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Modelling the Sgr A$^*$ and M87$^*$ shadows by using the Kerr-Taub-NUT metrics in the presence of a scalar field

T0 review · 5 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A new rotating black-hole metric family reproduces the M87* and Sgr A* shadow sizes.

desk verdict A plausible new Kerr-Taub-NUT-scalar metric family, but the exactness is unverified and the scalar-charge plus VLTI-bound inconsistencies must be fixed before the NUT limits are trusted. read the letter →

arxiv 2501.11692 v1 pith:OTW6XKER submitted 2025-01-20 gr-qc astro-ph.IM

khadije Jafarzade , Masoumeh Ghasemi-Nodehi , Fatemeh Sadeghi , Behrouz Mirza This is my paper · ORCID
classification gr-qcastro-ph.IM
keywords blackholeshadowsKerr-Taub-NUTmetricscalarfieldErnsttransformationNUTchargeEventHorizonTelescopeweakgravitationallensingGauss-Bonnettheorem
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

The paper derives a new exact family of rotating, Taub-NUT spacetimes coupled to a massless scalar field, called KTNS, by applying complex Ernst transformations to a known static scalar-field metric. It then asks whether these spacetimes can describe the supermassive objects whose shadows were imaged by the EHT. Using numerical ray tracing of null geodesics, it finds that the M87* circularity constraint is satisfied for NUT charge $n<0.5$ when the scalar parameter $ u=0$, and the Sgr A* Keck fractional-deviation bound for $n<0.41$, with the allowed windows shifting as rotation and $ u$ vary. The same family gives an analytic weak-deflection angle through the Gauss-Bonnet theorem, in which rotation, NUT charge, and scalar-field strength all increase light bending. If valid, the model places a quantitative upper bound on the gravomagnetic monopole charge of these supermassive objects.

What carries the argument

The load-bearing object is the KTNS line element, Eq. (40), built from the Ernst potential $\varepsilon'=1-2(m+in)/(r+i(a\cos\theta+n))$, a complex potential that encodes the metric functions in axisymmetric vacuum gravity. The construction assumes the Ernst equations remain valid in the presence of a scalar field, so the complex shift produces both rotation $a$ and NUT charge $n$ from a scalar-field seed metric; the metric functions $f'$, $\omega'$, and $\lambda'$ are then reconstructed and assembled with the scalar field (39). The observational part is carried by numerical ray tracing of null geodesics onto an image plane, with shadow radius, center, circularity deviation $\Delta C$, and fractional deviation $\delta$ extracted from the boundary. The deflection angle is obtained separately from the finite-distance Gauss-Bonnet optical geometry. Throughout, the NUT charge $n$ acts as the control parameter: small $n$ keeps shadows within EHT bounds, while large $n$ or $|\nu|$ pushes shadows and deflection angles away from Kerr behavior.

What would settle it

Substitute the metric (40) and the scalar field (39) directly into the field equations $R_{\alpha\rho}=\partial_\alpha\phi\,\partial_\rho\phi$ and $\nabla^\alpha\nabla_\alpha\phi=0$; if the equations do not reduce to identities, the claimed solution family is invalid. A complementary check is to recompute the shadow diameter and circularity for the M87* and Sgr A* parameter sets with an independent ray-tracing code: the paper's quoted windows ($n<0.5$, $n<0.41$, $n>0.34$) should be reproducible to within the published observational errors.

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Extended reading notes

Core claim

The paper's central claim is that Eq. (40) is a new exact family of Kerr-Taub-NUT metrics in the presence of a massless scalar field, obtained by applying the complex Ernst-potential transformation $r\to r+ia\cos\theta+in$, $m\to m+in$ to the static three-parameter scalar-field metrics of Ref. [20]. The authors assert that this is a legitimate solution of the Einstein-scalar system because, for their action, the Ernst equations are unchanged by the scalar field and only the wave equation $\square\phi=0$ is added. They then treat the metrics as models for M87* and Sgr A* and, using numerical ray tracing of null geodesics, impose the observed shadow sizes together with the EHT constraints on circularity deviation $\Delta C<0.1$ and fractional deviation $\delta$. The result is that the M87* circularity constraint is met for $n<0.5$ when $\nu=0$, the Sgr A* Keck bound for $n<0.41$, and the VLTI bound for $n>0.34$, with the allowed windows changing as rotation and the scalar parameter vary. A weak-field deflection angle computed from the Gauss-Bonnet theorem completes the model, and the paper shows that rotation, NUT charge, and scalar-field strength all increase the bending angle.

