Thermodynamic analysis of the static spherically symmetric field equations in Rastall theory

The restrictions on the Rastall theory due to apply the Newtonian limit to the theory are derived. In addition, we use the zero-zero component of the Rastall field equations as well as the unified first law of thermodynamics to find the Misner-Sharp mass content confined to the event horizon of the spherically symmetric static spacetimes in the Rastall framework. The obtained relation is calculated for the Schwarzschild and de-Sitter back holes as two examples. Bearing the obtained relation for the Misner-Sharp mass in mind together with recasting the one-one component of the Rastall field equations into the form of the first law of thermodynamics, we obtain expressions for the horizon entropy and the work term. Finally, we also compare the thermodynamic quantities of system, including energy, entropy and work, with their counterparts in the Einstein framework to have a better view about the role of the Rastall hypothesis on the thermodynamics of system.

The restrictions on the Rastall theory due to apply the Newtonian limit to the theory are derived. In addition, we use the zero-zero component of the Rastall field equations as well as the unified first law of thermodynamics to find the Misner-Sharp mass content confined to the event horizon of the spherically symmetric static spacetimes in the Rastall framework. The obtained relation is calculated for the Schwarzschild and de-Sitter back holes as two examples. Bearing the obtained relation for the Misner-Sharp mass in mind together with recasting the one-one component of the Rastall field equations into the form of the first law of thermodynamics, we obtain expressions for the horizon entropy and the work term. Finally, we also compare the thermodynamic quantities of system, including energy, entropy and work, with their counterparts in the Einstein framework to have a better view about the role of the Rastall hypothesis on the thermodynamics of system.

I. INTRODUCTION
For the first time, Jacobson could use thermodynamics to derive the Einstein field equations [1]. The Bekenstein entropy is the backbone of the Jackobson's approach confirming that the Einstein theory corresponds with the Bekenstein entropy [1]. Generalization of his approach to f (R) gravity shows that the terms other than the Einstein tensor in the gravitational field equations produce entropy and therefore modify the Bekenstein limit of the horizon entropy [2]. In fact, such terms and thus their corresponding entropy production terms are the signals of the non-equilibrium thermodynamic aspects of spacetime [2].
In order to study the mutual relation between gravity and thermodynamics, we need a proper energy definition, and it seems that the generalization of the Misner-Sharp mass in various theories is a suitable candidate for this aim [3][4][5][6][7][8][9][10]. It is also shown that, in various gravitational theory, if the gravitational field equations are considered as the first law of thermodynamics, then we can find an expression for the horizon entropy in the spherically symmetric static spacetimes [7][8][9][10]. This approach is used to investigate the mutual relation between the gravitational field equations and the system thermodynamic properties, such as entropy etc., in various gravitational theories [11][12][13][14][15][16][17][18][19][20][21][22][23]. In all of the above mentioned attempts , authors have only studied theories in which geometry and matter fields are coupled to each other in a minimal way, and therefore, the energy-momentum conservation law is met in their studies. In fact, in their studies, the system lagrangian is equal to the sum of the its constitutes lagrangian including the geometry and the matter fields.
Rastall [24] and curvature-matter coupling [25][26][27][28][29] theories are two gravitational theories in which geometry and matter fields are coupled to each other in a non-minimal way. In these theories, the lagrangian of a gravitational system is not only a simple sum of the geometry and the matter fields lagrangians [29,30]. For these theories, the energy-momentum conservation law is not always valid, and in fact, the divergence of the energy-momentum tensor is proportional with the derivative of Ricciscalar. This mutual relation between the divergence of the energy-momentum tensor and Ricciscalar is the backbone of the Rastall theory allowing a flux of energy between the energy-momentum source and the geometry. The Rastall hypothesis also modifies the Einstein field equations, a result that increases the hopes to describe the current phase of the universe expansion [31]. From classical point of view, the Rastall's theory may be supported by the matter production process in the cosmos [32][33][34][35]. The mutual relation between the Rastall cosmology and thermodynamics has recently been studied [36]. More studies on this theory can also be found in Refs. [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51].
Here, we are interested in studying the mutual relation between the thermodynamics first law and the Rastall field equations. In fact, we want to study the effects of the Rastall hypothesis on the thermodynamic properties of the spherically symmetric static spacetimes. In order to achieve this goal, generalizing the Misner-Sharp mass of the spherically symmetric static spacetimes to Rastall theory, we find an expression for the horizon entropy of the spherically symmetric static spacetimes. We also compare the thermodynamic quantities of system, including energy, entropy and work, with their counterparts in the Einstein framework to have a better view about the role of the Rastall hypothesis on the thermodynamics of system. The G = = c = 1 units are considered in this paper.
The paper is organized as follows. In the next section, after referring to the Rastall theory and the Newtonian limit constraints on this theory parameters, we use the unified first law of thermodynamics together with the zerozero component of the Rastall field equations to find the Misner-Sharp mass confined to en event horizon of the static spherically symmetric spacetimes in Rastall theory. Thereinafter, in the third section, recasting the one-one component of the Rastall field equations into the form of the first law of thermodynamics, we find an expression for the horizon entropy. The comparison of the obtained thermodynamic quantities with their counterparts in the Einstein general relativity is also addressed in this section. The last section is devoted to concluding remarks.

