ABSTRACT (original article): In recent years, there has been significant interest in the field of extended black hole thermodynamics, where the cosmological constant and/or other coupling parameters are treated as thermodynamic variables. Drawing inspiration from the Iyer-Wald formalism, which reveals the intrinsic and universal structure of conventional black hole thermodynamics, we illustrate that a proper extension of this formalism also unveils the underlying theoretical structure of extended black hole thermodynamics. As a remarkable consequence, for any gravitational theory described by a diffeomorphism invariant action, it is always possible to construct a consistent extended thermodynamics using this extended formalism.
CITATION (original article): Y. Xiao, Y. Tian and Y. X. Liu (2024), Extended Black Hole Thermodynamics from Extended Iyer-Wald Formalism, Phys. Rev. Lett. 132, no.2, 021401. https://doi.org/10.1103/PhysRevLett.132.021401
CITATION (original article): Y. Xiao, Y. Tian and Y. X. Liu (2024), Extended Black Hole Thermodynamics from Extended Iyer-Wald Formalism, Phys. Rev. Lett. 132, no.2, 021401. https://doi.org/10.1103/PhysRevLett.132.021401
Introduction
Introduction
In the above article, we have proposed a proper extension to the Iyer-Wald formalism, and show that it can produce the extended black hole thermodynamics. In addition to the theoretical analysis, we have studied several examples to demonstrate how to derive the extended black hole thermodynamics from our extended formalism. The examples include the Schwarzschild-AdS black hole, Kerr-AdS black hole, and the black holes in higher curvature gravity (with squared and cubic curvature terms) and Gauss-Bonnet gravity. We provide a Mathematica Notebook which includes all the detailed calculations for these examples, so the interested readers can readily repeat our results.
Note: The following Notebook uses the package “diffgeoM.m”, which can be downloaded from: https://github.com/bryango/diffgeoM
The package has significantly enhanced the convenience of programming on our Notebook. We express our heartfelt gratitude to the original author of this invaluable package.
The package has significantly enhanced the convenience of programming on our Notebook. We express our heartfelt gratitude to the original author of this invaluable package.
1. The calculations for AdS -Schwarzschild black hole
1. The calculations for AdS -Schwarzschild black hole
(0) The metric:
Quit[];
In[]:=
coord={t,r,θ,ϕ};metricsign=-1;metric=DiagonalMatrix-1++Λ,,,;$Assumptions=And[M>0,rh>0,r>0,θ≥0,θ≤π,ϕ≥0,ϕ<2π,Λ<0];<<"D:\\diffgeoM.wl"tDelta[var_]:=D[var,M]dM+D[var,Λ]dΛ;
2M
r
2
r
3
1
1--Λ
2M
r
2
r
3
2
r
2
r
2
Sin[θ]
(1) The computation of ∫Q on :
~
δ
S
r
xi={1,0,0,0};xiD=lower[xi];quvD=-1/(16Pi)2*antisymmetrize[covariant[xiD]];quv=raise[quvD,{1,2}];qXi=quv[[1,2]]*4Pir^2//Expand;dQ=tDelta[qXi]
dM
2
dΛ
3
r
6
(2) The computation of -∫ ξ·Θ on : ( the minus sign is considered implicitly below)
S
r
ddMetric=covariant[tDelta[metric]];1/(16Pi)(contract[ddMetric,{1,3}]-contract[ddMetric,{2,3}])//Simplify;xi**raise[%,{1}];xiTheta=%[[1,2]]*4Pir^2//Expand
dM
2
dΛ
3
r
3
(3) Adding the results from (1) and (2), one get -ξ·Θ for the Killing vector in the case of AdS-Schwarzschild black hole. Then regularize it by subtracting the pure AdS result. (Note that the substitution {M0,dM0} below gives the pure AdS result directly; alternatively, one can repeat the above calculation again directly starting from the metric of pure AdS.)
sQ
~
∫δ
ξ
t
dQ+xiTheta;%/.{M0,dM0};%%-%
dM
(4) Verification of the extended first law : dM=TdS+VdP. Below the expressions of M, T, S, V are denoted by mh, tem, entr and vol0. When verifying the extended first law, we use the outer horizon radius and the cosmological constant Λ as independent variables.
