Thermodynamic Analysis of Cosmological Black Hole

M.Akbar, Tayeb Brahimi, and S.M.Qaisar

E ff at University,Electrical and Computer Engineering Department,P.O.Box 34689,Jeddah,Saudi Arabia

1 Introduction

Black holes and cosmological metrics are obtained by solving Einstein’s field equations under different domains.Generally,the black metric represents a point mass covered by the horizons without expanding behavior of the universe.The main issue with the black holes is that these metrics disregard the expanding behavior of the universe.On the other hand,the cosmological metrics persist expanding universe without holding a point mass describing a black hole.However,the McVittie metric persists an integrated treatment holding a strongly gravitating central object,embedded in spatially fl at Friendmann-Robertson-Walker(FRW)universe.[1]This is one of the primary motivations of this study in order to integrate the thermodynamic treatment of the Mcvittie universe.Generally,the thermodynamics of the spacetime metrics are two folds:(i)Thermodynamics of black hole metrics at black hole horizons and(ii)Thermodynamics of the cosmological metrics at their respective horizons.In the case of black hole metrics,Padmanabhan[2]was the first who initiated the key development to launch relationship between the Einstein field equations and the first law of thermodynamics near the black hole horizons.It has been shown that in Einstein’s gravity as well as in a wider class of gravitational theories,the Einstein field equations establish the first law of thermodynamic T dS=dE+PdV.[2?3]Recently,various laws of thermodynamics have been studied[4?8]to explore deep relationship between gravity theory and thermodynamics.In the case of cosmological metrics,Cai and Kim[7]initiated and derived the Friendmann equations of FRW universe by employing the equilibrium Clausius relation at the apparent horizon.Later on,Akbar and Cai[9]formulated and recast the differential form of the Einstein field equations as a uni fi ed first law T dS=dE+W dV near the apparent horizon of FRW universe.This connection has also been extended for other gravity theories.[10?12]Due to this deep connection between gravity theory and the laws of ordinary thermodynamics,it has been argued[13]that thermal behavior of the Einstein field equations would be a generic property of the spacetime metrics and can be extended to any spacetime metric with horizons.In the present work,we consider a more general spacetime metric,called McVittie metric,to integrate and explore its thermodynamics.This metric has been modeled to present a family of spherically symmetric non-vacuum spacetime universe in which a black hole is embedded in a fl at FRW universe.In this study,we analyze qualitatively the roots of a cubic equation arising from the apparent horizon of the McVittie universe.The surface gravity of McVittie universe is discussed near apparent horizon and various conditions in terms of McVittie parameters are explored.The field equations of McVittie metric are obtained by applying the Uni fi ed first law of thermodynamics.Furthermore,heat capacity at constant pressure and generalized second law(GSL)are discussed at apparent horizon.

This paper is organized as follows.In Sec.2,we shall review brie fl y properties of McVittie metric.The horizons of McVittie universe shall be discussed in Sec.3.In Secs.4 and 5,we shall analyze conditions on surface gravity and heat capacity.The field equations will be derived via unifi ed first law in Sec.6.Furthermore we shall discuss GSL in Sec.7 while we shall conclude our results in Sec.8.

2 McVittie Universe

In his original work,McVitte was able to find[1]his well-known spacetime metric called McVittie universe and its line element is given in isotropic coordinates by

