Communications in Mathematical Physics

, Volume 358, Issue 2, pp 437–520 | Cite as

A Proof of Friedman’s Ergosphere Instability for Scalar Waves

  • Georgios Moschidis


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 \({\mathcal{H}^{+}}\). In Friedman (Commun Math Phys 63(3):243–255, 1978), Friedman observed that, on such spacetimes, there exist solutions \({\varphi}\) to the wave equation \({\square_{g}\varphi=0}\) such that their local energy does not decay to 0 as time increases. In addition, Friedman provided a heuristic argument that the energy of such solutions actually grows to \({+\infty}\). 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 et al. in (Phys Rev D 89(12):124008, 2014). 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. Applications of this estimate include a Morawetz-type bound for solutions \({\varphi}\) of \({\square_{g}\varphi=0}\) with frequency support bounded away from \({{\omega}=0}\) and \({{\omega}=\pm\infty}\).


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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA

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