The Stabilizing Effect of Spacetime Expansion on Relativistic Fluids With Sharp Results for the Radiation Equation of State

TL;DR

This paper demonstrates that spacetime expansion stabilizes relativistic fluids with certain equations of state, ensuring global existence of solutions, while showing instability and shock formation when specific conditions are not met, especially for the radiation case.

Contribution

It extends stability results of relativistic fluids on expanding spacetimes to more general scale factors and identifies conditions leading to shock formation in the radiation case.

Findings

Spacetime expansion stabilizes relativistic fluid solutions.

Explicit solutions remain close to perturbed initial data under expansion.

Failure of integrability condition leads to shock formation in the radiation case.

Abstract

In this article, we study the 1 + 3 dimensional relativistic Euler equations on a pre-specified conformally flat expanding spacetime background with spatial slices that are diffeomorphic to $\mathbb{R}^3.$ We assume that the fluid verifies the equation of state $p = c_s^2 \rho,$ where $0 \leq c_s \leq \sqrt{1/3}$ is the speed of sound. We also assume that the inverse of the scale factor associated to the expanding spacetime metric verifies a $c_s-$dependent time-integrability condition. Under these assumptions, we use the vectorfield energy method to prove that an explicit family of physically motivated, spatially homogeneous, and spatially isotropic fluid solutions is globally future-stable under small perturbations of their initial conditions. The explicit solutions corresponding to each scale factor are analogs of the well-known spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker family. Our nonlinear analysis, which exploits dissipative terms generated by the expansion, shows that the perturbed solutions exist for all future times and remain close to the explicit solutions. This work is an extension of previous results, which showed that an analogous stability result holds when the spacetime is exponentially expanding. In the case of the radiation equation of state $p = (1/3)\rho,$ we also show that if the time-integrability condition for the inverse of the scale factor fails to hold, then the explicit fluid solutions are unstable. More precisely, we show that arbitrarily small smooth perturbations of the explicit solutions' initial conditions can launch perturbed solutions that form shocks in finite time. The shock formation proof is based on the conformal invariance of the relativistic Euler equations when $c_s^2 = 1/3,$ which allows for a reduction to a well-known result of Christodoulou.