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

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.