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On the effect of the light bending phenomenon for a pulsar in a binary with a Kerr black hole

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

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Pith's one-line read For a pulsar in a binary with a stellar-mass Kerr black hole, the companion's spin changes light-bending delays by only nanoseconds — three orders below the microsecond delays themselves — so spin can be ignored in timing models unless…

desk verdict First full-Kerr treatment of pulsar light-bending delays, with a careful numerical core and a plausible central result that is broader than the computed parameter coverage supports. read the letter →

arxiv 2506.05783 v1 pith:PHAVZAMY submitted 2025-06-06 gr-qc astro-ph.HEastro-ph.SR

Jyotijwal Debnath , Manjari Bagchi This is my paper
classification gr-qcastro-ph.HEastro-ph.SR PACS 04.70.-s97.60.Gb98.62.Sb
keywords pulsartiminglightbendingdelayKerrblackholeframedraggingpulseprofiledistortionnullgeodesicsbinarygravitationallensing
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 asks whether the rotation of a black hole companion measurably alters the gravitational bending of a pulsar's radio beam. Solving the null geodesic equations in Kerr spacetime for a representative stellar-mass pulsar–black-hole binary, the authors find that the black hole's spin magnitude and spin-axis orientation change the longitudinal and latitudinal bending delays only at the nanosecond level, while mass, orbital period, eccentricity, and inclination shift the delays at the microsecond level. As a result, the beam distortion and pulse-profile changes are essentially the same as for a non-spinning companion, and the Schwarzschild-based bending-delay models remain sufficient unless timing precision reaches the nanosecond scale. The spin reveals itself only in the frame-dragging bending delay, the difference between the delays with and without spin, which shows nanosecond discontinuities at orbital phases where the light ray's direction reverses relative to the spin axis. For a pulsar around a $10^6$-solar-mass black hole, the bending delays are about three orders of magnitude larger and the pulse profile is strongly reshaped near superior conjunction, but again the spin's additional effect is imperceptible.

What carries the argument

The carrying mechanism is the numerical integration of the null geodesic equations in Kerr spacetime, recast in Boyer-Lindquist coordinates with the two conserved quantities $D_\lambda$ (the conserved angular momentum) and $D_q$ (the Carter-like constant), using the integral formalism that separates the radial and polar parts of the trajectory. The paper computes the initial photon direction in the black hole's frame from the pulsar-beam geometry, evaluates the conserved quantities, integrates the geodesics, and iteratively refines the grid of starting points to find the photon that reaches the observer along the line of sight. From the final direction of that photon it extracts the longitudinal and latitudinal bending delays, and it defines the frame-dragging bending delay as the difference between the Kerr result and the Schwarzschild result. The pulse profile is obtained by projecting the bent photon directions back onto the beam cross-section using a core-double-cone beam model.

What would settle it

Repeat the frame-dragging bending delay calculation for the same binary (M_c = 14.5 solar masses, spin parameter 0.9, inclination 87.5 degrees) but scan the pulsar orientation angles across the full allowed visibility range, for example $\alpha = 10^\circ$ or a beam without the central core component. If any configuration yields a maximum latitudinal or longitudinal frame-dragging delay above 0.1 microseconds, the claim that spin effects are negligible is falsified for that geometry.

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Extended reading notes

Core claim

The central claim is that for a pulsar with a stellar-mass rotating black hole companion, the spin of the black hole is effectively invisible in the light-bending phenomenon: the magnitude of the spin (spin parameter from 0 to 0.9) and the orientation of the spin axis change the longitudinal and latitudinal bending delays in the nanosecond range, whereas the delays themselves are in the microsecond range and respond at the microsecond level to changes in companion mass, orbital period, eccentricity, and inclination. Consequently the distortion of the beam and the resulting changes in the pulse shape are minimally influenced by the spin-related parameters. The spin's influence appears only through the frame-dragging bending delay, defined as the difference between the bending delay computed with spin and without spin; this difference exhibits nanosecond discontinuities at orbital phases where the line of sight, the pulsar's position vector, and the black hole's spin axis lie in the same plane, i.e., where the light ray switches from co-rotating to counter-rotating with respect to the spin. For a pulsar with a supermassive black hole companion, the bending delays become three orders of magnitude larger and the pulse profile near superior conjunction is strongly reshaped, yet the profiles computed for different spin parameters remain indistinguishable.

