The extremely-tilted fluid regime near asymptotically Kasner big bang singularities

TL;DR

This paper analyzes the behavior of relativistic fluids near Kasner big bang singularities, demonstrating global existence and expected asymptotics in the extremely-tilted regime with small sound speed, without symmetry or small data assumptions.

Contribution

It extends previous work by solving the Euler equations in the extremely-tilted regime, establishing global existence and asymptotic behavior without symmetry or smallness constraints.

Findings

Solutions exist globally towards the singularity.

Fluid particles approach the speed of light in the dominant Kasner direction.

Asymptotic behavior matches heuristic predictions.

Abstract

In this paper, we solve the relativistic Euler equations with a linear barotropic equation of state on a large class of background spacetimes with Kasner big bang asymptotics. Building on previous work in the asymptotically non-tilted regime, which applies when the speed of sound of the fluid is large in comparison to the Kasner exponents, we now consider the asymptotically extremely-tilted regime when the speed of sound is small. We solve the Cauchy problem for the Euler equations towards the big bang singularity and prove, without any symmetry assumptions or smallness conditions on the Cauchy data, that the solutions exist globally in time provided the mean curvature of the initial hypersurface is sufficiently large. Finally, we prove that the solutions exhibit the asymptotics expected from standard heuristic arguments in this regime; in particular, fluid particles are driven towards the speed of light in the direction of the largest Kasner exponent as the big bang singularity is approached.