Mixmaster Universe Research Articles

Scalar Waves in the Mixmaster Universe. I. The Helmholtz Equation in a Fixed Background

Solutions to the scalar wave equation in a fixed mixmaster background are presented. Separability conditions on the Laplace-Beltrami equation are related to the complete sets of invariant operators of the symmetry group S${\mathrm{O}}_{3}$. Solutions to the Helmholtz equation are equivalent to the quantum-mechanical problem of asymmetric rotors. The wave functions in the mixmaster space are the asymmetric-top wave functions. In Euler angle variables, they are expanded in terms of the symmetric-top wave functions (the wave functions in Taub space) possessing definite ($J, K, M$) symmetries, with a coupling in the intrinsic magnetic quantum number $K$; the case with $M=0$ can be expressed as a product of the Lam\'e functions in elliptic coordinates. Invariance of the mixmaster space under the four - group characterizes the wave functions into four symmetry species and causes the factorization of the energy matrix into four submatrices. Eigenvalues and expansion coefficients for the wave functions are calculated for some low-lying levels.

High-Frequency Sound Waves to Eliminate a Horizon in the Mixmaster Universe

From the linear wave equation for small-amplitude sound waves in a curved space-time, there is derived a geodesiclike differential equation for sound rays to describe the motion of wave packets. These equations are applied in the generic, nonrotating, homogeneous closed-model universe (the "mixmaster universe," Bianchi type IX). As for light rays described by Doroshkevich and Novikov (DN), these sound rays can circumnavigate the universe near the singularity to remove particle horizons only for a small class of these models and in special directions. Although these results parallel those of DN, different Hamiltonian methods are used for treating the Einstein equations.

Perturbations on the Mixmaster Universe

Mathematical formalisms for the separation and solution of the tensor perturbation equations in an empty, diagonal, type-IX space is developed, based upon group-symmetry properties of homogeneous spaces. Numerical results in sampling solutions of the "mixmaster universe" show damping amplitudes of perturbations as the universe expands, a behavior in qualitative accordance with earlier results on the Friedmann universe.

Rotation does not enhance mixing in the mixmaster universe

We investigate closed rotating cosmologies to determine if rotation leads to an enhancement of causal mixing proposed by Misner to guarantee the homogeneity of such models. We conclude that rotation cannot lead to significantly more efficient mixing than occurs in non-rotating models. Since arguments presented by Doroshkevich and Novikov and calculations made by Chitre give very small probability of mixing in nonrotating models, we therefore conclude that a plausible explanation of the homogeneity of the universe cannot be found within the framework of classical General Relativity. Such an explanation may lie in quantum effects on mixing near the singularity.

Mixmaster Universe Problem

IN recent months considerable interest has been aroused by a suggestion1 concerning certain general-relativistic spatially-homogeneous cosmological models of Bianchi type IX1,2. Misner1,3 has considered the simplest cases, in which matter flows orthogonally to the surfaces of homogeneity and in which the matter is dynamically insignificant near the (initial) singularity. I shall refer to these as simple Bianchi IX models. In general, according to Misner, these models had no particle horizons. Because information could be communicated from any observer to any other before any given time, Misner called such a model the “Mixmaster Universe”.

Action Functional of General Relativity as a Path Length. I. Closed Empty Universes

For the case of closed empty universes it is established (up to a uniqueness conjecture) that the action functional of general relativity is a Riemannian path length in superspace, an infinite-dimensional manifold whose points represent three-dimensional geometries. Each procedure for representing spacetimes by trajectories in superspace yields a corresponding superspace metric. These metrics are just the "geodesic sheaf" metrics found by DeWitt. A general expression for all of these metrics is obtained in terms of arbitrary coordinates on superspace. The expression requires an explicit and unique solution to the spacelike constraints (${G}^{0i}=0$, $i=1, 2, 3$) of general relativity. Supertrajectories are separated into "timelike" ones which are permitted because they have real actions and "spacelike" ones which are forbidden because their actions are imaginary. By varying the path-length action, one finds that solutions of Einstein's sourcefree field equations correspond to a class of timelike geodesics in superspace. This class of geodesics is subject to conserved constraints which are homogeneous and quadratic in time derivatives. The truncated superspace which contains the empty "mixmaster" universes studied by Misner is discussed as an application of the supergeometric approach. The approach is particularly useful for analyzing the maximum expansion stage of the universe. Universes at this stage of their evolution encounter a real singularity of superspace. Time-symmetric universes end their supertrajectories on this singularity, while all other universes penetrate it in a well-defined way. The singularity, together with the causal structure of superspace, limits the anisotropy of empty mixmaster universes which are a finite number of $e$-foldings from the stage of maximum expansion.