Topological Black Holes -- Outside Looking In

TL;DR

The paper develops and analyzes topological black holes formed by identifications in anti-de Sitter space, yielding horizons of nontrivial topology (genus $g\ge1$) across dimensions. It derives the exterior metrics, characterizes horizon structure and conserved charges, and discusses interior possibilities including inner (Cauchy) horizons and mass inflation. Through numerical study of a conformally coupled scalar field, it characterizes late-time radiative tails outside neutral TBHs and compares with AdS cases, showing exponential or power-law falloffs that depend on genus and the AdS scale, with implications for interior dynamics. The work highlights how topology and AdS asymptotics influence black hole interiors and offers insights relevant to quantum gravity and topology-changing processes, even as astrophysical applications remain indirect.

Abstract

I describe the general mathematical construction and physical picture of topological black holes, which are black holes whose event horizons are surfaces of non-trivial topology. The construction is carried out in an arbitrary number of dimensions, and includes all known special cases which have appeared before in the literature. I describe the basic features of massive charged topological black holes in $(3+1)$ dimensions, from both an exterior and interior point of view. To investigate their interiors, it is necessary to understand the radiative falloff behaviour of a given massless field at late times in the background of a topological black hole. I describe the results of a numerical investigation of such behaviour for a conformally coupled scalar field. Significant differences emerge between spherical and higher genus topologies.

Figures (13)

• Figure 3: Causal diagrams for topological black holes in the neutral, subextremal and extremal cases. The subextremal case can be that of a charged black hole or of a negative mass black hole.
• Figure 4: The potential ${\cal V}(x)$ in a Schwarzschild background and the resultant decay of a scalar wave. Prior to $t\approx 200$ the decay is accompanied by 'ringing' of the quasi-normal modes, after which the falloff rate is that of an inverse power-law.
• Figure 5: Potential functions $V(x)$ for the SAdS background. The six potentials are generated with the parameters (a) $\Lambda = \hbox{-} \, 10^{ \hbox{-} \, 4}, l = 2$, (b) $\Lambda = \hbox{-} \, 10^{ \hbox{-} \, 4}, l = 1$, (c) $\Lambda = \hbox{-} \, 10^{ \hbox{-} \, 4}, l = 0$, (A) $\Lambda = \hbox{-} \, 10^{ \hbox{-} \, 2}, l = 2$, (B) $\Lambda = \hbox{-} \, 10^{ \hbox{-} \, 2}, l = 1$, (C) $\Lambda = \hbox{-} \, 10^{ \hbox{-} \, 2}, l = 0$. The bottom diagram shows the $l=0$ scalar wave falloff pattern using Neumann condition.
• Figure 6: The $l=0$ (top) and $l=1$ (bottom) scalar wave falloff patterns using the Dirichlet condition.
• Figure 7: The $l=0$ (top) and $l=2$ (bottom) scalar wave falloff patterns using for large $|\Lambda|$.