# Which symmetry group for elementary particles with an electric charge today
and in the past?

G. de Saxcé
Univ. Lille, CNRS, Centrale Lille, UMR 9013 – LaMcube
Laboratoire de mécanique multiphysique multiéchelle,
F-59000, Lille, France, Email<EMAIL_ADDRESS>
###### Abstract

In this paper, we revisit the Kaluza-Klein theory from the perspective of the
classification of elementary particles based on the coadjoint orbit method.
The keystone conjecture is to consider the electric charge as an extra
momentum on an equal footing with the mass and the linear momentum. We study
the momentum map of the corresponding symmetry group $\hat{\mathbb{G}}_{1}$
which conserves the hyperbolic metric. We show that the electric charge is not
an invariant, i.e. it depends on the reference frame, which is in
contradiction with the experimental observations. In other words, it is not
the symmetry group of the Universe today as we know it. To avert this paradox,
we scale the fifth coordinate and consider the limit when the cylinder radius
$\omega$ vanishes. For the corresponding group $\hat{\mathbb{G}}_{0}$ also of
dimension 15, the charge is an invariant then independent of the frame of
reference and the observer. On this ground, we propose a cosmological scenario
in which the elementary particles of the early Universe are classified from
the momenta of the group $\hat{\mathbb{G}}_{1}$, next the three former
dimensions inflate quickly while the fifth one shrinks, leading to the 4D era
in which as today the particles are characterized by the momenta of the group
$\hat{\mathbb{G}}_{0}$. By this mechanism, the elementary particles can
acquire electric charge as a by-product of the $4+1$ symmetry breaking of the
Universe. This work opens the way to the geometric quantization of charged
elementary particles. We construct the corresponding
$\hat{\mathbb{G}}_{0}$-connections by pullback on the space-time. Imposing
that the linear 5-momentum is parallel-transported, we recover the
conservation of the charge and the equation of motion with Lorentz force. We
revisit the variational relativity and obtain the field equations for both the
gravitation and electromagnetic interactions with coupling terms which are
negligible at the Newtonian approximation, allowing to recover Maxwell
equations.

Keywords: Kaluza-Klein relativity, symplectic mechanics, cosmology, coadjoint
orbit method, elementary particles, electric charge.

MSC Codes  22E70; 37J15; 83E15; 83F05

##  1 Introduction

Kaluza-Klein is an extension of general relativity to unify the gravitation
and electromagnetism, considering a Riemannian manifold of dimension 5 by
completing the 4 coordinates $(X^{1},\cdots,X^{4})$ of the space-time with an
extra spacelike dimension coordinate $\hat{X}^{5}$. The pioneering paper,
Kaluza [1915], started out by interpreting the 1 + 5 metric as follows111Our
convention in this paper is that convention that Greek indices run from 1 to 5
and Latin ones run from 1 to 4

$[\hat{G}_{\mu\nu}]=\left[{{\begin{array}[]{cc}G_{ij}\hfill&2\,\alpha
A_{i}\hfill\\\ 2\,\alpha A_{j}\hfill&2\,\phi\hfill\\\ \end{array}}}\right]$

where $G_{ij}$ is the space-time metric, $A_{i}$ is the electromagnetic
potential, $\phi$ is a new field, often called dilaton, and $\alpha$ is a
coupling constant. Besides, Theodor Kaluza introduce the so-called cylinder
condition , i.e. the components of the metric and other tensors do not depend
on the fifth coordinate. Even so, the expression of the contravariant metric
$\hat{G}^{\alpha\beta}$ in terms of the $\hat{G}_{\alpha\beta}$ being complex,
he uses the linearized weak field approximation. The components of the Ricci
tensor read

$[\hat{R}_{\mu\nu}]=\left[{{\begin{array}[]{cc}\partial_{k}\Gamma^{k}_{ij}-\partial_{j}\Gamma^{k}_{ik}\hfill&-\alpha\,\partial_{k}F_{ki}\hfill\\\
-\alpha\,\partial_{k}F_{kj}\hfill&-\square\,\phi\hfill\\\
\end{array}}}\right]$

where $F_{ij}=\partial_{i}A_{j}-\partial_{j}A_{i}$ is the electromagnetic
field and $\square=G^{ij}\nabla_{i}\nabla_{j}$. With the energy-momentum
tensor of the dust matter
$\hat{T}_{\mu\nu}=\rho_{m}\hat{U}_{\mu}\hat{U}_{\nu}$ as source, Einstein
field equations extended to 5D

$\hat{R}_{ij}-\frac{1}{2}\,(\hat{G}^{kl}\hat{R}_{kl})\,\hat{G}_{ij}=-\kappa\,\hat{T}_{ij}$

allow to recover with $R_{ij}$ the field equations of the gravitation exactly,
Maxwell’s equations also with $\hat{R}_{i5}$ but at the cost of an hypothesis
that turned out, as remarked Einstein, to be untenable (grossly superluminal
velocity of electrons in the fifth dimension). In contrast, Lorentz force
appears naturally in the equation of motion of a particle but, because of the
above hypothesis, the last term of this equation and also the equation with
$\hat{R}_{55}$ are problematic.

After Kaluza’s promising paper but suffering several shortcomings, another
breakthrough is the work of Oskar Klein [1926a, b] who adopts also the
cylinder condition but with restrictions on the coordinate changes to
$X^{i}=H^{i}(X^{\prime 1},\cdots,X^{\prime 4}),\;\hat{X}^{5}=\hat{X}^{\prime
5}+h(X^{\prime 1},\cdots,X^{\prime 4})$ and then the following form the metric

$[\hat{G}_{\mu\nu}]=\left[{{\begin{array}[]{cc}G_{ij}+\phi^{2}A_{i}A_{j}\hfill&\phi^{2}A_{i}\hfill\\\
\phi^{2}A_{j}\hfill&\phi^{2}\hfill\\\ \end{array}}}\right]$

which leads to a more tractable expression of the contravariant metric. In
absence of source terms from the matter, the field equations are

$R_{ij}-\frac{1}{2}\,(G^{kl}R_{kl})\,G_{ij}=\frac{1}{2}\,\kappa\,\phi^{2}[F^{ik}F^{j}_{k}-\frac{1}{4}\,(F^{kl}F_{kl})\,G_{ij}]-\frac{1}{\phi}\,[\nabla_{i}\partial_{j}\phi-(\square\phi)\,G_{ij}]$
$\nabla_{k}F^{k}_{i}+\frac{3}{\phi}\,(\partial_{k}\phi)F^{k}_{i}=0$
$\square\phi=\frac{1}{2}\,\kappa^{3}(F^{kl}F_{kl})$

One of the interests of the formulation is that the electromagnetic stress-
energy tensor occurs naturally at the right hand member of the first equation.
Another interest is that the second equation provides Maxwell equation when
the dilaton $\phi$ is set to one, but if so the last equation is satisfied
only if the right hand member is null, i.e. under the too restrictive
constraint that the norms of the electric and magnetic fields are equal. In
contrast, if the dilaton varies, the field equations have no classical
interpretation. If we continue anyway to set $\phi$ to one, the equation of
geodesics reads

$\nabla U^{i}=U_{5}\,F^{i}_{j}U^{j}$

Owing to the cylinder condition, $U_{5}$ is constant along the geodesic, hence
the idea to put $U_{4}=q/m_{0}$ where $q$ is the electric charge and $m_{0}$
the rest mass, that leads to the equation of motion

$m_{0}\nabla U^{i}=q\,F^{i}_{j}U^{j}$

where the right hand member is just Lorentz force.

However, the discoveries of the starting quantum mechanics suggested to Klein
that the fifth dimension is curled up and microscopic. Using de Broglie
wavelength $\lambda^{5}=h/p^{5}$ where $h$ is Planck constant and $p^{5}$ is
the linear momentum in the fifth dimension, Klein found that the size $l_{K}$
of the space along the fifth dimension was about $10^{-30}$ cm, very small
with respect to the nuclear dimension (of the order of $10^{-13}$ cm) and
thereby gave an explanation for the cylinder condition in this small value.
For a comparison between Kaluza and Klein approaches, Sewards [2008] can be
consulted. Classical theory was completed in the 1940s and the full field
equations including the scalar field were obtained almost simultaneously
(Thiry [1948a, b], Jordan [1948], Scherrer [1949]).

Albert Einstein’s position with respect to Kaluza-Klein theory has changed
during his scientific career. In Einstein [1920], when giving his inaugural
lecture in Leyden, Einstein for the first time publicly commented positively
on this unification program. However, in a letter to H. Weyl [1922], he
regrets that it cannot produce non-singular rotation symmetric particle
solutions. Later on, his work with W. Mayer [1931] is a a first attempt to
propose an unified field theory. Commenting their work, É. Cartan [1934]
interprets the torsion of the 5D space as the electromagnetic field. The last
Einstein’s works on the topics are in collaboration with P. Bergmann [1938]
and later on with P. Bergmann and V. Bargmann [1941]. For more details, the
reader is referred to van Dongen [2000].

Despite of the astonishing breakthrough of the Kaluza-Klein theory (emergence
of Lorentz force, Maxwell equations and electromagnetic energy stress tensor
from the general relativity), often qualified of ”Kaluza miracles”, it suffers
of several serious shortfalls not yet rectified. Nonetheless this theory has
given rise to many developments that it is not possible to cite all. We are
focused on the topics relevant to the issues dealt with in the present paper.

In the modern language of differential geometry, Oskar Klein’s construction is
formalized as a principal $U(1)$-bundle. The electromagnetic potential of
covariant components $A_{i}$ can be interpreted as a connection 1-form valued
in the Lie algebra of the group. From the viewpoint of the physics of
elementary particles, a relevant generalization of the Kaluza-Klein theory
consists in substituting an arbitrary non-abelian group $G$ for $U(1)$. The
connection 1-form has coordinates $A^{a}_{j}$ and the curvature form in a
coordinate frame is

$F^{a}_{ij}=\frac{1}{2}\,(\partial_{i}A^{a}_{j}-\partial_{j}A^{a}_{i})+C^{a}_{bc}A^{b}_{i}A^{c}_{j}$

where $C^{a}_{bc}$ are the structure constants of the group $G$. It must
contain at least a $U(1)$ subgroup for the conservation of the electric charge
and a $SU(2)$ subgroup for the conservation of the isospin (Kerner [1981]).

Initially introduced by Bargmann [1954] to solve cohomologic difficulties to
construct group actions on quantum wave functions in the Galilean setting, the
Lie group that is named after him acts on a 5D space of which the fifth
coordinate has the physical meaning of a specific action (in the sense of
Hamilton least action principle), then it has not to be confused with Kaluza-
Klein space. Bargmannian geometric structures have been studies by several
authors (Duval et al. [1985, 1991], de Saxcé and Vallée [2010, 2016], Cardall
[2024]). Despite of the differences with Kaluza-Klein theory, some aspects of
the Bargmannian approach were a source of inspiration for this work.

Another issue which has grasped researcher attention is the Kaluza-Klein
cosmology. A generalized Friedmann-Lemaître-Robertson-Walker metric in 5D were
introduced by Chodos and Detweiler [1980]. The idea is that while the fifth
dimension has been shrinking, the other three spatial dimensions have been
expanding, that could explain the large ratio of the electromagnetic to
gravitational forces as a consequence of the age of the Universe, in agreement
with Dirac’s large-number hypothesis. Sahdev [1984] obtained improved
solutions of the field equations with perfect fluids added on the right-hand
side and found that this scenario can offer a resolution to the horizon
problem. However, instead of stabilizing at some small but finite value, as
any reasonable physics would require, the internal radius in the fifth
dimension tended to zero. Other models attempting to stabilize asymptotically
the internal radius were proposed by Matzner and Mezzacappa [1985], Copeland
and Toms [1985], Okada [1986], Giorgini and Kerner [1988]. These interesting
works did not necessarily receive the welcome they deserved, perhaps because
they were based on the theory of Kaluza-Klein whose weaknesses were known.

Finally, not developed from Kaluza-Klein theory but playing a key role in this
work, we have to point out Jean-Marie Souriau’s seminal work in which he
proposed for the classical description of an elementary particle to study a
family of homogeneous spaces for a symmetry group, the coadjoint orbits, and
to classify them (Souriau [1970, 1997b]). On this ground, he introduced the
geometric prequantization in Souriau [1966]. As Kaluza-Klein theory is based
on an $U(1)$-principal bundle of which the base manifold is the space-time,
the key idea of the prequantization is to work with a $U(1)$-principal bundle
of which the base space is a symplectic manifold, a coadjoint orbit of the
structure group. A tentative to combine this method with the 5D approach to
treat the mass and the electric charge on an equal footing is proposed by
Śniatycki [1980] but, in contrast to Kaluza-Klein theory, the fifth dimension
is timelike without physical justification. The reader not familiar with this
topic can consult, in addition to Souriau’s references previously cited,
Kostant [1970] who is with Souriau one of the pioneers of this method,
Kijowski [1977] and Woodhouse [1991].

The paper is structured as follows.

Sections 2 and 3 cover the background material needed to tackle the core of
the paper as well as to define the terminology and notations. The reader
familiar with one or several of these topics can skim through them. Section 2
is devoted to Euclidean spaces, pointing out adjoint maps, Hodge star operator
and a generalization of the vector product to arbitrary dimension. In Section
3, we recall useful main results of symplectic mechanics, namely the coadjoint
orbit method and the momentum map. The Section 4 is a brief reminder of the
application of this method to Poincaré group allowing to classify the
elementary particles in relativity.

After these preliminaries, the main body of the paper is a trilogy:
classification of the particles in Kaluza-Klein theory, pullback connection on
the space-time, extended variational relativity.

In Section 5, we apply the coadjoint orbit method to the symmetry group of
Kaluza-Klein 5D space, denoted $\hat{\mathbb{G}}_{1}$, which conserves the
hyperbolic metric. We show that the linear momentum along the fifth dimension,
interpreted as the electric charge, is not preserved by the group, then
depends on the reference frame, contradicting the observations. The aim of
Section 6 is to avert this paradox by using the fact that the fifth dimension
is microscopic. The result of the zoom in is to reveal a new symmetry group
denoted $\hat{\mathbb{G}}_{0}$ for which the charge is invariant. Section 7
begin with the presentation of a cosmological scenario in which the particles
in the early 5D Universe are classified according to the group
$\hat{\mathbb{G}}_{1}$ and, after a transition phase in which the three former
spatial dimensions inflate quickly while the fifth one shrinks, the particles
in the 4D era are classified thanks to $\hat{\mathbb{G}}_{0}$. By this
mechanism, the elementary particles can acquire electric charge as a by-
product of the 4 + 1 symmetry breaking of the Universe. Next, we apply the
coadjoint orbit method to the group $\hat{\mathbb{G}}_{\omega}$ for the
transition between $\hat{\mathbb{G}}_{1}$ and $\hat{\mathbb{G}}_{0}$. In 5D,
we generalize the spin polarisation in the form of a plane.

The second part of the trilogy is developed in Section 8. Our aim is to
construct the $\hat{\mathbb{G}}_{0}$-connections on the frame principal
bundle. As the zoom out leads to a singularity in the fifth dimension, we
build on the space-time a pullback connection. We claim that the motion of a
charged particle and the evolution of its charge are such that its linear
5-momentum is parallel-transported. The torsion free condition allows to show
that the charge is conserved along the trajectory and we recover the Lorentz
force.

The trilogy ends with Section 9 where we revisit Palatini variational
relativity by adding to the ten potentials of the metric the four
electromagnetic potentials. The generalized action depends on the matter and
the connection representing both the gravitation and electromagnetic
interactions. We discuss the field equations with coupling terms between the
two interactions. At the Newtonian approximation, the coupling is weak and we
recover Maxwell equations.

##  2 Euclidean spaces

The two former subsection are simple reminder to define the terminology and
notations used in the paper.

###  2.1 Metric

An Euclidean space $\mathcal{T}$ of dimension $n$ is a vector space equipped
with a (covariant) metric $\bm{G}$, i.e. a nondegenerate 2-covariant tensor
$\bm{G}$

$\forall\bm{U}\in\mathcal{T},\qquad\bm{G}(\bm{U},\bm{V})=0\quad\Leftrightarrow\quad\bm{V}=\bm{0}\
.$

The value of the metric tensor for $\bm{U}$ and $\bm{V}$ is called their
scalar product and denoted $\bm{U}\cdot\bm{V}$. Let $(\bm{e}_{i})$ be a basis
of $\mathcal{T}$. The symmetric regular matrix $G$ gathering the components
$G_{ij}=\bm{e}_{i}\cdot\bm{e}_{j}$ is called Gram’s matrix. Thus, it holds:

$\bm{G}(\bm{U},\bm{V})=\bm{U}\cdot\bm{V}=U^{T}G\,V\ ,$

where $U$ and $V$ are the columns gathering respectively the components of
$\bm{U}$ and $\bm{V}$. If $G$ is diagonal with element $+1$ or $-1$ on the
diagonal, the basis is called orthonormal. The number $p$ of positive number
on the diagonal is called the positive index of inertia. If $n\geq 2$ and
$p=1$, the space is said hyperbolic. To every vector $\bm{U}$ is associated
one and only one linear form $\bm{V}\mapsto\bm{G}(\bm{U},\bm{V})$ denoted
$\bm{U}^{*}$. The covariant components of $\bm{U}^{*}$ depends on the
contravariant components of $\bm{U}$ through the operation of lowering the
index: $U_{i}=G_{ij}U^{j}$. The elements $G^{ij}$ of the inverse $G^{-1}$ of
Gram’s matrix are the components of a $2$-contravariant tensor $\bm{G}^{-1}$
called contravariant metric, hence the reverse operation of raising the index:
$U^{i}=G^{ij}U_{j}$.

###  2.2 Adjoint of a linear map

Let $\bm{A}$ be a linear map from an Euclidean space $\mathcal{T}_{0}$ into
another one $\mathcal{T}$. Its adjoint (with respect to the scalar products)
is the linear map $\bm{A}^{*}$ from $\mathcal{T}$ into $\mathcal{T}_{0}$ such
that:

$\forall\bm{U}\in\mathcal{T},\quad\forall\bm{V}\in\mathcal{T}_{0},\qquad\bm{U}\cdot(\bm{A}\,\bm{V})=(\bm{A}^{*}\bm{U})\cdot\bm{V}\
.$

If $\bm{A}$ is represented by the matrix $A$ in bases of $\mathcal{T}_{0}$ and
$\mathcal{T}$, $\bm{A}^{*}$ is represented by:

$A^{*}=G_{0}^{-1}A^{T}G\ .$ (1)

We verify that:

$(\bm{A}+\bm{A}^{\prime})^{*}=\bm{A}^{*}+\bm{A}^{\prime*},\qquad(\bm{A}\,\bm{B})^{*}=\bm{B}^{*}\bm{A}^{*},\qquad(\bm{A}^{*})^{*}=\bm{A}\
.$

In particular, if $\mathcal{T}_{0}=\mathbb{R}$, the linear map
$\bm{U}:\mathbb{R}\rightarrow\mathcal{T}:\lambda\mapsto\lambda\,\bm{U}$ can be
identified to the vector $\bm{U}$ and $\bm{U}^{*}$ is the unique linear form
associated to $\bm{U}$ with respect to the metric since (1) degenerates into

$U^{*}=U^{T}G$ (2)

which is the matrix form of index lowering. Then the scalar product of two
vectors reads: $\bm{U}\cdot\bm{V}=U^{*}\,V$. Another particular case of
interest is when $\mathcal{T}_{0}=\mathcal{T}$ then (1) reads:

$A^{*}=G^{-1}A^{T}G$ (3)

We verify that $Tr(\bm{A}^{*})=Tr(\bm{A})$. The linear map is self-adjoint
(resp. anti-self-adjoint or skew-adjoint) if:

$\bm{A}=\bm{A}^{*}\qquad(\mbox{resp.}\quad\bm{A}=-\bm{A}^{*})\ .$

###  2.3 Vector product

We consider an oriented Euclidean space $\mathcal{T}$ of dimension $n$, i.e.
there exists a volume form $vol\in\bigwedge^{n}\mathcal{T}^{*}$ such that
$vol(\bm{e}_{1},\ldots,\bm{e}_{n})=1$ for every orthonormal basis
$(\bm{e}_{i})$. We denote $vol(\bm{e}_{1},\ldots,\bm{e}_{q})$ the $(n-q)$-form
such that

$(vol(\bm{e}_{1},\ldots,\bm{e}_{q}))(\bm{e}_{q+1},\ldots,\bm{e}_{n})=vol(\bm{e}_{1},\ldots,\bm{e}_{q},\bm{e}_{q+1},\ldots,\bm{e}_{n})$

We call vector product of $(n-1)$ vectors $\bm{V}_{1},\ldots,\bm{V}_{n-1}$ the
vector $\mathcal{J}(\bm{V}_{1},\ldots,\bm{V}_{n-1})$ such that

$\mathcal{J}(\bm{V}_{1},\ldots,\bm{V}_{n-1})^{*}\bm{U}=vol(\bm{V}_{1},\ldots,\bm{V}_{n-1},\bm{U})$

In Souriau [1965, §26.B], it is proved that

######  Theorem 2.1 (Souriau)

Properties of the vector product:

  * $\diamondsuit$

$\mathcal{J}(\bm{V}_{1},\ldots,\bm{V}_{n-1})\neq\bm{0}$ if and only if
$\bm{V}_{1},\ldots,\bm{V}_{n-1}$ are linearly independent

