Phase Transition of Quantum-Corrected Schwarzschild Black Hole in Rainbow Gravity

Physics Letters B 784 (2018) 6–11

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Physics Letters B

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Phase transition of quantum-corrected Schwarzschild black hole in rainbow gravity

Md. Shahjalal

Department of Mathematics, Shahjalal University of Science and Technology, Sylhet-3114, Bangladesh a r t i c l e i n f o a b s t r a c t

Article history: The thermodynamic phase transition of Schwarzschild black hole, after employing a quantum correction Received 30 May 2018 to the space–time metric, in the gravity’s rainbow is explored. The rainbow gravity-inspired Hawking Received in revised form 15 July 2018 temperature of the quantum-corrected rainbow Schwarzschild black hole is calculated initially, and then Accepted 16 July 2018 the entropy, the local temperature, and the local energy of the black hole in an isothermal cavity Available online 20 July 2018 are derived. The off-shell free energy is calculated to further investigate the critical behavior, the Editor: B. Grinstein thermodynamic stability, and the phase transition of the black hole. It is evident that the rainbow gravity Keywords: determines the late fate of the black hole as leading to a remnant, the ﬁndings of this letter also show Rainbow gravity that a quantum correction to the metric reduces the remnant mass of the Schwarzschild black hole in Black hole thermodynamics comparison with the usual rainbow Schwarzschild black hole. Phase transition © 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 physical momentum at the Planck scale, or Lorentz invariance violation. The relativistic standard energy–momentum dispersion relation, The existence of minimum length scale deforms the dispersion E2 − p2 = m2 is the consequence of one of the most striking relation, and the theoretical framework that has the tendency to symmetries in nature, namely the Lorentz symmetry, which also modify the dispersion relation is the doubly special relativity the- holds in the infrared limit of the quantum gravitational regime. ory [25–30], which is the extension of the special relativity theory But various approaches suggest that in the ultra-violate limit, in such a way that Planck length, in any inertial reference frame, is the symmetry might be violated [1,2], implying a modification also an invariant quantity along with the light velocity. Neverthe- in the usual dispersion relation. It has been observed that such less, doubly special relativity is not an impeccable theory, where modification arises in a variety of occations, such as in discrete some approaches to the doubly special theory-particularly those space–times [3], models based on string field theory [4], cos- involving Hopf algebra, result in non-commutative space–times. mologically varying moduli scenarios [5], space–time foam [6,7], Again, the definition of the dual position space in the framework the semiclassical spin network calculations in loop quantum grav- suffers non-linearity in the momentum space. To fix these issues, ity [8,9], non-commutative geometry [10–12], ghost condensation doubly special relativity has been generalized to the curved space– [13], and Horava–Lifshitzˇ gravity [14,15]. Theories of quantum time – the doubly general relativity, otherwise widely known as gravity [16–21] possess an observer-independent minimum mea- the rainbow gravity [31], where a proposal has been introduced surable length scale, assumed to be the Planck length as a basis. that the geometry of space–time ‘runs’ with the energy scale at This minimal length works as a natural cutoff, which is expected to which the geometry is probed, i.e., the metric of the space–time resolve the known curvature singularities in general relativity. The varies with respect to the energy. Therefore, the modification re- theory of general relativity, based on a smooth manifold, breaks sults in a one-parameter family of metrics-like a rainbow of met- down when the space–time is probed at energies of the order of rics, instead of a single metric describing the space–time, depend- Planck energy [22,23]. Then it is natural to expect a radically new ing on the energy (momentum) of the test particle. Two principles formation of space–time theory, including the departure from the – the correspondence principle and the modified equivalence prin- standard energy–momentum dispersion relation [24], by redefining ciple govern the rainbow gravity. The gravity’s rainbow has been used to study inflation [32,33], and resolving the big bang singularity [34–36]. Some other fascinating investigations on rainbow E-mail address: [email protected]. gravity can be found in Refs. [37–46]. https://doi.org/10.1016/j.physletb.2018.07.032 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. Md. Shahjalal / Physics Letters B 784 (2018) 6–11 7

