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 usual four
of space and time and considered an important
precursor to string theory.
The five-dimensional theory developed in three
steps. The original hypothesis came from Theodor
Kaluza, who sent his results to Einstein in
1919, and published them in 1921, which detailed
a purely classical extension of general relativity
to five dimensions and includes 15 components.
Ten components are identified with the four-dimensional
spacetime metric, four components with the
electromagnetic vector potential, and one
component with an unidentified scalar field
sometimes called the "radion" or the "dilaton".
Correspondingly, the five-dimensional Einstein
equations yield the four-dimensional Einstein
field equations, the Maxwell equations for
the electromagnetic field, and an equation
for the scalar field. Kaluza also introduced
the "cylinder condition" hypothesis, that
no component of the five-dimensional metric
depends on the fifth dimension. Without this
assumption, the field equations of five-dimensional
relativity grow enormous in complexity. Standard
four-dimensional physics seems to manifest
the cylinder condition.
In 1926, Oskar Klein gave Kaluza's classical
five-dimensional theory a quantum interpretation,
to accord with the then-recent discoveries
of Heisenberg and Schrödinger. Klein introduced
the hypothesis that the fifth dimension was
curled up and microscopic, to explain the
cylinder condition. Klein suggested that the
geometry of the extra fifth dimension could
take the form of a circle, with the radius
of 10−30 cm. Klein also calculated a scale
for the fifth dimension based on the quantum
of charge.In the 1940s the classical theory
was completed, and the full field equations
including the scalar field were obtained by
three independent research groups:
Thiry,
working in France on his dissertation under
Lichnerowicz; Jordan, Ludwig, and Müller
in Germany, with critical input from Pauli
and Fierz; and Scherrer working alone in Switzerland.
Jordan's work led to the scalar-tensor theory
of Brans–Dicke; Brans and Dicke were apparently
unaware of Thiry or Scherrer. The full Kaluza
equations under the cylinder condition are
quite complex, and most English-language reviews
as well as the English translations of Thiry
contain some errors. The complete Kaluza equations
were evaluated using tensor algebra software
in 2015.
== Kaluza hypothesis ==
In his 1921 paper, Kaluza established all
the elements of the classical five-dimensional
theory: the metric, the field equations, the
equations of motion, the stress-energy tensor,
and the cylinder condition. With no free parameters,
it merely extends general relativity to five
dimensions. One starts by hypothesizing a
form of the five-dimensional metric
g
~
a
b
{\displaystyle {\widetilde {g}}_{ab}}
, where Latin indices span five dimensions.
Let one also introduce the four-dimensional
spacetime metric
g
μ
ν
{\displaystyle {g}_{\mu \nu }}
, where Greek indices span the usual four
dimensions of space and time; a 4-vector
A
μ
{\displaystyle A^{\mu }}
identified with the electromagnetic vector
potential; and a scalar field
ϕ
{\displaystyle \phi }
. Then decompose the 5D metric so that the
4D metric is framed by the electromagnetic
vector potential, with the scalar field at
the fifth diagonal. This can be visualized
as:
g
~
a
b
≡
[
g
μ
ν
+
ϕ
2
A
μ
A
ν
ϕ
2
A
μ
ϕ
2
A
ν
ϕ
2
]
{\displaystyle {\widetilde {g}}_{ab}\equiv
{\begin{bmatrix}g_{\mu \nu }+\phi ^{2}A_{\mu
}A_{\nu }&\phi ^{2}A_{\mu }\\\phi ^{2}A_{\nu
}&\phi ^{2}\end{bmatrix}}}
.One can write more precisely
g
~
μ
ν
≡
g
μ
ν
+
ϕ
2
A
μ
A
ν
,
g
~
5
ν
≡
g
~
ν
5
≡
ϕ
2
A
ν
,
g
~
55
≡
ϕ
2
{\displaystyle {\widetilde {g}}_{\mu \nu }\equiv
g_{\mu \nu }+\phi ^{2}A_{\mu }A_{\nu },\qquad
{\widetilde {g}}_{5\nu }\equiv {\widetilde
{g}}_{\nu 5}\equiv \phi ^{2}A_{\nu },\qquad
{\widetilde {g}}_{55}\equiv \phi ^{2}}
where the index
5
{\displaystyle 5}
indicates the fifth coordinate by convention
even though the first four coordinates are
indexed with 0, 1, 2, and 3. The associated
inverse metric is
g
~
a
b
≡
[
g
μ
ν
−
A
μ
−
A
ν
g
α
β
A
α
A
β
+
1
ϕ
2
]
{\displaystyle {\widetilde {g}}^{ab}\equiv
{\begin{bmatrix}g^{\mu \nu }&-A^{\mu }\\-A^{\nu
}&g_{\alpha \beta }A^{\alpha }A^{\beta }+{1
\over \phi ^{2}}\end{bmatrix}}}
.This decomposition is quite general and all
terms dimensionless. Kaluza then applies the
machinery of standard general relativity to
this metric. The field equations are obtained
from five-dimensional Einstein equations,
and the equations of motion from the five-dimensional
geodesic hypothesis. The resulting field equations
provide both the equations of general relativity
and of electrodynamics; the equations of motion
provide the four-dimensional geodesic equation
and the Lorentz force law, and one finds that
electric charge is identified with motion
in the fifth dimension.
