Stable space-like singularity formation for axi-symmetric and polarized near-Schwarzschild black hole interiors
Stable space-like singularity formation for axi-symmetric and polarized near-Schwarzschild black hole interiors
复制标题
轴对称和偏振近史瓦西黑洞内部的稳定类空间奇点形成
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
G. Fournodavlos
中科院分区：
文献类型：
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作者：
S. Alexakis;G. Fournodavlos
我们显示的史瓦西奇点（黑洞区域内）的爱因斯坦真空方程的稳定性结果。在极化轴对称类中，在超曲面${r=e}$，$e<<2 M $上诱导的Schwarzschild数据扰动下，证明了这个结果.我们的结果只是部分的稳定性结果，因为我们表明，虽然（类空间）奇点持续扰动下如上所述，度量接近奇点的行为是更多的参与比史瓦西解决方案。事实上，我们发现，该解决方案显示渐近速度项占主导地位的动力学和方法不同的卡斯纳解决方案在每个点的奇点。这些卡斯纳型渐近性与各向同性相距甚远，因为（如史瓦西）有两个收缩方向和一个扩张方向。我们的证明依赖于能量方法和一种新的方法，在轴对称的EVE，我们相信有更广泛的适用性：在这个对称类和适当的测地线规范下，EVE可以作为一个自由波耦合到（非线性）常微分方程，耦合的几何投影，2+1时空的自由波进行研究。爱因斯坦方程的非线性部分是由常微分方程描述的，这一事实是人们如何克服奇点所表现出的某种线性不稳定性的核心。
We show a stability result for the Schwarzschild singularity (inside the black hole region) for the Einstein vacuum equations. The result is proven in the class of polarized axial symmetry, under perturbations of the Schwarzschild data induced on a hypersurface ${r=e}$, $e<<2M$. Our result is only partly a stability result, in that we show that while a (space-like) singularity persists under perturbations as above, the behaviour of the metric approaching the singularity is much more involved than for the Schwarzschild solution. Indeed, we find that the solution displays asymptocially-velocity-term-dominated dynamics and approaches a different Kasner solution at each point of the singularity. These Kasner-type asymptotics are very far from isotropic, since (as in Schwarzschild) there are two contracting directions and one expanding one. Our proof relies on energy methods and on a new approach to the EVE in axial symmetry, which we believe has wider applicability: In this symmetry class and under a suitable geodesic gauge, the EVE can be studied as a free wave coupled to (nonlinear) ODEs, which couple the geometry of the projected, 2+1 space-time to the free wave. The fact that the nonlinear part of the Einstein equations is described by ODEs lies at the heart of how one can overcome a certain linear instability exhibited by the singularity.