{"id":"cca6dfe9-9d18-44bb-9d5b-2d1f62dac4d8","arxiv_id":"2501.00292","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Strong lensing by rotating regular black holes shifts lensing coefficients, image positions, magnifications, and time delays relative to Kerr, with M87* time delays differing by up to tens of hours while image position shifts remain below 10 microarcseconds.","lead":"This paper computes how a rotating regular black hole, one with no singularity at its center, would bend light and delay photon arrival times compared with a standard Kerr black hole. It finds the image shifts for M87* and SgrA* are too small for today's telescopes, but some time delays might be large enough for future observations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The rotating metric in Eq. (4) is assumed to be a physical rotating regular black hole, but the paper never verifies field equations, energy conditions, or regularity at Σ=0; every lensing prediction is conditional on that unverified premise.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: the Newman-Janis-generated metric of Eq. (4) is assumed to describe a physical rotating regular black hole without any check of field equations or energy conditions. I agree that this is the central issue. The strong-lensing computation itself appears internally consistent: the Bozza formalism is applied to the stated metric, the photon-sphere equations follow from the effective potential, and the numerical trends in the tables are broadly coherent with the stated mass function. However, all of these results inherit their physical meaning from the spacetime being a genuine rotating regular black hole. If Eq. (4) is merely a metric ansatz with no known matter source, then the time-delay predictions, which are the paper's most measurable signature, are predictions for an ad hoc geometry rather than for a regular black hole candidate. I also noticed apparent numerical typos in Tables IV and VIII, for example the δΔT2,1 entry -0.845 in Table IV for a=-0.1, g=0.4 should likely be -0.085, and the a=0.2, g=0.4 entry in Table VIII differs from the corresponding a=-0.2 row in magnitude. These typos reduce confidence in the tables but are not the decisive issue. The decisive issue remains the unverified physicality of the metric, for which the proposed Einstein-tensor and energy-condition computation would provide a direct test. Since the reader already assigned a CONDITIONAL verdict and my concern matches that condition, no change to the verdict is needed beyond making the physicality check an explicit requirement.","tokens_in":16491,"tokens_out":6253,"duration_ms":69530,"concrete_test":"Compute the Einstein tensor of Eq. (4) with m(r)=M exp(-g^n M^γ/r^n) for (γ=2/3,n=2) and (γ=1,n=3), using a symbolic tensor package such as xAct. Evaluate the effective stress-energy tensor in an orthonormal frame along the equatorial plane and the polar axis for the parameter values used in Tables I-VIII, e.g. a/M=-0.2,...,0.2 and g=0,0.3,0.4,0.48. Accept the metric as a physical rotating regular black hole only if the weak and null energy conditions hold and if curvature invariants remain finite as r→0 along all θ, including θ=π/2 where Σ can vanish. If either condition fails, the lensing observables in Section IV cannot be attributed to a physically viable rotating regular black hole.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing assumption is the physical status of Eq. (4). Section II obtains the rotating metric by applying the Newman-Janis algorithm (ref. [57]) to the static mass function (2), and every subsequent lensing observable is computed from this line element. However, the paper never checks that Eq. (4) solves the field equations of a concrete gravity theory with a reasonable matter source, nor that the Newman-Janis transform preserves the regularity and Minkowski-core property of the static seed. This matters because the central claims—measurable time-delay differences from Kerr and sub-10 μas image deviations—are predictions about astrophysical black holes. If Eq. (4) is only a metric ansatz with no known matter content, then the results are mathematical consequences for a particular line element rather than physically meaningful lensing predictions for regular black holes. The concern is not that the calculation is internally inconsistent; the strong-lensing formalism and numerics appear to follow from the stated metric. The problem is that the mapping from the static regular black hole of Eq. (1) to a physical rotating black hole is asserted rather than demonstrated. Notably, Section II cites the original Newman-Janis paper rather than a subsequent analysis showing that the resulting spacetime is a viable rotating regular black hole. The parameter-space plots in Fig. 1 only establish existence of horizons, not physical viability or regularity. Without a check of curvature invariants and energy conditions, the observable predictions in Section IV rest on an unverified premise.