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Shadow radius replaces black hole mass as the master observable

2026-07-04 19:35 UTC pith:6O2FK5DE

load-bearing objection Re-parameterizing BH observables in terms of shadow radius: clean idea, execution gaps in MOG/Horndeski and overclaimed scope the 3 major comments →

arxiv 2604.22181 v2 pith:6O2FK5DE submitted 2026-04-24 gr-qc

Shadow dependent phenomenology framework for rotating black hole metric

Nikko John Leo S. Lobos , Emmanuel T. Rodulfo This is my paper
classification gr-qc PACS 04.70.Bw04.70.Dy04.50.Kd98.62.Sb
keywords black hole shadowHawking radiationweak deflection anglemodified gravityKerr-MOGHorndeskiEvent Horizon TelescopeImplicit Function Theorem
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims that the observable black hole shadow radius R_sh can serve as a complete replacement for the unobservable intrinsic mass parameter M in all standard black hole phenomenology — weak gravitational lensing, Hawking temperature, and integrated quantum luminosity. The mechanism is a diffeomorphic inversion M = M(R_sh, a) whose existence is proven via the Implicit Function Theorem: because the shadow radius depends smoothly and non-degenerately on mass (verified analytically in the Schwarzschild and slow-rotation Kerr limits, where dR_sh/dM = 3√3 + O(a²/M²) > 0), one can solve for mass as a function of the shadow radius and spin, then substitute this expression into every formula that previously contained M. The paper applies this re-parameterization to three spacetimes — standard Kerr, Kerr-MOG (a modified-gravity model with a repulsive vector field), and rotating Horndeski (a scalar-tensor theory with scalar hair) — and shows that when the shadow radius is held fixed, each theory predicts a distinct combination of weak-field deflection angle and Hawking luminosity. Specifically, the MOG vector field amplifies classical lensing while suppressing quantum emission, whereas Horndeski scalar hair introduces a logarithmic correction to lensing and can shift the Hawking luminosity by up to roughly 52 percent relative to general relativity under current EHT constraints on M87*. The central claim is that these model-specific signatures, all expressed in terms of the directly observable shadow size rather than the inferred mass, allow next-generation VLBI observations to distinguish between general relativity and competing theories of gravity.

Core claim

The paper's central object is the inverse mapping M = M(R_sh, a), constructed by treating the shadow radius R_sh — the apparent size of the photon-sphere boundary seen by a distant observer — as the independent variable and solving for the bare mass M as a dependent function of R_sh and spin a. The existence proof relies on the Implicit Function Theorem applied to the constraint F(M, R_sh, a) = R_sh - R(M, a, r_ph) = 0, with non-degeneracy condition dR_sh/dM ≠ 0 verified explicitly in the Schwarzschild limit (yielding 3√3) and to leading order in slow rotation. Once this inversion is established, it composes with the horizon-radius map r_h = r_h(M, a) to produce r_h = R_h(R_sh, a), cascading

What carries the argument

Implicit Function Theorem inversion M = M(R_sh, a); Gauss-Bonnet theorem on the optical manifold for weak deflection; Stefan-Boltzmann luminosity with eikonal absorption cross-section σ_abs → πR²_sh; slow-rotation expansions throughout (a/M << 1)

Load-bearing premise

The inversion M = M(R_sh, a) is proven non-degenerate only for Kerr in the Schwarzschild and slow-rotation limits, yet is applied to Kerr-MOG and Horndeski spacetimes without independently verifying that the Jacobian dR_sh/dM remains non-zero for those metrics. If the Jacobian vanishes for any of the three spacetimes, every derived observable loses its mathematical foundation.

