# Implications for the hierarchy problem, inflation and geodesic motion from
fiber fabric of spacetime

Cao H. Nam<EMAIL_ADDRESS>Phenikaa Institute for Advanced
Study and Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha
Dong, Hanoi 12116, Vietnam

###### Abstract

In this paper, we represent a resolution for the hierarchy problem where the
inverse size of the extra dimension and the fundamental Planck scale would all
be of the order of the TeV scale by proposing a fiber fabric of spacetime. The
origin of the large hierarchy is essentially due to the $\cosh$ function which
is physically originated from the dynamics of the horizontal metric in the
vacuum of non-zero energy. In addition, the fiber fabric of spacetime allows
us to resolve elegantly and naturally the problems of the chirality fermions
and stabilizing potential for the size of the extra dimension, which are
usually encountered in the higher dimensional theories. Then, we explore the
inflation with the modulus of the extra dimension identified as the inflaton
where our slow-roll inflationary model belongs to the E-model class with
$n=1$. We calculate the main inflationary observables which are consistent
with the present experiments. Finally, we study how the geodesic motion of
neutral test particles gets modified from the extension of spactime. We
compute the radius of the photon sphere, the innermost stable circular orbit,
the perihelion shift, the light bending angle, and the observables of the
strong gravitational lensing and the retrolensing phenomenon. By comparing the
predicted values with the experimental observations, we determine the
constraints on the fiber fabric of spacetime.

##  I Introduction

General Relativity (GR) is one of the very successful theories of modern
physics, besides the Standard Model (SM) of elementary particles, which
describes many gravitational phenomena from the solar system scale upto the
cosmological one to a high precision. In GR, spacetime is a four dimensional
Lorentz manifold whose geometry is defined in terms of the metric field.
Although the predictions of GR are well compatible to the observations, there
are strong indications that the spacetime picture described by GR should be
broken down at (very) short distances or high energies. For example, GR
predicts the presence of Big Bang/black hole singularity, which was pointed by
Hawking and Penrose that the formation of this spacetime singularity is
inevitable under the certain conditions, or the quantization of gravity in GR
is perturbatively nonrenormalizable. In addition, GR cannot explain inflation,
dark matter, and dark energy relating to the late accelerating expansion of
universe.

The problems of GR thus motivate alternative theories of gravity or the
extensions of four dimensional spacetime. On the other hand, it is possible
that spacetime fundamentally possesses the extra dimensions and certain hidden
topological/geometric structures corresponding to new fields which become
significant as probing into the short distances or the high energies. The
generalization of GR to five dimensions were first proposed by Kaluza
Kaluza1921  and Klein Klein1926  in attempting to unify the electromagnetic
interaction with GR. However, the Kaluza-Klein (KK) theory encountered many
problems such as all zero modes corresponding to the observed particles to be
electrically neutral and hence it is inconsistent with the experimental
observations. At the end of the 20th century, the extra dimensions have
attracted enormous attention because they could provide a resolution for the
problems of particle physics Dienes1998 ; Antoniadis1990 ; Arkani-Hamed2000 .

In particular, introducing the extra dimensions could resolve the hierarchy
problem. In the scenario of the large extra dimensions proposed by Arkani-
Hamed, Dimopoulos and Dvali (well-known as the ADD model), the inverse size of
the extra dimensions is expressed as $R^{-1}=2\pi
M_{*}(M_{*}/M_{\text{Pl}})^{2/n}$ where $M_{*}$ is the fundamental Planck
scale, $M_{\text{Pl}}$ is the four dimensional or observed Planck scale
$\sim{O}(10^{18})$ GeV, and $n$ is the number of the extra dimensions Arkani-
Hamed1998 ; Antoniadis1998 . It is interesting that the fundamental Planck
scale $M_{*}$ which is relevant to the quantum gravity scale would be lowered
to the scales much lower than the observed Planck scale such as the TeV scale
and thus the effects of quantum gravity might be accessible at the future
colliding experiments. With the fundamental Planck scale $M_{*}$ of the order
of a few TeV, the large hierarchy between the quantum gravity scale and the
electroweak scale $\sim\mathcal{O}(1)$ TeV would no longer appear and thus the
hierarchy problem is resolved in the scenario of the large extra dimensions.
But, here the inverse size of the extra dimensions is so small compared to the
fundamental Planck scale. This means that it leads to a new hierarchy
problem.111An alternative for the hierarchy problem is based on the warped
extra dimensions Randall1999  where the fundamental Planck scale and the
inverse size of the extra dimensions would be of the order of the observed
Planck scale. As we will indicate in the present work, this new hierarchy
problem can be resolved with a fiber fabric of five dimensional spacetime
where the inverse size of the fifth dimension and the fundamental Planck scale
would all be of the order of the TeV scale. There is another resolution for
the hierarchy problem at which $M_{*}$ and $R^{-1}$ are also considred to be
of the order of the TeV scale Kaloper2000 . In this work, the authors
considered the compact hyperbolic extra dimensions where the volume of the
extra space is exponentially amplified as, $\sim e^{a}$, with the topological
parameter $a$ properly chosen without being fixed by a physical mechanism.

In the scenario of the large extra dimensions, all of the SM fields must live
on a $3$-brane because the SM has been well tested up to the TeV scale.
However, if the inverse size of the extra dimensions is of the order of the
TeV scale or above, the SM fields can propagate in the higher dimensional
spacetime Nath1999 ; Masip1999 ; Casalbuoni1999 ; Carone2000 ; Appelquist2001
. The problem arising here is that the higher dimensional theories give rise
the spectrum of vector-like fermions, while the fermions in the SM are in fact
chiral, which means that the left-handed and right-handed fermions transform
in a different way under the gauge symmetry group. The problem of the
chirality fermions could be overcome by introducing the orbifolds with
singularities such as $S^{1}/Z_{2}$ and imposing the boundary conditions
consistent with the fields Csaki2004 . But, it requires many technical
assumptions which make the higher dimensional theories less nature. Therefore,
it is worth investigating alternatives which are more elegant and nature.

There is another important problem of the higher dimensional theories based on
the generalization of GR to the higher dimensions: what is the mechanism to
fix the size of the extra dimensions or the mass of the radion field? In such
higher dimensional theories, there is no the potential for the radion field
because its potential which respects to the higher dimensional general
covariance is constructed by contracting the indices of the metric to those of
its dual, which would lead to a cosmological constant. As a result, the radion
field is massless and thus it is inconsistent with the observations.

The inflation scenario provides an explanation for the horizon, flatness, and
magnetic monopole problems Starobinsky1980 ; Guth1981 ; Linde1983 ;
Albrecht1982 . In addition, it provides a mechanism to generate the primordial
density fluctuations in the early universe which are the origin of the
formation of the large scale structure as seen in the today universe. In order
to solve the above problems, the inflation scenario predicts that the early
universe undergoes a short period of accelerating expansion, which is simply
driven by a scalar field called the inflaton. What is the physical origin of
the inflaton or more explicitly whether it comes from particle physics or
alternative theories of gravity? This still remains an open question and an
active field of research. There have been the investigations which explore the
possibility of the radion field as a natural candidate for the inflaton
Cline2000 ; Kolb2003 ; Sundrum2010 ; Trudeau2012 ; Fukazawa2013 . Here, the
potential of the radion field is effectively generated by coupling to the
matter fields or quantum effects and breaks explicitly the higher dimensional
general covariance. The potential of the radion field in the present work is
different from the previous works as follows: it exists even at the tree level
and respects to the higher dimensional general covariance.

The different theories alternative to GR can be tested by investigating the
geodesic motion of the massless and massive test particles around the
gravitational bodies. Particularly, the observations in the regime of the
strongly gravitational field provide the promising opportunities to probe the
viability range of GR and the hidden structures of spacetime. There exists the
photon sphere, around the supermassive black holes, which is a set of the
circular orbits of the photon. The photon sphere has been explored in the
different black hole solutions Bahamonde2019 ; GuoMiao2020 ; Boonserm2020 ;
Turimov2020 ; XXZeng2020 ; GuoLi2020 . In addition, the observations are
obtained from the investigation of star motion around the supermassive black
holes, such as the innermost stable circular orbit (ISCO) or the perihelion
shift. ISCO of the massive particles is an important property for accretion
disc formed around a black hole and it is the subject of special interest
because its observation can derive the estimation on the parameters of the
black hole as well as impose the constraints on alternative theories of
gravity. The perihelion shift and the light bending angle in the solar system
are much small compared to $2\pi$, however, one expects them to be large in
the case of the strong gravitational bodies.

The present paper is structured as follows. In Sec. II, we propose a five
dimensional extension of spacetime with a fiber fabric which is nontrivial in
general. Then, we represent the intrinsic properties of spacetime and explore
its implications for the hierarchy problem, the stabilizing potential for the
size of the extra dimension, and the problem of the chirality fermions. In
Sec. III, we investigate the inflation with the modulus of the extra space
identified as the inflaton. In Sec. IV, we determine the geodesic equation and
study its astrophysical implications. We consider the photon sphere, the
radius of ISCO, the oscillating point of the near-circular orbits, the
perihelion shift, the bending angle of the light in the weak field limit, and
the strong gravitational lensing corresponding to the gravitational deflection
of light in the strong field limit. Finally, we conclude in Sec. V.

Throughout the present paper, we use the natural system of units ($\hbar=c=1$)
and the signature of the metric as $(+,-,-,-,\cdots,-)$.

##  II Fundamental spacetime

In the extra dimension theories which are the generalization of GR to $4+d$
dimensions with $d\geq 1$ to be the number of extra dimensions and respecting
to $4+d$ dimensional general covariance, one starts with $4+d$ dimensional
(Lorentz) spacetime manifold $\mathcal{M}_{4+d}$ which can be decomposed as

$\displaystyle\mathcal{M}_{4+d}=\bigcup_{i}U^{(4+d)}_{i},$ (1)

where every $U^{(4+d)}_{i}$ is a portion (a local region) of the manifold
$\mathcal{M}_{4+d}$ and looks like $\mathbb{R}^{4+d}$. Next, we assume that
the ground state of the system which is obtained by the compactification has
the topology $\mathcal{M}_{4}\times K_{d}$ (where $\mathcal{M}_{4}$ refers to
the standard four dimensional spacetime manifold and $K_{d}$ is a $d$
dimensional compact extra-space), rather than $\mathcal{M}_{4+d}$ Salam1982 .
But, as we mentioned in introduction, the extension in this way usually leads
to

  * (i)

What is the stabilizing potential for the size of the extra dimension or the
mass of the radion field.

  * (ii)

The problem of the chirality fermions if they propagate in the bulk.

In order to introduce the higher dimensional extension of spacetime and
overcome these problems in an elegant and natural way, we do not consider
spacetime manifold as decomposed by (1), but with the following decomposition

$\displaystyle\mathcal{M}_{4+d}=\bigcup_{i}\left[U^{(4)}_{i}\times G\right],$
(2)

where every $U^{(4)}_{i}\times G$ is a portion of $\mathcal{M}_{4+d}$,
$U^{(4)}_{i}$ looks like $\mathbb{R}^{4}$, and $G$ is a Lie group manifold of
$d$ dimensions. Such a spacetime manifold is realized as as a fiber bundle
with the fibers to be identified as $G$. For the mathematical details, we
refer the reader to Ref. Nakahara2003 . The fiber bundle structure allows us
to solve the problems (ii), whereas the fibers to be the Lie group allows us
to solve the problem222This will be clearly realized when one considers the
Lie group to be non-Abelian. (i).

In the present paper, we explore various aspects of the fiber fabric of
spacetime and its phenomenological implications for the case of five
dimensions, for simplicity. Our five dimensional spacetime $\mathcal{M}_{5}$
is a fiber bundle manifold whose typical fiber is the Lie group $U(1)$.

###  II.1 General coordinate transformation

By definition, one local region of spacetime $\mathcal{M}_{5}$ looks like
$\mathbb{R}^{4}\times U(1)$ and hence it allows us to define the local
coordinates for an arbitrary point in $\mathcal{M}_{5}$ as
$\left(x^{\mu},e^{i\theta}\right)$ where $\\{x^{\mu}\\}\in\mathbb{R}^{4}$ and
$e^{i\theta}\in U(1)$ with $\theta$ to be dimensionless and real. When two
local coordinate systems $U_{i}\times U(1)$ and $U_{j}\times U(1)$ are
assigned to a same point in spacetime $\mathcal{M}_{5}$, we have the general
coordinate transformation as

$\displaystyle x^{\mu}$ $\displaystyle\longrightarrow$ $\displaystyle
x^{\prime\mu}=x^{\prime\mu}(x),$ $\displaystyle e^{i\theta}$
$\displaystyle\longrightarrow$ $\displaystyle
e^{i\theta^{\prime}}=h(x)e^{i\theta},\ \ \textrm{or}\ \
\theta\longrightarrow\theta^{\prime}=\theta+\alpha(x).$ (3)

This is the general coordinate transformation under which the theory
constructed is required to be covariant.

We can consider the fundamental domain for the fiber coordinate $\theta$ as
$\theta\in[-\pi,\pi]$ where the point with the fiber coordinates $\theta=-\pi$
and $\theta=\pi$ is clearly as the same.333One can consider other fundamental
domains for the fiber coordinate $\theta$, such as $\theta\in[0,2\pi]$,
without changing the physical results. Accordingly, a field $\Phi(x,\theta)$
on spacetime $\mathcal{M}_{5}$ should satisfy the boundary condition.

$\displaystyle\Phi(x,-\pi)=\Phi(x,\pi).$ (4)

###  II.2 Intrinsic properties

Let $T_{p}\mathcal{M}_{5}$ be the tangent space to the spacetime manifold
$\mathcal{M}_{5}$ at an arbitrary point $p\in\mathcal{M}_{5}$. There is a one
dimensional subspace of $T_{p}\mathcal{M}_{5}$ which consists of all tangent
vectors passing through $p\in\mathcal{M}_{5}$ which are tangent to the fiber
or run along the fiber, called the vertical tangent subspace at
$p\in\mathcal{M}_{5}$ and denoted as $V_{p}\mathcal{M}_{5}$. In this sense,
$T_{p}\mathcal{M}_{5}$ can always be decomposed into a direct sum of the
vertical tangent subspace and its complement without reference to the local
coordinate system as follows

$T_{p}\mathcal{M}_{5}=H_{p}\mathcal{M}_{5}\oplus V_{p}\mathcal{M}_{5},$ (5)

where $H_{p}\mathcal{M}_{5}$ has dimension $4$ and is called the horizontal
tangent subspace. Clearly, $H_{p}\mathcal{M}_{5}$ consists of all tangent
vectors (at $p\in\mathcal{M}_{5}$) which are transversal to the fiber. Note
that, the directions to be transversal to the fibres are identified as the
directions in the usually four dimensional spacetime.

It is easy to see that the horizontal tangent subspace $H_{p}M_{5}$ is not
spanned by $\\{\partial_{\mu}\\}$ because the ordinary partial derivative
$\partial_{\mu}$ does not transform as a four dimensional vector under the
general coordinate transformation (3). But, it is spanned by the following
basis

$\displaystyle\Big{\\{}\partial_{\mu}-g_{{}_{X}}X_{\mu}\partial_{\theta}\Big{\\}}\equiv\left\\{\hat{\partial}_{\mu}\right\\},$
(6)

where the gauge field $X_{\mu}$ transforms under the general coordinate
transformation (3) as

$\displaystyle X_{\mu}\longrightarrow X^{\prime}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{\partial x^{\nu}}{\partial
x^{\prime\mu}}\left(X_{\nu}-\frac{1}{g_{{}_{X}}}\partial_{\nu}\alpha(x)\right).$
(7)

and $g_{{}_{X}}$ is the corresponding gauge coupling. It is important to note
here that the presence of the gauge field $X_{\mu}$ does not have the origin
from the metric because the line element has not been introduced so far, but
it arises from the non-triviality of the fiber fabric of spacetime. Whereas,
the vertical tangent subspace $V_{p}\mathcal{M}_{5}$ is spanned by the basis
vector $\partial_{\theta}$ which transforms as a one dimensional vector under
the general coordinate transformation (3).

The space dual to the tangent space $T_{p}\mathcal{M}_{5}$ is called the
cotangent space and denoted as $T^{*}_{p}\mathcal{M}_{5}$. Because of the
intrinsic decomposition of $T_{p}\mathcal{M}_{5}$ as analyzed above, the
cotangent space $T^{*}_{p}\mathcal{M}_{5}$ can always decomposed in a direct
sum as follows

$\displaystyle T^{*}_{p}\mathcal{M}_{5}=H^{*}_{p}\mathcal{M}_{5}\oplus
V^{*}_{p}\mathcal{M}_{5},$ (8)

where $H^{*}_{p}\mathcal{M}_{5}$ and $V^{*}_{p}\mathcal{M}_{5}$ are horizontal
and vertical cotangent subspaces which are dual to $H_{p}\mathcal{M}_{5}$ and
$V_{p}\mathcal{M}_{5}$, respectively. The basis which span
$H^{*}_{p}\mathcal{M}_{5}$ and $V^{*}_{p}\mathcal{M}_{5}$ are $\\{dx^{\mu}\\}$
and $\\{d\theta+g_{{}_{X}}X_{\mu}dx^{\mu}\\}$ which transform as four
dimensional and one dimensional covectors (one-forms) under the general
coordinate transformation (3), respectively.

