Hypersurfaces in Einstein-Riemann space and their compatibility conditions

AbstractThe following article deals with the differential geometry of those three-dimensional hypersurfaces Σ which are associated with the boundaries of moving bodies in the general theory of relativity. Compatibility conditions of the second and third orders are formulated for arbitrary coordinate systems under the assumption that second order discontinuities in the metric structure occur over the hypersurfaces Σ. The space-time representations of the various general results are considered with particular attention to the case of stationary surfaces since it is anticipated that the theory of discontinuities over such surfaces will provide the mathematical groundwork for physically significant new developments