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Generalized junction conditions for discontinuous metrics

J. A. Silva, F. C. Carvalho, Antonio R. G. Garcia

TL;DR

This work tackles the problem of formulating junction conditions when the spacetime metric is discontinuous across a hypersurface, a situation where classical distribution theory fails due to nonlinear curvature terms. It adopts Colombeau algebras to regularize and manipulate singular quantities, deriving a generalized Darmois-Israel framework that accommodates $δ(l)$, $δ'(l)$, and $δ^2(l)$-type contributions to the Einstein equations. The resulting energy-momentum content includes the standard surface term $S_{ab}$ plus new terms $W_{ab}$ and $J_{ab}$, reflecting additional geometric degrees of freedom introduced by the metric discontinuity. These results recover the traditional conditions in the smooth limit and provide a mathematically consistent platform for exploring non-smooth spacetime boundaries, with potential relevance to geometric boundaries, domain walls, and modified gravity theories. $[g_{ab}] \neq 0$ leads to a richer, yet well-defined, distributional structure that extends the classical thin-shell formalism into the realm of genuinely discontinuous geometries.

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.

Generalized junction conditions for discontinuous metrics

TL;DR

This work tackles the problem of formulating junction conditions when the spacetime metric is discontinuous across a hypersurface, a situation where classical distribution theory fails due to nonlinear curvature terms. It adopts Colombeau algebras to regularize and manipulate singular quantities, deriving a generalized Darmois-Israel framework that accommodates , , and -type contributions to the Einstein equations. The resulting energy-momentum content includes the standard surface term plus new terms and , reflecting additional geometric degrees of freedom introduced by the metric discontinuity. These results recover the traditional conditions in the smooth limit and provide a mathematically consistent platform for exploring non-smooth spacetime boundaries, with potential relevance to geometric boundaries, domain walls, and modified gravity theories. leads to a richer, yet well-defined, distributional structure that extends the classical thin-shell formalism into the realm of genuinely discontinuous geometries.

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.
Paper Structure (9 sections, 2 theorems, 84 equations)

This paper contains 9 sections, 2 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic Notions of the Colombeau Formalism
  4. Formulation of the discontinuous spacetime metric
  5. Geometric implications of metric discontinuity
  6. Deduction of junction conditions for discontinuous metrics
  7. Possible Physical Interpretations of Singular Terms on $\Sigma$
  8. Summary and Conclusions
  9. Geometric and distributional properties of $\delta'(l)$

Key Result

Lemma 1

The function $f(u) = \int_{-\infty}^{u}\varphi(s)\, ds - \dfrac{1}{2},$ where $\varphi_\varepsilon(x) = \dfrac{1}{\varepsilon}\varphi\!\left(\dfrac{x}{\varepsilon}\right)$ is a delta net, is an odd function.

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof