Necessary Conditions for Apparent Horizons and Singularities in Spherically Symmetric Initial Data

TL;DR

This paper derives simple necessary conditions for the formation of apparent horizons and singularities in spherically symmetric initial data in general relativity, linking energy density and support size, with implications for understanding gravitational collapse.

Contribution

It introduces new necessary inequalities involving energy density and support size that prevent apparent horizons and singularities under certain conditions in spherically symmetric initial data.

Findings

Bound on energy density and support size prevents apparent horizons.

Inequalities depend weakly on gauge choices.

Method uses Poincaré inequalities for spatial scalars.

Abstract

We establish necessary conditions for the appearance of both apparent horizons and singularities in the initial data of spherically symmetric general relativity when spacetime is foliated extrinsically. When the dominant energy condition is satisfied these conditions assume a particularly simple form. Let $\rho_{Max}$ be the maximum value of the energy density and $\ell$ the radial measure of its support. If $\rho_{Max}\ell^2$ is bounded from above by some numerical constant, the initial data cannot possess an apparent horizon. This constant does not depend sensitively on the gauge. An analogous inequality is obtained for singularities with some larger constant. The derivation exploits Poincar\'e type inequalities to bound integrals over certain spatial scalars. A novel approach to the construction of analogous necessary conditions for general initial data is suggested.