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REVIEW 3 major objections 5 minor 75 references

Time Delay of Pulsar Signals in Astrophysical Black Hole Spacetimes

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Pulsar signal time delays computed along null geodesics differ from Kerr by about 0.12 seconds in a deformed Kerr spacetime and by about 15 seconds in a rotating Janis-Newman-Winicour naked singularity, giving a phase-dependent observable…

desk verdict The deformed Kerr time-delay predictions rest on an invalid Hamilton-Jacobi separation; the rotating JNW part needs a closer look. read the letter →

arxiv 2506.06583 v1 pith:7BFBYFTY submitted 2025-06-06 gr-qc

Parth Bambhaniya , Viraj Kalsariya , Saurabh , Elisabete M. de Gouveia Dal Pino , Ivan De Martino , Riccardo Della Monica , Mariafelicia De Laurentis This is my paper · ORCID
classification gr-qc MSC 83C5783C10 PACS 04.70.-s04.20.-q
keywords pulsartiminggravitationaltimedelaynullgeodesicsKerrspacetimedeformedmetricJanis-Newman-Winicournakedsingularityblackholemimickers
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 that the fully relativistic propagation delay of pulses from a pulsar in a tight circular orbit around a supermassive black hole carries an orbital-phase pattern that can tell a Kerr black hole apart from two alternative spacetimes: a deformed Kerr geometry and a rotating Janis-Newman-Winicour naked singularity. It derives the null geodesic equations in all three spacetimes, solves the emitter-observer problem for equatorial circular orbits, and computes time-delay differences relative to Kerr for a Galactic Center-sized black hole. The predicted differences reach about 0.12 seconds for deformed Kerr and about 15 seconds for the rotating JNW case, while changing the spin from $0.1M$ to $0.9M$ moves the curves only slightly. These delay curves give a concrete observational target for next-generation pulsar timing searches near the Galactic Center.

What carries the argument

The machinery is the emitter-observer problem solved through the Hamilton-Jacobi separation of null geodesics. In Kerr spacetime, Carter's constant makes the equations of motion separable, giving first-order equations in the Mino parameter $\gamma$; the paper assumes the same separability for the deformed Kerr metric and for the rotating JNW metric. For each spacetime it forms the ratio of the $t$ and $\phi$ equations to the radial equation, yielding integrals for the observer's angle $\phi_0-\phi_e$ and the propagation time $t_0-t_e$ as functions of the impact parameter $\lambda$. A grid search over $\lambda$ is combined with the angular relation $\cos(\phi_0-\phi_e)=-\sin i\,\sin(\omega+\phi)$ to assign each time delay to the pulsar's orbital phase $\phi$, producing the delay-versus-phase curves. Direct photons (pulsar in front) and indirect photons (pulsar behind, with a radial turning point $r_{\min}$) are treated separately, and the placement of the deformation term in the deformed-Kerr equations follows the correction by Wang et al. (2025).

What would settle it

Integrate the full null geodesic equations for the deformed Kerr metric with $h(r,\theta)=\epsilon M^3 r/\rho^4$ and for the rotating JNW metric directly, without assuming a Carter constant, and compare the resulting delay-versus-phase curves with the paper's Eqs. (51)-(52) and (68)-(69); disagreements at the 0.1 s or 15 s level would mean the paper's central numbers are not the spacetimes' actual delays. Observationally, timing a pulsar on a roughly $100M$ circular orbit around a $4\times10^6\,M_\odot$ black hole across at least one orbit, with sub-0.1 second accuracy, would test whether the residuals reproduce the sharp feature at $\phi=\pi$ and the predicted model-to-model offsets.

