# Kaluza-Klein Dimensional Reduction From Elasticity Theory of Crumpled Paper

Mokhtar Adda-Bedia<EMAIL_ADDRESS>Université de Lyon, Ecole
Normale Supérieure de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon,
France  Eytan Katzav<EMAIL_ADDRESS>The Racah Institute of
Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel

###### Abstract

During the last century, two independent theories using the concept of
dimensional reduction have been developed independently. The first, known as
Föppl-von Kármán theory, uses Riemannian geometry and continuum mechanics to
study the shaping of thin elastic structures which could become as complex as
crumpled paper. The second one, known as Kaluza-Klein theory, uses Minkowskian
geometry and general relativity to unify fundamental interactions and gravity
under the same formalism. Here we draw a parallel between these two theories
in an attempt to use concepts from elasticity theory of plates to recover the
Einstein-Maxwell equations. We argue that Kaluza-Klein theory belongs to the
same conceptual group of theories as three-dimensional elasticity, which upon
dimensional reduction leads to the Föppl-von Kármán theory of two-dimensional
elastic plates. We exploit this analogy to develop an alternative Kaluza-Klein
formalism in the framework of elasticity theory in which the gravitational and
electromagnetic fields are respectively associated with stretching-like and
bending-like deformations. We show that our approach of dimensional reduction
allows us to retrieve the Lagrangian densities of both gravitational,
electromagnetic and Dirac spinors fields as well as the Lagrangian densities
of mass and charge sources.

##  I Introduction

In their seminal work, Kaluza Kaluza (1921) and Klein Klein (1926) provided a
scheme to unify electromagnetism and gravity by supplementing general
relativity with an additional fifth dimension of spacetime. The ideas of
cylindrical boundary condition in Kaluza and compactification in Klein state
that the extra dimension, involving the electromagnetic field, would be
physically undetectable even though it would provide a means of unification.
Since then, the Kaluza-Klein (KK) paradigm is often at the basis of physical
theories that aim to unify the known fundamental interactions
(electromagnetic, weak nuclear and strong nuclear) with gravity as well as to
generate elementary particles from common first principles. Such a quest has
seen a revival with the emergence of superstring and supergravity theories Cho
(1975); Witten (1981); Mecklenburg (1984); Duff et al. (1986); Bailin and Love
(1987); Overduin and Wesson (1997); Wesson (1999, 2006); Zee (2013), which are
based on the idea that the fundamental building blocks of nature are vibrating
modes of strings and membranes in high dimensions.

Dimensional reduction is not unique to KK-like theories. In mechanics of
continuous media, a similar procedure is performed to model geometrical and
mechanical response of elastic membranes from elasticity equations of three-
dimensional bulk materials. While the general geometrical foundations were
developed by Gauss Millman and Parker (1977), the formal basis for
understanding strongly deformed plates came from the work of Föppl Föppl
(1921) and von Kármán Von Kármán (1956). The constraints of mechanical
equilibrium and differential geometry were exploited to reduce the degrees of
freedom of a sheet onto two scalar fields associated with the curvature tensor
and the two-dimensional stress tensor, respectively. The former mediates the
out-of-plane bending deformations and the latter the in-plane stretching
deformations. The nonlinear coupling between the two fields in the resulting
Föppl-von Kármán (FvK) equations often leads to localized buckling events
which are precursors to patterns similar to those formed in crumpled paper Ben
Amar and Pomeau (1997); Lobkovsky and Witten (1997). For the last three
decades, various aspects related to FvK equations and their extensions have
been the area of intense research from both mechanical, physical and
mathematical perspective Kramer and Witten (1997); Witten (2007); Efrati et
al. (2009); Audoly and Pomeau (2010).

Although the underlying geometries are different, Riemannian geometry for
elastic plates and Minkowski geometry for general relativity, we seek here to
relate these two areas in which the concept of dimensional reduction was
developed. First, we identify a scale separation between gravitational and
electromagnetic modes of the Einstein-Maxwell (EM) action, which is similar to
that between stretching and bending energy densities involved in describing
the elastic response of thin plates. Consequently, we argue that KK formalism
is of the same kind as the reduction of three-dimensional (3D) elasticity
equations leading to the FvK equations of 2D elastic plates. Then, we exploit
this analogy to develop a KK formalism in the framework of elasticity theory
of thin plates in which the gravitational and electromagnetic fields are
respectively associated with stretching-like and bending-like deformations.

This paper is organized as follow. We start by drawing the similarity between
FvK and EM formalisms which allows us to propose an elastic-like approach to
KK theory. To remain self-consistent we relegate the introduction of basic
elements of elasticity theory of thin plates to Appendix A. Using this
analogy, we perform a dimensional reduction of the action of 5D gravitational
and matter fields which we consider as different entities. The resulting 4D
action involves a single additional term which reflects an explicit
interaction between gravitational and electromagnetic fields and which
contributes only at scales within the matter content. To verify the validity
of our approach at quantum scales, we also apply the same scheme of
dimensional reduction to the action of Dirac spinor field. Finally, we
conclude by pointing out the main results and possible extensions of our
approach.

To keep it concise, most of the algebraic computations are relegated to
Appendices. Moreover, even if it is not adequate stricto sensu, we will
sometimes use a language of mechanics in general relativity to reinforce the
analogy elasticity-general relativity. Finally, throughout the whole article
including the appendices, we shall adopt notational conventions to distinguish
the quantities defined in 5D from their 4D equivalent. Unless otherwise
specified, functions defined in five dimensions are noted by the same letters
used for four ordinary space time with a circumflex accent. We employ the
alphabetic convention that letters $M,N,..$ (resp. $\mu,\nu,...$) will denote
world indices taking the values $0,1,2,3,4$ (resp. $0,1,2,3$). Finally, when
using the tetrad representation the indices from the early alphabet
$\mathrm{A},\mathrm{B},...$ (resp. $\mathrm{a},\mathrm{b},...$) shall be the
frame labels taking the values $0,1,2,3,4$ (resp. $0,1,2,3$).

##  II Similarity of Föppl-von Kármán and Einstein-Maxwell formalisms

In 4D spacetime, the actions that corresponds to Einstein gravity and Maxwell
electromagnetism are given by Landau and Lifshitz (1980)

$\displaystyle{{S}_{\mathrm{E}}}$ $\displaystyle=$
$\displaystyle-\frac{c^{3}}{16\pi G}\int\mathrm{d}V\sqrt{-g}R\;,$ (1)
$\displaystyle{{S}_{\mathrm{M}}}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi
c}\int\mathrm{d}V\sqrt{-g}F^{\mu\nu}F_{\mu\nu}\;,$ (2)

where $c$ is the speed of light, $G$ is the gravitational constant,
$\mathrm{d}V$ is the 4D volume element in hyperspace, $R$ is the Ricci
(scalar) curvature of the Lorentzian metric $g_{\mu\nu}$, and $F_{\mu\nu}$ is
the electromagnetic field tensor given by $F_{\mu\nu}=A_{\nu;\mu}-A_{\mu;\nu}$
where $A_{\mu}$ denotes the vector potential field. While the two theories
have been developed separately, the classical Einstein-Maxwell action
${{S}_{\mathrm{EM}}}$ is simply defined as the sum of the two
${S}_{\mathrm{E}}+{S}_{\mathrm{M}}$, namely Landau and Lifshitz (1980)

${{S}_{\mathrm{EM}}}=-\frac{c^{3}}{16\pi
G}\int\mathrm{d}V\sqrt{-g}\left(R+\frac{G}{c^{4}}F^{\mu\nu}F_{\mu\nu}\right)\;.$
(3)

KK type theories aim to recover the action given by Eq. (3) from an underlying
unified theory involving both gravitational and electromagnetic potential
fields. Similarly, the total action of mass and electromagnetic current is the
sum of the actions of two source terms Landau and Lifshitz (1980)

${{S}_{m}}=\int\limits\mathrm{d}V\sqrt{-g}\rho_{m}c-\frac{1}{c}\int\limits\mathrm{d}V\sqrt{-g}\rho_{e}u^{\mu}A_{\mu}\;,$
(4)

where $u^{\mu}$ is the velocity vector along the worldline of the matter,
$\rho_{m}$ and $\rho_{e}$ are mass and charge density distributions per unit
space three-volume. Any definitive unifying theory should also justify the
additivity of the two source terms leading to Eq. (4). Such requisite is
usually a discriminating condition because it is more difficult to achieve.
Here we argue that we can achieve such a goal using an approach borrowed from
elasticity theory of continuous media. In particular, we state that the KK
compactification procedure is reminiscent of the dimensional reduction of
elasticity equations from three-dimensional (3D) bulk materials to two-
dimensional (2D) plates yielding FvK equations. For reasons of self-
consistency, the details of the latter theory are summarized in Appendix A.

Let us start by deriving scaling properties associated with the different
terms involved in Eqs. (3)-(4). Without loss of generality, we assume that
both mass and charge densities follow the same spatial distribution, which is
a realistic assumption for localised matter content Landau and Lifshitz
(1980). Therefore, we can define a mass scale $m$ and a charge scale $q$
associated to constituents of matter such that $\rho_{e}/\rho_{m}=q/m$. This
allows us to define characteristic length scales given by

$\mathrm{r}_{m}=\frac{Gm}{c^{2}}\;,\quad\mathrm{r}_{e}=\frac{\rho_{e}q}{\rho_{m}c^{2}}=\frac{q^{2}}{mc^{2}}\;,\quad\sqrt{\mathrm{r}_{m}\mathrm{r}_{e}}=\sqrt{\frac{Gq^{2}}{c^{4}}}\;,$
(5)

where $\mathrm{r}_{m}$ and $\mathrm{r}_{e}$ are length scales associated with
the content of matter and its nature (mass and charge). In addition, we can
define the geometric length scale $L$ associated with the range over which the
4D gravitational and potential fields vary. $L$ can be seen as the
characteristic extension of the 4D hyperspace, and should satisfy
$L\gg\\{\mathrm{r}_{m},\mathrm{r}_{e}\\}$ as long as matter consists of
localized sources. As a first approximation, one can assume that variations of
the metric $g$ and the potential $A_{\mu}$ are induced independently by the
mass $m$ and the charge $q$ respectively. In this case, dimensional analysis
allows us to deduce that these two fields obey scaling properties such that
$g\sim\mathrm{r}_{m}/L$ and $A_{\mu}\sim q/L$. Consequently, the Ricci scalar
and the field tensor scale as $R\sim\mathrm{r}_{m}/L^{3}$ and $F_{\mu\nu}\sim
q/L^{2}$. Therefore, using Eq. (5) one shows that the two components of the
action in Eq. (3) satisfy

$\frac{G}{c^{4}}F^{\mu\nu}F_{\mu\nu}\sim\left(\frac{\mathrm{r}_{e}}{L}\right)R\ll
R\;.$ (6)

Eq. (6) shows that there is a scale separation between gravitational and
electromagnetic effects as long as $\mathrm{r}_{e}\ll L$.

At this stage we draw an analogy with elasticity theory of plates. The action
${S}_{\mathrm{EM}}$ given by Eq. (3) has the same structure as the total
elastic energy of a plate of thickness $t$ which involves two distinct
contributions quantifying its stretching and bending energies. Whereas in
elastic plates the ratio of stretching over bending energies is quadratic in
$(t/L)\ll 1$ (see Appendix A), here the gravitational over electromagnetic
Lagrangian densities scale linearly in $(\mathrm{r}_{e}/L)\ll 1$. Except for
this difference in the strength of scale separation, KK dimensional reduction
of 5D gravity to 4D gravity and electromagnetism recalls FvK dimensional
reduction of 3D elasticity equations. In view of this analogy, the
gravitational term $R$ can be interpreted as a stretching-like response of a
5D gravity field within a 4D hypersurface and the electromagnetic term
$\frac{G}{c^{4}}F^{\mu\nu}F_{\mu\nu}$ as bending-like response. In other
words, the 4D metric field $g_{\mu\nu}$ describes internal 4D deformations
while the potential field $A_{\mu}$ encodes deformations along the fifth
dimension.

In the following, we lay the groundwork for our approach using the analogy
described above. It consists of assuming that the potential field $A_{\mu}$
arises as a perturbation through the fifth dimension around a 4D ground state
that involves only the 4D gravitational field. Specifically, we perform an
expansion of a 5D metric whose leading order yields pure gravitational
Einstein equations and the electromagnetic field show up as a first order
perturbation whose magnitude is proportional to the “thickness” of the fifth
direction of spacetime. Notice that although our approach resembles KK
dimensional reduction, it is different from classical KK theory in that it
does not assume neither compactification nor cylindrical boundary condition.

