Commun. math. Phys.

43, 199—220 (1975)© by Springer-Verlag 1975

Particle Creation by Black Holes

S.W. Hawking

Department of Applied Mathematics and Theoretical Physics,

University of Cambridge,

Cambridge, England

Received

April 12, 1975 Abstract.

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“In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies… This thermal emission leads to a slow decrease in the mass of the black hole and to its eventual disappearance: any primordial black hole of mass less than about 10 to the power of 15 g would have evaporated by now. Although these quantum effects violate the classical law that the area of the event horizon of a black hole cannot decrease, there remains a Generalized Second Law:

S+^A never decreases where S is the entropy of matter outside black holes and A is the sum of the surface areas of the event horizons. This shows that gravitational collapse converts the baryons and leptons in the collapsing body into entropy. It is tempting to speculate that this might be the reason why the Universe contains so much entropy per baryon.”

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“As the mass of the black hole decreases, the area of the event horizon would have to go down, thus violating the law that, classically, the area cannot decrease. This violation must, presumably, be caused by a flux of negative energy across the event horizon which balances the positive energy flux emitted to infinity. One might picture this negative energy flux in the following way. Just outside the event horizon there will be virtual pairs of particles, one with negative energy and one with positive energy. The negative particle is in a region which is classically forbidden but it can tunnel through the event horizon to the region inside the black hole where the Killing vector which represents time translations is space-like. In this region the particle can exist as a real particle with a time-like momentum vector even though its energy relative to infinity as measured by the time translation Killing vector is negative. The other particle of the pair, having a positive energy, can escape to infinity where it constitutes a part of the thermal emission described above. The probability of the negative energy particle tunneling through the horizon is governed by the surface gravity K since this quantity measures the gradient of the magnitude of the Killing vector or, in other words, how fast the Killing vector is becoming space-like. Instead of thinking of negative energy particles tunneling through the horizon in the positive sense of time one could regard them as positive energy particles crossing the horizon on past-directed world-lines and then being scattered on to future-directed world-lines by the gravitational field. It should be emphasized that these pictures of the mechanism responsible for the thermal emission and area decrease are heuristic only and should not be taken too literally.”

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“Although there has been a lot of work in the last fifteen years, I think it would be fair to say that we do not yet have a fully satisfactory and consistent quantum theory of gravity. At the moment classical General Relativity still provides the most successful description of gravity. In classical General Relativity one has a classical metric which obeys the Einstein equations, the right hand side of which is supposed to be the energy momentum tensor of the classical matter fields. However, although it may be reasonable to ignore quantum gravitational effects on the grounds that these are likely to be small, we know that quantum mechanics plays a vital role in the behavior of the matter fields. One therefore has the problem of defining a consistent scheme in which the space-time metric is treated classically but is coupled to the matter fields which are treated quantum mechanically. Presumably such a scheme would be only an approximation to a deeper theory (still to be found) in which space-time itself was quantized. However one would hope that it would be a very good approximation for most purposes except near space-time singularities.”

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Credit: S.W. Hawking; excerpts from “Particle Creation By Black Holes” (1975)

2017/7/22