{"id":"05d86f67-8d86-48b3-bf81-9e93114c2d30","arxiv_id":"2412.10299","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For pulsars orbiting stellar or intermediate-mass black holes, standard post-Newtonian timing formulas can differ from a full general-relativistic computation by 10^-7 to 10^-4 seconds, enough to affect future SKA timing.","lead":"This paper calculates how much the usual approximate formulas for pulsar timing can miss when a pulsar orbits a black hole, by comparing them with a full general-relativistic calculation. It finds timing errors of about 0.1 microseconds to 0.1 milliseconds for systems that future radio telescopes like SKA are designed to find.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The abstract's 10^-7 s at 10^16 cm for 100 M_sun contradicts the paper's own aR^{-4/3} scaling; one set of numbers must be wrong.","rationale":"The reader's chosen weakest assumption, the one-body metric that ignores the pulsar's mass in the photon propagation, is a real and acknowledged limitation (Section VII). It is most serious for comparable-mass systems and could alter the claimed dependence of Delta_DIF on aR and Mbar. However, the most load-bearing problem for the central claim is internal: the abstract's flagship numbers are incompatible with the paper's own scaling law. Even before questioning the one-body approximation, the manuscript contains two mutually exclusive quantitative claims. The concrete test above would settle which one is correct. I would keep the reader's CONDITIONAL verdict rather than escalate to REJECT, because the qualitative warning about post-Newtonian accuracy in closer pulsar-BH binaries may survive after correcting the numbers, but the abstract and Section VI must be reconciled and the numerical grid made available before the claimed magnitudes can be accepted.","tokens_in":20324,"tokens_out":10938,"duration_ms":90886,"concrete_test":"Recompute Delta_DIF with the stated algorithm for Mbar = 100 M_sun, aR = 10^16 cm, i = 90 deg, eR = 0, and compare the result with the value obtained by taking the Table I entry (Mbar = 100 M_sun, aR = 10^6 M, Delta_DIF = 2 x 10^-6 s) and applying the claimed Delta_DIF proportional to Mbar aR^{-4/3} scaling. If the direct computation returns about 10^-7 s, then the fitted scaling law is not valid over this range. If it returns about 3 x 10^-10 s, then the abstract overstates the effect by roughly a factor of 300. The authors should also release the interpolation grid behind Figure 3 so that the fitted trend can be checked point by point.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section VI states a fitted trend Delta_DIF ~ k_M aR^{-4/3}, with k_M depending only on the black-hole mass and with Delta_DIF growing linearly with Mbar for fixed aR. The abstract and Section VI then claim that for Mbar = 100 M_sun, Delta_DIF is about 10^-7 s even at aR ~ 10^16 cm. These statements are mutually inconsistent. Using the authors' own Table I for Mbar = 100 M_sun, i = 90 deg, aR = 10^6 M (about 1.48 x 10^13 cm), one finds Delta_DIF = 2 x 10^-6 s. Scaling that entry to aR = 10^16 cm with the stated aR^{-4/3} law reduces Delta_DIF to about 3 x 10^-10 s, roughly three orders of magnitude below the abstract's 10^-7 s. The same discrepancy appears if one starts from the Mbar = 10 M_sun, aR = 10^6 M value and applies linear-in-Mbar plus aR^{-4/3} scaling. Since the abstract's headline numbers are what motivate the conclusion that post-Newtonian formulas must be replaced for SKA, one of the two numerical claims is wrong. This concern is independent of the one-body approximation flagged by the reader: even granting the companion-only metric, the quantitative result is not internally consistent as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20603,"tokens_out":9978,"duration_ms":89148,"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":[{"comment":"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.","section":"Abstract; Section VI; Section VII; Table I"},{"comment":"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.","section":"Section IV; Section VI; Figure 3"},{"comment":"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.","section":"Section II; Section VII"}],"minor_comments":[{"comment":"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'.","section":"Section VI; Section VII"},{"comment":"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.","section":"Figure 