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 quantum 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. Note that the next leading order is logarithmic = ≈ / + 1 πη ( / ) choosing n 2 in the rainbow functions (1.2) without loss as S A 4G0 2 ln A 4 , which is reminiscent of the of generality. Then the energy for the massless particle can quantum correction to the entropy [58–62]. From these, one be solved as can see that the rainbow metric contributes to the quantum corrected metric. ωp ω = . (2.12) At this stage, it is appropriate to comment on the choice η + 2 ω2 = r+ p of n 2 in the rainbow functions (1.2), which makes us analytically solve the MDR (1.1) and eventually gives the Therefore, one can rewrite the black hole temperature (2.10) logarithmic correction to the entropy. If not, it would be dif- as ficult to solve the MDR first, and then have another forms of correction without the logarithmic term to the entropy [32]. = r+ 0, TH TH (2.13) 2 r+ + ηG0 3 Schwarzschild–AdS black hole thermodynamics in ω2 = / η = rainbow gravity with p 1 G0. When 0, it is just the standard Hawk- ing temperature of the Schwarzschild–AdS black hole.√ Note Now, let us study Schwarzschild–AdS black hole thermody- that as r+ → 0, it becomes finite as TH = 1/(4π ηG0). This result implies that the standard divergent Hawking tem- namics in rainbow gravity. In order for the study to be focused = perature of the Schwarzschild–AdS black hole could be reg- and feasible, we take the rainbow functions (1.2) with n 2. ularized in the rainbow gravity as like in the Schwarzschild We would like to consider a local observer who is at rest at case [53]. In Fig. 1, we have plotted the modified and the the radius r, which we mean to introduce a spherical cavity standard Hawking temperatures showing that the former is enclosing the Schwarzschild–AdS black hole. Then the local finite at r+ = 0 due to the rainbow gravity effect while the temperature Tloc seen by the local observer can be obtained: latter blows up. 2 It seems appropriate to comment that the modified Hawk- 1 1 + 3r+ r+ 4π r+ 2 2 +η ing temperature has its minimum TH l0 r+ G0 Tloc = √ = (3.1) −g √ 00 r2 2 1 − r+ 1 + + + r 3 3ηG r l2 l2 T m = 1 − 0 (2.14) 0 0 H π 2 2 l0 l0 by implementing the redshift factor of the metric [63]. η = √l0 − 6 G0 Before proceeding further, let us briefly look into the limits at r+ 1 2 . Moreover, the upper bound for the 3 l0 of the modified Hawking temperature (2.13) and local tem- η< 2/ parameter l0 6G0 is required for r+ being real. In Fig. perature (3.1) of the Schwarzschild–AdS black hole in the η = 2/ 1, the thick line is for l0 6G0 where r+ is zero and the rainbow gravity enclosed in a cavity. First, by turning off the curve has its minimum. rainbow gravity, in the Schwarzschild–AdS limit of η → 0, 123 557 Page 4 of 11 Eur. Phys. J. C (2016) 76:557 the standard Hawking and local temperatures will be 3b. The equality occurs if the temperature is the minimum at r+ = 2r/3. Below the minimum temperature there is no r+ SAdS = 1 1 + 3 , black hole solution. These can be reconfirmed by analyzing TH (3.2) 4π r+ l2 0 constant temperature slices [64] of the Schwarzschild black hole confined in a cavity in Fig. 4a, b, which are obtained by 1 1 + 3r+ 4π r+ l2 solving Eq. (3.5) in terms of r+ at a constant temperature. In SAdS = 0 , Tloc (3.3) the diagram, one can easily see that thermodynamic stabil- r2 2 − r+ + + + r ity is in between 2r/3 < r+ < r. The turning points of the 1 r 1 2 2 √ l0 l0 curves satisfying with r+Tloc = 3/4π remain on the line = / 2 →∞ η → r+ 2r 3. Two solutions of the Schwarzschild black hole and in the Schwarzschild limit of l0 as well as 0, those become in the cavity are possible to the right of this point, one above the line and one below it. They would meet if a given temper- 1 T Sch = , ature equals the minimum temperature. For a higher cavity H π (3.4) 4 r+ temperature, the constant temperature