Bar-mode instability of a 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 $6\ensuremath{\le}d\ensuremath{\le}8$. 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 $d=5$ [M. Shibata and H. Yoshino, Phys. Rev. D 81, 021501(R) (2010).]. The critical values of a nondimensional spin parameter for the onset of the instability are $q\ensuremath{\mathrel{:=}}a/{\ensuremath{\mu}}^{1/(d\ensuremath{-}3)}\ensuremath{\approx}0.74$ for $d=6$, $\ensuremath{\approx}0.73$ for $d=7$, and $\ensuremath{\approx}0.77$ for $d=8$ where $\ensuremath{\mu}$ 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. Long-term 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 time scale, 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.