pith. machine review for the scientific record. sign in

arxiv: 2512.15578 · v2 · submitted 2025-12-17 · ✦ hep-th · gr-qc

Recognition: 2 theorem links

· Lean Theorem

Scalar, vector and tensor fields on dS₃ with arbitrary sources: harmonic analysis and antipodal maps

Geoffrey Comp\`ere , S\'ebastien Robert
Authors on Pith no claims yet

Pith reviewed 2026-05-16 21:36 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords de Sitter spacetimespherical harmonicsantipodal mapsasymptotic datawave equationstensor decompositionvector fields
0
0 comments X

The pith

The asymptotic data for scalar, vector and tensor fields on three-dimensional de Sitter spacetime are related by antipodal maps between past and future infinity even with sources.

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

The paper defines scalar, vector and tensor spherical harmonics on dS3 and examines their behavior near past and future infinity. It establishes explicit antipodal relationships between the two sets of asymptotic data on the sphere, which can be non-local. A procedure extracts these data when sources are present, linking independent sets at the two boundaries. Several theorems decompose vectors and tensors obeying inhomogeneous wave equations, such as expressing them locally in terms of symmetric transverse traceless tensors. These findings support the analysis of fields in related spacetime contexts.

Core claim

Each harmonic defines two sets of asymptotic data on the two-sphere close to both past and future infinity on de Sitter spacetime. For each case, the antipodal relationship of both sets of asymptotic data between past and future infinity is made explicit, and it can be non-local. A procedure extracts these asymptotic data in the presence of sources, providing the relationship between two independent sets of data defined on the sphere in the asymptotic future with the corresponding data in the asymptotic past. Theorems prove that a large class of tensors obeying an inhomogeneous wave equation can be expressed locally in terms of a symmetric transverse traceless tensor.

What carries the argument

Spherical harmonics for scalar, vector and tensor fields on dS3, together with antipodal maps relating their asymptotic data sets at past and future infinity.

If this is right

  • The relationship between future and past asymptotic data holds for each class of propagating field.
  • Asymptotic data can be extracted even when sources are arbitrary.
  • A large class of tensors satisfying inhomogeneous wave equations decomposes locally into symmetric transverse traceless tensors.
  • These relations aid the description of interacting four-dimensional asymptotically flat fields at spatial infinity.

Where Pith is reading between the lines

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

  • The non-local antipodal maps suggest possible connections to radiation memory effects when mapping to four-dimensional flat-space problems at infinity.
  • The decomposition results may simplify explicit solutions for sourced higher-spin fields on other constant-curvature backgrounds.

Load-bearing premise

The fields satisfy the appropriate linear or inhomogeneous wave equations on dS3 with asymptotic expansions valid in standard Bondi-like coordinates on the conformal boundary.

What would settle it

Finding a sourced scalar, vector or tensor field on dS3 where the extracted asymptotic data at past and future infinity do not satisfy the stated antipodal relationship.

read the original abstract

The scalar, vector and tensor spherical harmonics on three-dimensional de Sitter spacetime are defined and analyzed. Each harmonic defines two sets of asymptotic data on the two sphere in the asymptotic expansion close to both the past and the future of de Sitter spacetime. For each case, we make explicit the antipodal relationship of both sets of asymptotic data between past and future infinity, which can be non-local. A procedure is defined to extract these asymptotic data in the presence of sources. This provides for each class of propagating field on de Sitter the relationship between two independent sets of data defined on the sphere in the asymptotic future with the corresponding data defined in the asymptotic past. We also provide several theorems on the decomposition of vector and tensors on de Sitter such as one proving that a large class of tensors obeying an inhomogeneous wave equation can be expressed locally in terms of a symmetric transverse traceless tensor. These results are instrumental in the description of interacting four-dimensional asymptotically flat fields at spatial infinity.

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

1 major / 2 minor

Summary. The paper defines scalar, vector, and tensor spherical harmonics on three-dimensional de Sitter spacetime. For each class it constructs the associated asymptotic data on the two-sphere at past and future infinity, derives the explicit (possibly non-local) antipodal map relating the two sets of data, and supplies a procedure for extracting the data when sources are present. It also proves several decomposition theorems, including that a broad class of tensors satisfying an inhomogeneous wave equation can be written locally in terms of a symmetric transverse-traceless tensor. The constructions are motivated by the need to relate independent data sets for propagating fields on dS3 and are presented as tools for the study of four-dimensional asymptotically flat fields at spatial infinity.

