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 \(\kappa >0\). We prove their linear instability when \(\kappa \) is 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.

我们考虑了巨大的，渐近平坦的，球形对称的爱因斯坦-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。这样的稳态被嵌入到由其中心红移\（\ kappa> 0 \）参数化的一参数族中。当\（\ kappa \）时，我们证明了它们的线性不稳定性足够大，即当它们是高度相对论时，并证明不稳定性是由增长模式驱动的。我们的工作证实了Zel'dovich＆Podurets（针对爱因斯坦-弗拉索夫系统）和Harrison，Thorne，Wakano和Wheeler（针对Einstein-Euler系统）在1960年代提出的动态不稳定情形。我们的结果与相应的非相对论牛顿环境形成鲜明对比。我们对上述稳态周围的线性动力学进行了仔细的分析，并证明了指数三分法结果和稳定/不稳定不变空间的相应指数定理。最后，