Kaluza–Klein Metric
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Kaluza–Klein theory In physics, Kaluza–Klein theory (KK theory) is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to ...
, a unification of
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
and
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
, the five-dimensional Kaluza–Klein metric is the generalization of the four-dimensional metric tensor. It additionally includes a
scalar field In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
called
graviscalar In theoretical physics, the hypothetical particle called the graviscalar or radion emerges as an excitation of general relativity's metric tensor, i.e. gravitational field, but is indistinguishable from a scalar in four dimensions, as shown in Ka ...
(or radion) and a
vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
called
graviphoton In theoretical physics and quantum physics, a graviphoton or gravivector is a hypothetical particle which emerges as an excitation of the metric tensor (i.e. gravitational field) in spacetime dimensions higher than four, as described in Kaluza–K ...
(or gravivector), which correspond to
hypothetical particles This is a list of hypothetical subatomic particles in physics. Elementary particles Some theories predict the existence of additional elementary bosons and fermions that are not found in the Standard Model. Particles predicted by supersy ...
. The Kaluza–Klein metric is named after
Theodor Kaluza Theodor Franz Eduard Kaluza (; 9 November 1885 – 19 January 1954) was a German mathematician and physicist known for the Kaluza–Klein theory, involving field equations in five-dimensional space-time. His idea that fundamental forces can b ...
and
Oskar Klein Oskar Benjamin Klein (; 15 September 1894 – 5 February 1977) was a Swedish theoretical physics, theoretical physicist. Oskar Klein is known for his work on Kaluza–Klein theory, which is partially named after him. Biography Klein was born ...
.


Definition

The ''Kaluza–Klein metric'' is given by: : \widetilde_ :=\begin g_+\phi^2A_\mu A_\nu & \phi^2A_\mu \\ \phi^2A_\nu & \phi^2 \end. Its
inverse matrix In linear algebra, an invertible matrix (''non-singular'', ''non-degenarate'' or ''regular'') is a square matrix that has an inverse. In other words, if some other matrix is multiplied by the invertible matrix, the result can be multiplied by an ...
is given by: : \widetilde^ =\begin g^ & -A^\mu \\ -A^\nu & g_A^\mu A^\nu+\phi^ \end. Defining an extended gravivector A_a=(A_\mu,1) shortens the definition to: : \widetilde_ =\operatorname(g_,0) +\phi^2A_aA_b, which also shows that the radion \phi cannot vanish as this would make the metric
singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular or sounder, a group of boar, see List of animal names * Singular (band), a Thai jazz pop duo *'' Singula ...
.


Properties

* A
contraction Contraction may refer to: Linguistics * Contraction (grammar), a shortened word * Poetic contraction, omission of letters for poetic reasons * Elision, omission of sounds ** Syncope (phonology), omission of sounds in a word * Synalepha, merged ...
directly shows the passing from four to five dimensions: *: g^g_=4, *: \widetilde^\widetilde_=5. * If \mathrms^2 =g_\mathrmx^\mu\mathrmx^\nu is the four-dimensional and \mathrm\widetilde^2 =\widetilde_\mathrm\widetilde^a\mathrm\widetilde^b is the five-dimensional
line element In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc ...
, then there is the following relation resembling the
Lorentz factor The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in sev ...
from
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
:Pope, Equation (1.7) *: \frac =\sqrt. * The determinants \widetilde:=\det(\widetilde_) and g:=\det(g_) are connected by:Pope, Equation (1.14) *: \widetilde =\phi^2g \Leftrightarrow \sqrt =\phi\sqrt. : Although the above expression \widetilde_ =\operatorname(g_,0) +\phi^2A_aA_b fits the structure of the
matrix determinant lemma In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, uvT, of a column vector u and a row vector vT. Statement Suppose A is an invertible ...
, it cannot be applied since the former term is singular. * Analogous to the metric tensor, but additionally using the above relation \widetilde=\phi^2g, one has: *: \widetilde^\partial_c\widetilde_ =\partial_c\ln(-\widetilde) =\partial_c\ln(-\phi^2g).


Literature

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References

{{DEFAULTSORT:Kaluza-Klein Metric Theories of gravity Particle physics Physical cosmology String theory Physics beyond the Standard Model