  * $\heartsuit$

$\mathcal{J}(\bm{V}_{1},\ldots,\bm{V}_{n-1})$ is orthogonal to every argument
$\bm{V}_{i}$ of $\mathcal{J}$

  * $\spadesuit$

The linear map
$\mathcal{J}(\bm{V}_{1},\ldots,\bm{V}_{n-2}):\mathcal{T}\rightarrow\mathcal{T}:\bm{V}\mapsto\mathcal{J}(\bm{V}_{1},\ldots,\bm{V}_{n-2},\bm{V})$
is one-to-one and skew-adjoint

Denoting $vol_{i_{1}\ldots i_{n}}$ the covariant components of the volume form
of an oriented Euclidean space of dimension $n$ and $V^{i}_{r}$ the $i$-th
component of the vector $\bm{V}_{r}$, the $j$-th component of the vector
product is given by

$(\mathcal{J}(\bm{V}_{1},\ldots,\bm{V}_{n-1}))^{j}=\sum_{i_{1}\ldots
i_{n-1}}\,vol_{i_{1}\ldots i_{n-1}}\,^{j}\,\,V^{i_{1}}_{1}\ldots
V^{i_{n-1}}_{n-1}$

where we rise the last index of the volume form with the metric. The element
of the matrix $\mathcal{J}(\bm{V}_{1},\ldots,\bm{V}_{n-2})$ at the
intersection of the $j$-th row and the $k$-th column is

$(\mathcal{J}(\bm{V}_{1},\ldots,\bm{V}_{n-2}))^{j}_{k}=\sum_{i_{1}\ldots
i_{n-1}}\,vol_{i_{1}\ldots i_{n-2}k}\,^{j}\,\,V^{i_{1}}_{1}\ldots
V^{i_{n-2}}_{n-2}$

Now, we would like to obtain a recursive formula to calculate the vector
product when we add an extra dimension. If
$V,V_{i}\in\mathcal{T}=\mathbb{R}^{n}$ , we put

$\hat{V}=\left[{{\begin{array}[]{cc}V\hfill\\\ v\hfill\\\
\end{array}}}\right],\hat{V}_{i}=\left[{{\begin{array}[]{cc}V_{i}\hfill\\\
v_{i}\hfill\\\ \end{array}}}\right]\in\hat{\mathcal{T}}=\mathbb{R}^{n+1}$

Then, it can be proved that

$\hat{\mathcal{J}}(\hat{V}_{1},\ldots,\hat{V}_{n})=\left[\sum^{n}_{k=1}(-1)^{n-k+1}\,v_{k}\,\mathcal{J}(V_{1},\ldots,V_{k-1},V_{k+1},\ldots,V_{n})^{*},vol(V_{1},\ldots,V_{n})\right]^{*}$
(4)

In the particular case of the classical positive Euclidean space of dimension
3 ($n=q=3$), we use the notation $j$ instead of $\mathcal{J}$ and we recover
the cross product $j(u,v)=u\times v$ and the linear map $u\mapsto j(u)$ is
one-to-one from $\mathbb{R}^{3}$ into the space of $3\times 3$ skew-symmetric
matrices. In the sequel, the notation $\mathcal{J}$ will be reserved to the
dimension 4 and $\hat{\mathcal{J}}$ to the dimension 5. In $\mathbb{R}^{4}$
equipped with the metric of Gram’s matrix $G=diag(1,-1,-1,-1)$, the vector
product of 3 vectors

$\Pi_{k}=\left[{{\begin{array}[]{cc}m_{k}\hfill\\\ p_{k}\hfill\\\
\end{array}}}\right],\qquad m_{k}\in\mathbb{R},\quad
p_{k}\in\mathbb{R}^{3}\qquad(1\leq k\leq 3)$

itemizes as

$\mathcal{J}(\Pi_{1},\Pi_{2},\Pi_{3})=\left[{{\begin{array}[]{cc}vol(p_{1},p_{2},p_{3})\hfill\\\
m_{1}j(p_{2},p_{3})-m_{2}j(p_{1},p_{3})+m_{3}j(p_{1},p_{2})\hfill\\\
\end{array}}}\right]$ (5)

from which we deduce

$\mathcal{J}(\Pi_{1},\Pi_{2})=\left[{{\begin{array}[]{cc}0\hfill&j(p_{1},p_{2})^{T}\hfill\\\
j(p_{1},p_{2})\hfill&m_{1}j(p_{2})-m_{2}j(p_{1})\hfill\\\
\end{array}}}\right]$ (6)

In $\mathbb{R}^{5}$ equipped with the metric of Gram’s matrix
$\hat{G}=diag(1,-1,-1,-1,-\omega^{2})$, let us consider 3 vectors

$\hat{\Pi}_{k}=\left[{{\begin{array}[]{cc}\Pi_{k}\hfill\\\ q_{k}\hfill\\\
\end{array}}}\right],\qquad\Pi_{k}\in\mathbb{R}^{4},\quad
q_{k}\in\mathbb{R}\qquad(1\leq k\leq 3)$

Then

$\hat{\mathcal{J}}(\hat{\Pi}_{1},\hat{\Pi}_{2},\hat{\Pi}_{3})=\left[{{\begin{array}[]{cc}q_{1}\mathcal{J}(\Pi_{2},\Pi_{3})-q_{2}\mathcal{J}(\Pi_{1},\Pi_{3})+q_{3}\mathcal{J}(\Pi_{1},\Pi_{2})\hfill&-j(\Pi_{1},\Pi_{2},\Pi_{3})\hfill\\\
-\omega^{-2}j(\Pi_{1},\Pi_{2},\Pi_{3})^{*}\hfill&0\hfill\\\
\end{array}}}\right]$ (7)

###  2.4 Hodge operator

Let us consider the oriented Euclidean space $\mathcal{T}=\mathbb{R}^{n}$ of
positive index of inertia $p$. Owing (2), the skew-adjoint map $M$ can be
identified to the 2-form $A_{M}$ through

$U^{*}M\,V=U^{T}A_{M}V=A_{M}(U,V)\quad\mbox{with}\quad A_{M}=G\,M$

Then $M=\mathcal{J}(V_{1},\ldots,V_{n-2})$ can be identified to the 2-form
$A_{M}=-vol(V_{1},\ldots,V_{n-2})$. The vector space
$\bigwedge^{q}\mathcal{T}$ of $q$-forms is an Euclidean space for the scalar
product

$G_{q}(A,B)=\frac{1}{q!}\,\sum_{\begin{array}[]{cc}i_{1}\ldots i_{q}\\\
j_{1}\ldots j_{q}\\\ \end{array}}G^{i_{1}j_{1}}\ldots
G^{i_{q}j_{q}}\,A_{i_{1}\ldots i_{q}}B_{j_{1}\ldots j_{q}}$

The adjoint222To do not confuse with the adjoint of a linear map. of a
$q$-form $A$ is the $(n-q)$-form $*A$ such that

$(*A)(V_{1},\ldots,V_{n-q})=(-1)^{q\,(n-q)}G_{q}(A,vol(V_{1},\ldots,V_{n-q}))$

The linear map
$*:\bigwedge^{q}\mathcal{T}\rightarrow\bigwedge^{n-q}\mathcal{T}:A\mapsto*A$
is called Hodge operator. The following result can be proved:

###### Theorem 2.2

Properties of the Hodge operator:

  * $\diamondsuit$

$M=\mathcal{J}(V_{1},\ldots,V_{n-2})$ is identified to the 2-form

$A_{M}=*(V^{*}_{1}\wedge\ldots\wedge V^{*}_{n-2})=-vol(V_{1},\ldots,V_{n-2})$

  * $\heartsuit$

If $A$ is a $q$-form, $*(*A)=(-1)^{q(n-1)+n-p}\,A$

  * $\spadesuit$

$*vol(V_{1},\ldots,V_{n-2})=(-1)^{n-p}\,V^{*}_{1}\wedge\ldots\wedge
V^{*}_{n-2}$

  * $\natural$

$*(V^{*}\wedge U^{*})=-vol(U,V)$

In the special case of interest : $n=4,q=2$, if $A_{M}$ is a 2-form, $*A_{M}$
is also a 2-form identified to an skew-adjoint map denoted $*M$. If
$V^{*}_{1}\wedge V^{*}_{2}=A_{M}^{\prime}$, then
$M^{\prime}=G^{-1}(V^{*}_{1}\otimes V^{*}_{2}-V^{*}_{2}\otimes
V^{*}_{1})=V_{1}V^{*}_{2}-V_{2}V^{*}_{1}$. On the other hand, $\diamondsuit$
reads $A_{\mathcal{J}(V_{1},V_{2})}=*A_{M}^{\prime}=A_{*M^{\prime}}$ then:

$*(V_{1}V^{*}_{2}-V_{2}V^{*}_{1})=\mathcal{J}(V_{1},V_{2})$ (8)

that implies, owing to $\heartsuit$

$*\mathcal{J}(V_{1},V_{2})=(-1)^{n-p}(V_{1}V^{*}_{2}-V_{2}V^{*}_{1})$ (9)

##  3 Coadjoint orbit method to classify the elementary particles

This Section is a simple reminder of concepts of symplectic mechanics. For
more details on this topics, the reader can consult for instance Souriau
[1970], Abraham and Marsden [1978], Guillemin and Sternberg [1984], Libermann
and Marle [1987], Souriau [1997b]. In the sequel, all the considered Lie
groups are matrix groups. Any Lie group $G$ left linearly acts on its Lie
algebra $\mathfrak{g}$ by the adjoint representation

$Ad\left(a\right):\mathfrak{g}\rightarrow\mathfrak{g}:Z^{\prime}\mapsto
Z=aZ^{\prime}a^{-1}$ (10)

$G$ left linearly acts on the dual $\mathfrak{g}^{*}$ of $\mathfrak{g}$ by the
coadjoint representation $Ad^{*}$ such that

$\forall
Z\in\mathfrak{g},\forall\mu\in\mathfrak{g}^{*},\qquad\left(Ad^{*}(a)\mu\right)\left(Z\right)=\mu\left(Ad\left(a^{-1}\right)Z\right)$
(11)

Let $\left(\mathcal{M},\varpi\right)$ be a symplectic manifold, a symplectic
action of a Lie group $G$

$G\times\mathcal{M}\rightarrow\mathcal{M}:\left(a,x\right)\mapsto
x^{\prime}=a\cdot x$

of a Lie group $G$ on $\mathcal{M}$

$L^{*}_{a}\varpi=\varpi$

and a momentum mapping $\psi:\mathcal{M}\rightarrow\mathfrak{g}^{*}$

$\forall Z\in\mathfrak{g},\,\forall dx\in
T_{x}\mathcal{M},\quad\varpi\left(Z\cdot
x,dx\right)=-d\left(\psi\left(x\right)Z\right)$

If the manifold $\mathcal{M}$ is connected, two momentum mappings differ by a
constant. All the symplectic manifolds considered in the sequel are assumed to
be connected. We consider now $\mathfrak{g}^{*}$ as equipped with the
associated structure of affine space. The momentum mappings belong to the
affine space of the mappings from $\mathcal{M}$ into $\mathfrak{g}^{*}$,
called the momentum mapping space. Any element $A$ of the affine group
$\mathbb{GA}\left(\mathfrak{g}^{*}\right)$ of $\mathfrak{g}^{*}$ is of the
form $A\left(\mu\right)=P\mu+\mu_{0}$ where
$P\in\mathbb{GL}\left(\mathfrak{g}^{*}\right)$ and
$\mu_{0}\in\mathfrak{g}^{*}$. The action of $G$ on $\mathcal{M}$ induces a
smooth right action on the momentum mapping space

$\forall x\in\mathcal{M},\forall a\in G,\quad\left(\psi\cdot
a\right)\left(x\right)=\psi\left(a\cdot x\right)$

######  Theorem 3.1 (Souriau)

Let $\left(\mathcal{M},\varpi\right)$ be a connected symplectic manifold and a
symplectic action $\left(a,x\right)\mapsto x^{\prime}=a\cdot x$ of a Lie group
$G$. Then, the induced action
$\left(a,\psi\right)\mapsto\psi^{\prime}=\psi\cdot a$ has the form

$\psi\cdot a=Ad^{*}\left(a\right)\psi+cocs\left(a\right)$ (12)

where $cocs\left(a\right)$ does not depends on $x$. The mapping
$cocs:G\rightarrow\mathfrak{g}^{*}$ is called a symplectic cocycle.

The demonstration can be found for instance in Souriau [1970, 1997b, Theorem
(11.34)] and [Abraham and Marsden, 1978, Section 4.2]. In the sequel, we
consider only symmetry groups for which the consideration of symplectic
cocycles is not relevant, then the group acts linearly.

The symplectic structure of the coadjoint orbits is revealed by the following
result.

######  Theorem 3.2 (Kirillov-Kostant-Souriau)

Let $G$ be a Lie group and an orbit of the coadjoint representation
$orb\,(\mu)\subset\mathfrak{g}^{*}$. Then:

  * $\diamondsuit$

The inclusion map $orb\,(\mu)\to\mathfrak{g}^{*}$ is a regular embedding. A
vector $d\mu\in T_{\mu}\mathfrak{g}^{*}$ is tangent to the orbit if there
exists $Z_{d}\in\mathfrak{g}$ such that:

$d\,\mu=\mu\,\circ\,ad\,(Z_{d})=-ad^{*}(Z_{d})\mu$

  * $\heartsuit$

The orbit $orb\,(\mu)$ is a symplectic manifold of which the symplectic form
is defined by:

$\varpi_{KKS}(d\mu,\delta\mu)=\mu\,\left[{Z_{d},Z_{\delta}}\right]$

The dimension of the orbit is even.

  * $\spadesuit$

$G$ is a symplectic group and any $\mu\in\mathfrak{g}^{*}$ is its own
momentum.

The reader can find a demonstration in Souriau [1970, 1997b, Theorem (11.34)].

In physical terms, every observer is working in a reference frame. The
symmetries $a\in G$ are changes of reference frames or, equivalently, of
observers. According to the mechanistic description of elementary particles
proposed by Souriau, an isolated dynamical system is said to be elementary
when the symmetry group acts transitively on the space of its motions, i.e. it
is the set of all possible motions of the system. The momentum map of its
action is then a symplectic diffeomorphism of this space onto a coadjoint
orbit of the group. For him, the so defined elementary systems are
mathematical models for elementary particles of physicists.

##  4 Elementary particles in Poincaré relativity

In special relativity, the Universe $\mathcal{U}$ is represented by a 4D
space333In general relativity, the Universe is a manifold and we have to
consider the tangent space regarding to the elementary particles.. An event
$\bm{X}\in\mathcal{U}$ occuring in a reference frame at time $t$ and position
$x\in\mathbb{R}^{3}$ is represented by its coordinates

$X=\left[{{\begin{array}[]{cc}t\hfill\\\ x\hfill\\\ \end{array}}}\right]$

$\mathcal{U}$ is equipped with the Minkowski 1 + 3 metric represented in an
orthonormal basis by Gram’s matrix

$G=\left[{{\begin{array}[]{cc}1\hfill&0\hfill\\\
0\hfill&-1_{\mathbb{R}^{3}}\hfill\\\ \end{array}}}\right]$ (13)

then in this basis the speed of the light $c$ is equal to 1. The Lorentz
transformations are the linear transformations $P$ of $\mathbb{R}^{4}$ which
conserve the 1 + 3 metric, then such

$P^{*}P=1_{\mathbb{R}^{4}}$

In terms of the boost (or velocity of transport) $v\in\mathbb{R}^{3}$ and a
rotation $R\in\mathbb{SO}(3)$, using the factor $\gamma=1/\sqrt{1-\parallel
v\parallel^{2}}$, the structure of a Lorentz transformation is

$P=\left[{{\begin{array}[]{cc}\gamma\hfill&\gamma\,v^{T}R\hfill\\\
\gamma\,v\hfill&\left[1_{\mathbb{R}^{4}}+\frac{\gamma^{2}}{\gamma+1}\,v\,v^{T}\right]\,R\hfill\\\
\end{array}}}\right]$

In the sequel, we consider only the special Lorentz transformations i.e. that
belong to the connected component of the identity $\mathbb{SO}^{+}(1,3)$, a
Lie group of dimension 10. The Poincaré group $\mathbb{G}$ is the set of
affine transformations of $\mathbb{R}^{4}$

$X^{\prime}\mapsto X=C+P\,X^{\prime}$ (14)

of which the linear part $P$ is a Lorentz transformation and that we denote
$a=(C,P)$. The translation is decomposed into a clock change $\tau$ and a
space translation $k$

$C=\left[{{\begin{array}[]{cc}\tau\hfill\\\ k\hfill\\\ \end{array}}}\right]$

$\mathbb{G}$ is a Lie group of dimension 10. The elements of its Lie algebra
$\mathfrak{g}$ are characterized by

$Z\in\mathfrak{g}\quad\Leftrightarrow\quad Z=\delta a=(\delta C,\delta
P)\;\mbox{such that}\;\delta P\;\mbox{is skew-adjoint}$

then

$\delta C=\left[{{\begin{array}[]{cc}\delta\tau\hfill\\\ \delta k\hfill\\\
\end{array}}}\right],\qquad\delta P=\left[{{\begin{array}[]{cc}0\hfill&\delta
v^{T}\hfill\\\ \delta v\hfill&j(\delta\theta)\hfill\\\ \end{array}}}\right]$
(15)

where $\delta R=j(\delta\theta)$ is an infinitesimal rotation around the
identity of $\mathbb{R}^{3}$. The corresponding momenta $\mu$ are linear forms
on $\mathfrak{g}$

$\mu(Z)=-\Pi^{*}\delta C-\frac{1}{2}Tr(M\,\delta P)$ (16)

characterized by

$\mu\in\mathfrak{g}^{*}\quad\Leftrightarrow\quad\mu=(\Pi,M)\;\mbox{such
that}\;M\;\mbox{is skew-adjoint}$

For physical purpose, we introduce the 1 + 3 block decomposition

$\Pi=\left[{{\begin{array}[]{cc}m\hfill\\\ p\hfill\\\
\end{array}}}\right],\qquad
M=\left[{{\begin{array}[]{cc}0\hfill&r^{T}\hfill\\\ r\hfill&j(l)\hfill\\\
\end{array}}}\right]$ (17)

where $m\in\mathbb{R}$ is the mass (equal to the energy because $c=1)$,
$p,r,l\in\mathbb{R}^{3}$ are respectively the linear momentum, the passage and
the angular momentum. Then (16) itemizes into

$\mu(Z)=l\cdot\delta\theta-r\cdot\delta v+p\cdot\delta k-m\,\delta\tau$

The coadjoint representation reads

$\mu=Ad(a)\,\mu^{\prime}\quad\Leftrightarrow\quad\Pi=P\,\Pi^{\prime},\quad
M=P\,M^{\prime}P^{*}+C\,(P\,\Pi^{\prime})^{*}-(P\,\Pi^{\prime})\,C^{*}$ (18)

In Souriau [1970, 1997b, Chapter 3, §14], Souriau uses the type (timelike or
lightlike) of the 4-momentum vector (or energy-momentum vector) $\Pi$ for a
classification of elementary systems. By these means, he obtains a large part
of the physicists’ classification of elementary particles. Below, briefly
summarized, his results are presented.

Let us suppose that $\Pi$ is timelike and define the spin momentum

$M_{0}=M+\Pi\,X^{*}-X\,\Pi^{*}$ (19)

from which we deduce of (18) its transformation law

$M_{0}=P\,M^{\prime}_{0}P^{*}$ (20)

and two properties

  * (a)

$M_{0}$ is skew-adjoint

  * (b)

The set of $X\in\mathbb{R}^{4}$ such that $M_{0}\,\Pi=0$ is a straight line
$\mathcal{D}$ parallel to $\Pi$

Owing to (20), $M_{0}$ does not depends on $X$ when $X$ runs over
$\mathcal{D}$. Taking into account that a non null vector orthogonal to a
timelike vector is spacelike (see Souriau [1965, §28]), $I,J\in\mathbb{R}^{4}$
such that

$\Pi=m_{0}\,I,\qquad I^{*}I=1,\qquad J^{*}J=-1,\qquad J^{*}I=0$ (21)

On this ground, we put

$M_{0}=s\,\mathcal{J}(I,J)$ (22)

verifying the properties (a) and (b) because of Theorem 2.1 $\heartsuit$ and
$\spadesuit$. The momentum $\mu=(\Pi,M)$ is characterized by $\mathcal{D}$ and
$J$ or, equivalently, by $X,I,J$. Owing to (8), we have

$*M=*M_{0}+*(X\,\Pi^{*}-\Pi\,X^{*})=*M_{0}+\mathcal{J}(X,\Pi)$

Let us introduce the polarization

$W=(*M)\,\Pi$ (23)

Then, taking into account Theorem 2.1 $\heartsuit$, (22) and (9)

$W=(*M_{0})\,\Pi+\mathcal{J}(X,\Pi)\,\Pi=s\,m_{0}\,(*\mathcal{J}(X,\Pi))\,I=s\,m_{0}\,(J\,I^{*}-I\,J^{*})\,I$

or, taking into account (21)

$W=s\,m_{0}\,J$

The straight line of direction $J$ is called the polarization line.

