Correlations between emission events in Rainbow Gravity
CCorrelations between emission events in RainbowGravity
D. A. Gomes, F. C. E. Lima, C. A. S. Almeida
Universidade Federal do Cear´a (UFC), Departamento de F´ısica, Campus do Pici,Fortaleza - CE, C.P. 6030, 60455-760 - Brazil
Abstract
In this work, we study emission correlations in Rainbow Gravity (RG)black holes known to have black hole remnants. Non-thermal correctionsare responsible for generating non-zero correlations between particle emis-sion events during the black hole evaporation process. With this in mind,we calculate the temperatures considering back-reaction effects. Then, weobtain the correlations between emission events for a RG metric proposedby Magueijo and Smolin. We also discuss through a numerical analysis theemission correlations behavior in the last stages of evaporation of the blackhole.
Keywords:
Rainbow Gravity, Black hole remnants, Correlation
1. Introduction
Due to experimental evidence [1] and the efficiency of theoretical results,new researchers and admirers have become adept at studying cosmologi-cal objects such as black holes. These objects are structures predicted byEinstein’s theory of relativity. Initially, with the arising of this theory, manyresearchers believed that black holes were structures that only absorbed mat-ter. However, around the 1970s, the well-known physicist Steven Hawkingchanged the paradigms of the theory and demonstrated that black holes emitradiation, which was named after him. Hawking’s theory has shown that due
Email addresses: [email protected] (D. A. Gomes), [email protected] (F. C. E. Lima), [email protected] (C. A. S.Almeida)
Preprint submitted to Elsevier March 3, 2020 a r X i v : . [ phy s i c s . g e n - ph ] F e b o quantum effects, black holes emit radiation at temperatures proportionalto their surface gravity. This seemingly strange result opened new horizonsfor the study of these amazing and interesting structures.Since the 1970’s, many scholars have tried to understand and accuratelydescribe such objects. We can easily see this from the works in the literature.For instance, in 1973, Bardeen, Carten, and Hawking [2] attempted to dis-cuss in the paper The Four Laws of Black Hole Mechanics the laws governingblack hole dynamics and their analogies with the laws of thermodynamics.Meanwhile, as early as 1973, Bekenstein sought to understand and interpretthe study of black hole entropy. Subsequently, Bekenstein himself seeks ageneralization to the second law of thermodynamics in order to give an accu-rate description of such structures. Building on many of these theories thatemerged in the mid-1970s, new studies and new lines of research eventuallyemerged over time. These studies include: information loss and entropy con-servation in quantum corrected Hawking radiation [3]; the study of Hawkingradiation with logarithmic correction tunneling [4]; the study of quasinormalfrequencies of self-dual black holes [5], etc.One new area of research in black hole theory involves the violation of theLorentz Symmetry. Such symmetry is fundamental for two of the most suc-cessful theories in the last century: General Relativity and Standard Model.It has been one of the most tested symmetries of Nature, and to the better ofour knowledge there is no experimental evidence disproving its validity [6, 7].In spite of this, Lorentz symmetry breaking could reveal interesting aspectsof nature. For instance, it is believed that a fundamental theory involvingGeneral Relativity and Quantum Mechanics would hold at the Planck scale.However, it is not possible to test this hypothesis with current technology.One way to address this problem is to identify Planck suppressed signals ofthis underlying theory at low energy scales, via Lorentz Symmetry Violation[8, 9].One of the approaches to violate the Lorentz invariance is to modify thestandard dispersion relation E − p = m , which is valid in the ultravioletlimit [10, 11, 12]. This leads to the so-called modified dispersion relations(MDR), which are often associated with the existence of a maximum energyscale. In this context, the special relativity can be extended to include MDR,leading to the Doubly Special Relativity (DSR) [13, 14, 15]. The term doublycomes from the fact that DSR has two universal constants: the speed of light c and the Planck energy E P .The extension fo DSR in curved spaces, proposed by Magueijo and Smolin,2s called Doubly General Relativity