Pith. sign in

REVIEW 6 minor 72 references

Foundations of Direct Waves in Schwarzschild Ringdown

T0 review · 0 major / 6 minor · reviewed 2026-07-10 · grok-4.5

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

Pith's one-line read Direct waves in black-hole ringdown come from the anti-causal near-horizon source, not from light-cone or horizon modes, and do not vanish in Schwarzschild.

desk verdict Clean first-principles derivation that the direct wave is the non-vanishing anti-causal filter-pole integral, not a light-cone artifact, and it matches the filtered waveform to ~0.1%. read the letter →

arxiv 2607.06677 v1 pith:Q3G3GFLL submitted 2026-07-07 gr-qc astro-ph.HEhep-th

Sizheng Ma , Hai-Yang Wang This is my paper · ORCID
classification gr-qcastro-ph.HEhep-th
keywords black-holeringdowndirectwavesGreen'sfunctionquasinormalmodesrationalfiltersSchwarzschildnear-horizondynamicsZerilliequation
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

After a black-hole merger, the ringdown signal is usually described by quasinormal modes, but filtered waveforms still show several decaying oscillatory cycles near the strain peak, called direct waves. Earlier work linked those cycles to source motion near the horizon, yet lacked a first-principles derivation, and a recent argument suggested they might vanish in Schwarzschild. This paper derives them from the causal structure of the filtered Green's function for the Zerilli equation. It shows that the filter introduces an anti-causal piece sourced by the trajectory segment that has already plunged toward the future horizon; that piece alone reproduces the filtered waveform for plunging particles to about 0.1 percent residual, while prompt response and tail contribute negligibly. The resulting signal is controlled by the near-horizon orbital phase and radial infall, not by a pure horizon mode or an instantaneous light-cone contribution. The result supplies a theoretical basis for reading near-horizon dynamics out of filtered ringdown, complementary to spectroscopy of quasinormal modes.

What carries the argument

Causal decomposition of the filtered Green's function into anti-causal (upper-half-plane filter poles for u < -r*), prompt, and tail windows; the anti-causal integral (Eq. 15) over the near-horizon segment is what produces the direct wave.

What would settle it

Compute the same rationally filtered Zerilli waveform for a plunging particle and check whether the anti-causal integral alone still matches the full filtered signal to ~0.1 percent, or whether the residual grows once the large-arc assumption is relaxed or the contour is evaluated differently.

Watch

Extended reading notes

Core claim

The direct wave does not vanish in Schwarzschild spacetime. It is exactly the anti-causal contribution of the rationally filtered Green's function, an integral over the near-horizon trajectory segment of the source, and that contribution alone accounts for the filtered Zerilli waveform of plunging particles; its instantaneous frequency and decay are set by the source's near-horizon phase and radial velocity.

Load-bearing premise

The large-arc pieces of the complex-frequency contours vanish, so the filtered waveform is fully determined by the anti-causal poles, the small arc, and the branch cuts.

Editorial extensions

If this is right

  • Direct waves remain available as a probe of near-horizon source dynamics even in non-spinning black holes.
  • They should not be modeled as pure horizon modes; their frequency and decay track the plunge trajectory.
  • The same anti-causal mechanism can be extended to Kerr and to residual post-merger distortions treated as effective sources on a remnant black hole.
  • Horizon signatures enter the observable only gradually through source redshift and frame-dragging, not as isolated horizon-mode rings.

Reading between the lines

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

  • If the anti-causal construction survives in Kerr, filtered NR ringdowns of real mergers could be used to constrain near-horizon plunge kinematics without relying on an effective-one-body picture.
  • Collective cancellation of individual horizon-mode poles may be a general feature of unit-modulus rational filters, so similar rearrangements could appear whenever QNMs are filtered from other linear wave equations.
  • The same contour split might isolate prompt-response and tail pieces cleanly enough to test whether nonlinearities after common-horizon formation still leave a clean anti-causal imprint.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 6 minor

