Generalized Misner-Sharp energy in the generalized Rastall theory

TL;DR

This paper derives a generalized Misner-Sharp mass and horizon entropy within the generalized Rastall theory, showing differences from Einstein and Rastall theories, and confirms the second law of thermodynamics in a flat FRW universe.

Contribution

It introduces a new formulation of the Misner-Sharp mass and horizon entropy in the generalized Rastall theory, extending thermodynamic relations beyond Einstein gravity.

Findings

The generalized Misner-Sharp mass differs from Einstein and Rastall theories.

The horizon entropy matches the Rastall theory and reduces to Bekenstein entropy in Einstein limit.

The second law of thermodynamics holds in the flat FRW universe under this framework.

Abstract

Employing the unified first law of thermodynamics and the field equations of the generalized Rastall theory, we get the generalized Misner-Sharp mass of spacetimes for which $g_{tt}=-g^{rr}=-f(r)$. The obtained result differs from those of the Einstein and Rastall theories. Moreover, using the first law of thermodynamics, the obtained generalized Misner-Sharp mass and the field equations, the entropy of the static spherically symmetric horizons is also addressed in the framework of the generalized Rastall theory. In addition, by generalizing the study to the flat FRW universe, the apparent horizon entropy is also calculated. Considering the effects of applying the Newtonian limit to the field equations on the coupling coefficients of the generalized Rastall theory, our study indicates $i$) the obtained entropy-area relation is the same as that of the Rastall theory, and $ii$) the Bekenstein entropy is recovered when the generalized Rastall theory reduces to the Einstein theory. The validity of the second law of thermodynamics is also investigated in the flat FRW universe.

Equations

$$T_{;\mu }^{\mu \nu }=(\lambda R{)}^{;\nu },$$

$$G_{\mu \nu }+\kappa \lambda g_{\mu \nu }R=\kappa T_{\mu \nu }.$$

$$\frac{\kappa }{4\kappa \lambda -1}(3\kappa \lambda -\frac{1}{2})={\kappa }_G,$$

$$dE\equiv A{\Psi }_{a}d{x}^a+WdV.$$

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

$$dE=4\pi r^2\rho dr.$$

$$\displaystyle dE=\frac{4\pi}{\kappa}\big{[}1-\frac{d(rf(r))}{dr}+$$

$$\displaystyle\kappa\lambda[\frac{d(r^{2}f^{\prime}(r))}{dr}-2(1-\frac{d(rf(r))}{dr})]\big{]}dr.$$

$$\displaystyle E=\frac{4\pi}{\kappa}\big{[}r\big{(}1-f(r)\big{)}+$$

$$\displaystyle\kappa\int\lambda[\frac{d(r^{2}f^{\prime}(r))}{dr}-2(1-\frac{d(rf(r))}{dr})]\big{]}dr.$$

$$\displaystyle E_{h}=\frac{4\pi}{\kappa}\bigg{[}r_{h}+$$

$$\displaystyle\kappa\bigg{(}\int\lambda[\frac{d(r^{2}f^{\prime}(r))}{dr}-2(1-\frac{d(rf(r))}{dr})]dr\bigg{)}_{r=r_{h}}\bigg{]},$$

$$d{E}_h=E_{r_h+d{r}_h}-E_{r_h}=\frac{d{E}_h}{d{r}_h}d{r}_h=$$

$$\displaystyle\frac{4\pi}{\kappa}\big{[}dr_{h}+\kappa\lambda[d(r^{2}f^{\prime}(r))_{r=r_{h}}-2\big{(}dr_{h}-d(rf(r))_{r=r_{h}}\big{)}]\big{]}.$$

$$\displaystyle dE=\frac{4\pi}{2\kappa_{G}}\big{[}1-\frac{d(rf(r))}{dr}\big{]}dr\Rightarrow E=\frac{r}{2G}[1-f(r)],$$

$$\displaystyle P(r)=\frac{1}{\kappa}\big{(}\frac{1}{r^{2}}\left[rf^{\prime}(r)-1+f(r)\right]$$

$$\displaystyle-\frac{\kappa\lambda}{r^{2}}[r^{2}f^{\prime\prime}(r)+4rf^{\prime}(r)-2+2f(r)]\big{)},$$

$$\displaystyle P(r_{h})=\frac{1}{\kappa r_{h}^{2}}\big{(}r_{h}f^{\prime}(r_{h})-1$$

$$\displaystyle-\kappa\lambda[r_{h}^{2}f^{\prime\prime}(r_{h})+4r_{h}f^{\prime}(r_{h})-2]\big{)},$$