3 caption"},{"comment":"There are several typos: 'Hamilton-Jabobi' should be 'Hamilton-Jacobi', 'Schawrzschild' should be 'Schwarzschild', and 'geodetic equations' should be 'geodesic equations'.","section":"Section II"},{"comment":"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.","section":"References"},{"comment":"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.","section":"Figure 3 caption"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely question and the numerical machinery appears to be a genuine independent benchmark, but the internal inconsistency between the abstract's headline numbers and the Table I/Fig. 3 scaling is a serious quantitative flaw that must be fixed. I would also ask the authors to provide convergence tests and to confront the one-body approximation quantitatively for the mp/Mbar ratios they actually use; without these, the paper's central extrapolated claims are not reliable. No concerns about novelty or scope: the topic is appropriate for a pulsar-timing or GR journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know about this paper is that it asks a timely question—whether the standard post-Newtonian timing formulas will be good enough for pulsar–black hole binaries that SKA is expected to find—and the qualitative answer is probably right: for edge-on binaries with massive companions, the PN delay can differ from a full geodesic computation at levels that matter. But the quantitative case is undermined by a discrepancy between the abstract and the paper's own scaling law. Using their Table I, a 100 M_sun black hole at aR = 10^6 M (about 1.5e13 cm) gives Delta_DIF = 2e-6 s. Scaling that to aR = 1e16 cm with their claimed aR^{-4/3} law gives roughly 3e-10 s, not the 1e-7 s stated in the abstract. One of those numbers has to be wrong, and the abstract's headline is what supports the SKA conclusion.\n\nWhat the paper does well: it extends the exact-geodesic timing method from earlier work on Sgr A* to stellar and intermediate-mass black holes, includes the pulsar's finite mass in the orbital motion (though not in the metric), and adds 1PN orbital corrections. The comparison with the post-Newtonian formulas is an external benchmark, with no parameters fitted to the target result, which is a genuine strength. The authors also clearly acknowledge the one-body approximation and its limits—the Section VII caveat that it is \"substantially wrong\" for comparable masses is honest.\n\nThe soft spots, in order: (1) the internal inconsistency above is the one that matters; it's independent of the one-body approximation and affects the paper's central quantitative claim. (2) There are no error bars or convergence checks, and the scaling law is fitted to the same data and then used for extrapolation. (3) For the fiducial 2 M_sun pulsar and 10 M_sun black hole, the mass ratio is 0.2, so the pulsar's gravity might not be negligible; the authors acknowledge this but do not estimate its effect on Delta_DIF. (4) The neglect of the retardation effect is mentioned but not mitigated.\n\nI would send this to peer review rather than desk reject it, because the question is important and the method is worth serious examination. But it needs major revision: the numbers have to be reconciled, and the numerical support needs to be much stronger. For your own reading, treat the qualitative warning as plausible and the specific numbers as provisional.\n\nRecommendation: engage with it, but don't cite the numbers until the revision appears.","headline":"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.","tokens_in":21149,"tokens_out":3927,"would_cite":false,"duration_ms":615575,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C10","83C25","83C57","70F05"],"pacs":["04.25.Nx","04.70.-s","97.60.Gb","95.30.Sf"],"model":"deepseek-v4-flash","headline":"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.","keywords":["pulsar timing","propagation delay","general relativity","post-Newtonian approximation","Shapiro delay","black hole companions","SKA","null geodesics"],"falsifier":"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.","tokens_in":20109,"feed_emoji":"🕐","tokens_out":14080,"duration_ms":106678,"temperature":0.7,"pith_summary":"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.","feed_headline":"Pulsar timing can be off by up to 100 microseconds near a black hole","feed_subtitle":"Full general-relativistic delay exceeds post-Newtonian formulas at SKA precision for pulsar-black hole binaries.