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TheKasner metric (developed by and named for the American mathematicianEdward Kasner in 1921)^[2] is anexact solution toAlbert Einstein's theory ofgeneral relativity. It describes an anisotropicuniverse withoutmatter (i.e., it is avacuum solution). It can be written in anyspacetimedimension${\displaystyle D>3}$ and has strong connections with the study of gravitationalchaos.

Themetric 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 theKasner 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${\displaystyle \mathrm{\Delta }{x}^j}$ are separated by a physical distance${\displaystyle t^{p_j}\mathrm{\Delta }{x}^j}$.

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

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

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

The first condition defines aplane, theKasner plane, and the second describes asphere, theKasner 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${\displaystyle S^{D-3}}$.

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 aBig Bang or aBig Crunch, depending on the sense of${\displaystyle t}$

• Isotropicexpansion 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

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

TheFriedmann–Lemaître–Robertson–Walker metric employed incosmology, 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 actuallycontracting.
• The Kasner metric is a solution to the vacuum Einstein equations, and so theRicci tensor always vanishes for any choice of exponents satisfying the Kasner conditions. The fullRiemann 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${\displaystyle x_j^{\prime }=t\sinh x_j}$.

• BKL singularity
• Mixmaster universe

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

• Exact solutions in general relativity
• Metric tensors

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