curve shifts to the left, and for a lower cavity temperature it shifts to the right. Sch = 1 . Tloc (3.5) Compared with the standard Schwarzschild black hole, π − r+ 4 r+ 1 r which is asymptotically flat, the temperatures and specific heats of the Schwarzschild–AdS black hole, which is not For the sake of a detailed discussion, their corresponding asymptotic flat, are depicted in Figs. 2 and 3c, d. When com- specific heats can also be written down as pared the Hawking temperature in Fig. 2c with the local tem- 2 2 2 2πr+ l + 3r+ perature in Fig. 2b, one can see that they show the same CSAdS =− 0 , 2 2 (3.6) behavior with each other except that in the case of Fig. 2c G0 l − 3r+ 0 the cavity seems to be located at infinity. Thus, the asymp- SAdS Cloc totically AdS geometry naturally plays the role of a cavity. 4πr 2 (r − r+)(l2 + r 2 + rr+ + r 2 )(l2 + 3r 2 ) The real positive solutions of the Schwarzschild–AdS black = + 0 + 0 + 4 2 2 4 2 2 2 2 + ≥ / π G0 r+(3l + 2l r+ + 3r+) − 2r(l + r )(l − 3r+) hole can be obtained when r TH 1 2 , where the equality√ 0 0 0 0 = / (3.7) takes place if the temperature is the minimum at r+ l0 3 (η = 0 case). The larger solution is locally stable and has a for the Schwarzschild–AdS black hole, and positive specific heat, while the smaller solution is unstable 2 2πr+ and has a negative specific heat as in Figs. 2, 3c. CSch =− , (3.8) G If the Schwarzschild–AdS black hole is enclosed in a cav- 0 2 ity, the local temperature (3.3) and specific heat (3.7)are 4πr+ r − r+ CSch =− depicted as in Figs. 2, 3d. As r+ approaches the radius r of the loc − (3.9) G0 2r 3r+ cavity, the local temperature goes up to infinity, and the spe- for the Schwarzschild black hole. Here, the subscript ‘loc’ cific heat to zero. This is also the case for the Schwarzschild indicates that the specific heat is obtained from a black hole black hole enclosed in the cavity, which can be seen through enclosed in a cavity. the factor of C ∼ (r − r+) in Eqs. (3.7) and (3.9), while the Figure 2a shows the standard Hawking temperature (3.4) specific heat of the standard Schwarzschild–AdS black hole of the Schwarzschild black hole which becomes zero as r+ diverges as in Fig. 3c. goes to infinity, while it diverges as r+ goes to zero. If it Now, let us go back to the local temperature (3.1), which is enclosed in a cavity, the local temperature at the cavity is seen by the observer at r, is depicted in Fig. 5 where the (3.5) goes up to infinity. This cavity makes the black hole local temperature of the Schwarzschild–AdS black hole in thermodynamically well defined. Then this thermodynamic the rainbow gravity enclosed in a cavity was also drawn for stability can easily be seen if one examines the specific heat the purpose of comparison, which is divergent as r+ → 0. of T (dS/dT) as in Fig. 3.InFig.3a, one sees the negative Figure 5 is also different from Fig. 1 in that this is the local specific heat as given in Eq. (3.8), which indicates that the temperature seen by a local observer at r but not the modi- Schwarzschild black hole is unstable. On the other hand, as fied Hawking temperature seen by an observer at the asymp- seen in Fig. 3b, the Schwarzschild black hole in the cavity totic infinity. One can see in the figure that for a fixed r the has an asymptote at r+ = 2r/3, which is obtained from the local temperature is divergent as r+ approaches r, which is condition of dTloc/dr+ = 0, and if r+ is larger than√ 2r/3, the same as the local temperature of the Schwarzschild–AdS the black hole is stable. More precisely, if r+Tloc ≥ 3/4π, black hole. We also observe that there exist a global minimum there are two possible solutions in which the larger one is sta- temperature Tm at r = rm, and a local maximum tempera- ble while the smaller one is unstable as shown in Figs. 2b and ture TM at r = rM . Moreover, the temperature remains finite 123 Eur. Phys. J. C (2016) 76:557 Page 5 of 11 557

