Probing Lorentz Symmetry Violation through Lensing Observables of Rotating Black Holes

Arun Kumar; Shafqat Ul Islam; Sushant G. Ghosh

REVIEW 5 major objections 5 minor 1 cited by

Probing Lorentz Symmetry Violation through Lensing Observables of Rotating Black Holes

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that strong and weak lensing by a rotating Bumblebee black hole carries measurable imprints of Lorentz symmetry breaking, with Einstein-ring data currently bounding the deviation parameter $\ell$ below about $10^{-6}$.

desk verdict Full-rotation strong lensing of the Bumblebee black hole is a useful project, but three load-bearing derivation errors break the paper's quantitative claims. read the letter →

arxiv 2509.00127 v1 pith:66FIKJT2 submitted 2025-08-29 gr-qc

Arun Kumar , Shafqat Ul Islam , Sushant G. Ghosh This is my paper · ORCID

classification gr-qc MSC 83C5783C10 PACS 04.70.Bw95.30.Sf98.62.Sb

keywords LorentzviolationBumblebeegravitygravitationallensingstrongdeflectionlimitblackholeshadowrotatingEventHorizonTelescopeEinsteinring

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 asks whether gravitational lensing can expose a background violation of Lorentz symmetry. It works with a rotating black hole in Bumblebee gravity, a theory where a vector field acquires a nonzero vacuum expectation value and spontaneously breaks Lorentz invariance, and it claims the single deviation parameter $\ell$ changes the event horizon, the photon sphere, and every strong-lensing observable relative to the Kerr spacetime. Comparing its predictions with the Event Horizon Telescope shadow measurements of M87* and Sgr A* places a significant portion of the (spin, $\ell$) parameter space inside the $1\sigma$ observational band, while an Einstein-ring observation of the galaxy ESO325-G004 yields an upper bound $\ell \lesssim 10^{-6}$. If the claims hold, gravitational lensing becomes a near-term observational test of Lorentz symmetry in the strong-field regime.

What carries the argument

The engine of the analysis is the RBBH metric, a Kerr-like axisymmetric spacetime built by applying a modified Newman–Janis algorithm to a static Bumblebee seed metric; it carries mass $M$, spin $a$, and the Lorentz-violating parameter $\ell$, and reduces to Kerr at $\ell = 0$. The metric determines the null geodesics, whose radial effective potential fixes the unstable photon-orbit radius $r_m$ and the critical impact parameter $u_m$, and those in turn feed the strong-deflection-limit formalism: the deflection integral $I(r_0)$ is split into divergent and regular parts, giving $\alpha_D(\theta) = -\bar a \log(\theta D_{OL}/u_m - 1) + \bar b$, with coefficients $\bar a$ and $\bar b$ that generate all the strong-lensing observables ($\theta_\infty$, $s$, $r_{\rm mag}$, $\Delta T_{2,1}$). The weak-field analysis uses a post-Kerr expansion of the same integral to produce a deflection series and, through the improved lens equation, the angular radius of the Einstein ring used for the $\ell$ bound.

What would settle it

The decisive check is to insert the metric (9), with the Bumblebee vector-field configuration assumed in the construction, into the field equations (5)–(6): if the equations are not satisfied identically for generic $\ell$, the RBBH is not a genuine solution and every lensing observable derived from it is unphysical. On the observational side, a future photon-ring measurement of Sgr A* with roughly one microarcsecond accuracy would distinguish the allowed $(a, \ell)$ windows from the Kerr prediction, since the paper's own ranges put $\theta_\infty$ up to several microarcseconds away from Kerr.

