On Diffeomorphism Invariance and Black Hole Entropy

TL;DR

This paper investigates the relationship between diffeomorphism invariance, Noether charges, and black hole entropy in general relativity, showing that the algebra of diffeomorphisms has no central extension near the horizon, challenging certain approaches to black hole entropy.

Contribution

It provides a covariant analysis of the Noether-charge and Hamiltonian realizations of the diffeomorphism algebra, clarifying their roles in black hole mechanics and the structure of the algebra near horizons.

Findings

No central extension of the diffeomorphism algebra at the horizon.

The Virasoro algebra cannot be obtained from vector fields near the horizon.

The mass formula corresponds to a vanishing Noether charge in vacuum black holes.

Abstract

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.