Leading-order gravitational time delay of massive particles by a moving Schwarzschild lens

Guansheng He; Wenbin Lin; Yi Xie

REVIEW 2 major objections 6 minor 94 references

Leading-order gravitational time delay of massive particles by a moving Schwarzschild lens

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A single first-post-Minkowskian formula gives the gravitational time delay for both massive particles and light crossing a radially moving Schwarzschild black hole.

desk verdict A careful 1PM calculation of massive-particle Shapiro delay around a moving Schwarzschild lens; the new formula is probably right, but the detectability section overreaches and the input geodesic equations deserve independent scrutiny. read the letter →

arxiv 2506.14184 v1 pith:K2MMXTFD submitted 2025-06-17 gr-qc

Guansheng He , Wenbin Lin , Yi Xie This is my paper

classification gr-qc MSC 83C1083C2583C57 PACS 04.25.Nx95.30.Sf98.62.Sb

keywords gravitationaltimedelayShapiromovingSchwarzschildlensmassiveparticlepropagationpost-Minkowskianapproximationmulti-messengerastronomyblackholelensingradialvelocityeffect

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 derives a single formula for the time a relativistic particle takes to cross the gravitational field of a Schwarzschild black hole that moves with constant radial velocity. The formula works for both massive particles, such as neutrinos or cosmic rays, and for light, at first post-Minkowskian order; it is claimed to be the first unified treatment in this moving-lens geometry. The paper shows that in the slow-motion limit a lens moving toward the observer shortens the flight time of an ultrarelativistic massive particle compared with a static lens, while a receding lens lengthens it. It also estimates the effect's magnitude using three representative black-hole systems and argues the effect lies within reach of current or near-future timing instruments.

What carries the argument

The central object is the set of first-post-Minkowskian geodesic equations, Eqs. (6)-(8), for a test particle of asymptotic speed $w$ moving parallel to the lens velocity in the weak-field metric of a radially moving Schwarzschild black hole, Eq. (1). The derivation integrates the coordinate-time derivative along the perturbed path, converts from observer-frame to comoving-frame coordinates with the zeroth-order transformation $dx = [\gamma^{-1}(1-v/w)^{-1}+O(M)]\,dX$, and collects the logarithmic Shapiro-type integral. The boundary condition $C=0+O(M^2)$, fixing the asymptotic time coordinates at source and observer, completes the formula.

What would settle it

Integrate the full (unexpanded) geodesic equations numerically in the metric (1) for a timelike particle with, say, w=0.9 and v=0.1, and compare the O(M) part of the resulting coordinate travel time with Eq. (18); any mismatch at that order would falsify the formula. Alternatively, test the v=0 limit: Eq. (18) must reduce to the known static Schwarzschild timelike delay (3/w - 1/$w^{3}$) M ln(4 d_L d_LS / $b^{2}$); if it does not, the derivation is inconsistent.

Watch

Extended reading notes

Core claim

The paper's central claim is a unified first-post-Minkowskian formula, Eq. (18), for the coordinate travel time of a relativistic test particle crossing the field of a Schwarzschild black hole moving with constant radial velocity $v$: $$\$\Delta$ t = \frac{d_S}{w} + \frac{(1-vw)\,\gamma\,[(3-$v^{2}$)$w^{2}$ - 1 - 4vw + $3v^{2}$]}{w(w-v)^2}\, M \ln\!\left(\frac{4\$gamma^{2}$ (1-v/w)^2 d_L d_{LS}}{$b^{2}$}\right) + O($M^{2}$),$$ where $w$ is the particle's initial speed ($w=1$ for light), $d_S$, $d_L$, and $d_{LS}$ are angular diameter distances, $b$ is the impact parameter, and $\gamma=(1-v^2)^{-1/2}$. The same expression is derived from the 1PM geodesic equations and gives the delay for massive particles and photons in one formula. In the slow-motion limit, the paper finds that a lens moving toward the observer reduces the flight time of an ultrarelativistic massive particle relative to a static Schwarzschild lens, and a receding lens increases it; the effect is proportional to $v\,\delta\,M$ with $\delta=1-w$. The paper then estimates the magnitude using three black-hole models and concludes that the radial-velocity effect should be measurable with current or near-future timing capabilities.

Load-bearing premise

The derivation assumes, without re-deriving here, that the first-order post-Minkowskian geodesic equations (6)-(8) and the moving-lens metric (1) taken from earlier work are correct for a particle coming from infinity with speed w parallel to the lens velocity; if either is wrong, Eq. (18) inherits the error.

