Three Roads To Quantum Gravity

174 THREE ROADS TO QUANTUM GRAVITY
Argument for the Bekenstein bound
Let us suppose that The Thing is big enough to be described
both in terms of an exact quantum description and in terms of
an averaged, macroscopic description. We shall argue by
contradiction, which means that we ®rst assume the opposite
of what we are trying to show. Thus we assume that the amount
of information required to describe The Thing is much larger
than the area of The Screen. For simplicity, we assume that The
Screen is spherical.
We know that The Thing is not a black hole, because we
know that the entropy of any black hole that can ®t into The
Screen must be equivalent to an area less than that of the
screen. But in this case its entropy must be lower than the area
of the screen, in Planck units. If we assume that the entropy
of a black hole counts the number of its possible quantum
states, this is much less than the information contained in The
Thing.
It then follows (from a theorem of classical general relativity)
that The Thing has less energy than a black hole that would just
®t inside The Screen. Now, we can slowly add energy to The
Thing by dripping it slowly through the screen. We shall reach
some point by which we shall have given it so much energy
that, by that same theorem, it must collapse to a black hole. But
then we know that its entropy is given by one-quarter of the
area of the screen. Since that is lower than the entropy of The
Thing initially, we have managed to lower the entropy of a
system. This contradicts the second law of thermodynamics.
We dripped the energy in slowly to ensure that nothing
surprising happens outside The Screen which might increase
the entropy strongly somewhere else. There seem to be no
loopholes in this argument. Therefore, if we believe the
second law of thermodynamics, we must believe that the
most entropy that we, outside the Screen, can attribute to The
Thing is one-quarter of the area of The Screen. And because
entropy is a count of answers to yes/no questions, this implies
the Bekenstein bound as we have stated it.