Axisymmetric evolution of Einstein equations and mass conservation

TL;DR

This paper proves the existence of a gauge in axisymmetric Einstein evolution where the total mass remains conserved and can be expressed as a positive definite integral, aiding in understanding global existence.

Contribution

It introduces a gauge fixing that ensures mass conservation as a positive definite integral in axisymmetric Einstein equations, with implications for global existence.

Findings

Mass integral is positive definite and conserved

Mass controls extrinsic curvature and metric derivatives

Application to global existence in axial symmetry

Abstract

For axisymmetric evolution of isolated systems, we prove that there exists a gauge such that the total mass can be written as a positive definite integral on the spacelike hypersurfaces of the foliation and the integral is constant along the evolution. The conserved mass integral controls the square of the extrinsic curvature and the square of first derivatives of the intrinsic metric. We also discuss applications of this result for the global existence problem in axial symmetry.