Decoupling the momentum constraints in general relativity

TL;DR

This paper introduces a novel 2+1 decomposition method for vacuum initial conditions in general relativity, simplifying the momentum constraints and enabling explicit solutions, including initial data with trapped surfaces and Kerr generalizations.

Contribution

It provides a new approach to decouple and solve momentum constraints in general relativity using a 2+1 decomposition and quasi isotropic coordinates, with explicit solution techniques.

Findings

Decoupling of momentum constraints in specific coordinates

Solution of the constraints via quadrature and integral expressions

Construction of initial data with trapped surfaces and Kerr generalizations

Abstract

We present a 2+1 decomposition of the vacuum initial conditions in general relativity. For a constant mean curvature one of the momentum constraints decouples in quasi isotropic coordinates and it can be solved by quadrature. The remaining momentum constraints are written in the form of the tangential Cauchy-Riemann equation. Under additional assumptions its solutions can be written in terms of integrals of known functions. We show how to obtain initial data with a marginally outer trapped surface. A generalization of the Kerr data is presented.