Abstract
We initiate the study of dynamical instabilities of higher-dimensional black holes using the blackfold approach, focusing on asymptotically flat boosted black strings and singly-spinning black rings in D ≥ 5. We derive novel analytic expressions for the growth rate of the Gregory-Laflamme instability for boosted black strings and its onset for arbitrary boost parameter. In the case of black rings, we study their stability properties in the region of parameter space that has so far remained inaccessible to numerical approaches. In particular, we show that very thin (ultraspinning) black rings exhibit a Gregory-Laflamme instability, giving strong evidence that black rings are unstable in the entire range of parameter space. For very thin rings, we show that the growth rate of the instability increases with increasing non-axisymmetric mode m while for thicker rings, there is competition between the different modes. However, up to second order in the blackfold approximation, we do not observe an elastic instability, in particular for large modes m ≫ 1, where this approximation has higher accuracy. This suggests that the Gregory-Laflamme instability is the dominant instability for very thin black rings. Additionally, we find a long-lived mode that describes a wiggly time-dependent deformation of a black ring. We comment on disagreements between our results and corresponding ones obtained from a large D analysis of black ring instabilities.
Highlights
In this paper we initiated a systematic study of the dynamical stability of black holes in D ≥ 5 in the blackfold limit and applied it to asymptotically flat boosted black strings and black rings
Despite our analysis including second order corrections to the blackfold approximation, we have not been able to identify an elastic instability of black rings, as that found numerically in [12] for m = 2 and D = 5.15 We have identified divergences in the dispersion relations for hydrodynamic and elastic modes that manifestly break the expansion when m = 3 for D = 5, m = 2 for D = 6 and m = 1 for all D ≥ 5
A qualitative comparison of the growth rates of potential Gregory-Laflamme and elastic instabilities for D = 5 and m = 2 found here with those numerically obtained in [6, 12] suggest that the analysis we have carried out is not valid for m = 2 and D = 5.16 On the other hand, the analysis performed here is more accurate for large modes m 1 for which, within this approach and up to second order, no elastic instability is found in any dimension D
Summary
We briefly review the essential aspects of the blackfold approach required for the purposes of this work. We discuss the blackfold equations up to second order in a long-wavelength expansion which determine the equilibrium configurations that we wish to perturb. We derive new general formulae for linearised perturbations of the equilibrium blackfold equations, focusing on the case of 2-dimensional worldvolumes which describe black strings and black rings. These results will be used in the remaining sections in order to study the hydrodynamic and elastic stability of these later two cases
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