Slicing conditions for axisymmetric gravitational collapse of Brill waves

Anton Khirnov; Tomas Ledvinka

REVIEW 3 major objections 5 minor 1 cited by

Slicing conditions for axisymmetric gravitational collapse of Brill waves

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Adding a maximal-slicing source term to the 1+log lapse cures the slicing blowups that stop near-critical Brill wave collapse evolutions.

desk verdict Quasi-maximal slicing is a plausible new cure for Brill-wave gauge failures; the A=5 evidence is thinner than the abstract suggests, but the claim likely holds and deserves review. read the letter →

arxiv 1908.06034 v1 pith:BB27AD5O submitted 2019-08-16 gr-qc

Anton Khirnov , Tomas Ledvinka This is my paper · ORCID

classification gr-qc

keywords numericalrelativityBrillwavesquasi-maximalslicing1+logmovingpuncturegaugegravitationalcollapseblackholeformationBSSNformulation

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 tries to establish that a small modification to the standard '1+log' lapse condition removes the coordinate singularity that has prevented numerical evolutions of near-critical Brill wave data from running to completion. The modification adds a source term derived from the maximal slicing condition, producing what the authors call quasi-maximal slicing. If the claim holds, existing moving-puncture codes can evolve strong gravitational wave packets long enough to see apparent horizons, event horizons, and the settling of the newly formed black hole to Schwarzschild, without excision or a different evolution formulation.

What carries the argument

The load-bearing object is the quasi-maximal slicing condition (22), $$(\partial_t-\mathcal{L}_\$\beta$)\$\alpha$=-2\$\alpha$ K+\kappa W_a,$$ which appends to the 1+log slicing law a source term $W_a$ constructed as an approximate solution of the elliptic equation (21), derived by differentiating the maximal slicing condition in time. In practice $W_a$ is computed by a pseudo-spectral elliptic solver run once per coarse time step, linearly extrapolated and interpolated between solves, and multiplied by a low-pass filter (25) that damps basis functions once they are no longer resolved by the finite-difference grid. Because the auxiliary equation has trivial principal part $\Delta W=0$, the added term does not change the principal part of the evolution system linearized around flat space, so hyperbolicity is inherited from 1+log slicing while the source term pushes $K$ toward zero.

What would settle it

Run the $A=5$ Brill wave with the low-pass filter removed and the pseudo-spectral solver for $W_a$ made substantially more accurate; if the resolution-dependent kink in the normal derivative of $\bar\rho^2$ reappears before $t\approx10$, the cure is an artifact of the approximate solver rather than a property of quasi-maximal slicing.

Watch

Extended reading notes

Core claim

Brill wave data with amplitude $A=5$ evolve under the moving puncture gauge for only about $t\approx4.9$ before the constant-time slices develop a cusp-like nonsmooth feature, visible as a resolution-dependent kink in the normal derivative of the circumferential radius $\bar\rho^2$; the simulation then fails. The central discovery is that replacing the 1+log slicing law $(\partial_t-\mathcal{L}_\beta)\alpha=-2\alpha K$ with the quasi-maximal condition $(\partial_t-\mathcal{L}_\beta)\alpha=-2\alpha K+\kappa W_a$, where $W_a$ is a rough, low-pass-filtered, time-interpolated solution of the elliptic equation (21), removes this pathology. With zero shift the $A=5$ run passes $t=10$, an apparent horizon of mass $M_{\mathrm{AH}}\approx0.56$ appears, and the spacetime can be described by gauge-invariant quantities; for $A=5.5$ the $\Gamma$-driver shift works and mass estimates converge to the apparent horizon mass as the hole settles to Schwarzschild.

Load-bearing premise

The cure relies on the assumption that a rough, low-pass-filtered, time-interpolated solution $W_a$ of the auxiliary elliptic equation stays close enough to true maximal slicing to suppress the singularity at the near-critical amplitude where 1+log fails, a behaviour checked numerically at low amplitude but not proven generally.

Editorial extensions

If this is right

Near-critical Brill wave data can be evolved with BSSN finite-difference moving-puncture codes far enough to form and track an apparent horizon, which 1+log slicing alone does not allow.
Quasi-maximal slicing with fixed solver parameters converges at fourth order in the time integrator to the same invariant spacetime, so the approximate source term does not alter the physics being simulated.
For supercritical amplitude $A=5.5$, late-time gauge-invariant mass estimates approach the final apparent horizon mass, showing the collapsed object settles to a Schwarzschild black hole.
Even away from criticality the gauge allows compactified conformal diagrams of the collapse, and reveals an early event horizon that is disk-like with a non-smooth rim, and transiently toroidal for $A=4.8$.

