# Slow-rotating charged black hole solution in dynamical Chern-Simons modified
gravity

G. G. L. Nashed<EMAIL_ADDRESS>Centre for Theoretical Physics, The British
University, P.O. Box 43, El Sherouk City, Cairo 11837, Egypt  Shin’ichi Nojiri
<EMAIL_ADDRESS>Department of Physics, Nagoya University,
Nagoya 464-8602, Japan
&
Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University, Nagoya 464-8602, Japan

(August 27, 2024)

###### Abstract

The Chern-Simons (CS) gravity is a modified theory of Einstein’s general
relativity (GR). The CS theory arises from the low energy limit of string
theory which involves anomaly correction to the Einstein-Hilbert action. The
CS term is given by the product of the Pontryagin density with a scalar field.
In this study, we derive a charged slowly rotating black hole (BH) solution.
The main incentives of this BH solution are axisymmetric and stationary and
form distortion of the Kerr-Newman BH solution with a dipole scalar field.
Additionally, we investigate the asymptotic correction of the metric with the
inverse seventh power of the radial distance to the BH solution, This
indicates that it will escape any meaningful constraints from weak-field
experiments. To find this kind of BHs by observations, we investigate the
propagation of the photon near the BH and we show that the difference between
the left-rotated polarization and the right-handed one could be observed as
stronger than the case of the Kerr-Newman BH. Finally, we derived the
stability condition using the geodesic deviations.

###### pacs:

04.50.Kd,97.60.Lf,04.25.-g,04.50.Gh

##  I Introduction

Is Einstein’s theory of general relativity (GR) still right? The solar system
Will:2014kxa ; Bertotti:2003rm  and the binary pulsar observations
Stairs:2003eg  have affirmed that GR could be right to ultra-precision in the
weak field domain, where the gravitational field is weak and non-dynamical.
The new gravitational wave observations detected by the advanced Laser
Interferometer Gravitational wave Observatory (LIGO) Abramovici:1992ah ;
LIGOScientific:2007fwp ; Harry:2010zz ; LIGOScientific:2007fwp  and the
advanced Virgo detector Caron:1997hu ; Giazotto:1988gw ; VIRGO:2014yos  have
ensured that GR appears to be on a solid base, even in the rather strong
gravity region, where the gravitational interaction is highly dynamical
Berti:2018cxi ; Berti:2018vdi .

According to these observations, one might wonder if additional tests of GR
are still urgent. The main motivations for continuous checking are
observational and theoretical. According to the observation viewpoint, the
late-time acceleration of the universe as inferred through supernova
SupernovaSearchTeam:2004lze , the rotation curves of galaxies Rubin:1970zza ,
and different other observations Helbig:1999cu ; Clowe:2006eq ; SDSS:2006lmn ;
Planck:2018vyg  point to anomalies that might be either solved by mentioning
the presence of dark energy Huterer:1998qv  and dark matter Carr:1994ci ,
otherwise, by amending GR on large scales Scherrer:2004au . According to a
theoretical viewpoint, the inherent inconsistency of GR with quantum mechanics
has given rise to the emergence of amendment theories that try a
reconciliation Ashtekar:2004eh ; Rovelli:2004tv . In any case of the
argumentation, additional testing could supply extra hints that might help in
solving some of these oddities.

The gravitational parity violation as evolved in the dynamical Chern-Simons
(dCS) gravity is the specific modification of GR that has attracted some
interests Jackiw:2003pm ; Alexander:2009tp . This theory amends the Einstein
GR by a dynamical (pseudo) scalar field that couples non-minimally with
curvature by the Pontryagin density. The magnitude of deviation in dCS theory
from GR is ruled by the size of its dimensional coupling parameter $\xi$.
Nowadays, dCS gravity theory is understood as an effective model, which is
valid in a region where the energy scale or curvature is small compared with
some cut-off scale, i.e., when the coupling $\xi$ is small. The reason is the
fact that the dCS theory is motivated by heterotic string theory at
4-dimensional compactification and a low-curvature expansion Alexander:2004xd
, from effective field theories of inflation Weinberg:2008hq , and loop
quantum gravity at the promotion of the Barbero-Immirzi parameter to the field
in the existence of matter Taveras:2008yf ; Calcagni:2009xz .

The gravitational parity violation in dCS theory appears in the systems that
break parity by the presence of a preferred axis like the one prescribed by
angular momentum in the dynamical system Yunes:2008ua ; Alexander:2007zg ;
Smith:2007jm ; Alexander:2007vt ; Nakamura:2018yaw ; Yagi:2013mbt . An example
is the isolated spinning BH, whose solutions in the dCS theory are known up to
the fifth order in the slow-rotation expansion Yunes:2009hc ; Yagi:2012ya ;
Maselli:2017kic , and in the near-maximum McNees:2015srl ; Chen:2018jed .
Another example is the spinning planet or star, whose behaviors in the solar
system have been investigated and observed for approximate horn. Therefore, we
expect that any dCS deformation from GR forecasts in the spin dynamics of the
solar system may be employed to limit the theory.

The CS theory is classified into the non-dynamical type and the dynamical
type. In the dynamical case, the (pseudo) scalar field evolves in time by
following the field equations. In the non-dynamical type, there is no kinetic
term for the scalar field and therefore the variation with respect to the
scalar field gives the constraint that the gravitational Pontryagin density
$R\tilde{R}$ identically vanishes although the Chern-Simons term gives non-
trivial-contributions. The two types present amended equations of motion
compared to general relativity field equations.

