The Klein–Gordon equation on the toric AdS-Schwarzschild black hole

Abstract

We consider the Klein–Gordon equation on the exterior of the toric anti de-Sitter Schwarzschild black hole with Dirichlet, Neumann and Robin boundary conditions at We define a non-degenerate energy for the equation which controls the renormalised H 1 norm of the field. We then establish both decay and integrated decay of this energy through vector field methods. Finally, we demonstrate the necessity of 'losing a derivative' in the integrated energy estimate through the construction of a Gaussian beam staying in the exterior of the event horizon for arbitrarily long coordinate time.