Stability and instability of self-gravitating relativistic matter distributions

Abstract

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 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.

93 pages; several proofs are revised and some previous errors corrected. In particular, Lemma 4.4, which is an important tool in the proof of Theorem 4.3 but was in part incorrect, also in the published version, has been corrected

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• Stability and instability results for equilibria of a (relativistic) self-gravitating collisionless gas -- A review
• Static solutions to the spherically symmetric Einstein-Vlasov system: a particle-number-Casimir approach
• EVStabilityNet: Predicting the Stability of Star Clusters in General Relativity
• A Birman-Schwinger Principle in General Relativity: Linearly Stable Shells of Collisionless Matter Surrounding a Black Hole
• Einstein-Vlasov system with equal-angular momenta in ${\text{AdS}}_5$
• Numerical experiments on stationary, oscillating, and damped spherical galaxy models