The hypothesis for the metric implies an invariant
five-dimensional length element
d
s
{\displaystyle \operatorname {d} \!s}
:
d
s
2
≡
g
~
a
b
d
x
a
d
x
b
=
g
μ
ν
d
x
μ
d
x
ν
+
ϕ
2
(
A
ν
d
x
ν
+
d
x
5
)
2
{\displaystyle \operatorname {d} \!s^{2}\equiv
{\widetilde {g}}_{ab}\operatorname {d} \!x^{a}\operatorname
{d} \!x^{b}=g_{\mu \nu }dx^{\mu }\operatorname
{d} \!x^{\nu }+\phi ^{2}(A_{\nu }\operatorname
{d} \!x^{\nu }+\operatorname {d} \!x^{5})^{2}}
== Field equations from the Kaluza hypothesis
==
The field equations of the 5-dimensional theory
were never adequately provided by Kaluza or
Klein, mainly regarding the scalar field.
The full Kaluza field equations are generally
attributed to Thiry, who obtained vacuum field
equations, although Kaluza originally provided
a stress-energy tensor for his theory and
Thiry included a stress-energy tensor in his
thesis. But as described by Gonner, several
independent groups worked on the field equations
in the 1940s and earlier. Thiry is perhaps
best known only because an English translation
was provided by Applequist, Chodos, & Freund
in their review book. Applequist et al. also
provided an English translation of Kaluza's
paper. There are no English translations of
the Jordan papers.To obtain the 5D field equations,
the 5D connections
Γ
~
b
c
a
{\displaystyle {\widetilde {\Gamma }}_{bc}^{a}}
are calculated from the 5D metric
g
~
a
b
{\displaystyle {\widetilde {g}}_{ab}}
, and the 5D Ricci tensor
R
~
a
b
{\displaystyle {\widetilde {R}}_{ab}}
is calculated from the 5D connections.
The classic results of Thiry and other authors
presume the cylinder condition:
∂
g
~
a
b
∂
x
5
=
0
{\displaystyle {\partial {\widetilde {g}}_{ab}
\over \partial x^{5}}=0}
.Without this assumption, the field equations
become much more complex, providing many more
degrees of freedom that can be identified
with various new fields. Paul Wesson and colleagues
have pursued relaxation of the cylinder condition
to gain extra terms that can be identified
with the matter fields, for which Kaluza otherwise
inserted a stress-energy tensor by hand.
It has been an objection to the original Kaluza
hypothesis to invoke the fifth dimension only
to negate its dynamics. But Thiry argued that
the interpretation of the Lorentz force law
in terms of a 5-dimensional geodesic militates
strongly for a fifth dimension irrespective
of the cylinder condition. Most authors have
therefore employed the cylinder condition
in deriving the field equations. Furthermore,
vacuum equations are typically assumed for
which
R
~
a
b
=
0
{\displaystyle {\widetilde {R}}_{ab}=0}
where
R
~
a
b
≡
∂
c
Γ
~
a
b
c
−
∂
b
Γ
~
c
a
c
+
Γ
~
c
d
c
Γ
~
a
b
d
−
Γ
~
b
d
c
Γ
~
a
c
d
{\displaystyle {\widetilde {R}}_{ab}\equiv
\partial _{c}{\widetilde {\Gamma }}_{ab}^{c}-\partial
_{b}{\widetilde {\Gamma }}_{ca}^{c}+{\widetilde
{\Gamma }}_{cd}^{c}{\widetilde {\Gamma }}_{ab}^{d}-{\widetilde
{\Gamma }}_{bd}^{c}{\widetilde {\Gamma }}_{ac}^{d}}
and
Γ
~
b
c
a
≡
1
2
g
~
a
d
(
∂
b
g
~
d
c
+
∂
c
g
~
d
b
−
∂
d
g
~
b
c
)
{\displaystyle {\widetilde {\Gamma }}_{bc}^{a}\equiv
{1 \over 2}{\widetilde {g}}^{ad}(\partial
_{b}{\widetilde {g}}_{dc}+\partial _{c}{\widetilde
{g}}_{db}-\partial _{d}{\widetilde {g}}_{bc})}
The vacuum field equations obtained in this
way by Thiry and Jordan's group are as follows.