##  III An elastic-like approach to Kaluza-Klein theory

Following Adda-Bedia (2022), we consider a general multidimensional pseudo-
Riemannian metric built upon a 5D Lorentzian spacetime metric $\hat{g}$ and an
infinite number of Euclidean-like diagonal components such that the distance
element along a worldline $\mathrm{d}s$ is given by

$\mathrm{d}s^{2}=\hat{g}_{MN}\mathrm{d}x^{M}\mathrm{d}x^{N}-\sum_{i=5}^{\infty}\mathrm{d}z^{i}\mathrm{d}z^{i}\;.$
(7)

Here, $\hat{g}_{MN}$ depends on the coordinates $x^{M}$ only. We refer to
$x^{M}$ as the active coordinates and $z^{i}$ ($i\geq 5$) as the passive ones.
We define the action $\hat{S}_{G}$ of the gravitational field associated with
the metric $\hat{g}$ as a generalization of the Einstein-Hilbert action to a
5D spacetime given by Overduin and Wesson (1997); Adda-Bedia (2022)

$\hat{S}_{\mathrm{G}}=-\frac{{{c^{3}\ell^{-1}}}}{{16\pi
G}}\int\limits{\mathrm{d}\hat{V}\sqrt{\left|\hat{g}\right|}\hat{R}}\;,$ (8)

where $\hat{R}$ is the Ricci curvature of the metric given by Eq. (7),
$\mathrm{d}\hat{V}$ is the volume element of the 5D hyperspace associated with
the metric $\hat{g}$, and $\ell$ is a length scale needed to render the action
carry the physical dimensions $[M][L]^{2}[T]^{-1}$.

Using the analogy with dimensional reduction of 3D elasticity theory to 2D
thin plate elasticity, yielding the FvK equations, we aim at developing a KK-
like formalism that yields Einstein-Maxwell equations starting from a 5D
description of gravity. For this, we assume that the ground state results from
a purely “stretching-induced” gravitational field described by a 4D active
spacetime. Then, we perform “out-of-plane” perturbations around this zeroth
order state that are induced by $x^{4}$-dependent fluctuations of the
gravitational field.

The leading order of such an expansion should recover the classical 4D
Einstein-Hilbert action given by Eq. (1), with $\zeta\equiv x^{4}$ behaving as
a passive coordinate. The latter condition imposes that the zeroth order
components of the metric $\hat{g}$ should satisfy $\hat{g}_{M4}=-\delta_{M4}$
and $g_{\mu\nu}$ should be independent of $\zeta$. Such a metric yields
$\hat{R}=R$ and $\left|\hat{g}\right|=\left|g\right|$ and therefore, to
retrieve $\hat{S}_{G}=S_{E}$ from Eq. (8) one should impose $\int
d\zeta=\ell$. This result holds for any partition of the hyperspace between
active and passive components Adda-Bedia (2022). The Einstein-Hilbert action
generalized to any spacetime dimension introduces a single physical constant
$\ell$ characterizing the extension of the hyperspace in each passive
direction of the metric of Eq. (7). The magnitude of $\ell$ is unknown since
scaling arguments do not invoke any quantum property that could allow us to
identify $\ell$ with the Planck length $\ell_{P}$.

Again, using the analogy with the theory of elasticity of slender structures,
the ground state should be translationally invariant in the direction of the
fifth spatial dimension. It is described by a 4D spacetime metric $g$ that
lies on a centroid: a hypersurface defined by a worldline $\zeta=0$. Due to
the finite “thickness” $\ell$ of the hyperspace, perturbations around this
ground state are induced by variations of a 5D metric such that
$\hat{g}(x^{M})=g(x^{\mu})+\sum_{n}{}^{\mbox{\tiny$(n)$}}\\!g(x^{\mu})\zeta^{n}$,
with $|\zeta|<\ell/2$. That is, we assume that the perturbations of the metric
across the thickness do not modify the “geometrical” structure of the
hyperspace. This allows us to expand to any desired order in $\zeta$ the
gravitational action (8) which can be rewritten as

$\hat{S}_{\mathrm{G}}=-\frac{{{c^{3}\ell^{-1}}}}{{16\pi
G}}\int\limits_{-\ell/2}^{\ell/2}{\mathrm{d}\zeta\int\limits{\mathrm{d}V\sqrt{\left|\hat{g}\right|}\hat{R}}}\;.$
(9)

In the following, we start from a general excpansion of $\hat{g}$ in powers of
$\zeta$ and compute the corresponding expansion of the action
$\hat{S}_{\mathrm{G}}$. Next, we identify the specific expansion of the 5D
metric that allows us to get as close as possible to the usual EM action
$S_{\mathrm{EM}}$ given by Eq. (3) from the gravitational action
$\hat{S}_{\mathrm{G}}$ given by Eq. (9). Next, we apply the resulting metric
to 5D matter and Dirac spinor fields to determine the conditions under which
their corresponding 4D definitions could be retrieved.

###  III.1 Dimensional reduction of the gravitational field

We aim at determining the most general metric ${\hat{g}_{MN}}$ such that the
integrand of the action $\hat{S}_{\mathrm{G}}$ in Eq. (9) would be of second
order in $\zeta$. To this purpose we start with the expansion of the metric

${\hat{g}_{MN}}=\left({\begin{array}[]{*{20}{c}}{{g_{\mu\nu}}+a{\zeta^{2}}{A_{\mu}}{A_{\nu}}+\bar{a}{\zeta^{4}}{{A^{2}}}{A_{\mu}}{A_{\nu}}}&{b\zeta{A_{\mu}}+\bar{b}{\zeta^{3}}{{A^{2}}}{A_{\mu}}}\\\
{b\zeta{A_{\mu}}+\bar{b}{\zeta^{3}}{{A^{2}}}{A_{\mu}}}&{-1+c{\zeta^{2}}{{A^{2}}}+\bar{c}{\zeta^{4}}{{{A^{4}}}}}\end{array}}\right)+{\cal
O}\left(\zeta^{5}\right)\;.$ (10)

where $g_{\mu\nu}$ is the 4D metric tensor, $A_{\mu}$ is an electromagnetic-
like potential field ($A^{2}=A_{\mu}A^{\mu}$) and
$(a,b,c,\bar{a},\bar{b},\bar{c})$ are physical constants. Notice that the
constants used in this Section and Appendix B should not be confused with
other physical constants and indices. Eq. (10) is the most general expansion
of the 5D metric that involves a single vector field and whose leading order
reduces to a 4D metric with a fifth-dimensional passive component. Notice that
the metric elements are dimensionless and the physical dimensions of $A_{\mu}$
are $[e][L]^{-1}$. Thus, considering only powers of the combination $\zeta
A_{\mu}$ allows us to avoid introducing characteristic length scales in the
metric elements (the charge scale can be absorbed in the multiplicative
constants), apart from the geometrical scale $\ell$ implied by the passive
components of hyperspace.

In Appendix B we perform expansions in powers of $\zeta$ of the determinant of
$\hat{g}_{MN}$, the 5D Christoffel symbols $\hat{\Gamma}_{MN}^{R}$, the
curvature tensor $\hat{R}_{MN}$ and the Ricci scalar $\hat{R}$. We find that
the first correction to the usual $\sqrt{-g}R$ in the integral term of the
action $\hat{S}_{\mathrm{G}}$ is of order $\zeta^{2}$. A careful analysis
shows that if one requires that under the integral sign of the action there
must stand an expression quadratic in the field $A_{\mu}$ which involves only
the derivatives $A_{\mu;\nu}$, then one should impose $a=\bar{a}=0$, as well
as $c={b^{2}}$. It also turns out that neither $\bar{b}$ nor $\bar{c}$
contribute to the second order in $\zeta$ in the integrand of the action.
Hence, in the following we restrict our discussion to the simpler 5D metric

${\hat{g}_{MN}}=\left({\begin{array}[]{*{20}{c}}{{g_{\mu\nu}}}&{b\zeta{A_{\mu}}}\\\
{b\zeta{A_{\mu}}}&{-1+{b^{2}}{\zeta^{2}}{A^{2}}}\end{array}}\right)+{\cal
O}\left(\zeta^{5}\right)\;,$ (11)

which leaves a single unknown parameter $b$. The determinant of this metric
obeys (see Appendix B)

$\sqrt{\left|\hat{g}\right|}=\sqrt{-g}+{\cal O}\left(\zeta^{3}\right)\;,$ (12)

while its Ricci scalar is

$\hat{R}=R+\tfrac{3}{4}{b^{2}}{\zeta^{2}}{F^{\mu\nu}}{F_{\mu\nu}}-{b^{2}}{\zeta^{2}}{R_{\mu\nu}}{A^{\mu}}{A^{\nu}}+{\cal
O}\left(\zeta^{3}\right)\;.$ (13)

Notice that the total derivatives in the expression for $\hat{R}$ have been
discarded assuming that boundary terms do not contribute to the action. After
integration over $\zeta$, Eq. (9) transforms into

$\hat{S}_{\mathrm{G}}=-\frac{{{c^{3}}}}{{16\pi
G}}\int\limits\mathrm{d}V\sqrt{-g}R-\frac{b^{2}\ell^{2}c^{3}}{{256\pi
G}}\int\limits{\mathrm{d}V\sqrt{-g}\left({{F^{\mu\nu}}{F_{\mu\nu}}-\frac{4}{3}{R_{\mu\nu}}{A^{\mu}}{A^{\nu}}}\right)}+{\cal
O}\left(\ell^{3}\right)\;.$ (14)

The sign of the coefficient in front of $F^{\mu\nu}F_{\mu\nu}$ is negative
definite just like for the Maxwell action $S_{\mathrm{M}}$ in Eq. (2).
Imposing an equality between the coefficients of $F^{\mu\nu}F_{\mu\nu}$ in Eq.
(2) and Eq. (14) yields

$b\ell=\frac{4\sqrt{G}}{c^{2}}=\frac{4\sqrt{\alpha}}{e}\,{\ell_{p}}\;,$ (15)

where $\alpha=e^{2}/\hbar c$ is the fine structure constant and
${{\ell}_{p}}=\sqrt{{\hbar G}/{{{c}^{3}}}\;}$ is the Planck length. With this
result, all the coefficients introduced in the expansion (10) of the metric
are determined. Eq. (15) shows that $b\ell\propto\ell_{P}$, which is not
sufficient to identify $\ell$ with $\ell_{P}$. However, it shows that $|b\zeta
A_{\mu}|<b\ell\sqrt{A^{2}}\sim\ell_{p}/L\ll 1$, where $L$ is the length scale
associated with the variation of the vector field $A_{\mu}$. This result
justifies a posteriori our perturbative approach and shows that it is not
constrained by the hypothesis of a slender fifth dimension since the ratio
$\ell/\ell_{P}$ is not constrained.

Our approach produces an additional term in the action when compared with Eq.
(3) which is proportional to ${R_{\mu\nu}}{A^{\mu}}{A^{\nu}}$ and breaks the
gauge invariance that is usually associated with the vector potential
$A_{\mu}$. The existence of this term is justified thanks to the identity
(65), which allows to rewrite it as a sum of terms that involve covariant
derivatives of $A_{\mu;\nu}$ and total derivatives. This explicit
gravitational-electromagnetic interaction term does not appear neither in
Einstein-Maxwell action (3) nor in the classical KK theory. However, it obeys
the weak principle of equivalence and thus does not violate the gauge
invariance at the level of special relativity Li (2016).

Before discussing further the results of this section, we consider the
perturbation of a 5D gravitational source field within a metric given by Eq.
(11).

