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-Nordström 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.
Highlights
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 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
For the Gibbons-Hawking metrics, the anti-self-dual two form field strength derived from the Killing vector is expanded in the triplet of anti-self-dual two forms and we show that these coefficients determine the Weyl tensor of the metric via two terms — the first being a direct tensor doubling formula as in section 2, and the second involving derivatives of the Abelian field strength
Summary
(The Killing vector-defined field strength may satisfy a self-duality condition or have this imposed.) We will study examples of both situations below. In the first case, where we construct F from the Killing vector K, we note that KμFμν is closed. This follows from application of Cartan’s formula for the Lie derivative: LK F = iK dF + diK F (2.5). Generically there are two principal null directions satisfying (2.11) which implies that the spacetime satisfies the conditions for falling within the type D class in the appropriate higher-dimensional classification [99] (see the overview [100]). The spacetime is said to fall into the type II class
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