Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

High-energy interactions of charged black holes in full general relativity I: Zoom-whirl orbits and universality with the irreducible mass

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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

Pith's one-line read Charged black hole scattering obeys a single universal length scale.

desk verdict First charged BBH zoom-whirl study with a plausible but not yet nailed-down claim of Mirr-universality; worth reviewing, but convergence and limited λ coverage need scrutiny. read the letter →

arxiv 2411.11960 v1 pith:5HFNV5PP submitted 2024-11-18 gr-qc hep-phhep-th

M A. M. Smith , Vasileios Paschalidis , Gabriele Bozzola This is my paper · ORCID
classification gr-qchep-phhep-th
keywords chargedblackholeszoom-whirlorbitsirreduciblemassnumericalrelativityhigh-energyscatteringimpactparameterthresholdsEinstein-Maxwelltheoryapparenthorizon
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 simulates high-energy collisions of equal-mass, like-charged, nonspinning black holes in full Einstein-Maxwell theory, varying the impact parameter for charge-to-mass ratios up to 0.6. It establishes that zoom-whirl orbits survive the repulsive Coulomb force, and that the impact parameters separating immediate merger, zoom-whirl, and scattering change with charge when measured in units of the ADM mass. The central discovery is that those thresholds become independent of charge when the impact parameter is normalized by the sum of the black holes' initial irreducible masses. This suggests the horizon's areal radius, not the gravitational mass, sets the fundamental length scale for close encounters in the strong-field regime.

What carries the argument

The key quantity is the irreducible mass M_irr, defined through the apparent-horizon areal radius r_A = $\sqrt$(A/4π) = 2 M_irr, which the paper computes per black hole via the isolated-horizon formalism. The argument works by normalizing the impact parameter by the sum of the initial M_irr values, which collapses the charge-dependent thresholds onto a single curve. The paper tests and rejects alternative scalings such as (1-$λ^{2}$), ($γ^{2}$-$λ^{2}$), and $\sqrt$(1-$λ^{2}$) with b normalized by M_ADM, M_irr, or the individual gravitational mass, leaving b/M_irr as the only universal combination among those probed.

What would settle it

Re-run the λ=0.1 and λ=0.4 scattering sequences at twice the finest resolution and check whether b_scat/M_irr and b*/M_irr move by more than roughly 0.004–0.03, the uncertainties reported in Table II.

Watch

Extended reading notes

Core claim

In full general relativity, the paper finds that for boosted, equal-mass black holes with equal charge, the immediate-merger threshold b* and the scattering threshold b_scat both decrease as the charge-to-mass ratio λ increases when normalized by the ADM mass. But when b is divided by the sum of the initial irreducible masses, the thresholds become universal: for λ=0.1 and λ=0.4 the paper obtains b*/M_irr between 5.08 and 5.12 and b_scat/M_irr between 5.15 and 5.16, consistent with the uncharged results from earlier work. The authors interpret this as the first explicit demonstration that the irreducible mass, which is proportional to the horizon areal radius, acts as a fundamental gauge-invariant length scale governing horizon-scale scattering in dynamical strong-field spacetimes.

Load-bearing premise

The threshold values are computed with a single numerical resolution per charge-to-mass ratio, with the convergence study deferred to a companion paper, so a resolution-dependent shift larger than the quoted uncertainties would undermine the claimed universality.

Editorial extensions

If this is right

  • Zoom-whirl orbits persist for charged binaries at least up to λ=0.6, so Coulomb repulsion does not suppress this relativistic phenomenon.
  • Charge leaves measurable imprints on the scattering thresholds at Lorentz factor ~1.52, a regime where head-on charged collisions behave like uncharged ones.
  • The universal thresholds b*/M_irr ≈ 5.1 and b_scat/M_irr ≈ 5.15–5.16 match the uncharged high-energy results, suggesting a unified scaling across charges.
  • Predicting merger and scattering outcomes for charged binaries requires knowing the irreducible masses of the black holes, not just their gravitational masses.