Watch

Extended reading notes

Core claim

The paper's central claim is that the time delay of photons traveling from a pulsar on a circular equatorial orbit at radius $100M$ to a distant observer is a precise function of the spacetime geometry, and that the phase-resolved delay difference with respect to Kerr is large enough to be a potential discriminator. For the deformed Kerr spacetime, with deformation function $h(r,\theta)=\epsilon M^3 r/\rho^4$, the delay difference reaches the ~0.1 second level and is antisymmetric under $\epsilon\rightarrow -\epsilon$. For the rotating JNW naked singularity, parametrized by the scalar-charge parameter $\nu$, the delay difference grows as $\nu$ decreases from 1, reaching about 15 seconds for $\nu=0.6$. In both cases the largest signals come from indirect photons that pass behind the black hole and experience strong lensing near the turning point, producing a sharp feature at orbital phase $\phi=\pi$. The paper concludes that these delay curves offer a potential observable distinguishing feature of black hole geometries, supplementing shadow and accretion-disk observations.

Load-bearing premise

The load-bearing premise is that photon motion in the deformed Kerr and rotating JNW spacetimes separates into independent radial and angular pieces with a Carter constant, an assumption the paper uses without proving it from the metric; if that separation fails, the computed delay curves for both mimickers do not describe the actual photon trajectories in those spacetimes.

Editorial extensions

If this is right

  • The sharp peak in the delay difference at orbital phase $\phi=\pi$, produced by indirect photons passing behind the black hole, is the feature most sensitive to which spacetime is present.
  • For the rotating JNW naked singularity, the delay difference relative to Kerr increases as the scalar-charge parameter $\nu$ decreases from 1 toward 0.6, reaching roughly 15 seconds.
  • For the deformed Kerr spacetime, the delay difference is antisymmetric in the deformation parameter $\epsilon$ and reaches the ~0.1 second level, a smaller signal that is nonetheless part of the paper's claimed observable distinction.
  • Changes in spin from $0.1M$ to $0.9M$ shift the delay curves by only small amounts, so the distinguishing power comes chiefly from the spacetime structure rather than the rotation rate.
  • The fully relativistic delay curves can be added to standard pulsar timing residuals to fit or exclude the deformation and scalar-charge parameters once a pulsar near the Galactic Center is found.

Reading between the lines

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

  • Because the JNW delay difference reaches about 15 seconds while millisecond pulsars can be timed to far better than a microsecond, a discovered Galactic Center pulsar could test the naked-singularity model with only a few orbital cycles, provided interstellar scattering is overcome at high observing frequencies.
  • The antisymmetry of the deformed-Kerr delay under $\epsilon\rightarrow-\epsilon$ provides a built-in way to separate the deformation parameter from spin effects, since spin changes do not produce the same sign-flipping pattern.
  • The same grid-search emitter-observer method could be extended to eccentric or inclined pulsar orbits by solving for two impact parameters, which would add periastron-phase structure to the delay curves and may separate the models even more sharply.
  • Adding the omitted solar-system and Earth-orbit terms through standard weak-field formulas would make the predicted delay curves directly comparable to real timing residuals without altering the strong-field signatures.
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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

3 major / 5 minor

Summary. The paper computes fully relativistic propagation time delays for photons emitted by a pulsar in a circular equatorial orbit around a supermassive black hole, treating the Kerr metric, the deformed Kerr (Johannsen-Psaltis) metric, and the rotating Janis-Newman-Winicour naked-singularity metric as competing spacetime models. For each spacetime, the authors solve the null-geodesic equations, solve the emitter-observer problem by a grid search over impact parameter, and compare the resulting time delay as a function of pulsar orbital phase, with Sgr A* parameters. The reported predictions are time-delay differences up to about 0.12 s for deformed Kerr versus Kerr and about 15 s for rotating JNW versus Kerr, depending on phase and on the metric parameters. The Kerr/Schwarzschild comparison is presented as a check of the method.

