Bar-mode instability of rapidly spinning black hole in higher dimensions: Numerical simulation in general relativity

TL;DR

This study extends bar-mode stability analysis to rapidly spinning Myers–Perry black holes in $d=6$--$8$ via nonlinear numerical relativity, confirming a nonaxisymmetric bar-mode instability above a dimensionless spin threshold $q_{\rm crit}\approx 0.74$ (d=6), $0.73$ (d=7), and $0.77$ (d=8). The instability drives spontaneous quadrupole gravitational-wave emission, spinning the BH down to $q<q_{\rm crit}$ and leaving a moderately spinning remnant ($q_f\sim 0.6$--$0.7$ depending on dimension and initial spin). The growth rate scales roughly linearly with $(q-q_{\rm crit})$, and the saturation is governed by gravitational radiation reaction; the emission predominantly alters angular momentum rather than mass. The work also discusses the role of superradiance, compares nonaxisymmetric bar-mode instability with axisymmetric and fragmentation scenarios, and outlines implications for mini black hole evolution in TeV-scale gravity scenarios, including possible fragmentation at ultra-high spins and the interplay with Hawking radiation. These results constrain the classical evolution of rapidly spinning higher-dimensional BHs and inform phenomenological models of mini-BH formation and decay in particle colliders.

Abstract

Numerical-relativity simulation is performed for rapidly spinning black holes (BHs) in a higher-dimensional spacetime of special symmetries for the dimensionality $6 \leq d \leq 8$. We find that higher-dimensional BHs, spinning rapidly enough, are dynamically unstable against nonaxisymmetric bar-mode deformation and spontaneously emit gravitational waves, irrespective of $d$ as in the case $d=5$ \cite{SY09}. The critical values of a nondimensional spin parameter for the onset of the instability are $q:=a/μ^{1/(d-3)} \approx 0.74$ for $d=6$, $\approx 0.73$ for $d=7$, and $\approx 0.77$ for $d=8$ where $μ$ and $a$ are mass and spin parameters. Black holes with a spin smaller than these critical values ($q_{\rm crit}$) appear to be dynamically stable for any perturbation. Longterm simulations for the unstable BHs are also performed for $d=6$ and 7. We find that they spin down as a result of gravitational-wave emission and subsequently settle to a stable stationary BH of a spin smaller than $q_{\rm crit}$. For more rapidly spinning unstable BHs, the timescale, for which the new state is reached, is shorter and fraction of the spin-down is larger. Our findings imply that a highly rapidly spinning BH with $q > q_{\rm crit}$ cannot be a stationary product in the particle accelerators, even if it would be formed as a consequence of a TeV-gravity hypothesis. Its implications for the phenomenology of a mini BH are discussed.

Figures (9)

• Figure 1: (a) Evolution of deformation parameter $\eta$ for $d=6$ and for the initial spin $q_i=a/\mu^{1/3} \approx 1.039$, 0.986, 0.933, 0.878, 0.821, 0.801, 0.781, 0.761, 0.750, 0.740, 0.718, and 0.674 (from the upper to lower curves) with $A=0.005$. (b) The growth rate of $\eta$, $1/\tau$, in units of $\mu^{-1/3}$ as a function of $q$ (solid curve). The dashed curve denotes $\Omega_{\rm H}/2\pi$. For $q_i \agt 0.75$, the value of $\eta$ increases exponentially with time, and otherwise, an exponential damping is seen. For $q_i = 0.750$ (thick solid curve in panel (a)) and 0.740 (below the curve of $q_i=0.750$), the growth and damping rates of $\eta$ are quite small, indicating that these BHs are close to the marginally stable state.
• Figure 2: (a) Evolution of deformation parameter $\eta$ for $d=6$ and for $q_i=0.801$ with initial perturbation amplitude $A=0.02$, 0.005, 0.001, and $10^{-6}$ (from the upper to lower curves) and with $N=30$ (dotted curve), 40 (dashed curves), and 50 (solid curves), respectively. (b) The same as (a) but for $q_i=0.821$, 0.801, and 0.780 (from left to right) with $A=0.02$ and with $N=40$ (dashed curves) and 50 (solid curves). (c) The same as (b) but for $q_i=1.039$, 0.986, 0.933, and 0.878 (from the upper to lower curves) with $A=0.005$. (d) The maximum value of $\eta$ as a function of $q_i$. The dashed line and solid curve denote $\eta_{\rm max}=2(q_i - q_{\rm crit})$ and relation (\ref{['etamaxf']}), respectively (see Sec. VI B for an approximate derivation of these relations).
• Figure 3: (a) The same as Fig. \ref{['FIG1']} but for $d=7$ and for $q_i=a/\mu^{1/4}=0.960$, 0.903, 0.844, 0.813, 0.783, 0.767, 0.751, 0.735, and 0.719 (from the upper to lower curves) with $N=50$. For $q_i \agt 0.73$, the value of $\eta$ increases exponentially with time, otherwise, an exponential damping is seen. For $q=0.735$, the growth (or damping) rate of $\eta$ is close to zero, implying that this BH is close to the marginally stable one. (b) The growth rate of $\eta$, $1/\tau$, in units of $\mu^{-1/4}$ as a function of $q$ (solid curve). The dashed curve denotes $\Omega_{\rm H}/2\pi$. (c) The same as panel (a) but for long runs with $q_i=1.017$, 0.960, 0.903, 0.873, 0.844, and 0.813.
• Figure 4: (a) and (b): $+$ and $\times$ modes of gravitational waveform (solid curve) from an unstable BH for $d=6$ and for $q_i = 0.801$ as a function of a retarded time defined by $t-r$ where $r$ is the coordinate distance from the center. We also plot $\eta/2$ as a function of $t$ (dashed curve). The initial condition is $A=0.005$ and the result for the grid resolution of $N=50$ is plotted. (c) and (d): The same as (a) and (b) but for $d=6$ and for $q_i=0.986$ with $A=0.005$.
• Figure 5: $C_p/C_e$ as a function of $q=a/\mu^{1/(d-3)}$ for $d=5$ -- 8.