Three Roads To Quantum Gravity

BLACK HOLES ARE HOT
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time, because things can fall into a black hole but nothing can
come out of one. This turns out to have a consequence, ®rst
discovered by Stephen Hawking, for the area of the horizon of
a black hole. He showed by a very elegant proof that the area
of the horizon of a black hole can never decrease in time. So it
was natural to suggest that the area of the horizon of a black
hole is analogous to entropy, in that it is a quantity that can
only increase in time. The great insight of Bekenstein was that
this was not just an analogy. He argued that a black hole has
real entropy, which he conjectured is proportional to the area
of its horizon and measures the amount of information
trapped beyond that horizon.
You may wonder why ®fteen thousand other physicists
were not able to take this step if it was based on a simple
analogy, apparent to anyone who looked at the problem. The
reason is that the analogy is not quite complete. For if nothing
can come out of a black hole, then it has zero temperature.
This is because temperature measures the energy in random
motion, and if there is nothing in a box, there can be no
motion of any kind, random or otherwise. But ordinarily a
system cannot have entropy without it being hot. This is
because the missing information results in random motion,
which means there is heat. So, if black holes have entropy,
there is a violation of the laws of thermodynamics.
So this seemed to be not a brilliant move, but the kind of
misuse of analogy that characterizes the thinking of novices in
any ®eld. But a few people did take Bekenstein seriously,
including Stephen Hawking, Paul Davies and Bill Unruh. The
mystery was solved ®rst by Hawking, who realized that if
black holes were hot there was no contradiction with the laws
of thermodynamics. By following a chain of reasoning
roughly like the story above, he was able to show that an
observer outside a black hole would see it to be at a ®nite
temperature. Expressed in Planck units, the temperature T of
a black hole is inversely proportional to its mass, m. This is a
third law, Hawking's law:
T = k/m
The constant k is very small in normal units. As a result,