In the non-dynamical CS model, Alexander and Yunes Alexander:2007vt  have
shown that the gravitomagnetic sector of the metric is amended in the solar
system, yielding a new parameterized post-Newtonian parameter Alexander:2007zg
. Similarly, Smith, et al. Smith:2007jm  have calculated the correction for
the spin-precession using a uniform distribution of density objects which
uniform rotation and compared their result to the observations using the
Gravity Probe B (GPB) experiment Everitt:2011hp  and the LAGEOS satellites to
put a constraint on the non-dynamical theory. Soon, Ali-Haïmoud and Chen Ali-
Haimoud:2011zme  studied the dynamical theory and computed the extra
correction to the gravitomagnetic of the metric and put an approximate
constraint on the theory of
$\xi_{\mathrm{CS}}^{1/4}\lesssim\mathcal{O}\left(10^{8}\,\mathrm{km}\right)$.
Moreover, the possibility of comparable constraints using quantum/Sagnac
interferometry has been investigated in Okawara:2012bi ; Okawara:2013wc ;
Kikuchi:2014mva . Soon, observing gravitational waves emitted by the rotating
BHs will provide us with an eight-degree improvement in such constraints
Loutrel:2018rxs , once these discoveries are made that are powerful enough to
break degeneracies between the spins of the objects and the dCS deformation.
Rotating black hole solutions in the $(3+1)$-dimensional Chern-Simons modified
gravity theory are investigated by taking account of perturbation around the
Schwarzschild solution Konno:2007ze . A detailed study of the BH solutions in
the Chern-Simons gravity has revealed that at least two different limits of
the Kerr BH are solutions to the modified field equations Grumiller:2007rv .
In the extended Chern-Simons modified gravity, an investigation of a rotating
BH has been carried out Konno:2009kg . A study of the null geodesics
corresponding to a slowly rotating BH in Chern Simons gravity with a small
coupling constant revealed that the photon orbits are separable as in the Kerr
geometry Amarilla:2010zq . A study of the null geodesics corresponding to a
slowly rotating BH in Chern Simons gravity, with a small coupling constant has
shown that the photon orbits are separable as in the Kerr geometry
Brihaye:2016lsx . Four-dimensional homogeneous static and rotating black
strings, with and without torsion in the dCS modified gravity, are presented
Cisterna:2018jsx .

Yunes and Pretorius Yunes:2007ss  have derived a non-trivial uncharged black
solution in the dCS theory that generates the parity violation. The purpose of
the present study is to derive a new weakly charged rotating BH solution in
the framework of the dCS gravitational theory.

The construction of this study is as follows: In Sec. II, we present the main
tools of CS-modified gravity. In Sec. III, we solve the field equations of the
dCS gravity for the line element prescribing slowly rotating BH which is valid
for small CS coupling constants. In Sec. IV, we consider some of the related
physics to the derived solutions to understand its physical properties. In
Sec. V, we study the motion of the CS BH presented in this study and derive
the conserved quantities of this BH and its orbital period. In Sec. VI, we
investigate the propagation of the photon near the BH and we consider the
difference of the polarization from that of the Kerr-Newman BH. In Sec. VII,
we conclude the main results of this study and present possible future work.

The following conventions are used throughout the present study: We use four-
dimensional space-times that have the following signature $(-,+,+,+)$
Misner:1973prb , square brackets and parentheses mean anti-symmetrization and
symmetrization, respectively, i.e.,
$T_{[\mu\nu]}=\frac{1}{2}(T_{\mu\nu}-T_{\nu\mu})$ and
$T_{(\mu\nu)}=\frac{1}{2}(T_{\mu\nu}+T_{\nu\mu})$. The partial derivatives are
refereed by commas (e.g. $\partial\varphi/\partial
r=\partial_{r}\varphi=\varphi_{,r}$). The Einstein summation
$A_{\mu}B^{\mu}=\sum_{\mu=0,1,2,3}A_{\mu}B^{\mu}$ is applied and geometrized
units $G=c=1$ are used.

##  II CS modified gravitational theory

In this section, we will prescribe the subjects which give a full construction
of the CS-modified gravitational theory and present some notation
Alexander:2009tp .

###  II.1 ABC of CS gravitational theory

The action of the CS gravitational theory is given as,

$\displaystyle S=S_{1}+S_{2}+S_{3}+S_{4}+S_{5}\,,$ (1)

where $S_{1}$ is the Einstein Hilbert action defined by,

$\displaystyle S_{1}=\frac{1}{2\kappa^{2}}\int_{V}d^{4}x\sqrt{-g},,$ (2)

$S_{2}$ is the Chern-Simon action given by,

$\displaystyle S_{2}=\frac{\sigma}{4}\int_{V}d^{4}x\sqrt{-g}\varphi
R\tilde{R}\,,$ (3)

$S_{3}$ is the action of the (pseudo) scalar field $\varphi$ with the
potential $U(\varphi)$ defined by,

$\displaystyle
S_{3}=-\frac{\sigma_{1}}{2}\int_{V}d^{4}x\sqrt{-g}\left[g^{\mu\nu}\left(\nabla_{\mu}\varphi\right)\left(\nabla_{\nu}\varphi\right)+2U(\varphi)\right]\,,$
(4)

$S_{4}$ is the action of the electromagnetic field given by,

$\displaystyle
S_{4}=\frac{\sigma_{1}}{4}\int_{V}d^{4}x\sqrt{-g}F^{2}\quad\mathrm{where}\quad
F^{2}=F_{\mu\nu}F^{\mu\nu}\,,\quad\mbox{and}\quad
F_{\mu\nu}=A_{\mu,\nu}-A_{\nu,\mu}\,,$ (5)

and $S_{5}$ is the action of the matters defined by,

$\displaystyle S_{5}=\int_{V}d^{4}x\sqrt{-g}{L}_{\mathrm{mat}}.$ (6)

Here $V$ is the bulk region in the space-time manifold.

The following notations are used throughout the whole of the present study:
$\sigma$ and $\sigma_{1}$ are dimensional constants, $\kappa^{2}=8\pi G$,
$\nabla_{\rho}$ is the covariant derivative, $R$ is the Ricci scalar, and $g$
is the determinant of the metric. The term
$R\tilde{R}$ is the Pontryagin density defined by,

$\displaystyle R\tilde{R}=R^{\nu}_{\ \mu\rho\sigma}\tilde{R}^{\mu\
\rho\sigma}_{\ \nu}\,,$ (7)

where $\tilde{R}^{\mu\ \rho\sigma}_{\ \nu}$ is the dual Riemann tensor which
is defined by,

$\displaystyle\tilde{R}^{\mu\ \rho\sigma}_{\
\nu}=\frac{1}{2}\epsilon^{\rho\sigma\tau\eta}R^{\mu}_{\ \nu\tau\eta}\,,$ (8)

where $\epsilon^{\rho\sigma\tau\eta}$ is the 4-dimensional Levi-Civita tensor
which is a totally skew-symmetric tensor with $\epsilon^{0123}=-1$.

