Research ArticleEffect of Generalized Uncertainty Principle on Main-SequenceStars and White Dwarfs

Mohamed Moussa

Physics Department Faculty of Science Benha University Benha 13518 Egypt

Correspondence should be addressed to Mohamed Moussa mohamedibrahimfscbuedueg

Received 16 February 2015 Revised 26 April 2015 Accepted 26 April 2015

Academic Editor George Siopsis

Copyright copy 2015 Mohamed Moussa This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

This paper addresses the effect of generalized uncertainty principle emerged from different approaches of quantum gravity withinPlanck scale on thermodynamic properties of photon nonrelativistic ideal gases and degenerate fermions A modification inpressure particle number and energy density are calculated Astrophysical objects such as main-sequence stars and white dwarfsare examined and discussed as an application A modification in Lane-Emden equation due to a change in a polytropic relationcaused by the presence of quantum gravity is investigated The applicable range of quantum gravity parameters is estimated Thebounds in the perturbed parameters are relatively large but they may be considered reasonable values in the astrophysical regime

1 Introduction

Different approaches for the quantum gravity have beenproposed in string theory and black hole physics to providea set of predictions for a minimum measurable length andan essential modification of the Heisenberg uncertaintyprinciple (GUP) [1ndash11] and a modification in the funda-mental commutation relation [119909

119894 119901119895] According to string

theory it has been found that strings cannot interact atdistances smaller than their size which yields generalizeduncertainty principle [1] From black hole physics [2 3]the uncertainty principle Δ119909 sim ℏΔ119901 is modified at thePlanck energy scale when the corresponding Schwarzschildradius is approximately equal to Compton wavelength at thePlanck scale Higher energies result in a further increase ofthe Schwarzschild radius to yield Δ119909 asymp ℓ2

119875119897Δ119901ℏ The above

approaches alongwith a combination of thought experimentsand rigorous derivations suggest that the (GUP) holds at allscales and is represented by the following [1ndash11]

Δ119909119894Δ119901119894

geℏ

2[1 + 120573 ((Δ119901)

2+ ⟨119901⟩

2) + 2120573 (Δ119901

2

119894+ ⟨119901119894⟩2)]

(1)

where 1199012 = sum119895119901119895119901119895 120573 = 120573

0(119872119901119888)2= 1205730(ℓ2

119901ℏ2) 119872119901is

Planck mass and 1198721199011198882 is Planck energy It was shown in

[9] that inequality (1) is equivalent to the following modifiedHeisenberg algebra

[119909119894 119901119895] = 119894ℏ (120575

119894119895+ 1205731205751198941198951199012+ 2120573119901

119894119901119895) (2)

This form ensures via the Jacobi identity that [119909119894 119909119895] = 0 =

[119901119894 119901119895] [10]Differentminimal length scale scenarios inspired

by various approaches to the quantum gravity have beenreviewed in [12] Apparently this suggests a modification ofthe physical momentum [9ndash11]

119901119894= 1199010119894(1 + 120573119901

2

0) (3)

while 119909119894= 1199090119894with 119909

0119894 1199010119895satisfies the canonical commu-

tation relations [1199090119894 1199010119895] = 119894ℏ120575

119894119895 such that 119901

0119894= minus119894ℏ120597120597119909

0119894

where 119901 is the momentum satisfying (2) The upper boundson the parameter 120573 have been derived in [13 14] and itwas found that it could predict an intermediate length scalebetween Planck scale and electroweak scale It was suggestedthat these bounds can be measured using quantum opticstechniques and gravitational wave techniques in [15 16]which is considered as milestone in the quantum gravityphenomenology It is noteworthy that the thermodynamical

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 343284 9 pageshttpdxdoiorg1011552015343284

2 Advances in High Energy Physics

phenomenological implications of GUP in quantum gases arenumerous for example [17 18] In [19] the quantum gravityinfluences on the statistic properties classical nonrelativisticultrarelativistic and photon gases were addressed Small cor-rections to the energy and entropy at low temperatures werefound and some modifications to the equations of state areemployed The authors considered two configurations Thefirst is the white dwarf which is composed of a nonrelativisticcold nuclei the mass of the star was calculated It is foundthat there is a quantum gravity correction that depends onthe number density of the star that leads to an increase in themass of the star and the quantum gravity tends to resist thecollapse of the star The other configuration is that the staris almost composed of ultrarelativistic particles They foundthat the increase in the star energy leads to monotonicalblowup of Fermi pressure which resists the gravitationalcollapse

We will use this form of GUP (1) to investigate theimplications of the quantum gravity on statistical propertiesof quantum gases After redefining the new phase space wewill put the partition function in a form that is consistentwith GUP and then rewrite the thermodynamic propertiesfor quantum gases namely photons nonrelativistic idealgases and fermions in degenerate state Using these resultswe will follow the proposal which was investigated in [20] instudying the stability ofmain-sequence stars andwhite dwarfAlso a simple method is used to study the effect of quantumgravity in a different stage of stellar evolution and a modifiedmass-radius relation for the white dwarf is calculated andboundaries in the quantum gravity perturbative terms will bedetermined

2 Statistical Mechanics and GUP

In this section we investigate the GUP modification in theformalism of the grand-canonical ensemble If we consider119873 particles with energy states 119864

119894 for each state there is 119899

119894

particles in that stateThe grand-canonical partition functionis obtained from the sum over all of the states

Z = sum119899119894

prod119894

[119890(120583minus119864119894119870119861119879)]

119899119894 (4)

where 119879 is the temperature 119870119861is the Boltzmann constant

and 120583 is the chemical potential The connection to thermo-dynamics is obtained after introducing the partition functiondefined as [21]

lnZ = sum119894

ln [1 + 119886119911119890minus119864119894119870119861119879] (5)

where 119911 = exp(120583119870119861119879) and the parameter 119886 takes value

depending on the considered particles If they are fermions119886 takes +1 value and if the considered particles are bosons 119886takes minus1 value

By considering large volume the summation turns tobe sum119894rarr int119889

31199091198893119901((2120587ℏ)

3(1 + 120573119901

2)3) where we are

considering the invariant phase space in the existence ofthe GUP that has been derived in [22] We should put thegrand canonical partition function for these systems in a form

consistent with GUP frameworkTheGUP can be consideredin the phase space analysis by two equivalent pictures (a)considering deformed commutation relations (ie deformedthe measure of integration) simultaneously with the nonde-formed Hamiltonian function or (b) calculating canonicalvariables on the GUP-corrected phase space which satisfy thestandard commutative algebra (ie nondeformed standardmeasure of integration) but the Hamiltonian function nowgets deformed These two pictures are related to each otherby the Darboux theorem in which it is quite possible tofind canonical coordinates on the symplectic manifold whichsatisfy standard Heisenberg algebra [23 24] In this paper weare going to consider deformed measure of integration withnondeformedHamiltonian function Based on the new phasespace the partition function is

lnZ =119881

(2120587ℏ)3

119892

119886int ln [1 + 119886119911119890minus119864119896119861119879]

1198893119901

(1 + 1205731199012)3

(6)

with 119864 = (11990121198882 +11989821198884)12 Pressure number of particles andinternal energy will be determined by the relations

119875 = 119870119861119879120597

120597119881lnZ

119899 = 119870119861119879120597

120597120583lnZ|119879119881

119880 = 1198701198611198792 120597

120597119879lnZ|119911119881

(7)

We will expand all the terms that contain 120573 and wewill keep the terms that are proportional to sim120573 only Thisapproximation breaks down around maximum measurableenergies such that 1205731199012 asymp 1 In this case we need an exactsolution But one can trust the perturbative solution where1205731199012≪ 1 Now we will use the modified partition function

to calculate the thermodynamics of photons nonrelativisticideal gases and degenerate fermions

21 Photon Gas For a gas of photons 119892 = 2 119886 = minus1 and120583 = 0 The energy density is given using (7) by

119906 =119880

119881=

1

1205872ℏ3int

119864

119890119864119870119861119879 minus 1

1199012119889119901

(1 + 1205731199012)3 (8)

where119864 = 119901119888 Use the approximation 1(1+1205731199012)3 ≃ 1minus31205731199012one gets

119906 =119888

1205872ℏ3int

119889119901

119890119888119901119870119861119879 minus 1(1199013minus 3120573119901

5)

=4120590

1198881198794minus 120573

8

21

12058741198706

119861

ℏ311988851198796

(9)

where 120590 = 1205872119896460ℏ31198882 is the Stefan-Boltzmann constant

For 120573 = 0 one can recover the usual Stefan-Boltzmann law

Advances in High Energy Physics 3

And the pressure of system (7) is given by

119875 =119888

1205872ℏ3int

119889119901

119890119888119901119870119879 minus 1[1

31199013minus3

51205731199015]

=4120590

31198881198794minus 120573

8

105

12058741198706

119861

ℏ311988851198796

(10)

It is clear that the modification in phase space decreasesthe available number ofmicrostates which holds the thermo-dynamic properties and hence decreases the internal energyand pressure of the system In order to obtain the pressure-energy density relation divide (10) by (9)

119875

119906=1

3+ 120573

16

21(119870119861120587

119888)2

1198792 (11)

If we remove the quantum gravity effect we can recoverthe usual relation 119906 = 3119875 In order to obtain the equation ofstate (EOS) 119875 = 119875(119906) we first solve (9) for the temperature toget

119879 = (119888

4120590)14

11990614+ 120573

10

7(120587119870119861

119888)2

(119888

4120590)34

11990634 (12)

Substitute (12) into (11) to get the first order correction to theEOS

119875 =119906

3+ 120573

16

21(151205872ℏ3

119888)

12

11990632 (13)

Equation (13) represents a modified equation of state dueto generalized uncertainty principle

22 Nonrelativistic Ideal Gases Let us now consider the effectof quantum gravity on nonrelativistic ideal gases In orderto do that the partition function for one particle with themodified phase space is given by

1199111=

119881

21205872ℏ3int 119890minus1199012(2119898119870119861119879)

1199012119889119901

(1 + 1205731199012)3 (14)

The total partition function is given by 119885 = (1119873)119911119873

1

by which we can derive the free energy for the system withthe equation 119865 = 119870

119861119879 ln119885 Using Stirlingrsquos formula ln119873 =

119873 ln119873 minus119873 we can determine the pressure

119875 = minus120597119865

120597119881

10038161003816100381610038161003816100381610038161003816119879119873=119873119870119861119879

119881 (15)

So we recover the usual EOS for nonrelativistic ideal gas119875 = 119899119870

119861119879 where 119899 = 119873119881 is the number density which

shows that the quantum gravity does not play here Thesame behavior is established in [23] Suffice calculation ofthe pressure at this point which we need and the otherthermodynamic properties of the nonrelativistic ideal gasescan be obtained easily by solving the integration in (14) andthen calculating the free energy

23 Degenerate Fermion Gas Fermions have a property thatatmost one can occupy each quantum state At119879 = 0 all stateswith energy less than Fermi energy level are occupied and allstates with greater energy are empty So the total energy ofthe system is the lowest possible consistent with the exclusionprincipleThe energy of the highest occupied state at absolutezero is the Fermi energy and corresponding momentum isreferred to as the Fermi momentum 119901

119891 In this case the

distribution function goes to step function According tothese assumptions and using (7) the particle number densityis given by

119899 =1

1205872ℏ3int119875119891

0

1199012119889119901

(1 + 1205731199012)3=

1

1205872ℏ3[1

31199013

119891minus 120573

3

51199015

119891] (16)

Solving this equation for Fermi momentum one gets

119901119891= (3120587

2ℏ3119899)13

+ 1205733

5(31205872ℏ3119899) (17)

The second term ensures that the Fermi momentumis increased due to the presence of quantum gravity andhence Fermi energy Also the energy density for fermions indegenerate state is given by

119906 =1

1205872ℏ3int119875119891

0

[11990121198882+ 11989821198884]12 119901

2119889119901

(1 + 1205731199012)3 (18)

Using the substitutional

119901 = 119898119888 sinh119909 (19)

the energy density integration will be

119906 =11989841198885

1205872ℏ3int119909119891

0

[sinh2119909 + (1 minus 312057311989821198882) sinh4119909

minus 312057311989821198882sinh6119909] 119889119909 = 119898

41198885

1205872ℏ3[1198651(119910) + (1

minus 312057311989821198882) 1198652(119910) minus 3120573119898

211988821198653(119910)]

(20)

where

119910 =119901119891

119898119888= sinh119909

119891(21)

with

1198651(119910) = minus

1

2ln [119910 + radic1 + 1199102] + 1

2119910radic1 + 1199102

1198652(119910) =

3

8ln [119910 + radic1 + 1199102]

+1

4119910radic1 + 1199102 [119910

2minus3

2]

1198653(119910) = minus

5

16ln [119910 + radic1 + 1199102]

+ 119910radic1 + 1199102 [5

16+ 1199102(1199102

6minus5

24)]

(22)

4 Advances in High Energy Physics

By the same way we can derive the pressure from (7)after making use of the integration by parts and functionexpansion one finds that

119875 =1198882

1205872ℏ3int119901119891

0

119889119901

[11990121198882 + 11989821198884]12[1

31199013minus3

51205731199015]

=11989841198885

1205872ℏ3[1

31198652(119910) minus

3

5120573119898211988821198653(119910)]

(23)

Equation (23) represents the ground state pressureThereis a considerable change due to the presence of the function1198653(119910) and 119910 itself depends on modified Fermi momentum

We should notice that the value of 119910 is increased due to theincreasing in Fermi momentum this leads to a considerablegrowing in the pressure (23) The change in pressure of theradiation and degenerate fermions will affect the polytropicrelations for astroobjects In the next section the modifiedpressure effect in the some stars stability and some otherapplications will be investigated exactly

3 Applications to Astrophysical Objects

31 Main-Sequence Stars Most main-sequence stars havea mass in the range about 10 to 50 solar mass They arecomposed by a mixture of nonrelativistic gas and radiationthat are stood in hydrostatic equilibrium by gravity [25] Thetotal pressure is 119875

119888= 119875119892+ 119875119903 where 119875

119888 119875119892 and 119875

119903are center

star gas and radiation pressure respectively We can write120573119888= 119875119892119875119888and 1 minus 120573

119888= 119875119903119875119888 By definition (1 minus 120573

119888) and

120573119888are the fractional contribution of radiation and gas to the

central pressure of the star and both are less than one then

119875119892

120573119888

=119875119903

1 minus 120573119888

(24)

The classical gas pressure can be expressed using (15) inthe form

119875119892=120588119888119870119861119879

119898 (25)

We used the relation 119899 = 120588119888119898 120588119888is the central density

of the star and 119898 is the average mass of the gas particle thatconstructs the star defined as 119898 = 119906

119873119898119873where 119906

119873is the

mean molecular weight and 119898119873is the nucleon mass Using

(10) (25) into (24) one finds that

120588119888119870119861

120573119888119898119879119888=

4120590

3119888 (1 minus 120573119888)1198794

119888minus8

105120573

12058741198706

119861

ℏ31198885 (1 minus 120573119888)1198796

119888 (26)

where 119879119888is the central temperature of the star We can find

the temperature by solving the above equation

119879119888= (

1 minus 120573119888

120573119888119898

3119888119870119861

4120590)

13

12058813

119888

+ 1205731

70

(1 minus 120573119888)

120573119888119898

12058741198707

119861

ℏ311988831205902120588119888

(27)

Using this expression to calculate the star pressure from therelation 119875

119888= 119875119892120573119888= (120588119888119870119861120573119888119898)119879119888one gets

119875119888= 119870112058843

119888+ 12057311987021205882

119888= 119875119904UP + 119901119904GUP (28)

1198701=(1 minus 120573

119888)13

(120573119888)43

(31198881198704

119861

41205901198984)

13

1198702=1

70

(1 minus 120573119888)

(120573119888)2

12058741198708

119861

ℏ3119888312059021198982

(29)

The first part of (28) is the ordinary polytropic relationwith a a polytropic exponent Γ = 43 (polytropic index 119899

119901=

3) [26] perturbed by a polytrope Γ = 2 (polytropic index 119899119901=

1) where 119899119901= 1(Γ minus 1)

The best example may be used to estimate GUP pertur-bation is our sun If we assume that 120573

119888is constant overall the

stars and its chemical composition is not changed so 119906119873is

constant For our sun 1 minus 120573119888= 10minus3 and 119906

119873= 0829 [27] and

central density 120588119888= 153 times 10

5 kgm3 then

119901119904GUP119875119904UP

= 18 times 10minus491205730 (30)

The perturbed pressure should not exceed the originalone such that 119901

119904GUP119875119904UP ≪ 1 so this equation suggests thatthe upper value of 120573

0should be 120573

0lt 1048 This bound is far

weaker than that set by electroweak measurementsAccording to (28) the quantum gravity modifies the

polytropic relation by adding a new term proportional to 1205882This suggests a new modification in Lane-Emden equation[28] The modified Lane-Emden equation is established inthe appendix Studying modified Lane-Emden equation andits solution is not the scope of this paper so results that areinvestigated by the this equationwill be postponed to anotherresearch

Main-Sequence Stars Stability In order to discuss the stabilityconcept the energy of a Newtonian polytrope systems withpressure 119875 can be expressed in terms of the internal energyand gravitational potential such that

119864 = 1198961119875119881 minus 119896

2

1198661198722

119877 (31)

where 1198961and 1198962are constants119866 is the gravitational constant

and 119872 and 119877 are the mass and radius of the star At hightdensities it is important to take into account the generalrelativistic effects in star stability To deal with it a generalrelativistic correction term should be added to (31) in lowestorder approximation namely 119864corr = minus119896

3(119866119888)21198727312058823

119888

where 1198963is constant that depends on the actual distribution

of matter According to [27] for 119899119875= 3 polytrope the

condition of stability is Γ gt 43 + 225(1198661198721198882119877) We can seethat addition of a relativistic correction contracts the regionof stability or increases the critical value of Γ In our case119866119872119888

2119877 sim 10

minus6 so this correction can be neglected So weexpect that the new term in a polytropic equation (28) with

Advances in High Energy Physics 5

119899119901= 1 which has a positive contribution will not affect the

problem of stability since Γ gt 2Using (28) into (31) the energy ofmain-sequence starswill

be

119864 = 119862111987212058813

119888+ 1205731198622119872120588119888minus 11989621198725312058813

119888 (32)

where 1198621and 119862

2are constants The mass119872 can be obtained

by using the equilibrium condition 120597119864120597120588119888= 0 the result is

119872 = [3

1198963

(1

31198621+ 120573119862212058823

119888)]

32

(33)

Then

119889 ln119872119889 ln 120588119888

= 12057331198622

1198621

12058823

119888 (34)

Equation (34) ensures that 119889 ln119872119889 ln 120588119888gt 0 which

is the stability condition [27] for main-sequence objectThese results show that the generalization process for theuncertainty principle does not change the stability of themain-sequence objects

Minimum and MaximumMasses for Main-Sequence Stars Inthis part we will discuss the evolution stage of main-sequencestars in the presence of quantum gravity to derive theminimum and maximum masses of these objects Initiallyin the stage of star forming the star must has mass enoughto generate a central temperature which is high enoughfor thermonuclear fusion At this stage the central pressurecomes by the electrons and ions which are forming an idealclassical gas their pressure can be expressed as in (25) Thestar will be made if this pressure is close to the pressureneeded to support the system It is known that the upper limitrelation between the pressure and density at the center of thestar is given by [29]

119875119888le (

120587

6)13

11986612058843

11988811987223 (35)

This relation is valid for any homogeneous star in whichthe mass is concentrated towards the center Equating (25)and (35) one can get the initial central temperature in thisstage

119870119861119879119888le (

120587

6)13

11989811986612058813

11988811987223 (36)

We can see that the quantum gravity does not appearin the picture As shown in this equation the central tem-perature will rise as the central density increases with starcontraction The contracting temperature continues to riseuntil the electrons at the center become fully degenerate andoccupy the lowest energy states in accordance with exclusionprinciple At this stage at the center the pressure comes fromthe electrons that are in degenerate state plus the pressurefrom the classical ions Quantum gravity will not change theclassical ions pressure according to (15) but a nonrelativisticdegenerate electrons pressure will change according to (23)

In nonrelativistic limit where 119910 ≪ 1 the degenerate pressure(23) will be

119875non-relat0

=1

15

1198984

1198901198884

1205872ℏ31199105minus3

35

1198986

1198901198887

1205872ℏ31199107 (37)

Use (17) (21) in above equation and add the result to (15)finally we can write the central pressure at this stage as

119875119888=

119886

5119898119890

11989953

119890+ 120573

121198862

35119898119890

11989973

119890+ 119899119894119870119861119879119888 (38)

where 119886 = (31312058723ℏ)2 The number density of the electrons119899119890and ions 119899

119894can be expressed in terms of central density

simply 119899119890= 119899119894= 120588119888119898119867 where 119898

119867is the mass of hydrogen

atom A hydrostatic pressure is achieved if this pressureequals the pressure needed to support the mass (35) Thisleads to the central temperature being as follows

119870119861119879119888le (

120587

6)13

1198661198981198731198722312058813

119888minus119886

5119898119890

(120588119888

119898119867

)

23

minus 120573121198862

35119898119890

(120588119888

119898119867

)

43

(39)

The first term is associated with classical ions and secondand third terms with degenerate electronsThe last two termsbecome important and are increasing with central densityuntil the temperature will cease to rise as the mass contractsFrom this equation we can find the maximum value oftemperature

10038161003816100381610038161198701198611198791198881003816100381610038161003816max le 119911119872

43[1 minus

60

712057311989811989011991111987243] (40)

119911 =5

4(120587

6)23 119898119890

11988611989883

1198671198662 (41)

Equation (40) proves that the maximum central tem-perature decreases due to quantum gravity The contractingmass achieves stardom if that maximum central temperaturereaches ignition temperature for the thermonuclear fusion ofhydrogen Denote this ignition temperature by 119879ign then theminimummass for the star will be given by

119872min ge (119870119861119879ign

119911)

34

[1 +45

7120573119898119890119870119861119879ign] (42)

This relation shows that the quantum gravity effectincreases theminimummass of themain-sequence starsTheapproximation has ameaning if the second term in (42) is lessthan unity If we take the ignition temperature for hydrogento be about 15 times 106 K this suggests that the upper value of1205730lt 1047

The situation is changed if the radiation pressure takesa part in this stage of revolution The star will be disruptedif radiation becomes the dominant source for the internalpressure This property will set a limit to the mass of a main-sequence starThe pressure at the center of hotmassive stars isdue to electrons and ions (as a classical gases) and radiation

6 Advances in High Energy Physics

the pressure that is produced by these species is derived in(28) Comparing this with the pressure needed to support thestar (35) to get the maximummass one finds that

(120587

6)13

11986611987223

max ge 1198701 + 120573119870212058823

119888 (43)

which means that the maximummass is increased due to thepresence of quantum gravity This increase depends on thecentral density Again the second term in the right-hand sideof inequality (43) should be less than the first this suggeststhat 120573

0lt 1048

32White Dwarfs Thewhite dwarfs consist ofHelium atomsthat are almost in an ionization state The temperature of thestar is about 107 K so the electrons are in a relativistic energyregime and can be considered in a complete degenerate stateand the Helium nuclei do not play a part in first orderapproximation in the dynamic of the system Now we willdetermine the effect of GUP on white dwarf radius accordingto pressure modification expressed in (23) If the number ofHeliumnuclei is1198732 then there are119873 electrons Suppose that119872 = 2119873119898

119901is the mass of the star so the particle density of

the electron is

119899119904=3119872

8120587119898119901

1

1198773 (44)

Using (17) (44) into (21) one gets

119910 =11987213

119877+3

51205731198982

1198901198882119872

1198773 (45)

119872 =9120587

8

119872

119898119901

119877 =119898119890119888

ℏ119877

(46)

where 119877 is the radius of the star and 119898119901is the proton mass

The equilibrium configuration of white dwarf is caused bythe pressure of the degenerate electrons against Newtoniangravitational collapse which is due to the nuclei whichmeansthat

1198750=120581

4120587

1198661198722

1198774= 11987010158401198722

1198774

1198701015840=120581119866

4120587(8119898119901

9120587)

2

(119898119890119888

ℏ)4

(47)

We have two extreme cases as follows(i) When the electron gas is in a low density nonrelativis-

tic dynamics may be used such that 119910 ≪ 1 and (23) will leadto

1198750=4

51198701199105minus36

351205731198982

11989011988821198701199107

119870 =1198981198901198882

121205872(119898119890119888

ℏ)3

(48)