Fig. 2 Temperatures for a the T T (a)H (b) loc Schwarzschild black hole, b the Schwarzschild black hole 0.10 enclosed in a cavity, c the 0.08 Schwarzschild–AdS black hole, 0.08 0.06 and d the Schwarzschild–AdS 0.06 black hole enclosed in a cavity 0.04 0.04 0.02 0.02 r r 2 4 6 8 10 2 4 6 8 10

Fig. 3 Speciﬁc heats for a the (a) C (b) Schwarzschild black hole, b the C Schwarzschild black hole 100 enclosed in a cavity, c the r 1000 Schwarzschild–AdS black hole, 100 2 4 6 8 10 500 and d the Schwarzschild–AdS black hole enclosed in a cavity 200 r 300 2 4 6 8 10 500 400 1000 500

(c)C (d) C

2000 800 1500 600 1000 400 500 200 r r 500 2 4 6 8 10 200 2 4 6 8 10 1000 400

Fig. 4 Constant temperature (a) (b) slices of the Schwarzschild r black hole conﬁned within a 10 cavity r r

8 Exterior Region 8 6 r 2 r 3 r 6 C 0 0.035 4 T T H 0.025 4 0.025 C 0 TH 0.035 2 0 5 10 r 0 r 0 2 4 6 8 10

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T loc To investigate thermodynamic stability of the 0.10 TM Schwarzschild–AdS black hole in the rainbow gravity, we 0.04564 T0 calculate the heat capacity as 0.08 0.04559 ∂ E 0.04554 C = tot 0.06 ∂Tloc 0 0.03 0.06 0.09 3 r rM 2 0 2 2 2 2 2 2 G 4πr+(r − r+) l + r + rr+ + r+ l + 3r+ 1 + η 0 0.04 0 0 r2 = + G0 H(r+, r,η,G0) Tm 0.02 (3.11) rm r 2 4 6 8 10 with Fig. 5 Local temperature of the Schwarzschild–AdS black hole in the ( , ,η, ) = 4 + 2 2 + 4 η = 1 rainbow gravity enclosed in a cavity (solid line) and the standard H r+ r G0 r+ 3l0 2l0r+ 3r+ local temperature of the Schwarzschild–AdS black hole (dotted line) = = − 2 + 2 2 − 2 with r 10, l0 7. Here, Tm is globally minimum, TM is locally 2r l0 r l0 3r+ = maximum, and T0 is the temperature at r+ 0 l4 −η 3 − 3 + 2 ( − ) − 0 . → = /( π η ( + 2/ 2)) G0 3 r+ 4r 6l0 r+ 2r (3.12) as r+ 0 with T0 1 4 G0 1 r l0 . Thus, as r+ a result of the introduction of the rainbow gravity, one can observe both the existence of the local maximum tempera- When η → 0, it becomes the specific heat (3.7) for the stan- ture near the origin and the finiteness of the local temperature dard Schwarzschild–AdS black hole in the cavity, and when at the origin, which were first shown in Ref. [53]. However, taking the radius r of the cavity to infinity, it becomes the we also note that these were originally absent in the modified specific heat (3.6) for the Schwarzschild–AdS black hole. Hawking temperature (2.13)asinFig.1. Moreover, when l0 →∞as well as η → 0, it reduces to Next, making use of the entropy (2.15) and the local tem- the specific heat (3.9) for the Schwarzschild black hole in the perature (3.1), the thermodynamic first law yields the energy cavity, and with the radius of the cavity asr →∞, it becomes Etot as the specific heat (3.8) for the standard Schwarzschild black

r+ hole. In Fig. 6 the specific heat is plotted as a function of Etot = TlocdS r+ and it shows three qualitatively different regions, two 0 stable and one unstable states of the black hole. Specifi- ⎛ ⎞ 3 cally, when r+ > r , the specific heat is positive so that r r 2 r+ 1 r m = ⎝ + − − + 2 − + ⎠ . 1 2 1 2 r the black hole is stable, which we will call the large stable G0 l0 r l0 r black hole (LSB), when rM < r+ < rm, it is negative, so (3.10) the black hole unstable, the intermediate unstable black hole

