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Symmetry-preserving matchings. (English) Zbl 1016.83011

In the literature, the matchings between spacetimes have been most of the time implicitly assumed to preserve some of the symmetries of the problem involved. But no definition for this kind of matching was given until recently.

The purpose of this paper is twofold: firstly, to motivate and present the definition of symmetry-preserving matching; and secondly, to show two immediate consequences on the algebraic and geometric properties of the group preserved by the matching. These results can lead to restrictions on the physical properties of the model. In fact, it is shown here that some of the ‘hidden’ assumptions can be derived from the junction conditions as a consequence of the preservation of the symmetry. Loosely speaking, the definition states that the matching hypersurface is restricted to be tangent to the orbits of a desired local group to be admitted by the whole matched spacetime. Its strict use leads to more general parametrizations than usual, which eventually manifests in more general ways of performing the matching, some of them with clear physical differences.

The definition also has geometric consequences. First, the matching hypersurface inherits the preserved symmetry and its algebraic type, and as a result, the algebraic type of the preserved group is necessarily kept across the matching. Second, the point-wise property that, when satisfied in an open set, ensures the orthogonal transitivity of any two-dimensional conformal ${G}_{2}$ group (including, of course, any ${G}_{2}$ group of isometries) is also kept on the matching hypersurface if the matching preserves that symmetry.

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 manifested.

##### MSC:
 83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems) 83C55 Macroscopic interaction of the gravitational field with matter (general relativity)