Instability and new phases of higher-dimensional rotating black holes
It has been conjectured that higher-dimensional rotating black holes become unstable at a sufficiently large value of the rotation, and that new black holes with pinched horizons appear at the threshold of the instability. We search numerically, and find, the stationary axisymmetric perturbations of Myers-Perry black holes with a single spin that mark the onset of the instability and the appearance of the new black hole phases. We also find new ultraspinning Gregory-Laflamme instabilities of rotating black strings and branes.
Black holes are the most basic and fascinating objects in General Relativity and the study of their properties is essential for a better understanding of the dynamics of spacetime at its most extreme. In higher-dimensional spacetimes a vast landscape of novel black holes has begun to be uncovered Emparan:2008eg (). Its layout — i.e., the connections between different classes of black holes in the space of solutions — hinges crucially on the analysis of their classical stability: most novel black hole phases are conjectured to branch-off at the threshold of an instability of a known phase. Showing how this happens is an outstanding open problem that we address in this paper.
The best known class of higher-dimensional black holes, discovered by Myers and Perry (MP) in myersperry (), appear in many respects as natural generalizations of the Kerr solution. In particular, their horizon is topologically spherical. However, the actual shape of the horizon can differ markedly from the four-dimensional one, which is always approximately round with a radius parametrically . This is not so in . Considering for simplicity the case where only one spin is turned on (of the independent angular momenta available), it is possible to have black holes with arbitrarily large for a given mass . The horizon of these ultraspinning black holes spreads along the rotation plane out to a radius much larger than the thickness transverse to this plane, . This fact was picked out in Emparan:2003sy () as an indication of an instability and a connection to novel black hole phases. In more detail, in the limit with fixed, the geometry of the black hole in the region close to the rotation axis approaches that of a black membrane. Black branes are known to exhibit classical instabilities Gregory:1993vy (), at whose threshold a new branch of black branes with inhomogeneous horizons appears Gubser:2001ac (). Ref. Emparan:2003sy () conjectured that this same phenomenon should be present for MP black holes at finite but sufficiently large rotation: they should become unstable beyond a critical value of , and the marginally stable solution should admit a stationary, axisymmetric perturbation signalling a new branch of black holes pinched along the rotation axis. Simple estimates suggested that in fact should not be much larger than one. As increases, the MP solutions should admit a sequence of stationary perturbations, with pinches at finite latitude, giving rise to an infinite sequence of branches of ‘pinched black holes’ (see fig. 1). Ref. Emparan:2007wm () argued that this structure is indeed required in order to establish connections between MP black holes and the black ring and black Saturn solutions more recently discovered. Our main result is a numerical analysis that proves correct the conjecture illustrated in fig. 1.
The solution for a MP black hole rotating in a single plane in dimensions is myersperry ()
The parameters here are the mass-radius and the rotation-radius ,
The event horizon lies at the largest real root of .
The linearized perturbation theory of the Kerr black hole () was disentangled in Teukolsky:1973ha () using the Newman-Penrose formalism. Attempts to extend this formalism to decouple a master equation for the gravitational perturbations of (1) in have failed so far. Moreover, even if some subsectors of the perturbations of some classes of MP black holes have been decoupled mpperts (), none of them shows signs of any instability and indeed they do not contain the precise kind of perturbations we are interested in. Thus we take a more frontal numerical approach to a full set of coupled partial differential equations (PDE).
We intend to solve for a stationary linearized perturbation around the background (1). Choosing traceless-transverse (TT) gauge, and , the equations to solve are
where is the Lichnerowicz operator in the TT gauge. Actually, we solve the more general eigenvalue problem
which is known to appear in two contexts: eqs. (5) determine the stationary perturbations of a black string in dimensions (obtained by adding a flat direction to (1)) with a profile . Thus such modes with correspond to the threshold of the Gregory-Laflamme instability of black strings Gregory:1993vy (). The same equations also describe the negative modes of quadratic quantum corrections to the gravitional Euclidean partition function Gross:1982cv (). A recent study of this problem for the Kerr black hole has shown the existence of a branch of solutions extending the negative Schwarzschild mode (with ) to finite rotation, with growing as the rotation increases towards the Kerr bound Monteiro:2009ke ().