###  III.2 Dimensional reduction of the matter field

We intend to generalize the elastic-like approach to KK theory in order to
model gravity and electromagnetism in the presence of matter fields. To this
aim, we need to specify the nature of the 5D energy density of source terms.
We assume that the corresponding action is associated with a matter component
only. Let us then start with an action $\hat{S}_{m}$ given by

${\hat{S}_{m}}=\int\limits_{-\ell/2}^{\ell/2}\mathrm{d}\zeta\int\limits\mathrm{d}V\sqrt{|\hat{g}|}\hat{\rho}c\;,$
(16)

where $\hat{\rho}$ is the mass density of the body, i.e. mass per unit space
four-volume. We emphasize that the absolute mass density should be defined per
unit proper four-volume, that is the volume in the reference system in which
the given portion of the body is at rest Landau and Lifshitz (1980). More
precisely, one has

$\hat{\rho}=\frac{\hat{\varrho}}{\sqrt{\hat{g}_{00}}}\frac{\mathrm{d}\hat{s}}{\mathrm{d}x^{0}}\;,$
(17)

where $\hat{\varrho}$ is the absolute mass density per unit space four-volume
Landau and Lifshitz (1980) and $\mathrm{d}\hat{s}$ is the infinitesimal line
element along the 5D worldline of the matter. We assume that mass is
distributed across the $\zeta$-direction such that
$\hat{\varrho}(x^{M})=\kappa(\zeta)\varrho(x^{\mu})$. Using
$\hat{g}_{00}\approx g_{00}$, which is correct to first order in $\zeta$ (see
Eq. 11), one gets

$\hat{\rho}(x^{M})=\kappa(\zeta)\,\rho_{m}(x^{\mu})\frac{\mathrm{d}\hat{s}}{\mathrm{d}s}\;,$
(18)

where $\kappa(\zeta)$ is an unknown function of physical dimension $[L]^{-1}$
that describes the mass distribution in the fifth dimension, $\rho_{m}$ is the
mass density distribution per unit space three-volume and $\mathrm{d}s$ is the
infinitesimal line element along the 4D worldline of the matter Landau and
Lifshitz (1980).

Eq. (18) shows that $\hat{\rho}(x^{M})$ depends explicitly on the field
$A_{\mu}$ through the term $\mathrm{d}\hat{s}/\mathrm{d}s$ which is given by

$\frac{\mathrm{d}\hat{s}}{\mathrm{d}s}=\sqrt{\frac{\hat{g}_{MN}\mathrm{d}x^{M}\mathrm{d}x^{M}}{\mathrm{d}s^{2}}}=\sqrt{1+2b\zeta
u^{5}u^{\mu}A_{\mu}-\left(1-b^{2}\zeta^{2}A^{2}\right)u^{5}u^{5}}\;,$ (19)

where $u^{\mu}=\frac{\mathrm{d}x^{\mu}}{\mathrm{d}s}$ and
$u^{5}=\frac{\mathrm{d}\zeta}{\mathrm{d}s}$ are the components of the
$5$-velocity field. Let us assume a priori that $u^{5}$ depends on the
component $\zeta$ only and satisfies $u^{5}=-\beta\zeta+O(\zeta^{3})$ with
$0<\beta\ell\ll 1$. Expanding Eq. (19) up to second order in $\zeta$ and
substituting in Eq. (16) yields

${\hat{S}_{m}}=\int\limits\mathrm{d}V\sqrt{-g}\rho_{m}c\int\limits_{-\ell/2}^{\ell/2}\mathrm{d}\zeta\kappa(\zeta)\left[1-\frac{1}{2}(2b\beta
u^{\mu}A_{\mu}+\beta^{2})\zeta^{2}+{\cal O}\left(\zeta^{3}\right)\right]\;.$
(20)

One should compare Eq. (20) with the usual 4D source terms given by Eq. (4).
It is straightforward to show that Eqs. (4) and (20) are equivalent if
$\kappa(\zeta)$ satisfies the identities

$\displaystyle\int\limits_{-\ell/2}^{\ell/2}\kappa(\zeta)\mathrm{d}\zeta$
$\displaystyle=$ $\displaystyle 1+\frac{\beta\rho_{e}}{2b\rho_{m}c^{2}}\;,$
(21)
$\displaystyle\int\limits_{-\ell/2}^{\ell/2}\zeta^{2}\kappa(\zeta)\mathrm{d}\zeta$
$\displaystyle=$ $\displaystyle\frac{\rho_{e}}{b\beta\rho_{m}c^{2}}\;.$ (22)

Eqs. (21)-(22) are physically relevant if both the mass $\rho_{m}(x^{\mu})$
and the charge $\rho_{e}(x^{\mu})$ densities follow the same distribution.
This is a realistic property which particularly allows us to define a
characteristic length scale $\mathrm{r}_{e}$ associated with the source fields
as proposed by Eq. (5). A constant uniform $\kappa(\zeta)$ satisfies Eqs.
(21,22), however such a solution is consistent only for $\beta\ell$ of order
unity, which contradicts the initial assumption $\beta\ell\ll 1$.
Nevertheless, a nonuniform $\kappa(\zeta)$ could satisfy Eqs. (21)-(22)
without selecting the parameter $\beta\ell$. An example for such a solution is
given by

$\kappa(\zeta)=\left(1+\frac{\beta\rho_{e}}{2b\rho_{m}c^{2}}\right)\delta(\zeta)+\frac{2\rho_{e}}{b\beta\ell^{2}\rho_{m}c^{2}}\left[\delta\left(\zeta-\ell/2\right)-2\delta(\zeta)+\delta\left(\zeta+\ell/2\right)\right]\;.$
(23)

Eq. (23) shows that $\kappa(\zeta)$ depends explicitly on $\beta\ell$ which is
still an unknown dimensionless quantity although we expect $\beta\ell\propto
b\ll 1$. Our approach only provides a relation between the two independent
quantities $\kappa(\zeta)$ and $u^{5}(\zeta)$ without indicating a principle
to select them independently. Despite this arbitrariness, we provide a
unifying scenario for the origin of two manifestations of real-world matter
(mass and charge): they arise from a single matter-like energy density
distribution in the 5D spacetime which manifests itself in the 4D spacetime
between mass and charge. In the example of Eq. (23), the first term results
from a monopole and the latter terms from a quadrupolar distribution.

###  III.3 Gravity and electromagnetic field equations

Combining the results of the dimensional reduction of 5D gravity and matter
fields, we arrive at a total action
$S_{\mathrm{T}}={\hat{S}_{\mathrm{G}}}+{\hat{S}_{m}}$ given by

$\displaystyle S_{\mathrm{T}}=$ $\displaystyle-\frac{{{c^{3}}}}{{16\pi
G}}\int\limits\mathrm{d}V\sqrt{-g}R+\int\limits\mathrm{d}V\sqrt{-g}\rho_{m}c$
(24) $\displaystyle-\frac{1}{16\pi
c}\int\limits{\mathrm{d}V\sqrt{-g}\left({{F^{\mu\nu}}{F_{\mu\nu}}-\frac{4}{3}{R_{\mu\nu}}{A^{\mu}}{A^{\nu}}}\right)}-\frac{1}{c^{2}}\int\limits\mathrm{d}V\sqrt{-g}J^{\mu}A_{\mu}\;,$

where $J^{\mu}=\rho_{e}cu^{\mu}$ is the 4D current vector. Eq. (24) recovers
the classical EM action in addition to a single term proportional to
${R_{\mu\nu}}{A^{\mu}}{A^{\nu}}$ which suggests an explicit interaction
between gravity and EM fields. Using similar scaling arguments as the ones
leading to Eq. (6), we can show that

${R_{\mu\nu}}{A^{\mu}}{A^{\nu}}\sim\left(\frac{\mathrm{r}_{m}}{L}\right)F^{\mu\nu}F_{\mu\nu}\ll
F^{\mu\nu}F_{\mu\nu}\;.$ (25)

Therefore, the new interaction term is always negligible compared to the
strength of both the electromagnetic Lagrangian density and the interaction
term $J^{\mu}A_{\mu}/c$.

To minimize the total action given by Eq. (24), we use again the analogy with
elastic plates. Without implying any geometric meaning, the fields
$g_{\mu\nu}$ and $A_{\mu}$ can be viewed as in-plane and out-of-plane fields
of a 4D manifold. However, in contrast to elastic plates which are Euclidean
surfaces constrained by Gauss’s Theorema Egregium (see Appendix A), these two
fields are independent variables since they operate within a Lorentzian
manifold. Therefore, the minimization of the action $S_{\mathrm{T}}$ should be
performed with respect to these two variables without additional constraints.
Since the electromagnetic terms depend on the metric, a full minimization of
the action with respect to $g_{\mu\nu}$ yields cumbersome modified Einstein
equations Li (2016). Nevertheless, the scale separation between gravitational
and electromagnetic contributions highlighted by Eqs. (6),(25) allows us to
settle for a perturbative scheme. First, we minimize the stretching terms of
$S_{\mathrm{T}}$ (the ones that do not involve $A_{\mu}$) with respect to
$g_{\mu\nu}$ and then the bending terms with respect to $A_{\mu}$. This
procedure gives the following two equations

$\displaystyle R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R=\frac{8\pi
G}{c^{3}}T^{\mu\nu}\;,$ (26) $\displaystyle
F^{\mu\nu}_{\;\;\;\;;\nu}-\frac{2}{3}R^{\mu\nu}A_{\nu}=-\frac{4\pi}{c}J^{\mu}\;,$
(27)

where $T^{\mu\nu}=\rho_{m}cu^{\mu}u^{\nu}$ is the energy momentum tensor of
pure matter content Landau and Lifshitz (1980). Eq. (26) is the usual Einstein
gravity equation without electromagnetic contributions which are neglected due
to scale separation. At this level of approximation, the electromagnetic field
is a slave of gravity, namely it does not affect gravity yet affected by it.
Indeed, Eq. (27) is a modified Maxwell equation that includes an explicit
interaction term with gravity and thus breaks gauge invariance of the
potential. However, Eq. (25) shows that this additional source-like term is
negligible compared to the current density. Moreover, Eq. (26) shows that
$R_{\mu\nu}=0$ out of matter allowing to recover the usual Maxwell equations
and to restore the gauge invariance in these regions. Therefore, the main
effect of this interaction term on the potential $A_{\mu}$ is confined to
regions filled with matter. It is then important at scales where quantum
effects dominate. To this purpose, we explore dimensional reduction of the
Dirac spinor field within the same approach.

##  IV Dimensional reduction of the Dirac spinor field

In this section, we account for Fermions and apply the same procedure of
dimensional reduction to the action involving a spinor field. In 5D curved
spacetime, the Dirac action $\hat{S}_{\mathrm{D}}$ associated with a kinetic-
like energy density of the source field is defined by Salam and Strathdee
(1982); Macias and Dehnen (1991); Pollock (2010)

${\hat{S}_{\mathrm{D}}}=-\frac{i\hbar}{2\ell}\int\limits_{-{\ell\mathord{\left/{\vphantom{\ell
2}}\right.\kern-1.2pt}2}}^{{\ell\mathord{\left/{\vphantom{h2}}\right.\kern-1.2pt}2}}{\mathrm{d}\zeta\int{\mathrm{d}V{\hat{\mathfrak{L}}_{\mathrm{D}}}}}\;,$
(28)

where $\hat{\mathfrak{L}}_{\mathrm{D}}$ is a Dirac Lagrangian density given by

${\hat{\mathfrak{L}}_{\mathrm{D}}}=\bar{\Psi}{\hat{\gamma}^{\mathrm{A}}}\hat{e}_{\left(\mathrm{A}\right)}^{M}{{\hat{D}}_{M}}\Psi-\left(\hat{e}_{\left(\mathrm{A}\right)}^{M}{{\hat{D}}_{M}}\bar{\Psi}\right){\hat{\gamma}^{\mathrm{A}}}\Psi\;.$
(29)

Here, $\Psi$ is a 4D spinor field ($\Psi$ is a four-components field both in
5D and 4D), $\hat{e}_{\left(\mathrm{A}\right)}^{M}$ are the vielbeins of the
tetrad representation of the metric $\hat{g}$ (sometimes referred to as the
fünfbeins Zee (2013)), $\hat{\gamma}^{\mathrm{A}}$ are the 5D Dirac matrices
and $\hat{D}_{M}$ is the 5D covariant derivative for Fermionic fields. The
detailed definitions of these quantities are given in Appendix C. Finally,
notice that the coefficient $1/\ell$ in front of the integral of Eq. (28)
stems from the same dimensional arguments as for the 5D action of the
gravitational field $\hat{S}_{\mathrm{G}}$.

