Black Hole Topology in $f(R)$ Gravity

TL;DR

This paper extends Hawking's black hole topology theorem to $f(R)$ gravity, establishing conditions under which the horizon cross-section must be spherical or toroidal in higher dimensions.

Contribution

It provides a differential condition on $f'(R)$ that constrains black hole horizon topology in $f(R)$ gravity across multiple dimensions.

Findings

Horizon topology restricted to $S^2$ or $S^1 imes S^1$ in $3+1$ dimensions.

Extension of topology restrictions to higher dimensions.

Restriction on the sign of the Yamabe invariant of the horizon cross-section.

Abstract

Hawking's topology theorem in general relativity restricts the cross-section of the event horizon of a black hole in $3+1$ dimension to be either spherical or toroidal. The toroidal case is ruled out by the topology censorship theorems. In this article, we discuss the generalization of this result to black holes in $f(R)$ gravity in $3+1$ and higher dimensions. We obtain a sufficient differential condition on the function $f'(R)$, which restricts the topology of the horizon cross-section of a black hole in $f(R)$ gravity in $3+1$ dimension to be either $S^2$ or $S^1 \times S^1$. We also extend the result to higher dimensional black holes and show that the same sufficient condition also restricts the sign of the Yamabe invariant of the horizon cross-section.