Kasner metric

The Kasner metric (developed by and named for the American mathematician Edward Kasner in 1921) is an exact solution to Albert Einstein's theory of general relativity. It describes an anisotropic universe without matter (i.e., it is a vacuum solution). It can be written in any spacetime dimension formula_1 and has strong connections with the study of gravitational chaos.
Metric and conditions.
The metric in formula_1 spacetime dimensions is
and contains formula_4 constants formula_5, called the "Kasner exponents." The metric describes a spacetime whose equal-time slices are spatially flat, however space is expanding or contracting at different rates in different directions, depending on the values of the formula_5. Test particles in this metric whose comoving coordinate differs by formula_7 are separated by a physical distance formula_8.
The Kasner metric is an exact solution to Einstein's equations in vacuum when the Kasner exponents satisfy the following "Kasner conditions,"
The first condition defines a plane, the "Kasner plane," and the second describes a sphere, the "Kasner sphere." The solutions (choices of formula_5) satisfying the two conditions therefore lie on the sphere where the two intersect (sometimes confusingly also called the Kasner sphere). In formula_12 spacetime dimensions, the space of solutions therefore lie on a formula_13 dimensional sphere formula_14.
Features.
There are several noticeable and unusual features of the Kasner solution: