Turning point principle for relativistic stars

TL;DR

This paper proves the turning point principle for relativistic stars, linking stability changes to extrema in mass-radius curves, and describes the linearized dynamics, showing the number of unstable modes increases with central redshift.

Contribution

It rigorously establishes the turning point principle for Einstein-Euler systems and details the linearized stability analysis for relativistic stars.

Findings

Spectral stability changes only at mass extrema.

Number of growing modes increases with redshift.

Provides a detailed linearized dynamics description.

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 [50] and Wheeler [22] 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.