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 D > 3 {displaystyle D>3} and has strong connections with the study of gravitational chaos. 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 D > 3 {displaystyle D>3} and has strong connections with the study of gravitational chaos. The metric in D > 3 {displaystyle D>3} spacetime dimensions is and contains D − 1 {displaystyle D-1} constants p j {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 p j {displaystyle p_{j}} . Test particles in this metric whose comoving coordinate differs by Δ x j {displaystyle Delta x^{j}} are separated by a physical distance t p j Δ x j {displaystyle t^{p_{j}}Delta x^{j}} . 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 p j {displaystyle p_{j}} ) satisfying the two conditions therefore lie on the sphere where the two intersect (sometimes confusingly also called the Kasner sphere). In D {displaystyle D} spacetime dimensions, the space of solutions therefore lie on a D − 3 {displaystyle D-3} dimensional sphere S D − 3 {displaystyle S^{D-3}} .