Kasner metric

The Kasner metric is an exact solution to 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 $D>3$ and has strong connections with the study of gravitational chaos.

The Metric and Kasner Conditions

The metric in $D>3$ spacetime dimensions is

:$ext\{d\}s^2\; =\; -\; ext\{d\}t^2\; +\; sum\_\{j=1\}^\{D-1\}\; t^\{2p\_j\}\; [\; ext\{d\}x^j]\; ^2$,

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 $Delta\; x^j$ 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 $S^\{D-3\}$.

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 $t\; o\; 0$ 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

::$sum\_\{j=1\}^\{D-1\}\; p\_j^2\; =\; frac\{1\}\{D-1\}\; e\; 1.$

: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.

ee also

*BKL singularity
*Mixmaster universe

References

*Misner, Thorne, and Wheeler, Gravitation.