Three Roads To Quantum Gravity

90 THREE ROADS TO QUANTUM GRAVITY
For simple black holes, which do not rotate and have no
electric charge, the values of the temperature and entropy can
be expressed very simply. The area of the horizon of a simple
black hole is proportional to the square of its mass, in Planck
units. The entropy S is proportional to this quantity. In terms
of Planck units, we have the simple formula
S = ˆA/h Å G
where A is the area of the horizon, and G is the gravitational
constant.
There is a very simple way to interpret this equation which
is due to Gerard 't Hooft, who did important work in elementary
particle physics ± for which he won the 1999 Nobel Prize
for Physics ± before turning his attention to the problem of
quantum gravity. He suggests that the horizon of a black
hole is like a computer screen, with one pixel for every four
Planck areas. Each pixel can be on or off, which means that it
codes one bit of information. The total number of bits of
information contained within a black hole is then equal to the
total number of such pixels that it would take to cover the
horizon. The Planck units are very, very tiny. It would take
10 66 Planck area pixels to cover a single square centimetre. So
an astrophysical black hole whose horizon has a diameter of
several kilometres can contain a stupendous amount of
information.
Entropy has another meaning besides being a measure of
information. If a system has entropy, it will act in ways that
are irreversible in time. This is because of the second law
of thermodynamics, which says that entropy can only be
created, not destroyed. If you shatter a teapot by dropping it
on the ¯oor, you have greatly increased its entropy ± it will be
very dif®cult to put it back together. In thermodynamics the
irreversibility of a process is measured by an increase of
entropy, because that measures the amount of information
lost to random motion. But such information, once lost, can
never be recovered, so the entropy cannot normally decrease.
This is one way of expressing the second law of thermodynamics.
Black holes also behave in a way that is not reversible in