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Towards an exact approach to pulsar timing

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Standard post-Newtonian formulas for pulsar timing underestimate the propagation delay by up to $10^{-4}$ seconds when the companion is a black hole, a discrepancy larger than the precision SKA is expected to reach.

desk verdict The paper's qualitative warning is plausible, but its headline numbers contradict its own scaling law, so the quantitative claims need fixing before the SKA implications can be trusted. read the letter →

arxiv 2412.10299 v1 pith:INJDI2RG submitted 2024-12-13 gr-qc astro-ph.HE

Amodio Carleo , Delphine Perrodin , Andrea Possenti This is my paper · ORCID
classification gr-qcastro-ph.HE MSC 83C1083C2583C5770F05 PACS 04.25.Nx04.70.-s97.60.Gb95.30.Sf
keywords pulsartimingpropagationdelaygeneralrelativitypost-NewtonianapproximationShapiroblackholecompanionsSKAnullgeodesics
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 argues that the standard post-Newtonian formulas used in pulsar timing are not accurate enough for binary pulsars with a stellar or intermediate-mass black hole companion. By numerically integrating null geodesics in the Schwarzschild spacetime of the companion, it finds that at superior conjunction the discrepancy between the full GR propagation delay and the sum of Roemer, lensed Shapiro, and geometric delays reaches about $10^{-7}$ seconds for a 100 solar-mass black hole even at wide separations, and about $10^{-4}$ seconds for a $10^5$ solar-mass black hole. These values exceed the roughly 10-nanosecond precision that the SKA telescope is projected to achieve, so if correct, the usual timing model would leave systematic residuals in exactly the systems SKA aims to discover. The paper also shows that adding first post-Newtonian corrections to the orbital motion changes the exact delay by up to about $3\times10^{-4}$ seconds in a single orbit, and that the effect should grow with observation time.

What carries the argument

The load-bearing object is the exact propagation delay obtained by integrating null geodesics in the companion's Schwarzschild spacetime, expressed through elliptic integrals of the first, second, and third kind. The delay is defined relative to the ascending node, $\Delta t_{\rm ex} = [T_r(\infty,\lambda_e) \pm T_r(r_e,\lambda_e)] - [T_r(\infty,\lambda_{\rm ref}) \pm T_r(r_{\rm ref},\lambda_{\rm ref})]$, and the comparison quantity is $\Delta_{\rm DIF} = |\Delta t_{\rm ex} - \Delta_R - \Delta_S^{(\rm lens)} - \Delta_{\rm geo} - \Delta_R^{(\rm BH)}|$ evaluated at superior conjunction. A two-module algorithm first solves the angular equation to obtain the impact parameter $\lambda_e$ and emission angle $\varphi_e$ for each orbital point, then evaluates the delay. The paper's quantitative claim is carried by the fitted scaling $\Delta_{\rm DIF} \sim k_{\bar M}\, a_R^{-4/3}$, while the first post-Newtonian orbital-correction scheme enters through the periastron-advance factor $q = 1 + 3\epsilon/(1-e_R^2)$ and its associated eccentricity shifts.

What would settle it

Compute the exact propagation delay for the fiducial edge-on circular orbit ($a_R = 10^6 M$, $\bar M = 10 M_\odot$, $m_p = 2 M_\odot$) using a metric that includes the pulsar's mass, such as a two-body or boosted metric, and compare with the paper's $\Delta_{\rm DIF} \sim 10^{-7}$ s; a difference of that order would show the central claim depends on the one-body approximation.

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Extended reading notes

Core claim

Starting from the exact geodesic computation, the paper claims that for a binary pulsar with a black hole companion in the mass range $5$ to $10^5\,M_\odot$, the maximum difference between the full GR propagation delay and the post-Newtonian sum of Roemer, lensed Shapiro, and geometric delays at superior conjunction obeys $\Delta_{\rm DIF} \sim k_{\bar M}\, a_R^{-4/3}$, where $k_{\bar M}$ grows linearly with the black hole mass and the pulsar mass does not matter. For a $10\,M_\odot$ black hole on a circular edge-on orbit at $a_R = 10^6 M \simeq 1.5\times10^{12}$ cm, the discrepancy is about $10^{-7}$ s, above the projected SKA precision; for $\bar M = 100\,M_\odot$ it is about $10^{-7}$ s even at $a_R\sim10^{16}$ cm, and it reaches about $10^{-4}$ s for $\bar M = 10^5\,M_\odot$. Including first post-Newtonian corrections to the orbital motion changes the exact delay by up to about $6.5\,G\bar M/c^2 \simeq 3.2\times10^{-4}$ s in one orbit for $a_R = 10^5 M$ and $\bar M = 10\,M_\odot$, an effect that grows secularly. The paper concludes that post-Newtonian timing formulas should be replaced by exact geodesic computations in these systems.