Load-bearing premise

The derivation assumes, on the authority of the cited literature, that adding a massless scalar field leaves the Ernst equations unchanged, so the complex shift $r\to r+ia\cos\theta+in$, $m\to m+in$ can be applied to the scalar-field seed; if that assumption fails, the KTNS metric is not a solution and the shadow bounds built on it do not follow.

Editorial extensions

If this is right

  • If Eq. (40) is a valid solution, EHT observations translate directly into quantitative NUT-charge bounds: $n<0.5$ from M87* circularity, $n<0.41$ from the Sgr A* Keck bound, and $n>0.34$ from the VLTI bound, giving a testable window for the gravomagnetic monopole parameter.
  • The same data favor fast rotation: for fixed $|\nu|$ and $n$, high-spin KTNS metrics satisfy the M87* shadow-size and circularity constraints more easily than slowly rotating ones.
  • The scalar-field parameter $\nu$ does not destroy the fit; larger $|\nu|$ shrinks the allowed M87* NUT range but enlarges the Sgr A* Keck-$\delta$ range, so shadow observations alone cannot yet isolate $\nu$.
  • In the weak-field limit, deflection angles grow monotonically with $a$, $n$, and $|\nu|$; for large $n$ or $|\nu|$ the prograde/retrograde bending asymmetry disappears, which would be a distinctive observable signature if confirmed.

Reading between the lines

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

  • An implication the authors leave implicit is that the same ray-tracing pipeline can be run on the $n=0$, $\nu=0$ Kerr limit to separate how much of the EHT agreement comes from rotation versus NUT charge.
  • Because the paper does not directly substitute Eq. (40) into the field equations, a symbolic verification of $R_{\alpha\rho}=\partial_\alpha\phi\,\partial_\rho\phi$ and $\square\phi=0$ would make explicit whether the solution family stands on the imported Ernst-invariance statement alone.
  • A natural extension is to test the NUT-induced gravitomagnetic effect using stellar-orbit astrometry around Sgr A*, which could constrain the NUT charge independently of the shadow window between $n\approx0.34$ and $n\approx0.41$.
  • The finite-distance deflection-angle formula predicts that larger $n$ or $|\nu|$ always bends light more, a monotonic trend that future very-long-baseline interferometry measurements near Sgr A* could distinguish from the predictions of pure Kerr spacetime.
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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

5 major / 5 minor

Summary. The paper claims to construct a new exact family of Kerr-Taub-NUT spacetimes with a minimally coupled scalar field (KTNS), given by Eq. (40), starting from a three-parameter static scalar-field metric and applying a complex Ernst transformation. The authors compute the Ricci scalar and admissible parameter regions, then use numerical ray tracing to test the shadow size and circularity against M87* and Sgr A* observations from the EHT, reporting bounds such as n<0.5 for M87*, n<0.41 for the Keck bound, and a contradictory VLTI bound in the abstract versus the body. They also derive a weak-deflection-angle formula via the Gauss-Bonnet theorem and conclude that all parameters increase the deflection angle.

Significance. If the exactness of Eq. (40) were established, the paper would provide a new rotating Taub-NUT scalar-field spacetime with observational consequences, and the EHT-based bounds on the NUT charge would be a useful contribution to tests of gravomagnetic monopoles and scalar hair. The ray-tracing results are compared with publicly reported EHT data, and the lensing formula is testable. However, the significance is conditional: the central exactness claim is not verified in the manuscript, one headline bound is stated with opposite inequalities in different sections, and the displayed initial-condition formula appears inconsistent with the metric's null condition.

