The Kasner metric (developed by and named for the American mathematician
Edward Kasner
Edward Kasner (April 2, 1878 – January 7, 1955) was an American mathematician who was appointed Tutor on Mathematics in the Columbia University Mathematics Department. Kasner was the first Jewish
Jews ( he, יְהוּדִים, , ...
in 1921)
[Kasner, E. "Geometrical theorems on Einstein’s cosmological equations." ''Am. J. Math.'' 43, 217–221 (1921).] is an
exact solution to
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's theory of
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. It describes an anisotropic
universe
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
without
matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...
(i.e., it is a
vacuum solution
In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or non ...
). It can be written in any
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
and has strong connections with the study of gravitational
chaos
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Arts, entertainment and media Fictional elements
* Chaos (''Kinnikuman'')
* Chaos (''Sailor Moon'')
* Chaos (''Sesame Park'')
* Chaos (''Warhammer'')
* Chaos, in ''Fabula Nova Crystallis Final Fantasy''
* Cha ...
.
Metric and conditions
The
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathema ...
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
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
, the ''Kasner plane,'' and the second describes a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
, 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
.
Features
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
The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
or a
Big Crunch
The Big Crunch is a hypothetical scenario for the ultimate fate of the universe, in which the expansion of the universe eventually reverses and the universe recollapses, ultimately causing the cosmic scale factor to reach zero, an event potential ...
, depending on the sense of
*
Isotropic
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
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
The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric is a metric based on the exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe tha ...
employed in
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount (lexicographer), Thomas Blount's ''Glossographia'', and in 1731 taken up in ...
, 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
, and the rest vanishing). Suppose we take the time coordinate
to increase from zero. Then this implies that while the volume of space is increasing like
, 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
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
always vanishes for any choice of exponents satisfying the Kasner conditions. The full
Riemann tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. I ...
vanishes only when a single
and the rest vanish, in which case the space is flat. The Minkowski metric can be recovered via the coordinate transformation
and
.
See also
*
BKL singularity
A Belinski–Khalatnikov–Lifshitz (BKL) singularity is a model of the dynamic evolution of the universe near the initial gravitational singularity, described by an anisotropic, chaotic solution of the Einstein field equation of gravitation ...
*
Mixmaster universe
Mixmaster may refer to:
Equipment and technology
* Sunbeam Mixmaster, an electric kitchen mixer that was the flagship product of Sunbeam Products
** Mix Diskerud, United States professional soccer player nicknamed after the mixer
* Mixmaster ano ...
Notes
References
*
{{Relativity
Exact solutions in general relativity
Metric tensors