Calculation of asymptotic charges at the critical sets of null infinity

TL;DR

The paper analyzes the matching of BMS supertranslation charges at the two null infinities by employing Friedrich's spatial infinity framework, which regularizes the otherwise singular behavior at i^0. By focusing on a spin-2 field on Minkowski space and then extending to full GR via the extended conformal field equations, it derives how charges at $\mathscr{I}^{\pm}$ depend on initial data and shows that logarithmic divergences arise unless specific regularity conditions are imposed. Under these regularity restrictions, the authors demonstrate antipodal matching of the harmonic components $Q_{l,m}$, i.e., $\mathcal{Q}_{l,m}|_{\mathcal{I}^{+}} = (-1)^{l}\mathcal{Q}_{l,m}|_{\mathcal{I}^{-}}$, thereby providing a concrete mechanism for Strominger's matching idea within a rigorous conformal framework. The work emphasizes that, for generic asymptotically flat spacetimes, matching cannot hold without imposing the identified regularity conditions on the initial data, guiding future investigations into the precise form of these conditions.

Abstract

The asymptotic structure of null and spatial infinities of asymptotically flat spacetimes plays an essential role in discussing gravitational radiation, gravitational memory effect, and conserved quantities in General Relativity. Bondi, Metzner and Sachs established that the asymptotic symmetry group for asymptotically simple spacetimes is the infinite-dimensional BMS group. Given that null infinity is divided into two sets: past null infinity $\mathscr{I}^{-}$ and future null infinity $\mathscr{I}^{+}$, one can identify two independent symmetry groups: $\text{BMS}^{-}$ at $\mathscr{I}^{-}$ and $\text{BMS}^{+}$ at $\mathscr{I}^{+}$. Associated with these symmetries are the so-called BMS charges. A recent conjecture by Strominger suggests that the generators of $\text{BMS}^{-}$ and $\text{BMS}^{+}$ and their associated charges are related via an antipodal reflection map near spatial infinity. To verify this matching, an analysis of the gravitational field near spatial infinity is required. This task is complicated due to the singular nature of spatial infinity for spacetimes with non-vanishing ADM mass. Different frameworks have been introduced in the literature to address this singularity, e.g., Friedrich's cylinder, Ashtekar-Hansen's hyperboloid and Ashtekar-Romano's asymptote at spatial infinity. This paper reviews the role of Friedrich's formulation of spatial infinity in the investigation of the matching of the spin-2 charges on Minkowski spacetime and in the full GR setting.

Figures (2)

• Figure 1: (a) The domain $\tilde{\mathcal{D}}$ containing spatial infinity, (b) The domain $\tilde{\mathcal{D}}$ on the conformal diagram of Minkowski spacetime.
• Figure 2: A diagram of the neighbourhood of spatial infinity in Friedrich's representation. In this representation, the spatial infinity point $i^0$ is blown up to a cylinder $\mathcal{I}$ connecting past null infinity $\mathscr{I}^{+}$ and future null infinity $\mathscr{I}^{-}$. The critical sets $\mathcal{I}^{\pm}$ represents the sets where $\mathcal{I}$ touches $\mathscr{I}^{\pm}$. The set $\mathcal{I}^{0}$ represents the intersection of the cylinder $\mathcal{I}$ with the initial hypersurface $\{ \tau =0\}$.

Theorems & Definitions (8)

• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• Proposition 1
• Remark 4
• Remark 5
• Remark 6