19

2. Rciprocal Property of Different Types of Lorentz

Transformations

Atikur Rahman Baizid, & Md. Shah Alam

20-35

3. Armenian Theory of Special Relativity

Robert Nazaryan, & Haik Nazaryan

36-42

4. Upper Limit of the Age of the Universe with

Cosmological Constant

Haradhan Kumar Mohajan

43-68

IJRSTP Publishes Online and Print Version Both



Non-Associativity of Lorentz Transformation

and Associative Reciprocal Symmetric

Transformation

Mushfiq Ahmad1*

,1Muhammad Shah Alam

2

1Department of Physics, Rajshahi University, Rajshahi-6205, Bangladesh 2Department of Physics, Shahjalal University, Sylhet, Bangladesh

ABSTRACT

Lorentz transformation is not associative. The non-associativity

makes it frame dependent; and it does not fulfill relativistic

requirements including reciprocity principle. The non

associativity also leads to ambiguities when three or more

velocities are involved. We have proposed an associative

Reciprocal Symmetric Transformation (RST) to replace Lorentz

transformation. RST is complex and is compatible with Pauli and

Dirac algebra.

PACS No.: 03.30.+p, 03.65.-w

Keywords: Lorentz boost; Non-associativity; Reciprocity

Principle; Reciprocal Symmetric Transformation; Thomas

Precession; Quaternionic transformation; Pauli quaternion; Spin;

Clifford algebra

1 INTRODUCTION

We have shown1 that Lorentz-Einstein law of addition of velocities is not

associative and that this law gives a frame dependent relative velocity. Frame

dependence contradicts2 the principle of relativity. Another implication of

*[email protected]

mailto:[email protected]



non-associativity is that it fails3 to fulfill reciprocity principle; when 2 or more

velocities are involved velocity BC (Fig.1) is not4 the negative of velocity CB

(Fig.2). Ungar5 recognizes that Einstein's law is “neither commutative nor

associative”4, but he attributes the failure of reciprocity principle to “the non-

commutativity of the relativistic composition of non-collinear velocities”4.

Wigner6 and Moller7 have tried to justify this failure by in invoking Thomas

Precession. Much later Oziewicz wrote, “There have been attempts8 [Ungar

1988] to explain the non-associativity, and also Mocanu paradox, as the

Thomas rotation, i.e. as non-transitivity of the parallelism of the spatial

frames. We consider this attempt not satisfactory.”9

Ungar10 has introduced gyrovectors to make Einstein's law of addition both

commutative and associative. The attempt is not satisfactory because:

(i) Non-commutativity is not a physical requirement. To try to make it

commutative is based on wrong diagnosis.

(ii) Inclusion of gyrovectors involves a series of arbitrary prescriptions.

As Ungar11 has observed, “The non-associativity of Einstein’s velocity addition

is not widely known". The popular belief remains that Einstein's law of addition

of velocities is associative, although it is an exercise in elementary algebra to

prove non-associativity (11)

Lorentz transformation may be put in a matrix form and matrix algebra is

associative. This has misled some people to think that Lorentz transformation

is associative. (This is clarified in Appendix A in section 10).

In this paper we intend to reiterate the non-associativity of Einstein's law of

addition of velocities and to show that (space-time) Lorentz transformation is

also not associative.

We want to present Reciprocal Symmetric Transformation (RST), in place of

LT as the solution. RST is complex. This is explained by fact that it obeys Pauli



quaternion algebra and is conform to Dirac electron theory which also obeys

Pauli quaternion algebra. The complex nature of RST makes it compatible

with quantum mechanics.

2. FAILURE OF RECIPROCITY

Consider three moving bodies A, B and C in relative motion with velocities u,

v and w as shown in Fig.1 We add (obeying Lorentz-Einstein law) velocities u

and v to get Lw where

2/1 cLL

vu

vuvuw

22 /1

)(

1

1

cc u

u

vu

vuu

(1)

2/1

1

cuu

(2)

L stand for Lorentz-Einstein addition. In Fig.2 we add velocities –v and –u

to get Lw'

)()(' uvw LL (3)

Lorentz-Einstein law of addition (1) gives

LL 'ww (4)

(4) contradicts the principle of reciprocity required by the principle of

relativity. Some12 authors have called the inequality a paradox and have

attributed it to “The non-commutativity and the non-associativity of the

composition law of the non-collinear velocities”13. Some4 others have



attributed it to non-commutativity only. We shall show below that non

commutativity does not contribute to it; the difficulty is due to non-

associativity only.

3. FRAME DEPENDENCE

Let u be the velocity of a moving body. An observer moving with velocity v

observers the relative velocity

uvz LL )( (5)

With respect to a second observer moving with velocity y, u and v become

uyu' L )( and vyv' L )( (6)

The relative velocity Lz becomes

')'(' uvz LL (7)

Using (1) we find5

LL zz ' (8)

Eq. (8) shows that relative motion is frame dependent and that (1) is not

relativistic.

4. AMBIGUITY

Consider a body moving with velocity u. An observer with velocity v

observers the velocity uvu L )(' . A second observer is traveling with

velocity y and



observers the velocity ')(" uyu L . This situation is represented by Fig.3

and eq. (9). Fig.4 and eq. (10) represents the situation in which velocities v

and y are added first to give m; yvm L . The observer moving with

velocity m observers ^u

uvyuyu LLL )()(')(" (9)

uyvumu LLL )}({)(^ (10)

The inequality

u^u " (11)

shows an ambiguity in the computation of the resultant velocity when 3 or

more velocities are added.

5. DIAGNOSIS AND THE SOLUTION OF THE PROBLEM

1. Intuitive approach

In the first triangle u comes first then v and they add up to w. In the second

triangle -v comes first then -u and they add up to w' . Instead of (4) we need

w'w

w'w or vuuv )()( (12)

In (12) we have used the notation (without suffix L) to represent an

associative addition.

2. Matrix Approach

Let uΜ be the matrix which takes to ' i.e. 'M(u) and

'u)M( . Then

)()( uMuM 1MuMuM (0))()(1 with )()( 1

uMuM

(13)



1Μ 0 is the identity matrix )M(0 . By repeated application of the

matrices we have

")()()( uvMuMvM (14)

Combining the matrices and their inverses we have

)0()()()()()()( Muv)MuvM(uMvMvMuM

(14)

Or

)uvM()uM(vMuM )()()()()( v (15)

(15) gives (12) corresponding to the familiar matrix rule 111 ABAB

3. Algebraic Approach

Theorem:

If stands for an associative addition

)()()( uvvu (16)

Proof:

Using associativity of and 0)( vv , we have

)}(){()( uvvu = )()}({ uvvu 0)( uu (17)

Therefore,

0)}(){()( uvvu (18)

Also we have

0)}({}{ vuvu (19)

Comparison between (14) and (15) shows

)( vu )()( uv (20)



6. PAULI QUATERNION

From 3-vectors u, v and w we construct 4-vectors

).1().1( 00 zzyyxx uuu σuu (21)

Where s have the following properties14 [Quaternion and Pauli Quaternion

differ by a factor of i].

xy zyx i with cyclic permutations (22)

x 0 xx 0 and 1xx and also for y and z (23)

We define quantities [in sections 6 and 7 we shall set 1c ].

).1( 0 σu uU (24)

Multiplying we get

VU )..1( 0 σu u ).1( 0 σv v ).1( 0 σw w W (24)

Where 2/1

/

c

ci

vu

vuvuw

and wvu )1( vu (25)

w of (24) and (25) agrees with (12). Multiplication of s is associative.

Therefore, is associative. Also

).1( 0 σv u ).1(. 0 σu u ).1( 0 σw w (26)

7. PAULI QUATERNION IN MATRIX FORM

Let s be matrices

10

010 ,

01

10x ,

0

0

i

iy and

10

01z (27)

zzyyxx uuu 00 .1.1 σu then becomes a 22 matrix. U is

now the matrix below

).1( 0 σu uU

zyx

yxz

u uiuu

iuuu

1

1 (28)

Corresponding to (24) we have



WVU (29)

The inverse matrix corresponding to (28) is

).1( 0

1σu uU (28)

1)(W

1)( VU 11 UV (30)

Note that the order is reversed as in (26). Lorentz transformation does not

have this property.