Now, in order to define the geometry of the spacetime manifold
$\mathcal{M}_{5}$, one introduces a metric or an inner product $G$ on it. The
intrinsic decomposition of any vector into the vertical and horizontal
components without reference to the local coordinate system suggests an inner
product $G$ on the spacetime manifold $\mathcal{M}_{5}$ as

$G(V_{1},V_{2})=G_{H}(V_{1H},V_{2H})+G_{V}(V_{1V},V_{2V}),$ (9)

where $V_{1H}$ ($V_{2H})$ and $V_{1V}$ ($V_{2V}$) refer to the horizon and
vertical components of the vector $V_{1}$ ($V_{2}$), respectively. In the
above inner product, $G_{H}$ refers to the horizontal metric and it is a
tensor field whose value at an arbitrary $p\in\mathcal{M}_{5}$ belongs to the
space $H^{*}_{p}\mathcal{M}_{5}\otimes H^{*}_{p}\mathcal{M}_{5}$, which is
generally given by

$G_{H}=g_{\mu\nu}(x,\theta)dx^{\mu}dx^{\nu}.$ (10)

While, $G_{V}$ refers to the vertical metric and it is a tensor field whose
value at an arbitrary $p\in\mathcal{M}_{5}$ belongs to the space
$V^{*}_{p}\mathcal{M}_{5}\otimes V^{*}_{p}\mathcal{M}_{5}$, which is generally
given as

$G_{V}=-T^{2}(x,\theta)\frac{(d\theta+g_{{}_{X}}X_{\mu}dx^{\mu})^{2}}{\Lambda^{2}},$
(11)

where $\Lambda$ refers to a constant of the energy dimension. Note that,
$\Lambda$ appears here because $d\theta+g_{{}_{X}}X_{\mu}dx^{\mu}$ is
dimensionless but $(d\theta+g_{{}_{X}}X_{\mu}dx^{\mu})/\Lambda$ has the
dimension of length and thus defines the infinitesimal length on the fiber.
One can check that, under the general coordinate transformation (3), the
metric fields $g_{\mu\nu}(x,\theta)$ and $T(x,\theta)$ transform as

$\displaystyle g_{\mu\nu}(x,\theta)$ $\displaystyle\longrightarrow$
$\displaystyle g^{\prime}_{\mu\nu}(x^{\prime},\theta^{\prime})=\frac{\partial
x^{\rho}}{\partial x^{\prime\mu}}\frac{\partial x^{\lambda}}{\partial
x^{\prime\nu}}g_{\rho\lambda}(x,\theta),$ $\displaystyle T(x,\theta)$
$\displaystyle\longrightarrow$ $\displaystyle
T^{\prime}(x^{\prime},\theta^{\prime})=T(x,\theta).$ (12)

In the frame
$\Big{\\{}\hat{\partial}_{\mu},\partial_{\theta}\Big{\\}}\equiv\left\\{\partial_{M}\right\\}$,
the coefficients of the Christoffel connection and the Riemann curvature
tensor take the form as follows

$\displaystyle\Gamma^{P}_{MN}$ $\displaystyle=$
$\displaystyle\frac{G^{PQ}}{2}\left(\partial_{M}G_{NQ}+\partial_{N}G_{MQ}-\partial_{Q}G_{MN}\right)+\frac{G^{PQ}}{2}\left(C^{O}_{QM}G_{ON}+C^{O}_{QN}G_{OM}\right)+\frac{C^{P}_{MN}}{2},$
(13) $\displaystyle\mathcal{R}^{O}_{MPN}$ $\displaystyle=$
$\displaystyle\partial_{N}[\Gamma^{O}_{PM}]-\partial_{P}[\Gamma^{O}_{NM}]+\Gamma^{Q}_{PM}\Gamma^{O}_{NQ}-\Gamma^{Q}_{NM}\Gamma^{O}_{PQ}+C^{Q}_{PN}\Gamma^{O}_{QM},$
(14)

where $C^{P}_{MN}$ are the non-holonomic functions and determine the
commutator of any two frame fields as

$\left[\partial_{M},\partial_{N}\right]=C^{P}_{MN}\partial_{P},$ (15)

and

$\displaystyle
G_{MN}=\textrm{diag}\left(g_{\mu\nu}(x,\theta),-\frac{T(x,\theta)^{2}}{\Lambda^{2}}\right),\
\
G^{MN}=\textrm{diag}\left(g^{\mu\nu}(x,\theta),-\frac{\Lambda^{2}}{T(x,\theta)^{2}}\right).$
(16)

###  II.3 Action

An action which describes the dynamics of the fields $X_{\mu}$, $g_{\mu\nu}$,
and $T$ can be written as a combination of the five dimensional Einstein-
Hilbert action and the classical (tree level) potential for the field $T$ as

$\displaystyle S_{\text{spacetime}}$ $\displaystyle=$ $\displaystyle
M^{3}_{*}\int
d^{4}xd\theta\sqrt{-G}\left[\frac{\mathcal{R}^{(5)}}{2}-f(T)\right]$ (17)
$\displaystyle=$ $\displaystyle\frac{M^{3}_{*}}{\Lambda}\int
d^{4}xd\theta\sqrt{-g}\left[\frac{T\hat{\mathcal{R}}}{2}+\frac{\Lambda^{2}}{8T}\left(\partial_{\theta}g^{\mu\nu}\partial_{\theta}g_{\mu\nu}+g^{\mu\nu}g^{\rho\lambda}\partial_{\theta}g_{\mu\nu}\partial_{\theta}g_{\rho\lambda}\right)\right.$
$\displaystyle\left.-\frac{g^{2}_{{}_{X}}T^{3}}{8\Lambda^{2}}g^{\mu\rho}g^{\nu\lambda}X_{\mu\nu}X_{\rho\lambda}+\frac{3}{T}g^{\mu\nu}\left(\hat{\partial}_{\mu}T\right)\left(\hat{\partial}_{\nu}T\right)-V(T)\right],$

where $\mathcal{R}^{(5)}$ is the scalar curvature of spacetime
$\mathcal{M}_{5}$ which is explicitly expanded in the appendix,
$\hat{\mathcal{R}}$ is denoted as

$\displaystyle\hat{\mathcal{R}}\equiv
g^{\mu\nu}(\hat{\partial}_{\nu}\Gamma^{\lambda}_{\lambda\mu}-\hat{\partial}_{\lambda}\Gamma^{\lambda}_{\nu\mu}+\Gamma^{\rho}_{\lambda\mu}\Gamma^{\lambda}_{\nu\rho}-\Gamma^{\rho}_{\nu\mu}\Gamma^{\lambda}_{\lambda\rho}),$
(18)

with
$\Gamma^{\rho}_{\mu\nu}\equiv\frac{g^{\rho\lambda}}{2}(\hat{\partial}_{\mu}g_{\lambda\nu}+\hat{\partial}_{\nu}g_{\lambda\mu}-\hat{\partial}_{\lambda}g_{\mu\nu})$,
$X_{\mu\nu}\equiv\partial_{\mu}X_{\nu}-\partial_{\nu}X_{\mu}$ is the field
strength tensor of the gauge field $X_{\mu}$ which is related to the
commutation of two horizontal basis vectors as

$\displaystyle[\hat{\partial}_{\mu},\hat{\partial}_{\nu}]=-g_{{}_{X}}X_{\mu\nu}\partial_{\theta},$
(19)

and $V(T)\equiv Tf(T)$ is the potential of the field $T$ at the tree level
which can be constructed due to the fact that $T$ transforms as a scalar under
the general coordinate transformation (3) and is generally given as follows

$\displaystyle
V(T)=\Lambda^{2}T\left[a_{0}+a_{1}T+a_{2}T^{2}+a_{3}T^{3}+a_{4}T^{4}+\cdots\right],$
(20)

with $a_{i}$ to be the dimensionless constants, which follows a Taylor series
expansion of $f(T)$. As seen below, when we normalize canonically $T$ as given
in Eq. (23), only the first two terms in the potential $V(T)$ have the
significant contribution as seen in Eq. (24). Whereas, the higher order terms
of the potential in terms of the canonically normalized field $T^{\prime}$
would be highly suppressed by the further powers of $1/M_{*}$ and hence they
are well-approximately ignored. This suggests that we can impose the
mathematical conditions on the potential $V(T)$ as follows

$\displaystyle\frac{\partial^{(n)}V}{T^{\prime(n)}}\Big{|}_{T^{\prime}=0}=0\ \
\ \ \text{with}\ \ n>4,$ (21)

which corresponds to the mathematical conditions on $f(T)$ as

$\displaystyle\frac{\partial^{(n)}\left(T^{\prime
2}f\right)}{T^{\prime(n)}}\Big{|}_{T^{\prime}=0}=0\ \ \ \ \text{with}\ \ n>4.$
(22)

Note that, from the action (17), we see that no kinetic term in the vertical
direction, which is proportional to
$G^{\theta\theta}\partial_{\theta}T\partial_{\theta}T$, is obtained from the
scalar curvature $\mathcal{R}^{(5)}$. This suggests that without loss of
generality we can consider the field $T$ independent on the coordinate
$\theta$, i.e. $T=T(x)$, in the following calculations.

Before proceeding, let us clarify the following point: the field $T$ is just
part of the vertical metric tensor given in Eq. (11); however, this field
appears as an independent field in the action $S_{\text{spacetime}}$. In the
expression of the vertical metric tensor, $T^{2}$ is its component in the
basis $\left\\{(d\theta+g_{{}_{X}}X_{\mu}dx^{\mu}\right)^{2}\\}$. When
changing the basis of the vertical metric tensor, its component or the field
$T$ transforms in the law given by Eq. (12). In this way, the vertical metric
tensor may be represented by the field $T$ equipped with the transformation
law (12). (In general, any tensor may be represented by its components
equipped with a transformation law which encodes the information of how its
components respond to the change of the basis). This means that, in order for
the terms related to the field $T$ in the action to be independent on the
choice of the basis, they must be invariant under the transformation law (12).

The field $T$ can be canonically normalized with the change of the field
variable as

$\displaystyle T^{\prime}=\left(\frac{24M^{3}_{*}}{\Lambda}T\right)^{1/2}.$
(23)

Then, the corresponding potential is

$\displaystyle V(T^{\prime})\simeq\mu^{2}_{T^{\prime}}T^{\prime
2}+\lambda_{T^{\prime}}T^{\prime 4},$ (24)

where

$\displaystyle\mu^{2}_{T^{\prime}}$ $\displaystyle\equiv$
$\displaystyle\frac{a_{0}\Lambda^{2}}{24},$
$\displaystyle\lambda_{T^{\prime}}$ $\displaystyle\equiv$
$\displaystyle\frac{a_{1}\Lambda^{3}}{576M^{3}_{*}}$ (25)

and the higher order terms in terms of $T^{\prime}$ in $V(T^{\prime})$ are
highly suppressed by the further power of $1/M_{*}$ and hence they are well-
approximately ignored. The nonzero vacuum expectation value (VEV) of the field
$T^{\prime}$ corresponding to the minimum of the potential $V(T^{\prime})$ is
obtained as

$\displaystyle\langle
T^{\prime}\rangle\simeq\sqrt{-\frac{\mu^{2}_{T^{\prime}}}{2\lambda_{T^{\prime}}}}.$
(26)

As a result, the size $R$ of the fifth dimension is physically determined as

$\displaystyle R\equiv\frac{\langle T\rangle}{\Lambda}=\frac{\langle
T^{\prime}\rangle^{2}}{24M^{3}_{*}}.$ (27)

We study the dynamics of the horizontal metric $g_{\mu\nu}$ in the vacuum
$T^{\prime}=\langle T^{\prime}\rangle$ at which the vacuum energy is non-zero
in general. Hence, first let us write the equation of motion for the
horizontal metric $g_{\mu\nu}$ in the presence of vacuum energy described by a
cosmological constant $\Lambda_{\text{cc}}$ as

$\displaystyle\bar{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\bar{R}+\frac{1}{4R^{2}}\left[g_{\mu\rho}g_{\nu\lambda}\partial^{2}_{\theta}g^{\rho\lambda}-\partial^{2}_{\theta}g_{\mu\nu}+2g^{\rho\lambda}\partial_{\theta}g_{\rho\lambda}\partial_{\theta}g_{\mu\nu}\right.$
$\displaystyle\left.+2\partial_{\theta}\left(g_{\mu\nu}g^{\rho\lambda}\partial_{\theta}g_{\rho\lambda}\right)-\frac{1}{2}g_{\mu\nu}\left(\partial_{\theta}g^{\mu\nu}\partial_{\theta}g_{\mu\nu}+g^{\mu\nu}g^{\rho\lambda}\partial_{\theta}g_{\mu\nu}\partial_{\theta}g_{\rho\lambda}\right)\right]+\Lambda_{\text{cc}}g_{\mu\nu}$
$\displaystyle=$ $\displaystyle 0,$ (28)

where the expressions of $\bar{R}_{\mu\nu}$ and $\bar{R}$ are same to the
conventional four dimensional Ricci and scalar curvatures, respectively, but
written in terms of $g_{\mu\nu}(x,\theta)$, and the cosmological constant
$\Lambda_{\text{cc}}$ is written more explicitly as

$\displaystyle\Lambda_{\text{cc}}=-24\lambda_{T^{\prime}}\langle
T^{\prime}\rangle^{2}+\cdots,$ (29)

with the first term coming from the value of the potential $V(T^{\prime})$ at
the minimum $T^{\prime}=\langle T^{\prime}\rangle$ and the ellipse referring
to the contribution expected from other sources such as the potential of the
scalar fields and zero-point energy caused by the quantum fluctuations. Note
that, we have assumed that the contributions coming from the fluctuation
around $\langle T^{\prime}\rangle$, the gauge field $X_{\mu}$ and other fields
are small compared to the vacuum energy and since they are negligible.

We solve this equation by separating the variables as
$g_{\mu\nu}(x,\theta)=\chi(\theta)\bar{g}_{\mu\nu}(x)$ where
$\bar{g}_{\mu\nu}(x)$ is identified as the effectively four dimensional
metric. From the boundary condition (4) on $g_{\mu\nu}(x,\theta)$, we find the
corresponding boundary condition on $\chi(\theta)$ as

$\displaystyle\chi(-\pi)=\chi(\pi).$ (30)

Then, we obtain the equations for $\bar{g}_{\mu\nu}(x)$ and $\chi(\theta)$ as

$\displaystyle\mathcal{R}^{(4)}_{\mu\nu}-\frac{1}{2}\bar{g}_{\mu\nu}\mathcal{R}^{(4)}+\frac{k}{4R^{2}}\bar{g}_{\mu\nu}$
$\displaystyle=$ $\displaystyle 0,$ (31) $\displaystyle
6\chi^{\prime\prime}(\theta)+4\frac{\chi^{\prime}(\theta)^{2}}{\chi(\theta)}-\frac{4\Lambda_{\text{cc}}}{\langle
G^{\theta\theta}\rangle}\chi(\theta)$ $\displaystyle=$ $\displaystyle k,$ (32)

where $\mathcal{R}^{(4)}_{\mu\nu}$ and $\mathcal{R}^{(4)}$ are the
conventional four dimensional Ricci and scalar curvatures corresponding to the
effectively four dimensional metric $\bar{g}_{\mu\nu}(x)$, respectively, and
$k$ is a constant. We have comments as follows:

  * 1.

The equation for $\chi(\theta)$ is a nonlinear differential equation and hence
there is no a general solution as the linear combination of partial solutions.

  * 2.

The vertical kinetic part of the horizontal metric leads to a contribution to
the vacuum energy in the four dimensional effective theory.

  * 3.

Eq. (31) is Einstein field equations in the four dimensional effective theory
with a cosmological constant $\Lambda^{(4)}_{\text{cc}}$ which is given by

$\displaystyle\Lambda^{(4)}_{\text{cc}}=\frac{k}{4R^{2}}.$ (33)

If $\Lambda^{(4)}_{\text{cc}}$ is identified as the observed cosmological
constant, by using the corresponding experimental value
$\Lambda^{(4)}_{\text{cc}}\approx 5.06\times 10^{-84}$ $\text{GeV}^{2}$
Tanabashi2018 , we determine $k$ in terms of the inverse size of the fifth
dimension as

$\displaystyle k$ $\displaystyle\simeq$ $\displaystyle 2\times
10^{-83}\times\frac{\text{GeV}^{2}}{R^{-2}}.$ (34)

This relation suggests that $k$ is positive and extremely small even $R^{-1}$
is of the order of the electroweak scale. Due to this fact in what follows we
shall consider $k\geq 0$.

We now find the solutions for Eq. (32) which satisfy the boundary condition
(30) for the following three cases:
_Case 1._ $\Lambda_{\text{cc}}<0$: To obtain the generally analytical solution
with $k\neq 0$ is a difficult task. However, one can find the perturbative
solution which has the following expansion

$\displaystyle\chi(\theta)=\chi^{(0)}(\theta)+\sum_{n=1}k^{n}\chi^{(n)}(\theta),$
(35)

where $\chi^{(0)}(\theta)$ is the solution at the leading order corresponding
to $k=0$ (corresponding to the exact vanishing of the cosmological constant in
the effectively four dimensional theory) and $\chi^{(n)}(\theta)$ are the
solutions at the subleading orders in $k$. Due to the fact that $k$ is
extremely small as indicated above, the solution can be well-approximated by
$\chi^{(0)}(\theta)$.