Summary. The paper derives a first-principles foundation for the 'direct wave' component of black-hole ringdown in Schwarzschild spacetime. Starting from the sourced Zerilli equation, the authors apply a rational filter that removes a prescribed set of QNMs and analyze the causal structure of the filtered Green's function via contour deformations in the complex frequency plane. They obtain a three-window decomposition (anti-causal, prompt, and tail; Eq. 12) and show that the filtered waveform for plunging point particles is dominated by the anti-causal contribution (Eq. 15), which is sourced by the near-horizon trajectory segment. For an ISCO plunge, this piece alone reproduces the filtered Zerilli waveform to ~0.1% residual (Fig. 3). An integration-by-parts reduction recovers the instantaneous frequency ω_G of Oshita et al., while clarifying that the direct wave is an accumulated anti-causal effect rather than a light-cone contribution subject to the cancellation argued in Kuntz et al. Horizon-mode contributions from individual filter poles cancel collectively, so the signal is controlled by near-horizon source dynamics.

Significance. If correct, the result places the direct-wave interpretation of filtered NR and GW250114 waveforms on a firm theoretical footing within linear Schwarzschild perturbation theory, and cleanly separates it from both ordinary QNMs and the vanishing light-cone modes of Ref. [56]. Strengths include a transparent contour analysis that reuses standard Green's-function machinery, an explicit integral formula (Eq. 15) that can be checked independently, a high-precision numerical match for the ISCO plunge, and a controlled phase-rescaling test (Fig. 5) that links the signal to near-horizon orbital motion. The work is complementary to QNM spectroscopy and supplies a concrete bridge from filtered waveforms to horizon-proximate source dynamics.

minor comments (6)
  1. The vanishing of the large-arc contributions (dotted contours in Fig. 1) is load-bearing for isolating the anti-causal piece. A short appendix or paragraph summarizing the estimates from Refs. [57–61] (or a brief self-contained argument for the filtered case) would make the paper more self-contained.
  2. Section III states that the anti-causal dominance holds for eccentric plunges, but only the ISCO case is shown. A single additional panel or a brief quantitative residual for one eccentric trajectory would strengthen the claim.
  3. In the reduction from Eq. (17) to (18)–(19), the dropped higher-order corrections are mentioned only in a footnote. A short estimate of their size near the horizon would clarify the domain of validity of ω_G.
  4. Figure 4 shows individual anti-causal poles decaying as e^{-κ u} while their sum does not; a sentence quantifying the degree of cancellation (e.g., relative residual at late u) would help the reader.
  5. Notation: the same symbol R is used for the particle trajectory and for homogeneous radial solutions (R_in, R_up, R_down). Distinct symbols would reduce occasional ambiguity in Sec. III.
  6. The discussion of the NR/close-limit interpretation (Sec. IV) is suggestive but qualitative. Framing it more clearly as an outlook rather than a derived claim would avoid over-reading the linear point-particle results.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-citations of rational filters and Green's-function contour tools; central anti-causal identification and non-vanishing result are independently derived and numerically verified.

  1. self citation load bearing [Sec. II, Eqs. (5)–(12) and Fig. 1]
    "building on the recently developed decomposition of the Schwarzschild Green's function [57–61]. … To remove a prescribed set of QNMs {ω_n}, we apply the rational filter [18, 30, 49] F(ω)=… The large arc vanishes as indicated by the dotted circles."

    The rational filter and the claim that large-arc contributions vanish (dotted contours) are taken from the authors' own prior papers. These tools are load-bearing for isolating the anti-causal poles, yet the isolation itself is not circular: the subsequent numerical evaluation of the anti-causal integral against the independently filtered waveform supplies an external check that does not reduce to the cited definitions.

