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

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.Comment: 25 pages, preprint YCTP-20-93, Revte

Asymptotic Behavior of the T3×RT^3 \times R Gowdy Spacetimes

We present new evidence in support of the Penrose's strong cosmic censorship conjecture in the class of Gowdy spacetimes with $T^3$ spatial topology. Solving Einstein's equations perturbatively to all orders we show that asymptotically close to the boundary of the maximal Cauchy development the dominant term in the expansion gives rise to curvature singularity for almost all initial data. The dominant term, which we call the ``geodesic loop solution'', is a solution of the Einstein's equations with all space derivatives dropped. We also describe the extent to which our perturbative results can be rigorously justified.Comment: 30 page