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ADM, BMS, and some puzzling interconnections

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper argues that the apparent mismatch between mechanical and radiated angular momentum in gravitational scattering disappears once the total loss is split into a radiative part and a static memory part computed in two different…

desk verdict A clear, honest perspective on the ADM-BMS angular momentum puzzle; the advertised resolution is conditional on a frame-existence assumption the author openly flags, and the new appendix calculation is the most concrete part. read the letter →

arxiv 2505.11937 v1 pith:A6KHKTMT submitted 2025-05-17 gr-qc

Gabriele Veneziano This is my paper · ORCID
classification gr-qc
keywords angularmomentumlossBondi-SachscoordinatesBMSsymmetryADMformalismgravitationalmemoryshearandnewstensorsscatteringnullinfinity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper addresses two puzzles at the boundary between the ADM and Bondi-Sachs descriptions of isolated gravitating systems. The main result is that the long-standing discrepancy in two-body gravitational scattering—mechanical angular-momentum loss starting at order $G^2$ while gravitational radiation starts at order $G^3$—is resolved by splitting the loss into a radiative part and a static memory part. The radiative part is computed in the canonical Bondi frame that is shear-free at past null infinity; the total mechanical loss requires comparing that frame with a different frame that is shear-free at future null infinity and adapted to the final two-body system. A supporting analysis shows that a claimed local, angle-by-angle energy conservation in scattering is a coordinate-matching effect, not an actual energy flow. If this reading is right, the angular momentum of the final binary, the static field, and the radiation can be separated unambiguously.

What carries the argument

The load-bearing object is the Bondi-Sachs metric near null infinity, whose asymptotic form is fixed by a mass aspect $M$, an angular-momentum aspect $N_A$, and the shear tensor $C_{AB}$; the news tensor $N_{AB}=\partial_u C_{AB}$ measures gravitational radiation. The BMS group of residual gauge symmetries (supertranslations and Lorentz transformations) makes $C_{AB}$ frame-dependent, and the paper exploits two particular frames: the canonical Bondi gauge, defined by $C_{AB}(u=-\infty)=0$, and the final frame, defined by $C_{AB}(u=+\infty)=0$. The identity that carries the argument is the decomposition of the mechanical angular-momentum loss into a radiative flux evaluated in the canonical gauge plus a static contribution built from the memory $S_{AB}$, with the two frames related by a supertranslation (and possibly a boost) that changes the bookkeeping of $N_A$ and $C_{AB}$.

What would settle it

Take a two-black-hole scattering at large impact parameter and compute the mechanical angular-momentum loss at $O(G^2)$ by two independent routes: directly from the equations of motion with radiation reaction, and from $\Delta J^{(\rm rad)}+\Delta J^{(\rm stat)}$ using the full memory tensor. If these disagree at $O(G^2)$, the frame-splitting resolution fails; the paper only verifies the match at the first non-trivial order in $G$.

Watch

Extended reading notes

Core claim

The paper's central claim is expressed by the balance equation $\Delta J^{(\rm mech)} = \Delta J^{(\rm rad)} + \Delta J^{(\rm stat)}$. Here $\Delta J^{(\rm rad)}$ is the Bondi angular-momentum flux in the canonical gauge, the unique frame with zero shear at $u=-\infty$, and $\Delta J^{(\rm stat)}$ is the difference between Bondi angular momenta evaluated at $u=+\infty$ in two frames: the canonical frame and a second frame chosen shear-free at $u=+\infty$ and attached to the center-of-mass motion of the final system. The static piece is controlled by the memory tensor $S_{AB}=C_{AB}(+\infty)-C_{AB}(-\infty)=\int_{-\infty}^{+\infty} du\, N_{AB}$, which is a BMS-invariant observable connected to zero-frequency gravitons. Because $\Delta J^{(\rm stat)}$ receives contributions at $O(G^2)$ while $\Delta J^{(\rm rad)}$ begins only at $O(G^3)$, the total mechanical loss can start one order in $G$ earlier than radiation, consistent with radiation-reaction formulas. The author frames this as a resolution of a recent saga on angular momentum, noting that the Poincaré subgroups of BMS selected at past and future null infinity are not the same.

