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

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.