8. COMPATIBILITY BETWEEN RST AND DIRAC THEORY

We write Dirac’s relativistic equation as15

02 mccE pα (31)

For the purpose of comparison with (21) we set 0m and we write (31) as

00 pσcE (32)

We multiply on the left by pσ cE 0 to get

pσ cE 0 22

0 ppσ cEcE (33)

(33) will be correct if s of (32) have the properties of (22) and (23). To see the

correspondence to spin consider the product of type

Bσ 0b . CBBCσCBCσ icbbcc 0 (34)

In the presence of electromagnetic field16, in Dirac theory, we find term like14

Bσ . CBσCBCσ i (35)

We get (35) from (34) by setting 0 cb . The last term of (34) or (35) gives

spin17. Therefore, the imaginary cross term of (25) corresponds to spin18.

9. CONCLUSION

We have seen that failure of reciprocity and ambiguity are consequence of the

non-associativity of Einstein's law of addition of velocities.

Reciprocal Symmetric Transformation (RST) we are proposing is associative.

RST is Clifford algebraic, and spin is innate in it as in Dirac equation.



REFERENCES

[1] M. Ahmad. M. S. Alam. Physics Essays. 22 164 (2009) [2] Z. Oziewicz. Ternary relative velocity. Physical Interpretation of

Relativity Theory, Moscow 2007. www.worldnpa.org/pdf/abstracts/abstracts_133.pdf

[3] C. I. Mocanu. Some difficulties within the Framework of Relativistic Electrodynamics. Arch. Elektrotech. 69 97-110 (1986)

[4] A. Ungar. Foundations of Physics 19, No. 11 (1989) [5] A. Ungar. The Relativistic Velocity Composition Paradox and the

Thomas Rotation, Foundations of Physics 30, No. 2 (2000) [6] E. P. Wigner, “On unitary representations of the inhomogeneous

Lorentz group,” Ann. Math. 40, 149–204 (1939). [7] C. Moller. The Theory of Relativity. Clarendon Press, Oxford (1952) [8] A. Ungar. Foundations of Physics Letters 1 (1) (1988) 57–89 [9] Z. Oziewicz. Ternary relative velocity. Physical Interpretation of

Relativity Theory, Moscow 2007. www.worldnpa.org/pdf/abstracts/abstracts_133.pdf

[10] A. Ungar. Beyond The Einstein Addition Law and its Gyroscopic Thomas Precession. Kluwer Academic Publishers. New York, Boston, Dordrecht, London, Moscow. 2002

[11] A. Ungar. Thomas precession: a kinematic effect of the algebra of Einstein's velocity addition law. Comments on 'Deriving relativistic momentum and energy: II. Three-dimensional case'. Eur. J. Phys. 27 (2006) L17–L20 doi:10.1088/0143-0807/27/3/L02

[12] C. I. Mocanu. Some difficulties within the Framework of Relativistic Electrodynamics. Arch. Elektrotech. 69 97-110 (1986)

[13] C. I. Mocanu. Foundation of Physics Letters, 5, No. 5, 1992I.

[14] P. Rastall. Reviews of Modern Physics. July 1964. P. 820-832 [15] L.I. Schiff (1970). Quantum Mechanics. Mc Graw-Hill Company. Third

Edition. [16] K. Potamianos. Relativistic Electron Theory; The Dirac Equation.

Mathematical Physics Project. Universite’ Libre De Bruxelles [17] M. Ahmad. Reciprocal Symmetry and the Origin of Spin. arXiv:math-

ph/0702043v1 [18] M. Ahmad, M. S. Alam, M.O.G. Talukder .Comparison between Spin

and Rotation Properties of Lorentz Einstein and Reflection Symmetric Transformations. arXiv:math-ph/0701067

http://www.worldnpa.org/pdf/abstracts/abstracts_133.pdf

http://www.worldnpa.org/pdf/abstracts/abstracts_133.pdf

http://arxiv.org/find/math-ph,math/1/au:+Ahmad_M/0/1/0/all/0/1

http://arxiv.org/abs/math-ph/0702043v1

http://arxiv.org/abs/math-ph/0702043v1

http://arxiv.org/find/math-ph/1/au:+Ahmad_M/0/1/0/all/0/1

http://arxiv.org/find/math-ph/1/au:+Alam_M/0/1/0/all/0/1

http://arxiv.org/find/math-ph/1/au:+Talukder_M/0/1/0/all/0/1

http://arxiv.org/abs/math-ph/0701067



APPENDIX

Non-Associativity and Matrix Representation

Consider the non-associativity of Einstein's addition relation

)( uvy LL uvy LL )( (36)

We want to see the matrix representation corresponding to (36). Consider the

quantity y (y slash) defined by

ccy

cy y /1

)/(1

/1

2y

y

(37)

mcvy mL /1 m (38)

where m and m are given by

2/1

/11/

c

yy

Lvy

yy

v.yvy

vym2

and 2/1. cvym yv (39)

To go to matrix form, from the 4 vector u we form the column vectors

uc

u u

/

1

u (40)

u may, therefore, stand for the row vector as in (37) as well as for the column

as in (40). We hope this ambiguity will not be a problem. With y and v

columns, to execute (38), we define, from Ly , the matrix

T

2

T

/11/1/

/1

/

1

yy

y

y

yy

c

c

cy

yyyLyL

(41)

Then (38) gives

cy

c

c

vyyy

vyL/

1

/11/1/

/1

T

2

T

vyy

y

y

cm m

/

1

m (42)



We now go to calculate

")( uuvy LL (43)

and

^)( uumuvy LLL (44)

Using (41) we have from (43) (dropping y , v and u )

T

2

T

/11/1/

/1

yy

y

y

yc

c

yy

ccv

c

c

vv /

1

/

1

/11/1/

/1

2 u"uvv

v

v

T

T

(45)

Since matrix multiplication is associative, we may re-arrange and re-write (45)

as below

ccv

c

c

yc

c

vvyy /"

1

/

1

/11/1/

/1

/11/1/

/1

22 uuvv

v

v

yy

y

y

T

T

T

T

(46)

(45) and (46) both represent (43), since matrix multiplication is associative.

We now go to find (44). Using (42) and (41) we have

ccm

c

cum u

mmumL

/^

1

/

1

/11/1/

/1

^T

2

T

uumm

m

m

(47)

where umu L^ corresponds to the right hand side of (36). ^" uu since

T

2

T

/11/1/

/1

mm

m

m

mc

c

mmmm

T

2

T

/11/1/

/1

yy

y

y

yc

c

yyvy

T

2

T

/11/1/

/1

vv

v

v

vc

c

vv (48)

Inequality (48) is responsible for non-associativity ^" uu .

----0----



Rciprocal Property of Different Types of

Lorentz Transformations

Atikur Rahman Baizid1, 2

& Md. Shah Alam2

1Department of Business Administration, Leading University, Sylhet, Bangladesh 2Department of Physics, Shahjalal University of Science and Technology, Sylhet, Bangladesh

ABSTRACT

Lorentz transformation is the basic tool for the study of

Relativistic mechanics. There are different types of Lorentz

transformations such as Special, Most general, Mixed number,

Geometric product, and Quaternion Lorentz transformations. In

this paper we have studied the reciprocal property of the above

mentioned Lorentz transformations.

Key Words: Lorentz transformation and Reciprocal Property.

PACS: 03.30. + p

1 INTRODUCTION

1.1 Special Lorentz transformation

Figure 1: The frame S is at rest and the frame S is moving with respect to

S with uniform velocity V along x-axis.



Consider two inertial frames of reference S and S where the frame S is at

rest and the frame S is moving along X-axis with velocity V with respect to

S frame. The space and time coordinates of S and S are (x, y, z, t) and (x,

y, z, t) respectively. Then the relation between the coordinates of S and

S is called special Lorentz transformation which can be written as

2

2

2

2

2

1

,,,

1c

V

c

xVt

tzzyy

c

V

Vtxx

(1)

and

)2(

1

,,,

12

2

2

2

2

c

V

c

xVt

tzzyy

c

V

tVxx

1.2 Most General Lorentz Transformation

Figure -2: The frame S is at rest and the frame S is moving with respect to

S with uniform velocity V

along any arbitrary direction.



