Black holes and entropy Black holes are objects so dense that the escape velocity from their surface exceeds the speed of light, c. Because of that, one would think that in a relativistic theory, outside observers performing classical experiments can never see their surfaces. As a rough-and-ready definition, we will call the surface defining the region where light itself can no longer escape from the gravitational attraction of a black hole, the event horizon. Nothing, in a classical theory, can be emitted from this horizon, though many things can fall through. Careful consideration of the theory of black holes in classical general relativity in the early 1970s led Jacob Bekenstein, Stephen Hawking, and others to a striking set of conclusions. They found that as a chargeless, non-rotating black hole accretes matter, its mass grows by an amount proportional to the strength of gravity at the black holes surface and the change in its surface area. Also, the black holes surface area (defined by its event horizon) cannot decrease under any circumstances, and usually increases in time. At a heuristic level, Bekenstein and Hawkings laws for black holes seem reminiscent of the laws of thermodynamics and statistical mechanics: The change in energy is proportional to the change in entropy and the entropy (a measure of disorder) of a system can only increase. This is no coincidence. The results of general relativity imply what they seem to: A black hole does carry an entropy proportional to its surface area, and, of course, it has an energy that grows with its mass. One mystery remains, however. In thermodynamics, the change in energy is proportional to the temperature times the change in entropy; and hot bodies radiate. Even though there is an analogous quantity—the surface gravity—in the black hole mechanics, no classical process can bring radiation through the horizon of a black hole. In a brilliant calculation in 1974, Stephen Hawking showed that,nevertheless, black holes radiate by a quantum process. This quantum effect occurs at just the right level to make the analogy between black hole thermodynamics and normal thermodynamics work perfectly. Hawkings calculation reinforces our belief that a black holes entropy should be proportional to its surface area. This is a bit confusing because most theories that govern the interactions of matter and force- carrying particles in the absence of gravity posit that entropy grows in proportion to the volume of the system. But in a gravity theory also containing these other degrees of freedom, if one tries to fill a region with enough particles so that their entropy exceeds the area bounding the region, one instead finds gravitational collapse into a black hole, whose entropy is proportional to its surface area. This means that at least in gravity theories, our naive idea that the entropy that can be contained in a space should scale with its volume must be incorrect.
Posted on: Tue, 28 Jan 2014 15:33:24 +0000
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