where d?2,M and a(t)are the line element on the unit 2-sphere,a positive constant which represents the mass of the black hole and an arbitrary positive function called the cosmic scale factor respectively.Throughout this paper,we adopt the units G=c=κ=~=1.In particular,the above metric(1)reduces to a spatially fl at FRW metric when M→0 while it reduces to the Schwarzschild black hole in isotropic coordinate at a(t)=constant.Apparently,McVittie metric seems to stand for a Schwarzschild black hole embedded in a homogeneous,spatially fl at FRW metric[14]but its physical interpretation is still under debate.[14?15]On the other hand,it has been argued in Ref.[15]that the McVittie metric,in general,can not represent a black hole embedded in a homogeneous,spatially fl at FRW universe because it becomes singular on the 2-sphere at r=M/2a and this singularity is spacelike.[16]Furthermore,it was found that the pressure of the matter density is in fi nite at r=M/2a while its energy density is if nite.It is also argued that the McVittie universe may illustrate a point mass situated at r=0 and embedded in a spatially fl at FRW universe.Moreover,in general,this point mass is covered by a singularity at r=M/2a.[16]In Ref.[17],Nolan found that it is a weak singularity in the sense that the volume of an object does not shrink to zero at a surface where r=M/2a,and therefore the energy density of the cosmic fl uid remains fi nite.On the other hand,the pressure of the cosmic fl uid and Ricci scalar diverge at r=M/2a.Also,Nolan rewrote McVittie metric in terms of R=ar(1+M/2ar)2and argued that the resulting metric behaves like a point-mass at R=0,covered by a singularity at R=2M.He also argued that if the present universe is expanding the surface at R=2M is covered by an anti-trapped region which admits white hole rather than a black hole.Meanwhile,other authors[18]discarded Nolan interpretation and suggested black hole interpretation by de fi ning ingoing radial null geodesics at a particular choice of the scale factor a(t).So,the McVittie metric represents some kind of strongly gravitating central object,embedded in fl at FRW universe.However,its physical interpretation is not totally clear and remains the subject of debate.

3 Apparent Horizon

Let us rewrite the McVittie metric(1)in spherical symmetric form

where R=a(t)r(1+M/2a(t)r)2is the time dependent areal radius of the universe,x0=t,x1=r and twodimensional metric

The apparent horizon is de fi ned by a marginally trapped surface with vanishing expansion.Thus mathematically,hab?aR?bR=0 fi xes the location of the apparent horizons,which after evaluating through McVittie metric(2),gives a cubic equation in terms of apparent radius RAgiven below

where H refers to the time dependent Hubble parameter and RAdenotes the apparent horizon radius.Particularly,at M=0,the apparent horizon reduces to the apparent horizon/cosmological horizon,RA=1/H,of fl at FRW universe,while at a(t)=constant i.e.H=0,it admits event horizon of Schwarzschild black hole located at RA=RE=2M.It has been argued in Ref.[19]that the apparent horizon described a causal horizon of dynamical spacetime holding gravitational entropy,surface gravity and other thermodynamical properties.It has been shown that these thermodynamic quantities associated with the apparent horizons obey the first law of thermodynamics in the Einstein gravity as well as a wider class of gravity theories.[20]In the present work,we study various thermodynamic characteristics associated with the apparent horizons of McVittie universe and analyze the process of energy fl ow through apparent horizon to provide the uni fi ed first law.Let us first proceed to find the apparent horizons,RA,of the McVittie universe by finding real positive roots of Eq.(3).It is convenient first to analyze Eq.(3)qualitatively and fi x conditions in terms of H and M for the existence of its horizons.In order to proceed this analysis,we de fi ne a real function f(RA)=H2?RA+2M of variable RA＞0 via Eq.(3).Mathematically,its extreme points are obtained by putting df(RA)/dRA=0,which admits two real points,RA±= ±1/H.Neglecting RA?<0 negative root,the root RA=RA+=1/H is a unique root representing extreme point.The second deriv√ative d2f(RA)/d＞0 at its extreme point RA=1/H,which implies that the extremal point corresponds to minimum and its minimumvalue is f(Rmin)= ?2/3H+2M at RA=Rmin=1/H.Also note that as RA→∞implies f(RA)→∞and f(RA=0)=2M.Since the function f(RA)is twice differentiable and positive at critical point,hence it is concave up.With reference to its minimum value,there are three cases;

Case 1 In this case,f(R)cuts the R-axis at two points admitting two real distinct roots RA1and RA2.Hence,there are two horizons inner and outer de fi ned by Eqs.(4)and(5)respectively.These roots are shown graphically by Fig.1.When f(Rmin)<0,McVittie metric admits two horizon provided M<1/3H.