or, more popularly, Gravity’s Rainbowand Rainbow Gravity [16]. In Rainbow Gravity, the spacetime is energy-dependent, i.e., particles with different energies would perceive the spacetimebackground differently. In this sense, we have a ”rainbow” of metrics definedby one parameter, the ratio between the energy of the test particle and thePlanck energy ( E/E P ). In addition to this, the MDR in Rainbow Gravityare modified by correction terms, known as rainbow functions, that dependon E/E P .Many interesting applications of Rainbow Gravity can be found in variouscontexts as inflation [17, 18], branes [19], wormholes, avoidance of the bigbang singularity [21, 22, 23], explanation for the absence of black hole detec-tion at LHC [24], among others. However, one of the most fruitful contextsin which Rainbow Gravity has been applied is black hole thermodynamics[25, 26, 27, 28, 29, 30, 31, 32], specially due the existence of the so calledblack hole remnants [33, 34, 35].The black hole remnants are particularly interesting because they repre-sent a possible solution for the information paradox. With this in mind, wewill work with the rainbow functions proposed by Magueijo e Smolin [15],which are known to produce black hole remnants [32]. This work aims toinvestigate the correlation between two successive particle emission eventsconsidering back-reaction effects.This paper is organized into four sections, starting by discussing the ther-modynamic properties for a black hole in a rainbow gravity scenario. In thenext section we investigate the study of correlations between particle emis-sion events. To conclude, we make a numerical analysis of the theoreticalresults presented in the previous sections and present the physical resultsfound.
2. The thermodynamic properties for the black hole in rainbowgravity
Motivated by the work of Magueijo and Smolin [16] we used the dispersionrelation given by E (1 − γE/E p ) − p = m , (1)3hich leads to the following metric: ds = (cid:18) − γ EE p (cid:19) (cid:18) − Mr (cid:19) dt − (cid:18) − Mr (cid:19) − dr − r ( dθ + sin θdφ ) . (2)Now, we will obtain the temperature and entropy for the metric above.For this purpose, we will make use of the tunneling method based on theworks [36, 37, 38, 39, 40]. For a non-rotating black hole, we can write themetric as follows ds = − f ( r ) dt + g ( r ) dr + h ( r )( dθ + sin θdφ ) . (3)The temperature for the black hole can be easily found via the expression T = (cid:112) f (cid:48) ( r + ) g (cid:48) ( r + )4 π , (4)where r + is the event horizon radius, while the entropy is obtained from therelation T dS = dM .In our case, we have f ( r ) = − (cid:18) − Mr (cid:19) (cid:18) − γ EE p (cid:19) ; g ( r ) = − (cid:18) − Mr (cid:19) . (5)Since the event horizon is the same as Schwarzschild’s, r H = 2 M , wearrived at the result: T = (cid:18) − γ EE p (cid:19) T , (6)where T = (8 πM ) − is the Schwarzschild temperature.Since Hawking radiation emission is a quantum process, the quanta emit-ted must obey the Heisenberg uncertainty principle, which must be valid inGR [41, 42], so that ∆ x ∆ p (cid:62)
1, in natural coordinates. We can obtain fromthe uncertainty principle a minimum energy value E (cid:62) / ∆ x , where E isthe particle energy emitted in the Hawking radiation process. Near the eventhorizon we have ∆ x ≈ r H = 2 M [34]. Then, E (cid:62) / M. (7)4eeping in mind that in natural coordinates we have E p = 1, we canrewrite the surface gravity and black hole temperature as [32] T = (cid:16) − γ M (cid:17) T . (8)From equation (8), it is easy to see that the rainbow γ parameter isresponsible for modifying temperature of the Schwarzschild black hole and,consequently, its surface gravity. In addition, the usual results are recoveredat the limit γ →
0. Note that since γ >
0, the obtained temperature islower than in the usual Schwarzschild case, as we will see numerically inlater sections.The entropy for this model is given by [32] S = 16 π (cid:90) M M − γ dM = S + 2 πγ [2 M + γ ln(2 M − γ )] , (9)where S = 4 πM is the Schwarzschild entropy.We can notice that S > S , in agreement with the decrease in tempera-ture. Furthermore, we see that at the limit γ →
0, the Schwarzschild entropy S is recovered, especially for large values of M , as shown in the numericalresult in section 4. We will also see that the entropy S becomes much largerthan S when M is approximately of the order of γ and that for large valuesof M , S does not differ much from S .
