Journal of Gravity

Volume 2016 (2016), Article ID 4504817, 7 pages

http://dx.doi.org/10.1155/2016/4504817

## On Gravitational Entropy of de Sitter Universe

^{1}Institute of Physics, University of Brasília, 70910-900 Brasília, DF, Brazil^{2}Department of Civil Engineering, FATECS, University Center of Brasília (UniCEUB), 70790-075 Brasília, DF, Brazil^{3}University Center of the Federal District (UDF), Department of Civil Engineering, 70390-045 Brasília, DF, Brazil

Received 13 March 2016; Accepted 23 June 2016

Academic Editor: Sergei D. Odintsov

Copyright © 2016 S. C. Ulhoa and E. P. Spaniol. 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.

#### Abstract

The paper deals with the calculation of the gravitational entropy in the context of teleparallel gravity for de Sitter space-time. In such a theory it is possible to define gravitational energy and pressure; thus we use those expressions to construct the gravitational entropy. We use the temperature as a function of the cosmological constant and write the first law of thermodynamics from which we obtain the entropy. In the limit we find that the entropy is proportional to volume, for a specific temperature’s choice; we find that as well. We also identify a phase transition in de Sitter space-time by analyzing the specific heat.

#### 1. Introduction

The idea of black hole thermodynamics started with the pioneering works of Bekenstein and Hawking [1, 2]. It was noted that the area of the event horizon behaves as an entropy. Together with such discovery it was also noted that it has a specific temperature. Thus it radiates and evaporates leading to a loss of the information inside the black hole, which apparently violates the second law of thermodynamics. It was the so-called information paradox [3]. Dolan has pointed out that such study was incomplete without the term in the first law of thermodynamics [4]. However the concept of gravitational pressure is difficult to establish as the very idea of gravitational energy. The matter of the definition of gravitational energy has a long story and yet it is a very controversial theme. The main approaches in this subject are Komar integrals [5], ADM formalism [6], and quasi-local expressions [7, 8]. In opposition to general relativity, in teleparallel gravity those quantities can be well defined.

Teleparallel Equivalent to General Relativity (TEGR) is an alternative theory of gravitation constructed out in terms of the tetrad field on Weitzenböck geometry. It was first proposed by Einstein in an attempt to derive a unified field theory [9]. Later it was revived with a paper entitled “New General Relativity” [10]; since then a lot of improvement has been made in the understanding of gravitational energy and the role of torsion [11, 12]. In the context of TEGR it is possible to define an expression for gravitational energy which is invariant under coordinates transformation and dependent on the reference frame. Those features are present in the special theory of relativity and there is no physical reason to abandon such ideas once one has dealt with a gravitational theory. Using the field equations of TEGR, it is possible to define an expression for the gravitational pressure. Therefore a natural extension is to define an expression for the gravitational entropy. The advantage of this procedure is defining an entropy in terms of purely thermodynamical quantities such as energy and pressure. This will be our main goal in this paper, for de Sitter Universe. This Universe model is important because it describes an expanding empty space. Thus it is possible to shed light on the vacuum energy and cosmological inflationary models.

The paper is organized as follows. In Section 2, we present the main ideas of teleparallel gravity. From field equation we derive the total energy and pressure. In Section 3, we calculate such quantities for de Sitter Universe, and then we use the first law of thermodynamics to get the gravitational entropy. To achieve such aim, we have interpreted the temperature of the system as a function of the cosmological constant. Finally we present our concluding remarks in Section 4.

*Notation.* Space-time indices and indices run from 0 to 3. Time and space indices are indicated according to . The tetrad field is denoted by and the determinant of the tetrad field is represented by . In addition we adopt units where , unless otherwise stated.