Over the last few decades, black hole thermodynamics has at- 2. A brief review on the rainbow gravity tracted a lot of physicists since the pioneering works of Bekenstein 2 2 and Hawking in the 1970s [47–50]. Later it was discovered that The modified dispersion relation has the form E F (E/E p) − 2 2 2 above certain temperature, pure thermal radiation in AdS space– p G (E/E p) = m , where E p is the Planck energy, and F(E/E p ) G time becomes unstable, and collapses to form black hole [51]. and (E/E p) are the correction terms-commonly known as the This is the well-known Hawking–Page phase transition between rainbow functions, which push the energy–momentum dispersion the Schwarzschild–AdS black hole and the thermal AdS space. This relation to the modification in ultra-violate limit. E and m are the phenomenon can be explained as the confinement/deconfinement energy and the mass of the probe particle, respectively. In this for- mulation, the two rainbow functions satisfy lim → F(E/E ) = phase transition of gauge field in the AdS/CFT correspondence [52]. E/E p 0 p 1, and lim → G(E/E ) = 1, so that the modified relation re- Since then phase transitions of various type of black holes have E/E p 0 p duces to the conventional one for a vanishing energy of the test been investigated from different perspectives. In the context of particle. Of the two governing principles of rainbow gravity, the rainbow gravity, the thermodynamic phase transition of the mod- correspondence principle can be stated as limE/E →0 gμν(E) = ified Schwarzschild black hole has been studied in Refs. [53,54]. p gμν , where the energy-dependent metric on the left is the rain- The effects of rainbow gravity on the thermal stability of charged bow metric, and that on the right is the general-relativistic metric. BTZ black hole [55,56], charged dilatonic black hole [57,58], Gauss– The modified equivalence principle demands a region of space– Bonnet black hole [59]havebeen analyzed recently. Thermody- time with curvature radius R such that R >> 1/E p , and declares namic stability, in massive gravity’s rainbow has been studied that freely-falling observers making measurements of particles and by Hendi et al. [60,61]. It has been observed that the energy- fields with energy E observe the laws of physics to be the same dependent space–time modifies the temperature of the black hole, so long as 1/R << E << E p . Therefore, they can be regarded as leading to a correction to the entropy of the black hole. The tem- inertial observers in rainbow space–time (upto the first order in perature starts to reduce after attaining a maximum value. Thus at 1/R). Their reference frames are a family of energy-dependent or- = F ˜ = a critical size, the temperature and the entropy vanish, leading to thonormal frames, locally given as e0 1/ E/E p e0, and ei ˜ a termination of the Hawking radiation – an indication in the ex- 1/G E/E p ei , where the tilde-quantities represent the energy- μν istence of a remnant mass [62,63]. It has been argued that such independent frame field. Then the flat metric is g(E) = η eμ ⊗ a remnant is subject to exist in case of any black object, and has eν , which varies with energy, and one gets a family of metrics. = been probed for Kerr, Kerr–Newman–dS, charged-AdS, higher di- These must satisfy the ‘rainbow Einstein equations’, Gμν(E) + mensional Kerr–AdS black holes, and for black Saturn in gravity’s 8π G(E)Tμν (E) gμν(E), where gravitational constant and the cosmological constant are dependent on energy. rainbow [64]. The rainbow functions, F(E/E ) and G(E/E ) have different In this letter, the phase transition and the stability of the p p forms emerged from different phenomenological motivations. Ac- quantum-corrected Schwarzschild black hole in rainbow gravity are n cording to Ref. [73], F(E/E p) = 1 and G(E/E p ) = 1 − η(E/E p) , analyzed. In other words, the