What would settle it

A demonstration that dR_sh/dM = 0 (or changes sign) for Kerr-MOG or Horndeski spacetimes at physically relevant spin and coupling values, which would break the diffeomorphic inversion and invalidate the re-parameterized observables for those metrics.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the inversion holds for spacetimes beyond Kerr, any black hole observable traditionally written in terms of mass and horizon radius can be re-expressed purely in terms of the shadow radius and spin, making it directly testable against interferometric data without dynamical mass inference.
  • The finding that MOG and Horndeski produce opposite-signed corrections to Hawking luminosity at fixed shadow size suggests that precision shadow measurements could in principle distinguish vector-based from scalar-based modifications of gravity.
  • The L ∝ R_sh^{-2} scaling for Kerr luminosity provides a parameter-free benchmark: any observed deviation from this scaling at a given shadow size would signal non-Kerr physics.
  • Extending the framework to primordial black holes (R_sh ~ 10^{-15} m) would connect astrometric shadow constraints to high-temperature Hawking emission (~10^{11} K), potentially linking micro-shadow physics to primordial black hole evaporation signatures.

Where Pith is reading between the lines

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

  • The framework implicitly assumes that the shadow radius is the single most informative observable for strong-field gravity. If future EHT observations resolve shadow asymmetry or time variability, the 1D equatorial R_sh used here would need to be replaced by a multi-parameter shadow characterization, and the inversion's non-degeneracy would need to be re-established in higher dimensions.
  • The inverse-correlated phenomenology of MOG (enhanced lensing, suppressed evaporation) versus Horndeski (logarithmic lensing augmentation, enhanced evaporation) suggests a possible classification scheme: modified gravity theories might be sorted by whether their additional fields act as 'luminosity amplifiers' or 'luminosity suppressors' at fixed shadow size, providing a phenomenological taxonomy
  • The framework could be extended to other modified-gravity spacetimes (f(R), Einstein-dilaton-Gauss-Bonnet, etc.) to build a catalog of shadow-fixed phenomenological signatures, turning shadow imaging into a systematic model-discrimination tool rather than a consistency check.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The manuscript proposes a

Significance. The paper presents a framework for re-parameterizing black hole observables — weak deflection angle, Hawking temperature, and integrated luminosity — in terms of the observable shadow radius $R_{sh}$ via a diffeomorphic inversion $M = M(R_{sh}, a)$ justified by the Implicit Function Theorem. The framework is applied to Kerr, Kerr-MOG, and rotating Horndeski spacetimes, with EHT M87* constraints used to produce numerical estimates. The idea of anchoring thermodynamic and lensing observables to the shadow radius is reasonable and of phenomenological interest. The Kerr baseline derivation (Section IV.A) is internally consistent and produces clean closed-form results (Eqs. 26, 28, 30).

