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Charged black holes in Kalb-Ramond gravity: Weak Deflection Angle, Shadow cast, Quasinormal Modes and Neutrino annihilation

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Kalb-Ramond black holes match EHT shadows with ℓ near zero

desk verdict New Dirac-mode and neutrino-annihilation results for a charged KR black hole are buried under a shadow analysis with a sign error that invalidates the advertised EHT constraints. read the letter →

arxiv 2505.17947 v1 pith:QBJJNSZE submitted 2025-05-23 gr-qc

Reggie C. Pantig , Ali Övgün , Ángel Rincón This is my paper
classification gr-qc PACS 95.30.Sf04.70.-s97.60.Lf04.50.+h
keywords Kalb-RamondgravityLorentzsymmetrybreakingblackholeshadowweakdeflectionanglequasinormalmodesneutrinoannihilationEventHorizonTelescopecharged
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes that electrically charged black holes in Kalb-Ramond gravity, a theory in which a background antisymmetric tensor field spontaneously breaks Lorentz symmetry, carry a single parameter ℓ that leaves imprints on every major observable: the weak deflection angle of light and massive particles, the black hole shadow radius, scalar and Dirac quasinormal-mode frequencies, and the energy deposited by neutrino pair annihilation. The authors derive analytic expressions for the deflection angle using the Gauss-Bonnet theorem and find ℓ-corrections that survive even for zero charge. For the shadow, they obtain a formula in which ℓ deforms the photon sphere and can reverse the sign of the charge contribution, and they use Event Horizon Telescope measurements of Sgr A* and M87* to bound ℓ to roughly [-0.19, 0.07] and [-0.17, 0.17], respectively, which they read as validation of the framework.

What carries the argument

The central object is the background Kalb-Ramond tensor field $B_{\mu\nu}$ with a fixed vacuum expectation value, which spontaneously breaks Lorentz invariance and, through the couplings $\xi_2,\xi_3$ and $\eta$, deforms the Reissner-Nordström metric into $F(r)=1/(1-\ell)-2M/r+Q^2/((1-\ell)^2 r^2)$; the effective Lorentz-violating parameter is $\ell=\tfrac12\xi_2 b^2$, and consistency of the field equations forces the electromagnetic coupling $\eta=\ell/(2b^2)$. The argument is carried by this single parameter $\ell$, which enters every observable: the Gauss-Bonnet theorem supplies the deflection angle in the non-asymptotically flat optical metric; the photon-sphere condition $A'(r)r^2-2A(r)r=0$ and the critical impact parameter give the shadow radius; the WKB and Pöschl-Teller methods convert the scalar and Dirac effective potentials into quasinormal frequencies; and the thermal-neutrino distribution, redshifted temperature, and angular factor $F(r)$ give the neutrino energy-deposition rate.

What would settle it

Compute the shadow radius by direct numerical integration of null geodesics in the metric Eq. (9) for, say, M=1, Q=0.1, ℓ=0.1, and compare with Eq. (39) and Eq. (38); the correct small-ℓ trend, enlargement versus shrinkage, decides whether the EHT bounds hold.

Watch

Extended reading notes

Core claim

The paper's central claim is that the charged Kalb-Ramond black hole, with metric function $F(r)=1/(1-\ell)-2M/r+Q^2/((1-\ell)^2 r^2)$, is a phenomenologically rich and observationally testable extension of Reissner-Nordström gravity. Working in the non-asymptotically flat optical geometry, the authors obtain a weak deflection angle $\hat{\alpha}\simeq \frac{4M}{b}-\frac{3\pi Q^2}{4b^2}-\big(\frac{\pi}{2}-\frac{Q^2}{2b^2}(1+\frac{9\pi}{8})+\frac{2M}{b}\big)\ell$ for photons, with analogous velocity-dependent formulas for massive particles. For the shadow, they argue that $R_{\rm sh}\simeq 3\sqrt{3}M\big(3-2\sqrt{1-\ell}\big)+\frac{2\sqrt{3}Q^2}{M}\big(\frac34-\frac1{\sqrt{1-\ell}}\big)$, a form whose charge term can change sign for small positive ℓ, marking a departure from Einstein-Maxwell behavior. They then compute scalar and Dirac quasinormal modes by WKB and Pöschl-Teller methods, finding all imaginary parts negative, hence stability, and they show that neutrino pair annihilation energy deposition is amplified for positive ℓ. The paper concludes that EHT shadow data bound ℓ within the intervals above, establishing the Kalb-Ramond framework as observationally testable.