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies strong gravitational lensing by two classes of rotating regular black holes with a Minkowski core, obtained by applying the Newman-Janis algorithm to the static regular black hole of Ref. [53]. The authors derive the strong-field-limit deflection angle for equatorial photons, then compute the lensing observables (asymptotic image position, angular separation, relative magnification, and time delays between relativistic images) under the assumption that the objects are M87* and SgrA*. The central claims are that the time-delay differences between these regular black holes and Kerr are potentially measurable, especially for M87*, while the deviations in image position and separation are below 10 microarcseconds and hence require next-generation Event Horizon Telescope capabilities.","tokens_in":16764,"tokens_out":9963,"duration_ms":100946,"significance":"If the rotating metric is taken as physically viable, the paper provides a concrete and falsifiable prediction: strong-lensing time delays can discriminate between Kerr and these regular black holes even when image-position differences are too small for current observations. The analytical derivation follows the standard Bozza strong-field-limit formalism, and the numerical values are internally plausible and consistent with the mass-scaling between M87* and SgrA*. The paper does not, however, ship machine-checked proofs or reproducible code, and no independent numerical verification or uncertainty estimates are presented. The main value is therefore as a model-dependent theoretical estimate rather than a demonstrated observational test.","major_comments":[{"comment":"The rotating metric in Eq. (4) is introduced as the Newman-Janis rotating counterpart of the static regular black hole, but the paper never checks whether this line element is a solution of any concrete gravitational field equations with a reasonable matter source, nor whether the regularity and Minkowski-core properties of the static seed survive rotation (e.g., by examining G_mu_nu, energy conditions, or curvature invariants at Sigma=0). Because every lensing observable in Sections III and IV is computed from Eq. (4), the astrophysical claims about M87* and SgrA* are conditional on this unverified premise. Please either supply the missing verification (or a reference that provides it) and discuss the resulting validity region, or explicitly reframe the conclusions as predictions of the metric ansatz (4).","section":"Section II, Eq. (4)"},{"comment":"The paper claims that the time-delay differences between the regular black holes and Kerr are 'measurable,' especially for M87*, but it does not provide a quantitative comparison with any observational uncertainty, such as expected EHT timing precision or mass and distance errors. The tables report predicted delays but no error bars, and the sub-10 microarcsecond image-position deviations are not translated into a signal-to-noise or detection-threshold statement. Please add a quantitative detectability discussion or temper the 'measurable' wording in the abstract and conclusion.","section":"Section IV.B, Tables I-VIII"}],"minor_comments":[{"comment":"In the a=-0.1, g=0.4 entry, the deviation delta Delta T_2,1 is printed as -0.845 min, while the difference of the preceding columns is -0.085 min; as printed this breaks the monotonicity of the g=0.4 column and should be corrected.","section":"Section IV.B, Table IV"},{"comment":"The last sentence refers to 'gamma=1,n=2', but the models studied in the paper are gamma=1,n=3; please correct the typo to match Eq. (2) and Fig. 2.","section":"Section III.A.2"},{"comment":"The displayed expression for x-dot appears corrupted in the version under review, containing non-mathematical characters; please ensure the formula is typeset correctly.","section":"Eq. (9)"},{"comment":"The conversion from the dimensionless rescaling x=r/(2M) used in Section III.A.1 to the physical hours and minutes reported in Tables I-VIII is not shown; please state the overall mass/distance prefactor explicitly.","section":"Section IV.A, Eqs. (30)-(36)"},{"comment":"The dimension of the regularity parameter g is not stated; for gamma=2/3,n=2 the exponent g^2 M^{2/3}/r^2 implies an unusual dimension that should be specified to avoid confusion about the plotted g/M values.","section":"Section II, Eq. (2)"},{"comment":"In the conclusion, 'Delta-tilde-T_2,1' should read 'Delta T_2,1' to match the notation in Tables I-IV; the two different time-delay quantities are otherwise easy to confuse.","section":"Section V"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a standard strong-lensing calculation whose main novelty is the application of Bozza's formalism to a specific Minkowski-core rotating metric. The two substantive issues are the unverified physical status of Eq. (4) and the absence of a quantitative detectability threshold for the claimed measurable time delays. I do not see a circularity problem: the lensing coefficients are computed, not fitted, and the mass and distance inputs are external EHT values. The self-citation to Ref. [53] is appropriate because that paper is the source of the static metric."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nQuick take: this is a solid application of the standard Bozza strong-lensing machinery to a specific rotating regular black hole family. The genuinely new content is the numerical evaluation of lensing observables for M87* and SgrA* using the rotating version of the Ling–Wu exponential mass function, in particular the time delays between relativistic images. The claim that for M87* the time-delay deviation from Kerr can reach tens of hours is concrete and, in principle, testable with next-generation interferometry. The paper also honestly reports that image position and separation deviations stay below ~10 μas, which tempers expectations for current EHT.