A Modi fi ed Thermodynamics Method to Generate Exact Solutions of Einstein Equations∗

Hong-Wei Tan(谭鸿威),Jin-Bo Yang(杨锦波),Tang-Mei He(何唐梅),and Jing-Yi Zhang(张靖仪)

Center for Astrophysics,Guangzhou University,Guangzhou 510006,China

1 Introduction

Since Bekenstein found the relationship between black hole dynamics and thermodynamics,[1]and Hawking presented Hawking radiation according to the quantum field theory in curved spacetime,[2]which is a pure thermodynamical radiation,the researchers have focused on the deep relationship between the theory of the gravitation and thermodynamics for a long time.

In fact,black hole thermodynamics can be viewed as spacetime thermodynamics,which means that the properties of the physical objects in black hole thermodynamics is global on a manifold which is equipped with a Lorentz metric,known as a spacetime.However,it is very difficult to construct thermodynamics in general situations for some common physical quantities such as mass,entropy and angular momentum which can not be well de fi ned.Moveover,in general spacetime,the thermodynamics is usually need to be considered as nonequilibrium state,which is very difficult to be dealt with even for ordinary matter.Though there are such difficulties to overcome,it does not stop the researchers from deriving the Einstein equations from thermodynamic laws.[3]

In 1995,Jacobson derived the Einstein equations from the basic equations of thermodynamics and the Raychaudhuri equations on the null hypersurface,[4]by using the local first law of equilibrium thermodynamics.In such work,the researchers used the assumption that the entropy is proportional to the area of the local Rindler horizon of an in fi nitely accelerated observer,and the Hawking-Unruh temperature,which had been exploited in Ref.[5],was treated as the temperature observed by such observer.Basing on such assumptions,the Einstein equations were derived.However,in that work,the researchers assumed that the spacetime is in a locally thermal equilibrium system,but as the equations that describe the evolution of all kinds of spacetime,Einstein equations are expected to be able to describe all kinds of spacetime’s evolution in principle naturally,including the spacetime that does not satisfy the locally thermal equilibrium assumption.In other words,the researchers obtained the equations that can describe general situations only based on a special assumption,which is unnatural in logic.[3]

For this reason,Ref.[3]put forward a new method to deal with this problem.In their paper,the researchers considered the spacetime equipped with spherically symmetry,whose metric ansatz is ds2=−f(r)dt2+h(r)dr2+r2dΩ2.In such spacetime,the energy of the gravitational field was de fi ned as the Misner–Sharp energy.[6]Firstly,the researchers applied the first law of equilibrium thermodynamics in an adiabatic system,dM=dW,to derive h(r).Deriving f(r)is a difficult task,to solve such problem,the researchers assumed that the surface gravity de fi ned in the traditional way is equal to the geometry surface de fi ned by the uni fi ed first law,[7]and then they generated several exact solutions of the Einstein equations.Furthermore,the authors also improved their work to high derivative gravity,and there is a mini review in Ref.[8].There is no doubt that the amazing results obtained in Ref.[3]provide a new way to study the gravitational thermodynamics.However,there is a limitation in this method,since such method requires the symmetry of the spacetime strictly because the Misner–Sharp energy can only be de fi ned in the spacetime with a spherically symmetry,a plane symmetry as well as a Pseudo spherically symmetry.[9−11]This difficulty motivates us to modify this method.

There are two steps of such modi fication introduced in our paper.Firstly,we replace the Misner–Sharp mass with only the Komar mass,[12]by using the first law of equilibrium thermodynamics in an adiabatic system just like the original method did,and then the results obtained here are similar with that obtained in the original method.Note that the Definition of the Komar mass only requires that the spacetime is stationary,it means that once there is a time-like Killing vector in the spacetime,then our method can be used in principle.In addition,the black hole solution surrounded by quintessence is also generated in this paper.Our another achievement is that we construct another Definition of the geometry surface gravity,which is de fi ned by the Komar mass.In the second step of our work,we use the ADM mass,[13]together with the Komar mass to complete such modi fication.If we do so,then we can also regenerate these exact solutions of Einstein equations.Furthermore,we modify the Definition of the ADM mass,and then the global monopole spacetime can be generated.

This paper is organized as follows:in Sec.2 we modify this method with only the Komar mass,and generate several exact solutions of Einstein equations.The geometry surface gravity de fi ned by Komar mass is also construct in this section.In Sec.3,we introduce the method modi fi ed by both the Komar mass and the ADM mass,and some comments on the situation that the spacetime with global monopole charge is arisen.In Secs.4 and 5,some discussion and conclusion are given.

