Kasner metric explained

D> 3 and has strong connections with the study of gravitational chaos.

Metric and conditions

The metric in

D> 3 spacetime dimensions isd s ^{2} = - d t ^{2} + \sum ^{D -1} _{j =1} t ^{2 p _{j}} [ 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 ^{D -1} _{j =1} p _{j} = 1 , \sum ^{D -1} _{j =1} p ^{2} _{j} = 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

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

• The volume of the spatial slices is always

O(t) . This is because their volume is proportional to \sqrt{ - g} , and\sqrt{ - g} = t ^{p _{1} + p _{2} + … + 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 ^{D -1} _{j =1} p ^{2} _{j} = (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

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 full Riemann tensor vanishes only when a single

p _{j=1} and the rest vanish, in which case the space is flat. The Minkowski metric can be recovered via the coordinate transformation t' = t \cosh x _{j} and x _{j'} = t \sinh x _{j} .

See also

• BKL singularity
• Mixmaster universe

References

• Book: Misner, Charles W. . Kip S. Thorne . John Archibald Wheeler . . . San Francisco . September 1973 . 0-7167-0344-0 .

Notes and References

1. Kasner, E. "Geometrical theorems on Einstein’s cosmological equations." Am. J. Math. 43, 217–221 (1921).