Mixmaster Universe - Metric

Metric

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

where 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. This relationship is reminiscent of the commutation relations for the lie algebra of SU(2). The group manifold for SU(2) is the three-sphere, and indeed the describe the metric of an that is allowed to distort anisotropically thanks to the presence of the .

Next 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 .