Pith. sign in

REVIEW 3 major objections 6 minor 71 references

Gravitational lensing in a plasma from worldlines

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read A worldline path-integral calculation produces a closed-form, next-to-leading-order deflection angle for light in an inhomogeneous plasma, generalizing previous specific-power-law results.

desk verdict A useful worldline derivation of the pure-plasma NLO deflection formula, cross-checked well, but the Schwarzschild scoping and the unshown Feynman rule need work before it is used as the full NLO bending angle. read the letter →

arxiv 2412.14126 v2 pith:HU752ZD6 submitted 2024-12-18 hep-th astro-ph.COgr-qc

Francesco Comberiati , Leonardo de la Cruz This is my paper · ORCID
classification hep-thastro-ph.COgr-qc
keywords gravitationallensingplasmaworldlineformalismdeflectionanglenext-to-leading-orderinhomogeneousSchwarzschildSyngeHamiltonian
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 aims to show that the worldline formalism—a path-integral technique borrowed from quantum field theory—can compute gravitational lensing in a plasma as cleanly as it computes vacuum light bending. It starts from Synge's Hamiltonian for geometrical optics in a medium and derives a position-space action in which the plasma acts as a position-dependent potential. For a homogeneous plasma the action reduces to that of a massive particle, recovering known deflection angles through a familiar analogy. For an inhomogeneous power-law plasma, the paper constructs time-domain Feynman rules and derives a single closed-form expression for the next-to-leading-order plasma-induced deflection angle, valid for all power-law exponents except half-integers. The result matches existing Gauss-Bonnet computations for h=1,2,3 and the authors' own Gauss-Bonnet calculations for h=4,5,6, and it gives nonvanishing angles for non-integer exponents such as h=1.1, which are relevant in astrophysical applications.

What carries the argument

The central object is the position-space worldline action derived from Synge's Hamiltonian, $S[x;g] = -\frac{1}{2\omega_0}\int d\tau \left(g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu + \frac{\omega_e(x)^2}{\omega_0^2}\right)$, in which the plasma enters as a position-dependent potential. The calculation expands the trajectory around a straight line $x_0^\mu = b^\mu + u^\mu \tau$, integrates out fluctuations with the retarded worldline propagator, and reads the impulse from the position fluctuation expectation value. The time-domain Feynman rules for the plasma interaction vertices, listed at leading and next-to-leading order in Eqs. (61) and (62), carry the argument; the NLO deflection angle emerges from a single two-vertex diagram whose time integrals evaluate to Gauss hypergeometric functions and finally to the closed form of Eq. (66).

What would settle it

Integrate the full null geodesic equation in the Synge effective metric for a Schwarzschild black hole surrounded by a power-law plasma with a non-integer index such as $h=1.5$, and compare the next-to-leading-order plasma-induced deflection with Eq. (66); a mismatch would show that the straight-line impulse and additivity assumptions miss required cross-terms.

Watch

Extended reading notes

Core claim

The central claim is that the purely plasma-induced next-to-leading-order deflection angle for a light ray passing a Schwarzschild black hole in a cold, non-magnetized plasma with electron density $N_E(r) = N_0 (R/r)^h$ is given by $\alpha_N^{(1)} = \frac{N_0^2 k_e^2}{2 \omega_0^4}\, \frac{\pi^{3/2} \sec(\pi h)}{\Gamma(1/2 - h)\,\Gamma(h - 1)} \left(\frac{R}{b}\right)^{2h}$, in agreement with earlier work for $h=1,2,3$ and with the authors' Gauss-Bonnet computations for $h=4,5,6$. The authors also claim that the worldline framework provides a systematic, diagrammatic way to compute such plasma effects, with the homogeneous case naturally reducing to a massive-particle probe whose deflection angle can be read off from known results.

Load-bearing premise

The deflection is computed as the sum of a pure-gravity part and a pure-plasma part, with the plasma impulse evaluated along the straight-line trajectory, so the quoted formula is the full next-to-leading-order angle only if mixed gravity-plasma cross-terms are suppressed at that order.

