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REVIEW 4 major objections 5 minor 25 references

Image of the time-dependent black hole

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A black hole with linearly growing mass produces an image whose brightness peak drifts outward while its angle stays fixed.

desk verdict Solid ray-tracing extension of the conformal Vaidya shadow work, but the time-evolution claim rests on an unjustified flux formula and a coordinate artifact. read the letter →

arxiv 2507.14565 v1 pith:UIM2ZRCB submitted 2025-07-19 gr-qc

Yu Liang , Sen Guo , Kai Lin , Yu-Hao Cui , Yu-Xiang Huang , Tao Yu This is my paper · ORCID
classification gr-qc MSC 83C5783C10
keywords Vaidyablackholeimagetime-dependentspacetimethinaccretiondiskraytracingconformaltransformationphotonsphereobservedflux
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 aims to show that a black hole whose mass grows linearly in time has an image that evolves: as a newly defined time coordinate $t_c$ increases, the observed shadow expands and the brightest part of the accretion-disk image drifts radially outward while keeping its angular position. The argument maps the time-dependent Vaidya metric through a conformal transformation to a static-looking metric, solves the null geodesics and ray-traced images there, and then transforms back to physical coordinates. If correct, the result gives a template for interpreting time-varying black hole images, such as the shifting brightness asymmetry reported for M87*, and separates variability that comes from the spacetime itself from variability that must come from the emitting source.

What carries the argument

The central machinery is the conformal transformation $v=r_0\Theta$, $r=R\Theta$ with $\Theta=e^{\nu_c/r_0}$, which turns the ingoing Vaidya metric with $m(v)=\mu v$ into a metric conformal to a static spacetime, together with the new time coordinate $t_c$. In these conformal coordinates the photon trajectories are controlled by the single function $G(U)=2M_0U^3-U^2+2U/r_0+1/b^2$, so the impact parameter $b$ organizes the rays into direct, lensed, and photon-ring images just as in static ray tracing. The Page-Thorne flux formula, applied to the conformal spacetime, gives an emitted flux $I(t_c,R)=I_0(R)e^{-4t_c}$ and a redshift factor $1+z=(1+z_0)e^{-t_c}$, so the observed flux $I_{\rm obs}=I_0/(1+z_0)^4$ is $t_c$-independent in conformal coordinates; time dependence re-enters only through the back-transformation, where the observer's conformal position $R_{\rm obs}$ moves toward $R_-$ as $t_c$ grows.

What would settle it

Solve the null geodesic equation directly in the original ingoing Vaidya coordinates $(v,r)$ and compute the thin-disk image using a non-stationary emission model derived from local energy-momentum conservation; if the peak flux stays at a fixed image radius for all $v$, or rotates instead of drifting, the paper's central claim is refuted.

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Extended reading notes

Core claim

The central claim is that, for a Vaidya black hole with mass $m(v)=\mu v$, an observer fixed at $r_{\rm obs}=10^5$ in the $t_c-r$ frame sees the thin-disk image change with $t_c$: the shadow grows, and the facula, the brightness peak, moves outward in the radial direction by an amount controlled by $\mu$, while its azimuthal position remains unchanged. This happens because the conformal factor $\Theta=e^{\nu_c/r_0}$ converts the mass increase into a time-dependent radial-coordinate relationship, so a physically fixed observer corresponds to an observer who gradually approaches the inner conformal Killing horizon in the conformal frame. In that frame the observed flux is independent of $t_c$; the time dependence reappears only after transforming back to the physical $(t_c,r)$ coordinates. The paper offers this as a first theoretical framework for time-dependent black-hole imaging and notes that for small $\mu$ the radial drift is slow, though it could become detectable over sufficiently long observing periods.

Load-bearing premise

The load-bearing premise is that the standard formula for light emitted by a thin disk around a stationary black hole also gives the correct emission when the black hole's mass is increasing with time.

Editorial extensions

If this is right

  • For an observer at fixed physical radius, a linearly accreting Vaidya black hole appears with a shadow that grows with time, equivalent to the observer approaching the inner conformal Killing horizon in the conformal frame.
  • The brightness peak of the thin-disk image drifts radially outward with $t_c$ and does not rotate: its azimuthal position stays fixed.
  • For small $\mu$ the radial drift is slow, so a rotating brightness asymmetry like the one observed in M87* is more naturally attributed to variability of the emission source than to the spacetime's mass evolution.
  • The maximum observed flux first increases slightly and then decreases rapidly as $\mu$ grows.
  • The semi-analytical ray-tracing pipeline for static black hole images, including direct and secondary images and photon-ring classification, extends to this time-dependent spacetime.

