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Conservation of asymptotic charges from past to future null infinity: Lorentz charges in general relativity

Kartik Prabhu, Ibrahim Shehzad

TL;DR

This work proves that, within a class of asymptotically flat spacetimes satisfying Ashtekar–Hansen conditions and additional regularity and continuity assumptions, the Lorentz charges at past and future null infinity match in the limit to spatial infinity, antipodally related. By combining a refined Ashtekar–Hansen construction with a careful fixing of supertranslation freedom, the authors show that all BMS charges—Lorentz and supertranslations—match between 𝓘^− and 𝓘^+ in the i^0 limit, extending Strominger’s conjecture to nonlinear general relativity under their hypotheses. Key steps include (i) isolating a pure Lorentz sector via gauge choices, (ii) establishing continuity of Weyl components at the boundaries of the null/spacelike cylinder 𝒞, and (iii) demonstrating conserved Lorentz charges on the space 𝔥, which ensures equality of charges across 𝒩^± and, hence, antipodal matching of 𝓘^± charges. Together with prior results on supertranslation matching, the paper confirms a diagonal asymptotic symmetry group and yields infinite conservation laws for gravitational scattering in this setting. The analysis clarifies the conditions under which flux conservation holds and highlights avenues for extending the framework beyond the assumed regularity class.

Abstract

We show that the asymptotic charges associated with Lorentz symmetries on past and future null infinity match in the limit to spatial infinity in a class of asymptotically-flat spacetimes. These are spacetimes that obey the Ashtekar-Hansen definition of asymptotic flatness at null and spatial infinity and satisfy an additional set of conditions which we lay out explicitly. Combined with earlier results on the matching of supertranslation charges, this shows that all Bondi-Metzner-Sachs (BMS) charges on past and future null infinity match in the limit to spatial infinity in this class of spacetimes, proving a relationship that was conjectured by Strominger. Assuming additional suitable conditions are satisfied at timelike infinities, this proves that the flux of all BMS charges is conserved in any classical gravitational scattering process in these spacetimes.

Conservation of asymptotic charges from past to future null infinity: Lorentz charges in general relativity

TL;DR

This work proves that, within a class of asymptotically flat spacetimes satisfying Ashtekar–Hansen conditions and additional regularity and continuity assumptions, the Lorentz charges at past and future null infinity match in the limit to spatial infinity, antipodally related. By combining a refined Ashtekar–Hansen construction with a careful fixing of supertranslation freedom, the authors show that all BMS charges—Lorentz and supertranslations—match between 𝓘^− and 𝓘^+ in the i^0 limit, extending Strominger’s conjecture to nonlinear general relativity under their hypotheses. Key steps include (i) isolating a pure Lorentz sector via gauge choices, (ii) establishing continuity of Weyl components at the boundaries of the null/spacelike cylinder 𝒞, and (iii) demonstrating conserved Lorentz charges on the space 𝔥, which ensures equality of charges across 𝒩^± and, hence, antipodal matching of 𝓘^± charges. Together with prior results on supertranslation matching, the paper confirms a diagonal asymptotic symmetry group and yields infinite conservation laws for gravitational scattering in this setting. The analysis clarifies the conditions under which flux conservation holds and highlights avenues for extending the framework beyond the assumed regularity class.

Abstract

We show that the asymptotic charges associated with Lorentz symmetries on past and future null infinity match in the limit to spatial infinity in a class of asymptotically-flat spacetimes. These are spacetimes that obey the Ashtekar-Hansen definition of asymptotic flatness at null and spatial infinity and satisfy an additional set of conditions which we lay out explicitly. Combined with earlier results on the matching of supertranslation charges, this shows that all Bondi-Metzner-Sachs (BMS) charges on past and future null infinity match in the limit to spatial infinity in this class of spacetimes, proving a relationship that was conjectured by Strominger. Assuming additional suitable conditions are satisfied at timelike infinities, this proves that the flux of all BMS charges is conserved in any classical gravitational scattering process in these spacetimes.

Paper Structure

This paper contains 12 sections, 189 equations, 1 figure.

Table of Contents

  1. Relating past and future null infinity: the construction
  2. Asymptotic symmetries at spatial infinity
  3. Behaviour of the BMS symmetries at $i^0$
  4. Spi symmetries on the space of null directions $\mathscr N^\pm$
  5. Fixing the supertranslation freedom at $i^0$
  6. Matching the Lorentz charges
  7. Limit to $\mathscr N^{\pm}$ along $\mathscr C$
  8. Limit to $\mathscr N^{\pm}$ along $\mathscr I^{\pm}$
  9. Discussion
  10. Spin-weighted spherical harmonics
  11. Solutions for reflection-odd $\boldsymbol{\beta}_{ab}$
  12. Affinity of $l^{a}$

Figures (1)

  • Figure 1: The space $\mathscr C$ of null and spatial directions $\vec{N}$ at $i^0$. The boundaries $\mathscr N^\pm \cong \mathbb S^2$, diffeomorphic to the space of generators of $\mathscr I^\pm$ respectively, represent the space of null directions. $\mathscr C \!\setminus\! \mathscr N^\pm$ is the space of rescaled spatial directions conformally diffeomorphic to the unit-hyperboloid $\mathscr H$. $\mathscr C$ depends on the choice of the rescaling function $\Sigma$ (defined below) and need not be a cylinder of unit radius in $Ti^0$ --- a "wiggly" cylinder has been drawn here to emphasize this.

Theorems & Definitions (4)

  • Definition 2.1: Ashtekar-Hansen structure Ash-in-Held
  • Remark 2.1: Conformal transformations of the asymptotic fields
  • Definition 2.2: Rescaling function $\Sigma$
  • Remark 4.1: Matching of Lorentz symmetries