Significance. If the stated theorems and maps are correct, the work supplies a concrete harmonic-analysis framework that relates past and future asymptotic data for linear fields on dS3, including in the presence of sources. The explicit antipodal relations and the local decomposition of inhomogeneous tensors into symmetric transverse-traceless pieces are technically useful results that can be directly imported into the analysis of radiation and boundary data in asymptotically flat spacetimes. The paper ships explicit constructions rather than abstract existence statements, which strengthens its utility for subsequent calculations.

major comments (1)
  1. The central claim that a large class of tensors obeying an inhomogeneous wave equation can be expressed locally in terms of a symmetric transverse-traceless tensor (abstract and §4) is load-bearing for the decomposition results. The proof sketch relies on the validity of the asymptotic expansions in Bondi-like coordinates and on the source term satisfying the appropriate fall-off; a concrete counter-example or an explicit statement of the precise regularity conditions on the source would be needed to confirm that the reduction remains local when the source is non-compact or has slower decay.
minor comments (2)
  1. Notation for the two independent sets of asymptotic data (e.g., the pair of coefficients in the expansion near I^- versus I^+) is introduced without a consolidated table; adding such a table would make the antipodal-map statements easier to track across scalar, vector, and tensor cases.
  2. The procedure for extracting asymptotic data in the presence of sources is described in general terms; an explicit worked example for at least one source profile (e.g., a point source or a compactly supported scalar source) would clarify the steps.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the single major comment below and have incorporated the requested clarification into the revised version.

read point-by-point responses

  1. Referee: The central claim that a large class of tensors obeying an inhomogeneous wave equation can be expressed locally in terms of a symmetric transverse-traceless tensor (abstract and §4) is load-bearing for the decomposition results. The proof sketch relies on the validity of the asymptotic expansions in Bondi-like coordinates and on the source term satisfying the appropriate fall-off; a concrete counter-example or an explicit statement of the precise regularity conditions on the source would be needed to confirm that the reduction remains local when the source is non-compact or has slower decay.

    Authors: We thank the referee for this observation. The decomposition theorem of §4 is formulated for sources that admit a Bondi-like asymptotic expansion with fall-off ensuring the locality of the reduction; specifically, we require the source to decay at least as O(r^{-3}) (in the coordinates of the paper) so that no non-local contributions arise from the integration. We have revised §4 to state these regularity conditions explicitly and have added a short remark clarifying that the result is local precisely under this decay (while slower decay or non-compact support may introduce non-local terms). We have not supplied a concrete counter-example, as the theorem is stated only for the class satisfying the stated fall-off. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained

full rationale

The paper defines scalar/vector/tensor spherical harmonics on dS3 via standard harmonic analysis, derives explicit (possibly non-local) antipodal maps between past and future asymptotic data for fields obeying linear or inhomogeneous wave equations, and proves decomposition theorems such as the local reduction of a large class of inhomogeneous tensors to symmetric transverse-traceless form. These steps rest directly on the wave equations and Bondi-like coordinate expansions without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations; the constructions are mathematically independent and externally verifiable from the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the assumption that the fields obey linear or inhomogeneous wave equations on a fixed dS3 background and that standard asymptotic expansions near the conformal boundaries are valid.

axioms (2)
  • domain assumption Scalar, vector, and tensor fields satisfy the appropriate wave equations (homogeneous or inhomogeneous) on dS3
    Invoked throughout the definitions of the harmonics and the extraction procedure.
  • domain assumption Asymptotic expansions near past and future infinity exist in the standard coordinates on the conformal boundary sphere
    Required for the two sets of asymptotic data to be well-defined.

pith-pipeline@v0.9.0 · 5478 in / 1439 out tokens · 56887 ms · 2026-05-16T21:36:22.144579+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    The scalar, vector and tensor spherical harmonics on three-dimensional de Sitter spacetime are defined and analyzed... antipodal relationship of both sets of asymptotic data between past and future infinity

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

  1. A proof of conservation laws in gravitational scattering: tails and breaking of peeling

    hep-th 2026-03 unverdicted novelty 6.0

    A proposed definition of asymptotically flat spacetimes enables proofs of antipodal matching conditions at spatial infinity for dual mass, shear tails, and peeling, expressed as boundary conservation laws.