Equating (47) and (48) we got the equation

1198773

minus4

5

119870

1198701015840119872minus13

1198772

minus48

351205731198982

11989011988821198701015840

11987011987213

= 0 (49)

The solution of that equation is

119877 ≃4

5

119870

1198701015840119872minus13

+15

71205731198982

11989011988821198701015840

119870119872 (50)

This equation shows that the radius of the star is increas-ing the elevation depends on the mass of the star The usedapproximation is true if the second term in the right-handside of (50) is much less than the first one this will put a limitin the upper value of 120573

0such that it should be 120573

0lt 1044 we

used a white dwarf mass119872 = 041119872⊙[30]

(ii) When the electron gas is in a high density thatrelativistic effect comes to play strongly such that 119910 ≫ 1 Wecan write

1198750= 119870 (119910

4minus 1199102) minus 120573

18

51198701198982

1198901198882(1199106

3minus1199104

4) (51)

Using (51) into (47) and simplifying then

1198774

+ [1198701015840

11987011987243

minus (1 minus3

101205731198982

1198901198882)11987223

]1198772

minus6

51205731198982

119890119888211987243

= 0

(52)

The solution of that equation keeping only the terms thatproportional to sim120573 is

119877 ≃ 1198770[1 + 120573

3

10

1198982

1198901198882

1198772

0

(211987243

1198772

0

minus1

211987223

)] (53)

where

1198772

0= 11987223

minus1198701015840

11987011987243

(54)

Using (46) into (54) and solving for 119877 and letting 119877 rarr

119877Ch one gets

119877Ch =(9120587)13

2

ℏ

119898119890119888(119872

119898119901

)

13

[1 minus (119872

119872Ch)

23

]

12

119872Ch =9 (3120587)

12

64120581(ℏ119888

1198661198982119901

)

32

119898119901

(55)

119877Ch is the unmodified white dwarf radius or Chan-drasekhar radius limit and 119872Ch is the Chandrasekhar limitmass of order 144119872

⊙ First part of (53) is the Chandrasekhar

radius and the second term is the radius correction due toGUP affect These analyses show that the increasing in starradius depends on [1 minus (119872119872Ch)

23]minus1 which formally goes

to infinity as119872 rarr 119872Ch It is clear that the quantum gravityeffect becomes significant when mass of the white dwarfgets very close to the Chandrasekhar limit in high density

Advances in High Energy Physics 7

stars Equations (23) (50) and (53) attract attention to anoticeable point that the growing in star radius ensures thatquantum gravity resists gravitational collapse by increasingthe electrons degeneracy pressure These results are consis-tentwith that established in [19]Our approximationwill havea meaning if the second term in the right-hand side in (53)is less than one this puts an upper limit for 120573

0such that

1205730lt 1043 We have used119872 = 059119872

⊙[30]

We should also notice that in the limit 119910 ≫ 1 inultrarelativistic regime the function 119865

3(119910) behaves like sim1199106

which almost goes like 1205882 This result is consistent withpolytrope behavior in which Γ = 2 that is correspondingto stability range of the star So we can conclude that thisapproach of quantum gravity does not produce any instabilityto white dwarfs

4 Discussion

Various approaches to quantum gravity such as string the-ory black hole physics and doubly special relativity pre-dict a considerable modification in Heisenberg uncertaintyprinciple to be a generalized uncertainty principle Thismodification leads to a change in the energy-momentumdispersion relation and in the physical phase space In thispaper we studied the modification due to a generalizeduncertainty principle (1) in the statistical and thermodynamicdynamic properties for photons nonrelativistic ideal gasesand degenerate fermions There is a considerable decreasein the number of accessible microstates in quantum gravitybackgroundThis approach leads to a decrease in the internalenergy and pressure of photon gas Pressure and internalenergy of fermions in degenerate state are increased andpressure of nonrelativistic ideal gases is not changed Theseresults are used in studying the stability of themain-sequencestars and white dwarfs It is found that this approach ofquantum gravity does not produce any changes in starsstability

We follow a very simple way to study the effect ofquantum gravity in a different stage of stellar evolution Itis found that there is no change in pressure or temperaturein the initial stage where the pressure is coming fromclassical ions In the second stage after stardom contractingwhere degenerate electrons control the central pressure thequantum gravity leads to a decrease in the central maximumtemperature and an increase in the minimum and maximummasses of the main-sequence stars This result shows that thequantum gravity has an effect on the building process of theuniverse and it will have a more strong effect in highly denseobjects

A mass-radius relation for white dwarf is also investi-gated The results show that there is an elevation in the starradius that is proportional to the mass of the star For thecase of high dense stars the elevation in the radius goesto infinity as the mass of the star reaches Chandrasekharlimit This means that quantum gravity resists gravitationalcollapse This result goes in parallel with that found in [19]Unfortunately quantum gravity behaves in a different wayfrom the prediction of the usual model for white dwarfThe current observation indicates that white dwarfs have

smaller radii than theoretical predictions [30ndash32] This is notconsistent with our results But this result may be reasonableif we add this perturbed term which comes from quantumgravity to the other perturbed terms that are coming fromother physical aspects such as Coulomb corrections latticeenergy correction due to rotation and magnetic field (whichmay be much larger than the correction due to GUP) andmore realistic density distribution for electronic gasThe sumof all these terms may be consistent with the current data

On the other hand implications of the GUP in variousfields such as high energy physics cosmology and blackholes have been studied In [13] potential experimentalsignatures in some familiar quantum systems are examined1205730is a numerical parameter that quantifies the modification

strength if we assume that parameter is of the order of unityThis assumption renders quantum gravity effects too small tobe measured [13] But the 120573

0dependent terms are important

when energies (momenta) are comparable to Planck energy(momenta) and length is comparable to the Planck lengthIf we cancel this choice the current experiments predictlarge upper bounds on it which are compatible with currentobservations It is agreed that such an intermediate lengthscale 119897inter ≃ 119897119875119897radic1205730 cannot exceed the electroweak lengthscale sim1017119897

119875119897[13] This means that 120573

0le 1034 The authors in

[13 14] determined a bound in 1205730parameter which proves

that GUP could be measured in a low energy system likeLandau levels Lamb shift potential barrier and potentialstep They calculated the intermediate length scales such that119897inter ≃ 10

18119897119875119897 1025119897119875119897 and 1010119897

119875119897of which the first two are

far bigger than the electroweak scale and the last is smallerbut may get further constrained with increased accuraciesIn this paper the boundary values of 120573

0parameter in cases

of main-sequence stars are 1048 1047 and 1048 and thosefromwhite dwarf are 1044 and 1043These can be converted tointermediate length scales 119897inter asymp 10

24119897119875119897for main-sequence

stars and 119897inter asymp 1022119897119875119897for white dwarf They are far bigger

than the electroweak scale but it is approximately compatiblewith the results addressed by [14] in one case and it can beconsidered reasonable values in the astrophysical regime

Appendix

In general the gravitational field inside the star can bedescribed by a gravitational potential 120601 which is a solutionof the Poisson equation

nabla2120601 = 4120587119866120588 (A1)

For spherical symmetry Poisson equation reduces to

1

1199032119889

119889119903(1199032 119889120601

119889119903) = 4120587119866120588 (A2)

In hydrostatic equilibrium besides (A2) it requires

119889119875

119889119903= minus

119889120601

119889119903120588 (A3)

8 Advances in High Energy Physics

For a homogeneous gaseous sphere if we assume thatthe pressure of the star can be calculated by the modifiedpolytropic relation

119875 = 1198701120588Γ1 + 120573119870

2120588Γ2 (A4)

The prototype constants 1198701and 119870

2are fixed and can be

calculated from nature constants If Γ1and Γ2are not equal to

unity use (A4) into (A3) and integrate with the boundaryconditions 120601 = 0 and 120588 = 0 at the surface of the sphere onegets

120601 = minus1198701(1 + 119899

1) 12058811198991 minus 120573119870

2(1 + 119899

2) 12058811198992 (A5)

Solving this equation for 120588 keeping only the termsproportional to 120573 and using 119899

119895= 1(Γ

119895minus 1) then

120588 = [minus120601

1198701(1 + 119899

1)]

1198991

minus 12057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A6)

where

119898 =1198991

1198992

+ 1198991minus 1 (A7)

Using 120588 from (A6) into (A2) one gets

1198892120601

1198891199032+2

119903

119889120601

119889119903

= 4120587119866[minus120601

1198701(1 + 119899

1)]

1198991

minus 412058711986612057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A8)

Now let us define

119911 = 119860119903

1198602=

4120587119866

1198701198991

1(1 + 119899

1)1198991(minus120601119888)1198991minus1

120596 =120601

120601119888

(A9)

where 120601119888is the potential at the center it can be defined like

(A5) as

120601119888= minus1198701(1 + 119899

1) 12058811198991

119888minus 1205731198702(1 + 119899

2) 12058811198992

119888 (A10)

Using (A5) (A10) to define 120596 as a function of densityone finds that

120596 =120601

120601119888

= (120588

120588119888

)

11198991

+ 1205731198702(1 + 119899

2)

1198701(1 + 119899

1)(1205881198991

1205881198992119888

)

111989911198992

sdot [1 minus (120588

120588119888

)

(1198992minus1198991)11989911198992

]

(A11)

Using (A9) into (A8) one gets

1198892120596

1198891199112+2

119911

119889120596

119889119911+ 1205961198991 + 120578120596

119898= 0 (A12)

where

120578 =12057311989911198702(1 + 119899

2)

120601119888

(minus120601119888

1198602

4120587119866)

11198992

(A13)

120578 is a dimensionless quantity Equation (A12) is the modifiedLane-Emden equationWe are interested in solutions that arefinite at the center 119911 = 0 So this equation can be solved withthe boundary conditions 120596(0) = 1 and 1205961015840(0) = 1 120596 can becalculated from (A12) and it can be used to determine themass density and pressure of the astrophysical objects as afunction of a radius

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by Benha university (httpwwwbuedueg) The author would like to thank the referees for theenlightening comments which helped to improve the paper

References

[1] D Amati M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1989

[2] M Maggiore ldquoA generalized uncertainty principle in quantumgravityrdquo Physics Letters B vol 304 no 1-2 pp 65ndash69 1993

[3] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999

[4] M Maggiore ldquoQuantum groups gravity and the generalizeduncertainty principlerdquo Physical Review D vol 49 no 10 pp5182ndash5187 1994

[5] MMaggiore ldquoThe algebraic structure of the generalized uncer-tainty principlerdquo Physics Letters B vol 319 no 1ndash3 pp 83ndash861993

[6] L J Garay ldquoQuantum gravity and minimum lengthrdquo Interna-tional Journal of Modern Physics A vol 10 no 2 pp 145ndash1661995

[7] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stoecker ldquoSignatures in the Planck regimerdquo PhysicsLetters B vol 575 no 1-2 pp 85ndash99 2003

[8] C Bambi and F R Urban ldquoNatural extension of the generalizeduncertainty principlerdquo Classical and Quantum Gravity vol 2Article ID 095006 2008

[9] A Kempf G Mangano and R B Mann ldquoHilbert spacerepresentation of the minimal length uncertainty relationrdquoPhysical Review D vol 52 no 2 pp 1108ndash1118 1995

[10] A Kempf ldquoNon-pointlike particles in harmonic oscillatorsrdquoJournal of Physics A Mathematical and General vol 30 no 6pp 2093ndash2101 1997

Advances in High Energy Physics 9

[11] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 Article ID 7691 1999

[12] S Hossenfelder ldquoMinimal length scale scenarios for quantumgravityrdquo Living Reviews in Relativity vol 16 no 2 2013

[13] S Das and E C Vagenas ldquoUniversality of quantum gravitycorrectionsrdquo Physical Review Letters vol 101 Article ID 2213014 pages 2008

[14] S Das and E C Vagenas ldquoPhenomenological implicationsof the generalized uncertainty principlerdquo Canadian Journal ofPhysic vol 87 no 3 pp 233ndash240 2009

[15] I Pikovski M R Vanner M Aspelmeyer M S Kim and CBrukner ldquoProbing planck-scale physics with quantum opticsrdquoNature Physics vol 8 no 5 pp 393ndash397 2012

[16] F Marin F Marino M Bonaldi et al ldquoGravitational bardetectors set limits to Planck-scale physics on macroscopicvariablesrdquo Nature Physics vol 9 no 2 pp 71ndash73 2013

[17] T V Fityo ldquoStatistical physics in deformed spaces withminimallengthrdquo Physics Letters A vol 372 no 37 pp 5872ndash5877 2008

[18] M Abbasiyan-Motlaq and P Pedram ldquoTheminimal length andquantum partition functionsrdquo Journal of Statistical MechanicsTheory and Experiment vol 2014 Article ID P08002 2014

[19] P Wang H Yang and X Zhang ldquoQuantum gravity effectson statistics and compact star configurationsrdquo Journal of HighEnergy Physics vol 2010 article 43 2010

[20] O Bertolami and C A D Zarro ldquoTowards a noncommutativeastrophysicsrdquo Physical Review D vol 81 no 2 Article ID025005 8 pages 2010

[21] L D Landau and E M Lifshitz Statistical Physics (Courseof Theoretical Physics Volume 5) Butterworth-HeinemannOxford UK 3rd edition 1980

[22] L N Chang D Minic N Okamura and T Takeuchi ldquoExactsolution of the harmonic oscillator in arbitrary dimensions withminimal length uncertainty relationsrdquo Physical Review D vol65 Article ID 125027 2002

[23] A F Ali and M Moussa ldquoTowards thermodynamics withgeneralized uncertainty principlerdquo Advances in High EnergyPhysics vol 2014 Article ID 629148 7 pages 2014

[24] M Moussa ldquoQuantum gases and white dwarfs with quantumgravityrdquo Journal of StatisticalMechanicsTheory andExperimentvol 2014 no 11 Article ID P11034 14 pages 2014

[25] S Chandrasekhar An Introduction to the Study of StellarStructure Dover Mineola NY USA 1967

[26] T PadmanabhanTheoretical Astrophysics Volume I Astrophys-ical Processes Cambridge University Press Cambridge UK2000

[27] T Padmanabhan Theoretical Astrophysics vol 2 of Stars andStellar Systems Cambridge University Press Cambridge UK2001

[28] R Kippenhahn and A Weigert Stellar Structure and EvolutionSpringer Berlin Germany 1994

[29] G W Collins Fundamentals of Stellar Astrophysics 2003[30] G J Mathews I-S Suh B OrsquoGorman et al ldquoAnalysis of white

dwarfs with strange-matter coresrdquo Journal of Physics G Nuclearand Particle Physics vol 32 no 6 pp 747ndash759 2006

[31] J L Provencal H L Shipman D Koester F Wesemael and PBergeron ldquoProcyon B outside the iron boxrdquo The AstrophysicalJournal vol 568 no 1 pp 324ndash334 2002

[32] A Camacho ldquoWhite dwarfs as test objects of Lorentz viola-tionsrdquo Classical and Quantum Gravity vol 23 no 24 ArticleID 7355 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

2 Advances in High Energy Physics

phenomenological implications of GUP in quantum gases arenumerous for example [17 18] In [19] the quantum gravityinfluences on the statistic properties classical nonrelativisticultrarelativistic and photon gases were addressed Small cor-rections to the energy and entropy at low temperatures werefound and some modifications to the equations of state areemployed The authors considered two configurations Thefirst is the white dwarf which is composed of a nonrelativisticcold nuclei the mass of the star was calculated It is foundthat there is a quantum gravity correction that depends onthe number density of the star that leads to an increase in themass of the star and the quantum gravity tends to resist thecollapse of the star The other configuration is that the staris almost composed of ultrarelativistic particles They foundthat the increase in the star energy leads to monotonicalblowup of Fermi pressure which resists the gravitationalcollapse

We will use this form of GUP (1) to investigate theimplications of the quantum gravity on statistical propertiesof quantum gases After redefining the new phase space wewill put the partition function in a form that is consistentwith GUP and then rewrite the thermodynamic propertiesfor quantum gases namely photons nonrelativistic idealgases and fermions in degenerate state Using these resultswe will follow the proposal which was investigated in [20] instudying the stability ofmain-sequence stars andwhite dwarfAlso a simple method is used to study the effect of quantumgravity in a different stage of stellar evolution and a modifiedmass-radius relation for the white dwarf is calculated andboundaries in the quantum gravity perturbative terms will bedetermined

2 Statistical Mechanics and GUP

In this section we investigate the GUP modification in theformalism of the grand-canonical ensemble If we consider119873 particles with energy states 119864

119894 for each state there is 119899

119894

particles in that stateThe grand-canonical partition functionis obtained from the sum over all of the states

Z = sum119899119894

prod119894

[119890(120583minus119864119894119870119861119879)]

119899119894 (4)

where 119879 is the temperature 119870119861is the Boltzmann constant

and 120583 is the chemical potential The connection to thermo-dynamics is obtained after introducing the partition functiondefined as [21]

lnZ = sum119894

ln [1 + 119886119911119890minus119864119894119870119861119879] (5)

where 119911 = exp(120583119870119861119879) and the parameter 119886 takes value

depending on the considered particles If they are fermions119886 takes +1 value and if the considered particles are bosons 119886takes minus1 value

By considering large volume the summation turns tobe sum119894rarr int119889

31199091198893119901((2120587ℏ)

3(1 + 120573119901

2)3) where we are

considering the invariant phase space in the existence ofthe GUP that has been derived in [22] We should put thegrand canonical partition function for these systems in a form

consistent with GUP frameworkTheGUP can be consideredin the phase space analysis by two equivalent pictures (a)considering deformed commutation relations (ie deformedthe measure of integration) simultaneously with the nonde-formed Hamiltonian function or (b) calculating canonicalvariables on the GUP-corrected phase space which satisfy thestandard commutative algebra (ie nondeformed standardmeasure of integration) but the Hamiltonian function nowgets deformed These two pictures are related to each otherby the Darboux theorem in which it is quite possible tofind canonical coordinates on the symplectic manifold whichsatisfy standard Heisenberg algebra [23 24] In this paper weare going to consider deformed measure of integration withnondeformedHamiltonian function Based on the new phasespace the partition function is

lnZ =119881

(2120587ℏ)3

119892

119886int ln [1 + 119886119911119890minus119864119896119861119879]

1198893119901

(1 + 1205731199012)3

(6)

with 119864 = (11990121198882 +11989821198884)12 Pressure number of particles andinternal energy will be determined by the relations

119875 = 119870119861119879120597

120597119881lnZ

119899 = 119870119861119879120597

120597120583lnZ|119879119881

119880 = 1198701198611198792 120597

120597119879lnZ|119911119881

(7)

We will expand all the terms that contain 120573 and wewill keep the terms that are proportional to sim120573 only Thisapproximation breaks down around maximum measurableenergies such that 1205731199012 asymp 1 In this case we need an exactsolution But one can trust the perturbative solution where1205731199012≪ 1 Now we will use the modified partition function

to calculate the thermodynamics of photons nonrelativisticideal gases and degenerate fermions

21 Photon Gas For a gas of photons 119892 = 2 119886 = minus1 and120583 = 0 The energy density is given using (7) by

119906 =119880

119881=

1

1205872ℏ3int

119864

119890119864119870119861119879 minus 1

1199012119889119901

(1 + 1205731199012)3 (8)

where119864 = 119901119888 Use the approximation 1(1+1205731199012)3 ≃ 1minus31205731199012one gets

119906 =119888

1205872ℏ3int

119889119901

119890119888119901119870119861119879 minus 1(1199013minus 3120573119901

5)

=4120590

1198881198794minus 120573

8

21

12058741198706

119861

ℏ311988851198796

(9)

where 120590 = 1205872119896460ℏ31198882 is the Stefan-Boltzmann constant

For 120573 = 0 one can recover the usual Stefan-Boltzmann law

Advances in High Energy Physics 3

And the pressure of system (7) is given by

119875 =119888

1205872ℏ3int

119889119901

119890119888119901119870119879 minus 1[1

31199013minus3

51205731199015]

=4120590

31198881198794minus 120573

8

105

12058741198706

119861

ℏ311988851198796

(10)

It is clear that the modification in phase space decreasesthe available number ofmicrostates which holds the thermo-dynamic properties and hence decreases the internal energyand pressure of the system In order to obtain the pressure-energy density relation divide (10) by (9)

119875

119906=1

3+ 120573

16

21(119870119861120587

119888)2

1198792 (11)

If we remove the quantum gravity effect we can recoverthe usual relation 119906 = 3119875 In order to obtain the equation ofstate (EOS) 119875 = 119875(119906) we first solve (9) for the temperature toget

119879 = (119888

4120590)14

11990614+ 120573

10

7(120587119870119861

119888)2

(119888

4120590)34

11990634 (12)

Substitute (12) into (11) to get the first order correction to theEOS

119875 =119906

3+ 120573

16

21(151205872ℏ3

119888)

12

11990632 (13)

Equation (13) represents a modified equation of state dueto generalized uncertainty principle

22 Nonrelativistic Ideal Gases Let us now consider the effectof quantum gravity on nonrelativistic ideal gases In orderto do that the partition function for one particle with themodified phase space is given by

1199111=

119881

21205872ℏ3int 119890minus1199012(2119898119870119861119879)

1199012119889119901

(1 + 1205731199012)3 (14)

The total partition function is given by 119885 = (1119873)119911119873

1

by which we can derive the free energy for the system withthe equation 119865 = 119870

119861119879 ln119885 Using Stirlingrsquos formula ln119873 =

119873 ln119873 minus119873 we can determine the pressure

119875 = minus120597119865

120597119881

10038161003816100381610038161003816100381610038161003816119879119873=119873119870119861119879

119881 (15)

So we recover the usual EOS for nonrelativistic ideal gas119875 = 119899119870

119861119879 where 119899 = 119873119881 is the number density which

shows that the quantum gravity does not play here Thesame behavior is established in [23] Suffice calculation ofthe pressure at this point which we need and the otherthermodynamic properties of the nonrelativistic ideal gasescan be obtained easily by solving the integration in (14) andthen calculating the free energy

23 Degenerate Fermion Gas Fermions have a property thatatmost one can occupy each quantum state At119879 = 0 all stateswith energy less than Fermi energy level are occupied and allstates with greater energy are empty So the total energy ofthe system is the lowest possible consistent with the exclusionprincipleThe energy of the highest occupied state at absolutezero is the Fermi energy and corresponding momentum isreferred to as the Fermi momentum 119901

119891 In this case the

distribution function goes to step function According tothese assumptions and using (7) the particle number densityis given by

119899 =1

1205872ℏ3int119875119891

0

1199012119889119901

(1 + 1205731199012)3=

1

1205872ℏ3[1

31199013

119891minus 120573

3

51199015

119891] (16)

Solving this equation for Fermi momentum one gets

119901119891= (3120587

2ℏ3119899)13

+ 1205733

5(31205872ℏ3119899) (17)

The second term ensures that the Fermi momentumis increased due to the presence of quantum gravity andhence Fermi energy Also the energy density for fermions indegenerate state is given by

119906 =1

1205872ℏ3int119875119891

0

[11990121198882+ 11989821198884]12 119901

2119889119901

(1 + 1205731199012)3 (18)

Using the substitutional

119901 = 119898119888 sinh119909 (19)

the energy density integration will be

119906 =11989841198885

1205872ℏ3int119909119891

0

[sinh2119909 + (1 minus 312057311989821198882) sinh4119909

minus 312057311989821198882sinh6119909] 119889119909 = 119898

41198885

1205872ℏ3[1198651(119910) + (1

minus 312057311989821198882) 1198652(119910) minus 3120573119898

211988821198653(119910)]

(20)

where

119910 =119901119891

119898119888= sinh119909

119891(21)

with

1198651(119910) = minus

1

2ln [119910 + radic1 + 1199102] + 1

2119910radic1 + 1199102

1198652(119910) =

3

8ln [119910 + radic1 + 1199102]

+1

4119910radic1 + 1199102 [119910

2minus3

2]

1198653(119910) = minus

5

16ln [119910 + radic1 + 1199102]

+ 119910radic1 + 1199102 [5

16+ 1199102(1199102

6minus5

24)]

(22)

4 Advances in High Energy Physics

By the same way we can derive the pressure from (7)after making use of the integration by parts and functionexpansion one finds that

119875 =1198882

1205872ℏ3int119901119891

0

119889119901

[11990121198882 + 11989821198884]12[1

31199013minus3

51205731199015]

=11989841198885

1205872ℏ3[1

31198652(119910) minus

3

5120573119898211988821198653(119910)]