Significance. If the underlying geodesic equations are correct, the paper is a useful methodological extension of fully relativistic pulsar timing calculations to two black-hole mimickers. A clear strength is the absence of circularity: no parameter is fitted to the time-delay output, the metric parameters are scanned, and the Kerr limit is explicitly checked against Schwarzschild. The Kerr part of the paper is standard and the numerical inversion scheme is appropriate. However, the deformed-Kerr results rest entirely on a Hamilton-Jacobi separation that is asserted but not valid as written, and the rotating-JNW equations are presented without derivation and with an apparent inconsistency with the stated metric. These are load-bearing issues because the deformed-Kerr and JNW time-delay predictions are the paper's main new claims. The paper would be acceptable only after the derivation is repaired or the claims are restricted to cases where the separation can be justified.

major comments (3)
  1. [§4.1, Eqs. (42)-(46)] The Hamilton-Jacobi reduction leading to Eqs. (43)-(46) is not a valid separation of variables. In Eq. (42), the coefficient of (∂S_r/∂r)^2 in the first brace is [(Δ+h a^2 sin^2θ)^2]/[Δ(1+h)], which depends on θ through h and sin^2θ, while the coefficient of (∂S_θ/∂θ)^2 in the second brace is (Δ+h a^2 sin^2θ)/Δ, which depends on r through Δ and h. Equating the two braces to a common Carter constant therefore assigns to each side a quantity that is not a function of a single coordinate. This is not repaired by Eqs. (43)-(44): Eq. (43) makes (∂S_r/∂r)^2 proportional to (1+h)/(Δ+h a^2 sin^2θ)^2 times R(r), and R(r) in Eq. (45) contains h(r,θ) through hρ^4/(1+h), so neither side is r-only. The text's statement that a term was moved to the radial part because it 'predominantly influences' the radial component is a heuristic, not an algebraic separation. Consequently, the first-order geodesic equations (47)-(50) and the time-delay integrals (51)-(52) do not follow from the stated Johannsen-Psaltis metric (38)-(41), and the deformed-Kerr time-delay differences in Figures 4-5, including the approximately 0.12 s deviations, are unsupported. The authors must either prove separability for h(r,θ)=ε M^3 r/ρ^4, or integrate the full geodesic equations without assuming a Carter constant, or restrict to a deformation for which the separation is known to hold.
  2. [§4.2, Eqs. (53)-(69)] The rotating-JNW geodesic equations are presented without the required derivation. Eq. (58) is stated as the result of solving the Hamilton-Jacobi equation for the metric (53)-(57), but no intermediate steps are shown, and the equation is not manifestly consistent with that metric: the cross term in Eq. (58) is written as -4aMK(r)/(rΔ) LE, while the metric's cross term is -4a f sin^2θ/ρ^2 with f defined in Eq. (54); the identity f = MK/r does not hold for general ν (it holds only at ν=1). In addition, because Eq. (56) defines Δ = r^2(1-2M/rν)+a^2 rather than r^2-2f+a^2, the usual Kerr-like relation between g_{tφ}, Δ, and ρ^2 needs to be verified. The authors should derive Eqs. (58)-(67) explicitly from Eqs. (53)-(57) and check the metric-inverse identities; until then the JNW time-delay curves in Figures 6-7 rest on an unproven separability assumption.
  3. [Abstract and §5] The claim that the differences are 'very small but detectable' is not backed by a detection or noise budget. The paper compares theoretical time delays for different metrics but does not estimate the timing precision needed to distinguish a roughly 0.1 s deformed-Kerr shift or a roughly 15 s JNW shift from Kerr, nor does it account for interstellar scattering, pulse jitter, or the expected number of pulses. Without such an estimate, 'detectable' and 'potential observable distinguishing feature' are statements about the theoretical magnitude only. The authors should either add a concrete observability analysis or soften the conclusions to say that the differences are, in principle, large enough not to be masked by the geometric model, while detectability remains to be assessed.
minor comments (5)
  1. [§2, below Eq. (13)] The text says 'separating functions of r and ϕ', but the separation is between r and θ; this should be corrected.
  2. [§3, Eq. (36)] The factor c appears in the denominator of the time-delay integrand; since geometrized units are used elsewhere, the paper should state explicitly how c is restored in the final seconds values.
  3. [§4.1, Figure 4 caption] The caption refers to green and purple lines, but the legend lists ε values; please align the color references with the actual plot.
  4. [§3 and §5] For reproducibility, please specify the numerical quadrature method, the tolerance of the grid search used to invert the emitter-observer relation, and the number of grid points in λ.
  5. [Data Availability] The statement that no new data were generated or analysed is unclear because Figures 2-7 are numerical products; please clarify whether the numerical output and code are available on request.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the time-delay curves are genuine model predictions with parameters scanned by hand; the main caveat is a correctness risk in the deformed-Kerr Hamilton-Jacobi separation, not a circularity.

