Doubly covariant action principle of singular hypersurfaces in general relativity and scalar-tensor theories

TL;DR

This paper develops a doubly covariant action principle for singular hypersurfaces in general relativity and scalar-tensor theories, allowing independent variations of hypersurface position and fields without symmetry assumptions.

Contribution

It introduces a doubly covariant variational principle for hypersurfaces in gravity theories, enabling independent variations of hypersurface position and fields.

Findings

Derivation of Einstein equations off the hypersurface

Formulation of Israel's junction conditions in a covariant manner

Equations of motion for matter and scalar fields

Abstract

An action principle of singular hypersurfaces in general relativity and scalar-tensor type theories of gravity in the Einstein frame is presented without assuming any symmetry. The action principle is manifestly doubly covariant in the sense that coordinate systems on and off a hypersurface are disentangled and can be independently specified. It is shown that, including variation of the metric, the position of the hypersurface and matter fields, the variational principle gives the correct set of equations of motion: the Einstein equation off the hypersurface, Israel's junction condition in a doubly covariant form and equations of motion of matter fields including the scalar fields. The position of the hypersurface measured from one side of the hypersurface and that measured from another side can be independently variated as required by the double covariance.