A particle with spin is characterized by two non vanishing 4-vectors, its
energy-momentum $\Pi$ (timelike) and its polarization $W$ (spacelike). The two
numbers (Casimirs)

$C_{2}=\Pi^{*}\Pi>0,\qquad C_{4}=W^{*}W<0$

are invariant of the orbit, then do not depend on the reference frame or the
observer. The calculus of the dimension of the isotropy group of the momentum
$\mu$ shows that they are only 2 independent invariants, then every invariant
is a combinations of the Casimirs. The numbers

$m_{0}=\sqrt{\Pi^{*}\Pi},\qquad s=\frac{\sqrt{-W^{*}W}}{\sqrt{\Pi^{*}\Pi}}$

are interpreted as the rest mass and the spin of the particle. In particular,
let

$\Pi^{\prime}=\left[{{\begin{array}[]{cc}m_{0}\hfill\\\ 0\hfill\\\
\end{array}}}\right]$

be the 4-momentum in the comoving frame (or proper reference frame). Applying
to it a boost $v$, i.e. a Lorentz transformation (4) where the rotation is the
identity, we obtain by (18) the mass and the linear momentum in the reference
frame of any observer

$m=m_{0}\gamma,\qquad p=m_{0}\gamma\,v$

This relation can be expressed also as

$\Pi=m_{0}\,U\quad\mbox{such that}\quad U^{*}U=1$ (24)

It is worth to remark that the straight line $\mathcal{D}$ is the trajectory
of the particle Souriau [1970, 1997b, Chapter 3, §14]. If it is parameterized
by the arc length $s$ (for the metric $G$), $U$ is the 4-velocity

$U=\frac{dX}{ds}=\left[\begin{array}[]{c}\gamma\hfill\\\ \gamma\,v\hfill\\\
\end{array}\right]$ (25)

They are two other kind of orbits characterizing a particle without spin and a
massless particle. For more details the reader is referred to Souriau [1970,
1997b, (14.24) and (14.29)].

We know that elementary particles are characterized by their mass, spin and
electric charge. This last property is missing from this formalism and is
usually reintroduced in the particle description as an external element to the
geometric approach (see Souriau [1970, 1997b, Chapter 3, §15]). Our aim now is
to develop a paradigm compatible with the experimental observations and in
which the charge naturally appears as a momentum by constructing a bridge
between the coadjoint orbit method and the Kaluza-Klein theory.

##  5 Elementary particles in Kaluza-Klein relativity

In this theory, the Universe $\hat{\mathcal{U}}$ is represented by a 5D
hyperbolic Euclidean space where the fifth dimension is curled up and
microscopic. An event $\hat{\bm{X}}\in\hat{\mathcal{U}}$ occuring in a
reference frame at time $t$, position $x$ and fifth coordinate $y$ is
represented by the 5-column

$\hat{X}=\left[{{\begin{array}[]{cc}X\hfill\\\ y\hfill\\\
\end{array}}}\right]=\left[{{\begin{array}[]{cc}t\hfill\\\ x\hfill\\\
y\hfill\\\ \end{array}}}\right]$

$\hat{\mathcal{U}}$ is equipped with a 1 + 4 metric represented in an
orthonormal basis by Gram’s matrix

$\hat{G}=\left[{{\begin{array}[]{cc}G\hfill&0\hfill\\\ 0\hfill&-1\hfill\\\
\end{array}}}\right]$ (26)

where $G$ is given by (13). The set $\hat{\mathbb{G}}_{1}$ of affine
transformations $\hat{a}=(\hat{C},\hat{P})$ of $\mathbb{R}^{5}$ of which the
linear part conserves the metric

$\hat{P}^{*}\hat{P}=1_{\mathbb{R}^{4}}$

is a Lie group of dimension 15. The structure of the transformations of
$\hat{\mathbb{G}}_{1}$ is given by

$\hat{C}=\left[{{\begin{array}[]{cc}C\hfill\\\ \xi\hfill\\\
\end{array}}}\right],\qquad\hat{P}=\left[{{\begin{array}[]{cc}P\hfill&\beta\,P^{*-1}b\hfill\\\
b^{*}\hfill&\beta\hfill\\\ \end{array}}}\right]$

where

$\displaystyle
b\in\mathbb{R}^{4},\quad\qquad\beta=\sqrt{1+b^{*}b}\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad$
$\displaystyle P=P_{L}B,\qquad P_{L}\;\;\mbox{is a Lorentz
transformation},\qquad B=1_{\mathbb{R}^{4}}+\frac{1}{\beta+1}\,b\,b^{*}$ (27)

Its restriction to $\mathbb{R}^{4}$ is Poincaré’s group. The elements of its
Lie algebra $\hat{\mathfrak{g}}_{1}$ are characterized by

$\hat{Z}\in\hat{\mathfrak{g}}_{1}\quad\Leftrightarrow\quad\hat{Z}=\delta\hat{a}=(\delta\hat{C},\delta\hat{P})\;\mbox{such
that}\;\delta\hat{P}\;\mbox{is skew-adjoint}$

then

$\delta\hat{C}=\left[{{\begin{array}[]{cc}\delta C\hfill\\\ \delta\xi\hfill\\\
\end{array}}}\right],\qquad\delta\hat{P}=\left[{{\begin{array}[]{cc}\delta
P\hfill&\delta b\hfill\\\ \delta b^{*}\hfill&0\hfill\\\ \end{array}}}\right]$

where $\delta P$ is skew-adjoint with respect to the metric (13).

Following Souriau, the momenta of the elementary particles can be obtained by
considering the group action on the coadjoint orbits. The momenta $\hat{\mu}$
of $\hat{\mathbb{G}}_{1}$ are of the form

$\hat{\mu}(\hat{Z})=-\hat{\Pi}^{*}\delta\hat{C}-\frac{1}{2}Tr(\hat{M}\,\delta\hat{P})$
(28)

characterized by

$\hat{\mu}\in\hat{\mathfrak{g}}^{*}_{1}\quad\Leftrightarrow\quad\hat{\mu}=(\hat{\Pi},\hat{M})\;\mbox{such
that}\;\hat{M}\;\mbox{is skew-adjoint}$

For convenience, we decompose the momenta as follows

$\hat{\Pi}=\left[{{\begin{array}[]{cc}\Pi\hfill\\\ q\hfill\\\
\end{array}}}\right],\qquad\hat{M}=\left[{{\begin{array}[]{cc}M\hfill&Q\hfill\\\
Q^{*}\hfill&0\hfill\\\ \end{array}}}\right]$ (29)

where the scalar $q$ and $Q\in\mathbb{R}^{4}$ are extra momenta with respect
to those of Poincaré’s group. The natural conjecture is to identify $q$ to the
electric charge with suitable physical units. This idea is mentioned in Petit
and D’Agostini [2014] but the extension of Poincaré’s group proposed by the
authors is only a subgroup of $\hat{\mathbb{G}}_{1}$ of dimension 11. In the
present work, the aim is to perform an in-depth investigation by examining the
consequences of this conjecture to verify if they fit the observations.

Then (28) itemizes into

$\hat{\mu}(\hat{Z})=-\Pi^{*}\delta C-\frac{1}{2}Tr(M\,\delta
P)-q\,\delta\xi-Q^{*}\delta b$ (30)

The coadjoint representation reads

$\hat{\mu}=Ad(\hat{a})^{*}\,\hat{\mu}^{\prime}\quad\Leftrightarrow\quad\hat{\Pi}=\hat{P}\,\hat{\Pi}^{\prime},\quad\hat{M}=\hat{P}\,\hat{M}^{\prime}\hat{P}^{*}+\hat{C}\,(\hat{P}\,\hat{\Pi}^{\prime})^{*}-(\hat{P}\,\hat{\Pi}^{\prime})\,\hat{C}^{*}$
(31)

The action of $\hat{\mathbb{G}}_{1}$ on the energy-momentum-charge vector
$\hat{\Pi}$ itemizes into

$\Pi=P\,\Pi^{\prime}+q^{\prime}\,\beta\,P^{*-1}b,\qquad
q=b^{*}\Pi+\beta\,q^{\prime}$

It results that the electric charge is dependent on the reference frame then
on the observer, totally at odds with the experimental observations. Likewise
for Poincaré’s group, we assume that $\hat{\Pi}$ is timelike. The only
invariant is

$\hat{\Pi}^{*}\hat{\Pi}=\Pi^{*}\Pi-q^{2}>0$

The value of the charge depends on the observer and may even be zero in
certain reference frames. The charge is not characteristic only of the
particle. Author’s opinion is that $\hat{\mathbb{G}}_{1}$ is not the symmetry
group of the Universe today as we know it but nonetheless must not be a priori
rejected. We have only to find the Physics that could admit it as symmetry
group. However before to discuss this point, our goal now is to find a more
appropriate symmetry group for the Physics today. It is what we shall be going
to see in the next Section. Latter, we shall come back to the analysis of the
structure of the momenta $\hat{\mu}$ of $\hat{\mathbb{G}}_{1}$ and the
exhaustive determination of the invariants of the motion.

##  6 Elementary particles in an Universe with a microscopic fifth dimension

###  6.1 Zoom in

To avert the previous paradox concerning the electric charge, it is worth to
remark that the fifth dimension is curled up and the estimate of the cylinder
radius (about $10^{-30}$ cm) is overwhelmingly small with respect to usual
lengths in nuclear physics. The idea is to take advantage of this fact by a
zoom in along the fifth coordinate and to consider the limit when the cylinder
radius $\omega$ vanishes. Then we consider the transformation law of a vector
$\hat{\bm{V}}$

$\hat{V}^{\prime}=\hat{P}^{-1}_{\omega}\hat{V}$ (32)

with the scaling

$\hat{P}_{\omega}=\left[{{\begin{array}[]{cc}1_{\mathbb{R}^{4}}\hfill&0\hfill\\\
0\hfill&\omega\hfill\\\ \end{array}}}\right]$ (33)

in such way that if the length was of the order of the cylinder radius in the
fifth coordinate, it is after scaling of the order of the unity in the new
coordinate (with a prime). Owing to the transformation law of 2-covariant
tensors,

$\hat{G}^{\prime}=\hat{P}^{T}_{\omega}\hat{G}\,\hat{P}_{\omega}$

the metric and its inverse are represented after the scaling by

$\hat{G}^{\prime}=\left[{{\begin{array}[]{cc}G\hfill&0\hfill\\\
0\hfill&-\omega^{2}\hfill\\\
\end{array}}}\right],\qquad\hat{G}^{\prime-1}=\left[{{\begin{array}[]{cc}G^{-1}\hfill&0\hfill\\\
0\hfill&-\omega^{-2}\hfill\\\ \end{array}}}\right]$ (34)

Next we omit the primes and consider the limits of these two matrices when
$\omega\rightarrow 0$

$\hat{G}_{0}=\left[{{\begin{array}[]{cc}G\hfill&0\hfill\\\ 0\hfill&0\hfill\\\
\end{array}}}\right],\qquad-\omega^{-2}\,\left[{{\begin{array}[]{cc}0\hfill&0\hfill\\\
0\hfill&1\hfill\\\
\end{array}}}\right]=-\omega^{-2}\,\left[{{\begin{array}[]{cc}0\hfill\\\
1\hfill\\\ \end{array}}}\right]\otimes\left[{{\begin{array}[]{cc}0\hfill\\\
1\hfill\\\ \end{array}}}\right]$ (35)

The first matrix represents a covariant semi-metric of signature $(+---0)$
while the second one represents a contravariant semi-metric of signature
$(0\,0\,0\,0-)$. After scaling, the Euclidean structure of $\hat{\mathcal{U}}$
is canceled but it remains two debris, a symmetric 2-covariant tensor that we
denote $\hat{\bm{G}_{0}}$ represented by $\hat{G}_{0}$ and a symmetric
2-contravariant tensor or, equivalently, a vector $\bm{\Omega}_{0}$
represented by the column

$\hat{\Omega}_{0}=\left[{{\begin{array}[]{cc}0\hfill\\\ 1\hfill\\\
\end{array}}}\right]$ (36)

The set $\hat{\mathbb{G}}_{0}$ of affine transformations
$\hat{a}=(\hat{C},\hat{P})$ of $\mathbb{R}^{5}$ of which the linear part
conserves the components of $\hat{\bm{G}_{0}}$ and $\bm{\Omega}_{0}$ is such
that

$\hat{C}=\left[{{\begin{array}[]{cc}C\hfill\\\ \xi\hfill\\\
\end{array}}}\right],\qquad\hat{P}=\left[{{\begin{array}[]{cc}P\hfill&0\hfill\\\
b^{*}\hfill&1\hfill\\\ \end{array}}}\right]$ (37)

where $\xi$ is a scalar, $C,b\in\mathbb{R}^{4}$ and $P$ is a Lorentz
transformation. As $\hat{\mathbb{G}}_{1}$, it is a Lie group of dimension 15.
Then the affine transformation
$\hat{X}^{\prime}\mapsto\hat{X}=\hat{C}+\hat{P}\,\hat{X}^{\prime}$ itemizes
into

$X=C+P\,X^{\prime},\qquad y=\xi+y^{\prime}+b^{*}X^{\prime}$ (38)

###  6.2 Gauge transformation

To understand the meaning of $b$, we change of point of view and work in the
framework of the general relativity in which $\hat{\bm{V}}$ is a tangent
vector to a the 5D manifold $\hat{\mathcal{U}}$. As in de Saxcé and Vallée
[2016], we hope to determine whether the G-structure in Kobayashi sense
(Kobayashi [1972]) is integrable by solving thanks to Frobenius method the PDE
system

$\frac{\partial\hat{X}}{\partial\hat{X}^{\prime}}=\hat{P},\qquad\hat{P}\in G$

For $\hat{\mathbb{G}}_{1}$, the manifold $\hat{\mathcal{U}}$ is Riemannian and
the the $\hat{\mathbb{G}}_{1}$-structure is not integrable if
$\hat{\mathcal{U}}$ is curved. If it is flat, the transition maps are of the
form $\hat{X}^{\prime}\mapsto\hat{X}=\hat{C}+\hat{P}\,\hat{X}^{\prime}$. For
$\hat{\mathbb{G}}_{0}$, the compatibility conditions of the system provides
the equation

$b=\mbox{grad}_{X}\,h$

where $X^{\prime}\mapsto h(X^{\prime})$ is a real function, that by
integration leads to the gauge transformation

$y=y^{\prime}+h(X^{\prime})$

The $\hat{\mathbb{G}}_{0}$-structure is integrable only if the manifold is
flat. Otherwise, it is convenient as in Souriau [1964, Chapter VII, (41.21)]
to consider the transition maps of standard charts

$\hat{X}=\hat{H}\,(\hat{X}^{\prime})=\left[{{\begin{array}[]{cc}H(X^{\prime})\hfill\\\
y^{\prime}+h(X^{\prime})\hfill\\\ \end{array}}}\right]$

In terms of 4-potential $A$ of the electromagnetism, the vector $b$ turns out
to be a gauge transformation

$A^{\prime}\mapsto A=A^{\prime}+b$

###  6.3 Coadjoint orbit

The elements of the Lie algebra $\hat{\mathfrak{g}}_{0}$ of
$\hat{\mathbb{G}}_{0}$ are characterized by

$\hat{Z}\in\hat{\mathfrak{g}}_{0}\quad\Leftrightarrow\quad\left\\{\begin{array}[]{l}\hat{Z}=\delta\hat{a}=(\delta\hat{C},\delta\hat{P})\;\mbox{such
that}\\\ \\\ \delta\hat{C}=\left[{{\begin{array}[]{cc}\delta C\hfill\\\
\delta\xi\hfill\\\
\end{array}}}\right],\quad\delta\hat{P}=\left[{{\begin{array}[]{cc}\delta
P\hfill&0\hfill\\\ \delta b^{*}\hfill&0\hfill\\\ \end{array}}}\right],\\\ \\\
\delta P\;\mbox{is skew-adjoint with}\\\ \mbox{respect to the metric}\;(\ref{G
=})\\\ \end{array}\right.$ (39)

The momenta $\hat{\mu}$ of $\hat{\mathbb{G}}_{0}$ are of the form

$\hat{\mu}(\hat{Z})=-\Pi^{*}\delta C-\frac{1}{2}Tr(M\,\delta
P)-q\,\delta\xi-Q^{*}\delta b$

The coadjoint representation $\hat{\mu}=Ad(\hat{a})^{*}\,\hat{\mu}^{\prime}$
is defined by the following transformation laws

$\Pi=P\,(\Pi^{\prime}-q^{\prime}\,b)$ (40)
$M=P\,M^{\prime}P^{*}+C\,(P\,(\Pi^{\prime}-q\,b))^{*}-(P\,(\Pi^{\prime}-q\,b))\,\hat{C}^{*}+(P\,b)(P\,Q^{\prime})^{*}-(P\,Q^{\prime})(P\,b)^{*}$
(41) $q=q^{\prime}$ (42) $Q=P\,Q^{\prime}+q^{\prime}C$ (43)

In contrast to what happened with $\hat{\mathbb{G}}_{1}$, the electric charge
is independent on the reference frame then on the observer. This include in
particular the invariance with respect to the gauge transformation. Besides,
owing to (38), the transformation law (43) is satisfied if we claim that

$Q=q\,X$ (44)

that provides the physical interpretation of the momentum $Q$ as the product
of the charge and the space-time position. We suggest to call it the
4-position-charge momentum. Let us remark also that, taking into account the
transformation law of $\Pi$, the one of $M$ is simplified as follow

$M=P\,M^{\prime}P^{*}+C\,\Pi^{*}-\Pi\,\hat{C}^{*}+(P\,b)(P\,Q^{\prime})^{*}-(P\,Q^{\prime})(P\,b)^{*}$
(45)

To calculate the number of independent invariants of the momentum of a group
$G$, we determine the isotropy group of $\hat{\mu}$ from which we deduce the
dimension of the coadjoint orbit

$\mbox{dim}\,(\mbox{orb}(\hat{\mu}))=\mbox{dim}G-\mbox{dim}(\mbox{iso}(\hat{\mu}))$

The number of independent invariant of the orbit is

$n_{I}=\mbox{dim}\,\mathfrak{g}-\mbox{dim}(\mbox{orb}(\hat{\mu}))=\mbox{dim}(\mbox{iso}(\hat{\mu}))$

Taking into account (40), (45 and (43), to determine the isotropy group of
$\hat{\mu}$, we need to solve with respect to the unknowns $C,P,b,\xi$ the
system of equations

$\Pi=P\,(\Pi-q\,b)$ (46)
$M=P\,MP^{*}+C\,(P\,(\Pi-q\,b))^{*}-(P\,(\Pi-q\,b))\,\hat{C}^{*}+(P\,b)(P\,Q)^{*}-(P\,Q)(P\,b)^{*}$
(47) $Q=P\,Q^{\prime}+q^{\prime}C$ (48)

the condition (42) being useless because the charge is an obvious invariant.

For a charged particle ($q\neq 0$), let us show that the number of independent
invariants is 3. Indeed, (46) and (48) give

$b=\frac{1}{q}\,(\Pi-P^{*}\Pi),\quad C=\frac{1}{q}\,(Q-P\,Q)$ (49)

Introducing these expressions of $b$ and $C$ into (47) leads to

$M_{0}=P\,M_{0}P^{*}$ (50)

where, taking into account (44)

$M_{0}=M+\frac{1}{q}\,(\Pi\,Q^{*}-Q\,\Pi^{*})=M+\Pi\,X^{*}-X\,\Pi^{*}$ (51)

can be interpreted as the spin momentum, by comparison to (19). As in Poincaré
relativity, the straight line $\mathcal{D}$ of equation $M_{0}\,\Pi=0$ is the
trajectory of the particle, parallel to $\Pi$ given by, owing to (24) and (25)

$\Pi=m_{0}\,U\quad\mbox{with}\quad U=\frac{dX}{ds}\quad\mbox{such that}\quad
U^{*}U=1$ (52)

First we consider a charged particle with spin ($q\neq 0,\Pi$ timelike,
$M_{0}\neq 0$). Equation (50) being non linear with respect to $P$ then
difficult to solve, we use an infinitesimal method by working with its
differential version in terms of Lie algebra. Differentiating it with respect
to $P$ at the identity and taking into account that $\delta P$ is skew-
adjoint, it holds

$\delta P\,M_{0}-M_{0}\delta P=0$

Putting

$\delta P=\left[{{\begin{array}[]{cc}0\hfill&\delta v^{T}\hfill\\\ \delta
v\hfill&j(\delta\theta)\hfill\\\ \end{array}}}\right],\qquad
M_{0}=\left[{{\begin{array}[]{cc}0\hfill&r^{T}_{0}\hfill\\\
r_{0}\hfill&j(l_{0})\hfill\\\ \end{array}}}\right]$

we obtain the system of equation

$l_{0}\times\delta\,v+r_{0}\times\delta\,\theta=0,\qquad
r_{0}\times\delta\,v-l_{0}\times\delta\,\theta=0$

In the general case ($l_{0}\times r_{0}\neq 0$), the solution is

$\delta\,\theta=\lambda\,r_{0}+\mu\,l_{0},\qquad\delta\,v=\mu\,r_{0}-\lambda\,l_{0}$

where $\lambda$ and $\mu$ are two independent scalars of arbitrary values.
Then the dimension of the isotropy group of $M_{0}$ is 2 because the couple
$(\delta\,\theta,\delta\,v)$ then $\delta\,P$ is defined by these 2
independent parameters. It is easy to verify that the dimension 2 is also
valid for the particular case. The value of $b$ and $C$ are fixed by (49). The
value of $\xi$ that does not occurs in the equations is free. Then the
dimension of the isotropy group of $\hat{\mu}$ and then the number of
independent invariants is 3.

