Stability and Instability of Self-Gravitating Relativistic Matter Distributions

Mahir Hadžić,Zhiwu Lin,Gerhard Rein

doi:10.1007/s00205-021-01647-2

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 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.

Highlights

Introduction and Main ResultsWe consider a smooth spacetime manifold M equipped with a Lorentzian metric gαβ with signature (− + + +)
The formal linearization around the steady state leads to a Hamiltonian partial differential equation, which comes with a rich geometric structure
In the case of the Einstein–Vlasov system the steady state can be interpreted as a critical point of the so-called energy-Casimir functional [32,40]

Summary

Introduction

Introduction and Main ResultsWe consider a smooth spacetime manifold M equipped with a Lorentzian metric gαβ with signature (− + + +). The Einstein equations read Gαβ = 8π Tαβ , (1.1). Where Gαβ is the Einstein tensor induced by the metric, and Tαβ is the energymomentum tensor given by the matter content of the spacetime; Greek indices run from 0 to 3, and we choose units in which the speed of light and the gravitational constant are equal to 1. We consider two matter models, namely a collisionless gas as described by the collisionless Boltzmann or Vlasov equation and an ideal fluid as described by the Euler equations. This results in the Einstein– Vlasov and Einstein–Euler systems, respectively. We study these systems under the assumption that the spacetime is spherically symmetric and asymptotically flat, but we first formulate them in general, together with our main results

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