$$P(r_h)dV=\frac{2f^{\prime }(r_h)}{\kappa }d(\frac{A}{4})$$

$$-\frac{4\pi }{\kappa }[1+\kappa \lambda (r_h^2{f}^{\prime \prime }(r_h)+4r_h{f}^{\prime }(r_h)-2)]dr,$$

$$[1+\kappa \lambda (r_h^2{f}^{\prime \prime }(r_h)+4r_h{f}^{\prime }(r_h)-2)]dr$$

$$=(1-2\kappa \lambda )d{r}_h+\kappa \lambda [d(r^2{f}^{\prime }(r){)}_{r=r_h}]+2\kappa \lambda r_h{f}^{\prime }(r)$$

$$\displaystyle=dr_{h}+\kappa\lambda[d(r^{2}f^{\prime}(r))_{r=r_{h}}-2\big{(}dr_{h}-d(rf(r))_{r=r_{h}}\big{)}],$$

$$P(r_h)dV=\frac{2f^{\prime }(r_h)}{\kappa }d(\frac{A}{4})-d{E}_h,$$

$$Td{S}_h=\frac{2f^{\prime }(r_h)}{\kappa }d(\frac{A}{4}),$$

$$d{S}_h=\frac{8\pi }{\kappa }d(\frac{A}{4})\Rightarrow S_h=\frac{8\pi }{\kappa }\frac{A}{4}.$$

$$d{s}^2=-d{t}^2+a^2\left(t\right)\left[d{r}^2+r^{2}d{\Omega }^2\right],$$

$${\tilde{r}}_A=\frac{1}{H},$$

$$\delta Q\equiv A{\Psi }_{a}d{x}^a,$$

$$\delta Q=-\frac{3V(\rho +p)H}{2}dt+\frac{A(\rho +p)}{2}(d\tilde{r}-\tilde{r}Hdt),$$

$$d{S}_A\equiv -\frac{\delta Q}{T}=6\pi V(\rho +p)dt,$$

$$\frac{dH}{dt}=\dot{H}=-\frac{\kappa }{2}(\rho +p),$$

$$d{S}_A=-\frac{12\pi V}{\kappa }dH.$$

$$S_A=\frac{8\pi }{\kappa }\frac{A}{4},$$

Full text

I Introduction

Based on the curvature-matter non-minimal coupling theories, the ordinary energy-momentum conservation law is not valid cmc ; cmc1 ; cmc2 , a hypothesis that returns to Rastall rastall . In this framework, energy can be exchanged between the matter source and geometry leading to a modified version of general relativity m1 that has interesting outcomes m1 ; m2 ; m3 ; m5 ; m6 ; m7 ; m8 ; m9 ; m10 ; m11 ; m12 ; m13 ; m14 and for example, lets baryonic matter to support traversable wormholes m4 . Recently, introducing a generalized version for the Rastall theory, it has been shown that such non-minimal coupling, and indeed, the geometry ability to couple with the matter fields in a non-minimal way, can theoretically describe inflation and the current accelerated universe without need for considering the dark energy candidates and inflationary fields epjc .

The profound connection between thermodynamics and gravitational theories haw ; GSL ; GSL0 ; jacob ; jacob1 ; lt ; cons ; cons1 ; cons2 ; cons3 ; Cai2 ; CaiKimt motivates physicists to look for the Misner-Sharp mass in the various gravitational theories for studying the thermodynamic aspects of the theories ms ; ref0 ; ref1 ; ref2 ; pad ; pad1 ; pad2 ; pad3 . The same analysis has been done in the Rastall framework rasah indicating that only when this theory reduces to the Einstein theory, the Bekenstein entropy is recovered in the static spherically symmetric spacetimes rasah , a result in agreement with those of the dynamic studies rasen ; plb .

In Ref. nepjc , studying some cosmological consequences of the generalized Rastall theory epjc , authors address some similarities between the generalized Rastall theory and other cosmological frameworks such as the Einstein gravity with particle creation mechanism pcp such that the Rastall parameter is related to the particle creation parameter. They also used the Misner-Sharp mass of the Einstein theory ms as well as the Cai-Kim temperature Cai2 in order to obtain the entropy of the apparent horizon of FRW spacetime. Based on their results, the generalized Rastall theory preserves the Bekenstein bound of entropy nepjc .

Here, we are interested in finding the generalized Misner-Sharp mass in the generalized Rastall theory by using the thermodynamic laws and the corresponding field equations. Our final goals are also to find the entropy corresponding to the horizons of the static spherically symmetric spacetime and the flat FRW universe. We present our analysis and results in the next section, where the requirements for the validity of the second law of thermodynamics will also been addressed. The last section is devoted to a summary. The unit of $c=\hbar=1$ is used throughout this paper.