Editorial extensions

If this is right

  • The homogeneous plasma deflection angle follows from the massive-particle analogy via the substitution $v \to \sqrt{1 - \omega_{e0}^2/\omega_0^2}$, so the full set of vacuum probe deflection results (including spin effects) carries over to the plasma case.
  • Equation (66) is a single closed-form expression covering all power-law exponents $h \ge 1$ except half-integers, replacing case-by-case computations for integer $h$.
  • For $h=1$ the NLO plasma-induced angle vanishes, while for $h=1.1$ it is nonzero, showing that the deflection is sensitive to the density-profile exponent in a non-monotonic way.
  • The separation of the impulse into vacuum plus plasma contributions at this order means the total deflection angle is the sum of the known Schwarzschild/Kerr vacuum angle and Eq. (66), with no additional mixed term needed at this order.

Reading between the lines

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

  • At the next order in the post-Minkowskian or plasma coupling expansion, gravity-plasma cross-terms will likely enter; the same worldline Feynman rules could be used to compute them, and the present additivity result should be checked against that computation.
  • The poles of Eq. (66) at half-integer $h$ suggest that the perturbative expansion breaks down for those density profiles; analytic continuation or a resummation might yield a finite angle there, which would be a testable extension.
  • Since the NLO angle scales as $(R/b)^{2h}$, frequency-dependent radio observations of lensing near the solar corona could in principle constrain the power-law index $h$ of the coronal density profile, provided this order is observationally accessible.
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.

Referee Report

3 major / 6 minor

Summary. Using Synge's Hamiltonian for a cold non-magnetized plasma, the paper constructs a first-quantized worldline action and computes deflection angles perturbatively. In the homogeneous case the action reduces to that of a massive particle, and the authors use this analogy to reproduce probe-limit deflection angles in Schwarzschild and to derive, via a field redefinition in Kerr-Schild gauge, the impulse and deflection in a Kerr background. In the inhomogeneous case with N_E(r) = N0 (R/r)^h, they introduce time-domain Feynman rules and obtain the leading and next-to-leading plasma-induced deflections; the NLO result, Eq. (66), is checked against Ref. [15] for h = 1, 2, 3 and against Gauss-Bonnet computations for h = 4, 5, 6.

Significance. The homogeneous-sector derivation is a clean demonstration of the massive-probe analogy in a worldline language, and the Kerr-Schild field-redefinition argument that reproduces the known 1PM and 2PM probe eikonal is a useful methodological contribution. The central new result is Eq. (66). It is stated in closed form for general h and has been cross-checked against two independent methods for integer h, which is a genuine strength. The main caveat is that the inhomogeneous calculation, as presented, computes the pure-plasma impulse in flat space; whether this constitutes the full NLO deflection in a Schwarzschild-plus-plasma system is not demonstrated. The paper would be significantly stronger if this distinction were made explicit, or if the mixed gravitational-plasma terms were addressed.

