Hawking radiation from the holographic screen Ying-Jie Zhao 1 ∗ 1Institute of Theoretical Physics, Beijing University of Technology, Beijing 100124, China 7 1 0 2 b e Abstract F 5 1 In this paper we generalize the Parikh-Wilczek scheme to a holographic screen in the framework of the ultraviolet self-complete quantum gravity. We calculate that the ] c tunneling probabilities of the massless and massive particles depend on their energies of q - the particles and the mass of the holographic screen. The radiating temperature has r g not been the standard Hawking temperature. On the contrary, the quantum unitarity [ principle always remains. 3 v 3 PACS Number(s): 04.60.-m, 04.70.Dy, 11.10.Nx 0 2 1 Keywords: Quantum gravity, quantum tunneling, Hawking temperature 0 . 1 0 7 1 : v i X r a ∗E-mail address: yj [email protected] According to the no-hair theorem, a black hole is generated from a celestial body’s collapse can becompletely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum. In the classical Einstein gravitational theory, other information of the celestial body is confined inside the black hole. However, in 1974 Hawking[1, 2] recovered that a black hole is not really black previously thought but radiates energy due to the quantum effects near the event horizon, and so that the black hole has a spectrum of a black body. The pure black body spectrum has no information and the total evaporation of the black hole give rise to the information loss paradox, which breaks the unitarity of quantum theory. In 1999, Parikh and Wilczek [3, 4] gave a semiclassical derivation of Hawking radiation as a tunneling process. They argued that with the continuous radiation of particles the energy of the black hole decreases and the contraction of radius of the horizon make the particles get across the classically forbidden trajectory, in other words, the potential barrier is created by the self-gravity of the system. They introduced a special coordinate system where the metric is not singular at the horizon, and then used the WKB method to compute the tunneling rate. In that article they worked out the emission rate of the massless and uncharged particles radiating from the Schwarzchild and R-N black holes, and calculated the Hawking temperature bycomparingtheexponentialpartoftheemissionratewithaBoltzmannfactorwhenneglecting the quadratic term of the energy. Inrecent years, theParikh-Wilczek methodhasbeengeneralizedtothemorecomplexcases. One case is that particles have mass or charge, or both, another case is that the spacetime is no longer so simple. For example, the noncommutativity idea [5] originating from the ultraviolet divergence elimination in loopquantum theory[6] has been introduced anda great many papers has been published[7]. All the noncommutative spacetimes predict the existence of a minimal length of the order of the Planck scale. Unfortunately, many sorts of spacetimes constructed depend on external noncommutativity parameters. In 2012, P. Nicolini and E. Spallucci[8, 9] derived a static, neutral, non-rotating black hole metric whose extremal configuration radius is equal to the Planck length in order to avoid introducing an additional principle to justify the existence of a minimal length to provide a UV cut off. Below the sub-Planckian scale the interior of the black hole loses any physical meaning thus no singularity in the origin. The authors named the particular black hole as holographic screen. Moreover, they discussed the thermodynamics of the holographic screen and pointed out that the area law is corrected by a logarithmic term anda minimal holographic screen corresponding to the zero entropy existed. Our aim is to generalize the Parikh-Wilczek method to the tunneling process of massless anduncharged particles fromholographicscreen. Our article isarrangedas follows. In Sections 2, a brief introduction about holographic screen is given. In section 3, we work out the tun- neling rate of the massless particles tunneling through the horizon of the holographic screen. 