In this thesis we have focused on semi-classical methods that enables us to obtain non-thermal spectra for the vast majority of black holes. This fact is due to taking into account the backreaction of the metric, when the black hole emits, imposing energy conservation. Specifically we have studied NS5 and Little String Theory (LST) black holes. We have calculated the Hawking radiation for both models, obtaining a non-thermal spectrum for NS5, whereas a purely thermal spectrum for LST. This last fact is due to the peculiar behavior of LST, whose temperature is independent of its mass. After a brief outline in Chapter 1 about properties of black holes, where we have introduced the information loss paradox, we have reviewed in the Chapter 2 how curved space-times, e.g. black hole backgrounds, create particles. Hawking demonstrated that black holes has temperature thus emit thermal radiation, and calculated its flux without taking into account the back-reaction of the metric. Afterwards we have presented two semi-classical methods, i.e. the tunneling approach and the complex path method, that somehow solve the information loss paradox stated by the work of Hawking. In Chapter 3 we have applied both semi-classical methods plus the covariant anomaly method in NS5 and Little String Theory (LST) black holes. We have calculated some thermodynamical quantities as the temperature and the entropy; furthermore, after reducing the ten-dimensional theory to a two-dimensional effective theor 16 Chapter 2. Semi-classical emission of Black Holes Now taking into account the relation (2.19) ∑ k (ik jk −ik jk) = ( e (!i+!j )  − 1 )∑ k ik jk =ij ; (2.34) and taking i =j ∑ k |ik|2 = 1 e 2!i  − 1 : (2.35) Actually the inverse process is needed, namely start with a positive frequency mode on the past null infinity I that propagates until it becomes a mixed positive and negative frequency mode on the future null infinity I +. The final result for the expectation value of the number of particles created and emitted to I + is hN iI+ = 1 e 2!  − 1 : (2.36) This result corresponds to a Planck distribution for black body radiation at the Hawking temperature TH = ~ 2 : (2.37) So far we have considered that all the thermal radiation emitted by the black hole arrives to the future null infinity I + without any change in the amplitude of the wave function. However, some emitted radiation will be partially scattered back to the event horizon. This fact is due to the gravitational potential barrier around the black hole, where some fraction of radiation will be reflected back to the hole, acting thus as a filter for the emitted radiation. Taking into account this effect we have to modify the orthonormal condition (2.19) by ∑ k (ik jk −ik jk) = Γ ; (2.38) where Γi is known as the greybody factor and it accounts for the deviation from pure black body spectrum, then the number of emitted particles will be hN iI+ = Γ e 2!  − 1 : (2.39) Greybody factors have a relevant importance because successful microscopic account of black hole thermodynamics should be able to predict them. For example, it is shown in [19] that D-branes provide an account of black hole microstates which is successful to predict the greybody factors. There exists a vast literature on how to compute greybody factors in the context of the quantum field theory in curved space-time, e.g. [20, 21, 22, 23, 24, 25, 26]. 3.3. Hawking radiation via tunneling 43 NS5-branes and LST. It is shown that once the near horizon limit is taken, i.e. LST, the emission is thermal even if back-reaction is taken into account. We remark that this fact is due to the LST mass-independent temperature. However, it is not the case for NS5, which shows a non-thermal emission and thus the possibility of recovering information through the correlations between the emitted particles. We motivate this study since a central issue in the black hole information puzzle is the problem of low-energy scattering for ordinary quanta by an extremal black hole with a subsequent absorption and Hawking reemission. From a semi-classical point of view the final radiation turns to be that of an exact black body [60, 61]. It has been argued, but not demonstrated, that departures from thermal emission could explain black hole evaporation without lost of information and hence reconcile quantum mechanics with general relativity. In most of the approaches in the literature the role of the black hole is similar to that of a soliton in field theory, being gravity treated as a non-perturbative field to be added to the game once the spectrum and quantization rules to the particle-like objects have been put down by quantum mechanics rules. Although this view suffices in a semi-classical picture it can be inappropriate when one probes Planck scales. One successful approach that overcomes partially this problem, incorporates the self-gravitation interaction in the radiation process [35]. The underlying idea in this model