Type D Spacetimes and the Weyl Double Copy

TL;DR

The paper develops the Weyl double copy, a curvature-based map $C_{ABCD} = \frac{1}{S} f_{(AB} f_{CD)}$ that links four-dimensional vacuum gravity of algebraic type D to Maxwell fields on flat space. By leveraging spinorial methods, it shows consistency with the Kerr-Schild double copy and supplies explicit mappings for the general Plebanski-Demianski type D family, including the C-metric where the gravity-to-gauge correspondence corresponds to a Liénard-Wiechert potential for accelerated charges. The Eguchi-Hanson instanton is analyzed as both a pure and a mixed Weyl double copy, illustrating how curvature data can arise from single or paired gauge-field solutions, and hinting at broader structures beyond type D. The work broadens the classical double copy program to exact solutions, offering new exact correspondences and guiding questions about higher dimensions, cosmological constants, and the inclusion of additional fields such as dilatons and two-forms.

Abstract

We study the double-copy relation between classical solutions in gauge theory and gravity, focusing on four-dimensional vacuum metrics of algebraic type D, a class that includes several important solutions. We present a double copy of curvatures that applies to all spacetimes of this type -- the Weyl double copy -- relating the curvature of the spacetime to an electromagnetic field strength. We show that the Weyl double copy is consistent with the previously known Kerr-Schild double copy, and in fact resolves certain ambiguities of the latter. The most interesting new example of the classical double copy presented here is that of the C-metric. This well-known solution, which represents a pair of uniformly accelerated black holes, is mapped to the Lienard-Wiechert potential for a pair of uniformly accelerated charges. We also present a new double-copy interpretation of the Eguchi-Hanson instanton.

Figures (1)

• Figure 1: The curved lines ($r=0$) represent two causally disconnected uniformly accelerated particles. The diagonal lines ($r=1/\alpha$) represent the acceleration horizons, which split the plot into four quadrants.