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the exact analytical propagation delay formula for Schwarzschild geodesics on which the algorithm's elliptic-integral evaluation is based.","marker":"[17]"},{"why":"Provides the elliptic-integral solution for the Kerr geodesic delay and the frame-dragging comparison that the present algorithm extends to stellar-mass companions.","marker":"[5]"},{"why":"Demonstrates the same geodesic-delay approach for a pulsar orbiting Sgr A*, setting the methodological template used here.","marker":"[6]"},{"why":"Gives the first post-Newtonian two-body motion scheme whose eccentricity and periastron corrections the paper uses for the relativistic orbital case.","marker":"[7]"},{"why":"Provides the relation between binary and component eccentricities and the relativistic delay formulas used in the orbital-motion corrections.","marker":"[9]"},{"why":"Defines the post-Newtonian Roemer, lensed Shapiro, geometric, and companion-motion delays that form the comparison baseline.","marker":"[35]"},{"why":"Projects the ~10 ns SKA timing precision that sets the significance threshold for the discrepancy.","marker":"[30]"},{"why":"Provides the SKA precision and distance-measurement prospects that motivate the need for exact timing models.","marker":"[39]"}],"fun_headline_variants":["Exact GR timing needed for pulsar-black hole binaries","Pulsar timing errors reach 100 microseconds near black holes","Post-Newtonian pulsar timing off by up to 0.1 ms near black holes","Full GR beats post-Newtonian formulas for pulsar timing","Pulsar timing approximations fail near black hole companions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Exact GR timing needed for pulsar-black hole binaries","Pulsar timing errors reach 100 microseconds near black holes","Post-Newtonian pulsar timing off by up to 0.1 ms near black holes","Full GR beats post-Newtonian formulas for pulsar timing","Pulsar timing approximations fail near black hole companions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000828,"raw_usage":{"total_tokens":3732,"prompt_tokens":1174,"completion_tokens":2558,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":790,"completion_tokens_details":{"reasoning_tokens":2468}},"tokens_in":790,"tokens_out":2558,"duration_ms":16640,"temperature":1.0,"reasoning_tokens":2468,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:59:03.989348+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Hackmann and A","cited_arxiv_id":null,"evidence_quote":"Supplies the exact analytical propagation delay formula for Schwarzschild geodesics on which the algorithm's elliptic-integral evaluation is based."},{"cited_title":"Ben-Salem and E","cited_arxiv_id":null,"evidence_quote":"Provides the elliptic-integral solution for the Kerr geodesic delay and the frame-dragging comparison that the present algorithm extends to stellar-mass companions."},{"cited_title":"However, at the moment the timing of SGR J1745-2900 has not yet reached sufficient levels of precision for this purpose, mainly due to the intrinsic variability of the source","cited_arxiv_id":null,"evidence_quote":"Demonstrates the same geodesic-delay approach for a pulsar orbiting Sgr A*, setting the methodological template used here."},{"cited_title":"Damour and N","cited_arxiv_id":null,"evidence_quote":"Gives the first post-Newtonian two-body motion scheme whose eccentricity and periastron corrections the paper uses for the relativistic orbital case."},{"cited_title":"Damour and N","cited_arxiv_id":null,"evidence_quote":"Provides the relation between binary and component eccentricities and the relativistic delay formulas used in the orbital-motion corrections."},{"cited_title":"Luo et al","cited_arxiv_id":null,"evidence_quote":"Defines the post-Newtonian Roemer, lensed Shapiro, geometric, and companion-motion delays that form the comparison baseline."},{"cited_title":"Kremer, S","cited_arxiv_id":null,"evidence_quote":"Projects the ~10 ns SKA timing precision that sets the significance threshold for the discrepancy."},{"cited_title":"Ridolfi, P","cited_arxiv_id":null,"evidence_quote":"Provides the SKA precision and distance-measurement prospects that motivate the need for exact timing models."}],"review_version":1}