A gravitational instability in higher dimensions

We explore a classical instability of spacetimes of dimension $D>4$. Firstly, we consider static solutions: generalised black holes and brane world metrics. The dangerous mode is a tensor mode on an Einstein base manifold of dimension $D-2$. A criterion for instability is found for the generalised Schwarzschild, AdS-Schwarzschild and topological black hole spacetimes in terms of the Lichnerowicz spectrum on the base manifold. Secondly, we consider perturbations in time-dependent solutions: Generalised dS and AdS. Thirdly, we show that any Ricci flat spactime is stabilised by embedding into a higher dimensional spacetime with cosmological constant. In particular, pure AdS black strings are stable. Finally, we study the stability of higher dimensional ``bubbles of nothing''.