is extremely simple: the full hole-particle system is reduced to an effective one- dimensional system and for that purpose all the degrees of freedom are truncated to two dimensional. In particular the model for emission/absorption is still only suitable for regions of low-curvature and exclusively tackles the s-wave part of the short-wavelength radiation. This fact allows to employ the WKB approximation that makes any calculation almost straightforward. All the studies pursued within the mentioned approach reveal so far that Hawking radiation is not purely thermal. These results, although encouraging to explain the Hawking effect, are distressing and it is not clear which is the ultimate reason that allows all the black holes to have a non-thermal emission independently of their nature. Our aim is to present some features of the semi-classical geometry and Hawking radiation in a family of black holes with strict thermal emission even if back-reaction effects are taken into account. We

A Secret Tunnel Through The Horizon Hawking radiation is often intuitively visualized as particles that have tunneled across the horizon. Yet, at first sight, it is not apparent where the barrier is. Here I show that the barrier depends on the tunneling particle itself. The key is to implement energy conservation, so that the black hole contracts during the process of radiation. A direct consequence is that the radiation spectrum cannot be strictly thermal. The correction to the thermal spectrum is of precisely the form that one would expect from an underlying unitary quantum theory. This may have profound implications for the black hole information puzzle. Published as: Int.J.Mod.Phys.D13:2351-2354,2004; Gen.Rel.Grav.36:2419-2422,2004 DOI: 10.1142/S0218271804006498 arXiv categories: hep-th astro-ph gr-qc [hep-th/0405160] References CU-TP-1114 A Secret Tunnel Through The Horizon Maulik Parikh 1 1 1 mkp@phys.columbia.edu Department of Physics, Columbia University, New York, NY 10027 Abstract Hawking radiation is often intuitively visualized as particles that have tunneled across the horizon. Yet, at first sight, it is not apparent where the barrier is. Here I show that the barrier depends on the tunneling particle itself. The key is to implement energy conservation, so that the black hole contracts during the process of radiation. A direct consequence is that the radiation spectrum cannot be strictly thermal. The correction to the thermal spectrum is of precisely the form that one would expect from an underlying unitary quantum theory. This may have profound implications for the black hole information puzzle. This essay was awarded First Prize in the 2004 Essay Competition of the Gravity Research Foundation. Classically, a black hole is the ultimate prison: anything that enters is doomed; there is no escape. Moreover, since nothing can ever come out, a classical black hole can only grow bigger with time. Thus it came as a huge shock to physicists when Stephen Hawking demonstrated that, quantum mechanically, black holes could actually radiate particles. With the emission of Hawking radiation, black holes could lose energy, shrink, and eventually evaporate completely. How does this happen? When an object that is classically stable becomes quantum-mechanically unstable, it is natural to suspect tunneling. Indeed, when Hawking first proved the existence of black hole radiation [ 1 ] , he described it as tunneling triggered by vacuum fluctations near the horizon. The idea is that when a virtual particle pair is created just inside the horizon, the positive energy virtual particle can tunnel out – no classical escape route exists – where it materializes as a real particle. Alternatively, for a pair created just outside the horizon, the negative energy virtual particle, which is forbidden outside, can tunnel inwards. In either case, the negative energy particle is absorbed by the black hole, resulting in a decrease in the mass of the black hole, while the positive energy particle escapes to infinity, appearing as Hawking radiation. This heuristic picture has obvious visual and intuitive appeal. But, oddly, actual derivations of Hawking radiation did not proceed in this way at all [ 1 , 2 ] . There were two apparent hurdles. For black holes, the energy and radius are related, and this means that the black hole has to shrink. It is this contraction that sets the scale: the horizon recedes from its original radius to a new, smaller radius. Moreover, the amount of contraction depends on the energy of the outgoing particle so, in a sense, it is the tunneling particle itself that secretly defines the barrier. Now, one might fear that a calculation of Hawking radiation in which energy conservation is critical would require a quantum theory of gravity because the metric must fluctuate to account for the contraction of the hole. Indeed, this is almost what is found. But, remarkably, an exact calculation [ 3 , 6 ] of the