In this study, the CS scalar field $\varphi$ is a function of the space-time
coordinates that parameterize the deviation from the Einstein GR. If
$\varphi=\mathrm{constant}$, the CS gravitational theory will be equivalent to
the Einstein GR theory because the Pontryagin density is the total divergence
of the CS topological current $K^{\mu}$ which is given by,

$\displaystyle\nabla_{\mu}K^{\mu}=\frac{1}{2}R\tilde{R}\,,$ (9)

where

$\displaystyle
K^{\mu}=\epsilon^{\mu\nu\rho\sigma}\Gamma^{\tau}_{\nu\eta}\left(\partial_{\rho}\Gamma^{\eta}_{\sigma\tau}+\frac{2}{3}\Gamma^{\eta}_{\rho\xi}\Gamma^{\xi}_{\sigma\tau}\right)\,,$
(10)

with $\Gamma^{\mu}_{\nu\rho}$ being the Christoffel connection. Eq. (9) tells
that $S_{2}$ can be rewritten in the form Yunes:2007ss :

$\displaystyle S_{2}=\frac{\sigma}{2}\int_{\partial V}dS_{\mu}\varphi
K^{\mu}-\frac{\sigma}{2}\int_{V}d^{4}x\sqrt{-g}\left(\nabla_{\mu}\varphi\right)K^{\mu}\,.$
(11)

Here $\partial V$ is the boundary hypersurface of the space-time manifold and
$dS_{\mu}$ is the infinitesimal area of the hypersurface $\partial V$. We
often neglect the first term in Eq. (11) because it is irrelevant to the
variation in the bulk space-time manifold when we derive the field equations.

The variations of the action (1) w.r.t. the metric, the scalar field, and the
electromagnetic field give the following field equations,

$\displaystyle R_{\mu\nu}+2\kappa^{2}\sigma_{1}C_{\mu\nu}=$
$\displaystyle\,\kappa^{2}\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T\right)\,,$
(12) $\displaystyle\sigma_{1}\square\varphi=$
$\displaystyle\,\sigma_{1}\frac{dV}{d\varphi}-\frac{\sigma}{4}R\tilde{R}\,,$
(13) $\displaystyle 0=$
$\displaystyle\,\partial_{\nu}\left(\sqrt{-g}F^{\mu\nu}\right)\,,$ (14)

where $R_{\mu\nu}$ is the Ricci tensor and $\square=\nabla_{\mu}\nabla^{\mu}$
is the D’Alembertian operator. The term $C_{\mu\nu}$ is the C-tensor which is
defined by,

$\displaystyle
C^{\mu\nu}=v_{\rho}\epsilon^{\rho\sigma\tau(\mu}\nabla_{\tau}R^{\nu)}_{\
\sigma}+v_{\rho\sigma}\tilde{R}^{\sigma(\mu\nu)\rho}\,,$ (15)

with

$\displaystyle v_{\mu}=\nabla_{\mu}\varphi\,,\qquad\mathrm{and}\qquad
v_{\mu\nu}=\nabla_{\mu}\nabla_{\nu}\varphi\,.$ (16)

As a final point, the total stress-energy tensor is written as,

$\displaystyle
T_{\mu\nu}=T^{\mathrm{mat}}_{\mu\nu}+T_{\mu\nu}^{\varphi}+T_{\mu\nu}^{\mathrm{EM}},$
(17)

with $T^{\mathrm{mat}}_{\mu\nu}$ being the matter stress-energy tensor (which
we will neglect in the present study), $T_{\mu\nu}^{\mathrm{EM}}$ and
$T_{\mu\nu}^{\varphi}$, are the stress-energy tensors of the electromagnetic
and scalar fields figured as,

$\displaystyle T_{\mu\nu}^{\mathrm{EM}}=$
$\displaystyle\,\frac{\sigma_{1}}{4\pi}\left[g_{\rho\sigma}F_{\nu}^{\
\rho}F_{\mu}^{\ \sigma}-\frac{1}{4}g_{\mu\nu}F^{2}\right]\,,$ $\displaystyle
T_{\mu\nu}^{\varphi}=$
$\displaystyle\,\sigma_{1}\left[\left(\nabla_{\mu}\varphi\right)\left(\nabla_{\nu}\varphi\right)-\frac{1}{2}g_{\mu\nu}\left(\nabla_{\rho}\varphi\right)\left(\nabla^{\rho}\varphi\right)-g_{\mu\nu}U(\varphi)\right]\,.$
(18)

In the realm of the CS gravitational theory, the strong equivalence principle,
i.e., $\left(\nabla^{\nu}T_{\mu\nu}^{\mathrm{mat}}=0\right)$, is verified
provided that the equation of motions of the scalar field $\varphi$, Eq. (13)
is satisfied. This is because if we consider the derivative of Eq. (12), the
first term on the l.h.s. vanishes due to the Bianchi identities, while the
second term is proportional to the Pontryagin density through the form,

$\displaystyle\nabla_{\mu}C^{\mu\nu}=-\frac{1}{8}v^{\nu}R\tilde{R}.$ (19)

The verification of Eq. (19) yields Eq. (13).

To finish this section, we are going to discuss the dimensions of the coupling
constants used in this study and the scalar field $\varphi$ and the
electromagnetic field $A_{\mu}$. By fixing the dimensions of
$(\sigma,\sigma_{1},\varphi,A_{\mu})$, the units of the other constants are
also fixed. For example, if the CS scalar field and the electromagnetic field
have the dimensions $[\varphi]=[A_{\mu}]=l^{s}$, then $[\sigma]=l^{2-s}$ and
$[\sigma_{1}]=l^{-2s}$, where $l$ expresses the dimension of length. The CS
scalar $\varphi$ and the electromagnetic field $A_{\mu}$ is often
dimensionless, which requires $[\sigma]=\left[A_{\mu}\right]=l^{2}$ and
$\sigma_{1}$ be dimensionless111In this study, we use the geometrical units
with $G=c=1$, and thus, the action has the units of $l^{2}$. Therefore, if
natural units are used where $\hbar=c=1$, then the action will be
dimensionless and therefore if $[\varphi]=[A_{\mu}]=l^{s}$ then
$[\sigma]==\left[A_{\mu}\right]l^{-s}$ and $[\sigma_{1}]=l^{-2s-2}$.. Other
selection is to put $\sigma=\sigma_{1}$, thus putting $S_{2}$, $S_{3}$, and
$S_{4}$ on equal footing and we have $[\varphi][A_{\mu}]=l^{-2}$. We will
leave this arbitrariness because the results of the previous studies are based
on the different choices of the unit.

