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Homogeneous solutions to the Einstein-matter equations with a magnetic field and a conformal gauge singularity

Ho Lee, Ernesto Nungesser, John Stalker, Paul Tod

TL;DR

The paper analyzes homogeneous cosmologies with a magnetic field under conformal gauge singularity, focusing on massless Einstein equations coupled to either a radiation fluid or kinetic matter. It shows that no conformal gauge singular solutions exist with a radiation fluid, but establishes well-posed, unique solutions for the Einstein–Vlasov and Einstein–Boltzmann systems in Bianchi I symmetry for a range of soft potentials; it also derives asymptotic expansions near the initial boundary and, in the kinetic cases, differentiability for sufficiently small magnetic fields. The method hinges on conformal rescaling and casting the equations as a Fuchsian system, enabling precise control of the singular behavior at $ au=0$ and the propagation of constraints. The results extend prior massless analyses to include a magnetic field, providing sharp conditions on initial data that guarantee existence, uniqueness, and detailed asymptotics, with potential implications for early-universe cosmology in magnetized, highly relativistic regimes.

Abstract

We study massless solutions to the Einstein equations coupled to different matter models with a magnetic field and a conformal gauge singularity assuming spatial homogeneity with three commuting spatial translations. We show that there are no solutions in the case that the matter model is a radiation fluid. If the matter is described via kinetic theory we obtain that there exist unique solutions to the Einstein-Vlasov system and the Einstein-Boltzmann system for a certain range of soft potentials. For both the Vlasov and the Boltzmann case we also obtain asymptotic expansions close to the initial conformal gauge singularity.

Homogeneous solutions to the Einstein-matter equations with a magnetic field and a conformal gauge singularity

TL;DR

The paper analyzes homogeneous cosmologies with a magnetic field under conformal gauge singularity, focusing on massless Einstein equations coupled to either a radiation fluid or kinetic matter. It shows that no conformal gauge singular solutions exist with a radiation fluid, but establishes well-posed, unique solutions for the Einstein–Vlasov and Einstein–Boltzmann systems in Bianchi I symmetry for a range of soft potentials; it also derives asymptotic expansions near the initial boundary and, in the kinetic cases, differentiability for sufficiently small magnetic fields. The method hinges on conformal rescaling and casting the equations as a Fuchsian system, enabling precise control of the singular behavior at and the propagation of constraints. The results extend prior massless analyses to include a magnetic field, providing sharp conditions on initial data that guarantee existence, uniqueness, and detailed asymptotics, with potential implications for early-universe cosmology in magnetized, highly relativistic regimes.

Abstract

We study massless solutions to the Einstein equations coupled to different matter models with a magnetic field and a conformal gauge singularity assuming spatial homogeneity with three commuting spatial translations. We show that there are no solutions in the case that the matter model is a radiation fluid. If the matter is described via kinetic theory we obtain that there exist unique solutions to the Einstein-Vlasov system and the Einstein-Boltzmann system for a certain range of soft potentials. For both the Vlasov and the Boltzmann case we also obtain asymptotic expansions close to the initial conformal gauge singularity.
Paper Structure (19 sections, 6 theorems, 186 equations)

This paper contains 19 sections, 6 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. The equations
  3. The metric and curvature
  4. The various sources
  5. The radiation perfect fluid
  6. The source-free magnetic field
  7. The kinetic part
  8. New time coordinates and conformal rescaling
  9. Behaviour of the metric and curvature
  10. Behaviour of the source-terms
  11. The rescaled Einstein equations
  12. The Einstein constraint equations
  13. Main results
  14. The Einstein-Vlasov system with a magnetic field
  15. The differentiability condition
  16. ...and 4 more sections

Key Result

Theorem 1

Let $f_0 \geq 0$ be a smooth function with compact support in ${\mathbb R}^3 \setminus \{ 0 \}$ and $h \in {\mathbb R}^3$. Suppose that $f_0$ is not identically zero and satisfies the constraints mconstraint, tildeH. Then, there exists a unique Bianchi I solution $\tilde{a}_{ i j } , \tilde{k}_{ i j where $a _ { 0 i j }$ is a constant which only depends on $f_0$ and $h$. If in addition $a_{0ij}h^i

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Theorem 3
  • Proposition 2
  • Theorem 4