The field equation for
ϕ
{\displaystyle \phi }
is obtained from
R
~
55
=
0
⇒
◻
ϕ
=
1
4
ϕ
3
F
α
β
F
α
β
{\displaystyle {\widetilde {R}}_{55}=0\Rightarrow
\Box \phi ={1 \over 4}\phi ^{3}F^{\alpha \beta
}F_{\alpha \beta }}
where
F
α
β
≡
∂
α
A
β
−
∂
β
A
α
{\displaystyle F_{\alpha \beta }\equiv \partial
_{\alpha }A_{\beta }-\partial _{\beta }A_{\alpha
}}
,
where
◻
≡
g
μ
ν
∇
μ
∇
ν
{\displaystyle \Box \equiv g^{\mu \nu }\nabla
_{\mu }\nabla _{\nu }}
, and where
∇
μ
{\displaystyle \nabla _{\mu }}
is a standard, 4D covariant derivative. It
shows that the electromagnetic field is a
source for the scalar field. Note that the
scalar field cannot be set to a constant without
constraining the electromagnetic field. The
earlier treatments by Kaluza and Klein did
not have an adequate description of the scalar
field, and did not realize the implied constraint
on the electromagnetic field by assuming the
scalar field to be constant.
The field equation for
A
ν
{\displaystyle A^{\nu }}
is obtained from
R
~
5
α
=
0
=
1
2
g
β
μ
∇
μ
(
ϕ
3
F
α
β
)
{\displaystyle {\widetilde {R}}_{5\alpha }=0={1
\over 2}g^{\beta \mu }\nabla _{\mu }(\phi
^{3}F_{\alpha \beta })}
It has the form of the vacuum Maxwell equations
if the scalar field is constant.
The field equation for the 4D Ricci tensor
R
μ
ν
{\displaystyle R_{\mu \nu }}
is obtained from
R
~
μ
ν
−
1
2
g
~
μ
ν
R
~
=
0
⇒
R
μ
ν
−
1
2
g
μ
ν
R
=
1
2
ϕ
2
(
g
α
β
F
μ
α
F
ν
β
−
1
4
g
μ
ν
F
α
β
F
α
β
)
+
1
ϕ
(
∇
μ
∇
ν
ϕ
−
g
μ
ν
◻
ϕ
)
{\displaystyle {\begin{aligned}{\widetilde
{R}}_{\mu \nu }-{1 \over 2}{\widetilde {g}}_{\mu
\nu }{\widetilde {R}}&=0\Rightarrow \\R_{\mu
\nu }-{1 \over 2}g_{\mu \nu }R&={1 \over 2}\phi
^{2}\left(g^{\alpha \beta }F_{\mu \alpha }F_{\nu
\beta }-{1 \over 4}g_{\mu \nu }F_{\alpha \beta
}F^{\alpha \beta }\right)+{1 \over \phi }\left(\nabla
_{\mu }\nabla _{\nu }\phi -g_{\mu \nu }\Box
\phi \right)\end{aligned}}}
where
R
{\displaystyle R}
is the standard 4D Ricci scalar.
This equation shows the remarkable result,
called the "Kaluza miracle", that the precise
form for the electromagnetic stress-energy
tensor emerges from the 5D vacuum equations
as a source in the 4D equations: field from
the vacuum. This relation allows the definitive
identification of
A
μ
{\displaystyle A^{\mu }}
with the electromagnetic vector potential.
Therefore, the field needs to be rescaled
with a conversion constant
k
{\displaystyle k}
such that
A
μ
→
k
A
μ
{\displaystyle A^{\mu }\rightarrow kA^{\mu
}}
.
The relation above shows that we must have
k
2
2
=
8
π
G
c
4
1
μ
0
=
2
G
c
2
4
π
ϵ
0
{\displaystyle {k^{2} \over 2}={8\pi G \over
c^{4}}{1 \over \mu _{0}}={2G \over c^{2}}{4\pi
\epsilon _{0}}}
where
G
{\displaystyle G}
is the gravitational constant and
μ
0
{\displaystyle \mu _{0}}
is the permeability of free space. In the
Kaluza theory, the gravitational constant
can be understood as an electromagnetic coupling
constant in the metric. There is also a stress-energy
tensor for the scalar field. The scalar field
behaves like a variable gravitational constant,
in terms of modulating the coupling of electromagnetic
stress energy to spacetime curvature. The
sign of
ϕ
2
{\displaystyle \phi ^{2}}
in the metric is fixed by correspondence with
4D theory so that electromagnetic energy densities
are positive. This turns out to imply that
the 5th coordinate is spacelike in its signature
in the metric.
