Generalized junction conditions for discontinuous metrics

TL;DR

This paper extends the Darmois-Israel junction formalism to discontinuous metrics using Colombeau algebras, allowing for consistent nonlinear operations with singular quantities and revealing new geometric degrees of freedom at space-time boundaries.

Contribution

It introduces a generalized junction condition framework for discontinuous metrics, incorporating higher-order singular terms and broadening the applicability of junction conditions in space-time geometry.

Findings

Includes higher-order singular terms in curvature and energy-momentum tensor.

Recovers traditional junction conditions as a special case.

Provides a mathematically consistent approach for discontinuous metrics.

Abstract

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.