We consider the stability of black holes within both classical general relativity and the semiclassical thermodynamic description. In particular, we study linearised perturbations and their contributions to the gravitational partition function. Exploring the connection between classical and thermodynamic stability, we find classical instabilities and new families of vacuum black holes.

We start by studying negative modes of black hole partition functions, which represent pathologies in the one-loop quantum corrections. In particular, we extend this study to charged black holes (Reissner-Nordstr¨om), using a method based on gauge-invariant perturbations, and to rotating black holes (Kerr-AdS), where a numerical technique is employed. In the both cases, we find a negative mode in the region where local thermodynamic stability fails, as expected.

We then present the first examples of linearised classical instabilities of vacuum asymptotically flat black holes. We analyse numerically perturbations of Myers-Perry solutions, both in the single spin and in the equal spins (odd D) cases. For sufficiently high rotation, in the so-called ultraspinning regime, new negative modes of the partition function may arise whose threshold marks both the onset of a classical instability of the black hole (not just of the associated black branes) and the bifurcation to a new family of black hole solutions. In the case of singly-spinning solutions, we find the threshold stationary modes signalling the instabilities, confirming a conjecture by Emparan and Myers. In the case of solutions with equal spins, we are able to find perturbations that grow exponentially in time in D = 9 (we believe that this extends to higher odd D).

Furthermore, the new family of solutions bifurcating at the onset of the instability should have a single rotational symmetry, saturating the rigidity theorem.