Extremal black holes in the Ho\v{r}ava-Lifshitz gravity

TL;DR

This paper investigates the near-horizon geometry of extremal black holes within the $z=3$ Hořava-Lifshitz gravity framework, revealing conditions under which the geometry is well-defined and analyzing entropy computation challenges.

Contribution

It provides a detailed analysis of extremal black hole horizons in Hořava-Lifshitz gravity, highlighting the limitations of the entropy function approach in this context.

Findings

For $ ext{lambda}>1/2$, near-horizon geometry is AdS$_2 imes S^2$ with variable radii.

For $1/3 ext{ to } 1/2$, the $S^2$ radius becomes negative, indicating ill-defined geometry.

The entropy function method fails to compute the entropy of extremal black holes in this theory.

Abstract

We study the near-horizon geometry of extremal black holes in the $z=3$ Ho\v{r}ava-Lifshitz gravity with a flow parameter $\lambda$. For $\lambda>1/2$, near-horizon geometry of extremal black holes are AdS$_2 \times S^2$ with different radii, depending on the (modified) Ho\v{r}ava-Lifshitz gravity. For $1/3\le \lambda \le 1/2$, the radius $v_2$ of $S^2$ is negative, which means that the near-horizon geometry is ill-defined and the corresponding Bekenstein-Hawking entropy is zero. We show explicitly that the entropy function approach does not work for obtaining the Bekenstein-Hawking entropy of extremal black holes.