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

We show that the recipe for computing the expansions ${\theta }_{\mathrm{\ell }}$ 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 ${\theta }_{\mathrm{\ell }}\to \kappa {\theta }_{\mathrm{\ell }}$, $\theta_n \to \kappa^{-1} \theta_n$\mathrm{.}Underthismapping,theproduct$\theta_\ell \theta_n$ is invariant, and thus the marginal surface computed from the vanishing of ${\theta }_{\mathrm{\ell }}$, 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.