In the presence of matter, the 5D vacuum condition
can not be assumed. Indeed, Kaluza did not
assume it. The full field equations require
evaluation of the 5D Einstein tensor
G
~
a
b
≡
R
~
a
b
−
1
2
g
~
a
b
R
~
{\displaystyle {\widetilde {G}}_{ab}\equiv
{\widetilde {R}}_{ab}-{1 \over 2}{\widetilde
{g}}_{ab}{\widetilde {R}}}
as seen in the recovery of the electromagnetic
stress-energy tensor above. The 5D curvature
tensors are complex, and most English-language
reviews contain errors in either
G
~
a
b
{\displaystyle {\widetilde {G}}_{ab}}
or
R
~
a
b
{\displaystyle {\widetilde {R}}_{ab}}
, as does the English translation of. See
for a complete set of 5D curvature tensors
under the cylinder condition, evaluated using
tensor algebra software.
== Equations of motion from the Kaluza hypothesis
==
The equations of motion are obtained from
the five-dimensional geodesic hypothesis in
terms of a 5-velocity
U
~
a
≡
d
x
a
/
d
s
{\displaystyle {\widetilde {U}}^{a}\equiv
dx^{a}/ds}
:
U
~
b
∇
~
b
U
~
a
=
d
U
~
a
d
s
+
Γ
~
b
c
a
U
~
b
U
~
c
=
0
{\displaystyle {\widetilde {U}}^{b}{\widetilde
{\nabla }}_{b}{\widetilde {U}}^{a}={d{\widetilde
{U}}^{a} \over ds}+{\widetilde {\Gamma }}_{bc}^{a}{\widetilde
{U}}^{b}{\widetilde {U}}^{c}=0}
This equation can be recast in several ways,
and it has been studied in various forms by
authors including Kaluza, Pauli, Gross & Perry,
Gegenberg & Kunstatter, and Wesson & Ponce
de Leon,
but it is instructive to convert it back to
the usual 4-dimensional length element
c
2
d
τ
2
≡
g
μ
ν
d
x
μ
d
x
ν
{\displaystyle c^{2}d\tau ^{2}\equiv g_{\mu
\nu }dx^{\mu }dx^{\nu }}
, which is related to the 5-dimensional length
element
d
s
{\displaystyle ds}
as given above:
d
s
2
=
c
2
d
τ
2
+
ϕ
2
(
k
A
ν
d
x
ν
+
d
x
5
)
2
{\displaystyle ds^{2}=c^{2}d\tau ^{2}+\phi
^{2}(kA_{\nu }dx^{\nu }+dx^{5})^{2}}
Then the 5D geodesic equation can be written
for the spacetime components of the 4velocity,
U
ν
≡
d
x
ν
/
d
τ
{\displaystyle U^{\nu }\equiv dx^{\nu }/d\tau
}
:
d
U
ν
d
τ
+
Γ
~
α
β
μ
U
α
U
β
+
2
Γ
~
5
α
μ
U
α
U
5
+
Γ
~
55
μ
(
U
5
)
2
+
U
μ
d
d
τ
ln
⁡
(
c
d
τ
d
s
)
=
0
{\displaystyle {dU^{\nu } \over d\tau }+{\widetilde
{\Gamma }}_{\alpha \beta }^{\mu }U^{\alpha
}U^{\beta }+2{\widetilde {\Gamma }}_{5\alpha
}^{\mu }U^{\alpha }U^{5}+{\widetilde {\Gamma
}}_{55}^{\mu }(U^{5})^{2}+U^{\mu }{d \over
d\tau }\ln \left({cd\tau \over ds}\right)=0}
The term quadratic in
U
ν
{\displaystyle U^{\nu }}
provides the 4D geodesic equation plus some
electromagnetic terms:
Γ
~
α
β
μ
=
Γ
α
β
μ
+
1
2
g
μ
ν
k
2
ϕ
2
(
A
α
F
β
ν
+
A
β
F
α
ν
−
A
α
A
β
∂
ν
ln
⁡
ϕ
2
)
{\displaystyle {\widetilde {\Gamma }}_{\alpha
\beta }^{\mu }=\Gamma _{\alpha \beta }^{\mu
}+{1 \over 2}g^{\mu \nu }k^{2}\phi ^{2}(A_{\alpha
}F_{\beta \nu }+A_{\beta }F_{\alpha \nu }-A_{\alpha
}A_{\beta }\partial _{\nu }\ln \phi ^{2})}
The term linear in
U
ν
{\displaystyle U^{\nu }}
provides the Lorentz force law:
Γ
~
5
α
μ
=
1
2
g
μ
ν
k
ϕ
2
(
F
α
ν
−
A
α
∂
ν
ln
⁡
ϕ
2
)