investigations on quantum deforma- originated from loop quantum gravity and non-commutative ge- tion of the Schwarzschild solution by Kazakov and Solodukhin [65], ometry, where η and n are dimensionless model parameter and and thermodynamic phase transition of a black hole in rainbow positive integer, respectively, are denoted by MDR1 for short. gravity by Zhong-Wen Feng and Shu-Zheng Yang [54]areamal- αE/E p F(E/E p) = e − 1 / αE/E p and G(E/E p ) = 1, where α is gamated here. For that, the generalized uncertainty principle is the model parameter, appeared to explain the hard spectra of utilized. It has been shown that in gravity’s rainbow, even though γ -ray bursts at cosmological distances, are called MDR2. And, the space–time depends on the energy, the usual uncertainty prin- F(E/E p) = 1/ 1 − γ E/E p and G(E/E p) = 1, which provide a ciple still holds [42,53]. According to Refs. [66–70], a photon near constancy in the light speed in rainbow gravity when the energy of the horizon is used to determine the position of a quantum parti- photons increases, γ is the model parameter, are defined as MDR3. cle of energy E, which can be related to the energy of the particle In this letter, the rainbow functions in MDR3 has been used. emitted in Hawking radiation, and according to the argument in Ref. [71], the uncertainty principle p ≥ 1/x can be translated to 3. Quantum correction to the Schwarzschild metric the lower bound E ≥ 1/x, and that stands as E ≥ 1/x 1/rH, where the value of x has its minimum value to be the horizon- The fact that to avoid the singularity problem, a successful radius, and this is probably the most elegant choice of length scale quantization theory is required is a cliché. Quantum corrections in the context of the geometry in the vicinity of the horizon. Be- are subject to a complete changed gravitational equations and the cause the space–time is probed at the lower-bounded energy E, space–time geometry at the Planck scale. The semi-classical ap- proach [74]results in some parts of the gravitational degrees of it can be regarded as the energy of the rainbow gravity. The im- freedom as classical [75], while other parts are describable by ex- pact of the generalized uncertainty principle on the Schwarzschild actly solvable quantum theory. The deformed metric due to the black hole thermodynamics has been examined in Ref. [66], and sph quantum excitation has the form [65] gˆ = g + h , where for many other type of black holes in Refs. [23,69,72]. μν μν μν sph The rest of the letter has such organization: in Sec. 2, a short gμν is the spherically symmetric term containing no propagating note on the gravity’s rainbow has been provided. Sec. 3 has briefly mode, and hμν is the non-spherically symmetric perturbative devi- discussed the quantum correction to the Schwarzschild metric. Ac- ation term with propagation modes, which is assumed to be small. tually, these two sections are for maintaining the self-consistency According to Ref. [65], the lapse function of the quantum- of this article. Then in Sec. 4, the central topics – the temperature, corrected Schwarzschild metric is the heat capacity, and the off-shell free energy of the quantum- r 2GM 1 corrected Schwarzschild black hole in the rainbow gravity have (r) =− + U(ρ)dρ r r been calculated, and the stability along with the phase transition √ have been scrutinized. Finally, Sec. 5 has contained some discus- 2GM r2 − a2 sion with concluding remarks. =− + , r r 8 Md. Shahjalal / Physics Letters B 784 (2018) 6–11 = −ρ −2ρ − 4 where the renormalized potential U (ρ) e / e π G R , G R = G ln(μ/μ0), G is the universal gravitational coupling con- 2 stant, μ is the scale parameter, and a = 4G R /π . In case of U (ρ) = 1, the usual Schwarzschild metric is recovered. The radial coordinate is restricted to r > a, and the event horizon has the 2 2 radius rH = (2GM) + a .