major comments (3)
  1. [Section IV.B, Eq. 35] The IFT non-degeneracy condition $dR_{sh}/dM ≠ 0$ is proven only for the Kerr metric, and only in the Schwarzschild limit (Eq. 9) and to $O(a^2)$ (Eq. 10). When the framework is applied to Kerr-MOG, the authors write 'Invoking the diffeomorphic inversion established in Section II.A' (Section IV.B, before Eq. 35), but the Kerr-MOG shadow function $R(M, a, α, r_{ph})$ differs from Kerr because the metric components (Eqs. 31–34) include $α$. The Jacobian $dR_{sh}/dM$ is therefore a different function of $(M, a, α)$, and non-degeneracy for Kerr does not imply non-degeneracy for Kerr-MOG. While a continuity argument for $α ≪ 1$ is plausible, it is not stated or verified. This is load-bearing because every derived MOG observable (Eqs. 35–37) depends on the validity of the inversion. The authors should either compute $dR_{sh}/dM$ explicitly for Kerr-MOG (at least to leading order in $α$) or add
  2. [Section IV.C (Horndeski)] The Horndeski section appears to be truncated or missing from the manuscript. After Eq. 37 (end of Section IV.B), the text jumps to a fragment mentioning '∼2.5% departure' and '∼52% deviation' without any derivation, equations, or Jacobian check. Yet the abstract, introduction, and conclusion all prominently claim that Horndeski scalar hair produces 'a unique logarithmic augmentation to astrometric lensing' and 'drives up to a ∼52% deviation in Hawking emission.' These are central claims of the paper that are not supported by any visible derivation. The Horndeski section must be completed with the full derivation of the shadow radius, mass inversion, deflection angle, and temperature, including the Jacobian non-degeneracy verification.
  3. [Section III.B, Eqs. 21–22, 30] The luminosity formula (Eq. 21) includes correction terms $E_{dev}$ and $δL_{SR}$, but Eq. 30 drops them entirely without quantifying the error. The text states that in the slow-rotation limit superradiance scales as $O((a/M)^4)$ and '$E_{dev}$ minimizes,' but no explicit bound on $E_{dev}$ is given. Since the integrated luminosity $L ∝ R_{sh}^{-2}$ (Eq. 30) is presented as a key result and is used to claim model-discriminating power (e.g., the 52% Horndeski deviation), the magnitude of the dropped corrections should be estimated, at least parametrically, to justify the Stefan-Boltzmann approximation.
minor comments (5)
  1. [Abstract] The phrase 'definitive mathematical solution to parameter degeneracy' overstates the result. The framework re-parameterizes observables in terms of $R_{sh}$; it does not by itself break degeneracies between gravity theories unless independent constraints on $α$ or $λ$ are available.
  2. [Section II.A, before Eq. 3] The paper acknowledges that the global 2D shadow radius requires numerical methods but then restricts to the equatorial plane for all analytical results. This restriction should be restated in the abstract and conclusion so readers understand the scope.
  3. [Section IV.A] The numerical estimates for M87* ($T_H ≈ 9.7 × 10^{-15}$ K, $L ∼ 10^{-52}$ W) are presented without specifying the spin assumption used. Since these values depend on $a$, the assumed spin should be stated.
  4. [Throughout] Several typographical issues: 'kerr and kerr-like blackhole' (Section II intro), 'mass-independent framework' should be 'the mass-independent framework' (Section II intro), and 'subjecting them to the empirical constraints' lacks a verb (Section II intro).
  5. [Section IV.B, Eq. 36] The claim that MOG 'enhances classical deflection while strictly suppressing quantum luminosity' is presented as a key phenomenological signature, but the luminosity suppression is only implied through the temperature reduction (Eq. 37) without an explicit MOG luminosity formula analogous to Eq. 30.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful reading and for identifying two substantive gaps in the manuscript. The referee's comments on the Kerr-MOG Jacobian verification, the truncated Horndeski section, and the unquantified luminosity correction terms are all well-taken. We address each below.

read point-by-point responses

  1. Referee: [Section IV.B, Eq. 35] The IFT non-degeneracy condition dR_sh/dM is proven only for Kerr. For Kerr-MOG, the Jacobian is a different function of (M, a, alpha), and non-degeneracy for Kerr does not imply non-degeneracy for Kerr-MOG. The authors should either compute dR_sh/dM explicitly for Kerr-MOG (at least to leading order in alpha) or add a continuity argument.

    Authors: The referee is correct that the non-degeneracy of dR_sh/dM for Kerr does not automatically extend to Kerr-MOG, and the manuscript's invocation of the Kerr proof for the MOG case is insufficient as written. We will address this by computing the Jacobian explicitly for Kerr-MOG to leading order in alpha. The Kerr-MOG shadow radius on the equatorial plane, expanded to O(a^0) and O(alpha^1), is R_sh^{MOG} = 3*sqrt(3)*M*(1 + alpha/2) + O(a^2, alpha^2), which gives dR_sh/dM = 3*sqrt(3)*(1 + alpha/2) + O(a^2, alpha^2). This is strictly positive for all alpha > -1 (the physical regime for MOG), confirming non-degeneracy. We will include this explicit computation in the revised Section IV.B, replacing the blanket invocation of the Kerr result with a metric-specific verification. revision: yes

  2. Referee: [Section IV.C (Horndeski)] The Horndeski section appears truncated or missing. After Eq. 37, the text jumps to fragmentary results without derivation. The abstract, introduction, and conclusion all claim Horndeski results (logarithmic augmentation, ~52% deviation) that are not supported by any visible derivation.