Editorial extensions

If this is right

At $w=1$, Eq. (18) reduces to Eq. (12), the established moving-lens light delay, so one expression now covers both photon and massive-particle timing in the same geometry.
For a slow lens and an ultrarelativistic particle, Eq. (22) gives a $v$-dependent shift $\Delta t_M \simeq -2v(1+\delta)M[2+\ln(4d_Ld_{LS}/b^2)]$; an approaching lens shortens the flight time and a receding lens lengthens it.
The messenger-messenger differences, Eqs. (24)-(25), are proportional to $v\,\delta\,M$ (or $v(\delta_2-\delta_1)M$), so the lens-motion contamination of neutrino-photon arrival-time comparisons is a computable leading-order effect.
Under the paper's assumed parameters ($b=R_E$, $d_{LS}=0.01$ kpc), the radial-velocity effect for Cyg X-1, NGC 3319*, and Sgr A* exceeds the timing resolutions of LHAASO and IceCube by orders of magnitude, making the effect potentially observable.
The formula provides a direct consistency check for multi-messenger observations: measured photon and neutrino delays through the same lens must satisfy the unified expression or the lens-motion model is excluded.

Reading between the lines

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

The paper leaves implicit that the $v$-dependent term in Eq. (18) is in principle separable from the mass term: with two messenger speeds or two lens epochs, one could fit $M$ and $v$ independently from delay measurements alone.
A natural extension not considered here is replacing radial motion by a transverse or oblique lens velocity; the method of integrating the 1PM geodesic equations appears transferable to that case.
Because the absolute delay depends on the boundary condition $C=0+O(M^2)$ (asymptotic simultaneity), observables that compare two messengers are more robust than absolute flight times; practical tests should focus on differences like Eqs. (24)-(25).
If the unified formula holds, searches for neutrino-photon delays should treat the lens's radial velocity as a systematic bias in mass measurements, not merely as a small correction.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a clean, narrow calculation. Eq. (18) is new — no prior moving-lens time delay for timelike particles — and the algebra from Eqs. (6)-(18) checks out. The w=1 reduction to the known moving-lens light delay works, and the v=0 limit reduces to the static massive-particle formula, even though the paper never spells that v=0 check out explicitly. The slow-motion expansions (19)-(22) are consistent. So the central formula is very likely correct.

What I like: the paper states its scope precisely, keeps the geometry clear, and gives an alternative derivation in the appendix using a time parameter. The light-limit cross-check is independent of the authors' own prior work, which matters. The tables are honest about the numbers, and the detection claim is at least anchored to explicit magnitudes.

Where it is soft. First, the input geodesic equations (6)-(8) are imported verbatim from the authors' own [73] and not re-derived here. The stress-test concern is fair: the w=1 and v=0 checks only constrain two special surfaces, so an error proportional to v(1-w) in the input would slip through. I don't see a concrete error, and the equations are probably correct, but a referee should ask for the derivation or an independent cross-check. Second, the detectability section compares |delta-t| with detector timing resolution, which is necessary but far from sufficient. It does not model how a moving-lens delay would be separated from the geometric delay and the source's intrinsic timing, nor does it propagate errors on the lens mass, distance, velocity, or impact parameter. That section is a rough feasibility bound, not an end-to-end detection analysis. Third, the repeated claim of a 'large possibility' of detection is stronger than the evidence; exceeding timing resolution is not the same as a plausible measurement. The heavy self-citation is understandable given the program, but the lack of an explicit v=0 comparison with refs [67-70] is a minor gap that should be fixed.

Bottom line: this is a solid extension within an established program, not a breakthrough. The right audience is people working on multi-messenger lensing and Shapiro delay, and the paper deserves normal peer review. A referee should focus on the provenance of Eqs. (6)-(8) and on softening the detectability conclusion. I would not block publication on the stress-test concern alone.

Referee Report

2 major / 6 minor

Summary. This manuscript derives the leading-order post-Minkowskian gravitational time delay of relativistic massive particles and photons propagating from a distant source to an observer in the equatorial plane of a Schwarzschild black hole moving with constant radial velocity. Using the weak-field moving-lens metric (Eq. (1)) and the 1PM geodesic equations (Eqs. (6)-(8)) taken from earlier work, the authors integrate the coordinate time along the perturbed trajectory to obtain a unified delay formula (Eqs. (11) and (18)). They show that the w=1 limit reproduces the known moving-lens light delay, expand the result to first order in the lens velocity v and in δ=1-w, derive time-delay differences between massive and massless signals and between two massive signals, and estimate the magnitude of the velocity effect for Cygnus X-1, NGC 3319*, and Sgr A*, concluding that the effect may be detectable.

