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 that the instability is driven by a growing mode. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zeldovich & 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.
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