When the velocity V

of S with respect to S is not along X-axis i.e. the

velocity V

has three components Vx, Vy and Vz then the relation between the

coordinates of S and S is called Most general Lorentz Transformation

which can be written as [2]

2

2

1

2

2

2/1

2

22/1

2

2

2

.1

111.

c

XVt

c

Vt

c

Vt

c

V

V

VXVXX

(3)

And

2

2

1

2

2

2/1

2

22/1

2

2

2

.1

111.

c

XVt

c

Vt

c

Vt

c

V

V

VXVXX

(4)

Where VV

1.3 Mixed Number Lorentz Transformation

Mixed number [3-7] is the sum of a scalar x and a vector A

.

i.e. Ax

The product of two mixed numbers is defined as [3-7]

BAiAyBxBAxyByAx

. (5)

Taking 0 yx , we get from equation (5)

BAiBABA

.

(6)

This product is called mixed product [7] and the symbol is chosen for it. We can

generate a type of most general Lorentz transformation using this mixed product.



Using 1c and

2

2

1

1

c

V

in equations (1) and (2), we can write

Vxtt

Vtxx

(7)

And

xVtt

tVxx

(8)

Now from equation (7), we get

xtVxtxtor

VtxVxtxt

, (9)

Using (t' + x') = p' and (t + x) = p in equation (9), we can write

pVpp (10)

Now from equation (8), we get

xtVxtxtor

tVxxVtxt

, (11)

Using pxt and pxt in equation (11), we can write

Vppp (12)

In the case of the most general Lorentz transformation, the velocity V

of S

with respect to S is not along the X-axis, i.e., the velocity V

has three

components, Vx, Vy, and Vz. Let in this case Z and Z' be the space parts in the

S and S frames respectively. In this case equation (10) can be written as

VPPP

(13)

Where, ZtP and ZtP

are two mixed numbers [5].



Therefore,

VZtZtZt

VZtZtZtor

0, (14)

Using equation (6), we can write

VZiVtVZVZt

.0 (15)

From (14) and (15) we get,

VZiVtZVZtZtor

VZiVtVZZtZt

.,

.. (16)

Equating the scalar and vector part from the both sides of equation (16), we

can write

VZiVtZZ

VZtt

.

(17)

Similarly, we can show that

VZiVtZZ

VZtt

.

(18)

Equations (17) and (18) are the mixed-number Lorentz transformation.

1.4 Geometric Product Lorentz Transformation

Bidyut Kumar Datta and his co-workers defined the geometric product of

vectors as [8, 9]

BABABA

.

where A

and B

are two vectors. We use the symbol instead of the symbol

. Therefore, the geometric product of vectors can be written as

BABABA

. (19)



We can also generate a type of most general Lorentz transformation using this

geometric product.

In this case the velocity V

of S with respect to S has also three

components, Vx, Vy, and Vz as the most general Lorentz transformation. Let in

this case Z

and Z be the space parts in the S and S frames respectively.

Equation (14) is

VZVtZtZtor

VZtZtZt

,

0 (20)

From equation (19) the geometric product of two vectors is

BABABA

.

Therefore, we can write

VZVZVZ

. (21)

From (20) and (21) we get,

VZVtZVZtZtor

VZVtVZZtZt

.,

. (22)

Equating the scalar and vector part from the both sides of equation (22), we can write

VZVtZZ

VZtt

.

(23)


VZVtZZ

VZtt

.

(24)

Equations (23) and (24) are the geometric product Lorentz transformation.

1.5 Quaternion Lorentz Transformation

The quaternion can also be written as the sum of a scalar and a vector [10]

i.e., AaA



The multiplication of any two quaternions AaA

and BaB

is

given by [10-12]

BAAbBaBAabBbAaBA

. (25)

Taking a = b = 0, we get from equation (25)

BABABA

. (26)

This product is called quaternion product. We can also generate a type of

most general Lorentz transformation using this quaternion product.

In this case the velocity V

of S with respect to S has also three

components, Vx, Vy, and Vz as the most general Lorentz transformation. Let in

this case Z

and Z be the space parts in the S and S frames respectively.

In this case, equation (14) is

VZtZtZt

0

Here, Zt , Zt

and V

0 are three quaternions [9]

According to the product of two quaternions [10-12] we can write

VZVtVZVZt

.0 (27)

From (27) and (14) we get

VZVtZVZtZtor

VZVtVZZtZt

.,

.

Equating the scalar and vector part from the both sides of equation (28), we

can write

VZVtZZ

VZtt

.

(29)




VZVtZZ

VZtt

.

(30)

Equations (29) and (30) are the quaternion Lorentz transformation.

2 RECIPROCAL PROPERTY OF DIFFERENT LORENTZ TRANSFORMATIONS

2.1 Reciprocal property of Special Lorentz Transformation

The velocity addition formula for the Special Lorentz transformation can be

written as

2cVxt

Vtx

t

xW

(31)

Dividing numerator and denominator of equation by t we get

21c

UV

VUW

UV

VUWor

1, in unit of c (32)

If we replace U by P where 1PU then W will be change to W where

PV

VPW

1

Reciprocal property demands that if 1PU then

111

PV

VP

UV

VUWW

Now,

2

2

2

2

1

1

1

11

VUVPV

VVPUV

UPVUVPV

VVPUVUP

PV

VP

UV

VUWW



So,

1WW (33)

Consequently, the Special Lorentz transformation satisfies the reciprocal property.

2.2 Reciprocal property of Most General Lorentz Transformation

From the transformation equation of addition of velocities of Most general

Lorentz transformation [2] we have

2

2

.

1.

c

XVt

tV

VXVX

t

XW

Dividing numerator and denominator by t we get

UV

VVU

VU

W

.1

1.

2

in unit of c.

(34)

Where

2

2

1

1

cV

If we replace U

by P

where 1. PU

then W

will be changed into W

where

PV

VVP

VP

W

.1

1.

2

Reciprocal property demands that if 1. PU

then

1

.1

1.

..1

1.

.

22

PV

VVP

VP

UV

VVU

VU

WW

Now,



PV

VVP

VP

UV

VVU

VU

WW

.1

1.

..1

1.

.

22

PVUVPVUV

VVP

VVU

PVUVPVUV

VVU

PVV

VPVUPU

....1

1.

1.

....1

1.

.1.

..

2

22

2

22

So,

1. WW

(35)

Consequently, the Most general Lorentz transformation does not satisfy the

Reciprocal property [13].

2.3 Reciprocal Property of Mixed Number Lorentz Transformation

From the transformation equation of addition of velocities of mixed number

Lorentz Transformation [3] we have

VZt

VZiVtZ

t

ZW

.

VZt

VZiVtZ

.

in unit of c. (36)

Dividing numerator and denominator of equation (36) by t we get

)37(

1. VU

VUiVU

tVZ

tt

tVZi

tVt

tZ

W

If we replace U

by P

where 1. PU

then W


where



VP

VPiVPW

1


then

11

.1

.

VP

VPiVP

VU

VUiVUWW

Now,

VPVU

VPiVPVUiVU

VP

VPiVP

VU

VUiVUWW

11

.

1.

1.

VPVU

VPiVPVUiVPiVPVVPiVPU

11

...

[ Since 1. PU

, 0. VVU

, 0. PVU

, 0.. VPUVPV

]

VPVU

VPVUVPVVU

11

..1 2

[ Since BACCBA

.. ]

VPVU

PVUVVPVVU

11

..1 2

[ Since CBACABCBA

.. ]

VPVU

PVVUVVPVVU

VPVU

PVUPUVVVPVVU

11

...1

11

...1

22

2

[Since 1. PU

and PVVP

.. ]

VPVU

VPVUVPVU

11

...1



So,

111

11.

VPVU

VPVUWW

(38)

Similarly

If we replace V

by Q

where 1. QV

then W


where

1. WW

Consequently, the Mixed number Lorentz transformation satisfies the

reciprocal property [13].