Load-bearing premise

The resolution assumes that in radiating spacetimes that are sufficiently stationary in the asymptotic past, the canonical-gauge Bondi angular momentum at $u=-\infty$ equals the ADM angular momentum, and that uniquely defined shear-free Bondi frames exist at $u=+\infty$ and beyond leading order in $G$; the paper itself flags the latter as an open point.

Editorial extensions

If this is right

  • A scattering binary's mechanical angular-momentum loss can be computed as a sum of a radiation flux and a memory term, so the two effects are not in competition and both are needed for balance.
  • Initial and final angular-momentum bookkeeping live in different Bondi frames, with different Poincaré subgroups, so a single gauge choice cannot serve both ends of the scattering.
  • Local angle-by-angle equality of past and future Bondi mass aspects at spatial infinity is a statement about how the shear adjusts near $i^0$, not a statement about where energy flows.
  • The $O(G^2)$ mechanical angular-momentum loss is consistent with the $O(G^3)$ onset of gravitational radiation, removing the apparent tension with linear-response back-reaction formulas.
  • Once the frames are fixed, the total initial angular momentum splits unambiguously among the binary system, the static gravitational field, and the radiation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the split is universal, gravitational-wave observables such as the final spin of a merged binary should show a memory-induced angular-momentum deficit that is not present in the radiated flux; comparing radiation-reaction-inferred spins with directly measured final spins would test it.
  • The same frame-splitting mechanism likely applies to boost charges and center-of-mass charges, where long-range static fields can create order-$G^2$ bookkeeping shifts that are not radiation.
  • The requirement of a final frame with uniquely defined shear and boost suggests that fully gauge-invariant definitions of angular momentum may need an enlarged asymptotic symmetry algebra rather than the original BMS group.
  • A direct numerical two-body simulation at high post-Minkowskian order could verify the decomposition by extracting $\Delta J^{(\rm stat)}$ from the memory and comparing it with the gap between mechanical and radiated losses.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper is a personal perspective on the relation between ADM and Bondi-Sachs/BMS formalism. It reviews BS coordinates, the BMS group, charges, and flux-balance laws, and then addresses two puzzles. The first concerns the claimed local-in-angle conservation of energy in gravitational scattering, which the author argues is a matching of mass aspects at spatial infinity under antipodal identification rather than an actual angle-by-angle energy flow. The main topic is the discrepancy between an O(G^2) mechanical angular-momentum loss in two-body scattering and an O(G^3) radiative angular-momentum loss. The proposed resolution, summarized in Eq. (19), splits the mechanical loss into a radiative part computed in the canonical Bondi gauge and a static part associated with the gravitational memory and with the difference between the final intrinsic Bondi frame (shear-free at u=+infinity) and the canonical frame evolved from u=-infinity. An appendix presents an explicit low-order check of the antipodal matching of mass aspects.

Significance. If the decomposition in Eq. (19) is correct, it would resolve a long-standing puzzle and give a clean physical meaning to the split between radiation and memory in angular-momentum balance, connecting ADM quantities at spatial infinity to BMS charges at null infinity. The manuscript's strengths are its accurate recapitulation of standard Bondi-Sachs/BMS material, its candid admission of open technical points, and its reliance on and citation of independent work by Ashtekar, Damour, Riva-Vernizzi-Wong, Javadinezhad-Porrati, and others. The appendix provides a concrete, though under-derived, consistency check. The value of the paper is primarily interpretive and synthetic rather than a new proof; the central claim is clearly stated but rests on assumptions that the text itself flags as open.