For a charged particle with spin, a set of 3 independent invariants of
$\hat{\mu}$ is the electric charge, the rest mass and the spin of the particle

$q,\qquad m_{0}=\sqrt{\Pi^{*}\Pi},\qquad
s=\frac{\sqrt{-W^{*}W}}{\sqrt{\Pi^{*}\Pi}}$

where the polarization $W$ is defined by (23).

For a charged particle without spin ($q\neq 0,\Pi$ timelike, $M_{0}=0$),
equation (50) is satisfied for every $P$. The number independent invariants is
7. The invariants are $q$ and the 6 independent components of $M_{0}$ which
are null.

For a particle without charge ($q=0,Q=0$), the coadjoint representation of
$\hat{\mathbb{G}}_{0}$ is reduced to (18) and we recover the classification of
elementary particles in Poincaré’s relativity.

To author’s knowledge, the Physics of the symmetry group
$\hat{\mathbb{G}}_{0}$ has not been considered in the literature. By the way,
there exists a group introduced by Lévy-Leblond [1965] and called by him
Carroll’s group444Lewis Carroll, the author of Alice’s Adventures in
Wonderland, although mathematician, had nothing to do with the creation of
this group., of which the elements have the same mathematical structure as the
ones of $\hat{\mathbb{G}}_{0}$. However the physical meaning of Carroll’s
group is very different from the one of $\hat{\mathbb{G}}_{0}$: Carroll’s
group linearly acts on a space of dimension 4 (not 5) and the dimension
analogous to the fifth of $\hat{\mathbb{G}}_{0}$ is timelike (not spacelike).
Carroll’s group was constructed as a degeneracy of Poincaré’s group (similar
but distinct from Galileo’s group). Despite of these discrepancies concerning
the physical aspects, Carroll’s group may be considered as a forunner of the
group $\hat{\mathbb{G}}_{0}$. For more details on Carroll’s group, the reader
is referred to Duval et al. [1985, 1991, 2014], Bergshoeff et al. [2014,
2023], Lévy-Leblond [2023].

##  7 A cosmological scenario for the evolution of elementary particle
structure

One of the important problems in Kaluza-Klein theories is how to explain the
large separation of the scale of our 3D space and that of the extra dimension.
This issue was addressed in Okada [1986] with a Robertson-Walker metric in our
four dimensions and a $d$-dimensional sphere $S^{d}$ in the extra dimensions.
For our concern, $d=1$. Under certain assumptions, this approach predicts
stable cosmological models for the extra dimension and, using Kasner solutions
for approximations when $t\rightarrow 0$, a possible evolution of the scale
factors $a$ for our 3D space and $\omega$ for the fifth dimension given by

$a(t)\cong a_{0}\,t^{1/2},\qquad\omega(t)\cong\omega_{0}\,t^{-1/6}$

Although the assumptions are affected by many uncertainties, it seems
reasonable to expect a cosmological scenario in which:

  * •

the elementary particles of the early 5D Universe are classified from the
momenta of the group $\hat{\mathbb{G}}_{1}$,

  * •

next the three former space dimensions inflate quickly while the fifth one
shrinks,

  * •

leading to the 4D era in which as today the particles are characterized by the
momenta of the group $\hat{\mathbb{G}}_{0}$.

By this mechanism, the elementary particles can acquire electric charge as a
by-product of the $4+1$ symmetry breaking of the Universe.

To describe it, we consider the metric $\hat{\bm{G}}_{\omega}$ at scale
$\omega$. We omit the prime for the scaled coordinates. Owing to (34), the
metric is represented by Gram’s matrix

$\hat{G}_{\omega}=\left[{{\begin{array}[]{cc}G\hfill&0\hfill\\\
0\hfill&-\omega^{2}\hfill\\\ \end{array}}}\right]$

The set $\hat{\mathbb{G}}_{\omega}$ of affine transformations
$\hat{a}=(\hat{C},\hat{P})$ of $\mathbb{R}^{5}$ of which the linear part
conserves the metric $\hat{G}_{\omega}$ is a Lie group of dimension 15. The
structure of the transformations of $\hat{\mathbb{G}}_{\omega}$ is given by

$\hat{C}=\left[{{\begin{array}[]{cc}C\hfill\\\ \xi\hfill\\\
\end{array}}}\right],\qquad\hat{P}=\left[{{\begin{array}[]{cc}P\hfill&\omega^{2}\beta\,P^{*-1}b\hfill\\\
b^{*}\hfill&\beta\hfill\\\ \end{array}}}\right]$

where

$\displaystyle
b\in\mathbb{R}^{4},\quad\qquad\beta=\sqrt{1+\omega^{2}b^{*}b}\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\quad$
$\displaystyle P=P_{L}B,\qquad P_{L}\;\;\mbox{is a Lorentz
transformation},\qquad
B=1_{\mathbb{R}^{4}}+\frac{\omega^{2}}{\beta+1}\,b\,b^{*}$ (53)

When $\omega=0$, we recover the group $\hat{\mathbb{G}}_{0}$. We would like to
classify the elementary particles with charge at scale $\omega>0$. In
particular, we shall obtain the classification for the early Universe
($\omega=1$) of which the study was only sketched out in Section 5. The other
limit case of the Universe today ($\omega=0)$ that is singular was already
studied in Section 6.

The elements $\hat{a}=(\delta\hat{C},\delta\hat{P})$ of the Lie algebra
$\hat{\mathfrak{g}}_{\omega}$ of $\hat{\mathbb{G}}_{\omega}$ are such that

$\delta\hat{C}=\left[{{\begin{array}[]{cc}\delta C\hfill\\\ \delta\xi\hfill\\\
\end{array}}}\right],\qquad\delta\hat{P}=\left[{{\begin{array}[]{cc}\delta
P\hfill&\omega^{2}\delta b\hfill\\\ \delta b^{*}\hfill&0\hfill\\\
\end{array}}}\right]$ (54)

where $\delta P$ is skew-adjoint with respect to the metric (13). The momenta
$\hat{\mu}$ of $\hat{\mathbb{G}}_{\omega}$ are of the form (28). We decompose
the momenta as follows

$\hat{\Pi}=\left[{{\begin{array}[]{cc}\Pi\hfill\\\ q\hfill\\\
\end{array}}}\right],\qquad\hat{M}=\left[{{\begin{array}[]{cc}M\hfill&Q\hfill\\\
\omega^{-2}Q^{*}\hfill&0\hfill\\\ \end{array}}}\right]$ (55)

transforming (28) into (30). The coadjoint representation is defined by (31)
or, in details

$\Pi=P\,\Pi^{\prime}+\omega^{2}q^{\prime}\,\beta\,P^{*-1}b$ (56)
$\displaystyle M$ $\displaystyle=$ $\displaystyle
P\,M^{\prime}P^{*}+\beta\,\left[(P^{*-1}b)(P\,Q^{\prime})^{*}-(P\,Q^{\prime})(P^{*-1}b)^{*}\right]$
(57)
$\displaystyle+\,C\,(P\,\Pi^{\prime}+\omega^{2}q^{\prime}\,\beta\,P^{*-1}b)^{*}-(P\,\Pi^{\prime}+\omega^{2}q^{\prime}\,\beta\,P^{*-1}b)\,C^{*}$
$q=b^{*}\Pi+\beta\,q^{\prime}$ (58) $\displaystyle Q$ $\displaystyle=$
$\displaystyle\beta\,P\,Q^{\prime}-\omega^{2}\,\left[P\,M^{\prime}b+\beta\,(b^{*}Q^{\prime})\,P^{*-1}b\right]$
(59)
$\displaystyle+\,\omega^{2}\,\left[\xi\,(P\,\Pi^{\prime}+\omega^{2}q^{\prime}\,\beta\,P^{*-1}b)-(b^{*}\Pi^{\prime}+\beta\,q^{\prime})\,C\right]$

By analogy with (19), we define the spin momentum

$\hat{M}_{0}=\hat{M}+\hat{\Pi}\,\hat{X}^{*}-\hat{X}\,\hat{\Pi}^{*}$

from which we deduce of (31) its transformation law

$\hat{M}_{0}=\hat{P}\,\hat{M}^{\prime}_{0}\hat{P}^{*}$ (60)

and two properties

  * (i)

$\hat{M}_{0}$ is skew-adjoint

  * (ii)

The set of $\hat{X}\in\mathbb{R}^{5}$ such that $\hat{M}_{0}\,\hat{\Pi}=0$ is
a straight line $\hat{\mathcal{D}}$ parallel to $\hat{\Pi}$

###  7.1 Polarization plane and map

At this stage, it is worth to remark an important difference between Poincaré
relativity, in which $A_{M_{0}}$ and its adjoint $*A_{M_{0}}$ are both
2-forms, and Kaluza-Klein one for which $A_{\hat{M}_{0}}$ is a 3-form while
$*A_{\hat{M}_{0}}$ is a 2-form, that has important consequences from the
geometrical and physical viewpoints. Owing to (60), $\hat{M}_{0}$ does not
depends on $\hat{X}$ when $\hat{X}$ varies over $\hat{\mathcal{D}}$. Let
$\hat{I},\hat{J}_{1},\hat{J}_{2}\in\mathbb{R}^{5}$ such that

$\hat{\Pi}=m_{0}\,\hat{I},\qquad\hat{I}^{*}\hat{I}=1,\qquad\hat{J}^{*}_{1}\hat{J}_{1}=\hat{J}^{*}_{2}\hat{J}_{2}=-1,\qquad\hat{J}^{*}_{1}\hat{I}=\hat{J}^{*}_{2}\hat{I}=\hat{J}^{*}_{1}\hat{J}_{2}=0$
(61)

Over $\hat{\mathcal{D}}$, we put

$\hat{M}_{0}=s\,\mathcal{J}(\hat{I},\hat{J}_{1},\hat{J}_{2})$ (62)

verifying the properties (i) and (ii) because of Theorem 2.1 $\heartsuit$ and
$\spadesuit$. The momentum $\hat{\mu}=(\hat{\Pi},\hat{M})$ is characterized by
$\mathcal{D}$ and $\hat{J}_{1},\hat{J}_{2}$ or, equivalently, by
$\hat{X},\hat{I},\hat{J}_{1},\hat{J}_{2}$. In Poincaré relativity, there is a
polarization straight line of direction $J$ while in Kaluza-Klein relativity,
there is a polarization plane spanned by the couple
$(\hat{J}_{1},\hat{J}_{2})$ that is one of its orthonormal basis. Owing to
Theorem 2.2 $\natural$, we have

$*A_{\hat{M}}=*A_{\hat{M}_{0}}+*(\hat{X}^{*}\wedge\hat{\Pi}^{*})=*A_{\hat{M}_{0}}+vol(\hat{\Pi},\hat{X})$

Then

$*A_{\hat{M}}(\hat{\Pi},\hat{J}_{k})=*A_{\hat{M}_{0}}(\hat{\Pi},\hat{J}_{k})+vol(\hat{\Pi},\hat{X},\hat{\Pi},\hat{J}_{k})=s\,*A_{\hat{\Omega}}(\hat{\Pi},\hat{J}_{k})\qquad(k=1,2)$

that leads to investigate the properties of the 2-form $*A_{\hat{\Omega}}$.
Applying Theorem 2.1 $\heartsuit$ and $\spadesuit$, its adjoint is the 3-form

$*A_{\hat{\Omega}}=-*vol(\hat{I},\hat{J}_{1},\hat{J}_{2})=-\hat{I}^{*}\wedge\hat{J}^{*}_{1}\wedge\hat{J}^{*}_{2}$

from which we deduce

$\hat{J}_{1}=-(*A_{\hat{\Omega}}(\hat{I},\hat{J}_{2}))^{*},\qquad\hat{J}_{2}=(*A_{\hat{\Omega}}(\hat{I},\hat{J}_{1}))^{*}$
(63)

In terms of the momentum $\hat{\mu}=(\hat{\Pi},\hat{M})$, introducing the
polarization map

$\mbox{pol}_{\hat{\mu}}:\mathbb{R}^{5}\rightarrow\mathbb{R}^{5}:\hat{U}\mapsto\hat{V}=\mbox{pol}_{\hat{\mu}}(\hat{U})=(*A_{\hat{M}}(\hat{\Pi},\hat{U}))^{*}$

the relations (63) are recast as

$s\,m_{0}\,\hat{J}_{1}=-\mbox{pol}_{\hat{\mu}}(\hat{J}_{2}),\qquad
s\,m_{0}\,\hat{J}_{2}=\mbox{pol}_{\hat{\mu}}(\hat{J}_{1})$

Now, we have to address the issues of the existence and (non) uniqueness of
the orthonormal basis $(\hat{J}_{1},\hat{J}_{2})$ to define the polarization
map:

  * •

Existence. Let us $\hat{K}_{1},\hat{K}_{2}$ two vectors of $\mathbb{R}^{5}$
that form with $\hat{I},\hat{J}_{1},\hat{J}_{2}$ an orthonormal basis of
$\mathbb{R}^{5}$ in which we decompose any vector

$\hat{U}=\lambda\hat{I}+\mu_{1}\hat{J}_{1}+\mu_{2}\hat{J}_{2}+\nu_{1}\hat{K}_{1}+\nu_{2}\hat{K}_{2}$

Then

$\mbox{pol}_{\hat{\mu}}(\hat{U})=s\,m_{0}\,(\mu_{1}\hat{J}_{2}-\mu_{2}\hat{J}_{1})$

that shows the image of the polarization map $\mbox{pol}_{\hat{\mu}}$ is the
polarization plane $\mathcal{P}$. This gives a method to determine a basis of
$\mathcal{P}$. We span $\mathbb{R}^{5}$ and apply the polarization map to find
two linearly independent vectors. They form a basis of $\mathcal{P}$. Then we
find an orthonormal basis by using the Gram–Schmidt process.

  * •

Non uniqueness. Let $(\hat{J}_{1},\hat{J}_{2})$ and
$(\hat{J}^{\prime}_{1},\hat{J}^{\prime}_{2})$ be two orthonormal bases such
that

$\mathcal{J}(\hat{I},\hat{J}_{1},\hat{J}_{2})=\mathcal{J}(\hat{I},\hat{J}^{\prime}_{1},\hat{J}^{\prime}_{2})$

Owing to Theorem 2.1 $\heartsuit$,

$\mathcal{J}(\hat{I},\hat{J}_{1},\hat{J}_{2})\,\hat{J}^{\prime}_{k}=\mathcal{J}(\hat{I},\hat{J}^{\prime}_{1},\hat{J}^{\prime}_{2})\,\hat{J}^{\prime}_{k}=0\qquad(k=1,2)$

then $\hat{J}^{\prime}_{1}$ and $\hat{J}^{\prime}_{2}$ are linear combinations
of $\hat{J}_{1}$ and $\hat{J}_{2}$. This ensures the uniqueness of the
polarization plane. In contrast, the orthonormal basis
$(\hat{J}_{1},\hat{J}_{2})$ is defined only modulo an orthogonal
transformation of $\mathcal{P}$

###  7.2 Invariants of the motion

For a charged particle ($q\neq 0$), let us verify on a simple case that the
number of independent invariants is 2. Taking into account (31), we determine
the isotropy group of the momentum by solving the system of equations

$\hat{\Pi}=P\,\hat{\Pi},\qquad\hat{M}=\hat{P}\,\hat{M}\hat{P}^{*}+\hat{C}\,(\hat{P}\,\hat{\Pi})^{*}-(\hat{P}\,\hat{\Pi})\,\hat{C}^{*}$

with respect to $\hat{a}=(\hat{C},\hat{P})$. Like in Section 6.3, we
differentiate these equations with respect to $\hat{a}$ at the identity and we
take into account that $\hat{M}$ is skew-adjoint, that leads to solve the
linear system

$\delta\hat{P}\,\hat{\Pi}=0,\qquad\delta\hat{P}\,\hat{M}-\hat{M}\,\delta\hat{P}+\delta\hat{C}\,\hat{\Pi}^{*}-\hat{\Pi}\,\delta\hat{C}^{*}=0$
(64)

with respect to $\delta\hat{a}=(\delta\hat{C},\delta\hat{P})$. Let us pick up

$\hat{I}=\left[{{\begin{array}[]{cc}I\hfill\\\ \kappa\hfill\\\
\end{array}}}\right]=\left[{{\begin{array}[]{cc}\mu\hfill\\\ 0\hfill\\\
\kappa\hfill\\\
\end{array}}}\right],\qquad\hat{J}_{k}=\left[{{\begin{array}[]{cc}J_{k}\hfill\\\
0\hfill\\\ \end{array}}}\right]=\left[{{\begin{array}[]{cc}0\hfill\\\
p_{k}\hfill\\\ 0\hfill\\\ \end{array}}}\right],\qquad(k=1,2)$

that satisfy conditions (61). Owing to (54) and (55), the first equation of
(64) itemizes into

$\delta P\,\Pi+\omega^{2}q\,\delta b=0,\qquad\delta b^{*}\Pi=0$ (65)

of which the solution is, owing to (15) and (17)

$\delta b=\left[{{\begin{array}[]{cc}\delta b_{0}\hfill\\\
\delta\bar{b}\hfill\\\
\end{array}}}\right]=\left[{{\begin{array}[]{cc}0\hfill\\\
-\frac{\mu}{\omega^{2}\kappa}\,\delta v\hfill\\\ \end{array}}}\right]$ (66)

Besides, owing to (5) and (6), we have

$\mathcal{J}(I,J_{1},J_{2})=\left[{{\begin{array}[]{cc}0\hfill\\\
\mu\,\psi\hfill\\\
\end{array}}}\right],\qquad\mathcal{J}(J_{1},J_{2})=\left[{{\begin{array}[]{cc}0\hfill&\psi^{T}\hfill\\\
\psi\hfill&0\hfill\\\ \end{array}}}\right]$

with $\psi=p_{1}\times p_{2}$, from which we deduce, taking into account (62)
and (7)

$\hat{M}_{0}=s\hat{\mathcal{J}}(\hat{I},\hat{J}_{1},\hat{J}_{2})=s\,\left[{{\begin{array}[]{cc}\kappa\,\mathcal{J}(J_{1},J_{2})\hfill&-\mathcal{J}(I,J_{1},J_{2})\hfill\\\
-\omega^{2}\mathcal{J}(I,J_{1},J_{2})^{*}\hfill&0\hfill\\\
\end{array}}}\right]$
$\hat{M}_{0}=s\,\left[{{\begin{array}[]{ccc}0\hfill&\kappa\,\psi^{T}\hfill&0\hfill\\\
\kappa\,\psi\hfill&0\hfill&-\mu\,\psi\hfill\\\
0\hfill&\mu\omega^{-2}\psi^{T}&0\hfill\\\ \end{array}}}\right]$

the second equation of (64) itemizes into

$s\,(\kappa\,j(\delta\theta)+\mu\,\delta
b_{0}1_{\mathbb{R}^{3}})\,\psi+m_{0}\,\mu\,\delta k=0,$ $(\kappa\,\delta
v+\mu\,\delta\bar{b})\times\psi=0,$
$-s(\mu\,j(\delta\theta)+\kappa\,\omega^{2}\delta
b_{0}1_{\mathbb{R}^{3}})\psi-m_{0}\,\omega^{2}\kappa\,\delta k=0$
$-s(\mu\,\delta
v+\kappa\,\omega^{2}\delta\bar{b})\cdot\psi+m_{0}\,\omega^{2}(\mu\,\delta\xi-\kappa\,\delta\tau_{0})=0$
(67)

Taking into account (66), the solution of this system of equation is of the
form

$\delta\tau=\alpha,\qquad\delta
k=0,\qquad\delta\xi=\frac{\kappa}{\mu}\,\alpha$ $\delta
v=\beta\,\psi,\qquad\delta\theta=\mu\,\psi,\qquad\delta
b_{0}=0,\qquad\delta\bar{b}=-\beta\,\frac{\mu}{\omega^{2}\kappa}\,\psi$

where $\alpha,\beta$ are two scalars of arbitrary values. Then the dimension
of the isotropy group of the momentum then the number of its independent
invariant is 2. We can take as independent invariants the rest mass and the
spin

$m_{0}=\sqrt{\hat{\Pi}^{*}\hat{\Pi}},\qquad
s=\frac{\sqrt{-(\mbox{pol}_{\hat{\mu}}(\hat{J}_{1}))^{*}\mbox{pol}_{\hat{\mu}}(\hat{J}_{1})}}{\sqrt{\hat{\Pi}^{*}\hat{\Pi}}\,\sqrt{-(\hat{J}_{1})^{*}\hat{J}_{1}}}$

It is worth to remark that the electric charge is the linear momentum along
the fifth dimension but is not an invariant of the motion in the early
Universe.

##  8 Constructing the connection associated to the group
$\hat{\mathbb{G}}_{0}$

At every symmetry group $G$ is attached a family of covariant derivatives of
which the connection matrices are defined on the $G$-principal bundle of the
$G$-frames and valued in its Lie algebra $\mathfrak{g}$ (see for instance
Kobayashi and Nomizu [1963]). We call them $G$-connections. For the group
$\hat{\mathbb{G}}_{1}$, we recover the Levi-Civita free torsion connection
used in Kaluza-Klein theory. A key-point of interest is to determine the
$\hat{\mathbb{G}}_{0}$-connections, the difficulty being that the manifold
$\hat{\mathcal{U}}$ is not Riemannian, and to use it to deduce the equation of
motion of a particle in gravitation and electromagnetic fields.

