Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Asymptotic Flatness and Lorentz Charges

Geoffrey Compère, François Dehouck, Amitabh Virmani

TL;DR

This work analyzes four-dimensional asymptotically flat spacetimes at spatial infinity within the Beig–Schmidt framework. It reveals that the six Lorentz charges are encoded in two mutually dual symmetric divergence-free tensors $V_{ab}$ and $W_{ab}$ and shows that integrability of Einstein's equations suffices to establish the equivalence between counter-term charges and Ashtekar–Hansen charges without requiring a parity condition on the mass aspect $\sigma$. By recasting second-order equations into dual SDT systems and interpreting Beig’s integrability conditions as vanishing Noether charges for a scalar field on $dS_3$, the paper unifies different charge constructions and clarifies their dependence on boundary data. The results strengthen the theoretical foundation of conserved charges in asymptotically flat gravity and provide exact links between counter-term, covariant phase space, and Ashtekar–Hansen formalisms, with implications for holography and dualities in flat space.

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.

On Asymptotic Flatness and Lorentz Charges

TL;DR

This work analyzes four-dimensional asymptotically flat spacetimes at spatial infinity within the Beig–Schmidt framework. It reveals that the six Lorentz charges are encoded in two mutually dual symmetric divergence-free tensors and and shows that integrability of Einstein's equations suffices to establish the equivalence between counter-term charges and Ashtekar–Hansen charges without requiring a parity condition on the mass aspect . By recasting second-order equations into dual SDT systems and interpreting Beig’s integrability conditions as vanishing Noether charges for a scalar field on , the paper unifies different charge constructions and clarifies their dependence on boundary data. The results strengthen the theoretical foundation of conserved charges in asymptotically flat gravity and provide exact links between counter-term, covariant phase space, and Ashtekar–Hansen formalisms, with implications for holography and dualities in flat space.

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.

Paper Structure

This paper contains 10 sections, 5 theorems, 107 equations.

Table of Contents

  1. Introduction and Summary of Results
  2. Asymptotic Flatness, Variational Principle, and Stress Tensor
  3. Equations of Motion and Integrability
  4. Equations of motion at second order
  5. Integrability conditions
  6. Mutually Dual Conserved Tensors
  7. Counter-term and Ashtekar-Hansen Charges Revisited
  8. Properties of Tensors on $dS_3$
  9. Properties of Killing Vectors on $dS_3$
  10. Classification of Symmetric and Divergence-free Tensors

Key Result

Lemma 1

On the three-dimensional hyperboloid, any traceless curl-free divergence-free symmetric tensor $T_{ab}$ such that $\square T_{ab}= 3 T_{ab}$ can be written as with $\square \Phi + 3\Phi = 0$. The scalar $\Phi$ is determined up to the ambiguity of adding a combination of the four functions zetas.

Theorems & Definitions (5)

  • Lemma 1: Ashtekar-Hansen
  • Lemma 2: Beig-Schmidt
  • Lemma 3
  • Lemma 4
  • Lemma 5