Nonaxisymmetric instability of rapidly rotating black hole in five dimensions

Short version: this is the paper that turned Emparan and Myers' thermodynamic hunch into a dynamical result, and the core claim holds up. Anyone working on higher-dimensional black hole formation should know it exists and should cite it for the instability itself, not for the precise number 0.87. What is new: the first full nonlinear evolution of a 5D Myers-Perry hole with a non-axisymmetric seed. Below q about 0.86 the bar deformation decays; above q about 0.88 it grows exponentially at a rate independent of seed amplitude; 0.87 sits in between. Two independently written codes agree, and the resolution study is concrete rather than hand-waved: horizon area drift of 0.2% at dx = 0.015 mu^(1/2) for q = 0.8, with the required spacings stated explicitly at 0.85 and 0.88. The A = 0.005, 0.01, 0.02 scaling check is the right control. They also say plainly that the final fate is unresolved, fragmentation versus saturation by radiation, and do not oversell it. Soft spots, in order of weight. First, the growth evidence at q = 0.89 comes from runs the authors themselves say are not accurate to t = 50; they stop when horizon area error reaches 3%. The cleanest instability signal therefore sits at the edge of the resolution envelope. q = 0.88 is better but still demanding. This does not sink the claim, since the trend across 0.86, 0.87, 0.88 is monotone and consistent, but the threshold is quoted to two digits with no error bar and no growth-rate-versus-q fit. Second, the cartoon reduction: d/dchi is a Killing vector of the background and of the seed, so only the chi-independent bar sector is probed. The text does say stability against a bar deformation, so this is signalled rather than hidden, but the abstract does not carry the qualifier. A faster-growing chi-dependent mode would push the threshold down; the existence claim is unaffected, the number is an upper bound. Third, convergence is asserted, not shown; there is no order plot. The thermodynamic aside around Eq. (7) is heuristic and reads that way, which is fine as motivation. Recommendation: send it out. The result is real and the controls are the right ones. Referees should push on the threshold uncertainty and the chi-sector caveat, not on whether the instability exists.

Sec. II states that 'a rotational Killing vector (∂/∂χ)^µ is present where χ = tan^{-1}(w/z)' and uses this to justify the cartoon reduction to a 3D (x,y,z) grid. The cartoon method is exact only if the full evolved spacetime — background plus perturbation — is invariant under ∂/∂χ. The seed in Eq. (4), W = W0[1 + A µ^{-1}(x²-y²) exp(-r²/2r_h²)], is manifestly χ-independent, so every run probes only the m_χ = 0 (bar) sector. The paper says it studies 'the nonaxisymmetric stability against a bar deformation,' which signals the restriction, but the abstract (abstract:s1) reports 'the critical value of spin parameter for onset of the instability is ≈ 0.87' without qualification. If any χ-dependent mode has a positive growth rate at q < 0.87, that number is an overestimate of the true threshold. The paper gives no argument that m_χ ≠ 0 modes are stable or slower-growing; the only supporting evidence is the qualitative analogy to rotating-star bar modes and the thermodynamic estimate of Emparan-Myers, neither of which addresses the χ sector. Crucially, this does not threaten the existence claim: growth in any sector proves instability. The concern is the precision and generality of the threshold, not the qualitative result.

The paper's qualitative claim is well supported. Two independently written codes (the nonuniform-grid code of [20] and SACRA5D) agree; the growth rate is reported to be independent of the seed amplitude A = 0.005, 0.01, 0.02; the resolution study is concrete (0.2% horizon-area drift at ∆x = 0.015 µ^{1/2} for q = 0.8, 1.2% at 0.0225, tighter spacings as q rises); and the a = 0 frequency f ≈ 0.15/µ^{1/2} matches the perturbation analysis of [25]. The threshold ≈ 0.87 is close to the Emparan-Myers thermodynamic estimate ≈ 0.85, which is independent support. The paper is also candid about what it cannot do: it states that the final fate is unresolved and that q ≥ 0.89 runs are limited to t < 50 µ^{1/2} with 3% horizon-area error. (The classifier record lists no stated limitations; that is a record artifact — the full text does state them.) The load-bearing soft spot is the one the reader identified: the cartoon reduction restricts the evolved sector to m_χ = 0. The paper gestures at this ('against a bar deformation') but does not quantify it, and the abstract's 'critical value' reads as a general threshold. This is a real qualification, not a refutation: the existence of the instability is proven by growth in any sector, and the threshold is already stated as approximate. A secondary, smaller concern is that the threshold is quoted to two digits from a scan with ~0.01 spacing in q and no fit of growth rate versus q, so the '≈ 0.87' carries more precision than the data strictly support. Neither concern moves the verdict: the reader's ACCEPT with the cartoon caveat already noted is the right call. I would keep the verdict at ACCEPT and ask that the abstract qualify 'critical value' as the threshold for the χ-independent bar sector.

