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Instability of spherical naked singularities of a scalar field under gravitational perturbations
Journal of Differential Geometry ( IF 2.5 ) Pub Date : 2022-01-01 , DOI: 10.4310/jdg/1641413698
Junbin Li 1 , Jue Liu 2
Affiliation  

In this paper, we initiate the study of the instability of naked singularities without symmetries. In a series of papers, Christodoulou proved that naked singularities are not stable in the context of the spherically symmetric Einstein equations coupled with a massless scalar field. We study in this paper the next simplest case: a characteristic initial value problem of this coupled system with the initial data given on two intersecting null cones, the incoming one of which is assumed to be spherically symmetric and singular at its vertex, and the outgoing one of which has no symmetries. It is shown that, arbitrarily fixing the initial scalar field, the set of the initial conformal metrics on the outgoing null cone such that the maximal future development does not have any sequences of closed trapped surfaces approaching the singularity, is of first category in the whole space in which the shear tensors are continuous. Such a set can then be viewed as exceptional, although the exceptionality is weaker than the at least 1 co-dimensionality in spherical symmetry. Almost equivalently, it is also proved that, arbitrarily fixing an incoming null cone Cε to the future of the initial incoming null cone, the set of the initial conformal metrics such that the maximal future development has at least one closed trapped surface before Cε, contains an open and dense subset of the whole space. Since the initial scalar field can be chosen such that the singularity is naked if the initial shear is set to be zero, we may say that the spherical naked singularities of a self-gravitating scalar field are not stable under gravitational perturbations. This in particular gives new families of nonspherically symmetric gravitational perturbations different from the original spherically symmetric scalar perturbations given by Christodoulou.

中文翻译:

引力扰动下标量场球面裸奇点的不稳定性

在本文中,我们开始研究非对称裸奇点的不稳定性。在一系列论文中,Christodoulou 证明了在球对称爱因斯坦方程与无质量标量场耦合的情况下,裸奇点是不稳定的。我们在本文中研究了下一个最简单的情况:这个耦合系统的特征初始值问题,初始数据在两个相交的空锥上给出,其中一个假设在其顶点处是球对称和奇异的,而出其中之一没有对称性。结果表明,任意固定初始标量场,输出空锥上的初始保形度量集,使得最大的未来发展没有任何接近奇点的封闭捕获表面序列,是剪切张量连续的整个空间中的第一类。然后可以将这样的集合视为例外,尽管例外性弱于球对称中的至少 1 个共维性。几乎等价地,还证明了,将传入空锥 Cε 任意固定到初始传入空锥的未来,初始共形度量的集合使得最大未来发展在 Cε 之前具有至少一个封闭的俘获表面,包含整个空间的一个开放而密集的子集。由于可以选择初始标量场,使得如果初始剪切设置为零,奇点是裸奇点,我们可以说自引力标量场的球形裸奇点在引力扰动下是不稳定的。
更新日期:2022-01-01
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