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Extended reading notes

Core claim

The central claim is that the rotating Bumblebee black hole (RBBH) is quantitatively distinguishable from Kerr as a gravitational lens for any $\ell \neq 0$, with the sign of $\ell$ fixing the direction of the deviation: $\ell > 0$ contracts the horizons and photon sphere relative to Kerr, suppressing the deflection angle, whereas $\ell < 0$ enlarges them and enhances deflection. Using the strong-deflection-limit expansion, the paper derives analytic forms for the deflection angle, the angular position $\theta_\infty$ of the packed relativistic images, their angular separation $s$, the flux ratio $r_{\rm mag}$, and the time delay $\Delta T_{2,1}$, and tabulates these for supermassive black holes modeled on Sgr A* and M87*: for example $\theta_\infty$ ranges over 18.25–33.3 $\mu$as for Sgr A* and 13.71–25.02 $\mu$as for M87*, and the time delay between the first two images reaches roughly 15 minutes for Sgr A* and 370 hours for M87*. In the weak-field regime the paper derives a post-Kerr deflection series and the angular radius of the Einstein ring, and it uses the galaxy ESO325-G004 as a lens to place the bound $\ell \lesssim 4.472 \times 10^{-6}$ at $1\sigma$ and $\ell \lesssim 8.084 \times 10^{-6}$ at $2\sigma$. The paper's stated conclusion is that within $1\sigma$ a substantial part of the parameter space agrees with the EHT results for M87* and Sgr A*, so strong and weak lensing provide a feasible probe of Lorentz symmetry breaking in extreme gravity.

Load-bearing premise

The metric in Eq. (9), produced by the modified Newman–Janis algorithm from the static seed (8), is assumed to be an exact solution of the Bumblebee field equations (5)–(6) even though the paper does not substitute it back into those equations to verify this.

Editorial extensions

If this is right

For $\ell > 0$ the deflection angle, the photon-ring radius, and the image magnification all sit below their Kerr values, while for $\ell < 0$ they sit above, so the sign of the Lorentz-violating parameter is read off from the direction of the shift.
Using Sgr A* and M87* as lenses, the predicted angular position of the packed images falls in 18.25–33.3 $\mu$as and 13.71–25.02 $\mu$as respectively, overlapping the EHT $1\sigma$ shadow windows over a substantial region of the $(a, \ell)$ plane.
Time delays between the first two relativistic images reach about 15 minutes for Sgr A* and 370 hours for M87*, providing a Kerr-independent discriminator that is, in principle, measurable.
The Einstein ring of the galaxy ESO325-G004 bounds the Lorentz-violating parameter to $\ell \lesssim 4.472 \times 10^{-6}$ at $1\sigma$ and $\ell \lesssim 8.084 \times 10^{-6}$ at $2\sigma$, i.e., $\ell \lesssim O(10^{-6})$.
At $\ell = 0$ the metric, the strong-deflection coefficients, and the weak-field deflection series reduce exactly to the Kerr ones, so the framework is a one-parameter extension that observations can continuously rule out.

Reading between the lines

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

A next-generation very-long-baseline campaign aimed at the Sgr A* photon ring could test the allowed $\ell$ windows directly, since the paper's strong-lensing shifts sit at the few-microarcsecond level; the paper notes the need but does not simulate such an observation.
Applying the paper's weak-field deflection formula (53) to a catalog of galaxy-scale Einstein rings, rather than the single ESO325-G004 system, would tighten the $\ell$ bound statistically through averaging.
If the RBBH is an exact solution, the same Lorentz-violating coupling should modify quasinormal-mode frequencies and gravitational-wave ringdowns; computing those would extend the test beyond lensing into the dynamical regime, which the paper does not do.
The EHT comparison uses only the shadow's angular diameter; using the full two-dimensional shadow shape could break the degeneracy between spin $a$ and $\ell$ that diameter matching leaves open.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Kumar, Islam, and Ghosh apply Bozza's strong-deflection formalism to the rotating Bumblebee black hole and push beyond the earlier slow-rotation studies in Refs. [106,107]. That is the paper's genuine contribution: full-rotation strong-lensing observables, time-delay tables, and a weak-lensing Einstein-ring bound on the Lorentz-violating parameter. The organization is clear, the prior literature is engaged honestly, and the idea of using EHT shadow sizes plus the ESO325-G004 Einstein ring to constrain ell is reasonable.

Unfortunately the quantitative results are not trustworthy as written. The stress-test and the reader's report are correct on the central checks. Equation (29), the photon-sphere condition, does not reduce to Schwarzschild's r=3M when a=ell=0; it reduces to r(-2r^2+5r-6)=0, whose only real root is r=0. Since rm and um feed the deflection angle, image positions, magnifications, and the EHT constraints, this one equation undermines Section III. The time-delay formula in Eq. (46) says Delta T_{2,1} = pi u_m, but the numbers in Table I correspond to 2 pi u_m; a factor-of-two conflict between a stated formula and the table. The weak-deflection expansion has its own problem: in the Kerr limit Eq. (53) gives a second-order coefficient 15 pi/64 rather than the standard 15 pi/4, so the Einstein-ring bound in Section IV inherits the error. These are not typo-level slips; each sits in a load-bearing position. The claim that a significant portion of parameter space agrees with EHT within 1 sigma is unsupported by the equations shown, even if the metric in Eq. (9) is taken as given.

