The two-body problem in 2+1 spacetime dimensions with negative cosmological constant: two point particles

TL;DR

This paper analyzes the two-body problem in 2+1-dimensional gravity with a negative cosmological constant, deriving analytic expressions for system parameters and exploring conditions for black hole formation, binary systems, or closed universes.

Contribution

It provides the first analytic expressions for total mass and spin in a two-body system in 2+1D AdS space, and conjectures the black hole formation threshold based on these parameters.

Findings

Derived analytic formulas for total mass and spin of two-body systems.

Conjectured black hole formation threshold at total mass equals total spin magnitude.

Classified global geometries as black holes, binary systems, or closed universes.

Abstract

We work towards the general solution of the two-body problem in 2+1-dimensional general relativity with a negative cosmological constant. The BTZ solutions corresponding to black holes, point particles and overspinning particles can be considered either as objects in their own right, or as the exterior solution of compact objects with a given mass $M$ and spin $J$, such as rotating fluid stars. We compare and contrast the metric approach to the group-theoretical one of characterising the BTZ solutions as identifications of 2+1-dimensional anti-de Sitter spacetime under an isometry. We then move on to the two-body problem. In this paper, we restrict the two objects to the point particle range $|J|-1\le M<-|J|$, or their massless equivalents, obtained by an infinite boost. (Both anti-de Sitter space and massless particles have $M=-1$, $J=0$). We derive analytic expressions for the total mass $M_\text{tot}$ and spin $J_\text{tot}$ of the system in terms of the six gauge-invariant parameters of the two-particle system: the rest mass and spin of each object, and the impact parameter and energy of the orbit. Based on work of Holst and Matschull on the case of two massless, nonspinning particles, we conjecture that the black hole formation threshold is $M_\text{tot}=|J_\text{tot}|$. The threshold solutions are then extremal black holes. We determine when the global geometry is a black hole, an eternal binary system, or a closed universe.