Advances in High Energy Physics

Volume 2016, Article ID 9813582, 21 pages

http://dx.doi.org/10.1155/2016/9813582

## Gravity’s Rainbow and Its Einstein Counterpart

^{1}Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran^{2}Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441 Maragha, Iran^{3}Physics Department, Shahid Beheshti University, Tehran 19839, Iran

Received 20 June 2016; Revised 21 July 2016; Accepted 21 July 2016

Academic Editor: Rong-Gen Cai

Copyright © 2016 S. H. Hendi et al. 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 SCOAP^{3}.

#### Abstract

Motivated by UV completion of general relativity with a modification of a geometry at high energy scale, it is expected to have an energy dependent geometry. In this paper, we introduce charged black hole solutions with power Maxwell invariant source in the context of gravity’s rainbow. In addition, we investigate two classes of gravity’s rainbow solutions. At first, we study energy dependent gravity without energy-momentum tensor, and then we obtain gravity’s rainbow in the presence of conformally invariant Maxwell source. We study geometrical properties of the mentioned solutions and compare their results. We also give some related comments regarding thermodynamical behavior of the obtained solutions and discuss thermal stability of the solutions.

#### 1. Introduction

An accelerated expansion of the universe was confirmed by various observational evidences. The luminosity distance of Supernovae type Ia [1, 2], the anisotropy of cosmic microwave background radiation [3], and also wide surveys on galaxies [4] confirm such accelerated expansion. On the other hand, baryon oscillations [5], large scale structure formation [6], and weak lensing [7] also propose such an accelerated expansion of the universe.

After discovery of such an expansion in , understanding its theoretical reasons presents one of the fundamental open questions in physics. Identifying the cause of this late time acceleration is a challenging problem in cosmology. Physicists are interested in considering this accelerated expansion in a gravitational background and they proposed some candidates to explain it. For example, a positive cosmological constant leads to an accelerated expansion, but it is plagued by the fine tuning problem [8–12]. In other words, the left hand side of Einstein equations can modify by the cosmological constant as a geometrical modification or it can be interpreted as a kinematic term on the right hand side with the equation of state parameter . By considering for a source term, it is possible to further modify this approach. This consideration has interpretation of dark energy which has been investigated in literature [13–19]. In dark energy models, the acceleration expansion of the universe is due to an unknown ingredient added to the cosmic pie. The effect of this unknown ingredient is extracted by modifying the stress energy tensor of the Einstein equation with a matter which is different from the usual matter and radiation components.

On the other hand, it is proposed that the presence of accelerated expansion of the universe indicates that the standard general relativity requires modification. To do so, one can generalize the Einstein field equations to obtain a modified version of gravity. There are different branches of modified gravity with various motivations, such as brane world cosmology [20–22], Lovelock gravity [23–27], scalar-tensor theories [28–35], and gravity [36–51].

Modifying general relativity opens a new way to a large class of alternative theories of gravity ranging from higher dimensional physics [52–54] to non-minimally coupled (scalar) fields [55–58]. Regarding various models of modified gravity, we will be interested in gravity [59–65] based on replacing the scalar curvature with a generic analytic function in the Hilbert-Einstein action. Some viable functional forms of gravity may be reconstructed starting from data and physically motivated issues.

However, the field equations of gravity are complicated fourth-order differential equations, and it is not easy to find exact solutions. In addition, adding a matter field to gravity makes the field equations much more complicated. On the other hand, regarding constant curvature scalar model as a subclass of general gravity can simplify the field equations. Also, one can extract exact solutions of gravity coupled to a traceless energy-momentum tensor with constant curvature scalar [66]. For example, considering exact solutions of gravity with conformally invariant Maxwell (CIM) field as a matter source has been investigated [67, 68].

General relativity coupled to a nonlinear electrodynamics attracts significant attentions because of its specific properties in gauge/gravity coupling. Interesting properties of various models of the nonlinear electrodynamics have been investigated before [69–81]. Power Maxwell invariant (PMI) theory is one of the interesting branches of the nonlinear electrodynamics and its Lagrangian is an arbitrary power of Maxwell Lagrangian [82–84]. The PMI theory is more interesting with regard to Maxwell field, and for unit power it reduces to linear Maxwell theory. These nonlinear electrodynamics enjoy conformal invariancy when the power of Maxwell invariant is a quarter of space-time dimensions, and this is one of the attractive properties of this theory. In other words, one can obtain traceless energy-momentum tensor for a special case “” which leads to conformal invariancy. It is notable to mention that this idea has been considered to take advantage of the conformal symmetry to construct the analogues of the four-dimensional Reissner-Nordström solutions with an inverse square electric field in arbitrary dimensions [85].

From the gravitational point of view, it is possible to show that the electric charge and cosmological constant can be extracted, simultaneously, from pure gravity (without matter field: ) [51]. In this paper, we are going to obtain -dimensional charged black hole solutions from gravity’s rainbow and pure gravity’s rainbow as well as gravity’s rainbow with CIM source and compare them to obtain their direct relation.

In order to build up special relativity from Galilean theory, one has to take into account an upper limit for velocity of particles. The same method could be used to restrict particles from obtaining energies no more than specific energy, the so-called Planck energy scale. This upper bound of energy may modify dispersion relation which is known as double special relativity [86]. Generalization of this doubly special relativity to incorporate curvature leads to gravity’s rainbow [87]. In gravity’s rainbow, space-time is a combination of the temporal and spatial coordinates as well as energy functions. The existence of such energy functions indicates that the particle probing the space-time can acquire specific range of energies which in essence leads to formation of a rainbow of energy.