Reading between the lines

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

  • The universality probably extends beyond the values probed: since M_irr already absorbs the Reissner-Nordström relation between mass and charge, higher values of λ closer to extremal might keep the same thresholds until the horizon shrinks significantly relative to the gravitational radius.
  • A similar normalization might apply to spinning black holes, where M_irr also encodes the spin-dependent horizon area, but this has not yet been tested.
  • If the universal thresholds hold, they could be used to calibrate analytical models of two-body dynamics near the scattering threshold without needing direct numerical simulation for every charge-to-mass ratio.
  • The convergence assumption is the most likely place for the claimed universality to break, since a resolution study could shift the thresholds and reveal a residual dependence on charge.
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

3 major / 5 minor

Summary. This paper presents full Einstein-Maxwell numerical relativity simulations of equal-mass, nonspinning, like-charged black holes at initial Lorentz factor 1.520, with impact parameter varied for charge-to-mass ratios lambda = 0.1, 0.4, and (for small initial separation) 0.6. The authors report three main findings: zoom-whirl orbits persist for charged binaries at least up to lambda = 0.6; the immediate-merger threshold b* and scattering threshold b_scat decrease with lambda when normalized by the ADM mass; and these thresholds become universal, i.e., independent of lambda, when normalized by the sum of the initial irreducible masses Mirr, so that the horizon areal radius emerges as the fundamental length scale for horizon-scale strong-field scattering. The central quantitative evidence is Table II, which gives b*/Mirr = 5.10 +/- 0.02 and b_scat/Mirr = 5.15 +/- 0.01 for lambda = 0.1, and b*/Mirr = 5.09 +/- 0.03 and b_scat/Mirr = 5.157 +/- 0.006 for lambda = 0.4, together with a comparison to the uncharged thresholds of Ref. [5] converted to Mirr units.

Significance. If the universality claim holds, this is a significant result: it would identify a gauge-invariant, horizon-based length scale (proportional to the areal radius) as the controlling scale for the threshold impact parameters in high-energy black-hole scattering, and it would extend zoom-whirl phenomenology to charged black holes, where previous head-on studies had found charge effects to be negligible. The paper has clear strengths: it uses a constraint-satisfying charged Bowen-York-type initial data solver, brackets the thresholds by sampling impact parameter space, compares directly with the established uncharged results of Ref. [5], and uses publicly available numerical infrastructure. The authors also test several alternative normalizations and explicitly acknowledge the post-hoc nature of the Mirr choice. However, the universality claim is currently supported by only two measured charge values and by threshold estimates without a numerical convergence study; the quoted errors are bracket half-widths, not full error budgets.

major comments (3)
  1. [Sec. II B and Table II] The central universality claim is not backed by a convergence study. Section II B states that each run has about 33 grid points across the smallest apparent horizon and that the finest resolutions differ among charge ratios (Mp/91, Mp/98, Mp/114 for lambda = 0.1, 0.4, 0.6), and the Conclusions explicitly defer a convergence study to Paper II [31]. The uncertainties quoted in Table II are half the bracket spacing in b, not full numerical-error budgets. Since the claimed universality rests on agreement between b_scat/Mirr = 5.15 +/- 0.01 and 5.157 +/- 0.006, a resolution-dependent shift of order 0.01 in b/Mirr could either create or destroy the apparent agreement. Moreover, the differing resolutions across lambda mean that the numerical truncation error is not held constant across the very data sets used for the comparison. The authors should include a convergence test for at least the threshold-determining runs, or explicitly soften the claim to a tentative, resolution-dependent statement.
  2. [Sec. III B, Table II] The universality statement is empirically underdetermined: it involves only two measured charge values, lambda = 0.1 and 0.4, plus a conversion of the uncharged thresholds from Ref. [5]. No uncharged threshold is measured in this paper, and the comparison with Ref. [5] uses a different code, grid setup, gauge choices, and resolution. With only two nonzero sampled values, many functions of lambda can appear approximately constant within the quoted errors, especially because the choice of Mirr normalization was selected after testing several alternatives (footnote 1). To support the claim that b/Mirr is independent of lambda, the authors should either measure a lambda = 0.0 threshold with the same methods, add a third nonzero lambda at the large separation used for Table II, or restrict the claim to consistency between lambda = 0.1 and 0.4.
  3. [Sec. III C and Fig. 4] The determination of the immediate-merger threshold b* is weakened by the ambiguous case at b/MADM = 3.29 shown in Fig. 4. The authors state that this binary shows neither the clear repeating features of immediate merger nor a clearly defined secondary peak, and then report b*/MADM = 3.30 +/- 0.01 as the midpoint of the 3.29-3.31 bracket. The quoted error only reflects the bracket spacing, not the classification ambiguity. For b*/Mirr, which is a key part of the universality claim, the authors should state explicitly how the threshold would shift if b/MADM = 3.29 were classified as immediate merger or as a zoom-whirl case, and should include that uncertainty in the reported error.
minor comments (5)
  1. [Sec. IV] In the Conclusions, 'scattering treshold' should be 'scattering threshold'.
  2. [Abstract and header] The phrase 'full gen eral relativity' has a spacing error; it should read 'full general relativity'.
  3. [Sec. III A] The sentence 'We did not include an investigation of lambda = 0.0 with impact parameter' is ambiguous; it should read 'with varying impact parameter', since a single lambda = 0.0 run with non-zero impact parameter was indeed performed.
  4. [Table I caption] The caption says 'charge-to-mass ratio of each BHs'; it should be 'of each BH' (singular).
  5. [Sec. III C] The text says 'The log of the magnitude of Psi_4 in this phase is a linear curve with respect to time'; a linear curve is better described as a straight line or a linear function.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the thresholds are directly measured from numerical evolutions and the Mirr normalization is an empirical scaling choice, not a fitted input disguised as a prediction.

