††institutetext: Bülent Ecevit University,
Physics Department, 67100, Zonguldak, Turkey

# A black hole solution of higher-dimensional Weyl-Yang-Kaluza-Klein theory by
the Wu-Yang ansatz

Halil Kuyrukcu<EMAIL_ADDRESS>
###### Abstract

We consider the Weyl-Yang gauge theory of gravitation in a $(4+3)$-dimensional
curved space-time within the scenario of the non-Abelian Kaluza-Klein theory
for the source and torsion-free limits. The explicit forms of the field
equations containing a new spin current term and the energy-momentum tensors
in the usual four dimensions are obtained by the well-known dimensional
reduction procedure. In this limit, these field equations admit (anti)-dyon
and magnetic (anti)-monopole solutions as well as a non-Einsteinian solution
in the presence of a generalized Wu-Yang ansatz and some specific scalar
potentials when the extra dimensions associated with the round and squashed
three-sphere $S^{3}$ are, respectively, included. The (anti)-dyonic solution
has similar properties to those of the Reissner-Nordström-de Sitter black
holes of the Einstein-Yang-Mills system. The cosmological constant naturally
appears in this approach, and that it associates with the constant dilaton
field as well as the three-sphere radius. It is demonstrated that the
squashing parameter $\ell$ not only behaves as the constant charge, but also
its sign can determine whether the solution is a dyon/monopole or an anti-
dyon/anti-monopole. It is also shown that whether the source term reduces to
Camenzind’s current density, and that the existence of non-constant dilaton
fields is essential for finding new black hole solutions in the Weyl-Yang
gauge theory.

###### Keywords:

Modified Theories of Gravity, Kaluza-Klein Theories, Exact Solutions,
Classical Black Holes

##  1 Introduction

The Weyl-Yang alternative gravitational theory with field equations containing
second derivatives of the affine connection and third derivatives of the
metric tensor obtained by the standard Palatini formalism palatini ; misner ;
Tsamparlis ; ferraris  with vanishing torsion was first considered by
Fairchild in the remarkable studies fairchild76 ; fairchild77 . The work was
inspired by early works conducted by Weyl weyl , Lanczos lanczos , Utiyama
utiyama , Lichnerowicz lichnerowicz1 , Stephenson stephenson , Higgs higgs ,
Kilmister and Newman kilmister , Loos Loos , Loos and Treat Loos1 , and Yang
yang . The theory was investigated and developed further by many different
authors such as Ramaswamy and Yasskin Ramaswamy , Tseytlin tseytlin ,
Szczyrbac Szczyrba , Hehl et al. hehl89 , Maluf Maluf91 , Rose Rose , and
Guilfoyle and Nolan Guilfoyle , and more recently by Vassiliev Vassiliev ,
Mielke and Maggiolo Maggiolo , Shen Shen09 , and Pasic and Barakovic Pasic .
It was also extended$/$generalized to the Abelian$/$non-Abelian Kaluza-Klein
theories of unified gravity Kaluza ; Klein ; Mandel ; Appelquist ; Bailin ;
Duff94 ; Overduin  by Başkal and Kuyrukcu$/$Kuyrukcu halil13 $/$halil14 , and
its ability to solve various cosmological problems was discussed by Gerard
gerard , Cook cook , Gonzalez gonzalez , Chen chen , Chen et al. chen1 , and
Yang and Yeung yeung . This Yang-Mills-type theory actually may also be
thought of as a particular case of the Poincaré Gauge theory hehl76 ; hehl78 ;
Goenner ; Garecki85 ; Maluf ; Sardanashvily ; hehl95 ; Gronwald ; Gronwald97 ;
Hehl99 ; daum ; Blagojevic13  “that can be considered as the standard theory
of gravity with torsion” according to Hehl et al. hehl13  (for more historical
notes, see e.g., Dean Dean ).

It is not surprising that the Weyl-Yang theory overcomes certain problems with
the linearized field equations aragone , but it also suffers from various
difficulties such a non-acceptable Newtonian limit for weak-field
approximation fairchild76 , the non-existence of a Birkhoff theorem
Blagojevic13 , uncertainty concerning the source of matter fairchild76 ;
fairchild77 ; Camenzind , and a negative residue in the graviton propagator
leeneeman . Nevertheless, this alternative gravitational theory still deserves
further investigation and exploration as it has some interesting features as a
model theory, although the Einstein field equations are sufficient for the
large scale of gravitational physics.

The most important and motivating feature of the theory lies in the fact that
the third-order Einsteinian equations are physically more contained, i.e.,
they possess a much richer structure than their the second-order counterparts
within the well-established gravity theory of Einstein yang . The master
equation of the theory can be written as

$\displaystyle D{\star{\mathcal{R}_{\mu\nu}}}\equiv
E_{\mu}\wedge{\star{D}}\mathcal{R}_{\nu}-E_{\nu}\wedge{\star{D}}\mathcal{R}_{\mu}=0.$
(1)

Note that, $\mathcal{R}_{\mu\nu}$, $\mathcal{R}_{\mu}$, $\star$, $E_{\mu}$ and
$D$ refer to the four-dimensional ($4$D) curvature 2-form, the Ricci 1-form
$\mathcal{R}_{\mu}=\iota_{X_{\lambda}}{\mathcal{R}}^{\lambda}\,_{\mu}$ where
$\iota_{X_{\lambda}}$ is the inner product, the Hodge star operator, the co-
frame 1-form and the covariant derivative, respectively. Thus, the pure
gravitational equations of motion (1) not only include all of the classes of
the solutions to Einstein’s vacuum theory (i.e., $\mathcal{R}_{\mu}=0$ and
$\mathcal{R}_{\mu}=\Lambda E_{\mu}$ for all constants $\Lambda$) but also have
a physical and un-physical family of non-Einsteinian solutions with or without
a spin current term, see e.g., pavelle2 ; gerard . Another physically
attractive property of the considered model is the corresponding gravitational
energy-momentum tensor, $\tau_{X_{\lambda}}$, which is symmetric, trace-free,
and covariantly conservative

$\displaystyle\tau_{X_{\lambda}}\equiv\frac{1}{2}(\iota_{X_{\lambda}}\mathcal{R}_{\mu\nu}\wedge{\star{\mathcal{R}^{\mu\nu}}}-\mathcal{R}_{\mu\nu}\wedge\iota_{X_{\lambda}}{\star{\mathcal{R}^{\mu\nu}}}).$
(2)

This tensor is associated with the Bel-Robinson superenergy tensor bel ;
robinson ; mashhoon99 , and it has a formal analogy with Maxwell’s stress-
energy tensor of classical electrodynamics jackson99  as well as that of the
Yang-Mills gauge theory yangmills . Furthermore, the non-trivial Lagrangian of
the theory is simple, purely quadratic, and homogeneous of degree $2$ in
curvature, and it only includes the Kretschmann invariant Kretschmann .
Therefore, it may be thought as the most natural model in the literature by
Weyl weyl . We can write the 4D Weyl-Yang action with the coupling constant
$\kappa$ as:

$\displaystyle
S=\int_{M_{4}}{\mathcal{L}}=-\frac{1}{2\kappa^{2}}\int_{M_{4}}\mathcal{R}_{\mu\nu}\wedge{\star{\mathcal{R}^{\mu\nu}}}.$
(3)

This curvature squared term also appears in higher-derivative gravity theories
such as superstring theory Candelas ; Gross85 ; Gross86 , quantum gravity
lausher2002  and loop quantum cosmology cognola2008 , which are perturbatively
renormalized Utiyama1 ; stelle1 ; Tomboulis1  just as the Yang-Mills theories
are Slavnov ; Hooft , as well as asymptotically free Tomboulis2 ; Fradkin  and
capable of ameliorating ghost modes Salam1 ; Stelle2 ; Antoniadis .

Despite its higher-order curvature-squared action and its system of third-
order coupled non-linear field equations, various exact solutions to the
theory have been obtained and discussed in the works of several authors. These
include the static spherically symmetric solutions of Pavelle pavelle1 ;
pavelle2 ; pavelle3 ; pavelle4 , Thompson thompson , Barrent et al. Barrent ,
Baekler et al. baekler1 ; baekler2 ; baekler , and Hsu and Yeung Hsu  (see
also Ni ni ); the double-dual solutions of Benn et al. benn81 ; the anti-self-
dual solutions of Mielke mielke81 ; and the pp-wave solutions of Başkal baskal
and Kuyrukcu kuyrukcu  (also see Pavelle pavelle3 ). Thus, it is quite natural
to discover exact solutions especially in more than $4$D Weyl-Yang theory
because of the fact that higher-dimensional models are already a standard
assumption in high energy physics. Although this technique offers promising
new features, to find exact solutions to the higher-dimensional models is much
more difficult Emparan . In this scheme, we study static spherically symmetric
solutions to the non-Abelian Weyl-Yang-Kaluza-Klein gravity model halil14
(which is the straightforward full generalization of the previous work halil13
to higher, $N$, dimensions) in the presence of a well-known generalized (not
rotating but both electrically- and magnetically-charged) $SU(2)$ Wu-Yang
monopole wuyang ; wuyang75 ; Actor ; carmeli  and the classical $(4+3)$D non-
Abelian Kaluza-Klein metric ansatz, without loss of generality. The Wu-Yang
monopoles (actually Wu-Yang solutions Chu ) may be interpreted as non-Abelian
versions of the Dirac magnetic monopole dirac  (for a review, see e.g.,
Goddard ; Shnir ), and they are a very important gauge class for quantum
chromodynamics with instanton-like configurations Konishi . It should also be
emphasized that the magnetic monopole solutions (i.e., topological solitons
Preskill ) together with the dyon, instanton, and meron solutions play very
significant roles, not only in $4$D Einstein-Yang-Mills-Higgs (EYMH) theories
Bais ; Cho ; Yaskin ; Volkov  but also in (non)-Abelian Kaluza-Klein theories
Einstein ; Witt ; Witt1 ; sorkin ; grossperry ; Lee1 ; Lee2 ; Perry ; Angus ;
Cotaescu , since they show explicitly the interaction between the
gravitational, gauge, and dilatonic fields, although they are not
experimentally observed. On the other hand, despite being the simplest
acceptable non-Abelian models Arık , the $6$D and $7$D theories are not as
physically attractive as the $5$D Abelian and $11$D supergravity and
superstring models. However, the considered model not only contains both the
graviton and the gauge boson together with scalars but also the three-sphere,
$S^{3}$, which being a compact $3$D maximally symmetric internal space also
has advantages for $SU(2)$ gauge theories Lunev . Moreover, the three-sphere
is simple, non-trivial, and the best-known $3$D manifold along with pseudo-
sphere, and it possesses more symmetry than the other space Okada86 .

The outline of our article is as follows. In section 2, the Weyl-Yang version
of the non-Abelian Kaluza-Klein theory is formulated. We investigate the
reduced structures of the field equations, which include the new spin current
term in terms of various combinations of $2$-form field strength as well as
Yang-Mills force density, and the stress-energy-momentum tensor, which
consists of the well-known and lesser-known $4$D energy-momentum tensors of
many different fields, including the Weyl-Yang and Yang-Mills theories in the
usual $4$D space-time without considering the dilaton fields, i.e.,
$\varphi(r)=1$. Moreover, we show that Camenzind’s current density appears in
the reduced field equations of the 5D Weyl-Yang theory if we consider the
specific assumptions. Section 3 is concerned with constructing exact solutions
to the third-order Weyl-Yang gravity model by taking into account the $7$D
classical non-Abelian Kaluza-Klein metric ansatz whose $4$D part is
spherically symmetric but stationary in time and whose $3$D part is the
standard round and squashed three-sphere, $S^{3}$, coupled with the non-
vanishing dilaton scalar fields, respectively. Then, different analytic
solutions are obtained. One of them is the non-trivial (anti)-dyonic black
hole, which carries both constant (one unit or $\ell$) electric and magnetic
charges and has similar properties to those of the Reissner-Nordström-de
Sitter-type (RNdS) black holes of the Einstein-Yang-Mills system. It is
demonstrated that the cosmological constant is just an integration constant
that comes, interestingly, out of the constant dilaton field and the principal
internal radii, which guarantee that its unit is the inverse of the square of
a length. This resulting solution is also compared with the $SO(3)$ magnetic
monopole solutions of the $6$D Perry  and $7$D Angus  Einsteinian models,
respectively. The other solutions that we found can be classified as non-
Einsteinian solutions for the $J(r)\neq 0$ case and new magnetic
(anti)-monopole solutions for the $J(r)=0$ case. It seems that obtaining a
dyon/monopole or an anti-dyon/anti-monopole solution depends on the sign of
the internal space squashing parameter, $\ell$. In this fashion, we also
obtain the energy-momentum components which are given in detail corresponding
to these exact solutions. It is also shown that the existence of non-constant
dilaton fields is essential for finding new black hole solutions in the Weyl-
Yang gauge theory. We finally present our conclusions, remarks, and future
plans in the last section.

Throughout this study, we will employ the Maurer-Cartan exterior forms (which
are compact-coordinate-free notations), together with geometric units (i.e.,
an unspecified length scale in which $c=1$, $\hbar=1$, and $8\pi G=1$) with
unit coupling constants. The differential forms are known to be very useful
for defining the basis of non-trivial $S^{3}$ manifolds and finding exact
solutions, rather than a coordinate basis, if there are symmetry conditions on
the metric tensor.

##  2 Gravitational field equations and stress-energy-momentum tensor

By taking into account a particular generalization of Einstein’s gravity
theory, the general exterior form of the Stephenson-Kilmister-Yang-Fairchild
(SKYF) action, which is quadratic in the Riemann-Christofell components
proportional to the real coupling constant, $\hat{\kappa}$, on the $(4+3)$D
space-time manifold, $(M_{7},\hat{G})$ may be simply defined as

$\displaystyle\hat{S}[\hat{E},\hat{\Gamma}]=\int_{M_{7}}\hat{\mathcal{L}}=\int_{M_{7}}-\frac{1}{2\hat{\kappa}^{2}}\hat{\mathcal{R}}_{AB}\wedge{\hat{\star}{\hat{\mathcal{R}}^{AB}}}+\hat{\Lambda}{\hat{\star}\mathbf{1}}+\hat{\mathcal{L}}_{m}[\hat{\Gamma}],$
(4)

which includes a cosmological constant, $\hat{\Lambda}$, and a matter-
radiation field, $\hat{\mathcal{L}}_{m}[\hat{\Gamma}]$, which is coupled to
the affine connection. The quantities in the $7$D whole space (ordinary $4$D
external space) are denoted by the symbols with (without) the hat symbol as a
superscript. The direct product of the usual $4$D Minkowski space-time and a
$3$D internal space gives the manifold, $M_{7}$, i.e., $M_{7}=M_{4}\times
M_{3}$. Here, the curved sub-space $M_{3}$ is taken to be the usual round
three-sphere $M_{3}=\mathcal{G}/H=S^{3}$ with a radius $l=1$, and therefore,
the isometry group of $S^{3}$ is defined by a special unitary group of degree
$2$ as $\mathcal{G}=SU(2)\times SU(2)\simeq SO(4)$. We conveniently describe
the $7$D Hodge dual operator $\hat{\star}$, which is a linear map,
$\hat{\star}:\mho_{p}(M_{7})\rightarrow\mho_{7-p}(M_{7})$ with respect to the
Loretzian metric tensor
$\hat{G}=\hat{\eta}_{AB}\hat{E}^{A}\otimes\hat{E}^{B}$, where
$\hat{\eta}_{AB}=$diag$(-1,+1,+1,+1,+1,+1,+1)$, $\mho_{p}(M_{7})$ is the
vector space of p-forms, $\hat{E}^{A}$ are the orthonormal basis, $\otimes$
denotes the symmetric tensor product, and the Latin capital vielbein indices
$A,B,...=0,1,2,3,5,6,7$ refer to the entire space, $M_{7}$. The invariant
tensor $\hat{\eta}^{AB}$/ $\hat{\eta}_{AB}$ is used to raise/lower the indices
appearing on $7$D quantities. Moreover, $\hat{\star}\mathbf{1}$ is the
oriented volume $7$-form as
$\hat{\star}\mathbf{1}={\star\mathbf{1}}\wedge\\#\mathbf{1}$, where
${\star\mathbf{1}}=\hat{E}^{0}\wedge\hat{E}^{1}\wedge\hat{E}^{2}\wedge\hat{E}^{3}$
is the usual $4$-form invariant volume element with a $4$D Hodge map,
$\star:\mho_{p}(M_{4})\rightarrow\mho_{4-p}(M_{4})$, whereas
$\\#\mathbf{1}=\hat{E}^{5}\wedge\hat{E}^{6}\wedge\hat{E}^{7}$ is the $3$-form
invariant volume element with a $3$D Hodge map,
$\\#:\mho_{p}(S^{3})\rightarrow\mho_{3-p}(S^{3})$ with the wedge product
$\wedge$. The anti-symmetric Levi-Civita tensor is also chosen to be
$\varepsilon_{0123567}=+1$ in this study.

Let us now investigate the coupled gravitational field equations by taking
into account the first-order Einstein-Palatini variation formalism which
assumes that two fundamental variables the $1$-form co-frames (vielbein
fields), $\hat{E}^{A}$, and the Levi-Civita connections,
$\hat{\Gamma}^{A}\,_{B}$ are independent each other, and “they are invariant
under both space-time diffeomorphisms and local frame rotations,” as noted by
Daum and Reuter daum . Hence, we have

$\delta\hat{S}[\hat{E},\hat{\Gamma}]=\int_{M_{7}}\delta\hat{E}^{A}\wedge\frac{\delta\hat{\mathcal{L}}}{\delta\hat{E}^{A}}+\delta\hat{\Gamma}^{AB}\wedge\frac{\delta\hat{\mathcal{L}}}{\delta\hat{\Gamma}^{AB}}.$
(5)

Thus, we evaluate the infinitesimal variation of the generalized SKYF
Lagrangian density, $\hat{\mathcal{L}}$, in equation (4) by assuming that
variations of the fields vanish on the boundary as follows:

$\displaystyle\delta\hat{S}[\hat{E},\hat{\Gamma}]$ $\displaystyle=$
$\displaystyle\int_{M_{7}}\frac{1}{2\hat{\kappa}^{2}}\delta\hat{E}^{A}\wedge\Bigl{(}\hat{\iota}_{\hat{X}_{A}}\hat{\mathcal{R}}_{BC}\wedge{\hat{\star}{\hat{\mathcal{R}}^{BC}}}-\hat{\mathcal{R}}_{BC}\wedge\hat{\iota}_{\hat{X}_{A}}{\hat{\star}{\hat{\mathcal{R}}^{BC}}}\Bigl{)}+\hat{\Lambda}\delta\hat{E}^{A}\wedge{\hat{\star}\hat{E}_{A}}-$
(6)
$\displaystyle-\frac{1}{\hat{\kappa}^{2}}\delta\hat{\Gamma}^{AB}\wedge\hat{D}{\hat{\star}{\hat{\mathcal{R}}_{AB}}}+\delta\hat{\Gamma}^{AB}\wedge{{\hat{S}_{BA}}},$

with inner multiplication,
$\hat{\iota}_{\hat{X}_{A}}:\mho_{p}(M_{7})\rightarrow\mho_{p-1}(M_{7})$. Here
the canonical term $\hat{S}_{AB}$ is the spin-density of all matter fields as
a complementary term of the theory, which is determined by the variation of
the matter Lagrangian $\hat{\mathcal{L}}_{m}$ with respect to the gauge
potentials
$\hat{S}_{BA}\equiv\delta\hat{\mathcal{L}}_{m}/\delta\hat{\Gamma}_{AB}$, and
$\hat{D}\equiv\hat{E}^{A}\hat{D}_{\hat{X}_{A}}$ stands for the covariant
exterior derivative. The variation principle $\delta\hat{S}=0$ gives the
desired two sets of field equations. If the cosmological constant term
$\hat{\Lambda}$ vanishes, then the first of these equations is the generalized
Yang equation with source term

$\displaystyle\hat{D}{\hat{\star}{\hat{\mathcal{R}}_{AB}}}=-\hat{\kappa}^{2}\hat{\star}{\hat{S}_{AB}}.$
(7)

This relationship is the master equation of the considered model with
gravitational current 1-form (${\hat{S}_{AB}}=\hat{S}_{ABC}\hat{E}^{C}$),
which are antisymmetric under interchange of $A$ and $B$ indices, as expected.
It is easy to verify from equation (7) and the gravitational Bianchi
identities, the first $\hat{\mathcal{R}}_{AB}\wedge\hat{E}^{B}=0$ and the
second $\hat{D}\hat{\mathcal{R}}_{AB}=0$, that the gauge current term
precisely satisfies the conservation property
$\hat{D}{\hat{\star}{\hat{S}_{AB}}}=0$ and the cyclic identity
$\hat{\iota}_{\hat{X}^{B}}\hat{\iota}_{\hat{X}^{A}}\hat{\star}{\hat{S}_{AB}}=0$,
or more simply, $\hat{S}_{AB}\wedge\hat{E}^{A}\wedge\hat{E}^{B}=0$. On the
other hand, the other fundamental field equations that are obtained from the
co-frame variation lead to

$\displaystyle\hat{\tau}_{\hat{X}_{A}}\equiv\frac{1}{2}(\hat{\iota}_{\hat{X}_{A}}\hat{\mathcal{R}}_{BC}\wedge{\hat{\star}{\hat{\mathcal{R}}^{BC}}}-\hat{\mathcal{R}}_{BC}\wedge\hat{\iota}_{\hat{X}_{A}}{\hat{\star}{\hat{\mathcal{R}}^{BC}}}),$
(8)

where ${\hat{\tau}_{\hat{X}_{A}}}=\hat{T}_{AB}{\hat{\star}\hat{E}^{B}}$ is
interpreted as the canonical stress-energy-momentum tensor, which is
determined by
$\hat{\tau}_{\hat{X}_{A}}\equiv\hat{\kappa}^{2}(\delta\hat{\mathcal{L}}/\delta\hat{E}^{A})$.
By taking the covariant exterior derivative of both sides of the energy
definition (8), we obtain

$\displaystyle\hat{D}\hat{\tau}_{\hat{X}_{A}}=-\hat{\iota}_{\hat{X}_{A}}\hat{\mathcal{R}}_{BC}\wedge{\hat{\star}{\hat{S}^{BC}}}.$
(9)

It can be seen above that $\hat{\tau}_{\hat{X}_{A}}$ is covariantly
conservative, provided that ${\hat{S}_{BC}}=0$. One can also derive that the
trace of the stress-energy tensor $\hat{\tau}_{\hat{X}_{A}}\wedge\hat{E}^{A}$
on the manifold $(M_{7},\hat{G})$ turns to be

$\displaystyle\hat{\tau}_{\hat{X}_{A}}\wedge\hat{E}^{A}=-\frac{3}{2}\hat{\mathcal{R}}_{BC}\wedge{\hat{\star}{\hat{\mathcal{R}}^{BC}}}.$
(10)

This is completely trace-free only in the $4$D limit.

