Thorny Spheres and Black Holes with Strings

TL;DR

This paper introduces thorny spheres with conical singularities to generate new spherically symmetric solutions of Einstein's equations describing spacetimes with multiple cosmic strings, establishing conditions for their equilibrium configurations.

Contribution

It demonstrates how thorny spheres can be used to model spacetimes with multiple cosmic strings and derives the equilibrium conditions for their configurations.

Findings

Arbitrary string configurations are possible under certain equilibrium conditions.

The equilibrium condition can be expressed as a force-balance in a flat 3-space.

For small tensions, the constraints simplify to linear relations between string directions.

Abstract

We consider thorny spheres, that is 2-dimensional compact surfaces which are everywhere locally isometric to a round sphere $S^2$ except for a finite number of isolated points where they have conical singularities. We use thorny spheres to generate, from a spherically symmetric solution of the Einstein equations, new solutions which describe spacetimes pierced by an arbitrary number of infinitely thin cosmic strings radially directed. Each string produces an angle deficit proportional to its tension, while the metric outside the strings is a locally spherically symmetric solution. We prove that there can be arbitrary configurations of strings provided that the directions of the strings obey a certain equilibrium condition. In general this equilibrium condition can be written as a force-balance equation for string forces defined in a flat 3-space in which the thorny sphere is isometrically embedded, or as a constraint on the product of holonomies around strings in an alternative 3-space that is flat except for the strings. In the case of small string tensions, the constraint equation has the form of a linear relation between unit vectors directed along the string axes.