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Asymptotic symmetries and charges at spatial infinity in general relativity

Kartik Prabhu, Ibrahim Shehzad

TL;DR

This work develops a fully covariant treatment of asymptotic symmetries and charges at spatial infinity in four-dimensional general relativity using the Ashtekar–Hansen framework. By working on the space of spatial directions $\mathscr H$ and allowing a general conformal factor, it derives (i) the infinite-dimensional Spi algebra as a semidirect product of Lorentz transformations with supertranslations, (ii) integrable Spi charges for supertranslations, and (iii) a Wald–Zoupas-based prescription to define integrable Lorentz charges with controlled conformal dependence. A key result is that supertranslation charges reproduceADM/Bondi momenta, while Lorentz charges can be made integrable and transform consistently under conformal changes, enabling potential matching with null infinity and Strominger’s conservation program. The framework sets the stage for connecting spatial-infinity data to null-infinity data, with implications for gravitational memory, soft theorems, and possibly quantum asymptotic dynamics on the unit hyperboloid $\mathscr H$.

Abstract

We analyze the asymptotic symmetries and their associated charges at spatial infinity in $4$-dimensional asymptotically-flat spacetimes. We use the covariant formalism of Ashtekar and Hansen where the asymptotic fields and symmetries live on the $3$-manifold of spatial directions at spatial infinity, represented by a timelike unit-hyperboloid (or de Sitter space). Using the covariant phase space formalism, we derive formulae for the charges corresponding to asymptotic supertranslations and Lorentz symmetries at spatial infinity. With the motivation of, eventually, proving that these charges match with those defined on null infinity -- as has been conjectured by Strominger -- we do not impose any restrictions on the choice of conformal factor in contrast to previous work on this problem. Since we work with a general conformal factor we expect that our charge expressions will be more suitable to prove the matching of the Lorentz charges at spatial infinity to those defined on null infinity, as has been recently shown for the supertranslation charges.

Asymptotic symmetries and charges at spatial infinity in general relativity

TL;DR

This work develops a fully covariant treatment of asymptotic symmetries and charges at spatial infinity in four-dimensional general relativity using the Ashtekar–Hansen framework. By working on the space of spatial directions and allowing a general conformal factor, it derives (i) the infinite-dimensional Spi algebra as a semidirect product of Lorentz transformations with supertranslations, (ii) integrable Spi charges for supertranslations, and (iii) a Wald–Zoupas-based prescription to define integrable Lorentz charges with controlled conformal dependence. A key result is that supertranslation charges reproduceADM/Bondi momenta, while Lorentz charges can be made integrable and transform consistently under conformal changes, enabling potential matching with null infinity and Strominger’s conservation program. The framework sets the stage for connecting spatial-infinity data to null-infinity data, with implications for gravitational memory, soft theorems, and possibly quantum asymptotic dynamics on the unit hyperboloid .

Abstract

We analyze the asymptotic symmetries and their associated charges at spatial infinity in -dimensional asymptotically-flat spacetimes. We use the covariant formalism of Ashtekar and Hansen where the asymptotic fields and symmetries live on the -manifold of spatial directions at spatial infinity, represented by a timelike unit-hyperboloid (or de Sitter space). Using the covariant phase space formalism, we derive formulae for the charges corresponding to asymptotic supertranslations and Lorentz symmetries at spatial infinity. With the motivation of, eventually, proving that these charges match with those defined on null infinity -- as has been conjectured by Strominger -- we do not impose any restrictions on the choice of conformal factor in contrast to previous work on this problem. Since we work with a general conformal factor we expect that our charge expressions will be more suitable to prove the matching of the Lorentz charges at spatial infinity to those defined on null infinity, as has been recently shown for the supertranslation charges.

Paper Structure

This paper contains 19 sections, 2 theorems, 190 equations.

Table of Contents

  1. Asymptotic-flatness at spatial infinity: Ashtekar-Hansen structure
  2. Maxwell fields: symmetries and charges at $i^0$
  3. Gravitational fields and Einstein equations at $i^0$
  4. The universal structure at $i^0$
  5. Metric perturbations and symplectic current at $i^0$
  6. Asymptotic symmetries at $i^0$: The $\mathfrak{spi}$ algebra
  7. Spi-charges
  8. Charges for supertranslations: Spi-supermomentum
  9. Lorentz charges with $\boldsymbol{B}_{ab} = 0$
  10. Transformation of charges under conformal changes
  11. Discussion
  12. Coordinates, universal structure and asymptotic expansions near $i^0$
  13. Some useful relations on $\mathscr H$
  14. Killing vector fields
  15. Symmetric tensors
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $\boldsymbol{T}_{ab}$ be a symmetric tensor on $\mathscr H$ with vanishing curl, i.e, $\boldsymbol{D}_{[c} \boldsymbol{T}_{a]b} = 0$ then there exists a function $\boldsymbol{t}$ on $\mathscr H$ such that

Theorems & Definitions (12)

  • Definition 2.1: Ashtekar-Hansen structure Ash-in-Held
  • Remark 2.1: Conformal freedom
  • Remark 3.1: Special choices of gauge
  • Remark 3.2: Logarithmic gauge transformations
  • Remark 4.1: Conformal transformations of the asymptotic fields
  • Remark 4.2: Logarithmic translations
  • Remark 6.1: Translation vectors at $i^0$
  • Remark 6.2: Conformal transformation of Spi symmetries
  • Remark 6.3: Special choices of conformal factor
  • Lemma 2.1
  • ...and 2 more