Let us now obtain reduced forms of the field equations and stress-energy
tensors in the usual 4D space-time by making use of a dimensionality reduction
procedure. First, we assume that the gravitational source terms vanish, and
that the fields are independent of the extra-coordinates in higher dimensions.
It is convenient to also write the (4+3)D metric tensor $\hat{G}(x,y)$ in the
form:

$\displaystyle\hat{G}(x,y)=G(x)+\delta_{ij}(A^{i}\otimes A^{j}+A^{i}\otimes
E^{j}+E^{i}\otimes A^{j})+G(y).$ (11)

We introduce 7D coordinates as $z^{A}=(x^{\mu},y^{i})$, where (and henceforth
in this work), the Greek vierbein indices $\mu,\nu,...=0,1,2,3$ refer to the
$4$D external space $M_{4}$ with the Minkowskian signature metric,
$G(x)=\eta_{\mu\nu}E^{\mu}(x)\otimes E^{\nu}(x)$, while the Latin dreibein
indices $i,j,...=5,6,7$ refer to the curved internal $3$D compact sub-space
$S^{3}$ with the Euclidean signature metric $G(y)=\delta_{ij}E^{i}(y)\otimes
E^{j}(y)$ and $\delta_{ij}=$diag$(+1,+1,+1)$. In addition, Yang-Mills gauge
field $1$-form can be defined as $A^{i}=A^{i}(x)=A^{i}_{\mu}(x)E^{\mu}(x)$.

On account of the full Lorentzian metric tensor
$\hat{G}(x,y)=\hat{\eta}_{AB}\hat{E}^{A}\otimes\hat{E}^{B}$ and equation (11),
the appropriate orthonormal co-frames take the simplified forms

$\displaystyle\hat{E}^{\mu}(x)=E^{\mu}(x),\qquad\qquad\qquad\hat{E}^{i}(x,y)=A^{i}(x)+E^{i}(y).$
(12)

It follows that we immediately obtain the components of interior product
operator $\hat{\iota}_{\hat{X}_{A}}$ as

$\displaystyle\hat{\iota}_{\hat{X}_{\mu}}(x,y)=\iota_{X_{\mu}}-A^{i}_{\mu}(x)\iota_{X_{i}},\qquad\qquad\qquad\hat{\iota}_{\hat{X}_{i}}(y)=\iota_{X_{i}}.$
(13)

Here, $\\{\hat{X}_{A}\\}$ are frame fields dual to the co-frames
$\\{\hat{E}^{A}\\}$, together with
$\hat{E}^{B}(\hat{X}_{A})=\hat{\iota}_{\hat{X}_{A}}\hat{E}^{B}=\hat{\delta}_{A}\,^{B}$,
and $\iota_{X_{\mu}}$/$\iota_{X_{i}}$ are the $4$D/$3$D contraction operators,
which are dual to co-frames $E^{\mu}$/$E^{i}$, i.e.,
$\iota_{X_{\mu}}E^{\nu}=\delta_{\mu}\,^{\nu}$/$\iota_{X_{i}}E^{j}=\delta_{i}\,^{j}$.
It is worthwhile to recall that, the first Cartan-Maurer structure equation is
represented by

$\displaystyle\hat{D}\hat{E}^{A}\equiv\hat{d}\hat{E}^{A}+\hat{\Gamma}^{A}\,_{B}\wedge\hat{E}^{B}=\hat{T}^{A},$
(14)

with the seven-exterior derivative
$\hat{d}\equiv\hat{E}^{A}\hat{d}_{\hat{X}_{A}}$. Equation (14) can be solved
for the constraint equation
$\hat{T}^{A}\equiv\frac{1}{2}\hat{T}_{BC}\,^{A}\hat{E}^{B}\wedge\hat{E}^{C}=0$,
as

$\displaystyle\hat{\Gamma}_{AB}=\frac{1}{2}[\hat{\iota}_{\hat{X}_{B}}\hat{d}\hat{E}_{A}-\hat{\iota}_{\hat{X}_{A}}\hat{d}\hat{E}_{B}+(\hat{\iota}_{\hat{X}_{A}}\hat{\iota}_{\hat{X}_{B}}\hat{d}\hat{E}_{C})\hat{E}^{C}],$
(15)

where $\hat{\Gamma}_{AB}$ are the $1$-form Levi-Civita connections, i.e.,
$\hat{\Gamma}_{(AB)}=0$, which means that the metric compatibility condition
$\hat{D}\hat{G}(x,y)=0$ is valid for the torsion-free case benntucker . Hence,
substituting equations (12) and (13) into the equation (15), we directly
obtain the following components of the spin connection $\hat{\Gamma}_{AB}$:

$\displaystyle\hat{\Gamma}^{\mu}\,_{\nu}$ $\displaystyle=$
$\displaystyle\Gamma^{\mu}\,_{\nu}-\frac{1}{2}F^{i\mu}\,_{\nu}\hat{E}_{i},$
$\displaystyle\hat{\Gamma}_{\mu}\,^{j}$ $\displaystyle=$
$\displaystyle-\hat{\Gamma}^{j}\,_{\mu}=-\frac{1}{2}F^{j}_{\mu\nu}E^{\nu},$
(16) $\displaystyle\hat{\Gamma}^{i}\,_{j}$ $\displaystyle=$
$\displaystyle\Gamma^{i}\,_{j}-\frac{1}{2}\varepsilon^{i}\,_{jk}A^{k}.$

By assuming that the $T_{i}$ are the generators of the group
$SU(\mathcal{N})$, the corresponding Lie algebra is usually defined as
$[T_{i},T_{j}]=f^{k}\,_{ij}T_{k}$ with real structure constants $f^{k}\,_{ij}$
and $T_{i}=\frac{1}{2}\sigma_{i}$, where $\sigma_{i}$ are the Pauli matrices.
We also have $f^{k}\,_{ij}=\varepsilon^{k}\,_{ij}$ for the gauge group $SU(2)$
and the little group $SO(3)$. Hence, we can prove that the co-frames of
internal $S^{3}$ space satisfy the right-invariant structure equations (see
e.g., Bailin ; Lee83 )

$\displaystyle dE^{i}(y)=\frac{1}{2}\varepsilon^{i}\,_{jk}E^{j}(y)\wedge
E^{k}(y),$ (17)

meaning that $\Gamma^{i}\,_{j}=\frac{1}{2}\varepsilon^{i}\,_{jk}E^{k}$.
Moreover, the $SU(2)$-valued $2$-form field strength is explicitly determined
by $F^{i}\equiv dA^{i}+\frac{1}{2}e\varepsilon^{i}\,_{jk}A^{j}\wedge
A^{k}=\frac{1}{2}F^{i}_{\mu\nu}E^{\mu}\wedge E^{\nu}$. Here, $e$ is nothing
but the coupling constant, and for convenience, it can be taken as unity,
$e=1$.

We now consider the second Maurer-Cartan structure equation,

$\displaystyle\hat{D}\hat{\Gamma}^{A}\,_{B}\equiv\hat{d}\hat{\Gamma}^{A}\,_{B}+\hat{\Gamma}^{A}\,_{C}\wedge\hat{\Gamma}^{C}\,_{B}=\hat{\mathcal{R}}^{A}\,_{B},$
(18)

to obtain the dimensionally-reduced curvature 2-form
$\hat{\mathcal{R}}^{A}\,_{B}$, where
$\hat{\mathcal{R}}^{A}\,_{B}\equiv\frac{1}{2}\hat{R}^{A}\,_{BCD}\hat{E}^{C}\wedge\hat{E}^{D}$.
Substituting gauge potential $1$-form (2) into equation (18) and performing
some algebra, the first component, $\hat{\mathcal{R}}_{\mu\nu}$, becomes (see,
e.g., dereli90 )

$\displaystyle\hat{\mathcal{R}}^{\mu}\,_{\nu}=\Psi^{\mu}\,_{\nu}+\Omega^{\mu}\,_{\nu}+\Sigma^{\mu}\,_{\nu},$
(19)

where the $2$-form terms are

$\displaystyle\Psi^{\mu}\,_{\nu}$ $\displaystyle=$
$\displaystyle\mathcal{R}^{\mu}\,_{\nu}-\frac{1}{2}F^{i\mu}\,_{\nu}F_{i}-\frac{1}{4}\iota_{X^{\mu}}F^{i}\wedge\iota_{X_{\nu}}F_{i},$
$\displaystyle\Omega^{\mu}\,_{\nu}$ $\displaystyle\equiv$
$\displaystyle\Omega^{i\mu}\,_{\nu}\wedge\hat{E}_{i}=-\frac{1}{2}\mathcal{D}F^{i\mu}\,_{\nu}\wedge\hat{E}_{i},$
(20) $\displaystyle\Sigma^{\mu}\,_{\nu}$ $\displaystyle\equiv$
$\displaystyle\Sigma^{ij\mu}\,_{\nu}\hat{E}_{i}\wedge\hat{E}_{j}=\frac{1}{4}(F^{i\mu\lambda}F^{j}_{\lambda\nu}-\varepsilon^{ij}\,_{k}F^{k\mu}\,_{\nu})\hat{E}_{i}\wedge\hat{E}_{j}.$

Here, we also make use of the ordinary curvature 2-form,
$\mathcal{R}_{\mu\nu}$, of the external space, $M_{4}$, and the Yang-Mills
covariant derivative,
$\mathcal{D}F^{i}_{\mu\nu}=DF^{i}_{\mu\nu}+e\varepsilon^{i}\,_{jk}A^{j}F^{k}_{\mu\nu}$
in the adjoint representation. It follows that the remaining non-vanishing
components can be written as

$\displaystyle\hat{\mathcal{R}}_{\mu}\,^{j}$ $\displaystyle=$
$\displaystyle-\hat{\mathcal{R}}^{j}\,_{\mu}=\Omega^{j}\,_{\mu\nu}\wedge
E^{\nu}-\Sigma^{ij}\,_{\mu\nu}E^{\nu}\wedge\hat{E}_{i},$
$\displaystyle\hat{\mathcal{R}}^{i}\,_{j}$ $\displaystyle=$
$\displaystyle\Sigma^{i}\,_{j\mu\nu}E^{\mu}\wedge
E^{\nu}+\frac{1}{4}\hat{E}^{i}\wedge\hat{E}_{j}.$ (21)

It is not difficult to conjecture that, in order to find the desired field
equations and stress-energy tensors, we need not only the curvature 2-form
(19)–(21) but also the reduced forms of various Hodge duality identities,
which are found to be

$\displaystyle{\hat{\star}{\hat{E}^{\mu}}}={\star{E^{\mu}}}\wedge\\#1,\qquad\qquad\qquad{\hat{\star}{\hat{E}^{i}}}=\\#E^{i}\wedge{\star
1},$
$\displaystyle{\hat{\star}({\hat{E}^{\mu}}\wedge{\hat{E}^{\nu}})}={\star({E^{\mu}}\wedge{E^{\nu}})}\wedge\\#1,$
$\displaystyle{\hat{\star}({\hat{E}^{\mu}}\wedge{\hat{E}^{j}})}=-{\hat{\star}({\hat{E}^{j}}\wedge{\hat{E}^{\mu}})}=-{\star{E^{\mu}}}\wedge\\#E^{j},$
$\displaystyle{\hat{\star}({\hat{E}^{i}}\wedge{\hat{E}^{j}})}=\\#({E^{i}}\wedge{E^{j}})\wedge{\star
1}.$ (22)

###  2.1 The reduced field equations

Taking into account the definition of a covariant exterior derivative, the
second gravitational Bianchi identity is manifestly written as

$\displaystyle\hat{D}{{\hat{\mathcal{R}}^{A}\,_{B}}}=\hat{d}{{\hat{\mathcal{R}}^{A}\,_{B}}}+\hat{\Gamma}^{A}\,_{C}\wedge{{\hat{\mathcal{R}}^{C}\,_{B}}}-\hat{\Gamma}^{C}\,_{B}\wedge{{\hat{\mathcal{R}}^{A}\,_{C}}}=0,$
(23)

enabling us to arrive at the source-free field equations (7) in the following
form:

$\displaystyle\hat{D}{\hat{\star}{\hat{\mathcal{R}}^{A}\,_{B}}}=\hat{d}{\hat{\star}{\hat{\mathcal{R}}^{A}\,_{B}}}+\hat{\Gamma}^{A}\,_{C}\wedge{\hat{\star}{\hat{\mathcal{R}}^{C}\,_{B}}}-\hat{\Gamma}^{C}\,_{B}\wedge{\hat{\star}{\hat{\mathcal{R}}^{A}\,_{C}}}=0.$
(24)

After some lengthy but tedious calculations and remembering that the
antisymmetry property of the wedge product gives some extra terms
(${\hat{E}^{A}}\wedge{\hat{E}^{B}}\equiv{\hat{E}^{A}}\otimes{\hat{E}^{B}}-{\hat{E}^{B}}\otimes{\hat{E}^{A}}$),
we obtain six reduced equations including the gravitational and the gauge
fields, which are determined by three field equations,
$\hat{D}{\hat{\star}{\hat{\mathcal{R}}^{\mu}\,_{v}}}=0$,
$\hat{D}{\hat{\star}{\hat{\mathcal{R}}^{\mu}\,_{j}}}=0$, and
$\hat{D}{\hat{\star}{\hat{\mathcal{R}}^{i}\,_{j}}}=0$. Their final forms are
explicitly written in a Cartesian chart as:

$\displaystyle
D_{\lambda}{{R^{\lambda}\,_{\sigma\mu\nu}}}=\frac{1}{2}\big{[}F_{i\mu\nu}J^{i}_{\sigma}+\frac{1}{2}(F_{i\sigma\nu}J^{i}_{\mu}-F_{i\sigma\mu}J^{i}_{\nu})+\mathcal{D}_{\nu}(F_{i\mu}\,^{\lambda}F^{i}_{\lambda\sigma})-\mathcal{D}_{\mu}(F_{i\nu}\,^{\lambda}F^{i}_{\lambda\sigma})\big{]},$
(25)
$\displaystyle\mathcal{D}_{\lambda}\mathcal{D}^{\lambda}F^{i}_{\mu\nu}+R_{\mu\nu\lambda\rho}F^{i\lambda\rho}+F^{i}_{\mu\nu}-\frac{1}{2}(F^{i}_{\lambda\rho}F^{j\lambda\rho}F_{j\mu\nu}-F^{i}_{\mu\lambda}F^{j\lambda\rho}F_{j\rho\nu}+F^{i}_{\nu\lambda}F^{j\lambda\rho}F_{j\rho\mu})+$
$\displaystyle+\varepsilon^{i}\,_{kj}(F^{k}\,_{\mu\lambda}F^{j\lambda}\,_{\nu}-F^{j}\,_{\mu\lambda}F^{k\lambda}\,_{\nu})=0,$
(26)
$\displaystyle\mathcal{D}_{\lambda}\mathcal{D}_{\mu}F^{i\lambda}\,_{\sigma}+R_{\mu\lambda\sigma\rho}F^{i\lambda\rho}+\frac{1}{2}F^{i}_{\mu\sigma}-\frac{1}{4}(F^{i}_{\lambda\rho}F^{j\lambda\rho}F_{j\mu\sigma}+2F^{i}_{\sigma\lambda}F^{j\lambda\rho}F_{j\rho\mu})-$
$\displaystyle-\varepsilon^{i}\,_{jk}F^{j}_{\mu\lambda}F^{k\lambda}\,_{\sigma}=0,$
(27)
$\displaystyle\mathcal{D}_{\mu}(F^{i}_{\lambda\rho}F^{j\lambda\rho})+F^{i}_{\mu}\,{}^{\lambda}J^{j}_{\lambda}-\varepsilon^{ij}\,_{k}J^{k}_{\mu}=0,$
(28) $\displaystyle
F^{j\lambda}\,_{\mu}J_{\lambda}^{i}-F^{i\lambda}\,_{\mu}J_{\lambda}^{j}-2\varepsilon^{ij}\,_{k}J^{k}_{\mu}=0,$
(29)
$\displaystyle\varepsilon^{ik}\,_{l}F^{j}_{\lambda\rho}F^{l\lambda\rho}-\varepsilon^{jk}\,_{l}F^{i}_{\lambda\rho}F^{l\lambda\rho}+2\varepsilon^{ij}\,_{l}F^{k}_{\lambda\rho}F^{l\lambda\rho}=0,$
(30)

where $J^{i}_{\nu}=\mathcal{D}_{\mu}F^{i\mu}\,_{\nu}$ is the Yang-Mills
current $1$-form (i.e., $\mathcal{D}\star{F^{i}}=\star{J^{i}}$) and the right
side of equation (25) can be described as a 4D source term of the matter field
as

$\displaystyle
S_{\mu\nu\sigma}=\frac{1}{2}\big{[}F_{i\mu\nu}J^{i}_{\sigma}+\frac{1}{2}(F_{i\sigma\nu}J^{i}_{\mu}-F_{i\sigma\mu}J^{i}_{\nu})+\mathcal{D}_{\nu}(F_{i\mu}\,^{\lambda}F^{i}_{\lambda\sigma})-\mathcal{D}_{\mu}(F_{i\nu}\,^{\lambda}F^{i}_{\lambda\sigma})\big{]}.$
(31)

By taking into account the equation (1) and the following identity

$\displaystyle[\mathcal{D}_{\mu},\mathcal{D}_{\nu}]F^{i\lambda}\,_{\sigma}=R^{\lambda}\,_{\tau\mu\nu}F^{i\tau}\,_{\sigma}-R^{\tau}\,_{\sigma\mu\nu}F^{i\lambda}\,_{\tau}+\varepsilon^{i}\,_{jk}F^{j}_{\mu\nu}F^{k\lambda}\,_{\sigma},$
(32)

the reduced field equations above can be written in terms of the reduced
components of the $1$-form Ricci tensor ${\hat{\mathcal{R}}}_{A}$,
${\hat{\mathcal{R}}}_{A}=\hat{\iota}_{\hat{X}_{B}}{\hat{\mathcal{R}}}^{B}\,_{A}=\hat{R}_{AB}\hat{E}^{B}$,
as

