General relativity 


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). It can be written in any spacetime dimension and has strong connections with the study of gravitational chaos.
The metric in spacetime dimensions is
and contains constants , called the Kasner exponents. The metric describes a spacetime whose equaltime 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 are separated by a physical distance .
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 ) satisfying the two conditions therefore lie on the sphere where the two intersect (sometimes confusingly also called the Kasner sphere). In spacetime dimensions, the space of solutions therefore lie on a dimensional sphere .
There are several noticeable and unusual features of the Kasner solution: