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

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.

25 pages, preprint YCTP-20-93, Revtex

References in corpus (1)

• Asymptotic Behavior of the $T^3\times R$ Gowdy Spacetimes

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• Asymptotic Behavior of Polarized and Half-Polarized U(1) symmetric Vacuum Spacetimes
• Strong cosmic censorship in vacuum space--times with compact, locally homogeneous Cauchy surfaces
• Numerical Evidence that the Singularity in Polarized U(1) Symmetric Cosmologies on $T^3\times R$ is Velocity Dominated
• Exact U(1) symmetric cosmologies with local Mixmaster dynamics
• Signature for local Mixmaster dynamics in U(1) symmetric cosmologies
• Adiabatic invariants and Mixmaster catastrophes
• Chern-Simons functional and the no-boundary proposal in Bianchi IX quantum cosmology
• On the Nature of the Generic Big Bang