2.4 Reciprocal Property of Geometric Product Lorentz Transformation

From the transformation equations of addition of velocities of Geometric

product Lorentz Transformation [3] we have,

VZt

VZVtZ

t

ZW

.

VZt

VZVtZ

.

in unit of c.

(39)


VU

VUVU

tVZ

tt

tVZ

tVt

tZ

W

1.

(40)

If we replace U

by P

where 1. PU

then W


where

VP

VPVPW

1


then

11

.1

.

VP

VPVP

VU

VUVUWW



Now, VP

VPVP

VU

VUVUWW

1.

1.

VPVU

VPVPVUVU

11

.

VPVU

VPVPVUVPVPVVPVPU

11

...

[Since 1. PU

, 0. VVU

, 0. PVU

, 0.. VPUVPV

]

VPVU

VPVUVPVVU

11

..1 2

[Since BACCBA

.. ]

VPVU

PVUVVPVVU

11

..1 2

[ Since CBACABCBA

.. ]

VPVU

PVVUVVPVVU

VPVU

PVUPUVVVPVVU

11

...1

11

...1

22

2

[Since 1. PU

and PVVP

.. ]

VPVU

VPVUVVPVU

11

..2.1 2

So,

1. WW

(41)

Consequently, the Geometric product Lorentz transformation does not

satisfy the reciprocal property [13].



2.5 Reciprocal Property of Quaternion Lorentz Transformation

From the transformation equation of addition of velocities of Quaternion

Lorentz transformation [3] we have

VZt

VZVtZ

t

ZW

.

VZt

VZVtZ

.

in unit of c.

(42)


VU

VUVU

tVZ

tt

tVZ

tVt

tZ

W

1.

(43)

If we replace U

by P

where 1. PU

then W


where

VP

VPVPW

1


then

11

.1

.

VP

VPVP

VU

VUVUWW

Now,

VPVU

VPVPVUVU

VP

VPVP

VU

VUVUWW

11

.

1.

1.

VPVU

VPVPVUVPVPVVPVPU

11

...

[Since 1. PU

, 0. VVU

, 0. PVU

, 0.. VPUVPV

]

VPVU

VPVUVPVVU

11

..1 2



[ Since BACCBA

.. ]

VPVU

PVUVVPVVU

11

..1 2

[ Since CBACABCBA

.. ]

VPVU

PVVUVVPVVU

VPVU

PVUPUVVVPVVU

11

...1

11

...1

22

2

[ Since 1. PU

and PVVP

.. ]

VPVU

VPVUVVPVU

11

..2.1 2

So, 1. WW

(44)

Consequently, the Quaternion Lorentz transformation does not satisfy the

reciprocal property [13].

3 CONCLUSION

The reciprocal property of different Lorentz transformations has been

discussed and we have obtained that

1. Special and Mixed number Lorentz transformations satisfy the reciprocal

property.

2. Most general, Geometric product and Quaternion Lorentz transformations

do not satisfy the reciprocal property.



REFERENCES

[1] Resnick, Robert. Introduction to special relativity, Wiley Eastern Limited New age international limited - 1994

[2] Moller C. The Theory of Relativity, Oxford University Press-1972 [3] Md. Shah Alam and Khurshida Begum. Different Types of Lorentz

Transformations, Jahangirnagar Physics studies-2009. [4] Alam, M.S. 2000. Study of Mixed Number. Proc. Pakistan Acad. of Sci.

37(1): 119-122. [5] Alam, M.S. 2001. Mixed product of vectors, Journal of Theoretics, 3(4).

http://www.journaloftheoretics.com/ [6] Alam, M.S. 2003. Comparative study of mixed product and quaternion

product, Indian J. Physics A 77: 47-49. [7] Alam, M.S. 2001. Mixed product of vectors, Journal of Theoretics, 3(4).

http://www.journaloftheoretics.com/ [8] Datta, B.K., De Sabbata V. and Ronchetti L. 1998. Quantization of gravity

in real space time, Il Nuovo Cimento, 113B. [9] Datta, B.K., Datta R. 1998. Einstein field equations in spinor formalism,

Foundations of Physics letters, 11, 1. [10] Kyrala, A. 1967. Theoretical Physics, W.B. Saunders Company.

Philadelphia & London, Toppan Company Limited. Tokyo, Japan. [11] http://mathworld.wolfram.com/Quaternion.html [12] http://www.cs.appstate.edu/~sjg/class/3110/mathfestalg2000/quate

rnions1.html [13] Atikur Rahman Baizid, Md. Shah Alam, “Properties of different types of

Lorentz transformations” American Journal of Mathematics and Statistics 2013, 3(3): 105-123

----0----

http://www.journaloftheoretics.com/

http://www.journaloftheoretics.com/

http://mathworld.wolfram.com/Quaternion.html

http://www.cs.appstate.edu/~sjg/class/3110/mathfestalg2000/quaternions1.html

http://www.cs.appstate.edu/~sjg/class/3110/mathfestalg2000/quaternions1.html



Armenian Theory of Special Relativity

Robert Nazaryan1 & Haik Nazaryan

2

1Physics Department, Yerevan State University, Yerevan 0025, Armenia

2Physics and Astronomy Department, California State University, Northridge, USA

ABSTRACT

By using the principle of relativity (first postulate), together with new

defined nature of the universal speed (our second postulate) and

homogeneity of time-space (our third postulate), we derive the most

general transformation equations of relativity in one dimensional

space. According to our new second postulate, the universal (not

limited) speed c in Armenian Theory of Special Relativity is not the

actual speed of light but it is the speed of time which is the same in all

inertial systems. Our third postulate: the homogeneity of time-space is

necessary to furnish linear transformation equations. We also state that

there is no need to postulate the isotropy of time-space. Our article is

the accumulation of all efforts from physicists to fix the Lorentz

transformation equations and build correct and more general

transformation equations of relativity which obey the rules of logic

and fundamental group laws without internal philosophical and

physical inconsistencies.

PACS: 03.30. p, 04.20.Fy

Keywords: Relativity, Relativistic, Transformations

INTRODUCTION

On the basis of the previous works of different authors,[2, 3, 4, 5] a sense of hope was

developed that it is possible to build a general theory of Special Relativity without

using light phenomena and its velocity as an invariant limited speed of nature.



The authors also explore the possibility to discard the postulate of isotropy

time-space.[1, 4]

In the last five decades, physicists gave special attention and made numerous

attempts to construct a theory of Special Relativity from more general

considerations, using abstract and pure mathematical approaches rather than

relying on so called experimental facts.[6]

After many years of research we came to the conclusion that previous authors

did not get satisfactory solutions and they failed to build the most general

transformation equations of Special Relativity even in one dimensional space,

because they did not properly define the universal invariant velocity and did

not fully deploy the properties of anisotropic time-space.

However, it is our pleasure to inform the scientific community that we have

succeeded to build a mathematically solid theory which is an unambiguous

generalization of Special Relativity in one dimensional space.

The principle of relativity is the core of the theory relativity and it requires

that the inverse time-space transformations between two inertial systems

assume the same functional forms as the original (direct) transformations. The

principle of homogeneity of time-space is also necessary to furnish linear

time-space transformations respect to time and space.[2, 3, 5]

There is also no need to use the principle of isotropy time-space, which is the

key to our success.

To build the most general theory of Special Relativity in one physical

dimension, we use the following three postulates:

1. All physical laws have the same mathematical functional forms in all inertial systems.

(1) 2. There exists a universal, not limited and invariant boundary

speed c, which is the speed of time. 3. In all inertial systems time and space are homogeneous

(Special Relativity).



