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 can only expand or contract: the only dynamical parameter, parameterized by the scale factor that preserve its volume and also maintain a constant Ricci curvature 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.

Metric

The metric studied by Misner (very slightly modified from his notation) is given by,

where

, considered as differential forms, are defined by

. These satisfy

is the exterior derivative the wedge product 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 , by

1. .

, defined as

.

Applications to cosmology

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.

Applications to black holes

Before the 1970s, most physicists believed that the interior of a Schwarzschild black hole curved inwards towards a sharp point at the singularity. However, in the late 1960s, Soviet physicists Vladimir Belinskii, Isaak Khalatnikov, and Evgeny Lifshitz discovered that this model was only true when the spacetime inside the black hole had not been perturbed. Any perturbations, such as those caused by matter or radiation falling in, would cause space to oscillate chaotically near the singularity according to Mixmaster dynamics. Any matter falling in would experience intense tidal forces rapidly changing in direction, all while being compressed into an increasingly small volume.

See also

• Bianchi classification

• BKL singularity

References