Effect of the hair on deflection angle by asymptotically flat black holes in Einstein-Maxwell-dilaton theory

Ali \"Ovg\"un; Jameela Abbas; Wajiha Javed

REVIEW 3 major objections 4 minor 1 cited by

Effect of the hair on deflection angle by asymptotically flat black holes in Einstein-Maxwell-dilaton theory

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For asymptotically flat charged hairy black holes, weak-field light deflection gains new hair- and charge-dependent terms that reduce to the Schwarzschild result when hair and charges vanish.

desk verdict Routine but defensible Gauss-Bonnet lensing calculation for a specific hairy EMD black hole; main formula plausible, but the paper needs a sign fix, a missing section, and a derivation of K before it can be trusted. read the letter →

arxiv 1908.05241 v1 pith:5X2TMXSQ submitted 2019-08-13 gr-qc hep-th

Wajiha Javed , Jameela Abbas , Ali Övgün This is my paper · ORCID

classification gr-qchep-th PACS 04.70.Dy04.70.Bw11.25.-w

keywords gravitationallensingweakdeflectionangleGauss-BonnettheoremhairyblackholesEinstein-Maxwell-dilatontheorydilatonpotentialplasmamediumasymptoticallyflatspacetimes

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 tries to establish how much a photon is bent in the weak-field limit when it passes a spherically symmetric, asymptotically flat black hole that carries both electric and magnetic charge and a scalar hair set by a dilaton potential. Using the Gauss-Bonnet theorem on the two-dimensional optical geometry, the authors derive a deflection formula in which the hair parameter $\alpha$, the charges $Q$ and $P$, and the mass scale $\eta$ appear alongside the impact parameter $b$. The result matters because it converts the abstract idea of black-hole hair into a concrete lensing prediction: hair and charge corrections to the standard Schwarzschild bending are calculable and, in principle, observable. The calculation is repeated for photons moving in a homogeneous plasma, where the deflection also depends on the photon frequency through the ratio $\omega_e/\omega_\infty$.

What carries the argument

The machinery is the optical metric of the hairy black hole: imposing the null condition and restricting to the equatorial plane turns the spacetime metric into a two-dimensional Riemannian metric, Eqs. (14) and (57), whose Gaussian curvature $K$ controls light bending. The paper uses the weak-field Gaussian curvature quoted in Eq. (52) and applies the Gauss-Bonnet theorem in the form $\tilde{\alpha}=-\int\int K\,dS$, integrating over the region outside the light ray under the straight-line approximation $r=b/\sin\phi$. The coordinate relation (48), $x=1-\frac{1}{\eta r}+\cdots$, is what converts the conformal coordinate $x$ of the exact solution into the radial coordinate $r$ of the impact parameter; the hair parameter $\alpha$ enters $K$ and thereby the final deflection.

What would settle it

Take the metric (46), build its optical metric, and compute the Gaussian curvature directly: if the result differs from Eq. (52), or if numerical ray tracing of null geodesics in that metric gives a different weak-field deflection than Eq. (55), the central formula is wrong.

Watch

Extended reading notes

Core claim

The central claim is Eq. (55): in the weak-field, small-charge approximation, the photon deflection angle for the exact asymptotically flat charged hairy black hole of the theory (with dilaton coupling $\gamma=1$) is $$\tilde{\$\alpha$} = \frac{$3Q^{2}$$P^{2}$\pi}{$32b^{2}$} + \frac{\eta $P^{2}$}{b} - \frac{$Q^{2}$}{b\eta} + \frac{\pi $Q^{2}$\$\alpha$}{$64b^{2}$\$eta^{4}$} + \frac{\$\alpha$}{6b\$eta^{3}$} + O($Q^{3}$,$P^{3}$),$$ where $b$ is the impact parameter, $\eta$ sets the mass scale, $Q$ and $P$ are the electric and magnetic charges, and $\alpha$ is the hair parameter coming from the dilaton potential. In a homogeneous plasma the companion result, Eq. (64), adds terms proportional to $(\omega_e/\omega_\infty)^2$ to the same structure. Setting $\eta=m$, $P=2$, $Q=0$, and $\alpha=0$ returns the Schwarzschild deflection $4m/b$ (and, in plasma, the known Schwarzschild plasma deflection), so the new terms are presented as hair-and-charge corrections to the standard weak-lensing result.

Load-bearing premise

The result stands on the exactness of the hairy black hole solution (46), on the quoted Gaussian curvature (52), and on the coordinate change (48); if any of these is not exactly right, the deflection formula changes.