full rationale

The paper's central claims are empirical results from full Einstein-Maxwell numerical relativity simulations, not derivations from a postulated normalization. The immediate-merger threshold b* and scattering threshold bscat are identified by binary outcome classification and gravitational-wave morphology (Sec. III C), not by fitting to the irreducible mass. The universality in b/Mirr is presented as a discovered scaling: after measuring bscat/MADM and b*/MADM, the authors test several normalizations and find that Mirr from Eq. (4) collapses the two probed charge ratios. This is post-hoc model selection rather than a fitted parameter renamed as a prediction, and the paper is transparent about having tested alternative scalings. The external uncharged data of Sperhake et al. [5] provide an independent anchor for the Mirr normalization, and the b* agreement is found after the normalization search was based on bscat, giving a partially independent consistency check. The companion-paper citation [31] is used only for deferred convergence studies and follow-up metrics, not to support the central claim, and no uniqueness theorem or ansatz is imported from the authors' prior work. The lack of a convergence study is a numerical-error/correctness concern about whether the reported thresholds are fully converged, but it is not a circularity of the derivation chain. Therefore no circular step is identified.

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

The paper introduces no new free parameters or invented entities. It relies on several domain assumptions of numerical relativity: constraint-satisfying initial data, reliable horizon diagnostics, and, critically, convergence at the chosen resolution, which is explicitly deferred to a companion paper.

assumptions (4)
  • domain assumption TwoChargedPunctures initial data solve the Einstein-Maxwell constraint equations and represent two boosted, charged black holes with the intended quasilocal masses and charges.
    Invoked in Section II A; all subsequent threshold measurements depend on the initial data being physical and accurate.
  • domain assumption The isolated-horizon diagnostics (QuasiLocalMeasuresEM) give reliable quasilocal mass, charge, and irreducible mass during dynamical evolution.
    Invoked in Section II C; the normalization b/Mirr and the reported lambda values come from these diagnostics.
  • domain assumption The numerical evolution at the chosen resolution is converged, with errors smaller than the quoted threshold uncertainties.
    Section II B chooses roughly 33 points across the smallest horizon and defers the convergence study to Paper II [31]; the claimed precision in Table II assumes this.
  • domain assumption The uncharged thresholds from Sperhake et al. [5] can be converted from b/MADM to b/Mirr using a gamma = 1.520 relation, and this conversion is valid.
    Section III B compares the new universality directly to [5] after conversion; the conversion details are not shown in this paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of High-energy interactions of charged black holes in full general relativity I: Zoom-whirl orbits and universality with the irreducible mass." pith.science (2026). https://pith.science/paper/5HFNV5PP

@misc{pith2026241111960,
  author       = {Pith},
  title        = {Pith review of: High-energy interactions of charged black holes in full general relativity I: Zoom-whirl orbits and universality with the irreducible mass},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5HFNV5PP}},
  note         = {Machine review of arXiv:2411.11960}
}
abstract