Load-bearing premise

The EHT bounds on ℓ inherit from the shadow-radius formula Eq. (39); if that formula carries the ℓ-sign error suggested by its disagreement with Eq. (38) and with the cited shrinkage trend, the quoted constraints would change.

Editorial extensions

If this is right

  • The Lorentz-violating parameter ℓ appears in the deflection angle even when the black hole is neutral, so precise lensing measurements can probe spontaneous Lorentz breaking independently of charge.
  • If the shadow formula Eq. (39) survives scrutiny, EHT shadow sizes put ℓ in [-0.19, 0.07] for Sgr A* and [-0.17, 0.17] for M87*, tightening the allowed window for the Kalb-Ramond framework.
  • The negative imaginary parts across the WKB and Pöschl-Teller tables mean the charged Kalb-Ramond black hole is stable under both scalar and Dirac perturbations, and its ringdown spectrum carries an ℓ-dependent shift.
  • Neutrino pair annihilation near the hole deposits more energy for positive ℓ, which would make this model relevant for gamma-ray burst energy budgets.
  • Slow massive particles experience a 1/v^2 enhancement in deflection, opening a low-velocity lensing channel to test ℓ.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A sign error in the small-ℓ shadow limit would not erase the deflection-angle or quasinormal-mode predictions, but it would change the quoted EHT parameter window; comparing Eq. (39) against direct geodesic integration is the fastest way to settle it.
  • The solar-system bounds on ℓ, near $10^{-10}$ or tighter, are orders of magnitude smaller than the EHT-derived window, so a combined fit would either push ℓ very close to zero or force large charge contributions.
  • The predicted sign flip in the charge term of the shadow radius is a distinctive Kalb-Ramond signature that future high-resolution black-hole images could test.
  • Because the same ℓ governs lensing, shadow, ringdown, and neutrino energy, multi-messenger observations could overdetermine ℓ and thereby falsify the single-parameter picture.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper studies the charged, spherically symmetric black hole solution of Kalb-Ramond gravity with Lorentz-violating parameter ℓ given in Eq. (9). It computes weak deflection angles for light and massive particles using the Gauss-Bonnet theorem, derives a shadow radius and uses Event Horizon Telescope measurements of Sgr A* and M87* to bound ℓ, calculates scalar and Dirac quasinormal modes with WKB and Pöschl-Teller methods, and evaluates neutrino-pair-annihilation energy deposition. The central claim is that EHT shadow data constrain ℓ to intervals of order ten percent and thereby validate the KR framework.

Significance. Should the shadow derivation be correct, the paper would offer a broad and useful phenomenology package for a charged KR black hole, including extensive QNM tables cross-checked by two independent approximations and an estimate of neutrino annihilation energy deposition. The WKB/Pöschl-Teller comparison and the explicit deflection-angle expansions are concrete and, in structure, reproducible. However, the load-bearing shadow formula is internally inconsistent, and the derived EHT bounds disagree with the Solar System limits quoted in the paper's own Table I; the claimed observational validation does not follow from the presented analysis.