Fig.1 Function f(R)for M=1 and H=1/6 is shown which admits two horizons when M<1/3H.

Case 2This gives repeated roots and identi fi ed as an extremal case.In this case f(R)touches R-axis at RA=Rminas shown in Fig.2.Furthermore,f(Rmin)=0,admits the condition,M=1/3H.

Fig.2 Repeated ho√rizons of McVittie Universe with M=1 and H=1/3 satisfying the condition M=1/3H.

Case 3 This case involves naked singularity and hence there are no real roots when f(Rmin)＞0.The graph of f(R)lies above the R-axis as shown in Fig.3.Beside the qualitative analysis of Eq.(3),one can obtain its exact roots.Since it is a cubic equation in RAand there are various methods available in literature to solve such an equation.However we follow the procedure given by Nickalls[21]to find its roots.These roots are given below,

where sin3θ =3MH(t).Obviously,after neglecting negative root R3<0,we have two real positive roots admitting two apparent horizons,RA1and RA2of McVittie universe.These two horizons RA1and RA2exist provided 0

Fig.3 No horizons of McVittie Universe with M=1 and H=1/2 satisfying M＞1/3H.

4 Horizon Thermodynamics

This section deals with the various thermodynamic quantities associated with the apparent horizons of McVittie universe.It has been shown by Hawking that a black hole emits thermal radiation at its horizon with a temperature proportional to the surface gravity and with an entropy proportional to the horizon area.These notions of temperature and entropy are not limited with the black holes horizons but also extended with other horizons of various spacetime geometries.Among these horizons,apparent horizon has been argued to be a causal horizon associated with the notions of temperature and entropy.Thus for our purpose,it would be suitable to study thermodynamic properties of McVittie universe at its apparent horizons.Let us de fi ne entropy SHassociated with apparent horizon of McVittie universe which is proportional to surface area A of the horizon

where RAis the horizon radius.The temperature THassociated with the apparent horizon is proportional to the surface gravity κ through relation TH= κ/2π,where κ is given by[8]

where?h is the determinant of hab=diag(g00,g11)with g00and g11t-r components of McVittie metric(1).Using above Eq.(8),the surface gravity κ of the McVittie universe turns out

where an over-dot denotes derivative with respect to the cosmic time t.Particularly,as M→0,the surface gravity of the McVittie universe reduces to the fl at FRW universe which admits

where in this case R=a(t)r is the apparent radius of the fl at FRW universe.It is evident from the above Eq.(10)that the surface gravity of FRW universe is positive,zero and negative provided/H2

wherePis the pressure of the perfect fl uid.The surface gravity of McVittie universe is positive,zero and negative provide the mass M＞(/4)(H2? 8πP),M=(/4)(H2? 8πP),and M<(/4)(H2? 8πP)respectively.On the other hand,One can easily check when the scale factor a(t)=constant andP=0,the surface gravity of Schwarzschild black hole at the event horizon rE=2M takes the form,κ=1/2rE.In case of matter dominated universe where the matter is given by the dust particles with no pressure,P=0,the surface gravity of the McVittie universe takes the simple form;κ=M/R2?H2R/4.Let us consider a special case of Kottler Schwarzschild de-sitter metric for which a(t)=In this case surface gravity reduces to