Reading between the lines

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

The same 'quasi-' construction should extend to other elliptic gauge conditions, for example deriving a source term from harmonic or minimal-distortion conditions, to stabilize hyperbolic gauges in other strong-field regimes; the paper does not test this.
Because a low-pass filter confining $W_a$ to the central region suffices to cure the failure, the 1+log singularity appears to originate in lapse dynamics inside the strong-field core rather than in outer-boundary gauge behavior.
The shift shock seen at $A=5$ suggests the remaining obstacle to full moving-puncture capability for near-critical data sits in the shift condition, so an analogous quasi-maximal driver for the shift is the natural next step.
The dimensionless invariant $\chi$, whose first negative region at Brill waves points 'the wrong way' relative to Schwarzschild, could be monitored over successive echo periods in a future critical-collapse search, since a discretely self-similar spacetime should repeat its range of values; the paper notes the possibility but does not pursue it.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a real advance, probably right, and worth sending to referees. The genuinely new piece is the quasi-maximal slicing condition (22), with a source term W_a obtained from the time derivative of the maximal slicing equation. That is not in the prior literature. The paper shows that this condition keeps K smaller than 1+log slicing, and at A=5 it avoids the cusp-like slicing singularities that kill the moving puncture gauge. It also produces the first long-lived axisymmetric Brill-wave evolutions of this kind, including conformal diagrams and a claimed toroidal event horizon. That combination matters.

Strengths: the evidence is honest and layered. At A=0.1 the maximum norm of K goes to zero with time step, which directly checks the intended target ∂_tt K = 0. At A=1, gauge-invariant quantities converge at fourth order under grid refinement. The reproduction of the known 1+log failure for A=5 is consistent with [11], and the apparent horizon masses agree with the independent pseudospectral results in [13]. The code is available. The authors also disclose the shift-shock problem and the limitations of their spectral solver rather than hiding them. Citation practice is fair: Hilditch et al. are credited and compared throughout.

Soft spots, in proportion: the headline A=5 claim rests essentially on one filtered, time-extrapolated run, with no resolution study or filter-sensitivity check at that amplitude. The stress-test note has a real point here. The filter, the approximate elliptic solve, and the empirically chosen switch time could all be doing stabilizing work that is not exactly the intended quasi-maximal driver. That said, the same approximate driver demonstrably pushes K down at A=1, so the mechanism is at least connected. And the A=5 run reaches t≈25, forms an apparent horizon, and gives a mass compatible with [13]; I do not think the result is a numerical mirage, just less well supported than it could be.

Other soft spots are minor. The toroidal event horizon is reported almost in passing for A=4.8, apparently without a dedicated convergence check; I would treat it as suggestive rather than established. Well-posedness of the modified gauge is argued only from the linearized principal part, which is acceptable for a numerical paper but not a proof. The zero-shift workaround is a clear limitation, and the authors say so.

Bottom line: anyone working on vacuum critical collapse or on gauge conditions in numerical relativity will get value from this. It deserves a serious referee, and I would engage with it myself.

Referee Report

3 major / 5 minor

Summary. The paper proposes a new 'quasi-maximal' slicing condition (Eq. 22), obtained by adding a source term W_a--an approximate solution of the elliptic equation (21)--to the 1+log slicing condition, in order to cure coordinate singularities that appear in moving-puncture evolutions of near-critical Brill wave data. The authors implement this in a BSSN code with the analytic Cartoon method, computing W_a with a pseudospectral elliptic solver, time extrapolation/interpolation, and a low-pass filter (25), and switching back to 1+log at an empirically chosen time t0 (26). They reproduce the known failure of 1+log slicing at A=5, show that max|K| tends to zero with the time step for A=0.1, that K is reduced at A=1, and that gauge-invariant quantities converge to fourth order under grid refinement at A=1. Their main result is a single A=5 run with zero shift that reaches t≈25, forms an apparent horizon, and allows construction of an event horizon and conformal diagrams; they also report long runs at A=4.6, 4.65, 4.8, 5.0, and 5.5. The paper acknowledges that with the Γ-driver the A=5 shift develops a shock and that the A=5 success is therefore limited to zero shift.

