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Concept
Mixmaster universe
Summary
The Mixmaster universe (named after Sunbeam Mixmaster, a brand of Sunbeam Products electric kitchen mixer) is a solution to Einstein field equations of general relativity studied by Charles Misner in an effort to better understand the dynamics of the early universe. He hoped to solve the horizon problem in a natural way by showing that the early universe underwent an oscillatory, chaotic epoch.
The model is similar to the closed Friedmann–Lemaître–Robertson–Walker universe, in that spatial slices are positively curved and are topologically three-spheres . However, in the FRW universe, the can only expand or contract: the only dynamical parameter is overall size of the , parameterized by the scale factor . In the Mixmaster universe, the can expand or contract, but also distort anisotropically. Its evolution is described by a scale factor as well as by two shape parameters . Values of the shape parameters describe distortions of the that preserve its volume and also maintain a constant Ricci curvature scalar. Therefore, as the three parameters assume different values, homogeneity but not isotropy is preserved.
The model has a rich dynamical structure. Misner showed that the shape parameters act like the coordinates of a point mass moving in a triangular potential with steeply rising walls with friction. By studying the motion of this point, Misner showed that the physical universe would expand in some directions and contract in others, with the directions of expansion and contraction changing repeatedly. Because the potential is roughly triangular, Misner suggested that the evolution is chaotic.
The metric studied by Misner (very slightly modified from his notation) is given by,
where
and the , considered as differential forms, are defined by
In terms of the coordinates . These satisfy
where is the exterior derivative and the wedge product of differential forms. The 1-forms form a left-invariant co-frame on the Lie group SU(2), which is diffeomorphic to the 3-sphere , so the spatial metric in Misner's model can concisely be described as just a left-invariant metric on the 3-sphere; indeed, up to the adjoint action of SU(2), this is actually the left-invariant metric.
Official source
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