# First order perturbations of hypersurfaces of arbitrary causal character

## TL;DR

This paper develops a formalism for analyzing first order perturbations of hypersurfaces with arbitrary causal character, extending previous work limited to specific causal types, and applies it to spacetime matching problems.

## Contribution

It generalizes the perturbation framework to hypersurfaces of any causal character, broadening the applicability of previous methods.

## Key findings

- Extended perturbation formalism to arbitrary causal hypersurfaces
- Applied the formalism to spacetime matching problems
- Provided a unified approach for first order perturbations

## Abstract

In this work we study the problem of first order perturbations of a general hypersurface, i.e. with arbitrary causal character at each point. We extend the framework by Mars (Class. Quantum Grav. 22 3325 (2005)) where this problem was studied to second order for everywhere timelike or spacelike hypersurfaces, and we adapt it to cover the general case. We apply the formalism to the matching of spacetimes across a general hypersurface to first order in perturbation theory.