r
h
The expressions of the thermodynamic quantities are listed below: (As we emphasized in the paper, we do not need these expressions in deriving the extended 1st law, in sharp contrast to other approaches relying on the manipulations of thermodynamic relations. Whatever, one needs these expressions to do verification, in order to remove possible doubts.)
mh=-Λ;tem=;entr=Pirh^2;-2PiIntegrate[rg,{θ,0,Pi},{r,rh,ifty}];%/.{m0,rh0};vol0=%%-%//Simplify
rh
2
3
rh
6
1-Λ
2
rh
4πrh
4π
3
rh
3
The verification:
va={rh,Λ};Fori=1,i≤2,i++,PrintSimplifyD[mh,va[[i]]]-tem*D[entr,va[[i]]]-(vol0)*D-,va[[i]]0
Λ
8Pi
True
True
2. The calculations for AdS-Kerr black hole
2. The calculations for AdS-Kerr black hole
(0) The metric: (The scale l is the AdS radius, with =- and in D=4 dimensions.)
2
l
3
Λ
dl=dΛ
3
2l
2
Λ
Quit[];
metricKerr=(-(+)-r(+(-2m+r))+(-))((+)),0,0,-(a(r(-2m+r+)+(+)))((-)(+)),0,+,0,0,0,0,+,0,-(a(r(-2m+r+)+(+)))((-)(+)),0,0,1---2mr+(+)1+(+);
2
a
2
l
2
r
3
r
2
l
2
a
2
l
4
a
2
Cos[θ]
2
Sin[θ]
2
l
2
r
2
a
2
Cos[θ]
2
l
2
a
3
r
2
a
2
a
2
r
2
Cos[θ]
2
Sin[θ]
2
a
2
l
2
r
2
a
2
Cos[θ]
2
r
2
a
2
Cos[θ]
-2mr+(+)1+
2
a
2
r
2
r
2
l
2
r
2
a
2
Cos[θ]
1-
2
a
2
Cos[θ]
2
l
2
l
2
a
3
r
2
a
2
a
2
r
2
Cos[θ]
2
Sin[θ]
2
a
2
l
2
r
2
a
2
Cos[θ]
2
Sin[θ]
2
(+)
2
a
2
r
2
a
2
Cos[θ]
2
l
2
a
2
a
2
r
2
r
2
l
2
Sin[θ]
2
-1+
2
a
2
l
2
r
2
a
2
Cos[θ]
coord={t,r,θ,ϕ};metricsign=-1;metric=metricKerr;$Assumptions=And[m>0,rh>0,r>0,θ≥0,θ≤π,ϕ≥0,ϕ<2π,Λ<0,l>a,a>0];<<"D:\\diffgeoM.wl"
tDelta[var_]:=D[var,m]dm+D[var,a]da+D[var,l]dl;tDelta1[var_]:=D[var,m]dm;tDelta2[var_]:=D[var,a]da;tDelta3[var_]:=D[var,l]dl;
(1) The computation of ∫Q on for the Killing vector=d/dt -d/dϕ conjugate to the energy. Below the contributions from the vector d/dt and the vector d/dϕ are calculated separately and are added together eventually.
~
δ
S
r
ξ
t
a
2
l
xi1={1,0,0,0};xiD=lower[%];quvD=-1/(16Pi)2*antisymmetrize[covariant[%]];quv=raise[%,{1,2}];qXi1=2PiIntegrate[%[[1,2]]*rg,{θ,0,Pi}];dQ1=tDelta[%]//Simplify;
xi2={0,0,0,1};xiD=lower[%];quvD=-1/(16Pi)2*antisymmetrize[covariant[%]];quv=raise[%,{1,2}];qXi2=2PiIntegrate[%[[1,2]]*rg,{θ,0,Pi}];dQ2=tDelta[%];dQ=dQ1-dQ2//Simplify
a
l^2
1
2
3
(a-l)
3
(a+l)
6
l
3
l
3
r
4
a
3
a
3
r
2
l
2
a
3
r
2
l
2
l
3
r
2
l
(2) The computation of -∫ ξ· Θ: the minus sign is considered implicitly below. The three parts proportional to are calculated separately, and afterwards we add them together.