major comments (2)
  1. [Section IV, Eqs. (38)-(41)] For Q=0, the metric (9) has f(r)=1/(1-ℓ)-2M/r. The photon-sphere condition gives r_ps=3M(1-ℓ), the critical impact parameter is b_crit=3√3M(1-ℓ)^{3/2}, and with the observer factor √A(r_obs→∞) one obtains R_sh=3√3M(1-ℓ). This is the leading term of Eq. (38) with c²=1/(1-ℓ). Equation (39) instead expands to 3√3M(3-2√(1-ℓ))≈3√3M(1+ℓ), which has the opposite ℓ-dependence and is about 22% larger than the correct value at ℓ=0.1. Since Eq. (41) is written as ℓ=-±δ√3/(9M), it corresponds to R_sh=3√3M-3√3Mℓ, i.e. to the sign in Eq. (38), not to Eq. (39); substituting Eq. (39) into Eq. (40) would give ℓ=+δ√3/(9M). The EHT intervals in Section IV are therefore not derived from the displayed final shadow formula, and the discussion of a charge-sign reversal based on Eq. (39) is unsupported.
  2. [Section IV, Table I and Eq. (41)] The EHT intervals quoted after Eq. (41), ℓ∈[-0.1899,0.0701] for Sgr A* and ℓ∈[-0.1699,0.1699] for M87*, place |ℓ| at the 10^{-1} level. Table I of the same paper reports Solar System limits from Mercury perihelion advance, gravitational light bending, and Shapiro delay that are all in the range 10^{-11} to 10^{-9}. Because the same ℓ appears in the metric (9), the two sets of bounds are mutually inconsistent by roughly eight to ten orders of magnitude. The manuscript does not discuss this tension, and the conclusion that the EHT bounds corroborate the theoretical consistency and observational relevance of the model is therefore not supported.
minor comments (6)
  1. [Section II] The sentence beginning 'In Ref. []' contains a blank citation placeholder; either supply the intended reference or remove the placeholder.
  2. [Section IV] The quantities c² and q are used in Eqs. (34)-(38) without being defined in terms of ℓ and Q; the derivation would be checkable if the paper stated c²=1/(1-ℓ) and q=Q/(1-ℓ).
  3. [Section IV, Eq. (34)] Equation (34) has a last term 4q²/r, but as written it is not the photon-sphere condition for the metric (9), and the quadratic solution in Eq. (35) does not satisfy it; the displayed formula should presumably be 2c²r²-6Mr+4q²=0.
  4. [Section V, Table II] At ℓ=0.40, n=2 the WKB imaginary part is not monotonic in lb, with |Im| for lb=8 larger than for lb=9, and at ℓ=0.50, n=2 the real part for lb=6 lies far below the eikonal trend; the authors should comment on the reliability of these entries.
  5. [Section VI] The conclusion refers to 'Fig. 1' for the neutrino annihilation ratio, but the corresponding figure is the fifth figure in the paper; the cross-reference should be corrected.
  6. [Throughout] There are numerous typographical issues, including 'Schrdinger', 'Poschl-Teller', 'ReissnerNordstrm', and 'WBK'; a careful proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper applies standard formalisms to an imported KR metric and derives constraints from EHT data rather than fitting parameters to the predicted quantity.

full rationale

The paper's derivation chain is self-contained in the sense relevant to circularity. The charged Kalb-Ramond black hole metric, Eq. (9), is imported from external prior work (Refs. [4,40] in the reference list), not derived from the shadow observables or from the EHT bounds. The deflection angle, shadow radius, quasinormal modes, and neutrino annihilation rate are computed by applying standard, established formalisms (Gauss-Bonnet theorem, photon-sphere/critical-impact-parameter method, WKB and Poschl-Teller approximations, and the Salmonson-Wilson formula respectively). The EHT analysis is a constraint exercise: Eq. (39) expresses R_sh as a function of (M,Q,ell), and Eqs. (40)-(41) invert the EHT shadow-radius bounds to place intervals on ell. No parameter is fitted to a subset of data and then 'predicted' for the same subset, and the model is not defined in terms of the shadow radius it claims to compute. The self-citations that appear (e.g., Refs. [10,42]) provide methodological background and prior context; they are not used as an unverified uniqueness theorem or as the sole justification for a load-bearing claim. The paper does contain an internal algebraic inconsistency: Eq. (38) implies a small-ell shadow correction of order 3*sqrt(3)*M*ell with one sign, while Eq. (39) yields the opposite sign, and the derived EHT ell bounds also conflict with the Solar System limits quoted in Table I. However, that is a correctness or sign-error concern, not a circular definition, a fitted-input-called-prediction, or a self-citation chain that forces the result. Therefore no circularity step is identified and the circularity score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper's results rest on the imported KR metric and its coupling relations, standard perturbation theory assumptions, and the GBT finite-distance framework. No new entities are invented; ℓ is an existing model parameter. The main fragility is not circularity but the incorrect shadow reduction.

