标量波Friedman能层不稳定性的证明
A proof of Friedman's ergosphere instability for scalar waves
摘要：
设$(\mathcal{M}^{3+1},g)$是一个实解析的、平稳的、渐近平坦的时空，它具有非空的能区$\mathscr{E}$和没有未来事件视界$\mathcal{H}^{+}$。在这样的时空中，弗里德曼提出了一个启发性的论证，即$\square_{g}ϕ=0$的某些解$ϕ$的能量随着时间的增加而增长到$+\infty$。本文给出了Friedman不稳定性的一个严格证明。事实上，我们的设定更一般。我们考虑光滑时空$(\mathcal{M}^{d+1},g)$，对于任何$d\ge2$，不一定是全局实解析的。我们只施加一个独特的连续条件的波动方程的边界$\partial\mathscr{E}$的$\mathscr{E}$上的一个小邻域的点$p\in\partial\mathscr{E}$。如果$(\mathcal{M},g)$在$p$的那个邻域内解析，则这个条件总是成立，但它也可以在$(\mathcal{M},g)$具有第二个Killing场$Φ$，使得$Φ$和定态Killing场$T$的跨度在$\partial\mathscr{E}$上是类时的情况下推断出来。我们也允许所考虑的时空$(\mathcal{M},g)$拥有一个（可能是空的）未来事件视界$\mathcal{H}^{+}$，这样，然而，$\mathcal{H}^{+}\cap\mathscr{E}=\emptyset$（因此，不包括克尔外部家族）。作为我们的定理的应用，我们推断的不稳定性结果的声波方程的流体动力学涡，奥利维拉，Cardoso和Crispino的数值研究的现象。此外，作为一个侧面的好处，我们的证明，我们提供了一个推导，完全基于向量场方法，一个Carleman型估计的外部的遍历一般类的平稳和渐近平坦的时空。
英文摘要：
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 $\mathcal{H}^{+}$. On such spacetimes, Friedman provided a heuristic argument that the energy of certain solutions $ϕ$ of $\square_{g}ϕ=0$ 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 $(\mathcal{M}^{d+1},g)$, 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 $Φ$ such that the span of $Φ$ 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, 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.
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