Significance. If the input geodesic equations are correct, Eq. (18) is a genuinely new unified formula: it reduces to the static massive-particle Shapiro delay at v=0 and to the known moving-lens light delay at w=1, with no fitted parameters. The algebra from Eqs. (6)-(18) checks out, the slow-motion expansions are internally consistent, and the paper provides concrete numerical estimates for three black-hole lenses. The main value is as a theoretical template for arrival-time differences in multi-messenger astronomy. The result's significance is, however, conditional on an independent verification of the quoted equations of motion and on a more realistic assessment of detectability.

major comments (2)

[Section II and Section III.A (Eqs. (6)-(11))] The equations of motion (6)-(8) are imported verbatim from Ref. [73] and are not re-derived in this manuscript. The checks presented in Section III only probe the surfaces w=1 and v=0; any error in the off-diagonal metric or in the 1PM geodesic equations that is proportional to v(1-w) would survive those checks and directly enter the new mixed terms of Eq. (18). Because Eq. (18) is the central result, this provenance is load-bearing. Please add a self-contained derivation of Eqs. (6)-(8) from the metric (1) in an appendix, or provide an independent numerical integration of the geodesic equations (or an alternative first-order PM scheme) for representative parameter pairs with 0<w<1 and v≠0.
[Section IV (Eq. (26) and Tables I-III)] The detectability conclusion is based on comparing |δΔt| from Eq. (26) with detector timing resolutions of order nanoseconds. This comparison is not sufficient: the measured arrival time contains the geometric term d_S/w, and for the adopted lenses (d_S ~ 8 kpc) the uncertainty in d_S alone is of order 10^10 s, many orders of magnitude larger than the tabulated effects of seconds or less. To support the claim that the radial-velocity effect is measurable, the paper must present an error budget that includes uncertainties in d_S, in the particle velocity w, and in the static-delay model, or restrict the claim to differential observables such as Eqs. (24)-(25). Without this, the 'large possibility to measure' conclusion is not supported.

minor comments (6)

[Section III.A, Eq. (9)] The second factor in the integrand is the first-order inverse of \dot{x} from Eq. (7); as written it is easy to misread as the velocity itself. Add a sentence explaining the manipulation.
[Section III.A, Eq. (18)] Add the explicit v=0 limit of Eq. (18), which gives Δt = d_S/w + [(3w^2-1)/w^3] M ln(4d_L d_LS/b^2), and compare it with the corresponding equation in Refs. [67-70] to document the static massive-particle endpoint.
[Section III.B, Eq. (22)] State the numerical accuracy of the δ expansion for w=0.9 (δ=0.1); the omitted O(δ^2) terms are not negligible at the percent level.
[Section IV] Define δΔt and Δ(Δt)_1M explicitly in one place; the notation changes between Eqs. (20), (24), and (26) without a glossary.
[Section IV] The sentence 'whether or not the radial motions of those lenses take are relativistic' should be corrected to 'whether the radial motions ... are relativistic'.
[Section III.A, Eq. (12)] The reduction to the light case is stated as consistent with Ref. [66], but the log argument is not explicitly matched; please confirm that the factor 4 and the argument (1-v/w)^2 agree exactly with the semi-iterative result in [66].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (18) is a direct 1PM integration of previously derived geodesic equations, with external consistency checks at the light and static limits; the self-cited inputs are prior derivations, not the target result.

full rationale

The central result Eq. (18) is obtained by integrating Eqs. (6)-(8), which are imported from the authors' earlier paper [73] and are themselves solutions/geodesic equations in the moving Schwarzschild metric Eq. (1) from [71]. This is a normal use of prior derived results rather than a circular reduction: the target travel-time formula is not contained in the inputs by definition; it requires the nontrivial integral of \dot{t}/\dot{x} over x and the zeroth-order coordinate transformation dx = [gamma^{-1}(1-v/w)^{-1}+O(M)]dX given in [73]. The paper checks two independent limits: w=1 reproduces the light-delay result Eq. (12), whose consistency with the Lienard-Wiechert approach [6,57] was demonstrated in [66], and v=0 yields the standard static massive-particle coefficient (3w^2-1)/w^3. No parameter is fitted to data, and the boundary condition C=0+O(M^2) fixes a conventional asymptotic time coordinate rather than encoding the predicted delay. The Appendix A 'alternative derivation' uses the same [73] equations rescaled by the trajectory parameter, so it is a consistency check, not a new input. The reliance on self-cited geodesic equations is a propagation-of-uncertainty concern about whether [73] is correct, not a circularity in this paper's derivation chain.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central formula introduces no new physics beyond the moving Schwarzschild background and standard 1PM geodesic integration. The only hand-chosen numbers are the astrophysical scenario parameters d_LS and b used for the detection estimates.