full rationale

The derivation chain begins from the sourced Zerilli equation, constructs the filtered Green's function via the rational filter F(ω), deforms contours in the three causal windows (u < -r*, -r* < u < |r*|, u > |r*|), and isolates the anti-causal poles (Eq. 11) as the sole source of the residual after QNM removal. Insertion of a point-particle source then yields the explicit near-horizon integral (Eq. 15). This integral is shown by direct numerical evaluation (Fig. 3) to reproduce the fully filtered Zerilli waveform to ~0.1 % residual for an ISCO plunge (and is stated to hold for eccentric plunges). The match is an independent computation, not a fit or a redefinition. The recovered instantaneous frequency ω_G (Eq. 19) coincides with the earlier heuristic of Oshita et al., but is obtained here by integration by parts of the anti-causal integral rather than by steepest-descent assumption. Self-citations appear for the rational filter (Refs. 18, 30, 49, 50) and for the contour-decomposition technology (Refs. 57–61, including the authors' own prompt-response paper). These supply standard machinery whose large-arc vanishing is assumed; they do not force the non-vanishing of the anti-causal piece or its identification with the direct-wave residual. No parameter is fitted to data and then re-predicted, no uniqueness theorem is imported to forbid alternatives, and no known empirical pattern is merely renamed. The result is therefore self-contained against the paper's own numerical benchmark within linear Schwarzschild perturbation theory.

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

The central claim rests on standard linear black-hole perturbation theory, the known analytic structure of the Schwarzschild Green's function, and the definition of the rational filter. No new free parameters are fitted to produce the non-vanishing result; trajectory choices and filter order N are demonstration parameters. No new physical entities are postulated.

free parameters (2)
  • filter order N (overtones removed) = 5
    Chosen by hand (N=5 in the main demonstration); the qualitative dominance of the anti-causal piece is stated to hold for other choices, but N is not derived.
  • phase-rescaling parameter α = varied 0.985–2
    Hand-varied in Fig. 5 to illustrate frequency dependence on near-horizon orbital phase; not fitted to data.
assumptions (4)
  • domain assumption The retarded Green's function of the Zerilli operator admits the standard frequency-domain representation with poles at the QNMs (zeros of A_inc) and a branch cut along the negative imaginary axis.
    Invoked throughout Sec. II; standard in the literature (Leaver and subsequent works).
  • domain assumption Large-arc contributions to the filtered frequency integrals vanish in the half-planes indicated by the dotted contours of Fig. 1.
    Taken from prior contour analyses [57–61] and used to isolate poles, small arc, and branch cuts as the only surviving pieces.
  • standard math The rational filter F(ω) has unit modulus on the real axis and merely redistributes existing source information in time.
    Stated in Sec. IV; follows from the product form (Eq. 5).
  • domain assumption A plunging point-particle source of the form (14) is sufficient to capture the direct-wave mechanism under study.
    Sec. III; used as a controlled proof-of-principle model for the phenomenon previously seen in NR.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Foundations of Direct Waves in Schwarzschild Ringdown." pith.science (2026). https://pith.science/paper/Q3G3GFLL

@misc{pith2026260706677,
  author       = {Pith},
  title        = {Pith review of: Foundations of Direct Waves in Schwarzschild Ringdown},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q3G3GFLL}},
  note         = {Machine review of arXiv:2607.06677}
}
read the original abstract

Recent studies have identified a new component in black-hole ringdown from merging binaries, termed the \emph{direct wave}. This component was argued to be tied to the dynamical source evolution near the black-hole horizon, and thus to encode horizon information. Yet a firm theoretical foundation for the direct wave has been lacking. Here we fill this gap by deriving direct waves from first principles in Schwarzschild spacetime, using the causal structure of the Green's function. We show that the direct wave does not vanish and is governed by the near-horizon source dynamics. Our results establish a theoretical basis for direct waves as a probe of near-horizon dynamics, complementary to quasinormal modes.

Figures

Figures reproduced from arXiv: 2607.06677 by the authors.