{\displaystyle {\widetilde {\Gamma }}_{5\alpha
}^{\mu }={1 \over 2}g^{\mu \nu }k\phi ^{2}(F_{\alpha
\nu }-A_{\alpha }\partial _{\nu }\ln \phi
^{2})}
This is another expression of the "Kaluza
miracle". The same hypothesis for the 5D metric
that provides electromagnetic stress-energy
in the Einstein equations, also provides the
Lorentz force law in the equation of motions
along with the 4D geodesic equation. Yet correspondence
with the Lorentz force law requires that we
identify the component of 5-velocity along
the 5th dimension with electric charge:
k
U
5
=
k
d
x
5
d
τ
→
q
m
c
{\displaystyle kU^{5}=k{dx^{5} \over d\tau
}\rightarrow {q \over mc}}
where
m
{\displaystyle m}
is particle mass and
q
{\displaystyle q}
is particle electric charge. Thus, electric
charge is understood as motion along the 5th
dimension. The fact that the Lorentz force
law could be understood as a geodesic in 5
dimensions was to Kaluza a primary motivation
for considering the 5-dimensional hypothesis,
even in the presence of the aesthetically-unpleasing
cylinder condition.
Yet there is a problem: the term quadratic
in
U
5
{\displaystyle U^{5}}
.
Γ
~
55
μ
=
−
1
2
g
μ
α
∂
α
ϕ
2
{\displaystyle {\widetilde {\Gamma }}_{55}^{\mu
}=-{1 \over 2}g^{\mu \alpha }\partial _{\alpha
}\phi ^{2}}
If there is no gradient in the scalar field,
the term quadratic in
U
5
{\displaystyle U^{5}}
vanishes. But otherwise the expression above
implies
U
5
∼
c
q
/
m
G
1
/
2
{\displaystyle U^{5}\sim c{q/m \over G^{1/2}}}
For elementary particles,
U
5
>
10
20
c
{\displaystyle U^{5}>{\rm {10}}^{20}c}
. The term quadratic in
U
5
{\displaystyle U^{5}}
should dominate the equation, perhaps in contradiction
to experience. This was the main shortfall
of the 5-dimensional theory as Kaluza saw
it, and he gives it some discussion in his
original article.
The equation of motion for
U
5
{\displaystyle U^{5}}
is particularly simple under the cylinder
condition. Start with the alternate form of
the geodesic equation, written for the covariant
5-velocity:
d
U
~
a
d
s
=
1
2
U
~
b
U
~
c
∂
g
~
b
c
∂
x
a
{\displaystyle {d{\widetilde {U}}_{a} \over
ds}={1 \over 2}{\widetilde {U}}^{b}{\widetilde
{U}}^{c}{\partial {\widetilde {g}}_{bc} \over
\partial x^{a}}}
This means that under the cylinder condition,
U
~
5
{\displaystyle {\widetilde {U}}_{5}}
is a constant of the 5-dimensional motion:
U
~
5
=
g
~
5
a
U
~
a
=
ϕ
2
c
d
τ
d
s
(
k
A
ν
U
ν
+
U
5
)
=
c
o
n
s
t
a
n
t
{\displaystyle {\widetilde {U}}_{5}={\widetilde
{g}}_{5a}{\widetilde {U}}^{a}=\phi ^{2}{cd\tau
\over ds}(kA_{\nu }U^{\nu }+U^{5})={\rm {constant}}}
== Kaluza's hypothesis for the matter stress-energy
tensor ==
Kaluza proposed a 5D matter stress tensor
T
~
M
a
b
{\displaystyle {\widetilde {T}}_{M}^{ab}}
of the form
T
~
M
a
b
=
ρ
d
x
a
d
s
d
x
b
d
s
{\displaystyle {\widetilde {T}}_{M}^{ab}=\rho
{dx^{a} \over ds}{dx^{b} \over ds}}
where
ρ
{\displaystyle \rho }
is a density and the length element
d
s
{\displaystyle ds}
is as defined above.