3. Correlation between emission events
We will now calculate the temperature back-reaction correction and thecorrelation between emission events for the Schwarzschild metric proposedby Magueijo and Smolin [16]. The self-gravitational effects in the quantumtunneling formalism were investigated in Refs [43, 44].We must remember that S ( M ) = 4 πM + 2 πγ [2 M + γ ln(2 M − γ )] . (10)Consequently, the entropy for the black hole after the emission of a particlewith energy ω is given by S ( M − ω ) = 4 π ( M − ω ) + 2 πγ [2( M − ω ) + γ ln(2( M − ω ) − γ )] . (11)5herefore, the variation of the entropy before and after the emission is∆ S = 4 πω ( ω − M ) + 2 πγ (cid:20) − ω + γ ln (cid:18) M − ω ) − γ M − γ (cid:19)(cid:21) . (12)Since the probability Γ of tunneling a particle with energy ω is given byexp[∆ S ] [44], and using the relation Γ = e − k B β , the corrected temperaturewill be given by T = (cid:20) π (2 M − ω + γ ) − πγ ω − ln (cid:18) M − ω ) − γ M − γ (cid:19)(cid:21) − , (13)where we made k B = 1. Note that when the rainbow parameter tends to zero( γ → T = [8 π ( M − ω/ − , as expected.The correlation between emission events is given by C ( ω + ω ; ω , ω ) = ln Γ( M, ω + ω ) − ln[Γ( M, ω )Γ( M, ω )] . (14)For our case, we haveΓ( M, ω + ω ) = exp (cid:26) π [ − (2 M + γ )( ω + ω ) + ( ω + ω ) ]+ 2 πγ ln (cid:20) M − ω − ω ) − γ M − γ (cid:21)(cid:27) . (15)Therefore, C = 8 πω ω + 2 πγ ln (cid:26) [2( M − ω − ω ) − γ ][2 M − γ ][2( M − ω ) − γ ][2( M − ω ) − γ ] (cid:27) . (16)Note that, in the limit γ →
0, we have recovered the Schwarzschild blackhole correlation C = 8 πω ω . Also, for small values of γ , the correlations C and C become approximately equal as the mass M becomes very large.
4. Numerical Results and Discussions
In this section, we will turn our attention to the study of numerical andgraphical results for a black hole in the rainbow gravity scenario. In ourresults, we will consider only values of mass greater than γ/
2, which corre-sponds to the remnant radius of the black hole. Using the analytical results6 M T Γ =0.5 Γ =1.0 Γ =1.5 Γ =2.0 M T (cid:144) T Figure 1: Behavior of temperature and temperature ratio for various values of the rainbowparameter. expressed in (1) and (2), we immediately obtain the results presented in therespective Fig. [1] and Fig. [2].From Fig. [1] we clearly notice that the temperature tends to zero when M → γ/
2. The same behaviour is observed for
T /T . Meanwhile, the tem-perature value is maximal in M = γ and tends to zero as M → ∞ , inde-pendently of the rainbow parameter γ . This is exactly what is expected fora Schwarzschild black hole and happens due to the fact that Schwarzschild’stemperature T drops faster than T . From this, we can clearly see that therainbow parameter directly influences the behavior of the thermodynamicproperties of the model.For the entropy, in agreement with the behavior of the temperature, werealized that, as showed in Fig. [2], the entropy S tends to zero before Mbecomes γ/
2. Along with this behaviour of the entropy, the heat capacity isalso zero for M = γ/
2, as shown in Ref. [32]. This means that the black holestops radiating for this value of mass, which indicates that the model has aremnant. Also, as the temperature decreases, the model entropy increasesmonotonically, as expected.We now turn our attention to the behavior of the entropy variation for7 =0.5 Γ =1.0 Γ =1.5 Γ =2.0 M S M S (cid:144) S Figure 2: Behavior of entropy and entropy ratio for various values of the rainbow param-eter. various values of the rainbow parameter. The results obtained are presentedin the Figs. [3] and [4]. For a better understanding of our results, the nextgraphic begin in M = 0. Analyzing the entropy variation as a function of themass for several rainbow parameter values, we notice that when M >> S → M → + ∞ . As can be observed from Fig. [3], the back-reaction effectis considerable only for the γ < M < γ region. Also, it