Besides the postulates (1), for simplicity purposes we also need to use the

following initial conditions as well:

When

(2) Then origins of all inertial systems coincide each other,

Therefore

Because of the first and third postulates (1), time and space transformations

between two inertial systems are linear:

Direct transformations

and

Inverse transformations

(3)

ARMENIAN RELATIVISTIC KINEMATICS

Using our postulates (1) with the initial conditions (2) and implementing them

into the general form of transformation equations (3), we finally get the most

general transformation equations in one physical dimension, which we call -

Armenian transformation equations. Armenian transformation equations,

contrary to the Lorentz transformation equations, has two new constants (s

and g) which characterize anisotropy and homogeneity of time-space. Lorentz

transformation equations and all other formulas can be obtained from the

Armenian Theory of Special Relativity by substituting s = 0 and g = -1.


and


(4)

Relations between reciprocal and direct relative velocities are:

(5)



Armenian gamma functions for direct and reciprocal relative velocities, with

Armenian subscript letter are:

(6)

Relations between reciprocal and direct Armenian gamma functions are:

(7)

Armenian invariant interval (we are using Armenian letter ) has the

following expression:

(8)

Armenian formulas of time, length and mass changes in inertial

systems are:

(9)

Transformations formulas for velocities (addition and subtraction) and

Armenian gamma functions are.

(10)

If we in the K inertial system use the following notations for mirror reflection

of time and space coordinates:

(11)



Then the Armenian relation between reflected and normal time-

space coordinates of the same event is:

(12)

The ranges of velocity w for the free moving particle, depending on the

domains of time-space constants s and g, are:

(13)

Where is the fixed velocity value for g < 0, which equals to:

(14)

Table (13) shows that there exists four different and distinguished range of

velocities w for free moving particle, which are produced by different

domains of time-space constants s and g as shown in the table below:

(15)

Table (15) shows us that each distinct domains of (s, g) time-space constants

corresponds to its own unique range of velocities, so therefore we can suggest

that each one of them represents one of the four fundamental forces of nature

with different flavours (depending on domains of s).

ARMENIAN RELATIVISTIC DYNAMICS

Armenian formulas for acceleration transformations between

inertial systems are:

(16)



Armenian acceleration formula, which is invariant for given movement, we define as:

(17)

Armenian relativistic Lagrangian function for free moving particle with

velocity w is:

(18)

Armenian relativistic energy and momentum formulas for free moving

particle with velocity w are:

(19)

First approximation of the Armenian relativistic energy and momentum

formulas (19) is:

(20)

Where we denote as the Armenian rest mass, which equals to:

(21)

Armenian momentum formula for rest particle (w = 0), which is a very new

and bizarre result, is:

(22)

From (22) we obtain Armenian dark energy and dark mass formulas, with

Armenian subscript letter and they are:

(23)

Armenian energy and momentum transformation equations are:




and


(24)

From (24) we get the following invariant Armenian relation :

(25)

Armenian force components in inertial systems are (see full article):

(26)

From (26) it follows that Armenian force space components are also invariant:

(27)

As you can see (15), we are a few steps away to construct a unified field theory, which can change the face of modern physics as we know it now. But the final stage of the construction will come after we finish the Armenian Theory of Special Relativity in three dimensions. You can get our full article with all derivations, proofs and other amazing formulas (in Armenian language) via E-mail.

REFERENCES

[1] Edwards W F, 1963, Am. J. Phys. 31, 482-90. [2] Jean-Marc Levy-Leblond, 1976, Am. J. Phys. Vol. 44, No. 3. [3] Vittorio Berzi and Vittorio Gorini, 1969, J. Math. Phys., Vol. 10, No. 8. [4] Jian Qi Shen, Lorentz, Edwards transformations and the principle of

permutation invariance, (China, 2008). [5] Shan Gao, Relativiti without light: a further suggestion, (University of Sydney). [6] G. Stephenson and C. W. Kilminster, Special relativity for Physicists,

(Longmans, London, 1958), Ch. 1. ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯

© - USA Copyright Office Registration Numbers: TXu 1-843-370 and TXu 1-862-255 - To whom correspondence shoud be addressed: [email protected] © "Armenian Theory of Special Relativity - One Dimensional Movement" Letter In this letter we only introduce, without proof, our new results such as: Armenian transformation equations, Armenian gamma functions, Armenian interval, Armenian Lagrangian function, Armenian energy and momentum formulas, Armenian momentum formula for rest particle, Armenian dark energy formula, Armenian transformation equations for energy and momentum, Armenian mass, acceleration and force formulas. All new physical quantities has Armenian subscript letter



Upper Limit of the Age of the Universe with

Cosmological Constant

Haradhan Kumar Mohajan*2

Premier University, Chittagong, Bangladesh

ABSTRACT

The Friedmann, Robertson-Walker universe is based on the

assumption that the universe is exactly homogeneous and isotropic.

This model expresses that there is an all encompassing big bang

singularity in the past as the origin from which the universe

emerges in a very hot phase and continues its expansion as it cools.

Here homogeneous and isotropic assumptions of the observed

universe are not strictly followed to calculate the present age of the

universe. Einstein equation plays an important role in cosmology to

determine the present age of the universe. The determination of

present age and density of the universe are two very important

issues in cosmology, as they determine the future evolution and the

nature of the universe. An attempt has been taken here to find the

upper limit of the age of the universe with cosmological constant.

PACs: 98.80-Bp: Origin and formation of the universe

Keywords: Einstein equation, Geodesic, Hubble constant, Space-time manifold, Universe

1 INTRODUCTION

The Friedmann, Robertson-Walker (FRW) model indicates that the universe is

exactly homogeneous and isotropic around us. Even though the universe is

* [email protected]

mailto:[email protected]



clearly inhomogeneous at the local scales of stars and cluster of stars, it is

generally argued that an overall homogeneity will be achieved only at a large

enough scale of about 14 billion light years or so (undetermined scale), in a

statistical sense only. Here homogeneous means the universe is roughly same

at all spatial points and the matter is uniformly distributed all over the space

i.e., no part of the universe can be distinguished from any other, and isotropy

means that all spatial directions are equivalent. But there is no fundamental

physical justification that isotropy and homogeneity are strictly obeyed in all

regions of space and at all epochs of time. The global hyperbolicity space-time

gives homogeneous, inhomogeneous, isotropic and anisotropic universe

(Mohajan 2013d).

The development in modern astrophysics states that dark matter surrounds

the bright stars and galaxies, and constitutes the dominant material content of

our universe. At present time, the evidence of the dark matter seems to be (a)

low mass, faint stars (b) massive black holes and (c) massive neutrinos, axions

or particles predicted by super symmetry. Of these axions are the most

popular candidate as it seems to fit best the astrophysical requirements.

Axions could have been produced in the early universe, and if the axions have

small, non-zero rest masses, and then they would be gravitationally dominant

today. The existence of axions was originally invoked when Pecei and Quinn

(1977) explained the property of C–P (charge and parity) conservation of

strong interactions.

To discuss cosmological models we need the knowledge of general relativity

and we have discussed briefly which portion of it is related to our study. We

have highlighted on Schwarzschild geometry and FRW model and

Raychaudhury equation (Mohajan 2013a, b, c, d, e). In this paper we have

determined the age of the universe with a cosmological constant.



2 A BRIEF DISCUSSION OF GENERAL RELATIVITY

A contravariant vector 3,2,1,0 A and a covariant vector A in

coordinates x to x transform as follows:

A

x

xA

,

Ax

xA

. (1)

Similarly a mixed tensor of rank three can be transformed as,

Ax

x

x

x

x

xA

(2)

where we have used summation convention.

A metric is defined as;

dxdxgds 2

(3)

where g is an indefinite metric in the sense that the magnitude of non-zero

vector could be either positive, negative or zero. Then any vector pTX is

called timelike, null or spacelike if (Joshi 1996);

0, ,0, ,0, XXgXXgXXg . (4)

An indefinite metric divides the vectors in pT into three disjoint classes,

namely the timelike, null or spacelike vectors. The null vectors form a cone in

the tangent space pT which separates the timelike vectors from spaelike

vectors.

The covariant differentiations of vectors are defined as;

AAA ,; (5a)

AAA ,; (5b)

where semi-colon denotes the covariant differentiation and coma denotes the

partial differentiation.