##  III Rotating Charged BH solution in dynamical CS gravity

Now, we are going to study rotating charged BHs in the dynamical construction
of the modified CS theory. The study of stationary axisymmetric line elements
in the frame of the CS gravitational theory without doing any approximation in
the calculation will be a tedious task. Therefore, we will use a pair of
approximations. Thence, we will proceed to solve the modified CS equation of
motions to second order in perturbation expansion. In the following, we only
consider the case that the potential $U(\varphi)$ for the scalar field
$\varphi$ vanishes, $U(\varphi)=0$.

###  III.1 The process of approximation

Now we will use two approximation processes: a slow-rotation and small-
coupling approximations. The small-coupling process deals with the modified CS
term as a small distortion of GR, which allows expanding the metric and the
gauge potential $A_{\mu}$ (up to second order) as follows,

$\displaystyle g_{\mu\nu}=g_{\mu\nu}^{(0)}+\xi
g^{(1)}_{\mu\nu}(\varphi)+\xi^{2}g^{(2)}_{\mu\nu}(\varphi)\,,\quad
A_{a}=A_{\mu}^{(0)}+\xi A^{(1)}_{\mu}+\xi^{2}A^{(2)}_{\mu}\,$ (20)

where $g_{\mu\nu}^{(0)}$ and $A_{\mu}^{(0)}$ are the background metric and
charge which satisfy the Einstein GR field equations, such as the Kerr-Newmann
metric, while $g_{\mu\nu}^{(1)}(\varphi)$, $A^{(1)}_{\mu}$,
$g_{\mu\nu}^{(2)}(\varphi)$, and $A^{(2)}_{\mu}$ are the first and the second-
order perturbation coming from the CS corrections. The parameter $\xi$ refers
to the order of the small-coupling approximation, which we will define soon.

On the other hand, the slow-rotation approximation allows to re-expand the
background and the $\xi$-perturbations in powers of the Kerr-Newmann rotation
parameter $a_{\mathrm{KN}}$. Therefore, the background metric and the metric
perturbation yield the following form,

$\displaystyle g_{\mu\nu}^{(0)}=$
$\displaystyle\,\eta_{\mu\nu}^{(0,0)}+\epsilon
h_{\mu\nu}^{(1,0)}+\epsilon^{2}h_{\mu\nu}^{(2,0)}\,,$ $\displaystyle\xi
g_{\mu\nu}^{(1)}=$ $\displaystyle\,\xi h_{\mu\nu}^{(0,1)}+\xi\epsilon
h_{\mu\nu}^{(1,1)}+\xi\epsilon^{2}h_{\mu\nu}^{(2,1)},$
$\displaystyle\xi^{2}g_{\mu\nu}^{(2)}=$
$\displaystyle\,\xi^{2}h_{\mu\nu}^{(0,2)}+\xi^{2}\epsilon
h_{\mu\nu}^{(1,2)}+\xi^{2}\epsilon^{2}h_{\mu\nu}^{(2,2)}\,,$ $\displaystyle
A_{\mu}^{(0)}=$ $\displaystyle\,\eta_{\mu}^{(0,0)}+\epsilon
A_{\mu}^{(1,0)}+\epsilon^{2}A_{\mu}^{(2,0)}\,,$ $\displaystyle\xi
A_{\mu}^{(1)}=$ $\displaystyle\,\xi A_{\mu}^{(0,1)}+\xi\epsilon
A_{\mu}^{(1,1)}+\xi\epsilon^{2}A_{\mu}^{(2,1)}\,,$
$\displaystyle\xi^{2}A_{\mu}^{(2)}=$
$\displaystyle\,\xi^{2}A_{\mu}^{(0,2)}+\xi^{2}\epsilon
A_{\mu}^{(1,2)}+\xi^{2}\epsilon^{2}A_{\mu}^{(2,2)}\,,$ (21)

where the parameter $\epsilon$ stands for the order of the slow-rotation
expansion, which we will also define soon. We must remind ourself that the
notation $h^{(n,m)}_{\mu\nu}$ labels for terms of
$\mathcal{O}\left(n,m\right)$, which stands for a term of
$\mathcal{O}\left(\epsilon^{n}\right)$ and $\mathcal{O}\left(\xi^{m}\right)$.
As an example, in Eq. (III.1), $\eta_{\mu\nu}^{(0,0)}$ and
$\eta_{\mu}^{(0,0)}$ are the background metric when the rotation parameter
vanishing, i.e., $a_{\mathrm{KN}}=0$, whilst $h_{\mu\nu}^{(1,0)}$,
$h_{\mu\nu}^{(2,0)}$, $A_{\mu}^{(1,0)}$ and $A_{\mu}^{(2,0)}$ are first and
second-order perturbations of the background metric and charge in the spin
parameter.

Combine both approximation processes, we obtain a bivariate expansion in terms
of two independent parameters $\xi$ and $\epsilon$, which yields the second
perturbation order of the metric and the electromagnetic field in the
following forms,

$\displaystyle g_{\mu\nu}=$ $\displaystyle\,\eta_{\mu\nu}^{(0,0)}+\epsilon
h_{\mu\nu}^{(1,0)}+\xi h_{\mu\nu}^{(0,1)}+\epsilon\xi
h_{\mu\nu}^{(1,1)}+\epsilon^{2}h_{\mu\nu}^{(2,0)}+\xi^{2}h_{\mu\nu}^{(0,2)}\,,$
$\displaystyle A_{\mu}=$ $\displaystyle\,A_{\mu}^{(0,0)}+\epsilon
A_{\mu}^{(1,0)}+\xi A_{\mu}^{(0,1)}+\epsilon\xi
A_{\mu}^{(1,1)}+\epsilon^{2}A_{\mu}^{(2,0)}+\xi^{2}A_{\mu}^{(0,2)}\,.$ (22)

The first-order expressions, refers to expressions of
$\mathcal{O}\left(1,0\right)$ or $\mathcal{O}\left(0,1\right)$, while second-
order terms refers to $\mathcal{O}\left(2,0\right)$,
$\mathcal{O}\left(0,2\right)$, or $\mathcal{O}\left(1,1\right)$.

In this study, the slow-rotation process is the expansion of the Kerr-Newmann
parameter, $a_{\mathrm{KN}}$, and therefore its dimensionless expansion
parameter $\epsilon$ should be $\epsilon=a_{\mathrm{KN}}/M$.