\n\nThe calculation itself follows the established strong-field-limit expansion carefully. The deflection coefficients, critical impact parameters, and tables are internally consistent, aside from a couple of clear typos (e.g., in Table IV, the δ value of –0.845 min should almost certainly be –0.085; Table VIII has a similar transposition of 4.420 vs. 4.320). The parameter-space horizon plots are a nice touch.\n\nThe soft spot is the one the stress test flags: Eq. (4) is obtained by Newman–Janis and taken as a physical rotating regular black hole without ever checking field equations, energy conditions, or regularity at Σ=0. The paper cites the original 1965 Newman–Janis paper, not the large literature on when NJ transforms yield viable spacetimes. So all the lensing predictions are conditional on that unverified premise. That doesn't invalidate the math, but it does limit what the numbers mean physically. A referee should ask for a justification, or at least an explicit caveat that the metric is an effective ansatz.\n\nOther omissions are minor: no code or data artifacts, no uncertainty budget on the mass/distance inputs, and the discussion of time-delay observability is optimistic—it doesn't address how one would resolve individual relativistic images, which is a serious practical hurdle. The typos should be fixed but are not load-bearing.\n\nWho is this for? People working on strong lensing phenomenology of regular black holes will find the tables useful. It's an incremental contribution, not a breakthrough, but it's a legitimate and competently executed one. I'd send it to peer review with a clear request for revision: address the physicality of the metric, add uncertainty estimates, fix typos, and moderate the observability language.","headline":"A competent application of Bozza's strong-lensing formalism to a specific rotating regular black hole family; the new numerical time-delay predictions are interesting, but the physical status of the rotating metric is assumed rather than demonstrated.","tokens_in":17315,"tokens_out":3910,"would_cite":false,"duration_ms":36527,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For two rotating regular black holes with a Minkowski core, the time delays between relativistic images deviate from Kerr by up to tens of hours for M87* parameters, while image-position deviations remain below 10 microarcseconds.","keywords":["strong gravitational lensing","regular black holes","Minkowski core","Kerr black hole","relativistic images","time delay","M87*","Newman-Janis metric"],"falsifier":"Compute the effective energy\\textendash momentum tensor of the rotating metric in Eq. (4) and check whether it satisfies standard energy conditions; if no reasonable matter source exists, the lensing predictions lack an astrophysical basis. Observationally, a next-generation Event Horizon Telescope measurement of the time delay between the first and second relativistic images of M87* that matches the Kerr value to better than the predicted tens-of-hours deviation would rule out these two regular models.","tokens_in":16283,"feed_emoji":"🕳️","tokens_out":5921,"duration_ms":52649,"temperature":0.7,"pith_summary":"This paper asks whether two families of rotating regular black holes with a non-singular Minkowski core can be told apart from a Kerr black hole by the way they bend light. Using a strong-field deflection-angle expansion, it computes the relativistic image position, separation, magnification, and time delays for a Bardeen-type and a Hayward-type regular black hole, scaled to the masses and distances of M87* and SgrA*. It finds that the image-position deviations from Kerr stay below 10 microarcseconds, below current resolution, but the time delays between relativistic images deviate from Kerr by up to tens of hours for M87*, which could be measurable. The authors conclude that time-delay observations, not image positions, are the promising route for testing whether a supermassive black hole is regular.","feed_headline":"Time delay can reveal regular black holes at M87*","feed_subtitle":"Image positions differ by under 10 microarcseconds, but relativistic-image delays deviate by up to tens of hours.","key_machinery":"The machinery is the Bozza strong-deflection-limit expansion for the deflection angle, $\\alpha_D(u) = -\\bar a \\log(u/u_m - 1) + \\bar b$, with the strong-deflection coefficients $\\bar a$ and $\\bar b$ extracted from the metric functions $A, B, C, D$ of the rotating regular spacetime in the equatorial plane. The same expansion, augmented by the time-delay coefficients $\\tilde a$ and $\\tilde b$, yields the image position $\\theta_\\infty = u_m/D_{OL}$, the separation $s$, the magnification ratio $r_{\\rm mag}$, and the inter-image time delays. These coefficients carry the entire dependence on the regularity parameter $g$ and the spin $a$, and the paper evaluates them numerically for the two regular models to predict observable shifts from Kerr.","core_discovery":"The paper's central claim is that strong gravitational lensing by the two rotating regular black holes differs from Kerr in a characteristic, g-dependent way, and that the difference is concentrated in the time-delay channel rather than the image geometry. For both parameter families, increasing the regularity parameter g reduces the critical impact parameter, the asymptotic image radius $\\theta_\\infty$, and the relative magnification $r_{\\rm mag}$, while increasing the angular separation $s$; the same-side time delay $\\Delta T_{2,1}$ between the first and second images is shorter than in Kerr, and the prograde\\textendash