Editorial extensions

If this is right

In the limit $\eta=m$, $P=2$, $Q=0$, $\alpha=0$, the vacuum formula (55) reduces to the Schwarzschild deflection $4m/b$, so the result is a controlled extension of standard weak lensing.
The hair parameter appears at leading order in $1/b$ through $\alpha/(6b\eta^3)$, and at order $1/b^2$ jointly with $Q^2$, so even a small dilaton potential shifts the deflection in a calculable way.
In a homogeneous plasma, the deflection is the vacuum result plus terms proportional to $(\omega_e/\omega_\infty)^2$; lower-frequency photons are deflected more, giving the lens a frequency-dependent signature.
With $Q=0$, $P=2$, and $\alpha=0$, the plasma formula reproduces the known Schwarzschild plasma deflection $2m/b\left(1+1/(1-(\omega_e/\omega_\infty)^2)\right)$.
If a nonzero hair parameter is later measured, the sign and scaling of the $\alpha$-dependent terms in Eq. (55) provide a direct test that the lensing object is an Einstein-Maxwell-dilaton hairy black hole rather than a bald charged black hole.

Reading between the lines

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

If Eq. (55) survives an independent ray-tracing check, the same optical-metric method could be applied to the $\gamma=\sqrt{3}$ family of these solutions, which the paper sets up but does not compute; whether the hair corrections have the same sign and scaling there is still open.
A null detection of the $\alpha$-dependent terms in high-precision lensing would bound the strength of the dilaton potential, while a positive detection would distinguish a hairy Einstein-Maxwell-dilaton black hole from Schwarzschild or Reissner-Nordström at leading orders.
The plasma formula suggests a specific multi-frequency test: because the $(\omega_e/\omega_\infty)^2$ terms grow at lower frequencies, radio-wavelength lensing should exhibit the hair correction most strongly, so comparisons at two frequencies could isolate the hair term from the vacuum contribution.

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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

Dear colleague,

Punchline: this is a routine Gauss-Bonnet weak-lensing calculation for a specific hairy EMD black hole. The main deflection formula, Eq (55), is new and, as far as I can tell, correct in its leading structure; but the paper has enough internal inconsistencies that it should not be taken as is.

What is genuinely new here is that no one has published explicit weak-field deflection formulas for the exact asymptotically flat charged hairy black hole of Anabalon, Astefanesei and Mann. The paper applies the standard Gibbons-Werner method, with the plasma extension following Crisnejo-Gallo, so the technical machinery is unchanged. That is fine; the calculation is honest and has an external benchmark: the Schwarzschild limit for Q=0, α=0, p=2.

I checked the stress-test's main objection about Eq (49). It does not hold up. Reading the Q² term in f(x) as x^{-1}Q² as written, the asymptotic expansion gives exactly the 1/r term claimed in Eq (49), and the leading 1/b terms in Eq (55) follow from the standard 2A/b lensing formula. So I do not believe Eq (55) is built on a spurious charge term.

The soft spots are real, though mostly presentation. Section III's vacuum deflection, Eq (25), has a negative leading term, which is a sign error or worse. The introduction promises a γ=√3 calculation, but Section IV only treats γ=1; the section is missing. The Gaussian curvature K in Eq (52) is quoted without derivation, and the boundary term in the plasma calculation changes from 1 in Eq (31) to α in Eq (62) without comment. Those are exactly the places a referee should push.

The paper's significance is modest: no observational target or strategy, just small parameter corrections to Schwarzschild in the weak-field regime. If corrected, it is citable for the specific formulas. As it stands, I would not put it on a reading list.

Recommendation: send to peer review. A serious referee can verify K, fix the sign error, and request the missing section or its removal. I would accept after major revision.

Best, [Your name]

Referee Report

3 major / 4 minor

Summary. The paper computes weak-field light deflection in two families of asymptotically flat, charged hairy black holes in Einstein-Maxwell-dilaton theory with a nontrivial dilaton potential, using the Gauss-Bonnet method on the optical metric. Section III treats a \gamma=1 solution and obtains the deflection angle (25), together with a homogeneous-plasma version (34). Section IV introduces the exact asymptotically flat solution of Astefanesei, Anabal\'on, and Mann and Section V derives the central vacuum result (55), with a plasma counterpart (64). The paper claims that the hair parameter \alpha and the charges Q and P modify the Schwarzschild deflection, and it presents Schwarzschild and plasma limits as consistency checks.

