Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī

2013-02-26T15:51:27Z

41.34.31.158: /* Symbolic algebra */

{{General relativity|cTopic=[[Exact solutions in general relativity|Solutions]]}}
The '''Kasner metric''' is an [[Exact solutions in general relativity|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]] <math>D>3</math> and has strong connections with the study of gravitational [[Chaos theory|chaos]].

==The Metric and Kasner Conditions==
The [[Metric (mathematics)|metric]] in <math>D>3</math> spacetime dimensions is

:<math>\text{d}s^2 = -\text{d}t^2 + \sum_{j=1}^{D-1} t^{2p_j} [\text{d}x^j]^2</math>,

and contains <math>D-1</math> constants <math>p_j</math>, 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 <math>p_j</math>. Test particles in this metric whose [[comoving coordinate]] differs by <math>\Delta x^j</math> are separated by a physical distance <math>t^{p_j}\Delta x^j</math>.

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

:<math>\sum_{j=1}^{D-1} p_j = 1,</math>

:<math>\sum_{j=1}^{D-1} p_j^2 = 1.</math>

The first condition defines a [[plane (geometry)|plane]], the ''Kasner plane,'' and the second describes a [[sphere]], the ''Kasner sphere.'' The solutions (choices of <math>p_j</math>) satisfying the two conditions therefore lie on the sphere where the two intersect (sometimes confusingly also called the Kasner sphere). In <math>D</math> spacetime dimensions, the space of solutions therefore lie on a <math>D-3</math> dimensional sphere <math>S^{D-3}</math>.

==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 <math>t</math>. This is because their volume is proportional to <math>\sqrt{-g}</math>, and

::<math>\sqrt{-g} = t^{p_1 + p_2 + \cdots + p_{D-1}} = t</math>

:where we have used the first Kasner condition. Therefore <math>t\to 0</math> can describe either a [[Big Bang]] or a [[Big Crunch]], depending on the sense of <math>t</math>

*[[Isotropic]] [[metric expansion of space|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 <math>p_j = 1/(D-1)</math> to satisfy the first Kasner condition. But then the second Kasner condition cannot be satisfied, for

::<math>\sum_{j=1}^{D-1} p_j^2 = \frac{1}{D-1} \ne 1.</math>

:The [[FRW|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 <math>p_j=1</math>, and the rest vanishing). Suppose we take the time coordinate <math>t</math> to increase from zero. Then this implies that while the volume of space is increasing like <math>t</math>, 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 <math>p_j=1</math> 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==

*[[BKL singularity]]
*[[Mixmaster universe]]

==References==

*Misner, Thorne, and Wheeler, '''Gravitation.'''

{{Relativity}}

[[Category:Exact solutions in general relativity]]
[[Category:Metric tensors]]

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Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī