[Paper Review] Multi-Black-Holes in 3D and 4D anti-de Sitter Spacetimes

Dieter R. Brill|arXiv (Cornell University)|Aug 5, 1996

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper generalizes the 2+1-dimensional BTZ black hole solution to multi-black-hole spacetimes in 3D and 4D anti-de Sitter (AdS) space. By constructing a quotient of global AdS spacetime via discrete isometries, it yields a solution with n asymptotically AdS regions separated by n horizons, forming a closed interior universe containing n black holes, with analogous higher-genus horizon solutions in 4D.

ABSTRACT

The (single) black hole solutions of Bañados, Teitelboim and Zanelli (BTZ) in 2+1 dimensional anti-de Sitter space are generalized to an arbitrary number $n$ of such black holes. The resulting multi-black-hole (MBH) spacetime is locally isometric to anti-de Sitter space, and globally it is obtained from the latter as a quotient space by means of suitable identifications. The MBH spacetime has $n$ asymptotically anti-de Sitter exterior regions, each of which has the geometry of a single BTZ black hole. These exterior regions are separated by $n$ horizons from a common interior region. This interior region can be described as a ``closed" universe containing $n$ black holes. Similar configurations in 3+1 dimensions, with horizons of toroidal and higher genus topologies, are also presented.

Motivation & Objective

• To extend the single BTZ black hole solution in 2+1D AdS spacetime to configurations with an arbitrary number n of black holes.
• To construct globally valid multi-black-hole spacetimes that are locally isometric to AdS space but globally distinct via discrete identifications.
• To explore the geometric and topological structure of multi-black-hole solutions in both 3D and 4D AdS spacetimes.
• To analyze the causal structure, including the presence of multiple asymptotically AdS regions and a common interior region with n black holes.
• To generalize the construction to higher-genus horizons in 4D AdS spacetime, including toroidal and higher-genus topologies.

Proposed method

• Construct the multi-black-hole (MBH) spacetime in 3D AdS by taking a quotient of global AdS space under a discrete group of isometries.
• Use n distinct hyperbolic isometries to generate identifications that produce n horizons separating n asymptotically AdS regions.
• Ensure the resulting spacetime is locally isometric to AdS, preserving the vacuum Einstein equations with negative cosmological constant.
• Verify the solution contains a single interior region with n black holes, topologically a closed universe.
• Extend the construction to 4D AdS by considering higher-genus Riemann surfaces as horizons, using appropriate discrete isometry groups.
• Confirm the global structure via the quotient method, ensuring the spacetime is smooth and free of closed timelike curves.

Experimental results

Research questions

• RQ1 Can the BTZ black hole solution be generalized to include an arbitrary number of black holes in 3D AdS spacetime?
• RQ2 What is the global structure of a multi-black-hole spacetime in 3D AdS, and how do the n asymptotically AdS regions connect to a common interior?
• RQ3 How do the horizons of the n black holes separate the spacetime, and what is the topology of the interior region?
• RQ4 Can similar multi-black-hole configurations be constructed in 4D AdS spacetime with non-spherical horizon topologies?
• RQ5 What role do discrete isometry groups play in generating consistent multi-black-hole solutions via quotient constructions?

Key findings

• The multi-black-hole (MBH) spacetime in 3D AdS is constructed as a quotient of global AdS space by a discrete group of isometries, yielding n asymptotically AdS regions.
• Each asymptotically AdS region is locally indistinguishable from a single BTZ black hole, confirming the external geometry is BTZ-like.
• The n black holes are separated by n distinct event horizons, all meeting at a common interior region that forms a closed universe.
• The interior region contains n black holes and is globally compact, with no external asymptotic regions.
• The construction generalizes to 4D AdS spacetime, producing solutions with toroidal and higher-genus horizon topologies.
• The solutions are globally smooth and satisfy the vacuum Einstein equations with negative cosmological constant, confirming their physical consistency.