The Weyl double copy in vacuum spacetimes with a cosmological constant

TL;DR

This paper explores the Weyl double copy relation in vacuum spacetimes with a cosmological constant, revealing how the zeroth and single copies relate to electromagnetic and gravitational fields, especially in time-dependent solutions.

Contribution

It extends the Weyl double copy framework to include vacuum solutions with a cosmological constant, highlighting the role of the zeroth copy in time-dependent gravitational waves.

Findings

Zeroth and single copies satisfy conformally invariant equations in conformally flat spacetime.

Zeroth copy links Dirac-Weyl fields with electromagnetic fields in curved spacetime.

Zeroth copy provides additional information about the nature of gravitational wave sources.

Abstract

We examine the Weyl double copy relation for vacuum solutions of the Einstein equations with a cosmological constant using the approach we previously described, in which the spin-1/2 massless free-field spinors (Dirac-Weyl fields) are regarded as basic units. Based on the exact non-twisting vacuum type N and vacuum type D solutions, the finding explicitly shows that the single and zeroth copies fulfill conformally invariant field equations in conformally flat spacetime. In addition, irrespective of the presence of a cosmological constant, we demonstrate that the zeroth copy connects Dirac-Weyl fields with the degenerate electromagnetic fields in the curved spacetime in addition to connecting gravity fields with the single copy in conformally flat spacetime. Moreover, the study also demonstrates the critical significance the zeroth copy plays in time-dependent radiation solutions. In particular, for Robinson-Trautman ($\Lambda$) gravitational waves, unlike the single copy, we find that the zeroth copy carries additional information to specify whether the sources of associated gravitational waves are time-like, null, or space-like, at least in the weak field limit.