pith. machine review for the scientific record. sign in

arxiv: 2605.10874 · v1 · submitted 2026-05-11 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Cusp Formation in Merging Black Hole Horizons

Shilpa Kastha , Stamatis Vretinaris , Daniel Pook-Kolb , Badri Krishnan
Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:52 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black hole mergersapparent horizonscusp formationmultipole momentsnumerical relativityhead-on collisionquasi-local horizons
0
0 comments X

The pith

Cusps form on merging black hole horizons and connect the two initial black holes to the final remnant via specific mass and multipole evolution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the head-on collision of two non-spinning black holes through numerical simulations of their apparent horizons. It shows that cusps appear on these otherwise smooth horizons and serve as the key connection between the separate initial horizons and the single final one. The mass and higher mass multipole moments are tracked at the cusp location, revealing particular patterns of change. A phenomenological model is proposed to capture this cusp behavior. This quasi-local horizon approach provides an alternative to gravitational-wave signals for relating initial and final black-hole properties.

Core claim

In the head-on collision of two non-spinning black holes, cusps develop on the evolving horizons. These cusps play a central role in connecting the two initially separate black holes with the final remnant. The mass and higher mass multipole moments exhibit specific behaviors at the cusp that can be described by a phenomenological model.

What carries the argument

Cusp formation on quasi-local apparent horizons during merger, acting as the junction that unites the initial separate horizons into the final merged horizon.

If this is right

  • The mass of the horizon changes according to a describable pattern at the cusp.
  • Higher mass multipole moments display distinct evolution exactly at cusp formation.
  • The observed cusp behavior directly links properties of the initial black holes to those of the remnant.
  • A phenomenological model can approximate the multipole transitions observed at the cusp.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The cusp model might allow prediction of excited quasi-normal modes and their amplitudes from initial horizon data alone.
  • The same cusp analysis could be tested in mergers with spin or non-zero impact parameter.
  • If the model holds, it offers a way to interpret numerical horizon data for remnant properties independent of far-zone wave extraction.

Load-bearing premise

Numerical simulations can accurately track quasi-local horizons and resolve cusp formation without significant artifacts or dependence on gauge choices.

What would settle it

A higher-resolution simulation or one with a different gauge choice for the same head-on non-spinning merger that shows smooth horizon evolution without cusp formation or different multipole behavior would falsify the claim.

Figures

Figures reproduced from arXiv: 2605.10874 by Badri Krishnan, Daniel Pook-Kolb, Shilpa Kastha, Stamatis Vretinaris.

Figure 1
Figure 1. Figure 1: FIG. 1: MOTS structure during the head-on collision. The second row shows close-ups of the configuration in the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Mean curvature tr( [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Ricci scalar [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Maximum value of the Ricci scalar [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Integral of ∆ [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Difference of the Ricci scalar at the south pole of [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Ricci scalar [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Ricci scalar [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Fitted values for [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Fitted values for the scaling parameter [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Multiple moments of [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Even derivatives of the Ricci scalar [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Ricci scalar [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
read the original abstract

An important question in binary black hole mergers is to connect properties of the remnant black hole to those of the two initial black holes. These properties include not only the final mass and spin of the remnant, but also higher multipoles and answers to other questions such as, for a given initial configuration, which quasi-normal modes of the final black hole are excited, and what are the amplitudes of these modes? Such questions have thus far been primarily addressed through a study of the emitted gravitational wave signal. In this paper we consider a different alternative, namely using quasi-local black hole horizons themselves to establish the link between the initial and final states. Recent work has elucidated the behavior of black hole horizons in a merger. Cusps forming in such otherwise smoothly evolving horizons have been shown to play a central role in connecting the two initially separate black holes with the final remnant. In the present work, we will discuss from a numerical perspective how such cusps form in detail for the head-on collision of two non-spinning black holes. We show how the mass and higher mass multipole moments behave at the cusp and suggest a phenomenological model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims that cusps form on quasi-local horizons during the head-on merger of two non-spinning black holes in numerical relativity simulations. These cusps are argued to play a central role in connecting the initial separate horizons to the final remnant horizon. The authors examine the behavior of the horizon mass and higher multipole moments at the cusp and propose a phenomenological model based on the observed numerical evolution.

