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Thermodynamic Stability of Modified Schwarzschild–Ads Black Hole In Eur. Phys. J. C (2016) 76:557 DOI 10.1140/epjc/s10052-016-4393-1 Regular Article - Theoretical Physics Thermodynamic stability of modified Schwarzschild–AdS black hole in rainbow gravity Yong-Wan Kim1,a, Seung Kook Kim2,b, Young-Jai Park3,c 1 Research Institute of Physics and Chemistry, Chonbuk National University, Jeonju 54896, Korea 2 Department of Physical Therapy, Seonam University, Namwon 55724, Korea 3 Department of Physics, Sogang University, Seoul 04107, Korea Received: 29 July 2016 / Accepted: 20 September 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract In this paper, we have extended the previous where p, m, ωp are the momentum, the mass of the test study of the thermodynamics and phase transition of the particle, and the Planck energy, respectively. Thus, quanta of Schwarzschild black hole in the rainbow gravity to the different energies see different background geometry, which Schwarzschild–AdS black hole where metric depends on is referred to as a rainbow gravity. Since then many efforts the energy of a probe. Making use of the Heisenberg uncer- have been made devoted to the rainbow gravity related to the tainty principle and the modified dispersion relation, we have gravity and other stimulated work at the Planck scale [7–26]. obtained the modified local Hawking temperature and ther- In connection with black hole thermodynamics in the rain- modynamic quantities in an isothermal cavity. Moreover, we bow gravity, there have also been much work with the fol- carry out the analysis of constant temperature slices of a lowing rainbow functions [1,2,27–35]: black hole. As a result, we have shown that there also exists another Hawking–Page-like phase transition in which case a (ω/ω ) = , (ω/ω ) = − η ω/ω n, locally stable small black hole tunnels into a globally stable f p 1 g p 1 p (1.2) large black hole as well as the standard Hawking–Page phase transition from a hot flat space to a black hole. which belong to the most interesting MDRs related to quan- tum gravity phenomenology among several other types [5,15, 22,36–52]. Here, n is a positive integer, η a constant of order 1 Introduction unity, and these functions satisfy with limω→0 f (ω/ωp) = 1 and limω→0 g(ω/ωp) = 1 at low energies. In particular, Li The possibility that standard energy-momentum dispersion et al. [44] have obtained the Schwarzschild–AdS black hole relations are modified near the Planck scale is one of the solution in the framework of rainbow gravity with differ- scenarios in quantum gravity phenomenology [1,2]. Such a ent rainbow functions from Eq. (1.2), and investigated ther- modified dispersion relation (MDR) was also advocated by modynamic stability without the analysis of phase transi- the study of nonlinear deformed or doubly special relativ- tion. Recently, Gim and Kim (GK) [53] have shown that the ity (DSR) [3,4] in which the Planck length as well as the Schwarzschild black hole in the rainbow gravity in an isother- speed of light is an observer invariant quantity. In particular, mal cavity has an additional Hawking–Page phase transition Magueijo and Smolin [5,6] have extended the DSR to gen- near the event horizon apart from the standard one, which is eral relativity by proposing that the spacetime background of relevance to the existence of a locally small black hole. ω felt by a test particle would depend on its energy . Such an On the other hand, an approach of the higher dimensional energy of the test particle deforms the background geometry flat embedding is used to study a local temperature for a and consequently the MDR as freely falling observer outside black holes [54,55]. And very recently, we have shown that a local temperature seen by ω2 f (ω/ω )2 − p2g(ω/ω )2 = m2, (1.1) p p a freely falling observer depends only on g(ω/ωp) [56]so that the choice of f (ω/ωp) = 1 makes not only the time- a e-mail: [email protected] like Killing vector in the rainbow Schwarzschild black hole b e-mail: [email protected] as usual, but also it makes the local thermodynamic energy c e-mail: [email protected] independent of the test particle’s energy. 