To perform the expansion of ${\hat{\mathfrak{L}}_{\mathrm{D}}}$ in powers of
$\zeta$, we assume that the relevant metric $\hat{g}_{MN}$ from which the
vielbeins $\hat{e}_{\left(\mathrm{A}\right)}^{M}$ are derived is given by Eq.
(11). The lengthy computations are deferred to Appendix C and we just show the
final result for ${\hat{\mathfrak{L}}_{\mathrm{D}}}$ which is given by

${\hat{\mathfrak{L}}_{D}}={\bar{\Psi}\left({{\gamma^{a}}e_{\left(a\right)}^{\mu}{D_{\mu}}+2{\gamma^{5}}{\partial_{4}}-b\zeta{\gamma^{5}}{A^{\mu}}{D_{\mu}}}\right)\Psi-\mathrm{h.c.}}\;,$
(30)

where $\mathrm{h.c.}$ denotes the Hermitian conjugate. At this stage, one
should make assumptions regarding the spinor fields. A plausible assumption
would be that $\Psi$ does not depend on the fifth dimension, namely
$\Psi\left({x^{A}}\right)={{\Psi}}\left({{x}^{\mu}}\right)$ and thus
${{\partial}_{4}}\Psi={{\partial}_{4}}\bar{\Psi}=0$. In this case, the second
term in Eq. (30) drops out. The third term also cancels upon integration over
$\zeta$. Therefore, one ends up with

${\hat{S}_{D}}\equiv
S_{D}=-\frac{i\hbar}{2}\int{\mathrm{d}V\left[{{{\bar{\Psi}}}{\gamma^{\mathrm{a}}}e_{\left(\mathrm{a}\right)}^{\mu}{D_{\mu}}{\Psi}-\left(e_{\left(\mathrm{a}\right)}^{\mu}{D_{\mu}}{{\bar{\Psi}}}\right){\gamma^{\mathrm{a}}}{\Psi}}\right]}\;,$
(31)

namely the Dirac action $\hat{S}_{D}$ becomes identical to the 4D case $S_{D}$
without further contributions involving the electromagnetic field $A_{\mu}$.

The coupling of the spinor field with $A_{\mu}$ and the mass term in Dirac
equation can be retrieved by replacing in Eq. (4) mass and current densities
with their quantum counterparts. They are readily given by

$\rho_{m}=m\,\bar{\Psi}\Psi\;,\quad J^{\mu}=ec\,\bar{\Psi}\gamma^{\mu}\Psi\;,$
(32)

and the quantisation of the action $S_{m}$ reads

${{S}_{m}}=\int\limits\mathrm{d}V\sqrt{-g}\left(mc\,\bar{\Psi}\Psi-\frac{e}{c}\bar{\Psi}\gamma^{\mu}A_{\mu}\Psi\right)\;.$
(33)

Therefore, we end up with all terms involving Dirac spinor fields without
invoking additional assumptions. These results strengthen the dimensional
reduction approach inspired by elasticity theory of thin plates.

##  V Conclusion

In this paper, we identify a scale separation between the gravitational and
electromagnetic Lagrangian densities of EM action which is similar to that
between the stretching and bending energy densities involved in describing the
elastic response of thin plates. From this observation, we argue that the KK
compactification formalism belongs to the class of techniques as the
dimensional reduction of 3D elasticity leading to FvK equations. We exploit
this analogy to develop a KK formalism in the framework of elasticity theory
of thin plates in which the gravitational and electromagnetic fields are
associated with stretching-like and bending-like deformations respectively.
This starting point is different from the classical KK theory where the 5D
metric is independent of the fifth component $\zeta$ and the cylindrical or
compactification hypothesis is required. We show that the same procedure of
dimensional reduction allows us to retrieve the Lagrangian densities of the
different fields (gravitational, electromagnetic and Dirac spinors) and of the
source terms of matter content (gravitational and electromagnetic). When one
solves the equations using a proposed perturbation scheme under the assumption
of scale separation, one recovers the Einstein gravitational equations without
the electromagnetic energy momentum tensor and the current parts and Maxwell
equations with an additional explicit interaction term with gravity.

Our approach suggests that fields and matter are different physical entities,
the former being a manifestation of the latter, and both should independently
undergo dimensional reduction. Concerning the nature of matter, our scheme
suggests that mass and charge in 4D spacetime are two distinct manifestations
of the same 5D matter content. While the parameter $\beta\ell$ and 5D matter
distribution along the fifth dimension introduced in Sec. III.2 cannot be
fully determined within our approach, the interpretation of our results remain
robust as long as $\beta\ell\ll 1$. Moreover, the present study is not
dependent on the recent suggestion in Adda-Bedia (2022) that $\ell$ could be
as large as a cosmic length scale.

The dimensional reduction of 5D gravitational field induces an explicit term
$\propto R_{\mu\nu}A^{\mu}A^{\nu}$ that breaks gauge invariance of the
electromagnetic potential. However, such a term violates the strong principle
of equivalence but not the weak version of it Li (2016). Effectively,
$R_{\mu\nu}=0$ both in the limit of a Minkowski spacetime and in regions
devoid of matter, which restores the gauge invariance for situations where it
has been unequivocally established. Our approach produces an explicit
interaction between gravity and electromagnetic fields that is embedded in a
single term (e.g., by construction a term proportional to $RA^{2}$ is
discarded) for which both sign and amplitude are uniquely determined.

From the observation that the new interaction term comes into play within the
matter content, we extend our approach to Dirac spinor fields. We find that
the dimensional reduction of the corresponding 5D Lagrangian density retrieves
the 4D one without any additional terms such as anomalous interactions
discussed by Pauli Pollock (2010); Salam and Strathdee (1982). The terms
involving mass and coupling with the electromagnetic field are simply
recovered from the action of the matter by substituting the matter and current
densities with their quantum equivalent.

Finally, the question of a possible unified description of fundamental
interactions within general relativity is often intertwined with the
dimensionality of our physical world. In this context, Ref. Adda-Bedia (2022)
postulated that the current four dimensionality of our physical world results
from a dynamical evolution of the spacetime dimension of a matter dominated
expanding universe. This result gives a rationale to KK-like approaches:
fields mediating other types of interactions should be manifestations of
hidden dimensions. Our scheme is then consistent with the scenario that a 4D
spacetime is the stable dimension of our universe Adda-Bedia (2022) and that
the higher dimensions should appear as perturbations around this fundamental
structure. Indeed, one could generalise our approach to include perturbations
of any number of passive dimensions paving the way for including more than the
electromagnetic field into higher dimensional gravity. The relevance of our
approach should be confronted with the results of such a generalization.

###### Acknowledgements.

This work was supported by the International Research Project “Non-Equilibrium
Physics of Complex Systems” (IRP-PhyComSys, France-Israel).

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##  Appendix A On the elasticity theory of thin plates

This section briefly outlines the dimensional reduction of 3D bulk elasticity
to the case of thin plates. A detailed presentation of this subject can be
found in Landau and Lifshitz (1986); Kramer (1997); Audoly and Pomeau (2010);
Witten (2007).

Consider a 3D homogeneous isotropic solid body regarded as a continuous
medium. Under the action of applied forces, the medium undergoes elastic
deformations described by an embedding $\vec{r}(x)=[x_{i}+{u}_{i}(x)]$, where
$x_{i}$ ($i=1,2,3$) are Euclidean material points of the rest state and
$\vec{u}$ is the associated displacement vector field. Upon deformation, the
distance between nearby points is given by
$\mathrm{d}s^{2}=g_{ij}\mathrm{d}x_{i}\mathrm{d}x_{j}$, where

$g_{ij}=\frac{d\vec{r}}{dx_{i}}\cdot\frac{d\vec{r}}{dx_{j}}=\delta_{ij}+2u_{ij}\;,$
(34)

and $u_{ij}$ is the (dimensionless) symmetric strain tensor. Upon
deformations, the increase in elastic energy is assumed to depend only on the
distances between nearby pairs of points. In this case, the most general
energy functional ${\cal E}[\vec{r}]$ quadratic in the strains and consistent
with a homogeneous isotropic material is

${\cal
E}[\vec{r}]=\frac{1}{2}\int\mathrm{d}x^{3}\left[\lambda\left(u_{ii}\right)^{2}+2\mu
u_{ij}u_{ij}\right]\;,$ (35)

where the elastic properties of the material are encoded in the Lamé
coefficients $\lambda$ and $\mu$.

A thin plate is characterized by a very small extent (thickness) $t$ along one
dimension of the solid compared to the two other dimensions. The 3D geometry
of the plate is described using a 2D manifold associated with its center
surface, or centroid, using the embedding
$\vec{r}_{c}(x)=[x_{\alpha}+u_{\alpha}(x),w(x)]$ ($\alpha=1,2$) which
distinguishes the displacements within the surface ($u_{\alpha}$) from the one
normal to it ($\vec{r}_{c}\cdot\vec{n}$). In the weakly strained regime, the
curvature tensor of the centroid is defined by
$C_{\alpha\beta}\equiv\vec{n}\cdot(\partial^{2}\vec{r}_{c}/\partial
x_{\alpha}\partial x_{\beta})$ ($\alpha,\beta=1,2$). Dimensional reduction of
the elastic energy functional for the centroid is found by integrating out the
components of the strain tensors which are transverse to the long directions.
The mathematical analysis of these approximations and their range of validity
is the subject of the theory of elastic shells.

To derive the elastic energy functional for the plate, one starts from Taylor
expansion of the embedding $\vec{r}(x)$ in powers of $x_{3}$. After
integration of Eq. (35) across the thickness, the energy of such a sheet may
be expressed in terms of the in-surface strain tensor $u_{\alpha\beta}$ and
the curvature tensor $C_{\alpha\beta}$ of the centroid Landau and Lifshitz
(1986). To lowest order in these tensors, ${\cal E}[\vec{r}]$ takes the form

${\cal
E}[\vec{r}]=S[\mathbf{u}]+B[\mathbf{C}]=\frac{1}{2}\int\mathrm{d}x^{2}\left[\hat{\lambda}\left(u_{\alpha\alpha}\right)^{2}+2\hat{\mu}u_{\alpha\beta}u_{\alpha\beta}\right]+\frac{\kappa}{2}\int\mathrm{d}x^{2}\,C_{\alpha\beta}C_{\alpha\beta}\;,$
(36)

where

$\hat{\lambda}=\frac{E\nu
t}{1-\nu^{2}};\quad\hat{\mu}=\frac{Et}{2(1+\nu)};\quad\kappa=\frac{Et^{3}}{12(1-\nu^{2})}\;.$
(37)

The Young modulus $E$ and Poisson ratio $\nu$ are material constants related
to Lamé coefficients Landau and Lifshitz (1986). Eq. (36) shows that the
energy is the sum of a stretching energy $S[\mathbf{u}]$ involving only the
in-surface strain and a bending energy $B[\mathbf{C}]$ involving only the
curvature. Notice that Eq. (36) does not explicitly couple the strains to the
curvatures of the manifold and that the scale separation between $B$ and $S$
is mediated by the length scale $\sqrt{\kappa/(Et)}\sim t$.

The minimization of the elastic energy functional can be performed in two
ways. The first one uses in-plane and out-of-plane displacements as
independent variables. In this case, functional derivatives are taken with
respect to $u_{\alpha}$ and $w$ Landau and Lifshitz (1986). The second route,
which is more similar to the minimization of EM action, instead uses the field
variables of in-plane strains $u_{\alpha\beta}$ and intrinsic curvature
$C_{\alpha\beta}$ Kramer (1997). However, the strains and the curvatures are
implicitly coupled because they are both defined via derivatives of the
embedding of the centroid. Therefore, contrary to minimization of EM action,
one has to include the constraint imposed by Gauss’s Theorema Egregium Kramer
(1997). Of course, both schemes yield the correct FvK equations which can be
written in the form

$\displaystyle\kappa\Delta^{2}w$ $\displaystyle=$ $\displaystyle
2t\,[w,\chi]\;,$ (38) $\displaystyle\Delta^{2}\chi$ $\displaystyle=$
$\displaystyle-E\,[w,w]\;,$ (39)

where $\Delta$ is the 2D Laplacian, the brackets $[,]$ are defined by

$[a,b]=\frac{1}{2}a_{,x_{1}x_{1}}b_{,x_{2}x_{2}}+\frac{1}{2}a_{,x_{2}x_{2}}b_{,x_{1}x_{1}}-a_{,x_{1}x_{2}}b_{,x_{1}x_{2}}\;,$
(40)

and $\chi$ is the so-called Airy stress function Kramer (1997); Landau and
Lifshitz (1986).

In many cases, the scale separation between bending and stretching leads the
plate to deform in such a way as to bend at large scales and to localize
stretching at singular points or along ridges leading to the phenomenon of
stress focusing in elastic sheets Witten (2007) for which crumpled paper is
the archetype.