Significance. If the central claim is correct, the quasi-maximal slicing condition is a practically valuable and computationally cheap modification of 1+log slicing that enables long-term simulations of Brill wave collapse in a finite-difference moving-puncture code, and the gauge-invariant diagnostics (Kretschmann mass, circumferential radius, chi, conformal diagrams) are useful tools for studying the approach to critical collapse. The paper's strengths include a clearly motivated construction (W_a's origin in maximal slicing is stated explicitly, so the derivation is not circular), several nontrivial checks--Δt-convergence of K at A=0.1, reduced K at A=1, fourth-order convergence of invariants at A=1, and reproduction of the known A=5 failure--and the release of the simulation code. However, the headline A=5 claim rests on a single zero-shift run with no resolution or filter-sensitivity study at that amplitude, and the well-posedness of the modified gauge is justified only by linearization around flat space. The significance is therefore conditional on additional evidence at the amplitude where the claim is made.

major comments (3)

[Section 5, 'Near-critical case: A=5'] The central claim that quasi-maximal slicing cures the A=5 singularities is supported by a single run. The convergence evidence in Sec. 4.2 and Figs. 3-4 is for A=0.1 and A=1 only; no resolution study is reported for A=5. Because W_a is, in the authors' words, a 'very approximate' solution of (21), and because the low-pass filter (25) with J=36, d=6 and the switch time t0 in (26) are empirically chosen, the run's survival could be an artifact of dissipation, of the time extrapolation, or of the switching schedule rather than of the intended maximal-slicing driver. A resolution study at A=5 (grid spacing, CFL factor, filter parameters J and d, and t0) is required to tie the observed stability to the quasi-maximal source term.
[Section 3, after Eq. (22)] The well-posedness argument is insufficient for the strong-field regime. The statement that the source term 'does not modify the principal part of the PDEs linearized around flat space' only addresses perturbations of Minkowski spacetime; near-critical A=5 Brill waves develop large K and strong lapse dynamics, and the actual W_a is time-interpolated and low-pass filtered. No argument is given that the nonlinear system with this approximate, time-dependent driver remains well-posed. At minimum, the A=5 run should be accompanied by a report of Hamiltonian and momentum constraint violations under resolution, or another diagnostic demonstrating that the modified gauge does not introduce an instability.
[Section 5, 'Near-critical case: A=5'] The demonstration is limited to zero shift. The authors state that with the Γ-driver the shift develops a shock profile around t≈10, so they set the shift to zero; the moving-puncture failure reported in [11] concerns the full gauge, including the Γ-driver. Thus the paper does not establish that quasi-maximal slicing cures the moving-puncture failure for A=5; it establishes that the modified slicing improves the lapse sector in a zero-shift run. This limitation should be stated explicitly, and if the claim is intended for the full moving-puncture gauge, further work on the shift sector is needed.

minor comments (5)

[Section 4.1, Quasi-maximal slicing] In the last paragraph, 'rougly' should be 'roughly'.
[Section 2.3] The symbol K is used both for the trace of the extrinsic curvature (7b) and for the Kretschmann scalar (12); the same symbol in adjacent sections is confusing, and the Kretschmann invariant should be denoted differently (e.g., \mathcal{K}).
[Figure 2] The y-axis label 'Ln \bar\rho^2/\sigma' in the left panel is likely meant to denote the normal derivative L_n \bar\rho^2, but it reads like a natural logarithm; the subscript should be typeset clearly.
[Section 4.1, Quasi-maximal slicing] The description of the elliptic solver reuse (S=1024) does not state whether the LU decomposition is refreshed during the S solves; since the metric changes with time, the preconditioner quality will degrade, and a sentence on this would help reproducibility.
[Eq. (24a)] The basis uses \arccot without specifying the branch or range; for a spectral basis this convention matters and should be stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the quasi-maximal slicing construction is a designed gauge modification validated by independent numerical benchmarks, not a prediction derived from its inputs.