2 Modi fi ed with Only Komar Mass

In this section,the method modi fi ed with only the Komar mass will be introduced.Here,the metric ansatz of a spherically symmetric spacetime is

In a stationary spacetime,the Komar mass can be de fi ned as

where ϵabcdis the volume element of the four-dimensional spacetime and ξdis a time-like Killing vector field.According to the two formulas above,one can get the Komar mass in this metric ansatz as

In the spherically symmetric spacetime,according to the uni fi ed first law,the geometry surface gravity can be de fi ned as[7]

in which Mmsis the Misner–Sharp energy de fi ned as[6]

and ω is the work term de fi ned as[7]

where Iabis the inverse of the induced metric of the spacetime in the leading two dimensions whose line element reads

On the other hand,in Eq.(1),the surface gravity is

In Ref.[3],the researchers assumed that the surface gravity is equal to the geometry surface gravity

In this paper we will follow this assumption.According to Eqs.(3),(4),(8),and(9),we can obtain the relationship between the Komar mass and the Misner–Sharp energy as

2.1 The Schwarzschild Solution

Considering a vacuum spacetime and the first law of equilibrium thermodynamics in an adiabatic system,one can get

The energy-stress tensor is zero in the vacuum space,so the work term ω must be zero.Combining Eqs.(5),(10),and(11)together,we have

Solving this equation,the result reads

Substituting it into Eqs.(3)and(5),and combing with Eq.(10),f(r)is obtained as

If we choose the asymptotically fl at spacetime as the boundary condition,then

And the Komar mass reads

Finally the result can be written as

It is exactly the line element of the Schwarzschild spacetime.Now we can draw a conclusion that the Kormar mass describes an adiabatic process.Furthermore,combining Eqs.(3),(8),(9)together,one can obtain the geometry surface gravity de fi ned by the Komar mass as

2.2 The Schwarzschild-de Sitter Solution

Now let us deal with the situation that there is force works.Considering the first law of thermodynamics again

wherePdonates the pressure and V is the volume

the work term is[3]

where Λ can be viewed as the cosmological constant.Substituting it into Eq.(10),then the Komar mass reads

Based on Eqs.(19),(21),(22),we get

Letting the Λ =4πP,the results are read as

It is just the line element of the Schwarzschild de Sitter spacetime

2.3 The RN-de Sitter Solution

Furthermore,in the situation that there is an electric charge is considered,then

Reference[3]assumed that the work of the electric field can be written as(q/r)dq,however,we find that using this assumption can not derive the RN solution.Indeed,in Ref.[7]the work of the electric force is considered as(q2/r2)dr.Moreover,if we use this as the assumption and then the RN solution can be obtained,which will be expressed as follows.

To be more general,we should consider that there are both force and electric field doing work,so the work term is written as[3]

One can obtain the Komar mass in this situation as

So,we can get the equation as

Considering Λ =4πPand solving the equation above,the solution is obtained as

Substituting this into Eq.(3),we get

And therefore,the line element of RN-de Sitter spacetime is obtained,that is,

2.4 More General Situations

In more general situations,if it is assumed that the work term ω and the pressurePare both power functions of r,applying the first law of thermodynamics,for convenient,the equation should be expressed as

where a,b,c and d are all constants.The solution of this equation is

If it is assumed that a,b,c and d are not independent with each other but constrained by following conditions

then h can be rewritten as

Above formula can be inserted into Eq.(10),then one can get

and the solution is

Setting C2=0,one can

Rede fi ning a new parameter α as

then the line element of the spacetime is

where α can be viewed as the charge of the spacetime.For some speci fic examples,if α=0,then a=c=0,the Schwarzschild solution can be obtained,and if d=−1,then one arrives at the RN spacetime and α=q2,where q is the electric charge.

Noted that if the range of d is set as

then we arrive at the black hole solution surrounded by quintessence,which has been obtained by Kiselev in 2003.[14]

It should be careful that when d=1.In such situation the solution is

it seems that the global monopole spacetime is generated.However,if above is submitted into Eq.(3),then the Kormas can be obtained as

and then the thermodynamical relationship reads

which means that there is not any work in this situation.It requires that

in Eq.(36),then we just arrive at the Schwarzschild situation again.

3 Modi fi ed with both Komar Mass and ADM Mass

In an asymptotically fl at sapcetime,the ADM mass can be de fi ned as[13]

where the hijis the spatial component of the induced metric in the asymptotically Descartes coordinates.In our spacetime metric ansatz,the line element of the induced metric can be written as

Since what we consider now is an asymptotically fl at sapcetime,so it can be believed that

So the spatial line element can be written approximately as

After some calculations,the ADM mass can be written as

After the limitation has been taken,then

3.1 The Schwarzschild Solution

Applying again the first law of thermodynamics in a vacuum spacetime which is in an adiabatic system

then the following differential equation can be obtained

Solving this equation,the result reads

The condition of asymptotic fl at spacetime requires that C1=1.Submitting Eq.(56)into Eq.(53),then the result can be obtained as

Inserting above result into the Komar mass(3)and using the first law of thermodynamic

then the following equation can be obtained

Solving the above equation,the result is

If the integral constants are chosen as

then the solution can be written as follows

Combining Eqs.(3),(57),and(62)together,then the following result is obtained

This result suggests that our method is reasonable.Applying this result into Eq.(1),then the Schwarzschild solution can be obtained

3.2 A Comment on the Spacetime with a Global Monopole Charge

Let us consider Eq.(53)again,which is under the condition that Now,let us assume that Eq.(53)still works in the spacetime with a global monopole charge.However,such spacetime is not a spherically symmetry spacetime anymore.Speci fically,let us consider a global monopole spacetime,whose line element is

where η is a constant.This line element can be rescaled as

In this spacetime,the integral∫dS is not 4π but 4π(1−8πη2),see Ref.[15].So,in order to carry the information of the global charge,we de fi ne the ADM mass in such spacetime as

It should be noted that when η=0,the Definition above reduces to Eq.(53).Now we have generalized the definition of ADM mass in the spacetime with a global monopole charge,and let us call this mass as quasi ADM mass.

