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 2p_α(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 ${\displaystyle D>3}$ and has strong connections with the study of gravitational chaos.

Metric and conditions

The metric in ${\displaystyle D>3}$ spacetime dimensions is

${\displaystyle {\text{d}}s^{2}=-{\text{d}}t^{2}+\sum _{j=1}^{D-1}t^{2p_{j}}[{\text{d}}x^{j}]^{2}}$,

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

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

${\displaystyle \sum _{j=1}^{D-1}p_{j}=1,}$

${\displaystyle \sum _{j=1}^{D-1}p_{j}^{2}=1.}$

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

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

${\displaystyle \sum _{j=1}^{D-1}p_{j}^{2}={\frac {1}{D-1}}\neq 1.}$

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 ${\displaystyle t'=t\cosh x_{j}}$ and ${\displaystyle x_{j}'=t\sinh x_{j}}$.

See also

• BKL singularity
• Mixmaster universe

Notes

1. The expression for r is derived by logarithming the power coefficients in the metric: ln [t^2p_α(1/u)] = 2p_α(1/u) ln t.
2. Kasner, Edward (October 1921). "Geometrical Theorems on Einstein's Cosmological Equations". American Journal of Mathematics. 43 (4): 217. doi:10.2307/2370192.

References

• Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973). Gravitation (PDF). San Francisco: W. H. Freeman. ISBN 978-0-7167-0334-1.