(23)

Equation (23) represents the ground state pressureThereis a considerable change due to the presence of the function1198653(119910) and 119910 itself depends on modified Fermi momentum

We should notice that the value of 119910 is increased due to theincreasing in Fermi momentum this leads to a considerablegrowing in the pressure (23) The change in pressure of theradiation and degenerate fermions will affect the polytropicrelations for astroobjects In the next section the modifiedpressure effect in the some stars stability and some otherapplications will be investigated exactly

3 Applications to Astrophysical Objects

31 Main-Sequence Stars Most main-sequence stars havea mass in the range about 10 to 50 solar mass They arecomposed by a mixture of nonrelativistic gas and radiationthat are stood in hydrostatic equilibrium by gravity [25] Thetotal pressure is 119875

119888= 119875119892+ 119875119903 where 119875

119888 119875119892 and 119875

119903are center

star gas and radiation pressure respectively We can write120573119888= 119875119892119875119888and 1 minus 120573

119888= 119875119903119875119888 By definition (1 minus 120573

119888) and

120573119888are the fractional contribution of radiation and gas to the

central pressure of the star and both are less than one then

119875119892

120573119888

=119875119903

1 minus 120573119888

(24)

The classical gas pressure can be expressed using (15) inthe form

119875119892=120588119888119870119861119879

119898 (25)

We used the relation 119899 = 120588119888119898 120588119888is the central density

of the star and 119898 is the average mass of the gas particle thatconstructs the star defined as 119898 = 119906

119873119898119873where 119906

119873is the

mean molecular weight and 119898119873is the nucleon mass Using

(10) (25) into (24) one finds that

120588119888119870119861

120573119888119898119879119888=

4120590

3119888 (1 minus 120573119888)1198794

119888minus8

105120573

12058741198706

119861

ℏ31198885 (1 minus 120573119888)1198796

119888 (26)

where 119879119888is the central temperature of the star We can find

the temperature by solving the above equation

119879119888= (

1 minus 120573119888

120573119888119898

3119888119870119861

4120590)

13

12058813

119888

+ 1205731

70

(1 minus 120573119888)

120573119888119898

12058741198707

119861

ℏ311988831205902120588119888

(27)

Using this expression to calculate the star pressure from therelation 119875

119888= 119875119892120573119888= (120588119888119870119861120573119888119898)119879119888one gets

119875119888= 119870112058843

119888+ 12057311987021205882

119888= 119875119904UP + 119901119904GUP (28)

1198701=(1 minus 120573

119888)13

(120573119888)43

(31198881198704

119861

41205901198984)

13

1198702=1

70

(1 minus 120573119888)

(120573119888)2

12058741198708

119861

ℏ3119888312059021198982

(29)

The first part of (28) is the ordinary polytropic relationwith a a polytropic exponent Γ = 43 (polytropic index 119899

119901=

3) [26] perturbed by a polytrope Γ = 2 (polytropic index 119899119901=

1) where 119899119901= 1(Γ minus 1)

The best example may be used to estimate GUP pertur-bation is our sun If we assume that 120573

119888is constant overall the

stars and its chemical composition is not changed so 119906119873is

constant For our sun 1 minus 120573119888= 10minus3 and 119906

119873= 0829 [27] and

central density 120588119888= 153 times 10

5 kgm3 then

119901119904GUP119875119904UP

= 18 times 10minus491205730 (30)

The perturbed pressure should not exceed the originalone such that 119901

119904GUP119875119904UP ≪ 1 so this equation suggests thatthe upper value of 120573

0should be 120573

0lt 1048 This bound is far

weaker than that set by electroweak measurementsAccording to (28) the quantum gravity modifies the

polytropic relation by adding a new term proportional to 1205882This suggests a new modification in Lane-Emden equation[28] The modified Lane-Emden equation is established inthe appendix Studying modified Lane-Emden equation andits solution is not the scope of this paper so results that areinvestigated by the this equationwill be postponed to anotherresearch

Main-Sequence Stars Stability In order to discuss the stabilityconcept the energy of a Newtonian polytrope systems withpressure 119875 can be expressed in terms of the internal energyand gravitational potential such that

119864 = 1198961119875119881 minus 119896

2

1198661198722

119877 (31)

where 1198961and 1198962are constants119866 is the gravitational constant

and 119872 and 119877 are the mass and radius of the star At hightdensities it is important to take into account the generalrelativistic effects in star stability To deal with it a generalrelativistic correction term should be added to (31) in lowestorder approximation namely 119864corr = minus119896

3(119866119888)21198727312058823

119888

where 1198963is constant that depends on the actual distribution

of matter According to [27] for 119899119875= 3 polytrope the

condition of stability is Γ gt 43 + 225(1198661198721198882119877) We can seethat addition of a relativistic correction contracts the regionof stability or increases the critical value of Γ In our case119866119872119888

2119877 sim 10

minus6 so this correction can be neglected So weexpect that the new term in a polytropic equation (28) with

Advances in High Energy Physics 5

119899119901= 1 which has a positive contribution will not affect the

problem of stability since Γ gt 2Using (28) into (31) the energy ofmain-sequence starswill

be

119864 = 119862111987212058813

119888+ 1205731198622119872120588119888minus 11989621198725312058813

119888 (32)

where 1198621and 119862

2are constants The mass119872 can be obtained

by using the equilibrium condition 120597119864120597120588119888= 0 the result is

119872 = [3

1198963

(1

31198621+ 120573119862212058823

119888)]

32

(33)

Then

119889 ln119872119889 ln 120588119888

= 12057331198622

1198621

12058823

119888 (34)

Equation (34) ensures that 119889 ln119872119889 ln 120588119888gt 0 which

is the stability condition [27] for main-sequence objectThese results show that the generalization process for theuncertainty principle does not change the stability of themain-sequence objects

Minimum and MaximumMasses for Main-Sequence Stars Inthis part we will discuss the evolution stage of main-sequencestars in the presence of quantum gravity to derive theminimum and maximum masses of these objects Initiallyin the stage of star forming the star must has mass enoughto generate a central temperature which is high enoughfor thermonuclear fusion At this stage the central pressurecomes by the electrons and ions which are forming an idealclassical gas their pressure can be expressed as in (25) Thestar will be made if this pressure is close to the pressureneeded to support the system It is known that the upper limitrelation between the pressure and density at the center of thestar is given by [29]

119875119888le (

120587

6)13

11986612058843

11988811987223 (35)

This relation is valid for any homogeneous star in whichthe mass is concentrated towards the center Equating (25)and (35) one can get the initial central temperature in thisstage

119870119861119879119888le (

120587

6)13

11989811986612058813

11988811987223 (36)

We can see that the quantum gravity does not appearin the picture As shown in this equation the central tem-perature will rise as the central density increases with starcontraction The contracting temperature continues to riseuntil the electrons at the center become fully degenerate andoccupy the lowest energy states in accordance with exclusionprinciple At this stage at the center the pressure comes fromthe electrons that are in degenerate state plus the pressurefrom the classical ions Quantum gravity will not change theclassical ions pressure according to (15) but a nonrelativisticdegenerate electrons pressure will change according to (23)

In nonrelativistic limit where 119910 ≪ 1 the degenerate pressure(23) will be

119875non-relat0

=1

15

1198984

1198901198884

1205872ℏ31199105minus3

35

1198986

1198901198887

1205872ℏ31199107 (37)

Use (17) (21) in above equation and add the result to (15)finally we can write the central pressure at this stage as

119875119888=

119886

5119898119890

11989953

119890+ 120573

121198862

35119898119890

11989973

119890+ 119899119894119870119861119879119888 (38)

where 119886 = (31312058723ℏ)2 The number density of the electrons119899119890and ions 119899

119894can be expressed in terms of central density

simply 119899119890= 119899119894= 120588119888119898119867 where 119898

119867is the mass of hydrogen

atom A hydrostatic pressure is achieved if this pressureequals the pressure needed to support the mass (35) Thisleads to the central temperature being as follows

119870119861119879119888le (

120587

6)13

1198661198981198731198722312058813

119888minus119886

5119898119890

(120588119888

119898119867

)

23

minus 120573121198862

35119898119890

(120588119888

119898119867

)

43

(39)

The first term is associated with classical ions and secondand third terms with degenerate electronsThe last two termsbecome important and are increasing with central densityuntil the temperature will cease to rise as the mass contractsFrom this equation we can find the maximum value oftemperature

10038161003816100381610038161198701198611198791198881003816100381610038161003816max le 119911119872

43[1 minus

60

712057311989811989011991111987243] (40)

119911 =5

4(120587

6)23 119898119890

11988611989883

1198671198662 (41)

Equation (40) proves that the maximum central tem-perature decreases due to quantum gravity The contractingmass achieves stardom if that maximum central temperaturereaches ignition temperature for the thermonuclear fusion ofhydrogen Denote this ignition temperature by 119879ign then theminimummass for the star will be given by

119872min ge (119870119861119879ign

119911)

34

[1 +45

7120573119898119890119870119861119879ign] (42)

This relation shows that the quantum gravity effectincreases theminimummass of themain-sequence starsTheapproximation has ameaning if the second term in (42) is lessthan unity If we take the ignition temperature for hydrogento be about 15 times 106 K this suggests that the upper value of1205730lt 1047

The situation is changed if the radiation pressure takesa part in this stage of revolution The star will be disruptedif radiation becomes the dominant source for the internalpressure This property will set a limit to the mass of a main-sequence starThe pressure at the center of hotmassive stars isdue to electrons and ions (as a classical gases) and radiation

6 Advances in High Energy Physics

the pressure that is produced by these species is derived in(28) Comparing this with the pressure needed to support thestar (35) to get the maximummass one finds that

(120587

6)13

11986611987223

max ge 1198701 + 120573119870212058823

119888 (43)

which means that the maximummass is increased due to thepresence of quantum gravity This increase depends on thecentral density Again the second term in the right-hand sideof inequality (43) should be less than the first this suggeststhat 120573

0lt 1048

32White Dwarfs Thewhite dwarfs consist ofHelium atomsthat are almost in an ionization state The temperature of thestar is about 107 K so the electrons are in a relativistic energyregime and can be considered in a complete degenerate stateand the Helium nuclei do not play a part in first orderapproximation in the dynamic of the system Now we willdetermine the effect of GUP on white dwarf radius accordingto pressure modification expressed in (23) If the number ofHeliumnuclei is1198732 then there are119873 electrons Suppose that119872 = 2119873119898

119901is the mass of the star so the particle density of

the electron is

119899119904=3119872

8120587119898119901

1

1198773 (44)

Using (17) (44) into (21) one gets

119910 =11987213

119877+3

51205731198982

1198901198882119872

1198773 (45)

119872 =9120587

8

119872

119898119901

119877 =119898119890119888

ℏ119877

(46)

where 119877 is the radius of the star and 119898119901is the proton mass

The equilibrium configuration of white dwarf is caused bythe pressure of the degenerate electrons against Newtoniangravitational collapse which is due to the nuclei whichmeansthat

1198750=120581

4120587

1198661198722

1198774= 11987010158401198722

1198774

1198701015840=120581119866

4120587(8119898119901

9120587)

2

(119898119890119888

ℏ)4

(47)

We have two extreme cases as follows(i) When the electron gas is in a low density nonrelativis-

tic dynamics may be used such that 119910 ≪ 1 and (23) will leadto

1198750=4

51198701199105minus36

351205731198982

11989011988821198701199107

119870 =1198981198901198882

121205872(119898119890119888

ℏ)3

(48)

Equating (47) and (48) we got the equation

1198773

minus4

5

119870

1198701015840119872minus13

1198772

minus48

351205731198982

11989011988821198701015840

11987011987213

= 0 (49)

The solution of that equation is

119877 ≃4

5

119870

1198701015840119872minus13

+15

71205731198982

11989011988821198701015840

119870119872 (50)

This equation shows that the radius of the star is increas-ing the elevation depends on the mass of the star The usedapproximation is true if the second term in the right-handside of (50) is much less than the first one this will put a limitin the upper value of 120573

0such that it should be 120573

0lt 1044 we

used a white dwarf mass119872 = 041119872⊙[30]

(ii) When the electron gas is in a high density thatrelativistic effect comes to play strongly such that 119910 ≫ 1 Wecan write

1198750= 119870 (119910

4minus 1199102) minus 120573

18

51198701198982

1198901198882(1199106

3minus1199104

4) (51)

Using (51) into (47) and simplifying then

1198774

+ [1198701015840

11987011987243

minus (1 minus3

101205731198982

1198901198882)11987223

]1198772

minus6

51205731198982

119890119888211987243

= 0

(52)

The solution of that equation keeping only the terms thatproportional to sim120573 is

119877 ≃ 1198770[1 + 120573

3

10

1198982

1198901198882

1198772

0

(211987243

1198772

0

minus1

211987223

)] (53)

where

1198772

0= 11987223

minus1198701015840

11987011987243

(54)

Using (46) into (54) and solving for 119877 and letting 119877 rarr

119877Ch one gets

119877Ch =(9120587)13

2

ℏ

119898119890119888(119872

119898119901

)

13

[1 minus (119872

119872Ch)

23

]

12

119872Ch =9 (3120587)

12

64120581(ℏ119888

1198661198982119901

)

32

119898119901

(55)

119877Ch is the unmodified white dwarf radius or Chan-drasekhar radius limit and 119872Ch is the Chandrasekhar limitmass of order 144119872

⊙ First part of (53) is the Chandrasekhar

radius and the second term is the radius correction due toGUP affect These analyses show that the increasing in starradius depends on [1 minus (119872119872Ch)

23]minus1 which formally goes

to infinity as119872 rarr 119872Ch It is clear that the quantum gravityeffect becomes significant when mass of the white dwarfgets very close to the Chandrasekhar limit in high density

Advances in High Energy Physics 7

stars Equations (23) (50) and (53) attract attention to anoticeable point that the growing in star radius ensures thatquantum gravity resists gravitational collapse by increasingthe electrons degeneracy pressure These results are consis-tentwith that established in [19]Our approximationwill havea meaning if the second term in the right-hand side in (53)is less than one this puts an upper limit for 120573

0such that

1205730lt 1043 We have used119872 = 059119872

⊙[30]

We should also notice that in the limit 119910 ≫ 1 inultrarelativistic regime the function 119865

3(119910) behaves like sim1199106

which almost goes like 1205882 This result is consistent withpolytrope behavior in which Γ = 2 that is correspondingto stability range of the star So we can conclude that thisapproach of quantum gravity does not produce any instabilityto white dwarfs

4 Discussion

Various approaches to quantum gravity such as string the-ory black hole physics and doubly special relativity pre-dict a considerable modification in Heisenberg uncertaintyprinciple to be a generalized uncertainty principle Thismodification leads to a change in the energy-momentumdispersion relation and in the physical phase space In thispaper we studied the modification due to a generalizeduncertainty principle (1) in the statistical and thermodynamicdynamic properties for photons nonrelativistic ideal gasesand degenerate fermions There is a considerable decreasein the number of accessible microstates in quantum gravitybackgroundThis approach leads to a decrease in the internalenergy and pressure of photon gas Pressure and internalenergy of fermions in degenerate state are increased andpressure of nonrelativistic ideal gases is not changed Theseresults are used in studying the stability of themain-sequencestars and white dwarfs It is found that this approach ofquantum gravity does not produce any changes in starsstability

We follow a very simple way to study the effect ofquantum gravity in a different stage of stellar evolution Itis found that there is no change in pressure or temperaturein the initial stage where the pressure is coming fromclassical ions In the second stage after stardom contractingwhere degenerate electrons control the central pressure thequantum gravity leads to a decrease in the central maximumtemperature and an increase in the minimum and maximummasses of the main-sequence stars This result shows that thequantum gravity has an effect on the building process of theuniverse and it will have a more strong effect in highly denseobjects

A mass-radius relation for white dwarf is also investi-gated The results show that there is an elevation in the starradius that is proportional to the mass of the star For thecase of high dense stars the elevation in the radius goesto infinity as the mass of the star reaches Chandrasekharlimit This means that quantum gravity resists gravitationalcollapse This result goes in parallel with that found in [19]Unfortunately quantum gravity behaves in a different wayfrom the prediction of the usual model for white dwarfThe current observation indicates that white dwarfs have

smaller radii than theoretical predictions [30ndash32] This is notconsistent with our results But this result may be reasonableif we add this perturbed term which comes from quantumgravity to the other perturbed terms that are coming fromother physical aspects such as Coulomb corrections latticeenergy correction due to rotation and magnetic field (whichmay be much larger than the correction due to GUP) andmore realistic density distribution for electronic gasThe sumof all these terms may be consistent with the current data

On the other hand implications of the GUP in variousfields such as high energy physics cosmology and blackholes have been studied In [13] potential experimentalsignatures in some familiar quantum systems are examined1205730is a numerical parameter that quantifies the modification

strength if we assume that parameter is of the order of unityThis assumption renders quantum gravity effects too small tobe measured [13] But the 120573

0dependent terms are important

when energies (momenta) are comparable to Planck energy(momenta) and length is comparable to the Planck lengthIf we cancel this choice the current experiments predictlarge upper bounds on it which are compatible with currentobservations It is agreed that such an intermediate lengthscale 119897inter ≃ 119897119875119897radic1205730 cannot exceed the electroweak lengthscale sim1017119897

119875119897[13] This means that 120573

0le 1034 The authors in

[13 14] determined a bound in 1205730parameter which proves

that GUP could be measured in a low energy system likeLandau levels Lamb shift potential barrier and potentialstep They calculated the intermediate length scales such that119897inter ≃ 10

18119897119875119897 1025119897119875119897 and 1010119897

119875119897of which the first two are

far bigger than the electroweak scale and the last is smallerbut may get further constrained with increased accuraciesIn this paper the boundary values of 120573

0parameter in cases

of main-sequence stars are 1048 1047 and 1048 and thosefromwhite dwarf are 1044 and 1043These can be converted tointermediate length scales 119897inter asymp 10

24119897119875119897for main-sequence

stars and 119897inter asymp 1022119897119875119897for white dwarf They are far bigger

than the electroweak scale but it is approximately compatiblewith the results addressed by [14] in one case and it can beconsidered reasonable values in the astrophysical regime

Appendix

In general the gravitational field inside the star can bedescribed by a gravitational potential 120601 which is a solutionof the Poisson equation

nabla2120601 = 4120587119866120588 (A1)

For spherical symmetry Poisson equation reduces to

1

1199032119889

119889119903(1199032 119889120601

119889119903) = 4120587119866120588 (A2)

In hydrostatic equilibrium besides (A2) it requires

119889119875

119889119903= minus

119889120601

119889119903120588 (A3)

8 Advances in High Energy Physics

For a homogeneous gaseous sphere if we assume thatthe pressure of the star can be calculated by the modifiedpolytropic relation

119875 = 1198701120588Γ1 + 120573119870

2120588Γ2 (A4)

The prototype constants 1198701and 119870

2are fixed and can be

calculated from nature constants If Γ1and Γ2are not equal to

unity use (A4) into (A3) and integrate with the boundaryconditions 120601 = 0 and 120588 = 0 at the surface of the sphere onegets

120601 = minus1198701(1 + 119899

1) 12058811198991 minus 120573119870

2(1 + 119899

2) 12058811198992 (A5)

Solving this equation for 120588 keeping only the termsproportional to 120573 and using 119899

119895= 1(Γ

119895minus 1) then

120588 = [minus120601

1198701(1 + 119899

1)]

1198991

minus 12057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A6)

where

119898 =1198991

1198992

+ 1198991minus 1 (A7)

Using 120588 from (A6) into (A2) one gets

1198892120601

1198891199032+2

119903

119889120601

119889119903

= 4120587119866[minus120601

1198701(1 + 119899

1)]

1198991

minus 412058711986612057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A8)

Now let us define

119911 = 119860119903

1198602=

4120587119866

1198701198991

1(1 + 119899

1)1198991(minus120601119888)1198991minus1

120596 =120601

120601119888

(A9)

where 120601119888is the potential at the center it can be defined like

(A5) as

120601119888= minus1198701(1 + 119899

1) 12058811198991

119888minus 1205731198702(1 + 119899

2) 12058811198992

119888 (A10)

Using (A5) (A10) to define 120596 as a function of densityone finds that

120596 =120601

120601119888

= (120588

120588119888

)

11198991

+ 1205731198702(1 + 119899

2)

1198701(1 + 119899

1)(1205881198991

1205881198992119888

)

111989911198992

sdot [1 minus (120588

120588119888

)

(1198992minus1198991)11989911198992

]

(A11)

Using (A9) into (A8) one gets

1198892120596

1198891199112+2

119911

119889120596

119889119911+ 1205961198991 + 120578120596

119898= 0 (A12)

where

120578 =12057311989911198702(1 + 119899

2)

120601119888

(minus120601119888

1198602

4120587119866)

11198992

(A13)

120578 is a dimensionless quantity Equation (A12) is the modifiedLane-Emden equationWe are interested in solutions that arefinite at the center 119911 = 0 So this equation can be solved withthe boundary conditions 120596(0) = 1 and 1205961015840(0) = 1 120596 can becalculated from (A12) and it can be used to determine themass density and pressure of the astrophysical objects as afunction of a radius

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by Benha university (httpwwwbuedueg) The author would like to thank the referees for theenlightening comments which helped to improve the paper

References

[1] D Amati M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1989

[2] M Maggiore ldquoA generalized uncertainty principle in quantumgravityrdquo Physics Letters B vol 304 no 1-2 pp 65ndash69 1993

[3] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999

[4] M Maggiore ldquoQuantum groups gravity and the generalizeduncertainty principlerdquo Physical Review D vol 49 no 10 pp5182ndash5187 1994

[5] MMaggiore ldquoThe algebraic structure of the generalized uncer-tainty principlerdquo Physics Letters B vol 319 no 1ndash3 pp 83ndash861993

[6] L J Garay ldquoQuantum gravity and minimum lengthrdquo Interna-tional Journal of Modern Physics A vol 10 no 2 pp 145ndash1661995

[7] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stoecker ldquoSignatures in the Planck regimerdquo PhysicsLetters B vol 575 no 1-2 pp 85ndash99 2003

[8] C Bambi and F R Urban ldquoNatural extension of the generalizeduncertainty principlerdquo Classical and Quantum Gravity vol 2Article ID 095006 2008

[9] A Kempf G Mangano and R B Mann ldquoHilbert spacerepresentation of the minimal length uncertainty relationrdquoPhysical Review D vol 52 no 2 pp 1108ndash1118 1995

[10] A Kempf ldquoNon-pointlike particles in harmonic oscillatorsrdquoJournal of Physics A Mathematical and General vol 30 no 6pp 2093ndash2101 1997

Advances in High Energy Physics 9

[11] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 Article ID 7691 1999

[12] S Hossenfelder ldquoMinimal length scale scenarios for quantumgravityrdquo Living Reviews in Relativity vol 16 no 2 2013

[13] S Das and E C Vagenas ldquoUniversality of quantum gravitycorrectionsrdquo Physical Review Letters vol 101 Article ID 2213014 pages 2008

[14] S Das and E C Vagenas ldquoPhenomenological implicationsof the generalized uncertainty principlerdquo Canadian Journal ofPhysic vol 87 no 3 pp 233ndash240 2009

[15] I Pikovski M R Vanner M Aspelmeyer M S Kim and CBrukner ldquoProbing planck-scale physics with quantum opticsrdquoNature Physics vol 8 no 5 pp 393ndash397 2012

[16] F Marin F Marino M Bonaldi et al ldquoGravitational bardetectors set limits to Planck-scale physics on macroscopicvariablesrdquo Nature Physics vol 9 no 2 pp 71ndash73 2013

[17] T V Fityo ldquoStatistical physics in deformed spaces withminimallengthrdquo Physics Letters A vol 372 no 37 pp 5872ndash5877 2008

[18] M Abbasiyan-Motlaq and P Pedram ldquoTheminimal length andquantum partition functionsrdquo Journal of Statistical MechanicsTheory and Experiment vol 2014 Article ID P08002 2014

[19] P Wang H Yang and X Zhang ldquoQuantum gravity effectson statistics and compact star configurationsrdquo Journal of HighEnergy Physics vol 2010 article 43 2010

[20] O Bertolami and C A D Zarro ldquoTowards a noncommutativeastrophysicsrdquo Physical Review D vol 81 no 2 Article ID025005 8 pages 2010

[21] L D Landau and E M Lifshitz Statistical Physics (Courseof Theoretical Physics Volume 5) Butterworth-HeinemannOxford UK 3rd edition 1980