II Thermodynamics of the generalized Rastall theory

Based on the generalized Rastall theory epjc

[TABLE]

which finally leads to

[TABLE]

where $\kappa$ is an unknown constant, called the Rastall gravitational coupling constant, and $\lambda$ is the Rastall parameter. The field equations look very similar to those of the Rastall theory, and indeed, only one difference exists. Unlike the Rastall theory, the Rastall parameter is variable here. Applying the Newtonian limit to this field equations, one easily reaches

[TABLE]

in which $\kappa_{G}\equiv 4\pi G$, meaning that this generalized Rastall theory lets $G$ change, because $\lambda$ is not generally constant. As a result, one easily obtains that if we presume $\kappa\equiv 8\pi G$, then the Newtonian limit automatically indicates that $\lambda=0$ meaning that we are in the Einstein framework, and thus the Einstein results should be recovered. Eq. (3) is similar to that of the Rastall theory rastall ; rasah , a result due to the fact that both theories include the same additional geometrical term (the Ricci scalar) compared with the Einstein theory.

The unified first law of thermodynamics is written as ref1 ; ref2

[TABLE]

Here, $\Psi_{a}=T^{b}_{a}\partial_{b}r+W\partial_{a}r$ and $W=-\frac{h^{ab}T_{ab}}{2}$ denote the energy supply vector and the work density, respectively, where $h^{ab}$ is the metric on the hypersurface ($t,r$). Additionally, $A=4\pi r^{2}$ is the area of the system boundary located at radius $r$. For the FRW and static spherically symmetric spacetimes, the apparent and event horizons are proper causal boundary, respectively haw ; GSL ; GSL0 ; jacob ; jacob1 ; lt ; ms ; ref0 ; ref1 ; ref2 ; pad ; pad1 ; pad2 ; pad3 ; rasah ; cons ; cons1 ; cons2 ; cons3 ; Cai2 ; CaiKimt ; nepjc ; rasen ; plb .

II.1 Thermodynamic analysis of the static spherically symmetric spacetimes

Now, let us follow Refs. pad ; rasah and consider the spherically symmetric static spacetime

[TABLE]

while it is supported by an energy-momentum source with energy density $\rho$ and pressure $p$ (or equally the energy-momentum tensor $T^{\mu}_{\nu}=diag(-\rho,p,p,p)$), and its horizon is located at $r_{h}$ (leading to $f(r_{h})=0$). Indeed, field equations let us to interpret $T^{\mu}_{\nu}$ as the energy-momentum tensor produced (induced into the spacetime) by the (existence of) black hole with radius $r_{h}$. In this manner, Eq. (4) easily leads to

[TABLE]

Using the $t-t$ component of (2), one can get $\rho$ placed in the above equation to reach at

[TABLE]

This is in fact the differential of the Misner-Sharp mass content of the generalized Rastall theory introduced in epjc , and its integral leads to

[TABLE]

In general, one can find $E$ by knowing the exact from of $\lambda$. For the Misner-Sharp mass confined to the radius $r_{h}$, this equation leads to

[TABLE]

and therefore, the energy changes due to the hypothetical displacement of the horizon radius from $r_{h}$ to $r_{h}+dr_{h}$ is evaluated as

[TABLE]

At the $\lambda=0$ limit, the above equations reduce to

[TABLE]

nothing but the Misner-Sharp mass in the Einstein theory ms ; ref0 ; ref1 ; ref2 ; pad ; pad1 ; pad2 ; pad3 . As a proper result, it leads to $E_{h}=\frac{r_{h}}{2G}$ for the Misner-Sharp mass confined to the horizon located at $r_{h}$. Thus, the emergence of the Einstein result is parallel to assume $\kappa=2\kappa_{G}$ or equally $\lambda=0$. It is also easy to check the $\lambda=constant$ case recovers the result of the Rastall theory rasah .

Now, let us use the $r-r$ component of (2) to find pressure at radius $r$ as

[TABLE]

leading to

[TABLE]

on the event horizon where $f(r_{h})=0$. Assuming $dV=4\pi r^{2}dr$ (in accordance with Eq. (6) compatible with the $dE=\rho dV$ relation), and using Eq. (13) , we can reach

[TABLE]

on the event horizon ($f(r_{h})=0$). Additionally, since $f(r_{h})=0$, one writes

[TABLE]

nothing but $\frac{\kappa}{4\pi}dE_{h}$ compared with Eq. (10). Thus, Eq. (14) reduces to

[TABLE]

compared with the first law of thermodynamics ($PdV=TdS-dE$) ref0 ; ref1 ; ref2 ; pad ; pad1 ; pad2 ; pad3 to get

