Entropy and Energy of Static Spherically Symmetric Black Hole in $f(R)$ theory

TL;DR

This paper derives formulas for entropy and energy of static spherically symmetric black holes in $f(R)$ gravity and applies them to specific solutions, advancing understanding of black hole thermodynamics in modified gravity.

Contribution

It introduces a general framework for calculating black hole entropy and energy in $f(R)$ theories and applies it to various solutions, which was not previously done.

Findings

Derived general formulas for black hole entropy and energy in $f(R)$ gravity.

Calculated entropy and energy for specific black hole solutions in $f(R)$ models.

Enhanced understanding of thermodynamics in modified gravity theories.

Abstract

We consider the new horizon first law in $f(R)$ theory with general spherically symmetric black hole. We derive the general formulas to computed the entropy and energy of the black hole. For applications, some nontrivial black hole solutions in some popular $f(R)$ theories are investigated, the entropies and the energies of black holes in these models are first calculated.

Equations

$$\delta E=T\delta S-P\delta V,$$

$$\delta G=-S\delta T+V\delta P,$$

$$d{s}^2=-N(r)d{t}^2+\frac{d{r}^2}{N(r)}+r^{2}d{\Omega }^2,$$

$$P=\frac{T}{2r_{+}}-\frac{1}{8\pi r_{+}^2},$$

$$P=D(r_{+})+C(r_{+})T,$$

$$V\delta P=S\delta T+\delta G,$$

$$G$$

$$S={\int }^{r_{+}}{V}^{\prime }(r)C(r)dr,$$

$$E=-{\int }^{r_{+}}{V}^{\prime }(r)D(r)dr.$$

$$d{s}^2=-W(r)d{t}^2+\frac{d{r}^2}{N(r)}+r^{2}d{\Omega }^2,$$

$${\kappa }_K=\frac{\sqrt{W^{\prime }(r_{+})N^{\prime }(r_{+})}}{2},$$

$$T=\frac{{\kappa }_K}{2\pi }=\frac{\sqrt{W^{\prime }(r_{+})N^{\prime }(r_{+})}}{4\pi }.$$

$$I=\int {\text{d}}^{4}x\sqrt{-g}\left[\frac{f(R)}{2k^2}+L_{\mathrm{m}}\right],$$

$$G_{\mu \nu }\equiv R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R=k^2\left(\frac{1}{F}{T}_{\mu \nu }+\frac{1}{k^2}{\mathcal{T}}_{\mu \nu }\right),$$

$${\mathcal{T}}_{\mu \nu }=\frac{1}{F(R)}\left[\frac{1}{2}{g}_{\mu \nu }(f-RF)+{\nabla }_{\mu }{\nabla }_{\nu }F-g_{\mu \nu }\square F\right],$$

$$RF(R)-2f(R)+3\square F(R)=k^{2}T.$$

$$G_1^1=\frac{1}{r^2}(-1+\frac{rN{W}^{\prime }}{W}+N),$$

$${\mathcal{T}}_1^1=\frac{1}{F}\left[\frac{1}{2}(RF-f)+\frac{N}{2W}{W}^{\prime }{F}^{\prime }+\frac{2}{r}N{F}^{\prime }\right],$$

$$k^{2}P=-\left[\frac{F}{r^2}+\frac{1}{2}(f-RF)\right]+\sqrt{\frac{N^{\prime }}{W^{\prime }}}\left(\frac{F}{r}+\frac{1}{2}{F}^{\prime }\right)\sqrt{W^{\prime }{N}^{\prime }}+\frac{NF}{r^2}+\frac{2N{F}^{\prime }}{r}.$$

$$P=-\frac{1}{8\pi }\left[\frac{F}{r_{+}^2}+\frac{1}{2}\left(f-RF\right)\right]+\frac{1}{4}\sqrt{\frac{N^{\prime }}{W^{\prime }}}\left(\frac{2F}{r_{+}}+F^{\prime }\right)T.$$

$$D(r_{+})=-\frac{1}{8\pi }\left[\frac{F}{r_{+}^2}+\frac{1}{2}\left(f-RF\right)\right],$$

$$C(r_{+})=\frac{1}{4}\sqrt{\frac{N^{\prime }}{W^{\prime }}}\left(\frac{2F}{r_{+}}+F^{\prime }\right).$$