###  III.2 The slowly rotating charged BH solution

The slowly rotating expansion of the background metric can be formulated using
the Hartle-Thorne approximation Thorne:1984mz ; Hartle:1968si , where the line
element can be parameterized as follows,

$\displaystyle ds^{2}=$
$\displaystyle\,-h\left[1+h_{1}\left(r,\theta\right)\right]dt^{2}+\frac{1}{h}\left[1+h_{2}\left(r,\theta\right)\right]dr^{2}+r^{2}\left[1+h_{3}\left(r,\theta\right)\right]d\theta^{2}$
$\displaystyle\,+r^{2}\sin^{2}{\theta}\left[1+h_{4}\left(r,\theta\right)\right]\left[d\phi-\omega\left(r,\theta\right)dt\right]^{2}\,,$
(23)

where $h$ is defined as $h=1-\frac{2M}{r}+\frac{q^{2}}{r^{2}}$, which is given
by the Reissner-Nordström solution, with $M$ being the mass of the charged BH
and $q$ being the electric charge in the absence of the CS expression. In Eq.
(III.2), we use the Boyer-Lindquist coordinates, i.e., $(t,r,\theta,\phi)$ and
the perturbations of the metric are $h_{1}\left(r,\theta\right)$,
$h_{2}\left(r,\theta\right)$, $h_{3}\left(r,\theta\right)$,
$h_{4}\left(r,\theta\right)$, and $\omega\left(r,\theta\right)$.

The metric (III.2) is rewritten similar to the one presented in Thorne:1984mz
; Hartle:1968si , however, the metric perturbations should be expanded in a
series in both $\xi$ and $\epsilon$. By the second order expansion, we have,

$\displaystyle h_{1}\left(r,\theta\right)=$ $\displaystyle\,\epsilon
h_{1}^{(1,0)}+\epsilon\xi h_{1}^{(1,1)}+\epsilon^{2}h_{1}^{(2,0)}\,,$
$\displaystyle h_{2}\left(r,\theta\right)=$ $\displaystyle\,\epsilon
h_{2}^{(1,0)}+\epsilon\xi h_{2}^{(1,1)}+\epsilon^{2}h_{2}^{(2,0)}\,,$
$\displaystyle h_{3}\left(r,\theta\right)=$ $\displaystyle\,\epsilon
h_{3}^{(1,0)}+\epsilon\xi h_{3}^{(1,1)}+\epsilon^{2}h_{3}^{(2,0)}\,,$
$\displaystyle h_{4}\left(r,\theta\right)=$ $\displaystyle\,\epsilon
h_{4}^{(1,0)}+\epsilon\xi h_{4}^{(1,1)}+\epsilon^{2}h_{4}^{(2,0)}\,,$
$\displaystyle\omega\left(r,\theta\right)=$
$\displaystyle\,\epsilon\omega^{(1,0)}+\epsilon\xi\omega^{(1,1)}+\epsilon^{2}\omega^{(2,0)}\,.$
(24)

Equations (III.2) have no expressions of $\mathcal{O}(0,0)$ because those
terms are already involved in the Reissner-Nordström structure of Eq. (III.2).
Also, we assume that when the rotation parameter of Kerr-Newman vanishes,
i.e., $a_{\mathrm{KN}}\rightarrow 0$, we got Reissner-Nordström space-time as
a solution, which ensures that all terms of $\mathcal{O}\left(0,m\right)$
vanish. Therefore, the CS expression should be linear in the Kerr-Newmann spin
parameter $a_{\mathrm{KN}}$. Using the slow-rotation limit of the Kerr-Newman
metric in GR, the metric and charge perturbations proportional to $\xi^{0}$ in
the first order are given by,

$\displaystyle
h_{1}^{(1,0)}=h_{2}^{(1,0)}=h_{3}^{(1,0)}=h_{4}^{(1,0)}=0\,,\quad\omega^{(1,0)}=\frac{\left[2Mr-q^{2}\right]a_{\mathrm{KN}}}{r^{4}},\quad
A_{t}^{(1,0)}=\frac{q\sin^{2}\theta a_{\mathrm{KN}}}{r},$ (25)

and in the second order,

$\displaystyle h_{1}^{(2,0)}=$
$\displaystyle\,\frac{{a_{\mathrm{KN}}}^{2}\left(2Mr-q^{2}\right)}{hr^{4}}\left(\cos^{2}{\theta}+\frac{2Mr-q^{2}}{r^{2}}\sin^{2}{\theta}\right)\,,\quad
h_{2}^{(2,0)}=\frac{{a_{\mathrm{KN}}}^{2}}{r^{2}}\left(\cos^{2}{\theta}-\frac{1}{h}\right)\,,$
$\displaystyle h_{3}^{(2,0)}=$
$\displaystyle\,\frac{{a_{\mathrm{KN}}}^{2}}{r^{2}}\cos^{2}{\theta}\,,\quad
h_{4}^{(2,0)}=\frac{{a_{\mathrm{KN}}}^{2}}{r^{2}}\left(1+\frac{2Mr-q^{2}}{r^{2}}\sin^{2}{\theta}\right)\,,\quad
A_{t}^{(2,0)}=-\frac{q{{a_{\mathrm{KN}}}^{2}}}{r^{3}}\,,\quad\omega^{(2,0)}=0\,,$
(26)

whose expressions coincide with those in the Kerr solution when $q=0$
Yunes:2007ss . All the fields are expanded by small-coupling and slow-rotation
approximation parameters, including the CS field. To derive the leading-order
corrections for $\varphi$, we must yield to the evolution equation (13). From
Eq. (13), we obtain $\partial^{2}\varphi\sim(\sigma_{1}/\sigma)R\tilde{R}$,
where the Pontryagin density equal to zero up to order in $a_{\mathrm{KN}}/M$.
Thus, the first order behavior of the CS scalar field must be
$\varphi\sim\left(\sigma_{1}/\sigma\right)\left(a_{\mathrm{KN}}/M\right)$,
which is proportional to $\epsilon$. Additionally, the assumption that the
Reissner-Nordström metric is the unique charged solution with vanishing
angular momentum, we should have $\varphi^{(0,s)}=0$ for all $s$. The analysis
given in Yunes:2007ss  to derive a detorsion to the Kerr solution in the dCS
construction did not take into account the effect of the charge. On the other
hand, in the present study, we expand the analysis presented in Yunes:2007ss
to take into account the effect of the charge. By employing Eq. (13) to Eq.
(III.2) and by using Eq. (III.2), we obtain,

$\displaystyle\varphi=\epsilon\varphi^{(1,0)}\left(r,\theta\right)+\epsilon\xi\;\varphi^{(1,1)}\left(r,\theta\right)+\epsilon^{2}\varphi^{(2,0)}\left(r,\theta\right)\,.$
(27)

