Topology of Cosmological Black Holes

TL;DR

This paper investigates how the topology of black holes evolves in an inhomogeneous universe with a positive cosmological constant, showing that certain non-spherical surfaces expand exponentially and are restricted within black holes.

Contribution

It demonstrates that non-spherical incompressible surfaces expand exponentially under mean curvature flow and constrains black hole topology in such cosmologies.

Findings

Non-spherical incompressible surfaces expand exponentially with rate at least 8πG_NΛ.

No trapped surface or apparent horizon can be a non-spherical, incompressible surface.

Black hole interiors cannot contain non-spherical, incompressible surfaces.

Abstract

Motivated by the question of how generic inflation is, I study the time-evolution of topological surfaces in an inhomogeneous cosmology with positive cosmological constant $\Lambda$. If matter fields satisfy the Weak Energy Condition, non-spherical incompressible surfaces of least area are shown to expand at least exponentially, with rate $d \log A_{\rm min}/d\lambda \geq 8\pi G_N\Lambda$, under the mean curvature flow parametrized by $\lambda$. With reasonable assumptions about the nature of singularities this restricts the topology of black holes: (a) no trapped surface or apparent horizon can be a non-spherical, incompressible surface, and (b) the interior of black holes cannot contain any such surface.