Abstract
Upon specifying an equation of state, spherically symmetric steady states of the Einstein-Euler system are embedded in 1-parameter families of solutions, characterized by the value of their central redshift. In the 1960’s Zel’dovich (Voprosy Kosmogonii 9:157–170, 1963) and Harrison et al. (Gravitation Theory and Gravitational Collapse. The University of Chicago press, Chicago, 1965) formulated a turning point principle which states that the spectral stability can be exchanged to instability and vice versa only at the extrema of mass along the mass-radius curve. Moreover the bending orientation at the extrema determines whether a growing mode is gained or lost. We prove the turning point principle and provide a detailed description of the linearized dynamics. One of the corollaries of our result is that the number of growing modes grows to infinity as the central redshift increases to infinity.
Highlights
In our previous work [18] jointly with Rein, among other things we introduced the so-called separable Hamiltonian formulation of the linearized Einstein–Euler system, which highlights the symplectic structure in the problem
We look for compactly supported steady states of the EE-system (1.7)–(1.12) satisfying u = 0
At κmax the so-called fractional binding energy has a maximum and for κ > κmax the equilibria are dynamically unstable. This stability scenario is very different from the mass-radius turning point principle that we prove in Theorem 1.13, as in the case of stars stability can in principle be exchanged to instability, and back to stability [22,29], see Fig. 1
Summary
Where we recall the definition of α (1.51). Our goal is to derive an equation for yκ in the regime 0 < κ 1. Recall the definitions of g (1.20) and g0 (1.50) It follows from (2.13) that there exists a sufficiently small κ0 such that g(y) = k−α γ −1 γ α yα + Oy→0+ (yα+1) = g0(y) + Oy→0+ (yα+1), y ∈ [0, κ0],. (The small redshift limit) There exist κ∗, C > 0 such that for all 0 ≤ κ < κ∗ the following bound holds: yκ − y0 C1([0,∞) ≤ C κ,. A simple corollary of Lemma 2.2 are the following a priori bounds. It follows that the operator Sκ has finite dimensional eigenspaces for negative and zero eigenvalues, and Sκ is uniformly positive on the complement space. From the definition (1.41) it follows that there exists a constant C (κ) such that e−μκ −3λκ r 2 d 2r μκ + 1 dr eμκ +λκ d dκ yκ.
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