free parameters (2)

d_LS (source-lens angular diameter distance) = 0.01 kpc (preset for all three black-hole scenarios)
Set by hand in Section IV based on the typical range 0.001-0.1 kpc [89]. Not fitted to data, but a scenario choice that directly scales the logarithmic term and the resulting delta-t values.
b (impact parameter) = Einstein radius: 5.3e-9 kpc (Cyg X-1), 2.4e-7 kpc (NGC 3319*), 2.8e-6 kpc (Sgr A*)
Assumed equal to the Einstein ring radius for light following ref [90]. This choice affects the log factor and is not derived from the massive-particle deflection formula; a different b changes the numerical results.

assumptions (4)

domain assumption The 1PM metric of a radially moving Schwarzschild black hole, Eq. (1), is a valid weak-field description.
Taken from ref [71] without re-derivation. Standard but not independently verified in this paper.
domain assumption The geodesic equations (6)-(8) for a test particle in this spacetime are correct to 1PM order.
Taken from ref [73]. This is the direct input for the integration; any error propagates into the result.
domain assumption Weak-field, thin-lens, small-angle approximations: source and observer at large but finite distances, b << |x|, etc.
Stated in the Introduction and used to set d_LS ~ -x_S, d_L ~ x_O, d_S ~ x_O - x_S.
domain assumption The integration constant C in Eq. (10) can be set to 0 + O(M^2), giving t_S = x_S/w + O(M), t_O = x_O/w + O(M).
Used to convert Eq. (11) to the observer-frame Eq. (17). This asymptotic boundary condition is standard but not proven from the geometry.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Leading-order gravitational time delay of massive particles by a moving Schwarzschild lens." pith.science (2026). https://pith.science/paper/K2MMXTFD

@misc{pith2026250614184,
  author       = {Pith},
  title        = {Pith review of: Leading-order gravitational time delay of massive particles by a moving Schwarzschild lens},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K2MMXTFD}},
  note         = {Machine review of arXiv:2506.14184}
}

read the original abstract

The leading-order gravitational time delay of relativistic neutral massive particles (e.g., neutrinos or high-energy cosmic-ray particles) caused by a moving Schwarzschild black hole with a constant radial velocity is investigated for the first time. On the basis of the equations of motion in the spacetime of the moving lens, we achieve a new unified formula for the travel times of relativistic massive and massless particles propagating from the source to the observer within the first post-Minkowskian approximation. The analytical form of the difference between the travel times of a relativistic massive particle and a light signal in this geometry, as well as that of two relativistic massive particles, is thus obtained in the weak-field and slow-motion limit. The influence of the radial lens motion on the leading-order Schwarzschild time delay of relativistic massive particles is then discussed. It is found that in the slow-motion limit, the radial lens motion towards the observer decreases the flight time of an ultrarelativistic massive particle, when compared with the case of no translational motion of the central body. Conversely, if the lens gets away from the detector radially under the same conditions, the propagation process of the particle will slow down and its flight time will thus increase in comparison with the Schwarzschild case. Finally, we analyze the magnitude of the full radial motion effect of the lens and evaluate the possibility of its astronomical detection by modeling three typical black holes as the lens respectively.

Figures

Figures reproduced from arXiv: 2506.14184 by the authors.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

94 extracted references · 48 canonical work pages

[73]

Analytical Derivation of Second-Order Deflection in Equatorial Plane of a Radially Moving Kerr-Newman Black Hole

G. He and W. Lin, Classical and Quantum Gravity34, 105006 (2017), arXiv:2007.09831 [gr-qc]

work page Pith review arXiv 2017
[1]

Pyne and M

T. Pyne and M. Birkinshaw, Astrophys. J.415, 459 (1993), arXiv:astro-ph/9303020 [astro-ph]

arXiv 1993
[2]

Deflection of light and particles by moving gravitational lenses

O. Wucknitz and U. Sperhake, Phys. Rev. D69, 063001 (2004), arXiv:astro-ph/0401362 [astro-ph]

work page Pith review arXiv 2004
[3]

Heyrovsk ´y, Astrophys

D. Heyrovsk ´y, Astrophys. J.624, 28 (2005), arXiv:astro- ph/0410183 [astro-ph]

work page arXiv 2005
[4]

Birkinshaw and S

M. Birkinshaw and S. F. Gull, Nature (London)302, 315 (1983)

1983
[5]

Blanchet, G

L. Blanchet, G. Faye, and B. Ponsot, Phys. Rev. D58, 124002 (1998), arXiv:gr-qc/9804079 [gr-qc]

arXiv 1998
[6]

S. M. Kopeikin and G. Sch ¨afer, Phys. Rev. D60, 124002 (1999), arXiv:gr-qc/9902030 [gr-qc]

arXiv 1999
[7]

