Symmetry-preserving matchings

TL;DR

This paper defines and analyzes symmetry-preserving matchings in spacetimes, establishing conditions for matching hypersurfaces that maintain symmetry properties, with implications for modeling axially symmetric bodies in General Relativity.

Contribution

It introduces a formal definition for symmetry-preserving matchings and derives conditions ensuring the preservation of symmetry groups across the matching hypersurface.

Findings

The algebraic type of the preserved group must be consistent on both sides.

Orthogonal transitivity of conformal groups is preserved on the matching hypersurface.

Results are applicable to higher dimensions and different signatures.

Abstract

In the literature, the matchings between spacetimes have been most of the times implicitly assumed to preserve some of the symmetries of the problem involved. But no definition for this kind of matching was given until recently. Loosely speaking, the matching hypersurface is restricted to be tangent to the orbits of a desired local group of symmetries admitted at both sides of the matching and thus admitted by the whole matched spacetime. This general definition is shown to lead to conditions on the properties of the preserved groups. First, the algebraic type of the preserved group must be kept at both sides of the matching hypersurface. Secondly, the orthogonal transivity of two-dimensional conformal (in particular isometry) groups is shown to be preserved (in a way made precise below) on the matching hypersurface. This result has in particular direct implications on the studies of axially symmetric isolated bodies in equilibrium in General Relativity, by making up the first condition that determines the suitability of convective interiors to be matched to vacuum exteriors. The definition and most of the results presented in this paper do not depend on the dimension of the manifolds involved nor the signature of the metric, and their applicability to other situations and other higher dimensional theories is manifest.