We simulate high-energy scattering of equal-mass, nonspinning black holes endowed with like charges in full general relativity while varying the impact parameter $b$. We show that electrodynamics does not suppress zoom-whirl orbits for at least charge-to-mass ratios $\lambda = 0.1, 0.4, 0.6$. However, we find that as $\lambda$ increases, the immediate merger and scattering thresholds defining the zoom-whirl regime move to smaller impact parameter $b/M_{\rm ADM}$, with $M_{\rm ADM}$ designating the binary black hole gravitational mass. This demonstrates that charge leaves observable imprints in key properties at energy scales where charge has negligible influence in head-on collisions. Additionally, we find that these threshold impact parameters become universal, i.e., charge-independent, when we normalize $b$ by the sum of the initial BH irreducible masses in the binary ($b/M_{\rm irr}$). This is the first explicit demonstration that the irreducible mass, which is proportional to the black hole areal radius, defines a fundamental gauge-invariant length scale governing horizon scale scattering events in the strong-field, dynamical spacetime regime.

Figures

Figures reproduced from arXiv: 2411.11960 by the authors.

Figure 1
Figure 1. FIG. 1. A diagram depicting the set-up for our simulations, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Representative puncture trajectories from simulat [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Puncture trajectories for a [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The natural log of the magnitude of Ψ [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. High-energy interactions of charged black holes in full general relativity II: Near-extremal merger remnants and universality with the irreducible mass

    gr-qc 2024-12 conditional novelty 6.0 of 10

    Charged black hole collisions near the scattering threshold produce near-extremal remnants (Υ_f = 0.97), radiate up to 31% of the total mass, and show charge-independent thresholds when impact parameters are normalize...

Reference graph

Works this paper leans on

60 extracted references · 26 canonical work pages · cited by 1 Pith paper

  1. [5]

    Healy, I

    J. Healy, I. Ruchlin, C. O. Lousto, and Y. Zlochower, High energy collisions of black holes numerically revis- ited, Physical Review D 94, 104020 (2016)

    work page 2016

  2. [31]

    Gundlach, S

    C. Gundlach, S. Akcay, L. Barack, and A. Nagar, Criti- cal phenomena at the threshold of immediate merger in binary black hole systems: The extreme mass ratio case, Physical Review D 86, 084022 (2012) , publisher: Amer- ican Physical Society

    2012

  3. [1]

    Cardoso, L

    V. Cardoso, L. Gualtieri, C. Herdeiro, and U. Sperhake, Exploring New Physics Frontiers Through Numerical Relativity, Living Reviews in Relativity 18, 1 (2015)

    2015

  4. [2]

    Sperhake, V

    U. Sperhake, V. Cardoso, F. Pretorius, E. Berti, and J. A. Gonz´ alez, High-Energy Collision of Two Black Holes, Physical Review Letters 101, 161101 (2008)

    work page 2008

  5. [3]

    These results demonstrate that zoom-whirl behavior can be exhibited for at least λ = 0 .6, and that the elec- trostatic repulsion does not suppress zoom-whirl orbits

    The BHs in this binary merge t ∼ 230MADM after their first encounter. These results demonstrate that zoom-whirl behavior can be exhibited for at least λ = 0 .6, and that the elec- trostatic repulsion does not suppress zoom-whirl orbits. In a future work, we plan to probe even larger λ to test the limits of zoom-whirl behavior. B. Impact Parameter Threshold...

    work page

  6. [4]

    Shibata, H

    M. Shibata, H. Okawa, and T. Yamamoto, High-velocity collision of two black holes, Physical Review D 78, 101501 (2008)

    2008

  7. [6]

    Sperhake, E

    U. Sperhake, E. Berti, V. Cardoso, and F. Pretorius, Gravity-dominated unequal-mass black hole collisions, Physical Review D 93, 044012 (2016)

    work page 2016

  8. [7]

    Sperhake, V

    U. Sperhake, V. Cardoso, F. Pretorius, E. Berti, T. Hin- derer, and N. Yunes, Cross Section, Final Spin, and Zoom-Whirl Behavior in High-Energy Black-Hole Col- lisions, Physical Review Letters 103, 131102 (2009) . 8

    work page 2009

Show all 60 references
  1. [8]