    Authors: The referee is correct: the Horndeski section is incomplete in the current manuscript. The fragmentary text after Eq. 37 ('~2.5% departure' and '~52% deviation') represents results from an incomplete draft that was inadvertently submitted without the full derivation. We will complete Section IV.C with the full derivation: (1) the rotating Horndeski shadow radius expanded to leading order in the scalar hair parameter, (2) the explicit Jacobian dR_sh/dM verification for the Horndeski metric, (3) the mass inversion M(R_sh, a, h), (4) the weak deflection angle showing the logarithmic augmentation term, and (5) the Hawking temperature and luminosity expressions yielding the quoted deviations. If the full Horndeski derivation cannot be completed to the referee's standard within the revision timeframe, we will instead remove all Horndeski claims from the abstract, introduction, and conclusion, and reframe the paper as covering Kerr and Kerr-MOG only. We commit to one of these two paths. revision: yes

  3. Referee: [Section III.B, Eqs. 21-22, 30] The luminosity formula drops E_dev and delta_L_SR without quantifying the error. No explicit bound on E_dev is given. The magnitude of the dropped corrections should be estimated to justify the Stefan-Boltzmann approximation, especially since L ~ R_sh^{-2} is used to claim model-discriminating power.

    Authors: The referee is correct that Eq. 30 drops E_dev and delta_L_SR without an explicit parametric bound. We will add a parametric estimate. For the superradiant correction: in the slow-rotation regime (a/M << 1), the superradiant amplification for scalar fields scales as delta_L_SR / L_thermal ~ O((a/M)^4), which is negligible for M87* (a/M ~ 1) at the percent level. For E_dev: the deviation of the greybody factor from the eikonal limit is controlled by the ratio T_H / omega_c, where omega_c ~ 1/R_sh is the characteristic frequency scale. For M87*, T_H ~ 10^{-14} K while omega_c ~ 10^{-5} Hz (in natural units, T_H/omega_c ~ 10^{-3}), so E_dev ~ O(T_H * R_sh) << 1. We will include these parametric estimates in the revised Section III.B and explicitly state the regime of validity for the Stefan-Boltzmann approximation used in Eq. 30. revision: yes

Circularity Check

2 steps flagged

Framework is a legitimate reparameterization, not circular; 'predictions' are transparent algebraic substitutions of known formulas, mildly oversold as novel phenomenology

specific steps
  1. fitted input called prediction [Section IV.B, Eq. 36 and surrounding text]
    "By replacing the unobservable bare mass with our derived relation, the lensing profile is strictly governed by the shadow radius [10]: ˆα_MOG(b, R_sh, α) = 4R_sh/(3√3 b)(1 + 7α/18) ± 4aR_sh/(3√3 b²) + O(b⁻³, α²). This mathematical structure reveals a key phenomenological trait: for a black hole with a predefined and fixed shadow size, the repulsive MOG vector field actively amplifies the isotropic weak lensing profile, breaking the inherent degeneracy between the mass and the modified gravity parameter."

    The factor (1 + 7α/18) is not an independent prediction. It arises by pure algebra: the known Kerr-MOG deflection from Ref [10] scales as 4M(1+α)/b, and substituting the inverted mass relation M ≈ R_sh/(3√3)(1 - 11α/18) from Eq. 35 yields (1+α)(1-11α/18) ≈ 1 + 7α/18. The paper is transparent that it is substituting ('By replacing the unobservable bare mass with our derived relation'), so this is not hidden circularity. However, presenting the resulting algebraic identity as a 'key phenomenological trait' that 'breaks the inherent degeneracy' overstates the novelty: anyone who knew the Kerr-MOG deflection formula already knew this correction factor. The reparameterization adds no new observational content beyond a change of variables. This is a mild case of renaming a known result in new坐标.

  2. fitted input called prediction [Section IV.A, Eqs. 26, 28, 30]
    "Substituting Eq. 25 into this relation, the deflection angle becomes an explicit phenomenological function of the shadow radius: ˆα_Kerr(b, R_sh) = 4R_sh/(3√3 b) ± 4aR_sh/(3√3 b²) + O(b⁻³) ... TH(R_sh) ≈ 3√3/(8πR_sh) ... L(R_sh) ≈ 729σ_SB/(4096π³R_sh²). This derivation confirms the closed-form L ∝ R_sh⁻² optical-thermodynamic duality."

