Narrow Results

We show that the recipe for computing the expansions ${𝜃}_{\ell }$ and ${𝜃}_n$ of outgoing and ingoing null geodesics normal to a surface admits a covariance group with nonconstant scalar $\kappa (x)$, corresponding to the mapping ${𝜃}_{\ell }\to \kappa {𝜃}_{\ell }$, ${𝜃}_n\to {\kappa }^{-1}{𝜃}_n$. Under this mapping, the product ${𝜃}_{\ell }{𝜃}_n$ is invariant, and thus the marginal surface computed from the vanishing of ${𝜃}_{\ell }$, 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.