Resultado 3

In Ref. [71], rapidly-rotating Myers-Perry black holes with a single spin in D ≥ 6 were conjectured to be unstable for Gregory-Laflamme-type modes. Such black holes have quasi-extended event horizons for large values of the angular momentum compared to the mass (|J|/M^D−2D−3 1), acquiring some properties of black branes, which are Gregory-Laflamme-unstable. An order of magnitude estimate for the threshold of the black hole instability was given in [71] by considering the thermodynamic behaviour. For small rotations, the temperature decreases with the rotation for fixed mass, as it happens for the Kerr black hole, which is expected to be classically stable. However, after a critical value of the rotation, the temperature actually increases, as it happens for black branes.

Classical stability was conjectured to fail roughly after the critical rotation.

The critical rotation is actually a thermodynamic zero-mode of the Hessian (6.3), beyond which the black hole possesses two thermodynamic instabilities. To understand this, recall our discussion in Section 3.3. Since a traceless-transverse negative modehabis a regular tensor on the black hole backgroundg_ab, it follows thatT and Ω_iare left unchanged by hab. A negative mode, which is an off-shell perturbation in the path integral, occurs when the thermodynamic Hessian has a negative eigenvalue. However, this argument also applies if a negative mode is continuously connected through the variation of the black hole parameters to azero-mode (λ= 0), which is a classical perturbation of the black hole.

This will be the case if one of the eigenvalues of the thermodynamic Hessian changes from positive to negative continuously, marking the appearance of a new local thermodynamic instability. It is now more convenient to use the thermodynamic Hessian

−S_αβ ≡ −∂²S(x_γ)

∂xα∂x_β , xα = (M, Ji), (6.7) which is simply the inverse matrix of Wab (recall Appendix 2.A). If the zero-mode h_ab preserves T and Ω_i then, using the first law, it preserves (∂S/∂M)_J = 1/T and (∂S/∂Ji)M = −Ω_i/T, that is, it preserves ∂S/∂x^α. Hence it must correspond to an

eigenvector ofS_αβ with eigenvalue zero:

0 =δ(∂_αS) =δx^β∂_β∂_αS =S_αβδx^β. (6.8) Notice that the corresponding thermodynamic instability occurs only for a certain range of the rotation, beyond the zero-mode, while the thermodynamic instability shown to exist for all vacuum black holes in the last Section occurs for any value of the rotation. These instabilities correspond to distinct negative eigenvalues of the thermodynamic Hessian, and hence to distinct Gregory-Laflamme instabilities of the black branes. This is a refinement of the Gubser-Mitra conjecture.

Ref. [72] confirmed numerically this picture and the conjecture of Ref. [71], showing that a new negative mode appears at the critical rotation and, more importantly, that additional negative modes which are not thermodynamic in origin occur for higher rota-tions. These are thresholds forclassical instabilities of the black hole, as explicitly verified for the equal spin case in [68], on which Chapter 8 is based. Notice that the zero-modes which are thermodynamic in origin can be identified by the simpler Hessian matrix

Hij ≡ −

∂²S

∂Ji∂Jj

M

=−S_ij, (6.9)

due to the identity

det(−S_αβ) =− 1

(D−3)M T det(H_ij), (6.10)

valid for asymptotically flat vacuum black holes; for a proof of this identity, see Ap-pendix 6.A. It follows that, for a black hole parameterised by (M, Ji), additional negative eigenvalues of−S_αβ correspond precisely to negative eigenvalues ofH_ij.

In the Myers-Perry case, for fixedM, the eigenvalues ofH_ij are all positive for small enough angular momenta. However, as some or all of the angular momenta are increased, an eigenvalue ofH_ij may become negative. If we consider the space parameterised by J_i (for fixedM), there is some region containing the origin in which Hij is positive definite.

We define the boundary of this region to be theultraspinning surface. Following Ref. [72],

6.2. ZERO-MODES AND BLACK HOLE STABILITY 107 we shall say that a given black hole is ultraspinning if it lies outside the ultraspinning surface. From the above arguments, we know that as one crosses the ultraspinning surface, the black hole will develop a new negative mode, and the associated black branes will develop a new classical instability. This is in addition to the instability already present at low angular momenta. Furthermore, on the ultraspinning surface, the new negative mode must reduce to a stationary zero-mode that corresponds to a variation of parameters within the Myers-Perry family of solutions, since they are identified by the Myers-Perry equation of stateS(M, Ji).

Ref. [72] conjectured that classical instabilities whose threshold is a stationary and axisymmetric zero-mode occur only for rotations higher than a thermodynamic zero-mode, i.e. in the ultraspinning regime. We emphasise that our conjecture gives a necessary condition for an instability, not a sufficient one.

The intuition leading to the conjecture is that modes of lower symmetry are usually the most unstable ones. For instance, the original Gregory-Laflamme instability occurs for the “s-wave” of the transversal black hole. An additional classical instability will arise after a critical value of the rotation, and it will correspond to a “p-wave” of the transversal black hole. As the rotation is increased, higher order waves may become unstable. Now, if we consider a black hole, instead of a black brane, the “s-wave” and the “p-wave” are associated with the asymptotic charges, mass and angular momenta. Therefore they are associated to purely thermodynamic instabilities. Higher order waves, on the other hand, may become classically unstable as the rotation is increased, starting with the “d-wave”.

Notice that these waves do not affect the asymptotic charges.¹ The results of Refs. [68,72], described in the next two Chapters, clarify the harmonic structure underlying this problem.

Recall also that the thresholds of the classical instabilities should be associated with bifurcations to different black hole families, highlighting the connection between stability and uniqueness.

1Recall that, in Section 2.2.2, we discussed how the thermodynamic Hessian gave the stability with respect to mass/angular momenta exchanges with a “reservoir”. If the asymptotic charges are unaltered, no “reservoir” is invoked, and it is the internal stability of the black hole which is being probed.

Figure 6.1: Parameter space for MP black holes withD= 6 (left) andD= 7 (right). The black hole is labelled by the horizon radiusr+ and the spin parameters ai which we take to be positive for clarity. ForD= 6, the blue curve corresponds to extreme black holes.

In the red region, both eigenvalues of Hij are positive. In the blue region, corresponding to ultraspinning black holes, one eigenvalue is positive and the other is negative. For D= 7, the blue surface corresponds to extreme black holes. The ultraspinning surface is the red surface near the origin. Inside this surface, Hij is positive definite. The orange surface is where another eigenvalue ofHij vanishes. Between the red and orange surfaces, two eigenvalues of H_ij are positive and one is negative. Between the orange and blue surfaces, one eigenvalue ofH_ij is positive and two are negative. Ultraspinning black holes correspond to points between the red and blue surfaces. Note that the “cusp” where the red and orange surfaces meet has equal spins.