Weyl double copy and massless free fields in curved spacetimes

TL;DR

This paper explores the connections between gravity, gauge fields, and spinor formalism in curved spacetimes, extending the Weyl double copy to various Petrov types and revealing fundamental features of massless free fields.

Contribution

It introduces a map between Weyl and Dirac-Weyl fields, systematically rebuilds the Weyl double copy for specific spacetime types, and uncovers new links between gravity and gauge theories in curved backgrounds.

Findings

Higher spin massless fields can be constructed from Dirac-Weyl spinors.

The Weyl double copy is extended to Petrov types N, D, and III.

A deep connection between gravity and gauge fields is revealed.

Abstract

In spinor formalism, since any massless free-field spinor with spin higher than $1/2$ can be constructed with spin-1/2 spinors (Dirac-Weyl spinors) and scalars, we introduce a map between Weyl fields and Dirac-Weyl fields. We determine the corresponding Dirac-Weyl spinors in a given empty spacetime. Regarding them as basic units, other higher spin massless free-field spinors are then identified. Along this way, we find some hidden fundamental features related to these fields. In particular, for non-twisting vacuum Petrov type N solutions, we show that all higher spin massless free-field spinors can be constructed with one type of Dirac-Weyl spinor and the zeroth copy. Furthermore, we systematically rebuild the Weyl double copy for non-twisting vacuum type N and vacuum type D solutions. Moreover, we show that the zeroth copy not only connects the gravity fields with a single copy but also connects the degenerate Maxwell fields with the Dirac-Weyl fields in the curved spacetime, both for type N and type D cases. Besides, we extend the study to non-twisting vacuum type III solutions. We find a particular Dirac-Weyl scalar independent of the proposed map and whose square is proportional to the Weyl scalar. A degenerate Maxwell field and an auxiliary scalar field are then identified. Both of them play similar roles as the Weyl double copy. The result further inspires us that there is a deep connection between gravity theory and gauge theory.