$\displaystyle\mathcal{D}_{\mu}\mathcal{P}_{\sigma\nu}-\mathcal{D}_{\nu}\mathcal{P}_{\sigma\mu}+\frac{1}{4}(F_{i\sigma\mu}\mathcal{Q}^{i}_{\nu}-F_{i\sigma\nu}\mathcal{Q}^{i}_{\mu}-2F_{i\mu\nu}\mathcal{Q}^{i}_{\sigma})$
$\displaystyle=$ $\displaystyle 0,$ (33)
$\displaystyle\mathcal{D}_{\mu}\mathcal{Q}^{i}_{\nu}-\mathcal{D}_{\nu}\mathcal{Q}^{i}_{\mu}+F^{i\lambda}\,_{\nu}\mathcal{P}_{\lambda\mu}-F^{i\lambda}\,_{\mu}\mathcal{P}_{\lambda\nu}+\frac{1}{2}F_{j\mu\nu}\mathcal{U}^{ij}$
$\displaystyle=$ $\displaystyle 0,$ (34)
$\displaystyle\mathcal{D}_{\mu}\mathcal{Q}^{i}_{\sigma}+F^{i\lambda}\,_{\sigma}\mathcal{P}_{\lambda\mu}+\frac{1}{4}F_{j\sigma\mu}\mathcal{U}^{ij}$
$\displaystyle=$ $\displaystyle 0,$ (35)
$\displaystyle\mathcal{D}_{\mu}\mathcal{U}^{ij}+F^{i}_{\mu}\,{}^{\lambda}\mathcal{Q}^{j}_{\lambda}-\varepsilon^{ij}\,_{k}\mathcal{Q}^{k}_{\mu}$
$\displaystyle=$ $\displaystyle 0,$ (36) $\displaystyle
F^{j\lambda}\,_{\mu}\mathcal{Q}_{\lambda}^{i}-F^{i\lambda}\,_{\mu}\mathcal{Q}_{\lambda}^{j}-2\varepsilon^{ij}\,_{k}\mathcal{Q}^{k}_{\mu}$
$\displaystyle=$ $\displaystyle 0,$ (37)
$\displaystyle\varepsilon^{ik}\,_{l}\mathcal{U}^{jl}-\varepsilon^{jk}\,_{l}\mathcal{U}^{il}+2\varepsilon^{ij}\,_{l}\mathcal{U}^{kl}$
$\displaystyle=$ $\displaystyle 0,$ (38)

where the non-Abelian
$\\{\mathcal{P}_{\mu\nu},\mathcal{Q}^{i}_{\nu},\mathcal{U}^{ij}\\}$-set
becomes the following:

$\displaystyle\mathcal{P}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu}-\frac{1}{2}F^{i}_{\mu\lambda}F_{i\nu}\,^{\lambda},$
$\displaystyle\mathcal{Q}^{i}_{\nu}$ $\displaystyle=$
$\displaystyle\mathcal{D}_{\mu}F^{i\mu}\,_{\nu},$ (39)
$\displaystyle\mathcal{U}^{ij}$ $\displaystyle=$ $\displaystyle
F^{i}_{\mu\nu}F^{j\mu\nu}+\varepsilon^{i}\,_{kl}\varepsilon^{jkl},$

with $\varepsilon^{i}\,_{kl}\varepsilon^{jkl}=2\delta^{ij}$ for the 3D
internal space. One can see that equations (25)–(30) (and (33)–(38)) possess a
richer structure than those found in Einstein’s theory (2.1) since these
equations naturally contain Einsteinian solutions

$\displaystyle
R_{\mu\nu}=\frac{1}{2}F^{i}_{\mu\lambda}F_{i\nu}\,^{\lambda},\qquad\quad\mathcal{D}_{\mu}F^{i\mu}\,_{\nu}=0,\qquad\quad
F^{i}_{\mu\nu}F^{j\mu\nu}=2\delta^{ij},$ (40)

and it is for the following reasons. For instance, from equations (28) and
(29), we have

$\displaystyle f^{(ij)}_{\mu}=-2\mathcal{D}_{\mu}(F^{i}\wedge\star{F^{j}}),$
(41)

which means that the Lorentz force density $4$-form, $f^{ij}_{\mu}$
($f^{i}_{\mu i}=-\iota_{X_{\mu}}F^{i}\wedge\star{J_{i}}$), which usually
appears in the geodesic equations of non-Abelian Kaluza-Klein theories (see
e.g., kerner ; orzalesi ) are equal to the negative gradient of the
$F^{i}\wedge\star{F^{j}}$ term, which is a well-known invariant of non-Abelian
theory halil13 ; halil14 . Moreover, the usual $4$D continuity equation,
$\mathcal{D}\star{J^{i}}=0$, is easily obtained by help of the equations (2.1)
and (2.1).

####  2.1.1 Camenzind’s current density

By taking into account the 4D second gravitational Bianchi identity
$D\mathcal{R}_{\mu\nu}=0$ and the Einstein field equation
$\mathbb{G}_{\mu}\equiv\star{\mathcal{R}_{\mu}}-\frac{1}{2}\star{E_{\mu}}\mathcal{R}=\tau_{X_{\mu}}$
where the 4D Ricci scalar is
${\mathcal{R}}=\iota_{X_{\nu}}\iota_{X_{\mu}}{\mathcal{R}}^{\mu\nu}$,
Camenzind has defined the current term $\mathcal{S}_{\mu\nu\sigma}$ to clarify
the source-term problem of the theory fairchild76 ; fairchild77 ; Camenzind
without using the variational principle as follows:

$\displaystyle
D_{\lambda}{{R^{\lambda}\,_{\sigma\mu\nu}}}\equiv\mathcal{S}_{\mu\nu\sigma}=D_{\nu}(T_{\sigma\mu}-\frac{1}{2}g_{\sigma\mu}T)-D_{\mu}(T_{\sigma\nu}-\frac{1}{2}g_{\sigma\nu}T),$
(42)

where $T_{\sigma\mu}=\iota_{X_{\mu}}\star{\tau_{X_{\sigma}}}$ is the usual 4D
stress-energy tensor whose trace is $T$, i.e., $T=T_{\lambda}\,^{\lambda}$ cam
; cam1  (see also cook ). In this subsection, we try to show whether the
current density term (31) can be written in the form of equation (42). In that
respect, it is more convenient to consider the Abelian counterparts of the
equations (33)–(38) (and of the (4+N)D non-Abelian Weyl-Yang-Kaluza-Klein
gravity model halil14 ) because the non-Abelian Kaluza-Klein theories have a
well-known inconsistency problem for the reduced components of the Einstein
field equation $\hat{\mathbb{G}}_{A}=0$ Ellicott . Indeed, we have already
obtained the dimensionally reduced field equations of the 5D Weyl-Yang-Kaluza-
Klein gravity model halil13  with scalar field $\varphi=\varphi(x)$ (and
$\partial_{\mu}\varphi=\varphi_{\mu}$) as

$\displaystyle
D_{\mu}\mathcal{P}_{\sigma\nu}-D_{\nu}\mathcal{P}_{\sigma\mu}+\frac{1}{4}(F_{\sigma\mu}\mathcal{Q}_{\nu}-F_{\sigma\nu}\mathcal{Q}_{\mu}+2F_{\nu\mu}\mathcal{Q}_{\sigma})$
$\displaystyle=$ $\displaystyle 0,$ (43) $\displaystyle
D_{\mu}\mathcal{Q}_{\nu}-D_{\nu}\mathcal{Q}_{\mu}+F_{\mu}\,^{\lambda}\mathcal{P}_{\lambda\nu}-F_{\nu}\,^{\lambda}\mathcal{P}_{\lambda\mu}+2\varphi^{-1}F_{\mu\nu}\mathcal{U}$
$\displaystyle=$ $\displaystyle 0,$ (44) $\displaystyle
D_{\mu}\mathcal{Q}_{\nu}+\varphi^{-1}(\varphi_{\nu}\mathcal{Q}_{\mu}+2\varphi_{\mu}\mathcal{Q}_{\nu})-F_{\nu}\,^{\lambda}\mathcal{P}_{\lambda\mu}+\varphi^{-1}F_{\mu\nu}\mathcal{U}$
$\displaystyle=$ $\displaystyle 0,$ (45) $\displaystyle
D_{\mu}\mathcal{U}-\frac{1}{4}\varphi^{3}F_{\mu}\,^{\lambda}\mathcal{Q}_{\lambda}+\varphi^{\lambda}\mathcal{P}_{\lambda\mu}$
$\displaystyle=$ $\displaystyle 0,$ (46)

where the Abelian
$\\{\mathcal{P}_{\mu\nu},\mathcal{Q}_{\nu},\mathcal{U}\\}$-set now reads

$\displaystyle\mathcal{P}_{\mu\nu}$ $\displaystyle=$ $\displaystyle
R_{\mu\nu}-\frac{1}{2}\varphi^{2}F_{\mu\lambda}F_{\nu}\,^{\lambda}-\varphi^{-1}D_{\mu}\varphi_{\nu},$
$\displaystyle\mathcal{Q}_{\nu}$ $\displaystyle=$ $\displaystyle
D_{\mu}F^{\mu}\,_{\nu}+3\varphi^{-1}\varphi_{\mu}F^{\mu}\,_{\nu},$ (47)
$\displaystyle\mathcal{U}$ $\displaystyle=$ $\displaystyle
D_{\mu}\varphi^{\mu}-\frac{1}{4}\varphi^{3}F_{\mu\nu}F^{\mu\nu}.$

After some manipulations, the equations (43)–(46) can be written as

$\displaystyle
D_{\mu}\widetilde{\mathcal{P}}_{\sigma\nu}-D_{\nu}\widetilde{\mathcal{P}}_{\sigma\mu}+\frac{1}{2}\varphi^{-1}(g_{\sigma\nu}D_{\mu}\mathcal{U}-g_{\sigma\mu}D_{\nu}\mathcal{U})+$
$\displaystyle+\frac{1}{4}(F_{\sigma\mu}\mathcal{Q}_{\nu}-F_{\sigma\nu}\mathcal{Q}_{\mu}+2F_{\nu\mu}\mathcal{Q}_{\sigma})$
$\displaystyle=$ $\displaystyle 0,$ (48) $\displaystyle
D_{\mu}\mathcal{Q}_{\nu}-D_{\nu}\mathcal{Q}_{\mu}+F_{\mu}\,^{\lambda}\widetilde{\mathcal{P}}_{\lambda\nu}-F_{\nu}\,^{\lambda}\widetilde{\mathcal{P}}_{\lambda\mu}+3\varphi^{-1}F_{\mu\nu}\mathcal{U}$
$\displaystyle=$ $\displaystyle 0,$ (49) $\displaystyle
D_{\mu}\mathcal{Q}_{\nu}+\varphi^{-1}(\varphi_{\nu}\mathcal{Q}_{\mu}+2\varphi_{\mu}\mathcal{Q}_{\nu})-F_{\nu}\,^{\lambda}\widetilde{\mathcal{P}}_{\lambda\mu}+\frac{3}{2}\varphi^{-1}F_{\mu\nu}\mathcal{U}$
$\displaystyle=$ $\displaystyle 0,$ (50) $\displaystyle
D_{\mu}\mathcal{U}-\frac{1}{4}\varphi^{3}F_{\mu}\,^{\lambda}\mathcal{Q}_{\lambda}+\varphi^{\lambda}\widetilde{\mathcal{P}}_{\lambda\mu}+\frac{1}{2}\varphi^{-1}\varphi_{\mu}\mathcal{U}$
$\displaystyle=$ $\displaystyle 0,$ (51)

where the new term $\widetilde{\mathcal{P}}_{\mu\nu}$ is

$\displaystyle\widetilde{\mathcal{P}}_{\mu\nu}$ $\displaystyle=$
$\displaystyle R_{\mu\nu}-(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T).$ (52)

Here the energy-momentum tensor $T_{\mu\nu}$, which is covariantly
conservative ($D_{\mu}T^{\mu}\,_{\nu}=0$), and the trace $T$ respectively
become

$\displaystyle T_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\varphi^{2}(F_{\mu\lambda}F_{\nu}\,^{\lambda}-\frac{1}{4}g_{\mu\nu}F_{\lambda\rho}F^{\lambda\rho})+\varphi^{-1}(D_{\mu}\varphi_{\nu}-g_{\mu\nu}D_{\lambda}\varphi^{\lambda}),$
(53) $\displaystyle T$ $\displaystyle=$
$\displaystyle-3\varphi^{-1}D_{\lambda}\varphi^{\lambda}.$ (54)

Hence, it is not difficult to conjecture that we can write the reduced forms
of the Weyl-Yang field equations in terms of not only the
$\\{\mathcal{P}_{\mu\nu},\mathcal{Q}_{\nu},\mathcal{U}\\}$-set in (43)–(46)
but also the
$\\{\widetilde{\mathcal{P}}_{\mu\nu},\mathcal{Q}_{\nu},\mathcal{U}\\}$-set in
(2.1.1)–(51), which are the empty space solutions of the ordinary Kaluza-Klein
gravity model wesson99 , as

$\displaystyle{\hat{\mathcal{R}}}_{A}$ $\displaystyle=$ $\displaystyle
0\Rightarrow\begin{cases}\hat{R}_{\mu\nu}=0\Rightarrow\mathcal{P}_{\mu\nu}=0,\\\
\hat{R}_{\mu 5}=0\Rightarrow\mathcal{Q}_{\mu}=0,\\\
\hat{R}_{55}=0\Rightarrow\mathcal{U}=0,\end{cases}$ (55)
$\displaystyle\hat{\mathbb{G}}_{A}$ $\displaystyle=$ $\displaystyle
0\Rightarrow\begin{cases}\hat{G}_{\mu\nu}=0\Rightarrow
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=T_{\mu\nu}\Rightarrow\widetilde{\mathcal{P}}_{\mu\nu}=0,\\\
\hat{G}_{\mu 5}=0\Rightarrow\mathcal{Q}_{\mu}=0,\\\
\hat{G}_{55}=0\Rightarrow\mathcal{U}=0.\end{cases}$ (56)

Therefore, it appears that equation (42) can be valid in the framework of the
5D Weyl-Yang-Kaluza-Klein model if we consider the specific assumptions for
the equations (2.1.1)–(51), where $T_{\mu\nu}$ is the energy-momentum tensor
that comes from Einstein’s description of gravity. For example, if we assume
that $D_{\mu}\mathcal{U}=0$ (but $\mathcal{U}\neq 0$) and
$\mathcal{Q}_{\mu}=0$, the field equations (2.1.1)–(51) are reduced to

$\displaystyle D_{\lambda}R^{\lambda}\,_{\sigma\mu\nu}$ $\displaystyle=$
$\displaystyle
D_{\nu}(T_{\sigma\mu}-\frac{1}{2}g_{\sigma\mu}T)-D_{\mu}(T_{\sigma\nu}-\frac{1}{2}g_{\sigma\nu}T),$
(57) $\displaystyle
F_{\nu}\,^{\lambda}\widetilde{\mathcal{P}}_{\lambda\mu}-\frac{3}{2}\varphi^{-1}F_{\mu\nu}\mathcal{U}$
$\displaystyle=$ $\displaystyle 0,$ (58)
$\displaystyle\varphi^{\lambda}\widetilde{\mathcal{P}}_{\lambda\mu}+\frac{1}{2}\varphi^{-1}\varphi_{\mu}\mathcal{U}$
$\displaystyle=$ $\displaystyle 0.$ (59)

###  2.2 The reduced stress-energy-momentum tensor

We can also obtain dimensionally reduced components of the energy-momentum
tensor of the matter fields $\hat{\tau}_{\hat{X}_{A}}$ by taking into account
equation (8) together with the equations (19)–(2). These equations are not
only rather lengthy and complicated but also the include the well-known Weyl-
Yang (WY) energy-momentum tensor

$\displaystyle\tau^{{WY}}_{X_{\mu}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\iota_{X_{\mu}}\mathcal{R}_{\nu\lambda}\wedge{\star{\mathcal{R}^{\nu\lambda}}}-\mathcal{R}_{\nu\lambda}\wedge\iota_{X_{\mu}}{\star{\mathcal{R}^{\nu\lambda}}}),$
(60)

and that of Yang-Mills (YM) theory

$\displaystyle\tau^{{YM}}_{X_{\mu}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\iota_{X_{\mu}}F_{i}\wedge{\star{F^{i}}}-F_{i}\wedge\iota_{X_{\mu}}{\star{F^{i}}}),$
(61)

as well as those of the various constituent fields, such as those which are
non-minimal, cubic, quartic etc. For completeness, we explicitly list here the
reduced components of the energy-momentum $6$-form of
$\hat{\tau}_{\hat{X}_{A}}$ by virtue of
${\hat{\tau}_{\hat{X}_{\mu}}}=\hat{T}_{\mu\nu}{\hat{\star}\hat{E}^{\nu}}+\hat{T}_{\mu
j}{\hat{\star}\hat{E}^{j}}$ and
${\hat{\tau}_{\hat{X}_{i}}}=\hat{T}_{i\nu}{\hat{\star}\hat{E}^{\nu}}+\hat{T}_{ij}{\hat{\star}\hat{E}^{j}}$
in the $(4+3)$ decomposition. We can also check by employing the components
below, whether the full energy-momentum tensor, $\hat{T}_{AB}$, satisfies the
conservation property $\hat{D}_{A}\hat{T}^{A}\,_{B}=0$ or not. Then, direct
calculations give the first component of the energy tensor as

$\displaystyle\hat{T}_{\mu\nu}=R_{\mu\lambda\sigma\rho}R_{\nu}\,^{\lambda\sigma\rho}-\frac{1}{4}\eta_{\mu\nu}R_{\tau\lambda\sigma\rho}R^{\tau\lambda\sigma\rho}+\frac{3}{2}(F_{i\mu\lambda}F^{i}_{\nu}\,{}^{\lambda}-\frac{1}{4}\eta_{\mu\nu}F_{i\tau\lambda}F^{i\tau\lambda})+$
$\displaystyle+\frac{3}{2}F^{i\rho\sigma}[F_{i(\mu}\,^{\lambda}R_{\nu)\lambda\sigma\rho}-\frac{1}{4}\eta_{\mu\nu}F_{i}^{\tau\lambda}R_{\tau\lambda\sigma\rho}]+\frac{3}{2}\varepsilon_{ijk}[\frac{1}{2}F^{i}_{(\mu|\sigma}F^{j\sigma\rho}F^{k}_{\rho|\nu)}-\frac{1}{4}\eta_{\mu\nu}F^{i}_{\tau\sigma}F^{j\sigma\rho}F^{k}_{\rho}\,{}^{\tau}]+$
$\displaystyle+\frac{1}{4}(F_{i\mu\lambda}F_{j}^{\lambda\sigma}F^{i}_{\sigma\rho}F^{j\rho}\,_{\nu}-\eta_{\mu\nu}F_{i\tau\lambda}F_{j}^{\lambda\sigma}F^{i}_{\sigma\rho}F^{j\rho\tau})+\frac{3}{8}F^{i}_{\sigma\rho}F^{j\sigma\rho}(F_{i\mu\lambda}F_{j\nu}\,^{\lambda}-\frac{1}{4}\eta_{\mu\nu}F_{i\tau\lambda}F_{j}^{\tau\lambda})+$
$\displaystyle+\frac{1}{4}(F_{i\mu\lambda}F_{j}^{\lambda\sigma}F^{j}_{\sigma\rho}F^{i\rho}\,_{\nu}-\eta_{\mu\nu}F_{i\tau\lambda}F_{j}^{\lambda\sigma}F^{j}_{\sigma\rho}F^{i\rho\tau})-\frac{1}{16}\eta_{\mu\nu}\delta_{ij}\delta^{ij}+$
$\displaystyle+(\frac{1}{2}{\mathcal{D}}_{\sigma}F_{i\mu\rho}{\mathcal{D}}^{\sigma}F^{i}_{\nu}\,{}^{\rho}+\frac{1}{4}{\mathcal{D}}_{\mu}F_{i\sigma\rho}{\mathcal{D}}_{\nu}F^{i\sigma\rho}-\frac{1}{4}\eta_{\mu\nu}{\mathcal{D}}_{\sigma}F_{i\tau\rho}{\mathcal{D}}^{\sigma}F^{i\tau\rho}).$
(62)

It should be emphasized here that the $\eta_{\mu\nu}$ and $\delta_{ij}$ are
the vierbein and dreibein metrics of the external and internal manifolds,
respectively and that $\delta_{ij}\delta^{ij}=3$. The remaining components
correspondingly are given by