There are several features for gravity’s rainbow which highlight the importance of such generalization. Among them one can point modification in energy-momentum dispersion relation which is supported by studies that are conducted in string theory [88], loop quantum gravity [89], and experimental observation [90]. Also, existence of remnant for black holes [91] is proposed to be a candidate for solving the information paradox [92]. In addition, this theory admits the usual uncertainty principle [93, 94].

Recently, there has been a growing interest in energy dependent space-time [95–100]. Different classes of black holes have been investigated in the context of gravity’s rainbow [101–104]. The hydrostatic equilibrium equation of stars in the presence of gravity’s rainbow has been obtained [105]. Furthermore, a study regarding the gravity’s rainbow and compact stars has been done [106]. In [107], the effects of gravity’s rainbow for wormholes have been investigated. Moreover, the influences of gravity’s rainbow on gravitational force have been investigated [108]. Also, Starobinsky model of theory in gravity’s rainbow has been studied in [109]. In addition, gravity’s rainbow has interesting effects on the early universe [110–112].

The main motivations for studying black holes in the presence of gravity’s rainbow are given as follows. First of all, due to the high energy properties of the black holes, it is necessary to consider quantum corrections of classical perspectives. One of the methods to include quantum corrections of gravitational fields is by considering energy dependent space-time. In fact, it was shown that the quantum correction of gravitational systems could be observed in dependency of space-time on the energy of particles probing it which is gravity’s rainbow point of view [93, 113, 114]. Since we are modifying our point of view to energy dependent space-time, it is expected to find its effects on the properties of black holes, especially in the context of their thermodynamics. This is another motivation for considering gravity’s rainbow generalization. Also, there are specific achievements for gravity’s rainbow in the context of black holes and among them one can name the following: modified uncertainty principle [93, 94], existence of remnants for the black holes [91, 102], furnishing a bridge toward Horava-Lifshitz gravity [115], providing possible solution toward information paradox [91], and finally being UV completion of Einstein gravity [116]. In addition, as it was pointed out before, in the context of cosmology, it presents a possible solution toward Big Bang singularity problem [110–112]. On the other hand, gravity provides correction toward gravitational sector of the Einstein theory of gravity. The importance of this correction is highlighted in the context of black holes. In order to have better picture regarding the physical nature of the black holes, one may consider high energy regime effects as well (considering that the concepts of Hawking radiation were derived by studying black holes in semiclassical/quantum regime). Here, we apply gravity’s rainbow to find the effects of generalization as well as energy dependency of space-time on the black hole solutions.

The outline of our paper is as follows. In Section 2, we are going to investigate black hole solutions in Einstein gravity’s rainbow with PMI and CIM fields. Then, we want to investigate conserved and thermodynamic quantities of the solutions and check the first law of thermodynamics. In Section 3, we will obtain black hole solutions for gravity’s rainbow with CIM source and check the first law of thermodynamics. We also discuss thermal stability of these solutions and criteria governing stability/instability in Section 4. Then, we consider pure gravity’s rainbow and compare these solutions with gravity’s rainbow with CIM source and give some related comments regarding its thermodynamical behavior. Finally, we finish our paper with some conclusions.

#### 2. Einstein Gravity’s Rainbow in the Presence of PMI Field

Here, we are going to introduce -dimensional solutions of the Einstein gravity’s rainbow in the presence of PMI field with the following Lagrangian:where and are, respectively, the Ricci scalar and the cosmological constant. In (1), the Maxwell invariant is , where is the electromagnetic tensor field and is the gauge potential. As we mentioned before, in the limit with , Lagrangian (1) reduces to the Lagrangian of Einstein-Maxwell gravity. Since the Maxwell invariant is negative, henceforth we set , without loss of generality. Using the variation principle, we can find the field equations the same as those obtained in [117].

##### 2.1. Black Hole Solutions

Here, we will obtain charged rainbow black hole solutions with negative cosmological constant in -dimensions. It is notable that the charged rainbow black hole solutions in Einstein gravity coupled to nonlinear electromagnetic fields have been studied in [118]. In this paper, we want to extend the space-time to -dimensions and obtain black hole solutions in the presence of PMI field as a matter source. The rainbow metric for spherical symmetric space-time in -dimensions can be written aswhere

Using metric (2) with the field equations, one can obtain the following solutions for the metric function as well as gauge potential:where the consistent function is or for and , respectively. In addition, and are integration constants which are, respectively, related to the mass and electric charge of the black hole. We should also mention that we consider for obtaining well-behaved electromagnetic field. It is notable that, by replacing and , solutions (4) reduce to the following higher dimensional Reissner-Nordström black hole solutions:

In order to investigate the geometrical structure of these solutions, we first look for the essential singularity(ies). The Ricci scalar can be written asand the behavior of Kretschmann scalar istherefore, confirming that there is a curvature singularity at . In addition, the Kretschmann scalar is for , which confirms that the asymptotical behavior of the charged rainbow black hole is adS. It is worthwhile to mention that the asymptotical behavior of these solutions is independent of the rainbow functions and the power of PMI source. This independency comes from the fact that rainbow functions are sensible in high energy regime such as near-horizon and one expects to ignore its effects far from the black hole (we only consider high energy regime). In addition, at large distances, the electric field of Maxwell and PMI theories vanishes, and, therefore, one may expect to ignore the effects of electric charge far from the origin. In order to investigate the possibility of the horizon, we plot the metric function versus in Figures 1 and 2. It is evident that, depending on the choices of values for different parameters, we may encounter two horizons (inner and outer horizons), one extreme horizon and without horizon (naked singularity).