123 557 Page 2 of 11 Eur. Phys. J. C (2016) 76:557 In this paper, we would extend GK’s work of the dependent constants are related with the physical ones as Schwarzschild black hole in the rainbow gravity to the G0 Schwarzschild–AdS spacetime. In order to study efficiently, G(ω/ωp) = , g(ω/ω ) we shall describe thermodynamics by using an event hori- p 3 zon r+ as a variable instead of the mass M as in GK’s work, (ω/ω ) =− = g2(ω/ω ) . (2.4) p 2(ω/ω ) p 0 since in the Schwarzschild–AdS black hole it is difficult to l p solve for the event horizon as a function of the mass. In Sect. Note that the solution reduces to the usual Schwarzschild– 2, black hole temperature for the Schwarzschild–AdS black AdS vacuum solution hole in the rainbow gravity will be calculated from the defini- 1 tion of the standard surface gravity. Then, making use of the s2 =−N 2 t2 + r 2 + r 2 2 d d 2 d d (2.5) MDR and the Heisenberg uncertainty principle, the energy N dependence of a test particle in black hole temperature will be in the low-energy limit of ω/ωp → 0. The mass defined by properly rephrased. And the entropy will be derived from the N 2 = 0 is given by first law of thermodynamics. In Sect. 3, we will study local thermodynamic quantities including temperature, energy and 2 r+ r+ heat capacity in an isothermal cavity with their various limits M(r+) = 1 + (2.6) 2G 2 for each other’s comparison. Furthermore, in order to clearly 0 l0 reconfirm thermodynamic stability, we also analyze constant temperature slices of the Schwarzschild(–AdS) black hole in with the event horizon r+. T the rainbow gravity. In Sect. 4, we will study phase transi- Then the modified Hawking temperature H is obtained, tion between various black hole states and the hot flat space κH g(ω/ωp) through investigating free energies of the Schwarzschild– T = = T 0, (2.7) H 2π f (ω/ω ) H AdS black hole in the rainbow gravity. Finally, conclusion p and discussion will be given in Sect. 5. from the surface gravity κH at the event horizon as follows: 1 −g11 (g11) 2 Temperature and entropy of Schwarzschild–AdS κH =− lim . (2.8) 2 r→r+ g00 g00 black hole in rainbow gravity Here, the standard Hawking temperature T 0 of the Let us consider the modified Schwarzschild–AdS black hole H Schwarzschild–AdS black hole is given by in rainbow gravity described as [44] 2 1 1 3r+ 2 =− N 2 0 = + . ds dt TH (2.9) 2(ω/ω ) π + 2 f p 4 r l0 1 r 2 + r 2 + 2, 2 2 d 2 d (2.1) Making use of the explicit rainbow functions (1.2), the black g (ω/ωp)N g (ω/ωp) hole temperature can be written as where n 0 TH = 1 − η(ω/ωp) T (2.10) 2 H 2 = − 2G0 M + r . N 1 2 (2.2) r l0 so that the temperature depends on the energy ω of a probe. Now, in order to eliminate the ω dependence of the probe This is a spherically symmetric solution to the modified field in the modified Hawking temperature (2.10), one can use the equation in rainbow gravity of Heisenberg uncertainty principle (HUP) as in Ref. [53]. In the vicinity of the black hole surface, an intrinsic position Gμν(ω/ω ) + (ω/ω )gμν(ω/ω ) p p p uncertainty x of the probe of order of the event horizon r+ = 8πG(ω/ωp)Tμν(ω/ωp) (2.3) leads to a momentum uncertainty of order of p [57]as in the absence of matter. Here, Gμν(Tμν) is the Einstein 1 (ω/ω )((ω/ω )) p = p ∼ . (2.11) (energy-momentum) tensor, G p p is an energy- r+ dependent Newton (cosmological) constant, and G0(0 = − / 2) 3 l0 is the physical Newton (cosmological) constant at Plugging the momentum uncertainty into the MDR (1.1), the low-energy limit of ω/ωp → 0. Here, the energy ω- one can determine the energy ω. We explicitly show this by 123 Eur. Phys. J. C (2016) 76:557 Page 3 of 11 557 TH Next, from the first law of black hole thermodynamics, 0.15 one can obtain the entropy: dM πr+ r+ = = 2 + η + πη −1 . 0.10 S r+ G0 sinh TH G0 ηG0 (2.15) 0.05 Here, we have chosen the integration constant to be satisfied with S → 0asr+ → 0. This is the exactly same form with the entropy of the Schwarzschild black hole in the rainbow r gravity [53], but explicitly different in r+. This is nothing new 5 10 15 20 since we know that the standard Schwarzschild black hole Fig. 1 The modified Hawking temperatures of the Schwarzschild– and the Schwarzschild–AdS black hole also have the same η = 2 AdS black hole in the rainbow gravity for 1(solid line)andfor form of entropy of S = πr+, but different in r+. Moreover, in η = / = = the upper bound of 49 6(thick line) with l0 7, G0 1, and the the standard Schwarzschild–AdS limit of η = 0, the entropy standard Hawking temperature (dashed line)withl0 = 7 (2.15) becomes one-fourth of the area of the event horizon S = A/4G0.
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