Reference graph

Works this paper leans on

70 extracted references · 70 canonical work pages · cited by 1 Pith paper · 25 internal anchors

  1. [1]

    subtracted

    after using the properties (112). D. Additional properties of tensors Let us derive more general features of tensors ondS 3. First, there is two equivalence relationships between either the vector KG inner product or the scalar KG inner product and the charges (314) associated with a vector. For any tensor that can be written as Jab =D (aV J b) (319) with...

    work page

  2. [2]

    Regge and J

    T. Regge and J. A. Wheeler, Phys. Rev.108, 1063 (1957)

    work page 1957

  3. [3]

    F. J. Zerilli, Phys. Rev. D2, 2141 (1970)

    work page 1970

  4. [4]

    J. M. Bardeen, Phys. Rev. D22, 1882 (1980)

    work page 1980

  5. [5]

    Kodama and M

    H. Kodama and M. Sasaki, Progress of Theoretical Physics Supplement78, 1 (1984)

    work page 1984

  6. [6]

    Signature of Gravity Waves in Polarization of the Microwave Background

    U. Seljak and M. Zaldarriaga, Phys. Rev. Lett.78, 2054 (1997), arXiv:astro-ph/9609169

    work page internal anchor Pith review Pith/arXiv arXiv 2054

  7. [7]

    A Probe of Primordial Gravity Waves and Vorticity

    M. Kamionkowski, A. Kosowsky, and A. Stebbins, Phys. Rev. Lett.78, 2058 (1997), arXiv:astro-ph/9609132

    work page internal anchor Pith review Pith/arXiv arXiv 2058

  8. [8]

    Lifshitz, J

    E. Lifshitz, J. Phys. (USSR)10, 116 (1946)

    work page 1946

  9. [9]

    E. M. Lifshitz and I. M. Khalatnikov, Adv. Phys.12, 185 (1963)

    work page 1963

  10. [10]

    Mashhoon, Phys

    B. Mashhoon, Phys. Rev. D8, 4297 (1973)

    work page 1973

  11. [11]

    B. L. Hu, Journal of Mathematical Physics15, 1748 (1974)

    work page 1974

  12. [12]

    R. T. Jantzen, Journal of Mathematical Physics 19, 1163 (1978), https://pubs.aip.org/aip/jmp/article- pdf/19/5/1163/19149335/1163_1_online.pdf

    work page 1978

  13. [13]

    U. H. Gerlach and U. K. Sengupta, Phys. Rev. D18, 1773 (1978)

    work page 1978

  14. [14]

    Tomita, Progress of Theoretical Physics68, 310 (1982), https://academic.oup.com/ptp/article- pdf/68/1/310/5319939/68-1-310.pdf

    K. Tomita, Progress of Theoretical Physics68, 310 (1982), https://academic.oup.com/ptp/article- pdf/68/1/310/5319939/68-1-310.pdf

    work page 1982

  15. [16]

    Beig, Proceedings of the Royal Society of London

    R. Beig, Proceedings of the Royal Society of London. A. Math- ematical and Physical Sciences391, 295 (1984)

    work page 1984

  16. [17]

    A holographic reduction of Minkowski space-time

    J. de Boer and S. N. Solodukhin, Nucl. Phys. B665, 545 (2003), arXiv:hep-th/0303006

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  17. [18]

    Rigorous steps towards holography in asymptotically flat spacetimes

    C. Dappiaggi, V . Moretti, and N. Pinamonti, Rev. Math. Phys. 18, 349 (2006), arXiv:gr-qc/0506069

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  18. [19]

    R. B. Mann and D. Marolf, Class. Quant. Grav.23, 2927 (2006), arXiv:hep-th/0511096

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  19. [20]

    Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited

    G. Barnich and C. Troessaert, Phys. Rev. Lett.105, 111103 (2010), arXiv:0909.2617 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  20. [21]

    The BMS/GCA correspondence

    A. Bagchi, Phys. Rev. Lett.105, 171601 (2010), arXiv:1006.3354 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  21. [22]

    T. He, V . Lysov, P. Mitra, and A. Strominger, JHEP05, 151 (2015), arXiv:1401.7026 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  22. [23]

    A 2D Stress Tensor for 4D Gravity

    D. Kapec, P. Mitra, A.-M. Raclariu, and A. Strominger, Phys. Rev. Lett.119, 121601 (2017), arXiv:1609.00282 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  23. [24]

    4D Scattering Amplitudes and Asymptotic Symmetries from 2D CFT

    C. Cheung, A. de la Fuente, and R. Sundrum, JHEP01, 112 (2017), arXiv:1609.00732 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  24. [25]

    Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere

    S. Pasterski, S.-H. Shao, and A. Strominger, Phys. Rev. D96, 065026 (2017), arXiv:1701.00049 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  25. [26]

    A Conformal Basis for Flat Space Amplitudes

    S. Pasterski and S.-H. Shao, Phys. Rev. D96, 065022 (2017), arXiv:1705.01027 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  26. [27]