###  8.1 Zoom out

It is time now to restore the primes canceled for convenience in Section 6.1
after the zoom in. For instance, according to (35) and (36), the two tensors
$\hat{\bm{G}_{0}}$ and $\bm{\Omega}_{0}$ are represented at the microscopic
scale respectively by

$\hat{G}^{\prime}_{0}=\left[{{\begin{array}[]{cc}G^{\prime}_{0}\hfill&0\hfill\\\
0\hfill&0\hfill\\\
\end{array}}}\right],\qquad\hat{\Omega}^{\prime}_{0}=\left[{{\begin{array}[]{cc}0\hfill\\\
1\hfill\\\ \end{array}}}\right]$ (68)

with the new notation for the matrix (13)

$G^{\prime}_{0}=\left[{{\begin{array}[]{cc}1\hfill&0\hfill\\\
0\hfill&-1_{\mathbb{R}^{3}}\hfill\\\ \end{array}}}\right]$ (69)

At first glance, we could think to come back to the macroscopic scale by
zooming out with the inverse transformation law of (32)

$\hat{V}=\hat{P}_{\omega}\hat{V}^{\prime}$

where $\hat{P}_{\omega}$ is given by (33), that leads to the singularity
$\hat{V}^{5}=0$ when $\omega\rightarrow 0$.

###  8.2 Hypotheses on the pullback connection

In order to avert the singularity resulting from the zoom out without to lose
the relevant information in the fifth dimension, we propose to pullback over
the space-time $\mathcal{U}$ the $\hat{\mathbb{G}}_{0}$-connection
$\hat{\nabla}$ on the tangent bundle $T\hat{\mathcal{U}}$. They are an
infinite number of $\hat{\mathbb{G}}_{0}$-connections. The purpose here is not
to study their properties in generic terms and to classify them. We are
interested only by those which satisfy certain physical requirements on the
ground of hypotheses considered below, (H1) to (H4) on the underlying
geometric structure of the Universe and (H5) on the motion of a charged
elementary particle in both the gravitation and electromagnetic fields.

  * (H1)

the space-time $\mathcal{U}$ is a Riemannian manifold. The metric $\bm{G}$ is
represented in an orthonormal basis
$(\bm{e}^{\prime}_{1},\cdots,\bm{e}^{\prime}_{4})$ of $T_{\bm{X}}\mathcal{U}$
by the matrix $G^{\prime}_{0}$ of (69) of the Minkowski metric.

  * (H2)

$\pi:\hat{\mathcal{U}}\rightarrow\mathcal{U}$ is a principal $U(1)$-bundle of
fibers isomorphic to the unitary group $U(1)=\left\\{z=e^{i\,t},0\leq
t<2\,\pi\right\\}$. The vector flow
$t\mapsto\hat{\bm{X}}=e^{i\,t}\cdot\hat{\bm{X}}_{0}$ is generated by the
vector field $\hat{\bm{X}}\mapsto\hat{\bm{\Omega}}_{0}(\hat{\bm{X}})$. Let
$\hat{\bm{X}}$ a point of the fiber of $\hat{\mathcal{U}}$ over $\bm{X}$. We
say that a basis $(\hat{\bm{e}}_{1},\cdot,\hat{\bm{e}}_{5})$ of
$T_{\hat{\bm{X}}}\hat{\mathcal{U}}$ is a fibered basis if the set of
$\bm{e}_{i}=(T\pi)\hat{\bm{e}}_{i}$ for $1\leq i\leq 4$ is a basis of
$T_{\bm{X}}\mathcal{U}$ and $\hat{\bm{e}}_{5}$ is tangent to the fiber
($(T\pi)\hat{\bm{e}}_{5}=\bm{0}$).

Let us consider the fibered basis
$(\hat{\bm{e}}^{\prime}_{\alpha})=(\hat{\bm{e}}^{\prime}_{1},\cdots,\hat{\bm{e}}^{\prime}_{4},\bm{\hat{\Omega}}_{0})$
of $T_{\hat{\bm{X}}}\hat{\mathcal{U}}$ such as
$(\bm{e}^{\prime}_{1},\cdots,\bm{e}^{\prime}_{4})$ is an orthonormal basis as
for (H1). The pullback of the metric

$\hat{\bm{G}}_{0}=\pi^{*}\bm{G}$

is a field of symmetric 2-covariant tensors (but not a metric!).
$(\hat{\bm{e}}^{\prime}_{\alpha})$ is a $\hat{\mathbb{G}}_{0}$-basis, i.e. a
basis in which $\hat{\bm{G}_{0}}$ and $\bm{\Omega}_{0}$ are represented by
(68).

  * (H3)

$\hat{\nabla}$ is a $\hat{\mathbb{G}}_{0}$-connection. Let
$\pi:\mathcal{F}(\hat{\mathcal{U}})\rightarrow\hat{\mathcal{U}}$ be the frame
bundle of $\hat{\mathcal{U}}$, i.e. the bundle whose fiber over $\bm{\hat{X}}$
is the set of all basis of $T_{\bm{\hat{X}}}\hat{\mathcal{U}}$. A moving frame
is a smooth section $\bm{\hat{X}}\mapsto(\hat{\bm{e}}^{\prime}_{\alpha})$ of
$\mathcal{F}$. It is called a coordinate frame (or natural frame) if locally
the basis is associated to a coordinate system (or local chart)
$(\hat{X}^{\alpha})$ and if so it is denoted $(\hat{\bm{\partial}}_{\alpha})$.
The set $\mathcal{F}_{\hat{\bm{G}_{0}}}$ of $\hat{\mathbb{G}}_{0}$-bases is
called $\hat{\mathbb{G}}_{0}$-structure. It is a subbundle of
$\mathcal{F}(\hat{\mathcal{U}})$ and a principal $\hat{\mathbb{G}}_{0}$-bundle
that is endowed with an Ehresmann connection. A
$\hat{\mathbb{G}}_{0}$-connection $\hat{\nabla}$ is defined by the field of
$\mathfrak{g}_{0}$-valued connection 1-form
$(\hat{\bm{e}}^{\prime}_{\alpha})\mapsto\hat{\Gamma}^{\prime}$ on
$\mathcal{F}_{\hat{\bm{G}_{0}}}$ such that

$\hat{\nabla}\hat{\bm{e}}^{\prime}_{\alpha}=\hat{\Gamma}^{\prime\beta}_{\alpha}\hat{\bm{e}}^{\prime}_{\beta}$

According to (39), the value of the connection 1-form is the $5\times 5$
matrix

$\hat{\Gamma}^{\prime}=\left[\begin{array}[]{cc}\Gamma^{\prime}\hfill&0\hfill\\\
\Gamma^{\prime 5}\hfill&0\hfill\\\ \end{array}\right]$ (70)

where $\Gamma^{\prime}$ is a $4\times 4$ matrix of 1-forms, skew-adjoint with
respect to the metric (69), $\Gamma^{\prime 5}$ is a 4-row of 1-forms and the
last column of $\hat{\Gamma}^{\prime}$ vanishes. Because
$\hat{G}^{\prime}_{0}$ and $\hat{\Omega}^{\prime}_{0}$ are constant in a
moving $\hat{\mathbb{G}}_{0}$-frame
$\bm{\hat{X}}\mapsto(\hat{\bm{e}}^{\prime}_{a})$, we have

$d\hat{G}^{\prime}_{0}=0,\qquad d\hat{\Omega}^{\prime}_{0}=0$

Otherwise, these conditions must be replaced by

$\hat{\nabla}\hat{G}_{0}=0,\qquad\hat{\nabla}\hat{\Omega}_{0}=0$

the connection $\hat{\nabla}$ preserving the pullback $\hat{\bm{G}}_{0}$ of
the space-time metric and the vector $\hat{\Omega}_{0}$

$\hat{\nabla}\,\hat{\bm{G}}_{0}=\bm{0},\qquad\hat{\nabla}\,\hat{\Omega}_{0}=\bm{0}$

  * (H4)

Let $f$ be a section of the principal $U(1)$-bundle $\hat{\mathcal{U}}$, then
$\bm{G}=f^{*}\hat{\bm{G}}_{0}$. The pullback bundle
$f^{*}\,T\hat{\mathcal{U}}$ is a bundle over $\mathcal{U}$ of which the fiber
over $\bm{X}\in\mathcal{U}$ is
$(f^{*}\,T\hat{\mathcal{U}})_{\bm{X}}=T_{f(\bm{X})}\hat{\mathcal{U}}$. For any
section $\hat{\bm{W}}$ of $T\hat{\mathcal{U}}$, the pullback section is
$f^{*}\hat{\bm{W}}=\hat{\bm{W}}\circ f$. The space-time is endowed with the
pullback connection $f^{*}\hat{\nabla}$, determined uniquely by

$(f^{*}\hat{\nabla})_{\bm{U}}(f^{*}\hat{\bm{W}})=f^{*}(\hat{\nabla}_{(Tf)\bm{U}}\hat{\bm{W}})$

and torsion free, then for all smooth fields
$\bm{X}\mapsto\bm{U},\bm{X}\mapsto\bm{V}$ of tangent vectors to the space-time
$\mathcal{U}$

$\hat{\nabla}_{(Tf)\,\bm{U}}(Tf)\bm{V}-\hat{\nabla}_{(Tf)\,\bm{V}}(Tf)\bm{U}-(Tf)\,[\bm{U},\bm{V}]=\bm{0}$
(71)

The section of the fiber bundle $\pi:\hat{\mathcal{U}}\rightarrow\mathcal{U}$
is an embedding of $\mathcal{U}$ into $\hat{\mathcal{U}}$. The subspace
$H_{f(\bm{X})}=(Tf)T_{\bm{X}}\mathcal{U}$ of $T_{f(\bm{X})}\hat{\mathcal{U}}$
is isomorphic to $T_{\bm{X}}\mathcal{U}$ then of dimension 4 and the tangent
space to the fiber $\hat{\mathcal{U}}_{\bm{X}}$ at $f(\bm{X})$ is the kernel
of $T\pi$. Then $T_{f(\bm{X})}\hat{\mathcal{U}}=H_{f(\bm{X})}\oplus
T_{f(\bm{X})}\hat{\mathcal{U}}_{\bm{X}}$. A basis adapted to the section $f$
is a fibered basis $(\hat{\bm{e}}_{1},\cdot,\hat{\bm{e}}_{5})$ such that
$(\hat{\bm{e}}_{1},\cdot,\hat{\bm{e}}_{4})$ is a basis of $H_{f(\bm{X})}$. A
way to build such a basis is to choose a non null vector $\hat{\bm{e}}_{5}$
tangent to the fiber $\hat{\mathcal{U}}_{\bm{X}}$ and a basis
$(\bm{e}_{1},\cdot,\bm{e}_{4})$ of $T_{\bm{X}}\mathcal{U}$. Then, with the
convention that Greek indices run from 1 to 5 and Latin ones run from 1 to 4,
$(\hat{\bm{e}}_{1},\cdot,\hat{\bm{e}}_{4},\hat{\bm{e}}_{5})=((Tf)\bm{e}_{1},\cdot,(Tf)\hat{\bm{e}}_{4},\hat{\bm{e}}_{5})$
is a basis adapted to the section $f$ and

$\bm{U}=U^{i}\bm{e}_{i}\quad\Leftrightarrow\quad(Tf)\bm{U}=U^{i}\hat{\bm{e}}_{i}$
(72)

In terms of the dual bases, on has

$\bm{e}^{i}(\bm{U})=\hat{\bm{e}}^{i}((Tf)\bm{U})=U^{i},\quad\hat{\bm{e}}^{5}((Tf)\bm{U})=0$
(73)

Using Christoffel’s symbols, the elements of the connection 1-form in this
basis are
$\hat{\Gamma}^{\beta}_{\alpha}=\hat{\Gamma}^{\beta}_{\rho\alpha}\bm{e}^{\rho}$.
In particular, considering pairs of basis vectors $\bm{e}_{i},\bm{e}_{j}$, the
free torsion condition (71) reads

$\Gamma^{\mu}_{ij}-\Gamma^{\mu}_{ji}-\delta^{\mu}_{k}c^{k}_{ij}=0,\qquad$ (74)

where $c^{k}_{ij}$ are the structure coefficient of the moving frame.

  * (H5)

At this point, it is worth to remark that (40) and (42) show that the 5-row

$\hat{\Pi}=[\Pi^{*},q]$ (75)

represents a linear form $\hat{\bm{\Pi}}$ that we call linear 5-momentum. Let
$s\mapsto\bm{X}(s)$ be the trajectory of a particle in the space-time,
parameterized by the arc length $s$ (for the metric $\bm{G}$) and

$\bm{U}=\frac{d\bm{X}}{ds}$ (76)

is the unit tangent vector such that $\bm{U}^{*}\bm{U}=1$. We claim that the
motion of a charged particle and the evolution of its charge $s\mapsto q(s)$
are such that its linear 5-momentum is parallel-transported

$(f^{*}\hat{\nabla})_{\bm{U}}(f^{*}\hat{\bm{\Pi}})=\bm{0}$ (77)

###  8.3 Explicit form of the connection and the equation of motion

Our goal now is to determine explicitly the connection satisfying the
hypotheses (H3) to (H5). As we know, the $\hat{\mathbb{G}}_{0}$-structure
$\mathcal{F}_{\hat{\bm{G}_{0}}}$ is in general non integrable. Then, instead
of the moving frame
$(\hat{\bm{e}}^{\prime}_{1},\cdots,\hat{\bm{e}}^{\prime}_{5})$ of (H3),
section of $\mathcal{F}_{\hat{\bm{G}_{0}}}$, we would like to work with a
coordinate frame $(\bm{\partial}_{1},\cdots,\bm{\partial}_{4})$ associated to
a coordinate system $(X^{1},\cdots,X^{4})$ of the space-time $\mathcal{U}$ and
the corresponding fibered frame $(\hat{\bm{e}}_{1},\cdots,\hat{\bm{e}}_{5})$
adapted to the section $f$, such as
$\hat{\bm{e}}_{\alpha}=(Tf)\bm{\partial}_{\alpha}$ and obtained by the change
of basis

$\hat{\bm{e}}_{\alpha}=\hat{P}^{\beta}_{\alpha}\hat{\bm{e}}^{\prime}_{\beta}$
(78)

where $\hat{P}^{\beta}_{\alpha}$ is the element at the intersection of the
$\beta$-th row and the $\alpha$-th column of the transformation matrix
partitioned into blocks as follows

$\hat{P}=\left[\begin{array}[]{cc}P\hfill&0\hfill\\\
-2\,A^{*}\hfill&1\hfill\\\ \end{array}\right]$ (79)

where $P$ is a $4\times 4$ matrix, $A$ is a 4-column. For $\hat{P}$ being
regular, $P$ must be regular. As the moving frames occurring in (78) are
pulled back on the space-time $\mathcal{U}$, $P$ and $A$ are functions of the
coordinates $X=(X^{1},\cdots,X^{4})$. Owing to (68), the pullback
$\hat{\bm{G}}=\pi^{*}\bm{G}$ of the metric is represented in the new frame by

$\hat{G}=\hat{P}^{T}\hat{G}^{\prime}_{0}\,\hat{P}=\left[{{\begin{array}[]{cc}P^{T}G^{\prime}_{0}P\hfill&0\hfill\\\
0\hfill&0\hfill\\\ \end{array}}}\right]$

where the $4\times 4$ matrix

$G=P^{T}G^{\prime}_{0}P$ (80)

is regular. As $G^{\prime}_{0}$ is constant and $P$ is a function of the
coordinates $X$ on the time-space $\mathcal{U}$, $G$ is also a function of $X$
(but does not depends on $X^{5}$). In the new frame, the connection matrix is
given by

$\hat{\Gamma}=\hat{P}^{-1}(\hat{\Gamma}^{\prime}\,\hat{P}+d\hat{P})$

which, taking into account (70), leads to

$\hat{\Gamma}=\left[\begin{array}[]{cc}\Gamma\hfill&0\\\
\Gamma^{5}\hfill&0\hfill\\\ \end{array}\right]$ (81)

where

$\Gamma=P^{-1}(\Gamma^{\prime}\,P+dP),\qquad\Gamma_{5}=\Gamma^{\prime
5}\,P-2\,[dA^{*}-A^{*}P^{-1}(\Gamma^{\prime}\,P+dP)]$ (82)

Owing to (73), the column representing $(Tf)\bm{U}$ in the basis
$(\hat{\bm{e}}^{\prime}_{i})$ is

$U^{\prime}=P\,U=P_{i}U^{i}$

where $P_{i}$ is the $i$-th column of $P$. Then, taking into account (82) and
the linear dependence of $\Gamma^{\prime}$ with respect to $\hat{U}^{\prime}$

$\Gamma(U)=P^{-1}(\Gamma^{\prime}(U^{\prime})\,P+dP)=\Gamma_{i}U^{i}$

where occurs the $4\times 4$ matrix

$\Gamma_{i}=P^{-1}\left[\Gamma^{\prime}(P_{i})\,P+\partial_{i}P\right]$

where $\partial_{i}$ is the partial derivative with respect to $X^{i}$. Owing
to (80), we have

$G\,\Gamma_{i}=P^{T}G^{\prime}_{0}\Gamma^{\prime}(P_{i})\,P+P^{T}G^{\prime}_{0}\partial_{i}P$

As $\Gamma^{\prime}$ is skew-adjoint with respect to the metric (69), the
first term of the right hand member is skew-symmetric. Then

$G\,\Gamma_{i}+(G\,\Gamma_{i})^{T}=P^{T}G^{\prime}_{0}\partial_{i}P+\partial_{i}P^{T}\,G^{\prime}_{0}P=\partial_{i}G$

or

$\partial_{i}G-G\,\Gamma_{i}-(\Gamma_{i})^{T}G=0$

and with index notation

$\nabla_{i}G_{jk}=\partial_{i}G_{jk}-G_{jm}\Gamma^{m}_{ik}-G_{mk}\Gamma^{m}_{ij}=0$
(83)

Besides, as we are working in a coordinate frame ($c^{k}_{ij}=0$), the torsion
free condition (74) gives

$\Gamma^{k}_{ij}=\Gamma^{k}_{ji}$ (84)

Using the fundamental theorem of Riemannian geometry, both previous relations
show that $\nabla$ is Levi-Civita connection

$\Gamma^{k}_{ij}=\frac{1}{2}\,G^{mr}\left[\partial_{j}G_{ir}+\partial_{i}G_{jr}-\partial_{r}G_{ij}\right]$
(85)

Then, owing to (82),

$\hat{\Gamma}(U)=\left[\begin{array}[]{cc}\Gamma(U)\hfill&0\hfill\\\
\Gamma^{5}(U)\hfill&0\hfill\\\ \end{array}\right]$ (86)

where

$\Gamma^{5}(U)=\Gamma^{\prime 5}(P\,U)\,P-2\,\nabla_{U}A^{*}$ (87)

In the moving frame
$(\hat{\bm{e}}^{\prime}_{1},\cdots,\hat{\bm{e}}^{\prime}_{5})$ of (H3),
section of $\mathcal{F}_{\hat{\bm{G}_{0}}}$, the linear 5-momentum
$\hat{\bm{\Pi}}$ is represented by the 5-row
$\hat{\Pi}^{\prime}=[\Pi^{\prime*},q^{\prime}]$ where, restoring the primes,
(52) reads

$\Pi^{\prime}=m_{0}\,U^{\prime}\quad\mbox{with}\quad
U^{\prime}=\frac{dX}{ds}\quad\mbox{such that}\quad U^{\prime*}U^{\prime}=1$

In the fibered frame $(\hat{\bm{e}}_{1},\cdots,\hat{\bm{e}}_{5})$ adapted to
the section $f$, the linear 5-momentum is represented by
$\hat{\Pi}=[\Pi^{*},q]$ given by the transformation law of linear forms

$\hat{\Pi}=\hat{\Pi}^{\prime}\,\hat{P}$

with (79), that gives

$\Pi^{*}=m_{0}\,U^{\prime*}P-2\,q^{\prime}A^{*},\qquad q=q^{\prime}$

hence the former relation can be recast as

$\Pi^{*}=m_{0}\,U^{*}-2\,q\,A^{*}$ (88)

In the considered bases, the covariant derivative reads

$\hat{\nabla}_{U}\hat{\Pi}=\dot{\hat{\Pi}}-\hat{\Pi}\,\hat{\Gamma}(U)$

where $\dot{\hat{\Pi}}$ is the derivative of $\hat{\Pi}$ with respect to the
arc length $s$ that is the proper time of the particle because $c=1$.

According to hypothesis (H5), (77), (86) and (87) lead to

$\nabla_{U}\Pi^{*}-q\,(\Gamma^{\prime 5}(P\,U)\,P-2\,\nabla_{U}A^{*})=0$ (89)
$\dot{q}=0$ (90)

As expected, according to the experience, the electric charge $q$ is an
integral of the motion. Then, using (88), (89) reads

$\forall m_{0}\neq 0,\;\;\forall q,\;\;\forall U\;\mbox{such
that}\;U^{*}U=1,\qquad\nabla_{U}(m_{0}\,U^{*})-q\,\Gamma^{\prime
5}(P\,U)\,P=0$ (91)

In particular, if $q=0$, we have

$\forall m_{0}\neq 0,\;\;\forall U\;\mbox{such
that}\;U^{*}U=1,\qquad\dot{m}_{0}\,U^{*}+m_{0}\,\nabla_{U}U^{*}=0$

Right multiplying by $U$ gives

$\forall m_{0}\neq 0,\;\;\forall U\;\mbox{such
that}\;U^{*}U=1,\qquad\dot{m}_{0}+m_{0}\,(\nabla_{U}U^{*})\,U=0$

but as $U^{*}U=1$ and $\nabla_{U}G=0$, we have

$(\nabla_{U}U)^{*}U=(\nabla_{U}U^{*})\,U=-U^{*}(\nabla_{U}U)=-(\nabla_{U}U)^{*}U$

then $(\nabla_{U}U)^{*}U$ vanishes and, consequently

$(\nabla_{U}U^{*})\,U=0$ (92)

that entails

$\dot{m}_{0}=0$

As expected, according to the experience, the rest mass $m_{0}$ is an integral
of the motion555However, it is possible to extend the formalism to a particle
with time-varying mass by introducing a thrust force as in de Saxcé and Vallée
[2016]. Besides, (91) is reduced to

$\forall m_{0}\neq 0,\;\;\forall q,\;\;\forall U\;\mbox{such
that}\;U^{*}U=1,\qquad m_{0}\,\nabla_{U}U^{*}-q\,\Gamma^{\prime 5}(P\,U)\,P=0$
(93)

Owing to (92), right multiplying by $U$ gives for $q\neq 0$

$\Gamma^{\prime 5}(P\,U)\,P\,U=0$

The bilinear form

$\varphi(U,V)=\Gamma^{\prime 5}(P\,U)\,P\,V$

is such that $\varphi(U,U)=0$ for all $U$ then it is skew-symmetric. There
exists a $4\times 4$ matrix $F$ such as

$F^{T}=-F\qquad\varphi(U,V)=U^{T}F\,V,\qquad\Gamma^{\prime 5}(P\,U)\,P=U^{T}F$
(94)

Moreover, (87) becomes

$\Gamma^{5}(U)=U^{T}F-2\,\nabla_{U}A^{*}$

The $j$-th component of this 4-row is

$\Gamma^{5}_{j}(U)=U^{k}F_{kj}-2\,\nabla_{U}A_{j}$

that gives rise, $(e_{i})$ being the canonical basis of $\mathbb{R}^{4}$, to
the corresponding Christoffel’s symbols

$\Gamma^{5}_{ij}=\Gamma^{5}_{j}(e_{i})=\delta^{k}_{i}F_{kj}-2\,\nabla_{i}A_{j}=F_{ij}-2\,\nabla_{i}A_{j}$
(95)

The torsion free condition (74) gives

$\Gamma^{5}_{ij}-\Gamma^{5}_{ji}=F_{ij}-F_{ji}-2\,(\nabla_{i}A_{j}-\nabla_{j}A_{i})=2\,[F_{ij}-(\nabla_{i}A_{j}-\nabla_{j}A_{i})]=0$
(96)

where, because of the torsion free condition (84)

$\nabla_{i}A_{j}-\nabla_{j}A_{i}=\left(\partial_{i}A_{j}-\Gamma^{k}_{ij}A_{k}\right)-\left(\partial_{j}A_{i}-\Gamma^{k}_{ji}A_{k}\right)=\left(\partial_{i}A_{j}-\partial_{j}A_{i}\right)-(\Gamma^{k}_{ji}-\Gamma^{k}_{ij})\,A_{k}=\partial_{i}A_{j}-\partial_{j}A_{i}$

Then the skew-symmetric $4\times 4$ matrix $F$ of elements

$F_{ij}=\partial_{i}A_{j}-\partial_{j}A_{i}$ (97)

represents a skew-symmetric 2-covariant tensor (or 2-form) $\bm{F}$ and, owing
to (94), the equation (93) becomes

$m_{0}\,\nabla_{U}U^{*}=q\,U^{T}F$

by transposition and left multiplication by $G^{-1}$, we obtain the equation
of motion

$m_{0}\,\nabla_{U}U=-q\,\bar{F}\,U$ (98)

where the skew-adjoint matrix

$\bar{F}=G^{-1}F$

represents a 1-covariant and 1-contravariant tensor $\bar{\bm{F}}$ of
components $F^{i}_{j}$. The physical interpretation is that the vector
$\bm{A}$ of which the adjoint $\bm{A}^{*}$ is represented by the row $A^{*}$
is the electromagnetic 4-potential, $\bm{F}$ (resp. $\bar{\bm{F}}$) is the
2-covariant (resp. 1 contravariant and 1-covariant) electromagnetic field
666with the conventions used in Souriau [1970, 1997b, §15] and

$\bm{f}=-q\,\bar{\bm{F}}\cdot\bm{U}$

is the Lorentz force, represented by the right hand side of (98). We recover
the equation of motion of a charged particle in the gravitation and
electromagnetic fields that fits the observations today. It deserves to
remember that the equation (77) itemizes into a group of two equations, the
former one being the previous equation of motion and the later one (90) being
the conservation of the charge

$\dot{q}=0$

The condition (97) means that the 2-form $\bm{F}$ is the exterior derivative
of the 1-form $\bm{A}^{*}$. The 4-potential $\bm{A}$ and its adjoint
$\bm{A}^{*}$ being represented respectively by

$A=\left[\begin{array}[]{c}\phi\hfill\\\ \mathsf{A}\hfill\\\
\end{array}\right],\qquad A^{*}=[\phi,-\mathsf{A}^{T}]$

the electromagnetic fields $\bm{F}$ and $\bar{\bm{F}}$ are represented
respectively by

$F=\left[\begin{array}[]{cc}0\hfill&-E^{T}\\\ E\hfill&-j(B)\\\
\end{array}\right],\qquad\bar{F}=\left[\begin{array}[]{cc}0\hfill&-E^{T}\\\
-E\hfill&j(B)\\\ \end{array}\right]$

where

$E=-grad\,\phi-\partial_{t}\mathsf{A},\qquad B=curl\,\mathsf{A}$ (99)

where $\partial_{t}$ is the partial derivative to the time. Owing to (25), the
Lorentz force $\bm{f}$ is represented by

$f=-q\,\bar{F}\,U=\gamma\,q\,\left[\begin{array}[]{c}E\cdot v\hfill\\\
E+v\times B\hfill\\\ \end{array}\right]$

Remark 1. The equation (98) is the usual form of the equation of motion as
found in the literature. It has been obtained by a suitable choice of adapted
bases but it is not the most general form according to the covariant equation
(77). Of course, it is possible to make other choices but it is not
necessarily interesting.

Remark 2. It can be verified that, as expected, the connection $\hat{\nabla}$
preserves in the adapted bases the pullback $\hat{\bm{G}}_{0}$ of the space-
time metric and the vector $\hat{\bm{\Omega}}_{0}$

$\hat{\nabla}\,\hat{G}_{0}=0,\qquad\hat{\nabla}\,\hat{\Omega}_{0}=d\hat{\Omega}_{0}+\hat{\Gamma}\,\hat{\Omega}_{0}=0$

For $\hat{\bm{G}}_{0}$, it results from (83) while for $\hat{\bm{\Omega}}_{0}$
it is a consequence of (81) and

$\hat{\Omega}_{0}=\left[{{\begin{array}[]{cc}0\hfill\\\ 1\hfill\\\
\end{array}}}\right]$

resulting from (68) and (79).

Remark 3. it remains some freedom in the definition of the electromagnetic
potential. For instance, we could have erased the factor $1/2$ before $A^{*}$
in the definition (79) but in this case we should have replaced $q$ by $-q/2$
everywhere. Both variants are conventional. We kept the one we considered the
most convenient.

Remark 4. It is worth now to revisit the original Kaluza-Klein theory. In a
$\hat{\bm{G}_{1}}$-basis
$(\hat{\bm{e}}^{\prime}_{1},\cdots,\hat{\bm{e}}^{\prime}_{5})$, the metric is
represented by

$\hat{G}^{\prime}=\left[\begin{array}[]{cc}G^{\prime}_{0}\hfill&0\hfill\\\
0\hfill&-1\hfill\\\ \end{array}\right]$

As in general the Riemannian structures are non integrable, we consider a
coordinate frame $(\bm{\partial}_{1},\cdots,\bm{\partial}_{5})$ associated to
a coordinate system $(X^{1},\cdots,X^{5})$ of the 5 dimensional universe
$\hat{\mathcal{U}}$ obtained by the change of basis

$\bm{\partial}_{\alpha}=\hat{P}^{\beta}_{\alpha}\hat{\bm{e}}^{\prime}_{\beta}$

with the transformation matrix (79). The metric is represented in the new
frame by

$\hat{G}=\hat{P}^{T}\hat{G}^{\prime}\,\hat{P}=\left[{{\begin{array}[]{cc}G-4\,A^{*T}A^{*}\hfill&2\,A^{*T}\hfill\\\
2\,A^{*}\hfill&-1\hfill\\\ \end{array}}}\right]$

where $G$ given by (80). Using classical tensor notation, it holds

$[\hat{G}_{\alpha\beta}]=\left[{{\begin{array}[]{cc}\hat{G}_{ij}=G_{ij}-4\,A_{i}A_{j}\hfill&\hat{G}_{j5}=2\,A_{j}\hfill\\\
\hat{G}_{5i}=2\,A_{i}\hfill&\hat{G}_{55}=-1\hfill\\\ \end{array}}}\right]$

If the expression stands out the standard presentation of the metric in
Kaluza-Klein theory, it is matter of conventions. The factor $2$ before
$A_{i}$ and $A_{j}$, and $-4$ before $A_{i}A_{j}$ result from the definition
of $A^{*}$. We explained these things in the previous remark . The negative
value of $\hat{G}_{55}$ is consistent with the choice of the signature
$(+----)$. Restoring the positive value consists in adopting the signature
$(-++++)$.

Calculating the connection matrix, we have to take care that both $G$ and
$A^{*}$ are fields defined on $\hat{\mathcal{U}}$, then functions of the five
coordinates $\hat{X}^{1},\cdots,\hat{X}^{5}$, giving up Kaluza’s ”cylinder
condition”.

Let $s\mapsto\hat{\bm{X}}(s)$ be the trajectory of a particle in the space-
time, parameterized by the arc length $s$ (for the metric $\hat{\bm{G}}$) and

$\hat{\bm{U}}=\frac{d\hat{\bm{X}}}{ds}$

is the unit tangent vector such that $\hat{\bm{U}}^{*}\hat{\bm{U}}=1$. We
claim that the motion of a particle is such that its linear 5-momentum is
parallel-transported

$\hat{\nabla}_{\hat{\bm{U}}}\hat{\bm{\Pi}}=\bm{0}$ (100)

The linear 5-momentum is represented in the $\hat{\bm{G}_{1}}$-basis by the
5-row

$\hat{\Pi}^{\prime}=[m_{0}\,U^{\prime*}P,q^{\prime}]$

with $q^{\prime}=m_{0}U^{\prime 5}$ and in the coordinate basis by

$\hat{\Pi}=[m_{0}\,U^{*}-2\,q\,A^{*},q]$

with $U^{*}=U^{\prime*}P$ and $q=m_{0}U^{5}$. Hence the trajectory of the
particle is not a geodesic of the manifold.

Remark 5. Finally, it is worth to remark that, although the choice (79) of
$\hat{P}$ is not general, it is sufficient to guarantee the existence and
uniqueness of the connection in terms of the fields $G$ and $A^{*}$ thanks to
the free torsion condition. On the other hand, this choice depends on the
section $f$. It is what we are going to discuss now.

###  8.4 Gauge transformation

The coordinate basis $(\bm{\partial}_{i})$ and $\hat{\bm{e}}_{5}$ being given,
the adapted basis $(\hat{\bm{e}}_{\alpha})$ of $\mathcal{U}$ to $f$ is
uniquely defined by the section $f$. We would like to know to what extend the
equation of motion is independent on the choice of the section $f$. For this
aim, we consider another section $f^{\prime\prime}$ and the corresponding
adapted basis $(\hat{\bm{e}}^{\prime\prime}_{\alpha})$. As
$(T\pi)\hat{\bm{e}}_{i}=(T\pi)\hat{\bm{e}}^{\prime\prime}_{i}=\bm{\partial}_{i}$,
the vector $\hat{\bm{e}}^{\prime\prime}_{i}-\hat{\bm{e}}_{i}$ is tangent to
the fiber of $\mathcal{U}$ over $\bm{X}$ then

$\hat{\bm{e}}^{\prime\prime}_{i}-\hat{\bm{e}}_{i}=b_{i}\hat{\bm{e}}_{5},\qquad\hat{\bm{e}}^{\prime\prime}_{5}=\hat{\bm{e}}_{5}$

the change of basis from $(\hat{\bm{e}}_{\alpha})$ to
$(\hat{\bm{e}}^{\prime\prime}_{\alpha})$ is given by the transformation matrix

$\hat{P}_{f\rightarrow
f^{\prime\prime}}=\left[\begin{array}[]{cc}1_{\mathbb{R}^{4}}\hfill&0\hfill\\\
b^{*}\hfill&1\hfill\\\ \end{array}\right]$

where $b^{*}$ is the 4-row of components $b_{i}$. The set of these matrices is
an Abelian group $\hat{\mathbb{G}}_{b}$ for the matrix product. To determine
whether the $\hat{\mathbb{G}}_{b}$-structure is integrable, we have to solve
the PDE system

$\frac{\partial\hat{X}^{\prime\prime}}{\partial\hat{X}}=\hat{P}_{f\rightarrow
f^{\prime\prime}},\qquad\hat{P}_{f\rightarrow
f^{\prime\prime}}\in\hat{\mathbb{G}}_{b}$

that gives rise to

$\frac{\partial X^{\prime\prime}}{\partial
X}=1_{\mathbb{R}^{4}},\qquad\frac{\partial X^{\prime\prime}}{\partial
X^{5}}=0,\qquad\frac{\partial X^{\prime\prime 5}}{\partial
X}=-b,\qquad\frac{\partial X^{\prime\prime 5}}{\partial X^{5}}=1$

The two former equations lead to $X"=X+X_{0}$ where the constant $X_{0}$ may
be discarded without inconvenience

$X"=X$

The last equation gives

$X"^{5}=X^{5}+2\,h(X)$

where $h$ is an arbitrary smooth function. Finally, the third equation
provides the integrability condition

$b^{*}=-2\,\frac{\partial h}{\partial X}$

The linear form $\bm{\Phi}$ represented in the basis $(\hat{\bm{e}}_{\alpha})$
by the 5-arrow

$\Phi=[A^{*},1]$

is represented in the basis $(\hat{\bm{e}}^{\prime\prime}_{\alpha})$ by the
5-arrow $\Phi^{\prime\prime}=\Phi\,\hat{P}_{f\rightarrow f^{\prime\prime}}$,
leading to the gauge transformation

$A^{\prime\prime*}=A^{*}+\frac{\partial h}{\partial X}$ (101)

As the matrix (80) does not depends on the fifth coordinate, $G$ is not
modified by the coordinate change as well as, owing to (97), $\bar{F}$ and
$F=G^{-1}F$. Hence the choice of the section $f$ does not modify the equation
of motion (98).

##  9 Extended variational relativity

The goal of this section is to extend the general theorems of the relativity
based on Einstein-Hilbert functional (Hilbert [1915]). We use Palatini
formalism, considering the connection as an unknown independent field
(Palatini [1919]).

###  9.1 Stationary action principle

Inspired by the presentation given in Souriau [1964], we start from the
following principles

  * (P1)

To every physical phenomenon corresponds a field $\bm{z}$, represented in a
local chart by a $n_{(z)}$-column $z$, and a field

$(z,\partial_{i}z,G_{ij},A_{i})\mapsto
L_{(z)}(z,\partial_{i}z,G_{ij},A_{i})\in\mathbb{R}$

called presence of the phenomenon in Souriau [1964].

  * (P2)

For the Lagrangian

$L=\sum_{z}L_{(z)}$

the action

$S=\int_{\mathcal{D}}L\,vol=\int_{\mathcal{D}}L\,u\,\mbox{d}^{4}X=\sum_{z}S_{(z)}=\sum_{z}\int_{\mathcal{D}}L_{(z)}u\,\mbox{d}^{4}X$
(102)

is stationary for every variation of $G,A^{*}$ and the concomitant phenomena,
null on the boundary of an open domain $\mathcal{D}$ of the space-time, $vol$
being the Riemannian volume 4-form.

As the signature of Minkowski metric is $(+---)$, $\det G<0$, then the
component of $vol$ is

$u=\sqrt{-\det G}$

and the covariant divergence of any vector field $\bm{V}$ for Levi-Civita
connection is the scalar field

$\mbox{div}\,\bm{V}=\nabla_{i}V^{i}=\frac{1}{u}\,\partial_{i}(u\,V^{i})$ (103)

By differentiation of the presence, we have

$\delta L_{(z)}=p\,\delta z+\sigma^{j}\delta(\partial_{j}z)+E^{km}_{(z)}\delta
G_{km}+\tilde{T}^{k}_{(z)}\delta A_{k}$ (104)

where $p,\sigma^{j}$ are $n_{(z)}$-rows, $E^{km}_{(z)}=E^{mk}_{(z)}$ and
$\tilde{T}^{k}_{(z)}$ are the components of a vector field
$\tilde{\bm{T}}_{(z)}$. Taking into account

$\frac{\delta u}{u}=\frac{1}{2}\,G^{ij}\delta
G_{ij}=-\frac{1}{2}\,G_{ij}\delta G^{ij}$

we have

$\delta(L_{(z)}u)=[p\,\delta
z+\sigma^{j}\delta(\partial_{j}z)+\frac{1}{2}\,T^{km}_{(z)}\delta
G_{km}+\tilde{T}^{k}_{(z)}\delta A_{k}]\,u$

where

$T^{jk}_{(z)}=2\,E^{jk}_{(z)}+L_{(z)}G^{jk}$ (105)

are the components of a symmetric 2-contravariant tensor field $\bm{T}_{(z)}$.
Introducing the $n_{(z)}$-row of scalar field

$W_{(z)}=p-\frac{1}{u}\,\partial_{j}(u\,\sigma^{j})=p-\nabla_{j}\sigma^{j}$

the vector field $\bm{V}_{(z)}(\delta\bm{z})$ represented by the 4-column
$V_{(z)}(\delta z)$ of components

$V^{j}_{(z)}(\delta z)=\sigma^{j}\delta z$

and owing to (103),

$\delta(L_{(z)}u)=[W_{(z)}\,\delta z+\mbox{div}\,(V_{(z)}(\delta
z))+\frac{1}{2}\,T^{km}_{(z)}\delta G_{km}+\tilde{T}^{k}_{(z)}\delta
A_{k}]\,u$ (106)

or, the expression being independent of the choice of the local chart

$\delta(L_{(z)}u)=[W_{(z)}\,\delta
z+\mbox{div}\,(\bm{V}_{(z)}(\delta\bm{z}))+\frac{1}{2}\,\bm{T}_{(z)}:\delta\bm{G}+\delta\bm{A}^{*}\cdot\tilde{\bm{T}}_{(z)}]\,u$
(107)

that allows to prove

###### Theorem 9.1

The stationary action principle (P2) is equivalent to

  * (i)

$W_{(z)}=0\,$ for every phenomenon

  * (ii)

$\bm{T}=\sum_{z}\bm{T}_{(z)}=\bm{0}$

  * (ii)

$\tilde{\bm{T}}=\sum_{z}\tilde{\bm{T}}_{(z)}=\bm{0}$

Proof. Indeed, combining (102) and (107), the variation of the action reads

$\delta S=\int_{\mathcal{D}}\delta(L\,u)\,\mbox{d}^{4}X=\sum_{z}\delta
S_{(z)}$

with

$\delta
S_{(z)}=\int_{\mathcal{D}}\delta(L_{(z)}u)\,\mbox{d}^{4}X=\int_{\mathcal{D}}[W_{(z)}\,\delta
z+\mbox{div}\,(\bm{V}_{(z)}(\delta\bm{z}))+\frac{1}{2}\,\bm{T}_{(z)}:\delta\bm{G}+\delta\bm{A}^{*}\cdot\tilde{\bm{T}}_{(z)}]\,vol$
(108)

where, denoting $\mathcal{L}_{\bm{V}_{(z)}(\delta\bm{z})}$ the Lie derivative
along the vector field $\bm{V}_{(z)}(\delta\bm{z})$, one has

$\int_{\mathcal{D}}\mbox{div}\,(\bm{V}_{(z)}(\delta\bm{z}))\,vol=\int_{\mathcal{D}}\mathcal{L}_{\bm{V}_{(z)}(\delta\bm{z})}\,vol$

Denoting $d$ the exterior derivative of a differential form $\iota_{\bm{V}}$
and the interior product with the vector $\bm{V}$, owing to Cartan’s formula
and Stokes’ theorem

$\int_{\mathcal{D}}\mbox{div}\,(\bm{V}_{(z)}(\delta\bm{z}))\,vol=\int_{\mathcal{D}}d[\iota_{\bm{V}_{(z)}(\delta\bm{z})}vol]=\int_{\partial\mathcal{D}}\iota_{\bm{V}_{(z)}(\delta\bm{z})}vol$

Considering smooth functions $\delta\bm{z}$ on compact supports contained in
$\mathcal{D}$, then null on its boundary, this integral vanishes and it
remains

$\delta S_{(z)}=\int_{\mathcal{D}}[W_{(z)}\,\delta
z+\frac{1}{2}\,\bm{T}_{(z)}:\delta\bm{G}+\delta\bm{A}^{*}\cdot\tilde{\bm{T}}_{(z)}]\,vol$

and the stationarity of the action entails

$\delta S=\int_{\mathcal{D}}[\sum_{z}W_{(z)}\,\delta
z+\frac{1}{2}\,\bm{T}:\delta\bm{G}+\delta\bm{A}^{*}\cdot\tilde{\bm{T}}]\,vol=0$

Considering the only non vanishing variation of a particular phenomenon $z$
and applying the fundamental lemma of the calculus of variations, we prove
(i). Likewise, considering the non vanishing compactly supported smooth
variations of $\bm{G}$ (resp. $\bm{A}^{*}$), we prove (ii) (resp. (iii)).
$\blacksquare$

Remark. It is worth to remark that the number of field equations (i) and (ii),
fourteen, is the same as the number of unknowns (independent components of $G$
and $A^{*}$), that avoids being caught in the dilemma of classical Kaluza-
Klein theory: add the dilaton but one condition has no physical
interpretation, or set the dilaton to one to recover Maxwell equations but
this condition is violated.