The solution with $k=0$ is easily obtained as follows

$\displaystyle\chi(\theta)=C_{1}\cosh^{3/5}\left[\frac{\sqrt{10}}{3}\kappa_{1}(\theta+C_{2})\right],$
(36)

where $\kappa_{1}\equiv\sqrt{-\Lambda_{\text{cc}}/R^{-2}}$ and $C_{1,2}$ are
the constants. Note that, $C_{1}$ can be set to be unity by the fact that
$C_{1}$ can be absorbed into the effective metric $\bar{g}_{\mu\nu}(x)$.
While, the boundary condition (30) suggests $C_{2}=0$.

It is interesting that there is a particularly and non-trivially analytical
solution as

$\displaystyle\chi(\theta)=\frac{5k}{28\kappa^{2}_{1}}\left[\cosh\left(\sqrt{\frac{2}{5}}\kappa_{1}\theta\right)-1\right].$
(37)

This solution clearly respects to the boundary condition (30).
_Case 2._ $\Lambda_{\text{cc}}=0$: We can find the analytical solution is

$\displaystyle\chi(\theta)=\left\\{\begin{array}[]{ll}\left[-\chi^{5/3}{{}_{2}F_{1}}\left(\frac{1}{2},\frac{5}{7};\frac{12}{7};-\frac{k\chi^{7/3}}{7}\right)\right]^{(-1)}(\theta),&\hbox{$-\pi\leq\theta<0$;}\\\
\left[\chi^{5/3}{{}_{2}F_{1}}\left(\frac{1}{2},\frac{5}{7};\frac{12}{7};-\frac{k\chi^{7/3}}{7}\right)\right]^{(-1)}(\theta),&\hbox{$0\leq\theta\leq\pi$,}\\\
\end{array}\right.$ (40)

where ${}_{2}F_{1}(a,b;c;z)$ is the hypergeometric function and
$[\cdots]^{(-1)}$ is denoted the inverse function which means

$\displaystyle\theta=\mp\chi^{5/3}{{}_{2}F_{1}}\left(\frac{1}{2},\frac{5}{7};\frac{12}{7};-\frac{k\chi^{7/3}}{7}\right).$
(41)

Of course, the solution in this case satisfies the boundary condition (30).
_Case 3._ $\Lambda_{\text{cc}}>0$: In analogy to the case of
$\Lambda_{\text{cc}}<0$, the analytical solution at the leading order in $k$
is given as

$\displaystyle\chi(\theta)=\left\\{\begin{array}[]{ll}\cos^{3/5}\left(\frac{\sqrt{10}}{3}\kappa_{2}\theta\right),&\hbox{$-\frac{\pi}{2}+2m\pi\leq\frac{\sqrt{10}}{3}\kappa_{2}\theta<\frac{\pi}{2}+2m\pi$;}\\\
\cos^{3/5}\left(\frac{\sqrt{10}}{3}\kappa_{2}\theta-\pi\right),&\hbox{$\frac{\pi}{2}+2m\pi\leq\frac{\sqrt{10}}{3}\kappa_{2}\theta<\frac{3\pi}{2}+2m\pi$,}\\\
\end{array}\right.$ (44)

where $\kappa_{2}\equiv\sqrt{\Lambda_{\text{cc}}/R^{-2}}$ and
$m\in\mathbb{Z}$. The boundary condition (30) imposes on the possible values
of $\kappa_{2}$ as

$\displaystyle\kappa_{2}=\frac{3}{\sqrt{10}}n,\ \ \ \ n=1,2,3,\cdots.$ (45)

In addition, we find a particularly and non-trivially analytical solution as

$\displaystyle\chi(\theta)=\frac{5k}{28\kappa^{2}_{2}}\left[1+\cos\left(\sqrt{\frac{2}{5}}\kappa_{2}\theta\right)\right],$
(46)

where the parameter $\kappa_{2}$ is not quantized by (45), but by

$\displaystyle\kappa_{2}=\sqrt{\frac{5}{2}}n,\ \ \ \ n=1,2,3,\cdots.$ (47)

Here, we note that for two cases with $k=0$ and $k\neq 0$, there exists the
trivial solution as

$\displaystyle\chi(\theta)=\text{constant},$ (48)

which obviously satisfies the boundary condition (30) and can be set to be
unity.

From the above analysis, we can rewrite the effective action from
$S_{\text{spacetime}}$ as

$\displaystyle S^{\text{eff}}_{\text{spacetime}}$ $\displaystyle=$
$\displaystyle\int
d^{4}x\sqrt{-g}\left[\frac{\widetilde{T}^{2}}{24}\frac{\mathcal{R}^{(4)}}{2}-\frac{1}{4}X^{\mu\nu}X_{\mu\nu}\left(1+\cdots\right)+\frac{\lambda_{\widetilde{T},X}}{2}\widetilde{T}^{2}X^{\mu}X_{\mu}\right.$
(49)
$\displaystyle\left.+\frac{1}{2}\partial_{\mu}\widetilde{T}\partial^{\mu}\widetilde{T}-\lambda_{\widetilde{T}}\left(\widetilde{T}^{2}-\langle\widetilde{T}\rangle^{2}\right)^{2}\right],$

where we have redefined $T^{\prime}$ such that the corresponding kinetic term
is canonical as

$\displaystyle\widetilde{T}$ $\displaystyle=$ $\displaystyle
T^{\prime}\left[\int^{\pi}_{-\pi}d\theta\chi(\theta)\right]^{1/2},$
$\displaystyle\lambda_{\widetilde{T}}$ $\displaystyle=$
$\displaystyle\lambda_{T^{\prime}}\left[\int^{\pi}_{-\pi}d\theta\chi^{2}(\theta)\right]\left[\int^{\pi}_{-\pi}d\theta\chi(\theta)\right]^{-2},$
(50)

due to the integration on $\theta$, also in order to have the canonically
normalized action for the gauge field $X_{\mu}$ the gauge coupling
$g_{{}_{X}}$ has been fixed as

$\displaystyle g_{{}_{X}}$ $\displaystyle=$
$\displaystyle\frac{R^{-1}}{M_{\text{Pl}}}\left[\frac{1}{\pi}\int^{\pi}_{-\pi}d\theta\chi(\theta)\right]^{1/2},$
(51)

$\lambda_{T^{\prime},X}$ is given by

$\displaystyle\lambda_{T^{\prime},X}$ $\displaystyle=$
$\displaystyle\frac{R^{-2}}{4\pi
M^{2}_{\text{Pl}}}\int^{\pi}_{-\pi}d\theta\left[\chi^{\prime\prime}(\theta)-\frac{\chi^{\prime}(\theta)^{2}}{2\chi(\theta)}\right],$
(52)

and the ellipse refers to the terms relating to the fluctuation field around
the vacuum $\langle\widetilde{T}\rangle$. Note that, first in obtaining the
effective action $S^{\text{eff}}_{\text{spacetime}}$, we have used the fact
that $\chi(\theta)$ and $\chi^{\prime}(\theta)$ are even and odd functions
over the fundamental domain $[-\pi,\pi]$, respectively, which suggests

$\displaystyle\int^{\pi}_{-\pi}d\theta\mathcal{F}[\chi(\theta)]\chi^{\prime}(\theta)=0,$
(53)

where $\mathcal{F}[\chi(\theta)]$ is an arbitrary function of $\chi(\theta)$.
Second, we have considered the vanishing of the effective cosmological
constant due to its near-zero value. This situation is suitable to the present
observation of the cosmological constant as well as the inflation scenario
because the inflation caused by the energy coming from the potential of the
inflaton (which is identified as the field $\widetilde{T}$ in this work) stops
when the inflaton rolls down to the global minimum of the potential at which
its value is equal to or near zero. Third, the indexes appearing in the
effective action $S^{\text{eff}}_{\text{spacetime}}$ are contracted together
by the effectively four dimensional metric $\bar{g}_{\mu\nu}(x)$.

We observe from the effective action $S^{\text{eff}}_{\text{spacetime}}$ that
the gauge field $X_{\mu}$ can become massive in the effective theory with the
mass squared given as

$\displaystyle m^{2}_{X}$ $\displaystyle=$
$\displaystyle\lambda_{\widetilde{T},X}\langle\widetilde{T}\rangle^{2}$ (61)
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{27}{80\pi}\bar{\kappa}_{1}\left\\{\frac{56\sinh(\bar{\kappa}_{1}\pi)}{\cosh^{\frac{2}{5}}(\bar{\kappa}_{1}\pi)}-15\cosh^{\frac{8}{5}}(\bar{\kappa}_{1}\pi)\Re\left[i\,_{2}F_{1}\left(\frac{1}{2},\frac{4}{5};\frac{9}{5};\cosh^{2}(\bar{\kappa}_{1}\pi)\right)\right]\right\\}\frac{1}{R^{2}},&\hbox{(I);}\\\
\frac{9\times
10^{-83}}{7\pi\kappa_{1}}\left[\sqrt{\frac{5}{2}}\sinh\left(\sqrt{\frac{2}{5}}\kappa_{1}\pi\right)-\kappa_{1}\pi\right]\times
1\ \ \text{GeV}^{2},&\hbox{(II);}\\\
\frac{35R^{-2}}{\pi\chi^{2/3}}\left[\chi\,_{2}F_{1}\left(\frac{1}{7},\frac{1}{2};\frac{8}{7};-\frac{1}{7}k\chi^{7/3}\right)-\frac{9k\chi^{7/3}}{280}\,_{2}F_{1}\left(\frac{1}{2},\frac{5}{7};\frac{12}{7};-\frac{1}{7}k\chi^{7/3}\right)\right.\\\
\left.+\frac{21+8\chi}{28}\left(1+\frac{1}{7}k\chi^{7/3}\right)^{1/2}\right]\Big{|}^{\chi=\chi_{0}}_{\chi=0},&\hbox{(III);}\\\
-\frac{27n^{2}R^{-2}}{4\pi\cos^{2/5}\theta}\Big{|}_{\theta\rightarrow\pi/2},&\hbox{(IV);}\\\
-\frac{9\times 10^{-83}}{7}\times 1\ \ \text{GeV}^{2},&\hbox{(V).}\\\
0,&\hbox{(VI).}\\\ \end{array}\right.$

where $\bar{\kappa}_{1}\equiv\frac{\sqrt{10}}{3}\kappa_{1}$, $\chi_{0}$ is the
solution of the following equation

$\displaystyle\chi^{5/3}_{0}{{}_{2}F_{1}}\left(\frac{1}{2},\frac{5}{7};\frac{12}{7};-\frac{k\chi^{7/3}_{0}}{7}\right)=\pi,$
(62)

the cases (I), (II), (III), (IV), (V), and (VI) correspond to the solutions
given in Eqs. (36), (37), (40), (44), (46), and (48), respectively. Note here
that, we have used the identification between the observed Planck scale
$M_{\text{Pl}}$ and ${\langle\widetilde{T}\rangle}$ as
$M_{\text{Pl}}=\frac{\langle\widetilde{T}\rangle}{2\sqrt{6}}$ which allows us
to fix the VEV of the field $\widetilde{T}$ as

$\displaystyle\langle\widetilde{T}\rangle=2\sqrt{6}M_{\text{Pl}}.$ (63)

We make a few comments on the mass of the gauge field obtained from various
solutions of $\chi(\theta)$. It is easily to see that the mass-square of the
gauge field $X_{\mu}$ corresponding to the cases (III) and (IV) gets the
infinitely negative values. This suggests the inconsistency of the theory and
thus these cases should be excluded. In the case (V), the mass-square of the
gauge field $X_{\mu}$ in the effective theory is negative but its value is
extremely small and hence it can get other corrections to become positive.
However, in this case first the corresponding gauge coupling is
$g_{{}_{X}}=\sqrt{\frac{20}{7n^{2}}}\times\frac{10^{-41}\ \
\text{GeV}}{M_{\text{Pl}}}\lesssim 7\times 10^{-60}$ which is near zero and
hence it almost decouples the effective theory. Second, because $k$ is
extremely small it would lead to the very large hierarchy between the five
dimensional Planck scale $M_{*}$ and the inverse size of the extra dimension
in order to obtain the observed Planck scale $M_{\text{Pl}}$. This implies the
solution (46) less interesting.

###  II.4 Resolving the hierarchy problem

Now let us indicate that the present spacetime model would allow to resolve
the hierarchy problem where there is no the large hierarchy between the
fundamental Planck scale $M_{*}$, the electroweak scale, and the inverse size
of the extra dimension. In this scenario, unlike the ADD model the inverse
size of the extra dimension and the fundamental Planck scale would all be of
the order of the TeV scale.

First, let us identify the observed Planck scale $M_{\text{Pl}}$ from the
effective action $S^{\text{eff}}_{\text{spacetime}}$ as

$\displaystyle M^{2}_{\text{Pl}}$ $\displaystyle=$
$\displaystyle\frac{M^{3}_{*}}{R^{-1}}\int^{\pi}_{-\pi}d\theta\chi(\theta).$
(64)

Then, for the solution given by Eq. (36), the observed Planck scale
$M_{\text{Pl}}$ is expressed in terms of the five dimensional Planck scale
$M_{*}$, the inverse size of the extra dimension, and the parameter
$\kappa_{1}$ as

$\displaystyle M^{2}_{\text{Pl}}$ $\displaystyle=$
$\displaystyle\frac{3}{4\kappa_{1}}\sqrt{\frac{5}{2}}\cosh^{\frac{8}{5}}\left(\frac{\sqrt{10}}{3}\kappa_{1}\pi\right)\Re\left[i\,_{2}F_{1}\left(\frac{1}{2},\frac{4}{5};\frac{9}{5};\cosh^{2}\left(\frac{\sqrt{10}}{3}\kappa_{1}\pi\right)\right)\right]\frac{M^{3}_{*}}{R^{-1}},$
(65)

where $\Re$ is denoted the real part. Form this relation, we can address the
hierarchy problem why the observed Planck scale
$M_{\text{Pl}}\sim{O}(10^{18})$ GeV is so much larger than the electroweak
scale $\sim\mathcal{O}(1)$ TeV and the inverse size of the extra dimension
even they are the same scale at the fundamental level.

It is important to note that, because the SM has been tested up to the scale
of a few TeV, the five dimensional Planck scale $M_{*}$ must be above
$\mathcal{O}(1)$ TeV. Furthermore, the five dimensional Planck scale $M_{*}$
plays the role of the natural UV cutoff. This means that, from the definition
which the parameter $\kappa_{1}$ is the ratio of the root-square modulus of
the bulk cosmological constant to the inverse size of the extra dimension, the
maximal value of $\kappa_{1}$ should be of the order of $M_{*}R$.

If we take $R^{-1}\sim 0.3$ TeV and $M_{*}\sim 6$ TeV, one finds
$\kappa_{1}\sim 33.8$ in order to obtain the observed Planck scale. For more
explicitly, we show the behavior of the five dimensional Planck scale $M_{*}$
in terms of the parameter $\kappa_{1}$ for various values of the inverse size
of the extra dimension in Fig. 1.

---
Figure 1: The dependence of the five dimensional Planck scale $M_{*}$ in the
parameter $\kappa_{1}$ for various values of the inverse size of the extra
dimension.

From this figure, we can see that both the five dimensional Planck scale
$M_{*}$ and the inverse size of the extra dimension being of the order of the
TeV scale require the parameter $\kappa_{1}\sim\mathcal{O}(33.5)$. Here, the
five dimensional Planck scale $M_{*}$ is slightly larger than the inverse size
of the extra dimension. It is easily to realize that the origin of the large
hierarchy between the observed Planck scale $M_{\text{Pl}}$ and the
electroweak scale is due to the function of
$\cosh\left(\frac{\sqrt{10}}{3}\kappa_{1}\pi\right)$ which comes from the
dynamics of the horizontal metric in the vacuum of non-zero energy and grows
too fast in terms of increasing of the parameter $\kappa_{1}$.

We note that we have assumed that the Higgs field lives on the $3$-brane which
is localized around $\theta=0$ in order for $\chi(0)\sim 1$. Since the
physical electroweak scale does not get modified its order due to the
normalization of the Higgs field.

###  II.5 Fermions in bulk spacetime

As we mentioned in the introduction, if the SM fermions propagate in the
higher dimensional spacetime, the problem of the chirality fermions would
arise because it gives rise the spectrum of vector-like fermions whereas the
SM fermions are in fact chiral. In this subsection, we would like to indicate
that with the fiber fabric of spacetime the conventional left-handed and right
handed Weyl spinor fields can be in nature introduced even in five dimensional
spacetime and hence the problem of the chirality fermions would disappear.
Then, we shall determine the action which describes their propagation in
spacetime $\mathcal{M}_{5}$ and obtain their KK spectrum.

One can write the horizontal metric $G_{H}$ at the local flat form by a
$\text{GL}(4,\mathbb{R})$-rotation of the basis vectors as

$\displaystyle\hat{e}_{\alpha}$ $\displaystyle=$
$\displaystyle{e_{\alpha}}^{\mu}\hat{\partial}_{\mu},$
$\displaystyle\hat{e}^{\alpha}$ $\displaystyle=$
$\displaystyle{e^{\alpha}}_{\mu}dx^{\mu},$ (66)

where $\left\\{\hat{e}_{\alpha}\right\\}$ and
$\left\\{\hat{e}^{\alpha}\right\\}$ are the new bases of the horizontal
tangent and cotangent subspaces, respectively, ${e_{\alpha}}^{\mu}$ is called
vierbeins and ${e^{\alpha}}_{\mu}$ is its inverse, i.e.
${e^{\alpha}}_{\mu}{e_{\alpha}}^{\nu}={\delta_{\mu}}^{\nu}$ and
${e^{\alpha}}_{\mu}{e_{\beta}}^{\mu}={\delta^{\alpha}}_{\beta}$. In the new
basis, the horizontal metric becomes

$\displaystyle G_{H}=\eta_{\alpha\beta}\hat{e}^{\alpha}\hat{e}^{\beta},$ (67)

where $\eta_{\mu\nu}=\text{diag}\left(1,-1,-1,-1\right)$ and
${e^{\alpha}}_{\mu}$ satisfies

$\displaystyle{e^{\alpha}}_{\mu}{e^{\beta}}_{\nu}\eta_{\alpha\beta}=g_{\mu\nu},$
(68)

which implies
${e^{\alpha}}_{\mu}=\chi^{1/2}(\theta){\bar{e}^{\alpha}}_{\mu}(x)$ with
${\bar{e}^{\alpha}}_{\mu}(x)$ satisfying
${\bar{e}^{\alpha}}_{\mu}(x){\bar{e}^{\beta}}_{\nu}(x)\eta_{\alpha\beta}=\bar{g}_{\mu\nu}(x)$.