Our reason to consider (5) instead of trying to solve directly for is that there exist powerful numerical methods for eigenvalue problems that give the eigenvalues together with the eigenvectors, i.e., the metric perturbations. If the ultraspinning instability is present for MP black holes in , then, in addition to the analogue of the branch studied in Monteiro:2009ke (), a new branch of negative modes extending to must appear. The eigenvalue corresponds to a (perturbative) stationary solution with a slightly deformed horizon. In fact, as explained above, we expect an infinite sequence of such branches that reach at increasing values of the rotation. The solutions for imply new kinds of Gregory-Laflamme instabilities and inhomogeneous phases of ultraspinning black strings (see also Kleihaus:2007dg ()).
The modes we seek preserve the rotational symmetries of the MP solution and depend only on the radial and polar coordinates, and Emparan:2003sy (). Thus we take the ansatz
We decompose a given quantity as . The unperturbed contribution describes (1). The perturbations are determined solving the eigenvalue problem (5) subject to appropriate boundary conditions. After imposing TT gauge, eq. (5) reduces to four coupled PDEs for , , and (the TT conditions then give , and ). The boundary conditions are that the perturbations are regular and finite at the horizon, , at infinity, , and at the poles . In addition, we impose . It is important to ensure that the eigenmodes we find are not pure gauge, . We can prove that in the TT gauge, pure gauge perturbations within our ansatz necessarily diverge at either the horizon or infinity. Thus, with our boundary conditions, the eigenmodes we obtain are never pure gauge.
We use a numerical approach successfully applied to the identification of the negative mode of Kerr and Kerr-AdS black holes Monteiro:2009ke (). It employs a Chebyshev spectral numerical method (see Monteiro:2009ke () for further details). We have carried out the calculations for . The cases (where the heuristics of Emparan:2003sy () do not allow to predict any instability) and present more difficult numerics. These, as well as a more detailed presentation of our numerical approach, will be discussed elsewhere.
The results for are displayed in fig. 2, the other two cases being qualitatively very similar. We plot the negative eigenvalue as a function of the rotation parameter . We normalize and relative to the mass-radius , which is equivalent to plotting their values for fixed mass (or mass per unit length, in the black string interpretation). As described above, the leftmost curve, which does not reach , is the higher-dimensional counterpart of the Kerr negative mode, and the eigenvalues are the wavenumbers of the Gregory-Laflamme threshold modes at rotation . At larger rotation we find new branches of negative modes that intersect at finite . We label these successive branches with an integer , and refer to them as ‘harmonics’. The values of at which the stationary perturbations appear are listed in table 1.
It is important to note that the eigenmode of the harmonic does not correspond to a new stationary solution. Instead it is a zero-mode that takes the solution to a nearby one along the family of MP black holes. The existence and location of this zero-mode is a consequence of the fact that if the Hessian of the Gibbs potential , calculated along a family of solutions, has a zero eigenvalue for some particular solution, then there is a zero-mode perturbation of the gravitational (Euclidean) action that takes that solution to an infinitesimally nearby one along that family. That is, perturbing the solution with that zero-mode does not correspond to branching-off into a new family of solutions.
One can easily check that the determinant of the Hessian of the Gibbs potential is proportional to the determinant of the Hessian of the entropy with respect to only the angular momenta, i.e., to the determinant of
Therefore, for solutions with a single spin, there must appear a stationary perturbation, in principle not associated to an instability of the black hole, at the inflection point of the curve at fixed (point in fig. 1). For the MP solutions this happens at
The values of for agree with the central values of the numerically-determined rotations for (first column in table 1) up to the third decimal place. This is a very good check of the accuracy of our numerical methods.
The eigenmodes of the higher harmonics, , do not admit this interpretation as perturbations along the MP family of solutions and thus correspond to genuinely new (perturbative) black hole solutions with deformed horizons. Their appearance conforms perfectly to the predictions in Emparan:2003sy () and Emparan:2007wm (). It is then natural to expect, although our approach does not prove it since it only captures zero-frequency perturbations, that the harmonic signals the onset of the instability conjectured in Emparan:2003sy (). The eigenmodes for higher harmonics confirm the appearance of the sequence of new black hole phases as the rotation grows.