Load-bearing premise

The nanosecond negligibility of the spin rests on a single representative choice of pulsar orientation angles ($\alpha = 50^\circ$, $\eta_p = 45^\circ$, $\lambda_p = 130^\circ$) and a core-double-cone beam model; a substantially different beam geometry or pulsar orientation could make the spin effect larger.

Editorial extensions

If this is right

  • Existing Schwarzschild-based analytic bending-delay formulae suffice for timing models of pulsar–stellar-mass black hole binaries unless timing precision reaches the nanosecond level.
  • Black hole spin magnitude and orientation can be dropped from the parameter space when fitting pulse-profile distortions, simplifying the timing solution.
  • Near superior conjunction, microsecond irregularities in bending delays and the associated profile distortion can bias template-based time-of-arrival estimation, so those phases need special treatment.
  • For pulsar–supermassive black hole binaries, bending delays reach the millisecond scale and pulse profiles are strongly reshaped at and near superior conjunction, making timing significantly harder.

Reading between the lines

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

  • If timing precision ever reaches roughly a nanosecond, the frame-dragging delay discontinuity would become a geometric probe: measuring the orbital phase at which the jump occurs would locate the black-hole spin axis direction in the sky.
  • Within current and near-future timing precision, the spin of a stellar-mass black hole companion is effectively unmeasurable through light bending; spin might instead be constrained via Shapiro delay asymmetries or orbital precession.
  • The strong phase-dependent pulse reshaping around superior conjunction in the supermassive case implies that a single stable template per orbit would introduce systematic time-of-arrival errors, so profile-morphology fitting across orbital phase would be required.
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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 studies light-bending delays and pulse-profile distortion for a pulsar in a binary with a rotating (Kerr) black hole companion, extending the authors' earlier Schwarzschild treatment (Paper-I). The authors integrate null geodesics in Kerr spacetime using the Kapec-Lupsasca / Gralla-Lupsasca formalism, model the pulsar beam with a core-double-cone emission pattern, and compute longitudinal and latitudinal bending delays and their spin-dependent differences (FD bending delays) for a representative stellar-mass black hole binary. They also present a test case with a 10^6 solar-mass black hole companion. The central claim is that black-hole spin magnitude and orientation affect bending delays at the nanosecond level, whereas other parameters affect them at the microsecond level, so spin can be neglected for modeling light-bending delays unless timing precision reaches the nanosecond scale. The paper additionally describes nanosecond-scale discontinuities in FD delays and shows that near superior conjunction the pulse profile is strongly distorted for a supermassive black hole but minimally affected by the spin parameter.

Significance. If the spin-negligibility claim holds, the paper provides practical guidance for future pulsar-black-hole timing models: standard Schwarzschild-based bending-delay formulas would suffice for stellar-mass black holes except at potentially nanosecond-level precision. The work is technically ambitious: it combines full Kerr geodesic integration, a detailed beam model, iterative ray shooting with a reported 0.1 ns convergence after 15 iterations, and a cross-check of the a=0 limit against the Schwarzschild treatment. The supermassive-black-hole test case and the geometric explanation of strong bending near superior conjunction (Appendix) are valuable additions. However, the quantitative hierarchy between spin effects and other parameter effects is established only for a narrow set of configurations, and the numerical validation residual is comparable to the spin effects being isolated. The paper is a useful contribution, but the headline conclusion needs stronger parameter coverage and convergence evidence before it can be accepted as a general statement.

