The Mixmaster Spacetime, Geroch's Transformation and Constants of Motion

TL;DR

This paper investigates a constant of motion related to Geroch's transformation in U(1)-symmetric spacetimes, showing its positivity in most cases and exploring its implications for the Mixmaster universe's dynamics.

Contribution

It demonstrates the positivity of a Geroch-related constant of motion in U(1)-symmetric initial data and explores its role in generating new solutions with complex dynamics.

Findings

The constant of motion is strictly positive in an open subset of initial data.

Perturbative calculation shows the constant is proportional to the Hamiltonian for Mixmaster spacetime.

Applying Geroch's transformation under the zero constant assumption yields a new, non-homogeneous solution with Mixmaster-like behavior.

Abstract

We show that for $U(1)$-symmetric spacetimes on $S^3 \times R$ a constant of motion associated with the well known Geroch transformation, a functional $K[h_{ij},\pi^{ij}]$, quadratic in gravitational momenta, is strictly positive in an open subset of the set of all $U(1)$-symmetric initial data, and therefore not weakly zero. The Mixmaster initial data appear to be on the boundary of that set. We calculate the constant of motion perturbatively for the Mixmaster spacetime and find it to be proportional to the minisuperspace Hamiltonian to the first order in the Misner anisotropy variables, i.e. weakly zero. Assuming that $K$ is exactly zero for the Mixmaster spacetime, we show that Geroch's transformation, when applied to the Mixmaster spacetime, gives a new \mbox{$U(1)$-symmetric} solution of the vacuum Einstein equations, globally defined on \mbox{$S^2 \times S^1 \times R$},which is non-homogeneous and presumably exhibits Mixmaster-like complicated dynamical behavior.