$\displaystyle\hat{T}_{\mu j}$ $\displaystyle=$
$\displaystyle\frac{1}{2}R_{\mu\lambda\sigma\rho}{\mathcal{D}}^{\lambda}F_{j}^{\sigma\rho}+\frac{1}{2}\varepsilon_{jik}F^{i\sigma\rho}{\mathcal{D}}_{(\mu}F^{k}_{\rho)\sigma}+\frac{3}{4}F_{i\mu}\,^{\lambda}F^{i\sigma\rho}{\mathcal{D}}_{\sigma}F_{j\rho\lambda}-\frac{1}{2}F_{j}^{\sigma\lambda}F_{i\lambda}\,^{\rho}{\mathcal{D}}_{(\mu}F^{i}_{\sigma)\rho},$
$\displaystyle\hat{T}_{ij}$ $\displaystyle=$
$\displaystyle\frac{3}{8}[\varepsilon_{kli}F^{l}_{\tau\sigma}F_{j}^{\sigma\rho}F^{k}_{\rho}\,{}^{\tau}+\varepsilon_{klj}F^{l}_{\tau\sigma}F_{i}^{\sigma\rho}F^{k}_{\rho}\,{}^{\tau}-\delta_{ij}\varepsilon_{klm}F^{l}_{\tau\sigma}F^{m\sigma\rho}F^{k}_{\rho}\,{}^{\tau}]-$
(63)
$\displaystyle-\frac{1}{16}(\frac{1}{2}F_{i\tau\lambda}F_{k}^{\lambda\sigma}F_{j\sigma\rho}F^{k\rho\tau}+\delta_{ij}F_{l\tau\lambda}F_{k}^{\lambda\sigma}F^{l}_{\sigma\rho}F^{k\rho\tau})+$
$\displaystyle+\frac{1}{4}(\frac{5}{2}F_{i\tau\lambda}F_{k}^{\lambda\sigma}F^{k}_{\sigma\rho}F_{j}^{\rho\tau}-\delta_{ij}F_{l\tau\lambda}F_{k}^{\lambda\sigma}F^{k}_{\sigma\rho}F^{l\rho\tau})+$
$\displaystyle+\frac{1}{4}({\mathcal{D}}_{\sigma}F_{i\tau\rho}{\mathcal{D}}^{\sigma}F_{j}^{\tau\rho}-\delta_{ij}{\mathcal{D}}_{\sigma}F_{k\tau\rho}{\mathcal{D}}^{\sigma}F^{k\tau\rho})-\frac{3}{8}F_{i\lambda\rho}F_{j}^{\lambda\rho}-$
$\displaystyle-\frac{1}{4}\delta_{ij}(R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho}-\frac{3}{2}F^{k\mu\nu}F_{k}^{\lambda\rho}R_{\mu\nu\lambda\rho}+\frac{3}{8}F_{k\mu\nu}F^{k\mu\nu}F_{l\lambda\rho}F^{l\lambda\rho}-\frac{1}{4}).$

It is not difficult to conjecture that for the $4$D case, we only obtain the
source-free field equation $D{\star{\mathcal{R}_{\mu\nu}}}=0$ and the stress-
energy tensor of the $4$D Weyl-Yang theory, $\tau^{{WY}}_{X_{\mu}}$.

##  3 Ansatz and static black hole solutions

Now, we investigate the black hole solutions to the $(4+3)$D Weyl-Yang gravity
model by taking into account the generalized Wu-Yang ansatz (for a review of
higher-dimensional black hole solutions for different manners, see e.g.,
Emparan ). It is very difficult to obtain exact solutions of higher-derivative
gravity theories. In order to obtain computational advantage in solving third-
order non-linear partial differential equations, it is useful to consider the
symmetric version of the $4$D static spherically-symmetric space-time metric
which describes not only charged but also asymptotically (anti-) de Sitter
black holes. In the usual Schwarzschild coordinates, $x=(t,r,\theta,\phi)$,
the external metric can put into the form

$\displaystyle G(x)$ $\displaystyle=$ $\displaystyle-f(r)dt\otimes
dt+f(r)^{-1}dr\otimes dr+r^{2}d\theta\otimes d\theta+r^{2}\sin^{2}\theta
d\phi\otimes d\phi,$ (64)

where the metric function $f(r)$ depends only on the radial coordinate $r$.
Moreover, we choose the $3$D line element for the internal space as the unit
three-sphere $S^{3}$ in terms of Euler coordinates, $y=(\Theta,\Phi,\psi)$
Henderson , describing rotations, which are given by the line element in its
standard form:

$\displaystyle G(y)$ $\displaystyle=$ $\displaystyle d\Theta\otimes
d\Theta+\sin^{2}\Theta d\Phi\otimes d\Phi+(d\psi+\cos\Theta
d\Phi)\otimes(d\psi+\cos\Theta d\Phi).$ (65)

Here, the ranges are $0\leq\theta,\Theta\leq\pi$, $0\leq\phi,\Phi\leq 2\pi$,
$0\leq\psi\leq 4\pi$, and $0<r<r_{\infty}$, and
$\partial/\partial\Phi=\partial_{\Phi}$ is the usual Killing field associated
with the circle $S^{1}$ in the background of the Abelian Kaluza-Klein
theories. The internal metric $G(y)$ in equation (65) is actually a special
case of Berger sphere Berger61 , and it can be rewritten in the general
rescaled, i.e., “squashed" form

$\displaystyle G(y)=\sum_{i=5}^{7}l^{2}_{i}E^{i}\otimes E^{i},$ (66)

where the $E^{i}$ are the right-invariant $1$-form set by

$\displaystyle E^{5}(y)$ $\displaystyle=$ $\displaystyle\cos\psi
d\Theta+\sin\psi\sin\Theta d\Phi,$ $\displaystyle E^{6}(y)$ $\displaystyle=$
$\displaystyle-\sin\psi d\Theta+\cos\psi\sin\Theta d\Phi,$ (67) $\displaystyle
E^{7}(y)$ $\displaystyle=$ $\displaystyle d\psi+\cos\Theta d\Phi.$

Other choices of $E^{i}$ that satisfy the structure equation (17) are also
possible (see e.g., Taub51 ; Newman63 ; Misner63 ). The $l_{i}$,
$0<l_{i}<\infty$, can be defined as the curvature radii of the internal space,
and they are constants for a static background. We consider the symmetric case
where the three principal radii of curvature are equal, $l_{5}=l_{6}=l_{7}$,
corresponding to the usual round three-sphere with isometry group
$\mathcal{G}=SU(2)\times SU(2)\simeq SO(4)$ Shen87 . In that special case, the
$S^{3}$ manifold is completely invariant under the right action of the gauge
group, $SU(2)$ Duff . We also investigate the case where $l_{5}=l_{6}\neq
l_{7}$, which corresponds to the squashed sphere geometry with isometry
algebra $\mathcal{G}=SU(2)\times U(1)$. It possesses a circular isometry, and
it usually appears in the quantum field theory, especially in the mixmaster
Universe (see Dowker  and references therein). The $0$-form dilaton field is,
for the sake of simplicity, taken $\varphi=\varphi(r)=1$ in the equation (11),
but here it can be useful to use $\varphi(r)$ to investigate the additional
effects of this field on the non-trivial solutions. Thus the final form of the
full metric $\hat{G}(x,y)$ becomes

$\displaystyle\hat{G}(x,y)=G(x)+\varphi(r)\sum_{i=5}^{7}(l_{i}E^{i}+A^{i})\otimes(l_{i}E^{i}+A^{i}).$
(68)

We also introduce well-known, time-independent, generalized Wu-Yang monopole-
like gauge potentials. Thus, in general, the gauge field ansatz is given by
the following spherically symmetric electric and magnetic parts:

$\displaystyle
A^{i}=\mathcal{A}^{i}_{0}(\vec{x},t)dt+\mathcal{A}^{i}_{a}(\vec{x},t)dx^{a},$
(69)

where

$\displaystyle\mathcal{A}^{i}_{0}(\vec{x},t)=\frac{J(r)}{gr}\hat{x}^{i},\qquad\qquad\qquad\mathcal{A}^{i}_{a}(\vec{x},t)=\varepsilon^{i}\,_{ab}\frac{[K(r)-1]}{gr}\hat{x}^{b},$
(70)

which are free from any line singularity if we assume all Yang-Mills fields to
be independent of the extra internal coordinates. Here, the profile functions,
$J(r)$ and $K(r)$, are arbitrary real-valued functions depending on the
variable $r=\mid\vec{x}\mid$, to be determined by the equations of motion. As
usual, $g$ is the quantized monopole charge,
$\hat{x}^{a}=x^{a}/\mid\vec{x}\mid$ are the spatial unit vectors,
$i,j,...=1,2,3$ are the color indices, and $a,b,...=1,2,3$ are the space
indices, which mean that both indices are used in the same way. After some
manipulations in Minkowski space-time, the corresponding Yang-Mills field
strength 2-form, in which the electrically- and magnetically-charged but-not-
rotating particles moving with unit charge (i.e., $g=1$) and coupling constant
($e=1$), can explicitly be obtained as:

$\displaystyle F^{i}$ $\displaystyle=$
$\displaystyle\left\\{\delta^{i}_{a}\left(\frac{JK}{r^{2}}\right)+{x}^{i}{x}_{a}\left[\frac{rJ^{\prime}-J(K+1)}{r^{4}}\right]\right\\}dx^{a}\wedge
dt+$ (71)
$\displaystyle+\varepsilon_{ab}\,^{c}\left[\delta^{i}_{c}\left(\frac{K^{\prime}}{r}\right)-\frac{{x}^{i}{x}_{c}}{r^{3}}\left(K^{\prime}-\frac{K^{2}-1}{r}\right)\right]dx^{a}\wedge
dx^{b},$

(see also e.g., dereli93 ; Ngome ). Here, prime implies a derivative taken
with respect to $r$, as usual. If we define the electric-like field $1$-form
as $E^{i}\equiv E^{i}_{a}dx^{a}$ and the magnetic-like field $2$-form as
$B^{i}\equiv B^{i}_{a}\star(dx^{a}\wedge dt)$, then we can write
$F^{i}=E^{i}\wedge dt+B^{i}$. Consequently, the electric fields and the
magnetic fields can be obtained from equation (71) by taking into account the
relationships $E^{i}\equiv\iota_{X_{t}}F^{i}$ and
$B^{i}\equiv-\iota_{X_{t}}\star F^{i}$ benntucker1 .

By substituting the squashed version of the (4+3)D metric tensor (68) where
$l_{5}=l_{6}=l$ and $l_{7}=\ell$, together with the gauge potentials (70),
into the master field equation (7) (${\hat{S}_{AB}}=0$), we unfortunately
obtain complicated and highly non-linear sets of field equations due to the
nature of the Weyl-Yang model (our calculations have been checked by the Atlas
$2$ Atlas2 , which is the differential geometry package for Mathematica).
However, after we carefully analyze the full patterns of differential
equations which are completely independent of the internal coordinates,
$y=(\Theta,\Phi,\psi)$, we recognize the fact that there are non-trivial
choices that make the field equations solvable. Therefore, the two following
cases easily appear for finding exact solutions (for similar choices for
different manners, see e.g., Mondaini ; Singleton ).

###  3.1 The $K(r)=K$ case

Assume the non-trivial gauge charge $K(r)$ to be a constant, $K(r)=K$. Then,
the particles still carry both electric and magnetic charge unless $J(r)=0$ or
$K(r)=1$, and the particle-like solution is called a dyon solution by
Schwinger Schwinger  in Abelian theory and later by Julia-Zee Julia  in non-
Abelian theory (by taking into account the ’t Hooft-Polyakov monopole solution
Hooft1 ; Polyakov1 , which is the first well-known magnetic monopole solution
for the Yang-Mills-Higgs theory with gravity neglected in the literature). The
dyons can also be interpreted as the excited states of magnetic monopoles from
the point of view of modern physics. Then, we obtain $21$ very complicated
dyon equations, and the simplest non-vanishing one becomes the following:

$\displaystyle
l^{4}\varphi(r)(K-1)\biggl{\\{}g^{2}r^{2}\bigl{[}3J(r)^{2}-4f(r)(K-1)^{2}\bigr{]}-J(r)^{2}(K-1)^{2}\varphi(r)^{2}-$
$\displaystyle-J(r)^{2}\bigl{[}g^{2}r^{2}+(K-1)^{2}\varphi(r)^{2}\bigr{]}\cos
2\theta\biggr{\\}}+2(l^{2}-\ell^{2})r\cos^{2}\theta(K-1)\varphi(r)J(r)J^{\prime}(r)\times$
$\displaystyle\times\biggl{\\{}l^{2}\bigl{[}g^{2}r^{2}+(K-1)^{2}\varphi(r)^{2}\bigr{]}+3gl^{2}\ell(K-1)\varphi(r)^{2}+\ell^{2}\bigl{[}g^{2}r^{2}+(K-1)^{2}\varphi(r)^{2}\bigr{]}\biggr{\\}}+$
$\displaystyle+\ell\varphi(r)\Biggl{(}gl^{4}\biggl{\\{}4g^{2}r^{2}\bigl{[}J^{2}-2f(r)(K-1)^{2}\bigr{]}-3J(r)^{2}(K-1)^{2}\varphi(r)^{2}-$
$\displaystyle-3\cos 2\theta
J(r)^{2}(K-1)^{2}\varphi(r)^{2}\biggr{\\}}+\ell(K-1)\biggl{\\{}-8g^{4}l^{4}r^{2}f(r)-gl^{2}\ell(K-1)\bigl{[}4g^{2}r^{2}f(r)-$
$\displaystyle-3J(r)^{2}\varphi(r)^{2}-3J(r)^{2}\varphi(r)^{2}\cos
2\theta\bigr{]}+2\cos^{2}\theta\ell^{2}J(r)^{2}\bigl{[}g^{2}r^{2}+(K-1)^{2}\varphi(r)^{2}\bigr{]}\biggr{\\}}\Biggr{)}+$
$\displaystyle+4g^{2}l^{2}r^{3}\varphi^{\prime}(r)\Biggl{(}l^{2}(K-1)\bigl{[}-J(r)^{2}+f(r)(K-1)^{2}\bigr{]}+$
$\displaystyle+g\ell\biggl{\\{}l^{2}\bigl{[}-J(r)^{2}+2f(r)(K-1)^{2}\bigr{]}+\ell
f(r)(K-1)\bigl{[}2gl^{2}+\ell(K-1)\bigr{]}\biggr{\\}}\Biggr{)}=0.$ (72)

Here we can prove that if $\ell=\pm l$, the equation (3.1) is reduced to more
simpler form as

$\displaystyle(K-1\pm gl)\left[f(r)(K-1)(K-1\pm
2gl)-J(r)^{2}\right]\left[r\varphi^{\prime}(r)-\varphi(r)\right]=0.$ (73)

This gives the following conditions for the constant $K$:

$K=\begin{cases}1-gl&\mbox{if }\ell=l,\\\ 1+gl&\mbox{if }\ell=-l,\end{cases}$
(74)

where the both conditions, $\ell=l$ and $\ell=-l$, correspond actually to the
round three-sphere due to the $\ell^{2}=(\pm l)^{2}=l^{2}$, if

$\displaystyle f(r)(K-1)(K-1\pm 2gl)-J(r)^{2}\neq 0,$ (75) $\displaystyle
r\varphi^{\prime}(r)-\varphi(r)\neq 0.$ (76)

Then, we can symbolically write

$\displaystyle K=1-g\ell.$ (77)

After that, we immediately obtain from the other field equations

$\displaystyle[rJ^{\prime}(r)-J(r)]^{2}-g^{2}\ell^{2}=0,$ (78)

with the supplementary equation

$\displaystyle\varphi^{\prime}(r)=0.$ (79)

We easily achieve the following particular solutions of equations (78) and
(79), respectively:

$\displaystyle J(r)$ $\displaystyle=$ $\displaystyle j_{\infty}r\pm g\ell,$
(80) $\displaystyle\varphi(r)$ $\displaystyle=$ $\displaystyle\varphi_{0},$
(81)

where $j_{\infty}$ and $\varphi_{0}$ are the corresponding integration
constants of $J(r)$ and the scalar field $\varphi(r)$. $J(r)$ can be
explicitly written as

$J(r)=\begin{cases}j_{\infty}r\pm gl&\mbox{if }\ell=l,\\\ j_{\infty}r\mp
gl&\mbox{if }\ell=-l.\end{cases}$ (82)

In that respect, the remaining field equations reduce to following set of
radial differential equations:

$\displaystyle
f^{\prime\prime\prime}(r)+\frac{2}{r}f^{\prime\prime}(r)-\frac{2}{r^{2}}f^{\prime}(r)+\frac{4\varphi_{0}^{2}\ell^{2}}{r^{5}}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
f^{\prime\prime}(r)+\frac{2}{r}f^{\prime}(r)-\frac{\varphi_{0}^{2}\ell^{2}(l^{2}+\ell^{2})}{2l^{2}r^{4}}+\frac{1}{\ell^{2}\varphi_{0}^{2}}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
f^{\prime\prime}(r)-\frac{2}{r^{2}}\left[f(r)-\frac{\ell^{2}}{l^{2}}\right]-\frac{\varphi_{0}^{2}\ell^{2}(l^{4}+3\ell^{4})}{2l^{4}r^{4}}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
f^{\prime}(r)+\frac{1}{r}\left[f(r)-1\right]+\frac{\varphi_{0}^{2}\ell^{6}}{2l^{4}r^{3}}+\frac{r\ell^{2}}{2l^{4}\varphi_{0}^{2}}$
$\displaystyle=$ $\displaystyle 0.\ $ (83)

Now, it is possible to simplify differential expressions above due to the
$\ell^{n}=l^{n}$ only if $n$ is an even integer. Therefore, we have the
following equations for round $S^{3}$ internal metric:

$\displaystyle
f^{\prime\prime\prime}(r)+\frac{2}{r}f^{\prime\prime}(r)-\frac{2}{r^{2}}f^{\prime}(r)+\frac{4\varphi_{0}^{2}l^{2}}{r^{5}}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
f^{\prime\prime}(r)+\frac{2}{r}f^{\prime}(r)-\frac{\varphi_{0}^{2}l^{2}}{r^{4}}+\frac{1}{l^{2}\varphi_{0}^{2}}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
f^{\prime\prime}(r)-\frac{2}{r^{2}}\left[f(r)-1\right]-\frac{2\varphi_{0}^{2}l^{2}}{r^{4}}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
f^{\prime}(r)+\frac{1}{r}\left[f(r)-1\right]+\frac{\varphi_{0}^{2}l^{2}}{2r^{3}}+\frac{r}{2l^{2}\varphi_{0}^{2}}$
$\displaystyle=$ $\displaystyle 0.$ (84)

After integration, these differential sets admit an exact but singular
solution of the following form:

$\displaystyle f(r)=1-\frac{2M}{r}+\frac{Z^{2}}{r^{2}}-\frac{1}{3}\Lambda
r^{2},$ (85)

where we have defined

$\displaystyle\Lambda=\frac{1}{2(l\varphi_{0})^{2}},\qquad\qquad\qquad\qquad
Z^{2}=\frac{1}{4\Lambda}=\frac{(l\varphi_{0})^{2}}{2},$ (86)

and $2M>0$ is the integration constant, which is identified with the total
time-independent gravitational mass of the black hole, and the parameter $Z$
can be interpreted as the charge-like term of the space-time. Equation (85) is
the Reissner-Nordström-de Sitter (RNdS)-type black hole solution with a
positive cosmological constant, $\Lambda>0$. The RNdS solution is a charged
black hole immersed in an asymptotically de Sitter space-time. It was first
obtained by Carter Carter  and its properties are well-defined (for a detailed
discussion, see e.g., Romans ; Brill ). Moreover, RN-type solutions have been
previously discussed for the EYMH model Bais ; Cho ; Yaskin ; Volkov  (also
see e.g., Wang75 ; Nieuwenhuizen ; Cordero ; Benguria ; Comtet ; Chambers ;
stephani ).

In the present situation, the positive cosmological constant of the
asymptotically de Sitter black hole is associated with a constant dilaton
field, $\varphi_{0}$, and the radius of the three-sphere, $l$, and the de
Sitter cosmological radius can be written as $R=\sqrt{6}l\varphi_{0}$ if
$\Lambda=3/R^{2}$. We remark that if $l=1$, then the constant dilaton field
can behave as a radius of three-sphere dereli90 . The observational data
suggest that the dimensionful parameter $\Lambda$ is non-zero with an
accelerating universe Riess ; Perlmutter  and its value is approximately
$|\Lambda|\approx~{}10^{-52}m^{-2}$ Peebles  (and see also Kagramanova ).
Then, we have $\varphi_{0}l\approx~{}10^{26}m$, and the dimension of the
charge is $[Z]=\mbox{Length}$.