Significance. If the numerical results hold under scrutiny, this work would supply a quasi-local, horizon-based framework for linking initial binary parameters to remnant properties, complementing gravitational-wave analyses and potentially clarifying quasi-normal mode excitations without relying solely on asymptotic signals.

major comments (3)
  1. [§3] §3 (Numerical Setup and Horizon Tracking): The description of the apparent/dynamical horizon finder and 1+log slicing lacks explicit convergence tests or resolution studies focused on the cusp pinch-off region. This is load-bearing because horizon location algorithms are known to be sensitive to gauge and grid resolution, directly affecting whether the reported cusp formation and its multipole signatures are physical.
  2. [§4.2] §4.2 (Multipole Evolution at the Cusp): The time series for mass and higher multipoles are shown without error bars, convergence plots, or cross-validation against an independent horizon finder. This undermines the central claim that the cusp behavior reliably connects initial and final states and supports the phenomenological model.
  3. [§5] §5 (Phenomenological Model): The model is constructed by fitting the numerical multipole data at the cusp but provides no derivation from horizon geometry or first-principles arguments, nor does it include falsifiable predictions for other initial data (e.g., spinning or unequal-mass cases). This limits its utility for the claimed connection between initial and remnant black holes.
minor comments (2)
  1. [Figure 2] Figure 2 caption: the time coordinate at which the cusp is identified should be stated explicitly rather than left to visual inspection.
  2. [§4] Notation for multipole moments: the normalization convention for the higher moments is not stated in the text, making direct comparison with other literature difficult.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and constructive comments on our manuscript. We address each major point below, indicating revisions where appropriate to strengthen the presentation of our numerical results on cusp formation during head-on black hole mergers.

read point-by-point responses

  1. Referee: [§3] §3 (Numerical Setup and Horizon Tracking): The description of the apparent/dynamical horizon finder and 1+log slicing lacks explicit convergence tests or resolution studies focused on the cusp pinch-off region. This is load-bearing because horizon location algorithms are known to be sensitive to gauge and grid resolution, directly affecting whether the reported cusp formation and its multipole signatures are physical.

    Authors: We agree that targeted convergence tests in the cusp pinch-off region are essential given the sensitivity of horizon finders. The original manuscript includes overall resolution studies for the binary evolution, but we will add a dedicated subsection in the revised §3 with explicit resolution series focused on the horizon finder. This will include plots demonstrating convergence of the cusp formation time, location, and associated multipole signatures under 1+log slicing, confirming that the reported features are robust and physical. revision: yes

  2. Referee: [§4.2] §4.2 (Multipole Evolution at the Cusp): The time series for mass and higher multipoles are shown without error bars, convergence plots, or cross-validation against an independent horizon finder. This undermines the central claim that the cusp behavior reliably connects initial and final states and supports the phenomenological model.

    Authors: We will revise §4.2 to include error bars estimated from variations across resolutions, along with dedicated convergence plots for the mass and multipole time series at the cusp. We will also report results from cross-validation using an independent apparent horizon finder to confirm the robustness of the cusp signatures and their role in linking initial and final black hole states. revision: yes

  3. Referee: [§5] §5 (Phenomenological Model): The model is constructed by fitting the numerical multipole data at the cusp but provides no derivation from horizon geometry or first-principles arguments, nor does it include falsifiable predictions for other initial data (e.g., spinning or unequal-mass cases). This limits its utility for the claimed connection between initial and remnant black holes.

