Relaxing the Parity Conditions of Asymptotically Flat Gravity

TL;DR

This paper redefines four-dimensional asymptotically flat spacetimes at spatial infinity without parity restrictions, introduces an anomalous counter-term for the Einstein-Hilbert action, and clarifies the structure of asymptotic symmetries and charges.

Contribution

It removes parity conditions in the definition of asymptotically flat spacetimes and analyzes the resulting symmetry charges and anomalies.

Findings

Poincaré and supertranslation charges are finite and conserved.

Lorentz charges are nonlinear functionals of asymptotic fields.

Supertranslations are shown to be pure gauge in the covariant phase space.

Abstract

Four-dimensional asymptotically flat spacetimes at spatial infinity are defined from first principles without imposing parity conditions or restrictions on the Weyl tensor. The Einstein-Hilbert action is shown to be a correct variational principle when it is supplemented by an anomalous counter-term which breaks asymptotic translation, supertranslation and logarithmic translation invariance. Poincar\'e transformations as well as supertranslations and logarithmic translations are associated with finite and conserved charges which represent the asymptotic symmetry group. Lorentz charges as well as logarithmic translations transform anomalously under a change of regulator. Lorentz charges are generally non-linear functionals of the asymptotic fields but reduce to well-known linear expressions when parity conditions hold. We also define a covariant phase space of asymptotically flat spacetimes with parity conditions but without restrictions on the Weyl tensor. In this phase space, the anomaly plays classically no dynamical role. Supertranslations are pure gauge and the asymptotic symmetry group is the expected Poincar\'e group.