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arxiv: 2605.29831 · v1 · pith:UNOQNJBJnew · submitted 2026-05-28 · ✦ hep-th · gr-qc

Five-dimensional Geometry from Spinning Amplitudes

Lucile Cangemi , Iustin Surubaru This is my paper

Pith reviewed 2026-06-29 06:51 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords spinning amplitudesfive dimensionsblack holesMyers-Perry solutionclassical limitmultipole expansionHodge dualspinor-helicity
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The pith

To match five-dimensional black hole geometries, the classical limit of spinning amplitudes requires augmenting multipole structures with the Hodge dual of the spin tensor.

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

The paper examines the classical limit of three-point spinor-helicity amplitudes involving spinning particles in five dimensions. It finds that these amplitudes reproduce the multipole expansion of black hole spacetimes only after the structures are supplemented by the Hodge dual of the classical spin tensor. This construction reproduces the geometries of five-dimensional black holes, including the Myers-Perry solution. The same spacetimes can also be obtained from alternative spinning states or by allowing an internal spin shift in the amplitudes. A reader would care because the result links quantum scattering amplitudes directly to classical solutions of Einstein's equations in higher dimensions.

Core claim

In five dimensions the classical limit of three-point spinor-helicity amplitudes reproduces the multipole expansion of black hole spacetimes only after the multipole structures are augmented with the Hodge dual of the classical spin tensor. The resulting expressions match the geometries of five-dimensional black holes, including the Myers-Perry solution. Alternative spinning states not built from fully symmetric products of polarizations describe the same spacetimes. Relaxing the orthogonality condition on the spin tensor allows modeling of spacetimes with a single rotational isometry, which correspond to the classical limit of amplitudes that permit an internal spin shift.

What carries the argument

The Hodge dual of the classical spin tensor, which augments the multipole structures arising from the map between spinor invariants and spin-operator expectation values.

If this is right

  • A specific class of amplitudes describes the Myers-Perry black hole.
  • Amplitudes with an internal spin shift match spacetimes possessing only a single rotational isometry.
  • Spinning states constructed from non-symmetric polarization products still generate the same black-hole geometries.
  • The construction extends in principle to arbitrary dimensions.

Where Pith is reading between the lines

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

  • Similar dual structures may be required when matching amplitudes to geometries in six or more dimensions.
  • The approach could supply a systematic route to compute higher-order gravitational scattering involving rotating compact objects.
  • It raises the question of whether charged or other non-vacuum five-dimensional solutions admit analogous amplitude descriptions.

Load-bearing premise

The classical limit of three-point spinor-helicity amplitudes in five dimensions directly reproduces the multipole expansion of black hole spacetimes once the Hodge dual of the spin tensor is included.

What would settle it

Explicit computation of the multipole moments from the augmented amplitude expressions that fails to match the known moments of the five-dimensional Myers-Perry metric.

read the original abstract

Massive spinor-helicity variables in four dimensions are a useful tool for studying amplitudes between higher-spin fields and gravitons. Among them a special, simple set of amplitudes reproduces the linearized stress-energy tensor of a Kerr black hole in the classical limit. In this work we initiate the study of the classical limit of three-point spinor-helicity amplitudes in five dimensions. We introduce the map between the spinor invariants and the expectation values of spin operators and match the amplitude building blocks with those of the multipole expansion. Interestingly, in order to take the classical limit of a general amplitude, we need to augment the multipole structures with the Hodge dual of the classical spin tensor. We study the classical limit of alternative spinning states not described by fully-symmetric products of polarisations and conclude that they can describe the same spacetimes. Finally, by relaxing the orthogonality condition of the spin tensor we are able to model spacetimes with a single rotational isometry and match these to the classical limit of amplitudes allowing for an internal spin shift. Along the way we also identify the class of amplitudes describing the Myers-Perry black hole and comment on its generalization to arbitrary dimensions.

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

3 major / 2 minor

Summary. The paper extends massive spinor-helicity methods from four to five dimensions to compute the classical limit of three-point amplitudes involving higher-spin fields and gravitons. It introduces a map from spinor invariants to spin-operator expectation values, matches amplitude building blocks to the multipole expansion of the linearized stress-energy tensor, and reports that the Hodge dual of the classical spin tensor must be included to reproduce five-dimensional black-hole geometries, including the Myers-Perry solution. The work also examines alternative (non-fully-symmetric) spinning states, shows they can describe the same spacetimes, and relaxes the spin-tensor orthogonality condition to model single-isometry backgrounds via amplitudes that allow an internal spin shift. Along the way it identifies the subclass of amplitudes that correspond to the Myers-Perry black hole.

