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arxiv: 2604.22181 · v1 · submitted 2026-04-24 · 🌀 gr-qc

Recognition: unknown

Shadow dependent phenomenology framework for rotating black hole metric

Nikko John Leo S. Lobos , Emmanuel T. Rodulfo
Authors on Pith no claims yet

Pith reviewed 2026-05-08 10:53 UTC · model grok-4.3

classification 🌀 gr-qc
keywords shadow radiusblack hole thermodynamicsweak deflection angleHawking radiationKerr-MOGrotating HorndeskiEvent Horizon Telescopemodified gravity
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The pith

A diffeomorphic inversion re-parameterizes rotating black hole mass using the observable shadow radius to compute deflection, temperature, and luminosity without the bare mass.

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

The paper develops a thermodynamic-optical duality by inverting the mass parameter through the measured shadow radius R_sh. This lets the weak deflection angle, Hawking temperature, and semiclassical luminosity be written directly in terms of R_sh for Kerr, Kerr-MOG, and rotating Horndeski metrics. With shadow size fixed by EHT M87* data, the standard Kerr case gives a simple luminosity scaling while the modified models produce distinct signatures: MOG suppresses emission and Horndeski adds a logarithmic lensing term with up to 52 percent deviation in radiated power. The method therefore turns an interferometric observable into a direct test of strong-field gravity and the Kerr hypothesis.

Core claim

By employing a diffeomorphic inversion, the intrinsic black hole mass is re-parameterized entirely in terms of the observable shadow radius R_sh. This mapping formulates the classical weak deflection angle, Hawking temperature, and integrated semiclassical luminosity without reference to the bare mass. For a statistically fixed shadow radius constrained by EHT observations of M87*, the standard Kerr geometry yields L proportional to R_sh to the minus two; Kerr-MOG enhances deflection while suppressing luminosity; and rotating Horndeski introduces logarithmic augmentation to lensing with up to approximately 52 percent deviation in Hawking emission.

What carries the argument

The diffeomorphic inversion that re-expresses the black hole mass parameter directly in terms of the shadow radius R_sh, thereby linking optical and thermodynamic observables.

Load-bearing premise

The diffeomorphic inversion is assumed to remain valid and to preserve all physical content when applied to the Kerr-MOG and rotating Horndeski spacetimes without introducing extra model-dependent corrections at the shadow boundary.

What would settle it

An independent dynamical mass measurement for M87* that yields a value inconsistent with the mass inferred from the observed R_sh under the inversion, or a luminosity measurement that fails to match the predicted scaling or the 52 percent Horndeski deviation, would falsify the mapping.

read the original abstract

We establish a thermodynamic-optical duality that directly bridges the semiclassical quantum evaporation of black holes with their classical macroscopic geometry. By employing a diffeomorphic inversion, we re-parameterize the intrinsic black hole mass entirely in terms of the observable shadow radius $R_{sh}$. This mapping allows the formulation of the classical weak deflection angle, Hawking temperature, and integrated semiclassical luminosity, bypassing the unobservable bare mass. Applying this methodology to the standard Kerr, Kerr-MOG, and rotating Horndeski spacetimes, we reveal distinct, model-specific phenomenological signatures. For a statistically fixed shadow radius constrained by Event Horizon Telescope (EHT) observations of M87*, the standard Kerr geometry yields a baseline luminosity scaling of $L \propto R_{sh}^{-2}$. In modified gravity regimes, the duality breaks the degeneracy between bare mass and modified field strengths: the MOG repulsive vector field enhances classical deflection while strictly suppressing quantum luminosity, whereas Horndeski scalar hair introduces a unique logarithmic augmentation to astrometric lensing and drives up to a $\sim 52\%$ deviation in Hawking emission under current EHT limits. By strictly anchoring theoretical observables to empirical interferometric boundaries, this framework provides a computationally efficient avenue for testing the Kerr hypothesis and probing fundamental fields in strong-field gravity.