major comments (3)
  1. [Sec. IV.B, Eq. (54)] The total impulse is written as Delta p^mu = Delta p_E^mu + Delta p_N^mu, and Delta p_N^mu is then evaluated on the straight-line trajectory x0^mu = b^mu + u^mu tau using the free retarded propagator (21). In a Schwarzschild background this split is not automatic: the worldline propagator is modified by curvature, and the combined expansion in G and N0 generates mixed terms of order G times N0 (and higher) that are absent from Delta p_E + Delta p_N. If Eq. (66) is meant to be the full NLO deflection angle for Schwarzschild with plasma, those mixed contributions must be computed or bounded. If instead Eq. (66) is the pure-plasma flat-space contribution, then the wording of the title and of the opening sentence of Section IV.B ("an inhomogeneous medium in Schwarzschild") should be changed accordingly. The agreement with Ref. [15] for h = 1, 2, 3 and the Gauss-Bonnet checks for h = 4, 5, 6 support the pure-plasma coefficient, but they do not by themselves establish the additivity hypothesis.
  2. [Sec. IV.B, Eq. (66)] Equation (66) is written with an explicit sec(pi h) and Gamma(1/2 - h), and the text states that the formula holds for all h >= 1 except half-integers where the secant has poles. These poles are removable. Using Gamma(1/2 - h) Gamma(h + 1/2) = pi / cos(pi h), Eq. (66) simplifies to alpha_N^(1) = (N0^2 k_e^2)/(2 omega_0^4) sqrt(pi) Gamma(h + 1/2) / Gamma(h - 1) (R/b)^(2h), which is finite at h = 3/2, 5/2, and so on. The manuscript should either present the simplified form or explain the analytic continuation; the stated domain of validity should be corrected accordingly.
  3. [Appendix B] Appendix B evaluates only one master integral, I[1,2], and states that the remaining integrals can be computed similarly. Because Eq. (66) for general, and in particular non-integer, h rests on the full set of master integrals, the derivation is incomplete as printed. Please provide explicit results for the remaining integrals, or an ancillary computation, so that the non-integer h behavior (including the h = 1.1 remark) can be verified independently of the integer-h checks.
minor comments (6)
  1. [Sec. IV.B, Eqs. (61)-(62)] The Feynman rules are stated without derivation; a short derivation from the Taylor expansion of N(x0 + z) would improve the presentation.
  2. [Sec. IV.B, Eq. (63)] The diagram notation "circled times circled times omega = 0 z~^mu(omega)" is not defined; please explain the multiplicity factor and the impulse insertion.
  3. [Sec. II, Eq. (11)] The normalization connecting the phase-space action (10) to the position-space action (11) is not spelled out; please clarify the factors of omega_0 and 1/2.
  4. [Sec. IV.B, after Eq. (57)] The identification r^2 approx b^2 + tau^2 is made in the approximation omega_e0/omega_0 << 1; please state explicitly that tau is the affine parameter and not the coordinate time.
  5. [Throughout] There are typographical issues, including "Gauß-Bonet" for "Gauss-Bonnet" and "wordline" in footnote 1; a careful proofreading pass is needed.
  6. [Fig. 1] Figure 1 has no caption text in the manuscript; please add a descriptive caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (66) is derived from the worldline action and benchmarked against independent Gauss-Bonnet results, with no fitted parameter or definitional identity between input and output.

full rationale

The derivation chain is self-contained. In the inhomogeneous case, the NLO plasma-induced deflection angle Eq. (66) is obtained by computing the impulse from the action (52)-(56) and evaluating the NLO Feynman diagram (63)-(65); no parameter is fitted to the target result, and the input plasma profile N_E(r) is not defined in terms of the output deflection angle. The homogeneous case uses the massive-probe analogy through the map (15), with the massive-probe deflection angle independently computed in Eq. (31), so this is a legitimate reduction rather than a renaming or a fitted input called a prediction. The self-citations that appear, such as Refs. [34], [39], and [44], occur in methodological remarks about the worldline formalism and are not load-bearing for Eq. (66). The quoted agreement with Ref. [15] for h = 1, 2, 3 and with Gauss-Bonnet computations for h = 4, 5, 6 is an external benchmark, not a self-referential check. The possible omission of mixed gravitational-plasma terms in the split Eq. (54) is a physical scope or correctness concern about whether Eq. (66) is the complete next-order angle in Schwarzschild, not a circularity: nothing in the paper defines the predicted angle as being equal by construction to the input action or to a fitted coefficient. Therefore no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The calculation rests on Synge's Hamiltonian, the cold-plasma refraction index, the probe limit, and a power-law density model. No fitted parameters and no invented entities are introduced. The quantities N0, R, h, and ω0 are physical inputs, not free parameters tuned to data.