1 In addition, we derive the temperature of the holographic screen and study the extremal case when its mass are very large. In section 4, we discuss the massive particles tunneling through the horizon of the holographic screen. Finally in section 5, we present a conclusion. 1 A self-regular holographic screen The mass density of a point particle with mass M in spherical coordinates is proportional to the Delta function M ρ(r) = δ(r), (1) 4πr2 which can be generalized to a derivative of a smooth function h(r) [?] M d ρ(r) = h(r) ≡ T0 (2) 4πr2dr 0 to overcome the problem that at the sub-Planckian energy regime the Compton wave length of aparticleislargerthanaSchwarzschild blackholewiththesamemass. Takingtheconservation equation ∇ Tµν = 0 into consideration the stress tensor takes the form µ r r Tν = diag −ρ,p ,p + ∂ p ,p + ∂ p (3) µ r r 2 r r r 2 r r (cid:16) (cid:17) with p = −ρ. By substituting eq.(3) into Einstein equation and assuming that the form of the r lefthandsideoftheEinsteinequationremainsunchangedwegetthemetric(gravitational constantG = L2) p 2L2m(r) 2L2m(r) −1 ds2 = − 1− p dt2 + 1− p dr2 +r2dΩ2, (4) r r (cid:18) (cid:19) (cid:18) (cid:19) where the parameter m(r) takes the form r m(r) = 4π dr r 2ρ(r ). (5) ′ ′ ′ Z0 The particular form of the function h(r) must satisfy two rules[?]: i). Spacetime in the sub- Planckianregimehasnophysicalmeaning; ii). Thecharacteristicscaleofthesystemisprovided by the spacetime itself of the scale of the Planck length, not imposed as a external parameter. The most natural and algebraically assumption can be written as r2 h(r) = , (6) r2 +L2 p 2 where L is the Planckian length, and thus the smeared energy density ρ(r) is p ML2 ρ(r) = P . (7) 2 2πr r2 +L2 p Now we have found the modified metric o(cid:0)f hologr(cid:1)aphic screen 2L2m(r) 2L2m(r) −1 ds2 = − 1− p dt2 + 1− p dr2 +r2dΩ2 r r (cid:18) (cid:19) (cid:18) (cid:19) 2ML 2r 2ML 2r −1 = − 1− P dt2 + 1− P dr2 +r2dΩ2. (8) r2 +L 2 r2 +L 2 (cid:18) p (cid:19) (cid:18) p (cid:19) 2 The Parikh-Wilczek Tunneling Mechanism and mass- less particles Now we investigate the quantum tunneling through the holographic screen via Parikh-Wilczek Tunneling mechanism. To study the quantum tunneling through the holographic screen we considering the modified metric of the holographic screen 2ML 2r 2ML 2r −1 ds2 = − 1− P dt 2 + 1− P dr2 +r2dΩ2. (9) r2 +L 2 s r2 +L 2 (cid:18) p (cid:19) (cid:18) p (cid:19) The holographic screen admits two horizons provided M > L p r = ML2 ±L M2L2 −1 (10) p p p ± q determined by 2ML 2r 1− P ± = 0. (11) r2 +L 2 p At the beginning, it is necessary to choose the Painlev´e coordinate that having no singularity at the horizon. The suitable choice can be written as [10] (r2 +L 2)(r2 +L 2 −2ML 2r) p p p dt = dt± dr. (12) s q r2 +L 2 −2ML 2r p p where t is the Painlev´e time. After the above transformation the Painlev´e line element reads 2ML 2r 2ML 2r ds2 = − 1− p dt2 +2 p dtdr+dr2 +r2dΩ2. (13) (cid:18) r2 +Lp2(cid:19) sr2 +Lp2 3 The radial null geodesics are calculated as dr 2ML2r r˙ = = ±1− p , (14) dt r2 +L2 s p where the plus (minus) sign corresponding to outgoing (ingoing) geodesics. Nowweconsideramasslessparticleradiatingfromtheholographicscreenasamasslessshell through its horizon. In the Parikh-Wilczek tunneling mechanism, the effect of