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

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 Poincaré 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.

15 pages, revtex, to appear in Phys. Rev. D

References in corpus (2)

• Sufficient Conditions for Apparent Horizons in Spherically Symmetric Initial Data
• Geometric Bounds in Spherically Symmetric General Relativity

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• Analytical studies on the hoop conjecture in charged curved spacetimes
• Geometric Bounds in Spherically Symmetric General Relativity
• Investigations on hoop conjecture for horizonless spherical charged stars