4. Thermodynamics of the quantum corrected rainbow Schwarzschild black hole

From discussions in Sec. 2, the transformations of the infinites- imal coordinate elements to the rainbow gravity can be found as i i dt → dt/F(E/E p) for temporal axis, and dx → dx /G(E/E p) for all spatial axes. Then the metric of the quantum-corrected rainbow Schwarzschild black hole is 1 1 − ds2 =− (r)dt2 + 1(r)dr2 2 2 F (E/E p) G (E/E p) 1 = = = + r2d 2, (1) Fig. 1. Rainbow local temperature versus mass, setting r 20, G 1, γ 1.7, and 2 = G (E/E p) a 1. =−2GM + r2−a2 2 = 2 + 2 2 where (r) r r , and d dθ sin θdθ is the hole along with the rainbow parameter γ . The lower bound in the metric over 2-sphere. It should be specified that the energy E is mass is the so-called remnant mass of the black hole in gravity’s not the ADM mass of the space–time, rather the energy of the rainbow. For the quantum-corrected rainbow Schwarzschild black = = 2 − 2 probes measured at infinity by a freely falling inertial observer. hole, the critical or remnant mass is Mcr M√0 γ G a /2G, = The Hawking temperature of the quantum-corrected Schwarzschild and the critical rainbow parameter is γcr a/ G. Comparing with = = 1 − 1 ∇μ ν∇ = the remnant mass of the usual rainbow Schwarzschild black hole black hole is [76] TH κH/2π 2π 2 ξ μξν = = (a 0), it can be found that a quantum deformation √indeed de- r rH creases the remnant mass, i.e., γ 2G − a2/2G < γ /2 G. More- 1 4 r2 − a2, where is the surface gravity of the black / π H κH over, this deformation imposes a bound on the rainbow parame- hole, and ξμ are the time-like Killing vectors. Then in the rain- ter γ . bow gravity, the Hawking temperature of the quantum-corrected Fig. 1 shows the relationship between the local temperature Schwarzschild black hole is and the mass of the quantum-corrected rainbow Schwarzschild √ black hole, for a fixed value of the rainbow parameter γ = 1.7, the G E E 2GM 2 + a2 − G RG ( / p) ( ) γ quantum correction term a = 1, the gravitational constant G = 1, T = TH = , (2) H F 2 (E/E p) 8π GM (2GM) + a2 and the radial distance r = 20. As the mass decreases from M3 to √ M2, the local temperature also decreases, then from M2 to M1, the = 2 + 2 = where E 1/rH 1/ (2GM) a , and E p 1/ G in natural temperature raises inversely where the mass decreases. At M1, the units have been used. Considering a = 0, the temperature of the black hole attains a local maximum in temperature, then sharply RG 1 γ 2 2 usual rainbow Schwarzschild black hole T = 1 − √ goes for nullity at the mass M0 = γ G − a /2G, which is the H 8π GM 2 GM [54]can be recovered. Using the first law of black hole ther- remnant mass. The red solid line in Fig. 2 indicates the local temperature of the modynamics, the entropy of the quantum-corrected rainbow quantum-corrected rainbow Schwarzschild black hole, where the Schwarzschild black hole can be obtained as blue solid line represents that temperature for the usual rainbow −1 RG = RG Schwarzschild black hole. It is obvious that as the mass decreases, S TH dM both types of black hole show almost a similar temperature for a √ fixed mass value, until the local maximum point in the plot. And 2π 2 = γ G ln (2GM)2 + a2 − γ G G then on, the quantum corrected black hole decreases its temper- √ ature more quickly than its counterpart, upto a zero temperature. + γ G (2GM)2 + a2 + 2G2 M2 . (3) Fig. 2 also shows that a quantum correction diminishes the remnant mass of the black hole in comparison with the usual black RG = hole. For√ a null√ quantum correction, √ the entropy is S 4 GMπ GM + γ + 2πγ2 ln 2 GM − γ . To evaluate the Next, the local energy of the quantum-corrected rainbow Schwarzschild black hole can be calculated explicitly using the first phase transition of the black hole at a finite surface r > a, the law of black hole thermodynamics as local temperature [77,78] has to be calculated, which is S RG G(E/E p) T T RG = √H RG = RG RG local Elocal TlocaldS F(E/E p) √ √ S 0 r γ G √ = √ 1 − . (4) r 8π GM r2 − a2 − 2GM (2GM)2 + a2 = r2 − a2 − γ 2G − a2 G From the condition that the temperature has to be a positive quan- RG − r2 − a2 − 2GM , (5) tity, Tlocal imposes a lower bound on the mass M of the black Md. Shahjalal / Physics Letters B 784 (2018) 6–11 9