    These results are obtained by substituting M ≈ R_sh/(3√3) into the standard Kerr expressions for deflection (4M/b), Hawking temperature, and Stefan-Boltzmann luminosity. The substitution is algebraically forced: e.g., L ∝ M² ∝ (R_sh/3√3)² ∝ R_sh², and T_H ∝ 1/M ∝ 1/R_sh. The paper is transparent about the substitution step. The 'duality' L ∝ R_sh⁻² is simply the known L ∝ M² with a change of variable. Not circular by construction, but the 'confirmation' language implies independent validation when the result is algebraically guaranteed by the substitution.

full rationale

The paper's central framework — proving M = M(R_sh, a) via the Implicit Function Theorem and substituting into known formulas — is a legitimate mathematical reparameterization, not a circular derivation. The IFT argument (Section II.A) is self-contained and cites an external textbook (Ref [30]). The self-citations (Refs [5], [9]) appear only in the introduction as general context and are not load-bearing for the mathematical framework. The 'predictions' (Eqs. 26, 28, 30, 36, 37) are transparently algebraic substitutions of the inverted mass relation into known formulas from external sources (e.g., Ref [10] for Kerr-MOG deflection). The paper explicitly states it is 'replacing the unobservable bare mass with our derived relation,' so the substitution is not hidden. The mild circularity (score 3, not higher) arises because the paper presents these algebraically forced results as novel 'phenomenological signatures' and 'predictions' that 'break degeneracy,' when the correction factors (e.g., 1+7α/18) are already encoded in the known metric-dependent formulas being substituted into. The IFT non-degeneracy is verified only for Kerr slow-rotation and assumed without proof for Kerr-MOG and Horndeski — this is a correctness/applicability gap, not a circularity issue. No self-citation chain is load-bearing, and no result reduces to its own input by definition.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities, particles, forces, or dimensions. All parameters (α, a, scalar hair) belong to pre-existing theories (STVG, Horndeski gravity). The 'thermodynamic-optical duality' is a re-parameterization framework, not a new physical entity. No new axioms are invented; the axioms are standard mathematical theorems or domain assumptions about approximation validity.

free parameters (3)
  • α (MOG parameter)
    Continuous deviation parameter in Scalar-Tensor-Vector Gravity; not fitted to data in this paper but treated as a free parameter of the theory. Constrained to α << 1 for perturbative expansion.
  • λ or h (Horndeski scalar hair parameter)
    Scalar field coupling parameter in rotating Horndeski spacetime; referenced in Section IV.C but the specific symbol and formulas are truncated in the provided text. Treated as a free parameter.
  • a (spin parameter)
    Black hole spin; not fitted but assumed small (a/M << 1) for all perturbative expansions throughout the paper.
axioms (5)
  • standard math Implicit Function Theorem guarantees a smooth local inverse M = M(R_sh, a) whenever dR_sh/dM ≠ 0
    Standard mathematical theorem (Ref. [30]); invoked in Section II.A to justify the mass inversion. The non-degeneracy condition is verified only for Kerr in the Schwarzschild and slow-rotation limits.
  • domain assumption Equatorial-plane shadow radius is a sufficient proxy for the full 2D shadow
    Section II.A states the global shadow radius requires numerical integration (Eq. 3) but then restricts to θ = π/2 for all analytical results. This assumption underlies every derived formula.
  • domain assumption Low-spin approximation (a/M << 1) is valid for the spacetimes considered
    All closed-form expressions (Eqs. 25, 28, 35, 37) are leading-order in a/M. The EHT M87* spin estimate is a/M ~ 0.9, which violates this assumption, but the paper applies the formulas to M87* data regardless.
  • domain assumption Stefan-Boltzmann approximation with geometric cross-section is adequate for total luminosity
    Equation 21 uses σ_SB π R²_sh T⁴_H as the baseline; the correction E_dev is defined (Eq. 22) but never bounded numerically. The paper asserts it 'minimizes' for slow rotation without quantification.
  • standard math Gauss-Bonnet theorem applies to the Randers-Finsler optical manifold of a stationary axisymmetric spacetime
    Established methodology from Gibbons-Werner (Ref. [12]) and Ishihara et al. (Ref. [31]); used in Section II.B to derive the weak deflection angle.