[22] L N Chang D Minic N Okamura and T Takeuchi ldquoExactsolution of the harmonic oscillator in arbitrary dimensions withminimal length uncertainty relationsrdquo Physical Review D vol65 Article ID 125027 2002

[23] A F Ali and M Moussa ldquoTowards thermodynamics withgeneralized uncertainty principlerdquo Advances in High EnergyPhysics vol 2014 Article ID 629148 7 pages 2014

[24] M Moussa ldquoQuantum gases and white dwarfs with quantumgravityrdquo Journal of StatisticalMechanicsTheory andExperimentvol 2014 no 11 Article ID P11034 14 pages 2014

[25] S Chandrasekhar An Introduction to the Study of StellarStructure Dover Mineola NY USA 1967

[26] T PadmanabhanTheoretical Astrophysics Volume I Astrophys-ical Processes Cambridge University Press Cambridge UK2000

[27] T Padmanabhan Theoretical Astrophysics vol 2 of Stars andStellar Systems Cambridge University Press Cambridge UK2001

[28] R Kippenhahn and A Weigert Stellar Structure and EvolutionSpringer Berlin Germany 1994

[29] G W Collins Fundamentals of Stellar Astrophysics 2003[30] G J Mathews I-S Suh B OrsquoGorman et al ldquoAnalysis of white

dwarfs with strange-matter coresrdquo Journal of Physics G Nuclearand Particle Physics vol 32 no 6 pp 747ndash759 2006

[31] J L Provencal H L Shipman D Koester F Wesemael and PBergeron ldquoProcyon B outside the iron boxrdquo The AstrophysicalJournal vol 568 no 1 pp 324ndash334 2002

[32] A Camacho ldquoWhite dwarfs as test objects of Lorentz viola-tionsrdquo Classical and Quantum Gravity vol 23 no 24 ArticleID 7355 2006

Submit your manuscripts athttpwwwhindawicom

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Advances in High Energy Physics 3

And the pressure of system (7) is given by

119875 =119888

1205872ℏ3int

119889119901

119890119888119901119870119879 minus 1[1

31199013minus3

51205731199015]

=4120590

31198881198794minus 120573

8

105

12058741198706

119861

ℏ311988851198796

(10)

It is clear that the modification in phase space decreasesthe available number ofmicrostates which holds the thermo-dynamic properties and hence decreases the internal energyand pressure of the system In order to obtain the pressure-energy density relation divide (10) by (9)

119875

119906=1

3+ 120573

16

21(119870119861120587

119888)2

1198792 (11)

If we remove the quantum gravity effect we can recoverthe usual relation 119906 = 3119875 In order to obtain the equation ofstate (EOS) 119875 = 119875(119906) we first solve (9) for the temperature toget

119879 = (119888

4120590)14

11990614+ 120573

10

7(120587119870119861

119888)2

(119888

4120590)34

11990634 (12)

Substitute (12) into (11) to get the first order correction to theEOS

119875 =119906

3+ 120573

16

21(151205872ℏ3

119888)

12

11990632 (13)

Equation (13) represents a modified equation of state dueto generalized uncertainty principle

22 Nonrelativistic Ideal Gases Let us now consider the effectof quantum gravity on nonrelativistic ideal gases In orderto do that the partition function for one particle with themodified phase space is given by

1199111=

119881

21205872ℏ3int 119890minus1199012(2119898119870119861119879)

1199012119889119901

(1 + 1205731199012)3 (14)

The total partition function is given by 119885 = (1119873)119911119873

1

by which we can derive the free energy for the system withthe equation 119865 = 119870

119861119879 ln119885 Using Stirlingrsquos formula ln119873 =

119873 ln119873 minus119873 we can determine the pressure

119875 = minus120597119865

120597119881

10038161003816100381610038161003816100381610038161003816119879119873=119873119870119861119879

119881 (15)

So we recover the usual EOS for nonrelativistic ideal gas119875 = 119899119870

119861119879 where 119899 = 119873119881 is the number density which

shows that the quantum gravity does not play here Thesame behavior is established in [23] Suffice calculation ofthe pressure at this point which we need and the otherthermodynamic properties of the nonrelativistic ideal gasescan be obtained easily by solving the integration in (14) andthen calculating the free energy

23 Degenerate Fermion Gas Fermions have a property thatatmost one can occupy each quantum state At119879 = 0 all stateswith energy less than Fermi energy level are occupied and allstates with greater energy are empty So the total energy ofthe system is the lowest possible consistent with the exclusionprincipleThe energy of the highest occupied state at absolutezero is the Fermi energy and corresponding momentum isreferred to as the Fermi momentum 119901

119891 In this case the

distribution function goes to step function According tothese assumptions and using (7) the particle number densityis given by

119899 =1

1205872ℏ3int119875119891

0

1199012119889119901

(1 + 1205731199012)3=

1

1205872ℏ3[1

31199013

119891minus 120573

3

51199015

119891] (16)

Solving this equation for Fermi momentum one gets

119901119891= (3120587

2ℏ3119899)13

+ 1205733

5(31205872ℏ3119899) (17)

The second term ensures that the Fermi momentumis increased due to the presence of quantum gravity andhence Fermi energy Also the energy density for fermions indegenerate state is given by

119906 =1

1205872ℏ3int119875119891

0

[11990121198882+ 11989821198884]12 119901

2119889119901

(1 + 1205731199012)3 (18)

Using the substitutional

119901 = 119898119888 sinh119909 (19)

the energy density integration will be

119906 =11989841198885

1205872ℏ3int119909119891

0

[sinh2119909 + (1 minus 312057311989821198882) sinh4119909

minus 312057311989821198882sinh6119909] 119889119909 = 119898

41198885

1205872ℏ3[1198651(119910) + (1

minus 312057311989821198882) 1198652(119910) minus 3120573119898

211988821198653(119910)]

(20)

where

119910 =119901119891

119898119888= sinh119909

119891(21)

with

1198651(119910) = minus

1

2ln [119910 + radic1 + 1199102] + 1

2119910radic1 + 1199102

1198652(119910) =

3

8ln [119910 + radic1 + 1199102]

+1

4119910radic1 + 1199102 [119910

2minus3

2]

1198653(119910) = minus

5

16ln [119910 + radic1 + 1199102]

+ 119910radic1 + 1199102 [5

16+ 1199102(1199102

6minus5

24)]

(22)

4 Advances in High Energy Physics

By the same way we can derive the pressure from (7)after making use of the integration by parts and functionexpansion one finds that

119875 =1198882

1205872ℏ3int119901119891

0

119889119901

[11990121198882 + 11989821198884]12[1

31199013minus3

51205731199015]

=11989841198885

1205872ℏ3[1

31198652(119910) minus

3

5120573119898211988821198653(119910)]

(23)

Equation (23) represents the ground state pressureThereis a considerable change due to the presence of the function1198653(119910) and 119910 itself depends on modified Fermi momentum

We should notice that the value of 119910 is increased due to theincreasing in Fermi momentum this leads to a considerablegrowing in the pressure (23) The change in pressure of theradiation and degenerate fermions will affect the polytropicrelations for astroobjects In the next section the modifiedpressure effect in the some stars stability and some otherapplications will be investigated exactly

3 Applications to Astrophysical Objects

31 Main-Sequence Stars Most main-sequence stars havea mass in the range about 10 to 50 solar mass They arecomposed by a mixture of nonrelativistic gas and radiationthat are stood in hydrostatic equilibrium by gravity [25] Thetotal pressure is 119875

119888= 119875119892+ 119875119903 where 119875

119888 119875119892 and 119875

119903are center

star gas and radiation pressure respectively We can write120573119888= 119875119892119875119888and 1 minus 120573

119888= 119875119903119875119888 By definition (1 minus 120573

119888) and

120573119888are the fractional contribution of radiation and gas to the

central pressure of the star and both are less than one then

119875119892

120573119888

=119875119903

1 minus 120573119888

(24)

The classical gas pressure can be expressed using (15) inthe form

119875119892=120588119888119870119861119879

119898 (25)

We used the relation 119899 = 120588119888119898 120588119888is the central density

of the star and 119898 is the average mass of the gas particle thatconstructs the star defined as 119898 = 119906

119873119898119873where 119906

119873is the

mean molecular weight and 119898119873is the nucleon mass Using

(10) (25) into (24) one finds that

120588119888119870119861

120573119888119898119879119888=

4120590

3119888 (1 minus 120573119888)1198794

119888minus8

105120573

12058741198706

119861

ℏ31198885 (1 minus 120573119888)1198796

119888 (26)

where 119879119888is the central temperature of the star We can find

the temperature by solving the above equation

119879119888= (

1 minus 120573119888

120573119888119898

3119888119870119861

4120590)

13

12058813

119888

+ 1205731

70

(1 minus 120573119888)

120573119888119898

12058741198707

119861

ℏ311988831205902120588119888

(27)

Using this expression to calculate the star pressure from therelation 119875

119888= 119875119892120573119888= (120588119888119870119861120573119888119898)119879119888one gets

119875119888= 119870112058843

119888+ 12057311987021205882

119888= 119875119904UP + 119901119904GUP (28)

1198701=(1 minus 120573

119888)13

(120573119888)43

(31198881198704

119861

41205901198984)

13

1198702=1

70

(1 minus 120573119888)

(120573119888)2

12058741198708

119861

ℏ3119888312059021198982

(29)

The first part of (28) is the ordinary polytropic relationwith a a polytropic exponent Γ = 43 (polytropic index 119899

119901=

3) [26] perturbed by a polytrope Γ = 2 (polytropic index 119899119901=

1) where 119899119901= 1(Γ minus 1)

The best example may be used to estimate GUP pertur-bation is our sun If we assume that 120573

119888is constant overall the

stars and its chemical composition is not changed so 119906119873is

constant For our sun 1 minus 120573119888= 10minus3 and 119906

119873= 0829 [27] and

central density 120588119888= 153 times 10

5 kgm3 then

119901119904GUP119875119904UP

= 18 times 10minus491205730 (30)

The perturbed pressure should not exceed the originalone such that 119901

119904GUP119875119904UP ≪ 1 so this equation suggests thatthe upper value of 120573

0should be 120573

0lt 1048 This bound is far

weaker than that set by electroweak measurementsAccording to (28) the quantum gravity modifies the

polytropic relation by adding a new term proportional to 1205882This suggests a new modification in Lane-Emden equation[28] The modified Lane-Emden equation is established inthe appendix Studying modified Lane-Emden equation andits solution is not the scope of this paper so results that areinvestigated by the this equationwill be postponed to anotherresearch

Main-Sequence Stars Stability In order to discuss the stabilityconcept the energy of a Newtonian polytrope systems withpressure 119875 can be expressed in terms of the internal energyand gravitational potential such that

119864 = 1198961119875119881 minus 119896

2

1198661198722

119877 (31)

where 1198961and 1198962are constants119866 is the gravitational constant

and 119872 and 119877 are the mass and radius of the star At hightdensities it is important to take into account the generalrelativistic effects in star stability To deal with it a generalrelativistic correction term should be added to (31) in lowestorder approximation namely 119864corr = minus119896

3(119866119888)21198727312058823

119888

where 1198963is constant that depends on the actual distribution

of matter According to [27] for 119899119875= 3 polytrope the

condition of stability is Γ gt 43 + 225(1198661198721198882119877) We can seethat addition of a relativistic correction contracts the regionof stability or increases the critical value of Γ In our case119866119872119888

2119877 sim 10

minus6 so this correction can be neglected So weexpect that the new term in a polytropic equation (28) with

Advances in High Energy Physics 5

119899119901= 1 which has a positive contribution will not affect the

problem of stability since Γ gt 2Using (28) into (31) the energy ofmain-sequence starswill

be

119864 = 119862111987212058813

119888+ 1205731198622119872120588119888minus 11989621198725312058813

119888 (32)

where 1198621and 119862

2are constants The mass119872 can be obtained

by using the equilibrium condition 120597119864120597120588119888= 0 the result is

119872 = [3

1198963

(1

31198621+ 120573119862212058823

119888)]

32

(33)

Then

119889 ln119872119889 ln 120588119888

= 12057331198622

1198621

12058823

119888 (34)

Equation (34) ensures that 119889 ln119872119889 ln 120588119888gt 0 which

is the stability condition [27] for main-sequence objectThese results show that the generalization process for theuncertainty principle does not change the stability of themain-sequence objects

Minimum and MaximumMasses for Main-Sequence Stars Inthis part we will discuss the evolution stage of main-sequencestars in the presence of quantum gravity to derive theminimum and maximum masses of these objects Initiallyin the stage of star forming the star must has mass enoughto generate a central temperature which is high enoughfor thermonuclear fusion At this stage the central pressurecomes by the electrons and ions which are forming an idealclassical gas their pressure can be expressed as in (25) Thestar will be made if this pressure is close to the pressureneeded to support the system It is known that the upper limitrelation between the pressure and density at the center of thestar is given by [29]

119875119888le (

120587

6)13

11986612058843

11988811987223 (35)

This relation is valid for any homogeneous star in whichthe mass is concentrated towards the center Equating (25)and (35) one can get the initial central temperature in thisstage

119870119861119879119888le (

120587

6)13

11989811986612058813

11988811987223 (36)

We can see that the quantum gravity does not appearin the picture As shown in this equation the central tem-perature will rise as the central density increases with starcontraction The contracting temperature continues to riseuntil the electrons at the center become fully degenerate andoccupy the lowest energy states in accordance with exclusionprinciple At this stage at the center the pressure comes fromthe electrons that are in degenerate state plus the pressurefrom the classical ions Quantum gravity will not change theclassical ions pressure according to (15) but a nonrelativisticdegenerate electrons pressure will change according to (23)

In nonrelativistic limit where 119910 ≪ 1 the degenerate pressure(23) will be

119875non-relat0

=1

15

1198984

1198901198884

1205872ℏ31199105minus3

35

1198986

1198901198887

1205872ℏ31199107 (37)

Use (17) (21) in above equation and add the result to (15)finally we can write the central pressure at this stage as

119875119888=

119886

5119898119890

11989953

119890+ 120573

121198862

35119898119890

11989973

119890+ 119899119894119870119861119879119888 (38)

where 119886 = (31312058723ℏ)2 The number density of the electrons119899119890and ions 119899

119894can be expressed in terms of central density

simply 119899119890= 119899119894= 120588119888119898119867 where 119898

119867is the mass of hydrogen

atom A hydrostatic pressure is achieved if this pressureequals the pressure needed to support the mass (35) Thisleads to the central temperature being as follows

119870119861119879119888le (

120587

6)13

1198661198981198731198722312058813

119888minus119886

5119898119890

(120588119888

119898119867

)

23

minus 120573121198862

35119898119890

(120588119888

119898119867

)

43

(39)

The first term is associated with classical ions and secondand third terms with degenerate electronsThe last two termsbecome important and are increasing with central densityuntil the temperature will cease to rise as the mass contractsFrom this equation we can find the maximum value oftemperature

10038161003816100381610038161198701198611198791198881003816100381610038161003816max le 119911119872

43[1 minus

60

712057311989811989011991111987243] (40)

119911 =5

4(120587

6)23 119898119890

11988611989883

1198671198662 (41)

Equation (40) proves that the maximum central tem-perature decreases due to quantum gravity The contractingmass achieves stardom if that maximum central temperaturereaches ignition temperature for the thermonuclear fusion ofhydrogen Denote this ignition temperature by 119879ign then theminimummass for the star will be given by

119872min ge (119870119861119879ign

119911)

34

[1 +45

7120573119898119890119870119861119879ign] (42)

This relation shows that the quantum gravity effectincreases theminimummass of themain-sequence starsTheapproximation has ameaning if the second term in (42) is lessthan unity If we take the ignition temperature for hydrogento be about 15 times 106 K this suggests that the upper value of1205730lt 1047

The situation is changed if the radiation pressure takesa part in this stage of revolution The star will be disruptedif radiation becomes the dominant source for the internalpressure This property will set a limit to the mass of a main-sequence starThe pressure at the center of hotmassive stars isdue to electrons and ions (as a classical gases) and radiation

6 Advances in High Energy Physics

the pressure that is produced by these species is derived in(28) Comparing this with the pressure needed to support thestar (35) to get the maximummass one finds that

(120587

6)13

11986611987223

max ge 1198701 + 120573119870212058823

119888 (43)

which means that the maximummass is increased due to thepresence of quantum gravity This increase depends on thecentral density Again the second term in the right-hand sideof inequality (43) should be less than the first this suggeststhat 120573

0lt 1048

32White Dwarfs Thewhite dwarfs consist ofHelium atomsthat are almost in an ionization state The temperature of thestar is about 107 K so the electrons are in a relativistic energyregime and can be considered in a complete degenerate stateand the Helium nuclei do not play a part in first orderapproximation in the dynamic of the system Now we willdetermine the effect of GUP on white dwarf radius accordingto pressure modification expressed in (23) If the number ofHeliumnuclei is1198732 then there are119873 electrons Suppose that119872 = 2119873119898

119901is the mass of the star so the particle density of

the electron is

119899119904=3119872

8120587119898119901

1

1198773 (44)

Using (17) (44) into (21) one gets

119910 =11987213

119877+3

51205731198982

1198901198882119872

1198773 (45)

119872 =9120587

8

119872

119898119901

119877 =119898119890119888

ℏ119877

(46)

where 119877 is the radius of the star and 119898119901is the proton mass

The equilibrium configuration of white dwarf is caused bythe pressure of the degenerate electrons against Newtoniangravitational collapse which is due to the nuclei whichmeansthat

1198750=120581

4120587

1198661198722

1198774= 11987010158401198722

1198774

1198701015840=120581119866

4120587(8119898119901

9120587)

2

(119898119890119888

ℏ)4

(47)

We have two extreme cases as follows(i) When the electron gas is in a low density nonrelativis-

tic dynamics may be used such that 119910 ≪ 1 and (23) will leadto

1198750=4

51198701199105minus36

351205731198982

11989011988821198701199107

119870 =1198981198901198882

121205872(119898119890119888

ℏ)3

(48)

Equating (47) and (48) we got the equation

1198773

minus4

5

119870

1198701015840119872minus13

1198772

minus48

351205731198982

11989011988821198701015840

11987011987213

= 0 (49)

The solution of that equation is

119877 ≃4

5

119870

1198701015840119872minus13

+15

71205731198982

11989011988821198701015840

119870119872 (50)

This equation shows that the radius of the star is increas-ing the elevation depends on the mass of the star The usedapproximation is true if the second term in the right-handside of (50) is much less than the first one this will put a limitin the upper value of 120573

0such that it should be 120573

0lt 1044 we

used a white dwarf mass119872 = 041119872⊙[30]

(ii) When the electron gas is in a high density thatrelativistic effect comes to play strongly such that 119910 ≫ 1 Wecan write

1198750= 119870 (119910

4minus 1199102) minus 120573

18

51198701198982

1198901198882(1199106

3minus1199104

4) (51)

Using (51) into (47) and simplifying then

1198774

+ [1198701015840

11987011987243

minus (1 minus3

101205731198982

1198901198882)11987223

]1198772

minus6

51205731198982

119890119888211987243

= 0

(52)

The solution of that equation keeping only the terms thatproportional to sim120573 is

119877 ≃ 1198770[1 + 120573

3

10

1198982

1198901198882

1198772

0

(211987243

1198772

0

minus1

211987223

)] (53)

where

1198772

0= 11987223

minus1198701015840

11987011987243

(54)

Using (46) into (54) and solving for 119877 and letting 119877 rarr

119877Ch one gets

119877Ch =(9120587)13

2

ℏ

119898119890119888(119872

119898119901

)

13

[1 minus (119872

119872Ch)

23

]

12

119872Ch =9 (3120587)

12

64120581(ℏ119888

1198661198982119901

)

32

119898119901

(55)

119877Ch is the unmodified white dwarf radius or Chan-drasekhar radius limit and 119872Ch is the Chandrasekhar limitmass of order 144119872

⊙ First part of (53) is the Chandrasekhar

radius and the second term is the radius correction due toGUP affect These analyses show that the increasing in starradius depends on [1 minus (119872119872Ch)

23]minus1 which formally goes

to infinity as119872 rarr 119872Ch It is clear that the quantum gravityeffect becomes significant when mass of the white dwarfgets very close to the Chandrasekhar limit in high density

Advances in High Energy Physics 7

stars Equations (23) (50) and (53) attract attention to anoticeable point that the growing in star radius ensures thatquantum gravity resists gravitational collapse by increasingthe electrons degeneracy pressure These results are consis-tentwith that established in [19]Our approximationwill havea meaning if the second term in the right-hand side in (53)is less than one this puts an upper limit for 120573

0such that

1205730lt 1043 We have used119872 = 059119872

⊙[30]

We should also notice that in the limit 119910 ≫ 1 inultrarelativistic regime the function 119865

3(119910) behaves like sim1199106

which almost goes like 1205882 This result is consistent withpolytrope behavior in which Γ = 2 that is correspondingto stability range of the star So we can conclude that thisapproach of quantum gravity does not produce any instabilityto white dwarfs

4 Discussion

Various approaches to quantum gravity such as string the-ory black hole physics and doubly special relativity pre-dict a considerable modification in Heisenberg uncertaintyprinciple to be a generalized uncertainty principle Thismodification leads to a change in the energy-momentumdispersion relation and in the physical phase space In thispaper we studied the modification due to a generalizeduncertainty principle (1) in the statistical and thermodynamicdynamic properties for photons nonrelativistic ideal gasesand degenerate fermions There is a considerable decreasein the number of accessible microstates in quantum gravitybackgroundThis approach leads to a decrease in the internalenergy and pressure of photon gas Pressure and internalenergy of fermions in degenerate state are increased andpressure of nonrelativistic ideal gases is not changed Theseresults are used in studying the stability of themain-sequencestars and white dwarfs It is found that this approach ofquantum gravity does not produce any changes in starsstability

We follow a very simple way to study the effect ofquantum gravity in a different stage of stellar evolution Itis found that there is no change in pressure or temperaturein the initial stage where the pressure is coming fromclassical ions In the second stage after stardom contractingwhere degenerate electrons control the central pressure thequantum gravity leads to a decrease in the central maximumtemperature and an increase in the minimum and maximummasses of the main-sequence stars This result shows that thequantum gravity has an effect on the building process of theuniverse and it will have a more strong effect in highly denseobjects

A mass-radius relation for white dwarf is also investi-gated The results show that there is an elevation in the starradius that is proportional to the mass of the star For thecase of high dense stars the elevation in the radius goesto infinity as the mass of the star reaches Chandrasekharlimit This means that quantum gravity resists gravitationalcollapse This result goes in parallel with that found in [19]Unfortunately quantum gravity behaves in a different wayfrom the prediction of the usual model for white dwarfThe current observation indicates that white dwarfs have

smaller radii than theoretical predictions [30ndash32] This is notconsistent with our results But this result may be reasonableif we add this perturbed term which comes from quantumgravity to the other perturbed terms that are coming fromother physical aspects such as Coulomb corrections latticeenergy correction due to rotation and magnetic field (whichmay be much larger than the correction due to GUP) andmore realistic density distribution for electronic gasThe sumof all these terms may be consistent with the current data

On the other hand implications of the GUP in variousfields such as high energy physics cosmology and blackholes have been studied In [13] potential experimentalsignatures in some familiar quantum systems are examined1205730is a numerical parameter that quantifies the modification

strength if we assume that parameter is of the order of unityThis assumption renders quantum gravity effects too small tobe measured [13] But the 120573

0dependent terms are important

when energies (momenta) are comparable to Planck energy(momenta) and length is comparable to the Planck lengthIf we cancel this choice the current experiments predictlarge upper bounds on it which are compatible with currentobservations It is agreed that such an intermediate lengthscale 119897inter ≃ 119897119875119897radic1205730 cannot exceed the electroweak lengthscale sim1017119897

119875119897[13] This means that 120573

0le 1034 The authors in

[13 14] determined a bound in 1205730parameter which proves

that GUP could be measured in a low energy system likeLandau levels Lamb shift potential barrier and potentialstep They calculated the intermediate length scales such that119897inter ≃ 10

18119897119875119897 1025119897119875119897 and 1010119897

119875119897of which the first two are

far bigger than the electroweak scale and the last is smallerbut may get further constrained with increased accuraciesIn this paper the boundary values of 120573

0parameter in cases

of main-sequence stars are 1048 1047 and 1048 and thosefromwhite dwarf are 1044 and 1043These can be converted tointermediate length scales 119897inter asymp 10

