A proof of Friedman's ergosphere instability for scalar waves

TL;DR

This paper rigorously proves Friedman's heuristic instability for scalar waves in ergoregions of stationary, asymptotically flat spacetimes, extending the result to more general settings and applications like the hydrodynamic vortex.

Contribution

It provides a general proof of ergosphere instability for scalar waves, relaxing analyticity assumptions and including spacetimes with horizons, with applications to fluid dynamics models.

Findings

Proved ergosphere instability for scalar waves in general stationary spacetimes.

Extended instability results to non-analytic and horizon-inclusive spacetimes.

Derived a Carleman estimate using the vector field method for ergoregion exterior.

Abstract

Let $(\mathcal{M}^{3+1},g)$ be a real analytic, stationary and asymptotically flat spacetime with a non-empty ergoregion $\mathscr{E}$ and no future event horizon $\mathcal{H}^{+}$. On such spacetimes, Friedman provided a heuristic argument that the energy of certain solutions $\phi$ of $\square_{g}\phi=0$ grows to $+\infty$ as time increases. In this paper, we provide a rigorous proof of Friedman's instability. Our setting is, in fact, more general. We consider smooth spacetimes $(\mathcal{M}^{d+1},g)$, for any $d\ge2$, not necessarily globally real analytic. We impose only a unique continuation condition for the wave equation across the boundary $\partial\mathscr{E}$ of $\mathscr{E}$ on a small neighborhood of a point $p\in\partial\mathscr{E}$. This condition always holds if $(\mathcal{M},g)$ is analytic in that neighborhood of $p$, but it can also be inferred in the case when $(\mathcal{M},g)$ possesses a second Killing field $\Phi$ such that the span of $\Phi$ and the stationary Killing field $T$ is timelike on $\partial\mathscr{E}$. We also allow the spacetimes $(\mathcal{M},g)$ under consideration to possess a (possibly empty) future event horizon $\mathcal{H}^{+}$, such that, however, $\mathcal{H}^{+}\cap\mathscr{E}=\emptyset$ (excluding, thus, the Kerr exterior family). As an application of our theorem, we infer an instability result for the acoustical wave equation on the hydrodynamic vortex, a phenomenon first investigated numerically by Oliveira, Cardoso and Crispino. Furthermore, as a side benefit of our proof, we provide a derivation, based entirely on the vector field method, of a Carleman-type estimate on the exterior of the ergoregion for a general class of stationary and asymptotically flat spacetimes.