First and Second Order Perturbations of Hypersurfaces

TL;DR

This paper derives general formulas for the first and second order perturbations of the geometry of hypersurfaces in spacetimes, accounting for metric changes and hypersurface displacements, applicable in any dimension and signature.

Contribution

It provides the first comprehensive derivation of second order perturbations of hypersurface geometry, including hypersurface movement, in arbitrary-dimensional spacetimes.

Findings

Formulas for first and second order perturbations of induced metric and extrinsic curvature.

Application to perturbed matching theory between spacetimes.

Results are fully general, valid in arbitrary dimensions and signatures.

Abstract

In this paper we find the first and second order perturbations of the induced metric and the extrinsic curvature of a non-degenerate hypersurface $\Sigma$ in a spacetime $(M,g)$, when the metric $g$ is perturbed arbitrarily to second order and the hypersurface itself is allowed to change perturbatively (i.e. to move within spacetime) also to second order. The results are fully general and hold in arbitrary dimensions and signature. An application of these results for the perturbed matching theory between spacetimes is presented.