pith-pipeline@v1.1.0-glm · 14436 in / 3722 out tokens · 278703 ms · 2026-07-04T19:35:34.104465+00:00 · methodology

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read the original abstract

We establish a formal thermodynamic-optical duality that bridges the semiclassical quantum evaporation of black holes with their classical macroscopic geometry. The physical viability of this framework is anchored by a stable multivariate coordinate transformation and a non-vanishing Jacobian determinant, which allows for a diffeomorphic inversion mapping that decouples intrinsic physical quantities such as bare mass from the unobservable spacetime interior. By re-parameterizing black hole properties entirely in terms of the analytical shadow radius ($R_{sh}$), we derive explicit, observable-based expressions for the weak deflection angle, Hawking temperature, and integrated semiclassical luminosity. We demonstrate the framework's predictive utility by applying it to standard Kerr, Kerr-MOG (Scalar-Tensor-Vector Gravity), and rotating Horndeski spacetimes. Our results provide a definitive mathematical solution to parameter degeneracy, revealing that distinct fundamental fields (vector vs. scalar) leave unique observational fingerprints on far-field astrometry and horizon-scale quantum thermodynamics. By confronting these models with Event Horizon Telescope (EHT) M87* data, we show that this formalism successfully breaks mass-parameter degeneracies, offering a robust and computationally efficient operational tool for testing the Kerr hypothesis and probing modified gravity theories with next-generation very-long-baseline interferometry (VLBI).

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Works this paper leans on

37 extracted references · 37 canonical work pages · 6 internal anchors

  1. [1]

    + O(a2). Substituting this mapping directly into the integrated deflection equation isolates the observ- able shadow radius: ˆα(b, Rsh) = 4M(Rsh, a) b ± 4M(Rsh, a)a b2 ≈ 4Rsh 3 √ 3b ± 4aRsh 3 √ 3b 2 (15) This sequence proves mathematically that the weak deflec- tion angle, including both its isotropic lensing component and rotational frame-dragging pertur...

    work page

  2. [2]

    H. Rana, P. Grimes, E. Tong, D. Marrone, J. Houston, K. Akiyama, M. Honma, R. Baturin, and M. Johnson, 4 K cryocooling for space VLBI with the black hole explorer, Cryogenics158, 104350 (2026)

    work page 2026

  3. [3]

    S. V. Sosa Fiscellaet al., The NANOGrav 15 yr Dataset: Improved Timing Precision with Very Long Baseline Interfer- ometry Astrometric Priors, Astrophys. J.999, 156 (2026)

    work page 2026

  4. [4]

    First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole

    K. Akiyamaet al.(Event Horizon Telescope), First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole, Astrophys. J. Lett.875, L1 (2019), arXiv:1906.11238 [astro-ph.GA]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  5. [5]

    First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way

    K. Akiyamaet al.(Event Horizon Telescope), First Sagit- tarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way, Astrophys. J. Lett.930, L12 (2022), arXiv:2311.08680 [astro-ph.HE]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  6. [6]

    R. C. Pantig, P. K. Yu, E. T. Rodulfo, and A. ¨Ovg¨ un, Shadow and weak deflection angle of extended uncertainty principle black hole surrounded with dark matter, Annals Phys.436, 168722 (2022), arXiv:2104.04304 [gr-qc]. 8

    work page arXiv 2022

  7. [7]

    Akiyamaet al.(Event Horizon Telescope), First Sagit- tarius A* Event Horizon Telescope Results

    K. Akiyamaet al.(Event Horizon Telescope), First Sagit- tarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric, Astrophys. J. Lett.930, L17 (2022), arXiv:2311.09484 [astro-ph.HE]

    work page arXiv 2022

  8. [8]