retrograde delay $\\Delta\\tilde{T}_{1,1}$ is longer. For M87* parameters the time-delay deviations reach tens of hours (up to about 44 hours in the Bardeen-type case at $g=0.48$, $a=0.2$), whereas the deviations in $\\theta_\\infty$ and $s$ are at most a few microarcseconds. Hence the paper argues that current Event Horizon Telescope observations cannot resolve the regularity through image position, but next-generation instruments may test it through time delays.","pith_inferences":["The results suggest that time-delay measurements are a more sensitive probe of near-horizon regularity than shadow-size or image-position measurements; if so, future lensing campaigns should prioritize timing observations.","The calculation assumes equatorial prograde and retrograde photon orbits; off-equatorial or photon-ring autocorrelation observations could expose additional $g$-dependent signatures not captured here.","If the Newman\\textendash Janis metric does not correspond to a known matter source, the predicted time-delay deviations would still serve as a bound on what lensing can say about regularity; conversely, a positive detection would motivate finding a theory that produces this metric.","The trend that the Bardeen-type model ($\\gamma = 2/3, n=2$) shows larger deviations than the Hayward-type model ($\\gamma = 1, n=3$) suggests that the specific form of the mass function matters, and probing multiple regular models may be needed to avoid degeneracy with spin."],"forward_implications":["If the paper is right, the time delay between the first and second relativistic images of M87* is the most promising strong-lensing observable for distinguishing these regular black holes from Kerr, with deviations of up to tens of hours.","The regularity parameter $g$ shrinks the critical impact parameter and hence the apparent image radius, but the effect stays below about 10 microarcseconds, below current Event Horizon Telescope resolution.","The separation $s$ between the outermost and asymptotic images grows with $g$, so higher-resolution observations of image pairs could complement time-delay measurements.","For SgrA* the same trends hold but the time delays are only minutes and the deviations are correspondingly smaller, making M87* the better target."],"supporting_citations":[{"why":"Supplies the strong-field limit expansion for the deflection angle and the formulas for relativistic image observables.","marker":"[25]"},{"why":"Extends the strong-field expansion to general asymptotically flat spacetimes, underpinning the coefficient derivation.","marker":"[26]"},{"why":"Provides the time-delay formulas for relativistic images used to compute $\\Delta T_{2,1}$ and $\\Delta\\tilde{T}_{1,1}$.","marker":"[28]"},{"why":"Defines the static regular black hole with Minkowski core and the mass function that is the seed for the rotating metric.","marker":"[53]"},{"why":"Identifies the Bardeen black hole limit corresponding to $\\gamma = 2/3, n=2$.","marker":"[54]"},{"why":"Identifies the Hayward black hole limit corresponding to $\\gamma = 1, n=3$.","marker":"[55]"},{"why":"Supplies the Newman\\textendash Janis algorithm used to generate the rotating regular black hole metric.","marker":"[57]"},{"why":"Provides the M87* mass and distance used to evaluate the lensing observables.","marker":"[6]"},{"why":"Provides the SgrA* distance used to evaluate the lensing observables.","marker":"[62]"}],"fun_headline_variants":["Time-delay lensing could expose regular black holes at M87*","Gravitational time delays may discern black hole regularity","Lensing time delays at M87* might reveal non-singular cores","Regular black holes: time delays are the key observable","M87* lensing delays could test for non-singular interiors"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire calculation rests on the assumption that the rotating metric obtained by the Newman\\textendash Janis algorithm actually describes a real black hole in some theory of gravity; the paper never verifies that it solves any field equations or obeys energy conditions.","fun_headline_variants_meta":{"raw":{"variants":["Time-delay lensing could expose regular black holes at M87*","Gravitational time delays may discern black hole regularity","Lensing time delays at M87* might reveal non-singular cores","Regular black holes: time delays are the key observable","M87* lensing delays could test for non-singular interiors"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001069,"raw_usage":{"total_tokens":4494,"prompt_tokens":979,"completion_tokens":3515,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":595,"completion_tokens_details":{"reasoning_tokens":3426}},"tokens_in":595,"tokens_out":3515,"duration_ms":24679,"temperature":1.0,"reasoning_tokens":3426,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:54:18.720186+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the effective energy\\textendash momentum tensor of the rotating metric in Eq. (4) and check whether it satisfies standard energy conditions; if no reasonable matter source exists, the lensing predictions lack an astrophysical basis. Observationally, a next-generation Event Horizon Telescope measurement of the time delay between the first and second relativistic images of M87* that matches the Kerr value to better than the predicted tens-of-hours deviation would rule out these two regular models.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the Bardeen black hole limit corresponding to $\\gamma = 2/3, n=2$."}],"review_version":1}