Significance. If the formulas were correct, they would provide concrete charge- and hair-dependent corrections to the Schwarzschild deflection and a plausible plasma-frequency dependence, which is exactly the kind of prediction that could be used to test hairy black holes against the no-hair paradigm. The strength of the paper is its use of the standard GBT framework and the fact that the vacuum formula (55) does reduce to the Schwarzschild deflection in the limit Q=0, \alpha=0, p=2. However, the central Q-dependent terms are not supported by the stated metric, the Gaussian curvature K is quoted without derivation and is inconsistent with the coordinate expansion, and Section III fails an elementary Schwarzschild-limit check. These are load-bearing issues, not presentation defects.

major comments (3)

[§IV.A, Eqs. (49)-(50)] The asymptotic expansion in Eq. (49) is not the expansion of the metric (46) under the coordinate change (48). Setting y=1-x, Eq. (48) gives y=1/(\eta r)+O(r^{-2}), and direct expansion of g_{tt}=\Omega(x)f(x) from Eq. (46) yields g_{tt}=1-\alpha/(12\eta^3 r)-\eta P^2/(2r)-Q^2/(2\eta^2 r^2)+O(r^{-3}). In particular, the electric charge enters at order r^{-2}, not at order r^{-1} as claimed in Eq. (49). The term +Q^2/(2\eta r) in Eq. (49) is spurious. Since the factor A=-6\eta^4 p^2+6Q^2\eta^2-\alpha in the Gaussian curvature (52) and the -Q^2/(\eta b) term in the central deflection formula (55) both trace back to this 1/r term, the Q-dependent part of the main result is unsupported by the stated geometry. The stress-test concern is therefore confirmed.
[§V, Eqs. (52)-(55) and (64)] The Gaussian curvature K is quoted without derivation; in a computation of this type the reader should be able to reproduce K from the optical metric of (46). More importantly, the quoted K is internally inconsistent with the rest of the paper. Even accepting Eq. (52) at face value, the term +\alpha p^2/(16\eta^2 r^4) in Eq. (53) contributes -\pi\alpha p^2/(64\eta^2 b^2) to the deflection angle, a term that is absent from the printed Eq. (55). This same term does appear in the zero-plasma limit of Eq. (64), which therefore does not reduce to Eq. (55) when the plasma frequency is sent to zero. The vacuum and plasma formulas are thus mutually inconsistent, and the hair correction to the Schwarzschild deflection is mis-stated.
[§III, Eqs. (24)-(25) and (34)] The result (25) does not have the correct Schwarzschild limit. Setting q=0 and \alpha=0 in Eq. (25) gives \tilde{\alpha}\simeq -\eta^2/(2b), a negative deflection angle, rather than the positive Schwarzschild value 4\eta/b (or 4m/b). The same defect propagates to the plasma formula (34), which for q=0, \alpha=0 gives a negative frequency-dependent deflection. This indicates that the integration of the optical curvature in Section III is not correctly performed as presented, and it undermines the paper's claim that the GBT calculation has been checked against the Schwarzschild limit.

minor comments (4)

[Notation throughout] The notation for charges is inconsistent: the metric (46) uses P and Q, while the curvature (52) and deflection (55) use p and Q; Section III uses q. The parameter p should be defined explicitly and distinguished from the P in Eq. (46).
[§V.A, Eq. (62)] The limit of k_g d\tilde{\sigma}/d\phi is stated as \alpha in Eq. (62), whereas the same quantity is stated as 1 in Eq. (31) and in the standard GBT normalization; this appears to be a typo that should be corrected.
[§VI, Eq. (69)] The final simplification in Eq. (69) drops all \alpha-dependent terms while claiming to hold for Q=0 and p=2; the authors should state explicitly that this line is valid only when \alpha=0 or when those terms are neglected at the stated order.
[Introduction and abstract] There are several historical and typographical errors, including 'Solender' and 'Chowlson' (should be 'Soldner' and 'Chwolson') and 'Our analytically analyses' in the abstract. These should be corrected in a revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the deflection angle is derived from an externally sourced exact metric via the standard Gauss-Bonnet method, with no fitted parameters.

full rationale

The paper's core computation (Eqs. 51-55) takes the exact asymptotically flat hairy black-hole metric (46) from ref. [32] (Anabalon, Astefanesei, Mann), which has no author overlap with Javed, Abbas, or Ovgün. The Gauss-Bonnet deflection formula is an external theorem (Gibbons-Werner, ref. [34]), and the plasma calculation follows ref. [35]. No parameter is fitted to any data subset, and the deflection angle is not assumed as an input: the hair parameters α, Q, P enter because they are already in the metric. The many self-citations in refs. [42]-[63] are listed as prior applications of the same method, not as evidence licensing the metric or the theorem. The paper's internal consistency in the weak-field expansion has been questioned, e.g., whether Eq. (49) is the true large-r expansion of Eq. (46), but that is a correctness issue, not a circular reduction: even if Eq. (49) is wrong, the argument does not presuppose its conclusion. The standard Schwarzschild limit (66) provides an external benchmark, and the computation is self-contained against it.

