Flat foliations of spherically symmetric geometries

TL;DR

This paper investigates flat foliations in spherically symmetric spacetimes, demonstrating their consistency, bounded extrinsic curvature, and conditions for apparent horizon formation, with implications for gauge invariance.

Contribution

It provides explicit conditions for flat foliation evolution, bounds on extrinsic curvature, and criteria for apparent horizon formation in spherically symmetric general relativity.

Findings

Flat slices can be consistently evolved without singularities for finite sources.

Explicit bounds on the extrinsic curvature are established.

Conditions for apparent horizon formation are shown to be gauge invariant.

Abstract

We examine the solution of the constraints in spherically symmetric general relativity when spacetime has a flat spatial hypersurface. We demonstrate explicitly that given one flat slice, a foliation by flat slices can be consistently evolved. We show that when the sources are finite these slices do not admit singularities and we provide an explicit bound on the maximum value assumed by the extrinsic curvature. If the dominant energy condition is satisfied, the projection of the extrinsic curvature orthogonal to the radial direction possesses a definite sign. We provide both necessary and sufficient conditions for the formation of apparent horizons in this gauge which are qualitatively identical to those established earlier for extrinsic time foliations of spacetime, Phys. Rev. D56 7658, 7666 (1997) which suggests that these conditions possess a gauge invariant validity.