Of course, there are many alternative choices for
$\left\\{\hat{e}^{\alpha}\right\\}$ which yield the same local flat form (67)
for the horizontal metric $G_{H}$, by the local rotation as

$\displaystyle\hat{e}^{\alpha}\longrightarrow\hat{e}^{\prime\alpha}={\Lambda^{\alpha}}_{\beta}\hat{e}^{\beta},$
(69)

where ${\Lambda^{\alpha}}_{\beta}$ satisfies
${\Lambda^{\alpha}}_{\beta}{\Lambda^{\delta}}_{\gamma}\eta_{\alpha\delta}=\eta_{\beta\gamma}$
and thus ${\Lambda^{\alpha}}_{\beta}\in SO(3,1)$. Accordingly, we introduce
the (left-handed or right handed) Weyl spinor field $\Psi$ which transforms
under the local rotation (69) as
$\Psi\rightarrow\Psi^{\prime}=\rho({\Lambda^{\alpha}}_{\beta})\Psi$ where
$\rho({\Lambda^{\alpha}}_{\beta})$ is the (left-handed or right-handed) Weyl
spinor representation of ${\Lambda^{\alpha}}_{\beta}$.

Now we introduce the action describing the dynamics of the Weyl spinor field
$\Psi$ in spacetime $\mathcal{M}_{5}$ as

$\displaystyle S[\Psi]$ $\displaystyle=$ $\displaystyle\int
d^{4}xd\theta\sqrt{-G}\left[\bar{\Psi}i\gamma^{\alpha}{e_{\alpha}}^{\mu}\left(\hat{\partial}_{\mu}+\frac{i}{2}\omega^{\alpha\beta}_{\mu}\Sigma_{\alpha\beta}\right)\Psi+\frac{1}{2\Lambda}\left(\partial^{\theta}\bar{\Psi}\partial_{\theta}\Psi^{C}+\textrm{H.c.}\right)\right],$
(70)

where $\gamma^{\alpha}$ are the Dirac matrices satisfying the Clifford algebra
$\\{\gamma^{\alpha},\gamma^{\beta}\\}=2\eta^{\alpha\beta}$,
$\Sigma_{\alpha\beta}=\frac{i}{4}[\gamma_{\alpha},\gamma_{\beta}]$ are the
generators of the group $SO(3,1)$, $\omega^{\alpha\beta}_{\mu}$ is the spin
connection expressed as

$\displaystyle\omega^{\alpha\beta}_{\mu}={e^{\alpha}}_{\nu}\left(\hat{\partial}_{\mu}e^{\beta\nu}+e^{\beta\lambda}\Gamma^{\nu}_{\mu\lambda}\right),$
(71)

with $e^{\beta\nu}=\eta^{\beta\gamma}{e_{\gamma}}^{\nu}$, and
$\Psi^{C}=C\Psi^{*}$ is the charge-conjugate field with
$C=i\gamma^{2}\gamma^{0}$ and $\Psi^{*}$ to be the complex conjugate field.
The first and second terms in the action (70) describe the dynamics of $\Psi$
along the horizontal and vertical directions in spacetime $\mathcal{M}_{5}$,
respectively. It is important to note that if the field $\Psi$ transforms non-
trivially under a certain local gauge symmetry then the second term in the
action (70) is not be invariant and hence forbidden. On the other hand, such a
field has no the vertical kinetic term and thus it has no the KK counterparts.

In order to derive the KK spectrum of $\Psi$, let us consider its free
propagation corresponding to the propagation in the vacuum
$\bar{g}_{\mu\nu}=\eta_{\mu\nu}$ [or
${e_{\alpha}}^{\mu}=\chi^{-1/2}(\theta){\delta_{\alpha}}^{\mu}$], $X_{\mu}=0$,
and $\widetilde{T}=\langle\widetilde{T}\rangle$. The equation of motion for
the field $\Psi$ reads

$\displaystyle\chi^{-1/2}(\theta)i\gamma^{\mu}\partial_{\mu}\Psi+\frac{R^{-2}}{\Lambda\chi^{2}(\theta)}\partial_{\theta}\left[\chi^{2}(\theta)\partial_{\theta}\Psi^{C}\right]=0.$
(72)

We solve this equation by separating the variables as
$\Psi(x,\theta)=\chi^{-3/4}(\theta)\psi(x)Y(\theta)$ where $Y(\theta)$
satisfies the boundary condition

$\displaystyle Y(-\pi)=Y(\pi),$ (73)

which is obtained from Eq. (4). Then, we find

$\displaystyle i\gamma^{\mu}\partial_{\mu}\psi(x)-m\psi(x)^{C}$
$\displaystyle=$ $\displaystyle 0,$ (74) $\displaystyle
Y^{\prime\prime}(\theta)+\frac{\chi^{\prime}(\theta)}{2\chi(\theta)}Y^{\prime}(\theta)-\left[\frac{3\chi^{\prime\prime}(\theta)}{4\chi(\theta)}+\frac{3}{16}\left(\frac{\chi^{\prime}(\theta)}{\chi(\theta)}\right)^{2}-a\chi^{-1/2}(\theta)\right]Y(\theta)$
$\displaystyle=$ $\displaystyle 0,$ (75)

where $a$ is a dimensionless constant, $m=aR^{-2}/\Lambda$, and
$\psi(x)^{C}=C\psi(x)^{*}$. Eq. (75) can be rewritten in the form of Sturm-
Liouville equation as

$\displaystyle\frac{d}{d\theta}\left[p(\theta)\frac{d\bar{Y}(\theta)}{d\theta}\right]+\left[q(\theta)+aw(\theta)\right]\bar{Y}(\theta)=0,$
(76)

where $\bar{Y}(\theta)\equiv\chi^{-3/4}(\theta)Y(\theta)$,
$p(\theta)=\chi^{2}(\theta)$, $q(\theta)=0$, and
$w(\theta)=\chi^{3/2}(\theta)$. Because $p(\theta)$ and $w(\theta)$ are the
differentiable and positive functions over the fundamental domain
$\theta\in[-\pi,\pi]$, all eigenvalues $\\{a\equiv a_{n}\\}$ are real and non-
negative. In general, the solution of $\Psi(x,\theta)$ is given in terms of
the KK expansion as

$\displaystyle\Psi(x,\theta)=\frac{\chi^{-3/4}(\theta)}{\sqrt{2\pi
R}}\sum_{n}\psi_{n}(x)Y_{n}(\theta),$ (77)

where $\\{Y_{n}(\theta)\\}$ corresponding to $\\{a_{n}\\}$ forms a complete
set each of which satisfies Eq. (75). By using the boundary condition (73) and
Eq. (75) for $Y_{n}(\theta)$, one finds

$\displaystyle(a_{n}-a_{m})\int^{\pi}_{-\pi}d\theta
Y_{n}(\theta)Y_{m}(\theta)=0.$ (78)

Clearly, this relation suggests the orthogonality relation with respect to
$a_{n}\neq a_{m}$.

With the nontrivial solution of $\chi(\theta)$, we can find the analytical
solution for Eq. (75) in the case of the small values of the parameters
$\kappa_{1,2}$ (where we leave the hierarchy problem). In this situation, we
are only interested in the nontrivial solution (36), whereas the remaining
nontrivial solutions (37) and (46) are less interesting because they lead to
the very large hierarchy between the five dimensional Planck scale $M_{*}$ and
the inverse size of the extra dimension due to the extreme smallness of $k$.
Up to the order $\kappa^{2}_{1}$, the solution of Eq. (75) is given as

$\displaystyle Y(\theta)$ $\displaystyle=$ $\displaystyle
e^{-\frac{\kappa^{2}_{1}\theta^{2}}{12}\left(1+\sqrt{1+6a/\kappa^{2}_{1}}\right)}\left[CH\left(-\frac{4+\sqrt{1+6a/\kappa^{2}_{1}}-6a/\kappa^{2}_{1}}{2\sqrt{1+6a/\kappa^{2}_{1}}},\frac{\kappa_{1}\theta}{\sqrt{6}}(1+6a/\kappa^{2}_{1})^{1/4}\right)\right.$
(79)
$\displaystyle\left.+C^{\prime}{{}_{1}F_{1}}\left(\frac{4+\sqrt{1+6a/\kappa^{2}_{1}}-6a/\kappa^{2}_{1}}{4\sqrt{1+6a/\kappa^{2}_{1}}};\frac{1}{2};\frac{\kappa^{2}_{1}\theta^{2}}{6}\sqrt{1+6a/\kappa^{2}_{1}}\right)\right],$

where $C$ and $C^{\prime}$ are the constants which can be fixed from the
normalization condition, $H(a,z)$ and ${{}_{1}F_{1}}(a;b;z)$ are the Hermite
polynomial and the confluent hypergeometric function, respectively. The
relation (73) for $Y(\theta)$ imposes the possible values for $a$ as

$\displaystyle a\equiv
a_{n}=\frac{\kappa^{2}_{1}}{12}\left[9+8n(1+2n)+(1+4n)\sqrt{21+8n(1+2n)}\right],$
(80)

where $n=0$, $1$, $2$, $\cdots$ which are the non-negative integer numbers.
Interestingly, the zero mode corresponding to
$a_{0}=(9+\sqrt{21})\kappa^{2}_{1}/12$ has the non-zero Majorana mass as

$\displaystyle
m_{0}=\frac{(9+\sqrt{21})\kappa^{2}_{1}}{12}\frac{R^{-2}}{\Lambda}.$ (81)

With the trivial solution of $\chi(\theta)$ given in Eq. (48), it is
straightforward to obtain the solution for Eq. (75) as

$\displaystyle
Y(\theta)=C^{\prime}\cos(\sqrt{a}\theta)+C\sin(\sqrt{a}\theta),$ (82)

where $a\equiv a_{n}=n^{2}$ with $n=0$, $1$, $2$, $\cdots$. The
phenomenological implications for particle physics corresponding to this
solution are explored in Refs. Nam2019a ; Nam2020a ; Nam2020b .

##  III Inflation

In this section, we would like to indicate that our model provides a natural
inflation scenario where the field $\widetilde{T}$ which is related to the
geometry of the extra dimension plays the role of the inflaton. In this sense,
the presence of inflation is unavoidable because the inflaton field is
associated to the mathematical structure of spacetime.

We begin with writing the action for the effective metric field
$\bar{g}_{\mu\nu}(x)$ and the field $\widetilde{T}$ as follows

$\displaystyle S^{\text{eff}}_{\text{spacetime}}\supset\int
d^{4}x\sqrt{-g}\left[\frac{\widetilde{T}^{2}}{24}\frac{\mathcal{R}^{(4)}}{2}+\frac{1}{2}\partial_{\mu}\widetilde{T}\partial^{\mu}\widetilde{T}-\lambda_{\widetilde{T}}\left(\widetilde{T}^{2}-\langle\widetilde{T}\rangle^{2}\right)^{2}\right],$
(83)

which is in the so-called Jordan frame. The above action can be rewritten in
the so-called Einstein frame via the conformal transformation as

$\displaystyle\hat{g}_{\mu\nu}=\Omega^{2}\bar{g}_{\mu\nu},\ \ \ \
\Omega^{2}=\frac{\widetilde{T}^{2}}{24M^{2}_{\text{Pl}}}.$ (84)

This conformal transformation leads to a non-canonical kinetic term for the
inflaton. Hence, we redefine $\widetilde{T}$ as

$\displaystyle\widetilde{T}=\langle\widetilde{T}\rangle
e^{\frac{\chi}{\sqrt{30}M_{\text{Pl}}}},$ (85)

where $\chi$ is the field of the canonical kinetic term. Then, the potential
for the field $\chi$ takes the form as

$\displaystyle U(\chi)$ $\displaystyle=$
$\displaystyle\frac{\lambda_{\widetilde{T}}\left(\widetilde{T}^{2}-\langle\widetilde{T}\rangle^{2}\right)^{2}}{\Omega^{4}}$
(86) $\displaystyle=$ $\displaystyle
576\lambda_{\widetilde{T}}M^{4}_{\text{Pl}}\left(1-e^{-\frac{2\chi}{\sqrt{30}M_{\text{Pl}}}}\right)^{2}.$

This inflation potential $U(\chi)$ belongs to the slow-roll inflation class of
E-model ($n=1$) Planck2018  with the parameter $\alpha^{E}_{1}=5$. The global
minimum of the potential $U(\chi)$ is at $\langle\chi\rangle=0$ which
corresponds to VEV of the original field $\widetilde{T}$ as
$\langle\widetilde{T}\rangle$ as seen in Eq. (85).

The slow-roll parameters are obtained from the inflation potential $U(\chi)$
as

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{\text{Pl}}}{2}\left[\frac{U^{\prime}(\chi)}{U(\chi)}\right]^{2},$
$\displaystyle\eta$ $\displaystyle=$ $\displaystyle
M^{2}_{\text{Pl}}\frac{U^{\prime\prime}(\chi)}{U(\chi)},$
$\displaystyle\zeta^{2}$ $\displaystyle=$ $\displaystyle
M^{4}_{\text{Pl}}\frac{U^{\prime\prime\prime}(\chi)U^{\prime}(\chi)}{U^{2}(\chi)}.$
(87)

These parameters are related to the observed quantities which are the spectral
index $n_{s}$, the tensor-to-scalar ratio $r$, the running index
$\frac{dn_{s}}{d\ln k}$, and the scalar perturbation amplitude $A_{s}$ as

$\displaystyle n_{s}$ $\displaystyle\simeq$ $\displaystyle 1-6\epsilon+2\eta,$
$\displaystyle r$ $\displaystyle\simeq$ $\displaystyle 16\epsilon,$
$\displaystyle\frac{dn_{s}}{d\ln k}$ $\displaystyle\simeq$ $\displaystyle
16\epsilon\eta-24\epsilon^{2}-2\zeta^{2},$ $\displaystyle A_{s}$
$\displaystyle=$ $\displaystyle\frac{U}{24\pi^{2}M^{2}_{\text{Pl}}\epsilon},$
(88)

which are evaluated at the horizon crossing value of the inflaton. Here, the
observed values of the spectral and running indexes at $68\%$ confidence limit
(CL) are $n_{s}=0.9649\pm 0.0042$ and $\frac{dn_{s}}{d\ln k}=-0.0045\pm
0.0067$, the scalar perturbation amplitude is measured as $A_{s}\approx
2.2\times 10^{-9}$, whereas the upper bound for the tensor-to-scalar ratio at
$95\%$ CL is $r<0.063$, as reported by Planck 2018 for the pivot scale $0.05$
Mpc-1 Planck2018 . The number of e-foldings is given as

$\displaystyle
N=\int^{\chi_{\text{in}}}_{\chi_{\text{end}}}\frac{1}{M^{2}_{\text{Pl}}}\frac{U}{U^{\prime}(\chi)}d\chi,$
(89)

where $\chi_{\text{end}}$ is the value of the inflaton at which the inflation
ends and $\chi_{\text{in}}$ is the horizon crossing value of the inflaton. By
using $\epsilon\simeq 1$ at the end of the inflation, we obtain
$\chi_{\text{end}}\simeq 1.14M_{\text{Pl}}$.

It is straightforward to solve Eq. (89) to find $\chi_{\text{in}}$ and then
the inflation observables. In Table 1, we show our predictions for
$\chi_{\text{in}}$, the inflation observables, and the coupling
$\lambda_{\widetilde{T}}$ for various number of e-foldings.

$\ \ N\ \ $ | $\ \ \chi_{\text{in}}\ \ $ | $\ \ n_{s}\ \ $ | $\ \ r\ \ $ | $\ \ \frac{dn_{s}}{d\ln k}\ \ $ | $\ \ \lambda_{\widetilde{T}}\ \ $
---|---|---|---|---|---
$\ \ 58\ \ $ | $\ \ 8.14M_{\text{Pl}}\ \ $ | $\ \ 0.9681\ \ $ | $\ \ 0.0124\ \ $ | $\ \ -5.30\times 10^{-4}\ \ $ | $\ \ 7.79\times 10^{-13}\ \ $
$59$ | $8.18M_{\text{Pl}}$ | $0.9686$ | $0.0120$ | $-5.12\times 10^{-4}$ | $7.55\times 10^{-13}$
$60$ | $8.22M_{\text{Pl}}$ | $0.9692$ | $0.0117$ | $-4.96\times 10^{-4}$ | $7.32\times 10^{-13}$
$61$ | $8.26M_{\text{Pl}}$ | $0.9696$ | $0.0114$ | $-4.80\times 10^{-4}$ | $7.10\times 10^{-13}$
$62$ | $8.29M_{\text{Pl}}$ | $0.9701$ | $0.0110$ | $-4.65\times 10^{-4}$ | $6.89\times 10^{-13}$
Table 1: The inflationary predictions in our model for various number of
e-foldings.

