Shadow constraints of charged black hole with scalar hair and gravitational waves from extreme mass ratio inspirals

Lai Zhao; Meirong Tang; Zhaoyi Xu

REVIEW 2 major objections 5 minor 114 references

Shadow constraints of charged black hole with scalar hair and gravitational waves from extreme mass ratio inspirals

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The LISA detector could measure scalar hair on a black hole at one part in ten thousand, far tighter than Event Horizon Telescope shadows.

desk verdict The numerical work is largely sound, but the central claim of separate scalar-hair and charge detection is undercut by an exact Q^2+s degeneracy in the metric and waveforms. read the letter →

arxiv 2506.04971 v2 pith:2NTGJMK6 submitted 2025-06-05 gr-qc

Lai Zhao , Meirong Tang , Zhaoyi Xu This is my paper

classification gr-qc

keywords Einstein-Maxwell-conformalcoupledscalarhairblackholeshadowEventHorizonTelescopeextrememassratioinspiralLISAwaveformmismatch

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 show that a specific black hole solution with scalar hair, the charged black hole with conformally coupled scalar field (CBH-SH) from Einstein-Maxwell-conformal coupling, leaves a measurable imprint in the gravitational waves of extreme mass ratio inspirals (EMRIs). The authors first use Event Horizon Telescope shadow measurements of M87* and Sgr A* to bound the two parameters of the solution, charge $Q/M$ and scalar hair $s/M^2$, finding constraints no better than the $10^{-1}$ level. They then build one-year EMRI waveforms with the analytic kludge method, compare against Schwarzschild and Reissner-Nordström templates using LISA's noise curve, and claim that LISA would distinguish scalar hair at $10^{-4}$ and charge at $10^{-2}$ for a $10^6\,M_\odot$ central black hole. If right, EMRI observations will test the underlying scalar gravity theory two to three orders of magnitude more strongly than shadow images do.

What carries the argument

The key object is the two-parameter metric function $f(r)=1-2M/r+(Q^2+s)/r^2$, which encodes the theory's deviations from general relativity in a single combination $Q^2+s$ in the $1/r^2$ term. Two derived quantities carry the argument: the photon sphere radius $r_{\rm ph}=\frac{1}{2}(3M+\sqrt{9M^2-8(Q^2+s)})$, which sets the shadow radius used for the EHT bounds, and the subleading $(Q^2+s)$ corrections to the quadrupole flux formulas, which accumulate into a one-year orbital dephasing. The AAK (augmented analytic kludge) waveform construction turns those flux differences into matched-filter mismatches against the standard templates, and the threshold $\mathcal{M}_{\rm th}=0.00875$ from LISA's signal-to-noise requirement sets the detectable parameter boundary.

What would settle it

Recompute the same one-year EMRI mismatch analysis with a higher-fidelity waveform model (e.g., gravitational self-force or higher-order post-Newtonian fluxes) for $p_0=15$, $e_0=0.1$, $M=10^6M_\odot$; if the CBH-SH versus Schwarzschild mismatch at $s/M^2=10^{-4}$ drops below 0.00875, the claimed LISA sensitivity would not hold.

Watch

Extended reading notes

Core claim

The paper's central claim is that the charged black hole with scalar hair derived from the Einstein-Maxwell-conformal coupled scalar theory can be distinguished from ordinary Schwarzschild and Reissner-Nordström black holes by LISA observations of extreme mass ratio inspirals. Its metric function, $f(r)=1-2M/r+(Q^2+s)/r^2$, reduces to Schwarzschild when $s=-Q^2$ and to Reissner-Nordström when $s=0$, so the parameter space spans both 'RN-like' and 'mutated RN' geometries. Using Event Horizon Telescope shadow radii for M87* and Sgr A*, the authors constrain $s/M^2$ to $0\le s/M^2\le 0.4632$ and $Q/M$ to $0\le Q/M\le 0.6806$ from M87*, and $0\ge s/M^2\ge -0.0277$ from Sgr A* when $Q\to0$, placing EHT constraints at the $10^{-1}$ level at best. Then, evolving one-year EMRI orbits with quadrupole fluxes and analytic kludge waveforms and comparing against Schwarzschild and Reissner-Nordström templates with LISA's noise, they find the waveform mismatch crosses the $\mathcal{M}_{\rm th}=0.00875$ detection threshold at $s/M^2\sim10^{-4}$ and $Q/M\sim10^{-2}$ for a $10^6M_\odot$ primary. This is the quantitative sense in which EMRI gravitational waves test the scalar gravity theory far more sharply than shadows do.

