受限几何的非线性不稳定性
On the nonlinear instability of confined geometries
摘要：
反德西特时空的“弱湍流”不稳定性的发现支持了受限涨落最终坍缩成黑洞的观点，并表明类似的现象在渐近平坦的时空中也是可能的，例如在恒星的球对称振荡或超紧物体的非径向脉动的背景下。在这里，我们提出了一个详细的研究在一个空腔中的爱因斯坦-克莱因-戈登系统的演化，与不同类型的变形的频谱，包括一个质量项的标量和诺依曼条件的边界。我们提供的数值证据表明，引力坍缩总是会发生，至少对于振幅是三个数量级小于Choptuik的临界值，并对应于超过$10^5$的反射崩溃前。在小振幅极限下，崩溃时间与初始振幅的平方成反比。此外，我们发现，非共振谱领域崩溃早于在完全共振的情况下，这一结果是在与目前的理解过程。能量通过直接级联转移到高频率时，频谱是共振的，但我们观察到的非共振谱的直接和逆级联效应。我们的研究结果表明，一个完全共振的频谱可能不是一个关键的成分的湍流不稳定性和其他机制可能是相关的。我们讨论如何一个明确的答案，这个问题基本上是不可能的，在目前的框架。
英文摘要：
The discovery of a "weakly-turbulent" instability of anti-de Sitter spacetime supports the idea that confined fluctuations eventually collapse to black holes and suggests that similar phenomena might be possible in asymptotically-flat spacetime, for example in the context of spherically symmetric oscillations of stars or nonradial pulsations of ultracompact objects. Here we present a detailed study of the evolution of the Einstein-Klein-Gordon system in a cavity, with different types of deformations of the spectrum, including a mass term for the scalar and Neumann conditions at the boundary. We provide numerical evidence that gravitational collapse always occurs, at least for amplitudes that are three orders of magnitude smaller than Choptuik's critical value and corresponding to more than $10^5$ reflections before collapse. The collapse time scales as the inverse square of the initial amplitude in the small-amplitude limit. In addition, we find that fields with nonresonant spectrum collapse earlier than in the fully-resonant case, a result that is at odds with the current understanding of the process. Energy is transferred through a direct cascade to high frequencies when the spectrum is resonant, but we observe both direct- and inverse-cascade effects for nonresonant spectra. Our results indicate that a fully-resonant spectrum might not be a crucial ingredient of the conjectured turbulent instability and that other mechanisms might be relevant. We discuss how a definitive answer to this problem is essentially impossible within the present framework.
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