Stability and Instability of the Sub-extremal Reissner–Nordström Black Hole Interior for the Einstein–Maxwell–Klein–Gordon Equations in Spherical Symmetry
Stability and Instability of the Sub-extremal Reissner–Nordström Black Hole Interior for the Einstein–Maxwell–Klein–Gordon Equations in Spherical Symmetry
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球对称性爱因斯坦-麦克斯韦-克莱因-戈登方程的次极值 Reissner-Nordström 黑洞内部的稳定性和不稳定性
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发表时间：
2017
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通讯作者：
M. Moortel
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作者：
M. Moortel
摘要我们展示了带电黑洞内部球对称性的非线性稳定性和不稳定性结果——足够快地接近亚极值赖斯纳-诺德斯特罗姆背景——在存在巨大带电标量场的情况下，受到该环境中强烈的宇宙审查猜想的推动：
1.稳定性我们证明，Einstein-Maxwell-Klein-Gordon 方程的球对称特征初始数据接近 Reissner-Nordström 背景，事件视界上具有充分衰减的多项式衰减率，从而产生在类时间无穷大邻域中具有柯西视界的时空。此外，如果衰变更强，我们证明时空度量允许连续延伸到柯西视界。这概括了 Dafermos 在球对称性中关于爱因斯坦-麦克斯韦实标量场著名的稳定性结果。 2.不稳定性 我们证明，对于稳定性部分中考虑的时空类，其标量场另外服从事件视界上的多项式平均 L2（一致）下界，标量场服从横向于柯西视界的积分下界。因此，我们证明了在穿过柯西视界的任何零曲面上，非简并能量是无限的，并且测地矢量场的曲率在柯西视界处接近时间无穷大时爆炸。这概括了球对称中爱因斯坦-麦克斯韦实标量场由于 Luk 和 Oh 导致的不稳定性结果。
黑洞内部的这种不稳定性也可以被视为解决单端渐近平坦初始数据的 C2 强宇宙审查猜想的一步。
AbstractWe show non-linear stability and instability results in spherical symmetry for the interior of a charged black hole—approaching a sub-extremal Reissner–Nordström background fast enough—in presence of a massive and charged scalar field, motivated by the strong cosmic censorship conjecture in that setting:
1.Stability We prove that spherically symmetric characteristic initial data to the Einstein–Maxwell–Klein–Gordon equations approaching a Reissner–Nordström background with a sufficiently decaying polynomial decay rate on the event horizon gives rise to a space–time possessing a Cauchy horizon in a neighbourhood of time-like infinity. Moreover, if the decay is even stronger, we prove that the space–time metric admits a continuous extension to the Cauchy horizon. This generalizes the celebrated stability result of Dafermos for Einstein–Maxwell-real-scalar-field in spherical symmetry.2.Instability We prove that for the class of space–times considered in the stability part, whose scalar field in addition obeys a polynomial averaged-L2 (consistent) lower bound on the event horizon, the scalar field obeys an integrated lower bound transversally to the Cauchy horizon. As a consequence we prove that the non-degenerate energy is infinite on any null surface crossing the Cauchy horizon and the curvature of a geodesic vector field blows up at the Cauchy horizon near time-like infinity. This generalizes an instability result due to Luk and Oh for Einstein–Maxwell-real-scalar-field in spherical symmetry.
This instability of the black hole interior can also be viewed as a step towards the resolution of the C2 strong cosmic censorship conjecture for one-ended asymptotically flat initial data.