    Sperhake, E

    U. Sperhake, E. Berti, V. Cardoso, and F. Preto- rius, Universality, Maximum Radiation, and Absorp- tion in High-Energy Collisions of Black Holes with Spin, Physical Review Letters 111, 041101 (2013)

    2013

  2. [9]

    Sperhake, E

    U. Sperhake, E. Berti, V. Cardoso, F. Pre- torius, and N. Yunes, Superkicks in ultrarel- ativistic encounters of spinning black holes, Physical Review D 83, 024037 (2011)

    2011

  3. [10]

    Sperhake, W

    U. Sperhake, W. Cook, and D. Wang, High-energy collision of black holes in higher dimensions, Physical Review D 100, 104046 (2019)

    2019

  4. [11]

    W. G. Cook, U. Sperhake, E. Berti, and V. Car- doso, Black-hole head-on collisions in higher dimensions, Physical Review D 96, 124006 (2017)

    2017

  5. [12]

    Alcubierre, J

    M. Alcubierre, J. C. Degollado, and M. Sal- gado, The Einstein-Maxwell system in 3+1 form and initial data for multiple charged black holes, Physical Review D 80, 104022 (2009)

    2009

  6. [13]

    Andrade, P

    T. Andrade, P. Figueras, and U. Sperhake, Ev- idence for violations of Weak Cosmic Censor- ship in black hole collisions in higher dimensions, Journal of High Energy Physics 2022, 111 (2022)

    2022

  7. [14]

    Zilh˜ ao, V

    M. Zilh˜ ao, V. Cardoso, C. Herdeiro, L. Lehner, and U. Sperhake, Collisions of charged black holes, Physical Review D 85, 124062 (2012)

    2012

  8. [15]

    S. L. Liebling and C. Palenzuela, Electromagnetic Lumi- nosity of the Coalescence of Charged Black Hole Binaries, Physical Review D 94, 064046 (2016)

    2016

  9. [16]

    Bozzola and V

    G. Bozzola and V. Paschalidis, Initial data for gen- eral relativistic simulations of multiple electrically charged black holes with linear and angular momenta, Physical Review D 99, 104044 (2019)

    2019

  10. [17]

    Zilh˜ ao, V

    M. Zilh˜ ao, V. Cardoso, C. Herdeiro, L. Lehner, and U. Sperhake, Collisions of oppositely charged black holes, Physical Review D 89, 044008 (2014)

    2014

  11. [18]

    Bozzola, Does Charge Matter in High-Energy Collisions of Black Holes?, Physical Review Letters 128, 071101 (2022)

    G. Bozzola, Does Charge Matter in High-Energy Collisions of Black Holes?, Physical Review Letters 128, 071101 (2022)

    2022

  12. [19]

    Mukherjee, N

    S. Mukherjee, N. K. Johnson-McDaniel, W. Tichy, and S. L. Liebling, Conformally curved initial data for charged, spinning blac k hole binaries on arbitrary orbits (2022), arXiv:2202.12133 [astro-ph, physics:gr-qc]

    2022 arXiv

  13. [20]

    Bozzola and V

    G. Bozzola and V. Paschalidis, Numerical-relativity sim- ulations of the quasi-circular inspiral and merger of non-spinning, charged black holes: methods and comparison with approximate approaches, Physical Review D 104, 044004 (2021)

    2021

  14. [21]

    Bozzola and V

    G. Bozzola and V. Paschalidis, General Rel- ativistic Simulations of the Quasicircular In- spiral and Merger of Charged Black Holes: GW150914 and Fundamental Physics Implications, Physical Review Letters 126, 041103 (2021)

    2021

  15. [22]

    Bozzola and V

    G. Bozzola and V. Paschalidis, Can quasicircular mergers of charged black holes produce extremal black holes?, Physical Review D 108, 064010 (2023)

    2023

  16. [23]

    R. Luna, G. Bozzola, V. Cardoso, V. Paschalidis, and M. Zilh˜ ao, Kicks in charged black hole binaries, Physical Review D 106, 084017 (2022)

    2022

  17. [24]

    Bombelli and E

    L. Bombelli and E. Calzetta, Chaos around a black hole, Classical and Quantum Gravity 9, 2573 (1992)

    1992

  18. [25]

    Pretorius and D

    F. Pretorius and D. Khurana, Black hole mergers and unstable circular orbits, Classical and Quantum Gravity 24, S83 (2007)

