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

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 $κ(x)$, corresponding to the mapping $θ_\ell \to κθ_\ell$, $θ_n \to κ^{-1} θ_n$. 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.