The principle of stationary action can be used in different ways. For
instance, it is worth to notice that the equation (i)

$p-\nabla_{j}\sigma^{j}=0$ (109)

can be obtained considering only the stationarity of the corresponding part
$S_{(z)}$ of the action. It is this way that we are going to follow from now
on.

###### Corollary 1

$\mbox{div}\,\tilde{\bm{T}}_{(z)}=0\,$ for every phenomenon

Proof. According to the gauge transformation (101), we perform the
substitution $A_{i}\rightarrow A_{i}+\partial_{i}h$ in the Lagrangian
$L_{(z)}$ of the action $S_{(z)}$. Applying the equation (109) to the
phenomenon $h$ (then $p=0$ and $\sigma_{j}=\tilde{T}^{j}_{(z)}$) gives

$\nabla_{j}\tilde{T}^{j}_{(z)}=0$ (110)

that achieves the proof. $\blacksquare$

###  9.2 Laws of conservation

###### Theorem 9.2

The equations $W_{(z)}=0\,$ and $\mbox{div}\,\tilde{\bm{T}}_{(z)}=0\,$ entail
the law of conservation

$\mbox{div}\,\bar{\bm{T}}_{(z)}+\tilde{\bm{T}}_{(z)}\cdot\bm{F}=\bm{0}$ (111)

of the 1-contravariant and 1-covariant tensor $\bar{\bm{T}}_{(z)}$ of
components $(T_{(z)})^{i}_{j}=(T_{(z)})^{ik}G_{kj}$.

Proof. A smooth vector field $\delta\bm{X}$ of compact support contained in
$\mathcal{D}$ is represented in a local chart by a 4-column $\delta X$ and
leads to the variations $\mathcal{L}_{\delta X}\,z,\mathcal{L}_{\delta X}\,G$
and $\mathcal{L}_{\delta X}\,A^{*}$. Owing to (106), the corresponding
variation of the Lagrangian is given by

$\delta(L_{(z)}u)=[W_{(z)}\,\mathcal{L}_{\delta
X}\,z+\mbox{div}\,(V_{(z)}(\delta
z))+\frac{1}{2}\,T^{km}_{(z)}(\mathcal{L}_{\delta
X}\,G)_{km}+\tilde{T}^{j}_{(z)}(\mathcal{L}_{\delta X}\,A^{*})_{j}]\,u$

where, owing to (85)

$(\mathcal{L}_{\delta X}\,G)_{km}=\delta
X^{r}\partial_{r}G_{km}+G_{kr}\partial_{m}\delta
X^{r}+G_{rm}\partial_{k}\delta X^{r}=\nabla_{k}\delta X_{m}+\nabla_{m}\delta
X_{k}$

Then, taking into account the symmetry of the tensor $\bm{T}_{(z)}$

$\frac{1}{2}\,T^{km}_{(z)}(\mathcal{L}_{\delta
X}\,G)_{km}=T^{km}_{(z)}\,\nabla_{k}\delta
X_{m}=\nabla_{k}(T^{km}_{(z)}\,\delta X_{m})-(\nabla_{k}T^{km}_{(z)})\,\delta
X_{m}$ $\frac{1}{2}\,T^{km}_{(z)}(\mathcal{L}_{\delta
X}\,G)_{km}=\nabla_{k}((T_{(z)})^{k}_{m}\,\delta
X^{m})-(\nabla_{k}(T_{(z)})^{k}_{m})\,\delta X^{m}$

or, the expression being independent of the choice of the local chart

$\frac{1}{2}\,\bm{T}_{(z)}:\mathcal{L}_{\delta\bm{X}}\bm{G}=\mbox{div}\,(\bar{\bm{T}}_{(z)}\cdot\delta\bm{X})-(\mbox{div}\,\bar{\bm{T}}_{(z)})\cdot\delta\bm{X}$
(112)

Besides, one has

$(\mathcal{L}_{\delta X}\,A^{*})_{j}=\delta
X^{r}\partial_{r}A_{j}+A_{r}\partial_{j}\delta X^{r}$

then, integrating by part

$u\,\tilde{T}^{j}_{(z)}(\mathcal{L}_{\delta
X}\,A^{*})_{j}=\partial_{j}(u\,\tilde{T}^{j}_{(z)}A_{r}\delta
X^{r})-[\partial_{j}(u\,\tilde{T}^{j}_{(z)}A_{r})-u\,\tilde{T}^{j}_{(z)}\partial_{r}A_{j}]\,\delta
X^{r}$

Taking into account (97)

$u\,\tilde{T}^{j}_{(z)}(\mathcal{L}_{\delta
X}\,A^{*})_{j}=\partial_{j}(u\,\tilde{T}^{j}_{(z)}A_{r}\delta
X^{r})-[\partial_{j}(u\,\tilde{T}^{j}_{(z)})A_{r}+u\,\tilde{T}^{j}_{(z)}F_{jr})]\,\delta
X^{r}$

and, owing to (103) and (110), we obtain

$u\,\tilde{T}^{j}_{(z)}(\mathcal{L}_{\delta
X}\,A^{*})_{j}=[\nabla_{j}(\tilde{T}^{j}_{(z)}A_{r}\delta
X^{r})-\tilde{T}^{j}_{(z)}F_{jr}\,\delta X^{r}]\,u$

or, the expression being independent of the choice of the local chart

$(\mathcal{L}_{\delta\bm{X}}\bm{A}^{*})\cdot\tilde{\bm{T}}_{(z)}=\mbox{div}\,((\bm{A}^{*}\cdot\delta\bm{X})\,\tilde{\bm{T}}_{(z)})-\tilde{\bm{T}}_{(z)}\cdot\bm{F}\cdot\delta\bm{X}$
(113)

Introducing into (108), with the variations given by Lie derivatives, the
expression (112) and (113), and owing to Theorem 9.1 (i), we have

$\int_{\mathcal{D}}\mbox{div}\,[\bm{V}_{(z)}\,(\mathcal{L}_{\delta\bm{X}}\bm{z})+\bar{\bm{T}}_{(z)}\cdot\delta\bm{X}+(\bm{A}^{*}\cdot\delta\bm{X})\,\tilde{\bm{T}}_{(z)}]\,vol-\int_{\mathcal{D}}[\mbox{div}\,\bar{\bm{T}}_{(z)})+\tilde{\bm{T}}_{(z)}\cdot\bm{F}]\cdot\delta\bm{X}\,vol=0$

Considering smooth variations of compact support contained in $\mathcal{D}$
and reasoning as in the proof of Theorem 9.1, the first integral vanishes and,
applying the fundamental lemma of the calculus of variations, we obtain (111).
$\blacksquare$

Our aim now is to apply these general principle by constructing Lagrangians
for three phenomena, the matter, the gravitation and the electromagnetism.

###  9.3 Laws of conservation of the matter

Matter manifold

In Souriau [1964, §36], the motion of a continuum is described by a line
bundle $\pi_{A}:\mathcal{U}\rightarrow\mathcal{M}$ where $\mathcal{M}$ is a
Riemannian manifold of dimension 3 representing the matter and called the
matter manifold. Its metric is denoted $\bm{G}_{m}$. Each fiber is the
trajectory of a material particle $\bm{a}=\pi_{A}(\bm{X})$. In a local chart
of $\mathcal{M}$, it is represented by its Lagrange coordinates
$a\in\mathbb{R}^{3}$. As $a$ is an invariant of the motion, the material
derivative vanishes:

$\frac{d\bm{a}}{ds}=(T\pi_{A})\cdot\bm{U}=\bm{0}$

where $\bm{U}$ is tangent to the trajectory and defined by (76). In local
charts of $\mathcal{U}$ and $\mathcal{M}$,

$\frac{da}{ds}=\frac{\partial a}{\partial X}\,\frac{dX}{ds}=M\,U=0$

where, because the fiber is a line, $M$ is a $3\times 4$ matrix of full rank
and $U$ is given by (25).

To extend this formalism to the 5D universe $\hat{\mathcal{U}}$, we consider a
line bundle $\hat{\pi}_{A}:\hat{\mathcal{U}}\rightarrow\hat{\mathcal{M}}$
where $\hat{\mathcal{M}}$ is a manifold of dimension 4 representing the matter
in this universe. By analogy with the hypothesis (H2) of Section 8.2, we
suppose that $\pi_{M}:\hat{\mathcal{M}}\rightarrow\mathcal{M}$ is a principal
$U(1)$-bundle. Bearing in mind that we have to perform a pullback, we assume
that $\pi_{A}\circ\pi=\pi_{M}\circ\hat{\pi}_{A}$, then the following diagram
is commutative

$\begin{CD}\hat{\mathcal{U}}@>{\hat{\pi}_{A}}>{}>\hat{\mathcal{M}}\\\
@V{\pi}V{}V@V{}V{\pi_{M}}V\\\ \mathcal{U}@>{}>{\pi_{A}}>\mathcal{M}\end{CD}$

In fibered charts, we have

$a=\pi_{A}(\pi(\hat{X}))=\pi_{A}(\pi(\left[{{\begin{array}[]{cc}X\hfill\\\
X^{5}\hfill\\\ \end{array}}}\right]))=\pi_{A}(X)$

and in the other hand

$a=\pi_{M}(\hat{a})=\pi_{M}(\left[{{\begin{array}[]{cc}a\hfill\\\
a^{4}\hfill\\\
\end{array}}}\right])=\pi_{M}(\hat{\pi}_{A}(\hat{X}))=\pi_{M}(\left[{{\begin{array}[]{cc}\varphi(\hat{X})\hfill\\\
\varphi^{4}(\hat{X})\hfill\\\ \end{array}}}\right])=\varphi(\hat{X})$

that entails

$\varphi(\left[{{\begin{array}[]{cc}X\hfill\\\ X^{5}\hfill\\\
\end{array}}}\right])=\pi_{A}(X)$ (114)

The vector $\hat{U}$ tangent to the trajectory of a particle $\hat{\bm{a}}$ is
such that

$(T\hat{\pi}_{A})\cdot\hat{\bm{U}}=\bm{0}$

In fibered charts, owing to (114), this relation is represented by

$\hat{M}\,\hat{U}=\left[{{\begin{array}[]{cc}M\hfill&0\hfill\\\
\Phi\hfill&\mu\hfill\\\
\end{array}}}\right]\,\left[{{\begin{array}[]{cc}U\hfill\\\ U^{5}\hfill\\\
\end{array}}}\right]=\left[{{\begin{array}[]{cc}0\hfill\\\ 0\hfill\\\
\end{array}}}\right]$

where $\Phi$ is a 4-row and $\mu$ is a non null scalar (for $\hat{M}$ being of
full rank). If $\mu=0$, A non null solution is $U=0,\hat{U}^{5}\neq 0$, then
$\hat{U}$ is not timelike. In the sequel, we exclude this case.

Conformation tensor

All we have just done is also valid in fibered bases, not necessarily
associated to local coordinates. As in hypothesis (H2) of Section 8.2, let us
consider a fibered basis
$(\hat{\bm{e}}^{\prime}_{\alpha})=(\hat{\bm{e}}^{\prime}_{1},\cdots,\hat{\bm{e}}^{\prime}_{4},\bm{\hat{\Omega}}_{0})$
of $T_{\hat{\bm{X}}}\hat{\mathcal{U}}$ such that
$(\bm{e}^{\prime}_{1},\cdots,\bm{e}^{\prime}_{4})$ is an orthonormal basis and
in which the tangent map $T\hat{\pi}_{A}$ is represented by the matrix

$\hat{M}^{\prime}=\left[{{\begin{array}[]{cc}M^{\prime}\hfill&0\hfill\\\
\Phi^{\prime}\hfill&\mu^{\prime}\hfill\\\ \end{array}}}\right]$ (115)

where $M^{\prime}$ is full rank and $\mu^{\prime}\neq 0$. Let us verify now
that 7 is the number of independent invariants of $\hat{M}^{\prime}$ for the
action of $\hat{\mathbb{G}}_{0}$

$\hat{M}^{\prime}\mapsto\hat{M}^{\prime}\,\hat{P}$

We determine the isotropy group of $\hat{M}^{\prime}$ by solving the system of
equations $\hat{M}^{\prime}=\hat{M}^{\prime}\,\hat{P}$ with respect to
$\hat{P}$ given by (37), itemized as

$M^{\prime}=M^{\prime}\,P,\qquad\Phi^{\prime}=\Phi^{\prime}\,P+\mu^{\prime}\,b^{*},\qquad\mu^{\prime}=\mu^{\prime}$

where $P$ is a Lorentz transformation and the last equation is trivially
fulfilled. Like in Section 6.3, we differentiate these equations with respect
to $\hat{P}$ at the identity and we take into account that $P$ is skew-
adjoint, that leads to determine the non vanishing solutions $(\delta P,\delta
b)$ of the linear system

$M^{\prime}\,\delta P=0,\qquad\Phi^{\prime}\,\delta P+\mu^{\prime}\,\delta
b^{*}=0$ (116)

where $\delta P^{*}=-\,\delta P$. According to Theorem 2.1 $\diamondsuit$, if
$\delta P\neq 0$ there exist two linear independent vectors
$V_{1},V_{2}\in\mathbb{R}^{4}$ such that $\delta P=\mathcal{J}(V_{1},V_{2})$.
if $M^{\prime*}_{1},M^{\prime*}_{2},M^{\prime*}_{3}$ are the rows of
$M^{\prime}$, then the first equation is satisfied if
$M^{\prime*}_{k}\mathcal{J}(V_{1},V_{2})=-(\mathcal{J}(V_{1},V_{2})M_{k})^{*}=0$
for $1\leq k\leq 3$, that implies there exists a linear combination between
the rows of $M^{\prime}$ and $rank(M^{\prime})<3$, in contradiction with the
hypothesis of full rank, then $\delta P=0$. As $\mu\neq 0$, the second
equation (116) shows that $\delta b=0$. The isotropy group of $M^{\prime}$ is
reduce to the identity. The dimension of the orbit is

$\mbox{dim}\,(\mbox{orb}(\hat{M}^{\prime}))=\mbox{dim}\,\hat{\mathbb{G}}_{0}-\mbox{dim}(\mbox{iso}(\hat{M}^{\prime}))=10-0=10$

The dimension of the set $V_{M}$ of matrices (115) is $3\times 4+4+1=17$ (the
number of non vanishing components). The number of independent invariants of
the orbit is

$n_{I}=\mbox{dim}\,V_{M}-\mbox{dim}(\mbox{orb}(\hat{M}^{\prime}))=17-10=7$

A set of independent invariants are $\mu$ and the 6 independent elements of
the $3\times 3$ matrix

$H^{\prime}=M^{\prime}\,M^{\prime*}=M^{\prime}(G^{\prime}_{0})^{-1}M^{\prime
T}G_{m}$

in which $G^{\prime}_{0}$ is Minkowski’s metric (69) and $G_{m}$ is Gram’s
matrix of $\bm{G}_{m}$. As in Souriau [1964], we suppose that the behavior of
the matter is isotropic

$G_{m}=\mathbb{R}^{3},\qquad M^{*}=G^{-1}M^{T},\qquad
H=M\,M^{*}=M\,G^{-1}M^{T}$

The matrix $H$ represents in a local frame of $\mathcal{M}$ a skew-symmetric
tensor

$\bm{H}=T\pi_{A}(T\pi_{A})^{*}$

called conformation in Souriau [1964].

As in Section 8.3, we would like now to work with a coordinate frame
$(\bm{\partial}_{1},\cdots,\bm{\partial}_{4})$ and the corresponding fibered
frame adapted to the section $f$, obtained by the change of basis (78) with
the transformation matrix $\hat{P}$ given by (79) and in which
$T\hat{\pi}_{A}$ is represented by the matrix

$\hat{M}=\hat{M}^{\prime}\hat{P}=\left[{{\begin{array}[]{cc}M\hfill&0\hfill\\\
\Phi-2\,\mu\,A^{*}\hfill&\mu\hfill\\\ \end{array}}}\right]$

where $M=M^{\prime}P$, $\Phi=\Phi^{\prime}P$ and $\mu=\mu^{\prime}$. If
$\mu\neq 0$, the matrix $\hat{M}$ being known, the vector $\hat{U}$ tangent to
the trajectory is obtained by solving the equation $M\,U=0$, next by
calculating $\hat{U}^{5}=(\Phi-2\,\mu\,A^{*})\,U/\mu$. Later on, as we are
only interested by the trajectory in the space-time when we reduce the problem
by pullback as in Section 8.3, we can consider only the conformation as
invariant, assuming without loss of generality that $\mu=\partial_{5}a^{4}=0$,
$\Phi=2\,\mu\,A^{*}$, then the value of $\hat{U}^{5}$ is arbitrary.

Flux of a physical quantity

Now we follow the construction proposed in Souriau [1964, (36.7)]. Let
$\bm{a}\mapsto r_{0}(\bm{a})$ be a non-vanishing scalar field on $\mathcal{M}$
representing the density of an extensive physical quantity (for instance, the
mass or the electric charge), $vol$ be the Riemannian volume 4-form on the
space-time $\mathcal{U}$, $vol_{m}$ be the Riemannian volume 3-form on the
matter manifold $\mathcal{M}$, and $vol_{r_{0}}$ be the volume 3-form
$r_{0}\,vol_{m}$. We define on the space-time the vector field $\bm{J}_{r}$
such that

$\iota_{\bm{J}_{r}}vol=\pi^{*}_{A}(vol_{r_{0}})$ (117)

In local chart, for any $d_{1}X,d_{2}X,d_{3}X\in\mathbb{R}^{4}$

$vol(J_{r},d_{1}X,d_{2}X,d_{3}X)=vol_{r_{0}}(M\,d_{1}X,M\,d_{2}X,M\,d_{3}X)$
(118)

In Souriau [1965, Chapter 5, §14, Theorem (16)], it is proved that this vector
exists and is unique. As $r_{0}\neq 0$, (118) shows that $\bm{J}_{r}$ vanishes
only if the rank of $M$ is less than 3. Otherwise, in choosing
$Md_{1}X=J_{r}$, we see that

$\frac{\partial a}{\partial X}\,J_{r}=M\,J_{r}=0$ (119)

then $\bm{J}_{r}(\bm{X})$ is tangent to the trajectory passing by $\bm{X}$. In
Souriau [1964, (36.13)], it is proved that

###### Theorem 9.3

If $J_{r}$ is not lightlike

  * (i)

we have

$(J_{r})^{*}J_{r}=-(r_{0})^{2}\det{H}$

then the conformation matrix $H=M\,M^{*}$ is regular

  * (ii)

the orthogonal projection matrix onto $J_{r}$ is

$\frac{J_{r}(J_{r})^{*}}{(J_{r})^{*}J_{r}}=1_{\mathbb{R}^{4}}-M^{*}H^{-1}M$

Assuming that $J_{r}$ is timelike, we deduce from (i) that

$h=-\det{H}>0$

Introducing

$r=r_{0}\sqrt{h}$

the unique vector $\bm{J}_{r}$ satisfying (117) is the tangent vector to the
trajectory of which the length is given by $(J_{r})^{*}J_{r}=r^{2}$, then we
can put $J_{r}=r\,U$ with $U^{*}U=1$, representing the 4-flux of $r$

$\bm{J}_{r}=r\,\bm{U}=r\,\frac{d\bm{X}}{ds}$

where $r$ is a short form of
$r_{0}(\pi_{A}(\bm{X}))\,\sqrt{\det{(T\pi_{A}(T\pi_{A})^{*})}}$. Reasoning as
in the proof of Theorem 9.1 and owing to (117), we have

$\mbox{div}\,(\bm{J}_{r})\,vol=\mathcal{L}_{\bm{J}_{r}}\,vol=d[\iota_{\bm{J}_{r}}vol]=\pi^{*}_{A}(d(vol_{r_{0}}))=0$

because $d(vol_{r_{0}})$ is a 4-form field on $\mathcal{M}$ of dimension 3,
then we obtain the conservation identity of the 4-flux of $r$

$\mbox{div}\,\bm{J}_{r}=0$ (120)

Lagrangian of the matter

On this ground, our aim is to construct a Lagrangian $L_{M}$ for the matter
that is invariant for every diffeomorphism $\varphi$ of the space-time
$\mathcal{U}$

$L_{M}(\bm{a},T\pi_{A},\bm{G},\bm{A}^{*})=L_{M}(\bm{a},T\pi_{A}\circ
T\varphi,\varphi^{*}(\bm{G}),\varphi^{*}(\bm{A}^{*}))$

where $\varphi^{*}$ stands for the pullback by $\varphi$ and $\bm{A}^{*}$ is
the adjoint of $\bm{A}$. In local charts, for every transformation matrix $P$