    2007

  19. [26]

    Glampedakis and D

    K. Glampedakis and D. Kennefick, Zoom and whirl: Ec- centric equatorial orbits around spinning black holes and their evolution under gravitational radiation reaction, Physical Review D 66, 044002 (2002)

    2002

  20. [27]

    Levin, R

    J. Levin, R. O’Reilly, and E. J. Copeland, Grav- ity waves from homoclinic orbits of compact binaries, Physical Review D 62, 024023 (2000)

    2000

  21. [28]

    Gold and B

    R. Gold and B. Br¨ ugmann, Radiation from low-momentum zoom-whirl orbits, Classical and Quantum Gravity 27, 084035 (2010)

    2010

  22. [29]

    Healy, J

    J. Healy, J. Levin, and D. Shoemaker, Zoom-Whirl Orbits in Black Hole Binaries, Physical Review Letters 103, 131101 (2009)

    2009

  23. [30]

    Damour and P

    T. Damour and P. Rettegno, Strong-field scattering of two black holes: Numerical relativity meets post- minkowskian gravity, Phys. Rev. D 107, 064051 (2023)

    2023

  24. [32]

    G. t. Hooft, Graviton dominance in ultra-high-energy scattering, Physics Letters B 198, 61 (1987)

    1987

  25. [34]

    Banks and W

    T. Banks and W. Fischler, A Model for High Energy Scattering in Quantum Gravity (1999), arXiv:hep-th/9906038

    1999 arXiv

  26. [35]

    Amati, M

    D. Amati, M. Ciafaloni, and G. Veneziano, Superstring collisions at planckian energies, Physics Letters B 197, 81 (1987)

    1987

  27. [36]

    Dreyer, B

    O. Dreyer, B. Krishnan, D. Shoemaker, and E. Schnetter, Introduction to isolated horizons in numerical relativity , Physical Review D 67, 024018 (2003)

    2003

  28. [37]

    W. E. East and F. Pretorius, Ultrarelativistic black hole formation, Physical Review Letters 110, 101101 (2013)

    2013

  29. [38]

    S. R. Brandt, G. Bozzola, C.-H. Cheng, P. Diener, A. Dima, W. E. Gabella, M. Gracia-Linares, R. Haas, Y. Zlochower, M. Alcubierre, D. Alic, G. Allen, M. An- sorg, M. Babiuc-Hamilton, L. Baiotti, W. Benger, E. Bentivegna, S. Bernuzzi, T. Bode, B. Bren- dal, B. Bruegmann, M. Cam...

    2021

  30. [39]

    Ansorg, B

    M. Ansorg, B. Bruegmann, and W. Tichy, A single- domain spectral method for black hole puncture data, Physical Review D 70, 064011 (2004) , arXiv:gr- qc/0404056

    2004

  31. [40]

    Arnowitt, S

    R. Arnowitt, S. Deser, and C. W. Mis- ner, The Dynamics of General Relativity, General Relativity and Gravitation 40, 1997 (2008)

    2008

  32. [41]

    Bozzola, kuibit: Analyzing Ein- stein Toolkit simulations with Python, Journal of Open Source Software 6, 3099 (2021)

    G. Bozzola, kuibit: Analyzing Ein- stein Toolkit simulations with Python, Journal of Open Source Software 6, 3099 (2021)

    2021

  33. [42]

    Sperhake, Binary black-hole evolutions of excision and puncture data, Physical Review D 76, 104015 (2007)

    U. Sperhake, Binary black-hole evolutions of excision and puncture data, Physical Review D 76, 104015 (2007)

    2007

  34. [43]

    T. W. Baumgarte and S. L. Shapiro, Ch. 2: The 3+1 decomposition of Einstein’s equations, in Numerical Relativity: Solving Einstein’s Equations on the Computer (Cambridge University Press, 2010) pp. 23–53

    2010

  35. [44]

    Witek, M

    H. Witek, M. Zilhao, G. Bozzola, M. Elley, G. Fi- carra, T. Ikeda, N. Sanchis-Gual, and H. Silva, Canuda: a public numerical relativity library to probe fund amental physics (2021)

    2021

  36. [45]

    Zilh˜ ao, H

    M. Zilh˜ ao, H. Witek, and V. Cardoso, Nonlinear interactions between black holes and Proca fields, Classical and Quantum Gravity 32, 234003 (2015)

