Let $${(\mathcal{M}^{3+1},g)}$$ ( M 3 + 1 , g ) be a real analytic, stationary and asymptotically flat spacetime with a non-empty ergoregion $${\mathscr{E}}$$ E and no future event horizon $${\mathcal{H}^{+}}$$ H + . In Friedman (Commun Math Phys 63(3):243–255, 1978), Friedman observed that, on such spacetimes, there exist solutions $${\varphi}$$ φ to the wave equation $${\square_{g}\varphi=0}$$ □ 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 $${+\infty}$$ + ∞ . In this paper, we provide a rigorous proof of Friedman’s instability. Our setting is, in fact, more general. We consider smooth spacetimes $${(\mathcal{M}^{d+1},g)}$$ ( M d + 1 , g ) , for any $${d\ge2}$$ d ≥ 2 , not necessarily globally real analytic. We impose only a unique continuation condition for the wave equation across the boundary $${\partial\mathscr{E}}$$ ∂ E of $${\mathscr{E}}$$ E on a small neighborhood of a point $${p\in\partial\mathscr{E}}$$ p ∈ ∂ E . This condition always holds if $${(\mathcal{M},g)}$$ ( M , g ) is analytic in that neighborhood of p, but it can also be inferred in the case when $${(\mathcal{M},g)}$$ ( M , g ) possesses a second Killing field $${\Phi}$$ Φ such that the span of $${\Phi}$$ Φ and the stationary Killing field T is timelike on $${\partial\mathscr{E}}$$ ∂ E . We also allow the spacetimes $${(\mathcal{M},g)}$$ ( M , g ) under consideration to possess a (possibly empty) future event horizon $${\mathcal{H}^{+}}$$ H + , such that, however, $${\mathcal{H}^{+}\cap\,\,\mathscr{E}=\emptyset}$$ 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 $${\varphi}$$ φ of $${\square_{g}\varphi=0}$$ □ g φ = 0 with frequency support bounded away from $${{\omega}=0}$$ ω = 0 and $${{\omega}=\pm\infty}$$ ω = ± ∞ .
Communications in Mathematical Physics – Springer Journals
Published: Nov 4, 2017
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