Background ambiguity and the G\"odel double copy

TL;DR

This paper examines the assumptions behind the classical double copy in curved spacetimes, using the G"odel universe as a case study, and highlights the importance of background choice for interpreting single-copy fields.

Contribution

It introduces a new perspective on the background dependence of the double copy, especially for non-Kerr--Schild spacetimes like G"odel, and compares different approaches to single-copy interpretation.

Findings

Weyl double copy effectively captures G"odel's properties.

Flat-space single-copy interpretation is inconsistent with the curved spacetime.

Curved-space single copy aligns with gravitoelectromagnetism analogies.

Abstract

In this work, we investigate the assumptions regarding spacetime backgrounds underlying the classical double copy. We argue (contrary to the norm) that single-copy fields naturally constructed on the original curved background metric are only interpretable on a flat metric when such a well-defined limit exists, for which Kerr--Schild coordinates offer a natural choice. As an explicit example where such a distinction matters, we initiate an exploration of single-copies for the G\"odel universe. This metric lacks a (geodesic) Kerr--Schild representation yet is Petrov type-D, meaning the technology of the ``Weyl double copy" may be utilized. The Weyl derived single copy has many desirable features, including matching the defining properties of the spacetime, and being sourced by the mixed Ricci tensor just as Kerr--Schild single copies are. To compare, we propose a sourced flat-space single-copy interpretation for the G\"odel metric by leveraging its symmetries, and find that this proposal lacks the defining properties of the spacetime, and is not consistent with the flat limit of our curved-space single copy. Notably, this inconsistency does not occur in Kerr--Schild metrics. Our curved-space single copy also lead to the same electromagnetic analogue of the G\"odel universe found separately through tidal force analogies, opening a new avenue of exploration between the double copy and gravitoelectromagnetism programs.