Then, the spacetime component gives a typical
"dust" stress energy tensor:
T
~
M
μ
ν
=
ρ
d
x
μ
d
s
d
x
ν
d
s
{\displaystyle {\widetilde {T}}_{M}^{\mu \nu
}=\rho {dx^{\mu } \over ds}{dx^{\nu } \over
ds}}
The mixed component provides a 4-current source
for the Maxwell equations:
T
~
M
5
μ
=
ρ
d
x
μ
d
s
d
x
5
d
s
=
ρ
U
μ
q
k
m
c
{\displaystyle {\widetilde {T}}_{M}^{5\mu
}=\rho {dx^{\mu } \over ds}{dx^{5} \over ds}=\rho
U^{\mu }{q \over kmc}}
Just as the five-dimensional metric comprises
the 4-D metric framed by the electromagnetic
vector potential, the 5-dimensional stress-energy
tensor comprises the 4-D stress-energy tensor
framed by the vector 4-current.
== Quantum interpretation of Klein ==
Kaluza's original hypothesis was purely classical
and extended discoveries of general relativity.
By the time of Klein's contribution, the discoveries
of Heisenberg, Schrödinger, and de Broglie
were receiving a lot of attention. Klein's
Nature paper suggested that the fifth dimension
is closed and periodic, and that the identification
of electric charge with motion in the fifth
dimension be interpreted as standing waves
of wavelength
λ
5
{\displaystyle \lambda ^{5}}
, much like the electrons around a nucleus
in the Bohr model of the atom. The quantization
of electric charge could then be nicely understood
in terms of integer multiples of fifth-dimensional
momentum. Combining the previous Kaluza result
for
U
5
{\displaystyle U^{5}}
in terms of electric charge, and a de Broglie
relation for momentum
p
5
=
h
/
λ
5
{\displaystyle p^{5}=h/\lambda ^{5}}
, Klein obtained an expression for the 0th
mode of such waves:
m
U
5
=
c
q
G
1
/
2
=
h
λ
5
⇒
λ
5
∼
h
G
1
/
2
c
q
{\displaystyle mU^{5}={cq \over G^{1/2}}={h
\over \lambda ^{5}}\qquad \Rightarrow \qquad
\lambda ^{5}\sim {hG^{1/2} \over cq}}
where
h
{\displaystyle h}
is the Planck constant. Klein found
λ
5
∼
10
−
30
{\displaystyle \lambda ^{5}\sim {\rm {10}}^{-30}}
cm, and thereby an explanation for the cylinder
condition in this small value.
Klein's Zeitschrift für Physik paper of the
same year, gave a more-detailed treatment
that explicitly invoked the techniques of
Schroedinger and de Broglie. It recapitulated
much of the classical theory of Kaluza described
above, and then departed into Klein's quantum
interpretation. Klein solved a Schroedinger-like
wave equation using an expansion in terms
of fifth-dimensional waves resonating in the
closed, compact fifth dimension.
== Quantum field theory interpretation ==
== Group theory interpretation ==
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. Kaluza published his derivation
in 1921 as an attempt to unify electromagnetism
with Einstein's general relativity.
In 1926, Oskar Klein proposed that the fourth
spatial dimension is curled up in a circle
of a 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 construction
of this compact dimension 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). In Kaluza–Klein
theory this group suggests that gauge symmetry
is the symmetry of circular compact dimensions.
Once this geometrical interpretation is understood,
it is 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 is 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.
The construction can be outlined, roughly,
as follows. One starts by considering a principal
fiber bundle P with gauge group G over a manifold
M. Given a connection on the bundle, and a
metric on the base manifold, and a gauge invariant
metric on the tangent of each fiber, one can
construct a bundle metric defined on the entire
bundle. Computing the scalar curvature of
this bundle metric, one finds that it is constant
on each fiber: this is the "Kaluza miracle".
One did not have to explicitly impose a cylinder
condition, or to compactify: by assumption,
the gauge group is already compact. Next,
one takes this scalar curvature as the Lagrangian
density, and, from this, constructs the Einstein–Hilbert
action for the bundle, as a whole. The equations
of motion, the Euler–Lagrange equations,
can be then obtained by considering where
the action is stationary with respect to variations
of either the metric on the base manifold,
or of the gauge connection. Variations with
respect to the base metric gives the Einstein
field equations on the base manifold, with
the energy-momentum tensor given by the curvature
(field strength) of the gauge connection.
On the flip side, the action is stationary
against variations of the gauge connection
precisely when the gauge connection solves
the Yang-Mills equations. Thus, by applying
a single idea: the principle of least action,
to a single quantity: the scalar curvature
on the bundle (as a whole), one obtains simultaneously
all of the needed field equations, for both
the space-time and the gauge field.
