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

We show that the recipe for computing expansions [Formula: see text] and of outgoing ingoing null geodesics normal to a surface admits covariance group with nonconstant scalar text], corresponding mapping text]. Under this mapping, product is invariant, thus marginal computed from vanishing which used define apparent horizon, invariant. This naturally appears in comparing different choices coordinate system.