Reading between the lines

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

  • Extending the picture, any slowly accreting black hole should show a secular outward drift of its image brightness centroid; a long observational campaign that finds no such drift would constrain the mass-growth rate $\mu$ independently of other measurements.
  • The $t_c$-independence of the conformal-frame observed flux suggests a duality between mass growth in the physical frame and radial observer motion in the conformal frame; the same conformal trick could be tried on rotating or charged Vaidya metrics to see whether angular drift appears once rotation is included.
  • A direct test of the paper's weakest step would be to replace the stationary Page-Thorne emission law with a local radiative-transfer calculation for an accreting Vaidya disk; if that changes the $e^{-4t_c}$ scaling, the quantitative prediction for the radial drift would need revision even if the geometric shadow growth survives.
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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

4 major / 5 minor

Summary. The paper studies a Vaidya black hole with linearly growing mass m(v)=µv, introduces a conformal transformation and a new time coordinate t_c, and analyzes null geodesics, direct and secondary images, and the radiation flux from a thin accretion disk using the Page-Thorne formula. The central claim is that the bright spot ('facula') on the observable plane undergoes radial displacement as t_c increases, which the authors connect to time variability in black hole images such as M87*.

Significance. The conformal Vaidya construction and the reduction of null geodesics to quadratures are useful and follow recent work by Solanki and Perlick and by Tan; the paper correctly identifies that photon trajectories in conformal coordinates are independent of t_c and that the shadow in the conformal frame is static. The shadow images in Figs. 1–3 and the dependence of image size on µ and observer position are plausible and constitute a legitimate semi-analytical ray-tracing exercise. However, the claimed temporal evolution of the flux is not supported by the analysis: the flux computation rests on an unjustified application of the Page-Thorne formula to a non-stationary spacetime, and the observed radial displacement of the facula is largely a coordinate artifact of the conformal transformation. If the central claim were established, it would be of direct relevance to time-dependent black hole imaging, but as it stands the paper does not demonstrate a physically new time evolution.

major comments (4)
  1. [Section 4, Eq. (26)] The Page-Thorne flux formula is derived for stationary, axisymmetric, asymptotically flat spacetimes, but the metric (8) depends on t_c through Θ = exp[(t_c + F_c(R))/r_0] and is not asymptotically flat. The statement in Section 4 that the metric 'becomes asymptotically flat at R+' is incorrect: R+ is a finite coordinate boundary where f_c → 0, not spatial infinity. No derivation is given for the non-stationary case, so the use of Eq. (26) is not justified and the time-dependent flux obtained from it is unsupported.
  2. [Section 4, Eqs. (27)–(29)] The energy, angular momentum, and angular velocity of disk particles are taken from the stationary metric formulas, but ∂/∂t_c is only a conformal Killing vector. Consequently, E and L are conserved along null geodesics (Eqs. 11–12) but not along timelike circular orbits in the physical metric (8). Since the Page-Thorne derivation relies on conserved particle energy and angular momentum for stationary circular orbits, the application of Eqs. (27)–(29) to this spacetime is invalid without a separate derivation.
  3. [Section 4, Eqs. (31) and (34)] The exponents e^{-4t_c} and e^{-t_c} are dimensionally inconsistent because t_c has dimensions of length (v_c = t_c + F_c(R) and v has length dimensions in geometric units); they should read e^{-4t_c/r_0} and e^{-t_c/r_0}. Although the cancellation in Eq. (35) would survive this correction, the error signals that the time-dependence extraction was not checked dimensionally and weakens confidence in the derivation.
  4. [Section 5 and Figs. 7–8] The observed flux in the conformal frame is exactly time-independent, as shown in Eq. (35). Therefore, the radial displacement of the facula in the (t_c, r) frame is generated entirely by the coordinate transformation r = R Θ(t_c, R) of Eq. (4) and by the observer-position relation r_obs = R_obs Θ(t_c, R_obs), as used in Eq. (37). The central finding (Conclusion 4) is thus a restatement of the conformal transformation rather than an independent physical prediction about the time-dependent Vaidya image.
minor comments (5)
  1. [Abstract] The phrase 'we investigate the time-dependent of a Vaidya black hole' should read 'the time dependence of a Vaidya black hole'.
  2. [Section 4, Eq. (30)] The determinant expression g = -Θ^8 R^4 omits the sin²θ factor coming from the angular sector; for the equatorial plane this may be intentional, but it should be stated explicitly.
  3. [Section 5, first paragraph] The sentence 'Although the time parameter tc is not be equivalent to the physical time' contains a grammatical error and should read 'is not equivalent'.
  4. [Figure 6 caption] The line-style description 'solid-dot' is unclear; it should be 'dash-dot' or another unambiguous term.
  5. [Section 4] The phrase 'it becomes asymptotically flat at R+' is misleading and should be replaced by a precise statement about the boundary behavior of f_c at the finite radius R+.
Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim depends primarily on the freely chosen mass growth rate µ and observer position r_obs, plus the unverified extension of the stationary Page-Thorne flux formula to a non-stationary spacetime. No new particles or fields are introduced.