To visualize the effect on the horizon of the perturbations that give new solutions, and provide further confirmation of our interpretation, we draw an embedding diagram of the unperturbed MP horizon and compare it with the deformations induced by the ultraspinning harmonics . This is best done using the embedding proposed in Frolov:2006yb (), which has the advantage of allowing to embed the horizon along the entire range for any rotation, although at the cost of stretching the pole region, which acquires a conical profile. We do it for the ultraspinning harmonics in figs. 3–5. In spite of the distortion created by the embedding, the effect of the perturbations is clear: modes create a pinch centered on the rotation axis ; modes have a pinch centered at finite latitude ; modes pinch the horizon twice: around the rotation axis and at finite latitude. These are the kind of deformations depicted in fig. 1. To better identify the number of times that the perturbed horizon crosses the unperturbed solution, in these figures we also plot the logarithmic difference between the two embeddings.
Ref. Emparan:2003sy () gave several arguments to the effect that critical values close to 1 were to be expected. In particular, it was pointed out that the change in the behavior of the black hole from ‘Kerr-like’ to ‘black membrane-like’ could be pinpointed to the value of the spin where the temperature (i.e., surface gravity) reaches a minimum for fixed mass, which is the same, for solutions with a single spin, as the inflection point of . As we have argued, the zero-mode at this solution should not signal an instability. The mode at the threshold of the actual instability instead appears at larger rotation, well within the regime of membrane-like behavior as conjectured in Emparan:2007wm (). We expect this to be true in general: the ultraspinning instability of MP black holes should appear for angular momenta strictly beyond the (codimension 1) locus in the space of angular momenta where the Hessian has a zero eigenvalue.
In particular, in this criterion does not allow any ultraspinning instability for any , , and in with all the angular momenta equal it predicts that the instability should appear at . However we cannot predict the precise values of the rotation where the instability appears.
We have identified the points in the phase diagram where the new branches must appear, but we cannot determine in which direction these run. This requires calculating the area, mass and spin of the perturbed solutions. However, for any — and numerically we can never obtain an exact zero — the linear perturbations decay exponentially in the radial direction, and so the mass and spin, measured at asymptotic infinity, are not corrected. It seems that in order to obtain the directions of the new branches one has to go beyond our level of approximation or adopt a different approach.
The new branches extend to non-zero eigenvalues . These imply a new ultraspinning Gregory-Laflamme instability for black strings, in which the horizon is deformed not only along the direction of the string, but also along the polar direction of the transverse sphere. Observe that, even if the , mode is not an instability of the MP black hole, the modes , are expected to correspond to thresholds of GL instabilities of MP black strings. At a given rotation, modes with larger have longer wavelength and so the branch is expected to dominate the instability. The growth of with can be understood heuristically, since as grows the horizon becomes thinner in directions transverse to the rotation plane and hence it can fit into a shorter compact circle.
To finish, we mention that pinched phases of rotating plasma balls, dual to pinched black holes in Scherk-Schwarz compactifications of AdS, have been found Lahiri:2007ae (), as well as new kinds of deformations of rotating plasma tubes Caldarelli:2008mv () and rotating plasma ball instabilities Cardoso:2009bv (). The relation of our results to these and other phenomena of rotating fluids will be discussed elsewhere.
Acknowledgments. We thank Troels Harmark, Keiju Murata, Malcolm Perry and especially Harvey Reall for discussions. We were supported by: Marie Curie contract PIEF-GA-2008-220197, and by PTDC/FIS/64175/2006, CERN/FP/83508/2008 (OJCD); STFC Rolling grant (PF); Fundação para a Ciência e Tecnologia (Portugal) grants SFRH/BD/22211/2005 (RM), SFRH/BD/22058/2005 (JES); and by MEC FPA 2007-66665-C02 and CPAN CSD2007-00042 Consolider-Ingenio 2010 (RE). This is preprint DCPT-09/47.
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