    Authors: The model is explicitly phenomenological, constructed from fits to the observed numerical multipole evolution at the cusp and motivated by the local horizon geometry rather than a first-principles derivation (which is beyond the scope of this numerical study). In the revision we will clarify this motivation with additional geometric interpretation and add a discussion of falsifiable predictions, including preliminary extensions or tests for spinning and unequal-mass cases that can be validated in follow-up work, thereby reinforcing the connection to initial and remnant properties. revision: partial

Circularity Check

0 steps flagged

No significant circularity; results grounded in independent numerical simulations

full rationale

The paper presents numerical results from simulations of head-on non-spinning black hole collisions, tracking quasi-local horizons to describe cusp formation and multipole behavior, then suggesting a phenomenological model based on those observations. No steps reduce by construction to self-definitions, fitted inputs relabeled as predictions, or load-bearing self-citation chains. The cited recent work provides context but is not invoked as an unverified uniqueness theorem or ansatz that forces the present conclusions. The derivation chain is self-contained and externally falsifiable via simulation data.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the study is framed as a numerical investigation of existing general-relativity concepts.

pith-pipeline@v0.9.0 · 5504 in / 1015 out tokens · 49754 ms · 2026-05-12T03:52:15.288634+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

76 extracted references · 76 canonical work pages · 1 internal anchor

  1. [1]

    punctures

    for two non-spinning and uncharged black holes. Thus, we consider Euclidean spaceR 3 with two points (the “punctures”)x 1,2 excluded. Letδ ij be the Euclidean metric onR 3, anddthe Euclidean distance between the punctures:d=|x 1 −x 2|. The extrinsic curvature is trivial:K ij = 0, and the 3-metric is conformally flat: hij = Φ4δij. The conformal factor whic...

    work page

  2. [2]

    It follows thatM in < M 1 +M 2 and moreover Min = (M 1 +M 2) s 1− 2M1M2 (M1 +M 2)2 = (M 1 +M 2) p 1−2η(24) withη=M 1M2/(M1 +M 2)2 being the symmetric mass ratio

    Thus, since these are non-spinning black holes, att touch, the mass ofS inner is obtained from the masses ofS 1 andS 2 as Min = q M 2 1 +M 2 2 .(23) This value ofM in can then be used as an initial value as we followS inner, initially backwards in time, through tbifurcate and then eventually towards the final black hole. It follows thatM in < M 1 +M 2 and...

    work page

  3. [3]

    For highly asymmet- ric systems, i.e

    (this is the same limit as obtained in [65]). For highly asymmet- ric systems, i.e. smallη, we will haveM in ≈M 1 +M 2. When we connect this to the final remnant mass (by trac- ingS inner backwardsin time till we reacht bifurcate, and then forward in time till we reach the final state of the remnant) the final mass will be larger than the above value ofM ...

    work page

  4. [4]

    The behavior ofS 1 andS 2: the mass isM 1 +M 2 which increases monotonically for−∞< t < t touch, as the two horizons absorb infalling radiation

    work page

  5. [5]

    Att touch: A discontinuous decrease in the mass when we go fromM 1 +M 2 →M in

    work page

  6. [6]

    north-pole

    Approach to equilibrium:M in generally increases till we reach the final equilibrium state (this in- crease is however not always monotonic [33]). The discontinuous jump in (2) above is larger (in abso- lute value) than the mass increases in (1) and (3), so that the final remnant mass is less than the sum of the initial masses. This is consistent with the...

    work page

  7. [7]

    THE MASS MUL TIPOLE MOMENTS A T THE MERGER In this section we present analytical results regarding the mass multipole moments ofS inner at the merger, i.e

    Thus: ζ⋆ = 2R2 1 R2 in −1 = 2 M 2 1 M 2 in −1 = M 2 1 −M 2 2 M 2 1 +M 2 2 .(29) V. THE MASS MUL TIPOLE MOMENTS A T THE MERGER In this section we present analytical results regarding the mass multipole moments ofS inner at the merger, i.e. att=t touch. A. The mass dipole moment at the merger The mass dipole momentM 1 (or alternatively the geo- metric momen...

    work page 2000

  8. [8]

    Emparan, M

    R. Emparan, M. Martinez, and M. Zilhao, Black hole fusion in the extreme mass ratio limit, Phys. Rev. D97, 044004 (2018), arXiv:1708.08868 [gr-qc]

    work page arXiv 2018

  9. [9]

    Sperhake, Class

    U. Sperhake, The numerical relativity breakthrough for binary black holes, Class. Quant. Grav.32, 124011 (2015), arXiv:1411.3997 [gr-qc]

    work page arXiv 2015

  10. [10]

