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arxiv: 2606.26848 · v1 · pith:NQS25SQTnew · submitted 2026-06-25 · 🌀 gr-qc · hep-th

Conservation law of super-Lorentz charges

Geoffrey Comp\`ere , S\'ebastien Robert This is my paper

Pith reviewed 2026-06-26 04:05 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords super-Lorentz chargesspatial infinityconservation lawBondi-Sachs fieldsasymptotic symmetriesgravitational scatteringsupertranslations
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The pith

Super-Lorentz charges conserve between future and past spatial infinity and are non-locally fixed by Bondi-Sachs fields.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that, under assumptions matching generic gravitational scattering, the vacuum gravitational field near spatial infinity is fixed at leading order by its supermomentum, dual supermomentum, and global supertranslation frame. At the next order the field is fixed by three further sets of charges, one of which consists of the super-Lorentz charges. The authors supply supertranslation-invariant expressions for these charges in terms of Bondi-Sachs data and an invariant version using Beig-Schmidt fields. They then use properties of homogeneous and inhomogeneous wave equations on the de Sitter boundary to prove that the super-Lorentz charges are the same when evaluated at future and past spatial infinity. The resulting super-Lorentz aspects turn out to be non-local functionals of the asymptotic fields.

Core claim

Under assumptions compatible with generic gravitational scattering, the vacuum relativistic gravitational field is entirely determined at leading order in the large radius expansion at spatial infinity by its supermomentum, its dual supermomentum and its global supertranslation frame. At subleading order, the gravitational field is determined by three additional sets of charges: the super-Lorentz charges, the leading tail charges and the leading peeling-breaking charges. The authors give a supertranslation-invariant definition of these charges in terms of asymptotic Bondi-Sachs fields and derive the conservation law of super-Lorentz charges between the future and past of spatial infinity, ob

What carries the argument

Super-Lorentz charges, obtained from solutions of wave equations on the boundary de Sitter spacetime and expressed invariantly through the asymptotic Bondi-Sachs fields.

If this is right

  • The super-Lorentz aspects cannot be extracted locally from the asymptotic fields but require a non-local construction.
  • The same charges admit an invariant definition in the Beig-Schmidt formalism that is also invariant under logarithmic translations.
  • The leading-order field is fixed solely by the supermomenta and the global supertranslation frame.
  • Subleading data include separate leading tail charges and leading peeling-breaking charges in addition to the super-Lorentz set.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The conservation relation supplies a global constraint that any complete scattering map must satisfy.
  • The non-local definition may link the super-Lorentz sector to integrated memory effects measured at null infinity.
  • The same wave-equation techniques could be applied to derive conservation statements for the tail and peeling-breaking charges.

Load-bearing premise

The vacuum relativistic gravitational field is entirely determined at leading order in the large radius expansion at spatial infinity by its supermomentum, its dual supermomentum and its global supertranslation frame.

What would settle it

An explicit evaluation of the super-Lorentz charges constructed from the same Bondi-Sachs data at past and future spatial infinity that yields unequal values.

Figures

Figures reproduced from arXiv: 2606.26848 by Geoffrey Comp\`ere, S\'ebastien Robert.

Figure 1
Figure 1. Figure 1: Gravitational onion structure of asymptotically flat spacetimes. Following the methodology of [31–33] (see also earlier work [34–43]), asymptotically flat spacetimes are defined from asymptotic fall-off conditions at their five infinities: fu￾ture and past null infinities I ±, spatial infinity i 0 and future and past timelike infinities i ±, together with fall-off condi￾tions on the codimension 1 fields de… view at source ↗
read the original abstract

Under assumptions compatible with generic gravitational scattering, the vacuum relativistic gravitational field is entirely determined at leading order in the large radius expansion at spatial infinity by its supermomentum, its dual supermomentum and its global supertranslation frame. At subleading order, the gravitational field is determined by three additional sets of charges: the super-Lorentz charges, the leading tail charges and the leading peeling-breaking charges. In this work we provide a supertranslation-invariant definition of these charges in terms of asymptotic Bondi-Sachs fields as well as a corresponding supertranslation and logarithmic translation invariant definition of these charges in terms of Beig-Schmidt fields. Using the properties of homogeneous and inhomogeneous solutions to relevant wave equations over the boundary de Sitter spacetime at spatial infinity, we derive the conservation law of super-Lorentz charges between the future and past of spatial infinity. We obtain that the super-Lorentz aspects are non-locally defined from the Bondi-Sachs fields.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims that, under assumptions compatible with generic gravitational scattering, the vacuum relativistic gravitational field is entirely determined at leading order in the large-radius expansion at spatial infinity by its supermomentum, dual supermomentum, and global supertranslation frame. At subleading order the field is determined by three additional sets of charges (super-Lorentz, leading tail, and leading peeling-breaking). It supplies supertranslation-invariant definitions of these charges in Bondi-Sachs and Beig-Schmidt gauges and, using properties of homogeneous and inhomogeneous solutions to wave equations on the boundary de Sitter spacetime, derives the conservation law of the super-Lorentz charges between future and past of spatial infinity, obtaining that the super-Lorentz aspects are non-locally defined from the Bondi-Sachs fields.