self-gravitation the particles tunnel out of a holographic screen and its energy decreases because of the total energy conversation, which makes the mass of the holographic screen decline and the horizon of the black hole shrink smaller. Naturally, the metric of the holographic screen must be edited. Here we fix the total mass M and denote the energy evaporating from the holographic screen as ω and use M → M −ω, the spacetime metric can be written as 2(M −ω)L 2r 2(M −ω)L 2r ds2 = − 1− p dt2 +2 p dtdr+dr2 +r2dΩ2, (15) (cid:20) r2 +Lp2 (cid:21) s r2 +Lp2 and in the same way, the radial null geodesics are calculated as dr 2(M −ω)L2r r˙ = = ±1− p . (16) dt s r2 +L2 p The characteristic length of the massless particle described as a spherically symmetric massless shellisinfinitesimalnearthehorizononaccountoftheinfiniteblue-shift, hencethewavenumber inclines to infinity. In accordance with the WKB method, the imaginary part of the action that an s-wave massless particle traveling on the radial null geodesics tunnel across the outer horizon r from the initial position r = ML2 + L M2L2 −1 to the final position r = + in p p p out (M −ω)L2 +L (M −ω)2L2 −1 can be exhibited as follows p p p p q rout rout pr ImI = Im p dr = Im dp dr (17) r r′ Zrin Zrin Z0 with the Hamilton equation dH dp = , (18) r′ r˙ and H = M −ω . We can deform the contour of the r integral around the pole at the horizon ′ in order to ensure the positive energy solutions decay in time (choose the lower half ω plane), ′ 4 so the imaginary part of the action is worked out as ω rout dr ImI = −Im dω ′ Z0 Zrin 1− 2(Mr2−+ωL′)2L2pr p q ω 2(M −ω′)L2p (M −ω′)Lp + (M −ω′)2L2p −1 = π dω h q i ′ Z0 (M −ω′)2L2p −1 q = πω(2M −ω)L2 +πL M M2L2 −1−πL (M −ω) (M −ω)2L2 −1 p p p p p ML + M2qL2 −1 q +πln p p (19) (M −ω)L +p(M −ω)2L2 −1 p p q The tunneling rate is Γ ∼ exp(−2ImI). (20) Expanding Γ with respect ω 4πω L2M+ L3pM2 2πω2 L2 L5pM3 + 2L3pM Γ ≈ e− " p (L2pM2−1)1/2#− "− p−(L2pM2−1)3/2 (L2pM2−1)3/2#, (21) and comparing the first order of ω with the Boltzmann factor e ω, we obtain the temperature −T of the holographic screen M2L2 −1 T = p , (22) 4πML2 M2L2 −1+ML p p p p which demonstrates the temperature of(cid:0)tphe holographic scre(cid:1)en is no longer equivalent to the Schwartzschild metric’s. We depict the the curves of the radiation temperature T versus the mass M in Figure 1 for comparisons. The entropy has a logarithmic correction, M dM S = = 2π L2M2 −1+L M L2M2 −1+ln L M + L2M2 −1 . (23) ZL−p1 T h p p q p (cid:16) p q p (cid:17)i The difference of the entropy after and before the radiation ∆S = S(M −ω)−S(M) = 2π (ω −2M)ωL2 +(M −ω)L (M −ω)2L2 −1−ML L2M2 −1 p p p p p h q q (M −ω)L + (M −ω)2L2 −1 p p +ln . (24) ML +qM2L2 −1  p p p  We depict the the curves of the entropy S versus the mass M in Figure 2 for comparisons. M ≫ L 1, the action can be simplified as −p 5 Figure 1: The radiation temperature T versus the mass M. The black curve stands for the temperature of Schwartzschildblackhole,andthe redcurvestandsforthe temperature ofholographicscreen. Hereweuse the Planck units. Figure 2: The entropy S versus the mass M. The black curve stands for the entropy of Schwartzschild black hole, and the red curve stands for the entropy of holographic screen. Here we use the Planck units. 