Instabilities of thin black rings: closing the gap

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 singlyspinning 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.Besides having proved to be extremely useful in finding new black hole solutions [14][15][16][17][18] in asymptotically flat space, we demonstrate here that the blackfold approach [7, 19] is a powerful tool for studying hydrodynamic (i.e. Gregory-Laflamme) and elastic instabilities of higher-dimensional black holes in the ultraspinning regime and away from it. 1 In this context, one first finds a stationary solution, modelled as a fluid confined to a surface, corresponding to the black hole solution whose stability one wishes to study. The fundamental fluid variables and the geometric properties of the surface describing the equilibrium configuration of the fluid are subsequently perturbed and the stability properties of black holes are found by studying the propagation of hydrodynamic and elastic modes.Black rings can be classified as thin 0 ≤ ν < 1/2 or as fat 1/2 ≤ ν < 1 where for very thin rings ν = r 0 /R is a measure of the ring thickness. Studying Penrose inequalities, the fat branch of black rings in D = 5 has been shown to be unstable [20][21][22] while for the thin branch in D = 5, the instability of black rings relies on numerical studies [6,12]. However, these numerical studies, due to lack of accuracy, have only established the existence of instabilities for ν ≥ 0.144 [6] and for ν ≥ 0.15 [12]. The region ν < 0.144 is left unknown, with the only suggestive arguments of [7,8] being applicable in the strict case of ν = 0, for which there is barely any distinction between the black ring and the boosted black string. Additionally, the numerical studies of [6,12] have not consid...

Black hole interactions at large D: brane blobology

“…One might use blob rings to study black ring dynamics at large D as has been done for Myers-Perry black holes. It is known that thin rings are unstable to Gregory-Laflamme (GL) type non-axisymmetric fluctuations [5,46,47]. In the large D limit, the same instability was found in the effective theory on the ring coordinate [25].…”

“…The elastic instability numerically found in D = 5 [5] is another interesting phenomenon, which deforms the ring keeping its thickness almost unchanged, in contrast to the GL instability. Recently, it was shown that thin black rings are free of this type of instability within the scope of the blackfold approximation [47]. 15 It is interesting to see if blob rings admit the elastic-type instability with or without the non-perturbative correction caused by the blob-blob interaction.…”

“…Here, the terms 'thin' and 'fat' are used regarding the applicability of the blob approximation 15. The elastic instability found in the large D ring[27] was questioned in ref [47]…”

On matched asymptotic expansions of backreacting metastable anti-branes

“…rings with small curvature radius, but is not observed for thin rings, i.e. rings with large curvature radius, [40,41].…”

Comments on the stability of the KPV state

“…Subsequently, we present the derivation of the blackfold perturbation equations used in the main text. For further discussion on embedding geometry and blackfold perturbation equation, see [27][28][29].…”