action for a tunneling spherically symmetric particle yields Γ ∼ exp ⁡ ( − 8 ​ π ​ M ​ E ​ ( 1 − E 2 ​ M ) ) . similar-to Γ 8 𝜋 𝑀 𝐸 1 𝐸 2 𝑀 \Gamma\sim\exp\left(-8\pi ME\left(1-{E\over 2M}\right)\right)\;. (5) If one neglects the E / 2 ​ M 𝐸 2 𝑀 E/2M term in the expression, it does take the form e − β ​ E superscript 𝑒 𝛽 𝐸 e^{-\beta E} with precisely the inverse of the temperature that Hawking found. So at this level we have confirmed that Hawking radiation can be viewed as tunneling particles and, furthermore, we have verified Hawking’s thermal formula. But, unlike traditional derivations, we have also taken into account the conservation of energy and this yields a correction, the additional term E / 2 ​ M 𝐸 2 𝑀 E/2M . Thus the spectrum is not precisely thermal! This is exciting news because arguments that information is lost during black hole evaporation rely in part on the assumption of strict thermality of the spectrum [ 7 ] . That the spectrum is not precisely thermal may open the way to looking for information-carrying correlations in the spectrum – work on this continues. For quantum theory teaches us that the rate for a process is expressible as the square of the amplitude multiplied by the phase space factor. In turn, the phase space factor is obtained by summing over final states and averaging over initial states. But, for a black Γ superscript amplitude 2 phase space factor similar-to superscript 𝑒 subscript 𝑆 final superscript 𝑒 subscript 𝑆 initial Δ 𝑆 \Gamma=|{\rm amplitude}|^{2}\times(\mbox{phase space factor})\sim{e^{S_{\rm final}}\over e^{S_{\rm initial}}}=\exp(\Delta S)\;. (7) Quantum mechanics, we observe, is in perfect agreement with our answer. References [1] S. W. Hawking, “Particle Creation by Black Holes,” Commun. Math. Phys. 43 (1975) 199. [2] G. W. Gibbons and S. W. Hawking, “Action integrals and partition functions in quantum gravity,” Phys. Rev. D15 (1977) 2752. [3] M. K. Parikh, “Energy Conservation and Hawking Radiation,” hep-th/0402166 . [4] P.

The construction of the conformal scalar propagator which has been obtained in the preceding two projects as an analytic function of the Schwarzschild black-hole space-time is completed with a boundary condition imposed by the physical context through contour integration in the exterior vicinity of the event horizon. It is shown that, as a consequence of the semi-classical character which the emitted quanta have in that exterior vicinity, the particle production by the Schwarzschild black hole which was formally established in the preceding project is identical to thermal Hawking radiation. By extension, it is established that such a particle production corresponds to a spectrum which detracts from thermality by the amount predicted by Parikh and Wilczek if energy conservation is properly imposed as a constraint on scalar propagation. The results obtained herein support the case made by S. Hawking on the relation between quantum propagation and observation of particles produced by a black hole.

mass via Hawking radiation. The presence of a black hole can be inferred through its interaction with matter and electromagnetic radiation such as visible A black hole is an astronomical body so compact that its gravity prevents anything, including light, from escaping. Albert Einstein's theory of general relativity, which describes gravitation as the curvature of spacetime, predicts that any sufficiently compact mass will form a black hole. The boundary of no escape is called the event horizon. In general relativity, crossing a black hole's event h A… If Hawking's theory of black hole radiation is correct, then black holes are expected to shrink and evaporate over time as they lose mass by the emission of photons and other particles. The temperature of this thermal spectrum (Hawking temperature) is proportional to the surface gravity of the black hole, which is inversely proportional to the mass. Hence, large black holes emit less radiation than small black holes. A stellar black hole of 1 M☉ has a Hawking temperature of 62 nanokelvins. This is far less than the 2.7 K temperature of the cosmic microwave background radiation. Stellar-mass or larger black holes receive more mass from the cosmic microwave background than they emit through Hawking radiation and thus will grow instead of shrinking. To have a Hawking temperature larger than 2.7 K (and be able to evaporate), a black hole would need a mass less than the Moon. Such a black hole would have a diameter of less than a tenth of a millimetre. The Hawking radiation for an astrophysical black hole is predicted to be very weak and would thus be exceedingly difficult to detect from Earth. A possible exception is the microsecond-long burst of gamma rays emitted in the last stage of the evaporation of primordial black holes. Extensive searches for such radiation have proven unsuccessful and provide upper limits on the possibility of existence of low mass primordial black holes. When…