full rationale

I walked the paper's derivation chain and found no step in which a claimed prediction or first-principles result reduces to its own inputs by construction. The quasi-maximal slicing condition (22) is explicitly constructed by adding a source term Wa derived from the maximal slicing condition (9) to the 1+log slicing (10). The paper openly states the design motivation: if Wa were an exact solution of (21), the system would reproduce maximal slicing. This is a design statement, not a hidden prediction; no scientific claim is disguised as an output of the construction. The central claim, that quasi-maximal slicing cures the singularities seen with 1+log slicing for A = 5 Brill waves, is an empirical numerical result supported by running the modified system and comparing it with independently reported behavior: the paper reproduces the 1+log slicing failure from [11] and compares horizon masses with [13], both by different author groups. The approximate elements of the numerical implementation, such as time extrapolation of spectral coefficients, the low-pass filter (25) with empirically chosen J = 36 and d = 6, and the empirically timed switch back to 1+log via (26), are presented as numerical techniques rather than as consequences of the gauge condition. The validation section checks convergence of the elliptic solver toward ∂ttK = 0 at A = 0.1 and convergence of spacetime invariants at A = 1; while these checks do not fully cover the near-critical A = 5 regime, that is a limitation or correctness risk, not circularity. The paper also explicitly acknowledges that it cannot yet evolve the A = 5 data indefinitely and that shift pathologies remain, which further indicates the claim is based on observed numerical behavior rather than on a self-referential argument. There is no load-bearing self-citation: the cited failure case and comparison masses come from other authors, and the paper does not invoke a uniqueness theorem or prior result of its own authors to force its choice of gauge. I therefore conclude that the derivation is self-contained and the finding is no significant circularity.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard BSSN evolution and standard gauge conditions, plus the new quasi-maximal slicing. The load-bearing assumptions are: (i) the modified gauge (22) remains well-posed, argued only via a linearized flat-space principal part analysis; (ii) the approximate, filtered, time-interpolated Wa computed by the pseudo-spectral solver is close enough to the exact solution of (21) to drive the slicing toward maximal slicing; (iii) the low-pass filter and the switch to 1+log at late times do not alter the physical spacetime. The free parameters are empirically chosen but disclosed, and the authors report qualitative insensitivity. No new physical entities are introduced.

free parameters (5)

Switch time t0 in (26) = Determined per simulation, roughly when apparent horizon forms or when largest peak in K forms/dissolves
Controls when quasi-maximal term is switched off; chosen empirically; authors state exact value does not affect results qualitatively.
Filter power d in (25) = 6
Empirically chosen so transition area is not smaller than roughest step size; results reportedly insensitive.
Filter amplitude J in (25) = 36
Chosen so exponential evaluates to double-precision epsilon at r=r0; part of low-pass filter for Wa.
Compactification scale L for initial data = 3 (in units of sigma)
Empirically determined basis scaling; affects initial data solver resolution at infinity.
Elliptic solver reuse period S = 1024
Number of solves between LU decompositions of the pseudo-spectral matrix; affects accuracy of Wa and performance.

assumptions (5)

standard math BSSN evolution equations (8) with constraints (6) are a valid strongly hyperbolic formulation of Einstein's equations.
The entire simulation rests on this standard formulation, cited as [15,16].
standard math Maximal slicing, with K=0 enforced, is well-posed (following [18]), and the derivation of (21) from the time derivative of (9) is valid.
Used to derive the source term Wa; the paper cites [18] for well-posedness.
domain assumption The modified slicing (22) remains well-posed because the added source term does not change the principal part linearized around flat space.
Stated in Section 3; no general proof is given, only this linear argument plus numerical evidence.
domain assumption The low-pass filtering (25) and the time interpolation/extrapolation of Wa do not change the physical spacetime, and any difference is a gauge effect that converges away.
Convergence of invariants is shown for A=1; for near-critical A=5 this is assumed without a dedicated convergence study.
domain assumption The apparent horizon (MOTS) tracking and event horizon tracing backwards provide an accurate approximation of the true event horizon.
Standard techniques; error of order 1e-3 claimed for late-time MOTS areas.

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Cite this review

Pith. "Pith review of Slicing conditions for axisymmetric gravitational collapse of Brill waves." pith.science (2026). https://pith.science/paper/BB27AD5O

@misc{pith2026190806034,
  author       = {Pith},
  title        = {Pith review of: Slicing conditions for axisymmetric gravitational collapse of Brill waves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BB27AD5O}},
  note         = {Machine review of arXiv:1908.06034}
}

read the original abstract

In numerical relativity, spacetimes involving compact strongly gravitating objects are constructed as numerical solutions of Einstein's equations. Success of such a process strongly depends on the availability of appropriate coordinates, which are typically constructed dynamically. A very robust coordinate choice is a so-called moving puncture gauge, commonly used for numerical simulations of black hole spacetimes. Nevertheless it is known to fail for evolving near-critical Brill wave data. We construct a new `quasi-maximal' slicing condition and demonstrate that it exhibits better behavior for such data. This condition is based on the 1+log slicing with an additional source term derived from maximal slicing. It is relatively simple to implement in existing moving puncture codes and computationally inexpensive. We also illustrate the properties of constructed spacetimes based on gauge-independent quantities in compactified spacetime diagrams. These invariants are also used to show how created black holes settle down to a Schwarzschild black hole.