As an approach to the unification of the forces,
it is 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
and warped models). For example, on the simplest
of principles, one might expect to have standing
waves in the extra compactified dimension(s).
If a spatial extra dimension is of radius
R, the invariant mass of such standing waves
would be Mn = nh/Rc with n an integer, h being
Planck's constant and c the speed of light.
This set of possible mass values is often
called the Kaluza–Klein tower. Similarly,
in Thermal quantum field theory a compactification
of the euclidean time dimension leads to the
Matsubara frequencies and thus to a discretized
thermal energy spectrum.
However, Klein's approach to a quantum theory
is flawed and, for example, leads to a calculated
electron mass in the order of magnitude of
the Planck mass .
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.
== 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 Space–Time–Matter
Consortium. In this version of the theory,
it is noted that solutions to the equation
R
~
a
b
=
0
{\displaystyle {\widetilde {R}}_{ab}=0}
may be re-expressed so that in four dimensions,
these solutions satisfy Einstein's equations
G
μ
ν
=
8
π
T
μ
ν
{\displaystyle G_{\mu \nu }=8\pi T_{\mu \nu
}\,}
with the precise form of the Tμν following
from the Ricci-flat condition on the five-dimensional
space. In other words, the cylinder condition
of the previous development is dropped, and
the stress–energy now comes from the derivatives
of the 5D metric with respect to the fifth
coordinate. Because the energy–momentum
tensor is normally understood to be due to
concentrations of matter in four-dimensional
space, the above result is interpreted as
saying that four-dimensional matter is induced
from geometry in five-dimensional space.
In particular, the soliton solutions of
R
~
a
b
=
0
{\displaystyle {\widetilde {R}}_{ab}=0}
can be shown to contain the Friedmann–Lemaître–Robertson–Walker
metric in both radiation-dominated (early
universe) and matter-dominated (later universe)
forms. The general equations can be shown
to be sufficiently consistent with classical
tests of general relativity to be acceptable
on physical principles, while still leaving
considerable freedom to also provide interesting
cosmological models.
== Geometric interpretation ==
The Kaluza–Klein theory has a particularly
elegant presentation in terms of geometry.
In a certain sense, it looks just like ordinary
gravity in free space, except that it is phrased
in five dimensions instead of four.
=== Einstein equations ===
The equations governing ordinary gravity in
free space can be obtained from an action,
by applying the variational principle to a
certain action. Let M be a (pseudo-)Riemannian
manifold, which may be taken as the spacetime
of general relativity. If g is the metric
on this manifold, one defines the action S(g)
as
S
(
g
)
=
∫
M
R
(
g
)
v
o
l
(
g
)
{\displaystyle S(g)=\int _{M}R(g)\mathrm {vol}
(g)\,}
where R(g) is the scalar curvature and vol(g)
is the volume element. By applying the variational
principle to the action
δ
S
(
g
)
δ
g
=
0
{\displaystyle {\frac {\delta S(g)}{\delta
g}}=0}
one obtains precisely the Einstein equations
for free space:
R
i
j
−
1
2
g
i
j
R
=
0
{\displaystyle R_{ij}-{\frac {1}{2}}g_{ij}R=0}
Here, Rij is the Ricci tensor.
=== Maxwell equations ===
By contrast, the Maxwell equations describing
electromagnetism can be understood to be the
Hodge equations of a principal U(1)-bundle
or circle bundle π: P → M with fiber U(1).
That is, the electromagnetic field F is a
harmonic 2-form in the space Ω2(M) of differentiable
2-forms on the manifold M. In the absence
of charges and currents, the free-field Maxwell
equations are
dF = 0 and d*F = 0.where * is the Hodge star.
=== Kaluza–Klein geometry ===
To build the Kaluza–Klein theory, one picks
an invariant metric on the circle S1 that
is the fiber of the U(1)-bundle of electromagnetism.
In this discussion, an invariant metric is
simply one that is invariant under rotations
of the circle. Suppose this metric gives the
circle a total length of Λ. One then considers
metrics
g
^
{\displaystyle {\widehat {g}}}
on the bundle P that are consistent with both
the fiber metric, and the metric on the underlying
manifold M. The consistency conditions are:
The projection of
g
^
{\displaystyle {\widehat {g}}}
to the vertical subspace
Vert
p
P
⊂
T
p
P
{\displaystyle {\mbox{Vert}}_{p}P\subset T_{p}P}
needs to agree with metric on the fiber over
a point in the manifold M.