From this table we find that our predictions are in good agreement with the
Planck 2018 data. Compared to the inflationary models of the Starobinsky-type
potential, our model predicts a larger tensor-to-scalar ratio which is around
$\sim 0.011$ and thus can be tested in the near future.

In Table 1, the effective coupling $\lambda_{\widetilde{T}}$ is determined
from the observed value of the scalar perturbation amplitude $A_{s}$. For the
number of e-foldings $N\sim 60$, the effective coupling
$\lambda_{\widetilde{T}}$ is around $\sim 7\times 10^{-13}$ which is very
small. In order to see the smallness of the effective coupling
$\lambda_{\widetilde{T}}$, let us first rewrite $\lambda_{\widetilde{T}}$ in
as

$\displaystyle\lambda_{\widetilde{T}}$ $\displaystyle=$
$\displaystyle\frac{\bar{a}_{1}}{576}\frac{R^{-2}}{M^{2}_{\text{Pl}}}\left[\int^{\pi}_{-\pi}d\theta\chi^{2}(\theta)\right]\left[\int^{\pi}_{-\pi}d\theta\chi(\theta)\right]^{-1},$
(90)

where $\bar{a}_{1}\equiv a_{1}\langle T\rangle^{3}$ and $\chi(\theta)$
corresponds to the solution (36). This relation implies that the effective
coupling $\lambda_{\widetilde{T}}$ would be suppressed by the observed Planck
scale. By using Eq. (64), we can determine the relation between the parameter
$\bar{a}_{1}$ and the inverse size of extra dimension for the ratio
$R^{-1}/M_{*}$ kept fixed. Such a relation is shown in Fig. 2. We observe that
$R^{-1}$ increases with the growth of $\bar{a}_{1}$, which suggests the large
hierarchy between $M_{*}$, $R^{-1}$ and the electroweak scale in the region of
the sufficiently large/small $\bar{a}_{1}$. Otherwise, no the large hierarchy
appears in the intermediate region of $\bar{a}_{1}$. In addition, for
$R^{-1}/M_{*}$ given, $\bar{a}_{1}$ and $R^{-1}$ cannot exceed a certain
value, which is due to the behavior of the function $\chi(\theta)$ and the
constraints coming from the identification of the observed Planck scale, and
the observed value of the scalar perturbation amplitude. For example, with
$R^{-1}/M_{*}=0.0125$ corresponding to the orange curve in Fig. 2, we find
$\bar{a}_{1}\lesssim 1.356\times 10^{-3}$ and $R^{-1}\lesssim 1.41\times
10^{15}$ GeV.

---
Figure 2: The dependence of the inverse size $R^{-1}$ of extra dimension in
the parameter $\bar{a}_{1}$ for various reference points of ratio
$R^{-1}/M_{*}$ with $\lambda_{\widetilde{T}}=7.32\times 10^{-13}$.

At the end of the inflation, the inflaton oscillates around the global minimum
or VEV $\langle\chi\rangle$ and corresponds to a mass

$\displaystyle m_{\chi}$ $\displaystyle=$ $\displaystyle
16\sqrt{\frac{3}{5}\lambda_{\widetilde{T}}}M_{\text{Pl}}.$ (91)

With the number of e-foldings $N\sim 60$, one finds that the mass of the
inflaton is about $\sim 2.6\times 10^{13}$ GeV.

##  IV Geodesic equation and Observations

In this section, we shall derive the geodesic equation which describes the
motion of neutral test particles in spacetime $\mathcal{M}_{5}$. Then, we
consider the phenomenological implications of the geodesic equation for the
motion of neutral test particles. We are interested in investigating the
photon sphere and the bound orbits, which have the important differences
between GR and our scenario about the qualitative and quantitative features
and hence would allow us to distinguish our scenario from other alternative
theories of gravity. In addition, we calculate the perihelion shift and the
bending angle of light in the weak field limit and then we compare the values
predicted by our spacetime $\mathcal{M}_{5}$ with the experimental
observations to find the corresponding constraints. With respect the
gravitational deflection of light in the strong field limit, we study the
impact of spacetime $\mathcal{M}_{5}$ on the strong gravitational lensing.

The worldline $\\{x^{\mu}(\lambda),\theta(\lambda)\\}\equiv X^{M}(\lambda)$ of
a neutral test particle moving in spacetime $\mathcal{M}_{5}$ satisfies the
geodesic equation

$\displaystyle\frac{d^{2}X^{M}}{d\lambda^{2}}+\Gamma^{M}_{PQ}\frac{dX^{P}}{d\lambda}\frac{dX^{Q}}{d\lambda}=0,$
(92)

where $\lambda$ is the Affine parameter. In addition, the line element of
spacetime $\mathcal{M}_{5}$ implies the following relation

$\displaystyle\bar{g}_{\mu\nu}(x)\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}=\frac{1}{\chi(\theta)}\left[k+\frac{\widetilde{T}^{4}}{\Delta^{2}}\left(\frac{d\theta}{d\lambda}\right)^{2}\right],$
(93)

where $k=0(1)$ for the null(timelike) geodesic and

$\displaystyle\Delta$ $\displaystyle\equiv$ $\displaystyle
24M^{3}_{*}\int^{\pi}_{-\pi}d\theta\chi(\theta).$ (94)

The geodesic equation is more specifically expanded as

$\displaystyle\frac{dv^{\mu}}{d\lambda}+\bar{\Gamma}^{\mu}_{\nu\rho}v^{\nu}v^{\rho}+\frac{g_{{}_{X}}\chi^{\prime}(\theta)}{2\chi(\theta)}\left[\frac{X^{\mu}}{\chi(\theta)}\left(k+\frac{\widetilde{T}^{4}}{\Delta^{2}}(v^{\theta})^{2}\right)-2v^{\mu}v^{\nu}X_{\nu}\right]$
$\displaystyle+\left[\frac{\chi^{\prime}(\theta)}{\chi(\theta)}{\delta^{\mu}}_{\nu}+\frac{g_{{}_{X}}\widetilde{T}^{4}}{\Delta^{2}\chi(\theta)}{X^{\mu}}_{\nu}\right]v^{\nu}v^{\theta}+\frac{2\widetilde{T}^{3}\partial^{\mu}\widetilde{T}}{\Delta^{2}\chi(\theta)}(v^{\theta})^{2}$
$\displaystyle=$ $\displaystyle 0,$ (95)
$\displaystyle\frac{dv^{\theta}}{d\lambda}+\frac{\chi^{\prime}(\theta)}{2\chi(\theta)}\left[\frac{\Delta^{2}}{\widetilde{T}^{4}}k+(v^{\theta})^{2}\right]-4\frac{\partial_{\mu}\widetilde{T}}{\widetilde{T}}v^{\mu}v^{\theta}$
$\displaystyle=$ $\displaystyle 0,$ (96)

where $v^{\mu}\equiv dx^{\mu}/d\lambda$, $v^{\theta}\equiv d\theta/d\lambda$,

$\displaystyle\bar{\Gamma}^{\mu}_{\nu\rho}$ $\displaystyle=$
$\displaystyle\frac{\bar{g}^{\mu\lambda}(x)}{2}\left[\partial_{\nu}\bar{g}_{\lambda\rho}(x)+\partial_{\rho}\bar{g}_{\lambda\nu}(x)-\partial_{\lambda}\bar{g}_{\nu\rho}(x)\right],$
(97)

$X^{\mu}\equiv\bar{g}^{\mu\nu}(x)X_{\nu}$,
${X^{\mu}}_{\nu}\equiv\bar{g}^{\mu\rho}(x)X_{\rho\nu}$, and
$\partial^{\mu}\equiv\bar{g}^{\mu\nu}(x)\partial_{\nu}$.

For the spherically-symmetric and static solution of the system described by
the effective action (49), the ansatz for the effectively four dimensional
metric $\bar{g}^{\mu\nu}(x)$, the gauge field $X_{\mu}$, and the field
$\widetilde{T}$ are given as

$\displaystyle\bar{g}_{\mu\nu}(x)dx^{\mu}dx^{\nu}$ $\displaystyle=$
$\displaystyle
f(r)dt^{2}-\frac{dr^{2}}{g(r)}-r^{2}(d\varphi_{1}^{2}+\sin^{2}\varphi_{1}d\varphi_{2}^{2}),$
$\displaystyle X_{\mu}dx^{\mu}$ $\displaystyle=$ $\displaystyle\phi(r)dt.$
$\displaystyle\widetilde{T}$ $\displaystyle=$ $\displaystyle\psi(r).$ (98)

It should be noted that the massive objects in astronomy are in fact rotating
and thus the spacetime geometry of these objects are described by the Kerr
metric Kerr1963 . In this work, we consider the massive objects to be the Sun
and black holes. The Sun rotates very slowly with its rotation rate varying
between approximately $13.8/$year to $7.7/$year at the equator and the poles,
respectively Beck2000 . Whereas, most black holes are born to have the
rotation rate to be very slowly with the rotation parameter $a\equiv J/M^{2}$
(with $J$ and $M$ to be the anglular momentum and the mass of black hole)
about $10^{-2}$ Fuller2019 . For black holes which are compact components in
low-mass X-ray binaries (LMXBs) which are of the most important astrophysical
laboratories, all current methods of spin measurement of these black holes
need to employ the accretion disk to infer their spin rate. Due to the
difficulty in modeling accretion disk, the spin rate measurement sometimes
leads to conflicting results. On the other hand, the origin of the black hole
spin in LMXBs can be accounted for by the matter accreted onto the black hole
Fragos2015 . In addition, if the accreation is chaotic then the matter
accreated on the black holes may not make increasing the black hole spin rate
Miler2014 . With respect to non-accreting stellar-mass black holes whose
growth is through mergers with other black holes, successive additions of
random spins of merging black holes tend to spin-down the resultant black
hole, unless alignment of their spin is very efficient Berti2008 . The spin
measurements of these black holes which are now available for mergers detected
by LIGO have pointed to that most of them are consistent with the low spin
Roulet2018 ; LIGO2018 .

Because of $a\ll 1$ in these situations, we can consider the zero order in the
rotation parameter $a$, which corresponds to the no rotation approximation of
very slowly rotating massive objects.

With the ansatz (98), Eqs. (95) and (96) become

$\displaystyle\frac{dv^{0}}{d\lambda}+\frac{f^{\prime}(r)}{f(r)}v^{0}v^{1}+\frac{g_{{}_{X}}\chi^{\prime}(\theta)}{2\chi(\theta)}\phi(r)\left[\frac{1}{\chi(\theta)}\left(k+\frac{\psi^{4}(r)}{\Delta^{2}}(v^{\theta})^{2}\right)-2(v^{0})^{2}\right]$
$\displaystyle+\left[\frac{\chi^{\prime}(\theta)}{\chi(\theta)}v^{0}-\frac{g_{{}_{X}}\psi^{4}(r)}{\Delta^{2}\chi(\theta)}\frac{\phi^{\prime}(r)}{f(r)}v^{1}\right]v^{\theta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{dv^{1}}{d\lambda}+\frac{g(r)f^{\prime}(r)}{2}(v^{0})^{2}-\frac{g^{\prime}(r)}{2g(r)}(v^{1})^{2}-rg(r)\left[(v^{2})^{2}+\sin^{2}\varphi_{1}(v^{3})^{2}\right]-\frac{g_{{}_{X}}\chi^{\prime}(\theta)}{\chi(\theta)}\phi(r)v^{0}v^{1}$
$\displaystyle+\left[\frac{\chi^{\prime}(\theta)}{\chi(\theta)}v^{1}-\frac{g_{{}_{X}}\psi^{4}(r)}{\Delta^{2}\chi(\theta)}g(r)\phi^{\prime}(r)v^{0}\right]v^{\theta}-\frac{2g(r)}{\Delta^{2}\chi(\theta)}\psi^{3}(r)\psi^{\prime}(r)(v^{\theta})^{2}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{dv^{2}}{d\lambda}+\frac{2}{r}v^{1}v^{2}-\frac{\sin
2\varphi_{1}}{2}(v^{3})^{2}-\frac{g_{{}_{X}}\chi^{\prime}(\theta)}{\chi(\theta)}\phi(r)v^{0}v^{2}+\frac{\chi^{\prime}(\theta)}{\chi(\theta)}v^{2}v^{\theta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{dv^{3}}{d\lambda}+\frac{2}{r}v^{1}v^{3}+\cot\varphi_{1}v^{2}v^{3}-\frac{g_{{}_{X}}\chi^{\prime}(\theta)}{\chi(\theta)}\phi(r)v^{0}v^{3}+\frac{\chi^{\prime}(\theta)}{\chi(\theta)}v^{3}v^{\theta}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\frac{dv^{\theta}}{d\lambda}+\frac{\chi^{\prime}(\theta)}{2\chi(\theta)}\left[\frac{\Delta^{2}}{\psi^{4}(r)}k+(v^{\theta})^{2}\right]-\frac{4\psi^{\prime}(r)}{\psi(r)}v^{r}v^{\theta}$
$\displaystyle=$ $\displaystyle 0.$ (99)

From the equation for $dv^{2}/d\lambda$, one sees that we always restrict the
equatorial motion of test particles, i.e. $\varphi_{1}=\pi/2$, without loss of
generality.

We consider the propagation of neutral test particles in the low energy
region. This implies $\widetilde{T}\simeq\langle\widetilde{T}\rangle$ because
as indicated above the fluctuation field around the vacuum
$\langle\widetilde{T}\rangle$ is very massive and thus decouples at the low
energy. Also, determining the analytical solution for these geodesic equations
with $\chi(\theta)$ which is nontrivial is a difficult task. Thus in the
following we consider the trivial solution of $\chi(\theta)$ which is given by
Eq. (48). As a result, the spherically-symmetric and static spacetime
background given in Eq. (98) reads

$\displaystyle f(r)=g(r)$ $\displaystyle=$ $\displaystyle
1-\frac{2M}{r}+\frac{Q^{2}_{X}}{r^{2}},$ $\displaystyle\phi(r)$
$\displaystyle=$ $\displaystyle-\frac{Q_{X}}{r},$ $\displaystyle\psi(r)$
$\displaystyle=$ $\displaystyle\langle\widetilde{T}\rangle,$ (100)

where $M$ is the ADM mass of the system and $Q_{X}$ is the electric-like
charge corresponding to the gauge field $X_{\mu}$. Because the charge $Q_{X}$
would be small, we safely neglect the contribution of the term
$Q^{2}_{X}/r^{2}$ compared to the term $2M/r$ in the expression of $f(r)$ and
$g(r)$. The Schwarzschild radius thus is approximately given by $2M$. Note
that, here (and in whats follows) we have considered the unit system where the
observed gravitational constant $G$ is given as $G=1$.

Then, by integrating the last expression in Eq. (99), one finds

$\displaystyle v^{\theta}=\frac{c_{\theta}}{R},$ (101)

where $c_{\theta}$ is a constant which means that the velocity along the
vertical direction is always a constant. Substituting this result into the
first and fourth lines of Eq. (99) and then integrating them, we obtain

$\displaystyle v^{0}$ $\displaystyle=$
$\displaystyle\frac{1}{f(r)}\left[E+\frac{c_{\theta}}{\sqrt{4\pi}}\phi(r)\right],$
$\displaystyle v^{3}$ $\displaystyle=$ $\displaystyle\frac{L}{r^{2}},$ (102)

where $E$ and $L$ are the constants which are at large distance identified as
the specific energy and the specific angular momentum of the massive particles
(while for the massless particles like the photon $E$ and $L$ can be thought
of as the energy and angular momentum, respectively) measured by a stationary
observer at large radius. With this result and using Eq. (93), we obtain

$\displaystyle(v^{1})^{2}=\left[E+\frac{c_{\theta}}{\sqrt{4\pi}}\phi(r)\right]^{2}-f(r)\left(k+c^{2}_{\theta}+\frac{L^{2}}{r^{2}}\right).$
(103)

This equation can be rewritten as

$\displaystyle(v^{1})^{2}=\left(E-V_{-}\right)\left(E-V_{+}\right),$ (104)

where the effective potentials $V_{\pm}$ are defined as follows

$\displaystyle
V_{\pm}=-\frac{c_{\theta}}{\sqrt{4\pi}}\phi(r)\mp\sqrt{f(r)\left(k+c^{2}_{\theta}+\frac{L^{2}}{r^{2}}\right)}.$
(105)

###  IV.1 Photon Sphere

Eq. (103) is written for the photon ($k=0$) which moves along the null paths
in the background given in (100) as

$\displaystyle\left(\frac{dr}{d\lambda}\right)^{2}=\left(E+\frac{\alpha}{r}\right)^{2}-\left(1-\frac{2M}{r}\right)\left(c^{2}_{\theta}+\frac{L^{2}}{r^{2}}\right),$
(106)

where

$\displaystyle\alpha\equiv-\frac{c_{\theta}Q_{X}}{\sqrt{4\pi}},$ (107)

which is positive or negative depending on the sign of the charge $Q_{X}$.
Here, we observe that the structure of spacetime $\mathcal{M}_{5}$ affects the
orbital dynamics of the photon through the additional terms related to the
quantities $\alpha$ and $c^{2}_{\theta}$. In order to see their effects on the
orbital dynamics of the photon, we investigate separately their role
corresponding to two cases which are $c^{2}_{\theta}\simeq 0$ and
$\alpha\simeq 0$ (or $Q_{X}\simeq 0$).

