Kasner metric

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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[edit]

The metric in D>3 spacetime dimensions is

\text{d}s^2 = -\text{d}t^2 + \sum_{j=1}^{D-1} t^{2p_j} [\text{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[edit]

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\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_{j=1}^{D-1} p_j^2 = \frac{1}{D-1} \ne 1.
The FLRW 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.

See also[edit]

References[edit]

  • Misner, Thorne, and Wheeler, Gravitation.