Spatially homogeneous black hole solutions in $z=4$ Ho\v{r}ava-Lifshitz gravity in $(4+1)$ dimensions with Nil geometry and $H^2\times R$ horizons

TL;DR

This paper introduces two new families of spatially homogeneous black hole solutions in five-dimensional Hořava-Lifshitz gravity with specific horizon geometries, analyzing their thermodynamics and stability properties.

Contribution

It presents novel black hole solutions with Nil and $H^2\times R$ horizons in five-dimensional Hořava-Lifshitz gravity, including thermodynamic analysis and stability conditions.

Findings

Solutions have non-spherical, non-hyperbolic horizons modeled on Thurston geometries.

Entropy receives negative corrections proportional to Hořava-Lifshitz parameters.

Solutions can be locally stable or unstable depending on parameter choices.

Abstract

In this paper, we present two new families of spatially homogeneous black hole solution for $z=4$ Ho\v{r}ava-Lifshitz Gravity equations in $(4+1)$ dimensions with general coupling constant $\lambda$ and the especial case $\lambda=1$, considering $\beta=-1/3$. The three-dimensional horizons are considered to have Bianchi types $II$ and $III$ symmetries, and hence the horizons are modeled on two types of Thurston $3$-geometries, namely the Nil geometry and $H^2\times R$. Being foliated by compact 3-manifolds, the horizons are neither spherical, hyperbolic, nor toroidal, and therefore are not of the previously studied topological black hole solutions in Ho\v{r}ava-Lifshitz gravity. Using the Hamiltonian formalism, we establish the conventional thermodynamics of the solutions defining the mass and entropy of the black hole solutions for several classes of solutions. It turned out that for both horizon geometries the area term in the entropy receives two non-logarithmic negative corrections proportional to Ho\v{r}ava-Lifshitz parameters. Also, we show that choosing some proper set of parameters the solutions can exhibit locally stable or unstable behavior.