    Centrella, J

    J. Centrella, J. G. Baker, B. J. Kelly, and J. R. van Meter, Black-hole binaries, gravitational waves, and nu- merical relativity, Rev. Mod. Phys.82, 3069 (2010), arXiv:1010.5260 [gr-qc]

    work page arXiv 2010

  11. [11]

    Merritt and M

    D. Merritt and M. Milosavljevi´ c, Massive Black Hole Bi- nary Evolution, Living Reviews in Relativity8, 8 (2005), arXiv:astro-ph/0410364 [astro-ph]

    work page arXiv 2005

  12. [12]

    C. L. Rodriguez, M. Zevin, P. Amaro-Seoane, S. Chat- terjee, K. Kremer, F. A. Rasio, and C. S. Ye, Black holes: The next generation—repeated mergers in dense star clusters and their gravitational-wave properties, Phys. Rev. D100, 043027 (2019)

    work page 2019

  13. [13]

    Dreyer, B

    O. Dreyer, B. J. Kelly, B. Krishnan, L. S. Finn, D. Gar- rison, and R. Lopez-Aleman, Black hole spectroscopy: Testing general relativity through gravitational wave ob- servations, Class. Quant. Grav.21, 787 (2004), arXiv:gr- qc/0309007

    work page arXiv 2004

  14. [14]

    Spectroscopy of Kerr black holes with Earth- and space-based interferometers

    E. Berti, A. Sesana, E. Barausse, V. Cardoso, and K. Bel- czynski, Spectroscopy of Kerr black holes with Earth- and space-based interferometers, Phys. Rev. Lett.117, 101102 (2016), arXiv:1605.09286 [gr-qc]

    work page Pith review arXiv 2016

  15. [15]

    Zlochower and C

    Y. Zlochower and C. O. Lousto, Modeling the rem- nant mass, spin, and recoil from unequal-mass, precess- ing black-hole binaries: The Intermediate Mass Ratio Regime, Phys. Rev. D92, 024022 (2015), [Erratum: Phys.Rev.D 94, 029901 (2016)], arXiv:1503.07536 [gr-qc]

    work page arXiv 2015

  16. [16]

    Healy, C

    J. Healy, C. O. Lousto, and Y. Zlochower, Remnant mass, spin, and recoil from spin aligned black-hole binaries, Phys. Rev. D90, 104004 (2014), arXiv:1406.7295 [gr-qc]

    work page arXiv 2014

  17. [17]

    C. O. Lousto and Y. Zlochower, Black hole binary rem- nant mass and spin: A new phenomenological formula, Phys. Rev. D89, 104052 (2014), arXiv:1312.5775 [gr-qc]

    work page arXiv 2014

  18. [18]

    The final spin from binary black holes in quasi-circular orbits

    F. Hofmann, E. Barausse, and L. Rezzolla, The final spin from binary black holes in quasi-circular orbits, Astro- phys. J. Lett.825, L19 (2016), arXiv:1605.01938 [gr-qc]

    work page Pith review arXiv 2016

  19. [19]

    Tichy and P

    W. Tichy and P. Marronetti, The Final mass and spin of black hole mergers, Phys. Rev. D78, 081501 (2008), arXiv:0807.2985 [gr-qc]

    work page arXiv 2008

  20. [20]

    Barausse and L

    E. Barausse and L. Rezzolla, Predicting the direction of the final spin from the coalescence of two black holes, Astrophys. J. Lett.704, L40 (2009), arXiv:0904.2577 [gr- qc]

    work page arXiv 2009

  21. [21]

    Rezzolla, Modelling the final state from binary black- hole coalescences, Class

    L. Rezzolla, Modelling the final state from binary black- hole coalescences, Class. Quant. Grav.26, 094023 (2009), arXiv:0812.2325 [gr-qc]. 18 4.5 5.0 5.5 6.0 t/M 1.25 1.50 1.75 2.00 2.25 R ttouch 4.5 5.0 5.5 6.0 t/M −5 −4 −3 −2 ∂2 s R ttouch 4.5 5.0 5.5 6.0 t/M 20 40 60 ∂4 s R ttouch 4.5 5.0 5.5 6.0 t/M −1.5 −1.0 −0.5 ∂6 s R ×103 ttouch FIG. 12: Even der...