Load-bearing premise

The photon path is computed in the spacetime of the black hole companion alone, with the pulsar's mass omitted from the metric even though it enters the orbital motion; for the fiducial 2 solar-mass pulsar and 10 solar-mass black hole the mass ratio is 0.2, so the pulsar's own gravity could change the delay.

Editorial extensions

If this is right

  • SKA-era timing of pulsar-black hole binaries will show systematic residuals at the $10^{-7}$ to $10^{-4}$ second level if the standard post-Newtonian model is used, since the discrepancy exceeds the projected ~10 ns precision.
  • For a 10 solar-mass black hole, the post-Newtonian formulas are practically indistinguishable from full GR only when the semi-major axis exceeds about $10^7 M$, i.e. roughly $10^{13}$ cm; closer systems need the exact treatment.
  • The need for full GR depends on inclination: at $a_R = 10^6 M$ with a 10 solar-mass companion, orbits with inclination between $75^\circ$ and $90^\circ$ may require the exact computation, while for a 100 solar-mass companion the discrepancy is relevant even at low inclination.
  • Relativistic corrections to the orbital motion strongly affect the exact delay: for $a_R = 10^5 M$ and a 10 solar-mass black hole the correction reaches about $3.2\times10^{-4}$ s in a single orbit, and being a periastron-advance effect it grows with observation time.
  • The fitted trend $\Delta_{\rm DIF} \sim k_{\bar M} a_R^{-4/3}$ gives a simple rule of thumb for deciding, before detailed modeling, whether a candidate pulsar-black hole binary needs exact geodesic delays.

Reading between the lines

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

  • The one-body metric approximation is the obvious next thing to test: the authors themselves call it 'substantially wrong' for comparable masses, so a two-body metric computation could shift both the numerical values and the claimed dependence of $\Delta_{\rm DIF}$ only on $a_R$ and $\bar M$.
  • The $a_R^{-4/3}$ scaling is extracted from log-log interpolation over a limited parameter grid; deriving that exponent from the geodesic equations, or checking more eccentricities and separations, would reveal whether it is robust or a feature of the chosen range.
  • The paper assumes zero spin; extending to spinning companions would bring in frame dragging, and earlier work cited in the paper suggests the post-Newtonian frame-dragging formula overestimates the effect near superior conjunction, so the discrepancy pattern may differ for spinning black holes.
  • The claim that the discrepancy 'exceeds SKA precision' is a comparison with nominal timing residuals, not with the bias on fitted parameters; an end-to-end simulation that fits synthetic times of arrival with both models would quantify how much the recovered masses and orbital elements actually shift.
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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 presents a numerical comparison between the full general-relativistic propagation delay for photons from a pulsar in the Schwarzschild/Kerr spacetime of a black hole companion and the standard post-Newtonian timing delays (Roemer, Shapiro, lensing, and geometric corrections). The intended target systems are pulsar binaries with stellar or intermediate-mass black holes, of the kind SKA may discover. The authors claim that the full GR-computed delay difference depends only on the binary semi-major axis aR and the black hole mass Mbar, that the maximum difference follows approximately Delta_DIF ~ k_Mbar aR^{-4/3}, and that for Mbar = 100 Msun the difference is significant (~1e-7 s) even at aR ~ 1e16 cm, growing to ~1e-4 s for Mbar = 1e5 Msun. They also study 1PN corrections to the orbital motion and find these can change the exact delay by ~1e-4 s for a compact eccentric orbit.

Significance. If the quantitative claims are correct, the paper would have immediate practical relevance for pulsar-BH timing with SKA-class telescopes, where timing residuals of order 10 ns are expected. The conceptual setup is sound in that the full GR geodesic integration is an independent benchmark rather than a fit to the post-Newtonian result, so the comparison is not circular. The authors are also transparent in Section VII about the one-body metric approximation and about the fact that Delta_DIF is not directly comparable to timing residuals without a full timing-model fit. The central limitation is that the headline numerical claims are not currently supported by the internally consistent numbers in the paper.