full rationale

The derivation chain is not circular. The pulsar time delay is obtained by numerically integrating Eqs. (35)-(36), (51)-(52), and (68)-(69) for fixed metric inputs (M, a, epsilon, nu); no parameter is fitted to the time-delay output, and the comparison curves are genuine predictions of each spacetime geometry. The reuse of the emitter-observer methodology from Kalsariya et al. (2024) and of the deformed-Kerr geodesic structure from Bambhaniya et al. (2021a) does not make the result circular, because those prior results are parameter-free, published, and do not contain the time-delay values claimed here; self-citation is not load-bearing in a definitional sense. The one substantive concern is correctness rather than circularity: Eqs. (43)-(44) assume a Carter-constant separation for the Johannsen-Psaltis metric with h(r,theta) = epsilon M^3 r / rho^4, but Eq. (43) carries an explicit theta-dependent prefactor (1+h)/(Delta + a^2 h sin^2 theta)^2, and the text's reassignment of E^2 h rho^4 / Delta(1+h) because it 'predominantly influences' the radial component is a heuristic, not a proof of separability. If that separation fails, the deformed-Kerr time delays are invalid, not tautological. The rotating-JNW equations (60)-(67) carry a similar unproven separation. These are validity risks to weigh in the correctness pass, not circular reductions, so the circularity score remains low.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the validity of the Kerr, deformed Kerr, and rotating JNW metrics as physical backgrounds, and on the separability of the Hamilton-Jacobi equation in the mimicker cases. The free parameters are scanned model inputs, not fitted to data; none of them are adjusted to the time-delay output.

free parameters (6)
  • spin parameter a = scanned: 0, 0.1, 0.4, 0.9 M
    Kerr spin chosen by hand; not fitted to any data; the results compare these values.
  • deformation parameter epsilon = scanned: -2, -1, 1, 2
    Johannsen-Psaltis deformation chosen by hand; affected by the separability assumption.
  • scalar charge parameter nu = scanned: 0.6, 0.75, 0.9
    Rotating JNW spacetime scalar charge chosen by hand.
  • pulsar orbital radius re = 100 M
    Circular orbit radius chosen for the analysis.
  • observer distance r0 = 4 x 10^10 M
    Distance to Earth chosen as a fixed observer position.
  • inclination i and periastron omega = i = pi/2, omega = pi/2
    Edge-on coplanar configuration chosen to maximize frame dragging effects.
assumptions (7)
  • domain assumption The Kerr metric with mass M and spin a describes the central black hole.
    Assumed in Section 2; the pulsar time delay is computed in this background.
  • domain assumption The Johannsen-Psaltis deformed Kerr metric (Eq. 38) is a valid description of the deformed black hole or naked singularity.
    Adopted from Johannsen and Psaltis 2011 in Section 4.1.
  • domain assumption The rotating JNW metric (Eq. 53) is a valid description of a rotating naked singularity.
    Adopted from Solanki et al. 2022 in Section 4.2.
  • ad hoc to paper The Hamilton-Jacobi equation for the deformed Kerr metric separates into radial and angular parts with a Carter constant.
    The paper does not prove separability or derive Eq. (42) from the metric; the validity of Eqs. (47)-(50) rests on this premise. This is the load-bearing assumption.
  • ad hoc to paper The Hamilton-Jacobi equation for the rotating JNW metric separates with a Carter constant.
    The paper assumes a separable ansatz for the rotating JNW metric; no proof is given.
  • domain assumption The pulsar is a test particle on a fixed circular equatorial orbit and the observer is coplanar on the equatorial plane.
    Stated in Section 5; reduces the emitter-observer problem to a single impact parameter lambda.
  • domain assumption Null geodesics and the test-particle approximation are sufficient for the strong-field propagation.
    Standard in the analysis, used throughout the paper.