Figure 1
Figure 1. Contour deformations of the filtered Green’s function in the three time regimes: [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Penrose diagram illustrating the causal decomposi [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. The ℓ = m = 2 Zerilli waveform for an ISCO plunge: unfiltered (gray) and filtered with overtones up to n = 5 removed (black). The decaying cycles over u ∈ [40, 60] correspond to direct waves. The anti-causal contribution alone [red dashed, Eq. (15)] reproduces the filtered waveform, up to a ∼ 0.1% residual (yellow dashed). Comparing Eqs. (16) and (15) is instructive. The inte￾gration domain of Eq. (16) shows that th… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Individual anti-causal poles ω¯n contributing to the direct wave [solid curves, Eq. (15)]. Each pole decays at late times as a horizon mode ∼ e −κu, shown by the red dashed line for reference. They cancel collectively in the total filtered waveform (black dashed). Equa…
Figure 5
Figure 5. Figure 5: Dependence of the direct wave on the near-horizon [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

72 extracted references · 72 canonical work pages

  1. [56]

    Vanishing of all redshift modes in Schwarzschild ringdown

    A. Kuntz and M. Della Rocca, (2026), arXiv:2606.07173 [gr-qc]

    work page Pith review arXiv 2026

  2. [1]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.116, 221101 (2016), [Erratum: Phys. Rev. Lett.121,no.12,129902(2018)], arXiv:1602.03841 [gr-qc]

    work page Pith review arXiv 2016

  3. [2]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. D100, 104036 (2019), arXiv:1903.04467 [gr-qc]

    work page Pith review arXiv 2019

  4. [3]

    Tests of General Relativity with GWTC-3

    R. Abbottet al.(LIGO Scientific, VIRGO, KAGRA), (2021), arXiv:2112.06861 [gr-qc]

    work page Pith review arXiv 2021

  5. [4]

    A. G. Abacet al.(LIGO Scientific, VIRGO, KAGRA), (2026), arXiv:2603.19019 [gr-qc]

    work page Pith review arXiv 2026

  6. [5]

    Quasinormal modes of black holes and black branes

    E. Berti, V. Cardoso, and A. O. Starinets, Class. Quant. Grav.26, 163001 (2009), arXiv:0905.2975 [gr-qc]

    work page Pith review arXiv 2009

  7. [6]

    Black hole spectroscopy: from theory to experiment

    J. Abediet al., (2025), arXiv:2505.23895 [gr-qc]

    work page Pith review arXiv 2025

  8. [7]

    Black Hole Spectroscopy: Testing General Relativity through Gravitational Wave Observations

    O. Dreyer, B. J. Kelly, B. Krishnan, L. S. Finn, D. Garri- son, and R. Lopez-Aleman, Class. Quant. Grav.21, 787 (2004), arXiv:gr-qc/0309007 [gr-qc]

    work page Pith review arXiv 2004

Show all 72 references
  1. [8]

    Buonanno, G

    A. Buonanno, G. B. Cook, and F. Pretorius, Phys. Rev. D75, 124018 (2007), arXiv:gr-qc/0610122 [gr-qc]

    2007 arXiv

  2. [9]

    Berti, V

    E. Berti, V. Cardoso, and C. M. Will, Phys. Rev.D73, 064030 (2006), arXiv:gr-qc/0512160 [gr-qc]

    2006 arXiv

  3. [10]

    Berti, K

    E. Berti, K. Yagi, H. Yang, and N. Yunes, Gen. Rel. Grav.50, 49 (2018), arXiv:1801.03587 [gr-qc]

    2018 arXiv

  4. [11]

    Giesler, M

    M. Giesler, M. Isi, M. Scheel, and S. Teukolsky, Phys. Rev.X9, 041060 (2019), arXiv:1903.08284 [gr-qc]

    2019 arXiv

  5. [12]

    Giesleret al., Phys

    M. Giesleret al., Phys. Rev. D111, 084041 (2025), arXiv:2411.11269 [gr-qc]

    2025 arXiv

  6. [13]

    M. Isi, M. Giesler, W. M. Farr, M. A. Scheel, and S. A. Teukolsky, Phys. Rev. Lett.123, 111102 (2019), arXiv:1905.00869 [gr-qc]

    2019 arXiv

  7. [14]

    Carullo, W

    G. Carullo, W. Del Pozzo, and J. Veitch, Phys. Rev.D99, 123029 (2019), [Erratum: Phys. Rev.D100,no.8,089903(2019)], arXiv:1902.07527 [gr-qc]