major comments (3)
  1. [Sec. 3, Figs. 8-11; Sec. 5] The central claim that spin-related parameters affect bending delays only at the nanosecond level is not supported over the full parameter space relevant to pulsar-black-hole binaries. The scans in Figs. 9 and 10 vary M_c, P_b, e, i, and eta_p at i=87.5 degrees, and the only edge-on cases (Fig. 8) use P_b=5.5 days. The authors do not combine i=90 degrees with short orbital periods, where the dominant ray impact parameter is the Einstein radius proportional to sqrt(M_c a) (Eq. 48) and the frame-dragging contribution relative to the monopole bending scales as P_b^{-2/3}. Given that the edge-on FD delays at P_b=5.5 days already reach tens to hundreds of nanoseconds (Fig. 8c,d), shorter periods at the same geometry could plausibly move FD delays into the microsecond regime, contradicting the abstract's nanosecond-level negligibility statement. The conclusion in Sec. 5 should be restricted to the tested parameter region, or new runs at i=90 degrees with P_b in the range 1-3 days (and a scan of pulsar orientation angles alpha, eta_p, lambda_p) should be added.
  2. [Sec. 3, Fig. 3] The validation of the Kerr code in the a=0 limit against the Schwarzschild treatment shows a residual difference of 0.9 ns in the longitudinal bending delay at superior conjunction and below 0.004 ns elsewhere. This residual is of the same order as the spin-induced FD delays that the paper aims to isolate at i=87.5 degrees (Figs. 6, 9, 10). The authors should demonstrate that the FD delays themselves are converged to well below the nanosecond level in the regions where the spin effect is claimed to be nanosecond-scale, for example by increasing N or the number of iterations and reporting the resulting change in the FD delays. Without this, the quoted nanosecond hierarchy at superior conjunction is not securely separated from numerical discretization error.
  3. [Sec. 3, Sec. 4] The paper uses a single representative set of unobservable pulsar orientation angles (alpha=50 degrees, eta_p=45 degrees, lambda_p=130 degrees) and a single beam model (core-double cone from Paper-I). The visibility condition (Eq. 25) restricts lambda_p to a narrow range for this alpha, but alpha itself is not scanned, and alternative beam geometries are not tested. Since the bending delays and FD delays depend on the beam geometry through the mapping of the line of sight onto the emission cone, the claim that spin effects are 'minimally influenced' by spin parameters could be a property of this particular geometry rather than a general result. The authors should scan alpha and, ideally, test at least one alternative beam model or explicitly argue that the conclusions are independent of the beam geometry.
minor comments (5)
  1. [Sec. 1] The text says 'In section 6, these features have been explained with a simplified geometry... The paper is ended with a short conclusive discussion in section 5.' This ordering is inconsistent: Section 5 is the conclusion and the simplified-geometry discussion is the Appendix. Please renumber or rephrase.
  2. [Captions of Figs. 9 and 10] The captions state 'we have chosen a=0.5 except for panel (f) where a is varied,' but panels (e) and (f) show variations of eta_p, and the spin-parameter variation is shown in Fig. 11. The caption and the text describing these panels should be corrected.
  3. [Sec. 4] The sentence 'They modelled the spin-spin and spin-curvature using with the help of with the help of Mathisson-Papapetrou-Dixon (MPD) equations' contains a duplicated phrase. Please fix.
  4. [Sec. 3, Fig. 9 caption] The caption says 'we set lambda_bh=i=87.5 degrees except for panel (c) where i is varied,' but panel (c) varies eccentricity, not inclination. The correct panel for inclination variation is (d).
  5. [Sec. 4] For the supermassive-black-hole test case, the paper states that omega changes by about 2 degrees per orbit and calls this 'very small' and ignorable, yet later notes that at t=P_b the delays differ from t=0 due to spin and orbital precession. The wording is contradictory; if the precession is included in the numerical runs, the text should say so explicitly, and if it is ignored, the statement about 2-degree change should be revised.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the nanosecond-versus-microsecond hierarchy is a computed numerical result, not a fitted or self-referential claim.

full rationale

The central numerical claim—that frame-dragging (spin) contributions to the bending delays are nanosecond-order while the bending delays themselves are microsecond-order—is obtained by explicit ray tracing through Kerr null geodesics, not by fitting or by definitional identity. Equations (28)-(29) define the FD delays as differences of independently computed delays, and the code is checked against the Schwarzschild limit (Fig. 3), which is an internal consistency test, not a circular reduction. The beam geometry and pulse-profile construction are inherited from Paper-I (Debnath, Bagchi, Basu 2023), a self-citation; however, this inheritance supplies the pulsar beam model and visibility conditions, and it does not itself determine the magnitude ordering of FD versus ordinary bending delays. No uniqueness theorem from the authors' prior work is invoked to force the Kerr formalism, and no parameter is fitted to the 'nanosecond vs microsecond' conclusion. The limited scan over inclination and beam models is a robustness limitation, which is outside the scope of circularity. Accordingly, no load-bearing circular step is identified; the paper's result is self-contained given its stated external geodesic formalism.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The central claims depend on standard GR geodesic separability in Kerr spacetime (external formalism), standard post-Newtonian evolution equations, and a set of arbitrarily chosen unobservable angles for the pulsar and black hole spin orientations. No new physical entities are introduced. The heaviest burden is the representativeness of the chosen angles and beam model, which is not exhaustively explored.