Now we are ready to explore what such generalization will give us.Firstly,we consider a global monopole spacetime.By using the Definition of the quasi ADM mass,then

Next,let us consider the global monopole spacetime with an electric charge,whose line element is

With calculation,the thermodynamical relationship can be obtained as

This result means that the global monopole charge results in a correction factor in the thermodynamical relationship.

Whatever,it is obvious that the first law of thermodynamics can be obtained in our Definition of quasi ADM mass,which suggests that such generalization is reasonable.

4 Discussion

There are several comments on our work introduced as follows:

(i)In Sec.3,we introduce the method that modi fi ed by both the Komar mass and the ADM mass.However,to be honest,only the Schwarzschild solution has been generated completely in our work.However,with some trick,some other exact solutions can also be regenerated.Let us take the RN solution as an example.Firstly,let us consider the thermodynamical relationship for ADM mass in this situation

and the solution reads

Submitting above into Eq.(3),and using the same thermodynamical relationship,then we have

the above equation is too difficult to be solved,but we can check that the following is one particular solution of this equation:

Here,the RN spacetime is generated though this trick is not strict enough.

(ii)Some analyses about the situation that the spacetime with global monopole charge are also given in Sec.3.However,we can consider the inverse logic.We assume that the thermodynamical relationship still works in this situation.In the vacuum,the thermodynamical relationship reads

and the solution reads

In this situation,the requirement of the asymptotically fl at sapcetime is loosen,so the integral constant can be chosen as C1=1−η,and the result reads

The f(r)can also be solved as

Then the global monopole spacetime has been generated.

5 Conclusion

In this paper,we modify the method to generate the exact solution of the Einstein equations with the laws of thermodynamics which was arisen in Ref.[3].In Ref.[3],the researchers used the Misner–Sharp energy and uni fi ed first law to derive several exact solutions of Einstein equations without involving it.However,the Misner–Sharp energy can only be de fi ned in the spacetime with a spherically symmetry,a plane symmetry as well as a Pseudo spherically symmetry,which limits this method to be generalized to more general situation.

This method is modi fi ed in two steps in this paper.Firstly,we use only the Komar mass to take the place of the Misner–Sharp energy to modify such method,and then several exact solutions of the Einstein equations are regenerated.Moreover,we obtain the geometry surface gravity de fi ned by the Komar mass in the specially symmetry spacetime.Since the Komar mass requires the symmetry less than the Misner–Sharp energy,means that method could be used in more situations general in principle.

Secondly,we modify this method with both the Komar mass and the ADM mass,some exact solutions of Einstein can also be regenerated.Moreover,the quasi ADM mass de fi ned in the spacetime with a global monopole charge and some thermodynamical properties of such mass are analyzed.We find that the first law of thermodynamics still works in such mass,and the global charge plays an important role in the relationship between the extra field and the work done by such extra field.

References

[1]J.D.Beckenstein,Phys.Rev.D 7(1973)2333.

[2]S.W.Hawking,Commun.Math.Phys.43(1975)199.

[3]H.Zhang,S.A.Hayward,X.H.Zhai,and X.Z.Li,Phys.Rev.D 89(2014)064052.

[4]T.Jacobson,Phys.Rev.Lett.75(1995)1260.

[5]W.G.Unruh,Phys.Rev.D 14(1976)870.

[6]C.w.Misner and D.H.Sharp,Phys.Rev.136(1964)B571.

[7]Hayward,Classical Quant.Grav.15(1998)3147.

[8]Hong-Sheng Zhang,The Universe 3(2015)30.

[9]H.Maeda and Nozawa,Phys.Rev.D 77(2008)064031.

[10]R.G.Cai,L.M.Cao,Y.P.Hu,and N.Ohta,Phys.Rev.D 80(2009)104019.

[11]H.Zhang,Y.Hu,and X.Z.Li,Phys.Rev.D 90(2014)024062.

[12]A.Komar,Phys.Rev.113(1959)934.

[13]R.Arnowitt,S.Deser,and C.Misner,Gen.Relativ.Grav.40(2008)1987.

[14]V.V.Kiselev,Classical Quant.Grav.20(2003)1187.

[15]Manuel Barriola and Alexander Vilenkin,Phys.Rev.Lett.63(1989)341.