Gravitation Singularities of the Caustic Type

TL;DR

This paper explores the nature of gravitation singularities by analyzing their relation to caustics and foliations in space-time, proposing a criterion based on singularities of space-time distributions and their geometric properties.

Contribution

It introduces a novel criterion for gravitation singularities using the framework of foliations, caustics, and Lagrange maps, linking singularities to geometric and topological features.

Findings

Singularities resemble foliation singularities of level surfaces of a real function.

Multi-valued functions lead to caustics via Lagrange map projections.

The approach connects gravitational singularities with geometric structures like caustics and foliations.

Abstract

In view of the well-known correspondence between gravitational fields and space-time distributions on a world manifold X, the criterion of gravitation singularities as singularities of these distributions is suggested. In the germ terms, singularities of a (3+1) distribution look locally like singularities of a foliation whose leaves are level surfaces of a real function f on X. If f is a single-valued function, changes of leave topology at critical points of f take place. In case of a multi-valued function f, one can lift the foliation to the total space of the cotangent bundle over X, then extend it over branch points of f and project this extension onto X. Singular points of this projection constitute a Lagrange map caustic by Arnol'd.