Fig. 4. Relationship between free energy of quantum-corrected rainbow Fig. 2. Comparison in the local temperature between the usual rainbow Schwarzschild black hole and local temperature, setting r = 20, G = 1, γ = Schwarzschild black hole and the quantum-corrected rainbow Schwarzschild black 1.7, and a = 1. hole, setting r = 20, G = 1, γ = 1.7, and a = 1. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.) stability. Therefore, the quantum-corrected rainbow Schwarzschild black hole has two stable regions and one unstable. The stable regions are [M0, M1], [M2, M3], and the unstable region is [M1, M2]. Clearly, the small black hole (SBH) and the large black hole (LBH) are stable, while an intermediate black hole (IBH) state is unstable, which quickly decays into the SBH or the LBH state. The off-shell free energy of the black hole in an isothermal cavity is defined as Foff = Elocal − Tlocal S. Substituting the relevant expressions from Eq. (3), Eq. (4), and Eq. (5), the quantum-corrected rainbow free energy of the Schwarzschild black hole can be found as √ r F RG = r2 − a2 − γ 2G − a2 − r2 − a2 − 2GM off G √ √ r γ G − √ 1 − 2 2 − 2 − (2GM)2 + a2 4G M r a 2GM √ 2 × γ G ln (2GM)2 + a2 − γ G Fig. 3. Heat capacity versus mass of the quantum-corrected rainbow Schwarzschild √ black hole, setting r = 20, G = 1, γ = 1.7, and a = 1. + γ G (2GM)2 + a2 + 2G2 M2 . where T RG = dM/dSRG is used, and the integration is taken over H Existence of black hole remnants demands the consideration of the interval [M , M], M is the critical or remnant mass of the 0 0 the hot curved space (HCS) instead of the hot flat space – abox black hole. The stability analysis in thermodynamics is the issue of → HCS = filled with thermal graviton, such that M M0, Foff 0[76]. the heat capacity. Using the temperature expression in Eq. (4), and In Fig. 4, the off-shell free energies of the HCS and the three the energy expression in Eq. (5), the quantum-corrected rainbow black hole states have been shown against the local temperature. heat capacity can be read off at a fixed isothermal cavity distance T0 = 0is the temperature corresponding to the remnant mass r as M . The inflection point T is the transition temperature between 0 1 SBH and IBH, and T2, between IBH and LBH. Three Hawking–Page RG = RG RG 1 2 3 RG C ∂ Elocal/∂ Tlocal type critical points are T , T , and T . As F demonstrates the r c c c off √ characteristic ‘swallow tail’ behavior, a first-order phase transition 8π GM2 r2 − a2 − 2GM First-order = point Tc can be found. It is a well-known fact that the √ . RG 5/2 2 2 2 state with lower value in F is more probable to exist. There- 4G γ M r −a −2GM r2−a2−3GM γ G off − 1− 1 2 3/2 2 2 fore, for T0 < T < T , the HCS is more probable than the three (2GM) +a2 2− 2− (2GM) +a c r a 2GM black holes. In the region T < T < T First-order, the IBH and the (6) 0 c LBH decay into the SBH, and the IBH and the SBH collapse into the = First-order 1 1 2 2 3 Fig. 3 shows that the heat capacity vanishes before M 0, LBH for Tc < T < Tc . Then for Tc < T < Tc , Tc < T < Tc , 3 implying the existence of a remnant of the quantum-corrected and Tc < T < T1, the LBH is more likely to exist; so that the sta- rainbow Schwarzschild black hole. If a black hole has its heat ca- ble SBH and the unstable IBH collapse into the LBH in the whole 1 pacity as positive, then it is in a stable state with the outside Tc < T < T1 region. These scenarios are exactly similar with the space–time region, and a negative heat capacity indicates the un- usual rainbow Schwarzschild black hole [54]. 10 Md. Shahjalal / Physics Letters B 784 (2018) 6–11

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