Kasner metric

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The Kasner metric is an exact solution to Einstein's theory of general relativity. It describes an anisotropic universe without matter (e.g., it is a vacuum solution). It can be written in any spacetime dimension $D > 3$ and has strong connections with the study of gravitational chaos (See also BKL oscillations and the Mixmaster universe).

1 The Metric and Kasner Conditions
2 Features of the Kasner Metric
3 See Also
4 References

[edit] The Metric and Kasner Conditions

The metric in $D > 3$ spacetime dimensions is

$\text{d}s^2 = -\text{d}t^2 + \sum_{j=1}^{D-1} t^{2p_j} [\text{d}x^j]^2$ ,

and contains $D - 1$ constants $p j$ , 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 $p j$ . Test particles in this metric whose comoving coordinate differs by $Δ x j$ are separated by a physical distance $t^{p_j}\Delta x^j$ .

The Kasner metric is an exact solution to Einstein's equations in vacuum when the Kasner exponents satisfy the following Kasner conditions,

$\sum_{j=1}^{D-1} p_j = 1,$

$\sum_{j=1}^{D-1} p_j^2 = 1.$

The first condition defines a plane, the Kasner plane, and the second describes a sphere, the Kasner sphere. The solutions (choices of $p j$ ) satisfying the two conditions therefore lie on the sphere where the two intersect (sometimes confusingly also called the Kasner sphere). In $D$ spacetime dimensions, the space of solutions therefore lie on a $D - 3$ dimensional sphere $S D - 3$ .

[edit] Features of the Kasner Metric

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

The volume of the spatial slices always goes like $t$ . This is because their volume is proportional to $\sqrt{-g}$ , and

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

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

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 $p j = 1 / (D - 1)$ to satisfy the first Kasner condition. But then the second Kasner condition cannot be satisfied, for

$\sum_{j=1}^{D-1} p_j^2 = \frac{1}{D-1} \ne 1.$

The FRW 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 $p j = 1$ , and the rest vanishing). Suppose we take the time coordinate $t$ to increase from zero. Then this implies that while the volume of space is increasing like $t$ , 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 Riemann tensor vanishes only when a single $p j = 1$ and the rest vanish. This has the interesting consequence that this particular Kasner solution must be a solution of any extension of general relativity in which the field equations are built from the Riemann tensor.