A Proof of Friedman's Ergosphere Instability for Scalar Waves

Let be a real analytic, stationary and asymptotically flat spacetime with a non-empty ergoregion and no future event horizon . In Friedman (Commun Math Phys 63(3):243-255, 1978), Friedman observed that, on such spacetimes, there exist solutions to the wave equation 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 . In this paper, we provide a rigorous proof of Friedman's instability. Our setting is, in fact, more general. We consider smooth spacetimes , for any , not necessarily globally real analytic. We impose only a unique continuation condition for the wave equation across the boundary of on a small neighborhood of a point . This condition always holds if is analytic in that neighborhood of p, but it can also be inferred in the case when possesses a second Killing field such that the span of and the stationary Killing field T is timelike on . We also allow the spacetimes under consideration to possess a (possibly empty) future event horizon , such that, however, (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 of with frequency support bounded away from omega = 0 and omega = +/-infinity.