The Entropy of Black Holes

In ordinary physics, thermal properties always arise from the motion of the constituents. For example, the temperature of the air is related to the average speed of the air molecules. There is a closely related concept, called "entropy." The entropy is the amount of disorder associated to the motion of all the constituents. The entropy is related to the temperature by the laws of thermodynamics, so it can be computed without knowing the microscopic details of the system. Hawking and Bekenstein showed that the entropy of a black hole is the same as the area of the horizon divided by the square of the Planck length, where l_Planck = 10^–33 cm. For a macroscopic black hole, this is an enormous entropy. It turns out that the laws of thermodynamics continue to be valid if the black hole contribution to the entropy is included. These are extremely puzzling results since it is not at all clear what the "constituents" of a black hole really are. The black hole is a hole in space-time so finding its constituents is intimately related to finding the most fundamental constituents of space-time geometry.

It is very interesting that the entropy of a black hole is proportional to its area and not its volume. In the early 1990's Hooft and Susskind proposed that in a theory that includes quantum mechanics and gravity, the number of constituents that are necessary to describe a system cannot be bigger than the area of a surface that encloses it. This implies that space-time is rather different from an ordinary solid since in the latter case, the number of constituents (the atoms) grows like the volume. For most practical purposes this entropy bound is not very stringent, but it has interesting theoretical implications, since it suggests that a region of space-time can be described in terms of constituents that live on the boundary of this region.

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