$$V(r_{+})$$

$$S$$

$$E=\frac{1}{2}{\int }^{r_{+}}\sqrt{\frac{W^{\prime }}{N^{\prime }}}\left[\frac{F}{r^2}+\frac{1}{2}(f-RF)\right]r^{2}dr.$$

$$2M=r_{+}-\frac{1}{12}{R}_0{r}_{+}^3.$$

$$f(R)=R-f_0{\int }_0^R{e}^{-\alpha \frac{R_1^{2n}}{{\left(x-R_1\right)}^{2n}}-f_0\frac{x}{{\Lambda }_i}}dx,$$

$$\frac{f_0}{{\Lambda }_i}>\frac{2n{R}_1^{2n}}{{\left(R-R_1\right)}^{2n+1}}.$$

$$R_0+f_0{R}_0{e}^{-\alpha \frac{R_1^{2n}}{{\left(R_0-R_1\right)}^{2n}}-f_0\frac{R_0}{{\Lambda }_i}}=2f_0{\int }_0^R{e}^{-\alpha \frac{R_1^{2n}}{{\left(x-R_1\right)}^{2n}}-f_0\frac{x}{{\Lambda }_i}}dx.$$

$$S=\pi r_{+}^2\left[1-f_0{e}^{-\alpha \frac{R_1^{2n}}{{\left(R-R_1\right)}^{2n}}-f_0\frac{R}{{\Lambda }_i}}\right].$$

Full text

I Introduction

There is a deep connection between thermodynamics and gravity. For a black hole, its area can be regarded as the entropy Bekenstein:1973ur , and four laws for dynamics like in thermodynamics were suggested Bardeen:1973gs . For diffeomorphism invariance of gravitational theory, the entropy of a black hole can be seen as a Noether charge Wald:1993nt ; Iyer:1994ys . The Einstein equations had been derived from the first law of thermodynamics Jacobson:1995ab , which was generalized to the non-equilibrium thermodynamics of spacetime Eling:2006aw . This connection was also investigated in modified gravity theories such as Lancos-Lovelock gravity Paranjape:2006ca , $f(R)$ theory Elizalde:2008pv , and the scalar-Gauss-Bonnet gravity Bamba:2009gq . It was shown that the equation of motion for generalized gravity theory is equivalent to the thermodynamical relationship $\delta Q=T\delta S$ Brustein:2009hy . In spherically symmetric spacetime Padmanabhan presented a general form for understanding the thermodynamics of the horizon Padmanabhan:2002sha . For a time-dependent evolution horizon or a fixed axis-symmetric horizon, the Einstein equations on horizon can be rewritten as a thermodynamic identity Kothawala:2007em . In cubic and quartic quasi-topological gravity, the field equations on $n+1$ dimensional topological black holes horizon with constant curvature can be expressed like the form of the first thermodynamical law Sheykhi:2014rka . In Yang:2011sx ; Yang:2014kna , one can obtain the thermal entropy density of any spacetime from Einstein equations without assuming the temperature or the horizon.

In higher-order theories of gravity, issues about the entropy and energy of black hole are important. Especially the problems related to the energy of black hole are problematic, some attempts have been made to find a satisfactory answer to this question Cognola:2011nj ; Deser:2002jk ; Deser:2007vs ; Abreu:2010sc ; Cai:2009qf . Using the first law of the horizon, one can derived the entropy and energy of black hole in Einstein gravity Padmanabhan:2002sha . Recently, as shown in Zheng:2018fyn , the entropy and energy of black hole in $f(R)$ theories can be obtained simultaneously though the new horizon first law which was proposed in Hansen:2016gud . Here we will consider the more complicated case: for the general spherically symmetric black hole we hope to obtain the energy and the entropy of black hole in $f(R)$ theory, which can reduce to the results in Zheng:2018fyn ; Cognola:2011nj for some special cases.

The rest of this article is organized as follows. In Sec. II, we briefly review the new horizon first law. In Sec. III, we investigate the entropy and energy of black holes in $f(R)$ Theories. In Sec. IV, we calculate the energies and entropies for black holes in some $f(R)$ theories for applications. Conclusion and discussion are given in Sec. V.