    Donnay, A

    L. Donnay, A. Fiorucci, Y . Herfray, and R. Ruzziconi, (2022), arXiv:2212.12553 [hep-th]

    work page arXiv 2022

  27. [28]

    Higuchi, J

    A. Higuchi, J. Math. Phys.28, 1553 (1987), [Erratum: J.Math.Phys. 43, 6385 (2002)]

    work page 1987

  28. [29]

    Chodos and E

    A. Chodos and E. Myers, Annals of Physics156, 412 (1984)

    work page 1984

  29. [30]

    M. A. Rubin and C. R. Ordóñez, Journal of Mathematical Physics26, 65 (1985)

    work page 1985

  30. [31]

    Group Averaging for de Sitter free fields

    D. Marolf and I. A. Morrison, Class. Quant. Grav.26, 235003 (2009), arXiv:0810.5163 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  31. [32]

    Ashtekar and R

    A. Ashtekar and R. O. Hansen, J. Math. Phys.19, 1542 (1978)

    work page 1978

  32. [33]

    Ashtekar and J

    A. Ashtekar and J. D. Romano, Class. Quant. Grav.9, 1069 (1992)

    work page 1992

  33. [34]

    Herberthson and M

    M. Herberthson and M. Ludvigsen, Gen. Rel. Grav.24, 1185 (1992)

    work page 1992

  34. [35]

    Perng, Journal of Mathematical Physics40, 1923–1950 (1999)

    S.-M. Perng, Journal of Mathematical Physics40, 1923–1950 (1999)

    work page 1923

  35. [37]

    The BMS4 algebra at spatial infinity

    C. Troessaert, Class. Quant. Grav.35, 074003 (2018), arXiv:1704.06223 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  36. [38]

    On BMS Invariance of Gravitational Scattering

    A. Strominger, JHEP07, 152 (2014), arXiv:1312.2229 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  37. [39]

    Asymptotic Symmetries of Yang-Mills Theory

    A. Strominger, JHEP07, 151 (2014), arXiv:1308.0589 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  38. [40]

    Henneaux and C

    M. Henneaux and C. Troessaert, Journal of High Energy Physics2019(2019), 10.1007/jhep05(2019)147

    work page doi:10.1007/jhep05(2019)147 2019

  39. [41]

    The asymptotic structure of electromagnetism in higher spacetime dimensions

    M. Henneaux and C. Troessaert, Phys. Rev. D99, 125006 (2019), arXiv:1903.04437 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  40. [42]

    Fuentealba and M

    O. Fuentealba and M. Henneaux, JHEP03, 081 (2025), arXiv:2412.05088 [gr-qc]

    work page arXiv 2025

  41. [43]

    Fuentealba and M

    O. Fuentealba and M. Henneaux, JHEP07, 112 (2025), arXiv:2504.05385 [hep-th]

    work page arXiv 2025

  42. [44]

    Asymptotic structure of a massless scalar field and its dual two-form field at spatial infinity

    M. Henneaux and C. Troessaert, JHEP05, 147 (2019), arXiv:1812.07445 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  43. [45]

    Relaxing the Parity Conditions of Asymptotically Flat Gravity

    G. Compère and F. Dehouck, Class. Quant. Grav.28, 245016 (2011), [Erratum: Class.Quant.Grav. 30, 039501 (2013)], arXiv:1106.4045 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  44. [46]

    Quantum Field Theory in Curved Spacetime

    R. M. Wald, “Quantum field theory in curved spacetime,” (1995), arXiv:gr-qc/9509057 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  45. [47]

    Bousso, A

    R. Bousso, A. Maloney, and A. Strominger, Physical Review D65(2002), 10.1103/physrevd.65.104039

    work page doi:10.1103/physrevd.65.104039 2002

  46. [48]

    R. M. Wald,Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics, Chicago Lectures in Physics (University of Chicago Press, Chicago, IL, 1995)

    work page 1995

  47. [49]

    The mathematical functions site,

    “The mathematical functions site,” (1998-2025)

    work page 1998

  48. [50]

    Marolf and I

    D. Marolf and I. A. Morrison, Classical and Quantum Gravity 26, 235003 (2009)

    work page 2009

  49. [51]

    Higuchi and R

    A. Higuchi and R. H. Weeks, Classical and Quantum Gravity 20, 3005–3021 (2003)

    work page 2003

  50. [52]

    Fuentealba, M

    O. Fuentealba, M. Henneaux, and C. Troessaert, JHEP02, 248 (2023), arXiv:2211.10941 [hep-th]

    work page arXiv 2023

  51. [53]