Load-bearing premise

The prediction rests on the approximate EMRI waveform model being accurate enough that a computed mismatch of 0.00875 really separates the scalar-hair black hole from the ordinary ones; if the waveform model itself carries comparable errors, the claimed LISA sensitivities would shift.

Editorial extensions

If this is right

For a $10^6\,M_\odot$ central black hole, LISA should bound scalar hair $s/M^2$ at the $10^{-4}$ level, about a thousand times tighter than the best EHT shadow constraint.
The charge parameter $Q/M$ should be bounded at the $10^{-2}$ level, an order of magnitude better than the shadow constraints.
Smaller central black hole masses improve the detection sensitivity, so EMRI events around lower-mass primaries are the preferred targets for scalar hair searches.
Initial orbital eccentricity has a weak effect on the mismatch, meaning the forecast applies across a range of plausible EMRI orbits.

Reading between the lines

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

If LISA indeed reaches these sensitivities, a measured non-zero $s$ would constitute direct evidence against the no-hair theorem, since the CBH-SH is an explicit scalar-hair counterexample.
The same mismatch pipeline could be applied to rotating black hole spacetimes, where spin-induced multipole moments will partially mimic the dephasing from $s$ and $Q$; disentangling them would require higher harmonics and longer waveforms.
Because the forecast uses quadrupole-flux orbital evolution, replacing it with self-consistent self-force fluxes is a natural testable extension that could shift the claimed $10^{-4}$ threshold in either direction.

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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

Honest read: the paper's math is mostly fine, but the headline conclusion overreaches. The metric (8) depends on Q and s only through q2 ≡ Q^2+s. Photon sphere, shadow, orbital energy, frequencies, and fluxes—everything in the EMRI waveform—depends only on that sum. So LISA cannot distinguish scalar hair from charge: (Q=0, s=10^-4) and (Q=10^-2, s=0) are the same gravitational-wave signal. The claimed O(10^-4) sensitivity to s/M^2 and O(10^-2) to Q/M require an unstated prior, say Q=0. The EHT constraints in Fig. 3 have the same issue: the 1σ regions are strips of constant q2, so the quoted ranges are projections, not independent bounds. The paper never acknowledges this exact degeneracy, so the central claim of separately testing scalar hair is not supported.

Credit where it is due: the geodesic and shadow derivations are clean, the AAK mismatch computation is a reasonable first-pass sensitivity estimate for the combination q2, and the order-of-magnitude numbers (q2/M^2 ~ 10^-4 for LISA, ~10^-1 for EHT) are new for this specific CBH-SH metric. That is genuinely useful for the EMRI and modified-gravity communities, as long as it is read as a bound on q2.

The weaker spots, in order. First, the degeneracy above is the load-bearing problem; without a prior or a second observable, the 'scalar hair detection' claim is just a re-labeling of the combined-parameter sensitivity. Second, the AAK kludge uses weak-field quadrupole fluxes and is not checked against a higher-fidelity model; that is a minor issue for an order-of-magnitude forecast, and the D/2ρ^2 threshold with D=7 is itself rough. Third, the manuscript has serious LaTeX contamination (Chinese placeholder text) and a sign typo: the Schwarzschild limit is s=-Q^2, not 'Q^2=s' as written.

Bottom line: this paper deserves a serious referee, but it needs major revision—reframe the conclusions in terms of q2, or bring in a prior—and a cleanup before it can be published. If I were an editor, I would send it to review rather than desk reject, because the underlying calculation is honest and the sensitivity numbers are the kind of thing the EMRI community will want to cite.