is worth to mentionthe existence of a formation law for entropy variation as a function of therainbow parameter, given by∆ S ( M ) ∝ tan (cid:18) M πγ (cid:19) , para 0 < M < γ, (17)for all γ .In order to understand how the energy variation of the emitted particleschanges the entropy variation and, consequently, the tunneling probability,we will investigate how ∆ S behaves when ω varies. Keeping this in mind,we obtain the graphical result shown in Fig. [4]. From this figure, we canobserve that the entropy variation and the tunneling probability become more8 .0 0.5 1.0 1.5 2.0 (cid:45) (cid:180) (cid:45) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) M (cid:68) S (cid:72) M (cid:76) (cid:45) (cid:180) (cid:45) (cid:45) (cid:180) (cid:45) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) Γ =0.5 Γ =1.0 Γ =1.5 Γ =2.0 Figure 3: Entropy variation as a function of mass for several values of rainbow parameterand particle with same energy. significant for particles with higher energies. Furthermore, as M → γ/ ω gets higher and higher as the weapproach the remnant state of the black hole.Finally, we analyze the graphical behavior of the correlation functionof the emitted particles shown in fig. [5]. If the emitted particles havethe same energy, the correlation will always be well located around M = γ/
2. This effect is not observed for particles with different energies since,in this case, the correlation diverges for M → γ/
2. We also observe thatthe correlation increases as we increase the value of the rainbow parameter.This indicates that the rainbow parameter plays an important role on thecorrelation between emitted particles.9 (cid:61) (cid:180) (cid:45) Ω (cid:61) (cid:180) (cid:45) Ω (cid:61) (cid:180) (cid:45) Ω (cid:61) (cid:180) (cid:45) Γ =2.0 M (cid:68) S (cid:72) M (cid:76) Figure 4: Entropy variation as function of mass for a constant rainbow parameter andparticles with different energies. \ Γ =0.5 Γ =1.0 Γ =1.5 Γ =2.0 M C o rr e l a ti on Figure 5: The correlations of particle emission. . Concluding remarks Motivated by the study of black holes in the Rainbow Gravity scenario,we studied analytically and numerically the behavior of the thermodynamicproperties such as temperature, entropy and particle emission correlationas function of a rainbow parameter. For this purpose, we worked with therainbow model proposed by Magueijo and Smolin.From the numerical results, we can have an idea of how meaningful therainbow parameter is for the quantities obtained. It not only modifies thesequantities, but also contributes with new consequences, specially when itcomes to the correlation between emitted particles. As observed in the lastsection, these consequences are particularly significant in the limit M → γ/ M >>
1, which indicates that therainbow gravity effects are more perceptible near the last stages of evapora-tion of the black hole.Notably, the correlation has its maximal value in M = γ/
2, suggestingthat the particles emitted from the black hole have highly correlated momentsbefore the black hole stops radiating. This is a striking result because, sincethe Rainbow Gravity restrains the complete evaporation of the black hole,we would expect the emitted particles to be uncorrelated, allowing the blackhole information to be stored in its remnant. We may conclude from thisthat the Rainbow Gravity allows some of the black hole information to flowduring the evaporation process, while maintaining the rest of it stored inits remnant. How this information is carried away from the black hole orwhether these results are valid for other Rainbow models would require moreinvestigation.
Acknowledgments
The authors thank the Coordena¸c˜ao de Aperfei¸coamento de Pessoal deN´ıvel Superior (CAPES), and the Conselho Nacional de DesenvolvimentoCient´ıfico e Tecnol´ogico (CNPq) for financial support. CASA thanks CNPQfor his grant No. 308638/2015-8.
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