24119897119875119897for main-sequence

stars and 119897inter asymp 1022119897119875119897for white dwarf They are far bigger

than the electroweak scale but it is approximately compatiblewith the results addressed by [14] in one case and it can beconsidered reasonable values in the astrophysical regime

Appendix

In general the gravitational field inside the star can bedescribed by a gravitational potential 120601 which is a solutionof the Poisson equation

nabla2120601 = 4120587119866120588 (A1)

For spherical symmetry Poisson equation reduces to

1

1199032119889

119889119903(1199032 119889120601

119889119903) = 4120587119866120588 (A2)

In hydrostatic equilibrium besides (A2) it requires

119889119875

119889119903= minus

119889120601

119889119903120588 (A3)

8 Advances in High Energy Physics

For a homogeneous gaseous sphere if we assume thatthe pressure of the star can be calculated by the modifiedpolytropic relation

119875 = 1198701120588Γ1 + 120573119870

2120588Γ2 (A4)

The prototype constants 1198701and 119870

2are fixed and can be

calculated from nature constants If Γ1and Γ2are not equal to

unity use (A4) into (A3) and integrate with the boundaryconditions 120601 = 0 and 120588 = 0 at the surface of the sphere onegets

120601 = minus1198701(1 + 119899

1) 12058811198991 minus 120573119870

2(1 + 119899

2) 12058811198992 (A5)

Solving this equation for 120588 keeping only the termsproportional to 120573 and using 119899

119895= 1(Γ

119895minus 1) then

120588 = [minus120601

1198701(1 + 119899

1)]

1198991

minus 12057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A6)

where

119898 =1198991

1198992

+ 1198991minus 1 (A7)

Using 120588 from (A6) into (A2) one gets

1198892120601

1198891199032+2

119903

119889120601

119889119903

= 4120587119866[minus120601

1198701(1 + 119899

1)]

1198991

minus 412058711986612057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A8)

Now let us define

119911 = 119860119903

1198602=

4120587119866

1198701198991

1(1 + 119899

1)1198991(minus120601119888)1198991minus1

120596 =120601

120601119888

(A9)

where 120601119888is the potential at the center it can be defined like

(A5) as

120601119888= minus1198701(1 + 119899

1) 12058811198991

119888minus 1205731198702(1 + 119899

2) 12058811198992

119888 (A10)

Using (A5) (A10) to define 120596 as a function of densityone finds that

120596 =120601

120601119888

= (120588

120588119888

)

11198991

+ 1205731198702(1 + 119899

2)

1198701(1 + 119899

1)(1205881198991

1205881198992119888

)

111989911198992

sdot [1 minus (120588

120588119888

)

(1198992minus1198991)11989911198992

]

(A11)

Using (A9) into (A8) one gets

1198892120596

1198891199112+2

119911

119889120596

119889119911+ 1205961198991 + 120578120596

119898= 0 (A12)

where

120578 =12057311989911198702(1 + 119899

2)

120601119888

(minus120601119888

1198602

4120587119866)

11198992

(A13)

120578 is a dimensionless quantity Equation (A12) is the modifiedLane-Emden equationWe are interested in solutions that arefinite at the center 119911 = 0 So this equation can be solved withthe boundary conditions 120596(0) = 1 and 1205961015840(0) = 1 120596 can becalculated from (A12) and it can be used to determine themass density and pressure of the astrophysical objects as afunction of a radius

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by Benha university (httpwwwbuedueg) The author would like to thank the referees for theenlightening comments which helped to improve the paper

References

[1] D Amati M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1989

[2] M Maggiore ldquoA generalized uncertainty principle in quantumgravityrdquo Physics Letters B vol 304 no 1-2 pp 65ndash69 1993

[3] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999

[4] M Maggiore ldquoQuantum groups gravity and the generalizeduncertainty principlerdquo Physical Review D vol 49 no 10 pp5182ndash5187 1994

[5] MMaggiore ldquoThe algebraic structure of the generalized uncer-tainty principlerdquo Physics Letters B vol 319 no 1ndash3 pp 83ndash861993

[6] L J Garay ldquoQuantum gravity and minimum lengthrdquo Interna-tional Journal of Modern Physics A vol 10 no 2 pp 145ndash1661995

[7] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stoecker ldquoSignatures in the Planck regimerdquo PhysicsLetters B vol 575 no 1-2 pp 85ndash99 2003

[8] C Bambi and F R Urban ldquoNatural extension of the generalizeduncertainty principlerdquo Classical and Quantum Gravity vol 2Article ID 095006 2008

[9] A Kempf G Mangano and R B Mann ldquoHilbert spacerepresentation of the minimal length uncertainty relationrdquoPhysical Review D vol 52 no 2 pp 1108ndash1118 1995

[10] A Kempf ldquoNon-pointlike particles in harmonic oscillatorsrdquoJournal of Physics A Mathematical and General vol 30 no 6pp 2093ndash2101 1997

Advances in High Energy Physics 9

[11] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 Article ID 7691 1999

[12] S Hossenfelder ldquoMinimal length scale scenarios for quantumgravityrdquo Living Reviews in Relativity vol 16 no 2 2013

[13] S Das and E C Vagenas ldquoUniversality of quantum gravitycorrectionsrdquo Physical Review Letters vol 101 Article ID 2213014 pages 2008

[14] S Das and E C Vagenas ldquoPhenomenological implicationsof the generalized uncertainty principlerdquo Canadian Journal ofPhysic vol 87 no 3 pp 233ndash240 2009

[15] I Pikovski M R Vanner M Aspelmeyer M S Kim and CBrukner ldquoProbing planck-scale physics with quantum opticsrdquoNature Physics vol 8 no 5 pp 393ndash397 2012

[16] F Marin F Marino M Bonaldi et al ldquoGravitational bardetectors set limits to Planck-scale physics on macroscopicvariablesrdquo Nature Physics vol 9 no 2 pp 71ndash73 2013

[17] T V Fityo ldquoStatistical physics in deformed spaces withminimallengthrdquo Physics Letters A vol 372 no 37 pp 5872ndash5877 2008

[18] M Abbasiyan-Motlaq and P Pedram ldquoTheminimal length andquantum partition functionsrdquo Journal of Statistical MechanicsTheory and Experiment vol 2014 Article ID P08002 2014

[19] P Wang H Yang and X Zhang ldquoQuantum gravity effectson statistics and compact star configurationsrdquo Journal of HighEnergy Physics vol 2010 article 43 2010

[20] O Bertolami and C A D Zarro ldquoTowards a noncommutativeastrophysicsrdquo Physical Review D vol 81 no 2 Article ID025005 8 pages 2010

[21] L D Landau and E M Lifshitz Statistical Physics (Courseof Theoretical Physics Volume 5) Butterworth-HeinemannOxford UK 3rd edition 1980

[22] L N Chang D Minic N Okamura and T Takeuchi ldquoExactsolution of the harmonic oscillator in arbitrary dimensions withminimal length uncertainty relationsrdquo Physical Review D vol65 Article ID 125027 2002

[23] A F Ali and M Moussa ldquoTowards thermodynamics withgeneralized uncertainty principlerdquo Advances in High EnergyPhysics vol 2014 Article ID 629148 7 pages 2014

[24] M Moussa ldquoQuantum gases and white dwarfs with quantumgravityrdquo Journal of StatisticalMechanicsTheory andExperimentvol 2014 no 11 Article ID P11034 14 pages 2014

[25] S Chandrasekhar An Introduction to the Study of StellarStructure Dover Mineola NY USA 1967

[26] T PadmanabhanTheoretical Astrophysics Volume I Astrophys-ical Processes Cambridge University Press Cambridge UK2000

[27] T Padmanabhan Theoretical Astrophysics vol 2 of Stars andStellar Systems Cambridge University Press Cambridge UK2001

[28] R Kippenhahn and A Weigert Stellar Structure and EvolutionSpringer Berlin Germany 1994

[29] G W Collins Fundamentals of Stellar Astrophysics 2003[30] G J Mathews I-S Suh B OrsquoGorman et al ldquoAnalysis of white

dwarfs with strange-matter coresrdquo Journal of Physics G Nuclearand Particle Physics vol 32 no 6 pp 747ndash759 2006

[31] J L Provencal H L Shipman D Koester F Wesemael and PBergeron ldquoProcyon B outside the iron boxrdquo The AstrophysicalJournal vol 568 no 1 pp 324ndash334 2002

[32] A Camacho ldquoWhite dwarfs as test objects of Lorentz viola-tionsrdquo Classical and Quantum Gravity vol 23 no 24 ArticleID 7355 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in High Energy Physics

By the same way we can derive the pressure from (7)after making use of the integration by parts and functionexpansion one finds that

119875 =1198882

1205872ℏ3int119901119891

0

119889119901

[11990121198882 + 11989821198884]12[1

31199013minus3

51205731199015]

=11989841198885

1205872ℏ3[1

31198652(119910) minus

3

5120573119898211988821198653(119910)]

(23)

Equation (23) represents the ground state pressureThereis a considerable change due to the presence of the function1198653(119910) and 119910 itself depends on modified Fermi momentum

We should notice that the value of 119910 is increased due to theincreasing in Fermi momentum this leads to a considerablegrowing in the pressure (23) The change in pressure of theradiation and degenerate fermions will affect the polytropicrelations for astroobjects In the next section the modifiedpressure effect in the some stars stability and some otherapplications will be investigated exactly

3 Applications to Astrophysical Objects

31 Main-Sequence Stars Most main-sequence stars havea mass in the range about 10 to 50 solar mass They arecomposed by a mixture of nonrelativistic gas and radiationthat are stood in hydrostatic equilibrium by gravity [25] Thetotal pressure is 119875

119888= 119875119892+ 119875119903 where 119875

119888 119875119892 and 119875

119903are center

star gas and radiation pressure respectively We can write120573119888= 119875119892119875119888and 1 minus 120573

119888= 119875119903119875119888 By definition (1 minus 120573

119888) and

120573119888are the fractional contribution of radiation and gas to the

central pressure of the star and both are less than one then

119875119892

120573119888

=119875119903

1 minus 120573119888

(24)

The classical gas pressure can be expressed using (15) inthe form

119875119892=120588119888119870119861119879

119898 (25)

We used the relation 119899 = 120588119888119898 120588119888is the central density

of the star and 119898 is the average mass of the gas particle thatconstructs the star defined as 119898 = 119906

119873119898119873where 119906

119873is the

mean molecular weight and 119898119873is the nucleon mass Using

(10) (25) into (24) one finds that

120588119888119870119861

120573119888119898119879119888=

4120590

3119888 (1 minus 120573119888)1198794

119888minus8

105120573

12058741198706

119861

ℏ31198885 (1 minus 120573119888)1198796

119888 (26)

where 119879119888is the central temperature of the star We can find

the temperature by solving the above equation

119879119888= (

1 minus 120573119888

120573119888119898

3119888119870119861

4120590)

13

12058813

119888

+ 1205731

70

(1 minus 120573119888)

120573119888119898

12058741198707

119861

ℏ311988831205902120588119888

(27)

Using this expression to calculate the star pressure from therelation 119875

119888= 119875119892120573119888= (120588119888119870119861120573119888119898)119879119888one gets

119875119888= 119870112058843

119888+ 12057311987021205882

119888= 119875119904UP + 119901119904GUP (28)

1198701=(1 minus 120573

119888)13

(120573119888)43

(31198881198704

119861

41205901198984)

13

1198702=1

70

(1 minus 120573119888)

(120573119888)2

12058741198708

119861

ℏ3119888312059021198982

(29)

The first part of (28) is the ordinary polytropic relationwith a a polytropic exponent Γ = 43 (polytropic index 119899

119901=

3) [26] perturbed by a polytrope Γ = 2 (polytropic index 119899119901=

1) where 119899119901= 1(Γ minus 1)

The best example may be used to estimate GUP pertur-bation is our sun If we assume that 120573

119888is constant overall the

stars and its chemical composition is not changed so 119906119873is

constant For our sun 1 minus 120573119888= 10minus3 and 119906

119873= 0829 [27] and

central density 120588119888= 153 times 10

5 kgm3 then

119901119904GUP119875119904UP

= 18 times 10minus491205730 (30)

The perturbed pressure should not exceed the originalone such that 119901

119904GUP119875119904UP ≪ 1 so this equation suggests thatthe upper value of 120573

0should be 120573

0lt 1048 This bound is far

weaker than that set by electroweak measurementsAccording to (28) the quantum gravity modifies the

polytropic relation by adding a new term proportional to 1205882This suggests a new modification in Lane-Emden equation[28] The modified Lane-Emden equation is established inthe appendix Studying modified Lane-Emden equation andits solution is not the scope of this paper so results that areinvestigated by the this equationwill be postponed to anotherresearch

Main-Sequence Stars Stability In order to discuss the stabilityconcept the energy of a Newtonian polytrope systems withpressure 119875 can be expressed in terms of the internal energyand gravitational potential such that

119864 = 1198961119875119881 minus 119896

2

1198661198722

119877 (31)

where 1198961and 1198962are constants119866 is the gravitational constant

and 119872 and 119877 are the mass and radius of the star At hightdensities it is important to take into account the generalrelativistic effects in star stability To deal with it a generalrelativistic correction term should be added to (31) in lowestorder approximation namely 119864corr = minus119896

3(119866119888)21198727312058823

119888

where 1198963is constant that depends on the actual distribution

of matter According to [27] for 119899119875= 3 polytrope the

condition of stability is Γ gt 43 + 225(1198661198721198882119877) We can seethat addition of a relativistic correction contracts the regionof stability or increases the critical value of Γ In our case119866119872119888

2119877 sim 10

minus6 so this correction can be neglected So weexpect that the new term in a polytropic equation (28) with

Advances in High Energy Physics 5

119899119901= 1 which has a positive contribution will not affect the

problem of stability since Γ gt 2Using (28) into (31) the energy ofmain-sequence starswill

be

119864 = 119862111987212058813

119888+ 1205731198622119872120588119888minus 11989621198725312058813

119888 (32)

where 1198621and 119862

2are constants The mass119872 can be obtained

by using the equilibrium condition 120597119864120597120588119888= 0 the result is

119872 = [3

1198963

(1

31198621+ 120573119862212058823

119888)]

32

(33)

Then

119889 ln119872119889 ln 120588119888

= 12057331198622

1198621

12058823

119888 (34)

Equation (34) ensures that 119889 ln119872119889 ln 120588119888gt 0 which

is the stability condition [27] for main-sequence objectThese results show that the generalization process for theuncertainty principle does not change the stability of themain-sequence objects

Minimum and MaximumMasses for Main-Sequence Stars Inthis part we will discuss the evolution stage of main-sequencestars in the presence of quantum gravity to derive theminimum and maximum masses of these objects Initiallyin the stage of star forming the star must has mass enoughto generate a central temperature which is high enoughfor thermonuclear fusion At this stage the central pressurecomes by the electrons and ions which are forming an idealclassical gas their pressure can be expressed as in (25) Thestar will be made if this pressure is close to the pressureneeded to support the system It is known that the upper limitrelation between the pressure and density at the center of thestar is given by [29]

119875119888le (

120587

6)13

11986612058843

11988811987223 (35)

This relation is valid for any homogeneous star in whichthe mass is concentrated towards the center Equating (25)and (35) one can get the initial central temperature in thisstage

119870119861119879119888le (

120587

6)13

11989811986612058813

11988811987223 (36)

We can see that the quantum gravity does not appearin the picture As shown in this equation the central tem-perature will rise as the central density increases with starcontraction The contracting temperature continues to riseuntil the electrons at the center become fully degenerate andoccupy the lowest energy states in accordance with exclusionprinciple At this stage at the center the pressure comes fromthe electrons that are in degenerate state plus the pressurefrom the classical ions Quantum gravity will not change theclassical ions pressure according to (15) but a nonrelativisticdegenerate electrons pressure will change according to (23)

In nonrelativistic limit where 119910 ≪ 1 the degenerate pressure(23) will be

119875non-relat0

=1

15

1198984

1198901198884

1205872ℏ31199105minus3

35

1198986

1198901198887

1205872ℏ31199107 (37)

Use (17) (21) in above equation and add the result to (15)finally we can write the central pressure at this stage as

119875119888=

119886

5119898119890

11989953

119890+ 120573

121198862

35119898119890

11989973

119890+ 119899119894119870119861119879119888 (38)

where 119886 = (31312058723ℏ)2 The number density of the electrons119899119890and ions 119899

119894can be expressed in terms of central density

simply 119899119890= 119899119894= 120588119888119898119867 where 119898

119867is the mass of hydrogen

atom A hydrostatic pressure is achieved if this pressureequals the pressure needed to support the mass (35) Thisleads to the central temperature being as follows

119870119861119879119888le (

120587

6)13

1198661198981198731198722312058813

119888minus119886

5119898119890

(120588119888

119898119867

)

23

minus 120573121198862

35119898119890

(120588119888

119898119867

)

43

(39)

The first term is associated with classical ions and secondand third terms with degenerate electronsThe last two termsbecome important and are increasing with central densityuntil the temperature will cease to rise as the mass contractsFrom this equation we can find the maximum value oftemperature

10038161003816100381610038161198701198611198791198881003816100381610038161003816max le 119911119872

43[1 minus

60

712057311989811989011991111987243] (40)

119911 =5

4(120587

6)23 119898119890

11988611989883

1198671198662 (41)

Equation (40) proves that the maximum central tem-perature decreases due to quantum gravity The contractingmass achieves stardom if that maximum central temperaturereaches ignition temperature for the thermonuclear fusion ofhydrogen Denote this ignition temperature by 119879ign then theminimummass for the star will be given by

119872min ge (119870119861119879ign

119911)

34

[1 +45

7120573119898119890119870119861119879ign] (42)

This relation shows that the quantum gravity effectincreases theminimummass of themain-sequence starsTheapproximation has ameaning if the second term in (42) is lessthan unity If we take the ignition temperature for hydrogento be about 15 times 106 K this suggests that the upper value of1205730lt 1047

The situation is changed if the radiation pressure takesa part in this stage of revolution The star will be disruptedif radiation becomes the dominant source for the internalpressure This property will set a limit to the mass of a main-sequence starThe pressure at the center of hotmassive stars isdue to electrons and ions (as a classical gases) and radiation

6 Advances in High Energy Physics

the pressure that is produced by these species is derived in(28) Comparing this with the pressure needed to support thestar (35) to get the maximummass one finds that

(120587

6)13

11986611987223

max ge 1198701 + 120573119870212058823

119888 (43)

which means that the maximummass is increased due to thepresence of quantum gravity This increase depends on thecentral density Again the second term in the right-hand sideof inequality (43) should be less than the first this suggeststhat 120573

0lt 1048

32White Dwarfs Thewhite dwarfs consist ofHelium atomsthat are almost in an ionization state The temperature of thestar is about 107 K so the electrons are in a relativistic energyregime and can be considered in a complete degenerate stateand the Helium nuclei do not play a part in first orderapproximation in the dynamic of the system Now we willdetermine the effect of GUP on white dwarf radius accordingto pressure modification expressed in (23) If the number ofHeliumnuclei is1198732 then there are119873 electrons Suppose that119872 = 2119873119898

119901is the mass of the star so the particle density of

the electron is

119899119904=3119872

8120587119898119901

1

1198773 (44)

Using (17) (44) into (21) one gets

119910 =11987213

119877+3

51205731198982

1198901198882119872

1198773 (45)

119872 =9120587

8

119872

119898119901

119877 =119898119890119888

ℏ119877

(46)

where 119877 is the radius of the star and 119898119901is the proton mass

The equilibrium configuration of white dwarf is caused bythe pressure of the degenerate electrons against Newtoniangravitational collapse which is due to the nuclei whichmeansthat

1198750=120581

4120587

1198661198722

1198774= 11987010158401198722

1198774

1198701015840=120581119866

4120587(8119898119901

9120587)

2

(119898119890119888

ℏ)4

(47)

We have two extreme cases as follows(i) When the electron gas is in a low density nonrelativis-

tic dynamics may be used such that 119910 ≪ 1 and (23) will leadto

1198750=4

51198701199105minus36

351205731198982

11989011988821198701199107

119870 =1198981198901198882

121205872(119898119890119888

ℏ)3

(48)

Equating (47) and (48) we got the equation

1198773

minus4

5

119870

1198701015840119872minus13

1198772

minus48

351205731198982

11989011988821198701015840

11987011987213

= 0 (49)

The solution of that equation is

119877 ≃4

5

119870

1198701015840119872minus13

+15

71205731198982

11989011988821198701015840

119870119872 (50)

This equation shows that the radius of the star is increas-ing the elevation depends on the mass of the star The usedapproximation is true if the second term in the right-handside of (50) is much less than the first one this will put a limitin the upper value of 120573

0such that it should be 120573

0lt 1044 we

used a white dwarf mass119872 = 041119872⊙[30]

(ii) When the electron gas is in a high density thatrelativistic effect comes to play strongly such that 119910 ≫ 1 Wecan write

1198750= 119870 (119910

4minus 1199102) minus 120573

18

51198701198982

1198901198882(1199106

3minus1199104

4) (51)

Using (51) into (47) and simplifying then

1198774

+ [1198701015840

11987011987243

minus (1 minus3

101205731198982

1198901198882)11987223

]1198772

minus6

51205731198982

119890119888211987243

= 0

(52)

The solution of that equation keeping only the terms thatproportional to sim120573 is

119877 ≃ 1198770[1 + 120573

3

10

1198982

1198901198882

1198772

0

(211987243

1198772

0

minus1

211987223

)] (53)

where

1198772

0= 11987223

minus1198701015840

11987011987243

(54)

Using (46) into (54) and solving for 119877 and letting 119877 rarr

119877Ch one gets

119877Ch =(9120587)13

2

ℏ

119898119890119888(119872

119898119901

)

13

[1 minus (119872

119872Ch)

23

]

12

119872Ch =9 (3120587)

12

64120581(ℏ119888

1198661198982119901

)

32

119898119901

(55)

119877Ch is the unmodified white dwarf radius or Chan-drasekhar radius limit and 119872Ch is the Chandrasekhar limitmass of order 144119872

⊙ First part of (53) is the Chandrasekhar

radius and the second term is the radius correction due toGUP affect These analyses show that the increasing in starradius depends on [1 minus (119872119872Ch)

23]minus1 which formally goes

to infinity as119872 rarr 119872Ch It is clear that the quantum gravityeffect becomes significant when mass of the white dwarfgets very close to the Chandrasekhar limit in high density

Advances in High Energy Physics 7

stars Equations (23) (50) and (53) attract attention to anoticeable point that the growing in star radius ensures thatquantum gravity resists gravitational collapse by increasingthe electrons degeneracy pressure These results are consis-tentwith that established in [19]Our approximationwill havea meaning if the second term in the right-hand side in (53)is less than one this puts an upper limit for 120573

0such that

1205730lt 1043 We have used119872 = 059119872

⊙[30]

We should also notice that in the limit 119910 ≫ 1 inultrarelativistic regime the function 119865

3(119910) behaves like sim1199106

which almost goes like 1205882 This result is consistent withpolytrope behavior in which Γ = 2 that is correspondingto stability range of the star So we can conclude that thisapproach of quantum gravity does not produce any instabilityto white dwarfs

4 Discussion

Various approaches to quantum gravity such as string the-ory black hole physics and doubly special relativity pre-dict a considerable modification in Heisenberg uncertaintyprinciple to be a generalized uncertainty principle Thismodification leads to a change in the energy-momentumdispersion relation and in the physical phase space In thispaper we studied the modification due to a generalizeduncertainty principle (1) in the statistical and thermodynamicdynamic properties for photons nonrelativistic ideal gasesand degenerate fermions There is a considerable decreasein the number of accessible microstates in quantum gravitybackgroundThis approach leads to a decrease in the internalenergy and pressure of photon gas Pressure and internalenergy of fermions in degenerate state are increased andpressure of nonrelativistic ideal gases is not changed Theseresults are used in studying the stability of themain-sequencestars and white dwarfs It is found that this approach ofquantum gravity does not produce any changes in starsstability

We follow a very simple way to study the effect ofquantum gravity in a different stage of stellar evolution Itis found that there is no change in pressure or temperaturein the initial stage where the pressure is coming fromclassical ions In the second stage after stardom contractingwhere degenerate electrons control the central pressure thequantum gravity leads to a decrease in the central maximumtemperature and an increase in the minimum and maximummasses of the main-sequence stars This result shows that thequantum gravity has an effect on the building process of theuniverse and it will have a more strong effect in highly denseobjects

