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

Abstract

Numerical-relativity simulation is performed for rapidly spinning black holes (BHs) in a higher-dimensional spacetime of special symmetries for the dimensionality $6\le d\le 8$. 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 $q:=a/{\mu }^{1/(d-3)}\approx 0.74$ for $d=6$, $\approx 0.73$ for $d=7$, and $\approx 0.77$ for $d=8$ where $\mu$ and $a$ are mass and spin parameters. Black holes with a spin smaller than these critical values ($q_{\mathrm{crit}}$) 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 $q_{\mathrm{crit}}$. 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 $q>q_{\mathrm{crit}}$ 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.

21 pages, 9 figures, accepted for publication in PRD

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