The Geometry of Warped Product Singularities

TL;DR

This paper investigates the structure of degenerate warped products of singular semi-Riemannian manifolds, demonstrating their semi-regularity and applications to cosmology and black hole models with singularities.

Contribution

It establishes conditions under which degenerate warped products are semi-regular and expresses their geometric properties in terms of factor manifolds, with applications to Einstein's equations.

Findings

Warped products can be semi-regular under certain conditions.

Connection and curvature are expressed via factor manifolds.

Applications to cosmological and black hole singularities.

Abstract

In this article the degenerate warped products of singular semi-Riemannian manifolds are studied. They were used recently by the author to handle singularities occurring in General Relativity, in black holes and at the big-bang. One main result presented here is that a degenerate warped product of semi-regular semi-Riemannian manifolds with the warping function satisfying a certain condition is a semi-regular semi-Riemannian manifold. The connection and the Riemann curvature of the warped product are expressed in terms of those of the factor manifolds. Examples of singular semi-Riemannian manifolds which are semi-regular are constructed as warped products. Applications include cosmological models and black holes solutions with semi-regular singularities. Such singularities are compatible with a certain reformulation of the Einstein equation, which in addition holds at semi-regular singularities too.