II The new horizon first law

Usually if the radial component of the stress-energy tensor acts as the thermodynamic pressure, $P=T^{r}_{~{}r}|_{r_{+}}$, which is a special case of the assumption firstly proposed in Yang:2014kna , and assume an horizon equation of state $P=P(V,T)$ with the temperature identified as the Hawking temperature and a geometric volume $V=V(r_{+})$ assigned to the horizon Parikh:2005qs , then the radial Einstein equation can be written as the first law of thermodynamics, which can be rewritten as a horizon first law by considering a imaginary displacement of the horizon Padmanabhan:2002sha

[TABLE]

where $E$ is the quasilocal energy and $S$ is the horizon entropy of the black hole. In Einstein gravity, $E$ proves to be the Misner-Sharp energy Misner:1964je . The horizon first law (1) is a special case of the ‘unified first law’ Hayward:1997jp .

There are two problems in this procedure that are needed to be further examined. The first problem is that the thermodynamic variables was vague in the original derivation, which require further determination. Secondly because both $S$ and $V$ are functions of only $r_{+}$, which makes the horizon first law (1) to be of a function only depending on $r_{+}$. As a matter of fact, the equation (1) can be rewritten as $\delta E=\left(TS^{\prime}+PV^{\prime}\right)\delta r_{+}$, where the primes stands for the derivative with respected to $r_{+}$. This makes the terms ‘heat’ and ‘work’ confused and results to a ‘vacuum interpretation’ of the first law (1) Hansen:2016gud . To avoid the problems mentioned above, a new horizon first law was proposed in Hansen:2016gud , by varying the horizon equation of state with the temperature $T$ and the pressure $P$ as independent thermodynamic quantities

[TABLE]

which obviously depends on both $T$ and $P$, meaning that $T$ and $P$ can vary independently. The energy, and therefore the standard horizon first law (1) can be obtained via a degenerate Legendre transformation $E=G+TS-PV$. The Gibbs free energy $G$ and the horizon entropy $S$ are derived concepts for specified volume. This new horizon first law has practical utility and provides further evidence for the connection between gravity and thermodynamics.

In 4-dimensional Einstein gravity, we briefly review this approach to explain how it works Hansen:2016gud . Considering the geometry of a static spherically symmetric black hole, which is given by

[TABLE]

The horizon is local at $r=r_{+}$ where $N(r_{+})=0$ with $N^{\prime}(r_{+})\neq 0$.

Supposing minimal coupling to the matter and that the thermal sources are also the gravitational sources Yang:2014kna , we have $P=T^{r}_{~{}r}$ and identify the Hawking temperature as the thermal temperature $T$ Padmanabhan:2002sha , the $(^{1}_{1})$ component of the Einstein equation can be written as at the horizon

[TABLE]

which is the horizon equation of state, where we take the units $G=c=\hbar=1$.

Since the identification of the Hawking temperature as the thermal temperature does not fall back on any gravitational field equations Hansen:2016gud . According to the conjecture suggested in Yang:2014kna , the definition of the pressure is identified as the $(^{r}_{r})$ component of the matter stress-energy, which is also independent of any gravitational theories. So it is reasonable to speculate that the radial component of the gravitational field equations under consideration takes the following form Hansen:2016gud

[TABLE]

where $C$ and $D$ are some functions of $r_{+}$, generally depending on the gravitational theory under consideration. Varying the generalized equation of state (5) and multiplying the volume $V(r_{+})$ of black hole, it is easy to get

[TABLE]

where

[TABLE]

and

[TABLE]

by using the integration by parts. Taking the degenerate Legendre transformation $E=G+TS-PV$, we finally derive the energy as

[TABLE]

Hypothesizing that $T$, $P$, and $V$ can be identified as the temperature, the pressure, and the volume, we can find that $G$, $S$, and $E$ are the Gibbs free energy, the entropy, and the energy of the black hole, respectively. For Einstein gravity in four dimensions, it is easily to have $C(r_{+})=\frac{1}{2r_{+}}$ and $D(r_{+})=-\frac{1}{8\pi r^{2}_{+}}$ from (4), substituting them into Eq. (8) and Eq. (9), we obtain $S=\pi r^{2}_{+}$ and $E=\frac{r_{+}}{2}$. It has been shown that the new horizon first law still works in $f(R)$ theories Zheng:2018fyn .

