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Generalized junction conditions for discontinuous metrics

In this work, the Darmois-Israel junction formalism is extended to the case of discontinuous metrics within the framework of Colombeau algebras of generalized functions. This formulation provides a mathematically consistent treatment of nonlinear operations involving singular quantities, such as products and derivatives of distributions. By relaxing the usual continuity condition on the metric, the generalized junction conditions naturally include higher-order singular terms in the curvature and in the surface energy-momentum tensor. These additional contributions represent new geometric degrees of freedom associated with genuine discontinuities in the space-time geometry. The resulting formalism recovers the traditional Darmois-Israel conditions as a limiting case, while offering a coherent extension applicable to geometric boundaries and abrupt transitions in space-time.