Kasner metric

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.[1] 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)[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.

  1. ^ The expression for r is derived by logarithming the power coefficients in the metric: ln [t2pα(1/u)] = 2pα(1/u) ln t.
  2. ^ Kasner, E. "Geometrical theorems on Einstein’s cosmological equations." Am. J. Math. 43, 217–221 (1921).

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