Kasner metric

Features

There are several noticeable and unusual features of the Kasner solution:

• The volume of the spatial slices is always ${\displaystyle O(t)}$. This is because their volume is proportional to ${\displaystyle \sqrt{-g}}$, and

${\displaystyle \sqrt{-g}=t^{p_1+p_2+\cdots +p_{D-1}}=t}$

where we have used the first Kasner condition. Therefore ${\displaystyle t\to 0}$ can describe either a Big Bang or a Big Crunch, depending on the sense of ${\displaystyle t}$

• Isotropic expansion or contraction of space is not allowed. If the spatial slices were expanding isotropically, then all of the Kasner exponents must be equal, and therefore ${\displaystyle p_j=1/(D-1)}$ to satisfy the first Kasner condition. But then the second Kasner condition cannot be satisfied, for

${\displaystyle \sum _{j=1}^{D-1}{p}_j^2=\frac{1}{D-1}\ne 1.}$

The Friedmann–Lemaître–Robertson–Walker metric employed in cosmology, by contrast, is able to expand or contract isotropically because of the presence of matter.

• With a little more work, one can show that at least one Kasner exponent is always negative (unless we are at one of the solutions with a single ${\displaystyle p_j=1}$, and the rest vanishing). Suppose we take the time coordinate ${\displaystyle t}$ to increase from zero. Then this implies that while the volume of space is increasing like ${\displaystyle t}$, at least one direction (corresponding to the negative Kasner exponent) is actually contracting.
• The Kasner metric is a solution to the vacuum Einstein equations, and so the Ricci tensor always vanishes for any choice of exponents satisfying the Kasner conditions. The full Riemann tensor vanishes only when a single ${\displaystyle p_j=1}$ and the rest vanish, in which case the space is flat. The Minkowski metric can be recovered via the coordinate transformation ${\displaystyle t^{\prime }=t\cosh x_j}$ and ${\displaystyle x_j^{\prime }=t\sinh x_j}$.