The Mixmaster universe (named after Sunbeam Mixmaster, a brand of Sunbeam Products electric kitchen mixer)[1] 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.
[2] 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
can only expand or contract: the only dynamical parameter is overall size of the
Values of the shape parameters describe distortions of the
that preserve its volume and also maintain a constant Ricci curvature scalar.
assume different values, homogeneity but not isotropy is preserved.
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
form a left-invariant co-frame on the Lie group SU(2), which is diffeomorphic to the 3-sphere
As the metric evolves via Einstein's equation, the geometry of this
which measure the volume of spatial slices, as well as "shape parameters"
, there should only be two free functions, which Misner chooses to be
, defined as The evolution of the universe is then described by finding
Misner hoped that the chaos would churn up and smooth out the early universe.
Also, during periods in which one direction was static (e.g., going from expansion to contraction) formally the Hubble horizon
in that direction is infinite, which he suggested meant that the horizon problem could be solved.
While an interesting example of gravitational chaos, it is widely recognized that the cosmological problems the Mixmaster universe attempts to solve are more elegantly tackled by cosmic inflation.