Singular sources in gravity and homotopy in the space of connections

TL;DR

This paper develops a geometric approach using homotopy theory to analyze singularities in gravitational theories, especially in the context of distributional geometries, providing new insights into the uniqueness and structure of such solutions.

Contribution

It introduces a novel geometric construction that represents singular field configurations with smooth fields, clarifying the role of homotopy in the space of connections and distributional Lagrangians.

Findings

Distributional Lagrangians can be uniquely characterized via smooth approximations.

Homotopy theory helps analyze the structure of singularities in gravitational theories.

Application to general relativity reveals new insights into distributional geometries.

Abstract

Suppose a Lagrangian is constructed from its fields and their derivatives. When the field configuration is a distribution, it is unambiguously defined as the limit of a sequence of smooth fields. The Lagrangian may or may not be a distribution, depending on whether there is some undefined product of distributions. Supposing that the Lagrangian is a distribution, it is unambiguously defined as the limit of a sequence of Lagrangians. But there still remains the question: Is the distributional Lagrangian uniquely defined by the limiting process for the fields themselves? In this paper a general geometrical construction is advanced to address this question. We describe certain types of singularities, not by distribution valued tensors, but by showing that the action functional for the singular fields is (formally) equivalent to another action built out of \emph{smooth} fields. Thus we manage to make the problem of the lack of a derivative disappear from a system which gives differential equations. Certain ideas from homotopy and homology theory turn out to be of central importance in analyzing the problem and clarifying finer aspects of it. The method is applied to general relativity in first order formalism, which gives some interesting insights into distributional geometries in that theory. Then more general gravitational Lagrangians in first order formalism are considered such as Lovelock terms (for which the action principle admits space-times more singular than other higher curvature theories).