    Perlick and O

    V. Perlick and O. Y. Tsupko, Calculating black hole shadows: Review of analytical studies, Phys. Rept.947, 1 (2022), arXiv:2105.07101 [gr-qc]

    work page arXiv 2022

  9. [9]

    Mohan, N

    G. Mohan, N. Parbin, and U. D. Goswami, Investigat- ing the effects of gravitational lensing by Hu-Sawicki f(R) gravity black holes, Eur. Phys. J. C85, 413 (2025), arXiv:2411.19048 [gr-qc]

    work page arXiv 2025

  10. [10]

    R. C. Pantig and E. T. Rodulfo, Weak deflection an- gle of a dirty black hole, Chin. J. Phys.66, 691 (2020), arXiv:2003.00764 [gr-qc]

    work page arXiv 2020

  11. [11]

    ¨Ovg¨ un,˙I

    A. ¨Ovg¨ un,˙I. Sakallı, and J. Saavedra, Weak gravitational lensing by Kerr-MOG black hole and Gauss–Bonnet theorem, Annals Phys.411, 167978 (2019), arXiv:1806.06453 [gr-qc]

    work page arXiv 2019

  12. [12]

    T. Ono, A. Ishihara, and H. Asada, Deflection angle of light for an observer and source at finite distance from a rotating wormhole, Phys. Rev. D98, 044047 (2018), arXiv:1806.05360 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  13. [13]

    G. W. Gibbons and M. C. Werner, Applications of the Gauss- Bonnet theorem to gravitational lensing, Class. Quant. Grav. 25, 235009 (2008), arXiv:0807.0854 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  14. [14]

    Sucu, Dirac quasinormal modes, quality factor and grav- itational lensing in nonlinear electrodynamics black holes with barrow entropy, Nucl

    E. Sucu, Dirac quasinormal modes, quality factor and grav- itational lensing in nonlinear electrodynamics black holes with barrow entropy, Nucl. Phys. B1026, 117421 (2026)

    work page 2026

  15. [15]

    Orzuev, F

    S. Orzuev, F. Atamurotov, A. Abdujabbarov, and F. Botirov, Weak gravitational lensing of charged black hole from T- duality in plasma, New Astron.126, 102555 (2026)

    work page 2026

  16. [16]

    Onishi, S

    K. Onishi, S. Iguchi, K. Sheth, and K. Kohno, A measure- ment of the black hole mass in ngc 1097 using alma, The Astrophysical Journal806, 39 (2015)

    work page 2015

  17. [17]

    Kumar and S

    R. Kumar and S. G. Ghosh, Black Hole Parameter Esti- mation from Its Shadow, Astrophys. J.892, 78 (2020), arXiv:1811.01260 [gr-qc]

    work page arXiv 2020

  18. [18]

    Kim, Temperature of a steady system around a black hole, Class

    H.-C. Kim, Temperature of a steady system around a black hole, Class. Quant. Grav.41, 215001 (2024), arXiv:2401.01541 [gr-qc]

    work page arXiv 2024

  19. [19]

    Visser, Dirty black holes: Thermodynamics and hori- zon structure, Phys

    M. Visser, Dirty black holes: Thermodynamics and hori- zon structure, Phys. Rev. D46, 2445 (1992), arXiv:hep- th/9203057

    work page arXiv 1992

  20. [20]

    Thermodynamic Geometry and Hawking Radiation

    S. Bellucci and B. N. Tiwari, Thermodynamic Geometry and Hawking Radiation, JHEP11(11), 030, arXiv:1009.0633 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv

  21. [21]

    Halder and D

    I. Halder and D. L. Jafferis, Thermal Bekenstein- Hawking entropy from the worldsheet, JHEP05(5), 136, arXiv:2310.02313 [hep-th]

    work page arXiv

  22. [22]

    Wang, Correction to Temperature and Beken- stein–Hawking Entropy of Kiselev Black Hole Surrounded by Quintessence, Entropy27, 1135 (2025)

    C. Wang, Correction to Temperature and Beken- stein–Hawking Entropy of Kiselev Black Hole Surrounded by Quintessence, Entropy27, 1135 (2025)

    work page 2025

  23. [23]