major comments (3)
  1. [Sec. 4.3, Eqs. (17)-(19)] The central decomposition delta_J_mech = delta_J_rad + delta_J_stat rests on two assumptions that are stated but not established in this manuscript: first, that the Bondi angular momentum at u=+infinity in a frame made shear-free at u=+infinity equals the final ADM angular momentum of the radiating system, and second, that a supertranslation connecting this final intrinsic frame to the canonical frame evolved from u=-infinity exists and is unique beyond leading order in G. The text itself concedes that 'it was far from obvious whether and how one could extend the definition of the intrinsic gauge to higher orders in G or ... whether any such gauge existed beyond the leading order.' The cited works [32]-[35], [11], and [43] are invoked but not shown to close this gap. Since the split into radiative and static losses is gauge-dependent unless the l=0,1 kernel of the elliptic equation determining the supertranslation is fixed by additional physical conditions, Eq. (19) is not yet a fully supported physical statement. The author should either supply the existence and uniqueness argument or state the conclusion as conditional, with the status of the conjecture made explicit.
  2. [Appendix A, Eq. (27)] The massless limit leading to Eq. (27) is asserted with 'By doing so one finds' and a numerical check, but the appendix is presented as an explicit resolution of the local-energy-conservation paradox. Because the cancellation in Eq. (24) relies on the exact delta-function form of Eq. (27), a distributional derivation, or at least a precise statement of the limiting procedure and the normalization convention (28), should be included. As written, the load-bearing step is reduced to an unpublished calculation cited as reference [17].
  3. [Sec. 4.3, discussion after Eq. (20)] The sentence claiming that both delta_J_rad and delta_J_stat 'have a BMS invariant meaning by referring to some uniquely specified Bondi frames' overstates what is demonstrated. The uniqueness of those frames is exactly the open point conceded earlier in the same section. Unless existence and uniqueness are proved or explicitly assumed with justification, the two quantities cannot be claimed to be invariant. The manuscript should either provide the argument or clearly qualify this statement as an assumption.
minor comments (4)
  1. [Sec. 4.1, paragraph after Eq. (13)] The sentence 'It has been argued [10] that not only M±, but also M±, satisfy the above-mentioned matching' appears garbled; the two objects being compared should be named distinctly, for example the mass aspect M±(u,theta) and the integrated Bondi mass M±(u).
  2. [Appendix A, Eq. (25)] The symbol v denotes both the advanced time coordinate in Eqs. (21)-(24) and the center-of-mass velocity in Eq. (25); this clash is confusing and the velocity should be renamed, for example w or beta.
  3. [References] Reference [52] appears twice, with the duplicate label on the Prabhu entry; the numbering should be corrected.
  4. [Appendix A, footnote 11] A numerical check is not a proof; the footnote should either describe an analytic derivation or state explicitly that the delta-function limit is verified numerically and is being relied upon as a plausible identity.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction found: the central radiative/static split is an algebraic identity with physically supported inputs, and the admitted open intrinsic-gauge existence question is a stated limitation rather than a circular step.

full rationale

The paper is a perspective that assembles prior results rather than fitting parameters and renaming them predictions. The central decomposition, Eq. (19), Delta J_mech = Delta J_rad + Delta J_stat, is an algebraic identity once the two Bondi frames are specified; the physical content is not in the identity but in the claims that Delta J_rad (computed in the canonical shear-free-at-past gauge) is O(G^3) and that Delta J_stat, the remainder between the final intrinsic frame and the evolved canonical frame, is the memory/static contribution entering at O(G^2). Those order-of-magnitude statements are supported by the flux formula (15), the canonical-gauge condition C_AB(-infinity)=0 in Eq. (16), and by the cited independent calculations of zero-frequency gravitons and memory ([38],[48]), not by the defining equation itself. The paper explicitly flags the load-bearing open point: 'some doubts remained: it was far from obvious whether and how one could extend the definition of the intrinsic gauge to higher orders in G or, even worse, whether any such gauge existed beyond the leading order' (Sec. 4.3). This is an admitted existence/uniqueness gap, which is a correctness or rigor risk, not a circular step: the final frame is not defined in terms of the predicted loss, and the loss is not fitted to reproduce the split. The Appendix's check of local energy conservation is a self-contained calculation against the Bondi mass-aspect equations and the known shear (25), providing an external benchmark. Self-citations are present and the narrative leans on [18] for the intrinsic-gauge idea, but the key existence and uniqueness claims are also attributed to independent groups (Ashtekar et al. [32]-[34], Compere et al. [39],[45], Chen-Wang-Yau [40],[41], Javadinezhad-Porrati [43], Riva-Vernizzi-Wong [11]). Thus any self-citation is not load-bearing in the sense required for a circularity finding. Score 2 reflects only the minor presence of self-citation, not a circular derivation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The 'intrinsic' and 'canonical' gauges are coordinate choices, not new degrees of freedom. The free-parameter list is empty because the paper is a theoretical derivation/check, not a fit to data. The axioms listed are the standard assumptions of the Bondi-Sachs/BMS framework plus the specific matching assumptions that anchor the ADM-BMS connection.