A mass-radius relation for white dwarf is also investi-gated The results show that there is an elevation in the starradius that is proportional to the mass of the star For thecase of high dense stars the elevation in the radius goesto infinity as the mass of the star reaches Chandrasekharlimit This means that quantum gravity resists gravitationalcollapse This result goes in parallel with that found in [19]Unfortunately quantum gravity behaves in a different wayfrom the prediction of the usual model for white dwarfThe current observation indicates that white dwarfs have

smaller radii than theoretical predictions [30ndash32] This is notconsistent with our results But this result may be reasonableif we add this perturbed term which comes from quantumgravity to the other perturbed terms that are coming fromother physical aspects such as Coulomb corrections latticeenergy correction due to rotation and magnetic field (whichmay be much larger than the correction due to GUP) andmore realistic density distribution for electronic gasThe sumof all these terms may be consistent with the current data

On the other hand implications of the GUP in variousfields such as high energy physics cosmology and blackholes have been studied In [13] potential experimentalsignatures in some familiar quantum systems are examined1205730is a numerical parameter that quantifies the modification

strength if we assume that parameter is of the order of unityThis assumption renders quantum gravity effects too small tobe measured [13] But the 120573

0dependent terms are important

when energies (momenta) are comparable to Planck energy(momenta) and length is comparable to the Planck lengthIf we cancel this choice the current experiments predictlarge upper bounds on it which are compatible with currentobservations It is agreed that such an intermediate lengthscale 119897inter ≃ 119897119875119897radic1205730 cannot exceed the electroweak lengthscale sim1017119897

119875119897[13] This means that 120573

0le 1034 The authors in

[13 14] determined a bound in 1205730parameter which proves

that GUP could be measured in a low energy system likeLandau levels Lamb shift potential barrier and potentialstep They calculated the intermediate length scales such that119897inter ≃ 10

18119897119875119897 1025119897119875119897 and 1010119897

119875119897of which the first two are

far bigger than the electroweak scale and the last is smallerbut may get further constrained with increased accuraciesIn this paper the boundary values of 120573

0parameter in cases

of main-sequence stars are 1048 1047 and 1048 and thosefromwhite dwarf are 1044 and 1043These can be converted tointermediate length scales 119897inter asymp 10

24119897119875119897for main-sequence

stars and 119897inter asymp 1022119897119875119897for white dwarf They are far bigger

than the electroweak scale but it is approximately compatiblewith the results addressed by [14] in one case and it can beconsidered reasonable values in the astrophysical regime

Appendix

In general the gravitational field inside the star can bedescribed by a gravitational potential 120601 which is a solutionof the Poisson equation

nabla2120601 = 4120587119866120588 (A1)

For spherical symmetry Poisson equation reduces to

1

1199032119889

119889119903(1199032 119889120601

119889119903) = 4120587119866120588 (A2)

In hydrostatic equilibrium besides (A2) it requires

119889119875

119889119903= minus

119889120601

119889119903120588 (A3)

8 Advances in High Energy Physics

For a homogeneous gaseous sphere if we assume thatthe pressure of the star can be calculated by the modifiedpolytropic relation

119875 = 1198701120588Γ1 + 120573119870

2120588Γ2 (A4)

The prototype constants 1198701and 119870

2are fixed and can be

calculated from nature constants If Γ1and Γ2are not equal to

unity use (A4) into (A3) and integrate with the boundaryconditions 120601 = 0 and 120588 = 0 at the surface of the sphere onegets

120601 = minus1198701(1 + 119899

1) 12058811198991 minus 120573119870

2(1 + 119899

2) 12058811198992 (A5)

Solving this equation for 120588 keeping only the termsproportional to 120573 and using 119899

119895= 1(Γ

119895minus 1) then

120588 = [minus120601

1198701(1 + 119899

1)]

1198991

minus 12057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A6)

where

119898 =1198991

1198992

+ 1198991minus 1 (A7)

Using 120588 from (A6) into (A2) one gets

1198892120601

1198891199032+2

119903

119889120601

119889119903

= 4120587119866[minus120601

1198701(1 + 119899

1)]

1198991

minus 412058711986612057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A8)

Now let us define

119911 = 119860119903

1198602=

4120587119866

1198701198991

1(1 + 119899

1)1198991(minus120601119888)1198991minus1

120596 =120601

120601119888

(A9)

where 120601119888is the potential at the center it can be defined like

(A5) as

120601119888= minus1198701(1 + 119899

1) 12058811198991

119888minus 1205731198702(1 + 119899

2) 12058811198992

119888 (A10)

Using (A5) (A10) to define 120596 as a function of densityone finds that

120596 =120601

120601119888

= (120588

120588119888

)

11198991

+ 1205731198702(1 + 119899

2)

1198701(1 + 119899

1)(1205881198991

1205881198992119888

)

111989911198992

sdot [1 minus (120588

120588119888

)

(1198992minus1198991)11989911198992

]

(A11)

Using (A9) into (A8) one gets

1198892120596

1198891199112+2

119911

119889120596

119889119911+ 1205961198991 + 120578120596

119898= 0 (A12)

where

120578 =12057311989911198702(1 + 119899

2)

120601119888

(minus120601119888

1198602

4120587119866)

11198992

(A13)

120578 is a dimensionless quantity Equation (A12) is the modifiedLane-Emden equationWe are interested in solutions that arefinite at the center 119911 = 0 So this equation can be solved withthe boundary conditions 120596(0) = 1 and 1205961015840(0) = 1 120596 can becalculated from (A12) and it can be used to determine themass density and pressure of the astrophysical objects as afunction of a radius

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by Benha university (httpwwwbuedueg) The author would like to thank the referees for theenlightening comments which helped to improve the paper

References

[1] D Amati M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1989

[2] M Maggiore ldquoA generalized uncertainty principle in quantumgravityrdquo Physics Letters B vol 304 no 1-2 pp 65ndash69 1993

[3] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999

[4] M Maggiore ldquoQuantum groups gravity and the generalizeduncertainty principlerdquo Physical Review D vol 49 no 10 pp5182ndash5187 1994

[5] MMaggiore ldquoThe algebraic structure of the generalized uncer-tainty principlerdquo Physics Letters B vol 319 no 1ndash3 pp 83ndash861993

[6] L J Garay ldquoQuantum gravity and minimum lengthrdquo Interna-tional Journal of Modern Physics A vol 10 no 2 pp 145ndash1661995

[7] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stoecker ldquoSignatures in the Planck regimerdquo PhysicsLetters B vol 575 no 1-2 pp 85ndash99 2003

[8] C Bambi and F R Urban ldquoNatural extension of the generalizeduncertainty principlerdquo Classical and Quantum Gravity vol 2Article ID 095006 2008

[9] A Kempf G Mangano and R B Mann ldquoHilbert spacerepresentation of the minimal length uncertainty relationrdquoPhysical Review D vol 52 no 2 pp 1108ndash1118 1995

[10] A Kempf ldquoNon-pointlike particles in harmonic oscillatorsrdquoJournal of Physics A Mathematical and General vol 30 no 6pp 2093ndash2101 1997

Advances in High Energy Physics 9

[11] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 Article ID 7691 1999

[12] S Hossenfelder ldquoMinimal length scale scenarios for quantumgravityrdquo Living Reviews in Relativity vol 16 no 2 2013

[13] S Das and E C Vagenas ldquoUniversality of quantum gravitycorrectionsrdquo Physical Review Letters vol 101 Article ID 2213014 pages 2008

[14] S Das and E C Vagenas ldquoPhenomenological implicationsof the generalized uncertainty principlerdquo Canadian Journal ofPhysic vol 87 no 3 pp 233ndash240 2009

[15] I Pikovski M R Vanner M Aspelmeyer M S Kim and CBrukner ldquoProbing planck-scale physics with quantum opticsrdquoNature Physics vol 8 no 5 pp 393ndash397 2012

[16] F Marin F Marino M Bonaldi et al ldquoGravitational bardetectors set limits to Planck-scale physics on macroscopicvariablesrdquo Nature Physics vol 9 no 2 pp 71ndash73 2013

[17] T V Fityo ldquoStatistical physics in deformed spaces withminimallengthrdquo Physics Letters A vol 372 no 37 pp 5872ndash5877 2008

[18] M Abbasiyan-Motlaq and P Pedram ldquoTheminimal length andquantum partition functionsrdquo Journal of Statistical MechanicsTheory and Experiment vol 2014 Article ID P08002 2014

[19] P Wang H Yang and X Zhang ldquoQuantum gravity effectson statistics and compact star configurationsrdquo Journal of HighEnergy Physics vol 2010 article 43 2010

[20] O Bertolami and C A D Zarro ldquoTowards a noncommutativeastrophysicsrdquo Physical Review D vol 81 no 2 Article ID025005 8 pages 2010

[21] L D Landau and E M Lifshitz Statistical Physics (Courseof Theoretical Physics Volume 5) Butterworth-HeinemannOxford UK 3rd edition 1980

[22] L N Chang D Minic N Okamura and T Takeuchi ldquoExactsolution of the harmonic oscillator in arbitrary dimensions withminimal length uncertainty relationsrdquo Physical Review D vol65 Article ID 125027 2002

[23] A F Ali and M Moussa ldquoTowards thermodynamics withgeneralized uncertainty principlerdquo Advances in High EnergyPhysics vol 2014 Article ID 629148 7 pages 2014

[24] M Moussa ldquoQuantum gases and white dwarfs with quantumgravityrdquo Journal of StatisticalMechanicsTheory andExperimentvol 2014 no 11 Article ID P11034 14 pages 2014

[25] S Chandrasekhar An Introduction to the Study of StellarStructure Dover Mineola NY USA 1967

[26] T PadmanabhanTheoretical Astrophysics Volume I Astrophys-ical Processes Cambridge University Press Cambridge UK2000

[27] T Padmanabhan Theoretical Astrophysics vol 2 of Stars andStellar Systems Cambridge University Press Cambridge UK2001

[28] R Kippenhahn and A Weigert Stellar Structure and EvolutionSpringer Berlin Germany 1994

[29] G W Collins Fundamentals of Stellar Astrophysics 2003[30] G J Mathews I-S Suh B OrsquoGorman et al ldquoAnalysis of white

dwarfs with strange-matter coresrdquo Journal of Physics G Nuclearand Particle Physics vol 32 no 6 pp 747ndash759 2006

[31] J L Provencal H L Shipman D Koester F Wesemael and PBergeron ldquoProcyon B outside the iron boxrdquo The AstrophysicalJournal vol 568 no 1 pp 324ndash334 2002

[32] A Camacho ldquoWhite dwarfs as test objects of Lorentz viola-tionsrdquo Classical and Quantum Gravity vol 23 no 24 ArticleID 7355 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 5

119899119901= 1 which has a positive contribution will not affect the

problem of stability since Γ gt 2Using (28) into (31) the energy ofmain-sequence starswill

be

119864 = 119862111987212058813

119888+ 1205731198622119872120588119888minus 11989621198725312058813

119888 (32)

where 1198621and 119862

2are constants The mass119872 can be obtained

by using the equilibrium condition 120597119864120597120588119888= 0 the result is

119872 = [3

1198963

(1

31198621+ 120573119862212058823

119888)]

32

(33)

Then

119889 ln119872119889 ln 120588119888

= 12057331198622

1198621

12058823

119888 (34)

Equation (34) ensures that 119889 ln119872119889 ln 120588119888gt 0 which

is the stability condition [27] for main-sequence objectThese results show that the generalization process for theuncertainty principle does not change the stability of themain-sequence objects

Minimum and MaximumMasses for Main-Sequence Stars Inthis part we will discuss the evolution stage of main-sequencestars in the presence of quantum gravity to derive theminimum and maximum masses of these objects Initiallyin the stage of star forming the star must has mass enoughto generate a central temperature which is high enoughfor thermonuclear fusion At this stage the central pressurecomes by the electrons and ions which are forming an idealclassical gas their pressure can be expressed as in (25) Thestar will be made if this pressure is close to the pressureneeded to support the system It is known that the upper limitrelation between the pressure and density at the center of thestar is given by [29]

119875119888le (

120587

6)13

11986612058843

11988811987223 (35)

This relation is valid for any homogeneous star in whichthe mass is concentrated towards the center Equating (25)and (35) one can get the initial central temperature in thisstage

119870119861119879119888le (

120587

6)13

11989811986612058813

11988811987223 (36)

We can see that the quantum gravity does not appearin the picture As shown in this equation the central tem-perature will rise as the central density increases with starcontraction The contracting temperature continues to riseuntil the electrons at the center become fully degenerate andoccupy the lowest energy states in accordance with exclusionprinciple At this stage at the center the pressure comes fromthe electrons that are in degenerate state plus the pressurefrom the classical ions Quantum gravity will not change theclassical ions pressure according to (15) but a nonrelativisticdegenerate electrons pressure will change according to (23)

In nonrelativistic limit where 119910 ≪ 1 the degenerate pressure(23) will be

119875non-relat0

=1

15

1198984

1198901198884

1205872ℏ31199105minus3

35

1198986

1198901198887

1205872ℏ31199107 (37)

Use (17) (21) in above equation and add the result to (15)finally we can write the central pressure at this stage as

119875119888=

119886

5119898119890

11989953

119890+ 120573

121198862

35119898119890

11989973

119890+ 119899119894119870119861119879119888 (38)

where 119886 = (31312058723ℏ)2 The number density of the electrons119899119890and ions 119899

119894can be expressed in terms of central density

simply 119899119890= 119899119894= 120588119888119898119867 where 119898

119867is the mass of hydrogen

atom A hydrostatic pressure is achieved if this pressureequals the pressure needed to support the mass (35) Thisleads to the central temperature being as follows

119870119861119879119888le (

120587

6)13

1198661198981198731198722312058813

119888minus119886

5119898119890

(120588119888

119898119867

)

23

minus 120573121198862

35119898119890

(120588119888

119898119867

)

43

(39)

The first term is associated with classical ions and secondand third terms with degenerate electronsThe last two termsbecome important and are increasing with central densityuntil the temperature will cease to rise as the mass contractsFrom this equation we can find the maximum value oftemperature

10038161003816100381610038161198701198611198791198881003816100381610038161003816max le 119911119872

43[1 minus

60

712057311989811989011991111987243] (40)

119911 =5

4(120587

6)23 119898119890

11988611989883

1198671198662 (41)

Equation (40) proves that the maximum central tem-perature decreases due to quantum gravity The contractingmass achieves stardom if that maximum central temperaturereaches ignition temperature for the thermonuclear fusion ofhydrogen Denote this ignition temperature by 119879ign then theminimummass for the star will be given by

119872min ge (119870119861119879ign

119911)

34

[1 +45

7120573119898119890119870119861119879ign] (42)

This relation shows that the quantum gravity effectincreases theminimummass of themain-sequence starsTheapproximation has ameaning if the second term in (42) is lessthan unity If we take the ignition temperature for hydrogento be about 15 times 106 K this suggests that the upper value of1205730lt 1047

The situation is changed if the radiation pressure takesa part in this stage of revolution The star will be disruptedif radiation becomes the dominant source for the internalpressure This property will set a limit to the mass of a main-sequence starThe pressure at the center of hotmassive stars isdue to electrons and ions (as a classical gases) and radiation

6 Advances in High Energy Physics

the pressure that is produced by these species is derived in(28) Comparing this with the pressure needed to support thestar (35) to get the maximummass one finds that

(120587

6)13

11986611987223

max ge 1198701 + 120573119870212058823

119888 (43)

which means that the maximummass is increased due to thepresence of quantum gravity This increase depends on thecentral density Again the second term in the right-hand sideof inequality (43) should be less than the first this suggeststhat 120573

0lt 1048

32White Dwarfs Thewhite dwarfs consist ofHelium atomsthat are almost in an ionization state The temperature of thestar is about 107 K so the electrons are in a relativistic energyregime and can be considered in a complete degenerate stateand the Helium nuclei do not play a part in first orderapproximation in the dynamic of the system Now we willdetermine the effect of GUP on white dwarf radius accordingto pressure modification expressed in (23) If the number ofHeliumnuclei is1198732 then there are119873 electrons Suppose that119872 = 2119873119898

119901is the mass of the star so the particle density of

the electron is

119899119904=3119872

8120587119898119901

1

1198773 (44)

Using (17) (44) into (21) one gets

119910 =11987213

119877+3

51205731198982

1198901198882119872

1198773 (45)

119872 =9120587

8

119872

119898119901

119877 =119898119890119888

ℏ119877

(46)

where 119877 is the radius of the star and 119898119901is the proton mass

The equilibrium configuration of white dwarf is caused bythe pressure of the degenerate electrons against Newtoniangravitational collapse which is due to the nuclei whichmeansthat

1198750=120581

4120587

1198661198722

1198774= 11987010158401198722

1198774

1198701015840=120581119866

4120587(8119898119901

9120587)

2

(119898119890119888

ℏ)4

(47)

We have two extreme cases as follows(i) When the electron gas is in a low density nonrelativis-

tic dynamics may be used such that 119910 ≪ 1 and (23) will leadto

1198750=4

51198701199105minus36

351205731198982

11989011988821198701199107

119870 =1198981198901198882

121205872(119898119890119888

ℏ)3

(48)

Equating (47) and (48) we got the equation

1198773

minus4

5

119870

1198701015840119872minus13

1198772

minus48

351205731198982

11989011988821198701015840

11987011987213

= 0 (49)

The solution of that equation is

119877 ≃4

5

119870

1198701015840119872minus13

+15

71205731198982

11989011988821198701015840

119870119872 (50)

This equation shows that the radius of the star is increas-ing the elevation depends on the mass of the star The usedapproximation is true if the second term in the right-handside of (50) is much less than the first one this will put a limitin the upper value of 120573

0such that it should be 120573

0lt 1044 we

used a white dwarf mass119872 = 041119872⊙[30]

(ii) When the electron gas is in a high density thatrelativistic effect comes to play strongly such that 119910 ≫ 1 Wecan write

1198750= 119870 (119910

4minus 1199102) minus 120573

18

51198701198982

1198901198882(1199106

3minus1199104

4) (51)

Using (51) into (47) and simplifying then

1198774

+ [1198701015840

11987011987243

minus (1 minus3

101205731198982

1198901198882)11987223

]1198772

minus6

51205731198982

119890119888211987243

= 0

(52)

The solution of that equation keeping only the terms thatproportional to sim120573 is

119877 ≃ 1198770[1 + 120573

3

10

1198982

1198901198882

1198772

0

(211987243

1198772

0

minus1

211987223

)] (53)

where

1198772

0= 11987223

minus1198701015840

11987011987243

(54)

Using (46) into (54) and solving for 119877 and letting 119877 rarr

119877Ch one gets

119877Ch =(9120587)13

2

ℏ

119898119890119888(119872

119898119901

)

13

[1 minus (119872

119872Ch)

23

]

12

119872Ch =9 (3120587)

12

64120581(ℏ119888

1198661198982119901

)

32

119898119901

(55)

119877Ch is the unmodified white dwarf radius or Chan-drasekhar radius limit and 119872Ch is the Chandrasekhar limitmass of order 144119872

⊙ First part of (53) is the Chandrasekhar

radius and the second term is the radius correction due toGUP affect These analyses show that the increasing in starradius depends on [1 minus (119872119872Ch)

23]minus1 which formally goes

to infinity as119872 rarr 119872Ch It is clear that the quantum gravityeffect becomes significant when mass of the white dwarfgets very close to the Chandrasekhar limit in high density

Advances in High Energy Physics 7

stars Equations (23) (50) and (53) attract attention to anoticeable point that the growing in star radius ensures thatquantum gravity resists gravitational collapse by increasingthe electrons degeneracy pressure These results are consis-tentwith that established in [19]Our approximationwill havea meaning if the second term in the right-hand side in (53)is less than one this puts an upper limit for 120573

0such that

1205730lt 1043 We have used119872 = 059119872

⊙[30]

We should also notice that in the limit 119910 ≫ 1 inultrarelativistic regime the function 119865

3(119910) behaves like sim1199106

which almost goes like 1205882 This result is consistent withpolytrope behavior in which Γ = 2 that is correspondingto stability range of the star So we can conclude that thisapproach of quantum gravity does not produce any instabilityto white dwarfs

4 Discussion

Various approaches to quantum gravity such as string the-ory black hole physics and doubly special relativity pre-dict a considerable modification in Heisenberg uncertaintyprinciple to be a generalized uncertainty principle Thismodification leads to a change in the energy-momentumdispersion relation and in the physical phase space In thispaper we studied the modification due to a generalizeduncertainty principle (1) in the statistical and thermodynamicdynamic properties for photons nonrelativistic ideal gasesand degenerate fermions There is a considerable decreasein the number of accessible microstates in quantum gravitybackgroundThis approach leads to a decrease in the internalenergy and pressure of photon gas Pressure and internalenergy of fermions in degenerate state are increased andpressure of nonrelativistic ideal gases is not changed Theseresults are used in studying the stability of themain-sequencestars and white dwarfs It is found that this approach ofquantum gravity does not produce any changes in starsstability

We follow a very simple way to study the effect ofquantum gravity in a different stage of stellar evolution Itis found that there is no change in pressure or temperaturein the initial stage where the pressure is coming fromclassical ions In the second stage after stardom contractingwhere degenerate electrons control the central pressure thequantum gravity leads to a decrease in the central maximumtemperature and an increase in the minimum and maximummasses of the main-sequence stars This result shows that thequantum gravity has an effect on the building process of theuniverse and it will have a more strong effect in highly denseobjects

A mass-radius relation for white dwarf is also investi-gated The results show that there is an elevation in the starradius that is proportional to the mass of the star For thecase of high dense stars the elevation in the radius goesto infinity as the mass of the star reaches Chandrasekharlimit This means that quantum gravity resists gravitationalcollapse This result goes in parallel with that found in [19]Unfortunately quantum gravity behaves in a different wayfrom the prediction of the usual model for white dwarfThe current observation indicates that white dwarfs have

smaller radii than theoretical predictions [30ndash32] This is notconsistent with our results But this result may be reasonableif we add this perturbed term which comes from quantumgravity to the other perturbed terms that are coming fromother physical aspects such as Coulomb corrections latticeenergy correction due to rotation and magnetic field (whichmay be much larger than the correction due to GUP) andmore realistic density distribution for electronic gasThe sumof all these terms may be consistent with the current data

On the other hand implications of the GUP in variousfields such as high energy physics cosmology and blackholes have been studied In [13] potential experimentalsignatures in some familiar quantum systems are examined1205730is a numerical parameter that quantifies the modification

strength if we assume that parameter is of the order of unityThis assumption renders quantum gravity effects too small tobe measured [13] But the 120573

0dependent terms are important

when energies (momenta) are comparable to Planck energy(momenta) and length is comparable to the Planck lengthIf we cancel this choice the current experiments predictlarge upper bounds on it which are compatible with currentobservations It is agreed that such an intermediate lengthscale 119897inter ≃ 119897119875119897radic1205730 cannot exceed the electroweak lengthscale sim1017119897

119875119897[13] This means that 120573

0le 1034 The authors in

[13 14] determined a bound in 1205730parameter which proves

that GUP could be measured in a low energy system likeLandau levels Lamb shift potential barrier and potentialstep They calculated the intermediate length scales such that119897inter ≃ 10

18119897119875119897 1025119897119875119897 and 1010119897

119875119897of which the first two are

far bigger than the electroweak scale and the last is smallerbut may get further constrained with increased accuraciesIn this paper the boundary values of 120573

0parameter in cases

of main-sequence stars are 1048 1047 and 1048 and thosefromwhite dwarf are 1044 and 1043These can be converted tointermediate length scales 119897inter asymp 10

24119897119875119897for main-sequence

stars and 119897inter asymp 1022119897119875119897for white dwarf They are far bigger

than the electroweak scale but it is approximately compatiblewith the results addressed by [14] in one case and it can beconsidered reasonable values in the astrophysical regime

Appendix

In general the gravitational field inside the star can bedescribed by a gravitational potential 120601 which is a solutionof the Poisson equation

nabla2120601 = 4120587119866120588 (A1)

For spherical symmetry Poisson equation reduces to

1

1199032119889

119889119903(1199032 119889120601

119889119903) = 4120587119866120588 (A2)

In hydrostatic equilibrium besides (A2) it requires

119889119875

119889119903= minus

119889120601

119889119903120588 (A3)

8 Advances in High Energy Physics

For a homogeneous gaseous sphere if we assume thatthe pressure of the star can be calculated by the modifiedpolytropic relation

119875 = 1198701120588Γ1 + 120573119870

2120588Γ2 (A4)