II. RASTALL FIELD EQUATIONS AND THE MISNER-SHARP MASS
For the Rastall's original field equations, we have [24] which finally leads to where λ and κ are the Rastall's parameter and Rastall's gravitational coupling constant, respectively. As Rastall has been shown [24], this equation leads to R(4κλ − 1) = T , indicating that, since T is not always zero, the κλ = 1 4 case is not allowed in this theory. Besides, it is also shown that if we use the Newtonian limit and define the Rastall dimensionless parameter γ = κλ, then for the Rastall gravitational coupling constant (κ) and the Rastall original parameter (λ) we have indicating that the Einstein result (κ = 8π) is obtainable in the appropriate limit λ = 0 which is parallel to the γ = 0 limit [51]. It is useful to mention here that as this equation shows, since the Rastall gravitational coupling constant diverges at γ = 1 6 , the γ = 1 6 case is not also allowed. Finally, the Rastall's field equations can be written as [51] G which leads to R(6γ −1) = 8πT meaning that, in agreement with (3), the γ = κλ = 1 6 case is not allowed in this theory. Moreover, as it is obvious from (3), λ diverges at γ = 1 4 , and thus, the γ = 1 4 case is not also allowed. Therefore, Newtonian limit indicates that, in fact, both the γ = 1 6 and γ = 1 4 cases are not allowed. In order to study the mutual relation between the first law of thermodynamics and the Rastall field equations, we need to generalize the Misner-Sharp energy in this theory [3,5,6]. It is worthwhile mentioning that one can either use the conserved charge method or the unified first law of thermodynamics in order to find the Misner-Sharp mass in a gravitational theory [5,6]. Here, since we are interested in having a fully thermodynamic analysis, we use the unified first law of thermodynamics to obtain an expression for the Misner-Sharp mass. Consider a spherically symmetric static spacetime, with a horizon located at r h , described by and filling by a source of T µ ν , the work density and energy supply vector are defined as Ψ a = T b a ∂ b r + W ∂ a r, respectively. In the above equations, dΩ 2 is the line element on the two-dimensional sphere with radius r, and h ab is metric on the two-dimensional hypersurface of (t, r). In order to generalize the Misner-Sharp mass definition to the Rastall theory, we follow the approach of [6], and assume that the unified first law of thermodynamics (UFL) is valid meaning that dE ≡ AΨ a dx a + W dV, in which x a denotes the coordinate on a two-dimensional hypersurface of metric h ab , and A is the area of the system boundary and therefore A = 4πr 2 . It is straightforward to show that Using the zero-zero component of Eq. (4) in rewriting this equation, we obtain which finally leads to where ′ denotes derivative with respect to r, for the energy confined to the radius r. In obtaining this result, we used rf ′ (r) = d(rf (r)) dr − f (r). For a black hole with an event horizon located at r h , since f (r h ) = 0, we find In fact, it is the Misner-Sharp mass content confined to the mentioned horizon in the Rastall framework. The above equation can also be written as It is also worthwhile mentioning that, in the γ → 0 limit, we haveẼ ≡ E 0 Γ → 0 leading to E → E 0 which is nothing but the Misner-Sharp mass content of the Einstein theory [3,7], meaning that, as expected, the Einstein result is recovered in the appropriate limit of γ → 0. The E 0 Γ term comes from the Rastall original hypothesis (1) that admits a mutual energy exchange between spacetime and energy-momentum source supporting the geometry. A hypothesis which leads to coupling the geometry and matter fields to each other in a non-minimal way and thus, leads to the covariantly non-conservation of Rastall gravity. As two examples, calculations for the value of Γ in the Schwarzschild and de-Sitter spacetimes lead to respectively. Finally, it is worthwhile mentioning that, in order to have positive energy, we should have Γ ≥ −1 condition for the Rastall dimensionless parameter.

