A Secret Tunnel Through the Horizon

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.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, whenHawking 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. The first was technical: in order to do a tunneling computation one needed to have a coordinate system that was wellbehaved at the horizon; none of the well-known coordinate systems were suitable. The second hurdle was conceptual: there didn't seem to be any barrier!

Charged particles' tunnelling from the Kerr–Newman black hole

“…They treat Hawking radiation as a tunnelling process, and the WKB method is used [2,3]. In this way a corrected spectrum, which is accurate to a first approximation, is given.…”

Black hole entropy, log corrections and quantum ergosphere

“…When no back reaction effects nor quantum 3 A similar logarithmic correction to the entropy-area law has also emerged from the calculation of one-loop effects of the (quantum) matter fields near the black hole [16]. 4 We now switch from k = = c = G = 1 units of the previous sections to k = = c = 1 to keep track of the Planck-scale suppressed terms. 5 In [17], for example, it is shown that the emergence of such a logarithmic correction can be related to Planck-scale modifications of a particle's quantum localization limit.…”

“…Parikh and Wilczek [3] (see also [4,5]) showed how such a description of black hole radiance is possible if one considers the emission as a transition between states with the same energy. In this way the lowering of the mass of the black hole during the process and the related change in the radius set the barrier through which the particle tunnels.…”

Non-equilibrium Landauer transport model for Hawking radiation from a Reissner—Nordstrom black hole

“…Now it is believed that Hawking radiation arises from the production of virtual particle pairs spontaneously near the inner of horizon due to the vacuum fluctuation. When the negative energy virtual particle tunnels inwards, the positive energy virtual particle materializes as a real particle and escapes to infinity [10][11][12][13][14][15][16][17][18][19][20][21]. But, how does the positive energy particle run?…”