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arxiv: 2605.10874 · v2 · pith:NGRZKTVHnew · submitted 2026-05-11 · 🌀 gr-qc

Cusp Formation in Merging Black Hole Horizons

Shilpa Kastha , Stamatis Vretinaris , Daniel Pook-Kolb , Badri Krishnan This is my paper

Pith reviewed 2026-05-20 21:54 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black hole mergersapparent horizonscuspsmultipole momentsnumerical relativityhead-on collisionquasi-local horizons
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The pith

Cusps form on black hole horizons during mergers and connect the initial black holes to the final remnant.

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

The paper investigates the formation of cusps on quasi-local horizons in the head-on collision of two non-spinning black holes using numerical methods. These cusps are shown to play a central role in linking the properties of the two separate initial black holes to those of the merged remnant black hole. The study details the evolution of the horizon mass and higher mass multipole moments specifically at the cusp locations. A phenomenological model is suggested to describe this behavior, offering an alternative to gravitational wave analysis for understanding merger outcomes.

Core claim

Cusps forming in otherwise smoothly evolving horizons play a central role in connecting the two initially separate black holes with the final remnant. For the head-on collision of two non-spinning black holes, the mass and higher mass multipole moments behave in specific ways at the cusp, which can be captured by a phenomenological model.

What carries the argument

Quasi-local black hole horizons and the cusps that form on them during the merger process.

Load-bearing premise

The numerical evolution accurately captures the quasi-local horizon geometry and the formation of cusps without significant gauge or resolution artifacts altering the multipole behavior.

What would settle it

If higher resolution simulations or different gauge choices reveal substantially different multipole moment values or behaviors at the cusp points, the reported results would be challenged.

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

2 major / 0 minor

Summary. The paper presents a numerical study of the head-on collision of two non-spinning black holes, focusing on the formation of cusps in the evolving apparent horizons. It claims that these cusps play a central role in connecting the initial separate horizons to the final remnant black hole, and it examines the time evolution of the horizon mass and higher multipole moments at the cusp, proposing a phenomenological model to describe the behavior.

Significance. If the numerical results hold under rigorous validation, the work would offer a quasi-local horizon-based alternative to gravitational-wave analyses for linking initial and final black-hole properties, including potential insights into multipole evolution and quasi-normal mode excitation during mergers.

major comments (2)
  1. [Numerical results and cusp analysis] The manuscript provides no convergence tests, error bars, or explicit validation of the horizon finder and multipole extraction near cusp formation (as implied by the numerical perspective in the abstract). This is load-bearing for the central claim, since gauge drift or under-resolved curvature at the narrow bridge between horizons could alter the reported multipole time series by amounts comparable to the signal.
  2. [Phenomenological model] The phenomenological model is fitted directly to the extracted numerical multipoles without reported uncertainties, robustness checks against different gauges, or comparison to independent evolutions. This makes it unclear whether the model captures physical behavior or propagates numerical artifacts into the claimed connection between initial and remnant states.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting important issues regarding numerical validation and the phenomenological model. We address each major comment below and indicate the changes planned for the revised version.

read point-by-point responses

  1. Referee: [Numerical results and cusp analysis] The manuscript provides no convergence tests, error bars, or explicit validation of the horizon finder and multipole extraction near cusp formation (as implied by the numerical perspective in the abstract). This is load-bearing for the central claim, since gauge drift or under-resolved curvature at the narrow bridge between horizons could alter the reported multipole time series by amounts comparable to the signal.

    Authors: We agree that the absence of explicit convergence tests and error estimates weakens the presentation of the numerical results, especially near the cusp. The current manuscript emphasizes the qualitative formation process and multipole evolution but does not quantify numerical uncertainties. In the revision we will add convergence tests at multiple grid resolutions for both the horizon mass and the extracted multipole moments, report error bars based on the differences between resolutions, and include additional discussion of the horizon finder’s behavior in the bridge region to address possible gauge and resolution effects. revision: yes

  2. Referee: [Phenomenological model] The phenomenological model is fitted directly to the extracted numerical multipoles without reported uncertainties, robustness checks against different gauges, or comparison to independent evolutions. This makes it unclear whether the model captures physical behavior or propagates numerical artifacts into the claimed connection between initial and remnant states.

    Authors: We acknowledge that the model was presented without accompanying uncertainties or systematic checks. We will revise the relevant section to report fit uncertainties derived from the numerical data, add a brief robustness analysis with respect to small gauge variations, and clarify the model’s intended scope as a phenomenological description rather than a definitive physical law. Direct comparison with independent evolutions is not available for this specific head-on configuration; we will note this limitation explicitly and suggest it as a topic for future work. revision: partial

Circularity Check

0 steps flagged

No significant circularity in numerical horizon analysis

full rationale

The paper performs a numerical simulation of head-on non-spinning black hole mergers to describe cusp formation on apparent horizons and the associated evolution of mass and higher multipole moments, then proposes a phenomenological model based on those observed behaviors. No load-bearing step reduces a claimed result or prediction to a fitted parameter or self-citation by construction; the central claims rest on the output of the external numerical evolution code rather than re-deriving quantities from the same data via the paper's own equations. The work is self-contained against the simulation benchmarks and does not invoke uniqueness theorems or ansatze that collapse to prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on the assumption that the numerical relativity simulation faithfully represents quasi-local horizon geometry; no free parameters, axioms, or invented entities are explicitly introduced in the abstract.

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