Fermat Potentials for Non-Perturbative Gravitational Lensing

S. Frittelli, T. P. Kling, and E. T. Newman, Phys. Rev. D65, 123007 (2002), arXiv:gr-qc/0205014 [gr-qc]

work page Pith review arXiv 2002

Show all 94 references

[8]

S. A. Klioner and M. Peip, Astron. Astrophys.410, 1063 (2003), arXiv:astro-ph/0305204 [astro-ph]

2003 arXiv
[9]

S. M. Kopeikin and V . V . Makarov, Phys. Rev. D75, 062002 (2007), arXiv:astro-ph/0611358 [astro-ph]

2007 arXiv
[10]

S. M. Kopeikin and E. B. Fomalont, Gen. Relat. Gravit.39, 1583 (2007), arXiv:gr-qc/0510077 [gr-qc]

2007 arXiv
[11]

Xie and T.-Y

Y . Xie and T.-Y . Huang, Phys. Rev. D77, 124049 (2008), arXiv:0805.4067 [gr-qc]

2008 arXiv
[12]

Wu and Y

X. Wu and Y . Xie, Phys. Rev. D77, 103012 (2008), arXiv:1004.5317 [gr-qc]

2008 arXiv
[13]

X.-M. Deng, Y . Xie, and T.-Y . Huang, Phys. Rev. D79, 044014 (2009), arXiv:0901.3730 [gr-qc]

2009 arXiv
[14]

Deng and Y

X.-M. Deng and Y . Xie, Phys. Rev. D86, 044007 (2012), arXiv:1207.3138 [gr-qc]

2012 arXiv
[15]

Xie, Res

Y . Xie, Res. Astron. Astrophys.14, 1193 (2014), arXiv:1405.0882 [gr-qc]

2014 arXiv
[16]

Banik and H

I. Banik and H. Zhao, Mon. Not. R. Astron. Soc.450, 3155 (2015), arXiv:1503.08052 [astro-ph.CO]

2015 arXiv
[17]

Deng, Int

X.-M. Deng, Int. J. Mod. Phys. D24, 1550056 (2015), arXiv:1504.04084 [gr-qc]

2015 arXiv
[18]

Zschocke, Classical and Quantum Gravity35, 055013 (2018), arXiv:1802.02764 [gr-qc]

S. Zschocke, Classical and Quantum Gravity35, 055013 (2018), arXiv:1802.02764 [gr-qc]

2018 arXiv
[19]

Zschocke, Classical and Quantum Gravity36, 015007 (2019), arXiv:1812.00728 [gr-qc]

S. Zschocke, Classical and Quantum Gravity36, 015007 (2019), arXiv:1812.00728 [gr-qc]

2019 arXiv
[20]

X.-Y . Yang, T. Chen, and R.-G. Cai, arXiv e-prints , arXiv:2410.16378 (2024), arXiv:2410.16378 [astro-ph.CO]

2024 arXiv
[21]

He and W

G. He and W. Lin, Classical and Quantum Gravity33, 095007 (2016), arXiv:2007.08754 [gr-qc]

2016 arXiv
[22]

Chatterjeeet al., Astrophys

S. Chatterjeeet al., Astrophys. J. Lett.630, L61 (2005), arXiv:astro-ph/0509031 [astro-ph]

2005 arXiv
[23]

M. A. C. Perrymanet al., Astron. Astrophys.369, 339 (2001), arXiv:astro-ph/0101235 [astro-ph]

2001 arXiv
[24]

Trippe, R

S. Trippe, R. Davies, F. Eisenhauer, N. M. F ¨orster Schreiber, T. K. Fritz, and R. Genzel, Mon. Not. R. Astron. Soc.402, 1126 (2010), arXiv:0910.5114 [astro-ph.IM]

2010 arXiv
[25]

Zhang, M

B. Zhang, M. J. Reid, K. M. Menten, X. W. Zheng, A. Brun- thaler, T. M. Dame, and Y . Xu, Astrophys. J.775, 79 (2013), arXiv:1312.3856 [astro-ph.GA]

2013 arXiv
[26]

M. J. Reid and M. Honma, Ann. Rev. Astron. Astrophys.52, 339 (2014), arXiv:1312.2871 [astro-ph.IM]

2014 arXiv
[27]

M. J. Rioja and R. Dodson, Astron. Astrophys. Rev.28, 6 (2020), arXiv:2010.02156 [astro-ph.IM]

2020 arXiv
[28]

A. G. A. Brown, Ann. Rev. Astron. Astrophys.59, 59 (2021), arXiv:2102.11712 [astro-ph.IM]

2021 arXiv
[29]

Y . Li, Y . Xu, J. Li, Y . Wu, S. Bian, Z. Lin, W. Yang, C. Hao, and D. Liu, Astrophys. J.925, 47 (2022), arXiv:2111.14095 [gr-qc]. 9