By (5b) we can write;

ARAA ;;;; , (6)

where

;;R (6a)

is a tensor of rank four and called Riemann curvature tensor. From (6) we

observe that the curvature tensor components are expressed in terms of the

metric tensor and its second derivatives. From (6a) we get;

0R

. (7)

Taking inner product of both sides of (6a) with g one gets covariant

curvature tensor;

xx

g

xx

g

xx

g

xx

gR

2222

2

1+

g . (8)

Contraction of curvature tensor (6) gives Ricci tensor;

RgR . (9)

Further contraction of (9) gives Ricci scalar;

RgR ˆ

. (10)

From which one gets Einstein tensor as;

RRG

2

1

(11)

where 0;

GGdiv .

The space-time gM , is said to have a flat connection iff;

0R

. (12)

This is necessary and sufficient condition for a vector at a point p to remain

unaltered after parallel transported along an arbitrary closed curve through p.



This is because all such curves can be shrunk to zero, in which case the space-

time is simply connected (Hawking and Ellis 1973).

The energy momentum tensor T is defined as;

uuT 0 (13)

where 0 is the proper density of matter, and if there is no pressure. A perfect

fluid is characterized by pressure xpp , then;

pguupT . (14)

The principle of local conservation of energy and momentum states that;

0; T

. (15)

According to the Newton’s law of gravitation, the field equations in the

presence of matter are;

G 42 (16)

where is the gravitational potential, is the scalar density of matter, G is

the gravitational constant.

If classical equation (16) is generalized for the relative theory of gravitation

then this must be expressed as a tensor equation satisfying following

conditions;

the tensor equation should not contain derivatives of g higher than

the second order,

it must be linear in the second differential coefficients, and

its covariant divergence must vanish identically.

The most appropriate tensor of the form required is the Einstein’s tensor

which is given by (16); then Einstein’s field equation can be written as;

T

c

GRgR

4

8

2

1

. (17)



where 21311 skgm10673.6 G is the gravitational constant and 810c m/s

is the velocity of light. Einstein introduced a cosmological constant 0 for

static universe solutions as;

T

c

GgRgR

4

8

2

1

. (18)

In relativistic unit G = c = 1, hence in relativistic units (18) becomes;

TRgR 82

1

. (19)

It is clear that divergence of both sides of (18) and (19) is zero. For empty

space 0T then gR , then;

0R for 0 (20)

which is Einstein’s law of gravitation for empty space.

3 SCHWARZSCHILD SPACE-TIME MANIFOLD

The Schwarzschild metric which represents the outside metric for a star is

given by;

2 2sin2221

212

212 ddrdr

r

mdt

r

mds

(21)

If 0r is the boundary of a star then 0rr gives the outside metric as in (21). If

there is no surface, (21) represents a highly collapsed object viz. a black hole of

mass m (will be discussed later). The metric (21) has singularities at r = 0 and r

= 2m, so it represents patches mr 20 or rm2 . If we consider the

patches mr 20 then it is seen that as r tends to zero, the curvature scalar,

6

248

r

mRR

tends to and it follows that r = 0 is a genuine curvature singularity i.e.,

space-time curvature components tend to infinity (Mohajan 2013a, e).



At r = 2m the curvature scalars are well behaved at this point, so it is a

singularity due to inappropriate choice of coordinates. The maximal extension

of the manifold (21) with rm2 was obtained by Kruskal (1960) and

Szekeres (1960). We now discuss about this, which uses suitably defined

advanced and retarded null coordinates. For null geodesics (21) takes the

form,

2

1

2 21

21 dr

r

mdt

r

m

drmr

rt

2

1

2log2

m

rmr

constant

*rt +constant. (22)

The null coordinates u and v are defined by;

*rtu , *rtv

2

* uvr

(23)

Now;

2

1

2 21

21 dr

r

mdt

r

m

2*2 12

2drdt

m

r

r

m

dudveer

mm

uvm

r

2

22

. 2 22222

drdudveer

mds m

uvm

r

(24)

Here ;



2222 sin ddd .

As mr 2 corresponds to u or v , we define new coordinates U

and V by;

mu

eU 4 , m

v

eV 4

dudvem

dUdV muv

16

14

2

.

Hence (24) becomes;

. 32 222

32

drdUdVe

r

mds m

r

(25)

Hence there is no singularity at U = 0 or V = 0 which corresponds to the value

at r = 2m.

Let us take a final transformation by;

2

VUT

and

2

UVX

, then (25) becomes;

22222

32 32

drdXdTer

mds m

r

(26)

which is Kruskal-Szekeres form of Schwarzschild metric. Then transformation

rt, to XT , becomes;

1

2 2222

m

reeUVTX mrmuv

(27)

m

t

X

T

4tanh

X

Tmt 1tanh4

. (28)

From (28), 0r gives 122 TX . The physical singularity at r = 0 gives

21

2 1 TX , and we observe that there is no singularity now at r = 2m.



4 FRIEDMANN, ROBERTSON–WALKER (FRW) MODEL

The FRW model plays an important role in Cosmology. This model is

established on the basis of the homogeneity and isotropy of the universe as

described above. The current observations give a strong motivation for the

adoption of the cosmological principle stating that at large scales the universe

is homogeneous and isotropic and, hence, its large-scale structure is well

described by the FRW metric. The FRW geometries are related to the high

symmetry of these backgrounds. Due this symmetry numerous physical

problems are exactly solvable and a better understanding of physical effects in

FRW models could serve as a handle to deal with more complicated

geometries.

In ,,, rt coordinates the Robertson-Walker line element is given by;

2 2sin2

1 2

2

2222 ddr

kr

drtSdtds

(29)

where k is a constant which denotes the spatial curvature of the three-space

and could be normalized to the values +1, 0, –1. When k = 0 the three-space is

flat and (29) is called Einstein de-Sitter static model, when k = +1 and k = –1

the three-space are of positive and negative constant curvature; these

incorporate the closed and open Friedmann models respectively (Figure 1).

Let us assume the matter content of the universe as a perfect fluid then by (14)

and (15), solving (29) we get;

0343

pS

S

, and (30)

03

83

22

2

S

k

S

S

(31)



where we have considered 0 . If 0 and 0p then 0S . So S =

constant and 0S indicates the universe must be expanding, and 0S

indicates contracting universe. The observations by Hubble of the red-shifts of

the galaxies were interpreted by him as implying that all of them are receding

from us with a velocity proportional to their distances from us that is why the

universe is expanding. For expanding universe 0S , so by (30) and (31) we

get 0S . Hence S is a decreasing function and at earlier times the universe

must be expanding at a faster rate as compared to the present rate of

expansion. But if the expansion be constant rate as like the present expansion

rate at all times then,

Figure 1: The behavior of the curve S(t) for the three values k = –1, 0, +1; the

time 0tt is the present time and 1tt

is the time when S(t) reaches zero

again for k = +1 .

0

0

HS

S

tt

. (32)

Now 1

0

H implies a global upper limit for the age of any type of Friedmann

models. So the age of the universe will be less than 1

0

H . The quantity 0H is



called Hubble constant and at any given epoch it measures the rate of

expansion of the universe. By observation 0H has a value somewhere in the

range of 50 to 120 kms–1Mpc–1.

At S = 0, the entire three-surface shrinks to zero volume and the densities and

curvatures grow to infinity. Hence by FRW models there is a singularity at a

finite time in the past. This curvature singularity is called the big bang. Now

we have a basic qualitative difference between the Schwarzschild singularity

and that occurring in FRW models. The Schwarzschild singularity could be

the final result of a gravitationally collapsing of massive star. However FRW

singularity must be interpreted as the catastrophic event from which the

entire universe emerged and where all the known physical laws breakdown

in such a way that we cannot know what was before this singularity. The

existence of a strong curvature singularity at t = 0 indicated by the FRW

models imply the existence of a very hot, dense and radiation dominated

region in the very early phase of the evolution of the universe (Islam 2002,

Hawking and Ellis 1973).

5 RAYCHAUDHURI EQUATION AND GRAVITATIONAL FOCUSING

Now let us consider the Raychaudhuri equation (Raychaudhuri 1955), (for

null case similar equation holds with 3

1 is replaced by

2

1)

222 223

1

VVR

dt

d

(33)

which describes the rate of change of the volume expansion as one moves

along the timelike geodesic curves in the congruence (Mohajan 2013a). Here

0 is expansion, 0 is shear and is rotation tensors which are

defined as follows:

hhV ;



h3

1

;Vhh

.

