Covariance group for null geodesic expansion calculations, and its application to the apparent horizon

Abstract

We show that the recipe for computing the expansions [Formula: see text] and [Formula: see text] of outgoing and ingoing null geodesics normal to a surface admits a covariance group with nonconstant scalar [Formula: see text], corresponding to the mapping [Formula: see text], [Formula: see text]. Under this mapping, the product [Formula: see text] is invariant, and thus the marginal surface computed from the vanishing of [Formula: see text], 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.