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

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

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

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

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.

References

Misner, Thorne, and Wheeler, Gravitation.

Relativity

Special
relativity

Background	Principle of relativity Introduction to special relativity Special relativity

Foundations	Relative motion Frame of reference Speed of light Maxwell's equations

Formulation	Galilean relativity Galilean transformation Lorentz transformation

Consequences	Time dilation Relativistic mass Mass–energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Relativistic disks

Spacetime	Minkowski spacetime World line Spacetime diagrams Light cone

General
relativity

Background	Introduction Mathematical formulation Resources

Fundamental concepts	Special relativity Equivalence principle World line Riemannian geometry Spacetime diagrams Spacetime in General relativity

Phenomena	Kepler problem Lenses Waves Frame-dragging Geodetic effect Event horizon Singularity Black hole

Equations	Linearized gravity Post-Newtonian formalism Einstein field equations Geodesic equation Friedmann equations ADM formalism BSSN formalism Hamilton–Jacobi–Einstein equation

Advanced theories	Kaluza–Klein Quantum gravity

Solutions	Schwarzschild Reissner–Nordström Gödel Kerr Kerr–Newman Kasner Taub-NUT Milne Robertson–Walker pp-wave van Stockum dust

Scientists

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Kasner metric

The Metric and Kasner Conditions

Features of the Kasner Metric

See also

References