By Einstein equation (19) we can write (Joshi 1996, Kar and SenGupta 2007);

TVVTVVR

2

18

. (34)

The term VVT is the energy density measured by a timelike observer

with the unit tangent four velocity of the observer, V . In classical physics;

0 VVT

. (35)

Such an assumption is called the weak energy condition (the matter density

observed by the corresponding observers is always non-negative i.e. 0

and 0 p ). Now let us consider (Joshi 2013);

TVVT2

1

. (36)

Such an assumption is called the strong energy condition (the trace of the tidal

tensor measured by the corresponding observers is always non-negative i.e.,

0 p and 03 p ) which implies from (34) for all timelike vectors V ,

0 VVR

. (37)

Both the strong and weak energy condition will be valid for perfect fluid

provided energy density 0 and there are no large negative pressures.

An additional energy condition required often by the singularity theorems is

the dominant energy condition which states that in addition to the weak

energy condition, the pressure of the medium must not exceed the energy

density (i.e., p ). The dominant energy condition also states that

VT is

non-spacelike and future-directed. Equation (37) implies that the effect of



matter on space-time curvature causes a focusing effect in the congruence of

timelike geodesics due to gravitational attraction.

Let us suppose is a timelike geodesic. Then two points p and q along are

called conjugate points if there exists Jacobi field along which is not

identically zero but vanishes at p and q. If infinitesimally nearby null

geodesics of the congruence meet again at some other point q in future, then p

and q are called conjugate to each other, where at q (Figure 2). We

can define conjugate point another way as follows (Mohajan 2013c):

Let S be a smooth spacelike hypersurface in M which is an embedded three

dimensional sub-manifold. Consider a congruence of timelike geodesics

orthogonal to S. Then a point p along a timelike geodesic of the congruence

is called conjugate to S along if there exists a Jacobi vector field along

which is non-zero at S but vanishes at p, which means that there are two

infinitesimally nearby geodesics orthogonal to S which intersect at p (Figure

3). Again we face

Figure 2: Infinitesimally separated null geodesics cross at p and q, which are conjugate points along the curve .

equivalent condition that the expansion for the congruence orthogonal to S

tends to at p. If V denotes the normal to S, then the extrinsic

curvature of S is defined as;



V (38)

which is evaluated at S.

So, 0

VV . Since S is orthogonal to the congruence this implies

0 , hence .

The trace of the extrinsic curvature, is denoted by , and is given by;

h (39)

where is expansion of the congruence orthogonal to S.

Figure 3: A point p conjugate to the spacelike hypersurface S. The timelike geodesic is orthogonal to S, which is intersected by another infinitesimally nearby timelike geodesic.

Let us consider the situation when the space-time satisfies the strong energy

condition and the congruence of timelike geodesics is hypersurface

orthogonal, then 0 implies 02 then (33) gives;

3

2

d

d

(40)

which means that the volume expansion parameter must be necessarily

decreasing along the timelike geodesics. Let us denote 0 as initial expansion

then integrating (40) we get;



c3

1

. (41)

Initially 0 then (41) becomes;

0

1

3

1

. (42)

By (42) we confirm that if the congruence is initially converging and 0 is

negative then within a proper time distance 0

3

, provided can

be extended to that value of the proper time.

Now suppose 0 , and further, it is bounded above by a negative value

m ax , so all the timelike curves of the congruence orthogonal to S will contain

a point conjugate to S within a proper time distance m ax

3

, provided the

geodesics can be extended to that value of the proper time.

By the above results the existence of space-time singularities in the form of

geodesic incompleteness. Now we introduce the gravitational focusing effect

for the congruence of null geodesics orthogonal to a spacelike two surfaces as

follows (Joshi 1996):

Let M be a space-time satisfying 0 KKR for all null vectors K and

be a null geodesic of the congruence. If the convergence of null geodesic

from some point p is 00 at some point q along , then within an

affine distance less than or equal to 0

2

from q the null geodesic will

contain a point conjugate to p, provided that it can be extended to that affine

distance.



6 UPPER LIMIT OF THE AGE OF THE UNIVERSE

6.1 MATHEMATICAL FORMULATION

For 0 (33) becomes;

(43)

A timelike geodesic t will be orthogonal to 0S provided the expansion

along t satisfies i

i at 0S , where ij is the second fundamental form

of the spacelike hypersurface. Let dt

dz

z.

1 with 3xz then (43) becomes

(Joshi 1996);

02

2

xtHds

xd (44)

where 223

1

VVRtH .

Now we have to find a point p conjugate to 0S along t , that is to find a

solution x(t) to equation (44) which vanishes at p . So that initially for some

constant ;

,0 x and

xdt

dx

3

1

,

3

1

0tdt

dx

(45)

which vanishes at p.

To solve equation (44) we use Sturn comparison theorem which compares the

distribution zeros of the solutions u(t) and v(t) of the equations (Joshi 1996) ;

012

2

utGdt

ud

, (46)

022

2

vtGdt

vd

.23

1 22

VVRdt

d



where 21 GG in an interval (a, b). The theorem then shows that if u(t) has m

zeros in bta then v(t) has at least m zeros in the same interval and the kth

zero of v(t) is must be earlier than the kth zero of u(t).

Now let,

22 23

1minmin

VVRtHA

and consider the equation;

02

2

2

xAdt

xd

. (47)

If we apply the Sturm theorem to equations (44) and (47) we observe that if

the solution to equation (47) satisfying the initial conditions (45) has a zero in

the interval 10 tt , then the solution of equation (24) defined by the same

initial conditions must have a zero in the same interval, which must occur

before the zero of the solution of equation (47). Now the general solution of

(47) can be written as;

AtCCx 21 sin . (48)

Let us choose the initial condition as;

2

122

10

A

x

,

21

220 A

dt

dx

t

. (49)

Since the universe is expanding everywhere so 0 on 0S . The universe

may contract or it may expand at some places and may contract in some other

places, but we shall not consider such possibilities here, instead we consider



only expanding behavior. Using initial conditions (49), solution (48) can be

written as;

AtA

x sin1

(50)

with

21

22

1sin

A

A

.

We have 2

0

and a zero for x must occur within the interval;

20

At

At

20

, (51)

i.e., if t is any timelike curve geodesic orthogonal to 0S , then there must be

a point p on t , conjugate to 0S , within the above interval where

0min2 tHA .

No timelike curve from 0S can be extended into the past beyond the proper

time length A2

. Let q be an event on 0S

and be a past directed, endless

timelike curve from 0q . Let can be extended to arbitrary values of

proper time in the past, then choose

Ap

2

to be an event on this

trajectory. Then there exists a timelike from p orthogonal to 0S along which

the proper time lengths of all non-spacelike curves from p to 0S

are

maximized and further, does not contain any conjugate point to 0S

between p and 0S . Again, we have shown that any timelike geodesic t



must contain a point conjugate to 0S within the proper time length

A2

. But

this is impossible and we can say that timelike curve form can be extended

into the past beyond the proper time length A2

i.e.,

At

2max

.

Now the above results can be applied to obtain general upper bounds to the

age of a globally hyperbolic universe in the following manner.

By using (14) and (18) we can write,

TVVTGVVR

2

18

GPGVPGVVR 448

24

PGVVR 34

. (52)

6.2 LIMIT OF THE AGE OF THE UNIVERSE

We assume that energy density of the present universe is mainly contributed

by the non-relativistic free gas of neutrinos for which P then (52)

becomes;

GVVR 4

GVVRA 3

42

3

1min 22

where is the present density of the universe. Hence the maximum possible

age of the universe m axt , is given by;

21

21

max16

3

4

3

22

GGAt

(53)

with the basis of general globally hyperbolic space-time (Figure 4).

In radiation dominated models we can write 3

1P , then (52) becomes;



(54)

Then (33) becomes;

21

max 32

3

Gt

. (55)

Figure 4: No timelike curve from the surface S extends the maximum limit

m axt in the past and must encounter a space-time singularity before this epoch.

Both (53) and (55) give upper limits to the age of the universe irrespective of

whether or not the distribution of whether on 0S is isotropic and

homogeneous which does not assume.