assumptions (8)
  • domain assumption Synge's Hamiltonian H = 1/2 \bar{g}^{μν} p_μ p_ν with \bar{g}^{μν} = g^{μν} - (1 - 1/n^2) V^μ V^ν describes light in a non-dispersive medium.
    Invoked at Eqs. (4) to (6); the paper explicitly assumes a non-dispersive medium so that \bar{g}^{μν} can be treated as a metric.
  • domain assumption The cold non-magnetized plasma has refraction index n_e^2 = 1 - ω_e(x)^2/ω(x)^2.
    Used at Eqs. (7) and (8) to connect the plasma frequency to the electron density.
  • domain assumption The spacetime is static and the medium rest frame is V^μ = g00^{-1/2}(1,0,0,0).
    Stated in Section II before Eq. (7); needed to identify the photon frequency and the plasma potential.
  • domain assumption The probe limit: the black hole is fixed and does not recoil.
    Adopted in Section III to compute deflection against a fixed background metric, standard in probe-limit scattering.
  • domain assumption A photon in a homogeneous plasma behaves like a massive particle with effective mass and velocity u_eff^2 = 1 - ω_e0^2/ω_0^2.
    Used at Eqs. (13) to (15) to map known massive-probe deflection angles to the homogeneous plasma case.
  • domain assumption The electron density follows the power-law model N_E(r) = N0(R/r)^h.
    Introduced at Eq. (57) as a typical model for elliptical galaxies [16]; the NLO result is obtained for this family.
  • standard math The tree-level worldline path integral with retarded propagator yields the classical impulse.
    The classical limit from worldline QFT is taken via Wick contractions and tree-level diagrams, following Refs. [29,47].
  • standard math Analytic regularization and continuation in the power-law index h are valid for the NLO integrals.
    Used in Appendix B, where integrals are expressed through hypergeometric functions and expanded around integer h using HypExp.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Gravitational lensing in a plasma from worldlines." pith.science (2026). https://pith.science/paper/HU752ZD6

@misc{pith2026241214126,
  author       = {Pith},
  title        = {Pith review of: Gravitational lensing in a plasma from worldlines},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HU752ZD6}},
  note         = {Machine review of arXiv:2412.14126}
}
read the original abstract

We study the deflection of light rays in a cold, non-magnetized plasma using the worldline framework. Starting from Synge's Hamiltonian formalism, we construct a position-space action and use it perturbatively to calculate light bending angles. In the homogeneous case, the action reduces to that of a massive particle, allowing us to extract the bending angle of light in the presence of the medium using a well-known analogy. For the inhomogeneous case, we consider a power law model and construct Feynman rules in time to compute the purely plasma-induced corrections to the bending angle at Next-to-Leading-Order (NLO).

Figures

Figures reproduced from arXiv: 2412.14126 by the authors.

Figure 1
Figure 1. α as a function of ωe/ω in the homogeneous case. B. The inhomogeneous plasma Let us now consider the case of an inhomogeneous medium in Schwarzschild. In this case, the second term in the (11) also contributes. Specifically, upon the back￾ground expansion (12), we obtain SN = −i ke 2ω0 Z dτ [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

71 extracted references · 17 canonical work pages

  1. [15]

    Crisnejo and E

    G. Crisnejo and E. Gallo, Weak lensing in a plasma medium and gravitational deflection of massive particles using the Gauss-Bonnet theorem. A unified treatment, Phys. Rev. D97 (2018) 124016 [1804.05473]

    arXiv 2018

  2. [1]

    (58) Noticethattheintegraloverthelineartermin τ vanishes

    Choosing the power law model we have ∆pµ N = N0ke 2ω0 Z ∞ −∞ dτ N ′ E(r) r xµ 0 =bµh N0ke 2ω0 Z ∞ −∞ dτ 1 r2+h . (58) Noticethattheintegraloverthelineartermin τ vanishes. We thus find ∆pµ N = N0ke ω0 √πΓ((h + 1)/2) Γ(h/2) bµ b R b h . (59) The scattering angle can be computed from the impulse as α(0) N = |b · ∆pN| bp0 = N0ke ω2 0 √πΓ((h + 1)/2) Γ(h/2) R b...

    work page

  3. [2]

    Gordon,Zur Lichtfortpflanzung nach der Relativität- stheorie, Annalen Phys

    W. Gordon,Zur Lichtfortpflanzung nach der Relativität- stheorie, Annalen Phys. 377 (1923) 421

    work page 1923

  4. [3]

    Ehlers, Zum Übergang von der wellenoptik zur ge- ometrischen optik in der allgemeinen relativitätstheorie, Zeitschrift für Naturforschung A22 (1967) 1328

    J. Ehlers, Zum Übergang von der wellenoptik zur ge- ometrischen optik in der allgemeinen relativitätstheorie, Zeitschrift für Naturforschung A22 (1967) 1328

    work page 1967

  5. [4]