The conditions for the photon moving on a circular orbit are that the right
hand side of Eq. (106) and its first-order derivative in $r$ vanish both. For
the first case of $c_{\theta}\simeq 0$ which hence we ignore the term relating
to $c^{2}_{\theta}$ in Eq. (106) and then we find

$\displaystyle L_{\pm}$ $\displaystyle=$ $\displaystyle\pm
r\frac{\sqrt{1-2M/r}}{r-3M}\alpha,$ $\displaystyle E_{\pm}$ $\displaystyle=$
$\displaystyle-\frac{\alpha}{r}\pm L_{\pm}\frac{\sqrt{f(r)}}{r}$ (108)
$\displaystyle=$ $\displaystyle\frac{M\alpha}{r(r-3M)},$

We note that the energy and the angular momentum of the photon are positive
both. First, with respect to the angular momentum of the photon, the positive
condition corresponding to the positive branch $L_{+}$ suggests the constraint
on $r$ and $\alpha$ as $r>3M$ and $\alpha>0$ or an alternative $r<3M$ and
$\alpha<0$. Corresponding to the negative branch $L_{-}$, the positive
condition leads to $r>3M$ and $\alpha<0$ or $r<3M$ and $\alpha>0$. Whereas,
applying the positive condition on the energy of the photon, one finds the
same constraint as the case of the positive branch for the angular momentum.
This means that only the positive branch $L_{+}$ of the angular momentum is
suitable to the expression of energy.

By solving the first equation (corresponding to $L_{+}$) in Eq. (108), one
determines the radius for the photon sphere as

$\displaystyle r^{+}_{ph}$ $\displaystyle=$
$\displaystyle\frac{3L^{2}/\alpha^{2}-1+\sqrt{1+3L^{2}/\alpha^{2}}}{L^{2}/\alpha^{2}-1}M,$
$\displaystyle r^{-}_{ph}$ $\displaystyle=$
$\displaystyle\frac{3L^{2}/\alpha^{2}-1-\sqrt{1+3L^{2}/\alpha^{2}}}{L^{2}/\alpha^{2}-1}M.$
(109)

Here, the positive and negative branches of the solution are always larger and
smaller than $3M$ and hence would correspond to $\alpha>0$ and $\alpha<0$,
respectively. We show the behavior of the radius $r_{ph}$ for the photon
sphere as a function of the angular momentum $L$ in Fig. 3.

---
Figure 3: The scaled radius $r_{ph}/M$ for the photon sphere as a function of
the angular momentum $L$ at $|\alpha|=1/10$. The blue and red curves
correspond to $r^{+}_{ph}$ ($\alpha=1/10$) and $r^{-}_{ph}$ ($\alpha=-1/10$),
respectively. The horizontal dashed black line refers to the prediction of GR,
$r_{ph}=3M$.

Let us now analyze the features of the radius $r_{ph}$ for the photon sphere
derived above.

  * (1)

In GR and almost alternative theories of gravity Bahamonde2019 ; GuoMiao2020 ;
Boonserm2020 ; Turimov2020 ; XXZeng2020 ; GuoLi2020  as well as references
therein, the radius $r_{ph}$ for the photon sphere is fixed by the geometry of
spacetime. However, in our work, besides the dependence of the geometry of
spacetime $r_{ph}$ is also dependent on the angular momentum $L$ of the
photon.

  * (2)

For the case $|\alpha|\ll L$, the radius $r_{ph}$ for the photon sphere given
in Eq. (109) is expanded as

$\displaystyle
r_{ph}=3M\left[1+\frac{\alpha}{\sqrt{3}L}+\mathcal{O}(\alpha^{2}/L^{2})\right].$
(110)

It means that, in the limit of the large angular momentum of the photon, the
radius $r_{ph}$ for the photon sphere reduces to $3M$ which is the
corresponding value for the Schwarzschild spacetime geometry and hence our
prediction for $r_{ph}$ is the same as that of GR. This can be seen in Fig. 3.

  * (3)

Our prediction for $r_{ph}$ deviates significantly from GR about the
qualitative and quantitative features in the region of the small angular
momentum of photon at which the magnitude of the quantity $\alpha$ can be
comparable to $L$ or is much larger than $L$, as seen in Fig. 3.

  * (4)

The positive branch $r^{+}_{ph}$ predicts the radius for the photon sphere
larger than the prediction of GR. Also, the radius for the photon sphere
corresponding to this branch exists only for the region $L\in(\alpha,+\infty)$
(because the radius for the photon sphere is negative for $L<\alpha$) and
blows up as $L\rightarrow\alpha$. Whereas, the radius for the photon sphere
predicted by the negative branch $r^{-}_{ph}$ lies in the region $(2M,3M)$ and
hence it yields a smaller radius for the photon sphere compared to the
prediction of GR.

For the second case of $\alpha\simeq 0$, from the conditions for the photon
moving on a circular orbit in analogy to what we have done in the first case,
we find two radii as

$\displaystyle r^{\pm}_{ph}$ $\displaystyle=$
$\displaystyle\frac{L^{2}}{2c^{2}_{\theta}M}\left(1\pm\sqrt{1-\frac{12c^{2}_{\theta}M^{2}}{L^{2}}}\right),$
(111)

but only if $L^{2}\geq 12c^{2}_{\theta}M^{2}$. The radii $r^{+}_{ph}$ and
$r^{-}_{ph}$ correspond to the stable and unstable circular orbits,
respectively. Because the fact that the quantity $c_{\theta}$ is small, the
radius $r^{+}_{ph}$ of the stable circular orbit which is not predicted in GR
is very large and approaches infinity when $c_{\theta}$ goes to zero. This
means that the stable circular orbit of the photon manifests significantly for
$c_{\theta}$ sufficiently large. Here, we have several comments in order:

  * (1)

In the case of $L=12c^{2}_{\theta}M^{2}$, the radii $r^{+}_{ph}$ and
$r^{-}_{ph}$ coincide and in which case it leads to ISCO for the photon with
radius $r_{c}=6M$ which is equal to the ISCO radius of the massive particle in
the Schwarzschild geometry.

  * (2)

For the photon with angular momentum $L$ below $12c^{2}_{\theta}M^{2}$
corresponding to the fact that the effective potential has no minimum and
maximum, there is no turning point to hit and since the photon keeps moving
and never returns.

  * (3)

Because the presence of the stable and unstable circular orbits, there exists
the orbits for the photon which is bound but non-circular with the suitable
energy.

###  IV.2 Bound orbits of massive particles

First, we analyze the circular orbits of the massive particle by investigating
Eq. (103) with $k=1$, given as follows

$\displaystyle\left(\frac{dr}{d\lambda}\right)^{2}=\left(E+\frac{\alpha}{r}\right)^{2}-\left(1-\frac{2M}{r}\right)\left(1+c^{2}_{\theta}+\frac{L^{2}}{r^{2}}\right)\equiv
V(r),$ (112)

where $c_{\theta}$ must be realized to be the velocity along the vertical
direction of the massive particle and the parameter $\alpha$ is defined the
same as in Eq. (107). Note that, here the value of $\alpha$ is generally
different to the photon case due to the difference in the velocity along the
vertical direction (relating to $c_{\theta}$) and the $Q_{X}$ charge caused by
two different sources. The conditions for the massive particle moving on a
circular orbit are $V(r)=0$ and $V^{\prime}(r)=0$, which lead to the following
equation

$\displaystyle(3M-r)L^{2}+(1+c^{2}_{\theta})Mr^{2}+\alpha\sqrt{r(r-2M)\left[L^{2}+(1+c^{2}_{\theta})r^{2}\right]}=0,$
(113)

which determines radii for the circular orbits of the massive particle. It is
not easily to obtain analytical solutions for Eq. (113). In fact $\alpha$ is
small because the predictions of GR agree with the experimental observations,
in particular, in the solar system. Hence one can solve perturbatively Eq.
(113), given as

$\displaystyle
r_{\pm}=r_{0\pm}-\frac{\sqrt{r_{0\pm}(r_{0\pm}-2M)\left[L^{2}+(1+c^{2}_{\theta})r_{0\pm}^{2}\right]}}{2(1+c^{2}_{\theta})Mr_{0\pm}-L^{2}}\alpha+\mathcal{O}(\alpha^{2}),$
(114)

where $r_{0\pm}$ are radii of two circular orbits at the leading order
corresponding to $\alpha=0$ (or $Q_{X}=0$) and are given as

$\displaystyle
r_{0\pm}=\frac{L^{2}}{2(1+c^{2}_{\theta})M}\left(1\pm\sqrt{1-\frac{12M^{2}(1+c^{2}_{\theta})}{L^{2}}}\right),$
(115)

but only if $L\geq 2\sqrt{3(1+c^{2}_{\theta})}M$. The radii $r_{-}$ and
$r_{+}$ correspond to the unstable and stable circular orbits, respectively.
For $\alpha>0$, the radii $r_{-}$ and $r_{+}$ are enhanced and lowered by the
correction coming from the fibre fabric of spacetime, respectively. On the
contrary, the correction with $\alpha<0$ makes the radii $r_{-}$ and $r_{+}$
decreasing and increasing, respectively.

The radius $r_{c}$ for ISCO of the massive particle happens when the radii
$r_{-}$ and $r_{+}$ coincide. As a result, we find radius $r_{c}$ for ISCO of
the massive particle as

$\displaystyle
r_{c}=6M-2\sqrt{\frac{2}{1+c^{2}_{\theta}}}\alpha+\mathcal{O}(\alpha^{2}),$
(116)

at

$\displaystyle
L=2\sqrt{3(1+c^{2}_{\theta})}M+\sqrt{\frac{2}{3}}\alpha+\mathcal{O}(\alpha^{2}).$
(117)

Note that, the value $r_{c}=6M$ is predicted by GR for the Schwarzschild
spacetime geometry, which is independent on $c_{\theta}$ in the case of
$\alpha=0$ (or $Q_{X}=0$). Obviously, the radius $r_{c}$ for ISCO of the
massive particle reduces (enlarges) compared to the prediction of GR if
$\alpha>0$ ($\alpha<0$).

We arrive at studying the bound but non-circular orbits for the massive
particle. From the second line of Eq. (102) and Eq. (112), one finds the
equation describing the orbital dynamics of a massive particle as

$\displaystyle\left(\frac{dy}{d\varphi_{2}}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{L^{2}}\left[\left(E+\frac{\alpha}{r_{+}}\right)^{2}-\left(1-\frac{2M}{r_{+}}\right)\left(1+c^{2}_{\theta}+\frac{L^{2}}{r^{2}_{+}}\right)\right]+\frac{2}{L^{2}}\left[\frac{L^{2}}{r^{2}_{+}}(3M-r_{+})+M(1+c^{2}_{\theta})\right.$
(118)
$\displaystyle\left.+\alpha\left(E+\frac{\alpha}{r_{+}}\right)\right]y-\left(1-\frac{6M}{r_{+}}-\frac{\alpha^{2}}{L^{2}}\right)y^{2}+2My^{3},$

where $y\equiv\frac{1}{r}-\frac{1}{r_{+}}$ which describes the deviation from
the stable circular orbit. For the orbit which is nearly circular and hence
$y$ is small, we can ignore the term in $y^{3}$ and the solution has the
following form

$\displaystyle y=y_{0}+A\cos(k\varphi_{2}+\varphi_{20}),$ (119)

where the constants $y_{0}$, $A$, and $k$ are given as

$\displaystyle y_{0}$ $\displaystyle=$
$\displaystyle\frac{L^{2}(3M-r_{+})+r^{2}_{+}\left[M(1+c^{2}_{\theta})+\alpha(E+\alpha/r_{+})\right]}{r_{+}\left[L^{2}(r_{+}-6M)-\alpha^{2}r_{+}\right]},$
$\displaystyle k$ $\displaystyle=$
$\displaystyle\left(1-\frac{6M}{r_{+}}-\frac{\alpha^{2}}{L^{2}}\right)^{1/2},$
$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{1}{k}\left\\{\frac{1}{L^{2}}\left[\left(E+\frac{\alpha}{r_{+}}\right)^{2}-\left(1-\frac{2M}{r_{+}}\right)\left(1+c^{2}_{\theta}+\frac{L^{2}}{r^{2}_{+}}\right)\right]+k^{2}y^{2}_{0}\right\\}^{1/2},$
(120)

and $\varphi_{20}$ is the initial phase which can be set to be zero for
simplicity. We have two remarks as follows:

  * (1)

In GR and almost alternative theories of gravity, $y_{0}$ vanishes because
$r_{+}$ satisfies the equation obtained from the vanishing of the first-order
derivative of the effective potential. Thus, the bound orbits which is nearly
circular will oscillate around the radius of the stable circular orbit.
However, in our case $y_{0}$ is generally nonzero unless $\alpha=0$ (or
$Q_{X}=0$) and dependent on the specific energy and angular momentum of the
particle. As a result, the orbit oscillates not about $y=0$ but about
$y=y_{0}$ which is not fixed by the geometry of spacetime.

  * (2)

The perihelion shift per orbit is

$\displaystyle\Delta\varphi_{2}=\frac{2\pi}{k}-2\pi\simeq\frac{6\pi
M}{\hat{r}_{+}}\left(1+\frac{\sqrt{\hat{r}_{+}(\hat{r}_{+}-2M)(L^{2}+\hat{r}_{+}^{2}})}{\hat{r}_{+}(2M\hat{r}_{+}-L^{2})}\alpha+\frac{M\hat{r}_{+}}{L^{2}\sqrt{1-12M^{2}/L^{2}}}c^{2}_{\theta}\right).$
(121)

where

$\displaystyle\hat{r}_{+}=\frac{L^{2}}{2M}\left(1+\sqrt{1-\frac{12M^{2}}{L^{2}}}\right),$
(122)

is the prediction of GR for the radius of the stable circular orbit. The first
term in the final expression of Eq. (121) is the GR contribution, whereas the
second and third terms are the corrections coming from spacetime
$\mathcal{M}_{5}$ up to the second order in $c_{\theta}$. The measured value
for the perihelion shift of Mercury is $42.98\pm 0.04^{\prime\prime}$ per a
century CMWill2006 . Using $\hat{r}_{+}\approx 5.550\times 10^{7}$ km,
$M=M_{\odot}\approx 1.474$ km where $M_{\odot}$ is the mass of the Sun, and
each orbit of Mercury around the Sun takes approximately $87.97$ days, one
finds the value as predicted by GR for the perihelion shift of Mercury as
$\approx 42.75^{\prime\prime}$ per a century. Then, from the observed and GR
values and $L^{2}\simeq M_{\odot}\hat{r}_{+}$, we obtain the constraint on the
charge $Q_{X}$ of the Sun and and $c_{\theta}$ corresponding to Mercury as

$\displaystyle\sqrt{4\pi}\left(1.47c_{\theta}-\frac{10^{-2}}{c_{\theta}}\right)\lesssim\frac{Q_{X}}{\text{km}}\lesssim
5.2c_{\theta}.$ (123)

These upper and lower bounds are depicted in Fig. 4 where the white region
refers to the allowed parameter region.

---
Figure 4: The allowed parameter region (white region) for the charge $Q_{X}$
of the Sun and and $c_{\theta}$ corresponding to Mercury. The red line and
blue curve correspond to the upper and lower bounds given in Eq. (123).

###  IV.3 Gravitational deflection of light

We determine the deflection angle of light when it passes through a central
massive object in our spacetime $M_{5}$, given as

$\displaystyle\Delta\varphi_{2}=2\int^{\infty}_{r_{0}}dr\frac{L}{r^{2}}\left[\left(E+\frac{\alpha}{r}\right)^{2}-\left(1-\frac{2M}{r}\right)\left(c^{2}_{\theta}+\frac{L^{2}}{r^{2}}\right)\right]^{-1/2}-\pi,$
(124)

where $r_{0}$ is the point of closest approach to the central massive object
which is determined by $dr/d\lambda=0$ corresponding to

$\displaystyle\left(E+\frac{\alpha}{r_{0}}\right)^{2}-\left(1-\frac{2M}{r_{0}}\right)\left(c^{2}_{\theta}+\frac{L^{2}}{r^{2}_{0}}\right)=0.$
(125)

In the approximation of the weak deflection, i.e. $\frac{M}{r}\ll 1$, and the
small corrections coming from spacetime $M_{5}$, i.e. $\frac{\alpha}{L}\ll 1$
or $c_{\theta}\ll 1$, we obtain the deflection angle of light to the
subleading order approximation of $c_{\theta}$ as

$\displaystyle\Delta\varphi_{2}$ $\displaystyle\simeq$ $\displaystyle
2\left[\left(1+\frac{15M^{2}}{4b^{2}}+\frac{2M\alpha}{bL}+\frac{48M^{3}\alpha}{b^{3}L}\right)\arcsin(by)-\left(2+\frac{64M^{2}}{3b^{2}}+\frac{15My}{4}+\frac{32M^{2}y^{2}}{3}\right.\right.$
(126) $\displaystyle\left.\left.+\frac{15M\alpha}{bL}+\frac{48M^{2}\alpha
y}{bL}\right)M\sqrt{\frac{1}{b^{2}}-y^{2}}\right]\Big{|}^{y=1/b}_{y=0}-\pi$
$\displaystyle\simeq$ $\displaystyle\frac{4M}{b}+\frac{15\pi
M^{2}}{4b^{2}}+\frac{128M^{3}}{3b^{3}}+2\left(\pi+\frac{15M}{b}+\frac{24\pi
M^{2}}{b^{3}}\right)\frac{M\alpha}{bL},$

where $y$ is related to $r$ as
$y^{2}\equiv\frac{1}{r^{2}}\left(1-\frac{2M}{r}\right)-\frac{\alpha}{bLr}$ and
$b\equiv\frac{L}{E}$ is the impact parameter. The first three terms in the
second line are the prediction of GR where the first term $4M/b$ is the
leading contribution or the standard GR light deflection angle which is
usually considered in the literature GRbook-Schutz . Whereas, the terms
concerning $\alpha/L$ are the corrections coming from spacetime
$\mathcal{M}_{5}$. We note here that, in obtaining the above deflection angle
of light, we have expanded up to $\frac{M^{3}}{r^{3}}$ to compare the
subleading contributions of GR with the the $\alpha/L$ contributions. This
should allow to impose the strong constraint on the corrections coming from
our model, as seen later.