Figures

Figures reproduced from arXiv: 1908.06034 by the authors.

**Figure 1.** A diagram of the interpolation-extrapolation scheme for evaluating Wa. Crosses are the Cns — the solutions obtained from the pseudo-spectral solver. Circles are the C0 ns — the predicted coefficients obtained from Cns by extrapolation (dashed lines). Solid lines denote the evaluation of Wa(t) by linear interpolation. then become invisible to the pseudospectral solver, strongly reducing the accuracy of the solution. … view at source ↗

**Figure 2.** Failure of the 1+log slicing for A = 5 Brill waves. The shift is set to zero. Left: plotted is the derivative of ρ¯ 2 along the normal to the spatial surfaces on the x-axis at times {4.625, 4.6875, 4.75, 4.8125, 4.875} (bottom to top). The inset compares the data on the t = 4.875 slice for runs with 2048 (black) and 512 (gray) grid points. Circles/crosses indicate the positions of the grid points. Right: the same qu… view at source ↗

**Figure 3.** Convergence of the quasi-maximal slicing with the time step ∆t. Left: A = 0.1 Brill waves, plotted is the maximum norm of K at simulation time t = 1 512 (one full time step for the coarsest run) as a function of the time step ∆t. Right: A = 1 Brill waves, time evolution of the maximum norm of K over the finest refinement level for (top to bottom): 1+log slicing and quasi-maximal slicing with ∆x/∆t = 1 4 , 1 8 , 1 1… view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Convergence of spacetime invariants with quasi-maximal slicing for A = 1 Brill waves. Invariant quantities at ρ¯ = 1 — left: ρ¯ 02 , right: K — are computed as functions of the proper time τ at a given location for simulations with {256, 384, 512} grid points per refin…

**Figure 5.** Figure 5: Formation of the event horizon for A = 5 Brill waves, zero shift. It is constructed by evolving the apparent horizon section from the t = 25 slice as a null surface back in time. Left: solid lines show cross-sections of the event horizon in the x−z plane at simulation …

**Figure 6.** Figure 6: Comparison of the A = 5 runs with 1+log / quasi-maximal slicings — respectively the left and right half of each panel — in the double-null coordinates {U, V } in the equatorial plane. Color denotes the value of K (left panel) and α (right panel), dotted regions are not…

**Figure 7.** Figure 7: Evolution of mass estimates for A = 5.5 Brill waves. Left: evolution of MK (thin line) and Mρ¯ (thick line) at constant spatial coordinate {x = 20, z = 0} with the simulation time t. Right: Mρ¯ along the x-axis at times t = {0, 50, 100} (thin to thick). In both picture…

**Figure 8.** Figure 8: Evolution of χ in the equatorial plane for Brill waves with A = 4.6 (left) and A = 5.5 (right). 0.0 0.5 1.0 1.5 2.0 2.5 x 0.0 0.5 1.0 1.5 2.0 2.5 z 1−χ ρ¯2σ−2 1 3 −6 −3 0 3 6 0.0 0.5 1.0 1.5 2.0 2.5 x 0.0 0.5 1.0 1.5 2.0 2.5 z sign(K) [PITH_FULL_IMAGE:figures/full…

**Figure 9.** Figure 9: Invariants in the x−z plane at coordinate time when they reach their highest values (t = 11); A = 4.65 data, zero shift [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]

**Figure 10.** Figure 10: Conformal diagrams for the equatorial plane of the spacetimes produced by evolving the Brill waves. The values of A are {4.0, 4.65, 4.8, 5.0} for, respectively, top left, top right, bottom left, and bottom right. The top two spacetimes are sub-critical, while the bott…

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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

Comparing twist-free axisymmetric gravitational waves near the black hole threshold
gr-qc 2026-07 conditional novelty 5.0 of 10

Independent codes (bamps, prague, sphGR) agree on near-threshold vacuum wave collapse, confirming family-dependent scaling and quasi-universal strong-field features.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [11]

Hilditch D, Baumgarte T W, Weyhausen A, Dietrich T, Br ¨ugmann B, Montero P J and M¨uller E 2013 Collapse of nonlinear gravitational waves in moving-puncture coordinates Phys. Rev. D 88 103009

work page 2013

[2] [13]

Hilditch D, Weyhausen A and Br ¨ugmann B 2017 Evolutions of centered brill waves with a pseudospectral method Phys. Rev. D 96 104051

work page 2017

[3] [1]

Eppley K 1977 Evolution of time-symmetric gravitational waves: Initial data and apparent horizons Phys. Rev. D 16 1609–1614

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