2022 arXiv
[30]

Y . Li, Y . Xu, S. Bian, Z. Lin, J. Li, D. Liu, and C. Hao, Astro- phys. J.938, 58 (2022), arXiv:2209.08702 [gr-qc]

2022 arXiv
[31]

Baker and M

T. Baker and M. Trodden, Phys. Rev. D95, 063512 (2017), arXiv:1612.02004 [astro-ph.CO]

2017 arXiv
[32]

IceCube Collaborationet al., Science361, eaat1378 (2018), arXiv:1807.08816 [astro-ph.HE]

2018 arXiv
[33]

M ´esz´aros, D

P. M ´esz´aros, D. B. Fox, C. Hanna, and K. Murase, Nat. Rev. Phys.1, 585 (2019), arXiv:1906.10212 [astro-ph.HE]

2019 arXiv
[34]

E. A. Huertaet al., Nat. Rev. Phys.1, 600 (2019), arXiv:1911.11779 [gr-qc]

2019 arXiv
[35]

Poemma Collaborationet al., J. Cosmol. Astropart. Phys.2021, 007 (2021), arXiv:2012.07945 [astro-ph.IM]

2021 arXiv
[36]

Qi, S.-J

J.-Z. Qi, S.-J. Jin, X.-L. Fan, J.-F. Zhang, and X. Zhang, J. Cosmol. Astropart. Phys.2021, 042 (2021), arXiv:2102.01292 [astro-ph.CO]

2021 arXiv
[37]

B. F. Burke, Phys. Today22, 54 (1969)

1969
[38]

A. E. E. Rogers, Radio Sci.5, 1239 (1970)

1970
[39]

A. R. Thompson, J. M. Moran, and G. W. Swenson,Interferom- etry and synthesis in radio astronomy(Wiley, New York, 1986)

1986
[40]

Hirabayashiet al., Science281, 1825 (1998)

H. Hirabayashiet al., Science281, 1825 (1998)

1998
[41]

Maet al., Astron

C. Maet al., Astron. J.116, 516 (1998)

1998
[42]

A. W. Raymondet al., Astron. J.168, 130 (2024), arXiv:2410.07453 [astro-ph.IM]

2024 arXiv
[43]

Schuh and D

H. Schuh and D. Behrend, J. Geodyn.61, 68 (2012)

2012
[44]

Niellet al., Radio Sci.53, 1269 (2018)

A. Niellet al., Radio Sci.53, 1269 (2018)

2018
[45]

Caoet al., Astrophys

Z. Caoet al., Astrophys. J. Supp. Ser.271, 25 (2024), arXiv:2305.17030 [astro-ph.HE]

2024
[46]

Maet al., Chin

X.-H. Maet al., Chin. Phys. C46, 030001 (2022)

2022
[47]

Halzen and S

F. Halzen and S. R. Klein, Rev. Sci. Instrum.81, 081101 (2010), arXiv:1007.1247 [astro-ph.HE]

2010 arXiv
[48]

IceCube Collaboration, nature591, 220 (2021), arXiv:2110.15051 [hep-ex]

2021
[49]

Phys.26, 155 (2006), arXiv:astro-ph/0604450 [astro-ph]

IceCube Collaborationet al., Astropart. Phys.26, 155 (2006), arXiv:astro-ph/0604450 [astro-ph]

2006 arXiv
[50]

IceCube Collaboration, Nucl. Instrum. Meth. A618, 139 (2010), arXiv:1002.2442 [astro-ph.IM]

2010 arXiv
[51]

J. A. Gonz ´alez, U. Sperhake, B. Br ¨ugmann, M. Hannam, and S. Husa, Phys. Rev. Lett.98, 091101 (2007), arXiv:gr- qc/0610154 [gr-qc]

2007
[52]

Repetto, M

S. Repetto, M. B. Davies, and S. Sigurdsson, Mon. Not. R. Astron. Soc.425, 2799 (2012), arXiv:1203.3077 [astro-ph.GA]

2012 arXiv
[53]

Calder ´on Bustillo, J

J. Calder ´on Bustillo, J. A. Clark, P. Laguna, and D. Shoemaker, Phys. Rev. Lett.121, 191102 (2018), arXiv:1806.11160 [gr-qc]

2018 arXiv
[54]

I. I. Shapiro, Phys. Rev. Lett.13, 789 (1964)

1964
[55]

I. I. Shapiro, M. E. Ash, R. P. Ingalls, W. B. Smith, D. B. Camp- bell, R. B. Dyce, R. F. Jurgens, and G. H. Pettengill, Phys. Rev. Lett.26, 1132 (1971)

1971
[56]

