Poisson Algebra of Diffeomorphism Generators in a Spacetime Containing a Bifurcation

TL;DR

This paper investigates whether the Poisson algebra of diffeomorphism generators near a black hole bifurcation admits a nontrivial central extension, analyzing boundary conditions and their impact on boundary terms and algebra structure.

Contribution

It provides a detailed analysis showing that under two different boundary conditions, no nontrivial central extension arises in the algebra.

Findings

No nontrivial central extension under fixed bolt metric boundary condition.

No nontrivial central extension under fixed surface gravity boundary condition.

Boundary terms depend on boundary conditions and influence algebra structure.

Abstract

In this article we will analyze the possibility of a nontrivial central extension of the Poisson algebra of the diffeomorphism generators, which respect certain boundary conditions on the black hole bifurcation. The origin of a possible central extension in the algebra is due to the existence of boundary terms in the in the canonical generators. The existence of such boundary terms depend on the exact boundary conditions one takes. We will check two possible boundary conditions i.e. fixed bolt metric and fixed surface gravity. In the case of fixed metric the the action acquires a boundary term associated to the bifurcation but this is canceled in the Legendre transformation and so absent in the canonical generator and so in this case the possibility of a nontrivial central extension is ruled out. In the case of fixed surface gravity the boundary term in the action is absent but present in the Hamiltonian. Also in this case we will see that there is no nontrivial central extension, also if there exist a boundary term in the generator.