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arxiv: 2604.05094 · v2 · submitted 2026-04-06 · ✦ hep-th · gr-qc

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Weak-Field Limits of Black Hole Metrics from the KMOC formalism: Schwarzschild, Kerr, Reissner-Nordstr\"om, and Kerr-Newman

Jacobo Hern\'andez C.
Authors on Pith no claims yet

Pith reviewed 2026-05-12 00:45 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords black hole metricsKMOC formalismscattering amplitudesweak-field limitKerr-Newman metricmomentum impulsegeodesic motionexponential spin structure
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The pith

Three-point amplitudes with exponential spin yield the weak-field limits of Schwarzschild, Kerr, Reissner-Nordström and Kerr-Newman metrics via the KMOC impulse formula.

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

The paper shows that the leading weak-field expansions of four standard black hole metrics emerge directly from quantum scattering amplitudes. Three-point amplitudes carrying exponential spin factors for gravity and electromagnetism are used to build four-point amplitudes; the KMOC formula then extracts the momentum impulse. This impulse is matched to the geodesic equation in a general weak-field metric ansatz, fixing the metric components order by order in G, spin parameter a and charge-squared Q². In the Kerr-Newman case the same procedure produces an extra interference term proportional to Q² a / r³ in the g_{tφ} component that is absent when gravity or electromagnetism is considered alone.

Core claim

Starting from three-point amplitudes with exponential spin structure for both gravitational and electromagnetic interactions, four-point scattering amplitudes are computed and the momentum impulse is extracted via the KMOC formula. Matching the impulse to geodesic motion in a general metric reconstructs the metric components to leading order in G, a and Q². For Kerr-Newman, gravitational-electromagnetic interference produces a Q² a / r³ contribution to g_{tφ} that does not appear in the separate Kerr or Reissner-Nordström limits.

What carries the argument

The KMOC formula that converts the classical limit of the four-point amplitude into a momentum impulse, followed by direct matching of that impulse to the geodesic equation in a general weak-field metric.

If this is right

  • The weak-field Schwarzschild metric follows from the spinless gravitational three-point amplitude alone.
  • Exponential spin factors in the three-point amplitude generate the leading spin-dependent corrections that reproduce the Kerr metric.
  • Electromagnetic three-point amplitudes with charge factors reproduce the Reissner-Nordström weak-field limit.
  • Interference between gravitational and electromagnetic four-point diagrams supplies the mixed Q² a term in the Kerr-Newman g_{tφ} component.

Where Pith is reading between the lines

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

  • The same amplitude-to-metric pipeline could be extended to higher orders in G by including loop contributions or higher-point amplitudes.
  • The method supplies a scattering-amplitude route to classical observables that bypasses direct solution of the Einstein-Maxwell equations.
  • Because the reconstruction relies only on the impulse, it can be tested against other classical limits such as the deflection angle or periastron advance without constructing the full metric.

Load-bearing premise

The chosen three-point amplitudes with exponential spin structure correctly encode the classical gravitational and electromagnetic interactions of black holes, and matching the resulting impulse to geodesic motion fully determines the metric components.

What would settle it

A numerical or analytic computation of the momentum transfer in the weak-field limit that disagrees with the known geodesic motion in the standard Schwarzschild, Kerr, Reissner-Nordström or Kerr-Newman metric to order G, a or Q².

read the original abstract

The KMOC (Kosower-Maybee-O'Connell) formalism establishes a bridge between quantum scattering amplitudes and classical observables in gravitational systems. In this work, we show how the weak-field limits of the four classical black hole metrics - Schwarzschild, Kerr, Reissner-Nordstrom, and Kerr-Newman - can be reproduced within this formalism. Starting from three-point amplitudes with exponential spin structure for both gravitational and electromagnetic interactions, we compute four-point scattering amplitudes and extract the momentum impulse via the KMOC formula. Matching these results with geodesic motion in a general metric allows us to reconstruct the metric components to leading order in G, a, and Q^2. For the Kerr-Newman case, we include interference terms between gravitational and electromagnetic interactions, which produce a Q^2 a/r^3 contribution to g_{t\phi} that does not appear in the Kerr or Reissner-Nordstrom weak-field limits separately. Our results are consistent with those of arXiv:1907.00431 [hep-th], where the Kerr-Newman metric is derived from minimal coupling amplitudes using the KMOC formalism arXiv:1908.04342 [hep-th]. All results are verified through their consistency with the well-known full metrics, though we emphasize that the KMOC formalism as applied here reproduces only the weak-field expansions, not the complete non-linear solutions.

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 / 3 minor

Summary. The paper claims that the weak-field limits of the Schwarzschild, Kerr, Reissner-Nordström, and Kerr-Newman metrics can be reconstructed from the KMOC formalism. It computes four-point amplitudes from three-point amplitudes with exponential spin structure for gravitational and electromagnetic interactions, extracts the classical momentum impulse, and matches the result to geodesic motion in a general weak-field metric ansatz, recovering the metric components to leading order in G, a, and Q². Interference terms are explicitly included for the Kerr-Newman case, producing a Q²a/r³ contribution to g_{tφ}.