Furthermore when a(t)=a0t2/3,the surface gravity is positive and zero accordingly

and negative when

5 Heat Capacity

Another important thermodynamic quantity is that of heat capacity of a thermal system.The heat capacity of a black hole has been determined to study its stability conditions.[22]The heat capacity of a cosmological metric is de fi ned via enthalpyN=E+PV of the thermal system enclosed by the apparent horizon,whereE,Pand V are the internal energy of the system,pressure and volume enclosed by the horizon respectively.We assume the Misner–Sharp Mms=E.Hence the enthalpy of the system is expressed as a function of horizon radius RAof McVittie universe asN=RA/2+(4/3)πP.Similarly the temperature in Eq.(18)is written as a function of RAby elimination M through Eq.(3).Hence the heat capacity CPof the thermal system at constant pressure is de fi ned via enthalpy as

Using the above Definition of heat capacity along with the horizon temperature

in terms of horizon radius,the heat capacity Cpof McVittie universe enclosed by apparent horizon reads

Obviously whenP＞0,the heat capacity of the universe is positive,negative and divergent accordingly RA＞and RA=From these conditions,one can easily conclude 8πP? 3H2＞0.The divergence of the heat capacity at RA=indicate the universe undergo second order phase transition.[22]It is clear from Eq.(15)that the heat capacity Cpof the universe is always negative for the matter dominated phase havingP=0 and is positive provided the pressureP＞(2+3H2R2)/8πR2＞0.Obviously whenP<0,Cp＞0 providedP< ?1/8πR2.Furthermore,the horizon temperature(15)TH→∞as RA→0 and TH→0 as RA→The case whenP＞0,the temperature THof the universe attains its minimum value Tmin=1/2πRminat Rmin=Interestingly,the heat capacity diverges at Rminwhere temperature is minimum.

6 Uni fi ed First Law of Thermodynamics

In this section,we shall apply uni fi ed first law of thermodynamics to derive Einstein’s field equation which demonstrates a deep connection between gravity theory and laws of ordinary thermodynamics.Uni fi ed first law was first proposed by Hayward to handle thermodynamics associated with the trapping horizon of a dynamical black hole.[23]Consequently he was able to derive Einstein’s field equations with the application of uni fi ed first law.In this paper,we shall apply a similar procedure to extract Einstein’s field equations of McVittie metric from uni fi ed first law.The Einstein field equations,Rμν?Rgμν/2=8πTμν,admit the following non-zero components arising from McVittie metric,

where=pand= ?p are the energy density and pressure of the fl uid respectively and Tμν=(p+P)UμUν?Pgμνis the stress energy tensor of the perfect fl uid.Applying the stress-energy conservation=0,we get

Note that other components of Einstein’s field equations satisfyThe temperature associated with the apparent horizon is determined via TH= κ/2π which admits,

The horizon temperature THin terms of apparent horizon RAcan be written as

The entropy SHassociated with horizon is given by

where A is the horizon area.The Misner–Sharp energy[24]of a spherically symmetric spacetime geometry is de fi ned by 1? 2Mms/R=gab?aR?bR.This energy Mmsreveals total matter energy enclosed by the sphere of radius R and is constructed from spacetime metric gab.In addition to this,various Definitions of energy are given in general relativity,such as,ADM mass,Bondi–Sachs energy,Brown–York energy and others.[25]A detail comparison of various energy Definitions in general relativity has been given in references.[24]However,the Misner-Sharp energy is purely geometric quantity and is related with a spacetime structure.Therefore we consider Misner–Sharp energy as total matter energy in order to derive the Einstein field equations via uni fi ed first law.The Misner–Sharp energy for the McVittie metric is given by

The unified first law is defined by[23]

where dE=dMmsis the change of energy within the volume enclosed by the apparent horizon.This change occurs due to the crossing of energy through the apparent horizon.W is the work density and V=4π/3 is volume of the spherical system bounded by the apparent horizon.The first term on the right side of the above Eq.(23)could be presented by the energy supply term while the second term can be interpreted as work done by the energy content to support this state.Let us turn to de fi ne two invariant quantities,the work density W and energy supply vector Ψ through stress-energy tensorSo the work-density associated with the stress energy tensor is given by

where haband Tabare the 2×2 components of the McVittie metric and stress-energy tensor respectively.Energy supply vector is de fi ned by

which after simpli fication,admits

whereandrepresent derivatives with respect to cosmic time t and radius r respectively.Thus the scalar Ψ is given by