The prototype constants 1198701and 119870

2are fixed and can be

calculated from nature constants If Γ1and Γ2are not equal to

unity use (A4) into (A3) and integrate with the boundaryconditions 120601 = 0 and 120588 = 0 at the surface of the sphere onegets

120601 = minus1198701(1 + 119899

1) 12058811198991 minus 120573119870

2(1 + 119899

2) 12058811198992 (A5)

Solving this equation for 120588 keeping only the termsproportional to 120573 and using 119899

119895= 1(Γ

119895minus 1) then

120588 = [minus120601

1198701(1 + 119899

1)]

1198991

minus 12057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A6)

where

119898 =1198991

1198992

+ 1198991minus 1 (A7)

Using 120588 from (A6) into (A2) one gets

1198892120601

1198891199032+2

119903

119889120601

119889119903

= 4120587119866[minus120601

1198701(1 + 119899

1)]

1198991

minus 412058711986612057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A8)

Now let us define

119911 = 119860119903

1198602=

4120587119866

1198701198991

1(1 + 119899

1)1198991(minus120601119888)1198991minus1

120596 =120601

120601119888

(A9)

where 120601119888is the potential at the center it can be defined like

(A5) as

120601119888= minus1198701(1 + 119899

1) 12058811198991

119888minus 1205731198702(1 + 119899

2) 12058811198992

119888 (A10)

Using (A5) (A10) to define 120596 as a function of densityone finds that

120596 =120601

120601119888

= (120588

120588119888

)

11198991

+ 1205731198702(1 + 119899

2)

1198701(1 + 119899

1)(1205881198991

1205881198992119888

)

111989911198992

sdot [1 minus (120588

120588119888

)

(1198992minus1198991)11989911198992

]

(A11)

Using (A9) into (A8) one gets

1198892120596

1198891199112+2

119911

119889120596

119889119911+ 1205961198991 + 120578120596

119898= 0 (A12)

where

120578 =12057311989911198702(1 + 119899

2)

120601119888

(minus120601119888

1198602

4120587119866)

11198992

(A13)

120578 is a dimensionless quantity Equation (A12) is the modifiedLane-Emden equationWe are interested in solutions that arefinite at the center 119911 = 0 So this equation can be solved withthe boundary conditions 120596(0) = 1 and 1205961015840(0) = 1 120596 can becalculated from (A12) and it can be used to determine themass density and pressure of the astrophysical objects as afunction of a radius

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by Benha university (httpwwwbuedueg) The author would like to thank the referees for theenlightening comments which helped to improve the paper

References

[1] D Amati M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1989

[2] M Maggiore ldquoA generalized uncertainty principle in quantumgravityrdquo Physics Letters B vol 304 no 1-2 pp 65ndash69 1993

[3] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999

[4] M Maggiore ldquoQuantum groups gravity and the generalizeduncertainty principlerdquo Physical Review D vol 49 no 10 pp5182ndash5187 1994

[5] MMaggiore ldquoThe algebraic structure of the generalized uncer-tainty principlerdquo Physics Letters B vol 319 no 1ndash3 pp 83ndash861993

[6] L J Garay ldquoQuantum gravity and minimum lengthrdquo Interna-tional Journal of Modern Physics A vol 10 no 2 pp 145ndash1661995

[7] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stoecker ldquoSignatures in the Planck regimerdquo PhysicsLetters B vol 575 no 1-2 pp 85ndash99 2003

[8] C Bambi and F R Urban ldquoNatural extension of the generalizeduncertainty principlerdquo Classical and Quantum Gravity vol 2Article ID 095006 2008

[9] A Kempf G Mangano and R B Mann ldquoHilbert spacerepresentation of the minimal length uncertainty relationrdquoPhysical Review D vol 52 no 2 pp 1108ndash1118 1995

[10] A Kempf ldquoNon-pointlike particles in harmonic oscillatorsrdquoJournal of Physics A Mathematical and General vol 30 no 6pp 2093ndash2101 1997

Advances in High Energy Physics 9

[11] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 Article ID 7691 1999

[12] S Hossenfelder ldquoMinimal length scale scenarios for quantumgravityrdquo Living Reviews in Relativity vol 16 no 2 2013

[13] S Das and E C Vagenas ldquoUniversality of quantum gravitycorrectionsrdquo Physical Review Letters vol 101 Article ID 2213014 pages 2008

[14] S Das and E C Vagenas ldquoPhenomenological implicationsof the generalized uncertainty principlerdquo Canadian Journal ofPhysic vol 87 no 3 pp 233ndash240 2009

[15] I Pikovski M R Vanner M Aspelmeyer M S Kim and CBrukner ldquoProbing planck-scale physics with quantum opticsrdquoNature Physics vol 8 no 5 pp 393ndash397 2012

[16] F Marin F Marino M Bonaldi et al ldquoGravitational bardetectors set limits to Planck-scale physics on macroscopicvariablesrdquo Nature Physics vol 9 no 2 pp 71ndash73 2013

[17] T V Fityo ldquoStatistical physics in deformed spaces withminimallengthrdquo Physics Letters A vol 372 no 37 pp 5872ndash5877 2008

[18] M Abbasiyan-Motlaq and P Pedram ldquoTheminimal length andquantum partition functionsrdquo Journal of Statistical MechanicsTheory and Experiment vol 2014 Article ID P08002 2014

[19] P Wang H Yang and X Zhang ldquoQuantum gravity effectson statistics and compact star configurationsrdquo Journal of HighEnergy Physics vol 2010 article 43 2010

[20] O Bertolami and C A D Zarro ldquoTowards a noncommutativeastrophysicsrdquo Physical Review D vol 81 no 2 Article ID025005 8 pages 2010

[21] L D Landau and E M Lifshitz Statistical Physics (Courseof Theoretical Physics Volume 5) Butterworth-HeinemannOxford UK 3rd edition 1980

[22] L N Chang D Minic N Okamura and T Takeuchi ldquoExactsolution of the harmonic oscillator in arbitrary dimensions withminimal length uncertainty relationsrdquo Physical Review D vol65 Article ID 125027 2002

[23] A F Ali and M Moussa ldquoTowards thermodynamics withgeneralized uncertainty principlerdquo Advances in High EnergyPhysics vol 2014 Article ID 629148 7 pages 2014

[24] M Moussa ldquoQuantum gases and white dwarfs with quantumgravityrdquo Journal of StatisticalMechanicsTheory andExperimentvol 2014 no 11 Article ID P11034 14 pages 2014

[25] S Chandrasekhar An Introduction to the Study of StellarStructure Dover Mineola NY USA 1967

[26] T PadmanabhanTheoretical Astrophysics Volume I Astrophys-ical Processes Cambridge University Press Cambridge UK2000

[27] T Padmanabhan Theoretical Astrophysics vol 2 of Stars andStellar Systems Cambridge University Press Cambridge UK2001

[28] R Kippenhahn and A Weigert Stellar Structure and EvolutionSpringer Berlin Germany 1994

[29] G W Collins Fundamentals of Stellar Astrophysics 2003[30] G J Mathews I-S Suh B OrsquoGorman et al ldquoAnalysis of white

dwarfs with strange-matter coresrdquo Journal of Physics G Nuclearand Particle Physics vol 32 no 6 pp 747ndash759 2006

[31] J L Provencal H L Shipman D Koester F Wesemael and PBergeron ldquoProcyon B outside the iron boxrdquo The AstrophysicalJournal vol 568 no 1 pp 324ndash334 2002

[32] A Camacho ldquoWhite dwarfs as test objects of Lorentz viola-tionsrdquo Classical and Quantum Gravity vol 23 no 24 ArticleID 7355 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in High Energy Physics

the pressure that is produced by these species is derived in(28) Comparing this with the pressure needed to support thestar (35) to get the maximummass one finds that

(120587

6)13

11986611987223

max ge 1198701 + 120573119870212058823

119888 (43)

which means that the maximummass is increased due to thepresence of quantum gravity This increase depends on thecentral density Again the second term in the right-hand sideof inequality (43) should be less than the first this suggeststhat 120573

0lt 1048

32White Dwarfs Thewhite dwarfs consist ofHelium atomsthat are almost in an ionization state The temperature of thestar is about 107 K so the electrons are in a relativistic energyregime and can be considered in a complete degenerate stateand the Helium nuclei do not play a part in first orderapproximation in the dynamic of the system Now we willdetermine the effect of GUP on white dwarf radius accordingto pressure modification expressed in (23) If the number ofHeliumnuclei is1198732 then there are119873 electrons Suppose that119872 = 2119873119898

119901is the mass of the star so the particle density of

the electron is

119899119904=3119872

8120587119898119901

1

1198773 (44)

Using (17) (44) into (21) one gets

119910 =11987213

119877+3

51205731198982

1198901198882119872

1198773 (45)

119872 =9120587

8

119872

119898119901

119877 =119898119890119888

ℏ119877

(46)

where 119877 is the radius of the star and 119898119901is the proton mass

The equilibrium configuration of white dwarf is caused bythe pressure of the degenerate electrons against Newtoniangravitational collapse which is due to the nuclei whichmeansthat

1198750=120581

4120587

1198661198722

1198774= 11987010158401198722

1198774

1198701015840=120581119866

4120587(8119898119901

9120587)

2

(119898119890119888

ℏ)4

(47)

We have two extreme cases as follows(i) When the electron gas is in a low density nonrelativis-

tic dynamics may be used such that 119910 ≪ 1 and (23) will leadto

1198750=4

51198701199105minus36

351205731198982

11989011988821198701199107

119870 =1198981198901198882

121205872(119898119890119888

ℏ)3

(48)

Equating (47) and (48) we got the equation

1198773

minus4

5

119870

1198701015840119872minus13

1198772

minus48

351205731198982

11989011988821198701015840

11987011987213

= 0 (49)

The solution of that equation is

119877 ≃4

5

119870

1198701015840119872minus13

+15

71205731198982

11989011988821198701015840

119870119872 (50)

This equation shows that the radius of the star is increas-ing the elevation depends on the mass of the star The usedapproximation is true if the second term in the right-handside of (50) is much less than the first one this will put a limitin the upper value of 120573

0such that it should be 120573

0lt 1044 we

used a white dwarf mass119872 = 041119872⊙[30]

(ii) When the electron gas is in a high density thatrelativistic effect comes to play strongly such that 119910 ≫ 1 Wecan write

1198750= 119870 (119910

4minus 1199102) minus 120573

18

51198701198982

1198901198882(1199106

3minus1199104

4) (51)

Using (51) into (47) and simplifying then

1198774

+ [1198701015840

11987011987243

minus (1 minus3

101205731198982

1198901198882)11987223

]1198772

minus6

51205731198982

119890119888211987243

= 0

(52)

The solution of that equation keeping only the terms thatproportional to sim120573 is

119877 ≃ 1198770[1 + 120573

3

10

1198982

1198901198882

1198772

0

(211987243

1198772

0

minus1

211987223

)] (53)

where

1198772

0= 11987223

minus1198701015840

11987011987243

(54)

Using (46) into (54) and solving for 119877 and letting 119877 rarr

119877Ch one gets

119877Ch =(9120587)13

2

ℏ

119898119890119888(119872

119898119901

)

13

[1 minus (119872

119872Ch)

23

]

12

119872Ch =9 (3120587)

12

64120581(ℏ119888

1198661198982119901

)

32

119898119901

(55)

119877Ch is the unmodified white dwarf radius or Chan-drasekhar radius limit and 119872Ch is the Chandrasekhar limitmass of order 144119872

⊙ First part of (53) is the Chandrasekhar

radius and the second term is the radius correction due toGUP affect These analyses show that the increasing in starradius depends on [1 minus (119872119872Ch)

23]minus1 which formally goes

to infinity as119872 rarr 119872Ch It is clear that the quantum gravityeffect becomes significant when mass of the white dwarfgets very close to the Chandrasekhar limit in high density

Advances in High Energy Physics 7

stars Equations (23) (50) and (53) attract attention to anoticeable point that the growing in star radius ensures thatquantum gravity resists gravitational collapse by increasingthe electrons degeneracy pressure These results are consis-tentwith that established in [19]Our approximationwill havea meaning if the second term in the right-hand side in (53)is less than one this puts an upper limit for 120573

0such that

1205730lt 1043 We have used119872 = 059119872

⊙[30]

We should also notice that in the limit 119910 ≫ 1 inultrarelativistic regime the function 119865

3(119910) behaves like sim1199106

which almost goes like 1205882 This result is consistent withpolytrope behavior in which Γ = 2 that is correspondingto stability range of the star So we can conclude that thisapproach of quantum gravity does not produce any instabilityto white dwarfs

4 Discussion

Various approaches to quantum gravity such as string the-ory black hole physics and doubly special relativity pre-dict a considerable modification in Heisenberg uncertaintyprinciple to be a generalized uncertainty principle Thismodification leads to a change in the energy-momentumdispersion relation and in the physical phase space In thispaper we studied the modification due to a generalizeduncertainty principle (1) in the statistical and thermodynamicdynamic properties for photons nonrelativistic ideal gasesand degenerate fermions There is a considerable decreasein the number of accessible microstates in quantum gravitybackgroundThis approach leads to a decrease in the internalenergy and pressure of photon gas Pressure and internalenergy of fermions in degenerate state are increased andpressure of nonrelativistic ideal gases is not changed Theseresults are used in studying the stability of themain-sequencestars and white dwarfs It is found that this approach ofquantum gravity does not produce any changes in starsstability

We follow a very simple way to study the effect ofquantum gravity in a different stage of stellar evolution Itis found that there is no change in pressure or temperaturein the initial stage where the pressure is coming fromclassical ions In the second stage after stardom contractingwhere degenerate electrons control the central pressure thequantum gravity leads to a decrease in the central maximumtemperature and an increase in the minimum and maximummasses of the main-sequence stars This result shows that thequantum gravity has an effect on the building process of theuniverse and it will have a more strong effect in highly denseobjects

A mass-radius relation for white dwarf is also investi-gated The results show that there is an elevation in the starradius that is proportional to the mass of the star For thecase of high dense stars the elevation in the radius goesto infinity as the mass of the star reaches Chandrasekharlimit This means that quantum gravity resists gravitationalcollapse This result goes in parallel with that found in [19]Unfortunately quantum gravity behaves in a different wayfrom the prediction of the usual model for white dwarfThe current observation indicates that white dwarfs have

smaller radii than theoretical predictions [30ndash32] This is notconsistent with our results But this result may be reasonableif we add this perturbed term which comes from quantumgravity to the other perturbed terms that are coming fromother physical aspects such as Coulomb corrections latticeenergy correction due to rotation and magnetic field (whichmay be much larger than the correction due to GUP) andmore realistic density distribution for electronic gasThe sumof all these terms may be consistent with the current data

On the other hand implications of the GUP in variousfields such as high energy physics cosmology and blackholes have been studied In [13] potential experimentalsignatures in some familiar quantum systems are examined1205730is a numerical parameter that quantifies the modification

strength if we assume that parameter is of the order of unityThis assumption renders quantum gravity effects too small tobe measured [13] But the 120573

0dependent terms are important

when energies (momenta) are comparable to Planck energy(momenta) and length is comparable to the Planck lengthIf we cancel this choice the current experiments predictlarge upper bounds on it which are compatible with currentobservations It is agreed that such an intermediate lengthscale 119897inter ≃ 119897119875119897radic1205730 cannot exceed the electroweak lengthscale sim1017119897

119875119897[13] This means that 120573

0le 1034 The authors in

[13 14] determined a bound in 1205730parameter which proves

that GUP could be measured in a low energy system likeLandau levels Lamb shift potential barrier and potentialstep They calculated the intermediate length scales such that119897inter ≃ 10

18119897119875119897 1025119897119875119897 and 1010119897

119875119897of which the first two are

far bigger than the electroweak scale and the last is smallerbut may get further constrained with increased accuraciesIn this paper the boundary values of 120573

0parameter in cases

of main-sequence stars are 1048 1047 and 1048 and thosefromwhite dwarf are 1044 and 1043These can be converted tointermediate length scales 119897inter asymp 10

24119897119875119897for main-sequence

stars and 119897inter asymp 1022119897119875119897for white dwarf They are far bigger

than the electroweak scale but it is approximately compatiblewith the results addressed by [14] in one case and it can beconsidered reasonable values in the astrophysical regime

Appendix

In general the gravitational field inside the star can bedescribed by a gravitational potential 120601 which is a solutionof the Poisson equation

nabla2120601 = 4120587119866120588 (A1)

For spherical symmetry Poisson equation reduces to

1

1199032119889

119889119903(1199032 119889120601

119889119903) = 4120587119866120588 (A2)

In hydrostatic equilibrium besides (A2) it requires

119889119875

119889119903= minus

119889120601

119889119903120588 (A3)

8 Advances in High Energy Physics

For a homogeneous gaseous sphere if we assume thatthe pressure of the star can be calculated by the modifiedpolytropic relation

119875 = 1198701120588Γ1 + 120573119870

2120588Γ2 (A4)

The prototype constants 1198701and 119870

2are fixed and can be

calculated from nature constants If Γ1and Γ2are not equal to

unity use (A4) into (A3) and integrate with the boundaryconditions 120601 = 0 and 120588 = 0 at the surface of the sphere onegets

120601 = minus1198701(1 + 119899

1) 12058811198991 minus 120573119870

2(1 + 119899

2) 12058811198992 (A5)

Solving this equation for 120588 keeping only the termsproportional to 120573 and using 119899

119895= 1(Γ

119895minus 1) then

120588 = [minus120601

1198701(1 + 119899

1)]

1198991

minus 12057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A6)

where

119898 =1198991

1198992

+ 1198991minus 1 (A7)

Using 120588 from (A6) into (A2) one gets

1198892120601

1198891199032+2

119903

119889120601

119889119903

= 4120587119866[minus120601

1198701(1 + 119899

1)]

1198991

minus 412058711986612057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A8)

Now let us define

119911 = 119860119903

1198602=

4120587119866

1198701198991

1(1 + 119899

1)1198991(minus120601119888)1198991minus1

120596 =120601

120601119888

(A9)

where 120601119888is the potential at the center it can be defined like

(A5) as

120601119888= minus1198701(1 + 119899

1) 12058811198991

119888minus 1205731198702(1 + 119899

2) 12058811198992

119888 (A10)

Using (A5) (A10) to define 120596 as a function of densityone finds that

120596 =120601

120601119888

= (120588

120588119888

)

11198991

+ 1205731198702(1 + 119899

2)

1198701(1 + 119899

1)(1205881198991

1205881198992119888

)

111989911198992

sdot [1 minus (120588

120588119888

)

(1198992minus1198991)11989911198992

]

(A11)

Using (A9) into (A8) one gets

1198892120596

1198891199112+2

119911

119889120596

119889119911+ 1205961198991 + 120578120596

119898= 0 (A12)

where

120578 =12057311989911198702(1 + 119899

2)

120601119888

(minus120601119888

1198602

4120587119866)

11198992

(A13)

120578 is a dimensionless quantity Equation (A12) is the modifiedLane-Emden equationWe are interested in solutions that arefinite at the center 119911 = 0 So this equation can be solved withthe boundary conditions 120596(0) = 1 and 1205961015840(0) = 1 120596 can becalculated from (A12) and it can be used to determine themass density and pressure of the astrophysical objects as afunction of a radius

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by Benha university (httpwwwbuedueg) The author would like to thank the referees for theenlightening comments which helped to improve the paper

References

[1] D Amati M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1989

[2] M Maggiore ldquoA generalized uncertainty principle in quantumgravityrdquo Physics Letters B vol 304 no 1-2 pp 65ndash69 1993

[3] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999

[4] M Maggiore ldquoQuantum groups gravity and the generalizeduncertainty principlerdquo Physical Review D vol 49 no 10 pp5182ndash5187 1994

[5] MMaggiore ldquoThe algebraic structure of the generalized uncer-tainty principlerdquo Physics Letters B vol 319 no 1ndash3 pp 83ndash861993

[6] L J Garay ldquoQuantum gravity and minimum lengthrdquo Interna-tional Journal of Modern Physics A vol 10 no 2 pp 145ndash1661995

[7] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stoecker ldquoSignatures in the Planck regimerdquo PhysicsLetters B vol 575 no 1-2 pp 85ndash99 2003

[8] C Bambi and F R Urban ldquoNatural extension of the generalizeduncertainty principlerdquo Classical and Quantum Gravity vol 2Article ID 095006 2008

[9] A Kempf G Mangano and R B Mann ldquoHilbert spacerepresentation of the minimal length uncertainty relationrdquoPhysical Review D vol 52 no 2 pp 1108ndash1118 1995

[10] A Kempf ldquoNon-pointlike particles in harmonic oscillatorsrdquoJournal of Physics A Mathematical and General vol 30 no 6pp 2093ndash2101 1997

Advances in High Energy Physics 9

[11] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 Article ID 7691 1999

[12] S Hossenfelder ldquoMinimal length scale scenarios for quantumgravityrdquo Living Reviews in Relativity vol 16 no 2 2013

[13] S Das and E C Vagenas ldquoUniversality of quantum gravitycorrectionsrdquo Physical Review Letters vol 101 Article ID 2213014 pages 2008

[14] S Das and E C Vagenas ldquoPhenomenological implicationsof the generalized uncertainty principlerdquo Canadian Journal ofPhysic vol 87 no 3 pp 233ndash240 2009

[15] I Pikovski M R Vanner M Aspelmeyer M S Kim and CBrukner ldquoProbing planck-scale physics with quantum opticsrdquoNature Physics vol 8 no 5 pp 393ndash397 2012

[16] F Marin F Marino M Bonaldi et al ldquoGravitational bardetectors set limits to Planck-scale physics on macroscopicvariablesrdquo Nature Physics vol 9 no 2 pp 71ndash73 2013

[17] T V Fityo ldquoStatistical physics in deformed spaces withminimallengthrdquo Physics Letters A vol 372 no 37 pp 5872ndash5877 2008

[18] M Abbasiyan-Motlaq and P Pedram ldquoTheminimal length andquantum partition functionsrdquo Journal of Statistical MechanicsTheory and Experiment vol 2014 Article ID P08002 2014

[19] P Wang H Yang and X Zhang ldquoQuantum gravity effectson statistics and compact star configurationsrdquo Journal of HighEnergy Physics vol 2010 article 43 2010

[20] O Bertolami and C A D Zarro ldquoTowards a noncommutativeastrophysicsrdquo Physical Review D vol 81 no 2 Article ID025005 8 pages 2010

[21] L D Landau and E M Lifshitz Statistical Physics (Courseof Theoretical Physics Volume 5) Butterworth-HeinemannOxford UK 3rd edition 1980

[22] L N Chang D Minic N Okamura and T Takeuchi ldquoExactsolution of the harmonic oscillator in arbitrary dimensions withminimal length uncertainty relationsrdquo Physical Review D vol65 Article ID 125027 2002

[23] A F Ali and M Moussa ldquoTowards thermodynamics withgeneralized uncertainty principlerdquo Advances in High EnergyPhysics vol 2014 Article ID 629148 7 pages 2014

[24] M Moussa ldquoQuantum gases and white dwarfs with quantumgravityrdquo Journal of StatisticalMechanicsTheory andExperimentvol 2014 no 11 Article ID P11034 14 pages 2014

[25] S Chandrasekhar An Introduction to the Study of StellarStructure Dover Mineola NY USA 1967

[26] T PadmanabhanTheoretical Astrophysics Volume I Astrophys-ical Processes Cambridge University Press Cambridge UK2000

[27] T Padmanabhan Theoretical Astrophysics vol 2 of Stars andStellar Systems Cambridge University Press Cambridge UK2001

[28] R Kippenhahn and A Weigert Stellar Structure and EvolutionSpringer Berlin Germany 1994

[29] G W Collins Fundamentals of Stellar Astrophysics 2003[30] G J Mathews I-S Suh B OrsquoGorman et al ldquoAnalysis of white

dwarfs with strange-matter coresrdquo Journal of Physics G Nuclearand Particle Physics vol 32 no 6 pp 747ndash759 2006

[31] J L Provencal H L Shipman D Koester F Wesemael and PBergeron ldquoProcyon B outside the iron boxrdquo The AstrophysicalJournal vol 568 no 1 pp 324ndash334 2002

[32] A Camacho ldquoWhite dwarfs as test objects of Lorentz viola-tionsrdquo Classical and Quantum Gravity vol 23 no 24 ArticleID 7355 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 7

stars Equations (23) (50) and (53) attract attention to anoticeable point that the growing in star radius ensures thatquantum gravity resists gravitational collapse by increasingthe electrons degeneracy pressure These results are consis-tentwith that established in [19]Our approximationwill havea meaning if the second term in the right-hand side in (53)is less than one this puts an upper limit for 120573