800

2000 rM 600 0 2000 400 0 0.02 0.04 200 rm rU r 2 4 6 8 10 200

400

600

Fig. 6 Heat capacity of the Schwarzschild–AdS black hole in the η = 1 rainbow gravity enclosed in the cavity with r = 10, l0 = 7, G0 = 1, where rU is the upper bound of the event horizon r+

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10 r r I r 8 0.08 II 0.06 6 0.04 C 0 C 0 0.02 C 0 4 r C 0 10. III

2 TH 0.04564 r 0.492966 r r 2 4 6 8 10

Fig. 7 Constant temperature slice of the Schwarzschild–AdS black hole in the rainbow gravity conﬁned within a cavity. Here, we choose T = 0.04564 as an external cavity temperature where there exist three black hole states

2 (IUB), and when 0 < r+ < rM , it is again positive, and + 3r+ r+ 1 2 we call it as the small stable black hole (SSB). Note that l0 − the SSB appears in the very vicinity of the vanishing event 3 − r+ + 1 2 − r+ horizon. 4G0 1 r 2 r r l0 In Fig. 7, it shows constant temperature slices of the ⎛ ⎞ Schwarzschild–AdS black hole in the rainbow gravity con- −1 r+ ⎜ ηG0 sinh η ⎟ ⎜ G0 ⎟ fined in the cavity. Compared with Fig. 4, one can reconfirm × ⎝1 + ⎠ , (4.1) η that there is also a fine structure near the vanishing event 2 + G0 r+ 1 2 horizon due to the appearance of the local maximum tem- r+ perature as well as the finiteness of the temperature at the where S is the entropy in Eq. (2.15). When η = 0, it becomes origin, which is enlarged separately in the right-hand side of the on-shell free energy of the Schwarzschild–AdS black hole the diagram. From this figure, we see that the thermodynamic enclosed in a cavity. Furthermore, when l0 →∞, it recov- states of the Schwarzschild–AdS black hole in the rainbow ers the on-shell free energy of the Schwarzschild black hole gravity confined in the cavity are divided into three regions; enclosed in a cavity. In order to understand the phase transi- region I with positive specific heat which corresponds to the tion, we also need to introduce off-shell free energy, which LSB, region II with negative specific heat to the IUB, and is composed of a set of saddle points of the on-shell free region III again with positive specific heat to the SSB, respec- energies as tively.

Foff = Etot − TS. (4.2)

4 Free energy and phase transition Here, Etot and S are the same as before, while T is an external temperature of heat reservoir to control the phase transition. Now, let us study the thermodynamic phase transition [65– In Fig. 8, we have plotted both the on-shell Fon(r+) and 67]. The on-shell free energy of the Schwarzschild–AdS the off-shell Foff (r+) free energies as a function of r+.This black hole in the rainbow gravity enclosed in a cavity is picture is helpful when one considers thermodynamic stabil- obtained by the use of the local temperature Tloc in Eq. (3.1) ity. On the other hand, in Fig. 9, we have drawn the on-shell and the thermodynamic energy Etot in Eq. (3.10), explicitly free energy as a function of the temperature T , which helps us as to understand phase as temperature changes. Some regions including near the vanishing event horizon are magniﬁed to = − Fon Etot Tloc S show ﬁne details in Figs. 8 and 9. ⎛ ⎞ 3 First of all, the situation is mainly divided by two as shown r r 2 r+ 1 r = ⎝ + − − + 2 − + ⎠ in Fig. 8b, c, respectively. First, in Fig. 8c, which describes 1 2 1 2 r G0 l0 r l0 r the well-known Hawking–Page phase transition, when T < 123 557 Page 8 of 11 Eur. Phys. J. C (2016) 76:557

F (b) Fon 1. 10 6 (a) F T 3 5. 10 7 0 2 r Tm 0.01 0.02 0.03 0.04 0.05 1 1 7 T 5. 10 2 c TM T r F c (c) T 2 4 6 8 10 0.4 m T 1 1 0.2 c r 2 0.2 2 4 6 8 10 0.4 3 2 0.6 T0, Tc , TM 0.8 4 F 1.0 Fon on

(1) (2) Fig. 8 On-shell (solid line) and off-shell (dashed line) free energies minimum, TM local maximum temperature, T0 at r+ = 0, and Tc , Tc F(r+) of the Schwarzschild–AdS black hole in the η = 1 rainbow grav- phase transition temperatures ity enclosed in the cavity withr = 10,l0 = 7, G = 1, where Tm is global

0.2

T 1 m Tc 0.0 T 0.01 0.02 0.03 0.04

0.2 F 4 10 7 0.4 7 2 2 10 Tc TM T T 2 10 7 0 0.045641 0.6 4 10 7

0.8

Fig. 9 On-shell free energy F(T ) in terms of T with r = 10, l0 = 7, G0 = 1, η = 1. The circle is magniﬁed to show ﬁne detail which structure comes from the rainbow gravity

Tm, there are no black holes but a pure thermal radiation hole evaporation it would reduce its free energy. In brief, phase which has zero free energy. When T = Tm, the free the two small and large black hole states in this temperature energy exhibits an inflexion point in Fig. 8catr+ = rm range are less probable than a pure thermal radiation. When (1) where the specific heat is ill-defined, but a single unstable the temperature becomes Tc , the large black hole is at local black hole is formed which eventually decays into a pure minimum with zero free energy, while the small black hole thermal radiation. As the temperature goes up in between keeps in locally unstable. When the temperature is in the (1) (1) Tm < T < Tc , there are two black holes that can be in region of Tc < T < T0, there are still two black holes equilibrium with a thermal radiation, the small black hole is that the large black hole is now in globally stable, which formed at a local maximum of the off-shell free energy, while has both positive heat capacity and negative free energy, but the large black hole is formed at a local minimum. However, the small black hole has negative heat capacity and positive the small black hole, which corresponds to the IUB with free energy so that it is unstable to decay the globally stable the negative specific heat, is locally unstable and so decays large black hole state. In short, below the phase transition (1) either into a thermal radiation or to the large black hole. On temperature of Tc , it is more probably a thermal radiation, (1) the other hand, the large black hole corresponding to the LSB while above Tc it is more probably a large black hole so with positive free energy is not globally stable; it is locally that there exists a Hawking–Page phase transition between (1) stable in between Tm < T < Tc though, so by the black them.

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Second, in Fig. 8b, which shows off-shell free energies perature is quite different from the standard Hawking tem- near the vanishing event horizon, one observes a new mixed perature of the Schwarzschild–AdS black hole in a cavity as phase transition of the Schwarzschild–AdS black hole in the shown in Fig. 5 where the former is finite even at r = r+ rainbow gravity enclosed in a cavity. Note that compared while the latter is divergent. As a result, it is shown that there with the previous case, the phenomenon occurs in relatively exists an additional stable tiny black hole together with the (2) extremely tiny range of temperature between T0 < Tc < small black hole above T0. Moreover, it is also shown that (2) TM where three black hole states exist. One of them is the there exists an additional critical temperature Tc at which SSB and the others are the same as before, the IUB and LSB. the locally stable tiny black hole tunnels into the large sta- (2) When T0 < T < Tc , the IUB, of which the free energy is ble black hole with the finite transition probability seen from at local maximum with positive value and negative specific Figs. 8 and 9 of the on-shell and off-shell free energies. heat, is unstable so it decays either into a thermal radiation Finally, it is appropriate to comment that in the presence (2) or to the SSB/LSB. When the temperature is Tc ,theIUB of the cosmological constant the HUP in Eq. (2.11), which is has zero free energy. When the temperature is in between used to eliminate the energy dependence of the probe in the (2) Tc < T < TM , all the three black holes are in stable states. modified Hawking temperature, is corrected by the extended When T = TM , two stable black holes remains. However, uncertainty relation [68,69]. Therefore, as a further investi- regardless how stable the SSB is, it is the LSB with much gation, it will be interesting to analyze the thermodynamic lower free energy so that the SSB eventually decays into the quantities and phase transition in the rainbow gravity by using LSB. Finally, when T > TM , only the LSB exists. the generalized uncertainty principle [70–74].