free parameters (1)
  • ℓ (Lorentz-violating parameter) = bounded by EHT: [-0.1899, 0.0701] for Sgr A*, [-0.1699, 0.1699] for M87*; Solar System: ~1e-10
    The paper treats ℓ as a free parameter of the KR model and derives constraints from EHT shadow data (Eq. 41) and Solar System deflection (Eq. 33). It is not fitted by the paper itself, but the derived bounds are central claims.
assumptions (4)
  • domain assumption The charged KR metric Eq. (9) is a valid solution of the action Eqs. (1)-(2)
    Imported from Yang et al. [40] and Duan et al. [4]; the paper does not re-derive the solution and assumes its consistency.
  • domain assumption The KR field background has a non-vanishing VEV with constant norm b^2 and the coupling relation η = ℓ/(2b^2) holds
    These conditions are taken from the prior KR gravity literature and are necessary for the metric to solve the field equations.
  • domain assumption The Gauss-Bonnet theorem applies to the optical geometry of the non-asymptotically flat spacetime
    Section III uses GBT with F(∞) = 1/(1 - ℓ); the finite-distance formulation is assumed valid.
  • standard math Effective potentials for scalar and Dirac perturbations are single-peaked so WKB and Pöschl-Teller approximations are valid
    Standard WKB requirement; the paper states the potentials have well-defined peaks for the parameter ranges considered but provides no formal proof.

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Cite this review

Pith. "Pith review of Charged black holes in Kalb-Ramond gravity: Weak Deflection Angle, Shadow cast, Quasinormal Modes and Neutrino annihilation." pith.science (2026). https://pith.science/paper/QBJJNSZE

@misc{pith2026250517947,
  author       = {Pith},
  title        = {Pith review of: Charged black holes in Kalb-Ramond gravity: Weak Deflection Angle, Shadow cast, Quasinormal Modes and Neutrino annihilation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QBJJNSZE}},
  note         = {Machine review of arXiv:2505.17947}
}
abstract

In this paper, we investigate the phenomenology of electrically charged black holes in a Lorentz-violating gravitational framework mediated by a background Kalb-Ramond (KR) antisymmetric tensor field. Employing the Gauss-Bonnet theorem in a non-asymptotically flat geometry, we derive analytic expressions for the weak deflection angle of light and massive particles, revealing persistent corrections due to the Lorentz-violating parameter $ \ell $. Scalar and Dirac perturbations are also studied using both the WKB approximation and the Poschl-Teller approximation approach to verify the stability of the solution against these types of perturbations. Shadow analysis further uncovers a nontrivial deformation of the photon sphere and critical impact parameter, with KR-induced effects modifying the charge contribution in a manner incompatible with standard Einstein-Maxwell theory. Constraints derived from Event Horizon Telescope data for Sgr A* and M87* validate the model and provide stringent bounds on $ \ell $, establishing the KR framework as an observationally testable extension of General Relativity.

Figures

Figures reproduced from arXiv: 2505.17947 by the authors.

Figure 1
Figure 1. FIG. 1. Effective potential barrier for massless scalar perturbations against the radial coordinate for the parameters shown in the [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Effective potential barrier for massless Dirac perturbations against the radial coordinate for the parameters shown in the [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Quasinormal modes for massless scalar perturbations. [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Quasinormal modes for massless Dirac perturbations. [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The solid, dotted and dashed curves of the ratio [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]

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Forward citations

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Reviewed August 7, 2026 · model on record in the stance chip above.