    Synge,Relativity: The General Theory, North-Holland series in physics

    J. Synge,Relativity: The General Theory, North-Holland series in physics. North-Holland Publishing Company, 1960

    work page 1960

  6. [5]

    D. O. Muhleman and I. D. Johnston,Radio propagation in the solar gravitational field, Phys. Rev. Lett.17 (1966) 455

    work page 1966

  7. [6]

    K. S. Virbhadra and G. F. R. Ellis, Schwarzschild black hole lensing , Phys. Rev. D 62 (2000) 084003 [astro-ph/9904193]

    arXiv 2000

  8. [7]

    K. S. Virbhadra, Relativistic images of Schwarzschild black hole lensing , Phys. Rev. D 79 (2009) 083004 [0810.2109]

    arXiv 2009

Show all 71 references
  1. [8]

    G. S. Bisnovatyi-Kogan and O. Y. Tsupko,Gravitational Lensing in Presence of Plasma: Strong Lens Systems, Black Hole Lensing and Shadow, Universe 3 (2017) 57 [1905.06615]

    2017 arXiv

  2. [9]

    P. V. P. Cunha and C. A. R. Herdeiro, Shadows and strong gravitational lensing: a brief review, Gen. Rel. Grav. 50 (2018) 42 [1801.00860]

    2018 arXiv

  3. [10]

    Perlick and O

    V. Perlick and O. Y. Tsupko,Calculating black hole shad- ows: Review of analytical studies, Phys. Rept.947 (2022) 1 [2105.07101]

    2022 arXiv

  4. [11]

    G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Gravita- tional radiospectrometer, Grav. Cosmol. 15 (2009) 20 [0809.1021]

    2009 arXiv

  5. [12]

    G. S. Bisnovatyi-Kogan and O. Y. Tsupko,Gravitational lensing in a non-uniform plasma,Mon. Not. Roy. Astron. Soc. 404 (2010) 1790 [1006.2321]

    2010 arXiv

  6. [13]

    Perlick, ed.,Ray Optics, Fermat’s Principle, and Ap- plications to General Relativity

    V. Perlick, ed.,Ray Optics, Fermat’s Principle, and Ap- plications to General Relativity. Springer Berlin, Heidel- berg, 2000

    2000

  7. [14]

    G. W. Gibbons and M. C. Werner,Applications of the Gauss-Bonnet theorem to gravitational lensing, Class. Quant. Grav. 25 (2008) 235009 [0807.0854]

    2008 arXiv

  8. [16]

    Crisnejo, E

    G. Crisnejo, E. Gallo and K. Jusufi,Higher order cor- rections to deflection angle of massive particles and light rays in plasma media for stationary spacetimes using the Gauss-Bonnet theorem, Phys. Rev. D100 (2019) 104045 [1910.02030]

    2019 arXiv

  9. [17]

    Er and S

    X. Er and S. Mao, Effects of plasma on gravitational lensing, Mon. Not. Roy. Astron. Soc. 437 (2014) 2180 [1310.5825]

    2014 arXiv

  10. [18]

    Kimpson, K

    T. Kimpson, K. Wu and S. Zane, Spatial dispersion of light rays propagating through a plasma in Kerr space–time,Mon. Not. Roy. Astron. Soc.484(2019)2411 [1901.03733]

    2019 arXiv

  11. [19]

    Kimpson, K

    T. Kimpson, K. Wu and S. Zane, Pulsar timing in extreme mass ratio binaries: a general relativistic ap- proach, Mon. Not. Roy. Astron. Soc. 486 (2019) 360 [1903.08258]

    2019 arXiv

  12. [20]

    Er and S

    X. Er and S. Mao, The effects of plasma on the mag- nification and time delay of strongly lensed fast radio bursts, Mon. Not. Roy. Astron. Soc. 516 (2022) 2218 [2208.08208]

    2022 arXiv

  13. [21]

    N. E. J. Bjerrum-Bohr, P. H. Damgaard, L. Plante and P. Vanhove,The SAGEX review on scattering amplitudes Chapter 13: Post-Minkowskian expansion from scattering amplitudes, J. Phys. A55 (2022) 443014 [2203.13024]

