论文标题
用于球形对称的爱因斯坦 - 标准场系统的大约自相似裸奇异点的结构
A construction of approximately self-similar naked singularities for the spherically symmetric Einstein-scalar field system
论文作者
论文摘要
在这项工作中，我们研究了一类裸奇异性空间的稳定性和不稳定性。球形对称的爱因斯坦 - 斯卡尔场野外系统中对裸奇异性形成的首次严格研究是归功于Christodoulou，他构建了一个家庭$（\ overline {g} _k，\ \ overlineDlinex_k）$ k $ -k $ -ssim-similar similar Solutions的$ k^2 \ in in in $ k^2 \ in（0）我们扩展了构造，以产生赤裸裸的空间内部和外部区域的示例，而本地建模为$（\ overline {g} _k，\ overlineDlinex_k）$，而无需确切的自我相似。 主要结果是在微调数据扰动下进行全球稳定性声明，用于满足自相似界限的一类裸奇异性空间。鉴于在爱因斯坦 - 赛车场模型中适当规则裸奇异的蓝光不稳定，我们需要在数据扰动上进行非大规模条件。特别是，标量字段沿着奇异点$ \ Mathcal {o} $的过去灯光扰动在$ \ MATHCAL {O} $附近消失。技术困难来自背景解决方案的奇异行为，以及在奇异性的轴线和过去的灯笼上的规律性考虑。内部区域是通过向后稳定性参数构建的，从而避免激活蓝光不稳定性。外观区域的扩展被视为$ \ Mathcal {O} $的未来的全球存在问题，Rodnianski的适应技术和Shlapentokh-Rothman在真空度上进行了。
In this work we investigate the stability and instability properties of a class of naked singularity spacetimes. The first rigorous study of naked singularity formation in the spherically symmetric Einstein-scalar field system was due to Christodoulou, who constructed a family $(\overline{g}_k, \overlineϕ_k)$ of $k$-self-similar solutions, for any $k^2 \in (0,\frac{1}{3})$. We extend the construction to produce examples of interior and exterior regions of naked singularity spacetimes locally modeled on the $(\overline{g}_k, \overlineϕ_k)$, without requiring exact self-similarity. The main result is a global stability statement under fine-tuned data perturbations, for a class of naked singularity spacetimes satisfying self-similar bounds. Given the well-known blueshift instability for suitably regular naked singularities in the Einstein-scalar field model, we require non-generic conditions on the data perturbations. In particular, the scalar field perturbation along the past lightcone of the singular point $\mathcal{O}$ vanishes to high order near $\mathcal{O}$. Technical difficulties arise from the singular behavior of the background solution, as well as regularity considerations at the axis and past lightcone of the singularity. The interior region is constructed via a backwards stability argument, thereby avoiding activating the blueshift instability. The extension to the exterior region is treated as a global existence problem to the future of $\mathcal{O}$, adapting techniques of Rodnianski and Shlapentokh-Rothman for vacuum spacetimes.
