Kasner metric

Solution of Einstein field equations 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 2pα(1/u)τ where τ is logarithmic time: τ = ln t. Shrinking along the axes is linear and uniform (no chaoticity). 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 {\displaystyle D>3} and has strong connections with the study of gravitational chaos. Metric and conditions The metric in D > 3 {\displaystyle D>3} spacetime dimensions is d s 2 = − d t 2 + ∑ j = 1 D − 1 t 2 p j [ d x j ] 2 {\displaystyle {\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 {\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 i