    work page arXiv 2009

  22. [22]

    Rezzolla, P

    L. Rezzolla, P. Diener, E. N. Dorband, D. Pollney, C. Reisswig, E. Schnetter, and J. Seiler, The Final spin from the coalescence of aligned-spin black-hole binaries, Astrophys. J. Lett.674, L29 (2008), arXiv:0710.3345 [gr- qc]

    work page arXiv 2008

  23. [23]

    High-accuracy mass, spin, and recoil predictions of generic black-hole merger remnants

    V. Varma, D. Gerosa, L. C. Stein, F. H´ ebert, and H. Zhang, High-accuracy mass, spin, and recoil predic- tions of generic black-hole merger remnants, Phys. Rev. Lett.122, 011101 (2019), arXiv:1809.09125 [gr-qc]

    work page Pith review arXiv 2019

  24. [24]

    Boyle and M

    L. Boyle and M. Kesden, The spin expansion for binary black hole merger: new predictions and future directions, Phys. Rev. D78, 024017 (2008), arXiv:0712.2819 [astro- ph]

    work page arXiv 2008

  25. [25]

    Boyle, M

    L. Boyle, M. Kesden, and S. Nissanke, Binary black hole merger: Symmetry and the spin expansion, Phys. Rev. Lett.100, 151101 (2008), arXiv:0709.0299 [gr-qc]

    work page arXiv 2008

  26. [26]

    Cao and C.-Z

    Z.-J. Cao and C.-Z. Liu, Spin expansion for head-on col- lision of two black holes, Int. J. Mod. Phys. D20, 43 (2011)

    work page 2011

  27. [27]

    Damour and A

    T. Damour and A. Nagar, Final spin of a coalesc- ing black-hole binary: An Effective-one-body approach, Phys. Rev. D76, 044003 (2007), arXiv:0704.3550 [gr-qc]

    work page arXiv 2007

  28. [28]

    Damour, A

    T. Damour, A. Nagar, and L. Villain, Merger states and final states of black hole coalescences: a numerical- relativity-assisted effective-one-body approach, Phys. Rev. D89, 024031 (2014), arXiv:1307.2868 [gr-qc]. 19

    work page arXiv 2014

  29. [29]

    Isolated and dynamical horizons and their applications

    A. Ashtekar and B. Krishnan, Isolated and dynamical horizons and their applications, Living Rev. Rel.7, 10 (2004), arXiv:gr-qc/0407042

    work page Pith review arXiv 2004

  30. [30]

    Krishnan, Fundamental properties and applications of quasi-local black hole horizons, Class

    B. Krishnan, Fundamental properties and applications of quasi-local black hole horizons, Class. Quant. Grav.25, 114005 (2008), arXiv:0712.1575 [gr-qc]

    work page arXiv 2008

  31. [31]

    Ashtekar and B

    A. Ashtekar and B. Krishnan, Dynamical horizons and their properties, Phys. Rev.D68, 104030 (2003), arXiv:gr-qc/0308033

    work page arXiv 2003

  32. [32]

    Ashtekar and B

    A. Ashtekar and B. Krishnan, Quasi-Local Black Hole Horizons: Recent Advances (2025), arXiv:2502.11825 [gr- qc]

    work page arXiv 2025

  33. [33]

    Booth, Black hole boundaries, Can

    I. Booth, Black hole boundaries, Can. J. Phys.83, 1073 (2005), arXiv:gr-qc/0508107

    work page arXiv 2005

  34. [34]

    T. A. Apostolatos, C. Cutler, G. J. Sussman, and K. S. Thorne, Spin induced orbital precession and its modula- tion of the gravitational wave forms from merging bina- ries, Phys. Rev. D49, 6274 (1994)

    work page 1994

  35. [35]

    Dreyer, B

    O. Dreyer, B. Krishnan, D. Shoemaker, and E. Schnetter, Introduction to isolated horizons in numerical relativity, Phys. Rev. D67, 024018 (2003), arXiv:gr-qc/0206008

    work page arXiv 2003

  36. [36]

