Weyl doubling

TL;DR

This paper explores algebraic relations between Weyl curvature and Abelian gauge fields in various spacetimes, extending the double copy concept beyond flat backgrounds to curved geometries and brane systems.

Contribution

It introduces generalized Weyl doubling formulas for curved spacetimes, including Gibbons-Hawking and Spin(7) manifolds, and applies these to Einstein-Maxwell and supergravity brane solutions.

Findings

Weyl curvature can be expressed in terms of Abelian field strengths in certain curved spacetimes.

Generalized doubling formulas involve derivatives and are necessary for multi-center Gibbons-Hawking spaces.

A similar doubling relation applies to brane solutions in supergravity, linking field strengths and spacetime curvature.

Abstract

We study a host of spacetimes where the Weyl curvature may be expressed algebraically in terms of an Abelian field strength. These include Type D spacetimes in four and higher dimensions which obey a simple quadratic relation between the field strength and the Weyl tensor, following the Weyl spinor double copy relation. However, we diverge from the usual double copy paradigm by taking the gauge fields to be in the curved spacetime as opposed to an auxiliary flat space. We show how for Gibbons-Hawking spacetimes with more than two centres a generalisation of the Weyl doubling formula is needed by including a derivative-dependent expression which is linear in the Abelian field strength. We also find a type of twisted doubling formula in a case of a manifold with Spin(7) holonomy in eight dimensions. For Einstein Maxwell theories where there is an independent gauge field defined on spacetime, we investigate how the gauge fields determine the Weyl spacetime curvature via a doubling formula. We first show that this occurs for the Reissner-Nordstrom metric in any dimension, and that this generalises to the electrically-charged Born-Infeld solutions. Finally, we consider brane systems in supergravity, showing that a similar doubling formula applies. This Weyl formula is based on the field strength of the p-form potential that minimally couples to the brane and the brane world volume Killing vectors.