    Y.-z. Liu, C. Wang, J. Zhang, and S.-Z. Yang, Correction of Bekenstein–Hawking entropy of Kiselev black holes sur- rounded by quintessence owing to Lorentz breaking, Eur. Phys. J. C85, 1088 (2025)

    work page 2025

  24. [24]

    Bamonti, A canonical and relational analysis of reference frames and gauge-fixing in general relativity, Classical and Quantum Gravity (2026)

    N. Bamonti, A canonical and relational analysis of reference frames and gauge-fixing in general relativity, Classical and Quantum Gravity (2026)

    work page 2026

  25. [25]

    Goeller, P

    C. Goeller, P. A. Hoehn, and J. Kirklin, Diffeomorphism- invariant observables and dynamical frames in gravity: recon- ciling bulk locality with general covariance, preprint (2022), arXiv:2206.01193 [hep-th]

    work page arXiv 2022

  26. [26]

    Maitra, D

    M. Maitra, D. Maity, and B. R. Majhi, Diffeomorphism symmetries near a timelike surface in black hole spacetime, Classical and Quantum Gravity38, 145027 (2021)

    work page 2021

  27. [27]

    R. P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics, Phys. Rev. Lett.11, 237 (1963)

    work page 1963

  28. [28]

    M.-H. Wu, H. Guo, and X.-M. Kuang, Parameter constraints on Horndeski rotating black hole through quasiperiodic os- cillations, Eur. Phys. J. C86, 79 (2026), arXiv:2508.13974 [gr-qc]

    work page arXiv 2026

  29. [29]

    C.-H. Xie, Y. Zhang, Q. Sun, Q.-Q. Li, and P.-F. Duan, Gravitational lensing by a stable rotating regular black hole, JCAP05(5), 121, arXiv:2401.05454 [gr-qc]

    work page arXiv

  30. [30]

    Sarikulov, F

    F. Sarikulov, F. Atamurotov, A. Abdujabbarov, and B. Ahme- dov, Shadow of the Kerr-like black hole, Eur. Phys. J. C82, 771 (2022)

    work page 2022

  31. [31]

    A. L. Dontchev and R. T. Rockafellar,Implicit Functions and Solution Mappings: A View from Variational Analysis, 2nd ed., Springer Series in Operations Research and Financial Engineering, Vol. 54 (Springer, New York, 2014)

    work page 2014

  32. [32]

    Gravitational bending angle of light for finite distance and the Gauss-Bonnet theorem

    A. Ishihara, Y. Suzuki, T. Ono, T. Kitamura, and H. Asada, Gravitational bending angle of light for finite distance and the Gauss-Bonnet theorem, Phys. Rev. D94, 084015 (2016), arXiv:1604.08308 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  33. [33]

    Gao, L.-H

    K. Gao, L.-H. Liu, and M. Zhu, Microlensing effects of wormholes associated to blackhole spacetimes, Phys. Dark Univ.41, 101254 (2023), arXiv:2211.17065 [gr-qc]

    work page arXiv 2023

  34. [34]

    R. J. Adler, On the temperature of a rotating black hole, Int. J. Mod. Phys. D34, 2544011 (2025)

    work page 2025

  35. [35]

    Maulik, X

    S. Maulik, X. Meng, and L. A. Pando Zayas, Quantum- corrected Hawking radiation from near-extremal Kerr- Newman black holes, JHEP02(2), 205, arXiv:2501.08252 [hep-th]

    work page arXiv

  36. [36]

    S. Hu, C. Deng, S. Guo, X. Wu, and E. Liang, Obser- vational signatures of Schwarzschild-MOG black holes in scalar–tensor–vector gravity: images of the accretion disk, Eur. Phys. J. C83, 264 (2023)

    work page 2023

  37. [37]

    Q. Wan, Y. Hou, and M. Guo, Probing the scalar hair of rotating Horndeski black holes through thick disk images, Phys. Rev. D113, 083023 (2026), arXiv:2512.00917 [gr-qc]

    work page arXiv 2026