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Cite this review

Pith. "Pith review of Time Delay of Pulsar Signals in Astrophysical Black Hole Spacetimes." pith.science (2026). https://pith.science/paper/7BFBYFTY

@misc{pith2026250606583,
  author       = {Pith},
  title        = {Pith review of: Time Delay of Pulsar Signals in Astrophysical Black Hole Spacetimes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7BFBYFTY}},
  note         = {Machine review of arXiv:2506.06583}
}
read the original abstract

In this paper, we investigate the fully relativistic time delay of pulsar signals propagating in the vicinity of a rotating black hole and its potential mimickers, including a deformed Kerr black hole and the Janis-Newman-Winicour naked singularity. We aim to compute and compare the pulsar time delays caused by different spacetime geometries to explore possible observational signatures that distinguish between black holes and their alternatives. We begin by solving the equations of motion for null geodesics in these background geometries. Subsequently, we address the emitter-observer problem to compute the time delay of pulsar signals in Kerr, deformed Kerr, and JNW spacetimes. A comparative analysis between Schwarzschild and Kerr black holes allows us to observe the effect of spin on propagation delay in pulsar timing. Further, we examine the impact of the deformation parameter in the deformed Kerr black hole and the influence of the scalar field on the rotating JNW spacetime. Our study considers both direct and indirect photons emitted by a source in the equatorial circular orbit. We find that the variations in the spin parameter show very small but detectable changes when we compare the time delay cases of a Kerr black hole with deformed Kerr and rotating JNW spacetimes. Our pulsar time delay results suggest a potential observable distinguishing feature of these astrophysical black hole geometries which could be useful for the forthcoming observational facilities such as Square Kilometer Array Observatory, Five-hundred-meter Aperture Spherical Telescope and Event Horizon Telescope.

Figures

Figures reproduced from arXiv: 2506.06583 by the authors.

Figure 1
Figure 1. The schematic diagram illustrates a pulsar orbiting a rotating central compact object (C) at a radial distance re (blue plane) with an inclination angle i to the plane of sky (gray plane). Here ω is the argument of periastron. The spin of the central compact object is along the z-axes for real orbit in the blue plane. The pulsar’s positions along its orbit are labelled P1 to P4. Photons following the red-dotted puls… view at source ↗
Figure 2
Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. This figure shows the time delay difference between same spinning deformed Kerr and Kerr spacetimes in seconds as a function of the mean anomaly φ. The comparison is made for two different spin values: a = 0.9M (left) and a = 0.1M (right). The coloured lines represent different deformation parameters ϵ, as indicated in the legend. The inset figures are the zoomed in portion of the squared part. ϵ = 2 ϵ = 1 ϵ = -1 ϵ … view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: The figure on the left shows the difference between (tDKerr − tKerr)a=0.9M and (tDKerr − tKerr)a=0.1M, while the figure on the right shows the zoomed-in view for the pulsar orbit from 0 to 2π rotation. Carter’s constant C. Then we find  ∂S r ∂r 2 = (1 + h) (∆ + ha2 s…
Figure 6
Figure 6. Figure 6: This figure represents the propagation time delay between rotating JNW spacetime and Kerr black hole for the same spin. The left-hand side plot is for spin a = 0.9M, while the right-hand side plot is for spin a = 0.1M. The different deformation parameters ϵ, are indica…
Figure 7
Figure 7. Figure 7: The figure represents the difference between (tJNW − tKerr)a=0.9M and (tJNW − tKerr)a=0.1M along with a zoomed-in figure on right hand side for a pulsar orbit from 0 to 2π rotation. similarly, from Equations 64 and 67, we derive the equation to solve the time delay in …

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