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 manuscript proposes a thermodynamic-optical duality for rotating black holes in which a diffeomorphic inversion re-expresses the bare mass M solely as a function of the observable shadow radius R_sh. This mapping is used to rewrite the weak deflection angle, Hawking temperature, and integrated semiclassical luminosity directly in terms of R_sh for the Kerr, Kerr-MOG, and rotating Horndeski metrics. For an EHT-constrained R_sh corresponding to M87*, the standard Kerr case yields L ∝ R_sh^{-2}; Kerr-MOG produces enhanced deflection but suppressed luminosity due to the repulsive vector field; rotating Horndeski introduces logarithmic corrections to lensing and up to a ~52% deviation in Hawking emission.

Significance. If the inversion is rigorously shown to preserve the physical content of the temperature and luminosity formulas without model-specific corrections to the photon sphere or surface gravity, the framework would supply a computationally direct route to translate EHT shadow measurements into constraints on modified-gravity parameters via their imprint on semiclassical evaporation. The explicit model comparisons and the reported scaling relations could serve as falsifiable predictions for future high-precision shadow and multi-messenger observations.

major comments (3)
  1. [Abstract and §3] Abstract and §3 (thermodynamic-optical duality): The ~52% deviation in Hawking emission for rotating Horndeski is asserted for EHT limits on R_sh, yet no explicit inverted expression for T_H(R_sh, a, ξ) or the integrated luminosity is supplied, nor is the numerical evaluation (including the range of the Horndeski parameter ξ) shown; without these steps the deviation cannot be verified as arising from the duality rather than from an arbitrary choice of ξ.
  2. [§4.2–4.3] §4.2–4.3 (applications to Kerr-MOG and rotating Horndeski): The substitution M = M(R_sh, a, α) into the surface-gravity formula assumes that all α-dependence is already encoded in the algebraic shadow-radius relation; however, the effective potential for null geodesics and the Killing-vector normalization in these spacetimes contain non-factorizable α terms, so additional corrections to the shadow boundary itself may be required before the thermodynamic quantities can be re-expressed solely in R_sh.
  3. [§3] §3, Eq. (inversion mapping): The claim that the duality “bypasses the unobservable bare mass” is load-bearing for all subsequent results, but the manuscript provides neither the explicit functional form of the inversion nor a demonstration that it remains invertible and one-to-one when the spin a and extra parameters vary within the EHT uncertainty band on R_sh.
minor comments (2)
  1. The phrase “diffeomorphic inversion” is introduced without a precise mathematical definition or a short proof that the map is indeed a diffeomorphism on the parameter space; a one-paragraph clarification would remove ambiguity.
  2. No uncertainty propagation from the EHT measurement error on R_sh is reported when quoting the 52% deviation or the L ∝ R_sh^{-2} scaling; adding error bands would strengthen the phenomenological claims.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions, which have prompted us to strengthen the presentation of the thermodynamic-optical duality. We address each major comment below, providing clarifications and committing to revisions where the manuscript requires additional explicit derivations or demonstrations.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (thermodynamic-optical duality): The ~52% deviation in Hawking emission for rotating Horndeski is asserted for EHT limits on R_sh, yet no explicit inverted expression for T_H(R_sh, a, ξ) or the integrated luminosity is supplied, nor is the numerical evaluation (including the range of the Horndeski parameter ξ) shown; without these steps the deviation cannot be verified as arising from the duality rather than from an arbitrary choice of ξ.

    Authors: We agree that the explicit inverted expressions and numerical details were insufficiently detailed for independent verification. In the original §3 the duality is constructed by solving the shadow-radius equation for M and substituting into the surface-gravity and luminosity formulas, but the intermediate algebraic steps for the Horndeski case were omitted. In the revised manuscript we will insert the closed-form inversion M(R_sh, a, ξ), the resulting T_H(R_sh, a, ξ) = (1/2π) κ(M(R_sh, a, ξ), a, ξ), and the integrated luminosity expression. We will also add a table listing the percentage deviation in Hawking emission for ξ values within the EHT-allowed interval for M87*, confirming that the reported ~52% shift originates directly from the duality mapping rather than from an arbitrary parameter choice. revision: yes

  2. Referee: [§4.2–4.3] §4.2–4.3 (applications to Kerr-MOG and rotating Horndeski): The substitution M = M(R_sh, a, α) into the surface-gravity formula assumes that all α-dependence is already encoded in the algebraic shadow-radius relation; however, the effective potential for null geodesics and the Killing-vector normalization in these spacetimes contain non-factorizable α terms, so additional corrections to the shadow boundary itself may be required before the thermodynamic quantities can be re-expressed solely in R_sh.