The projection of
g
^
{\displaystyle {\widehat {g}}}
to the horizontal subspace
Hor
p
P
⊂
T
p
P
{\displaystyle {\mbox{Hor}}_{p}P\subset T_{p}P}
of the tangent space at point p ∈ P must
be isomorphic to the metric g on M at π(p).The
Kaluza–Klein action for such a metric is
given by
S
(
g
^
)
=
∫
P
R
(
g
^
)
vol
(
g
^
)
{\displaystyle S({\widehat {g}})=\int _{P}R({\widehat
{g}})\;{\mbox{vol}}({\widehat {g}})\,}
The scalar curvature, written in components,
then expands to
R
(
g
^
)
=
π
∗
(
R
(
g
)
−
Λ
2
2
|
F
|
2
)
{\displaystyle R({\widehat {g}})=\pi ^{*}\left(R(g)-{\frac
{\Lambda ^{2}}{2}}\vert F\vert ^{2}\right)}
where π* is the pullback of the fiber bundle
projection π: P → M. The connection A on
the fiber bundle is related to the electromagnetic
field strength as
π
∗
F
=
d
A
{\displaystyle \pi ^{*}F=\mathrm {d} A}
That there always exists such a connection,
even for fiber bundles of arbitrarily complex
topology, is a result from homology and specifically,
K-theory. Applying Fubini's theorem and integrating
on the fiber, one gets
S
(
g
^
)
=
Λ
∫
M
(
R
(
g
)
−
1
Λ
2
|
F
|
2
)
vol
(
g
)
{\displaystyle S({\widehat {g}})=\Lambda \int
_{M}\left(R(g)-{\frac {1}{\Lambda ^{2}}}\vert
F\vert ^{2}\right)\;{\mbox{vol}}(g)}
Varying the action with respect to the component
A, one regains the Maxwell equations. Applying
the variational principle to the base metric
g, one gets the Einstein equations
R
i
j
−
1
2
g
i
j
R
=
1
Λ
2
T
i
j
{\displaystyle R_{ij}-{\frac {1}{2}}g_{ij}R={\frac
{1}{\Lambda ^{2}}}T_{ij}}
with the stress–energy tensor being given
by
T
i
j
=
F
i
k
F
j
l
g
k
l
−
1
4
g
i
j
|
F
|
2
,
{\displaystyle T^{ij}=F^{ik}F^{jl}g_{kl}-{\frac
{1}{4}}g^{ij}\vert F\vert ^{2},}
sometimes called the Maxwell stress tensor.
The original theory identifies Λ with the
fiber metric g55, and allows Λ to vary from
fiber to fiber. In this case, the coupling
between gravity and the electromagnetic field
is not constant, but has its own dynamical
field, the radion.
=== Generalizations ===
In the above, the size of the loop Λ 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 technique of introducing
compact dimensions to obtain a higher-dimensional
manifold is referred to as compactification.
Compactification does not produce group actions
on chiral fermions except in very specific
cases: the dimension of the total space must
be 2 mod 8 and the G-index of the Dirac operator
of the compact space must be nonzero.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 super-symmetric
Yang–Mills theory.
== Empirical tests ==
No experimental or observational signs of
extra dimensions have been officially reported.
Many theoretical search techniques for detecting
Kaluza–Klein resonances have been proposed
using the mass couplings of such resonances
with the top quark. However, until the Large
Hadron Collider (LHC) reaches full operational
power, observation of such resonances are
unlikely. An analysis of results from the
LHC in December 2010 severely constrains theories
with large extra dimensions.The observation
of a Higgs-like boson at the LHC establishes
a new empirical test which can be applied
to the search for Kaluza–Klein resonances
and supersymmetric particles.
The loop Feynman diagrams that exist in the
Higgs interactions allow any particle with
electric charge and mass to run in such a
loop. Standard Model particles besides the
top quark and W boson do not make big contributions
to the cross-section observed in the H → γγ
decay, but if there are new particles beyond
the Standard Model, they could potentially
change the ratio of the predicted Standard
Model H → γγ cross-section to the experimentally
observed cross-section. Hence a measurement
of any dramatic change to the H → γγ cross
section predicted by the Standard Model is
crucial in probing the physics beyond it.
Another more recent paper from July 2018 does
bode some hope for this theory; In the paper
they dispute that gravity is leaking into
higher dimensions as in brane theory. However
the paper does demonstrate that EM and Gravity
share the same number of dimensions and this
fact lends support to Kaluza–Klein theory,
whether the number of dimensions is really
3+1 or in fact 4+1 is the subject of further
debate.
== See also ==
Classical theories of gravitation
Complex spacetime
DGP model
Quantum gravity
Randall–Sundrum model
String theory
Supergravity
Superstring theory
== Notes