A softer point: the rotating metric is imported from Ref. [113] via a modified Newman-Janis algorithm rather than verified here. That is standard practice in this literature and not fatal, but it makes the solution status depend entirely on the cited derivation.

Bottom line: the paper's topic is timely and the extension is worth doing, but the current manuscript fails basic consistency checks. I would not cite it in this form, and I would not take the constraints on ell from it. A corrected version, in which Eq. (29) is re-derived, the time-delay formula is reconciled with the table, and the weak-deflection expansion is redone, could be a legitimate contribution. As it stands, rejection or a major revision is the right call.

Referee Report

5 major / 5 minor

Summary. The manuscript studies strong and weak gravitational lensing by a rotating Bumblebee black hole (RBBH) with a Lorentz-violating parameter ℓ. The authors present a Kerr-like metric (Eq. 9) obtained via a modified Newman–Janis algorithm from the static Bumblebee seed (Eq. 8), compute the photon-sphere radius and critical impact parameter, and use Bozza's strong-deflection formalism to obtain the angular position θ∞, separation s, magnification ratio rmag, and time delay ΔT21 of relativistic images. These observables are evaluated for Sgr A* and M87* and compared with EHT shadow-size measurements to constrain the parameters (a, ℓ). The paper also derives a weak-deflection angle, computes the Einstein ring of the galaxy ESO325-G004, and claims an upper bound ℓ ≲ O(10^-6). The central claims are that ℓ > 0 suppresses and ℓ < 0 increases deflection relative to Kerr, and that current observations leave a significant allowed region of parameter space.

Significance. The topic is timely: gravitational lensing and EHT shadow observations are active probes of deviations from the Kerr paradigm, and the Bumblebee model is a standard framework for spontaneous Lorentz violation. The paper contains a substantial amount of numerical work, including tables of observables for Sgr A* and M87* and a parameter-space diagram, and the limiting values in Table I (e.g., θ∞ for a=0, ℓ=0 close to the Schwarzschild photon-ring value) indicate that the numerical code may have used correct geodesic equations. If the results were correct, the paper would provide a useful extension of strong-lensing constraints to Lorentz-violating gravity. However, the quantitative claims are not supported by the equations as written: the displayed photon-sphere condition fails the Schwarzschild limit, the weak-deflection expansion has a wrong second-order coefficient, and the time-delay formula disagrees with the table by a factor of two. Since these equations feed directly into the EHT and Einstein-ring constraints, the manuscript's main conclusions cannot be accepted in its present form.

major comments (5)

[III, Eq. (29)] Setting a=0 and ℓ=0 in Eq. (29) gives r(-2r^2+5r-6)=0, whose only real root is r=0; the Schwarzschild photon sphere r=3M is not recovered. This is a decisive consistency failure because Eq. (29) is presented as the solution of the photon-sphere conditions (28) and is the analytic input for rm and um, which in turn determine θ∞, s, rmag, and the EHT constraints in Section III. The numerical entries in Table I (e.g., θ∞≈26.33 μas for a=0, ℓ=0) are consistent with the correct um≈3√3M, so the authors appear to have computed with a different equation than the one displayed; they must derive and display the correct photon-sphere equation and recompute or verify all strong-lensing results against it.
[IV, Eq. (53)] For ℓ=0 and a=0, Eq. (53) reduces to α(b)=4M/b + (15π/64)(M/b)^2 + O(b^-3), whereas the standard Schwarzschild weak-deflection limit is 4M/b + (15π/4)(M/b)^2 + ... . The displayed coefficient is a factor of 16 too small. The inconsistency is compounded by Eq. (57), which contains the standard 15π/4 coefficient, so the Einstein-ring equation does not follow from the expansion (53). All weak-lensing results in Section IV, including the bound ℓ≲4.472×10^-6, rest on this erroneous expansion and must be re-derived.
[III, Eq. (46) and Table I] Equation (46) states ΔT21≈πu_m, but the values in Table I correspond to 2πu_m: for Sgr A* at a=0, ℓ=0, the tabulated delay of 11.5 min is approximately 2πu_m (with u_m≈5.3M and M_time≈19.7 s), not πu_m. One of the two must be corrected; as written, the time-delay observable, which is one of the paper's proposed Kerr discriminators, is internally inconsistent.
[II, Eq. (9)] The paper asserts, with reference to Ref. [113], that Eq. (9) is the rotating counterpart of the static Bumblebee solution (8), but it does not demonstrate that this metric satisfies the Bumblebee field equations (5)-(6). The modified Newman–Janis algorithm is a solution-generating technique only when the field equations are actually checked; all subsequent lensing calculations are computed from Eq. (9). The authors should either provide a direct verification of Eq. (9) as a solution, or state precisely what has been verified in the cited literature, so that the lensing results are not conditional on an unproven metric.
[IV, Eq. (59) and ESO325-G004] The Einstein-ring analysis applies the point-mass RBBH deflection formula to the galaxy ESO325-G004, whose mass is stated to include dark and luminous matter inside the ring; an extended mass distribution does not generally produce the same Einstein radius as a point mass, so the bound ℓ≲O(10^-6) depends on an unvalidated model choice. In addition, the distances in Eq. (59) are not angular diameter distances: dS=2.863×10^4 Mpc for z_s=1.141 is far too large, and cz(1+z)/H0 is not the correct distance measure for lensing at this redshift. The weak-lensing constraint should be recomputed with proper angular diameter distances and a realistic mass model.