Referee Report

2 major / 5 minor

Summary. The paper studies the charged black hole with scalar hair (CBH-SH) of Einstein–Maxwell–conformal-scalar theory, with metric function f(r)=1−2M/r+(Q^2+s)/r^2. It derives the photon-sphere and shadow radius, uses EHT measurements of M87* and Sgr A* to constrain the (Q/M, s/M^2) parameter space, and then constructs EMRI waveforms with the AAK kludge method driven by Peters–Mathews quadrupole fluxes. Comparing CBH-SH waveforms to Schwarzschild and Reissner–Nordström templates, it evaluates waveform mismatches against LISA's expected sensitivity and concludes that LISA can detect scalar hair at O(10^-4) and charge at O(10^-2), two orders of magnitude better than EHT shadow constraints.

Significance. The geodesic, shadow, and flux derivations are explicit and self-contained, and the mismatch pipeline is transparent: the paper provides concrete analytic expressions for the photon sphere, the shadow radius, the orbital energy/angular momentum, the secular frequencies, and the leading post-Newtonian fluxes. If the claimed two-parameter sensitivity were valid, this would be a valuable forecast for testing EMCS theory with future LISA data. However, the paper's central quantitative claim is undermined by an exact degeneracy: every computed observable depends only on q2 = Q^2 + s, so the EHT and LISA results constrain only this combination. The advertised separate sensitivities to s/M^2 and Q/M are projections under implicit priors, not independent measurements. The underlying q2 sensitivity, once stated correctly, remains a useful and credible forecast.

major comments (2)

[III, IV.B] The observables depend only on q2 = Q^2 + s. The metric function (8), photon-sphere radius (23), shadow radius (22), orbital energy and angular momentum (29)-(30), frequencies (33)-(34), and Peters–Mathews fluxes (37)-(38) all contain only Q^2+s; the AAK waveforms (46)-(50) are built exclusively from these quantiies. Therefore the EHT 1σ regions in Fig. 3 and the mismatch maps in Figs. 5-8 constrain only q2/M^2. The quoted ranges 0≤s/M^2≤0.4632 and 0≤Q/M≤0.6806 from M87* are projections rather than independent bounds, and similarly Sgr A*'s quoted negative-hair range is a projection. The claimed LISA sensitivities s/M^2 ~ 10^-4 and Q/M ~ 10^-2 are the same statement about q2/M^2 ~ 10^-4 under the priors Q=0 and s=0; indeed (Q=0, s=10^-4 M^2) and (Q=10^-2 M, s=0) produce exactly identical waveforms and identical shadow radii. The paper must be reframed around the measurable combination q2, with separate s and Q sensitivities stated only under explicit priors.
[IV.A, IV.B] The LISA detection forecast is computed with the AAK kludge using quadrupole-order Peters–Mathews fluxes (37)-(38) and Barack–Cutler harmonic amplitudes (48)-(50), but the accuracy of this kludge is not validated against a higher-fidelity EMRI model. The mismatch threshold M_th = D/(2ρ^2) = 0.00875 is reached through phase accumulation over roughly 1.5×10^4 cycles for M=10^6 M_sun, and a fractional flux error of order (Q^2+s)/p ~ 10^-6 is comparable to the effect being measured. Such an error could shift the apparent detection boundary in Figs. 5-8. The paper's own conclusion states that only preliminary parameter estimates are possible, yet the headline numbers are presented without an associated systematic uncertainty. A calibration of the kludge fluxes against a self-consistent EMRI trajectory or a discussion of the resulting error budget is needed before the O(10^-4)/O(10^-2) sensitivities can be taken as quantitative forecasts.

minor comments (5)

