Exact U(1) symmetric cosmologies with local Mixmaster dynamics

TL;DR

This paper constructs explicit inhomogeneous vacuum cosmological solutions with U(1) symmetry that exhibit BKL oscillatory behavior, extending previous numerical results through solution-generating techniques applied to Mixmaster models.

Contribution

It introduces a method to generate exact, spatially inhomogeneous solutions with U(1) symmetry from Mixmaster solutions, demonstrating BKL oscillations analytically.

Findings

Solutions exhibit BKL oscillatory behavior similar to numerical simulations.

Explicit construction of inhomogeneous solutions from homogeneous Mixmaster models.

Behavior aligns qualitatively with previous numerical studies of U(1) symmetric spacetimes.

Abstract

By applying a standard solution generating technique, we transform an arbitrary vacuum Mixmaster solution on $S^3 \times {\bf R}$ to a new solution which is spatially inhomogeneous. We thereby obtain a family of exact, spatially inhomogeneous, vacuum spacetimes which exhibit Belinskii, Khalatnikov, and Lifshitz (BKL) oscillatory behavior. The solutions are constructed explicitly by performing the transformations on numerically generated, homogeneous Mixmaster solutions. Their behavior is found to be qualitatively like that seen in previous numerical simulations of generic U(1) symmetric cosmological spacetimes on $T^3 \times {\bf R}$.