    2022 arXiv

  14. [22]

    D. A. Kosower, R. Monteiro and D. O’Connell, The SAGEX review on scattering amplitudes Chapter 14: Classical gravity from scattering amplitudes, J. Phys. A 55 (2022) 443015 [2203.13025]

    2022 arXiv

  15. [23]

    Fabbrichesi, R

    M. Fabbrichesi, R. Pettorino, G. Veneziano and G. A. Vilkovisky, Planckian energy scattering and surface terms in the gravitational action, Nucl. Phys. B 419 (1994) 147 [hep-th/9309037]

    1994 arXiv

  16. [24]

    Schubert, Perturbative quantum field theory in the string inspired formalism, Phys

    C. Schubert, Perturbative quantum field theory in the string inspired formalism, Phys. Rept. 355 (2001) 73 [hep-th/0101036]

    2001 arXiv

  17. [25]

    F. Bastianelli, Path integrals in curved space and the worldline formalism, in8th International Conference on Path Integrals from Quantum Information to Cosmology, 8, 2005, hep-th/0508205

    2005 arXiv

  18. [26]

    J. P. Edwards and C. Schubert, Quantum mechanical path integrals in the first quantised approach to quantum field theory, 12, 2019,1912.10004. 10

    2019

  19. [27]

    Bastianelli and C

    F. Bastianelli and C. Schubert,Worldline path integrals and quantum field theory. Cambridge University Press, book in preparation

  20. [28]

    R. A. Porto, The effective field theorist’s approach to gravitational dynamics, Phys. Rept. 633 (2016) 1 [1601.04914]

    2016 arXiv

  21. [29]

    Levi, Effective Field Theories of Post-Newtonian Gravity: A comprehensive review, Rept

    M. Levi, Effective Field Theories of Post-Newtonian Gravity: A comprehensive review, Rept. Prog. Phys.83 (2020) 075901 [1807.01699]

    2020 arXiv

  22. [30]

    Mogull, J

    G. Mogull, J. Plefka and J. Steinhoff,Classical black hole scattering from a worldline quantum field theory, JHEP 02 (2021) 048 [2010.02865]

    2021 arXiv

  23. [31]

    G. U. Jakobsen, G. Mogull, J. Plefka and J. Steinhoff, Classical Gravitational Bremsstrahlung from a World- line Quantum Field Theory, Phys. Rev. Lett.126 (2021) 201103 [2101.12688]

    2021 arXiv

  24. [32]

    G. U. Jakobsen, G. Mogull, J. Plefka and J. Steinhoff, SUSY in the sky with gravitons, JHEP 01 (2022) 027 [2109.04465]

    2022 arXiv

  25. [33]

    Shi and J

    C. Shi and J. Plefka,Classical double copy of worldline quantum field theory, Phys. Rev. D 105 (2022) 026007 [2109.10345]

    2022 arXiv

  26. [34]

    G. U. Jakobsen, G. Mogull, J. Plefka and B. Sauer,All things retarded: radiation-reaction in worldline quantum field theory, JHEP 10 (2022) 128 [2207.00569]

    2022 arXiv

  27. [35]

    Bastianelli, F

    F. Bastianelli, F. Comberiati and L. de la Cruz,Light bending from eikonal in worldline quantum field theory, JHEP 02 (2022) 209 [2112.05013]

    2022 arXiv

  28. [36]

    W. D. Goldberger and I. Z. Rothstein,An Effective field theory of gravity for extended objects, Phys. Rev. D 73 (2006) 104029 [hep-th/0409156]

    2006 arXiv

  29. [37]

    Kälin and R

    G. Kälin and R. A. Porto,Post-Minkowskian Effective Field Theory for Conservative Binary Dynamics, JHEP 11 (2020) 106 [2006.01184]

    2020 arXiv

  30. [38]

    Z. Liu, R. A. Porto and Z. Yang,Spin Effects in the Ef- fective Field Theory Approach to Post-Minkowskian Con- servative Dynamics, JHEP 06 (2021) 012 [2102.10059]

    2021 arXiv

  31. [39]