6 M ImI ≈ 2πω(2M −ω)L2 +πln . (25) p M −ω And the tunneling rate is Γ ∼ e−8πML2pω−4πω2(2/M2−L2p). (26) The temperature degenerates to the conventional results of the Schwartzschild black hole, 1 T ≈ , (27) 8πML2 p while the entropy reduces to S ≈ 4πL2M2 +2πln(2L M). (28) p p 3 Massive particles In the section we discuss massive particles radiating from the holographic screen as a shell through its horizon. A massive particle because of the particles have which makes the mass of the holographic screen decline and the horizon of the black hole shrink smaller. A massive particle has no longer moved along a null geodesics but a time-like trajectory determined by the Lagrangian, m dxµdxν L(xµ,τ) = g , (29) µν 2 dτ dτ where τ stands for the proper time. 2ML 2r 2ML 2r ds2 = − 1− p dt2 +2 p dtdr+dr2 +r2dΩ2. (30) (cid:18) r2 +Lp2(cid:19) sr2 +Lp2 The fact that the Lagrangian does not contain canonical coordinate t implies in its correspond- ing Euler-Lagrange equation the corresponding canonical momentum of t, also the particle’s energy is conserved (t˙= dt,r˙ = dr), dτ dτ ∂L 2ML 2r 2ML 2r p = = m − 1− p t˙ + p r˙ = −ω, (31) t ∂t˙ " (cid:18) r2 +Lp2(cid:19) p sr2 +Lp2 # 7 here the minus sign before ω results from the positivity of the energy of the tunneling particle. For a time-like trajectory, we also have dxµdxν g = −1, (32) µν dτ dτ i.e., 2ML 2r 2ML 2r − 1− p t˙2 +2 p t˙r˙ +r˙2 = −1, (33) (cid:18) r2 +Lp2(cid:19) sr2 +Lp2 and in consequence of Eq.(31) and (33) the particle’s trajectory along the radial direction is dr −g g m2 +ω2 = 00 00 . (34) dt ω g2 −g +g g m2 +ω2 01 0p0 01 00 where p p 2ML 2r 2ML 2r g = − 1− p , g = p . (35) 00 (cid:18) r2 +Lp2(cid:19) 01 sr2 +Lp2 By repeating the same steps in Section 3 , the imaginary part of the action that a massive particle along the radial direction tunnels across the outer horizon r from r = ML2 + + in p L M2L2 −1 to r = (M −ω)L2 +L (M −ω)2L2 −1 can be exhibited as follows p p out p p p q p rout ω′ dH ImI = Im dr r˙ Zrin Zm ω rout ω′ g021 −g00 +g01 g00m2 +ω′2 dr = −Im dω ′ (cid:16) p −g g mp2 +ω2 (cid:17) Zm Zrin 00 00 ′ ω rout (r2 +L2p) ω′ pg021 −g00 +g01 g00m2 +ω′2 dr = −Im dω , (36) ′ ((cid:16)r −pr )(r −r ) g pm2 +ω2 (cid:17) Zm Zrin + 00 ′ − where the two horizons are p r = (M −ω )L2 ±L (M −ω )2L2 −1, H = M −ω . (37) ′ p p ′ p ′ ± q Clearly, there exists a pole at r = (M−ω )L2+L (M −ω )2L2 −1, and hence we can deform ′ p p ′ p the contour of the r integral around the pole atqthe horizon in order to ensure the positive energy solutions decay in time (choose the lower half ω plane), so the imaginary part of the ′ 8 action is expressed as ω r2 +L2 ImI = 2π + pdω r −r ′ Z0 + − = πM2L2 +πL M − M2L2 −1+πln ML + M2L2 −1 p p p p p q h q i −π(M −ω)2L2 −πL (M −ω) (M −ω)2L2 −1 p p p q −πln (M −ω)L + (M −ω)2L2 −1 (38) p p h q i The tunneling rate is Γ ∼ exp(−2ImI). (39) Expanding Γ with respect ω 4πL2M LM +1 ω Γ ≈ e− √L2M2−1 (cid:18) (cid:19) 2πL2(4L5M5−9L3M3−7L2M2√L2M2−1+√L2M2−1+4L4M4√L2M2−1+4LM) ω2 e (L2M2−1)3/2(√L2M2−1+LM)2 (40) and comparing the first order of ω with the Boltzmann factor e ω, we obtain the temperature −T of the holographic screen M2L2 −1 T = p , (41) 4πML2 M2L2 −1+ML p p p p which demonstrates the temperature of(cid:0)tphe holographic scre(cid:1)en is no longer equivalent to the Schwartzschild metric’s. 4 Conclusion Inthispaper, wehave appliedthetheParikh-Wilczek scheme onaholographicscreen, analyzed the tunneling process of the massless and massive particles and derived their tunneling rates. Wehave pointedout that theradiationspectra arenotpurely thermal, andthetemperatures of theholographicscreen arenot equal to thestandard Hawking temperature oftheSchwarzschild black hole. Moreover, we have also noticed that the changes of the entropies ∆S = −2ImI thus the unitarity principle remains in the tunneling process of the holographic screen model. In our future work the approach in this article will be developed and massive charged particles emitting from the holographic screen will be studied. It is expected that several interesting results will be gained. 9