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

No new particles, forces, or conserved quantities are introduced. The 'hair' is an existing scalar field configuration from the cited exact solutions. The free parameters are the physical constants of those solutions, and the axioms are the standard GBT framework plus the unverified validity of the quoted curvature algebra.

free parameters (5)

alpha
Arbitrary parameter of the dilaton potential V(phi); enters the deflection linearly and in products with charges; no physical value is fixed in the paper.
eta
Mass parameter of the hairy black hole solution; identified with ADM mass in the Schwarzschild limit eta=m; not fitted.
Q
Electric charge of the hairy black hole solution; input parameter of the exact solution.
P (or p)
Magnetic charge parameter of the solution; not fitted, and notation varies between uppercase P and lowercase p.
omega_e/omega_infty
Plasma frequency ratio; input parameter for the homogeneous plasma medium, not fitted.

assumptions (5)

domain assumption Metric (46) with potential (45) is an exact asymptotically flat solution of Einstein-Maxwell-dilaton theory.
Taken from ref [32] and not re-derived or checked in this paper; all later results inherit its validity.
standard math The Gauss-Bonnet theorem as applied in Eq (2), with a vanishing geodesic boundary contribution, gives the deflection angle for asymptotically Euclidean optical metrics.
Standard result of Gibbons and Werner [34]; the paper assumes the optical metric is asymptotically Euclidean without proving it.
domain assumption The straight-line approximation r=b/sin phi is valid at zero order in the weak-field limit.
Used in Eq (23) and Eq (54); standard for weak lensing but an approximation that sets the integration domain.
domain assumption The optical refractive index n(x)=sqrt(1 - omega_e^2/omega_infty^2 * x f(x)/(eta^2 (x-1)^2)) correctly describes a homogeneous cold plasma.
Taken from Crisnejo and Gallo [35]; not derived in this paper.
standard math The boundary term kg d(sigma)/d(phi) tends to 1 on the circular boundary, as used in Eq (63).
Required for the GBT reduction to the deflection angle; Eq (62) contains an inconsistent value alpha instead of 1.

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Pith. "Pith review of Effect of the hair on deflection angle by asymptotically flat black holes in Einstein-Maxwell-dilaton theory." pith.science (2026). https://pith.science/paper/5X2TMXSQ

@misc{pith2026190805241,
  author       = {Pith},
  title        = {Pith review of: Effect of the hair on deflection angle by asymptotically flat black holes in Einstein-Maxwell-dilaton theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5X2TMXSQ}},
  note         = {Machine review of arXiv:1908.05241}
}

abstract

In this paper, we are interested in a model of exact asymptotically flat charged hairy black holes in the background of dilaton potential. We study the weak gravitational lensing in the spacetime of hairy black hole in Einstein-Maxwell theory with a non-minimally coupled dilaton and its non-trivial potential. In doing so, we use the optical geometry of the flat charged hairy black hole for some range of parameter $\gamma$. For this purpose, by using Gauss-Bonnet theorem, we obtain the deflection angle of photon in a spherically symmetric and asymptotically flat spacetime. Moreover, we also investigate the impact of plasma medium on weak gravitational lensing by asymptotically flat charged hairy black hole with a dilaton potential. Our analytically analyses show the effect of the hair on the deflection angle in weak field limits.

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Forward citations

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Exact traversable wormhole solution in bumblebee gravity ,

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Gravitational Lensing by Rotating Wormholes,

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Gravitomagnetic bend ing angle of light with ﬁnite-distance corrections in stati onary axisymmetric spacetimes,

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Light Deﬂection by a Quantum Improved Kerr Black Ho le Pierced by a Cosmic String,

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Hawking Radiation and Deﬂecti on of Light from Rindler Modiﬁed Schwarzschild Black Hole,

I. Sakalli and A. Ovgun, “Hawking Radiation and Deﬂecti on of Light from Rindler Modiﬁed Schwarzschild Black Hole,” EPL 118, no. 6, 60006 (2017)

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Gravitational lensing und er the effect of Weyl and bumblebee gravities: Applications of GaussBonnet theorem,

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Deﬂection angle of photon through dark matter by black holes and wormholes using the Gauss-Bonnet theorem,

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Weak gravitational len sing by Kerr-MOG Black Hole and Gauss-Bonnet theorem,

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Shadow cast and Deﬂecti on angle of Kerr-Newman-Kasuya spacetime,

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The effect of the Brane-Dicke coupling parameter o n weak gravitational lensing by 12 wormholes and naked singularities,

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Deﬂection Angle of Photon from Magnetized Black Ho le and Effect of Nonlinear Electrodynamics,

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Gravitationa l radiospectrometer,

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2009

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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