    Kälin, J

    G. Kälin, J. Neef and R. A. Porto,Radiation-reaction in the Effective Field Theory approach to Post-Minkowskian dynamics, JHEP 01 (2023) 140 [2207.00580]

    2023 arXiv

  32. [40]

    Comberiati and L

    F. Comberiati and L. de la Cruz,Classical off-shell cur- rents, JHEP 03 (2023) 068 [2212.09259]

    2023 arXiv

  33. [41]

    P. H. Damgaard, E. R. Hansen, L. Planté and P. Van- hove, The relation between KMOC and worldline for- malisms for classical gravity , JHEP 09 (2023) 059 [2306.11454]

    2023 arXiv

  34. [42]

    Capatti and M

    Z. Capatti and M. Zeng,Classical worldlines from scat- tering amplitudes, 2412.10864

    arXiv

  35. [43]

    D. A. Kosower, B. Maybee and D. O’Connell, Ampli- tudes, Observables, and Classical Scattering, JHEP 02 (2019) 137 [1811.10950]

    2019 arXiv

  36. [44]

    Maybee, D

    B. Maybee, D. O’Connell and J. Vines,Observables and amplitudes for spinning particles and black holes, JHEP 12 (2019) 156 [1906.09260]

    2019 arXiv

  37. [45]

    de la Cruz, B

    L. de la Cruz, B. Maybee, D. O’Connell and A. Ross, Classical Yang-Mills observables from amplitudes, JHEP 12 (2020) 076 [2009.03842]

    2020 arXiv

  38. [46]

    X. Feal, A. Tarasov and R. Venugopalan, QED as a many-body theory of worldlines: General formalism and infrared structure, Phys. Rev. D 106 (2022) 056009 [2206.04188]

    2022 arXiv

  39. [47]

    X. Feal, A. Tarasov and R. Venugopalan, QED as a many-body theory of worldlines. II. All-order S- matrix formalism, Phys. Rev. D 107 (2023) 096021 [2211.15712]

    2023 arXiv

  40. [48]

    Das,Field Theory: A Path Integral Approach

    A. Das,Field Theory: A Path Integral Approach. World Scientific Press, 2019, 10.1142/11339

    2019 doi

  41. [49]

    Bastianelli and A

    F. Bastianelli and A. Zirotti,Worldline formalism in a gravitational background, Nucl. Phys. B 642 (2002) 372 [hep-th/0205182]

    2002 arXiv

  42. [50]

    Bastianelli, P

    F. Bastianelli, P. Benincasa and S. Giombi,Worldline ap- proach to vector and antisymmetric tensor fields, JHEP 04 (2005) 010 [hep-th/0503155]

    2005 arXiv

  43. [51]

    Bastianelli and P

    F. Bastianelli and P. van Nieuwenhuizen,Path integrals and anomalies in curved space, Cambridge Monographs on Mathematical Physics. Cambridge University Press, 9, 2006, 10.1017/CBO9780511535031

    2006 doi

  44. [52]

    Crisnejo, E

    G. Crisnejo, E. Gallo and J. R. Villanueva, Gravita- tional lensing in dispersive media and deflection angle of charged massive particles in terms of curvature scalars and energy-momentum tensor, Phys. Rev. D100 (2019) 044006 [1905.02125]

    2019 arXiv

  45. [53]

    T. J. Hollowood and G. M. Shore,The Refractive index of curved spacetime: The Fate of causality in QED, Nucl. Phys. B 795 (2008) 138 [0707.2303]

    2008 arXiv

  46. [54]

    T. J. Hollowood and G. M. Shore,Causality and Micro- Causality in Curved Spacetime, Phys. Lett. B655 (2007) 67 [0707.2302]

    2007 arXiv

  47. [55]

    G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Gravita- tional Lensing in Plasmic Medium, Plasma Phys. Rep. 41 (2015) 562 [1507.08545]

    2015 arXiv

  48. [56]

    U. Kol, D. O’connell and O. Telem,The radial action from probe amplitudes to all orders, JHEP 03 (2022) 141 [2109.12092]

    2022 arXiv

  49. [57]

    Menezes and M

    G. Menezes and M. Sergola,NLO deflections for spin- ning particles and Kerr black holes, JHEP 10 (2022) 105 [2205.11701]

    2022 arXiv

  50. [58]

