It has long been suggested that the Cauchy horizon of dynamical black holes is subject to a weak null singularity, under the mass inflation scenario. We study in spherical symmetry the Einstein–Maxwell–Klein–Gordon equations and while we do not directly show mass inflation , we obtain a “mass inflation/ridigity” dichotomy. More precisely, we prove assuming (sufficiently slow) decay of the charged scalar field on the event horizon, that the Cauchy horizon emanating from time-like infinity $$\mathcal {CH}_{i^+}$$ CH i + can be partitioned as $$\mathcal {CH}_{i^+}= {\mathcal {D}} \cup {\mathcal {S}}$$ CH i + = D ∪ S for two (possibly empty) disjoint connected sets $${\mathcal {D}}$$ D and $${\mathcal {S}}$$ S such that $${\mathcal {D}}$$ D (the dynamical set) is a future set on which the Hawking mass blows up (mass inflation scenario). $${\mathcal {S}}$$ S (the static set) is a past set isometric to a Reissner–Nordström Cauchy horizon i.e. the radiation is zero on $${\mathcal {S}}$$ S . As a consequence of this result, we prove that the entire Cauchy horizon $$\mathcal {CH}_{i^+}$$ CH i + is globally $$\underline{C^2-{ inextendible}}$$ C 2 - inextendible ̲ , extending a previous local result established by the author. To this end, we establish a novel classification of Cauchy horizons into three types: dynamical ( $${\mathcal {S}}=\emptyset $$ S = ∅ ), static ( $${\mathcal {D}}=\emptyset $$ D = ∅ ) or mixed. As a side benefit, we prove that there exists a trapped neighborhood of the Cauchy horizon, thus the apparent horizon cannot cross the Cauchy horizon, which is a result of independent interest. Our main motivation is to prove the $$C^2$$ C 2 Strong Cosmic Censorship Conjecture for a realistic model of spherical collapse in which charged matter emulates the repulsive role of angular momentum. In our case, this model is the Einstein–Maxwell–Klein–Gordon system on space-times with one asymptotically flat end. As a consequence of the $$C^2$$ C 2 -inextendibility of the Cauchy horizon, we prove the following statements, in spherical symmetry: 1. Two-ended asymptotically flat space-times are $$C^2$$ C 2 -future-inextendible i.e. $$C^2$$ C 2 Strong Cosmic Censorship is true for Einstein–Maxwell–Klein–Gordon, assuming the decay of the scalar field on the event horizon at the expected rate. 2. In the one-ended case, under the same assumptions, the Cauchy horizon emanating from time-like infinity is $$C^2$$ C 2 -inextendible. This result suppresses the main obstruction to $$C^2$$ C 2 Strong Cosmic Censorship in spherical collapse. The remaining obstruction in the one-ended case is associated to “locally naked” singularities emanating from the center of symmetry, a phenomenon which is also related to the Weak Cosmic Censorship Conjecture.

中文翻译：

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质量膨胀和球对称带电标量场动力学黑洞的 $$C^2$$-不可扩展性

长期以来，人们一直认为，在大规模暴胀情景下，动力学黑洞的柯西视界会受到弱零奇点的影响。我们以球对称研究爱因斯坦-麦克斯韦-克莱因-戈登方程，虽然我们没有直接显示质量膨胀，但我们获得了“质量膨胀/精确”二分法。更准确地说，我们证明假设事件视界上带电标量场的衰减（足够慢），从类似时间的无穷大 $$\mathcal {CH}_{i^+}$$ CH i + 发出的柯西视界可以被划分为 $$\mathcal {CH}_{i^+}= {\mathcal {D}} \cup {\mathcal {S}}$$ CH i + = D ∪ S 对于两个（可能是空的）不相交的连接设置 $${\mathcal {D}}$$ D 和 $${\mathcal {S}}$$ S 使得 $${\mathcal {D}}$$ D（动态集）是霍金质量爆炸（大规模通货膨胀情景）。$${\mathcal {S}}$$ S（静态集）是与 Reissner-Nordström Cauchy 视界等距的过去集，即 $${\mathcal {S}}$$ S 上的辐射为零。作为这个结果的结果，我们证明整个柯西视界 $$\mathcal {CH}_{i^+}$$ CH i + 是全局的 $$\underline{C^2-{ inextendible}}$$ C 2 - 不可扩展 ̲ ，扩展了作者建立的先前局部结果。为此，我们将柯西视界分为三种类型：动态的（ $${\mathcal {S}}=\emptyset $$ S = ∅ ）、静态的（ $${\mathcal {D}}=\空集 $$ D = ∅ ) 或混合。作为附带的好处，我们证明了柯西视界存在一个被困邻域，因此视界不能跨越柯西视界，这是独立兴趣的结果。我们的主要动机是证明 $$C^2$$ C 2 强宇宙审查猜想，用于球形坍缩的现实模型，其中带电物质模拟角动量的排斥作用。在我们的例子中，这个模型是时空上的爱因斯坦-麦克斯韦-克莱因-戈登系统，具有一个渐近平端。由于柯西视界的 $$C^2$$ C 2 不可延展性，我们证明了以下球对称陈述： 1. 两端渐近平坦的时空是 $$C^2$$ C 2 -future-inextendible ie $$C^2$$ C 2 强宇宙审查对爱因斯坦-麦克斯韦-克莱因-戈登来说是正确的，假设事件视界上的标量场以预期的速率衰减。2. 在一端的情况下，在相同的假设下，从类似时间的无穷大发出的柯西视界是 $$C^2$$C 2 -不可扩展的。这一结果抑制了对球面坍缩中的 $$C^2$$ C 2 强宇宙审查的主要障碍。单端情况下的剩余障碍与从对称中心发出的“局部裸”奇点有关，这一现象也与弱宇宙审查猜想有关。