If correct, this is a genuinely important result: it removes the implicit assumption, widespread in the higher-dimensional black hole formation literature, that a rapidly rotating Myers-Perry hole is a stable endpoint of a high-energy collision. The paper's evidential structure is strong for a letter: two independently written codes (the nonuniform-grid code of Ref. [20] and SACRA5D) reach the same conclusion; the resolution requirement is calibrated against a physical quantity (drift of the apparent-horizon area, held to 1% at t = 50 μ^{1/2}); and the growth rate is checked to be independent of the seed amplitude A = 0.005, 0.01, 0.02. The result is falsifiable in a concrete way — a threshold near q ≈ 0.87, a growth rate, and a characteristic frequency that matches the known a = 0 quasinormal mode at f ≈ 0.15/μ^{1/2}. The connection drawn to the Emparan-Myers thermodynamic argument and to the Cm/Ce ≈ 0.38 oblateness criterion is a useful, testable organizing observation.

The paper asks a question that formation scenarios for miniature black holes had simply assumed away: is a rapidly rotating black hole in five dimensions stable? In four dimensions the answer is believed to be yes, so a spun-up remnant is treated as the quiet endpoint of a merger. The authors instead evolve the full five-dimensional Einstein equations, planting a small bar-shaped distortion on a Myers-Perry black hole and watching whether it fades or feeds on itself. For spin parameter $q=|a|/m^{1/2}$ at or below about 0.86 the distortion decays; at or above about 0.88 it grows exponentially, at a rate independent of how large the seed was, and the outgoing gravitational waves grow with it. The crossover sits near $q = 0.87$, which is close to the value a horizon-area thermodynamic argument had predicted years earlier.

What carries the argument. Two ingredients carry the argument. The first is the deformation parameter $, built from the proper lengths of horizon meridians at $, $, $, and $, which turns a statement about horizon shape into a single number that can be tracked in time. The second is the cartoon method: because the Myers-Perry background and the chosen seed perturbation both preserve the rotational Killing vector $ with $, the 4+1 evolution can be run on a three-dimensional $(x,y,z)$ grid. The background is the five-dimensional Myers-Perry solution in quasi-isotropic coordinates, evolved with a higher-dimensional BSSN scheme, fourth-order finite differencing, and puncture gauge conditions, and cross-checked between a fixed nonuniform-grid code and an adaptive-mesh code. Numerical control is set by requiring the apparent-horizon area of an unperturbed hole to stay constant within one percent out to $.

Core claim. The central claim is that five-dimensional rotating black holes above a critical spin are dynamically unstable against nonaxisymmetric deformation, with the critical value $q_c = |a|/m^{1/2} = 0.87$ for a single-spin Myers-Perry hole. The evidence is a horizon deformation parameter $: the spread among meridian circumferences measured at several angles in the rotation plane. For $q > 0.88$ this parameter rises without bound after an initial relaxation phase, and the $+$ mode of the gravitational waveform $h_+$ rises with it; for $q < 0.86$ it decays. The growth rate is the same for seed amplitudes $A = 0.005$, $0.01$, and $0.02$, which is what distinguishes a genuine instability from a slow start. The authors also note that the emitted wave frequency $f$, converted to $: the angular rate, $, falls below the horizon angular velocity $ for large spin, so that the first law of black hole mechanics gives $, allowing radiation reaction to spin the hole down. They read this as at least part of the mechanism, and they point out that the unstable holes are strongly oblate, with a meridional-to-equatorial circumference ratio of about 0.38 at threshold, against a maximum of 0.64 for any four-dimensional Kerr hole.

The premise it rests on. The evolution uses the cartoon method, which requires the perturbation to be independent of $ with $, so the search covers only the bar-shaped sector lying in the rotation plane; if a deformation that varies with $ grows faster, the reported threshold of 0.87 is an upper bound rather than the true critical spin.

A five-dimensional black hole formed with spin above roughly 0.87 cannot be treated as a stable final state; the endpoint of a high-energy merger must be sought in fragmentation into two holes, a black ring, or a spin-down by radiation.

The measured threshold near 0.87 sits close to the value of about 0.85 obtained from comparing horizon areas, which supports the horizon-area comparison as a usable stability diagnostic for rotating black holes in higher dimensions.

Because the emitted gravitational-wave frequency drops below the horizon angular velocity at large spin, radiation reaction can remove angular momentum from the hole, making it a plausible way for the instability to saturate rather than run away.

The authors propose an oblateness criterion: a black hole with meridional-to-equatorial circumference ratio below about 0.4 is expected to be nonaxisymmetrically unstable regardless of dimension, which can be checked for six or more dimensions where Myers-Perry holes can be flatter still.

The cartoon reduction restricts the search to perturbations that are independent of $, so the quoted 0.87 is most safely read as an upper bound on the true critical spin; a faster-growing mode with $ dependence would lower it, and that sector is not explored here.

If the instability saturates by radiating, a decaying five-dimensional black hole would produce a waveform with a frequency set by the quasinormal mode and an exponentially rising amplitude, a distinctive shape worth searching for in any higher-dimensional waveform template bank.

The oblateness rule, if it survives testing, would let one predict the stability of six- through ten-dimensional Myers-Perry holes from horizon geometry alone, without running evolutions.

The cost of the simulation climbs steeply with spin (grid spacing must fall below about $0.008\,m^{1/2}$ already at $q=0.89$), so settling the final fate likely needs a different formulation rather than more resolution.

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