##  Appendix B Determination of the appropriate 5D metric tensor $\hat{g}$

We are interested in the expansion of the Ricci scalar curvature $\hat{R}$ in
powers of $\zeta\equiv x^{4}$ up to order ${{\zeta}^{2}}$. To this purpose,
one needs to consider the perturbation of the metric tensor $\hat{g}_{MN}$ up
to order ${{\zeta}^{4}}$ as included in Eq. (10). The contravariant metric
tensor is obtained by using the identity
$\hat{g}^{MP}\hat{g}_{PN}=\hat{\delta}^{M}_{N}+{\cal O}(\zeta^{5})$. One shows
that $\hat{g}^{MN}$ is given by

${\hat{g}^{MN}}=\left({\begin{array}[]{*{20}{c}}{{g^{\mu\nu}}-\left({{b^{2}}+a}\right){\zeta^{2}}{A^{\mu}}{A^{\nu}}+\bar{d}{\zeta^{4}}\left({{A^{2}}}\right){A^{\mu}}{A^{\nu}}}&{b\zeta{A^{\mu}}+\bar{e}{\zeta^{3}}\left({{A^{2}}}\right){A^{\mu}}}\\\
{b\zeta{A^{\mu}}+\bar{e}{\zeta^{3}}\left({{A^{2}}}\right){A^{\mu}}}&{-1+\left({{b^{2}}-c}\right){\zeta^{2}}\left({{A^{2}}}\right)+\bar{f}{\zeta^{4}}{{\left({{A^{2}}}\right)}^{2}}}\end{array}}\right)+{\cal
O}(\zeta^{5})\;,$ (41)

with

$\displaystyle\bar{d}$ $\displaystyle=$
$\displaystyle-\bar{a}-2b\bar{b}+{b^{2}}\left({{b^{2}}+2a-c}\right)+{a^{2}}\;,\hfill$
(42) $\displaystyle\bar{e}$ $\displaystyle=$ $\displaystyle
b\left({-{b^{2}}-a+c}\right)+\bar{b}\;,\hfill$ (43) $\displaystyle\bar{f}$
$\displaystyle=$
$\displaystyle-\bar{c}-2b\bar{b}+{b^{2}}\left({{b^{2}}+a-2c}\right)+{c^{2}}\;.\hfill$
(44)

Now, we calculate the determinant of the metric tensor given by Eq. (10).
Notice that we are interested in the expansion of the determinant up to second
order in $\zeta$. We can achieve this by expanding the full metric tensor as
$\hat{g}={{\hat{g}}_{0}}+\zeta{{\hat{g}}_{1}}+{{\zeta}^{2}}{{\hat{g}}_{2}}$,
with

${\hat{g}_{0}}=\left({\begin{array}[]{*{20}{c}}{{g_{\mu\nu}}}&0\\\
0&{-1}\end{array}}\right)\;,\qquad{\hat{g}_{1}}=\left({\begin{array}[]{*{20}{c}}0&{b{A_{\mu}}}\\\
{b{A_{\mu}}}&0\end{array}}\right)\;,\qquad{\hat{g}_{2}}=\left({\begin{array}[]{*{20}{c}}{a{A_{\mu}}{A_{\nu}}}&0\\\
0&{c{A^{2}}}\end{array}}\right)\;.$ (45)

Using the identity $\det\hat{g}={{e}^{\mathrm{Tr}\ln\hat{g}}}$, which is true
for any non-singular matrix, one finds

$\det\hat{g}\approx\det{\hat{g}_{0}}\left\\{{1+\mathrm{Tr}\left({\hat{g}_{0}^{-1}{\hat{g}_{1}}}\right)\zeta+\left[{\mathrm{Tr}\left({\hat{g}_{0}^{-1}{\hat{g}_{2}}}\right)-\frac{1}{2}{\mathrm{Tr}\left({{{\left({\hat{g}_{0}^{-1}{\hat{g}_{1}}}\right)}^{2}}}\right)+\frac{1}{2}{{\left({\mathrm{Tr}\left({\hat{g}_{0}^{-1}{\hat{g}_{1}}}\right)}\right)}^{2}}}}\right]{\zeta^{2}}}\right\\}+{\cal
O}\left({{\zeta^{3}}}\right)\;,$ (46)

which yields

$\det\hat{g}=-\left(\det{{g}}\right)\left[{1+\left({a-c+{b^{2}}}\right){\zeta^{2}}A^{2}}\right]+{\cal
O}\left({{\zeta^{3}}}\right)\;.$ (47)

Therefore, the leading order correction to the determinant of the 4D metric
tensor is quadratic in $\zeta$. Using the notation $g\equiv\det g$, one gets

$\sqrt{\left|\hat{g}\right|}=\sqrt{-g}\left[{1+\frac{1}{2}\left({a-c+{b^{2}}}\right){\zeta^{2}}A^{2}}\right]+{\cal
O}\left({{\zeta^{3}}}\right)\;.$ (48)

Let us now work on the Christoffel symbols given by

$\hat{\Gamma}_{MN}^{R}=\frac{1}{2}{\hat{g}^{RP}}\left({{\partial_{M}}{\hat{g}_{PN}}+{\partial_{N}}{\hat{g}_{PM}}-{\partial_{P}}{\hat{g}_{MN}}}\right)\;,$
(49)

and expand each component of $\hat{\Gamma}$ up to the appropriate power of
$\zeta$ such that all the contributions to the Ricci scalar curvature up to
$\zeta^{2}$ are taken into account. After lengthy algebraic calculations, one
finds

$\displaystyle\hat{\Gamma}_{\mu\nu}^{\rho}$ $\displaystyle=$
$\displaystyle\Gamma_{\mu\nu}^{\rho}+\frac{1}{2}a{\zeta^{2}}{g^{\rho\sigma}}\left({{A_{\mu}}{F_{\nu\sigma}}+{A_{\nu}}{F_{\mu\sigma}}}\right)$
(50) $\displaystyle+$
$\displaystyle\frac{1}{2}\left({{b^{2}}+a}\right){\zeta^{2}}{A^{\rho}}\left({{A_{\mu;\nu}}+{A_{\nu;\mu}}}\right)-ab{\zeta^{2}}{A^{\rho}}{A_{\mu}}{A_{\nu}}+{\cal
O}(\zeta^{3})\;,$ $\displaystyle\hat{\Gamma}_{\mu 4}^{4}$ $\displaystyle=$
$\displaystyle\Gamma_{4\mu}^{4}=\frac{1}{2}{b^{2}}{\zeta^{2}}{A^{\nu}}{F_{\mu\nu}}+ab{\zeta^{2}}{A_{\mu}}\left({{A^{2}}}\right)-\frac{1}{2}c{\zeta^{2}}{\left({{A^{2}}}\right)_{;\mu}}+{\cal
O}(\zeta^{3})\;,$ (51) $\displaystyle\hat{\Gamma}_{44}^{\mu}$ $\displaystyle=$
$\displaystyle
b{A^{\mu}}+b\left({-{b^{2}}-a+c}\right){\zeta^{2}}{A^{\mu}}\left({{A^{2}}}\right)-\frac{1}{2}c{\zeta^{2}}{g^{\mu\nu}}{\left({{A^{2}}}\right)_{;\nu}}+{\cal
O}(\zeta^{3})\;,$ (52) $\displaystyle\hat{\Gamma}_{44}^{4}$ $\displaystyle=$
$\displaystyle\left({{b^{2}}-c}\right)\zeta\left({{A^{2}}}\right)+{\cal
O}(\zeta^{3})\;,$ (53) $\displaystyle\hat{\Gamma}_{\mu\nu}^{4}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}b\zeta\left({{A_{\nu;\mu}}+{A_{\mu;\nu}}}\right)-\frac{1}{2}\left({\bar{b}-b\left({{b^{2}}+a-c}\right)}\right){\zeta^{3}}\left({{A^{2}}}\right)\left({{A_{\nu;\mu}}+{A_{\mu;\nu}}}\right)$
(54) $\displaystyle+$ $\displaystyle
a\zeta{A_{\mu}}{A_{\nu}}+\left[{2\bar{a}-a\left({{b^{2}}-c}\right)}\right]{\zeta^{3}}\left({{A^{2}}}\right){A_{\mu}}{A_{\nu}}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\bar{b}{\zeta^{3}}\left[{{{\left({{A^{2}}}\right)}_{;\mu}}{A_{\nu}}+{{\left({{A^{2}}}\right)}_{;\nu}}{A_{\mu}}}\right]+\frac{1}{2}ab{\zeta^{3}}{A^{\lambda}}\left({{A_{\nu}}{F_{\mu\lambda}}+{A_{\mu}}{F_{\nu\lambda}}}\right)+{\cal
O}(\zeta^{4})\;,$ $\displaystyle\hat{\Gamma}_{4\nu}^{\mu}$ $\displaystyle=$
$\displaystyle
a\zeta{A^{\mu}}{A_{\nu}}+\left[{2\bar{a}+a\left({-{b^{2}}-a}\right)}\right]{\zeta^{3}}\left({{A^{2}}}\right){A^{\mu}}{A_{\nu}}+$
(55) $\displaystyle+$
$\displaystyle\frac{1}{2}b\zeta{g^{\mu\lambda}}{F_{\nu\lambda}}+\frac{1}{2}\bar{b}{\zeta^{3}}{g^{\mu\lambda}}\left({{A^{2}}}\right){F_{\nu\lambda}}+\frac{1}{2}b\left({-{b^{2}}-a}\right){\zeta^{3}}{A^{\mu}}{A^{\lambda}}{F_{\nu\lambda}}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\bar{b}{\zeta^{3}}{A^{\mu}}{\left({{A^{2}}}\right)_{;\nu}}-\frac{1}{2}\bar{b}{\zeta^{3}}{g^{\mu\lambda}}{\left({{A^{2}}}\right)_{;\lambda}}{A_{\nu}}+\frac{1}{2}bc{\zeta^{3}}{A^{\mu}}{\left({{A^{2}}}\right)_{;\nu}}+{\cal
O}(\zeta^{4})\;,$

where the definition of the field tensor $F_{\mu\nu}\equiv
A_{\nu;\mu}-A_{\mu;\nu}$ has been used. Notice that Eqs. (50-55) do not
involve terms proportional to $\bar{c}$ which appear at higher order in
$\zeta$.

Now let us turn to the computation of Ricci tensor defined as

${\hat{R}_{MN}}\equiv{\partial_{R}}\hat{\Gamma}_{MN}^{R}-{\partial_{N}}\hat{\Gamma}_{MR}^{R}+\hat{\Gamma}_{MN}^{R}\hat{\Gamma}_{RP}^{P}-\hat{\Gamma}_{MP}^{R}\hat{\Gamma}_{NR}^{P}\;.$
(56)

Using the expressions for the $\hat{\Gamma}$’s given by (50-55), one finds

$\displaystyle{{\hat{R}}_{\mu\nu}}$ $\displaystyle=$
$\displaystyle{R_{\mu\nu}}+a{A_{\mu}}{A_{\nu}}-\frac{1}{2}b\left({{A_{\nu;\mu}}+{A_{\mu;\nu}}}\right)+$
(57) $\displaystyle+$
$\displaystyle{\left[{\frac{1}{2}a{\zeta^{2}}{g^{\rho\sigma}}\left({{A_{\mu}}{F_{\nu\sigma}}+{A_{\nu}}{F_{\mu\sigma}}}\right)+\frac{1}{2}\left({{b^{2}}+a}\right){\zeta^{2}}{A^{\rho}}\left({{A_{\mu;\nu}}+{A_{\nu;\mu}}}\right)}\right]_{;\rho}}$
$\displaystyle-$ $\displaystyle
ab{\zeta^{2}}{A^{\rho}}_{;\rho}{A_{\mu}}{A_{\nu}}+\left[{6\bar{a}-a\left({a+2{b^{2}}-2c}\right)}\right]{\zeta^{2}}\left({{A^{2}}}\right){A_{\mu}}{A_{\nu}}$
$\displaystyle-$
$\displaystyle\frac{1}{2}\left({{b^{2}}+a-c}\right){\zeta^{2}}{\left({{A^{2}}}\right)_{;\mu;\nu}}+b\left({{b^{2}}+a-c}\right){\zeta^{2}}\left({{A^{2}}}\right)\left({{A_{\nu;\mu}}+{A_{\mu;\nu}}}\right)$
$\displaystyle+$
$\displaystyle\frac{3}{2}ab{\zeta^{2}}{A^{\rho}}\left({{A_{\mu}}{F_{\nu\rho}}+{A_{\nu}}{F_{\mu\rho}}}\right)+\frac{1}{2}{b^{2}}{\zeta^{2}}{g^{\rho\lambda}}\left({{A_{\nu;\rho}}{F_{\mu\lambda}}+{F_{\mu\rho}}{F_{\nu\lambda}}+{A_{\mu;\rho}}{F_{\nu\lambda}}}\right)$
$\displaystyle-$
$\displaystyle\frac{3}{2}\bar{b}{\zeta^{2}}\left\\{{{{\left[{\left({{A^{2}}}\right){A_{\nu}}}\right]}_{;\mu}}+{{\left[{\left({{A^{2}}}\right){A_{\mu}}}\right]}_{;\nu}}}\right\\}+{\cal
O}\left({{\zeta^{3}}}\right)\;,$ $\displaystyle{{\hat{R}}_{4\mu}}$
$\displaystyle=$ $\displaystyle
a\zeta{A_{\mu}}{A^{\rho}}_{;\rho}+\frac{1}{2}\zeta
b{\left({{g^{\rho\lambda}}{F_{\mu\lambda}}}\right)_{;\rho}}+\zeta
ab{A_{\mu}}\left({{A^{2}}}\right)-$ (58) $\displaystyle-$
$\displaystyle\frac{1}{2}\left({{b^{2}}+a}\right)\zeta{\left({{A^{2}}}\right)_{;\mu}}+\zeta\left({\frac{1}{2}{b^{2}}-a}\right){A^{\rho}}{F_{\mu\rho}}+{\cal
O}\left({{\zeta^{3}}}\right)\;,$ $\displaystyle{{\hat{R}}_{44}}$
$\displaystyle=$ $\displaystyle
b{A^{\rho}}_{;\rho}+b\left({-{b^{2}}-a+c}\right){\zeta^{2}}\left({{A^{2}}}\right){A^{\rho}}_{;\rho}-\frac{1}{2}c{\zeta^{2}}{g^{\rho\lambda}}{\left({{A^{2}}}\right)_{;\lambda;\rho}}$
(59) $\displaystyle-$ $\displaystyle
a\left({{A^{2}}}\right)+\left[{-6\bar{a}+a\left({2{b^{2}}+2a-c}\right)}\right]{\zeta^{2}}{\left({{A^{2}}}\right)^{2}}-\frac{1}{4}{b^{2}}{\zeta^{2}}{F^{\rho\sigma}}{F_{\rho\sigma}}$
$\displaystyle-$
$\displaystyle\frac{1}{2}b\left({{b^{2}}+a}\right){\zeta^{2}}{A^{\rho}}{\left({{A^{2}}}\right)_{;\rho}}+{\cal
O}\left({{\zeta^{3}}}\right)\;.$