We can analyze the curvature invariant, curvature scalar and roots of $f(r)$
to understand the general structure of RNdS-type black hole solutions, as
usual. As might be expected, the singularity behavior of the considered
solution is similar to that of the RNdS black hole. The Kretschmann scalar or
the Lagrangian density
$\hat{\mathcal{K}}=\hat{\star}{(\hat{\mathcal{R}}_{AB}\wedge\hat{\star}{\hat{\mathcal{R}}^{AB}})}$
are calculated as

$\displaystyle\hat{\mathcal{K}}=\frac{17}{24(l\varphi_{0})^{4}}+\frac{96M[Mr^{2}-(l\varphi_{0})^{2}r]+21(l\varphi_{0})^{4}}{4r^{8}},$

and the Ricci scalar
$\hat{\mathcal{R}}=\hat{\iota}_{\hat{X}_{B}}\hat{\iota}_{\hat{X}_{A}}\hat{\mathcal{R}}^{AB}$
behaves as

$\displaystyle\hat{\mathcal{R}}=\frac{7}{2(l\varphi_{0})^{2}}.$ (88)

Hence, the curvature invariant in equation (3.1) as well as the metric
function $f(r)$ in equation (85) exhibit a physically essential (curvature)
singularity at the origin, $r=0$, and the Ricci scalar $\hat{\mathcal{R}}$ is
finite everywhere. As $r\rightarrow\infty$, the curvature invariant becomes
$\hat{\mathcal{K}}=\frac{17}{24(l\varphi_{0})^{4}}$, and for $r\neq 0$ it is
also finite. On the other hand, the horizon function or the lapse function
$\Delta(r)$ can be written as

$\displaystyle\Delta(r)=-\frac{1}{3}\Lambda r^{4}+r^{2}-2Mr+Z^{2}.$ (89)

The real roots are determined by solving the quartic polynomial function
$\Delta(r)\equiv 0$, and they can be ordered as $r_{1}<0<r_{-}\leq
r_{+}<r_{c}$ if $Z\neq 0$. That is

$\displaystyle\Delta(r)=(r-r_{1})(r-r_{-})(r-r_{+})(r-r_{c}).$ (90)

Here, $r_{-}$ is the inner (Cauchy) horizon, $r_{+}$ is the black hole event
horizon, $r_{c}$ corresponds to the outer cosmological horizon, and the
negative root $r_{1}=-(r_{c}+r_{+}+r_{-})$ is nonphysical. From equation (89),
we can deduce the explicit forms of the zeros of $\Delta(r)$ in the following
forms:

$\displaystyle r_{\pm}=\Omega\pm\sqrt{\Psi^{(-)}},\qquad\qquad\qquad
r_{c,1}=-\Omega\pm\sqrt{\Psi^{(+)}},$ (91)

where

$\displaystyle\Omega=l\varphi_{0}\sqrt{1+\left[\left(\frac{3}{2}\frac{M}{l\varphi_{0}}\right)^{2}-1\right]^{1/3}},\qquad\qquad\quad\Psi^{(\pm)}=3(l\varphi_{0})^{2}\left(1\pm\frac{M}{\Omega}\right)-\Omega^{2}.$
(92)

If we want to obtain a space-time with horizons but without any un-physical
naked singularity, we must restrict the value of mass $M$ to the certain
limits by taking into account $l$ and $\varphi_{0}$. For instance, we suppose
that $\frac{9}{4}\frac{M^{2}}{l^{2}\varphi^{2}_{0}}>1$ to obtain a positive
cube root term, and obviously $\Omega>0$. This restriction is compatible with
Lake’s condition, $Z^{2}<\frac{9}{8}M^{2}$ Lake , as expected.

We can also construct the “extremal" or “cold" black hole solution where two
or three roots are equal to each other. For example, if the two horizons
coincide $r_{-}=r_{+}=r_{h}$ at $r_{h}$, then we have $\Psi^{(-)}=0$ which
means that the mass $M$ ($M<Z$) yields

$\displaystyle M=\frac{2}{3}l\varphi_{0}.$ (93)

Hence, $r_{h}$ becomes

$\displaystyle r_{h}=l\varphi_{0}.$ (94)

This is a charged Nariai-type black hole solution with a horizon at $r_{h}$
Bousso . It is easy to see that the cosmological horizon also satisfies
$r_{c}=r_{h}$. In other words, three horizons (rather than two, as assumed)
coincide at $r_{h}=r_{ucd}=l\varphi_{0}$, and we actually obtain the ultracold
black hole together with
$f(r_{ucd})=f^{\prime}(r_{ucd})=f^{\prime\prime}(r_{ucd})=0$,
$M=\frac{2}{3}r_{ucd}$, and $Z^{2}=\frac{1}{2}r^{2}_{ucd}$, which are all
compatible with the results of Romans  and Rahman  (also see e.g., Hartmann ).
Moreover, the surface gravity of the corresponding horizon takes a form in
terms of the slope of $f(r)$ at that horizon Hawking75 ; Wald  (for the
alternative definition, see Ridgway ), given by:

$\displaystyle\kappa_{(-,+,c)}=\frac{1}{2}\left|\frac{df(r)}{dr}\right|_{r=r_{(-,+,c)}}.$
(95)

For instance, the surface gravity associated with the event horizon
$\kappa_{+}$ can be written as

$\displaystyle\kappa_{+}=\frac{1}{2}\left|\frac{2M}{r_{+}^{2}}-\frac{r_{+}}{3(l\varphi_{0})^{2}}-\frac{(l\varphi_{0})^{2}}{r_{+}^{3}}\right|.$
(96)

Furthermore, the associated Hawking temperature, $T_{H}$, Hawking75 ; Hawking
can be calculated by using the relationship $T_{H}=\frac{|\kappa|}{2\pi}$,
i.e.,

$\displaystyle
T_{+}=\frac{1}{4\pi}\left|\frac{2M}{r_{+}^{2}}-\frac{r_{+}}{3(l\varphi_{0})^{2}}-\frac{(l\varphi_{0})^{2}}{r_{+}^{3}}\right|.$
(97)

In the extreme case, the RNdS black hole has zero temperature, and for the
$Z^{2}=M^{2}$ case which represents lukewarm black holes Ross95  the
temperature is given by

$\displaystyle T=\frac{1}{6\pi
l\varphi}\sqrt{\sqrt{3}-\frac{3}{2}}\approx\frac{0.026}{l\varphi_{0}},$ (98)

and the solutions are stable in this fashion. This temperature is also
directly obtained from the relationship

$\displaystyle
T=\frac{1}{2\pi}\sqrt{\frac{\Lambda}{3}\left(1-4M\sqrt{\frac{\Lambda}{3}}\right)},$
(99)

which is given in Mellor ; Romans .

We can also record the gauge potentials in the following forms:

$\displaystyle A^{i}=\ell\left[\left(\frac{j_{\infty}}{g\ell
r}\pm\frac{q}{r^{2}}\right){x}^{i}dt-\frac{p}{r^{2}}\varepsilon^{i}\,_{ab}{x}^{b}dx^{a}\right].$
(100)

Here, the constant terms $q=1$ and $p=1$ are placed into the general gauge
equation above for later convenience. In the first place, it is not difficult
to conjecture that $A_{(\ell=-l)}^{i}\neq-A_{(\ell=l)}^{i}$. However, by
taking into account Dirac’s quantization relation dirac  we require not only
that $e=g$, but also that $g=-\frac{1}{\ell}$, which satisfies
${\mathcal{D}}\star{F^{i}}=0$. Then, equation (100) can be explicitly written
as

$\displaystyle\left(\begin{array}[]{c}A^{1}\\\ A^{2}\\\ A^{3}\\\
\end{array}\right)=\ell\Bigl{[}\left(\pm\frac{q}{r}-j_{\infty}\right)\left(\begin{array}[]{c}\sin\theta\cos\phi\\\
\sin\theta\sin\phi\\\ \cos\theta\\\
\end{array}\right)dt-p\left(\begin{array}[]{c}\sin\phi\\\ -\cos\phi\\\ 0\\\
\end{array}\right)d\theta-p\sin\theta\left(\begin{array}[]{c}\cos\theta\cos\phi\\\
\cos\theta\sin\phi\\\ -\sin\theta\\\ \end{array}\right)d\phi\Bigr{]},$ (113)

which satisfies $A_{(\ell=-l)}^{i}=-A_{(\ell=l)}^{i}$. Here, the parameters
$j_{\infty}$, $q$, and $p$ determine the scale of $J(r)$ Lugo99 , and the
electric and magnetic charge, respectively.

For $SU(2)$ gauge fields, the dimensionless, global, gauge invariant non-
Abelian electric charge, ${\mathcal{Q}}^{{YM}}$, and the magnetic charge,
${\mathcal{P}}^{{YM}}$, may be defined by employing surface integrals over the
two-sphere $S^{2}$ at $r\rightarrow\infty$, as in Ashtekar1 ; Ashtekar2 ;
Corichi1 ; Corichi2  (also see Sudarsky ):

${\mathcal{Q}}^{{YM}}\equiv-\frac{g}{4\pi}\oint_{r\rightarrow\infty}\sqrt{\sum_{i}(\star
F^{i}_{\theta\phi})^{2}}d\theta
d\phi,\qquad\quad{\mathcal{P}}^{{YM}}\equiv-\frac{g}{4\pi}\oint_{r\rightarrow\infty}\sqrt{\sum_{i}(F^{i}_{\theta\phi})^{2}}d\theta
d\phi.$ (115)

Equation (115) is also useful for obtaining horizon charges and local charges
Kleihaus1 ; Kleihaus2 . Thus, following the Yang-Mills topological charges
from equation (115), we have the following:

$\displaystyle{\mathcal{Q}}^{{YM}}=\frac{|q\ell|}{\ell},\qquad\qquad\qquad\qquad{\mathcal{P}}^{{YM}}=\frac{|p(p-2)\ell|}{\ell}.$
(116)

That is

$\mathcal{Q}^{{YM}}=\begin{cases}+1&\mbox{if }\ell=l,\\\ -1&\mbox{if
}\ell=-l,\end{cases}\qquad\qquad\qquad{\mathcal{P}}^{{YM}}=\begin{cases}+1&\mbox{if
}\ell=l,\\\ -1&\mbox{if }\ell=-l.\end{cases}$ (117)

Thus, it is found that for the case $\ell=l$ the gauge field corresponds to
the positively-charged dyon solution whereas we have the negatively-charged
anti-dyon solution for the case $\ell=-l$. It seems that whether a dyon or
anti-dyon solution is obtained depends on the sign of the internal space
squashing parameter, $\ell$. In this fashion, the (anti)-dyonic black hole
carries one unit of electric charge and one unit of magnetic charge, and it is
appropriate to redefine parameter $Z^{2}$ in (86) as

$\displaystyle Z^{2}=\frac{{\mathcal{Q}}^{2}+{\mathcal{P}}^{2}}{8\Lambda},$
(118)

where the total (electric plus magnetic) time-independent gravitational charge
of the space-time is equal to

$\displaystyle{\mathcal{Q}}^{2}+{\mathcal{P}}^{2}=2.$ (119)

It is essential here to note that, for the $SU({\mathcal{N}})$ gauge field,
the only value of the magnetic charge ${\mathcal{P}}$ appearing in the black
hole solutions is given by (see e.g., Winstanley07 ; Winstanley08 ;
Winstanley12 )

$\displaystyle{\mathcal{P}}^{2}=\frac{1}{6}{\mathcal{N}}({\mathcal{N}}+1)({\mathcal{N}}-1).$
(120)

Hence, for the gauge group ${\mathcal{G}}=SU(2)$, we have
${\mathcal{P}}^{2}=1$, which we already obtained from equation (117). The
resulting dyon solution is actually relevant to the Abelian-like solution of
the EYMH model, which was previously given by Kasuya and Kamata Kasuya1 ;
Kasuya2 ; Kasuya3  (also see Lugo99 ; Perryy ) as a special solution.
Moreover, when $q=j_{\infty}=0$, the gauge potentials reduce to the well-known
Wu-Yang monopole with infinite classical energy wuyang ; stuller  (see also
the Bogomolnyi-Prasad-Sommerfield monopole solutions Bogomolnyi ; Prasad ) but
this case is not a solution of our model when $J(r)=0$.

It is useful a quickly review the main results of the remarkable studies of
Perry Perry  and Angus Angus . They considered a case where not only was the
field strength in a four-index form but also the gauge potentials were simpler
than the generalized Wu-Yang ansatz (70), by taking the $6$D and $7$D Einstein
equations into account together with a cosmological constant and the matter
fields, respectively. In the compact form, the gauge ansatz that was used was
$A^{i}=\delta^{i}_{3}A(r,\theta)$, which can be explicitly written as

$\displaystyle\left(\begin{array}[]{c}A^{1}\\\ A^{2}\\\ A^{3}\\\
\end{array}\right)=J(r)\left(\begin{array}[]{c}0\\\ 0\\\ -1\\\
\end{array}\right)dt+K(r)\left(\begin{array}[]{c}0\\\ 0\\\ \cos\theta\\\
\end{array}\right)d\phi.$ (130)

This condition is also known as the Abelian gauge, because of the its
similarity to the $U(1)$ (which is a subgroup of $SU(2)$) Abelian model
Marciano , keeping in mind that any Abelian solutions solve the non-Abelian
equations. Perry found the Abelian-like solution to the $6$D model with a
matter source from the Einstein action in the following form:

$\displaystyle f(r)=1-\frac{2M}{r}+\frac{Z^{2}}{r^{2}},$ (131)

where

$\displaystyle
Z^{2}=\frac{q^{2}+p^{2}}{8\Lambda}=\frac{p^{2}}{4\Lambda},\qquad\qquad\qquad\Lambda=\frac{1}{2\varphi_{0}^{2}},$
(132)

The gauge field and whose functions are

$\displaystyle A^{3}$ $\displaystyle=$
$\displaystyle\mp\frac{q}{r}dt+p\cos\theta d\phi,$ (133) $\displaystyle J(r)$
$\displaystyle=$ $\displaystyle\pm\frac{q}{r},\qquad\qquad\qquad\qquad\qquad
K(r)=p,$ (134)

where $q=p$ and $p$ is a constant. The topological charge, $p$, must be
$p=\frac{n}{2}$, where $n$ is an integer (for $Z_{2}$ monopoles
$p=\pm\frac{1}{2}$ Preskill ) to avoid string singularities. On the other
hand, the solutions of the $7$D model found by Angus are very problematic, and
only approximate solutions may be possible, due to more the complicated non-
Abelian Kaluza-Klein metric ansatz. The metric function, $f(r)$, can roughly
be written as

$\displaystyle f(r)=1-\frac{2M}{r}+\frac{Z^{2}}{r^{2}}+O(\frac{1}{r^{3}}),$
(135)

where

$\displaystyle
Z^{2}=\frac{q^{2}+p^{2}}{4\Lambda},\qquad\qquad\qquad\qquad\Lambda=\frac{1}{2l^{2}},$
(136)

for $\alpha_{0}=0$. Furthermore, the unknown profile functions and the gauge
field respectively are

$\displaystyle J(r)$ $\displaystyle=$
$\displaystyle\frac{q}{r}+O(\frac{1}{r^{2}}),\qquad\qquad\qquad\qquad\qquad
K(r)=p,$ (137) $\displaystyle A^{3}$ $\displaystyle=$
$\displaystyle-\left[\frac{q}{r}+O(\frac{1}{r^{2}})\right]dt+p\cos\theta
d\phi,$ (138)

where $q$ and $p$ are again constant. Hence, our solution (85) is not only in
a more desirable and suitable form than both Perry’s (131) and Angus’ (135)
solutions but also the spherically-symmetric gauge potentials (113) are more
general and non-Abelian than both (133) and (138).

Although we prefer to keep the radii of curvature $\ell$ out of the gauge
potentials in equation (113), we can also consider the case where $q=p=\ell$.
In fact, due to the quantization conditions, we can already write
$p=\frac{1}{2}nR$, where $R$ is the Kaluza-Klein circle radius of $S^{1}$ for
the 5D Abelian model Angus1 . Therefore, it is not surprising that the
magnetic monopole charge $p$ is related to the internal space parameter
$\ell$. Hence, we have the total $SU(2)$ charge as
${\mathcal{Q}}^{2}+{\mathcal{P}}^{2}=2l^{2}g^{2}$, and the parameter $Z^{2}$
and the new modified cosmological constant $\widetilde{\Lambda}$ in the
following forms:

$\displaystyle
Z^{2}=\frac{2l^{2}g^{2}}{8\widetilde{\Lambda}}=\frac{l^{2}g^{2}}{4\widetilde{\Lambda}},\qquad\qquad\qquad\widetilde{\Lambda}=l^{2}g^{2}\Lambda=\frac{g^{2}}{2\varphi_{0}^{2}}.$
(139)

It is not difficult to conjecture that these equations reduce to the previous
equations (86) if $g=-\frac{1}{\ell}$, where $\widetilde{\Lambda}=\Lambda$,
and Perry’s results in equation (132) for the case $g=1$, respectively. The
new positive cosmological term $\widetilde{\Lambda}$ depends on the constant
dilaton field as well as the charge parameter, $g$, in this case.

As mentioned before, the topological charges in equation (117) have very
special values with $q=p=1$. As was noted by Perry Perry , these types of
charges actually occur in the vacuum solutions of Einsteinian gravity (see
e.g., Cotaescu ), i.e., ${\hat{\mathcal{R}}}_{A}=\hat{\Lambda}\hat{E}_{A}$ for
all constant $\hat{\Lambda}$ where $\hat{\Lambda}=\Lambda$, which corresponds
to the Einstein manifold. We obtain the same results for the Weyl-Yang model,
due to the fact that the empty-space solutions exactly solve
$\hat{D}{\hat{\star}{\hat{\mathcal{R}}_{AB}}}=0$. However the master equation
$\hat{D}{\hat{\star}{\hat{\mathcal{R}}_{AB}}}=0$ also contains the physical
and un-physical families of non-Einsteinian solutions. We obtain the following
relations from the equation (73), which do not appear in the empty space
solutions of the Einstein field equation:

$\displaystyle f(r)=\frac{J^{2}(r)}{(K-1)(K-1+2g\ell)},$ (140)

where $K\neq 1$ and $K\neq 1-2g\ell$, or

$\displaystyle\varphi^{\prime}(r)-\frac{1}{r}\varphi(r)=0.$ (141)

The solution of dilaton equation (141) is found to be

$\displaystyle\varphi(r)=\varphi_{0}r.$ (142)

It is straightforward to insert solution (142) into the field equations to
investigate the exact analytic solutions. Doing this, one of the field
equations becomes the following:

$\displaystyle
rJ^{2}(r)\frac{f^{\prime}(r)}{f(r)}-(K-1)(K-1+2g\ell)f(r)-2rJ(r)J^{\prime}(r)+J^{2}(r)=0.$
(143)

It may be rewritten as

$\displaystyle\left[\frac{J^{2}(r)}{rf(r)}\right]^{\prime}-\frac{(K-1)(K-1+2g\ell)}{r^{2}}=0.$
(144)

Hence, we have

$\displaystyle f(r)=\frac{J^{2}(r)}{(K-1)(K-1+2g\ell)+f_{0}r},$ (145)

where $r\neq-(K-1)(K-1+2g\ell)/f_{0}$ and $f_{0}$ is an integration constant,
$f_{0}\neq 0$. However, it is very difficult to find the function $J(r)$, the
constant $f_{0}$, and therefore the metric function $f(r)$, which satisfies
all of the coupled field equations, even if we consider further assumptions
about the function $J(r)$. Thus, further investigations are needed to clarify
the physical meaning of both solutions (140) and (145), if they exist. For
instance, numerical analysis can be made possible by taking into account the
finite-energy dyon configurations.