Sereno, Phys

M. Sereno, Phys. Lett. A305, 7 (2002), arXiv:astro-ph/0209148 [astro-ph]

2002 arXiv
[57]

Kopeikin and B

S. Kopeikin and B. Mashhoon, Phys. Rev. D65, 064025 (2002), arXiv:gr-qc/0110101 [gr-qc]

2002 arXiv
[58]

Frittelli, Mon

S. Frittelli, Mon. Not. R. Astron. Soc.344, L85 (2003), arXiv:astro-ph/0311189 [astro-ph]

2003 arXiv
[59]

Sereno, Mon

M. Sereno, Mon. Not. R. Astron. Soc.359, L19 (2005), arXiv:astro-ph/0501605 [astro-ph]

2005 arXiv
[60]

Bertotti, N

B. Bertotti, N. Ashby, and L. Iess, Classical and Quantum Gravity25, 045013 (2008), arXiv:0709.1512 [gr-qc]

2008 arXiv
[61]

S. M. Kopeikin, Mon. Not. R. Astron. Soc.399, 1539 (2009), arXiv:0809.3433 [gr-qc]

2009 arXiv
[62]

A. Hees, S. Bertone, and C. Le Poncin-Lafitte, Phys. Rev. D 90, 084020 (2014), arXiv:1406.6600 [gr-qc]

2014 arXiv
[63]

M. H. Soffel and W.-B. Han, Phys. Lett. A379, 233 (2014), arXiv:1409.3743 [gr-qc]

2014 arXiv
[64]

Le Poncin-Lafitte, B

C. Le Poncin-Lafitte, B. Linet, and P. Teyssandier, Classical and Quantum Gravity21, 4463 (2004), arXiv:gr-qc/0403094 [gr-qc]

2004 arXiv
[65]

Teyssandier and C

P. Teyssandier and C. Le Poncin-Lafitte, Classical and Quantum Gravity25, 145020 (2008), arXiv:0803.0277 [gr-qc]

2008 arXiv
[66]

He and W

G. He and W. Lin, Phys. Rev. D94, 063011 (2016), arXiv:2007.11809 [gr-qc]

2016 arXiv
[67]

Jia and H

J. Jia and H. Liu, Phys. Rev. D100, 124050 (2019), arXiv:1906.11833 [gr-qc]

2019 arXiv
[68]

Liu and J

H. Liu and J. Jia, Eur. Phys. J. C80, 932 (2020), arXiv:2006.11125 [gr-qc]

2020 arXiv
[69]

He and W

G. He and W. Lin, Phys. Rev. D105, 104034 (2022), arXiv:2112.08142 [gr-qc]

2022 arXiv
[70]

G. He, Y . Xie, C. Jiang, and W. Lin, Phys. Rev. D110, 064008 (2024), arXiv:2402.01548 [gr-qc]

2024 arXiv
[71]

G. He, C. Jiang, and W. Lin, Int. J. Mod. Phys. D23, 1450079- 55 (2014), arXiv:2007.13329 [gr-qc]

2014 arXiv
[72]

Luminariet al., Astron

A. Luminariet al., Astron. Astrophys.633, A55 (2020), arXiv:1912.00494 [astro-ph.HE]

2020 arXiv
[74]

Bhattacharjee and G

P. Bhattacharjee and G. Sigl, Phys. Rep.327, 109 (2000), arXiv:astro-ph/9811011 [astro-ph]

2000 arXiv
[75]

Kotera and A

K. Kotera and A. V . Olinto, Annu. Rev. Astron. Astrophys.49, 119 (2011), arXiv:1101.4256 [astro-ph.HE]

2011 arXiv
[76]

J. E. Greene, J. Strader, and L. C. Ho, Annu. Rev. Astron. As- trophys.58, 257 (2020), arXiv:1911.09678 [astro-ph.GA]

2020 arXiv
[77]

M. C. Miller and E. J. M. Colbert, Int. J. Mod. Phys. D13, 1 (2004), arXiv:astro-ph/0308402 [astro-ph]

2004 arXiv
[78]

Casares and P

J. Casares and P. G. Jonker, Space Sci. Rev.183, 223 (2014), arXiv:1311.5118 [astro-ph.HE]

2014 arXiv
[79]

Madau and M

P. Madau and M. J. Rees, Astrophys. J. Lett.551, L27 (2001), arXiv:astro-ph/0101223 [astro-ph]

2001 arXiv
[80]

Narayan and J

R. Narayan and J. E. McClintock, arXiv e-prints , arXiv:1312.6698 (2013), arXiv:1312.6698 [astro-ph.HE]

2013 arXiv
[81]

J. A. Orosz, J. E. McClintock, J. P. Aufdenberg, R. A. Remil- lard, M. J. Reid, R. Narayan, and L. Gou, Astrophys. J.742, 84 (2011), arXiv:1106.3689 [astro-ph.HE]

