On "hard stars" in general relativity

TL;DR

This paper investigates static and dynamic properties of relativistic neutron star models called 'hard stars', demonstrating their stability, linearization, and existence of time-periodic solutions within Einstein-Euler equations.

Contribution

It establishes the existence and stability properties of 'hard star' solutions and analyzes their linearized dynamics, including time-periodic solutions.

Findings

Small stars are local minima of mass energy functional.

Energy boundedness is proven for linearized solutions.

Time-periodic solutions exist in the linearized system.

Abstract

We study spherically symmetric solutions to the Einstein-Euler equations which model an idealized relativistic neutron star surrounded by vacuum. These are barotropic fluids with a free boundary, governed by an equation of state which sets the speed of sound equal to the speed of light. We demonstrate the existence of a 1-parameter family of static solutions, or ''hard stars,'' and describe their stability properties: First, we show that small stars are a local minimum of the mass energy functional under variations which preserve the total number of particles. In particular, we prove that the second variation of the mass energy functional controls the ''mass aspect function.'' Second, we derive the linearisation of the Euler-Einstein system around small stars in ''comoving coordinates,'' and prove a uniform boundedness statement for an energy, which is exactly at the level of the variational argument. Finally, we exhibit the existence of time periodic solutions to the linearised system, which shows that energy boundedness is optimal for this problem.