On Diffeomorphism Invariance and Black Hole Entropy

Details On Diffeomorphism Invariance and Black Hole Entropy On Diffeomorphism Invariance and Black Hole Entropy

2003-04-09

arxiv.org/abs/gr-qc/0304037v1

Phys.Rev.D68:104006,2003

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

32 pages, 1 figure

Scientific paper

10.1103/PhysRevD.68.104006

The Noether-charge realization and the Hamiltonian realization for the $\diff({\cal M})$ algebra in diffeomorphism invariant gravitational theories are studied in a covariant formalism. For the Killing vector fields, the Nother-charge realization leads to the mass formula as an entire vanishing Noether charge for the vacuum black hole spacetimes in general relativity and the corresponding first law of the black hole mechanics. It is analyzed in which sense the Hamiltonian functionals form the $\diff({\cal M})$ algebra under the Poisson bracket and shown how the Noether charges with respect to the diffeomorphism generated by vector fields and their variations in general relativity form this algebra. The asymptotic behaviors of vector fields generating diffeomorphism of the manifold with boundaries are discussed. In order to get more precise estimation for the "central extension" of the algebra, it is analyzed in the Newman-Penrose formalism and shown that the "central extension" for a large class of vector fields is always zero on the Killing horizon. It is also checked whether the Virasoro algebra may be picked up by choosing the vector fields near the horizon. The conclusion is unfortunately negative.

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