A proof of Friedman's ergosphere instability for scalar waves

Georgios Moschidis

doi:10.1007/s00220-017-3010-y

paper

A proof of Friedman's ergosphere instability for scalar waves

arXiv:1608.02035 · doi:10.1007/s00220-017-3010-y

Abstract

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 $H^{+}$ . On such spacetimes, Friedman provided a heuristic argument that the energy of certain solutions $ϕ$ of $□_{g} ϕ = 0$ 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 $(M^{d + 1}, g)$ , for any $d \geq 2$ , not necessarily globally real analytic. We impose only a unique continuation condition for the wave equation across the boundary $\partial E$ of $E$ on a small neighborhood of a point $p \in \partial E$ . 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 $\partial E$ . We also allow the spacetimes $(M, g)$ under consideration to possess a (possibly empty) future event horizon $H^{+}$ , such that, however, $H^{+} \cap E = \emptyset$ (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.

72 pages

A proof of Friedman's ergosphere instability for scalar waves

Abstract

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