III The entropy and energy of black hole in $f(R)$ theory

We consider the general spherically symmetric and static black hole in $f(R)$ theories, whose geometry takes the form

[TABLE]

where $W(r)$ and $N(r)$ are general functions of the coordinate $r$. Taking the largest positive root of $N(r_{+})=0$, yields the event horizon which fulfils $N^{\prime}(r_{+})\neq 0$. The surface gravity is given by DiCriscienzo:2009hd

[TABLE]

which gives the temperature of the black hole as

[TABLE]

In four-dimensional spacetime, the general action of $f(R)$ gravity theories with source is

[TABLE]

where $k^{2}=8\pi$, $f(R)$ is a function only depending on the Ricci scalar $R$, and $L_{\rm m}$ is the matter Lagrangian. Physically $f(R)$ must fulfil stability conditions Pogosian:2007sw : (a) no ghosts, $df/dR>0$; (b) no tachyons, $d^{2}f/dR^{2}>0$ Dolgov:2003px ; (c) stable solutions, $df/dR/d^{2}f/dR^{2}>R$ Sawicki:2007tf . Additionally $f(R)$ must also satisfy $\lim_{R\rightarrow\infty}(f(R)-R)/R=0$ for the existence of an effective cosmological constant at high curvature and $\lim_{R\rightarrow\infty}d(f(R)-R)/dR=0$ for recovering general relativity at early time. Varying the action (13) with respect to the metric, yields the gravitational field equations

[TABLE]

where $F=\frac{df}{dR}$ and $T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta L_{\rm m}}{\delta g^{\mu\nu}}$ is the energy-momentum tensor for the matter. The tress-energy tensor for the effective curvature fluid, $\mathcal{T}_{\mu\nu}$, is given as following

[TABLE]

where $\Box=\nabla^{\gamma}\nabla_{\gamma}$. Taking the trace of the equation (13), gives the relation as follow

[TABLE]

For the general spherically symmetric and static black hole (10), the $(^{1}_{1})$ components of the Einstein tensor and the stress-energy tensor of the effective curvature fluid are respectively given by

[TABLE]

and

[TABLE]

where the prime denotes the derivative with respected to $r$. Substituting Eqs. (17), (18), and $T^{1}_{1}=P$ into the equation (14), we derive

[TABLE]

At the horizon, $r=r_{+}$, thinking of the temperature (12) and $N(r_{+})=0$, the equation (19) reduces to

[TABLE]

Comparing Eq. (5) with Eq. (20), we then have

[TABLE]

and

[TABLE]

The volume $V$ of the black hole (10) is given by Parikh:2005qs

[TABLE]

where we have used the relation $\frac{N(r_{+})}{W(r_{+})}=\frac{N^{\prime}(r_{+})}{W^{\prime}(r_{+})}$ DiCriscienzo:2009hd . Substituting Eqs. (23) and (19) into Eq. (8), we obtain the entropy as

[TABLE]

thought the black hole (10) has different temperature and volume, its entropy (24) has the same form for the entropy of black hole with $W(r)=N(r)$ obtained in Zheng:2018fyn . The formula (24) can be obtained by using the Euclidean semiclassical approach or the Wald entropy formula Dyer:2008hb ; Vollick:2007fh ; Iyer:1995kg . Using the degenerate Legendre transformation $E=G+TS-PV$, inserting equations (23) and (21) into Eq. (9), then the energy of the black hole is given as follow

[TABLE]

For $W(r)=N(r)e^{2\alpha(r)}$, the equation (25) reduces to the results obtained in Cognola:2011nj where the entropy was obtained by using the Wald method. For $W(r)=N(r)$, the equation (25) gives the result obtained in Zheng:2018fyn . Eqs. (24) and (25) show that the new horizon first law still work in $f(R)$ theories for the general spherically symmetric and static black hole.

IV Applications

In this section, we will use Eqs. (24) and (25) to calculate the entropies and energies of black holes (10) in some popular $f(R)$ theories to illustrate the method described above. These models allowe solutions with constant Ricci curvature (such as Schwarzschild-de Sitter or Schwarzschild solutions) or solutions with non-constant Ricci curvature. For a Schwarzschild-de Sitter solution, we have $N(r_{+})=1-2M/r-R_{0}r^{2}/12=0$ at the horizon Multamaki:2006zb , which gives

[TABLE]

This equation relates the mass $M$, the radius $r_{+}$, and the Ricci curvature $R_{0}$ together.

We firstly start with a $f(R)$ model which unifies inflation and cosmic acceleration under the same picture and was confirmed by the solar system tests Nojiri:2007as . This model takes the form

[TABLE]

where $R_{1}$ is a constant which is given by $f_{0}R_{1}\int_{0}^{1}e^{-\alpha/x^{2}}dx=R_{\rm now}$ with $0<f_{0}<1$ and $R_{\rm now}$ the Ricci scalar at present. The effective cosmological constant is $-f(-\infty)=\Lambda_{i}$ in the early universe and is $2R_{0}$ in the present era. The stability condition, $f^{\prime\prime}(R)>0$, gives

[TABLE]

Due to the fact that $f(0)=0$, this model allows a Schwarzschild solution. For a Schwarzschild-de Sitter solution, we obtain from Eq. (16),

[TABLE]

For the black hole (10), we calculate the entropy from Eq. (24) as

[TABLE]

The nonnegativity of the entropy gives additional limits on the parameters: $1>f_{0}e^{-\alpha\frac{R_{1}^{2n}}{\left(R-R_{1}\right)^{2n}}-f_{0}\frac{R}{\Lambda_{i}}}$. The entropy for Schwarzschild-de Sitter black hole is

[TABLE]

where $R_{0}$ must fulfil the equations (28) and (29).

For Schwarzschild black hole ($R_{0}=0$), the stability condition (28) reduces to $n<\frac{f_{0}R_{1}}{2\Lambda_{i}}$, and the entropy (31) reduces to

[TABLE]

Computing the energy by using Eq. (25), we have

[TABLE]

For the Schwarzschild-de Sitter black hole, equation (33) reduces to

[TABLE]

where equations (26) and (29) were used. For the Schwarzschild black hole, $R_{0}=0$, equation (IV) reduces to

[TABLE]

Since $S\geq 0$ and $E\geq 0$, Eqs. (32) and (35) give new constraints on the parameters: (a) $1>f_{0}e^{-\alpha\frac{R_{1}^{2n}}{\left(R_{0}-R_{1}\right)^{2n}}-f_{0}\frac{R_{0}}{\Lambda_{i}}}$ for Schwarzschild-de Sitter black hole; and (b) $1\geq f_{0}e^{-\alpha}$ for Schwarzschild black hole.

The second $f(R)$ model we consider here was recovered from the entropy of black holes Caravelli:2010be

[TABLE]

where $0\leq\epsilon\leq\frac{e}{4}(1+\frac{4}{e}\alpha)$ and $q=4a_{0}^{\beta}/c(\beta+1)$ with $\alpha\geq 0$, $\beta>0$, $a_{0}=l_{p}^{2}$ and $c$ a constant. Since $R\neq 0$, this type of $f(R)$ theory admits no Schwarzschild solution. Use the equation (16), we get the condition for the Schwarzschild-de Sitter solution

[TABLE]

From the Eq. (24), the entropy of the black hole (10) is computed as

[TABLE]

The nonnegativity of the entropy gives additional constraints on the parameters: $1>\left(\beta+1\right)q\alpha R^{\beta}+\left(\beta+1\right)q\epsilon R^{\beta}\ln\left(\frac{a_{0}^{\beta}R^{\beta}}{c}\right)$. For Schwarzschild-de Sitter black hole, the entropy is

[TABLE]

where the condition (37) was uesed. Since $S\geq 0$, gives new limits on the parameters: $2>(\alpha\beta-\alpha+2\epsilon)qR_{0}^{\beta}$ for $\beta>1$ and $2<(\alpha\beta-\alpha+2\epsilon)qR_{0}^{\beta}$ for $\beta<1$.

The energy of the black hole (10) is derived from Eq. (25) as

[TABLE]

For Schwarzschild-de Sitter black hole, equation (41) takes the form

[TABLE]

where the equations (37) and (26) were used. The nonnegativity of the energy gives new constraints on the parameters: $1+\beta>(\beta\epsilon+\alpha\beta^{2}+\epsilon-\alpha\beta)qR_{0}^{\beta}$ for $\beta>1$ and $1+\beta<(\beta\epsilon+\alpha\beta^{2}+\epsilon-\alpha\beta)qR_{0}^{\beta}$ for $\beta<1$.

