Thermodynamic Analysis of the Static Spherically Symmetric Field Equations in Rastall Theory

Moradpour, Hooman; Salako, Ines G.

doi:https://doi.org/10.1155/2016/3492796

Advances in High Energy Physics

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Research Article | Open Access

Volume 2016 | Article ID 3492796 | https://doi.org/10.1155/2016/3492796

Thermodynamic Analysis of the Static Spherically Symmetric Field Equations in Rastall Theory

Hooman Moradpour¹and Ines G. Salako^2,3,4

Academic Editor: Elias C. Vagenas

Received20 Jul 2016

Accepted19 Sept 2016

Published10 Oct 2016

Abstract

The restrictions on the Rastall theory due to application of 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.

1. Introduction

For the first time, Jacobson could use thermodynamics to derive the Einstein field equations [1]. The Bekenstein entropy is the backbone of Jackobson’s approach confirming that the Einstein theory corresponds with the Bekenstein entropy [1]. Generalization of his approach to 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 nonequilibrium 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 [3–10]. 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 [7–10]. This approach is used to investigate the mutual relation between the gravitational field equations and the system thermodynamic properties, such as entropy, in various gravitational theories [11–23]. In all of the above mentioned attempts [1–23], 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 lagrangian system is equal to total sum of its lagrangian constituents including the geometry and the matter fields.

Rastall [24] and curvature-matter coupling [25–29] theories are two gravitational theories in which geometry and matter fields are coupled to each other in a nonminimal way. In these theories, the lagrangian of a gravitational system is not just 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 to 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 a classical point of view, Rastall’s theory may be supported by the matter production process in the cosmos [32–35]. The mutual relation between the Rastall cosmology and thermodynamics has recently been studied [36]. More studies on this theory can also be found in [37–51].

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 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 zero-zero component of the Rastall field equations to find the Misner-Sharp mass confined to 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.

2. Rastall Field Equations and the Misner-Sharp Mass

For Rastall’s original field equations, we have [24]which finally leads towhere and are Rastall’s parameter and Rastall’s gravitational coupling constant, respectively. As Rastall has been shown [24], this equation leads to , indicating that, since is not always zero, the 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 , then for the Rastall gravitational coupling constant () and the Rastall original parameter () we haveindicating that the Einstein result () is obtainable in the appropriate limit which is parallel to the limit [51]. It is useful to mention here that as this equation shows, since the Rastall gravitational coupling constant diverges at , the case is not also allowed. Finally, Rastall’s field equations can be written as [51]which leads to meaning that, in agreement with (3), the case is not allowed in this theory. Moreover, as it is obvious from (3), diverges at , and thus, the case is not also allowed. Therefore, Newtonian limit indicates that, in fact, both the and 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 use either 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 , described byand filled by a source of energy-momentum , the work density and energy supply vector are defined asrespectively. In the above equations, is the line element on the two-dimensional sphere with radius , and is metric on the two-dimensional hypersurface of . 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 (UFL) of thermodynamics is valid meaning thatin which denotes the coordinate on a two-dimensional hypersurface of metric and is the area of the system boundary and therefore . It is straightforward to show thatUsing the zero-zero component of (4) in rewriting this equation, we obtainwhich finally leads towhere denotes derivative with respect to , for the energy confined to radius . In obtaining this result, we used . For a black hole with an event horizon located at , since , we findIn 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 , whereIt is also worthwhile mentioning that, in the limit, we have leading to 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 . The term comes from the Rastall original hypothesis (1) that admits a mutual energy exchange between spacetime and energy-momentum source supporting the geometry. This hypothesis couples the geometry and matter fields in a nonminimal way and leads to a covariant nonconservation of Rastall gravity. As two examples, calculations for the value of in the Schwarzschild and de-Sitter spacetimes lead torespectively. Finally, it is worthwhile mentioning that, in order to have positive energy, we should have leading to the condition for the Rastall dimensionless parameter.

3. Horizon Entropy

Here, following the approach of [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) yieldswhere is the radial pressure of the energy-momentum source [7], and therefore, it finally takes theform. Here, prime () denotes the derivative with respect to radius (). On the event horizon, and therefore,Multiplying this equation by , one getswhere . For the second term of the RHS of this equation we reachNow, bearing the condition in mind, since , one can simplify the RHS of the recent equation and take integral from that to get (11). Therefore, (17) can be written as follows:Since is the horizon temperature, comparing this equation with the first law of thermodynamics () [7], one gets which finally leads towhere is the Bekenstein entropy, for the horizon entropy. It is now obvious that, in the limit, we have , and thus the Einstein result (the Bekenstein entropy) is recovered. Indeed, as authors have shown in [2, 7–11], terms other than the Einstein tensor in modified gravities modify the Bekenstein limit of the system entropy in agreement with our result (). In addition, since entropy is a positive quantity, the Rastall dimensionless parameter should meet either or . 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 the and definitions in mind, one can rewrite (19) as is the amount of work done during a hypothetical displacement to the horizon in the Einstein framework [7]. Therefore, if we decompose the work term () as , then we reachwhich denotes the additional work done in the Rastall theory in comparison with the Einstein theory. Finally, it is also useful to mention here that, as a desired result, in the absence of the Rastall term (), we have meaning that the Einstein result is recovered [7].

4. Concluding Remarks

We saw that the Rastall theory of either or 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 a term other than the Einstein tensor in the gravitational field equations, the Rastall term (), modifies the Bekenstein limit. We also compared the obtained thermodynamic quantities, including entropy, energy, and work, with their counterparts in the Einstein case to have a better view about the obtained quantities. As we saw, in the limit, the results of the Einstein theory are obtainable.

Competing Interests

The authors declare no competing interests regarding the publication of this paper.

Acknowledgments

The work of H. Moradpour has been supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM) under research Project no. 1/4165-5. Ines G. Salako thanks African Institute for Mathematical Sciences (AIMS), Melrose Road, Muizenberg , South Africa, for partial support.

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Copyright

Copyright © 2016 Hooman Moradpour and Ines G. Salako. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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