A Proof of Friedman’s Ergosphere Instability for Scalar Waves

Let ( M 3 + 1 , g ) be a real analytic, stationary and asymptotically flat spacetime with a non-empty ergoregion E and no future event horizon H + . In Friedman (Commun Math Phys 63(3):243–255, 1978 ), Friedman observed that, on such spacetimes, there exist solutions φ to the wave equation □ g φ = 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 + ∞ . In this paper, we provide a rigorous proof of Friedman’s instability. Our setting is, in fact, more general. We consider smooth spacetimes ( M d + 1 , g ) , for any d ≥ 2 , not necessarily globally real analytic. We impose only a unique continuation condition for the wave equation across the boundary ∂ E of E on a small neighborhood of a point p ∈ ∂ E . This condition always holds if ( M , g ) is analytic in that neighborhood of p , but it can also be inferred in the case when ( M , g ) possesses a second Killing field Φ such that the span of Φ and the stationary Killing field T is timelike on ∂ E . We also allow the spacetimes ( M , g ) under consideration to possess a (possibly empty) future event horizon H + , such that, however, H + ∩ E = ∅ (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 φ of □ g φ = 0 with frequency support bounded away from ω = 0 and ω = ± ∞ .