Three Puzzles with Covariance and Supertranslation Invariance of Angular Momentum Flux (with Solutions)

TL;DR

This paper addresses three conceptual puzzles in the covariant and supertranslation-invariant formulations of angular momentum flux in asymptotically flat spacetimes, providing solutions that clarify the roles of auxiliary degrees of freedom and reference frames.

Contribution

It introduces covariant formulas that resolve ambiguities and clarify the relationship between supertranslation invariance, auxiliary degrees of freedom, and Lorentz charges.

Findings

Resolved the invariance under spacetime translations

Clarified the role of auxiliary degrees of freedom in flux formulas

Explained the distinction between covariant and canonical charges

Abstract

We describe and solve three puzzles arising in covariant and supertranslation-invariant formulas for the flux of angular momentum and other Lorentz charges in asymptotically flat spacetimes: 1) Supertranslation-invariance and covariance imply invariance under spacetime translations; 2) the flux depends on redundant auxiliary degrees of freedom that cannot be set to zero in all Lorentz frames without breaking Lorentz covariance; 3) supertranslation-invariant Lorentz charges do not generate the transformations of the Bondi mass aspect implied by the isometries of the asymptotic metric. In this letter, we solve the first two puzzles by presenting covariant formulas that unambiguously determine the auxiliary degrees of freedom and clarify the last puzzle by explaining the different role played by covariant and canonical charges. Our construction makes explicit the choice of reference frame underpinning seemingly unambiguous results presented in the current literature.