free parameters (7)
  • a_tilde (dimensionless black hole spin) = 0.1, 0.5, 0.9
    Chosen to represent low, intermediate, and near-extremal spin; not fitted to data. Affects FD bending delays (Fig. 11).
  • alpha (angle between pulsar spin and magnetic axes) = 50 deg
    Unobservable and chosen to satisfy the visibility condition Eq. (24); affects the beam-LoS geometry.
  • eta_p (azimuthal orientation of pulsar spin in sky plane) = 45 deg
    Unobservable, chosen arbitrarily; Figs. 9e-f and 10e-f show FD delays depend on it.
  • lambda_p (polar angle of pulsar spin with respect to LoS) = 130 deg
    Unobservable, chosen near the middle of the allowed range 125.49-134.51 deg for alpha=50 deg.
  • eta_bh (azimuthal orientation of BH spin in sky plane) = -90 deg (default), varied 0-360 deg
    Unobservable, chosen; determines the discontinuity location via Eq. (43) and the FD delay magnitude.
  • lambda_bh (polar angle of BH spin with respect to LoS) = i (87.5 or 90 deg) default, varied 0-180 deg
    Unobservable, chosen; FD delays are maximum when lambda_bh = i.
  • M_c for SMBH test case = 1e6 solar masses
    Chosen as a test case for a supermassive companion; all other parameters remain as in Table 1.
assumptions (5)
  • standard math Separability of null geodesics in Kerr spacetime and the integral form of the geodesic equations (Gralla & Lupsasca 2020; Kapec & Lupsasca 2020)
    Invoked in Sec. 2.6, Eqs. (30)-(31), to compute light-ray trajectories around the rotating black hole.
  • domain assumption The black hole is treated as stationary during the light-crossing time and its orbital motion is ignored
    Stated in Sec. 1: 'the motion of the gravitating body (the companion of the pulsar) is ignored', inherited from Paper-I.
  • domain assumption The black hole spin vector is aligned with the orbital angular momentum vector in the orbital evolution model
    Stated in Sec. 2.3; this neglects black-hole spin precession and is used when computing P_b, e, omega and the pulsar spin precession.
  • standard math Post-Newtonian equations of Peters (1964), Barker & O'Connell (1975), and Damour & Schafer (1988) describe the orbital and spin evolution
    Used in Sec. 2.3 to evolve the orbital elements and the pulsar spin vector.
  • domain assumption The core-double-cone beam model with the intensity distribution and emission height from Paper-I is a valid description of the pulsar beam
    Adopted in Sec. 2.1 and Sec. 3; the computed pulse profiles and their bending-induced distortion depend on this model.

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Pith. "Pith review of On the effect of the light bending phenomenon for a pulsar in a binary with a Kerr black hole." pith.science (2026). https://pith.science/paper/PHAVZAMY

@misc{pith2026250605783,
  author       = {Pith},
  title        = {Pith review of: On the effect of the light bending phenomenon for a pulsar in a binary with a Kerr black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PHAVZAMY}},
  note         = {Machine review of arXiv:2506.05783}
}
read the original abstract

We study the effect of light-bending on the signal of a pulsar in binaries with rotating black hole companions, focusing on stellar mass black holes. We show that the impacts of various parameters on the bending delays visually match with those observed for a non-rotating black holes, because the magnitude of the spin as well as the orientation of the spin axis of the black hole introduce changes in the nanosecond order and other parameters do so in the microsecond order. Consequently, the distortion of the beam and the resulting changes in the pulse shape are minimally influenced by spin-related parameters of the black hole. We also investigate the impact of various parameters on the difference of the delays with and without the spin of the black hole and notice nanosecond scale discontinuities at orbital phases where the path of the light ray changes its direction with respect to the direction of the spin of the black hole. Moreover, as in the Schwarzschild case, the bending delays become irregular (on the microsecond scale) near the superior conjunction. We also explore the effect of bending on the pulse profiles and bending delays if the companion of the pulsar is a rotating super-massive black hole. We find significant enhancement and change in the shape of the profiles at and near the superior conjunction in comparison to stellar mass black holes. Moreover, bending delays are about three orders of magnitude higher than those in case of the stellar mass black holes.

Figures

Figures reproduced from arXiv: 2506.05783 by the authors.