$L_{M}(a,M,G,A^{*})=L_{M}(a^{\prime},M^{\prime},G^{\prime},A^{\prime*}),\quad
a=a^{\prime},\quad M^{\prime}=M\,P,\quad G^{\prime}=P^{T}G\,P,\quad
A^{\prime*}=A^{*}P$

Particles without spin are characterized by two physical quantities, their
mass (or energy if $c=1$) and their electric charge. Hence, the Lagrangian
must at least depend on the densities of mass and charge. According to the
principle of stationary action (P2), we adopt the decomposition

$L_{M}(a,M,G,A^{*})=\kappa\,[L_{m}(a,M,G)+L_{e}(a,M,A^{*})]$

where $\kappa$ is a non vanishing coupling constant, $L_{m}$ is active for all
particles, charged or not, sensitive to the gravitation forces, while $L_{e}$
is active only for charged particles, sensitive also to the electromagnetic
forces. For $L_{m}$ being invariant, it must depend on $a,M,G$ through the
invariants $a$ and $H$. At this stage, we restrict our presentation to fluids
then the dependence on $H$ can be limited to a dependence on $h=-\det H$ only.
The expression

$L_{m}(a,M,G)=\rho_{m0}(a)\,[\sqrt{h}+\psi(h)]$

contains the mass density

$\rho_{m}=\rho_{m0}(a)\,\sqrt{h}$

that can be completed by an internal energy to modelize pressure effects
through the function $\psi$. For $L_{e}$ being invariant, it must depend on
the invariants constructed from $a,M,G,A^{*}$. We already know the invariants
$a$ and $h$ to which we have to add a joint invariant of $M$ and $A^{*}$. The
electric current density $\bm{J}_{\rho_{e}}$ is the 4-flux of the electric
charge density

$\rho_{e}=\rho_{e0}(a)\,\sqrt{h}$ (121)

defined by, according to (118)

$vol(J_{\rho_{e}},d_{1}X,d_{2}X,d_{3}X)=vol_{\rho_{e0}}(M\,d_{1}X,M\,d_{2}X,M\,d_{3}X)$
(122)

On the other hand,

$\rho_{e0}\,\det\left[\begin{array}[]{c}M\hfill\\\ A^{*}\hfill\\\
\end{array}\right]vol(J_{\rho_{e}},d_{1}X,d_{2}X,d_{3}X)$
$=-\rho_{e0}\,vol\left(\left[\begin{array}[]{c}A^{*}J_{\rho_{e}}\hfill\\\
M\,J_{\rho_{e}}\hfill\\\
\end{array}\right],\left[\begin{array}[]{c}A^{*}d_{1}X,\hfill\\\
M\,d_{1}X,\hfill\\\
\end{array}\right],\left[\begin{array}[]{c}A^{*}d_{2}X,\hfill\\\
M\,d_{2}X,\hfill\\\
\end{array}\right],\left[\begin{array}[]{c}A^{*}d_{3}X,\hfill\\\
M\,d_{3}X,\hfill\\\ \end{array}\right],\right)$

Taking into account the property (119), we have $M\,J_{\rho_{e}}=0$ then

$\rho_{e0}\,\det\left[\begin{array}[]{c}M\hfill\\\ A^{*}\hfill\\\
\end{array}\right]vol(J_{\rho_{e}},d_{1}X,d_{2}X,d_{3}X)=(A^{*}J_{\rho_{e}})\,vol_{\rho_{e0}}(M\,d_{1}X,M\,d_{2}X,M\,d_{3}X)$

Owing to (122), we construct a function

$L_{e}(a,M,A^{*})=\rho_{e0}(a)\,\det\left[\begin{array}[]{c}M\hfill\\\
A^{*}\hfill\\\ \end{array}\right]=-A^{*}J_{\rho_{e}}=-\rho_{e}\,A^{*}U$

which is invariant because $A^{\prime*}U^{\prime}=(A^{*}P)(P^{-1}U)=A^{*}U$
and the electric charge density is invariant.

In a nutshell, the Lagrangian of the matter reads

$L_{M}(a,M,G,A^{*})=\kappa\,[(\rho_{m0}-\rho_{e0}\,A^{*}U)\,\sqrt{h}+\rho_{m0}\,\psi(h)]$

which can be enriched to modelize more complex behaviors such as, for
instance, those of hyperelastic solids for which the lagrangian depends on the
invariant conformation $H$ in a general way, not only through $h$.

Principle of stationary action and law of conservation

Owing to (104), the components of the vector $\tilde{\bm{T}}_{M}$ read

$(\tilde{T}_{M})^{i}=-\kappa\,(J_{\rho_{e}})^{i}=-\kappa\,\rho_{e}U^{i}$ (123)

Applying the principle of stationary action, Corollary 1 gives

$\mbox{div}\bm{J}_{\rho_{e}}=0$ (124)

which is satisfied because of the conservation of the 4-flux of electric
charge density $\rho_{e}$, according to (120).

To obtain the law of conservation, we have to determine the expression of
$E^{jk}$ by differentiation the Lagrangian with respect to the metric, i.e.
through $h$, then denoting $\partial_{h}$ the partial derivative with respect
to $h$

$\delta L_{M}=\partial_{h}L_{M}\delta h=-\partial_{h}L_{M}\delta(\det
H)=h\,\partial_{h}L_{M}Tr(H^{-1}\delta H)$ $\delta
L_{M}=h\,\partial_{h}L_{M}Tr(H^{-1}M\delta(G^{-1})M^{T})=-h\,\partial_{h}L_{M}Tr(M^{T}H^{-1}MG^{-1}(\delta
G)G^{-1})$

Using Theorem 9.3) (ii) with $J_{\rho_{m}}=\rho_{m}U$, it holds

$\delta
L_{M}=-h\,\partial_{h}L_{M}Tr(G^{-1}(1_{\mathbb{R}^{4}}-U\,U^{*})G^{-1}\delta
G)=h\,\partial_{h}L_{M}Tr((U\,U^{T}-G^{-1})\delta G)$

or in tensor notation

$E^{jk}=h\,\partial_{h}L_{M}(G^{jk}-U^{j}U^{k})$

from which we deduce the components (105) of the tensor field $\bm{T}_{M}$

$(T_{M})^{ij}=2\,E^{ij}_{M}+L_{M}G^{ij}=\kappa\,[(\rho+p)U^{i}U^{j}-p\,G^{ij}]$
(125)

in which occur the density of energy

$\rho=L_{M}/\kappa=(\rho_{m0}-\rho_{e0}\,A^{*}U)\,\sqrt{h}+\rho_{m0}\,\psi(h)=\rho_{m}+\rho_{m0}\,\psi(h)-A^{*}J_{\rho_{e}}$

where the second and third terms represent respectively the internal energy
and the coupling with the electromagnetism, and the pressure

$p=(2\partial_{h}L_{M}-L_{M})/\kappa=\rho_{m0}\,(2\,h\,\partial_{h}\psi-\psi)$

The law of conservation (111) for the matter

$\nabla_{i}(T_{M})^{i}_{j}+(\tilde{T}_{M})^{k}F_{kj}=0$

itemizes, owing to (123) and (125), into

$\nabla_{i}[(\rho+p)U^{i}U_{j}-p\,\delta^{i}_{j}]-\rho_{e}U^{k}F_{kj}=0$ (126)

Interpretation of the law of conservation in terms of the
$\hat{\mathbb{G}}_{0}$-connection

Let a 1-contravariant and 1-covariant free divergence tensor field
$\hat{\mathcal{\bm{T}}}$ on the space-time $\mathcal{U}$, section of the
pullback bundle $f^{*}(T\hat{\mathcal{U}}\otimes T^{*}\hat{\mathcal{U}})$

$\hat{\nabla}_{i}\hat{\mathcal{T}}^{i}_{\beta}=\partial_{i}\hat{\mathcal{T}}^{i}_{\beta}+\hat{\Gamma}^{i}_{i\mu}\hat{\mathcal{T}}^{\mu}_{\beta}-\hat{\mathcal{T}}^{i}_{\mu}\hat{\Gamma}^{\mu}_{i\beta}=0$

with the convention that Greek indices run from 1 to 5 and Latin ones run from
1 to 4. For the $\hat{\mathbb{G}}_{0}$-connection (81),
$\hat{\Gamma}^{\mu}_{i5}=0$ then the previous equation itemizes into

$\hat{\nabla}_{i}\hat{\mathcal{T}}^{i}_{j}=\partial_{i}\hat{\mathcal{T}}^{i}_{j}+\Gamma^{i}_{im}\hat{\mathcal{T}}^{m}_{j}-\hat{\mathcal{T}}^{i}_{m}\Gamma^{m}_{ij}-\hat{\mathcal{T}}^{i}_{5}\Gamma^{5}_{ij}=\nabla_{i}\hat{\mathcal{T}}^{i}_{j}-\hat{\mathcal{T}}^{i}_{5}\Gamma^{5}_{ij}=0$
$\hat{\nabla}_{i}\hat{\mathcal{T}}^{i}_{5}=\partial_{i}\hat{\mathcal{T}}^{i}_{5}+\Gamma^{i}_{im}\hat{\mathcal{T}}^{m}_{5}=\nabla_{i}\hat{\mathcal{T}}^{i}_{5}=0$
(127)

where $\nabla$ is Levi-Civita connection. It is worth to point out that the
components $\mathcal{T}^{5}_{\beta}$ are absent of these expressions.
Mimicking the linear 5-momentum of an elementary particule defined by (75) and
(88), we claim that the components of the linear 5-momentum of a continuum are

$\hat{\Pi}_{j}=\rho U_{j}-2\,\rho_{e}A_{j},\qquad\hat{\Pi}_{5}=\rho_{e}$

Adding a projection term to take into account the effects of the pressure $p$,
we claim that

$\hat{\mathcal{T}}^{i}_{j}=U^{i}\hat{\Pi}_{j}+p\,(U^{i}U_{j}-\delta^{i}_{j})=U^{i}(\rho
U_{j}-2\,\rho_{e}A_{j})+p\,(U^{i}U_{j}-\delta^{i}_{j}),\qquad\hat{\mathcal{T}}^{i}_{5}=U^{i}\hat{\Pi}_{5}=\rho_{e}U^{i}$

Owing to (95), the equation

$\mbox{div}\,\hat{\mathcal{\bm{T}}}=\bm{0}$

is represented in local charts by (127)

$\nabla_{i}[(\rho+p)U^{i}U_{j}-p\,\delta^{i}_{j}]-\rho_{e}U^{i}F_{ij}=0,\qquad\nabla_{i}(\rho_{e}U^{i})=0$

then, by comparison to (124) and (126), is equivalent to the equations
provided by the principle of stationary action and the law of conservation

$\mbox{div}\,\tilde{\bm{T}}_{M}=0,\qquad\mbox{div}\,\bar{\bm{T}}_{M}+\tilde{\bm{T}}_{M}\cdot\bm{F}=\bm{0}$

###  9.4 Field equations of the gravitation and electromagnetism

In general relativity, the gravitation is represented by the geometry of the
space-time through the connection $\Gamma$ generated by 10 potentials, the
independent components of the metric $G$. In our unified theory of gravitation
and electromagnetism, the geometry of the space-time stems from the
$\hat{\mathbb{G}}_{0}$-connection $\hat{\Gamma}$ preserving the pullback
$\hat{\bm{G}}_{0}$ of the space-time metric and the vector
$\hat{\bm{\Omega}}_{0}$. It is generated by 14 potentials, the independent
components of $G$ and $A^{*}$.

Curvature tensor

Our starting point is to calculate the curvature tensor $\hat{\bm{R}}$ on the
space-time $\mathcal{U}$ endowed with the pullback connection. This tensor is
defined, for any smooth section $\hat{\bm{W}}$ of $T\hat{\mathcal{U}}$ and for
all smooth fields $\bm{X}\mapsto\bm{U},\bm{X}\mapsto\bm{V}$ of tangent vectors
to the space-time, by the relation

$\hat{\nabla}_{(Tf)\,\bm{U}}\,(\hat{\nabla}_{(Tf)\,\bm{V}}\,\hat{\bm{W}})-\hat{\nabla}_{(Tf)\,\bm{V}}\,(\hat{\nabla}_{(Tf)\,\bm{U}}\,\hat{\bm{W}})-\hat{\nabla}_{(Tf)\,[\bm{U},\bm{V}]}\,\hat{\bm{W}}=\hat{\bm{R}}(\bm{U},\bm{V})\,\hat{\bm{W}}$
(128)

Then it is a 1-contravariant and 3-covariant tensor, skew-symmetric with
respect to the arguments $\bm{U}$ and $\bm{V}$. With the convention that Greek
indices run from 1 to 5 and Latin ones run from 1 to 4, it is represented by
the components

$\hat{R}^{\alpha}_{ij\beta}=\hat{\Gamma}^{\alpha}_{i\mu}\hat{\Gamma}^{\mu}_{j\beta}-\hat{\Gamma}^{\alpha}_{j\mu}\hat{\Gamma}^{\mu}_{i\beta}+\partial_{i}\hat{\Gamma}^{\alpha}_{j\beta}-\partial_{j}\hat{\Gamma}^{\alpha}_{i\beta}$

skew-symmetric with respect to the indices $i$ and $j$. Owing to the form (86)
of the connection matrix, Christoffel’s symbols with a covariant index equal
to 5 vanish then the only non null curvature components are

$R^{p}_{ijk}=\hat{R}^{p}_{ijk}=\Gamma^{p}_{i\mu}\Gamma^{\mu}_{jk}-\Gamma^{p}_{j\mu}\Gamma^{\mu}_{ik}+\partial_{i}\Gamma^{p}_{jk}-\partial_{j}\Gamma^{p}_{ik}$
$\tilde{R}_{ijk}=\hat{R}^{5}_{ijk}=\Gamma^{5}_{i\mu}\Gamma^{\mu}_{jk}-\Gamma^{5}_{j\mu}\Gamma^{\mu}_{ik}+\partial_{i}\Gamma^{5}_{jk}-\partial_{j}\Gamma^{5}_{ik}$

The components $R^{p}_{ijk}$ are the ones of the classical curvature tensor of
space-time endowed with the metric $\bm{G}$ (hypothesis H1). Developing (95),
one has

$\Gamma^{5}_{ij}=-\partial_{i}A_{j}-\partial_{j}A_{i}+2\,\Gamma^{k}_{ij}A_{k}$

The additional curvature tensor components read

$\tilde{R}_{ijk}=\nabla_{k}F_{ji}+2A_{q}R^{q}_{ijk}$ (129)

Lagrangian of the geometry

According to Souriau [1964, Chapter 5, §14, (35.8) to (35.10)], the Lagrangian
of the geometry depends on the connection and its partial derivatives through
the curvature tensor

$L_{G}(\hat{\Gamma},\partial_{i}\hat{\Gamma},\bm{G},\bm{A}^{*})=L_{G}(\hat{\Gamma},\hat{\bm{R}},\bm{G},\bm{A}^{*})$

We assume the decomposition

$L_{G}(\hat{\Gamma},\hat{\bm{R}},\bm{G},\bm{A}^{*})=L_{g}(\Gamma,\bm{R},\bm{G})+L_{em}(\Gamma^{5},\tilde{\bm{R}},\bm{G},\bm{A}^{*})$

where $L_{g}$ describes the behavior of the gravitation alone and $L_{em}$ is
an extra term to take into account the one of the electromagnetism. The
Lagrangian must be invariant for every diffeomorphism of the space-time.
Introducing the Ricci tensor of components obtained by contraction

$R_{jk}=R^{p}_{pjk}$

the most simple invariant built from $R^{p}_{ijk}$ and $G^{ij}$ is the scalar
curvature

$R=G^{jk}R_{jk}$

The Hilbert-Einstein Lagrangian including the cosmological constant $\Lambda$
is

$L_{g}(\Gamma,\bm{R},\bm{R})=-\Lambda+\frac{1}{2}\,G^{ij}R_{ij}$

Besides, from $\tilde{R}_{ijk},G^{ij}$ and $A_{i}$, we can construct the
invariant

$\tilde{R}=A_{r}G^{ri}G^{jk}\tilde{R}_{ijk}=A_{r}\tilde{R}^{rk}_{\;\;\;\,k}$
(130)

Then we consider the Lagrangian of the electromagnetism

$L_{em}(\Gamma^{5},\tilde{\bm{R}},\bm{G},\bm{A}^{*})=-\tilde{k}\,A_{r}G^{ri}G^{jk}\tilde{R}_{ijk}$

where $\tilde{k}$ is a new coupling constant.

In a nutshell, the Lagrangian of the geometry reads

$L_{G}(\hat{\Gamma},\hat{\bm{R}},\bm{G},\bm{A}^{*})=-\Lambda+\frac{1}{2}\,G^{ij}R_{ij}-\tilde{k}\,A_{r}G^{ri}G^{jk}\tilde{R}_{ijk}$

from which we deduce the components (105) of the tensor field $\bm{T}_{G}$

$(T_{G})^{ij}=2\,(E_{G})^{ij}+L_{G}G^{ij}=-[R^{ij}-\frac{1}{2}\,R\,G^{ij}+\Lambda\,G^{ij}]+\tilde{k}\,[A^{(i}\,\tilde{R}^{j)\;\,k}_{\;\;\;k}+A_{r}\tilde{R}^{r(ij)}-\frac{1}{2}\,\tilde{R}\,G^{ij}]$
(131)

Besides, the components of the vector $\tilde{\bm{T}}_{G}$ are

$(\tilde{T}_{G})^{i}=-\tilde{k}\,\left(\nabla_{j}F^{ji}+2A_{q}R^{qij}_{\quad\;j}\right)$
(132)

Field equations

Applying the stationary action principle (Theorem 9.1, (ii) and (iii)), we
have

$\bm{T}=\bm{T}_{G}+\bm{T}_{M}=\bm{0},\qquad\tilde{\bm{T}}=\tilde{\bm{T}}_{G}+\tilde{\bm{T}}_{M}=\bm{0}$

which, owing to (123), (125), (131) and (132), leads to

$R^{ij}-\frac{1}{2}\,R\,G^{ij}+\Lambda\,G^{ij}-\tilde{k}\,[A^{(i}\,\tilde{R}^{j)\;\,k}_{\;\;\;k}+A_{r}\tilde{R}^{r(ij)}-\frac{1}{2}\,\tilde{R}\,G^{ij}]=\kappa\,[(\rho+p)U^{i}U^{j}-p\,G^{ij}]$
(133) $-\tilde{k}\,[\nabla_{j}F^{ji}+2A_{q}R^{qij}_{\quad
j}]=\kappa\,\rho_{e}U^{i}$ (134)

In the absence of the electromagnetic potential, the first equation is reduced
to

$R^{ij}-\frac{1}{2}\,R\,G^{ij}+\Lambda\,G^{ij}=\kappa\,[(\rho+p)U^{i}U^{j}-p\,G^{ij}]$

We recover Einstein’s equations for the gravitation. Einstein’s constant (with
$c=1$ and the gravitational constant $G_{N}$ with the index $N$ in Newton’s
honour)

$\kappa=8\,\pi\,G_{N}$

has been chosen such that the non-relativistic limit yields the usual form of
Newton’s law of universal gravitation.

Our aim now is to discuss the classical limits of these field equations,
starting with (134). First, we consider the first term. In the Galilean
approximation (Duval et al. [1985, 1991], Souriau [1997a], de Saxcé and Vallée
[2016]), the connection is associated to Galilei group and Christoffel’s
symbols $\Gamma^{m}_{ij}$ vanish except777with the convention that the index
$t$ corresponds to time and $a,b,c,d$ to the spatial coordinates.

$\Gamma^{a}_{tt}=-g^{a},\qquad\Gamma^{a}_{tb}=\Gamma^{a}_{bt}=\Omega^{a}_{b}\
,$ (135)

where occur the gravity $g^{a}$ and Coriolis’ effects given by
$\Omega^{a}_{b}$, a simplified notation for the elements of the $3\times 3$
skew-symmetric matrix $j(\Omega)$ such that $j(\Omega)\,v=\Omega\times v$ for
all 3-columns $\Omega$ and $v$. For a 2-contravariant tensor of components
$F^{ji}$, the covariant divergence reads

$\nabla_{j}F^{ji}=\partial_{j}F^{ji}+\Gamma^{j}_{jm}F^{mi}+F^{jm}\Gamma^{i}_{jm}$

that, owing to (135), itemizes respectively for $j=t$ and $j=a$ as

$\nabla_{j}F^{jt}=\partial_{j}F^{jt},\qquad\nabla_{j}F^{ja}=\partial_{j}F^{ja}-g^{a}F^{tt}+(F^{bt}+F^{tb})\Omega^{a}_{b}$

For the electromagnetic field, $F^{ij}=-F^{ji}$, this leads, at the Galilean
approximation, to the exact simplification

$\nabla_{j}F^{ji}=\partial_{j}F^{ji}$ (136)

Next, we would like to assess the order of magnitude of the second term with
respect to the first one in (134), i.e. the coupling term between gravitation
and electromagnetism containing

$A_{q}R^{qij}_{\quad j}=A_{q}G^{im}G^{jq}R^{q}_{\;\;mkj}$

As we are interested by an estimate, we consider only the gravity. At the
Newtonian approximation, the only non vanishing Christoffel’s symbol is
$\Gamma^{a}_{tt}=-g^{a}$ and

$G^{-1}\cong G\cong diag(1,-1,-1,-1)$

then