Consequently, the expansion in powers of $\zeta$ of the scalar curvature
$\hat{R}={\hat{g}^{MN}}{{\hat{R}}_{MN}}$ is given by

$\displaystyle\hat{R}$ $\displaystyle=$ $\displaystyle
R-\left({{b^{2}}+a}\right){\zeta^{2}}{A^{\mu}}{A^{\nu}}{R_{\mu\nu}}+\frac{3}{4}{b^{2}}{\zeta^{2}}{F^{\mu\nu}}{F_{\mu\nu}}$
(60) $\displaystyle+$ $\displaystyle
2a\left({{A^{2}}}\right)+4\left[{3\bar{a}-a\left({2{b^{2}}+a-c}\right)}\right]{\zeta^{2}}{\left({{A^{2}}}\right)^{2}}$
$\displaystyle+$ $\displaystyle
4b\left(b^{2}+a-c\right)\zeta^{2}\left({{A^{2}}}\right)\left({{A^{\mu}}_{;\mu}}\right)-{\left[{2b{A^{\rho}}+3\bar{b}{\zeta^{2}}\left({{A^{2}}}\right){A^{\rho}}}\right]_{;\rho}}$
$\displaystyle+$
$\displaystyle\left({{b^{2}}+a}\right){\zeta^{2}}{\left({{A_{\mu}}{F^{\mu\rho}}+{A^{\rho}}{A^{\mu}}_{;\mu}}\right)_{;\rho}}-\frac{1}{2}\left({{b^{2}}+a-2c}\right){\zeta^{2}}g^{\mu\rho}{\left({{A^{2}}}\right)}_{;\mu;\rho}+{\cal
O}\left({{\zeta^{3}}}\right)\;.$

Eq. (60) shows that the leading order correction to the 4D scalar curvature is
quadratic in $\zeta$.

Finally, we are interested in the the expansion of
$\sqrt{\left|\hat{g}\right|}\hat{R}$ up to order $\zeta^{2}$. Using Eq. (48)
and Eq. (60), one finds that

$\displaystyle\sqrt{\left|\hat{g}\right|}\,\hat{R}$ $\displaystyle=$
$\displaystyle\sqrt{-g}R\left[{1+\frac{1}{2}\left({{b^{2}}+a-c}\right)\left({{A^{2}}}\right){\zeta^{2}}}\right]-\left({{b^{2}}+a}\right){\zeta^{2}}{A^{\mu}}{A^{\nu}}{R_{\mu\nu}}+\frac{3}{4}{b^{2}}{\zeta^{2}}{F^{\mu\nu}}{F_{\mu\nu}}$
(61) $\displaystyle+$ $\displaystyle
2a\left({{A^{2}}}\right)+\left[{12\bar{a}-a\left({7{b^{2}}+3a-3c}\right)}\right]{\zeta^{2}}{\left({{A^{2}}}\right)^{2}}$
$\displaystyle+$ $\displaystyle
3b\left(b^{2}+a-c\right)\zeta^{2}\left({{A^{2}}}\right)\left({{A^{\mu}}_{;\mu}}\right)-{\left[{2b{A^{\rho}}+3\bar{b}{\zeta^{2}}\left({{A^{2}}}\right){A^{\rho}}}\right]_{;\rho}}$
$\displaystyle+$
$\displaystyle\left({{b^{2}}+a}\right){\zeta^{2}}{\left({{A_{\mu}}{F^{\mu\rho}}+{A^{\rho}}{A^{\mu}}_{;\mu}}\right)_{;\rho}}-\frac{1}{2}\left({{b^{2}}+a-2c}\right){\zeta^{2}}g^{\mu\rho}{\left({{A^{2}}}\right)}_{;\mu;\rho}+{\cal
O}\left({{\zeta^{3}}}\right)\;.$

Now, let us consider the action given by Eq. (9). Using Gauss theorem, terms
in Eq. (61) that involve total 4D derivatives can be transformed into an
integral over the hypersurface surrounding the whole 4D volume. Assuming that
these terms do not contribute to the action, they can thus be discarded from
Eq. (61) yielding

$\displaystyle\sqrt{\left|\hat{g}\right|}\,\hat{R}$ $\displaystyle=$
$\displaystyle\sqrt{-g}R\left[{1+\frac{1}{2}\left({{b^{2}}+a-c}\right)\left({{A^{2}}}\right){\zeta^{2}}}\right]-\left({{b^{2}}+a}\right){\zeta^{2}}{A^{\mu}}{A^{\nu}}{R_{\mu\nu}}+\frac{3}{4}{b^{2}}{\zeta^{2}}{F^{\mu\nu}}{F_{\mu\nu}}$
(62) $\displaystyle+$ $\displaystyle
2a\left({{A^{2}}}\right)+\left[{12\bar{a}-a\left({7{b^{2}}+3a-3c}\right)}\right]{\zeta^{2}}{\left({{A^{2}}}\right)^{2}}$
$\displaystyle+$ $\displaystyle
3b\left(b^{2}+a-c\right)\zeta^{2}\left({{A^{2}}}\right)\left({{A^{\mu}}_{;\mu}}\right)+{\cal
O}\left({{\zeta^{3}}}\right)\;.$

Notice that Eq. (62) is independent of $\bar{b}$. We aim at identifying the
vector field $A_{\mu}$ with the electromagnetic potential field. To this
purpose, we should build an action that allows for $A_{\mu}$ to satisfy
differential equations linear in $A_{\mu}$. Therefore, under the integral sign
for the action there must stand an expression quadratic in that field.
Moreover, we impose that the potentials enter into the expression of the
action $\hat{S}_{\mathrm{G}}$ only through their derivatives $A_{\mu;\nu}$
Landau and Lifshitz (1980). To fulfil these conditions, one should cancel the
undesirable terms in Eq. (62) by fixing some of the constants in the expansion
of the 5D metric tensor given by Eq. (10). These physical constraints allow us
to impose

$a=\bar{a}=0\;;\qquad c=b^{2}\;.$ (63)

Interestingly the condition $a=0$ implies that the field $A^{\mu}$ is
massless. Using these conditions, Eq. (62) is simplified into

$\frac{\sqrt{\left|\hat{g}\right|}}{\sqrt{-g}}\,\hat{R}=R+\frac{3}{4}{b^{2}}{\zeta^{2}}{F^{\mu\nu}}{F_{\mu\nu}}-{b^{2}}{\zeta^{2}}{A^{\mu}}{A^{\nu}}{R_{\mu\nu}}+{\cal
O}\left({{\zeta^{3}}}\right)\;.$ (64)

Eq. (64) is the main result of this Section and is reproduced in Eq. (13) in
the main text. Notice that using the identity
${A^{\mu}}R_{\mu\nu}={A^{\mu}}_{;\nu;\mu}-{A^{\mu}}_{;\mu;\nu}$, one has

$A^{\mu}{A^{\nu}}R_{\mu\nu}=\left({A^{\mu}}_{;\mu}\right)^{2}-{A^{\mu}}_{;\nu}{A^{\nu}}_{;\mu}+\left(A^{\nu}{A^{\mu}}_{;\nu}-A^{\mu}{A^{\nu}}_{;\nu}\right)_{;\mu}\;.$
(65)

showing explicitly that the third term in the right hand side of Eq. (64) can
indeed be rewritten as a quadratic function of the potential derivatives only
(up to total derivatives that do not contribute to the action).

The main result of these calculations is that the 5D metric that allows for an
expansion of the scalar curvature up to $\zeta^{2}$, with suitable properties,
involves a single unknown constant $b$. We will then focus on the 5D metric
tensor given by Eq. (11). Now, we summarize our results (up to the desired
order in $\zeta$) using the simple expression for $\hat{g}_{MN}$. First, one
has

${\hat{g}^{MN}}=\left({\begin{array}[]{*{20}{c}}{{g^{\mu\nu}}-{b^{2}}{\zeta^{2}}{A^{\mu}}{A^{\nu}}}&{b\zeta{A^{\mu}}}\\\
{b\zeta{A^{\mu}}}&{-1}\end{array}}\right)\;.$ (66)

Notice that $\hat{g}^{MN}$ has the same structure as the KK covariant metric
tensor Overduin and Wesson (1997). It can be verified that for this metric,
the determinant $\left|\hat{g}\right|$ satisfies Eq. (12) and the Christoffel
symbols become

$\displaystyle\hat{\Gamma}_{\mu\nu}^{\rho}$ $\displaystyle=$
$\displaystyle\Gamma_{\mu\nu}^{\rho}+\frac{1}{2}{b^{2}}{\zeta^{2}}{A^{\rho}}\left({{A_{\nu;\mu}}+{A_{\mu;\nu}}}\right)\;,$
(67) $\displaystyle\hat{\Gamma}_{\mu 4}^{\nu}$ $\displaystyle=$
$\displaystyle\hat{\Gamma}_{4\mu}^{\nu}=\frac{1}{2}b\zeta{g^{\nu\rho}}{F_{\mu\rho}}-\frac{1}{2}{b^{3}}{\zeta^{3}}{A^{\rho}}{A^{\nu}}{F_{\mu\rho}}+\frac{1}{2}{b^{3}}{\zeta^{3}}{A^{\nu}}{\left({{A^{2}}}\right)_{;\mu}}\;,$
(68) $\displaystyle\hat{\Gamma}_{\mu\nu}^{4}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}b\zeta\left({{A_{\nu;\mu}}+{A_{\mu;\nu}}}\right)\;,$
(69) $\displaystyle\hat{\Gamma}_{4\mu}^{4}$ $\displaystyle=$
$\displaystyle\hat{\Gamma}_{\mu
4}^{4}=\frac{1}{2}{b^{2}}{\zeta^{2}}{A^{\sigma}}{F_{\mu\sigma}}-\frac{1}{2}{b^{2}}{\zeta^{2}}{\left({{A^{2}}}\right)_{,\mu}}\;,$
(70) $\displaystyle\hat{\Gamma}_{44}^{\mu}$ $\displaystyle=$ $\displaystyle
b{A^{\mu}}-\frac{1}{2}{b^{2}}{\zeta^{2}}{g^{\mu\sigma}}{\left({{A^{2}}}\right)_{,\sigma}}\;,$
(71) $\displaystyle\hat{\Gamma}_{44}^{4}$ $\displaystyle=$ $\displaystyle
0\;.$ (72)

A corollary of the result for $\hat{\Gamma}_{\mu 4}^{\nu}$ is that
$\hat{\Gamma}_{4\rho}^{\rho}=\frac{1}{2}{b^{3}}{\zeta^{3}}{A^{\rho}}{\left({{A^{2}}}\right)_{;\rho}}$.
The Ricci tensor is simplified into