Significance. If the result holds, the work would establish a conservation law for super-Lorentz charges across spatial infinity and supply invariant definitions of subleading charges, extending the asymptotic-symmetry framework with potential relevance to gravitational scattering and memory effects. The use of wave-equation analysis on the de Sitter boundary provides a systematic route to matching future and past data.

major comments (2)
  1. [Abstract] Abstract, first sentence: the load-bearing claim that the vacuum field is 'entirely determined' at leading order by supermomentum, dual supermomentum, and global supertranslation frame must be accompanied by an explicit demonstration that the general solution of the linearized Einstein equations on the de Sitter boundary admits no additional independent homogeneous or inhomogeneous solutions beyond these three sets of data. Any incompleteness would directly undermine the subsequent supertranslation-invariant definitions of the subleading charges and the conservation law.
  2. [Wave-equation analysis] Derivation section (wave-equation analysis): the conservation law between future and past rests on the asserted properties of homogeneous and inhomogeneous solutions to the relevant wave equations over boundary de Sitter spacetime. The manuscript must supply the explicit boundary conditions, a complete characterization of the solution space, and error estimates confirming that the matching exhausts all degrees of freedom without residual contributions.
minor comments (1)
  1. [Definitions] Clarify the precise relation between 'charges' and 'aspects' in the definitions, as the terminology shifts between Bondi-Sachs and Beig-Schmidt formulations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting points that require clarification. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract, first sentence: the load-bearing claim that the vacuum field is 'entirely determined' at leading order by supermomentum, dual supermomentum, and global supertranslation frame must be accompanied by an explicit demonstration that the general solution of the linearized Einstein equations on the de Sitter boundary admits no additional independent homogeneous or inhomogeneous solutions beyond these three sets of data. Any incompleteness would directly undermine the subsequent supertranslation-invariant definitions of the subleading charges and the conservation law.

    Authors: We agree that the abstract claim is load-bearing and requires an explicit demonstration. The wave-equation analysis in the manuscript solves the linearized Einstein equations on the boundary de Sitter spacetime and shows that the leading-order data are fixed by the supermomentum, dual supermomentum, and global supertranslation frame, with the solution space exhausted by these quantities. To make this fully explicit as requested, we will add a dedicated paragraph in the derivation section (or a short appendix) that writes the general solution, identifies the independent data, and confirms the absence of additional homogeneous or inhomogeneous contributions at this order. This will directly support the subsequent supertranslation-invariant definitions. revision: yes

  2. Referee: [Wave-equation analysis] Derivation section (wave-equation analysis): the conservation law between future and past rests on the asserted properties of homogeneous and inhomogeneous solutions to the relevant wave equations over boundary de Sitter spacetime. The manuscript must supply the explicit boundary conditions, a complete characterization of the solution space, and error estimates confirming that the matching exhausts all degrees of freedom without residual contributions.

    Authors: The conservation law is derived from the properties of homogeneous and inhomogeneous solutions to the wave equations on the de Sitter boundary, with boundary conditions fixed by asymptotic flatness. The manuscript already uses these properties to match future and past data and obtain the non-local relation for the super-Lorentz aspects. We acknowledge that a more self-contained presentation would benefit readers. In the revision we will expand the relevant section to state the boundary conditions explicitly, give the complete characterization of the solution space (including how homogeneous and inhomogeneous parts are matched), and include a brief discussion of why the matching exhausts the degrees of freedom (with a qualitative error estimate based on the decay properties of the solutions). revision: yes

Circularity Check

0 steps flagged

No circularity: leading-order field determination stated as assumption; conservation law follows from wave-equation properties.

full rationale

The paper opens by stating the leading-order determination of the vacuum field by supermomentum, dual supermomentum and global supertranslation frame as an explicit assumption 'compatible with generic gravitational scattering.' The conservation law of super-Lorentz charges is then obtained from the properties of homogeneous and inhomogeneous solutions to wave equations on the de Sitter boundary at spatial infinity. No step reduces a claimed prediction to a fitted parameter, renames a known result, or rests on a self-citation whose content is itself unverified within the paper. The non-local definition of super-Lorentz aspects is presented as a derived output, not an input. The derivation chain is therefore self-contained against the stated assumptions and external wave-equation analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 3 invented entities

Review performed on abstract only; ledger entries extracted from the stated assumptions and introduced objects.

axioms (1)
  • domain assumption Assumptions compatible with generic gravitational scattering allow the vacuum field at spatial infinity to be fixed at leading order by supermomentum, dual supermomentum and global supertranslation frame.
    First sentence of abstract; this premise sets the stage for introducing the subleading charges.
invented entities (3)
  • super-Lorentz charges no independent evidence
    purpose: Subleading charges that complete the description of the gravitational field at spatial infinity
    Defined via asymptotic Bondi-Sachs fields; no independent evidence supplied in abstract.
  • leading tail charges no independent evidence
    purpose: Additional subleading charges
    Mentioned as part of the subleading description; no independent evidence in abstract.
  • leading peeling-breaking charges no independent evidence
    purpose: Additional subleading charges
    Mentioned as part of the subleading description; no independent evidence in abstract.

pith-pipeline@v0.9.1-grok · 5688 in / 1435 out tokens · 52308 ms · 2026-06-26T04:05:09.863882+00:00 · methodology

discussion (0)

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Reference graph

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