Gravitational instability in higher dimensions

TL;DR

This work identifies a concrete, base-manifold–driven stability mechanism for higher-dimensional Einstein spacetimes by analyzing transverse-traceless tensor perturbations and tying stability to the Lichnerowicz spectrum of the base $B$. It derives a Schrödinger-type radial equation for static spacetimes and extends the analysis to time-dependent backgrounds, obtaining a general instability criterion $ ilde{ au}_{min}<4-rac{(5-d)^2}{4}$ in the vanishing-$ ext{c}$ case and analogous conditions with a cosmological constant. The study applies these results to a range of geometries, including Schwarzschild–Tangherlini black holes, AdS and dS generalizations, topological black holes, brane-world metrics, and bubbles of nothing, finding that AdS embeddings typically stabilise base modes while AdS cosmologies are prone to instability and brane-worlds can be stable. The work also aggregates Lichnerowicz spectra for several Einstein bases, showing spheres are stabilizing, product bases can be destabilizing in lower dimensions, and negative-curvature bases tend to be stable, with implications for gravity-gauge dualities and higher-dimensional phenomenology.

Abstract

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, subject to the usual limitations of a linear analysis, any Ricci flat spacetime may be stabilised by embedding into a higher dimensional spacetime with cosmological constant. We apply our results to pure AdS black strings. Finally, we study the stability of higher dimensional ``bubbles of nothing''.