Multi-Black-Hole Geometries in (2+1)-Dimensional Gravity

TL;DR

This work addresses constructing multi-black-hole configurations in 2+1 dimensional gravity with a negative cosmological constant by extending the BTZ black hole to MBH geometries. The author develops initial data through hyperbolic geometry doublings, analyzes time evolution in a time-orthogonal gauge, and provides a procedure to generate MBHs with angular momentum for $n \ge 4$ using identification surfaces and twists. Key contributions include explicit $J=0$ MBH initial data described by a right-angled polygon in hyperbolic space, the interior MBH universe interpretation, and a framework for angular-momentum carrying MBHs with a set of consistency conditions. The results illuminate the rich global structure and horizon dynamics of MBH spacetimes in AdS$_3$ and offer a concrete method for constructing such configurations with multiple asymptotic regions.

Abstract

Generalizations of the Black Hole geometry of Bañados, Teitelboim and Zanelli (BTZ) are presented. The theory is three-dimensional vacuum Einstein theory with a negative cosmological constant. The $n$-black-hole solution has $n$ asymptotically anti-de Sitter ``exterior" regions that join in one ``interior" region. The geometry of each exterior region is identical to that of a BTZ geometry; in particular, each contains a black hole horizon that surrounds (as judged from that exterior) all the other horizons. The interior region acts as a closed universe containing $n$ black holes. The initial state and its time development are discussed in some detail for the simple case when the angular momentum parameters of all the black holes vanish. A procedure to construct $n$ black holes with angular momentum (for $n \geq 4$) is also given.