III. HORIZON ENTROPY
Here, following the approach of Ref. [7], we recast the one-one component of the Rastall field equations to the form of the first law of thermodynamics and use the result of previous section to get the horizon entropy. The one-one component of (4) yields where T 1 1 ≡ P (r) is the radial pressure of the energy-momentum source [7], and therefore, it finally takes the form. Here, prime (′) denotes the derivative with respect to radius (r). On the event horizon, f (r h ) = 0 and therefore, Multiplying this equation by dV = 4πr 2 h dr h , one gets where A = 4πr 2 h . For the second term of the RHS of this equation we reach 6γ − 1 2(4γ − 1) Now, bearing the f (r h ) = 0 condition in mind, since dr − f (r)) r=r h , one can simplify the RHS of the recent equation and take integral from that to get (11). Therefore, Eq. (17) can be written as follow Since T = f ′ (r h )/4π is the horizon temperature, comparing this equation with the first law of thermodynamics (P dV = T dS − dE) [7], one gets dS = 6γ−1 4γ−1 d( A 4 ) which finally leads to where S 0 = A 4 is the Bekenstein entropy, for the horizon entropy. It is now obvious that, in the γ → 0 limit, we havẽ S ≡ 2γ 4γ−1 S 0 → 0, and thus the Einstein result (the Bekenstein entropy) is recovered. Indeed, as authors have been shown in Refs. [2,[7][8][9][10][11], terms other than the Einstein tensor in modified gravities modify the Bekenstein limit of the system entropy in agreement with our result (S). In addition, since entropy is a positive quantity, the Rastall dimensionless parameter should meet either γ < 1 6 or γ > 1 4 . Here, the energy-momentum conservation law is not valid and therefore, the energy and entropy terms differ from those of the Einstein theory [7]. In order to have a better view about our results, we compare our results with those of the Einstein theory [7]. Bearing theS andẼ definitions in mind, one can rewrite Eq. (19) as T dS 0 − dE 0 ≡ dW 0 is the amount of work done during apply the hypothetical displacement dr h to the horizon in the Einstein framework [7]. Therefore, if we decompose the work term (P (r h )dV )as P (r h )dV = dW + dW 0 , then we reach which denotes the additional work done in the Rastall theory in comparison with the Esintein theory. Finally, it is also useful to mention here that, as a desired result, in the absence of the Rastall term (λ = γ = 0), we have dS = dẼ = dW = 0 meaning that the Einstein result is recovered [7].

IV. CONCLUDING REMARKS
We saw that the Rastall theory of either γ = 1 6 or γ = 1 4 is not allowed due to the fact that the Newtonian limit should be satisfied by the Rastall theory. Additionally, we used the unified first law of thermodynamics as well as the zero-zero component of the Rastall field equations to generalize the Misner-Sharp mass of the static spherically symmetric spacetimes to the Rastall theory.
Moreover, bearing the obtained Misner-Sharp mass in mind, we started from the one-one component of the Rastall field equations, and rewrote it as the first law of thermodynamics which helped us in finding an expression for the horizon entropy in this theory. Our study shows that the term other than the Einstein tensor in the gravitational field equations, the Rastall term (γg µν R), modifies the Bekenstein limit. We also compared the obtained thermodynamic quantities, including entropy, energy and work, with their counterparts in the Esintein case to have a better view about the obtained quantities. As we saw, in the γ = λ = 0 limit, the results of the Einstein theory are obtainable.