    2019 arXiv

  8. [15]

    Cotesta, G

    R. Cotesta, G. Carullo, E. Berti, and V. Cardoso, Phys. Rev. Lett.129, 111102 (2022), arXiv:2201.00822 [gr-qc]

    2022 arXiv

  9. [16]

    Isi and W

    M. Isi and W. M. Farr, (2022), arXiv:2202.02941 [gr-qc]

    2022 arXiv

  10. [17]

    Finch and C

    E. Finch and C. J. Moore, Phys. Rev. D106, 043005 (2022), arXiv:2205.07809 [gr-qc]

    2022 arXiv

  11. [18]

    S. Ma, L. Sun, and Y. Chen, Phys. Rev. D107, 084010 (2023), arXiv:2301.06639 [gr-qc]

    2023 arXiv

  12. [19]

    C. D. Capano, M. Cabero, J. Westerweck, J. Abedi, S. Kastha, A. H. Nitz, Y.-F. Wang, A. B. Nielsen, and B. Krishnan, Phys. Rev. Lett.131, 221402 (2023), arXiv:2105.05238 [gr-qc]

    2023 arXiv

  13. [20]

    Siegel, M

    H. Siegel, M. Isi, and W. M. Farr, Phys. Rev. D108, 064008 (2023), arXiv:2307.11975 [gr-qc]

    2023 arXiv

  14. [21]

    A. G. Abacet al.(LIGO Scientific, VIRGO, KAGRA), Astrophys. J. Lett.993, L25 (2025), arXiv:2507.08219 [astro-ph.HE]

    2025 arXiv

  15. [22]

    Wang, S.-P

    H.-T. Wang, S.-P. Tang, P.-C. Li, and Y.-Z. Fan, (2025), arXiv:2509.02047 [gr-qc]

    2025 arXiv

  16. [23]

    A. G. Abacet al.(LIGO Scientific, Virgo, KAGRA), Phys. Rev. Lett.135, 111403 (2025), arXiv:2509.08054 [gr-qc]

    2025 arXiv

  17. [24]

    A. G. Abacet al.(LIGO Scientific, Virgo, KAGRA), Phys. Rev. Lett.136, 041403 (2026), arXiv:2509.08099 [gr-qc]

    2026 arXiv

  18. [25]

    A. G. Abacet al.(LIGO Scientific, VIRGO, KAGRA), (2026), arXiv:2603.19021 [gr-qc]

    2026 arXiv

  19. [26]

    Dyer and C

    R. Dyer and C. J. Moore, Phys. Rev. Lett.136, 191403 (2026), arXiv:2510.13954 [gr-qc]

    2026 arXiv

  20. [27]

    Dyer and C

    R. Dyer and C. J. Moore, Phys. Rev. D113, 104031 (2026), arXiv:2510.11783 [gr-qc]

    2026 arXiv

  21. [28]

    Mitmanet al., Phys

    K. Mitmanet al., Phys. Rev. Lett.130, 081402 (2023), arXiv:2208.07380 [gr-qc]

    2023 arXiv

  22. [29]

    M. H.-Y. Cheunget al., Phys. Rev. Lett.130, 081401 (2023), arXiv:2208.07374 [gr-qc]

    2023 arXiv

  23. [30]

    S. Ma, K. Mitman, L. Sun, N. Deppe, F. Hébert, L. E. Kidder, J. Moxon, W. Throwe, N. L. Vu, and Y. Chen, Phys. Rev. D106, 084036 (2022), arXiv:2207.10870 [gr- qc]

    2022 arXiv

  24. [31]

    Khera, S

    N. Khera, S. Ma, and H. Yang, Phys. Rev. Lett.134, 211404 (2025), arXiv:2410.14529 [gr-qc]

    2025 arXiv

  25. [32]

    Ma and H

    S. Ma and H. Yang, Phys. Rev. D109, 104070 (2024), arXiv:2401.15516 [gr-qc]

    2024 arXiv

  26. [33]