$\displaystyle{{\hat{R}}_{\mu\nu}}$ $\displaystyle=$
$\displaystyle{R_{\mu\nu}}-\frac{1}{2}b\left({{A_{\nu;\mu}}+{A_{\mu;\nu}}}\right)+{\left[{\frac{1}{2}{b^{2}}{\zeta^{2}}{A^{\rho}}\left({{A_{\nu;\mu}}+{A_{\mu;\nu}}}\right)}\right]_{;\rho}}$
(73)
$\displaystyle+\frac{1}{4}{b^{2}}{\zeta^{2}}{g^{\rho\sigma}}\left[{{F_{\mu\sigma}}\left({{A_{\nu;\rho}}+{A_{\rho;\nu}}}\right)+{F_{\nu\rho}}\left({{A_{\sigma;\mu}}+{A_{\mu;\sigma}}}\right)}\right]\;,$
$\displaystyle{\hat{R}_{4\mu}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}b\zeta{\left({{g^{\rho\sigma}}{F_{\mu\sigma}}}\right)_{;\rho}}-\frac{1}{2}{b^{2}}\zeta{\left({{A^{2}}}\right)_{;\mu}}+\frac{1}{2}{b^{2}}\zeta{A^{\sigma}}{F_{\mu\sigma}}\;,$
(74) $\displaystyle{\hat{R}_{44}}$ $\displaystyle=$ $\displaystyle
b{A^{\rho}}_{;\rho}-\frac{1}{2}{b^{2}}{\zeta^{2}}{g^{\rho\sigma}}{\left({{A^{2}}}\right)_{;\sigma;\rho}}-\frac{1}{2}{b^{3}}{\zeta^{2}}{A^{\sigma}}{\left({{A^{2}}}\right)_{;\sigma}}-\frac{1}{4}{b^{2}}{\zeta^{2}}{F^{\mu\nu}}{F_{\mu\nu}}\;.$
(75)

Finally, the Ricci scalar curvature coincides with Eq. (13), up to total
derivatives.

##  Appendix C Expansion of the Lagrangian density of Dirac spinor field

This appendix is devoted to the expansion in powers of $\zeta$ of the Dirac
Lagrangian density given by Eq. (29). Here, we restrict the computations to
the metric given by Eq. (11). Before proceeding, let us define the different
quantities introduced in Eq. (29). First,
$\hat{e}_{\left(\mathrm{A}\right)}^{M}$ are the vielbeins which are determined
from the tetrad representation of the metric ${\hat{g}_{MN}}$ such that

${\hat{g}_{MN}}={\hat{\eta}_{\mathrm{AB}}}\hat{e}_{M}^{\left(\mathrm{A}\right)}\hat{e}_{N}^{\left(\mathrm{B}\right)}\;,$
(76)

where ${{\hat{\eta}}_{\mathrm{AB}}}=\mathrm{diag}\left(+1,-1,-1,-1,-1\right)$.
Then, $\hat{\gamma}^{\mathrm{A}}$ are the 5D Dirac matrices given by

$\hat{\gamma}^{0}=\gamma^{0}\;;\quad\hat{\gamma}^{1}=\gamma^{1}\;;\quad\hat{\gamma}^{2}=\gamma^{2}\;;\quad\hat{\gamma}^{3}=\gamma^{3}\;;\quad\hat{\gamma}^{4}=\gamma^{5}\equiv
i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}\;.$ (77)

While the ${{\hat{\gamma}}^{\mathrm{A}}}$ are defined in the tetrad frame,
hence they are constant and obey the anti-commutation relations given by

$\left\\{{{\hat{\gamma}^{\mathrm{A}}},{\hat{\gamma}^{\mathrm{B}}}}\right\\}=2{\hat{\eta}^{\mathrm{AB}}}\;.$
(78)

One can also define the curved-space Dirac gamma matrices $\hat{\gamma}^{M}$
as

${\hat{\gamma}^{M}}=\hat{e}_{\left(\mathrm{A}\right)}^{M}{\hat{\gamma}^{\mathrm{A}}}\;,$
(79)

which obey anti-commutation relations given by

$\left\\{{{{\hat{\gamma}}^{M}},{{\hat{\gamma}}^{N}}}\right\\}=2{\hat{g}^{MN}}.$
(80)

Finally, $\hat{D}_{M}$ is the 5D covariant derivative for Fermionic fields
defined by

${\hat{D}_{M}}={\hat{\partial}_{M}}-\frac{i}{4}\hat{\omega}_{M}^{\mathrm{AB}}{\hat{\sigma}_{\mathrm{AB}}}\;,$
(81)

where $\hat{\omega}_{M}^{\mathrm{AB}}$ are the spin connections and
$\hat{\sigma}_{\mathrm{AB}}$ are the spin operators. They are explicitly given
by

$\displaystyle\hat{\omega}_{M}^{\mathrm{AB}}$ $\displaystyle=$
$\displaystyle\hat{e}_{N}^{(\mathrm{A})}\hat{\Gamma}_{RM}^{N}{\hat{e}^{R\left(\mathrm{B}\right)}}+\hat{e}_{N}^{(\mathrm{A})}{\partial_{M}}{\hat{e}^{N\left(\mathrm{B}\right)}}\;,$
(82) $\displaystyle{\hat{\sigma}^{\mathrm{AB}}}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\left[{{\hat{\gamma}^{\mathrm{A}}},{\hat{\gamma}^{\mathrm{B}}}}\right]\equiv\frac{i}{2}\left({{\hat{\gamma}^{\mathrm{A}}}{\hat{\gamma}^{\mathrm{B}}}}-{{\hat{\gamma}^{\mathrm{B}}}{\hat{\gamma}^{\mathrm{A}}}}\right)\;.$
(83)

Note that the spin connections are antisymmetric with respect to the exchange
of $\mathrm{A}$ and $\mathrm{B}$.

To proceed with the dimensional reduction of the Dirac action for the spinor
field, we start by determining the tetrad representation of the metric
${\hat{g}_{MN}}$ given by Eq. (11). We start with a general form of the
vielbeins given by

$\hat{e}_{\left(\mathrm{A}\right)}^{M}=\left({\begin{array}[]{*{20}{c}}{e_{\left(\mathrm{a}\right)}^{\mu}}&{{P^{\mu}}}\\\
{{Q_{\left(\mathrm{a}\right)}}}&{\phi^{-1}}\end{array}}\right)\;,$ (84)

where $P^{\mu}$ and $Q_{\left(\mathrm{a}\right)}$ are arbitrary 4-vectors,
$\phi$ is an arbitrary scalar and $e_{\left(\mathrm{a}\right)}^{\mu}$ are the
$4\times 4$ vierbeins that give rise to the 4D metric ${{g}_{\mu\nu}}$. One
has

$\hat{e}_{\left(\mathrm{A}\right)}^{M}\hat{e}_{N}^{\left(\mathrm{A}\right)}=\delta_{N}^{M}\;;\qquad\hat{e}_{\left(\mathrm{A}\right)}^{M}\hat{e}_{M}^{\left(\mathrm{B}\right)}=\delta_{\mathrm{A}}^{\mathrm{B}}\;,$
(85)

with $\hat{e}_{M}^{\left(\mathrm{A}\right)}$ the corresponding $1$-forms of
the 5D vielbeins. Using Eq. (85) allows us to write the $1$-forms as

$\hat{e}_{M}^{\left(\mathrm{A}\right)}=\left({\begin{array}[]{*{20}{c}}{e_{\mu}^{\left(\mathrm{a}\right)}}&{-\phi{P^{\left(\mathrm{a}\right)}}}\\\
{-\phi{Q_{\mu}}}&\phi\end{array}}\right)\;.$ (86)

The representations of the vielbeins and the $1$-forms given by Eqs. (84),(86)
yield a metric ${\hat{g}_{MN}}$ that is given by

${\hat{g}_{MN}}=\left({\begin{array}[]{*{20}{c}}{{g_{\mu\nu}}-{\phi^{2}}{Q_{\mu}}{Q_{\nu}}}&{-\phi{P_{\mu}}+{\phi^{2}}{Q_{\mu}}}\\\
{-\phi{P_{\nu}}+{\phi^{2}}{Q_{\nu}}}&{-{\phi^{2}}+{\phi^{2}}{P^{2}}}\end{array}}\right)\;.$
(87)

Upon comparison of Eq. (87) with Eq. (11), one concludes that

$Q_{\mu}=0\;;\quad\phi=1\;;\quad{P_{\mu}}=-b\zeta{A_{\mu}}\;.$ (88)

Therefore, the components of the vielbeins and the $1$-forms associated with
the metric ${\hat{g}_{MN}}$ are given by

$\displaystyle\hat{e}_{\left(\mathrm{a}\right)}^{\mu}=e_{\left(\mathrm{a}\right)}^{\mu}\;;\quad\hat{e}_{\left(4\right)}^{\mu}=-b\zeta{A^{\mu}}\;;\quad\hat{e}_{\left(\mathrm{a}\right)}^{4}=0\;;\quad\hat{e}_{\left(4\right)}^{4}=1\;,$
(89)
$\displaystyle\hat{e}_{\mu}^{\left(\mathrm{a}\right)}=e_{\mu}^{\left(\mathrm{a}\right)}\;;\quad\hat{e}_{4}^{\left(\mathrm{a}\right)}=b\zeta{A^{\left(\mathrm{a}\right)}}\;;\quad\hat{e}_{\mu}^{\left(4\right)}=0\;;\quad\hat{e}_{4}^{\left(4\right)}=1\;.$
(90)

In the following, we will also use the vielbeins with all upper indices,
namely
${{\hat{e}}^{M\left(\mathrm{A}\right)}}={{\hat{\eta}}^{\mathrm{AB}}}\hat{e}_{\left(\mathrm{B}\right)}^{M}$.
They are given by

${\hat{e}^{\mu\left(\mathrm{a}\right)}}={e^{\mu\left(\mathrm{a}\right)}}\;;\quad{\hat{e}^{\mu\left(4\right)}}=b\zeta{A^{\mu}}\;;\quad{\hat{e}^{4\left(\mathrm{a}\right)}}=0\;;\quad{\hat{e}^{4\left(4\right)}}=-1\;.$
(91)

We can now calculate the spin connections $\hat{\omega}_{M}^{\mathrm{AB}}$
that are needed in the covariant derivatives of Fermionic fields. It is useful
to write down explicitly Eq. (82) to separate the 4D terms from the 5D ones,
namely

$\displaystyle\hat{\omega}_{\mu}^{\mathrm{ab}}=\hat{e}_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda\mu}^{\nu}{{\hat{e}}^{\lambda\left(\mathrm{b}\right)}}+\hat{e}_{4}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda\mu}^{4}{{\hat{e}}^{\lambda\left(\mathrm{b}\right)}}+\hat{e}_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{4\mu}^{\nu}{{\hat{e}}^{4\left(\mathrm{b}\right)}}+\hat{e}_{4}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{4\mu}^{4}{{\hat{e}}^{4\left(\mathrm{b}\right)}}+\hat{e}_{\nu}^{\left(\mathrm{a}\right)}{\partial_{\mu}}{{\hat{e}}^{\nu\left(\mathrm{b}\right)}}+\hat{e}_{4}^{\left(\mathrm{a}\right)}{\partial_{\mu}}{{\hat{e}}^{4\left(\mathrm{b}\right)}}\;,$
(92)
$\displaystyle\hat{\omega}_{\mu}^{\mathrm{a}4}=\hat{e}_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda\mu}^{\nu}{{\hat{e}}^{\lambda\left(4\right)}}+\hat{e}_{4}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda\mu}^{4}{{\hat{e}}^{\lambda(4)}}+\hat{e}_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{4\mu}^{\nu}{{\hat{e}}^{4\left(4\right)}}+\hat{e}_{4}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{4\mu}^{4}{{\hat{e}}^{4\left(4\right)}}+\hat{e}_{\nu}^{\left(\mathrm{a}\right)}{\partial_{\mu}}{{\hat{e}}^{\nu\left(4\right)}}+\hat{e}_{4}^{\left(\mathrm{a}\right)}{\partial_{\mu}}{{\hat{e}}^{4\left(4\right)}}\;,$
(93)
$\displaystyle\hat{\omega}_{4}^{\mathrm{ab}}=\hat{e}_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda
4}^{\nu}{{\hat{e}}^{\lambda\left(\mathrm{b}\right)}}+\hat{e}_{4}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda
4}^{4}{{\hat{e}}^{\lambda\left(\mathrm{b}\right)}}+\hat{e}_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{44}^{\nu}{{\hat{e}}^{4\left(\mathrm{b}\right)}}+\hat{e}_{\nu}^{\left(\mathrm{a}\right)}{\partial_{4}}{{\hat{e}}^{\nu\left(\mathrm{b}\right)}}+\hat{e}_{4}^{\left(\mathrm{a}\right)}{\partial_{4}}{{\hat{e}}^{4\left(\mathrm{b}\right)}}\;,$
(94)
$\displaystyle\hat{\omega}_{4}^{\mathrm{a}4}=\hat{e}_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda
4}^{\nu}{{\hat{e}}^{\lambda\left(4\right)}}+\hat{e}_{4}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda
4}^{4}{{\hat{e}}^{\lambda\left(4\right)}}+\hat{e}_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{44}^{\nu}{{\hat{e}}^{4\left(4\right)}}+\hat{e}_{\nu}^{\left(\mathrm{a}\right)}{\partial_{4}}{{\hat{e}}^{\nu\left(4\right)}}+\hat{e}_{4}^{\left(\mathrm{a}\right)}{\partial_{4}}{{\hat{e}}^{4\left(4\right)}}\;.$
(95)

where ${{\partial}_{4}}$ is the derivative with respect to the fifth dimension
$\zeta\equiv x^{4}$. Using the expressions of the vielbeins given by Eq. (91),
one gets