Significance. If the central matching procedure is valid, the work offers a systematic amplitude-based route to weak-field black-hole metrics, extending prior KMOC applications and confirming consistency with known results such as arXiv:1907.00431. It highlights the formalism's ability to capture interference effects between gravity and electromagnetism while remaining limited to perturbative expansions rather than full nonlinear solutions.

major comments (1)
  1. [Kerr-Newman reconstruction] In the Kerr-Newman reconstruction (abstract and the section presenting the combined gravitational-electromagnetic case), the full KMOC impulse—including the 2 Re(A_grav A_em*) interference—is matched directly to the momentum change along geodesics in a weak-field metric ansatz. This yields a Q² a / r³ term in g_{tφ}. The standard Kerr-Newman line element has g_{tφ} = −(2 M a r sin²θ)/Σ with Σ = r² + a² cos²θ and no Q dependence at this order; its weak-field expansion (harmonic or isotropic coordinates) therefore contains only M a / r and higher a³ terms. The manuscript must clarify the probe charge, whether the matching incorporates the Lorentz force from the electromagnetic field of the source, and demonstrate explicitly (via the ansatz and matching equations) that the procedure reproduces the known metric without introducing a spurious Q-dependent term or relying on an unstated gauge.
minor comments (3)
  1. The abstract states that results are verified by consistency with the full metrics, but the main text should include a brief table or explicit coefficient comparison for at least one component (e.g., g_{tt} or g_{tφ}) across all four spacetimes to make the verification transparent.
  2. Specify the coordinate gauge (harmonic, isotropic, etc.) in which the weak-field expansions are written when comparing to the standard metrics.
  3. The three-point amplitudes with exponential spin structure are central; a short appendix or reference to their explicit form would aid reproducibility of the four-point computation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for the constructive feedback. We address the major comment below and will incorporate the necessary clarifications and adjustments in a revised version.

read point-by-point responses

  1. Referee: In the Kerr-Newman reconstruction (abstract and the section presenting the combined gravitational-electromagnetic case), the full KMOC impulse—including the 2 Re(A_grav A_em*) interference—is matched directly to the momentum change along geodesics in a weak-field metric ansatz. This yields a Q² a / r³ term in g_{tφ}. The standard Kerr-Newman line element has g_{tφ} = −(2 M a r sin²θ)/Σ with Σ = r² + a² cos²θ and no Q dependence at this order; its weak-field expansion (harmonic or isotropic coordinates) therefore contains only M a / r and higher a³ terms. The manuscript must clarify the probe charge, whether the matching incorporates the Lorentz force from the electromagnetic field of the source, and demonstrate explicitly (via the ansatz and matching equations) that the procedure reproduces the known metric without introducing a spurious Q-dependent term or relying on an unstated gauge.

    Authors: We agree that the current presentation requires clarification and that the matching procedure must be made fully explicit. The goal of the metric reconstruction is to recover the spacetime geometry as probed by neutral test particles, for which the equations of motion reduce to geodesics in the metric (with no Lorentz force). Accordingly, only the gravitational amplitudes enter this matching; electromagnetic contributions and their interference with gravity are relevant exclusively for charged probes, where the full classical impulse includes both gravitational and electromagnetic forces. We will revise the manuscript to (i) state explicitly that the Kerr-Newman metric reconstruction uses neutral probes and gravitational amplitudes alone, (ii) remove the interference term from the metric-matching equations, (iii) display the general weak-field metric ansatz together with the explicit matching conditions that recover the standard expansion (containing only the Ma/r term in g_{tφ} at the order considered), and (iv) specify the coordinate system employed. These changes will eliminate the spurious Q-dependent term and address the referee’s concerns about probe charge and the Lorentz force. revision: yes

Circularity Check

0 steps flagged

No significant circularity: derivation proceeds from independent amplitudes via KMOC to metric reconstruction

full rationale

The paper computes four-point amplitudes from three-point amplitudes with exponential spin structure, extracts the classical impulse using the established KMOC formula, and equates the result to the impulse from geodesic motion in a general weak-field metric ansatz whose coefficients are then solved for. This constructs the metric components from QFT inputs rather than presupposing them. Results are verified by consistency with known full metrics and cited prior derivations, but the central chain does not reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The provided text contains no equations or steps where a claimed prediction equals its input by definition or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the validity of the KMOC formula and the assumption that three-point amplitudes with exponential spin structure capture the classical limit of black-hole interactions; these are standard domain assumptions imported from prior literature rather than derived in the paper.

axioms (2)
  • domain assumption The KMOC formula correctly extracts the classical momentum impulse from quantum scattering amplitudes.
    Invoked when converting four-point amplitudes into the observable used for metric reconstruction.
  • domain assumption Three-point amplitudes with exponential spin structure encode the relevant gravitational and electromagnetic interactions for spinning and charged particles.
    This is the starting point for all amplitude computations in the paper.

pith-pipeline@v0.9.0 · 5556 in / 1457 out tokens · 41699 ms · 2026-05-12T00:45:27.195151+00:00 · methodology

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Reference graph

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