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

TL;DR

This paper introduces a covariance group for null geodesic expansion calculations, showing that the apparent horizon remains invariant under certain transformations, which aids in comparing different coordinate system computations.

Contribution

It identifies a covariance group with a scalar function that preserves the apparent horizon definition under coordinate transformations.

Findings

The product of expansions $ heta_ ext{l} heta_ ext{n}$ is invariant.

The covariance group allows consistent comparison of expansion calculations across coordinate choices.

The apparent horizon remains unchanged under the covariance transformations.

Abstract

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.