#### 2. Teleparallel Equivalent to General Relativity (TEGR)

Teleparallel gravity is a theory entirely equivalent to general relativity; however it is formulated in the framework of Weitzenböck geometry rather than in terms of Riemann geometry. Weitzenböck geometry is endowed with the Cartan connection [13], given by , where is the tetrad field; thus the torsion tensor can be calculated in terms of this field bySuch a geometry keeps a relation to a Riemannian manifold; for instance, we note the Cartan connection and Christoffel () symbols are related by a mathematical identity, which readswhereis the contortion tensor. The tetrad field, which is the dynamical variable of the theory, is obtained from the metric tensor. Thus Weitzenböck geometry is less restrictive than Riemann geometry; for each metric tensor it is possible to construct an infinite number of tetrad fields. This apparent arbitrary behavior is amended once we recall the interpretation of the tetrad field. The component is associated with the four-velocity of the observer; then for each reference frame there exists only one tetrad field. In fact we have to make use of the acceleration tensor to completely settle the state of an observer [14], since it could be in rotation as well as in translation.

If one tries to construct the curvature from the Cartan connection, he/she will find out that it vanishes identically. Hence the Weitzenböck geometry is described by a vanishing curvature and the presence of torsion. The Riemann geometry, as is well known, has vanishing torsion and a nonvanishing curvature tensor. Therefore making use of identity (2) to construct the scalar curvature, it leads to where is the determinant of the tetrad field, , and is the scalar curvature constructed out in terms of such a field. It should be noted that the metric tensor alone does not establish a geometry. From the above identity, we see that it is possible to construct a tetrad field adapted to a specific reference frame which induces, for the same metric tensor, both a curvature in Riemannian manifold and torsion in a Weitzenböck geometry. By means of the very same above identity, it is possible to find the counterpart of Hilbert-Einstein Lagrangian density for teleparallel gravity; thus, up to a total divergence (which plays no role in the field equations), it readswhere and stands for the Lagrangian density for the matter fields. This Lagrangian density can be rewritten aswhere

The field equations can be derived from Lagrangian (6) using a variational derivative with respect to ; they readwhere . The field equations may be rewritten aswhere andIn view of the antisymmetry property , it follows thatSuch equation leads to the following continuity equation: Therefore we identify as the gravitational energy-momentum tensor [15, 16].

Then, as usual, the total energy-momentum vector is defined by [17]where is a volume of the three-dimensional space. It is important to note that the above expression is invariant under coordinate transformations and it transforms like a 4-vector under Lorentz transformations. The energy-momentum flux is given by the time derivative of (13); thus by means of (12) we find

If we assume a vacuum solution, for example, a vanishing energy-momentum tensor of matter fields, then we havewhich is the gravitational energy-momentum flux [18]. Using field equations (9), the total energy-momentum flux readsNow let us restrict our attention to the spatial part of the energy-momentum flux, that is, the momentum flux; we have wherewe note that the momentum flux is precisely the force; hence, since is an element of area, we see that represents the pressure, along the direction, over an element of area oriented along the direction [16]. It should be noted that all definitions presented in this section follow exclusively from field equations (9).

We point out that general relativity and teleparallel gravity are equivalent only concerning dynamical features. This means that both theories will predict the same behavior of a test particle around a mass distribution. In other words both of them will agree in the classical experimental tests such as Mercury’s perihelion deviation and the bending of light. However predictions concerning the gravitational field features are strictly different, such as gravitational energy, momentum, and angular momentum. For instance, there is no analogous tensor in general relativity equivalent to the gravitational energy-momentum tensor in (10). The main reason for this is that some of the tensorial quantities in teleparallel gravity are analogous to connections in Riemannian geometry which gives rise to pseudo-tensors describing gravitational energy and momentum. Particularly there is no tensorial form, in Riemannian geometry, of on Weitzenböck geometry. The problem of defining gravitational energy in general relativity is a long-standing one; as a consequence any thermodynamical attempt to define a gravitational entropy would be plagued by the same problems. On the other hand such a quantity is natural in teleparallel gravity; thus our approach has several advantages. In Table 1, we chart the features of general relativity and teleparallel gravity.