major comments (3)
  1. [Abstract; Section VI; Section VII; Table I] The headline value in the abstract, namely Delta_DIF ~ 1e-7 s at aR ~ 1e16 cm for Mbar = 100 Msun, contradicts the paper's own Table I and the stated scaling law. Table I (i = 90 deg, Mbar = 100 Msun, aR = 1e6 M, which is about 1.48e13 cm) gives Delta_DIF = 2e-6 s. Scaling that entry to aR = 1e16 cm with the stated Delta_DIF ~ k_Mbar aR^{-4/3} law gives approximately 3e-10 s, about three orders of magnitude below the abstract's value. The same contradiction appears starting from the Mbar = 10 Msun entry and applying the stated linear-in-Mbar plus aR^{-4/3} scaling. Since these numbers are the central quantitative motivation for replacing post-Newtonian formulas, this internal inconsistency must be resolved before the main claim can be assessed.
  2. [Section IV; Section VI; Figure 3] The extrapolation to aR ~ 1e16 cm rests on a fitted power-law trend with no reported convergence checks, error bars, or resolution study. Section IV states that the computation uses a discrete set of (lambda, phi_e) couples and interpolation, and Section VI says the trend was roughly extracted from a log-log plot. The exponent -4/3 is fitted to the same data that are then extrapolated, so the abstract's values at 10^16 cm are unsupported as presented. Please provide convergence tests in the number of lambda samples, root-finding tolerances, and interpolation errors, or explicitly restrict the conclusions to the computed range.
  3. [Section II; Section VII] The one-body metric approximation is load-bearing for the claimed dependence of Delta_DIF only on aR and Mbar. Section II states that 'if the companion is a black hole, it is sufficient to ignore the mass of the pulsar in the computation of the photon delay,' while Section VII concedes that this approximation 'is substantially wrong in cases where the two objects have comparable masses.' For the fiducial case mp = 2 Msun and Mbar = 10 Msun, the mass ratio is 0.2, which is not a small-parameter regime. The paper's own limitation statement therefore implies that the computed Delta_DIF and its claimed scaling could change once the pulsar's gravitational field is included, and Section VII indeed says the propagation delay should ultimately depend on the pulsar mass. Please estimate the size of the neglected pulsar-potential contribution, or otherwise justify why it cannot affect the claimed scaling and numerical values.
minor comments (5)
  1. [Section VI; Section VII] The expressions 'Delta_DIF ~ x10^-8 s' and 'Delta_DIF ~ x10^-9 s' are missing the numerical prefactor; they should presumably read '~1 x 10^-8 s' and '~1 x 10^-9 s'.
  2. [Figure 3 caption] The conversion factor 'G Mbar c^-3 ~ 4.4 x 10^-5 (Mbar/Msun)' appears to be wrong by a factor of 10; the correct value is G Msun/c^3 ~ 4.9 x 10^-6 s. This matters because Table I entries are converted with the correct value.
  3. [Section II] There are several typos: 'Hamilton-Jabobi' should be 'Hamilton-Jacobi', 'Schawrzschild' should be 'Schwarzschild', and 'geodetic equations' should be 'geodesic equations'.
  4. [References] References [8] and [9] list the same Damour-Deruelle paper; one should be removed or the citations should be disambiguated so that the 1985 paper on post-Newtonian motion and the 1986 paper on the timing formula are clearly distinguished.
  5. [Figure 3 caption] The caption's description of the blue dashed line is confusing: it says the line is for a generic mass and that the y-axis must be multiplied by G Mbar c^-3, but it is not clear which curves are the actual computed points and which are the interpolated fits. Please clarify the plotting conventions.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the full-GR delay is computed from geodesic equations and benchmarked against independent post-Newtonian formulas; the one self-citation is not load-bearing.

full rationale

The paper's central quantity, Delta_DIF, is obtained by evaluating the Kerr geodesic delay (Eqs. 3-14) and subtracting the standard post-Newtonian delays (Eqs. 19-22), so the comparison is an external benchmark rather than an input. The aR^{-4/3} and linear-in-M trends in Section VI are summaries of the computed grid, not parameters fitted to the target discrepancy. The analytical delay formula Eq. (11) is cited from [5,6,17]; although [6] shares an author, the same result is given in the independent works [5,17], so this citation does not force the conclusion. The one-body metric assumption is an acknowledged approximation (Section VII), not a definitional equivalence. The abstract's 10^-7 s at 10^16 cm claim appears inconsistent with the stated aR^{-4/3} scaling, but that is an internal consistency or correctness issue, not circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central estimates rest on four domain assumptions: one-body Kerr spacetime for the photon path, Keplerian/1PN orbits for the emission point, a coordinate mapping between PN and Kerr radial coordinates, and neglect of retardation and spin. The only fitted quantities are the power-law trend and its mass-dependent constant, extracted from the authors' own numerical output. No new physical entities are introduced.

