Physics Letters B 785 (2018) 274–283 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Thermodynamics of novel charged dilaton black holes in gravity’s rainbow M. Dehghani Department of Physics, Razi University, Kermanshah, Iran a r t i c l e i n f o a b s t r a c t Article history: In the present work, we have studied the thermodynamical properties of black holes arising as the Received 7 June 2018 solutions of the four-dimensional dilaton gravity coupled to Maxwell’s electrodynamics in gravity’s Received in revised form 6 August 2018 rainbow. This theory allows three classes of asymptotically non-flat and non-AdS black hole solutions. Accepted 8 August 2018 We showed that the self-interacting scalar function, as the solution to the scalar field equation, can Available online 30 August 2018 be written as the linear combination of three Liouville-type potentials. The thermodynamical quantities Editor: N. Lambert are identified and in particular, a generalized Smarr formula is derived. It is shown that, although Keywords: the thermodynamic quantities are affected by the rainbow functions, the validity of the black hole Four-dimensional black hole thermodynamical first law is supported. The thermodynamic stability of the solutions have been analyzed Charged dilaton black hole through the black hole heat capacity. We have shown that, even in the presence of the rainbow functions, Maxwell’s theory of electrodynamics the black holes can be locally stable in the sense that there exists a range of the horizon radiuses for Gravity’s rainbow which the heat capacity is positive. © 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction the energy and velocity that a particle can attain [14]. This theory is accomplished based on a nonlinear Lorentz transformation in One of the more outstanding problems in the context of the- the momentum space in such a way that the energy-momentum oretical physics is to combine the gravity with the quantum me- relation appears with the corrections in the order of the Planck chanics and to construct the theory of quantum gravity. Despite length and the Planck-scale corrected dispersion relation preserves the attempts made through the string theory [1,2], loop quan- a deformed Lorentz symmetry [11,15]. tum gravity [3], space time foam [4], Horava–Lifshitz gravity [5], The doubly special relativity has now been extended to the non-commutative geometry [6] and other approaches there is cur- curved space times and a doubly general relativity or gravity’s rain- rently no complete theory of quantum gravity and this problem bow has been proposed by Magueijo and Smolin [16,17]. In the is still open. A common feature of almost all of these alternative rainbow gravity theory, the geometry of space time depends on the approaches is a strong interest on the modification of the usual energy of the test particle. Thus, it seems different for the parti- energy-momentum dispersion relation at the Planck-scale regime cles having different amounts of energy and the energy dependent [7,8]. It seems that this modification plays an important role in metrics form a rainbow of metrics. This is why the double general establishing a full theory of quantum gravity, but it leads to the vi- relativity is named as gravity’s rainbow. The modified dispersion olation of Lorentz invariance, as the most fundamental symmetry relation can be written in the following general form [18,19] in the universe. One of the approaches to overcome this problem is the doubly or deformed special relativity [9,10]. The doubly spe- cial relativity, as a modified formalism of special relativity, has E2 f 2(ε) − p2 g2(ε) = m2, (1.1) been proposed to make the modified dispersion relation Lorentz invariant [11–13]. In the high energy formalism of the special rel- where, ε = E/E P , E P is the Planck-scale energy, E is the energy ativity, known as the doubly special relativity, the speed of light of the test particle and the functions f (ε) and g(ε) are known and the Planck energy are two Lorentz invariant quantities. Also, as the rainbow functions. The rainbow functions are satisfied the the Planck energy and the speed of light are the upper bounds of following conditions = = E-mail address: [email protected]. lim f (ε) 1, and lim g(ε) 1. (1.2) ε→0 ε→0 https://doi.org/10.1016/j.physletb.2018.08.045 0370-2693/© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. M. Dehghani / Physics Letters B 785 (2018) 274–283 275 By these requirements one is able to reproduce the standard dis- 2. The field equations in gravity’s rainbow persion relation in the limit of ε → 0. It must be noted that the functional form of the rainbow functions are not unique and there The action of four-dimensional Einstein–Maxwell-dilaton grav- are several expressions for them which are correspond to different ity theory can be written in the following general form [24,34] phenomenological motivations. Some of the proposed models for 1 √ − the temporal and spatial rainbow functions are as follows [20,21] I = −gd4x R − V (φ) − 2(∇φ)2 − Fe 2αφ . (2.1) 16π R f (ε) = 1, and g(ε) = 1 − ηεn, (1.3) Here, is the Ricci scalar, φ is a scalar field coupled to it- self via the functional form V (φ) and the last term denotes the eξε − 1 coupled scalar-electromagnetic lagrangian. The parameter α is the f (ε) = , and g(ε) = 1, (1.4) μν ξε scalar-electromagnetic coupling constant and F = F Fμν being the Maxwell invariant, Fμν = ∂μ Aν − ∂ν Aμ is the Maxwell’s elec- 1 f (ε) = g(ε) = , (1.5) tromagnetic tensor and Aμ denotes the electromagnetic potential. 1 − λε By varying the action (2.1)with respect to electromagnetic, gravi- tational and scalar fields, we get the related field equations which where, η, n, ξ and λ, known as the rainbow parameters, are di- are in the following forms mensionless constants of the order of unity. Nowadays, gravity’s rainbow, in which the quantum gravita- − ∇ e 2αφ F μν = 0, (2.2) tional effects are taken into account, has attracted a lot of interest μ and many papers have been appeared in which the physical prop- 1 −2αφ ρ 1 erties of the black holes are investigated in the presence of rain- Rμν − gμν V (φ) = 2∇μφ∇νφ + e 2Fμρ Fν − F gμν , 2 2 bow functions. Thermodynamics and phase transition of the modi- fied Schwarzschild black holes via gravity’s rainbow are studied in (2.3) refs. [19,22]. Thermal stability of nonlinearly charged BTZ and four- dV(φ) − 42φ = − 2αFe 2αφ,φ= φ(r). (2.4) dimensional rainbow black holes has been analyzed in refs. [23,24]. dφ The effects of rainbow functions on the rotating BTZ black holes are the subject of ref. [25]. Thermodynamics of Gauss–Bonnet black We would like to solve the above field equations in a four- holes in rainbow gravity has been discussed in refs. [26,27]. Also, dimensional spherically symmetric geometry with the following thermodynamics and stability of the black holes have been studied line element [18,33] in the context of massive gravity’s rainbow by Hendi et al. [28,29]. U(r) Study of the physics in the energy dependent space times has pro- ds2 =− dt2 2 vided many interesting results such as: black hole remnant [30,31], f (ε) nonsingular universe [32], etc. 1 1 + dr2 + r2 R2(r)(dθ 2 + sin2 θdϕ2) . (2.5) Thermodynamics and thermal stability of charged dilatonic BTZ g2(ε) U(r) black holes in gravity’s rainbow have been studied in ref. [18]in the presence of Coulomb and modified Coulomb fields, separately. It must be noted that the only nonvanishing component of F = The impacts of thermal fluctuations, as a quantum gravitational electromagnetic field is Ftr. Assuming as a function of r, − 2 2 2 phenomenon, on the thermodynamic phase transition of rainbow 2 f (ε)g (ε)(Ftr(r)) . In order to obtain the metric function U (r), dilaton black holes have been investigated in my last work [33]. we use Eq. (2.5)in the gravitational field equations (2.3). It leads Here, I tend to extend my previous works to the case of electrically to the following differential equations charged four-dimensional dilaton black holes. The main object is to g2 1 R r 1 obtain the new exact black hole solutions to the Einstein–Maxwell- (ε) ( ) ett = U (r) + 2 + U (r) + V (φ) dilaton theory in rainbow gravity and to investigate the thermody- 2 r R(r) 2 namic proprieties as well as the thermal stability or phase transi- − 2 2 2 −2αφ = f (ε)g (ε)Ftre 0, (2.6) tion of the new charged rainbow black holes. The plan of this paper is organized as follows: Sec. 2 is devoted R (r) 2R (r) 2 err = ett + 2 + + (φ (r)) U(r) = 0, (2.7) to solving the coupled gravitational, electromagnetic and scalar R(r) rR(r) field equations in an energy dependent space time. The solution 1 R (r) to the scalar field equation has been written as the linear com- eθθ = eϕϕ = + U (r) r R(r) bination of three Liouville-type potentials.