    P. H. Damgaard, J. Hoogeveen, A. Luna and J. Vines, Scattering angles in Kerr metrics, Phys. Rev. D 106 (2022) 124030 [2208.11028]

    2022 arXiv

  51. [59]

    Hoogeveen,Charged test particle scattering and effec- tive one-body metrics with spin, Phys

    J. Hoogeveen,Charged test particle scattering and effec- tive one-body metrics with spin, Phys. Rev. D108 (2023) 024049 [2303.00317]

    2023 arXiv

  52. [60]

    Scharf, Schwarzschild geodesics in terms of elliptic functions and the related red shift, J

    G. Scharf, Schwarzschild geodesics in terms of elliptic functions and the related red shift, J. Mod. Phys.2 (2011) 274 [1101.1207]

    2011 arXiv

  53. [61]

    Cheung, J

    C. Cheung, J. Parra-Martinez, I. Z. Rothstein, N. Shah and J. Wilson-Gerow,Effective Field Theory for Extreme Mass Ratio Binaries, Phys. Rev. Lett.132 (2024) 091402 [2308.14832]

    2024 arXiv

  54. [62]

    Bianchi, C

    M. Bianchi, C. Gambino and F. Riccioni,A Rutherford- like formula for scattering off Kerr-Newman BHs and subleading corrections , JHEP 08 (2023) 188 [2306.08969]

    2023 arXiv

  55. [63]

    Vines, Scattering of two spinning black holes in post-Minkowskian gravity, to all orders in spin, and effective-one-body mappings, Class

    J. Vines, Scattering of two spinning black holes in post-Minkowskian gravity, to all orders in spin, and effective-one-body mappings, Class. Quant. Grav. 35 (2018) 084002 [1709.06016]

    2018 arXiv

  56. [64]

    Arkani-Hamed, Y.-t

    N. Arkani-Hamed, Y.-t. Huang and D. O’Connell,Kerr black holes as elementary particles, JHEP 01 (2020) 046 [1906.10100]

    2020 arXiv

  57. [65]

    A. B. Aazami, C. R. Keeton and A. O. Petters,Lens- ing by Kerr Black Holes. II: Analytical Study of Quasi- Equatorial Lensing Observables,J. Math. Phys.52(2011) 102501 [1102.4304]

    2011 arXiv

  58. [66]

    S. V. Iyer and E. C. Hansen,Strong and Weak Deflection of Light in the Equatorial Plane of a Kerr Black Hole, 0908.0085. 11

    arXiv

  59. [67]

    G. U. Jakobsen and G. Mogull, Conservative and Radiative Dynamics of Spinning Bodies at Third Post-Minkowskian Order Using Worldline Quantum Field Theory, Phys. Rev. Lett. 128 (2022) 141102 [2201.07778]

    2022 arXiv

  60. [68]

    G. U. Jakobsen, G. Mogull, J. Plefka and B. Sauer,Tidal effects and renormalization at fourth post-Minkowskian order, Phys. Rev. D109 (2024) L041504 [2312.00719]

    2024 arXiv

  61. [69]

    Psaltis,The Influence of Gas Dynamics on Measur- ing the Properties of the Black Hole in the Center of the Milky Way with Stellar Orbits and Pulsars, Astrophys

    D. Psaltis,The Influence of Gas Dynamics on Measur- ing the Properties of the Black Hole in the Center of the Milky Way with Stellar Orbits and Pulsars, Astrophys. J. 759 (2012) 130 [1112.0026]

    2012 arXiv

  62. [70]

    Bezděková, O

    B. Bezděková, O. Y. Tsupko and C. Pfeifer,Deflection of light rays in a moving medium around a spherically symmetric gravitating object, Phys. Rev. D 109 (2024) 124024 [2403.16842]

    2024 arXiv

  63. [71]

    Huber and D

    T. Huber and D. Maitre, HypExp: A Mathematica package for expanding hypergeometric functions around integer-valued parameters, Comput. Phys. Commun.175 (2006) 122 [hep-ph/0507094]

    2006 arXiv

Pith tools

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