major comments (5)
  1. [Sec. II, Eq. (40)] The central claim that Eq. (40) is an exact Einstein-scalar solution is not demonstrated. The derivation relies on the assertion after Eq. (16) that, for the scalar-field action, the Ernst equations are unchanged and only □φ=0 is added, citing Refs. [92-94], and then applies the complex transformation (18). No direct substitution of the metric (40) and scalar field (39) into R_{μν}=∂_μφ ∂_νφ and □φ=0 is shown; the Ricci scalar in Eq. (41) only identifies curvature singularities. Because all shadow constraints in Sec. III are computed from this metric, the manuscript must provide an explicit field-equation verification (e.g., by computer algebra) or a precise proof that the transformation preserves the Einstein-scalar system in the (a,n,ν) region used for M87* and Sgr A*.
  2. [Abstract vs. Sec. III B and Sec. V] The abstract states that the VLTI bound on δ is fulfilled for n>0.34, while Sec. III B states that for the rotating FJNW metric the VLTI bound is satisfied for n≤0.34, and Sec. V repeats n≤0.34. These are opposite inequalities for a headline result. The correct bound, including the parameter values (spin, ν, inclination) at which it holds, must be stated consistently.
  3. [Eq. (8) and Eqs. (33)-(34), (39)] The seed scalar field in Eq. (8) is written ambiguously: it can be read either as sqrt((1-γ²-ν)/2) ln(1-2m/r) or as (1/2) sqrt(1-γ²-ν) ln(1-2m/r). For γ=1 these differ by a factor of sqrt(2), and only the first reading matches the coefficient c1=sqrt(-ν/2) used in Eqs. (33)-(34) and (39). Please write this expression unambiguously and, if the intended form is the second one, re-derive the subsequent scalar field, since the scalar charge enters the metric and all shadow predictions.
  4. [Eq. (46)] The initial condition for k^t_0 in Eq. (46) is written as a Euclidean norm of the spatial momentum components, with no metric coefficients. For the null condition g_{μν}k^μ k^ν=0 in the metric (40), k^t_0 must involve g_tt, g_tφ, g_rr, g_θθ, and g_φφ; the displayed formula is not the null condition except in flat spacetime. Since the shadow boundary is initialized from Eqs. (45)-(46), this should be corrected, or the actual relation used in the numerical code should be stated explicitly.
  5. [Sec. III A/B] The numerical bounds n<0.5, n<0.41, and n≤0.34 are quoted in the abstract and conclusions without specifying the full parameter tuple (spin a, scalar parameter ν, inclination, observer distance) or the error propagation from the EHT quantities in Eqs. (51) and (56). The figures show regions, but the text converts them into scalar inequalities; please state precisely for which parameter values each inequality holds and how the observational uncertainties are included.
minor comments (5)
  1. [Throughout] There are many typographical errors, including "T aub" in the title, "nessecay" and "dtermine" in Sec. III A, and "suppermassive" in Sec. V; a careful proofreading pass is needed.
  2. [Figs. 4-6] The axes labeled a* and n* are not defined in the captions; please define them (presumably dimensionless spin and NUT charge) and state the fixed values of m, ν, and inclination for each panel.
  3. [Ref. [24]] The reference lists "Harison" and should read "Harrison".
  4. [Sec. II, Eq. (41)] The Ricci scalar is used to locate singularities, but the text says "Due to the negativity of parameter ν"; please clarify the allowed sign and range of ν consistently with the conditions in Eq. (9), especially since ν is later used as a scalar-field parameter in the ray-tracing plots.
  5. [Sec. IV, Eq. (70)] The definitions of A and B in Eqs. (67) and (69) are stated without derivation; a short explanation of how they arise from the metric expansion would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the KTNS metric is derived by explicit Ernst transformations from published seed metrics, and the shadow constraints are comparisons with independent EHT data; self-citations are not load-bearing.

full rationale

The claimed derivation does not reduce to its inputs. The seed three-parameter scalar metric (4)-(8) is taken from the prior paper [20], and the assertion that the Ernst equations are unchanged in the presence of the scalar field is explicitly imported from external refs. [92-94], not from a self-referential loop; the new metric (40) is then obtained by explicit steps: the complex Ernst transformation (18)-(19), the reconstruction of omega' from Eqs. (23)-(27), the integrability condition (32) yielding the scalar field (33)-(34), and the integration for lambda' in Eq. (35) with the constant fixed by the n=0 and a=0 limits. No displayed equation in this chain is equal by construction to the final metric, and no fitted parameter is renamed as a prediction. The shadow analysis compares numerically ray-traced shadows with independent EHT observables (shadow size d ~ 11 +/- 1.5 for M87*, Delta C < 0.1, and the Keck/VLTI delta intervals for Sgr A*); the resulting NUT bounds are constraints imposed by external data, not outputs built into the model. The self-citations [20-24] provide seed solutions and limiting checks but are not the source of the central shadow constraints. The main weaknesses of the paper, namely the absence of a direct substitution of (40) into the field equations (2)-(3) and the imported Ernst-scalar assertion after Eq. (16), are correctness/falsifiability concerns rather than circular reductions; likewise the possible factor discrepancy between Eq. (8) and Eq. (39) is a consistency issue, not a circular one. Under the required standard, no circular step is exhibited.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The model has three free parameters (spin a, NUT charge n, scalar-field strength ν) plus the mass scale set by observations. The main axioms are the validity of the Ernst transformation in the presence of a minimally coupled scalar field, and the assumption that the shadow observables of these naked-singularity spacetimes can be compared directly to EHT images. No new particles or new forces are introduced; the NUT charge and scalar field are pre-existing concepts in the literature.

