Role of boundary conditions on Lifshitz spacetimes

TLDR: In this article, the authors consider a class of four-dimensional Lifshitz spacetimes with critical exponent $z = 2, including a hyperbolic and a spherical topological black hole, and use mode decomposition to split the equation of motion into a radial and into an angular component.

Abstract: On a class of four-dimensional Lifshitz spacetimes with critical exponent $z=2$, including a hyperbolic and a spherical Lifshitz topological black hole, we consider a real Klein-Gordon field. Using a mode decomposition, we split the equation of motion into a radial and into an angular component. As first step, we discuss under which conditions on the underlying parameters we can impose to the radial equation boundary conditions of Robin type and whether bound state solutions do occur. Subsequently, we show that, whenever bound states are absent, one can associate to each admissible boundary condition a ground and a KMS state whose associated two-point correlation function is of local Hadamard form.