DO BLACK HOLES HAVE NO HAIR? Stephen Hawking Lectures(Part-4)

Bekenstein's theorem implies that a large amount of information is lost in a gravitational collapse. For example, the final black hole state is independent of whether the body antimatter, or whether it was spherical or highly irregular in shape. In other words, a black hole of a given mass, angular momentum and electric charge could have been formed by the collapse of any one of a large number of different configurations of matter including any one of a large number of different types of stars. Indeed, if quantum effects are left aside, the number of potential configurations would be infinite, since the black hole could have been formed by the collapse of a cloud of an indefinitely large number of particles, of indefinitely low mass. But could the number of configurations really be infinite? This is where quantum effects come in.

                   The uncertainty principle of quantum mechanics implies that only particles with a wavelength smaller than that of the black hole itself could form a black hole. That means the range of potential wavelengths would be limited: it could not be infinite.

       It therefore appears that the number of configurations that could form a black hole of a given mass, angular momentum and electric charge, although very large, may also be finite. Jacob Bekenstein suggested that from this finite number one could derive the entropy of a black hole. This would be a measure of the amount of information that was irretrievably lost during the collapse when a black hole was created.

         The apparently fatal flaw in Bekenstein's suggestion was that if a black hole has a finite entropy that is proportional to the area of its event horizon, it also ought to have a finite temperature, which would be proportional to its surface gravity. This would imply that a respect to thermal radiation, at some temperature other than zero. Yet according to classical concepts, no such equilibrium is possible, since the black hole would absorb any thermal radiation that fell on it, but by definition would not be able to emit anything in return. It can't emit anything. it can't emit heat.

 This is a paradox. And it's one to which I am going to return in my next lecture, when I'll be exploring how black holes challenge the most basic principle about the predictability of universe, and the certainty of history and asking what would happen if you ever got sucked into one.