Figure 1
Figure 1. Reference frames, angles, and unit vectors used to model the ge￾ometry of the beam of the pulsar, as explained in the text. makes it appear to originate from a co-latitude 𝜁𝐿. The unit vectors 𝑚bI and b𝑛I can be expressed as: 𝑚bI = [cos Φ𝑚 sin 𝛼, sin Φ𝑚 sin 𝛼, cos 𝛼] (1) and b𝑛I = [cos Φ𝑝 sin 𝜁𝑝, sin Φ𝑝 sin 𝜁𝑝, cos 𝜁𝑝]. (2) From [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. The orbital geometry of a pulsar (denoted by a ★) in a binary system. The sky plane (the XsYs plane) is illustrated in light red, while the orbital plane (the XbYb plane) is shown in light green. The blue plane represents the X ′ I Y ′ I plane. The unit vector along the spin axis of the pulsar is denoted by 𝑆b𝑝 and the unit vector along the spin axis of the companion is denoted by 𝑆bbh, both have been shown after sh… view at source ↗
Figure 3
Figure 3. The differences in the bending delays calculated using two different solutions of the null geodesic around a non-rotating black hole, one setting the spin parameter 𝑎˜ = 0 in the Kerr solutions and the other using the Schwarzschild solution. For both of the cases, we use 𝜆bh = 𝑖 = 87.5 ◦ , and the values of all other relevant parameters as listed in [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (20 more)
Figure 4
Figure 4. Figure 4: The comparison of the bending delays for 𝑎˜ = 0 (black dashed line) and 𝑎˜ = 0.9 (blue triangles) over one full orbit for a hypothetical pulsar-black hole binary with 𝜆bh = 𝑖 = 87.5 ◦ . The values of all other relevant parameters are the same as those listed in [PITH_…
Figure 5
Figure 5. Figure 5: A schematic diagram showing how the directions of the light rays (orange lines) aligning with the LoS (the purple line) vary with respect to the spin of the black hole, represented by a grey dashed circle with arrows. Various positions of the pulsar in its orbit (the l…
Figure 6
Figure 6. Figure 6: The FD bending delays for 𝑎˜ = 0.9 over one full orbit for a hypothetical pulsar-black hole binary with 𝜆bh = 𝑖 = 87.5 ◦ . The values of all other relevant parameters are the same as those listed in [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: The FD bending delays for 𝑎˜ = 0.9 over one full orbit for a hypothetical pulsar-black hole binary. We have chosen 𝜂bh = −40◦ , 𝜆bh = 𝑖 = 87.5 ◦ , and the values of all other relevant parameters are the same as those listed in [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: The bending delays and the FD bending delays obtained for a hypothetical pulsar-black hole binary. The upper left panel (panel a) shows the latitudinal bending delay and the upper right panel (panel b) shows the longitudinal bending delay. The lower left panel (panel c…
Figure 9
Figure 9. Figure 9: The latitudinal FD bending delay curves for hypothetical pulsar-black hole binaries, changing one parameter in each panel and keeping all the other parameters the same as those mentioned in [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: The latitudinal FD bending delay curves for hypothetical pulsar-black hole binaries, changing one parameter in each panel and keeping all the other parameters the same as those mentioned in [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: The latitudinal (panel a) and the longitudinal (panel b) FD bending delay curves for hypothetical pulsar-black hole binaries for different values of 𝑎˜. We use 𝜆bh = 𝑖 = 87.5 ◦ and the values of all the other relevant parameters are taken the same as those mentioned i…
Figure 12
Figure 12. Figure 12: The impact of change in the values of 𝜂bh and 𝜆bh on the latitudinal and the longitudinal bending delays for 𝑎˜ = 0.9 for hypothetical pulsar-black hole binaries with 𝑖 = 90◦ . In each panel, the orbital phase is plotted in degrees along the abscissa only in the range…
Figure 13
Figure 13. Figure 13: The latitudinal FD bending delays for hypothetical pulsar-black hole binaries with 𝑎˜ = 0.5, for different orientations of the spin axis of the black hole. In each panel. Only one parameter is varied at a time. We take 𝑖 = 87.5 ◦ and other relevant parameters are kept…
Figure 14
Figure 14. Figure 14: The longitudinal FD bending delays for hypothetical pulsar-black hole binaries with 𝑎˜ = 0.5, for different orientations of the spin axis of the black hole. In each panel. Only one parameter is varied at a time. We take 𝑖 = 87.5 𝑐𝑖𝑟𝑐 and other relevant parameters are …
Figure 15
Figure 15. Figure 15: The photon distribution on the cross-section of the beam of a pulsar with a black hole companion with 𝜆bh = 𝑖 = 90◦ . The top-left panel is the photon distribution of the original beam (without any bending), the top-right panel is the photon distribution of the distor…
Figure 16
Figure 16. Figure 16: Comparison of the pulse profile of a pulsar with a black hole companion with 𝜆bh = 𝑖 = 90◦ and 𝐴𝑇 + 𝜔 = 90◦ . We show the profiles without bending (black dashed line) and with bending for various values of the spin parameter of the black hole, e.g., 𝑎˜ = 0 (solid oran…
Figure 17
Figure 17. Figure 17: Comparison of the pulse profile of a pulsar with a black hole companion with 𝜆bh = 𝑖 = 90◦ . We show the profiles without bending (black dashed line) and with bending for various values of the spin parameter of the black hole, e.g., 𝑎˜ = 0 (solid orange line), 𝑎˜ = 0.…
Figure 18
Figure 18. Figure 18: Comparison of the pulse profile of a pulsar with a black hole companion with 𝜆bh = 𝑖 = 85◦ , 𝐴𝑇 + 𝜔 = 90◦ . We show the profiles without bending (black dashed line) and with bending for various values of the spin parameter of the black hole, e.g., 𝑎˜ = 0 (solid orange…
Figure 19
Figure 19. Figure 19: Comparison of the pulse profile of a pulsar with a black hole companion with 𝜆bh = 𝑖 = 80◦ . We show the profiles without bending (black dashed line) and with bending for various values of the spin parameter of the black hole, e.g., 𝑎˜ = 0 (solid orange line), 𝑎˜ = 0.…
Figure 20
Figure 20. Figure 20: The bending delays over one full orbit for a hypothetical pulsar-super massive black hole binary with 𝜆bh = 𝑖 = 80◦ . The left panel shows the latitudinal bending delays and the right panel is for the longitudinal bending delay. In both of the cases, we have used diff…
Figure 21
Figure 21. Figure 21: The FD bending delays over one full orbit for a hypothetical pulsar-super massive black hole binary with 𝜆bh = 𝑖 = 80◦ . The left panel shows the latitudinal bending delays and the right panel is for the longitudinal bending delay. In both of the cases, we have used d…
Figure 22
Figure 22. Figure 22: A schematic diagram showing bending and propagation of light rays at the superior conjunction configuration. 𝐸 ′′. As the rays that were above the 𝑋𝑌 line before 𝐼 are below the 𝑋𝑌 line in the 𝑋3-plane (and vice versa), 𝐼 ′′ 2 might be termed as an inverted image of 𝐼…
Figure 23
Figure 23. Figure 23: Understanding strong bending for a pulsar - super massive black hole binary system when the pulsar is near the superior conjunction for different combinations of 𝑖 and 𝐴𝑇 + 𝜔 in the observer’s frame, where the centre of the beam moves along the magenta line ‘𝐴𝐵’ that …