δ
m,aandl
δ
δ
ddMetric=covariant[tDelta1[metric]];1/(16Pi)(contract[ddMetric,{1,3}]-contract[ddMetric,{2,3}]);xi1-xi2**raise[%,{1}];xiTheta1=2PiIntegrate[Simplify[%[[1,2]]*rg],{θ,0,Pi}];
a
l^2
ddMetric=covariant[tDelta2[metric]];1/(16Pi)(contract[ddMetric,{1,3}]-contract[ddMetric,{2,3}]);xi1-xi2**raise[%,{1}];xiTheta2=2PiIntegrate[Simplify[%[[1,2]]*rg],{θ,0,Pi}];
a
l^2
ddMetric=covariant[tDelta3[metric]];1/(16Pi)(contract[ddMetric,{1,3}]-contract[ddMetric,{2,3}]);xi1-xi2**raise[%,{1}];xiTheta3=2PiIntegrate[Simplify[%[[1,2]]*rg],{θ,0,Pi}];xiTheta=xiTheta1+xiTheta2+xiTheta3//Simplify;
a
l^2
The verification:
(0) The metric:
(We have solved the black hole solution perturbatively around the AdS-Schwarzschild metric in the given higher curvature theory. The integration constants have been chosen to fix M as the canonical energy of the system. This choice simplifies the subsequent analysis. Otherwise, M should be regarded as an ordinary mass parameter, and it requires some special attention to be paid to the integrability condition when studying the extended black hole thermodynamics. )
Above, af[r] and bf[r] are the concrete expressions of a[r] and b[r]. We do the substitution a → af, b → bf when it is needed.
In higher curvature gravity, the expressions for Q and Θ depend on the derivative of the Lagrangian with respective to Riemann curvature, which we shall denote as eof. We calculate its concrete expression for the present model as below.
The expressions of the thermodynamic quantities are listed below: (For the convenience of the readers, starting from the metric, we have derived the expressions of the mass, temperature and entropy from standard techniques. The entropy is computed from Wald entropy formula.)
(6) One can use the effective cosmological constant Λe to rewrite the first law.
From it, you can read off the expression of Λe as a function of Λ, a2, a3, up to 2nd order, which is given by
Conversely, you can also solve the expression of Λ as a function of Λe, a2, a3, up to 2nd order, which is given by
Thus, the term dΛ in the extended the first law can be replaced by the terms proportional to dΛe, da2, and da3:
4. The calculations for the spherically symmetric black hole in Lovelock gravity in D=5 dimensions
4. The calculations for the spherically symmetric black hole in Lovelock gravity in D=5 dimensions
(where only the Gauss-Bonnet term are involved):
(0) The metric:
In higher curvature gravity, the expressions for Q and Θ depend on the derivative of the Lagrangian with respective to Riemann curvature, which we shall denote as eof. Below we calculate its concrete expression for the present model.
The expressions of the thermodynamic quantities are listed below:
Below is another verification by comparing with existing literature. In the first line, we obtain the thermodynamic volume conjugate to α from eq.(2.23) in arXiv:1406.7015. In the second line, we present our result. One can observe that the outcomes are the same, omitting the unimportant constant factor.
(6) One can use the effective cosmological constant Λe (or the effective AdS radius denoted by le) to rewrite the first law.
Thus we get the relationship between the AdS radius l (its expression is denoted as lf) and the effective AdS radius le (its expression is denoted as lef ), as given by
If one would like to use Λe as the variable in the extended first law, dl can be replaced by terms proportional to dle and dα like this:
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Extended black hole thermodynamics from extended Iyer-Wald formalism
by Yong Xiao, Yu Tian & Yu-Xiao Liu
Wolfram Community, STAFF PICKS, March 20, 2024
https://community.wolfram.com/groups/-/m/t/3144602
by Yong Xiao, Yu Tian & Yu-Xiao Liu
Wolfram Community, STAFF PICKS, March 20, 2024
https://community.wolfram.com/groups/-/m/t/3144602