0such that

1205730lt 1043 We have used119872 = 059119872

⊙[30]

We should also notice that in the limit 119910 ≫ 1 inultrarelativistic regime the function 119865

3(119910) behaves like sim1199106

which almost goes like 1205882 This result is consistent withpolytrope behavior in which Γ = 2 that is correspondingto stability range of the star So we can conclude that thisapproach of quantum gravity does not produce any instabilityto white dwarfs

4 Discussion

Various approaches to quantum gravity such as string the-ory black hole physics and doubly special relativity pre-dict a considerable modification in Heisenberg uncertaintyprinciple to be a generalized uncertainty principle Thismodification leads to a change in the energy-momentumdispersion relation and in the physical phase space In thispaper we studied the modification due to a generalizeduncertainty principle (1) in the statistical and thermodynamicdynamic properties for photons nonrelativistic ideal gasesand degenerate fermions There is a considerable decreasein the number of accessible microstates in quantum gravitybackgroundThis approach leads to a decrease in the internalenergy and pressure of photon gas Pressure and internalenergy of fermions in degenerate state are increased andpressure of nonrelativistic ideal gases is not changed Theseresults are used in studying the stability of themain-sequencestars and white dwarfs It is found that this approach ofquantum gravity does not produce any changes in starsstability

We follow a very simple way to study the effect ofquantum gravity in a different stage of stellar evolution Itis found that there is no change in pressure or temperaturein the initial stage where the pressure is coming fromclassical ions In the second stage after stardom contractingwhere degenerate electrons control the central pressure thequantum gravity leads to a decrease in the central maximumtemperature and an increase in the minimum and maximummasses of the main-sequence stars This result shows that thequantum gravity has an effect on the building process of theuniverse and it will have a more strong effect in highly denseobjects

A mass-radius relation for white dwarf is also investi-gated The results show that there is an elevation in the starradius that is proportional to the mass of the star For thecase of high dense stars the elevation in the radius goesto infinity as the mass of the star reaches Chandrasekharlimit This means that quantum gravity resists gravitationalcollapse This result goes in parallel with that found in [19]Unfortunately quantum gravity behaves in a different wayfrom the prediction of the usual model for white dwarfThe current observation indicates that white dwarfs have

smaller radii than theoretical predictions [30ndash32] This is notconsistent with our results But this result may be reasonableif we add this perturbed term which comes from quantumgravity to the other perturbed terms that are coming fromother physical aspects such as Coulomb corrections latticeenergy correction due to rotation and magnetic field (whichmay be much larger than the correction due to GUP) andmore realistic density distribution for electronic gasThe sumof all these terms may be consistent with the current data

On the other hand implications of the GUP in variousfields such as high energy physics cosmology and blackholes have been studied In [13] potential experimentalsignatures in some familiar quantum systems are examined1205730is a numerical parameter that quantifies the modification

strength if we assume that parameter is of the order of unityThis assumption renders quantum gravity effects too small tobe measured [13] But the 120573

0dependent terms are important

when energies (momenta) are comparable to Planck energy(momenta) and length is comparable to the Planck lengthIf we cancel this choice the current experiments predictlarge upper bounds on it which are compatible with currentobservations It is agreed that such an intermediate lengthscale 119897inter ≃ 119897119875119897radic1205730 cannot exceed the electroweak lengthscale sim1017119897

119875119897[13] This means that 120573

0le 1034 The authors in

[13 14] determined a bound in 1205730parameter which proves

that GUP could be measured in a low energy system likeLandau levels Lamb shift potential barrier and potentialstep They calculated the intermediate length scales such that119897inter ≃ 10

18119897119875119897 1025119897119875119897 and 1010119897

119875119897of which the first two are

far bigger than the electroweak scale and the last is smallerbut may get further constrained with increased accuraciesIn this paper the boundary values of 120573

0parameter in cases

of main-sequence stars are 1048 1047 and 1048 and thosefromwhite dwarf are 1044 and 1043These can be converted tointermediate length scales 119897inter asymp 10

24119897119875119897for main-sequence

stars and 119897inter asymp 1022119897119875119897for white dwarf They are far bigger

than the electroweak scale but it is approximately compatiblewith the results addressed by [14] in one case and it can beconsidered reasonable values in the astrophysical regime

Appendix

In general the gravitational field inside the star can bedescribed by a gravitational potential 120601 which is a solutionof the Poisson equation

nabla2120601 = 4120587119866120588 (A1)

For spherical symmetry Poisson equation reduces to

1

1199032119889

119889119903(1199032 119889120601

119889119903) = 4120587119866120588 (A2)

In hydrostatic equilibrium besides (A2) it requires

119889119875

119889119903= minus

119889120601

119889119903120588 (A3)

8 Advances in High Energy Physics

For a homogeneous gaseous sphere if we assume thatthe pressure of the star can be calculated by the modifiedpolytropic relation

119875 = 1198701120588Γ1 + 120573119870

2120588Γ2 (A4)

The prototype constants 1198701and 119870

2are fixed and can be

calculated from nature constants If Γ1and Γ2are not equal to

unity use (A4) into (A3) and integrate with the boundaryconditions 120601 = 0 and 120588 = 0 at the surface of the sphere onegets

120601 = minus1198701(1 + 119899

1) 12058811198991 minus 120573119870

2(1 + 119899

2) 12058811198992 (A5)

Solving this equation for 120588 keeping only the termsproportional to 120573 and using 119899

119895= 1(Γ

119895minus 1) then

120588 = [minus120601

1198701(1 + 119899

1)]

1198991

minus 12057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A6)

where

119898 =1198991

1198992

+ 1198991minus 1 (A7)

Using 120588 from (A6) into (A2) one gets

1198892120601

1198891199032+2

119903

119889120601

119889119903

= 4120587119866[minus120601

1198701(1 + 119899

1)]

1198991

minus 412058711986612057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A8)

Now let us define

119911 = 119860119903

1198602=

4120587119866

1198701198991

1(1 + 119899

1)1198991(minus120601119888)1198991minus1

120596 =120601

120601119888

(A9)

where 120601119888is the potential at the center it can be defined like

(A5) as

120601119888= minus1198701(1 + 119899

1) 12058811198991

119888minus 1205731198702(1 + 119899

2) 12058811198992

119888 (A10)

Using (A5) (A10) to define 120596 as a function of densityone finds that

120596 =120601

120601119888

= (120588

120588119888

)

11198991

+ 1205731198702(1 + 119899

2)

1198701(1 + 119899

1)(1205881198991

1205881198992119888

)

111989911198992

sdot [1 minus (120588

120588119888

)

(1198992minus1198991)11989911198992

]

(A11)

Using (A9) into (A8) one gets

1198892120596

1198891199112+2

119911

119889120596

119889119911+ 1205961198991 + 120578120596

119898= 0 (A12)

where

120578 =12057311989911198702(1 + 119899

2)

120601119888

(minus120601119888

1198602

4120587119866)

11198992

(A13)

120578 is a dimensionless quantity Equation (A12) is the modifiedLane-Emden equationWe are interested in solutions that arefinite at the center 119911 = 0 So this equation can be solved withthe boundary conditions 120596(0) = 1 and 1205961015840(0) = 1 120596 can becalculated from (A12) and it can be used to determine themass density and pressure of the astrophysical objects as afunction of a radius

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by Benha university (httpwwwbuedueg) The author would like to thank the referees for theenlightening comments which helped to improve the paper

References

[1] D Amati M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1989

[2] M Maggiore ldquoA generalized uncertainty principle in quantumgravityrdquo Physics Letters B vol 304 no 1-2 pp 65ndash69 1993

[3] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999

[4] M Maggiore ldquoQuantum groups gravity and the generalizeduncertainty principlerdquo Physical Review D vol 49 no 10 pp5182ndash5187 1994

[5] MMaggiore ldquoThe algebraic structure of the generalized uncer-tainty principlerdquo Physics Letters B vol 319 no 1ndash3 pp 83ndash861993

[6] L J Garay ldquoQuantum gravity and minimum lengthrdquo Interna-tional Journal of Modern Physics A vol 10 no 2 pp 145ndash1661995

[7] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stoecker ldquoSignatures in the Planck regimerdquo PhysicsLetters B vol 575 no 1-2 pp 85ndash99 2003

[8] C Bambi and F R Urban ldquoNatural extension of the generalizeduncertainty principlerdquo Classical and Quantum Gravity vol 2Article ID 095006 2008

[9] A Kempf G Mangano and R B Mann ldquoHilbert spacerepresentation of the minimal length uncertainty relationrdquoPhysical Review D vol 52 no 2 pp 1108ndash1118 1995

[10] A Kempf ldquoNon-pointlike particles in harmonic oscillatorsrdquoJournal of Physics A Mathematical and General vol 30 no 6pp 2093ndash2101 1997

Advances in High Energy Physics 9

[11] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 Article ID 7691 1999

[12] S Hossenfelder ldquoMinimal length scale scenarios for quantumgravityrdquo Living Reviews in Relativity vol 16 no 2 2013

[13] S Das and E C Vagenas ldquoUniversality of quantum gravitycorrectionsrdquo Physical Review Letters vol 101 Article ID 2213014 pages 2008

[14] S Das and E C Vagenas ldquoPhenomenological implicationsof the generalized uncertainty principlerdquo Canadian Journal ofPhysic vol 87 no 3 pp 233ndash240 2009

[15] I Pikovski M R Vanner M Aspelmeyer M S Kim and CBrukner ldquoProbing planck-scale physics with quantum opticsrdquoNature Physics vol 8 no 5 pp 393ndash397 2012

[16] F Marin F Marino M Bonaldi et al ldquoGravitational bardetectors set limits to Planck-scale physics on macroscopicvariablesrdquo Nature Physics vol 9 no 2 pp 71ndash73 2013

[17] T V Fityo ldquoStatistical physics in deformed spaces withminimallengthrdquo Physics Letters A vol 372 no 37 pp 5872ndash5877 2008

[18] M Abbasiyan-Motlaq and P Pedram ldquoTheminimal length andquantum partition functionsrdquo Journal of Statistical MechanicsTheory and Experiment vol 2014 Article ID P08002 2014

[19] P Wang H Yang and X Zhang ldquoQuantum gravity effectson statistics and compact star configurationsrdquo Journal of HighEnergy Physics vol 2010 article 43 2010

[20] O Bertolami and C A D Zarro ldquoTowards a noncommutativeastrophysicsrdquo Physical Review D vol 81 no 2 Article ID025005 8 pages 2010

[21] L D Landau and E M Lifshitz Statistical Physics (Courseof Theoretical Physics Volume 5) Butterworth-HeinemannOxford UK 3rd edition 1980

[22] L N Chang D Minic N Okamura and T Takeuchi ldquoExactsolution of the harmonic oscillator in arbitrary dimensions withminimal length uncertainty relationsrdquo Physical Review D vol65 Article ID 125027 2002

[23] A F Ali and M Moussa ldquoTowards thermodynamics withgeneralized uncertainty principlerdquo Advances in High EnergyPhysics vol 2014 Article ID 629148 7 pages 2014

[24] M Moussa ldquoQuantum gases and white dwarfs with quantumgravityrdquo Journal of StatisticalMechanicsTheory andExperimentvol 2014 no 11 Article ID P11034 14 pages 2014

[25] S Chandrasekhar An Introduction to the Study of StellarStructure Dover Mineola NY USA 1967

[26] T PadmanabhanTheoretical Astrophysics Volume I Astrophys-ical Processes Cambridge University Press Cambridge UK2000

[27] T Padmanabhan Theoretical Astrophysics vol 2 of Stars andStellar Systems Cambridge University Press Cambridge UK2001

[28] R Kippenhahn and A Weigert Stellar Structure and EvolutionSpringer Berlin Germany 1994

[29] G W Collins Fundamentals of Stellar Astrophysics 2003[30] G J Mathews I-S Suh B OrsquoGorman et al ldquoAnalysis of white

dwarfs with strange-matter coresrdquo Journal of Physics G Nuclearand Particle Physics vol 32 no 6 pp 747ndash759 2006

[31] J L Provencal H L Shipman D Koester F Wesemael and PBergeron ldquoProcyon B outside the iron boxrdquo The AstrophysicalJournal vol 568 no 1 pp 324ndash334 2002

[32] A Camacho ldquoWhite dwarfs as test objects of Lorentz viola-tionsrdquo Classical and Quantum Gravity vol 23 no 24 ArticleID 7355 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

For a homogeneous gaseous sphere if we assume thatthe pressure of the star can be calculated by the modifiedpolytropic relation

119875 = 1198701120588Γ1 + 120573119870

2120588Γ2 (A4)

The prototype constants 1198701and 119870

2are fixed and can be

calculated from nature constants If Γ1and Γ2are not equal to

unity use (A4) into (A3) and integrate with the boundaryconditions 120601 = 0 and 120588 = 0 at the surface of the sphere onegets

120601 = minus1198701(1 + 119899

1) 12058811198991 minus 120573119870

2(1 + 119899

2) 12058811198992 (A5)

Solving this equation for 120588 keeping only the termsproportional to 120573 and using 119899

119895= 1(Γ

119895minus 1) then

120588 = [minus120601

1198701(1 + 119899

1)]

1198991

minus 12057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A6)

where

119898 =1198991

1198992

+ 1198991minus 1 (A7)

Using 120588 from (A6) into (A2) one gets

1198892120601

1198891199032+2

119903

119889120601

119889119903

= 4120587119866[minus120601

1198701(1 + 119899

1)]

1198991

minus 412058711986612057311989911198702(1 + 119899

2)

1198701(1 + 119899

1)[

minus120601

1198701(1 + 119899

1)]

119898

(A8)

Now let us define

119911 = 119860119903

1198602=

4120587119866

1198701198991

1(1 + 119899

1)1198991(minus120601119888)1198991minus1

120596 =120601

120601119888

(A9)

where 120601119888is the potential at the center it can be defined like

(A5) as

120601119888= minus1198701(1 + 119899

1) 12058811198991

119888minus 1205731198702(1 + 119899

2) 12058811198992

119888 (A10)

Using (A5) (A10) to define 120596 as a function of densityone finds that

120596 =120601

120601119888

= (120588

120588119888

)

11198991

+ 1205731198702(1 + 119899

2)

1198701(1 + 119899

1)(1205881198991

1205881198992119888

)

111989911198992

sdot [1 minus (120588

120588119888

)

(1198992minus1198991)11989911198992

]

(A11)

Using (A9) into (A8) one gets

1198892120596

1198891199112+2

119911

119889120596

119889119911+ 1205961198991 + 120578120596

119898= 0 (A12)

where

120578 =12057311989911198702(1 + 119899

2)

120601119888

(minus120601119888

1198602

4120587119866)

11198992

(A13)

120578 is a dimensionless quantity Equation (A12) is the modifiedLane-Emden equationWe are interested in solutions that arefinite at the center 119911 = 0 So this equation can be solved withthe boundary conditions 120596(0) = 1 and 1205961015840(0) = 1 120596 can becalculated from (A12) and it can be used to determine themass density and pressure of the astrophysical objects as afunction of a radius

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research is supported by Benha university (httpwwwbuedueg) The author would like to thank the referees for theenlightening comments which helped to improve the paper

References

[1] D Amati M Ciafaloni and G Veneziano ldquoCan spacetime beprobed below the string sizerdquo Physics Letters B vol 216 no 1-2pp 41ndash47 1989

[2] M Maggiore ldquoA generalized uncertainty principle in quantumgravityrdquo Physics Letters B vol 304 no 1-2 pp 65ndash69 1993

[3] F Scardigli ldquoGeneralized uncertainty principle in quantumgravity from micro-black hole gedanken experimentrdquo PhysicsLetters B vol 452 no 1-2 pp 39ndash44 1999

[4] M Maggiore ldquoQuantum groups gravity and the generalizeduncertainty principlerdquo Physical Review D vol 49 no 10 pp5182ndash5187 1994

[5] MMaggiore ldquoThe algebraic structure of the generalized uncer-tainty principlerdquo Physics Letters B vol 319 no 1ndash3 pp 83ndash861993

[6] L J Garay ldquoQuantum gravity and minimum lengthrdquo Interna-tional Journal of Modern Physics A vol 10 no 2 pp 145ndash1661995

[7] S HossenfelderM Bleicher S Hofmann J Ruppert S Schererand H Stoecker ldquoSignatures in the Planck regimerdquo PhysicsLetters B vol 575 no 1-2 pp 85ndash99 2003

[8] C Bambi and F R Urban ldquoNatural extension of the generalizeduncertainty principlerdquo Classical and Quantum Gravity vol 2Article ID 095006 2008

[9] A Kempf G Mangano and R B Mann ldquoHilbert spacerepresentation of the minimal length uncertainty relationrdquoPhysical Review D vol 52 no 2 pp 1108ndash1118 1995

[10] A Kempf ldquoNon-pointlike particles in harmonic oscillatorsrdquoJournal of Physics A Mathematical and General vol 30 no 6pp 2093ndash2101 1997

Advances in High Energy Physics 9

[11] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 Article ID 7691 1999

[12] S Hossenfelder ldquoMinimal length scale scenarios for quantumgravityrdquo Living Reviews in Relativity vol 16 no 2 2013

[13] S Das and E C Vagenas ldquoUniversality of quantum gravitycorrectionsrdquo Physical Review Letters vol 101 Article ID 2213014 pages 2008

[14] S Das and E C Vagenas ldquoPhenomenological implicationsof the generalized uncertainty principlerdquo Canadian Journal ofPhysic vol 87 no 3 pp 233ndash240 2009

[15] I Pikovski M R Vanner M Aspelmeyer M S Kim and CBrukner ldquoProbing planck-scale physics with quantum opticsrdquoNature Physics vol 8 no 5 pp 393ndash397 2012

[16] F Marin F Marino M Bonaldi et al ldquoGravitational bardetectors set limits to Planck-scale physics on macroscopicvariablesrdquo Nature Physics vol 9 no 2 pp 71ndash73 2013

[17] T V Fityo ldquoStatistical physics in deformed spaces withminimallengthrdquo Physics Letters A vol 372 no 37 pp 5872ndash5877 2008

[18] M Abbasiyan-Motlaq and P Pedram ldquoTheminimal length andquantum partition functionsrdquo Journal of Statistical MechanicsTheory and Experiment vol 2014 Article ID P08002 2014

[19] P Wang H Yang and X Zhang ldquoQuantum gravity effectson statistics and compact star configurationsrdquo Journal of HighEnergy Physics vol 2010 article 43 2010

[20] O Bertolami and C A D Zarro ldquoTowards a noncommutativeastrophysicsrdquo Physical Review D vol 81 no 2 Article ID025005 8 pages 2010

[21] L D Landau and E M Lifshitz Statistical Physics (Courseof Theoretical Physics Volume 5) Butterworth-HeinemannOxford UK 3rd edition 1980

[22] L N Chang D Minic N Okamura and T Takeuchi ldquoExactsolution of the harmonic oscillator in arbitrary dimensions withminimal length uncertainty relationsrdquo Physical Review D vol65 Article ID 125027 2002

[23] A F Ali and M Moussa ldquoTowards thermodynamics withgeneralized uncertainty principlerdquo Advances in High EnergyPhysics vol 2014 Article ID 629148 7 pages 2014

[24] M Moussa ldquoQuantum gases and white dwarfs with quantumgravityrdquo Journal of StatisticalMechanicsTheory andExperimentvol 2014 no 11 Article ID P11034 14 pages 2014

[25] S Chandrasekhar An Introduction to the Study of StellarStructure Dover Mineola NY USA 1967

[26] T PadmanabhanTheoretical Astrophysics Volume I Astrophys-ical Processes Cambridge University Press Cambridge UK2000

[27] T Padmanabhan Theoretical Astrophysics vol 2 of Stars andStellar Systems Cambridge University Press Cambridge UK2001

[28] R Kippenhahn and A Weigert Stellar Structure and EvolutionSpringer Berlin Germany 1994

[29] G W Collins Fundamentals of Stellar Astrophysics 2003[30] G J Mathews I-S Suh B OrsquoGorman et al ldquoAnalysis of white

dwarfs with strange-matter coresrdquo Journal of Physics G Nuclearand Particle Physics vol 32 no 6 pp 747ndash759 2006

[31] J L Provencal H L Shipman D Koester F Wesemael and PBergeron ldquoProcyon B outside the iron boxrdquo The AstrophysicalJournal vol 568 no 1 pp 324ndash334 2002

[32] A Camacho ldquoWhite dwarfs as test objects of Lorentz viola-tionsrdquo Classical and Quantum Gravity vol 23 no 24 ArticleID 7355 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 9

[11] F Brau ldquoMinimal length uncertainty relation and the hydrogenatomrdquo Journal of Physics A vol 32 Article ID 7691 1999

[12] S Hossenfelder ldquoMinimal length scale scenarios for quantumgravityrdquo Living Reviews in Relativity vol 16 no 2 2013

[13] S Das and E C Vagenas ldquoUniversality of quantum gravitycorrectionsrdquo Physical Review Letters vol 101 Article ID 2213014 pages 2008

[14] S Das and E C Vagenas ldquoPhenomenological implicationsof the generalized uncertainty principlerdquo Canadian Journal ofPhysic vol 87 no 3 pp 233ndash240 2009

[15] I Pikovski M R Vanner M Aspelmeyer M S Kim and CBrukner ldquoProbing planck-scale physics with quantum opticsrdquoNature Physics vol 8 no 5 pp 393ndash397 2012

[16] F Marin F Marino M Bonaldi et al ldquoGravitational bardetectors set limits to Planck-scale physics on macroscopicvariablesrdquo Nature Physics vol 9 no 2 pp 71ndash73 2013

[17] T V Fityo ldquoStatistical physics in deformed spaces withminimallengthrdquo Physics Letters A vol 372 no 37 pp 5872ndash5877 2008

[18] M Abbasiyan-Motlaq and P Pedram ldquoTheminimal length andquantum partition functionsrdquo Journal of Statistical MechanicsTheory and Experiment vol 2014 Article ID P08002 2014

[19] P Wang H Yang and X Zhang ldquoQuantum gravity effectson statistics and compact star configurationsrdquo Journal of HighEnergy Physics vol 2010 article 43 2010

[20] O Bertolami and C A D Zarro ldquoTowards a noncommutativeastrophysicsrdquo Physical Review D vol 81 no 2 Article ID025005 8 pages 2010

[21] L D Landau and E M Lifshitz Statistical Physics (Courseof Theoretical Physics Volume 5) Butterworth-HeinemannOxford UK 3rd edition 1980

[22] L N Chang D Minic N Okamura and T Takeuchi ldquoExactsolution of the harmonic oscillator in arbitrary dimensions withminimal length uncertainty relationsrdquo Physical Review D vol65 Article ID 125027 2002

[23] A F Ali and M Moussa ldquoTowards thermodynamics withgeneralized uncertainty principlerdquo Advances in High EnergyPhysics vol 2014 Article ID 629148 7 pages 2014

[24] M Moussa ldquoQuantum gases and white dwarfs with quantumgravityrdquo Journal of StatisticalMechanicsTheory andExperimentvol 2014 no 11 Article ID P11034 14 pages 2014

[25] S Chandrasekhar An Introduction to the Study of StellarStructure Dover Mineola NY USA 1967

[26] T PadmanabhanTheoretical Astrophysics Volume I Astrophys-ical Processes Cambridge University Press Cambridge UK2000

[27] T Padmanabhan Theoretical Astrophysics vol 2 of Stars andStellar Systems Cambridge University Press Cambridge UK2001

[28] R Kippenhahn and A Weigert Stellar Structure and EvolutionSpringer Berlin Germany 1994

[29] G W Collins Fundamentals of Stellar Astrophysics 2003[30] G J Mathews I-S Suh B OrsquoGorman et al ldquoAnalysis of white

dwarfs with strange-matter coresrdquo Journal of Physics G Nuclearand Particle Physics vol 32 no 6 pp 747ndash759 2006

[31] J L Provencal H L Shipman D Koester F Wesemael and PBergeron ldquoProcyon B outside the iron boxrdquo The AstrophysicalJournal vol 568 no 1 pp 324ndash334 2002

[32] A Camacho ldquoWhite dwarfs as test objects of Lorentz viola-tionsrdquo Classical and Quantum Gravity vol 23 no 24 ArticleID 7355 2006

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of