Covariance Group for Null Geodesic Expansion Calculations, and its Application to the Apparent Horizon

Abstract

We show that the recipe for computing the expansions ${\theta }_{\ell }$ and ${\theta }_n$ of outgoing and ingoing null geodesics normal to a surface admits a covariance group with nonconstant scalar $\kappa (x)$, corresponding to the mapping ${\theta }_{\ell }\to \kappa {\theta }_{\ell }$, ${\theta }_n\to {\kappa }^{-1}{\theta }_n$. Under this mapping, the product ${\theta }_{\ell }{\theta }_n$ is invariant, and thus the marginal surface computed from the vanishing of ${\theta }_{\ell }$, which is used to define the apparent horizon, is invariant. This covariance group naturally appears in comparing the expansions computed with different choices of coordinate system.

Latex, 6 pages

References in corpus (1)

• Quasi-local black hole horizons