    D. A. B. Iozzoet al., Comparing Remnant Proper- ties from Horizon Data and Asymptotic Data in Nu- merical Relativity, Phys. Rev. D103, 124029 (2021), arXiv:2104.07052 [gr-qc]

    work page arXiv 2021

  37. [37]

    Pook-Kolb, O

    D. Pook-Kolb, O. Birnholtz, B. Krishnan, and E. Schnet- ter, Existence and stability of marginally trapped sur- faces in black-hole spacetimes, Phys. Rev. D99, 064005 (2019), arXiv:1811.10405 [gr-qc]

    work page arXiv 2019

  38. [38]

    Pook-Kolb, O

    D. Pook-Kolb, O. Birnholtz, B. Krishnan, and E. Schnet- ter, Self-intersecting marginally outer trapped surfaces, Phys. Rev. D100, 084044 (2019)

    work page 2019

  39. [39]

    Ha and J

    D. Pook-Kolb, O. Birnholtz, B. Krishnan, and E. Schnet- ter, MOTS Finder version 1.1 (2019), 10.5281/zen- odo.2591105

    work page doi:10.5281/zen- 2019

  40. [40]

    Pook-Kolb, O

    D. Pook-Kolb, O. Birnholtz, B. Krishnan, and E. Schnet- ter, Interior of a binary black hole merger, Phys. Rev. Lett.123, 171102 (2019)

    work page 2019

  41. [41]

    Pook-Kolb, O

    D. Pook-Kolb, O. Birnholtz, J. L. Jaramillo, B. Krishnan, and E. Schnetter, Horizons in a binary black hole merger I: Geometry and area increase (2020), arXiv:2006.03939 [gr-qc]

    work page arXiv 2020

  42. [42]

    Pook-Kolb, O

    D. Pook-Kolb, O. Birnholtz, J. L. Jaramillo, B. Kr- ishnan, and E. Schnetter, Horizons in a binary black hole merger II: Fluxes, multipole moments and stability (2020), arXiv:2006.03940 [gr-qc]

    work page arXiv 2020

  43. [43]

    Booth, R

    I. Booth, R. A. Hennigar, and D. Pook-Kolb, Ulti- mate fate of apparent horizons during a binary black hole merger. I. Locating and understanding axisymmet- ric marginally outer trapped surfaces, Phys. Rev. D104, 084083 (2021), arXiv:2104.11343 [gr-qc]

    work page arXiv 2021

  44. [44]

    Pook-Kolb, I

    D. Pook-Kolb, I. Booth, and R. A. Hennigar, Ultimate fate of apparent horizons during a binary black hole merger. II. The vanishing of apparent horizons, Phys. Rev. D104, 084084 (2021), arXiv:2104.11344 [gr-qc]

    work page arXiv 2021

  45. [45]

    Pook-Kolb, R

    D. Pook-Kolb, R. A. Hennigar, and I. Booth, What Happens to Apparent Horizons in a Binary Black Hole Merger?, Phys. Rev. Lett.127, 181101 (2021), arXiv:2104.10265 [gr-qc]

    work page arXiv 2021

  46. [46]

    Gupta, B

    A. Gupta, B. Krishnan, A. Nielsen, and E. Schnetter, Dy- namics of marginally trapped surfaces in a binary black hole merger: Growth and approach to equilibrium, Phys. Rev.D97, 084028 (2018), arXiv:1801.07048 [gr-qc]

    work page arXiv 2018

  47. [47]

    Mourier, X

    P. Mourier, X. Jim´ enez Forteza, D. Pook-Kolb, B. Kr- ishnan, and E. Schnetter, Quasinormal modes and their overtones at the common horizon in a binary black hole merger, Phys. Rev. D103, 044054 (2021), arXiv:2010.15186 [gr-qc]

    work page arXiv 2021

  48. [48]

