Everything about Kaluza-klein Theory totally explained
In
physics,
Kaluza–Klein theory (or
KK theory, for short) is a model that seeks to unify the two
fundamental forces of
gravitation and
electromagnetism. The theory was first published in
1921 and was discovered by the mathematician
Theodor Kaluza who extended
general relativity to a five-dimensional spacetime. The resulting equations can be separated out into further sets of equations, one of which is equivalent to
Einstein field equations, another set equivalent to
Maxwell's equations for the
electromagnetic field and the final part an extra
scalar field now termed the "
radion".
Overview
A splitting of
five-dimensional spacetime into the
Einstein equations and Maxwell equations in four dimensions was first discovered by
Gunnar Nordström in 1914, in the context of
his theory of gravity, but subsequently forgotten. In
1926,
Oskar Klein proposed that the fourth spatial dimension is curled up in a
circle of very small
radius, so that a
particle moving a short distance along that axis would return to where it began. The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a
compact set, and the phenomenon of having a space-time with compact dimensions is referred to as
compactification.
In modern geometry, the extra fifth dimension can be understood to be the
circle group U(1), as
electromagnetism can essentially be formulated as a
gauge theory on a
fiber bundle, the
circle bundle, with
gauge group U(1). Once this geometrical interpretation is understood, it's relatively straightforward to replace
U(1) by a general
Lie group. Such generalizations are often called
Yang–Mills theories. If a distinction is drawn, then it's that Yang–Mills theories occur on a flat space-time, whereas Kaluza–Klein treats the more general case of curved spacetime. The base space of Kaluza–Klein theory need not be four-dimensional space-time; it can be any (
pseudo-)
Riemannian manifold, or even a
supersymmetric manifold or
orbifold or even a
noncommutative space.
As an approach to the unification of the forces, it's straightforward to apply the Kaluza-Klein theory in an attempt to unify gravity with the
strong and
electroweak forces by using the symmetry group of the
Standard Model,
SU(3) ×
SU(2) ×
U(1). However, an attempt to convert this interesting geometrical construction into a bona-fide model of reality flounders on a number of issues, including the fact that the
fermions must be introduced in an artificial way (in nonsupersymmetric models). Nonetheless, KK remains an important
touchstone in theoretical physics and is often embedded in more sophisticated theories. It is studied in its own right as an object of geometric interest in
K-theory.
Even in the absence of a completely satisfying theoretical physics framework, the idea of exploring extra, compactified, dimensions is of considerable interest in the
experimental physics and
astrophysics communities. A variety of predictions, with real experimental consequences, can be made (in the case of
large extra dimensions/
warped models). For example, on the simplest of principles, one might expect to have
standing waves in the extra compactified dimension(s). If an extra dimension is of radius
R, the
energy of such a standing wave would be
with
n an
integer,
h being
Planck's constant and
c the
speed of light. This set of possible energy values is often called the
Kaluza–Klein tower.
Examples of experimental pursuits include work by the
CDF collaboration, which has re-analyzed
particle collider data for the signature of effects associated with large extra dimensions/
warped models.
Brandenberger and Vafa have speculated that in the early universe,
cosmic inflation causes three of the space dimensions to expand to cosmological size while the remaining dimensions of space remained microscopic.
So far no evidence has been found to support the existence of extra dimensions yet the idea remains popular among theoretical scientists. The LHC is hoped to provide evidence for the existence of extra dimensions.
Space-time-matter theory
One particular variant of Kaluza–Klein theory is
space-time-matter theory or
induced matter theory, chiefly promulgated by
Paul Wesson and other members of the so-called
Space-Time-Matter Consortium
. In this version of the theory, it's noted that solutions to the equation
» , and allows
to vary from fiber to fiber. In this case, the coupling between gravity and the electromagnetic field isn't constant, but has its own dynamical field, the
radion.
Commentary and generalizations
In the above, the size of the loop
acts as a coupling constant between the gravitational field and the electromagnetic field. If the base manifold is four-dimensional, the Kaluza–Klein manifold
P is five-dimensional. The fifth dimension is a
compact space, and is called the
compact dimension. The phenomenon of having a higher-dimensional manifold where some of the dimensions are compact is referred to as
compactification.
The above development generalizes in a more-or-less straightforward fashion to general
principal G-bundles for some arbitrary
Lie group G taking the place of
U(1). In such a case, the theory is often referred to as a
Yang-Mills theory, and is sometimes taken to be synonymous. If the underlying manifold is
supersymmetric, the resulting theory is a supersymmetric Yang–Mills theory.
Further Information
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