Kasner metric

The Kasner metric (developed by and named for the American mathematician Edward Kasner in 1921)[2] 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).

and has strong connections with the study of gravitational chaos.

, 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

Test particles in this metric whose comoving coordinate differs by

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.

) satisfying the two conditions therefore lie on the sphere where the two intersect (sometimes confusingly also called the Kasner sphere).

spacetime dimensions, the space of solutions therefore lie on a

There are several noticeable and unusual features of the Kasner solution:

Figure 1. Dynamics of Kasner metrics eq. 2 in spherical coordinates towards singularity. The Lifshitz-Khalatnikov parameter is u =2 (1/ u =0.5) and the r coordinate is 2 p α (1/ u )τ where τ is logarithmic time: τ = ln t . [ 1 ] Shrinking along the axes is linear and uniform (no chaoticity).