Hypertranslations and Hyperrotations

TL;DR

This paper explores the asymptotic symmetries of Einstein gravity in flat space using a special double null gauge, revealing four new classes of diffeomorphisms called hypertranslations and hyperrotations, which extend the known BMS algebra.

Contribution

It introduces and classifies four new asymptotic diffeomorphisms in flat space gravity, expanding the BMS algebra with hypertranslations and hyperrotations.

Findings

Identifies four new asymptotic symmetry functions: hypertranslations and hyperrotations.

Proves the completeness of these symmetries under certain fall-off conditions.

Derives a four-fold infinite extension of the BMS algebra.

Abstract

We study the asymptotic symmetries of Einstein gravity in flat space. Instead of Bondi gauge, we work with the recently introduced special double null gauge, in which $\mathscr{I}^{+}$ and $\mathscr{I}^{-}$ are approached along null directions. We find four new functions worth of asymptotic diffeomorphisms beyond the familiar supertranslations and superrotations, which are of relevance in discussions of finite surface charges. Two of these arise from angle-dependent shifts in the $v$-coordinate near $\mathscr{I}^{+}$. We call these hypertranslations and sub-leading hypertranslations, with analogous statements in the $u$-coordinate near $\mathscr{I}^{-}$. There are also two Diff$(S^2)$ transformations, which we call hyperrotations, that are sub-leading to the Virasoro superrotations. With power law fall-offs in the null coordinate and the standard metric on the sphere at leading order, we prove that this is the exhaustive list of diffeomorphisms whose associated metric parameters can show up in the (finite) surface charges. We compute the algebra of the asymptotic Killing vectors under the Barnich-Troessaert bracket, and find a four-fold infinite generalization of the BMS algebra.