free parameters (3)
  • a (spin parameter) = scanned over 0 to about 1 in units of mass
    Rotation parameter entering metric (40); constrained only through the allowed regions in Figs. 4-6, not determined independently.
  • n (NUT charge) = reported bounds n<0.5 (M87*), n<0.41 (Keck), n<0.34 or n>0.34 (VLTI, text vs abstract)
    Gravomagnetic monopole charge; the paper's headline constraint. The exact bound depends on spin and ν and is stated inconsistently in different sections.
  • ν (scalar-field parameter) = scanned over 0, -1, -2, -3
    Controls the scalar hair through φ' and enters the metric through the factor [...]^ν in Eq. (40); not fixed by the theory and varied in the shadow plots.
assumptions (4)
  • domain assumption The Ernst equations remain valid for a minimally coupled massless scalar field, with only the scalar wave equation added.
    Invoked in Sec. II after Eq. (16) with citations [92-94]; if false, metric (40) is not a solution. This is the load-bearing premise of the derivation.
  • domain assumption The static three-parameter metric (4) with γ=1 is a valid seed solution of the Einstein-scalar equations.
    The derivation starts from this metric, taken from refs [20-24] by the same group.
  • domain assumption Shadow size and circularity of the KTNS spacetime can be compared directly to EHT images of M87* and Sgr A*.
    The paper does not require the spacetime to have an event horizon; it treats the 'singularity shadow' as the observable in Sec. III.
  • standard math The Gauss-Bonnet theorem and the OIA method correctly give the weak deflection angle for stationary axisymmetric spacetimes.
    Sec. IV uses the standard technique from refs [85,112]; this is a well-established method.

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Cite this review

Pith. "Pith review of Modelling the Sgr A$^*$ and M87$^*$ shadows by using the Kerr-Taub-NUT metrics in the presence of a scalar field." pith.science (2026). https://pith.science/paper/OTW6XKER

@misc{pith2026250111692,
  author       = {Pith},
  title        = {Pith review of: Modelling the Sgr A$^*$ and M87$^*$ shadows by using the Kerr-Taub-NUT metrics in the presence of a scalar field},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OTW6XKER}},
  note         = {Machine review of arXiv:2501.11692}
}
abstract

The recent unveiling of the images of Sgr A* and M87* has significantly advanced our understanding of gravitational physics. In this study, we derive a class of Kerr-Taub-NUT metrics in the presence of a scalar field (KTNS). Treating these metrics as models for supermassive objects, we constrain the parameters using shadow size estimates done by observations of M87* and Sgr A* from the Event Horizon Telescope (EHT). Comparing the obtained results with M87* data, we show an upper limit on the NUT charge $n$ such that the constraint on the shadow deviation from circularity ($ \Delta C $) will be fulfilled for $ n<0.5 $, and this allowed range changes with a variation in other parameters. Additionally, our findings reveal that fast-rotating KTNS metrics are better candidates for supermassive M87* than slowly rotating ones. We continue our study by estimating parameters using Keck and VLTI observations of Sgr A* and find that the constraint on the fraction deviation $ \delta $ is maintained within a certain range of the NUT charge such that the Keck bound is satisfied for $ n<0.41 $. In contrast, the VLTI bound can be fulfilled for $ n>0.34 $. Finally, we investigate weak gravitational lensing using the Gauss-Bonnet theorem and illustrate that all model parameters increase the deflection angle, causing light rays to deviate more significantly near fast-rotating KTNS objects.

Figures

Figures reproduced from arXiv: 2501.11692 by the authors.

Figure 1
Figure 1. The Ricci scalar function for a class of KTNS metrics with parameters [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. The admissible region (denoted by shaded areas) is displayed in (a): [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. The function R(ϕ) is characterized as the measurement of distance from every point along the boundary of the shadow to the center of that shadow. A. Testing corresponding singularities using the M87* observation In this subsection, we examine M87* within the framework of Kerr-Taub-NUT metrics in the presence of a scalar field, and we assess the limitations placed on the metric pa￾rameters through the application of … view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The density plot of the circularity deviation [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: The limitations imposed by the results of the EHT regarding the deviation of the [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: The limitations imposed by the results of the EHT regarding the deviation of the [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Deflection angle αpro (continuous curves) and αretro (dashed curves) vs impact parameter b for various values of a, n and ν. V. CONCLUSIONS Research in the field of imaging and the optical characteristics of massive objects has progressed significantly into an extensiv…

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Forward citations

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