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First order perturbations of hypersurfaces of arbitrary causal character

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 (2005 Class. Quantum Grav. 22 3325–48) 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.