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 1969 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.
Discussion The model is similar to the closed Friedmann–Lemaître–Robertson–Walker universe, in that spatial slices are positively curved and are topologically three-spheres S 3 {\displaystyle S^{3}} . However, in the FRW universe, the S 3 {\displaystyle S^{3}} can only expand or contract: the only dynamical parameter is overall size of the S 3 {\displaystyle S^{3}} , parameterized by the scale factor a ( t ) {\displaystyle a(t)} . In the Mixmaster universe, the S 3 {\displaystyle S^{3}} can expand or contract, but also distort anisotropically. Its evolution is described by a scale factor a ( t ) {\displaystyle a(t)} as well as by two shape parameters β ± ( t ) {\displaystyle \beta _{\pm }(t)} . Values of the shape parameters describe distortions of the S 3 {\displaystyle S^{3}} that preserve its volume and also maintain a constant Ricci curvature scalar. Therefore, as the three parameters a , β ± {\displaystyle a,\beta _{\pm }} assume different values, homogeneity but not isotropy is preserved. The model has a rich dynamical structure. Misner showed that the shape parameters β ± ( t ) {\displaystyle \beta _{\pm }(t)} 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.
Metric The metric studied by Misner (very slightly modified from his notation) is given by,
d s 2 = − d t 2 + ∑ k = 1 3 L k 2 ( t ) σ k ⊗ σ k {\displaystyle {\text{d}}s^{2}=-{\text{d}}t^{2}+\sum _{k=1}^{3}{L_{k}^{2}(t)}\sigma _{k}\otimes \sigma _{k}}
where
L k = R ( t ) e β k {\displaystyle L_{k}=R(t)e^{\beta _{k}}}
and the σ k {\displaystyle \sigma _{k}} , considered as differential forms, are defined by
σ 1 = sin ψ d θ − cos ψ sin θ d ϕ {\displaystyle \sigma _{1}=\sin \psi {\text{d}}\theta -\cos \psi \sin \theta {\text{d}}\phi }
σ 2 = cos ψ d θ + sin ψ sin θ d ϕ {\displaystyle \sigma _{2}=\cos \psi {\text{d}}\theta +\sin \psi \sin \theta {\text{d}}\phi }
σ 3 = − d ψ − cos θ d ϕ {\displaystyle \sigma _{3}=-{\text{d}}\psi -\cos \theta {\text{d}}\phi }
In terms of the coordinates ( θ , ψ , ϕ ) {\displaystyle (\theta ,\psi ,\phi )} . These satisfy
d σ i = 1 2 ϵ i j k σ j ∧ σ k {\displaystyle {\text{d}}\sigma _{i}={\frac {1}{2}}\epsilon _{ijk}\sigma _{j}\wedge \sigma _{k}}
… excerpt ends here. Continue reading the full article.