[IV.A, Fig. 5] The text repeatedly identifies s=Q^2 as the Schwarzschild limit. From Eq. (8), Schwarzschild requires Q^2+s=0, i.e. s=-Q^2 (for Q≠0); the dark-blue low-mismatch regions near the boundary of the forbidden zone correspond to s≈-Q^2, not s≈Q^2. This sign error should be corrected in the main text and in the discussion of Figure 5.
[Fig. 5 and Section III] The statement that the physically forbidden region satisfies '0>s>Q^2' is not an interval because Q^2≥0. The scalar field is imaginary for -Q^2 < s < 0 (with the boundary case at s=-Q^2 needing separate treatment), consistent with Region III defined in Section II; the captions and text should use this interval.
[IV.B] The choice D=7 for the number of independent variables in the mismatch threshold is not derived or referenced explicitly enough. If the CBH-SH model adds the parameter q2 relative to the Schwarzschild template, the effective number of independent parameters should be justified, and the sensitivity of the threshold to this choice should be discussed.
[II, Eq. (9)] Equation (9) is rendered unclearly; the scalar field expression should be written with an explicit denominator s+Q^2 and a discussion of its reality conditions, since the division of the (Q,s) plane into Regions III and IV depends on the sign of s/(s+Q^2).
[Captions of Figs. 4-8] Several figure captions contain Chinese placeholder text (e.g., '图 8:两行三列并排图片示例', '并排图片排列示例') that appears to be leftover LaTeX placeholder material. These should be removed before submission.

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed separate LISA sensitivities for scalar hair and charge reduce to the single combination Q^2+s; the quoted 10^-4 hair and 10^-2 charge levels are two labels for the same q2 sensitivity.

other [Sec. II Eq. (8); Sec. III Eq. (23) and Fig. 3; Sec. IV.B Figs. 5-8 and Abstract]
"f(r) = 1− 2M/r + (Q^2 +s)/r^2 ... rph = 1/2(3M+sqrt(9M^2−8Q^2−8s)) ... [T]he results indicate that for central BHs of M=10^6 M_sun, LISA is expected to detect scalar hair s/M^2 at the O(10^-4) level and charge Q/M at the O(10^-2) level."

The metric function contains Q and s only as q2=Q^2+s. The photon-sphere radius (23), orbital energy/angular momentum (29)-(30), frequencies (33)-(34), and Peters-Mathews fluxes (37)-(38) all depend only on Q^2+s; the AAK waveform inherits this dependence. Consequently the mismatch M(ha,hb) against Schwarzschild or RNBH is a function of q2 alone: the parameter pair (Q=0, s=10^-4) and (Q=10^-2, s=0) produce the identical f(r) and identical waveform. The quoted 'scalar hair s/M^2 at O(10^-4)' and 'charge Q/M at O(10^-2)' are therefore two readings of the same q2=10^-4 sensitivity, since (10^-2)^2=10^-4. The separate detection claims are the q2 detection relabeled, not independent outputs of the calculation.

full rationale

The paper's core computation is self-contained: the shadow radius formula is compared with external EHT radii, and the LISA forecast is a theoretical sensitivity estimate built from the metric, timelike geodesics, Peters-Mathews fluxes, and the AAK waveform, with no parameter fitted to the target observables. The self-citations (refs. [62,64,65]) are methodological context and are not load-bearing for the derivation. However, the central claim of separately detectable scalar hair and charge is not supported by the calculation: every derived quantity entering the waveform depends only on q2=Q^2+s, so the mismatch contours are level sets of q2. The paper's O(10^-4) hair sensitivity and O(10^-2) charge sensitivity are the same q2 sensitivity expressed in two variables, and the EHT constraint intervals are projections of one degenerate contour. This is a reduction-by-construction of the separate parameter claims rather than an independent two-parameter prediction. The AAK waveform fidelity is a separate accuracy concern, not circularity.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The central claims rest on the validity of the CBH-SH solution, the applicability of EHT shadow-radius values to a static model, and the accuracy of the AAK/Peters-Mathews waveform approximation. The chosen EMRI parameters (masses, p0, e0, distance, duration) and the threshold parameters (D=7, SNR=20) are hand-picked and affect the forecast.

free parameters (8)

central BH mass M = 1e6 M_sun
Chosen as representative supermassive BH for LISA EMRI; the sensitivity forecast depends on this choice.
small CO mass m = 10 M_sun
Chosen as a typical stellar-mass compact object; affects the signal amplitude and inspiral rate.
initial semi-latus rectum p0 = 15 M
Sets the initial orbit size; higher p reduces the effect of the hair because (Q^2+s)/p is smaller.
initial eccentricity e0 = 0.1
Low eccentricity chosen; the paper says eccentricity has weak influence, but it still enters the flux and harmonics.
luminosity distance DL = 1 Gpc
Sets the waveform amplitude and hence SNR; closer sources would be easier to detect.
integration time T = 1 year
The mismatch accumulates with observation time; longer T improves sensitivity.
number of independent parameters D = 7
Used in the mismatch threshold M_th = D/(2ρ^2); a different D changes the detection boundary.
LISA SNR threshold ρ = 20
Standard LISA design requirement; varies the mismatch threshold quadratically.