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Works this paper leans on

37 extracted references · 17 canonical work pages

  1. [1]

    Agazie G., et al., 2024, @doi [ ] 10.3847/1538-4357/ad36be , https://ui.adsabs.harvard.edu/abs/2024ApJ...966..105A 966, 105

    doi:10.3847/1538-4357/ad36be 2024

  2. [2]

    Bagchi M., 2010, @doi [ ] 10.1016/j.newast.2009.07.003 , https://ui.adsabs.harvard.edu/abs/2010NewA...15..126B 15, 126

    doi:10.1016/j.newast.2009.07.003 2010

  3. [3]

    Bagchi M., 2018, @doi [Universe] 10.3390/universe4020036 , 4, 36

    doi:10.3390/universe4020036 2018

  4. [4]

    F., 2014, @doi [Journal of Cosmology and Astroparticle Physics] 10.1088/1475-7516/2014/08/055 , 2014, 055

    Bagchi M., Torres D. F., 2014, @doi [Journal of Cosmology and Astroparticle Physics] 10.1088/1475-7516/2014/08/055 , 2014, 055

    doi:10.1088/1475-7516/2014/08/055 2014

  5. [5]

    M., O'Connell R

    Barker B. M., O'Connell R. F., 1975, @doi [ ] 10.1103/PhysRevD.12.329 , https://ui.adsabs.harvard.edu/abs/1975PhRvD..12..329B 12, 329

    doi:10.1103/physrevd.12.329 1975

  6. [6]

    Ben-Salem B., Hackmann E., 2022, @doi [ ] 10.1093/mnras/stac2337 , https://ui.adsabs.harvard.edu/abs/2022MNRAS.516.1768B 516, 1768

    work page doi:10.1093/mnras/stac2337 2022

  7. [7]

