On the role of boundary conditions within Ho\v{r}ava-Lifshitz gravity

TL;DR

This paper investigates boundary conditions for scalar fields in Lifshitz black hole spacetimes, establishing criteria for bound states and constructing physically meaningful quantum states when bound states are absent.

Contribution

It analyzes boundary conditions and bound state existence for Klein-Gordon fields in Lifshitz black holes, linking boundary choices to quantum state construction.

Findings

Bound states depend on boundary parameters.

Ground and KMS states can be constructed without bound states.

Two-point functions are of local Hadamard form.

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.