    2015

  37. [46]

    T. W. Baumgarte and S. L. Shapiro, On the Nu- merical Integration of Einstein’s Field Equations, Physical Review D 59, 024007 (1998)

    1998

  38. [47]

    Shibata and T

    M. Shibata and T. Nakamura, Evolution of three- dimensional gravitational waves: Harmonic slicing case, Physical Review D 52, 5428 (1995)

    1995

  39. [48]

    Schnetter, S

    E. Schnetter, S. H. Hawley, and I. Hawke, Evolutions in 3D numerical relativity using fixed mesh refinement, Classical and Quantum Gravity 21, 1465 (2004)

    2004

  40. [49]

    Hinder, A

    I. Hinder, A. Buonanno, M. Boyle, Z. B. Etienne, J. Healy, N. K. Johnson-McDaniel, A. Nagar, H. Nakano, Y. Pan, H. P. Pfeiffer, M. P¨ urrer, C. Reisswig, M. A. Scheel, E. Schnetter, U. Sperhake, B. Szil´ agyi, W. Tichy, B. Wardell, A. Zenginoglu, D. Alic, S. Bernuzzi, T. Bode, ...

    2013 arXiv

  41. [50]

    Newman and R

    E. Newman and R. Penrose, An Approach to Gravi- tational Radiation by a Method of Spin Coefficients, Journal of Mathematical Physics 3, 566 (1962)

    1962

  42. [51]

    Thornburg, A fast apparent horizon finder for three- dimensional Cartesian grids in numerical relativity, Classical and Quantum Gravity 21, 743 (2003)

    J. Thornburg, A fast apparent horizon finder for three- dimensional Cartesian grids in numerical relativity, Classical and Quantum Gravity 21, 743 (2003)

    2003

  43. [52]

    whirl” phase where the punc- tures have a close encounter is followed by the punctures “zoom

    for a review on extraction of gravitational waves in numerical relativity). Our Weyl Newman-Penrose scalar Ψ 4 is reported at r/Mp = 80. The ADM mass, and ADM angular momentum, JADM, of the spacetime are calculated by TwoChargedPunctures at the initial data level. The initial ...

  44. [53]

    Witek, V

    H. Witek, V. Cardoso, C. Herdeiro, A. Nerozzi, U. Sper- hake, and M. Zilh˜ ao, Black holes in a box: Toward the numerical evolution of black holes in AdS space-times, Physical Review D 82, 104037 (2010)

    2010

  45. [54]

    N. T. Bishop and L. Rezzolla, Extraction of Gravitational Waves in Numerical Relativ- ity, Living Reviews in Relativity 19, 2 (2016) , arXiv:1606.02532 [gr-qc]

    2016 arXiv

  46. [55]

    M. A. M. Smith, V. Paschalidis, and G. Bozzola, In preparation (2024)

    2024

  47. [56]

    S. A. Teukolsky, Rotating Black Holes: Separable Wave Equations for Gravitational and Electromagnetic Pertur- bations, Physical Review Letters 29, 1114 (1972)

    1972

  48. [57]

    D. N. Page, Can two ultrarelativistic objects lose almost all their energy to gravitational radiation?, Phys. Rev. D 107, 064057 (2023)

    2023

  49. [58]

    C. R. Harris, K. J. Millman, S. J. van der Walt, R. Gom- mers, P. Virtanen, D. Cournapeau, E. Wieser, J. Tay- lor, S. Berg, N. J. Smith, R. Kern, M. Picus, S. Hoyer, M. H. van Kerkwijk, M. Brett, A. Haldane, J. F. del R ´ ıo, M. Wiebe, P. Peterson, P. G´ erard-Marchant, K. She...

    2020

  50. [59]

    Virtanen, R

    P. Virtanen, R. Gommers, T. E. Oliphant, M. Haber- land, T. Reddy, D. Cournapeau, E. Burovski, P. Pe- terson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, I. Po- lat, Y....

    2020

  51. [60]

    HDF5 for Python , http://www.h5py.org/

  52. [61]

    J. D. Hunter, Matplotlib: A 2D Graphics Environment, Computing in Science & Engineering 9, 90 (2007) , con- ference Name: Computing in Science & Engineering

    2007

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.