The ratio between the observed value of light deflection angle and the
corresponding GR prediction has been found as Shapiro2004 ; Will2014

$\displaystyle\frac{\left(\Delta\varphi_{2}\right)_{\text{exp.}}}{\left(\Delta\varphi_{2}\right)_{\text{GR}}}\approx
1.0001\pm 0.0001.$ (127)

In solar system, the gravitational deflection of radio signals has been
measured using the Very Long Baseline Interferometry (VLBI) Robertson1991 .
With the numerical values, $M=M_{\odot}\approx 1.474$ km and $b\approx
6.96\times 10^{5}$ km, we find the constraint for $\alpha/L$ as

$\displaystyle 0\lesssim\frac{\alpha}{L}\lesssim 1.27\times 10^{-4}.$ (128)

Assume that the velocity of the massive particles and the photon along the
vertical direction or the direction of the internal space is the same order,
we combine the experimental bounds given by Eqs. (123) and (128), which is
shown in Fig. 5 for various values of the angular momentum $L$ of the photon.
We observe that the constraint of the light bending angle becomes more
strongly with the decreasing of the angular momentum of the photon. When the
angular momentum of the photon is low enough compared to the specific angular
momentum of the massive particles $L=\sqrt{M_{\odot}\hat{r}_{+}}$, the
constraint of the light bending angle is more stringent than that of the
perihelion shift. On the contrary, the allowed parameter region in the
$c_{\theta}-Q_{X}$ plane is determined by the constraint coming from the
perihelion shift.

---
Figure 5: The allowed parameter region (white region) for the charge $Q_{X}$
of the Sun and and the velocity $c_{\theta}$ of the particles along the
direction of the internal space. The red horizontal line and orange curve
correspond to the constraint (128), whereas, the blue curve corresponds to the
constraint (123). Top-left panel: $L=\sqrt{M_{\odot}\hat{r}_{+}}\approx
9032.44$ km. Top-right panel: $L=0.1\sqrt{M_{\odot}\hat{r}_{+}}$. Bottom-left
panel: $L=0.01\sqrt{M_{\odot}\hat{r}_{+}}$. Botom-right panel:
$L=0.001\sqrt{M_{\odot}\hat{r}_{+}}$.

###  IV.4 Strong gravitational lensing

In this subsection, we study the impact of spacetime $\mathcal{M}_{5}$ on the
observables of the strong gravitational lensing whose diagram is given in Fig.
6.

---
Figure 6: The diagram of the gravitational lensing.

The gravitational lens is considered in this work to be the supermassive black
hole (SMBH) of Sagittarius A∗ at the center of our galaxy (Milky Way) and the
galaxies which are Andromeda and Messier 87 (or Virgo A). The observable data
about the mass of Sagittarius A∗ SMBH and Andromeda and Messier 87 galaxies
and the distance (denoted $D_{OL}$) between them and Earth (the observer) are
given in Table 2.

Gravitational lens | Mass $(M_{\odot})$ |  $D_{OL}$ (Mpc)
---|---|---
Sagittarius A∗ Ghez2008  | $\ \ \ \ \ \ 4.1\times 10^{6}\ \ \ \ \ \ $ | $\ \ \ \ \ \ 8\times 10^{-3}\ \ \ \ \ \ $
Andromeda Kafle2018  | $\ \ \ \ \ \ 8.0\times 10^{11}\ \ \ \ \ \ $ | $\ \ \ \ \ \ 0.78\ \ \ \ \ \ $
Messier 87 Murphy2011  | $\ \ \ \ \ \ 5.7\times 10^{12}\ \ \ \ \ \ $ | $\ \ \ \ \ \ 16.4\ \ \ \ \ \ $
Table 2: The mass of the Sagittarius A∗ SMBH and the Andromeda and Messier
galaxies and the distance from them to the Earth.

The deflection angle of light as a function of the closest approach distance
$r_{0}$ is rewritten as follows

$\displaystyle\Delta\varphi_{2}(r_{0})=I(r_{0})-\pi,$ (129)

where $I(r_{0})$ is given as

$\displaystyle
I(r_{0})=\int^{\infty}_{r_{0}}\frac{2\sqrt{B(r)}dr}{\sqrt{C(r)}\sqrt{\frac{C(r)}{C_{0}}\frac{A_{0}}{A(r)}-1}},$
(130)

with

$\displaystyle A(r)$ $\displaystyle=$
$\displaystyle\left(1-\frac{2M}{r}\right)\left(1+\frac{c^{2}_{\theta}r^{2}}{L^{2}}\right)\left(1+\frac{\alpha
b}{Lr}\right)^{-2},$ $\displaystyle B(r)$ $\displaystyle=$
$\displaystyle\frac{1}{A(r)}\left(1+\frac{\alpha b}{Lr}\right)^{-2},$
$\displaystyle C(r)$ $\displaystyle=$ $\displaystyle r^{2},$ (131)

and $A_{0}\equiv A(r_{0})$, $C_{0}\equiv C(r_{0})$. The deflection angle of
light becomes large when the closest approach distance $r_{0}$ decreases. When
$r_{0}$ reaches the radius of the photon sphere, the deflection angle of light
is divergent (which is logarithmic in nature) or in other words the light is
captured by the supermassive body. In the strong field limit, the closest
approach distance $r_{0}$ of light is very close to the radius of the photon
sphere. This means that in this situation one can expand the deflection angle
of light around the radius of the photon sphere as Bozza2002

$\displaystyle\Delta\varphi_{2}(\theta)=-\bar{a}\log\left(\frac{\theta
D_{OL}}{b_{m}}-1\right)+\bar{b}+\mathcal{O}(r_{0}-r_{ph}),$ (132)

where $\theta$ is the incident angle to the observer, $\bar{a}$ and $\bar{b}$
are the coefficients of the strong gravitational lensing, and $b_{m}$ is the
minimal impact parameter corresponding to the photon sphere. In order to
calculate the coefficients $\bar{a}$ and $\bar{b}$, we follow the method
developed by Bozza Bozza2002 . They are given as

$\displaystyle\bar{a}$ $\displaystyle=$
$\displaystyle\frac{R(0,r_{ph})}{2\sqrt{\beta_{ph}}},$ $\displaystyle\bar{b}$
$\displaystyle=$
$\displaystyle-\pi+b_{R}+\bar{a}\log\frac{2\beta_{ph}}{A_{ph}},$ (133)

where

$\displaystyle\beta_{ph}$ $\displaystyle=$
$\displaystyle\frac{C_{ph}(1-A_{ph})^{2}\left[C^{\prime\prime}(r_{ph})A_{ph}-C_{ph}A^{\prime\prime}(r_{ph})\right]}{2A^{2}_{ph}C^{\prime}(r_{ph})^{2}},$
$\displaystyle b_{R}$ $\displaystyle=$
$\displaystyle\int^{1}_{0}\left[R(z,r_{ph})f(z,r_{ph})-R(0,r_{ph})f_{0}(z,r_{ph})\right]dz,$
(134)

with $C_{ph}\equiv C(r_{ph})$ and $A_{ph}\equiv A(r_{ph})$. In Eqs. (133) and
(134), the functions $R(z,r_{0})$, $f(z,r_{0})$ are defined as follows

$\displaystyle R(z,r_{0})$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{BA}}{CA^{\prime}}(1-A_{0})\sqrt{C_{0}},$
$\displaystyle f(z,r_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{A_{0}-[(1-A_{0})z+A_{0}]\frac{C_{0}}{C}}},$ (135)

with $z\equiv[A(r)-A_{0}]/[1-A_{0}]$ and the function $f_{0}(z,r_{0})$ is the
expansion of the argument of the square root in $f(z,r_{0})$ to the second
order in $z$ and given as

$\displaystyle f_{0}(z,r_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\alpha z+\beta z^{2}}},$ (136)

where

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{1-A_{0}}{C_{0}A^{\prime}(r_{0})}\left[C^{\prime}(r_{0})A_{0}-C_{0}A^{\prime}(r_{0})\right],$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{(1-A_{0})^{2}}{2C^{2}_{0}A^{\prime}(r_{0})^{3}}\left[2C_{0}C^{\prime}(r_{0})A^{\prime}(r_{0})^{2}+(C_{0}C^{\prime\prime}(r_{0})-2C^{\prime}(r_{0})^{2})A_{0}A^{\prime}(r_{0})\right.$
(137)
$\displaystyle\left.-C_{0}C^{\prime}(r_{0})A_{0}A^{\prime\prime}(r_{0})\right].$

Now we study the impact of spacetime $\mathcal{M}_{5}$ on the observational
quantities of the strong gravitational lensing. The angular separation between
the outermost and innermost images and the ratio of the magnification of the
outermost image to the others can be approximated in the situation that the
lens, source, and observer are highly aligned as Bozza2002

$\displaystyle s$ $\displaystyle=$
$\displaystyle\theta_{\infty}e^{\frac{\bar{b}-2\pi}{\bar{a}}},$
$\displaystyle\mathcal{R}$ $\displaystyle=$ $\displaystyle
2.5\log_{10}\left(e^{\frac{2\pi}{\bar{a}}}\right),$ (138)

where $\theta_{\infty}$ represents the asymptotic angular position where all
images except the outermost one are packed together and can expressed as
$\theta_{\infty}=b_{m}/D_{OL}$. Note that, the expression in the logarithm
function, i.e. $e^{\frac{2\pi}{\bar{a}}}$, is the ratio of the flux of the
outermost image to the others.

In Tables 3, 4, and 5, we show the numerical estimation of the observables of
the strong gravitational lensing to the subleading order approximation of
$c_{\theta}$ for the realistic cases which are given in Table 2. Here, the
predictions by GR and spacetime $\mathcal{M}_{5}$ correspond to $\alpha/L=0$
and $\alpha/L\neq 0$, respectively. From these tables, we observe that the
$\bar{a}$ and $\bar{b}$ coefficients and the relative magnification
$\mathcal{R}$ are well-approximately independent on where the light comes from
but only depend on the parameter $\alpha/L$. In addition, we find that the
angular position $\theta_{\infty}$ of the innermost image and the angular
separation $s$ between the outermost and innermost images grow with the
increasing of the parameter $\alpha/L$. This means that the size of Einstein
ring becomes bigger with the growth of the parameter $\alpha/L$. Whereas, the
relative magnification $\mathcal{R}$ decreases with the increasing of the
parameter $\alpha/L$. With the variation of the observables of the strong
gravitational lensing, namely $\theta_{\infty}$, $s$, and $\mathcal{R}$, in
the parameter $\alpha/L$, by measuring these quantities and comparing to the
predictions of GR, we can determine the corrections of spacetime $M_{5}$ on
the observables of the strong gravitational lensing. This could allow us to
detect the existence of spacetime $M_{5}$ at the more fundamental level.

$\ \ \ \ \frac{\alpha}{L}\times 10^{4}\ \ \ \ $ | $\ \ \ \ \bar{a}\ \ \ \ $ | $\ \ \ \ \bar{b}\ \ \ \ $ |  $\ \ \theta_{\infty}$ ($\mu$as) |  $\ \ s$ ($\mu$as) | $\ \ \ \ \mathcal{R}\ \ \ \ $
---|---|---|---|---|---
$0$ | $1$ | $\ \ -0.40023\ \ $ | $26.239$ | $0.032838$ | $6.82188$
$0.3$ | $\ \ 1.00002\ \ $ | $\ \ -0.400226\ \ $ | $26.2404$ | $0.0328437$ | $6.82176$
$0.6$ | $1.00003$ | $\ \ -0.400223\ \ $ | $26.2417$ | $0.0328493$ | $6.82165$
$0.9$ | $1.00005$ | $\ \ -0.400219\ \ $ | $26.2431$ | $0.0328549$ | $6.82153$
$1.2$ | $1.00007$ | $\ \ -0.400216\ \ $ | $26.2445$ | $0.0328605$ | $6.82141$
Table 3: The numerical estimation of the observables of the strong gravitational lensing for the case of Sagittarius A∗ SMBH. $\ \ \ \ \frac{\alpha}{L}\times 10^{4}\ \ \ \ $ | $\ \ \ \ \bar{a}\ \ \ \ $ | $\ \ \ \ \bar{b}\ \ \ \ $ |  $\ \ \theta_{\infty}$ (mas) |  $\ \ s$ (mas) | $\ \ \ \ \mathcal{R}\ \ \ \ $
---|---|---|---|---|---
$0$ | $1$ | $\ \ -0.40023\ \ $ | $52.5108$ | $0.0657171$ | $6.82188$
$0.3$ | $\ \ 1.00002\ \ $ | $\ \ -0.400226\ \ $ | $52.5136$ | $0.0657284$ | $6.82176$
$0.6$ | $1.00003$ | $\ \ -0.400223\ \ $ | $52.5163$ | $0.0657396$ | $6.82165$
$0.9$ | $1.00005$ | $\ \ -0.400219\ \ $ | $52.519$ | $0.0657509$ | $6.82153$
$1.2$ | $1.00007$ | $\ \ -0.400216\ \ $ | $52.5218$ | $0.0657622$ | $6.82141$
Table 4: The numerical estimation of the observables of the strong gravitational lensing for the case of Andromeda galaxy. $\ \ \ \ \frac{\alpha}{L}\times 10^{4}\ \ \ \ $ | $\ \ \ \ \bar{a}\ \ \ \ $ | $\ \ \ \ \bar{b}\ \ \ \ $ |  $\ \ \theta_{\infty}$ (mas) |  $\ \ s$ (mas) | $\ \ \ \ \mathcal{R}\ \ \ \ $
---|---|---|---|---|---
$0$ | $1$ | $\ \ -0.40023\ \ $ | $17.7945$ | $0.0222697$ | $6.82188$
$0.3$ | $\ \ 1.00002\ \ $ | $\ \ -0.400226\ \ $ | $17.7954$ | $0.0222735$ | $6.82176$
$0.6$ | $1.00003$ | $\ \ -0.400223\ \ $ | $17.7963$ | $0.0222773$ | $6.82165$
$0.9$ | $1.00005$ | $\ \ -0.400219\ \ $ | $17.7972$ | $0.0222811$ | $6.82153$
$1.2$ | $1.00007$ | $\ \ -0.400216\ \ $ | $17.7982$ | $0.022285$ | $6.82141$
Table 5: The numerical estimation of the observables of the strong
gravitational lensing for the case of Messier 87 galaxy.

###  IV.5 Retrolensing

In this subsection, we study the influence of spacetime $\mathcal{M}_{5}$ on
retrolensing of light rays emitted by the Sun, which is another situation of
strong gravitational lensing. The gravitational lens in the ordinary situation
of strong gravitational lensing is situated between the observer and the light
source. However, if one consider the light of the Sun which is reflected by a
black hole with photon sphere and then reaches the observer which is placed
between the gravitational lens and the light source, it can give rise the
images with the deflection angle closer to $(2n+1)\pi$ with $n$ to be
nonnegative integers. In this way, the black hole with photon sphere can
reflect the light ray like a mirror and is usually so-called retrolens.

In order investigate retrolensing, we introduce the Ohanian lens equation
given as Ohanian1987 ; Bozza2004

$\displaystyle\beta=\pi-\bar{\alpha}(\theta)+\theta+\bar{\theta},$ (139)

where the angle $\beta$ is defined as $\beta=\angle OLS$ (with $O$, $L$, $S$
referring to the observer, retrolens, and the Sun, respectively),
$\bar{\alpha}(\theta)=\Delta\varphi_{2}(\theta)-2\pi n$ is the effective
deflection angle with $n$ denoting the winding number of light ray around
retrolens before reaching the observer, and $\bar{\theta}$ denotes the angle
at $S$ between the light ray emitted from the Sun and the line $LS$. Ignoring
the small terms $\theta$ and $\bar{\theta}$, the positive solution of this
equation is approximately obtained as

$\displaystyle\theta_{+}(\beta)=\frac{b_{m}}{D_{OL}}\left[1+e^{\frac{\bar{b}-(2n+1)\pi+\beta}{\bar{a}}}\right].$
(140)

In addition, retrolens would produce the images in pairs due to the spherical
symmetry, called the retrolensing double image, which corresponds to the
occurrence of primary and secondary images on the opposite sides of retrolens.
On the other hand, there is a negative solution
$\theta_{-}(\beta)=-\theta_{+}(-\beta)$ of the lens equation. The separation
of retrolensing double image is given by

$\displaystyle\theta_{+}-\theta_{-}\simeq
2\theta_{+}=\frac{2b_{m}}{D_{OL}}\left[1+e^{\frac{\bar{b}-(2n+1)\pi+\beta}{\bar{a}}}\right],$
(141)

where the approximation appears when the Sun, observer, retrolens are aligned
to be near perfectly, i.e. $\beta\approx 0$. We define a quantity
$\Delta(\theta_{+}-\theta_{-})$ which is the relative difference between the
double image separation $\theta_{+}-\theta_{-}$ predicted by our scenario and
that predicted by GR. In Fig. 7, we show the relative difference
$\Delta(\theta_{+}-\theta_{-})$ with Sagittarius A∗ and Messier 87∗ (which is
SMBH at the center of Messier 87 galaxy) as retrolens. Here, the retrolens
masses and the distances from Earth to retrolens for Sagittarius A∗ and
Messier 87∗ are given as $M\approx 4.1\times 10^{6}M_{\odot}$, $D_{OL}\approx
8\times 10^{-3}$ Mpc (see Table 2) and $M\approx 6.5\times 10^{9}M_{\odot}$,
$D_{OL}\approx 16.8$ Mpc ETH2019 , respectively.

