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Concept# Kasner metric

Summary

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 D>3 and has strong connections with the study of gravitational chaos.
Metric and conditions
The metric in D>3 spacetime dimensions is
:\text{d}s^2 = -\text{d}t^2 + \sum_{j=1}^{D-1} t^{2p_j} [\text{d}x^j]^2,
and contains D-1 constants 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 p_j. Test particles in this metric whose comoving coordinate differs by \Delta x^j are separated by a physical distance t^

Official source

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