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.
Highlights
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
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
While we will not focus on the developments towards a quantum theory of gravitation, referring an interested reader to the recent review [2], we stress that different applications of these models have been discussed ranging from cosmology, to quantum critical systems [3], to condensed matter physics, see e.g., [4] and references therein, and to the AdS=CFT correspondence
Summary
In the quest of finding a quantization scheme for the gravitational field, Horava-Lifshitz gravity, [1], was first introduced as a theory in which the underlying spacetime possesses a time coordinate t and spatial counterparts x appearing with a different scaling behavior t ↦ δzt and x ↦ δx with z > 1 and δ > 0: ð1:1Þ. These boundary conditions are noteworthy because they imply that we are considering a closed system, namely the flux of energy and momentum across the Lifshitz boundary vanishes, see in particular [9] Inspired by these works, we consider a massive, real, scalar field, with an arbitrary coupling to scalar curvature and we investigate whether one can extend the class of admissible boundary conditions on a four dimensional Lifshitz spacetime with dynamical critical exponent z 1⁄4 2 and on its generalizations to a hyperbolic and to a spherical Lifshitz topological black hole, as determined in [16]. Several computations of this work are reproduced in a Mathematica notebook available at [20]
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