minor comments (5)

[Abstract and Table I] The abstract and text report rmag in μas, but rmag is a dimensionless flux ratio; the Table I column header 's (nas)' should also be made consistent with the μas values quoted in the text.
[Section II, Eq. (13)] The sentence preceding Eq. (13) repeats the same clause: 'The separation between the inner and outer horizons reads The separation between the horizons is'; please clean up the duplication.
[Section IV] The text refers to Fig. 8 for the weak-deflection deviation δαD(u), but Fig. 8 shows shadow angular diameters; the correct references should be to the δαD plots in Figs. 9 and 10.
[Section III, Eq. (39)] The citation keys 'Bozza:2018ev,Bozza:2001xd' appear in the text next to Eq. (39) and should be replaced by proper numbered references.
[Section I] The text 'Atkinson it et al.' should read 'Atkinson et al.'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lensing observables are forward-modeled from the RBBH metric and compared with external EHT and ESO325-G004 data.

full rationale

The derivation chain is not circular. The RBBH metric, Eq. (9), is taken from the externally cited construction of Ref. [113] via the Newman-Janis algorithm, and the strong-lensing observables (theta_infinity, s, r_mag, Delta T) are computed from the geodesic equations of that metric using the standard Bozza strong-deflection formalism. The EHT-based constraints on (a,ell) are obtained by comparing the forward-modeled shadow angular size 2*theta_infinity with observed values; the Einstein-ring bound from ESO325-G004 is likewise a forward prediction of theta_E from the weak-deflection expansion compared with measured data. No parameter is fitted to a subset of data and then renamed as a prediction. The paper's self-citations (e.g., Ref. [48] for the effective-potential form) are not load-bearing: the relevant equations are re-derived in the text from the Lagrangian and null condition. There is no uniqueness theorem imported from the authors' prior work, and the central metric input comes from an external source rather than from a self-citation chain. Concerns such as the apparent failure of Eq. (29) to reduce to the Schwarzschild photon sphere and the unusual coefficient in Eq. (53) are correctness or typographical issues, not circular reductions of outputs to inputs; similarly, the unverified status of Eq. (9) as an exact solution of the Bumblebee field equations is a missing-support/correctness concern, not a circularity. Thus the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the assumed exactness of the NJA-generated RBBH metric, the applicability of Bozza's strong-deflection formalism, and the identification of theta_infinity with the EHT shadow. The weak-lensing bound further assumes a galaxy can be modeled as a point-mass RBBH. The model parameter ell and spin a are the only free parameters; no new particles or fields are introduced by this paper.

free parameters (2)

Lorentz-violating parameter ell = constrained: ell less than about 4.472e-6 (1 sigma) and 8.084e-6 (2 sigma) from ESO325-G004; allowed intervals from…
Model parameter of the RBBH metric; varied in strong-lensing predictions and bounded by matching the observed Einstein ring size and EHT shadow diameters.
Spin parameter a = scanned over 0 to 0.95 M in figures; tabulated at 0, 0.3, 0.6 M
Dimensionless spin chosen for plots and tables; not independently measured here.

