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Black hole entropy is the Noether charge. (English) Zbl 0942.83512

Summary: A black hole is a region from which no energy can escape, classically, due to the high gravitational field and the mass-energy equivalence. In the late 1960s Roger Penrose pointed out that black holes could be used to convert thermal radiation into usable energy by lowering a box filled with the thermal radiation and releasing it into the black hole. If the box was lowered by winding out a spring, after releasing the radiation the spring would have more energy in it than had been spent on it. Thus we could get usable energy from unusable energy and violate the second law of thermodynamics. Bekenstein saved the law by postulating an entropy for the black hole which was proportional to its surface area. The thermal radiation absorbed by the black hole would increase its mass and hence its surface area. Thus the entropy reduction in the black hole’s surroundings would be (more than) compensated for by the increase in the black hole entropy. This conjecture received support from Hawking’s quantisation of scalar fields in a black hole background, which appeared to indicate that the black hole would radiate at a temperature proportional to its surface gravity.
Totally separate from black hole considerations, in the 1920s Emmy Noether proved that for every continuous transformation leaving the Lagrangian invariant there would be a conserved quantity. Such a quantity is called a Noether charge, by analogy with the electric charge which is conserved as a consequence of the invariance of the electromagnetic Lagrangian under the usual electromagnetic gauge transformation.
It had been earlier proved by the author that for any Hamiltonian theory of gravity a perturbation of the black hole causes an increase of the area and hence the black hole entropy is related to the Noether charge. However, as pointed out by him, the proof was deficient in that it did not demonstrate that the two were equal. In this paper that deficiency is removed. Using a Lagrangian \(n\)-form it is shown that the perturbation is proportional to the change of surface area of the black hole and further that the constant of proportionality is the surface gravity over \(2\pi\). Thus the temperature should be identified with this constant and (as the title of the paper says) the black hole entropy is (!) the Noether charge.

MSC:

83C57 Black holes
83C47 Methods of quantum field theory in general relativity and gravitational theory
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References:

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