Now we are going to apply the process described above to solve the above-
modified field equations, by stressing the evolution equation of the dCS
scalar. Up to the order $\mathcal{O}(1,0)$, we obtain the evolution equation
in the following form,

$\displaystyle
h\varphi^{(1,0)}_{,rr}+\frac{2}{r}\varphi^{(1,0)}_{,r}\left(1-\frac{M}{r}\right)+\frac{1}{r^{2}}\varphi^{(1,0)}_{,\theta\theta}+\frac{\cot{\theta}}{r^{2}}\varphi^{(1,0)}_{,\theta}=-\frac{24\sigma
M\left(Mr-q^{2}\right)\left(3Mr-2q^{2}\right)\cos{\theta}}{\sigma_{1}r^{9}}\frac{a_{\mathrm{KN}}}{M}\,.$
(28)

The solution of partial differential equation (28) is a linear composition of
the homogeneous and the particular solution:
$\varphi^{(1,0)}=\varphi^{(1,0)}_{\mathrm{Hom}}+\varphi^{(1,0)}_{\mathrm{Part}}$.
The variables of the homogeneous solution for the equation can be separated,

$\displaystyle\varphi^{(1,0)}_{\mathrm{Hom}}\left(r,\theta\right)=\varphi(r)\varphi(\theta)\,.$
(29)

Eq. (29) shows that the partial differential equation becomes a set of
ordinary differential equations for $\varphi(r)$ and $\varphi(\theta)$, whose
solutions take the following forms,

$\displaystyle\varphi(r)=$
$\displaystyle\,c_{1}\mathcal{H}\left[\left[\frac{s}{2},\frac{s}{2}\right],s,\frac{2\sqrt{H}}{r-M+\sqrt{H}}\right]\left(r-M+\sqrt{H}\right)^{-\frac{s}{2}}$
$\displaystyle\,+c_{2}\mathcal{H}\left[\left[\frac{s_{1}}{2},\frac{s_{1}}{2}\right],s_{1},\frac{2\sqrt{H}}{r-M+\sqrt{H}}\right]\left(M-r+\sqrt{H}\right)^{-\frac{s_{1}}{2}},\quad
M>q\,,$ $\displaystyle\varphi(\theta)=$
$\displaystyle\,c_{3}L\left(-\frac{s}{2},\,\cos\theta\right)+c_{4}L_{1}\left(-\frac{s}{2},\cos\theta\right)\,,$
(30)

where $H=M^{2}-q^{2}$, and $\mathcal{H}(\cdots)$ are generalized
hypergeometric functions222 The generalized hypergeometric function
$\mathcal{H}\left(\left[n_{1},n_{2},\cdots,n_{p}\right],\left[d_{1},d_{2},\cdots,d_{q}\right],z\right)$
is generally defined by,
$\mathcal{H}\left(\bm{n},\bm{d},z\right)=\sum_{k=0}^{\infty}\frac{\prod_{i=1}^{p}\mathrm{PS}\left(n_{i},k\right)}{\prod_{j=1}^{q}\mathrm{PS}\left(d_{j},k\right)}\frac{z^{k}}{k!}\,,$
where $\bm{n}=\left[n_{1},n_{2},\cdots,n_{p}\right]$,
$\bm{d}=\left[d_{1},d_{2},\cdots,d_{q}\right]$ and $\mathrm{PS}(n,k)$ is the
Pochhammer symbol, $\mathrm{PS}(n,k)\equiv\prod_{j=0}^{k-1}\left(n+j\right)$.
$H(\cdots)$’s in (III.2) correspond to $p=2$ and $q=1$. , $L(\cdot)$ is the
Legendre polynomial of the first kind333 The Legendre polynomial of the first
kind is defined by,
$L(b,z)=\mathcal{H}\left([-b,b+1],[1],\frac{1}{2}(1-z)\right)\,.$ ,
${L_{1}}(\cdot)$ is the Legendre polynomial of the second kind444 The Legendre
polynomial of the second kind is defined by,
$L_{1}(b,z)=\frac{\sqrt{\pi}\Gamma(1+b)\mathcal{H}\left(\left[1+\frac{b}{2},\frac{1}{2}+\frac{b}{2}\right],\left[\frac{3}{2}+b\right],\frac{1}{z^{2}}\right)}{2z^{1+b}\Gamma\left(\frac{3}{2}+b\right)2^{b}}\,.$
$c_{i}$, $i=1,\cdots,4$ are constants of integration, and the constants $s$
and $s_{1}$ are defined by,

$\displaystyle s=1-\sqrt{1-4c_{5}}\,,\quad s_{1}=1+\sqrt{1-4c_{5}},$ (31)

where $c_{5}$ is the constant of integration that arises through the
separation of variables.

We study the solution of $\varphi^{(1,0)}$ in detail to understand the physics
in the constants of integration that appear in it. For this purpose, we will
consider the behavior of the solution when $r\gg M$ and obtain,

$\displaystyle\varphi(r)\sim
c_{1}\left[1+\frac{Mr-q^{2}}{2r^{2}}s\right]r^{-\frac{s}{2}}+c_{2}\left[1+\frac{Mr-q^{2}}{2r^{2}}s_{1}\right]r^{-\frac{s_{1}}{2}}.$
(32)

Moreover, we require the scalar field $\varphi$ to have a real value, then the
constants $s$ and $s_{1}$ must be real, $s,\,s_{1}\in\Re$, which requires
$c_{5}<1/4$ as we find in (31). Moreover, if we also require the scalar field
$\varphi$ to have finite total energy, thence $\varphi$ must decrease to a
constant asymptotically faster than $1/r$, which tells $s>2$ and $s_{1}>2$.
The first constraint cannot be satisfied when $c_{5}<1/4$, therefore we find
$c_{1}=0$, and the second constraint yields $c_{5}<0$. Thus, the constraints
coming from the requirement of the finite total energy tell that $\varphi$
cannot be proportional to $\ln(h)$. By summarizing the above discussion, we
obtain,

$\displaystyle\varphi^{(1,0)}_{\mathrm{Hom}}=\mathrm{const}\,.$ (33)

We should note that the expression in (III.2) diverges at the horizon, where
$r-M+\sqrt{H}$ vanishes. The above arguments about the boundary conditions,
however, tell that the homogeneous solution must be a constant as shown in Eq.
(33), and therefore the homogeneous solution does not show the singularity
anywhere. As we will see soon in (III.2) and (III.2), the expressions of the
particular part of the solutions do not include the factor $r-M+\sqrt{H}$ even
in the integrands and therefore the scalar field $\varphi$ is also regular at
the horizon.