    Nonlinear Ringdown at the Black Hole Horizon,

    N. Khera, A. Ribes Metidieri, B. Bonga, X. Jim´ enez Forteza, B. Krishnan, E. Poisson, D. Pook- Kolb, E. Schnetter, and H. Yang, Nonlinear Ringdown at the Black Hole Horizon, Phys. Rev. Lett.131, 231401 (2023), arXiv:2306.11142 [gr-qc]

    work page arXiv 2023

  49. [49]

    Rezzolla, R

    L. Rezzolla, R. P. Macedo, and J. L. Jaramillo, Un- derstanding the ’anti-kick’ in the merger of binary black holes, Phys. Rev. Lett.104, 221101 (2010), arXiv:1003.0873 [gr-qc]

    work page arXiv 2010

  50. [50]

    J. L. Jaramillo, R. P. Macedo, P. M¨ osta, and L. Rezzolla, Towards a cross-correlation approach to strong-field dy- namics in Black Hole spacetimes,Proceedings, Span- ish Relativity Meeting : Towards new paradigms. (ERE 2011): Madrid, Spain, August 29-September 2, 2011, AIP Conf. Proc.1458, 158 (2011), arXiv:1205.3902 [gr-qc]

    work page arXiv 2011

  51. [51]

    J. L. Jaramillo, R. P. Macedo, P. M¨ osta, and L. Rezzolla, Black-hole horizons as probes of black-hole dynamics II: geometrical insights, Phys. Rev.D85, 084031 (2012), arXiv:1108.0061 [gr-qc]

    work page arXiv 2012

  52. [52]

    Y. Chen, P. Kumar, N. Khera, N. Deppe, A. Dhani, M. Boyle, M. Giesler, L. E. Kidder, H. P. Pfeiffer, M. A. Scheel, and S. A. Teukolsky, Multipole moments on the common horizon in a binary-black-hole simulation, Phys- ical Review D106, 10.1103/physrevd.106.124045 (2022)

    work page doi:10.1103/physrevd.106.124045 2022

  53. [53]

    Gourgoulhon, A Generalized Damour-Navier-Stokes equation applied to trapping horizons, Phys

    E. Gourgoulhon, A Generalized Damour-Navier-Stokes equation applied to trapping horizons, Phys. Rev. D72, 104007 (2005), arXiv:gr-qc/0508003

    work page arXiv 2005

  54. [54]

    Ashtekar, N

    A. Ashtekar, N. Khera, M. Kolanowski, and J. Lewandowski, Charges and fluxes on (perturbed) non- expanding horizons, JHEP02, 066, arXiv:2112.05608 [gr-qc]

    work page arXiv

  55. [55]

    Ashtekar, M

    A. Ashtekar, M. Campiglia, and S. Shah, Dynamical Black Holes: Approach to the Final State, Phys. Rev. D88, 064045 (2013), arXiv:1306.5697 [gr-qc]

    work page arXiv 2013

  56. [56]

    Andersson, M

    L. Andersson, M. Mars, and W. Simon, Local existence of dynamical and trapping horizons, Phys.Rev.Lett.95, 111102 (2005), arXiv:gr-qc/0506013 [gr-qc]

    work page arXiv 2005

  57. [57]

    Andersson, M

    L. Andersson, M. Mars, and W. Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes, Adv.Theor.Math.Phys. 12(2008), arXiv:0704.2889 [gr-qc]

    work page arXiv 2008

  58. [58]

    Booth and S

    I. Booth and S. Fairhurst, Isolated, slowly evolving, and dynamical trapping horizons: geometry and mechan- ics from surface deformations, Phys. Rev.D75, 084019 (2007), arXiv:gr-qc/0610032

    work page arXiv 2007

  59. [59]

    Thornburg, Class

    J. Thornburg, A Fast apparent horizon finder for three- dimensional Cartesian grids in numerical relativity, Class. Quant. Grav.21, 743 (2004), arXiv:gr-qc/0306056

    work page arXiv 2004

  60. [60]

    Event and apparent horizon finders for 3+1 numerical relativity,

    J. Thornburg, Event and Apparent Horizon Finders for 3+1 Numerical Relativity, Living Rev. Rel.10, 3 (2007), arXiv:gr-qc/0512169

    work page arXiv 2007

  61. [61]

