Structure of Malicious Singularities

TL;DR

This paper explores the structure of singularities in spacetime using noncommutative geometry and groupoid algebras, revealing how singularities relate to group representations and their induced systems.

Contribution

It introduces a novel approach to analyze spacetime singularities as noncommutative spaces via groupoid algebras and applies the Mackey theorem to understand singular fiber structures.

Findings

Establishes a correspondence between groupoid representations and systems of imprimitivity.

Uses Mackey theorem to analyze the structure of singular fibers.

Identifies subgroup $K$ as a measure of singularity complexity.

Abstract

We investigate spacetimes with their singular boundaries as noncommutative spaces. Such a space is defined by a noncommutative algebra on a transformation groupoid $\Gamma = E \times G$, where $E$ is the total space of the frame bundle over spacetime with its singular boundary, and $G$ its structural group. There is a bijective correspondence between unitary representations of the groupoid $\Gamma $ and the systems of imprimitivity of the group $G$. This allows us to apply the Mackey theorem, and deduce from it some information concerning singular fibres of the groupoid. A subgroup $K$ of $G$, from which -- according to the Mackey theorem -- the representation is induced to the whole of $G$, can be regarded as measuring the "richness" of the singularity structure.