Poisson algebra of diffeomorphism generators in a spacetime containing a bifurcation

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 canonical generators, which are necessary to make them differentiable. The existence of such boundary terms depends on the exact boundary conditions that one takes. We will check two possible boundary conditions on the black hole bifurcation: fixed metric and fixed surface gravity. In the case of a fixed metric of the bifurcation the action acquires a boundary term but this term is canceled in the Legendre transformation and is thus absent in the Hamiltonian, and so in this case the possibility of a central extension is ruled out. In the case of fixed surface gravity the boundary term in the action is absent but therefore present in the Hamiltonian. Also in this case we will see that there is no central extension, also if there exist boundary terms in the generators.