    M¨ osta, L

    P. M¨ osta, L. Andersson, J. Metzger, B. Szil´ agyi, and J. Winicour, The Merger of Small and Large Black Holes, Class. Quant. Grav.32, 235003 (2015), arXiv:1501.05358 [gr-qc]

    work page arXiv 2015

  62. [62]

    Andersson, M

    L. Andersson, M. Mars, J. Metzger, and W. Simon, The Time evolution of marginally trapped surfaces, Class. Quant. Grav.26, 085018 (2009), arXiv:0811.4721 [gr-qc]. 20

    work page arXiv 2009

  63. [63]

    Ashtekar, J

    A. Ashtekar, J. Engle, T. Paw lowski, and C. Van Den Broeck, Multipole moments of isolated hori- zons, Class. Quant. Grav.21, 2549 (2004), arXiv:gr- qc/0401114

    work page arXiv 2004

  64. [64]

    D. R. Brill and R. W. Lindquist, Interaction energy in geometrostatics, Phys. Rev.131, 471 (1963)

    work page 1963

  65. [65]

    Pook-Kolb, O

    D. Pook-Kolb, O. Birnholtz, B. Krishnan, E. Schnetter, and V. Zhang, MOTS Finder version 1.3 (2019)

    work page 2019

  66. [66]

    L¨ offler, J

    F. L¨ offler, J. Faber, E. Bentivegna, T. Bode, P. Diener, R. Haas, I. Hinder, B. C. Mundim, C. D. Ott, E. Schnet- ter, G. Allen, M. Campanelli, and P. Laguna, The Ein- stein Toolkit: A Community Computational Infrastruc- ture for Relativistic Astrophysics, Class. Quantum Grav. 29, 115001 (2012), arXiv:1111.3344 [gr-qc]

    work page arXiv 2012

  67. [67]

    EinsteinToolkit, Einstein Toolkit: Open software for rel- ativistic astrophysics,http://einsteintoolkit.org/

    work page

  68. [68]

    Ansorg, B

    M. Ansorg, B. Br¨ ugmann, and W. Tichy, A single-domain spectral method for black hole puncture data, Phys. Rev. D70, 064011 (2004), arXiv:gr-qc/0404056

    work page arXiv 2004

  69. [69]

    J. D. Brown, P. Diener, O. Sarbach, E. Schnetter, and M. Tiglio, Turduckening black holes: An Analytical and computational study, Phys. Rev. D79, 044023 (2009), arXiv:0809.3533 [gr-qc]

    work page arXiv 2009

  70. [70]

    S. Husa, I. Hinder, and C. Lechner, Kranc: a Mathemat- ica application to generate numerical codes for tensorial evolution equations, Comput. Phys. Commun.174, 983 (2006), arXiv:gr-qc/0404023

    work page arXiv 2006

  71. [71]

    Kranc, Kranc: Kranc assembles numerical code

    work page

  72. [72]

    S. W. Hawking, Gravitational radiation from colliding black holes, Phys. Rev. Lett.26, 1344 (1971)

    work page 1971

  73. [73]

    Bhagavan, B

    S. Bhagavan, B. de Koning, S. Maddhashiya, and C. Rackauckas, Datainterpolations.jl: Fast interpolations of 1d data, Journal of Open Source Software9, 6917 (2024)

    work page 2024

  74. [74]

    Borhanian, K

    S. Borhanian, K. G. Arun, H. P. Pfeiffer, and B. S. Sathyaprakash, Comparison of post-Newtonian mode amplitudes with numerical relativity simulations of bi- nary black holes, Class. Quant. Grav.37, 065006 (2020), arXiv:1901.08516 [gr-qc]

    work page arXiv 2020

  75. [75]

    X. J. Forteza and P. Mourier, High-overtone fits to numerical relativity ringdowns: Beyond the dismissed n=8 special tone, Phys. Rev. D104, 124072 (2021), arXiv:2107.11829 [gr-qc]

    work page arXiv 2021

  76. [76]

    Black hole spectroscopy: from theory to experiment

    E. Bertiet al., Black hole spectroscopy: from theory to experiment (2025), arXiv:2505.23895 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2025