Metric of Kasner

The metric of Kasner is a particular form of metric introduced by the physicist Kasner in 1921 to study the anisotropic models of universe.

The metric one is given by the following equation:

$g\_\; \{ij\}\; =\; ds^2\; =\; -\; dt^2\; +\; t^\; \{2\; p\_1\}\; dx^2\; +\; t^\; \{2\; p\_2\}\; dy^2\; +\; t^\; \{2\; p\_3\}\; dz^2$

The parameters of metric, $(p\_1,\; p\_2,\; p\_3)$ check the following conditions:

$p\_1\; +\; p\_2\; +\; p\_3\; =\; \{p\_1\}\; ^2\; +\; \{p\_2\}\; ^2\; +\; \{p\_3\}\; ^2\; =\; 1$

Immediate properties

The condition on the value of the parameters implies that one of them is negative (except in the commonplace case where one of them is equal to 1 and the two others to zero). Indeed: $(p\_1\; +\; p\_2\; +\; p\_3)\; ^2\; =\; \{p\_1\}\; ^2\; +\; \{p\_2\}\; ^2\; +\; \{p\_3\}\; ^2\; +\; 2\; p\_1\; p\_2\; +\; 2\; p\_1\; p\_3\; +\; 2\; p\_2\; p\_3$ from where: $p\_1\; p\_2\; +\; p\_1\; p\_3\; +\; p\_2\; p\_3\; =\; 0$, condition which cannot be carried out if all the $p\_i$ are positive. It is shown that (for example): .

The elementary element of volume in this metric has as a measurement $\backslash \; sqrt\; \{-\; G\}\; =\; T$. The universe describes by this metric is thus expanding. However, owing to the fact that at least of the $p\_i$ is negative, this expansion is transformed into contraction in one of the directions.

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