    Zhuet al., Phys

    H. Zhuet al., Phys. Rev. D109, 104050 (2024), arXiv:2401.00805 [gr-qc]

    2024 arXiv

  27. [34]

    Redondo-Yuste, G

    J. Redondo-Yuste, G. Carullo, J. L. Ripley, E. Berti, and V. Cardoso, Phys. Rev. D109, L101503 (2024), arXiv:2308.14796 [gr-qc]

    2024 arXiv

  28. [35]

    Bucciotti, L

    B. Bucciotti, L. Juliano, A. Kuntz, and E. Trincherini, Phys. Rev. D110, 104048 (2024), arXiv:2405.06012 [gr- qc]

    2024 arXiv

  29. [36]

    Bourg, R

    P. Bourg, R. Panosso Macedo, A. Spiers, B. Leather, B. Bonga, and A. Pound, Phys. Rev. Lett.134, 061401 (2025), arXiv:2405.10270 [gr-qc]

    2025 arXiv

  30. [37]

    Andersson, Phys

    N. Andersson, Phys. Rev. D55, 468 (1997), arXiv:gr- qc/9607064

    1997

  31. [38]

    Chavda, M

    A. Chavda, M. Lagos, and L. Hui, JCAP07, 084 (2025), arXiv:2412.03435 [gr-qc]

    2025 arXiv

  32. [39]

    De Amicis, E

    M. De Amicis, E. Cannizzaro, G. Carullo, and L. Sberna, (2025), arXiv:2506.21668 [gr-qc]

    2025

  33. [41]

    De Amiciset al., (2024), arXiv:2412.06887 [gr-qc]

    M. De Amiciset al., (2024), arXiv:2412.06887 [gr-qc]

    2024

  34. [42]

    Islam, G

    T. Islam, G. Faggioli, G. Khanna, S. E. Field, M. van de Meent, and A. Buonanno, Phys. Rev. D112, 024061 (2025), arXiv:2407.04682 [gr-qc]

    2025

  35. [43]

    De Amicis, S

    M. De Amicis, S. Albanesi, and G. Carullo, Phys. Rev. D110, 104005 (2024), arXiv:2406.17018 [gr-qc]

    2024 arXiv

  36. [44]

    S. Ma, M. A. Scheel, J. Moxon, K. C. Nelli, N. Deppe, L. E. Kidder, W. Throwe, and N. L. Vu, Phys. Rev. D 112, 024003 (2025), arXiv:2412.06906 [gr-qc]

    2025 arXiv

  37. [45]

    Cardoso, G

    V. Cardoso, G. Carullo, M. De Amicis, F. Duque, T. Katagiri, D. Pereniguez, J. Redondo-Yuste, T. F. M. Spieksma, and Z. Zhong, Phys. Rev. D109, L121502 (2024), arXiv:2405.12290 [gr-qc]

    2024 arXiv

  38. [46]

    Islam, G

    T. Islam, G. Faggioli, and G. Khanna, Phys. Rev. D113, 124025 (2026), arXiv:2511.21898 [gr-qc]

    2026 arXiv

  39. [47]

    Oshita, S

    N. Oshita, S. Ma, Y. Chen, and H. Yang, (2025), arXiv:2509.09165 [gr-qc]

    2025 arXiv

  40. [48]

    N. Lu, S. Ma, O. J. Piccinni, Y. Chen, and L. Sun, Nature 655, 71 (2026), arXiv:2510.01001 [gr-qc]

    2026 arXiv

  41. [49]

    S. Ma, L. Sun, and Y. Chen, Phys. Rev. Lett.130, 141401 7 (2023), arXiv:2301.06705 [gr-qc]

    2023 arXiv

  42. [50]

    N. Lu, S. Ma, O. J. Piccinni, L. Sun, and E. Finch, Phys. Rev. D112, 064047 (2025), arXiv:2505.18560 [gr-qc]

    2025 arXiv

  43. [51]

    Kankani and S

    A. Kankani and S. T. McWilliams, (2026), arXiv:2603.15474 [gr-qc]

    2026

  44. [52]