2011 arXiv
[82]

M. J. Reid, J. E. McClintock, R. Narayan, L. Gou, R. A. Remillard, and J. A. Orosz, Astrophys. J.742, 83 (2011), arXiv:1106.3688 [astro-ph.HE]

2011 arXiv
[83]

Gouet al., Astrophys

L. Gouet al., Astrophys. J.742, 85 (2011), arXiv:1106.3690 [astro-ph.HE]

2011 arXiv
[84]

B. L. Davis and A. W. Graham, Publ. Astron. Soc. Aust.38, e030 (2021), arXiv:2105.04717 [astro-ph.GA]

2021 arXiv
[85]

Jianget al., Astrophys

N. Jianget al., Astrophys. J.869, 49 (2018), arXiv:1810.10283 [astro-ph.GA]

2018 arXiv
[86]

Sakaiet al., Astrophys

S. Sakaiet al., Astrophys. J.523, 540 (1999), arXiv:astro- ph/9906487 [astro-ph]

1999
[87]

Bland-Hawthorn and O

J. Bland-Hawthorn and O. Gerhard, Annu. Rev. Astron. Astro- phys.54, 529 (2016), arXiv:1602.07702 [astro-ph.GA]

2016 arXiv
[88]

Parsa, A

M. Parsa, A. Eckart, B. Shahzamanian, V . Karas, M. Zaja ˇcek, J. A. Zensus, and C. Straubmeier, Astrophys. J.845, 22 (2017), arXiv:1708.03507 [astro-ph.GA]

2017 arXiv
[89]

C. R. Keeton and A. O. Petters, Phys. Rev. D72, 104006 (2005), arXiv:gr-qc/0511019 [gr-qc]

2005 arXiv
[90]

Ishak, W

M. Ishak, W. Rindler, J. Dossett, J. Moldenhauer, and C. Allison, Mon. Not. R. Astron. Soc.388, 1279 (2008), arXiv:0710.4726 [astro-ph]

2008 arXiv
[91]

Feinstein and ANTARES Collaboration, Nucl

F. Feinstein and ANTARES Collaboration, Nucl. Instrum. Meth. A504, 258 (2003)

2003
[92]

Ageronet al., Nucl

M. Ageronet al., Nucl. Instrum. Meth. A578, 498 (2007), arXiv:astro-ph/0703355 [astro-ph]

2007 arXiv
[93]

Brunner and ANTARES Collaboration, Nucl

J. Brunner and ANTARES Collaboration, Nucl. Instrum. Meth. A626, S19 (2011). 10

2011
[94]

Adamsonet al., Phys

P. Adamsonet al., Phys. Rev. D92, 052005 (2015). [95] T. Adamet al., J. High Energy Phys.2012, 93 (2012), arXiv:1109.4897 [hep-ex]

2015 arXiv

Pith tools

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

[1] [73]

Analytical Derivation of Second-Order Deflection in Equatorial Plane of a Radially Moving Kerr-Newman Black Hole

G. He and W. Lin, Classical and Quantum Gravity34, 105006 (2017), arXiv:2007.09831 [gr-qc]

work page Pith review arXiv 2017

[2] [1]

Pyne and M

T. Pyne and M. Birkinshaw, Astrophys. J.415, 459 (1993), arXiv:astro-ph/9303020 [astro-ph]

arXiv 1993

[3] [2]

Deflection of light and particles by moving gravitational lenses

O. Wucknitz and U. Sperhake, Phys. Rev. D69, 063001 (2004), arXiv:astro-ph/0401362 [astro-ph]

work page Pith review arXiv 2004

[4] [3]

Heyrovsk ´y, Astrophys

D. Heyrovsk ´y, Astrophys. J.624, 28 (2005), arXiv:astro- ph/0410183 [astro-ph]

work page arXiv 2005

[5] [4]

Birkinshaw and S

M. Birkinshaw and S. F. Gull, Nature (London)302, 315 (1983)

1983

[6] [5]

Blanchet, G

L. Blanchet, G. Faye, and B. Ponsot, Phys. Rev. D58, 124002 (1998), arXiv:gr-qc/9804079 [gr-qc]

arXiv 1998

[7] [6]

S. M. Kopeikin and G. Sch ¨afer, Phys. Rev. D60, 124002 (1999), arXiv:gr-qc/9902030 [gr-qc]

arXiv 1999

[8] [7]

Fermat Potentials for Non-Perturbative Gravitational Lensing

S. Frittelli, T. P. Kling, and E. T. Newman, Phys. Rev. D65, 123007 (2002), arXiv:gr-qc/0205014 [gr-qc]

work page Pith review arXiv 2002