Investigating the junction conditions for a discontinuous metric

In this work, the Darmois-Israel junction formalism is extended to the case of discontinuous metrics by employing the framework of Colombeau simplified generalized functions. This formalism provides a mathematically consistent treatment of nonlinear operations involving singular quantities, such as products of distributions, where Colombeau algebra acts as a differential algebra containing distributions as a subspace and C∞ as a faithful subalgebra. By relaxing the continuity conditions traditionally adopted by Darmois and Israel, the junction conditions naturally incorporate higher-order singular terms in the curvature and the surface energy-momentum tensor, representing nonlinear, singular geometric structures associated with genuine spacetime discontinuities. As a concrete application, we analyze Kerr-Schild spacetimes within this generalized setting, demonstrating how singular metrics and associated curvature tensors are rigorously accommodated. The resulting formalism recovers the traditional Darmois-Israel conditions as a limiting case, while providing a consistent framework applicable to geometric boundaries and abrupt transitions in spacetime.