Kasner metric

The Kasner metric (developed by and named for the American mathematician Edward Kasner in 1921) is an exact solution to 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 ${\displaystyle D>3}$ and has strong connections with the study of gravitational chaos.

The metric in ${\displaystyle D>3}$ spacetime dimensions is

and contains ${\displaystyle D-1}$ constants ${\displaystyle p_{j}}$, 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 ${\displaystyle p_{j}}$. Test particles in this metric whose comoving coordinate differs by ${\displaystyle \Delta x^{j}}$ are separated by a physical distance ${\displaystyle t^{p_{j}}\Delta x^{j}}$.

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