We thirdly investigate the following $f(R)$ theory which allows black hole solutions with non-constant Ricci curvature Canate:2015dda

[TABLE]

where $\alpha$ is a parameter of the model which is related to an effective cosmological constant and $a>0$ is a parameter with units [distance]-1. This model admits a static spherically symmetric solution (10) which takes the follow form Canate:2015dda

[TABLE]

where $Q$ is an integration constant. The event horizon of the black hole is located at: (a) $r_{+}=\sqrt{3\alpha+\alpha\sqrt{9+12\alpha Q}}/\alpha$ for $\alpha>0$ and $Q>0$; (b) $r_{+}=\sqrt{3\alpha-\alpha\sqrt{9+12\alpha Q}}/\alpha$ for $\alpha>0$, $Q<0$ and $\alpha Q>-3/4$; (c) $r_{+}=\sqrt{3/\alpha-\sqrt{9+12\alpha Q}/\alpha}$ for $\alpha<0$ and $Q<0$; and (d) $r_{+}=\sqrt{6/\alpha}$ for $\alpha>0$ and $Q=0$. The Ricci scalar evolves as

[TABLE]

Substituting $W^{\prime}(r_{+})=N^{\prime}(r_{+})$ into Eq. (24), the entropy is calculated as

[TABLE]

[TABLE]

[TABLE]

[TABLE]

From Eq. (25), the energy is given by

[TABLE]

[TABLE]

[TABLE]

[TABLE]

For $\alpha>0$ and $Q>0$, the nonnegativity of the energy gives additional constraints on the parameters: $\alpha Q<7/12$.

We finally consider a power-law of $F(r)$, for example: $F=\alpha r^{a}$, with constants $a$ and $\alpha$. The functions $W(r)$ and $N(r)$ in (10) are found to take forms Amirabi:2015aya

[TABLE]

and

[TABLE]

where $C_{1}$ is a integration constant. The event horizon is local at $r_{+}=\left[-\frac{(2+a)^{2}}{C_{1}(2a^{2}+2a+2)(2+2a-a^{2})}\right]^{-\frac{2+a}{2a^{2}+2a+2}}$. In this case, function $f(R)$ is given by

[TABLE]

where $\alpha_{1}=2\alpha(2-a)^{\frac{a}{2}-1}\left(\frac{3a}{2+2a+a^{2}}\right)^{\frac{a}{2}}$. The entropy (24) and the energy (25) respectively reads

[TABLE]

[TABLE]

For $a=0$, we obtain the results in Einstein’s gravity. The nonnegativity of the entropy and the energy give new constraints on the parameters: $\alpha\geq 0$ and $C_{1}(a-2)\geq 0$.

V Conclusions and discussions

We investigated whether the new horizon first law still holds in $f(R)$ theory with general spherically symmetric black hole (10). We derived the general formulas to computed the entropy and energy of the black hole. For black hole (10), its temperature, volume and entropy have different forms, but its entropy (24) has the same form for the entropy of black hole (3) obtained in Zheng:2018fyn . For applications, some nontrivial black hole solutions in some popular $f(R)$ theories are considered, the entropies and the energies of black holes in these models are first calculated. The nonnegativity of entropy or energy gives new constraints on the parameters of $f(R)$ theories, which may be useful for future researches. The formulas presented here can be used to calculate the energies and entropies of black holes in other types $f(R)$ theories, but cannot be used in the case discussed in Nashed:2019tuk , where $F(R)=0$. In Reference Bamba:2009id , a picture of equilibrium thermodynamics on the apparent horizon in the expanding cosmological background was obtained for a wide class of modified gravity theories. Whether this procedure can be applied to the apparent horizon in the expanding cosmological background worths further study

Acknowledgements.

This study is supported in part by National Natural Science Foundation of China (Grant No. 11273010), the Hebei Provincial Outstanding Youth Fund (Grant No. A2014201068), the Outstanding Youth Fund of Hebei University (No. 2012JQ02).