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

Florian Beyer

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.

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

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.
Paper Structure (19 sections, 4 theorems, 202 equations)

This paper contains 19 sections, 4 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Informal discussion of results in this paper
  4. Organisation of the paper
  5. Notation, conventions and background
  6. Background spacetimes with Kasner big bang asymptotics
  7. Analysis background
  8. The Euler equations for fluid solutions near the big bang
  9. Frauendiener-Walton formulation of the Euler equations and its $N+1$-decomposition
  10. Convenient variables and heuristics
  11. The asymptotically extremely-tilted fluid regime near the big bang
  12. The first main theorem
  13. Euler PDE systems for the asymptotically extremely-tilted case
  14. The main Fuchsian Euler PDE system and its local Cauchy problem
  15. The proof of Theorem \ref{['thm:Euler1']}
  16. ...and 4 more sections

Key Result

Theorem 1.1

Pick a background spacetime with Kasner big bang asymptotics in the sense of Definition def:asympKasner (which satisfies additional conditions below). There exists a constant $c_*^2\in [0,P)$ so that, given any speed of sound parameter $c_s^2$ in the range all solutions of the Euler equations launched from spacelike hypersurfaces whose mean curvature $H$ is sufficiently large exist globally towar

Theorems & Definitions (7)

  • Theorem 1.1: Informal version of Theorems \ref{['thm:Euler1']} and \ref{['thm:Euler2']}
  • Definition 2.1: Spacetimes with Kasner big bang asymptotics
  • Example 2.2: Kasner-scalar field spacetimes
  • Theorem 4.1
  • Proposition 4.2
  • proof : Proof of Proposition \ref{['prop:Fuchsiansystem']}
  • Theorem 5.1