Significance. If the central matching holds after the required technical clarifications, the paper supplies a concrete dictionary between five-dimensional spinor-helicity amplitudes and the multipole structure of higher-dimensional black holes. This would extend the four-dimensional Kerr-amplitude program to D=5 and furnish a systematic way to generate linearized solutions with one or two rotational isometries directly from on-shell data. The explicit identification of Myers-Perry amplitudes is a useful concrete result.

major comments (3)
  1. [Abstract] Abstract, paragraph beginning 'Interestingly, in order to take the classical limit...': the claim that augmentation by the Hodge dual *S_{\mu\nu} is required to match the classical limit is load-bearing for every subsequent geometric identification. No explicit derivation is supplied showing that the dual term is forced by the five-dimensional little-group structure or by the map from spinor invariants to spin-operator expectation values, rather than being an artifact of the chosen polarization basis that could be absorbed by redefinition.
  2. [Section on classical limit and multipole matching] Section describing the map between spinor invariants and spin operators (the paragraph that introduces the classical-limit matching): the paper states that the resulting expressions reproduce the multipole expansion of Myers-Perry and more general five-dimensional black holes, yet no explicit component-by-component comparison of the amplitude-derived stress-energy tensor with the linearized Myers-Perry metric is provided. Without this check it remains unclear whether the match is unique or holds only after the dual is inserted by hand.
  3. [Section on relaxed orthogonality and single-isometry spacetimes] Section on relaxing the orthogonality condition: the assertion that this relaxation models spacetimes with a single rotational isometry and matches amplitudes allowing an internal spin shift is central to the claim of broader applicability. The text does not demonstrate that the resulting stress-energy tensor is conserved or that its multipole moments agree with the known single-isometry solutions once the dual term is included.
minor comments (2)
  1. Notation for the Hodge dual *S_{\mu\nu} is introduced without an explicit definition of the five-dimensional Hodge star in the spinor-helicity language; a short appendix or footnote clarifying the conventions would improve readability.
  2. The statement that 'alternative spinning states not described by fully-symmetric products of polarisations' can describe the same spacetimes would benefit from a brief table comparing the leading multipole moments obtained from symmetric versus non-symmetric constructions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below and indicate where revisions will be made to strengthen the presentation.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph beginning 'Interestingly, in order to take the classical limit...': the claim that augmentation by the Hodge dual *S_{μ u} is required to match the classical limit is load-bearing for every subsequent geometric identification. No explicit derivation is supplied showing that the dual term is forced by the five-dimensional little-group structure or by the map from spinor invariants to spin-operator expectation values, rather than being an artifact of the chosen polarization basis that could be absorbed by redefinition.

    Authors: We agree that an explicit derivation of the Hodge dual term would clarify its origin. The dual arises necessarily from the five-dimensional little-group structure when mapping spinor invariants to classical spin-operator expectation values, as required to reproduce the correct multipole expansion for spinning sources in D=5; it is not removable by polarization redefinition. In the revised manuscript we will insert a dedicated paragraph in the classical-limit section deriving this requirement step by step from the little-group invariants. revision: yes

  2. Referee: [Section on classical limit and multipole matching] Section describing the map between spinor invariants and spin operators (the paragraph that introduces the classical-limit matching): the paper states that the resulting expressions reproduce the multipole expansion of Myers-Perry and more general five-dimensional black holes, yet no explicit component-by-component comparison of the amplitude-derived stress-energy tensor with the linearized Myers-Perry metric is provided. Without this check it remains unclear whether the match is unique or holds only after the dual is inserted by hand.

    Authors: The matching is established by equating the amplitude building blocks to the multipole structures of the linearized stress-energy tensor. To address the request for explicit verification we will add, in the revised version, a short appendix containing the component-by-component comparison for the Myers-Perry case, confirming that the match holds only after inclusion of the dual term and is unique within the chosen basis. revision: yes

  3. Referee: [Section on relaxed orthogonality and single-isometry spacetimes] Section on relaxing the orthogonality condition: the assertion that this relaxation models spacetimes with a single rotational isometry and matches amplitudes allowing an internal spin shift is central to the claim of broader applicability. The text does not demonstrate that the resulting stress-energy tensor is conserved or that its multipole moments agree with the known single-isometry solutions once the dual term is included.

    Authors: We will expand the relevant section to include an explicit check that the stress-energy tensor obtained after relaxing orthogonality remains conserved (via direct computation of its divergence) and that its multipole moments reproduce those of the known single-isometry solutions when the dual term is retained. This will be accompanied by a brief comparison with the internal-spin-shift amplitudes. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation proceeds from amplitudes to geometry without reduction to inputs by construction.

full rationale

The paper introduces a map from spinor invariants to spin-operator expectation values, augments multipole structures with the Hodge dual of the spin tensor to obtain the classical limit, and matches the resulting expressions to five-dimensional black-hole geometries including Myers-Perry. This direction runs from amplitude building blocks to linearized stress-energy and multipole expansions; no equation is shown to equal its own input by definition, no fitted parameter is relabeled as a prediction, and no load-bearing premise reduces to a self-citation chain. The augmentation is presented as required for matching rather than presupposed, and alternative spinning states are checked to reproduce the same spacetimes. The derivation therefore remains self-contained against external geometric benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; the central claim rests on the domain assumption that the classical limit of amplitudes reproduces black-hole multipoles and on the validity of the newly introduced map between spinor invariants and spin-operator expectation values.

axioms (1)
  • domain assumption The classical limit of three-point spinor-helicity amplitudes reproduces the multipole expansion of black-hole geometries once the Hodge dual of the spin tensor is included.
    This is the load-bearing link asserted in the abstract between the amplitude calculation and the target spacetime.

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