[TABLE]

where $T$ and $S_{h}$ denote the horizon temperature and entropy, respectively. Now, bearing the horizon temperature ($T=\frac{f^{\prime}(r_{h})}{4\pi}$) in mind ref0 ; ref1 ; ref2 ; pad ; pad1 ; pad2 ; pad3 , the horizon entropy is obtained as

[TABLE]

At first sight, it looks like the Bekenstein entropy, but in fact, it reduces to the Bekenstein entropy whenever $G$ is constant and $\lambda=0$ parallel to the $\kappa=8\pi G=constant$ constraint. Indeed, this is the same as the entropy of the Rastall theory, despite the fact that $\lambda=constant\neq 0$ (or equally $G$ is constant) in the Rastall theory rasah ; rasen . The latter may be due to that $i$) both the Rastall theory and its generalized version modify the Einstein field equations with the same geometrical term ($R$), and $ii$) in both theories, unlike the Rastall parameter, the Rastall gravitational coupling is constant.

II.2 Entropy of the apparent horizon of FRW universe

Now, consider a flat FRW universe with scale factor $a(t)$ and line element

[TABLE]

filled by an energy-momentum source specified by $T_{\mu}^{\nu}=\textmd{diag}(-\rho,p,p,p)$. Its apparent horizon, as the proper causal boundary cons ; cons1 ; cons2 ; cons3 ; Cai2 ; CaiKimt , is located at Cai2 ; CaiKimt

[TABLE]

where $H=\frac{\frac{da}{dt}}{a}$ is called the Hubble parameter, and has temperature $T=\frac{H}{2\pi}$ cons3 ; Cai2 ; CaiKimt . During the time interval $dt$, the energy flux crossing ($\delta Q$) the apparent horizon is evaluated as

[TABLE]

in which $A$ denotes the area of the apparent horizon Cai2 ; cons3 . Bearing the definition of $\Psi_{a}$ in mind, one can reach

[TABLE]

combined with the Clausius relation ($TdS_{A}=\delta Q$) and $V=\frac{4\pi}{3H^{3}}$ (the volume confined by the apparent horizon) to get

[TABLE]

where $S_{A}$ is the entropy corresponding to the apparent horizon (the causal boundary). Before proceeding, we use the field equations (2) for obtaining

[TABLE]

inserted into Eq. (23) to find

[TABLE]

The integral of this equation is straightforward leading to

[TABLE]

for the apparent horizon entropy in the generalized Rastall theory epjc . The same as the static spherically symmetric spacetimes, the obtained relation $i$) reduces to that of the Einstein theory at the approprite limit $\kappa=8\pi G$ (or equally $\lambda=0$), and $ii$) is the same as that of the Rastall theory rasah ; rasen . Finally, one can use Eqs. (23) and (24) to see that, during the cosmic evolution, for which $\dot{H}<0$, the second law of thermodynamics ($\frac{dS_{A}}{dt}\geq 0$) is met whenever the $\rho+p>0$ condition is satisfied which yields $\kappa>0$.

III Summary

Using the $t-t$ component of the field equations and the unified first law of thermodynamics, we could obtain the generalized Misner-Sharp mass, confined to the event horizon of the static spherically symmetric spacetime with metric (5), in the generalized Rastall theory. Thereinafter, combining this result with the $r-r$ component of the field equations, the entropy corresponding to the horizon has been calculated. Here, we only considered the special case for which $g_{tt}=-g^{rr}$ pad ; rasah ; ref2 , and the general metrics including the $g_{tt}g_{rr}\neq constant$ cases ref2 need more investigations considered as the future work. We also generalized our investigation to the flat FRW universe, and got the apparent horizon entropy. Applying the Newtonian limit to the Rastall field equations, relation between the Newtonian gravitational coupling, $\lambda$ and $\kappa$ has also been established which is similar to that of the Rastall theory rastall ; rasah due to the fact that both theories add the same geometrical term ($R$) to the Einstein field equations.

Our study shows that the horizon entropy in the generalized Rastall theory is the same as that of the Rastall theory rasah ; rasen which may have two reasons including $i$) both theories modify the Einstein theory with the same geometrical term, and $ii$) unlike the Rastall parameter, the Rastall gravitational coupling is constant in both theories. It is worthwhile mentioning that, the same as the Rastall theory and also other works rasah ; rasen ; plb ; epjc , the results indicate that the $S\propto A$ relation is valid in the generalized Rastall theory, a property preserved by the Bekenstein entropy. In both static and dynamic cases, the obtained entropy relation reduces to the Bekenstein entropy at the appropriate limit $\kappa=8\pi G$ (or equally $\lambda=0$), a desired result. Finally, we saw that if $\rho+p>0$, then the second law of thermodynamics is satisfied in this theory.

Acknowledgment

We are grateful to the anonymous referees for valuable comments.