    A., Matzner R

    Bunandar D., Caveny S. A., Matzner R. A., 2011, @doi [ ] 10.1103/PhysRevD.84.104005 , https://ui.adsabs.harvard.edu/abs/2011PhRvD..84j4005B 84, 104005

    work page doi:10.1103/physrevd.84.104005 2011

  8. [8]

    Oxford University Press, New York

    Chandrasekhar S., 1983, The Mathematical Theory of Black Holes . Oxford University Press, New York

    work page 1983

Show all 37 references
  1. [9]

    R., Bailes M., Broekgaarden F., 2021, @doi [Monthly Notices of the Royal Astronomical Society] 10.1093/mnras/stab973 , 504, 3682

    Chattopadhyay D., Stevenson S., Hurley J. R., Bailes M., Broekgaarden F., 2021, @doi [Monthly Notices of the Royal Astronomical Society] 10.1093/mnras/stab973 , 504, 3682

    2021 doi

  2. [10]

    Damour T., Schafer G., 1988, @doi [Nuovo Cimento B Serie] 10.1007/BF02828697 , https://ui.adsabs.harvard.edu/abs/1988NCimB.101..127D 101B, 127

    1988 doi

  3. [11]

    Debnath J., Bagchi M., Basu A., (Paper-I) 2023, @doi [ ] 10.1093/mnras/stad2147 , https://ui.adsabs.harvard.edu/abs/2023MNRAS.524.5411D 524, 5411

    2023 doi

  4. [12]

    V., Kopeikin S

    Doroshenko O. V., Kopeikin S. M., 1995, MNRAS, 274, 1029

    1995

  5. [13]

    H., 2017, @doi [ ] 10.3847/1538-4357/aa610e , https://ui.adsabs.harvard.edu/abs/2017ApJ...837...87E 837, 87

    Estes J., Kavic M., Lippert M., Simonetti J. H., 2017, @doi [ ] 10.3847/1538-4357/aa610e , https://ui.adsabs.harvard.edu/abs/2017ApJ...837...87E 837, 87

    2017 doi

  6. [14]

    Fuller J., Ma L., 2019, @doi [ ] 10.3847/2041-8213/ab339b , https://ui.adsabs.harvard.edu/abs/2019ApJ...881L...1F 881, L1

    2019 doi

  7. [15]

    E., Lupsasca A., 2020, @doi [Phys

    Gralla S. E., Lupsasca A., 2020, @doi [Phys. Rev. D] 10.1103/PhysRevD.101.044032 , 101, 044032

    2020 doi

  8. [16]

    Hu H., et al., 2022, @doi [ ] 10.1051/0004-6361/202244825 , 667, A149

    2022 doi

  9. [17]

    I., 2012, @doi [ ] 10.1111/j.1365-2966.2011.20238.x , https://ui.adsabs.harvard.edu/abs/2012MNRAS.420.2325J 420, 2325

    Jones D. I., 2012, @doi [ ] 10.1111/j.1365-2966.2011.20238.x , https://ui.adsabs.harvard.edu/abs/2012MNRAS.420.2325J 420, 2325

    2012

  10. [18]

    Kapec D., Lupsasca A., 2020, @doi [Class. Quant. Grav.] 10.1088/1361-6382/ab519e , 37, 015006

    2020 doi

  11. [19]

    Kopeikin S., Mashhoon B., 2002, @doi [ ] 10.1103/PhysRevD.65.064025 , https://ui.adsabs.harvard.edu/abs/2002PhRvD..65f4025K 65, 064025

    2002 doi

  12. [20]

    Laguna P., Wolszczan A., 1997, @doi [The Astrophysical Journal] 10.1086/310835 , 486, L27

    1997 doi

  13. [21]

    J., Wu K., Singh D., 2019, @doi [ ] 10.1093/mnras/stz389 , https://ui.adsabs.harvard.edu/abs/2019MNRAS.485.1053L 485, 1053

    Li K. J., Wu K., Singh D., 2019, @doi [ ] 10.1093/mnras/stz389 , https://ui.adsabs.harvard.edu/abs/2019MNRAS.485.1053L 485, 1053

    2019 doi

  14. [22]

    J., Wu K., Leung P

    Li K. J., Wu K., Leung P. K., Singh D., 2022, @doi [ ] 10.1093/mnras/stab2925 , https://ui.adsabs.harvard.edu/abs/2022MNRAS.511.3602L 511, 3602

    2022 doi

  15. [23]