Acknowledgments We would like to thank Prof. W. Kim and Mr. Y. 5 Discussion Gim for helpful discussions. Open Access This article is distributed under the terms of the Creative In this paper, we have studied local thermodynamics includ- Commons Attribution 4.0 International License (http://creativecomm ing its phase transition of the Schwarzschild–AdS black hole ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit in the rainbow gravity enclosed in a cavity subject to the to the original author(s) and the source, provide a link to the Creative MDR. The black hole temperature in the rainbow gravity Commons license, and indicate if changes were made. depends on the energy ω of a probe provided by the rainbow Funded by SCOAP3. functions (1.2), and making use of the HUP and deploying the MDR, we have derived the modified Hawking temperature which is finite at r+ = 0. It implies that the divergent standard Hawking temperature of the Schwarzschild–AdS References black hole is regularized in the rainbow gravity. Moreover, the parameter η is found to have the upper limit given by the 1. G. Amelino-Camelia, J.R. Ellis, N.E. Mavromatos, D.V.Nanopou- Newton and the cosmological constants where r+ is zero and los, Distance measurement and wave dispersion in a Liouville the modified Hawking temperature has its minimum value, string approach to quantum gravity. Int. J. Mod. Phys. A 12, 607 (1997). arXiv:hep-th/9605211 in contrast to the case of the Schwarzschild black hole in the 2. G. Amelino-Camelia, Quantum-spacetime phenomenology. Living rainbow gravity having no upper bound [53]. Rev. Rel. 16, 5 (2013). arXiv:0806.0339 [gr-qc] We have summarized in Fig. 2 the temperatures and spe- 3. G. Amelino-Camelia, Relativity in space-times with short distance cific heats of the Schwarzschild(–AdS) black hole with/ structure governed by an observer independent (Planckian) length scale. Int. J. Mod. Phys. D 11, 35 (2002). arXiv:gr-qc/0012051 without a cavity which are obtained from taking the limits 4. G. Amelino-Camelia, Doubly-special relativity: facts, myths and of the Schwarzschild–AdS black hole in the rainbow grav- some key open issues. Symmetry 2, 230 (2010). arXiv:1003.3942 ity enclosed in a cavity. As a result, from Figs. 2b, c and [gr-qc] 3b, c, we have observed again the well-known facts that the 5. J. Magueijo, L. Smolin, Generalized Lorentz invariance with an invariant energy scale. Phys. Rev. D 67, 044017 (2003). Schwarzschild–AdS black hole has a similar thermodynamic arXiv:gr-qc/0207085 behavior to the Schwarzschild black hole in a cavity; for a 6. J. Magueijo, L. Smolin, Gravity’s rainbow. Class. Quant. Grav. 21, given temperature of T > Tm, there are two black holes, the 1725 (2004). arXiv:gr-qc/0305055 larger one is stable and the small one is unstable, needless 7. S. Liberati, S. Sonego and M. Visser, Interpreting doubly special relativity as a modified theory of measurement. Phys. Rev. D 71, to say, including the Hawking–Page phase transition from a 045001 (2005). arXiv:gr-qc/0410113 (1) hot flat space to a black hole at the critical temperature, Tc 8. P. Galan, G. A. Mena Marugan, Quantum time uncertainty in a for our case. gravity’s rainbow formalism. Phys. Rev. D 70, 124003 (2004) However, for the Schwarzschild–AdS black hole in the arXiv:gr-qc/0411089 9. P. Galan, G.A. Mena Marugan, Length uncertainty in a grav- rainbow gravity, due to deformation of temperature near the ity’s rainbow formalism. Phys. Rev. D 72, 044019 (2005). vanishing event horizon, the modified local Hawking tem- arXiv:gr-qc/0507098 123 557 Page 10 of 11 Eur. Phys. J. C (2016) 76:557

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