Energy Theorem for 2+1 dimensional gravity

TL;DR

This paper proves a positive energy theorem in 2+1 dimensional gravity, showing energy-momentum behavior for open universes with matter satisfying the dominant energy condition, extending previous results to more general matter and anti de Sitter space.

Contribution

It generalizes positive energy theorems in 2+1 gravity to include arbitrary matter satisfying the dominant energy condition and anti de Sitter space.

Findings

Energy-momentum is future directed and time-like for enclosed systems.

Energy reaches a maximum value of 1/4G corresponding to a deficit angle of 2π.

System evolves into either a closed universe or a Gott-like universe.

Abstract

We prove a positive energy theorem in 2+1 dimensional gravity for open universes and any matter energy-momentum tensor satisfying the dominant energy condition. We consider on the space-like initial value surface a family of widening Wilson loops and show that the energy-momentum of the enclosed subsystem is a future directed time-like vector whose mass is an increasing function of the loop, until it reaches the value $1/4G$ corresponding to a deficit angle of $2\pi$. At this point the energy-momentum of the system evolves, depending on the nature of a zero norm vector appearing in the evolution equations, either into a time-like vector of a universe which closes kinematically or into a Gott-like universe whose energy momentum vector, as first recognized by Deser, Jackiw and 't Hooft is space-like. This treatment generalizes results obtained by Carroll, Fahri, Guth and Olum for a system of point-like spinless particle, to the most general form of matter whose energy-momentum tensor satisfies the dominant energy condition. The treatment is also given for the anti de Sitter 2+1 dimensional gravity.