    Authors: The shadow radius R_sh is obtained by solving the full null-geodesic effective potential, which already incorporates every non-factorizable α (or ξ) term present in the metric, including those arising from the Killing-vector normalization. Consequently, the algebraic inversion M(R_sh, a, α) encodes the complete parameter dependence, and direct substitution into the surface-gravity formula—itself derived from the same metric—yields the thermodynamically consistent quantities without requiring further corrections to the photon-sphere boundary. The observable R_sh remains fixed by EHT data; the duality simply re-expresses all other quantities in terms of that fixed observable. We will expand §4.2–4.3 to display the explicit effective-potential equation and the substitution steps, thereby clarifying that no additional boundary corrections are needed. revision: partial

  3. Referee: [§3] §3, Eq. (inversion mapping): The claim that the duality “bypasses the unobservable bare mass” is load-bearing for all subsequent results, but the manuscript provides neither the explicit functional form of the inversion nor a demonstration that it remains invertible and one-to-one when the spin a and extra parameters vary within the EHT uncertainty band on R_sh.

    Authors: The inversion is defined by solving the shadow-radius equation R_sh(M, a, α) = observed value for M, which for each metric yields an explicit algebraic expression (quadratic for Kerr, linear in M for Kerr-MOG after rearrangement, and involving the Horndeski parameter for the rotating case). We will include these closed-form inversions in the revised §3. To demonstrate invertibility and uniqueness within the EHT uncertainty band, we will add a short appendix showing that ∂R_sh/∂M > 0 throughout the relevant ranges of a and α (or ξ), with no critical points or branch crossings inside the 1σ EHT interval for M87*. This establishes that the mapping is one-to-one and that the duality indeed bypasses the bare mass without ambiguity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reparameterization is a change of variables with independent model content

full rationale

The paper introduces a diffeomorphic inversion to express black hole mass in terms of observed shadow radius R_sh, then substitutes into expressions for weak deflection angle, Hawking temperature, and luminosity for Kerr, Kerr-MOG, and rotating Horndeski metrics. This is a straightforward reparameterization anchored to EHT data for M87*. The baseline L ∝ R_sh^{-2} for Kerr follows from standard scaling relations (T ∝ 1/M, R_sh ∝ M) but is presented only as a reference point. Model-specific deviations (e.g., MOG suppression of luminosity or ~52% Horndeski shift) arise from the distinct R_sh(M, a, param) functional forms in each spacetime, which are independent of the inversion step itself. No load-bearing self-citation, fitted-input prediction, or self-definitional reduction is exhibited; the derivation chain remains self-contained against external benchmarks like EHT constraints and does not reduce derived quantities to the input R_sh by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The framework rests on a newly introduced diffeomorphic inversion whose validity is asserted rather than derived from first principles; standard black-hole thermodynamics and the EHT shadow measurement are taken as external inputs.

free parameters (1)
  • Shadow radius R_sh
    Fixed to EHT value for M87* to enable model comparison; all derived luminosities and temperatures are expressed in terms of this single observable.
axioms (1)
  • domain assumption Diffeomorphic inversion maps the bare mass to R_sh while preserving thermodynamic and optical relations for the listed spacetimes
    Invoked to bypass the unobservable mass and obtain all quantities from the shadow radius.
invented entities (1)
  • Thermodynamic-optical duality no independent evidence
    purpose: Bridges semiclassical evaporation with classical shadow geometry
    Newly postulated mapping that allows direct substitution of R_sh for mass in all formulas.

pith-pipeline@v0.9.0 · 5526 in / 1411 out tokens · 33748 ms · 2026-05-08T10:53:43.635112+00:00 · methodology

discussion (0)

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Reference graph

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