Unruh Radiation, Black Holes and Partial Waves. Note Quote.


It is well known that Hawking radiation from an asymptotically flat Schwarzschild black hole is dominated by low angular momentum modes. This is a consequence of the fact that a black hole of Hawking temperaure TH and Schwarzschild radius rs has TH rs ∼ 1, so that high angular momentum modes of energy TH are trapped behind a large barrier in the effective radial potential. Since a local observer is unlikely to encounter such quanta, one might then conclude that a (much-weakened) version of postulate “A freely falling observer experiences nothing out of the ordinary when crossing the horizon” might still hold in which the suppression is replaced by a fixed (1/area) power law. In addition, one would need to propose a mechanism through which these quanta would arise from the infalling perspective. This would appear to require that the infalling observer experience violations of local quantum field theory at this (power-law-suppressed) level.

This would already be a striking result: these quanta must appear quite close to the horizon and so violate the standard wisdom that the horizon is not a distinguished location. And they are not rare in the sense that their number is of the same order as the number of actual Hawking quanta.

As noted long ago by Unruh and Wald, it is possible to ‘mine’ energy from the modes trapped behind the effective potential. The basic procedure is to lower some object below the potential barrier, let the object absorb the trapped modes, and then raise the object back above the barrier. Unruh and Wald thought of the object as a box that could be opened to collect ambient radiation and then closed to keep the radiation from escaping. One may also visualize the object as a particle detector, though the two are equivalent at the level discussed here.

In the context of such a mining operation, one need only consider the internal state of the mining equipment to be part of the late-time Hawking radiation. In particular, postulate “outside the stretched horizon of a massive black hole, physics can be described to good approximation by a set of semi-classical field equations”, can be used to evolve the mode to be mined backward in time and to conclude for an old black hole that, even before the mining process takes place, the mode must be fully entangled with the early-time radiation. “A freely falling observer experiences nothing out of the ordinary when crossing the horizon” is then violated for these modes as well, suggesting that the infalling observer encounters a Planck density of Planck scale radiation and burns up. One might say that the black hole is protected by a Planck-scale firewall.

Note that this firewall need not be visible to any observer that remains outside the horizon. All that we have argued is that the infalling observer does not experience a pure state. There remains considerable freedom in the possible reduced density matrices that could describe a few localized degrees of freedom outside the black hole, so that this matrix might still agree perfectly with that predicted by Hawking. In this case any local signal that an external observer might hope to ascribe to the firewall at distance 1/ω cannot be disentangled from the Unruh radiation that results from probing this scale without falling into the black hole.


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