Finally, the components of the gravitational energy-momentum second-rank
tensor are given by employing the RNdS-like black hole solution and expressing
the relationship
$\hat{T}_{AB}=\hat{\iota}_{\hat{X}_{B}}\hat{\star}{\hat{\tau}_{\hat{X}_{A}}}$
in the following forms:

$\displaystyle\hat{T}_{tt}$ $\displaystyle=$
$\displaystyle\frac{3}{16(l\varphi_{0})^{4}}+\frac{2r^{4}+24(l\varphi_{0})^{2}(3r^{2}-8Mr)+63(l\varphi_{0})^{4}}{24r^{8}},$
$\displaystyle\hat{T}_{rr}$ $\displaystyle=$
$\displaystyle-\frac{3}{16(l\varphi_{0})^{4}}-\frac{10r^{4}+24(l\varphi_{0})^{2}(r^{2}-4Mr)+39(l\varphi_{0})^{4}}{24r^{8}},$
$\displaystyle\hat{T}_{\theta\theta}$ $\displaystyle=$
$\displaystyle\hat{T}_{\phi\phi}=-\frac{3}{16(l\varphi_{0})^{4}}+\frac{2r^{4}+16(l\varphi_{0})^{2}(r^{2}-3Mr)+15(l\varphi_{0})^{4}}{8r^{8}},$
$\displaystyle\hat{T}_{\Theta\Theta}$ $\displaystyle=$
$\displaystyle-\frac{5}{48(l\varphi_{0})^{4}}-\frac{96M\left[Mr^{2}-(l\varphi_{0})^{2}r\right]+(l\varphi_{0})^{4}\left(21-4\sin^{2}\theta\cos^{2}\phi\right)}{8r^{8}},$
$\displaystyle\hat{T}_{\Phi\Phi}$ $\displaystyle=$
$\displaystyle-\frac{5}{48(l\varphi_{0})^{4}}-\frac{96M\left[Mr^{2}-(l\varphi_{0})^{2}r\right]+(l\varphi_{0})^{4}\left(21-4\sin^{2}\theta\sin^{2}\phi\right)}{8r^{8}},$
$\displaystyle\hat{T}_{\psi\psi}$ $\displaystyle=$
$\displaystyle-\frac{5}{48(l\varphi_{0})^{4}}-\frac{96M\left[Mr^{2}-(l\varphi_{0})^{2}r\right]+(l\varphi_{0})^{4}(19-2\cos
2\theta)}{8r^{8}}.$ (146)

On the other hand, the flux-like components can be found to be

$\displaystyle\frac{\hat{T}_{t\Theta}}{\sin\theta\cos\phi}$ $\displaystyle=$
$\displaystyle\frac{\hat{T}_{t\Phi}}{\sin\theta\sin\phi}=\frac{\hat{T}_{t\psi}}{\cos\theta}=\frac{\ell\varphi_{0}\sqrt{f(r)}\left[6Mr-5(l\varphi_{0})^{2}\right]}{r^{7}},$
(147) $\displaystyle\frac{\hat{T}_{\Theta\Phi}}{\sin^{2}\theta\sin 2\phi}$
$\displaystyle=$ $\displaystyle\frac{\hat{T}_{\Theta\psi}}{\sin
2\theta\cos\phi}=\frac{\hat{T}_{\Phi\psi}}{\sin
2\theta\sin\phi}=\frac{(l\varphi_{0})^{4}}{4r^{8}}.$ (148)

One can also write the remaining components of the stress-energy tensor by
employing its symmetry properties. All other components vanish as well. Let us
finally mention that the all stress-energy tensors above are divergence at the
singular point $r=0$, and there are no proper values for $\theta$ and $\phi$
that make all off-diagonal components zero. However, the components in
equation (147) that depend only on the parameter $\ell$, among all the
components, vanish at the zeros of $f(r)$.

###  3.2 The $J(r)=0$ case

We can alternatively consider the case where the Coulomb part of gauge field
is chosen so that ${\mathcal{A}}^{i}_{0}(\vec{x})=0$. In other words, the
electric field vanishes at $J(r)\equiv 0$, and the Yang-Mills fields become
purely magnetic, rather than dyons, due to this restricted monopole
configuration. In this regard, the corresponding magnetic-monopole solutions
are dipole-like solutions, i.e., $F^{i}_{ab}\sim\frac{1}{r^{3}}$ (see the
equation (71)), and they vanish faster than the monopole-like solutions, i.e.,
$F^{i}_{ab}\sim\frac{1}{r^{2}}$. Then, the simplest field equation becomes

$\displaystyle
3g^{2}l^{2}\ell^{2}r^{2}f(r)\varphi(r)^{2}[K^{\prime}(r)]^{2}+2g^{2}l^{2}(l^{2}-\ell^{2})r^{2}f(r)\varphi(r)\varphi^{\prime}(r)[K(r)-1]K^{\prime}(r)-$
$\displaystyle-3[K(r)-1]^{2}\Biggl{(}-g^{2}l^{2}r^{2}+2gl^{2}\ell[K(r)-1]\varphi(r)^{2}+\ell^{2}\biggl{\\{}[K(r)-1]^{2}\varphi(r)^{2}+g^{2}r^{2}+$
$\displaystyle+4g^{2}l^{2}\varphi(r)^{2}+2g\ell[K(r)-1]\varphi(r)^{2}\biggr{\\}}\Biggr{)}=0.$
(149)

Again, we can assume that $\ell=\pm l$, then the equation (3.2) simply gives

$\displaystyle\varphi(r)^{2}\left\\{g^{2}l^{2}r^{2}f(r)\left[K^{\prime}(r)\right]{}^{2}-[K(r)-1]^{2}[K(r)-1\pm
2gl]^{2}\right\\}=0.$ (150)

Thus, the metric function $f(r)$ takes the form if $\varphi(r)\neq 0$ as:

$\displaystyle f(r)=\frac{[K(r)-1]^{2}[K(r)-1\pm
2gl]^{2}}{g^{2}l^{2}r^{2}\left[K^{\prime}(r)\right]{}^{2}}.$ (151)

However, we could not find any solution, which satisfies all of the coupled
field equations, for $K(r)\neq constant$. Hence, let us assume again that the
space component function of the gauge field is of the form $K(r)=K$, where $K$
is a constant and $K\neq 1$, to simplify the field equations. Then using
$\ell^{n}=l^{n}$ with $n$ being an even integer we obtain by employing the
equation (150):

$\displaystyle K=1-2g\ell.$ (152)

Here, we also prove that $K=1-g\ell$ is surprisingly not a solution anymore
for this considered case. The piecewise-defined constant $K$ can explicitly be
written in the following forms:

$K=\begin{cases}1-2gl&\mbox{if }\ell=l,\\\ 1+2gl&\mbox{if
}\ell=-l,\end{cases}$ (153)

for the round internal metric. The gauge potentials, which are electrically
neutral are found to be

$\displaystyle\left(\begin{array}[]{c}A^{1}\\\ A^{2}\\\ A^{3}\\\
\end{array}\right)=\ell\Bigl{[}-\widetilde{p}\left(\begin{array}[]{c}\sin\phi\\\
-\cos\phi\\\ 0\\\
\end{array}\right)d\theta-\widetilde{p}\sin\theta\left(\begin{array}[]{c}\cos\theta\cos\phi\\\
\cos\theta\sin\phi\\\ -\sin\theta\\\ \end{array}\right)d\phi\Bigr{]},$ (163)

which again satisfy $A_{(\ell=-l)}^{i}=-A_{(\ell=l)}^{i}$ with the constant
term $\widetilde{p}=2$. In the round limit, all components of the Yang-Mills
strength (71) vanish, and the $A_{(\ell=l)}^{i}$ is the so-called pure gauge.
However, there are two possibilities for the constant $g$ that satisfy the 4D
source-free Yang-Mills equations ${\mathcal{D}}\star F^{i}=0$ with $e=g$ for
this situation. These are

$\displaystyle g=-\frac{1}{\ell},\qquad\qquad\qquad\qquad g=-\frac{1}{2\ell}.$
(164)

The electric charge already vanishes, ${\mathcal{Q}}^{{YM}}=0$, in the absence
of $J(r)$, while the non-zero color magnetic charge is the same as equation
(116) for $g=-\frac{1}{\ell}$:

$\displaystyle{\mathcal{P}}^{{YM}}=\frac{|\widetilde{p}(\widetilde{p}-2)\ell|}{\ell},$
(165)

which gives zero charge in both the $\ell=l$ and $\ell=-l$ cases, due to the
$\widetilde{p}=2$

$\displaystyle{\mathcal{P}}^{{YM}}=0.$ (166)

On the other hand, for $g=-\frac{1}{2\ell}$, ${\mathcal{P}}^{{YM}}$ is
calculated as

$\displaystyle{\mathcal{P}}^{{YM}}=\frac{|\widetilde{p}(\widetilde{p}-4)\ell|}{4\ell}.$
(167)

Then we obtain the positively-charged monopole and the negatively-charged
anti-monopole as

$\mathcal{P}^{{YM}}=\begin{cases}+1&\mbox{if }\ell=l,\\\ -1&\mbox{if
}\ell=-l.\end{cases}$ (168)

This monopole solution, which has zero energy, was previously discussed by Wu
and Yang by taking the sourceless Yang-Mills equations into account. However
the anti-monopole solution is new, and corresponds to the infinite energy Wu-
Yang monopole wuyang .

Next, we turn to the investigation of the black hole solutions. Inserting
expression (152) into the field equations yields three coupled nonlinear
differential equations as follows:

$\displaystyle
f^{\prime\prime\prime}+\left[3\left(\frac{\varphi^{\prime}}{\varphi}\right)+\frac{2}{r}\right]f^{\prime\prime}-\left[3\left(\frac{\varphi^{\prime}}{\varphi}\right)^{2}+\frac{2}{r^{2}}\right]f^{\prime}=0,$
$\displaystyle
f^{\prime\prime}+3\left(\frac{\varphi^{\prime}}{\varphi}\right)f^{\prime}-2\left[3\left(\frac{\varphi^{\prime}}{\varphi}\right)^{2}+\frac{1}{r^{2}}\right]f+\frac{2\ell^{2}}{l^{2}r^{2}}=0,$
$\displaystyle\left(\frac{\varphi^{\prime}}{\varphi}\right)f^{\prime\prime}+\left[4\left(\frac{\varphi^{\prime\prime}}{\varphi}\right)+\left(\frac{\varphi^{\prime}}{\varphi}\right)^{2}+\frac{2}{r}\left(\frac{\varphi^{\prime}}{\varphi}\right)\right]f^{\prime}+2\Biggr{\\{}\left(\frac{\varphi^{\prime\prime\prime}}{\varphi}\right)+\left(\frac{\varphi^{\prime\prime}}{\varphi}\right)\left[\left(\frac{\varphi^{\prime}}{\varphi}\right)+\frac{2}{r}\right]-$
$\displaystyle-2\left(\frac{\varphi^{\prime}}{\varphi}\right)\left[\left(\frac{\varphi^{\prime}}{\varphi}\right)^{2}+\frac{1}{r^{2}}\right]\Biggr{\\}}f+\frac{1}{l^{2}\varphi^{2}}\left(\frac{\varphi^{\prime}}{\varphi}\right)=0.$
(169)

In the first place, we recognize the fact that there is no relationship
between the charge-like constant $\widetilde{p}$ and the field equations
above. Thus, this case may be considered as a kind of Brans-Dicke-type case.
It is not easy to obtain the general analytic solutions of two unknown
functions $f=f(r)$ and $\varphi=\varphi(r)$ from the field equations (3.2).
However, we can consider some specific dilaton ansatz not only for the sake of
simplicity but also deserving attention in the following subsections:

####  3.2.1 The $\varphi(r)=\varphi_{0}$ case

First, we can restrict ourselves to the simplest case where the scalar field
is just a constant, i.e., $\varphi(r)=\varphi_{0}$. Then, the differential
equations (3.2) reduce to two simple sets as

$\displaystyle
f^{\prime\prime\prime}(r)+\frac{2}{r}f^{\prime\prime}(r)-\frac{2}{r^{2}}f^{\prime}(r)$
$\displaystyle=$ $\displaystyle 0,$ (170) $\displaystyle
f^{\prime\prime}(r)-\frac{2}{r^{2}}f(r)+\frac{2}{r^{2}}$ $\displaystyle=$
$\displaystyle 0.$ (171)

It is not difficult to find exact solutions for the equations (170) and (171).
Hence, we obtain

$\displaystyle f(r)=1-\frac{2M}{r}-\frac{1}{3}\Lambda r^{2},$ (172)

where the two integration constants, $M$ and $\Lambda$, may be interpreted as
a mass and a cosmological constant of the black hole, respectively. This
solution is nothing but the famous neutral Schwarzschild black hole (Kottler)
solution in the de Sitter space-time if $\Lambda>0$. That is the simplest
black hole solution of the Einstein equations with a positive cosmological
constant Kottler ; Weyl1919 ; Trefftz . In the 4D Weyl-Yang limit, this
solution has already been obtained by several authors fairchild76 ; pavelle1 ;
pavelle3 ; ni ; Hsu , because of fact that the vacuum solutions of Einstein
equations, i.e., ${\hat{\mathcal{R}}}_{A}=0$ and
${\hat{\mathcal{R}}}_{A}=\hat{\Lambda}\hat{E}_{A}$, are also the first
integral solutions of the master equation (7) of the theory ($\hat{S}_{AB}=0$)
to be found in the natural way, as mentioned before. However, in our approach
the cosmological constant comes naturally out from the field equations as a
integration constant. This situation seems to be more natural than adding a
cosmological constant into the Einstein field equations by hand. The boundary
conditions of the (in)-homogeneous Universe determine the cosmological
constant, and fine-tuning may not be as serious of a problem as it is in
Einstein’s theory of relativity (for a detailed discussion, see cook ; chen1
).

On the other hand, the roots of the black hole solution (172) can be
classified into three cases:

$\begin{cases}\mbox{two real positive
zeros}~{}~{}r_{+}<r_{c}&\mbox{if}~{}~{}0<9M^{2}\Lambda<1,\\\ \mbox{the extreme
case}~{}~{}r_{+}=r_{c}=3M&\mbox{if}~{}~{}9M^{2}\Lambda=1,\\\ \mbox{no real
positive zeros}&\mbox{if}~{}9M^{2}\Lambda>1,\end{cases}$ (173)

where $M>0$ (see e.g., Molina ). Let us also remark that, if we assume that
$\Lambda<0$, we obtain the Schwarzschild-anti-de Sitter solution, and by
setting $\Lambda=0$ we just have the Schwarzschild solution. The invariants in
this case become

$\displaystyle\hat{\mathcal{K}}=\frac{3}{8(l\varphi_{0})^{4}}+\frac{4\Lambda^{2}}{3}+\frac{24M^{2}}{r^{6}},\qquad\qquad\qquad\qquad\hat{\mathcal{R}}=\frac{3}{2(l\varphi_{0})^{2}}+4\Lambda,$
(174)

and the nonzero components of the stress-energy tensor turn out to be

$\displaystyle\hat{T}_{tt}=-\hat{T}_{rr}=-\hat{T}_{\theta\theta}=-\hat{T}_{\phi\phi}$
$\displaystyle=$ $\displaystyle\frac{3}{16(l\varphi_{0})^{4}},$ (175)
$\displaystyle\hat{T}_{\Theta\Theta}=\hat{T}_{\Phi\Phi}=\hat{T}_{\psi\psi}$
$\displaystyle=$
$\displaystyle\frac{1}{16(l\varphi_{0})^{4}}-\frac{2\Lambda^{2}}{3}-\frac{12M^{2}}{r^{6}}.$
(176)

Therefore, the gravitational energy-momentum tensor $\hat{T}_{AB}$ is just
diagonal, and is formally similar to that of Maxwell’s theory of
electromagnetism. At the origin $r=0$, the Kretschmann invariant and the $3$D
parts of the stress-energy tensor have a pole, whereas the curvature scaler is
finite everywhere. We finish this subsection with one important remark: the
dilaton field must satisfy the condition $\varphi(r)\neq\mbox{constant}$,
otherwise, as we can see, one obtains trivial solutions (see also kuyrukcu ).
Therefore, the existence of non-constant dilaton fields in not only the
(anti)-monopole solutions, but also the (anti)-dyon solutions (see equations
(81) and (142)) is essential for finding new black hole solutions. Therefore,
the scalar fields play very important roles in the Weyl-Yang theory just as
they do in the supergravity theories.

####  3.2.2 The $\varphi(r)=\varphi_{0}r$ case

Let us search for black hole solutions by assuming that the scalar field is of
the form of $\varphi(r)=\varphi_{0}r$, where $\varphi_{0}$ is any constant,
taking inspiration from the non-Einsteinian condition in equation (142). Then,
we obtain from the equation (3.2)

$\displaystyle
f^{\prime\prime\prime}(r)+\frac{5}{r}f^{\prime\prime}(r)-\frac{5}{r^{2}}f^{\prime}(r)$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
f^{\prime\prime}(r)+\frac{3}{r}f^{\prime}(r)-\frac{8}{r^{2}}f(r)+\frac{2}{r^{2}}$
$\displaystyle=$ $\displaystyle 0,$ (177) $\displaystyle
f^{\prime\prime}(r)+\frac{3}{r}f^{\prime}(r)-\frac{8}{r^{2}}f(r)+\frac{1}{\varphi_{0}^{2}l^{2}r^{2}}$
$\displaystyle=$ $\displaystyle 0.$

One can easily see that this ansatz leads to the same coefficients of the
$f^{\prime}(r)$ and $f(r)$ terms as in the last two equations of (3.2). Thus,
the exact solution can be found as

$\displaystyle
f(r)=\frac{1}{4}+\left(\frac{Z^{2}}{2r^{2}}\right)^{2}-\frac{1}{3}\Lambda
r^{2},$ (178)

where $Z$ and $\Lambda$ are the arbitrary constants of integration. The
equation is formally equivalent to the purely charged (anti)-de Sitter metric,
and it seems to be a special case of the charged bulk black hole solution of
the 5D Einstein-Maxwell system Barcelo ; Maartens1 ; Maartens2 . We also
obtain the condition $\varphi_{0}=\pm\frac{1}{\sqrt{2}l}$, which means that
the dilatonic field is free from singularities

$\displaystyle\varphi(r)=\pm\left(\frac{1}{\sqrt{2}l}\right)r.$ (179)

Solution (178) has six distinct roots in the (anti)-de Sitter space-time. For
instance, the positive event horizon can be found for the anti-de Sitter case
as

$\displaystyle
r_{+}=\frac{1}{2}\sqrt{\frac{1}{\Lambda}\left(1+\Omega+\frac{1}{\Omega}\right)},$
(180)

where

$\displaystyle\Omega=\left[1+24Z^{4}\Lambda^{2}+4\sqrt{3Z^{4}\Lambda^{2}(1+12Z^{4}\Lambda^{2})}\right]^{\frac{1}{3}}.$
(181)

To avoid the space-like naked singularity, we must again be careful when
choosing the constants $Z$ and $\Lambda$. The Kretschmann and Ricci scalars
are calculated by means of the equation (178) as

$\displaystyle\hat{\mathcal{K}}=\frac{14\Lambda^{2}}{3}+\frac{9r^{8}+75Z^{8}}{4r^{12}},\qquad\qquad\qquad\qquad\hat{\mathcal{R}}=14\Lambda.$
(182)

The covariantly-conserved and diagonal energy-momentum tensor also reduces to
the following components:

$\displaystyle\hat{T}_{tt}=-\hat{T}_{rr}$ $\displaystyle=$
$\displaystyle\frac{9}{8r^{12}}(r^{8}-5Z^{8})+\Lambda^{3},$
$\displaystyle\hat{T}_{\theta\theta}=\hat{T}_{\phi\phi}$ $\displaystyle=$
$\displaystyle\frac{3}{8r^{12}}(r^{8}-21Z^{8})-\Lambda^{3},$ (183)
$\displaystyle\hat{T}_{\Theta\Theta}=\hat{T}_{\Phi\Phi}=\hat{T}_{\psi\psi}$
$\displaystyle=$
$\displaystyle-\frac{1}{8r^{12}}(5r^{8}+63Z^{8})-\Lambda^{3}.$

The curvature invariant and the energy-momentum tensors, again, are all finite
everywhere except at $r=0$. At $r\rightarrow\infty$, $\hat{\mathcal{K}}$ goes
to $\hat{\mathcal{K}}=\frac{14\Lambda^{2}}{3}$ and $\varphi(r)$ diverges. In
this regard, the non-trivial solution (178) is well-behaved, since the
curvature invariants and charge are finite at infinity.

Let us finally remark that it is quite natural to consider the case where the
time components of the gauge fields, as well as the scalar fields, survive and
the space components of the gauge fields vanish. Due to the coupled field
equations, there are no solutions in this case (see e.g., Winstanley121 ).