$\displaystyle\hat{\omega}_{\mu}^{\mathrm{ab}}$ $\displaystyle=$
$\displaystyle
e_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda\mu}^{\nu}{e^{\lambda\left(\mathrm{b}\right)}}+b\zeta{A^{\left(\mathrm{a}\right)}}\hat{\Gamma}_{\lambda\mu}^{4}{e^{\lambda\left(\mathrm{b}\right)}}+e_{\nu}^{\left(\mathrm{a}\right)}{\partial_{\mu}}{e^{\nu\left(\mathrm{b}\right)}}\;,$
(96) $\displaystyle\hat{\omega}_{\mu}^{\mathrm{a}4}$ $\displaystyle=$
$\displaystyle
e_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda\mu}^{\nu}b\zeta{A^{\lambda}}+b\zeta{A^{\left(\mathrm{a}\right)}}\hat{\Gamma}_{\lambda\mu}^{4}b\zeta{A^{\lambda}}-e_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{4\mu}^{\nu}-b\zeta{A^{\left(\mathrm{a}\right)}}\hat{\Gamma}_{4\mu}^{4}\;,$
(97) $\displaystyle\hat{\omega}_{4}^{\mathrm{ab}}$ $\displaystyle=$
$\displaystyle e_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda
4}^{\nu}{e^{\lambda\left(\mathrm{b}\right)}}+b\zeta{A^{\left(\mathrm{a}\right)}}\hat{\Gamma}_{\lambda
4}^{4}{e^{\lambda\left(\mathrm{b}\right)}}+e_{\nu}^{\left(\mathrm{a}\right)}{\partial_{4}}{e^{\nu\left(\mathrm{b}\right)}}\;,$
(98) $\displaystyle\hat{\omega}_{4}^{\mathrm{a}4}$ $\displaystyle=$
$\displaystyle e_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda
4}^{\nu}b\zeta{A^{\lambda}}+b\zeta{A^{\left(\mathrm{a}\right)}}\hat{\Gamma}_{\lambda
4}^{4}b\zeta{A^{\lambda}}-e_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{44}^{\nu}+be_{\nu}^{\left(\mathrm{a}\right)}{A^{\nu}}\left({{\partial_{4}}\zeta}\right)\;.$
(99)

Using ${{\partial}_{4}}\zeta=1$,
$e_{\nu}^{\left(\mathrm{a}\right)}{{\partial}_{4}}{{e}^{\nu\left(\mathrm{b}\right)}}={{\eta}^{\mathrm{ab}}}{{\partial}_{4}}$
(that is the 4D vierbeins do not depend on the fifth coordinate $\zeta$) and
$\omega_{\mu}^{\mathrm{ab}}=e_{\nu}^{(\mathrm{a})}\Gamma_{\sigma\mu}^{\nu}{{e}^{\sigma(\mathrm{b})}}+e_{\nu}^{(\mathrm{a})}{{\partial}_{\mu}}{{e}^{\nu(\mathrm{b})}}$,
one finds

$\displaystyle\hat{\omega}_{\mu}^{\mathrm{ab}}$ $\displaystyle=$
$\displaystyle\omega_{\mu}^{\mathrm{ab}}+b\zeta{A^{\left(\mathrm{a}\right)}}\hat{\Gamma}_{\lambda\mu}^{4}{e^{\lambda\left(\mathrm{b}\right)}}+e_{\nu}^{(\mathrm{a})}\left({\hat{\Gamma}_{\sigma\mu}^{\nu}-\Gamma_{\sigma\mu}^{\nu}}\right){e^{\sigma(\mathrm{b})}}\;,$
(100) $\displaystyle\hat{\omega}_{\mu}^{\mathrm{a}4}$ $\displaystyle=$
$\displaystyle b\zeta
e_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda\mu}^{\nu}{A^{\lambda}}+{b^{2}}{\zeta^{2}}{A^{\left(\mathrm{a}\right)}}\hat{\Gamma}_{\lambda\mu}^{4}{A^{\lambda}}-e_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{4\mu}^{\nu}-b\zeta{A^{\left(\mathrm{a}\right)}}\hat{\Gamma}_{4\mu}^{4}\;,$
(101) $\displaystyle\hat{\omega}_{4}^{\mathrm{ab}}$ $\displaystyle=$
$\displaystyle e_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda
4}^{\nu}{e^{\lambda\left(\mathrm{b}\right)}}+b\zeta{A^{\left(\mathrm{a}\right)}}\hat{\Gamma}_{\lambda
4}^{4}{e^{\lambda\left(\mathrm{b}\right)}}+{\eta^{\mathrm{ab}}}{\partial_{4}}\;,$
(102) $\displaystyle\hat{\omega}_{4}^{\mathrm{a}4}$ $\displaystyle=$
$\displaystyle b\zeta e_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{\lambda
4}^{\nu}{A^{\lambda}}+{b^{2}}{\zeta^{2}}{A^{\left(\mathrm{a}\right)}}\hat{\Gamma}_{\lambda
4}^{4}{A^{\lambda}}-e_{\nu}^{\left(\mathrm{a}\right)}\hat{\Gamma}_{44}^{\nu}+b{A^{\left(\mathrm{a}\right)}}\;.$
(103)

Now, using the Christoffel symbols as given by Eqs. (67)-(72) yields

$\displaystyle\hat{\omega}_{\mu}^{\mathrm{ab}}$ $\displaystyle=$
$\displaystyle\omega_{\mu}^{\mathrm{ab}}\;,$ (104)
$\displaystyle\hat{\omega}_{\mu}^{\mathrm{a}4}$ $\displaystyle=$
$\displaystyle\frac{1}{2}b\zeta
e_{\nu}^{\left(\mathrm{a}\right)}\left({2\Gamma_{\lambda\mu}^{\nu}{A^{\lambda}}+{g^{\nu\lambda}}{F_{\lambda\mu}}}\right)\;,$
(105) $\displaystyle\hat{\omega}_{4}^{\mathrm{ab}}$ $\displaystyle=$
$\displaystyle{\eta^{\mathrm{ab}}}{\partial_{4}}+\frac{1}{2}b\zeta{e^{\rho\left(\mathrm{a}\right)}}{e^{\lambda\left(\mathrm{b}\right)}}{F_{\lambda\rho}}\;,$
(106) $\displaystyle\hat{\omega}_{4}^{\mathrm{a}4}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}{b^{2}}{\zeta^{2}}{e^{\rho\left(\mathrm{a}\right)}}\left({{A^{\lambda}}{F_{\rho\lambda}}-{{\left({{A^{2}}}\right)}_{;\rho}}}\right)\;.$
(107)

We now turn to the 5D covariant derivative for Fermionic fields defined by Eq.
(81). For convenience, let us separate explicitly in Eq. (81) the 4D terms
from the 5D one. Using the anti-commutation rules of Dirac matrices, Eq. (81)
becomes

$\displaystyle{{\hat{D}}_{\mu}}$ $\displaystyle=$
$\displaystyle{\partial_{\mu}}+\frac{1}{4}\hat{\omega}_{\mu}^{\mathrm{ab}}{\gamma_{\mathrm{a}}}{\gamma_{\mathrm{b}}}-\frac{1}{2}\hat{\omega}_{\mu}^{\mathrm{a}4}{\gamma_{5}}{\gamma_{\mathrm{a}}}\;,$
(108) $\displaystyle{{\hat{D}}_{4}}$ $\displaystyle=$
$\displaystyle{\partial_{4}}+\frac{1}{4}\hat{\omega}_{4}^{\mathrm{ab}}{\gamma_{\mathrm{a}}}{\gamma_{\mathrm{b}}}-\frac{1}{2}\hat{\omega}_{4}^{\mathrm{a}4}{\gamma_{5}}{\gamma_{\mathrm{a}}}\;.$
(109)

Plugging into these equations the spin connections given by Eqs. (104)-(107),
the definition of the 4D Fermionic covariant derivative
${{D}_{\mu}}={{\partial}_{\mu}}-\frac{i}{4}\omega_{\mu}^{\mathrm{ab}}{{\sigma}_{\mathrm{ab}}}$
and using the identity
${{\eta}^{\mathrm{ab}}}{{\gamma}_{\mathrm{a}}}{{\gamma}_{\mathrm{b}}}={{\gamma}^{\mathrm{a}}}{{\gamma}_{\mathrm{a}}}=4{{I}_{4}}$,
one obtains

$\displaystyle{{\hat{D}}_{\mu}}$ $\displaystyle=$
$\displaystyle{D_{\mu}}-\frac{1}{4}b\zeta
e_{\nu}^{\left(\mathrm{a}\right)}\left({2\Gamma_{\lambda\mu}^{\nu}{A^{\lambda}}+{g^{\nu\lambda}}{F_{\lambda\mu}}}\right){\gamma_{4}}{\gamma_{\mathrm{a}}}\;,$
(110) $\displaystyle{{\hat{D}}_{4}}$ $\displaystyle=$ $\displaystyle
2{\partial_{4}}+\frac{1}{8}b\zeta{e^{\rho\left(\mathrm{a}\right)}}{e^{\lambda\left(\mathrm{b}\right)}}{\gamma_{\mathrm{a}}}{\gamma_{\mathrm{b}}}{F_{\lambda\rho}}+\frac{1}{4}{b^{2}}{\zeta^{2}}{e^{\rho\left(\mathrm{a}\right)}}\left[{{A^{\lambda}}{F_{\rho\lambda}}-{{\left({{A^{2}}}\right)}_{;\rho}}}\right]{\gamma_{4}}{\gamma_{\mathrm{a}}}\;.$
(111)

Using Eqs. (89),(110),(111), the Dirac operator
${{\gamma}^{\mathrm{A}}}\hat{e}_{\left(\mathrm{A}\right)}^{M}{{\hat{D}}_{M}}$
in Eq. (29) becomes

$\displaystyle{\gamma^{\mathrm{A}}}\hat{e}_{\left(\mathrm{A}\right)}^{M}{{\hat{D}}_{M}}$
$\displaystyle=$
$\displaystyle{{\gamma^{\mathrm{a}}}e_{\left(\mathrm{a}\right)}^{\mu}{D_{\mu}}+2{\gamma^{5}}{\partial_{4}}-b\zeta{\gamma^{5}}{A^{\mu}}{D_{\mu}}-}$
(112) $\displaystyle-$
$\displaystyle{\frac{1}{8}b\zeta{\gamma^{5}}e_{\nu}^{\left(\mathrm{b}\right)}{e^{\mu\left(\mathrm{a}\right)}}\left({4\Gamma_{\lambda\mu}^{\nu}{A^{\lambda}}+{g^{\nu\lambda}}{F_{\lambda\mu}}}\right){\gamma_{\mathrm{a}}}{\gamma_{\mathrm{b}}}-\frac{1}{2}{b^{2}}{\zeta^{2}}{\gamma^{\mathrm{a}}}e_{\left(\mathrm{a}\right)}^{\rho}{A_{\rho;\lambda}}{A^{\lambda}}}\;.$

Finally the Dirac Lagrangian density given by Eq. (29) is simplified into Eq.
(30). Indeed the two last terms in Eq. (112) do not involve derivatives of the
spinor field with respect to $D_{\mu}$ or $\partial_{4}$ and therefore they
cancel out.