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Concept
Kasner metric
Summary
The Kasner metric (developed by and named for the American mathematician Edward Kasner in 1921) 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 and has strong connections with the study of gravitational chaos.
The metric in spacetime dimensions is
and contains constants , 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 . Test particles in this metric whose comoving coordinate differs by are separated by a physical distance .
The Kasner metric is an exact solution to Einstein's equations in vacuum when the Kasner exponents satisfy the following Kasner conditions,
The first condition defines a plane, the Kasner plane, and the second describes a sphere, the Kasner sphere. The solutions (choices of ) satisfying the two conditions therefore lie on the sphere where the two intersect (sometimes confusingly also called the Kasner sphere). In spacetime dimensions, the space of solutions therefore lie on a dimensional sphere .
There are several noticeable and unusual features of the Kasner solution:
The volume of the spatial slices is always . This is because their volume is proportional to , and
where we have used the first Kasner condition. Therefore can describe either a Big Bang or a Big Crunch, depending on the sense of
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 to satisfy the first Kasner condition. But then the second Kasner condition cannot be satisfied, for
The Friedmann–Lemaître–Robertson–Walker metric employed in cosmology, by contrast, is able to expand or contract isotropically because of the presence of matter.
Official source
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