Stability and Instability of Self-Gravitating Relativistic Matter Distributions
Stability and Instability of Self-Gravitating Relativistic Matter Distributions
复制标题
自引力相对论物质分布的稳定性和不稳定性
DOI：
10.1007/s00205-021-01647-2
复制
发表时间：
2021
影响因子：
2.5
通讯作者：
Rein, Gerhard
中科院分区：
文献类型：
--
作者：
Hadžić, Mahir;Lin, Zhiwu;Rein, Gerhard
我们考虑大质量、渐近平坦、球对称爱因斯坦-弗拉索夫系统的稳态解，即星系或球状星团的相对论模型，以及爱因斯坦-欧拉系统的稳态解，即恒星的相对论模型。这种稳态被嵌入到由其中心红移参数化的单参数族中。当足够大时，即当它们具有强相对论性时，我们证明了它们的线性不稳定性，并证明了不稳定性是由增长模式驱动的。我们的工作证实了 Zel'dovich 和 Podurets（针对爱因斯坦-弗拉索夫系统）以及 Harrison、Thorne、Wakano 和 Wheeler（针对爱因斯坦-欧拉系统）在 20 世纪 60 年代提出的动态不稳定情景。我们的结果与相应的非相对论牛顿设置形成鲜明对比。我们对上述稳态周围的线性动力学进行了仔细分析，并证明了稳定/不稳定不变空间的指数三分结果和相应的指数定理。最后，在爱因斯坦-欧拉系统的情况下，我们证明了转折点原理的严格版本，该原理将沿单参数族的稳态稳定性与所谓的质量半径曲线的缠绕点联系起来。
We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein–Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein–Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parameterized by their central redshift. We prove their linear instability whenis sufficiently large, i.e., when they are strongly relativistic, and prove that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zel’dovich & Podurets (for the Einstein–Vlasov system) and by Harrison, Thorne, Wakano, & Wheeler (for the Einstein–Euler system). Our results are in sharp contrast to the corresponding non-relativistic, Newtonian setting. We carry out a careful analysis of the linearized dynamics around the above steady states and prove an exponential trichotomy result and the corresponding index theorems for the stable/unstable invariant spaces. Finally, in the case of the Einstein–Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve.