Cauchy problems for Einstein equations in three-dimensional spacetimes

TL;DR

This paper studies the existence and properties of solutions to Einstein equations in three-dimensional spacetimes with negative cosmological constant, introducing new data families and analyzing energy bounds related to angular and linear momentum.

Contribution

It constructs a new family of vacuum spacelike data parameterized by poles at the conformal boundary and reviews global Hamiltonian charges in this context.

Findings

Established lower bounds for energy in terms of angular momentum, linear momentum, and center of mass.

Constructed a new family of vacuum, spacelike data with poles at the boundary.

Analyzed the difficulties in defining global charges in three-dimensional spacetimes.

Abstract

We analyze existence and properties of solutions of two-dimensional general relativistic initial data sets with a negative cosmological constant, both on spacelike and characteristic surfaces. A new family of such vacuum, spacelike data parameterised by poles at the conformal boundary at infinity is constructed. We review the notions of global Hamiltonian charges, emphasising the difficulties arising in this dimension, both in a spacelike and characteristic setting. One or two, depending upon the topology, lower bounds for energy in terms of angular momentum, linear momentum, and center of mass are established.