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On the topology and area of higher dimensional black holes

Mingliang Cai, Gregory J. Galloway

TL;DR

The paper addresses the topology and area of higher-dimensional black holes by extending Hawking's topology theorem and Gibbons' entropy bound to arbitrary dimensions in a time-symmetric setting. It uses the σ-constant (Yamabe invariant) as the central topological invariant and a stability analysis of minimal horizons to show that, under $S = 2T_{00} + 2\Lambda \ge 0$, the horizon $\Sigma^{n-1}$ admits a metric of positive scalar curvature, enabling topology constraints that generalize the $3+1$ results. In particular, for $4+1$ dimensions the horizon is restricted to connected sums of $S^3$ and $S^2\times S^1$ (with a flat case yielding a 3-torus when Ricci-flat), and lower-area bounds are derived in terms of the σ-constant or Yamabe invariant, especially when $\sigma(\Sigma) \le 0$ and $S \ge -\kappa$. The bounds are expressed as $\mathrm{vol}(\Sigma^{n-1}) \ge \left|\frac{\lambda[g]}{\kappa}\right|^{\frac{n-1}{2}} \ge \left|\frac{\sigma(\Sigma)}{\kappa}\right|^{\frac{n-1}{2}}$, with refinements in asymptotically locally anti-de Sitter spaces. These results connect horizon geometry to global topological invariants in higher dimensions, with implications for string theory, AdS/CFT contexts, and the study of black strings and p-branes.

Abstract

Over the past decade there has been an increasing interest in the study of black holes, and related objects, in higher (and lower) dimensions, motivated to a large extent by developments in string theory. The aim of the present paper is to obtain higher dimensional analogues of some well known results for black holes in 3+1 dimensions. More precisely, we obtain extensions to higher dimensions of Hawking's black hole topology theorem for asymptotically flat ($Λ=0$) black hole spacetimes, and Gibbons' and Woolgar's genus dependent, lower entropy bound for topological black holes in asymptotically locally anti-de Sitter ($Λ<0$) spacetimes. In higher dimensions the genus is replaced by the so-called $σ$-constant, or Yamabe invariant, which is a fundamental topological invariant of smooth compact manifolds.

On the topology and area of higher dimensional black holes

TL;DR

The paper addresses the topology and area of higher-dimensional black holes by extending Hawking's topology theorem and Gibbons' entropy bound to arbitrary dimensions in a time-symmetric setting. It uses the σ-constant (Yamabe invariant) as the central topological invariant and a stability analysis of minimal horizons to show that, under , the horizon admits a metric of positive scalar curvature, enabling topology constraints that generalize the results. In particular, for dimensions the horizon is restricted to connected sums of and (with a flat case yielding a 3-torus when Ricci-flat), and lower-area bounds are derived in terms of the σ-constant or Yamabe invariant, especially when and . The bounds are expressed as , with refinements in asymptotically locally anti-de Sitter spaces. These results connect horizon geometry to global topological invariants in higher dimensions, with implications for string theory, AdS/CFT contexts, and the study of black strings and p-branes.

Abstract

Over the past decade there has been an increasing interest in the study of black holes, and related objects, in higher (and lower) dimensions, motivated to a large extent by developments in string theory. The aim of the present paper is to obtain higher dimensional analogues of some well known results for black holes in 3+1 dimensions. More precisely, we obtain extensions to higher dimensions of Hawking's black hole topology theorem for asymptotically flat () black hole spacetimes, and Gibbons' and Woolgar's genus dependent, lower entropy bound for topological black holes in asymptotically locally anti-de Sitter () spacetimes. In higher dimensions the genus is replaced by the so-called -constant, or Yamabe invariant, which is a fundamental topological invariant of smooth compact manifolds.

Paper Structure

This paper contains 4 sections, 6 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Topological restrictions
  3. Lower area bounds and the $\sigma$-constant
  4. Final remarks

Key Result

Theorem 1

Let $V^{n}$, $n\ge 3$, be a regular time symmetric spacelike hypersurface, with compact boundary horizon $\Sigma^{n-1}$, in a spacetime $M^{n+1}$, as described above. Suppose the scalar curvature of $V^{n}$ satisfies, $S = 2T_{00} + 2\Lambda\ge 0$ along $\Sigma^{n-1}$. Then either

Theorems & Definitions (6)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Theorem 6