free parameters (2)
  • mass growth rate µ = 5e-3, 1e-5, etc.
    Controls the rate of mass increase in m(v)=µv and the conformal horizon structure. The paper sets M0=µr0=1, so r0=1/µ; µ is chosen by hand, not fitted to data.
  • observer radial position r_obs = 10^5 or 0.9R+
    Fixed physical radius of the observer in the (t_c,r) frame. Because r = R Θ, the conformal coordinate R_obs(t_c) moves, producing the image evolution.
assumptions (5)
  • domain assumption The Vaidya metric with linearly growing mass m(v)=µv describes the time-dependent black hole spacetime.
    Motivated by the EHT time variability; the linear mass function is chosen for separability but is not derived from accretion physics.
  • standard math The conformal transformation (Eq. 4) preserves null geodesic structure, so ray trajectories can be computed in the conformal static frame.
    Null geodesics are conformally invariant as unparametrized curves; this is used throughout Section 2.
  • ad hoc to paper The Page-Thorne flux formula (Eq. 26) for stationary thin disks applies to the conformal Vaidya background.
    The physical spacetime is non-stationary; no derivation of the time-dependent flux formula is given.
  • domain assumption The accretion disk is geometrically thin, optically thin, and confined to the equatorial plane, with inner edge at the ISCO.
    Standard in black hole imaging; used to set the disk emission and the ISCO radius.
  • domain assumption An observer at fixed area radius r_obs in the (t_c,r) coordinate system is physically meaningful.
    The paper fixes r_obs and lets R_obs(t_c) evolve; the physical status of this observer in a non-stationary spacetime is not discussed.

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Cite this review

Pith. "Pith review of Image of the time-dependent black hole." pith.science (2026). https://pith.science/paper/UIM2ZRCB

@misc{pith2026250714565,
  author       = {Pith},
  title        = {Pith review of: Image of the time-dependent black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UIM2ZRCB}},
  note         = {Machine review of arXiv:2507.14565}
}
abstract

The Event Horizon Telescope's 2024 observations report a shift in the position angle of the brightness asymmetry in M87*, revealing time variability in the black hole's image. In this analysis, we investigate the time-dependent of a Vaidya black hole. By introducing a mass function that increases linearly with time, along with a conformal transformation, we derive the conformal Vaidya metric and define a new time coordinate $t_c$. Using the semi-analytical approach, we analyze the ray trajectories and radiation flux of the Vaidya black hole in the background of a thin accretion disk. We discuss how the observed flux in the Vaidya spacetime evolves as a function of the new time coordinate $t_c$. The results show that the facula on the observable plane undergoes radial displacement as $t_c$ increases, revealing the time-dependent evolution of black hole images.

Figures

Figures reproduced from arXiv: 2507.14565 by the authors.

Figure 1
Figure 1. Behavior of photons in the conformal Vaidya spacetime as a function of impact parameter [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The conformal coordinate system. The left [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. The direct (red) and secondary (blue) images of a thin accretion disk in the conformal Vaidya spacetime background. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The observed flux from the conformal Vaidya black hole, surrounded by thin accretion disk. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: The relationship between the peak radiation flux [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: The evolution of Robs as a function of tc/r0 is shown for an observer located at robs = 105 . The solid, dashed, dotted, dash-dotted, and solid-dot curves correspond to µ values of 5 × 10−3 , 10−3 , 5 × 10−4 , 10−4 , and 10−5 , respectively. The gray dotted line repres…
Figure 7
Figure 7. Figure 7: The observed flux of the Vaidya black hole [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: The evolution of the observed flux peak position as a function of tc for µ = 5×10−3 . The upper panel shows the radial evolution, while the lower panel illustrates the angular variation. It is observed that the radial distance increases with tc, whereas the angular pos…

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