##  4 Conclusions

In this paper, we have combined the Weyl-Yang gravitational gauge theory with
the Kaluza-Klein theories of unified gravity against the background of the
non-Abelian gauge fields. This is, from a theoretical point of view, a new and
rather appealing approach in the literature. The theoretical description was
reduced from seven to four dimensions by taking into account the exterior
differential forms that provide the more transparent equations along with
shortcut calculations. The corresponding equations of motion (25)–(30) were
not only mathematically more complicated than those of the non-Abelian version
of the Kaluza-Klein theory that comes from Einstein’s gravity, i.e., the
$\\{\mathcal{P}_{\mu\nu},\mathcal{Q}^{i}_{\nu},\mathcal{U}^{ij}\\}$-set in
(2.1) but also included additional physical information because of the more
general field equations. We could easily obtain, for instance, the Yang-Mills
force density $4$-form $f^{(ij)}_{\mu}$ in equation (41), which only appears
in geodesic equations kerner ; orzalesi , and the usual $4$D continuity
equation, ${\mathcal{D}}\star{J^{i}}=0$, with the help of equations (2.1) and
(2.1), without making any further assumptions. Furthermore, the right side of
the field equation (25), i.e., is the $S_{\mu\nu\sigma}$ term (31), may
conveniently be interpreted as the source term of the $4$D Weyl-Yang gauge
theory. We have explicitly shown that this source term contains Camenzind’s
current density for the Abelian case and it can be completely transformed to
that if $D_{\mu}\mathcal{U}=0$ and $\mathcal{Q}_{\mu}=0$. However, it is
possible to find an energy-momentum tensor different from (53) such that the
source term can be put into the desired form (42). The considered model also
has additional solutions that can not be obtained from Einstein’s theory.
After Pavelle obtained the static spherically symmetric solution of the 4D
Yang equation, he noted that pavelle1  “the solar experiments are presently
unable to distinguish between the Kilmister-Yang equations and Einstein’s
vacuum equations” but today, the situation is different. The new effects of
Yang equation on various physical issues such as the strong equivalence
principle gerard , the vacuum-energy cook , the cosmological constant cook ;
chen ; chen1 , dark radiation chen1 ; yeung , and dark matter gonzalez ; yeung
have already been discussed by several authors. Let us remark that the
reduction mechanism transformed the higher-dimensional sourceless Weyl-Yang
theory into the $4$D Weyl-Yang theory with a source term in the usual $4$D
space-time. In fact, the same situation has already been discussed for the
$5$D version of the presented theory with the $U(1)$ Abelian case halil13 . A
similar accomplishment has already been obtained by ordinary Kaluza-Klein
theories, in which the $5$D vacuum equations were reduced to the $4$D
Einstein-Maxwell-scalar system. This is sometimes called the Kaluza-Klein
miracle.

We have also constructed the exact static (anti)-dyon (85) and pure magnetic
(anti)-monopole solutions (172) and (178) with the generalized Wu-Yang ansatz
and some different types of scalar potentials, keeping in mind the spherically
symmetric stationary external and internal space-times with the product
topology $M_{4}\times S^{3}$. For the sake of simplicity, we concentrated only
on the pure internal symmetry group ${\mathcal{G}}=SU(2)$, which led to the
mixing of the space-time and color indices. These strong restrictions and
highly symmetric assumptions were necessary owing to the complex nature of the
considered model and the fact that it is already enormously difficult to find
a solution in higher dimensions because there is no general method for the
same. To analytically discover regular solutions, two special cases with
$K(r)=K$ and $J(r)=0$ were also considered as a type of strategy. In this
manner, we obtained the singular Wu-Yang-type (anti)-dyon solution that
corresponded to the RNdS-like black hole solution (85), which is in a more
desirable and suitable form than both Perry’s (131) and Angus’ (135)
solutions. In addition, its spherically symmetric gauge potentials (113) are
more general and non-Abelian than both solutions (133) and (138), and carry
unit or constant electric and magnetic charges. It was demonstrated in this
approach that the cosmological constant naturally appeared as an integration
constant (i.e., “it is a free parameter unrelated to the vacuum energy of
quantum fields” according to Cook cook ) and that it was associated with the
constant dilaton field as well as with the three-sphere radius. The general
structure of this regular black hole was also studied in detail by means of
its mathematical similarity to the well-known RNdS black hole solution. The
corresponding gravitational stress-energy tensors of these solutions were also
analyzed. On the other hand, the pure magnetic (anti)-monopole solution (178)
appears in the literature for the first time in connection with the Weyl-Yang-
Kaluza-Klein theory and the corresponding gravitational stress-energy tensor
(3.2.2). It is just diagonal and formally similar to that of Maxwell’s theory
of electromagnetism, which means that the coupled non-linear differential
equations (3.2) contain some mathematically interesting features. In
conclusion, the (4+3)D non-Abelian Weyl-Yang-Kaluza-Klein theory with the Wu-
Yang ansatz admits a non-trivial solution against the background of the static
spherically-symmetric metric with a three-sphere.

Whether the solution is a monopole or an anti-monopole may depend on the sign
of the massive scalar Higgs field Marciano ; Marciano1 , on the sign of the
Bogomol’nyi equations Rossi , or on the $\phi$-winding number, which can be a
positive or negative integer of the internal space Teh ; Teh1 . We have shown
in this study that the sign of the internal space-squashing parameter $\ell$
determines whether the solution is a dyon/monopole or an anti-dyon/anti-
monopole. Moreover, we have topologically presented non-trivial anti-dyon and
anti-monopole solutions, which do not solve the source-free Yang-Mills
equations in the Minkowski space-time, by taking into account the round
version of the internal space. Finally, the existence of non-constant dilaton
fields is essential for finding new black hole solutions in the Weyl-Yang
gauge theory.

It would be interesting to discover the Abelian solutions of the $5$D version
of this alternative gravitational model in both the ordinary $4$D and $5$D
space-times. We may consider the $U(1)$ Abelian ansatz (and its group
manifold, $S^{1}$) from the non-Abelian gauge by assuming that only the
${\mathcal{A}}^{i}_{a}(\vec{x})$ component of the gauge field survives, that
is

$\displaystyle{\mathcal{A}}^{i}_{a}(\vec{x})=\delta^{i}_{3}{\mathcal{A}}_{a}^{(Dirac)}(r,\theta).$
(184)

Obtaining the $4$D field equations from the Weyl-rescaled reduced action,

$\displaystyle\tilde{{S}}=-\frac{1}{2\kappa^{2}}\int_{M_{4}}d^{4}x\sqrt{-\tilde{\hat{g}}}\tilde{\hat{R}}_{ABCD}\tilde{\hat{R}}^{ABCD},$
(185)

and investigating whether this non-linear coupled system of partial
differential equations possesses a well-posed initial value formulation as
other higher-derivative gravity theories Noakes  are other important
mathematical challenges. A detailed discussion of these problems will be
presented in a forthcoming study.

