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Mixmaster universe

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 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.

DiscussionEdit

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.

MetricEdit

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 general left-invariant metric. As the metric evolves via Einstein's equation, the geometry of this   typically distorts anisotropically. Misner defines parameters   and   which measure the volume of spatial slices, as well as "shape parameters"  , by

 .


Since there is one condition on the three  , there should only be two free functions, which Misner chooses to be  , defined as

 

The evolution of the universe is then described by finding   as functions of  .

Applications to cosmologyEdit

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. Since the directions of expansion and contraction varied, presumably given enough time the horizon problem would get solved in every direction.

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. The metric Misner studied is also known as the Bianchi type IX metric.

See alsoEdit

ReferencesEdit

  1. ^ Barry R. Parker, Chaos in the Cosmos: The Stunning Complexity of the Universe, Springer, 2013, p. 257.
  2. ^ Charles W. Misner, "Mixmaster Universe", Physical Review Letters, Vol. 22, Issue 20 (May 1969), pp. 1071-1074, doi:10.1103/PhysRevLett.22.1071, Bibcode1969PhRvL..22.1071M. Mirror link. Also available as an entry in the Gravity Research Foundation's 1969 essay competition. Mirror link.