    A. K.-W. Chung, K. K.-H. Lam, A. Liu, and N. Yunes, (2026), arXiv:2606.06592 [gr-qc]

    2026 arXiv

  45. [53]

    R. Dyer, A. K.-W. Chung, and C. J. Moore, (2026), arXiv:2606.25021 [gr-qc]

    2026 arXiv

  46. [54]

    Mino and J

    Y. Mino and J. Brink, Phys. Rev. D78, 124015 (2008), arXiv:0809.2814 [gr-qc]

    2008 arXiv

  47. [55]

    Zimmerman and Y

    A. Zimmerman and Y. Chen, Phys. Rev. D84, 084012 (2011), arXiv:1106.0782 [gr-qc]

    2011 arXiv

  48. [57]

    Arnaudo, J

    P. Arnaudo, J. Carballo, and B. Withers, (2025), arXiv:2510.18956 [gr-qc]

    2025

  49. [58]

    J. Su, N. Khera, M. Casals, S. Ma, A. Chowdhuri, and H. Yang, (2026), arXiv:2601.22015 [gr-qc]

    2026

  50. [59]

    Kuntz, (2025), arXiv:2510.17954 [gr-qc]

    A. Kuntz, (2025), arXiv:2510.17954 [gr-qc]

    2025 arXiv

  51. [60]

    R. F. Rosato, M. De Amicis, and P. Pani, (2026), arXiv:2603.20490 [gr-qc]

    2026

  52. [61]

    Ma, (2026), arXiv:2604.08680 [gr-qc]

    S. Ma, (2026), arXiv:2604.08680 [gr-qc]

    2026 arXiv

  53. [62]

    E. W. Leaver, Phys. Rev. D34, 384 (1986)

    1986

  54. [63]

    Nagar, T

    A. Nagar, T. Damour, and A. Tartaglia, Class. Quant. Grav.24, S109 (2007), arXiv:gr-qc/0612096

    2007 arXiv

  55. [64]

    S. Ma, H. Yu, and Y. Chen, Phys. Rev. D101, 123020 (2020), arXiv:2003.02373 [gr-qc]

    2020 arXiv

  56. [65]

    Kankani and S

    A. Kankani and S. T. McWilliams, (2026), arXiv:2607.02380 [gr-qc]

    2026 arXiv

  57. [66]

    R. H. Price and J. Pullin, Phys. Rev. Lett.72, 3297 (1994), arXiv:gr-qc/9402039

    1994 arXiv

  58. [67]

    J. G. Baker, M. Campanelli, and C. O. Lousto, Phys. Rev. D65, 044001 (2002), arXiv:gr-qc/0104063

    2002 arXiv

  59. [68]

    D. A. Nichols and Y. Chen, Phys. Rev. D82, 104020 (2010), arXiv:1007.2024 [gr-qc]

    2010 arXiv

  60. [69]

    D. A. Nichols and Y. Chen, Phys. Rev. D85, 044035 (2012), arXiv:1109.0081 [gr-qc]

    2012 arXiv

  61. [70]

    R. J. Gleiser, C. O. Nicasio, R. H. Price, and J. Pullin, Class. Quant. Grav.13, L117 (1996), arXiv:gr- qc/9510049

    1996

  62. [71]

    R. J. Gleiser, C. O. Nicasio, R. H. Price, and J. Pullin, Phys. Rev. Lett.77, 4483 (1996), arXiv:gr-qc/9609022

    1996 arXiv

  63. [72]

    Directed acyclic graphs enabling au- tonomous research,

    H. Wanget al., “Directed acyclic graphs enabling au- tonomous research,” (2026), in preparation

    2026

  64. [73]

    T. J. Boerner, S. Deems, T. R. Furlani, S. L. Knuth, and J. Towns, inPractice and Experience in Advanced Research Computing 2023: Computing for the Common Good, PEARC ’23 (Association for Computing Machinery, New York, NY, USA, 2023) p. 173–176

    2023

Pith tools

Reviewed July 10, 2026 · model on record in the stance chip above.