assumptions (5)
  • domain assumption Asymptotic flatness and the Bondi-Sachs expansion (Eq. 2) hold with the stated fall-offs.
    Used throughout to define M, C_AB, N_A and derive the evolution equations (3).
  • domain assumption Continuity matching at spatial infinity: limits of Bondi quantities at i0 coincide with ADM quantities, with antipodal identification for angle-dependent fields.
    Invoked in Secs. 4.1 and 4.3, and tested in Appendix A; necessary for the claim that the canonical gauge at u=-infinity gives the ADM angular momentum.
  • domain assumption Sufficient stationarity in the asymptotic past and future to allow shear-free Bondi gauges and to identify Poincare subgroups at u=plus or minus infinity.
    Used in Sec. 4.3 to define canonical and final Bondi frames; the paper itself flags this as an open issue beyond leading order.
  • domain assumption The Bini-Damour linear-response formula [19] correctly relates loss of energy and angular momentum to back-reaction on the orbit.
    Used in Sec. 4.2 to argue that the J-loss must start at O(G^2), forcing the static/radiative split.
  • standard math Standard BMS transformation laws and the Dray-Streubel charge definition are correct.
    Assumed in Eqs. (7)-(9) and used to compute the flux-balance laws.

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Cite this review

Pith. "Pith review of ADM, BMS, and some puzzling interconnections." pith.science (2026). https://pith.science/paper/A6KHKTMT

@misc{pith2026250511937,
  author       = {Pith},
  title        = {Pith review of: ADM, BMS, and some puzzling interconnections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A6KHKTMT}},
  note         = {Machine review of arXiv:2505.11937}
}
read the original abstract

The precise connection between the ADM and BMS formalisms is still far from being fully understood. It leads superficially to some puzzles whose resolution can provide new interesting physical insights. One example concerns the claimed local-in-angle conservation of energy in a gravitational scattering process, whose physical meaning I will try to clarify in an explicit example. As my main topic, I will sketch my own understanding of how a recent saga on the definition of angular momentum, its loss, and its radiation, appears to have found its way to a happy ending by appealing to some non-trivial connections between ADM and BMS quantities in asymptotically flat and stationary spacetimes.

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. "Waveforms" at the Horizon

    gr-qc 2026-02 conditional novelty 6.0 of 10

    A probe scattering off a Schwarzschild black hole transfers a definite leading-order post-Minkowskian angular momentum to the horizon, given by new closed formulas (3.24b), (3.29), (3.36).

  2. Lorentz Covariant Supertranslation Frames for the Angular Momentum Aspect

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    Angular momentum in GR is made unambiguous by fixing supertranslation frames through the center-of-mass condition J_{\mu\nu}P^\nu=0, extended covariantly to the final state via P^i_\mu P^f_\nu J^{f\mu\nu}=0.

  3. Preface to Fields, Gravity, Strings and Beyond: In Memory of Stanley Deser

    hep-th 2025-09 unverdicted

    An editorial preface, not a research paper: it tributes Stanley Deser and catalogues the special issue's contributed articles in four thematic areas.

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Reviewed August 15, 2026 · model on record in the stance chip above.