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Asymptotic Symmetries at Spatial Infinity

Sharad Mishra, Kinjal Banerjee, Jishnu Bhattacharyya

TL;DR

This work extends the Beig-Schmidt-Ashtekar-Romano framework to describe asymptotic flatness at spatial infinity with relaxed smoothness, revealing an enlarged symmetry group that includes logarithmic supertranslations. By formulating unit hyperboloid completions and a completion-independent universal structure, the authors derive transformation rules, classify completions via ripples, and define the Spi group as a semidirect product of supertranslations and Lorentz transformations. They show completion independence for H-preserving vector fields under precise asymptotic conditions and decompose asymptotic symmetry generators into a supertranslation part and an asymptotic Lorentz/Killing part, clarifying the algebraic structure and the role of the mass aspect. The results bridge covariant geometric constructions with completion-sensitive gauge notions and point to future work on Noether charges, dynamics, and connections across spatial, null, and timelike infinities.

Abstract

We describe asymptotic symmetries at spatial infinity of asymptotically flat spacetimes within the context of a generalization of the Beig-Schmidt-Ashtekar-Romano-framework. We demonstrate that it is possible to relax certain smoothness requirements of the asymptotic transformations considered previously, without violating asymptotic flatness. This leads to an enhancement of the asymptotic symmetry group that includes logarithmic supertranslations at spatial infinity. Our results complement several recent results which confirm the existence of logarithmic supertranslations at spatial infinity in the Hamiltonian formalism.

Asymptotic Symmetries at Spatial Infinity

TL;DR

This work extends the Beig-Schmidt-Ashtekar-Romano framework to describe asymptotic flatness at spatial infinity with relaxed smoothness, revealing an enlarged symmetry group that includes logarithmic supertranslations. By formulating unit hyperboloid completions and a completion-independent universal structure, the authors derive transformation rules, classify completions via ripples, and define the Spi group as a semidirect product of supertranslations and Lorentz transformations. They show completion independence for H-preserving vector fields under precise asymptotic conditions and decompose asymptotic symmetry generators into a supertranslation part and an asymptotic Lorentz/Killing part, clarifying the algebraic structure and the role of the mass aspect. The results bridge covariant geometric constructions with completion-sensitive gauge notions and point to future work on Noether charges, dynamics, and connections across spatial, null, and timelike infinities.

Abstract

We describe asymptotic symmetries at spatial infinity of asymptotically flat spacetimes within the context of a generalization of the Beig-Schmidt-Ashtekar-Romano-framework. We demonstrate that it is possible to relax certain smoothness requirements of the asymptotic transformations considered previously, without violating asymptotic flatness. This leads to an enhancement of the asymptotic symmetry group that includes logarithmic supertranslations at spatial infinity. Our results complement several recent results which confirm the existence of logarithmic supertranslations at spatial infinity in the Hamiltonian formalism.
Paper Structure (21 sections, 6 theorems, 180 equations, 1 figure)

This paper contains 21 sections, 6 theorems, 180 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Spatial infinity as a boundary
  3. Remarks:
  4. Unit hyperboloid completions
  5. Remarks:
  6. Remarks:
  7. The group of supertranslations at spatial infinity
  8. Asymptotic symmetries of AM spacetimes
  9. Remarks:
  10. Asymptotic symmetry generators: basic properties
  11. Asymptotic symmetry generators: algebra
  12. Outlook
  13. Proofs etc.
  14. Proof of Theorem \ref{['thm:omega']}
  15. Three-metrics induced by $\mathsf{q}_{a b}$ and $\mathsf{q}'_{a b}$ on $\CMcal{H}$
  16. ...and 6 more sections

Key Result

Theorem 1

Given a unit hyperboloid completion $(\hat{\mathscr{M}}, \Omega)$ of an asymptotically Minkowskian spacetime $(\mathscr{M}, \mathsf{g})$, if a function $\omega$ is $C^{1}$ in a neighborhood $\mathscr{N}_{\CMcal{H}}$ of $\CMcal{H}$ but (possibly) singular on $\CMcal{H}$ in a way such that the conditi where $\sigma$ is a function of $\Omega$ only which is at least $C^{1}$ in $\mathscr{N}_{\CMcal{H}}

Figures (1)

  • Figure 1: Schematic diagram of transformations between completion functions. Each directed arrow represents a transformation and the label shows the corresponding '$\omega$-function'.

Theorems & Definitions (17)

  • Definition 1: Asymptote at spatial infinity
  • Definition 2: AFSI spacetime
  • Definition 3: AM spacetime
  • Theorem 1: Characterization of unit hyperboloid completion functions
  • Definition 4: Universal structure of an AM spacetime
  • Definition 5: $\CMcal{H}$-preserving vector fields of AM spacetimes
  • Theorem 2: Completion independence of $\CMcal{H}$-preserving vector fields
  • Definition 6: Asymptotic symmetry generator of an AM spacetime
  • Theorem 3: Characterization of asymptotic symmetry generators
  • Theorem 4: Basic characterization of asymptotic symmetry generators
  • ...and 7 more