    M., Lazio T

    Liu K., Wex N., Kramer M., Cordes J. M., Lazio T. J. W., 2012, @doi [ ] 10.1088/0004-637X/747/1/1 , https://ui.adsabs.harvard.edu/abs/2012ApJ...747....1L 747, 1

    2012 doi

  16. [24]

    R., Kramer M., 2004, Handbook of Pulsar Astronomy

    Lorimer D. R., Kramer M., 2004, Handbook of Pulsar Astronomy . Cambridge University Press, Cambridge

    2004

  17. [25]

    Maj \'a r J., 2009, @doi [ ] 10.1103/PhysRevD.80.104028 , https://ui.adsabs.harvard.edu/abs/2009PhRvD..80j4028M 80, 104028

    2009 doi

  18. [26]

    Maj \'a r J., Mik \'o czi B., 2012, @doi [ ] 10.1103/PhysRevD.86.064028 , https://ui.adsabs.harvard.edu/abs/2012PhRvD..86f4028M 86, 064028

    2012 doi

  19. [27]

    F., Ruba B., Oancea M

    Paganini C. F., Ruba B., Oancea M. A., 2016, @doi [arXiv e-prints] 10.48550/arXiv.1611.06927 , p. arXiv:1611.06927

    2016 doi arXiv

  20. [28]

    C., 1964, @doi [Physical Review] 10.1103/PhysRev.136.B1224 , https://ui.adsabs.harvard.edu/abs/1964PhRv..136.1224P 136, 1224

    Peters P. C., 1964, @doi [Physical Review] 10.1103/PhysRev.136.B1224 , https://ui.adsabs.harvard.edu/abs/1964PhRv..136.1224P 136, 1224

    1964 doi

  21. [29]

    R., Lai D., 2006, @doi [ ] 10.1103/PhysRevD.73.063003 , https://ui.adsabs.harvard.edu/abs/2006PhRvD..73f3003R 73, 063003

    Rafikov R. R., Lai D., 2006, @doi [ ] 10.1103/PhysRevD.73.063003 , https://ui.adsabs.harvard.edu/abs/2006PhRvD..73f3003R 73, 063003

    2006 doi

  22. [30]

    L., Tartaglia A., 2005, @doi [Phys

    Ruggiero M. L., Tartaglia A., 2005, @doi [Phys. Rev. D] 10.1103/PhysRevD.72.084030 , 72, 084030

    2005 doi

  23. [31]

    Schneider J., 1990, , 232, 62

    1990

  24. [32]

    E., 2014, @doi [ ] 10.1093/mnras/stu614 , https://ui.adsabs.harvard.edu/abs/2014MNRAS.441..800S 441, 800

    Singh D., Wu K., Sarty G. E., 2014, @doi [ ] 10.1093/mnras/stu614 , https://ui.adsabs.harvard.edu/abs/2014MNRAS.441..800S 441, 800

    2014 doi

  25. [33]

    L., Nagar A., 2005, @doi [Phys

    Tartaglia A., Ruggiero M. L., Nagar A., 2005, @doi [Phys. Rev. D] 10.1103/PhysRevD.71.023003 , 71, 023003

    2005 doi

  26. [34]

    L., Capolongo E., 2011, @doi [Advances in Space Research] 10.1016/j.asr.2010.10.023 , https://ui.adsabs.harvard.edu/abs/2011AdSpR..47..645T 47, 645

    Tartaglia A., Ruggiero M. L., Capolongo E., 2011, @doi [Advances in Space Research] 10.1016/j.asr.2010.10.023 , https://ui.adsabs.harvard.edu/abs/2011AdSpR..47..645T 47, 645

    2011 doi

  27. [35]

    M., 1999, @doi [ ] 10.1086/306933 , https://ui.adsabs.harvard.edu/abs/1999ApJ...514..388W 514, 388

    Wex N., Kopeikin S. M., 1999, @doi [ ] 10.1086/306933 , https://ui.adsabs.harvard.edu/abs/1999ApJ...514..388W 514, 388

    1999 doi

  28. [36]

    Yang X., Wang J., 2013, @doi [The Astrophysical Journal Supplement Series] 10.1088/0067-0049/207/1/6 , 207, 6

    2013 doi

  29. [37]

    Zhang F., Saha P., 2017, @doi [ ] 10.3847/1538-4357/aa8f47 , https://ui.adsabs.harvard.edu/abs/2017ApJ...849...33Z 849, 33

    2017 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.