assumptions (5)

domain assumption The rotating metric (9) obtained by the modified Newman-Janis algorithm from seed metric (8) is a solution of the Bumblebee field equations (5)-(6).
Paper invokes Ref. [113] and never verifies Eq (9) satisfies the field equations; the full rotating solution status is assumed.
domain assumption Bozza strong-deflection formalism applies to RBBH in the equatorial plane with asymptotically flat observer and source.
Sec III uses integral (31) and expansion (36)-(38) following Refs [26,29]; standard for static and stationary metrics, assumed valid here.
domain assumption The shadow angular diameter equals twice the innermost image position, theta_sh = 2 theta_infinity.
Sec III B uses this identification to map EHT ring measurements to model predictions.
ad hoc to paper The weak-field deflection formula (53) for an isolated BH point mass also describes lensing by galaxy ESO325-G004.
Sec IV A applies the RBBH point-mass deflection to a galaxy whose mass includes dark matter and luminous matter; this is not a black hole lens.
domain assumption The lens equation (39) with perfect alignment and the Einstein ring condition (54)-(55) apply.
Standard lensing geometry assumed, with source and observer at large distance.

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Pith. "Pith review of Probing Lorentz Symmetry Violation through Lensing Observables of Rotating Black Holes." pith.science (2026). https://pith.science/paper/66FIKJT2

@misc{pith2026250900127,
  author       = {Pith},
  title        = {Pith review of: Probing Lorentz Symmetry Violation through Lensing Observables of Rotating Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/66FIKJT2}},
  note         = {Machine review of arXiv:2509.00127}
}

abstract

We find a Kerr-like black hole solution-a rotating Bumblebee black hole (RBBH) with a Lorentz-violating parameter $\ell$ and examine the strong lensing by it. The parameter $\ell$ changes the event horizon radius and photon sphere, resulting in a different lensing signature compared to the Kerr black hole of general relativity. Using the strong deflection limit formalism, we compute key observables such as the angular positions of relativistic images, their separation, magnification, and time delays for supermassive black holes Sgr A* and M87*. Our results show that the parameter $\ell$ has a profound influence on these observables, with $\ell > 0$ suppressing and $\ell < 0$ increasing the deflection angle compared to the Kerr case. We compare RBBH observables with those of Kerr black holes, using Sgr A* and M87* as lenses to observe the effect of the Lorentz symmetry-breaking parameter $\ell$. For Sgr A*, the angular position $\theta_\infty$ in $\in~(18.25-33.3)~\mu as$, while for M87* $\in~(13.71-25.02)~\mu as$. The angular separation $s$, for supermassive black holes (SMBHs) Sgr A* and M87*, differs significantly, with values ranging $\in~(0.005-0.81)~\mu as$ for Sgr A* and $\in~(0.003-0.6)~\mu as$ for M87*. The relative magnitude $r_{\text{mag}}$ $\in~(3.04-8.15)~\mu as$. We also compared the time delays between the relativistic images in the SMBHs and found that RBBH can be quantitatively distinguished from Kerr black holes. Our analysis concludes that, within the 1$\sigma$ region, a significant portion of the parameter space agrees with the EHT results of M87* and Sgr A*. This demonstrates the feasibility of utilizing strong gravitational lensing to identify Lorentz symmetry violations in extreme gravity regimes. Weak lensing analysis and Einstein ring observations provide further constraints, producing an upper bound of $\ell \lesssim \mathcal{O}(10^{-6})$.

Figures

Figures reproduced from arXiv: 2509.00127 by the authors.

**Figure 2.** FIG. 2. Comparison of the event horizon ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Effective potential [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: FIG. 5. Deflection angle [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6. Strong lensing observables for supermassive black h [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 7.** Figure 7: FIG. 7. Flux magnification ratio [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: FIG. 8. Shadow angular diameter [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]

**Figure 9.** Figure 9: FIG. 9. The deviation in weak lensing deflection angle [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: FIG. 10. The deviation in weak lensing deflection angle [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

**Figure 11.** Figure 11: FIG. 11. The estimation of angular radius of Einstein ring [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]

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