On Asymptotic Flatness and Lorentz Charges

TL;DR

This paper clarifies the structure of Lorentz charges in asymptotically flat spacetimes and proves the equivalence of different charge definitions without requiring parity conditions, enhancing understanding of spatial infinity.

Contribution

It establishes that Lorentz charges are encoded in dual tensors and proves the equivalence of counter-term and Ashtekar-Hansen charges without parity assumptions.

Findings

Lorentz charges are encoded in dual symmetric tensors.

Equivalence of counter-term and Ashtekar-Hansen charges is proven.

Parity condition on the mass aspect is not needed.

Abstract

In this paper we establish two results concerning four-dimensional asymptotically flat spacetimes at spatial infinity. First, we show that the six conserved Lorentz charges are encoded in two unique, distinct, but mutually dual symmetric divergence free tensors that we construct from the equations of motion. Second, we show that integrability of Einstein's equations in the asymptotic expansion is sufficient to establish the equivalence between counter-term charges defined from the variational principle and charges defined by Ashtekar and Hansen. These results clarify earlier constructions of conserved charges in the hyperboloid representation of spatial infinity. In showing this, parity condition on the mass aspect is not needed. Along the way in establishing these results, we prove two lemmae on tensor fields on three dimensional de Sitter spacetime stated by Ashtekar-Hansen and Beig-Schmidt and state and prove three additional lemmae.