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

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

1997-09-24

arxiv.org/abs/gr-qc/9709065v1

Phys.Rev. D56 (1997) 7666-7673

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

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

Scientific paper

10.1103/PhysRevD.56.7666

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.

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