Although we have the homogenous solution of Eq. (28), we need a particular
solution $\varphi^{(1,0)}_{\mathrm{Part}}$ to find the full inhomogeneous
solution. The particular solution is given by,

$\displaystyle\varphi^{(1,0)}_{\mathrm{Part}}\left(r,\theta\right)=$
$\displaystyle\,\frac{48\epsilon\sigma{a_{\mathrm{KN}}}}{\sigma_{1}H}\left\\{\sqrt{H}\left(r-M\right)\int\frac{\left(Mr-q^{2}\right)\left(3Mr-2q^{2}\right)\left[\left(r-M\right)\tanh^{-1}\left({\frac{M-r}{\sqrt{H}}}\right)+\sqrt{H}\right]}{r^{7}}dr\right.$
$\displaystyle\,\left.+\left(\sqrt{H}\left(r-M\right)\tanh^{-1}\left({\frac{M-r}{\sqrt{H}}}\right)-H\right)\int\frac{\left(3r^{2}M^{2}-5q^{2}Mr+2q^{4}\right)\left(M-r\right)}{r^{7}}{dr}\right\\}\,.$
(34)

Eq. (III.2) which describes the particular solution gives the following form
when the charge $q$ vanishes,

$\displaystyle\left.\varphi^{(1,0)}_{\mathrm{Part}}\left(r,\theta\right)\right|_{q=0}=\frac{144\epsilon\sigma{a_{\mathrm{KN}}}}{M\sigma_{1}}$
$\displaystyle\,\left(\left[M-r\right]\left[\int\frac{\tanh^{-1}\left(\frac{M-r}{M}\right)\left[M-r\right]-M}{r^{5}}dr\right.\right.$
$\displaystyle\,\left.\left.-\left\\{\tanh^{-1}\left(\frac{M-r}{M}\right)-\frac{M}{M-r}\right\\}\int\frac{M-r}{r^{5}}{dr}\right]\right)\,.$

The asymptotic form of Eq. (III.2) gives Eq. (35) in Yunes:2007ss  , which is
derived for slowly uncharged rotating solution in dCS theory.

Because we have succeeded to present the solution of the dCS scalar field, we
try to derive dCS corrections for the metric perturbations. We stress that the
stress-energy tensor (II.1) of the dCS scalar field appears in the modified
field equations (12) up to $\mathcal{O}(2,1)$, and thus we neglect the
contributions in the metric perturbation. In such a case, the modified
Einstein equations are divided into two types: The first one constitutes a
closed system of the differential equations involving ${h_{1}}^{(1,1)}$,
${h_{2}}^{(1,1)}$, ${h_{3}}^{(1,1)}$, and ${h_{4}}^{(1,1)}$, which comes from
the components $(t,t)$, $(r,r)$, $\left(r,\theta\right)$, $(\theta,\theta)$
and $(\phi,\phi)$-components. The second set which comes from the modified
Einstein equations yields one differential equation for $\omega^{(1,1)}$,
which is the $(t,\phi)$-component of the modified Einstein equations.

The first type does not depend on the dCS field, $\varphi$, and therefore they
are not changed from the equations in GR. Therefore, we only consider the
second set, $(t,\phi)$-component, which gives,

$\displaystyle\,\frac{576\kappa^{2}\sigma^{2}a_{\mathrm{KN}}}{r^{7}H^{3}\sigma_{1}}\left\\{H_{1}hr^{2}\int\frac{\left(Mr-q^{2}\right)\left(3Mr-2q^{2}\right)\left[\sqrt{H}-\left(M-r\right)\tanh^{-1}\left(\frac{M-r}{\sqrt{H}}\right)\right]}{r^{7}}dr+\left\\{H_{1}hr^{2}\tanh^{-1}\left(\frac{M-r}{\sqrt{H}}\right)\right.\right.$
$\displaystyle\,\left.\left.-\left(\left[q^{2}+2M^{2}\right]r^{2}-6Mq^{2}r+3q^{4}\right)\sqrt{M^{2}-q^{2}}\right\\}\int\frac{\left(Mr-q^{2}\right)\left(3Mr-2q^{2}\right)\left(M-r\right)}{r^{7}}dr\right\\}$
$\displaystyle=$
$\displaystyle\,r^{2}h\omega_{rr}\left(r,\theta\right)+\omega_{\theta\theta}\left(r,\theta\right)+4rh{\omega}_{r}\left(r,\theta\right)+3\omega_{\theta}\left(r,\theta\right)\cot\theta\,,$
(36)

where $H_{1}=\left(q^{2}+2M^{2}\right)r-3Mq^{2}$. When $q=0$, Eq. (III.2)
gives the following form,

$\displaystyle\omega^{(1,1)}_{,\theta\theta}+3\cot\theta\omega^{(1,1)}_{,\theta}+4rh\omega^{(1,1)}_{,r}+r^{2}h\omega^{(1,1)}_{,rr}\approx\frac{15\kappa^{2}\mu^{2}a_{\mathrm{KN}}h}{\nu
r^{8}}\sin^{2}\theta\left(3r^{2}+8Mr+18M^{2}\right)\,,$ (37)

which coincides with the form derived in Yunes:2007ss  in the case of the
uncharged BH. Again, the general solution is a linear combination of a
homogeneous solution and a particular solution. The particular solution is
given by,

$\displaystyle\omega^{(1,1)}=$
$\displaystyle\,-\frac{576\kappa^{2}a_{\mathrm{KN}}\sigma^{2}}{r^{5}H^{3/2}h\sigma_{1}}\int\frac{1}{r^{4}}\left[\int\left\\{r^{2}hH_{1}\int\frac{\left(Mr-q^{2}\right)\left[\left(M-r\right)\tanh^{-1}\left(\frac{M-r}{\sqrt{H}}\right)-\sqrt{H}\right]\left(3Mr-2q^{2}\right)}{r^{7}}dr\right.\right.$
$\displaystyle\,-r^{2}hH_{1}\left[\tanh^{-1}\left(\frac{M-r}{\sqrt{H}}\right)-\frac{\left(\left(q^{2}+2M^{2}\right)r^{2}-3q^{2}\left[Mr-q^{2}\right]\right)\sqrt{H}}{r^{2}hH}\right]$
$\displaystyle\,\left.\left.\quad\times\int\frac{\left(Mr-q^{2}\right)\left(M-r\right)\left(3Mr-2q^{2}\right)}{r^{7}}{dr}\right\\}dr\right]dr\,.$
(38)