## References

  * (1) A. Palatini, Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton, Rend. Circ. Mat. Palermo 43 (1919) 203.
  * (2) C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, San Francisco (1973), chap. 21.
  * (3) M. Tsamparlis, On the Palatini method of variation, J. Math. Phys. 19 (1977) 555.
  * (4) M. Ferraris, M. Francaviglia and C. Reina, Variational formulation of general relativity from 1915 to 1925 Palatini’s method discovered by Einstein in 1925, Gen. Relativ. Gravit. 14 (1982) 243.
  * (5) E. E. Fairchild Jr., Gauge theory of gravitation, Phys. Rev. D 14 (1976) 384 [Erratum ibid 14 (1976) 2833].
  * (6) E. E. Fairchild Jr., Yang-Mills formulation of gravitational dynamics, Phys. Rev. D 16 (1977) 2438.
  * (7) H. Weyl, Gravitation and electricity, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1918 (1918) 465, ibid A New Extension of Relativity Theory, Ann. Phys. (Leipzig) 59 (1919) 101, ibid Über die physikalischen Grundlagen der erweiterten Relativitätstheorie Phys. Zeitschr. 22 (1921) 473.
  * (8) C. Lanczos, A Remarkable Property of the Riemann-Christoffel Tensor in Four Dimensions, Ann. Math. 39 (1938) 842, ibid Lagrangian Multiplier and Riemannian Spaces, Rev. Mod. Phys. 21 (1949) 497, ibid Electricity and General Relativity, Rev. Mod. Phys. 29 (1957) 337.
  * (9) R. Utiyama, Invariant Theoretical Interpretation of Interaction, Phys. Rev. 101 (1956) 1597.
  * (10) A. Lichnerowicz, Sur un procedé de quantification du champ de gravitation, C. R. Acad. Sci. Paris 247 (1958) 433.
  * (11) G. Stephenson, Quadratic lagrangians and general relativity, Il Nuovo Cimento 9 (1958) 263.
  * (12) P. W. Higgs, Quadratic lagrangians and general relativity, Il Nuovo Cimento 11 (1959) 816.
  * (13) C. W. Kilmister and D. J. Newman,  The use of algebraic structures in physics, Math. Proc. Cambridge Philos. Soc. 57 (1961) 851, ibid in Proceedings of the Workshop on Les théories relativistes de la gravitation, Royaumont (1959).
  * (14) H. G. Loos, Spin connection in general relativity, Ann. Phys. 25 (1963) 91.
  * (15) H. G. Loos and R. P. Treat, Conditional dynamic equivalence of free Yang-Mills fields and free gravitational fields, Phys. Lett. A 26 (1967) 91.
  * (16) C. N. Yang, Integral Formalism for Gauge Fields, Phys. Rev. Lett. 33 (1974) 445.
  * (17) S. Ramaswamy and P. B. Yasskin, Birkhoff theorem for an $R+R^{2}$ theory of gravity with torsion, Phys. Rev. D 19 (1979) 2264.
  * (18) A. A. Tseytlin, Poincaré and de Sitter gauge theories of gravity with propagating torsion, Phys. Rev. D 26 (1982) 3327.
  * (19) V. Szczyrbac, Stephenson-Kilmister-Yang theory of gravity and its dynamics, Phys. Rev. D 36 (1987) 351.
  * (20) F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne’eman, Progress in metric-affine gauge theories of gravity with local scale invariance, Found. Phys. 19 (1989) 1075.
  * (21) J. W. Maluf, On the absence of physical degrees of freedom in the Stephenson-Kilmister-Yang theory, J. Math. Phys. 32 (1991) 1556.
  * (22) B. Rose, Field equations for gravity quadratic in the curvature, Int. J. Theor. Phys. 31 (1992) 1283.
  * (23) B. S. Guilfoyle and B. C. Nolan, Yang’s Gravitational Theory, Gen. Relativ. Gravit. 30 (1998) 473.
  * (24) D. Vassiliev, Pseudoinstantons in Metric-Affine Field Theory, Gen. Relativ. Gravit. 34 (2002) 1239.
  * (25) E. W. Mielke and A. A. Rincón Maggiolo, Duality in Yang’s theory of gravity, Gen. Relativ. Gravit. 37 (2005) 997.
  * (26) J. Q. Shen, Gravitational Gauge Theory Developed Based on the Stephenson-Kilmister-Yang Equation, Int. J. Theor. Phys. 48 (2009) 1566.
  * (27) V. Pasic and E. Barakovic, PP-waves with torsion: A metric-affine model for the massless neutrino, Gen. Relativ. Gravit. 46 (2014) 1787.
  * (28) Th. Kaluza, On the unification problem in physics, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) K1 (1921) 966.
  * (29) O. Klein, Quantum theory and five-dimensional relativity, Z. Phys. 37 (1926) 895, ibid Nature 118 (1926) 516.
  * (30) H. Mandel, Zur Herleitung der Feldgleichungen in der allgemeinen Relativitätstheorie, Z. Phys. 39 (1926) 136.
  * (31) T. Appelquist, A. Chodos, and P. G. O. Freund, Modern Kaluza-Klein theories, Addison-Wesley (1987).
  * (32) D. Bailin and A. Love, Kaluza-Klein theories, Rep. Prog. Phys. 50 (1987) 1087.
  * (33) M. J. Duff, Kaluza-Klein Theory in Perspective [arXiv:9410046].
  * (34) J. M. Overduin and P. S. Wesson, Kaluza-Klein gravity, Phys. Rep. 283 (1997) 303.
  * (35) S. Başkal and H. Kuyrukcu, Kaluza-Klein reduction of a quadratic curvature model, Gen. Relativ. Gravit. 45 (2013) 359.
  * (36) H. Kuyrukcu, The non-Abelian Weyl-Yang-Kaluza-Klein gravity model, Gen. Relativ. Gravit. 46 (2014) 1751.
  * (37) J.-M. Gerard, The strong equivalence principle from a gravitational gauge structure, Class. Quantum Grav. 24 (2007) 1867.
  * (38) R. J. Cook, The Gravitational-Electromagnetic Analogy: A Possible Solution to the Vacuum-Energy and Dark-Energy Problems [arXiv:0810.4495].
  * (39) G. R. Gonzalez-Martin, The G Gravitational Parameter and the Concepts of Mass and Dark Matter [arXiv:0910.3380].
  * (40) P. Chen, Gauge theory of gravity with de sitter symmetry as a solution to the cosmological constant problem and the dark energy puzzle, Modern Phys. Lett. A 25 (2010) 2795.
  * (41) P. Chen, K. Izumi and N.-E. Tung, Natural emergence of cosmological constant and dark radiation from the Stephenson-Kilmister-Yang-Camenzind theory of gravity, Phys. Rev. D 88 (2013) 123006.
  * (42) Y. Yang and W. B. Yeung, Quadratic Gravitational Lagrangian with Torsion Can Give Possible Explanations of the Form of Galactic Rotation Curves, of the Amount of Intergalactic Lensings, and of the Accelerating Expansion of the Universe [arXiv:1111.7062]; Spontaneously Broken Erlangen Program Offers a Bridge Between the Einstein and the Yang-Mills Theories [arXiv:1205.2690]; A Yang-Mills Type Gauge Theory of Gravity and the Dark Matter and Dark Energy Problems [arXiv:1210.0529]; The GL(4 R) Yang-Mills Theory of Gravity Predicts An Inflationary Scenario For The Evolution of The Primordial Universe [arXiv:1303.3801]; A New Approach to the Yang-Mills Gauge Theory of Gravity and Its Applications [arXiv:1312.4528].
  * (43) F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, General relativity with spin and torsion: Foundations and prospects, Rev. Mod. Phys. 48 (1976) 393.
  * (44) F. W. Hehl, Y. Ne’eman, J. Nitsch, and P. von der Heyde, Short-range confining component in a quadratic poincaré gauge theory of gravitation, Phys. Lett. B 78 (1978) 102.
  * (45) H. Goenner and F. Müller-Hoissen, Spatially homogeneous and isotropic spaces in theories of gravitation with torsion, Class. Quantum Grav. 1 (1984) 651.
  * (46) J. Garecki, New physical interpretation of the Bel-Robinson tensor, Class. Quantum Grav. 2 (1985) 403.
  * (47) J. W. Maluf, The Stephenson-Kilmister-Yang Hamiltonian, Class. Quantum Grav. 5 (1988) L81.
  * (48) G. Sardanashvily and O. Zakharov, Gauge Gravitation Theory, World Scientific, Singapore (1992).
  * (49) F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne’eman, Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance, Phys. Rept. 258 (1995) 1.
  * (50) F. Gronwald and F. W. Hehl, On the Gauge Aspects of Gravity [arXiv:9602013].
  * (51) F. Gronwald, Metric-Affine Gauge Theory of Gravity I: Fundamental Structure and Field Equations, Int. J. Mod. Phys. D 6 (1997) 263.
  * (52) F. W. Hehl and A. Macias, Metric-Affine Gauge Theory of Gravity II: Exact Solutions, Int. J. Mod. Phys. D 8 (1999) 399.
  * (53) J.-E. Daum and M. Reuter, Einstein-Cartan gravity, Asymptotic Safety, and the running Immirzi parameter, Ann. Phys. 334 (2013) 351.
  * (54) M. Blagojević and F. W. Hehl, Gauge Theories of Gravitation: A Reader with Commentaries, Imperial College Press, London (2013).
  * (55) F. W. Hehl, Y. N. Obukhov and D. Puetzfeld, On Poincaré gauge theory of gravity, its equations of motion, and Gravity Probe B, Phys. Lett. A 377 (2013) 1775.
  * (56) B. H. Dean, Variational Results and Solutions in Gauge Gravity and a Bifurcation Analysis of Black Hole Orbital Dynamics [arXiv:1312.7501].
  * (57) C. Aragone and A. Restuccia, Dynamical structure of linearized GL(4) gravities, Gen. Relativ. Gravit. 9 (1978) 409.
  * (58) M. Camenzind, Theories of Gravity with Structure-Dependent $\gamma$’s, Phys. Rev. Lett. 35 (1975) 1188, ibid Weak and strong sources of gravity: An $SO(1,3)$-gauge theory of gravity Phys. Rev. D 18 (1978) 1068, ibid Homogeneous and isotropic world models in the Yang-Mills dynamics of gravity. The structure of the adiabats, J. Math. Phys. 19 (1978) 624.
  * (59) C. Y. Lee and Y. Ne’eman,  Renormalization of gauge-affine gravity, Phys. Lett. B 242 (1990) 59.
  * (60) R. Pavelle, Unphysical Solutions of Yang’s Gravitational-Field Equations, Phys. Rev. Lett. 34 (1975) 1114.
  * (61) L. Bel, Sur la radiation gravitationelle, C. R. Acad. Sci. Paris 247 (1958) 1094, ibid Introduction d’un tenseur du quatrieme ordre, C. R. Acad. Sci. Paris 248 (1959) 1297.
  * (62) L. Robinson, Kings College Lectures (unpublished) (1958), ibid On the Bel - Robinson tensor, Class. Quantum Grav. 14 (1997) A331.
  * (63) B. Mashhoon, J. C. McClune and H. Quevedo, On the gravitoelectromagnetic stress-energy tensor, Class. Quantum Grav. 16 (1999) 1137.
  * (64) J. D. Jackson, Classical Electrodynamics, John Wiley and Sons Inc., New York, (1999).
  * (65) C. N. Yang and R. L. Mills, Conservation of Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev. 96 (1954) 191.
  * (66) E. Kretschmann, Uber die prinzipielle Bestimmbarkeit der berechtigten Bezugssysteme beliebiger Relativitätstheorien (I), Ann. der Phys. 48 (1915) 907.
  * (67) P. Candelas, G. T. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B 258 (1985) 46.
  * (68) D. J. Gross, J. Harvey, E. Martinec and R. Rohm, Heterotic String, Phys. Rev. Lett. 54 (1985) 502.
  * (69) D. J. Gross and E. Witten, Superstring modifications of Einstein’s equations, Nucl. Phys. B 277 (1986) 1.
  * (70) O. Lauscher and M. Reuter, Flow equation of quantum Einstein gravity in a higher-derivative truncation, Phys. Rev. D 66 (2002) 025026.
  * (71) G. Cognola and et al., Class of viable modified $f(R)$ gravities describing inflation and the onset of accelerated expansion, Phys. Rev. D 77 (2008) 046009.
  * (72) R. Utiyama and B. S. DeWitt, Renormalization of a Classical Gravitational Field Interacting with Quantized Matter Fields, J. Math Phys. 3 (1962) 608.
  * (73) K. S. Stelle, Renormalization of higher-derivative quantum gravity, Phys. Rev. D 16 (1977) 953.
  * (74) E. T. Tomboulis, $1N$ expansion and renormalization in quantum gravity, Phys. Lett. B 70 (1977) 361.
  * (75) A. A. Slavnov, Ward identities in gauge theories, Theor. Math. Phys. 10 (1972) 99.
  * (76) G. ’t Hooft and M. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B 44 (1972) 189.
  * (77) E. T. Tomboulis, Renormalizability and asymptotic freedom in quantum gravity, Phys. Lett. B 97 (1980) 77.
  * (78) F. S. Fradkin and A. A. Tseytlin, Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B 201 (1982) 469.
  * (79) A. Salam and J. Strathdee, Remarks on high-energy stability and renormalizability of gravity theory, Phys. Rev. D 18 (1978) 4480.
  * (80) K. S. Stelle, Classical gravity with higher derivatives, Gen. Relativ. Gravit. 9 (1978) 353.
  * (81) I. Antoniadis and E. T. Tomboulis, Gauge invariance and unitarity in higher-derivative quantum gravity, Phys. Rev. D 33 (1986) 2756.
  * (82) R. Pavelle, Yang’s Gravitational Field Equations, Phys. Rev. Lett. 33 (1974) 1461.
  * (83) R. Pavelle, Unphysical Characteristics of Yang’s Pure-Space Equations, Phys. Rev. Lett. 37 (1976) 961.
  * (84) R. Pavelle, Mansouri-Chang Gravitation Theory, Phys. Rev. Lett. 40 (1978) 267.
  * (85) A. H. Thompson, Yang’s Gravitational Field Equations, Phys. Rev. Lett. 34 (1975) 507, ibid Geometrically Degenerate Solutions of the Kilmister-Yang Equations 35 (1975) 320.
  * (86) G. W. Barrent, L. J. Rose and A. E. Stuart, Minimal differential equations for the Kilmister-Yang model of gravity, Phys. Lett. A 60 (1977) 278.
  * (87) P. Baekler, F. W. Hehl and E. W. Mielke, in Proceedings of the Second Marcel Grossmann Meetings on General Relativity, North-Holland, Amsterdam, 1982 pg. 413.
  * (88) P. Baekler, F. W. Hehl and H. J. Lenzen, in Proceedings of the Third Marcel Grossmann Meetings on General Relativity, Science Press, Beijing and North-Holland, Amsterdam, 1983.
  * (89) P. Baekler and P. B. Yasskin, All torsion-free spherical vacuum solutions of the quadratic Poincaré gauge theory of gravity, Gen. Relativ. Gravit. 16 (1984) 1135.
  * (90) R. R. Hsu and W. B. Yeung, The Gravitational Field of a Spinless Point Source in the Poincaré Gauge Theory of Gravitation with a Quadratic Lagrangian, Chin. J. Phys. 25 (1987) 463.
  * (91) W.-T. Ni, Yang’s Gravitational Field Equations, Phys. Rev. Lett. 35 (1975) 319 [Erratum ibid 35 (1975) 1748].
  * (92) I. M. Benn, T. Dereli and R. W. Tucker, Double-dual solutions of generalized theories of gravitation, Gen. Relativ. Gravit. 13 (1981) 581.
  * (93) E. W. Mielke, On pseudoparticle solutions in Yang’s theory of gravity, Gen. Relativ. Gravit. 13 (1981) 175.
  * (94) S. Başkal, Radiation in Yang-Mills formulation of Gravity and a Generalized pp-Wave metric, Prog. Theor. Phys. 102 (1999) 803.
  * (95) H. Kuyrukcu, PP-wave solutions of the Weyl-Yang theory of gravity with a five-dimensional Kaluza-Klein ansatz, Class. Quantum Grav. 30 (2013) 155013.
  * (96) R. Emparan and H. S. Reall, Black Holes in Higher Dimensions, Living Rev. Relat. 11 (2008) 6.
  * (97) T. T. Wu and C. N. Yang, in Properties of Matter Under Unusual Conditions, Festschrift for the 60th birtday of E. Teller, Interscience, New York, 1968 pg. 349.
  * (98) T. T. Wu and C. N. Yang, Some remarks about unquantized non-Abelian gauge fields, Phys. Rev. D 12 (1975) 3843.
  * (99) A. Actor, Classical solutions of $SU(2)$ Yang-Mills theories, Rev. Mod. Phys. 51 (1979) 461.
  * (100) M. Carmeli, Kh. Huleihil and E. Leibowitz, Gauge fields, Classification and Equations of Motion, World Sci., Singapore (1989).
  * (101) C.-S. Chu, S.-L. Ko and P. Vanichchapongjaroen, Non-abelian self-dual string solutions, J. High Energy Phys. 2012 (2012) 18.
  * (102) P. A. M. Dirac, Quantised Singularities in the Electromagnetic Field, Proc. R. Soc. London Ser. A 133 (1931) 60.
  * (103) P. Goddard and D. I. Olive, Magnetic monopoles in gauge field theories, Rep. Prog. Phys. 41 (1978) 1357.
  * (104) Y. M. Shnir, Magnetic monopoles, Berlin, Germany (2005), pg. 532.
  * (105) K. Konishi and K. Takenaga, Confinement, chiral symmetry breaking and Faddeev-Niemi decomposition in QCD, Phys. Lett. B 508 (2001) 392.
  * (106) J. P. Preskill, Magnetic Monopoles, Annu. Rev. Nucl. Part. Sci. 34 (1984) 461.
  * (107) F. A. Bais and R. J. Russell, Magnetic-monopole solution of non-Abelian gauge theory in curved spacetime, Phys. Rev. D 11 (1975) 2692.
  * (108) Y. M. Cho and P. G. O. Freund, Gravitating ’t Hooft monopoles, Phys. Rev. D 12 (1975) 1588.
  * (109) P. B. Yasskin, Solutions for gravity coupled to massless gauge fields, Phys. Rev. D 12 (1975) 2212.
  * (110) M. S. Volkov and D. V. Galt’sov, Gravitating non-Abelian solitons and black holes with Yang-Mills fields, Phys. Rep. 319 (1999) 1.
  * (111) A. Einstein and P. Bergmann, On a Generalization of Kaluza’s Theory of Electricity, Ann. Math. 39 (1938) 683.
  * (112) B. S. De Witt, Relativity, Groups and Topology, New York: Gordon and Breach (1964), pg. 725.
  * (113) B. S. De Witt, Dynamical Theory of Groups and Fields, New York: Gordon and Breach (1965), pg. 139.
  * (114) R. D. Sorkin, Kaluza-Klein Monopole, Phys. Rev. Lett. 51 (1983) 87.
  * (115) D. J. Gross and M. J. Perry, Magnetic monopoles in Kaluza-Klein theories, Nucl. Phys. B 226 (1983) 29.
  * (116) H. M. Lee and S. C. Lee, Spherically symmetric Kaluza-Klein monopoles, Phys. Lett. B 149 (1984) 95.
  * (117) S. C. Lee, Kaluza-Klein dyons and the Toda lattice, Phys. Lett. B 149 (1984) 98, ibid Kaluza-Klein monopoles in six dimensions 149 (1984) 100.
  * (118) M. J. Perry, Non-abelian Kaluza-Klein monopoles, Phys. Lett. B 137 (1984) 171.
  * (119) I. G. Angus, Magnetic monopoles in nonabelian Kaluza-Klein theories, Nucl. Phys. B 264 (1986) 337.
  * (120) I. I. Cotăescu, Generalized Dirac monopoles in non-Abelian Kaluza-Klein theories, Nucl. Phys. B 719 (2005) 140.
  * (121) M. Arık, E. Hizel and A. Mostafazadeh, The Schwarzschild solution in non-Abelian Kaluza-Klein theory, Class. Quantum Grav. 7 (1990) 1425.
  * (122) F. A. Lunev, Three dimensional Yang-Mills theory in gauge invariant variables, Phys. Lett. B 295 (1992) 99.
  * (123) J. Okada, Symmetry breakings in the Kaluza-Klein theory, Class. Quantum Grav. 3 (1986) 221.
  * (124) I. M. Benn and R. W. Tucker, An Introduction to Spinors and Geometry with Applications in Physics, Bristol, Philadelphia (1987), pg. 215.
  * (125) H. C. Lee, An Introduction to Kaluza-Klein Theories: Proceedings of the Workshop on Kaluza-Klein Theories, Chalk River, Canada (1983), pg. 167.
  * (126) T. Dereli and G. Üçoluk, Direct-curvature Yang-Mills field couplings induced by the Kaluza-Klein reduction of Euler form actions in seven dimensions, Class. Quantum Grav. 7 (1990) 533.
  * (127) R. Kerner, Generalization of the Kaluza-Klein theory for an arbitrary non-Abelian group, Ann. Inst. Henri Poincaré 9 (1968) 143.
  * (128) C. A. Orzalesi and M. Pauri, Geodesic motion in multidimensional unified gauge theories, Il Nuovo Cimento B 68 (1982) 193.
  * (129) M. Camenzind, On the curvature dynamics for metric gravitational theories, J. Math. Phys. 16 (1975) 1023.
  * (130) M. Camenzind, On the Yang-Mills structure of gravitation: A new issue of the final state, Gen. Relativ. Gravit. 8 (1977) 103.
  * (131) P. Ellicott and D. J. Toms, Grassmannian Kaluza-Klein theory, Class. Quantum Grav. 6 (1989) 1033.
  * (132) P. S. Wesson, Space Time Matter: Modern Kaluza-Klein Theory, World Scientific Publishing Co. Pte. Ltd., Singapore (1999), pg. 149.
  * (133) D. W. Henderso, Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces, 2nd edition, Pearson Education (2001), chap. 20.
  * (134) M. Berger, Les variétés riemanniennes homogénes normales simplement connexes à courbure strictement positive, Ann. Scuola Norm. Sup. Pisa 15 (1961) 179.
  * (135) A. H. Taub, Empty Space-Times Admitting a Three Parameter Group of Motions, Ann. Math. 53 (1951) 472.
  * (136) E. Newman, L. Tamburino and T. Unti, Empty-Space Generalization of the Schwarzschild Metric, J. Math. Phys. 4 (1963) 915.
  * (137) C. W. Misner, The Flatter Regions of Newman, Unti, and Tamburino’s Generalized Schwarzschild Space, J. Math. Phys. 4 (1963) 924.
  * (138) T. C. Shen and J. Sobczyk, Higher-dimensional self-consistent solution with deformed internal space, Phys. Rev. D 36 (1987) 397.
  * (139) M. J. Duff, B. E. W. Nilsson and C. N. Pope, Kaluza-Klein supergravity, Phys. Rept. 130 1 (1986), pg. 38.
  * (140) J. S. Dowke, Effective actions on the squashed 3-sphere, Class. Quantum Grav. 16 (1999) 1937.
  * (141) S. Başkal and T. Dereli, Higher-derivative effective Yang-Mills theory and static spherically symmetric field configurations, J. Phys. G: Nucl. Part. Phys. 19 (1993) 477.
  * (142) J.-P. Ngome, (Super)symmetries of semiclassical models in theoretical and condensed matter physics [arXiv:1103.4876].
  * (143) I. M. Benn and R. W. Tucker, An Introduction to Spinors and Geometry with Applications in Physics, Bristol, Philadelphia (1987), pg. 192.
  * (144) Atlas 2: http://www.digi-area.com/Mathematica/atlas/.
  * (145) R. P. Mondaini and N. O. Santos, Conformally flat Einstein-Yang-Mills-Higgs solutions with spherical symmetry, Phys. Rev. D 28 (1983) 1872.
  * (146) D. Singleton, Exact Schwarzschild-like solution for Yang-Mills theories, Phys. Rev. D 51 (1995) 5911.
  * (147) J. Schwinger, A Magnetic Model of Matter, Science 165 (1969) 757.
  * (148) B. Julia and A. Zee, Poles with both magnetic and electric charges in non-Abelian gauge theory, Phys. Rev. D 11 (1975) 2227.
  * (149) G. ’t Hooft, Magnetic monopoles in unified gauge theories, Nucl. Phys. B 79 (1974) 276.
  * (150) A. M. Polyakov, Particle spectrum in quantum field theory, JETP Lett. 20 (1974) 194, ibid Isomeric states of quantum fields, Sov. Phys.-JETP 41 (1975) 988, ibid Compact gauge fields and the infrared catastrophe, Phys. Lett. B 59 (1975) 82.
  * (151) B. Carter, in Black Holes, Gordon and Breach, New York (1973).
  * (152) L. J. Romans, Supersymmetric, cold and lukewarm black holes in cosmological Einstein-Maxwell theory, Nucl. Phys. B 383 (1992) 395.
  * (153) D. R. Brill and S. A. Hayward, Global structure of a black hole cosmos and its extremes, Class. Quantum Grav. 11 (1994) 359.
  * (154) M. Y. Wang, A solution of coupled Einstein-SO(3) gauge field equations, Phys. Rev. D 12 (1975) 3069.
  * (155) P. van Nieuwenhuizen, D. Wilkinson and M. J. Perry, Regular solution of ’t Hooft’s magnetic monopole model in curved space, Phys. Rev. D 13 (1976) 778.
  * (156) P. Cordero, Canonical formulation of the spherically symmetric Einstein-Yang-Mills-Higgs system for a general gauge group, Ann. Phys. 108 (1977) 79.
  * (157) R. Benguria, P. Cordero and C. Teitelboim, Aspects of the Hamiltonian dynamics of interacting gravitational gauge and Higgs fields with applications to spherical symmetry, Nucl. Phys. B 122 (1977) 61.
  * (158) A. Comtet, Magnetic monopoles in curved spacetimes, Ann. Inst. Henri Poincaré 32 (1980) 283.
  * (159) C. M. Chambers, The Cauchy horizon in black hole de sitter space-times, Annals Israel Phys.Soc. 13 (1997) 33.
  * (160) H. Stephani, D. Karamer, M. A. H. Maccallum, C. Hoenselaers and E. Herlt, Exact Solutions of Einstein’s Field Equations, 2nd edition, Cambridge University Press, Cambridge, England (2003).
  * (161) A. G. Riess and et al., Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, Astron. J. 116 (1998) 1009.
  * (162) S. Perlmutter and et al., Measurements of $Ω$ and $Λ$ from $42$ High-Redshift Supernovae, Astrophys. J. 517 (1999) 565.
  * (163) P. J. E. Peebles and B. Ratra, The cosmological constant and dark energy, Rev. Mod. Phys. 75 (2003) 559.
  * (164) V. Kagramanova, J. Kunz and C. Lämmerzahl, Solar system effects in Schwarzschild-de Sitter space-time, Phys. Lett. B 634 (2006) 465.
  * (165) K. Lake, Reissner-Nordstörm-de Sitter metric, the third law, and cosmic censorship, Phys. Rev. D 19 (1979) 421.
  * (166) R. Bousso, Quantum global structure of de Sitter space, Phys. Rev. D 60 (1999) 063503.
  * (167) M. A. Rahman and M. H. Ali, Transverse wave propagation in relativistic two-fluid plasmas around Reissner-Nordström-de Sitter black hole, Gen. Relativ. Gravit. 42 (2010) 1623.
  * (168) B. Hartmann, Y. Brihaye and B. Bertrand, Spherically symmetric Yang-Mills solutions in a 5-dimensional (Anti-) de Sitter space-time, Phys. Lett. B 570 (2003) 137.
  * (169) R. M. Wald, General Relativity, The University of Chicago Press, Chicago and London (1984), chap. 12.
  * (170) S. W. Hawking, Particle creation by black holes, Commun Math. Phys. 43 (1975) 199.
  * (171) S. A. Ridgway and E. J. Weinberg, Static black hole solutions without rotational symmetry, Phys. Rev. D 52 (1995) 3440.
  * (172) G. W. Gibbons and S. W. Hawking, Cosmological event horizons, thermodynamics, and particle creation, Phys. Rev. D 15 (1977) 2738.
  * (173) R. B. Mann and S. F. Ross, Cosmological production of charged black hole pairs, Phys. Rev. D 52 (1995) 2254.
  * (174) F. Mellor and I. Moss, Black holes and quantum wormholes, Phys. Lett. B 222 (1989) 361, ibid Black holes and gravitational instantons, Class. Quantum Grav. 6 (1989) 1379.
  * (175) A. R. Lugo and F. A. Schaposnik, Monopole and dyon solutions in AdS space, Phys. Lett. B 467 (1999) 43.
  * (176) A. Ashtekar, C. Beetle and S. Fairhurst, Mechanics of isolated horizons, Class. Quantum Grav. 17 (2000) 253.
  * (177) A. Ashtekar, S. Fairhurst and B. Krishnan, Isolated horizons: Hamiltonian evolution and the first law, Phys. Rev. D 62 (2000) 104025.
  * (178) A. Corichi and D. Sudarsky, Mass of colored black holes, Phys. Rev. D. 61 (2000) 101501.
  * (179) A. Corichi, U. Nucamendi and D. Sudarsky, Einstein-Yang-Mills isolated horizons: Phase space, mechanics, hair, and conjectures, Phys. Rev. D (2000) 62 044046\.
  * (180) D. Sudarsky and R. M. Wald, Extrema of mass, stationarity, and staticity, and solutions to the Einstein-Yang-Mills equations, Phys. Rev. D. 46 (1992) 1453.
  * (181) B. Hartmann, B. Kleihaus and J. Kunz, Axially symmetric monopoles and black holes in Einstein-Yang-Mills-Higgs theory, Phys. Rev. D 65 (2002) 024027.
  * (182) S. F. Kleihaus, J. Kunz and F. Navarro-Lérida, Rotating Einstein-Yang-Mills black holes, Phys. Rev. D 66 (2002) 104001.
  * (183) J. E. Baxter, M. Helbling and E. Winstanley, Soliton and black hole solutions of $su(N)$ Einstein-Yang-Mills theory in anti-de Sitter space, Phys. Rev. D 76 (2007) 104017.
  * (184) J. E. Baxter and E. Winstanley, On the existence of soliton and hairy black hole solutions of ${\mathfrak{su}}(N)$ Einstein-Yang-Mills theory with a negative cosmological constant, Class. Quantum Grav. 25 (2008) 245014.
  * (185) B. L. Shepherd and E. Winstanley, Characterizing asymptotically anti-de Sitter black holes with abundant stable gauge field hair, Class. Quantum Grav. 29 (2012) 155004.
  * (186) M. Kasuya and M. Kamata, An exact dyon solution with the Reissner-Nordström metric, Il Nuovo Cimento B 66 (1981) 75.
  * (187) M. Kasuya, Gravitating Julia-Zee dyon and the cosmological term, Gen. Relativ. Gravit. 14 (1982) 293.
  * (188) M. Kasuya, Exact solution of a rotating dyon black hole, Phys. Rev. D 25 (1982) 995.
  * (189) M. J. Perry, Black holes are coloured, Phys. Lett. B 71 (1977) 234.
  * (190) R. L. Stuller, $SU(2)$ Yang-Mills Coulomb Green’s functions in the presence of a generalized Wu-Yang configuration, Phys. Rev. D 22 (1980) 2510.
  * (191) M. K. Prasad and C. M. Sommerfield, Exact Classical Solution for the ’t Hooft Monopole and the Julia-Zee Dyon, Phys. Rev. Letts. 35 (1975) 760.
  * (192) E. B. Bogomolnyi, The stability of classical solutions, Sov. J. Nucl. Phys. 24 (1976) 449.
  * (193) W. J. Marciano, Magnetic monopoles and non-Abelian gauge theories, Int. J. Theor. Phys. 17 (1978) 275.
  * (194) I. G. Angus, Spherically symmetric solutions in Abelian Kaluza-Klein theories, Nucl. Phys. B 264 (1986) 349.
  * (195) F. Kottler, Über die physikalischen Grunglagen der Einsteinschen Gravitationstheorie, Ann. Physik. 56 (361) (1918) 401.
  * (196) H. Weyl, Über die statischen, kugelsymmetrischen Lösungen von Einsteins kosmologischen Gravitationsgleichungen, Phys. Zeitschr. 20 (1919) 31.
  * (197) E. Trefftz, Das statische Gravitationsfeld zweier Massenpunkte in der Einsteinschen Theorie, Math. Ann. 86 (1922) 317.
  * (198) C. Molina, D. Giugno, E. Abdalla and A. Saa, Field propagation in de Sitter black holes, Phys. Rev. D 69 (2004) 104013.
  * (199) C. Barcelo and M. Visser, Living on the edge: cosmology on the boundary of anti-de Sitter space, Phys. Lett. B 482 (2000) 183.
  * (200) R. Maartens, Brane-World Gravity, Living Rev. Relat. 7 (2004) 7, pg. 43.
  * (201) R. Maartens and K. Koyama, Brane-World Gravity, Living Rev. Relat. 13 (2010) 5, pg. 47.
  * (202) B. C. Nolan and E. Winstanley, On the existence of dyons and dyonic black holes in Einstein-Yang-Mills theory, Class. Quantum Grav. 29 (2012) 235024.
  * (203) W. Marciano and H. Pagels, Quantum chromodynamics, Phys. Rep. 36 (1978) 137, pg. 244.
  * (204) P. Rossi, Exact results in the theory of non-Abelian magnetic monopoles Phys. Rep. 86 317 (1982), pg. 337.
  * (205) R. Teh and K.-M. Wong, Static monopoles and their anticonfigurations, Int. J. Mod. Phys. A 20 (2005) 4291.
  * (206) R. Teh, A Note on the Julia-Zee Dyon Solution [arXiv:0505058].
  * (207) D. R. Noakes, The initial value formulation of higher derivative gravity, J. Math. Phys. 24 (1983) 1846.