Substituting these quantities in Eq.(23)and evaluate AΨ+W dV which admits

From Misner–Sharp energy(22),one reads

Substituting Eqs.(28)and(29)in uni fi ed first law,dE=AΨ+W dV,and then first comparing the coefficients of dt on both side,we get

Substituting the value of RA/˙RAto the above Eq.(30)and simplifying,we get

which is exactlycomponent of the Einstein field equation.Similarly,comparing the coefficients of dr,we immediately getcomponent of the Einstein field equation.From the above analysis,we are able to extract the Einstein field equations from the Uni fi ed first law.

7 Generalized Second Law of Thermodynamics

Generalized Second Law(GSL)states that the sum of the entropy,SH,associated with geometrical horizon and the entropy,SRassociated with the matter and radiation fields within the horizon never decreases.Mathematically,GSL can be expressed as

which obviously express dynamical nature of the apparent horizon RA.It is straightforward to know that→∞as RA→ 1/letus now turn to differentiate equation(7)to find out TH

Since GSL veri fi es a special connection between thermodynamics,gravitation,and quantum theory,[26]therefore the validity of GSL has been investigated widely for black holes as well as cosmological spacetimes.[27]The purpose of this section is to find the conditions under which GSL will satisfy at the apparent horizon of McVittie universe.Let us first differentiate Eq.(3)with respect to cosmic time,we get

To ensure that the thermal system bounded by apparent horizon is in thermal equilibrium near the apparent horizon,we assume that the horizon temperature THshould equal to the temperature Tmof the perfect fl uid so that TH=Tm=T.Using Eqs.(34)and(35)together with assumption of the thermal equilibrium near apparent horizon,one can get

wherePis the pressure of the perfect fl uid.It is shown in the above section that the Einstein field equations satisfy uni fi ed first law dE=THdSH+PdW instead of usual first law dE=THdSH+PdV.However the matter energy densityEm=pV,the matter entropy Smand the temperature Tmof the matter field hold the Gibbs identity TmdSm=dEm+PdV on the apparent horizon.By solving Gibbs identity on the apparent horizon RAof McVittie universe,it turns out

By introducing horizon temperature THfrom Eq.(20)in the above equation,one can get

AssumingP＞0,it is easy to show that GSL holds near the apparent horizon only if˙RA＞0.

8 Conclusion

In this paper a cubic equation constructed from the apparent horizon of McVittie universe is analyzed qualitatively.We derived conditions in terms of McVittie parameters in order to obtain two,repeated,and complex roots and presented graphically.We derived surface gravity at apparent horizon of McVittie universe and particular cases are also presented.It is shown that for a particular case when M=0,earlier known surface gravity of fl at FRW universe is recovered.Furthermore,the heat capacity of the universe is obtained at the apparent horizon and discussed various cases.It is shown that the heat capacity diverges at the minimum value of apparent horizon Rminand the universe undergoes the second order phase transition.Also,in the case of matter dominated phase,the heat capacity is always negative.

In addition,the Einstein field equations arising from the McVittie universe are derived by using uni fi ed first law,dE=AΨ+W dV,of thermodynamics,where dEis the amount of energy crossing the apparent horizon and the terms AΨ and W dV are interpreted as the energy supply term and work done by the energy content to change the volume dV of the universe bounded by the apparent horizon.In fact,these thermodynamic identities delegate intrinsic thermodynamic properties of spacetime metrics.Also,we discussed GSL at the apparent horizon of McVittie universe.It is shown that GSL is respected when＞0 together withP＞0.

Acknowledgments

The authors are grateful to the referee for his/her useful comments which have signi ficantly improve quality of the paper.

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