    Compère, S

    G. Compère, S. E. Gralla, and H. Wei, Class. Quant. Grav.40, 205018 (2023), arXiv:2303.17124 [gr-qc]

    work page arXiv 2023

  52. [54]

    The general so- lution isH=c+ψ (q) 1 withψ (q) 1 a linear combination ofn=1, ℓ=0,1q-parity scalar harmonics

    More precisely, the conformal Killing equation is equivalent to DaDbH+q abH=cq ab, wherecis a constant. The general so- lution isH=c+ψ (q) 1 withψ (q) 1 a linear combination ofn=1, ℓ=0,1q-parity scalar harmonics. SinceHis defined up to an additive constant through the relation ˆχa =D aH, we may set c=0

    work page

  53. [55]

    Beig and B

    R. Beig and B. G. Schmidt, Communications in Mathematical Physics87, 65 (1982)

    work page 1982

  54. [56]

    Deser, Ann

    S. Deser, Ann. Inst. H. Poincare Phys. Theor. A7, 149 (1967). 31

    work page 1967

  55. [57]

    However, this constant is not relevant since DaDbC− 1 3 □Cq ab =0

    Technically, the equation is(□+3)φ (n) =S (n)φ +C, where Cis a constant. However, this constant is not relevant since DaDbC− 1 3 □Cq ab =0

    work page

  56. [58]

    Ashtekar, L

    A. Ashtekar, L. Bombelli, and O. Reula, (1990)

    work page 1990

  57. [59]

    BMS Group at Spatial Infinity: the Hamiltonian (ADM) approach

    M. Henneaux and C. Troessaert, JHEP03, 147 (2018), arXiv:1801.03718 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  58. [60]

    On Asymptotic Flatness and Lorentz Charges

    G. Compère, F. Dehouck, and A. Virmani, Class. Quant. Grav. 28, 145007 (2011), arXiv:1103.4078 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  59. [61]

    Prabhu and I

    K. Prabhu and I. Shehzad, JHEP08, 029 (2022), arXiv:2110.04900 [gr-qc]

    work page arXiv 2022

  60. [62]

    Compère and S

    G. Compère and S. Robert, to appear (2026)

    work page 2026

  61. [63]

    Asymptotic charges in massless QED revisited: A view from Spatial Infinity

    M. Campiglia and A. Laddha, JHEP05, 207 (2019), arXiv:1810.04619 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  62. [64]

    Atul Bhatkar, JHEP02, 082 (2021), arXiv:2007.03627 [hep- th]

    S. Atul Bhatkar, JHEP02, 082 (2021), arXiv:2007.03627 [hep- th]

    work page arXiv 2021

  63. [65]

    Compère, D

    G. Compère, D. Fontaine, and K. Nguyen, SciPost Phys. Core 8, 066 (2025), arXiv:2503.23937 [hep-th]

    work page arXiv 2025

  64. [66]

    Logarithmic Terms in the Soft Expansion in Four Dimensions

    A. Laddha and A. Sen, JHEP10, 056 (2018), arXiv:1804.09193 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  65. [67]

    Observational Signature of the Logarithmic Terms in the Soft Graviton Theorem

    A. Laddha and A. Sen, Phys. Rev. D100, 024009 (2019), arXiv:1806.01872 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  66. [68]

    Sahoo and A

    B. Sahoo and A. Sen, JHEP02, 086 (2019), arXiv:1808.03288 [hep-th]

    work page arXiv 2019

  67. [69]

    Briceño, H

    M. Briceño, H. A. González, M. Henneaux, and A. Pérez, (2025), arXiv:2510.21072 [hep-th]

    work page arXiv 2025

  68. [70]

    Henneaux and C

    M. Henneaux and C. Troessaert, Journal of High Energy Physics2018(2018), 10.1007/jhep05(2018)137

    work page doi:10.1007/jhep05(2018)137 2018

  69. [71]

    Massless QED hass=0 and leadingr=0 fields haven=−1 so thatn+1≤ℓfor anyℓ≥0. For General Relativity, first order SDT tensors satisfyn+1=0≤ℓ, second order SDT tensors (n+1=1)can not haveℓ=0 harmonics and third order SDT tensors(n+1=2)can not haveℓ=0,1 harmonics

    work page

  70. [72]

    Freidel, D

    L. Freidel, D. Pranzetti, and A.-M. Raclariu, JHEP02, 176 (2024), arXiv:2212.12469 [hep-th]

    work page arXiv 2024