---
Figure 7: The relative difference $\Delta(\theta_{+}-\theta_{-})$ as a
function of $\alpha$ for SMBHs of Sagittarius A∗ and Messier 87∗ as retrolens
and the winding number $n=0,1$.

From this figure, we find that the relative difference
$\Delta(\theta_{+}-\theta_{-})$ grows linearly with the increasing of the
parameter $\alpha$. The relative difference is of order $10^{-2}$ $\mu$as for
$\alpha\sim\mathcal{O}(10^{-4})$.

The magnification of the images determined by the image angles $\theta_{\pm}$
is given by Bozza2004

$\displaystyle\mu_{-}(\beta)\simeq-\mu_{+}(\beta)$ $\displaystyle=$
$\displaystyle\frac{D^{2}_{OS}}{D^{2}_{LS}}s(\beta)\theta_{+}\frac{d\theta_{+}}{d\beta}$
(142) $\displaystyle\simeq$
$\displaystyle\frac{D^{2}_{OS}}{D^{2}_{LS}}\frac{b^{2}_{m}}{D^{2}_{OL}}\frac{e^{\frac{\bar{b}-(2n+1)\pi}{\bar{a}}}}{\bar{a}}\left[1+e^{\frac{\bar{b}-(2n+1)\pi}{\bar{a}}}\right]s(\beta),$

where $s(\beta)=1/\beta$ is for the point source and for the fine-size
extension of source $s(\beta)$ is given by

$\displaystyle s(\beta)=\frac{D^{2}_{LS}}{\pi
R^{2}_{S}}\int_{D_{S}}d\beta^{\prime}d\phi,$ (143)

where $R_{S}$ is the radius of the Sun, $\beta^{\prime}$ and $\phi$ are the
reduced radial coordinate (rescaled by $1/D_{LS}$) and azimuthal coordinate on
the source plane, respectively, and $D_{S}$ is a uniform-luminous disk
describing the Sun on the source plane Witt1994 ; Nemiroff1994 ; Alcock1997 .
In general, the integral (143) can be expressed in the elliptic integrals of
first and second kind Bozza2004  or can be simplified by considering the
intersection point between the $\beta=0$ line (corresponding to the case of
perfect alignment) and the source plane as the origin of coordinates
Tsukamoto2017a ; Tsukamoto2017b . The total magnification of retrolensing
double image is found as

$\displaystyle\mu_{\text{tot}}(\beta)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\left[|\mu_{+}(\beta)|+|\mu_{-}(\beta)|\right]$
(144) $\displaystyle=$ $\displaystyle
2\frac{D^{2}_{OS}}{D^{2}_{LS}}\frac{b^{2}_{m}}{D^{2}_{OL}}\frac{e^{\frac{\bar{b}}{\bar{a}}}}{\bar{a}}\left[\frac{1}{2\sinh(\pi/\bar{a})}+\frac{e^{\frac{\bar{b}}{\bar{a}}}}{2\sinh(2\pi/\bar{a})}\right]|s(\beta)|.$

The maximal magnification $\mu_{\text{tot}}(0)$ corresponds to the situation
that the system of the Sun, observer, and retrolens is perfectly aligned with
$s(0)$ given by $s(0)=2D_{LS}/R_{S}$. The apparent magnitude is defined as
Geralico2003

$\displaystyle m(\beta)=m_{\odot}-2.5\log_{10}\mu_{\text{tot}}(\beta),$ (145)

where $m_{\odot}=-26.8$ is the optical visual magnitude of the solar
luminosity. The maximal apparent magnitude is determined by $m(0)$
corresponding to the maximal brightness of the image. In Fig. 8, we show the
relative difference $\Delta m(0)$ between the maximal apparent magnitude
predicted by our scenario and that predicted by GR.

---
Figure 8: The relative difference $\Delta m(0)$ as a function of $\alpha$.

Note that, $\Delta m(0)$ is almost independent on the retrolens mass,
$D_{OL}$, $D_{OS}$, and $D_{LS}$ due to the presence of logarithm function and
a fact that $\bar{a}$ and $\bar{b}$ are dependent only on the parameter
$\alpha$ at the first order of their expansion in $\alpha$. We observe here
that the relative difference $\Delta m(0)$ decreases monotonically as the
parameter $\alpha$ increases. This suggests that compared to the Schwarzschild
retrolens the maximal brightness of the image is highly amplified for the
region of negative $\alpha$ parameter.

##  V  Conclusion

It is possible that spacetime fundamentally has more than four dimensions and
possesses hidden topological/geometric structures. In this paper, we have
proposed five dimensional spacetime with a fiber fabric which is generally
nontrivial. This non-trivially topological structure of spacetime allows us to
resolve the hierarchy problem with the inverse size of the fifth dimension and
the fundamental Planck scale to be of the order of the TeV scale. The large
hierarchy between the observed Planck scale $\sim{O}(10^{18})$ GeV and the
electroweak scale $\sim{O}(1)$ TeV is due to the function of
$\cosh\left(\frac{\sqrt{10}}{3}\kappa_{1}\pi\right)$ where $\kappa_{1}$ is the
ratio of the root-square modulus of the bulk cosmological constant to the
inverse size of the fifth dimension. Here, this $\cosh$ function is physically
determined from the dynamics of the horizontal metric in the vacuum of non-
zero energy. The large hierarchy could be resolved with
$\kappa_{1}\sim\mathcal{O}(35)$. In addition, the fiber fabric of spacetime
allows us to resolve the problem of the chirality fermions in an elegant and
natural way, which usually appears in the higher dimension theories which are
the generalization of General Relativity (GR) to higher dimensions.

We have pointed out that the potential of the radion field in the present work
exists at the tree level and respects to the higher dimensional general
covariance. Therefore, the size of the fifth dimension or the radion mass is
physically determined. Interestingly, the radion field could be responsible
for a natural candidate of the inflaton. The effective action for the four
dimensional metric and the radion field obtained from the fundamental action
in five dimensions is naturally given in the so-called Jordan frame. In the
so-called Einstein frame obtained via a properly conformal transformation, the
potential for the redefined scalar field of the canonical kinetic term belongs
the slow-roll inflation model of E-model. We have calculated the main observed
quantities of the inflation which are the spectral index, the tensor-to-scalar
ratio, the running index, and the scalar perturbation amplitude $A_{s}$. With
the number of e-foldings about $\mathcal{O}(60)$, the inflationary observables
predicted by our model are consistent with the experimental observations.
Furthermore, our model predicts a sufficiently large tensor-to-scalar ratio
which is around $\sim 0.011$ and hence it can be tested in the near future.

Finally, we have obtained the geodesic equation which describes the motion of
neutral test particles in spacetime with the fiber fabric and tested it by
investigating the geodesic motion of massless and massive particles around the
central gravitational bodies. We have calculated the radius of the photon
sphere, the radius of the innermost stable circular orbit, the oscillating
point of the near-circular orbits, the perihelion shift, the light bending
angle in the weak field limit, and the observations of the strong
gravitational lensing and retrolensing (another situation of strong
gravitational lensing). These calculations exhibit the important differences
between GR and our spacetime about the qualitative and quantitative features
and therefore could allow us to distinguish it from other extensions of GR. We
have compared the computed results with the observations to determine the
constraints on the fiber fabric of spacetime.

## Acknowledgments

We would like to express sincere gratitude to two referees for their
constructive comments, suggestions and questions by which the quality of the
paper has been improved. This research is funded by Vietnam National
Foundation for Science and Technology Development (NAFOSTED) under grant
number 103.01-2019.353.

## Appendix

The coefficients of the Christoffel connection given in Eq. (13) are
explicitly expanded as

$\displaystyle\Gamma^{\rho}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{g^{\rho\lambda}}{2}(\hat{\partial}_{\mu}g_{\lambda\nu}+\hat{\partial}_{\nu}g_{\lambda\mu}-\hat{\partial}_{\lambda}g_{\mu\nu}).$
$\displaystyle\Gamma^{\theta}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{G^{\theta\theta}}{2}\partial_{\theta}g_{\mu\nu}+\frac{C^{\theta}_{\mu\nu}}{2}$
$\displaystyle=$
$\displaystyle\frac{\Lambda^{2}}{2T^{2}}\partial_{\theta}g_{\mu\nu}-\frac{g_{{}_{X}}}{2}X_{\mu\nu},$
$\displaystyle\Gamma^{\theta}_{\mu\theta}=\Gamma^{\theta}_{\theta\mu}$
$\displaystyle=$
$\displaystyle\frac{G^{\theta\theta}}{2}\hat{\partial}_{\mu}G_{\theta\theta}$
$\displaystyle=$ $\displaystyle-\frac{\hat{\partial}_{\mu}T}{T},$
$\displaystyle\Gamma^{\nu}_{\mu\theta}=\Gamma^{\nu}_{\theta\mu}$
$\displaystyle=$
$\displaystyle\frac{g^{\nu\lambda}}{2}\partial_{\theta}g_{\lambda\mu}+\frac{g_{{}_{X}}}{2\Lambda^{2}}T^{2}g^{\nu\lambda}X_{\lambda\mu},$
$\displaystyle\Gamma^{\mu}_{\theta\theta}$ $\displaystyle=$
$\displaystyle-\frac{g^{\mu\nu}}{2}\hat{\partial}_{\nu}G_{\theta\theta}$
$\displaystyle=$
$\displaystyle\frac{g^{\mu\nu}}{\Lambda^{2}}T\hat{\partial}_{\nu}T,$
$\displaystyle\Gamma^{\theta}_{\theta\theta}$ $\displaystyle=$
$\displaystyle\frac{G^{\theta\theta}}{2}\partial_{\theta}G_{\theta\theta}$
(146) $\displaystyle=$ $\displaystyle-\frac{\partial_{\theta}T}{T}$

The scalar curvature $\mathcal{R}^{(5)}$ of spacetime $\mathcal{M}_{5}$ is
explicitly given as

$\displaystyle\mathcal{R}^{(5)}=G^{MN}\mathcal{R}^{P}_{MPN}=g^{\mu\nu}\mathcal{R}^{P}_{\mu
P\nu}+G^{\theta\theta}\mathcal{R}^{P}_{\theta P\theta},$ (147)

where

$\displaystyle g^{\mu\nu}\mathcal{R}^{P}_{\mu P\nu}$ $\displaystyle=$
$\displaystyle
g^{\mu\nu}\left(\hat{\partial}_{\nu}\Gamma^{M}_{M\mu}-\hat{\partial}_{M}\Gamma^{M}_{\nu\mu}+\Gamma^{N}_{M\mu}\Gamma^{M}_{\nu
N}-\Gamma^{N}_{\nu\mu}\Gamma^{M}_{MN}+C^{N}_{M\nu}\Gamma^{M}_{N\mu}\right)$
(148) $\displaystyle=$ $\displaystyle
g^{\mu\nu}\left(\hat{\partial}_{\nu}\Gamma^{\lambda}_{\lambda\mu}-\hat{\partial}_{\lambda}\Gamma^{\lambda}_{\nu\mu}+\Gamma^{\rho}_{\lambda\mu}\Gamma^{\lambda}_{\nu\rho}-\Gamma^{\rho}_{\nu\mu}\Gamma^{\lambda}_{\lambda\rho}\right)+g^{\mu\nu}\left(\hat{\partial}_{\nu}\Gamma^{\theta}_{\theta\mu}+\Gamma^{\rho}_{\theta\mu}\Gamma^{\theta}_{\nu\rho}+\Gamma^{\theta}_{\lambda\mu}\Gamma^{\lambda}_{\nu\theta}+\Gamma^{\theta}_{\theta\mu}\Gamma^{\theta}_{\nu\theta}\right)$
$\displaystyle-g^{\mu\nu}\left(\partial_{\theta}\Gamma^{\theta}_{\nu\mu}-\Gamma^{\theta}_{\nu\mu}\Gamma^{\lambda}_{\lambda\theta}-\Gamma^{\rho}_{\nu\mu}\Gamma^{\theta}_{\theta\rho}-\Gamma^{\theta}_{\nu\mu}\Gamma^{\theta}_{\theta\theta}\right)+g^{\mu\nu}C^{\theta}_{\lambda\nu}\Gamma^{\lambda}_{\theta\mu},$
$\displaystyle G^{\theta\theta}\mathcal{R}^{P}_{\theta P\theta}$
$\displaystyle=$ $\displaystyle
G^{\theta\theta}\left(\partial_{\theta}\Gamma^{M}_{M\theta}-\partial_{M}\Gamma^{M}_{\theta\theta}+\Gamma^{N}_{M\theta}\Gamma^{M}_{\theta
N}-\Gamma^{N}_{\theta\theta}\Gamma^{M}_{MN}+C^{N}_{M\theta}\Gamma^{M}_{N\theta}\right)$
(149) $\displaystyle=$ $\displaystyle
G^{\theta\theta}\left(\partial_{\theta}\Gamma^{\mu}_{\mu\theta}-\hat{\partial}_{\mu}\Gamma^{\mu}_{\theta\theta}+\Gamma^{\nu}_{\mu\theta}\Gamma^{\mu}_{\theta\nu}+\Gamma^{\mu}_{\theta\theta}\Gamma^{\theta}_{\theta\mu}-\Gamma^{\mu}_{\theta\theta}\Gamma^{\nu}_{\nu\mu}-\Gamma^{\theta}_{\theta\theta}\Gamma^{\mu}_{\mu\theta}\right).$

In (148) and (149), we find the following combinations

$\displaystyle
g^{\mu\nu}\left(\hat{\partial}_{\nu}\Gamma^{\theta}_{\theta\mu}-\Gamma^{\rho}_{\nu\mu}\Gamma^{\theta}_{\theta\rho}\right)-G^{\theta\theta}\Gamma^{\mu}_{\theta\theta}\Gamma^{\theta}_{\theta\mu}$
$\displaystyle\equiv$ $\displaystyle\nabla_{M}Y^{M}_{1},$ $\displaystyle
g^{\mu\nu}\left(\partial_{\theta}\Gamma^{\theta}_{\nu\mu}+\Gamma^{\theta}_{\nu\mu}\Gamma^{\lambda}_{\lambda\theta}+\Gamma^{\theta}_{\nu\mu}\Gamma^{\theta}_{\theta\theta}\right)$
$\displaystyle\equiv$
$\displaystyle\nabla_{M}Y^{M}_{2}+\frac{G^{\theta\theta}}{2}\partial_{\theta}g^{\mu\nu}\partial_{\theta}g_{\mu\nu},$
$\displaystyle
G^{\theta\theta}\left(\partial_{\theta}\Gamma^{\mu}_{\mu\theta}-\Gamma^{\theta}_{\theta\theta}\Gamma^{\mu}_{\mu\theta}\right)-g^{\mu\nu}\Gamma^{\theta}_{\mu\nu}\Gamma^{\lambda}_{\lambda\theta}$
$\displaystyle\equiv$ $\displaystyle\nabla_{M}Y^{M}_{3},$ $\displaystyle
G^{\theta\theta}\left(\hat{\partial}_{\mu}\Gamma^{\mu}_{\theta\theta}+\Gamma^{\mu}_{\theta\theta}\Gamma^{\nu}_{\nu\mu}+\Gamma^{\mu}_{\theta\theta}\Gamma^{\theta}_{\theta\mu}\right)$
$\displaystyle\equiv$
$\displaystyle\nabla_{M}Y^{M}_{4}+\frac{g^{\mu\nu}}{2}\hat{\partial}_{\mu}G^{\theta\theta}\hat{\partial}_{\nu}G_{\theta\theta},$
(150)

where

$\displaystyle Y^{M}_{1}$ $\displaystyle\equiv$
$\displaystyle\left(g^{\mu\nu}\Gamma^{\theta}_{\theta\mu},0\right),$
$\displaystyle Y^{M}_{2}$ $\displaystyle\equiv$
$\displaystyle\left(0,g^{\mu\nu}\Gamma^{\theta}_{\nu\mu}\right),$
$\displaystyle Y^{M}_{3}$ $\displaystyle\equiv$
$\displaystyle\left(0,G^{\theta\theta}\Gamma^{\mu}_{\mu\theta}\right),$
$\displaystyle Y^{M}_{4}$ $\displaystyle\equiv$
$\displaystyle\left(G^{\theta\theta}\Gamma^{\mu}_{\theta\theta},0\right).$
(151)

Note that, the vectors $Y^{M}_{1}$ and $Y^{M}_{4}$ have the zero vertical
components, whereas the vectors $Y^{M}_{2}$ and $Y^{M}_{3}$ have the zero
horizontal components. The terms $\nabla_{M}Y^{M}_{i}$ with $i=1,2,3,4$ are
divergences which correspond to the boundary terms and hence vanish at
infinity.

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