assumptions (6)

domain assumption The CBH-SH metric is a valid solution of EMCS theory (Eq. 8 from Astorino 2013).
The paper adopts the exact solution from [40] without re-deriving it; if this solution is not physically applicable, all constraints are void.
standard math The shadow radius at infinity equals the critical impact parameter b_c = r_ph/sqrt(f(r_ph)).
Standard for static spherically symmetric metrics; ignores photon ring and finite-distance corrections.
domain assumption EHT shadow radius measurements (M87*: 5.5±0.75 M, Sgr A*: 4.885±0.335 M) can be applied to this static model.
The observed values are for rotating black holes; using them for a non-rotating model is an approximation.
domain assumption The Peters-Mathews quadrupole formula (Eqs. 35-38) accurately gives the gravitational wave energy and angular momentum losses for the CBH-SH background.
This ignores conservative self-force, non-quadrupole radiation, and the hair's effect on the emission.
domain assumption The AAK waveform generation (refs [98-100]) is accurate enough to compute mismatch at the 0.00875 level.
No validation of AAK for this metric is provided; AAK is a kludge designed for Kerr, not for exotic charged-hairy backgrounds.
domain assumption The LISA noise power spectral density (refs [110,111]) represents the detector sensitivity.
The forecast depends on the assumed LISA sensitivity curve.

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Cite this review

Pith. "Pith review of Shadow constraints of charged black hole with scalar hair and gravitational waves from extreme mass ratio inspirals." pith.science (2026). https://pith.science/paper/2NTGJMK6

@misc{pith2026250604971,
  author       = {Pith},
  title        = {Pith review of: Shadow constraints of charged black hole with scalar hair and gravitational waves from extreme mass ratio inspirals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2NTGJMK6}},
  note         = {Machine review of arXiv:2506.04971}
}

abstract

Black hole (BH) shadow observations and gravitational wave astronomy have become crucial approaches for exploring BH physics and testing gravitational theories in extreme environments. This paper investigates the charged black hole with scalar hair (CBH-SH) derived from the Einstein-Maxwell-conformal coupled scalar (EMCS) theory. We first constrain the parameter space $(Q/M, s/M^2)$ of the BH using the Event Horizon Telescope (EHT) observations of M87* and Sgr A*. The results show that M87* provides stronger constraints on positive scalar hair, constraining the scalar hair $s$ within $0\le s/M^2\le0.4632$ and the charge $Q$ within the range $0\le Q/M\le0.6806$. In contrast, Sgr A* imposes tighter constraints on negative scalar hair. When $Q$ approaches zero, $s$ is constrained within the range $0\geq s/M^2\geq-0.0277$. Overall, EHT observations can provide constraints at most on the order of $\mathcal{O}\left({10}^{-1}\right)$. Subsequently, we construct extreme mass ratio inspiral (EMRI) systems and calculate their gravitational waves to assess the detection capability of the LISA detector for these BHs. The results indicate that for central BHs of $M={10}^6M_\odot$, LISA is expected to detect scalar hair $s/M^2$ at the $\mathcal{O}\left({10}^{-4}\right)$ level and charge $Q/M$ at the $\mathcal{O}\left({10}^{-2}\right)$ level, with detection sensitivity far exceeding the current EHT capabilities. This demonstrates the immense potential of EMRI gravitational wave observations in testing EMCS theory.

Figures

Figures reproduced from arXiv: 2506.04971 by the authors.

**Figure 2.** FIG. 2: The effective potential of photon motion in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3: EHT observational constraints on the parameter space of the CBH-SH. The left panel shows the constraints [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 7.** Figure 7: FIG. 7: The waveform mismatch between the CBH-SH and the RNBH ( [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

**Figure 8.** Figure 8: FIG. 8: The mismatch between CBH-SH and the RNBH ( [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]

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