free parameters (2)
  • scaling exponent gamma in Delta_DIF ~ k_M aR^gamma = approx. -4/3
    Roughly extracted from a log-log interpolation of the numerical results in Fig. 3; used to extrapolate to large aR and high BH masses.
  • proportionality constant k_Mbar = not reported explicitly
    Mass-dependent constant fitted to discrete (Delta_DIF, aR) couples; the paper states it depends only on the black hole mass but does not give its value or units.
assumptions (6)
  • domain assumption The spacetime for photon propagation is the Kerr metric of the companion black hole with the pulsar mass neglected.
    Used in Section II to compute the propagation delay; acknowledged in Section VII to be substantially wrong for comparable masses.
  • domain assumption The pulsar moves on a Keplerian or 1PN Damour-Deruelle orbit, with eccentric anomaly parametrization and periastron advance encoded in q; the emission point coordinates are derived from this orbit.
    Section V introduces the 1PN quasi-Newtonian scheme of Damour and Deruelle; the relation between true anomaly and eccentric anomaly, Eq. (27), is Taylor expanded in epsilon.
  • domain assumption The relation between post-Newtonian harmonic coordinates and Kerr coordinates is r_PN -> r - M for zero spin.
    Stated in Section III as a prescription similar to Hackmann and Dhani [17]; the full GR and PN delays are compared in different coordinate systems, so this mapping affects Delta_DIF.
  • domain assumption The retardation effect, from the orbital motion of the companion during light crossing, is neglected in both the PN and full GR calculations.
    Section III states a boosted Kerr metric would be needed to include it and that it is beyond the scope; this effect can shift delay curves by about a_c/c.
  • standard math Observations are made by an observer at infinity; delays are computed relative to a reference point on the orbit to remove the infrared divergence.
    Section II, Eq. (14); this is the standard way to define finite propagation delays and is not unique to this paper.
  • domain assumption The black hole spin is set to zero for the numerical estimates, so frame-dragging is neglected.
    Section IV says 'we will assume a = 0'; frame-dragging delay is stated to be negligible at distances greater than about 1e6 M, but this is not shown in detail.

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Cite this review

Pith. "Pith review of Towards an exact approach to pulsar timing." pith.science (2026). https://pith.science/paper/INJDI2RG

@misc{pith2026241210299,
  author       = {Pith},
  title        = {Pith review of: Towards an exact approach to pulsar timing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/INJDI2RG}},
  note         = {Machine review of arXiv:2412.10299}
}
abstract

The pulsar timing technique, which compares the observed arrival times of electromagnetic radiation from a pulsar with the predicted arrival times derived from a theoretical model of the pulsar system, is used in pulsar astronomy to infer a multitude of physical information and to constrain possible corrections to General Relativity (GR). The propagation delay is usually computed using formulas based on a post-Newtonian approach, for both the light trajectory and the orbital motion. However, evidence has recently emerged that this approximation may no longer be sufficient when the companion object is a supermassive black hole; deviations from a full GR computation of the propagation delay can reach a few seconds. In this paper, we analyze the case of binary pulsars with a stellar or intermediate black hole companion, whose discovery and timing are key goals of SKA. With a numerical algorithm, we have found that in this case, the full GR value depends only on the semi-major axis of the relative orbit and on the mass of the black hole companion. If the mass of the latter is sufficiently large ($100 M_{\odot}$), the maximum difference between the two approaches is significant ($\sim10^{-7}$ s) even for large binaries ($\sim10^{16}$ cm), and increases up to $\sim 10^{-4}$ s when the mass is $10^5 M_{\odot}$. We also consider relativistic corrections to the orbital motion, and discover that they can strongly affect the value of the propagation delay. We conclude that in the future, post-Newtonian formulas should be replaced with a more accurate approach in these systems, especially in view of future discoveries made by new large telescopes such as SKA.

Figures

Figures reproduced from arXiv: 2412.10299 by the authors.

Figure 1
Figure 1. FIG. 1. Delay difference [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Delay difference [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Log-log plot of the maximum delay difference as a [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Difference between the exact delay with and without [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Time Delay of Pulsar Signals in Astrophysical Black Hole Spacetimes

    gr-qc 2025-06 conditional novelty 5.0 of 10

    Pulsar time delays are computed for Kerr, deformed Kerr (Johannsen-Psaltis), and rotating Janis-Newman-Winicour spacetimes, producing model-dependent signatures of up to about 15 seconds.

  2. Galactic Centre Pulsars with the SKAO

    astro-ph.HE 2026-07 accept novelty 4.0 of 10

    Updated SKA-MID sensitivity and multi-beam search strategies can detect up to ~84% of Galactic Centre pulsars (and ~60% of MSPs) under magnetar-like scattering, unlocking precision tests around Sgr A*.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.