The average mass density as indicated by the visible galaxies is about 10–30 gm

cm–3. The X-ray observations strongly favor the existence of a hot ionized

intergalactic gas within the cluster of galaxies whereas weakly interacting

massive neutrinos could be another source. If the microwave background

radiation (MBR) as having some kind of global origin, then MBR provides a

.8 GVVR



firm lower limit of the min sought for and general upper limit to the age of

the universe as given by (55) is;

122

1

max 102.3 32

3

Gt

years, (56)

3104.4 MBR gm cm–3.

The relationships (54) and (56) provide upper limits to the age even when

allowing for departures from homogeneity and isotropy. If we take the

contribution by matter into account, we have to choose an entire range of

densities as suggested by the above mentioned possibilities.

The average matter density arising from all possible sources is believed to be

between 3110 to 2810 gm cm–3.

For 3110 gm cm–3 in (56) we find;

102

1

max 1043.9 32

3

Gt

years, (57)

and for 2810 gm cm–3 in (56) we find;

102

1

max 1094.0 32

3

Gt

years. (58)

7 LOWER BOUNDS ON AXION REST MASS

In a general cosmology when the densities may vary on 0S and then m axt

comes from observations. Thus if obt denotes observed age, we can write

maxttob , then (53) becomes;

20

1.

16

3

obtG

(59)



which implies that a prescribed lower limit on the observed age of the

universe will provide an upper limit to the matter density. If F

0 be the

Friedmann density parameter then (59) becomes;

20

1.

16

3

ob

F

tG

. (60)

In the Friedmann model, the density due to axions, which produced in the

early universe, is a function of absolute temperature (Preskill et al. 1983);

QCDPl

aaF

m

fTmT

23

0

3

(61)

where am = axion mass, 21

/ GcmPl is the Plank mass and

MeVQCD 200 is the scale parameter in quantum chronomodynamics.

Now axion mass am is related to the vacuum expectation value af of the

scalar field that the spontaneously breaks the Peccei–Quinn symmetry

invoked to explain the C–P invariance of strong interactions, and is given by

(Weinberg 1978, Weilezek 1978)

a

af

GeVeVm

125 10

1024.1

. (62)

Particle physics does not specify the exact value of af ; it can lie anywhere

between the weak interaction scale and the mass scale of grand unification. If

dark matter is made up entirely of axions, then since FF

a T 0 , so;

2

0

1.

16

3

tGTF

a

. (63)

By (61) and (63) becomes;

max222

0

31024.1

1.

1..

16a

QCDPl

a fGeVtT

m

Gf

. (64)



If T= 2.73K be the present temperature of the universe and MeVQCD 200 then;

min

max

125 10

1024.1 a

a

a mf

GeVeVm

. (65)

obt (in Gyr) maxaf (in 1012GeV)

minam (in 10–5eV)

13 1.15 1.07

14 0.99 1.24

15 0.87 1.43

16 0.76 1.62

17 0.67 1.83

18 0.60 2.05

19 0.54 2.29

20 0.49 2.53

21 0.44 2.80

22 0.40 3.07

23 0.36 3.41

24 0.33 3.72

25 0.30 4.09

Table 1: Upper limits on af and lower limits on am for various values of the

observed age of the universe

Observationally, the best lower limits for the age of the universe come from

studies of globular clusters of stars in our galaxy, these are (13–19) Giga year

(1Gyr = 109 yr), (18.8–24.8) Gyr. The results are shown in the following Table

1.

8 UPPER LIMIT OF THE AGE OF THE UNIVERSE WITH COSMOLOGICAL

CONSTANT

If we include cosmological constant then for P << ρ then (52) takes the form

(Joshi 1996);

28 cGVVR m . (66)

By this we obtain the maximum possible age m axt ;



vmm GcGt

2

3

4

1

4

3

2 2max

(67)

where m is the material energy density and v is the vacuum energy, so the

critical energy density,

vmc . (68)

For a dust full universe, vP . Defining c

vv

,

c

mm

we get;

vcGt

3

1.

16

32

max

c

vcG

31

1.

16

3

(69)

vcG

31

1.

16

3

02

1

16

331

2

max0

2

max

tHtG c

v

(70)

3

1 v

. (71)

Clearly utt max , where ut = age of the oldest known objects in the universe.

Combining the above constraints we get;

3

1

2

11

3

12

0

v

utH

. (72)

Taking 10105.1 Ptu yrs, 11

00 100 MpckmshH (72) becomes;

3

121

3

122

0

v

ph. (73)



Positive lower bound on v is obtained for 20 Ph , hence for 10 h we get,

ut = 21 billion yrs; this implies a positive lower bound on v .

For radiation dominated age vmP 3

1 then (69) becomes;

vmGt

52

1.

16

32

max

(74)

Which gives the bound on v when 10 Ph . Thus it is possible to derive the

required bounds in terms of the age of the oldest objects in the universe.

9 CONCLUSIONS

In this paper we have tried to describe the general upper limit of the age of

the universe with cosmological constant. We have briefly described general

relativity, Schwarzschild geometry, FRW model, and Raychaudhury equation

to make the study easier to the common readers. The universe is

homogeneous and isotropic around us about 14 billion light years. In our

discussion we have not strictly followed the homogeneity and isotropy of the

universe to determine the age of the universe. In our stud we have found that

the age of the universe is around 1010 years. We have avoided difficult

mathematical calculations and have displayed diagrams where necessary.

REFERENCES

[1] Hawking, S.W. and Ellis, G.F.R. (1973), The Large Scale Structure of Space-time, Cambridge University Press, Cambridge.

[2] Islam, J.N. (2002), An Introduction to Mathematical Cosmology, 2nd ed., Cambridge University Press, Cambridge.

[3] Joshi, P.S. (1996), Global Aspects in Gravitation and Cosmology, 2nd ed., Clarendon Press, Oxford.

[4] Joshi P.S. (2013), Spacetime singularities, arXiv:1311.0449v1 [gr-qc] 3 Nov 2013.



[5] Kar, S. and SenGupta, S. (2007), The Raychaudhuri Equations: A Brief Review, Pramana Journal of Physics, Indian Academy of Sciences, 69(1): 49–76.

[6] Kruskal, M.D. (1960), Maximal Extension of Schwarzschild Metric, Physical Review, 119(5): 1743–1745.

[7] Mohajan, H.K. (2013a), Singularity Theorems in General Relativity, M. Phil. Dissertation, Lambert Academic Publishing, Germany.

[8] Mohajan, H.K. (2013b), Scope of Raychaudhuri Equation in Cosmological Gravitational focusing and space-time singularities, Peak Journal of Physical and Environmental Science Research, 1(7): 106–114.

[9] Mohajan, H.K. (2013c), Space-time Singularities and Raychaudhuri Equations, Journal of Natural Sciences, 1(2): 1–13.

[10] Mohajan, H.K. (2013d), Friedmann, Robertson-Walker (FRW) Models in Cosmology, Journal of Environmental Treatment Techniques, 1(3): 158–164.

[11] Mohajan, H.K. (2013e), Schwarzschild Geometry of Exact Solution of Einstein Equation in Cosmology, Journal of Environmental Treatment Techniques, 1(2): 69–75.

[12] Pecci, R. and Quinn, H. (1977), CP Conservation in the Presence of Pseudo Particles, Physical Review Letters, 38 (25): 1440–1443.

[13] Preskill, J.; Wise, M.B. and Wilczek, F. (1983), Cosmology of the Invisible Axion, Physics Letters, 120B(1, 2, 3): 127–132.

[14] Raychaudhuri, A.K. (1955), Relativistic Cosmology, Physical Review, 98(4): 1123–1126.

[15] Szekeres, G. (1960), On the Singularities of a Riemannian Manifold, Publicationes Mathematicae, Debrecen, 7: 285–301.

[16] Weinberg, S. (1978), A New Light Boson?, Phys. Rev. Lett., 40(4): 223–226. [17] Wilczek, F. (1978), Problems of Strong P and T Invarience of Instantons,

Phys. Rev. lett., 40: 279–282.

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