Spacetime constructed from a contact manifold with a degenerate metric

TL;DR

This paper constructs a novel four-dimensional spacetime from a contact manifold with a degenerate metric, leading to explicit Einstein solutions involving null dust and cosmic strings.

Contribution

It introduces a new method to build spacetimes from contact geometry with degenerate metrics, providing explicit Einstein solutions with arbitrary matter functions.

Findings

The constructed spacetime satisfies Einstein's equations with null dust and cosmic strings.

The spacetime is Petrov type D when cosmic strings are present, otherwise conformally flat.

Explicit solutions are analyzed for simple matter densities.

Abstract

We construct a four-dimensional spacetime using a three-dimensional contact manifold equipped with a degenerate metric. The degenerate metric is set to be compatible with the contact structure. The compatibility condition is defined in this paper. Our construction yields a Ricci tensor of a particularly simple form, which leads to a solution of the Einstein equation with a null dust and cosmic strings. The solution includes two arbitrary functions: the energy density of the null dust and the number density of the cosmic strings. When there exist the cosmic strings, the spacetime is of Petrov type D. Otherwise, the spacetime is conformally flat. For some simple matter densities, we examine the Einstein equation in detail.