The Kasner metric is an exact solution to Einstein's theory of general relativity. It describes an anisotropic universe without matter (e.g., 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 (See also BKL oscillations and the Mixmaster universe). Category: Mathematics stubs ... Einstein redirects here. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... Universe is a word derived from the Old French univers, which in turn comes from the Latin roots unus (one) and versus (a form of vertere, to turn). Based on observations of the observable universe, physicists attempt to describe the whole of space-time, including all matter and energy and... This article or section does not adequately cite its references or sources. ... A vacuum solution is a solution of a field equation in which the sources of the field are taken to be identically zero. ... In physics, spacetime is a mathematical model that combines space and time into a single construct called the space-time continuum. ... :For other senses of this word, see dimension (disambiguation). ... To meet Wikipedias quality standards, this article may require cleanup. ... A BKL singularity is a non-symmetric, chaotic, vacuum solution to Einsteins field equations conjectured to represent the actual interior geometry of a physical black hole formed by gravitational collapse. ... The Mixmaster Universe is a solution to Einsteins general relativity studied by Charles Misner in an effort to better understand the dynamics of the early universe [1]. He hoped to solve the horizon problem in a natural way by showing that the early universe underwent an oscillatory, chaotic epoch. ...

The Metric and Kasner Conditions

The metric in D > 3 spacetime dimensions is In mathematics a metric or distance is a function which assigns a distance to elements of a set. ...

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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 Δx^j 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 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}. In mathematics, a plane is the fundamental two-dimensional object. ... A sphere is a perfectly symmetrical geometrical object. ...

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 , 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 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

The FRW 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.

According to the Big Bang theory, the universe emerged from an extremely dense and hot state (singularity). ... In physical cosmology, the Big Crunch is the hypothesis that the universe will collapse upon itself after its expansion eventually stops — a counterpart to the Big Bang. ... Isotropic means independent of direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. ... The metric expansion of space is a key part of sciences current understanding of the universe, whereby space itself is described by a metric which changes over time. ... The Friedmann-Lemaître-Robertson-Walker (FLRW) metric describes a homogeneous, isotropic expanding/contracting universe. ... Cosmology, from the Greek: κοσμολογία (cosmologia, κόσμος (cosmos) order + λογια (logia) discourse) is the study of the Universe in its totality, and by extension, humanitys place in it. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...

See Also

• BKL singularity
• Mixmaster universe

A BKL singularity is a non-symmetric, chaotic, vacuum solution to Einsteins field equations conjectured to represent the actual interior geometry of a physical black hole formed by gravitational collapse. ... The Mixmaster Universe is a solution to Einsteins general relativity studied by Charles Misner in an effort to better understand the dynamics of the early universe [1]. He hoped to solve the horizon problem in a natural way by showing that the early universe underwent an oscillatory, chaotic epoch. ...

References

• Misner, Thorne, and Wheeler, Gravitation.