The asymptotic form of Eq. (III.2) has the following form,

$\displaystyle\omega^{(1,1)}\approx-\frac{9\kappa^{2}\sigma^{2}a_{\mathrm{KN}}M\left(42M^{4}+175M^{2}q^{2}+35q^{4}-80M^{3}r-40q^{2}Mr\right)}{35\sigma_{1}H^{3/2}r^{8}}=\frac{9\kappa^{2}\sigma^{2}a_{\mathrm{KN}}\,M}{35\sigma_{1}H^{3/2}r^{7}}\left(H_{2}-\frac{H_{3}}{r}\right)\,,$
(39)

where $H_{2}=40M\left(2M^{2}-q^{2}\right)$ and
$H_{3}=42M^{4}+175M^{2}q^{2}+35q^{4}$. Eq. (39) is different from the one
derived in Yunes:2007ss  when $q=0$. The reason why Eq. (39) is different from
the one derived in Yunes:2007ss  when $q=0$ comes from the terms including
$\tanh^{-1}\left({\frac{M-r}{\sqrt{H}}}\right)$. The homogeneous solution of
Eq. (III.2) is a sum of generalized hypergeometric functions, whose argument
is $r/(2M)$ and has some separation constant $c_{6}$. Although certain values
of such constant make the solution purely real, the solution diverges at the
spatial infinity. The other values of constant $c_{6}$ make the solution
infinite or complex. The aforementioned discussion forces us to choose the
integration constants, which are the coefficients of the hypergeometric
functions, to vanish. Therefore, Eq. (III.2) gives the full solution.

The full gravitomagnetic metric perturbation in the linear order with respect
to $\varepsilon$ and $\xi$ yields,

$\displaystyle\omega=$
$\displaystyle\,-\frac{2Mra_{\mathrm{KN}}-q^{2}}{r^{4}}$
$\displaystyle\,+\frac{576a_{\mathrm{KN}}\kappa^{2}\sigma^{2}}{r^{5}H^{3/2}h\sigma_{1}}\int\frac{1}{r^{4}}\left[\int\left\\{r^{2}hH_{1}\int\frac{\left(Mr-q^{2}\right)\left[\left(M-r\right)\tanh^{-1}\left(\frac{M-r}{\sqrt{H}}\right)-\sqrt{H}\right]\left(3Mr-2q^{2}\right)}{r^{7}}dr\right.\right.$
$\displaystyle\,-r^{2}hH_{1}\left[\tanh^{-1}\left({\frac{M-r}{\sqrt{H}}}\right)-\frac{\left(\left(q^{2}+2M^{2}\right)r^{2}-3q^{2}\left[Mr-q^{2}\right]\right)\sqrt{H}}{r^{2}hH}\right]$
$\displaystyle\,\quad\left.\left.\times\int\frac{\left(Mr-q^{2}\right)\left(M-r\right)\left(3Mr-2q^{2}\right)}{r^{7}}dr\right\\}{dr}\right]{dr}\,,$
(40)

whose asymptotic form is given by,

$\displaystyle\omega\approx-\frac{2Mra_{\mathrm{KN}}-q^{2}}{r^{4}}+\frac{9\kappa^{2}\sigma^{2}a_{\mathrm{KN}}M}{35\sigma_{1}H^{3/2}r^{7}}\left(H_{2}-\frac{H_{3}}{r}\right)\,.$
(41)

Eq. (III.2) constitutes the first charged slow-rotating BH solution in dCS
modified gravity. Note that the perturbation is highly suppressed in the far
field limit and decreases as $r^{-7}$, which suggests that its significance
can only be observed in the strong field regime. Now let us discuss the
solution (III.2) with/without the charge $q$. When $q=0$ the solution
decreases as $r^{-6}$ Yunes:2007ss  and when $q\neq 0$ the solution decreases
asymptotically as $r^{-7}$. This means that the solution with $q$ decreases
faster than the one without $q$.

We can verify that the approximated solution given by (III.2) is self-
consistent by calculating the next order correction to $\varphi$. Such
correction consists of $\varphi^{(2,0)}$ and $\varphi^{(1,1)}$, which can be
calculated by solving the evolution equation to the next order. Carrying out
such calculations, we find,

$\displaystyle\varphi^{(1,1)}=$
$\displaystyle\,\frac{13824a_{\mathrm{KN}}\kappa^{2}\alpha^{3}}{\beta^{2}H^{4}}\left(HH_{4}\left(r-M\right)\int\frac{\left(Mr-q^{2}\right)H_{4}}{r^{6}}\left\\{\int\frac{1}{r^{5}h}\left[H_{1}r^{2}h\sqrt{H}\int\frac{\left(q^{2}-Mr\right)H_{4}\left(3Mr-2q^{2}\right)}{r^{7}}dr\right.\right.\right.$
$\displaystyle\,-\int\frac{\left(M-r\right)\left(Mr-q^{2}\right)\left(3Mr-2q^{2}\right)}{r^{7}}dr$
$\displaystyle\,\quad\times\left.\left.\left(H_{1}r^{2}h\sqrt{H}\tanh^{-1}\left(\frac{M-r}{\sqrt{H}}\right)+H\left(\left[2M^{2}+q^{2}\right]r^{2}-6Mrq^{2}+3q^{4}\right)\right)\right]dr\right\\}dr$
$\displaystyle\,+\int\frac{\left(M-r\right)\left(Mr-q^{2}\right)}{r^{6}}\left\\{\int\frac{1}{r^{5}h}\left[H_{1}r^{2}h\sqrt{H}\int\frac{\left(q^{2}-Mr\right)H_{5}\left(3Mr-2q^{2}\right)}{r^{7}}{dr}\right.\right.$
$\displaystyle\,\quad+\int\frac{\left(r-M\right)\left(Mr-q^{2}\right)\left(3Mr-2q^{2}\right)}{r^{7}}dr$
$\displaystyle\,\quad\left.\left.\left.\times\left(H_{1}r^{2}h\sqrt{H}\tanh^{-1}\left(\frac{M-r}{\sqrt{H}}\right)+H\left(\left[2M^{2}+q^{2}\right]r^{2}-6Mrq^{2}+3q^{4}\right)\right)\right]dr\right\\}dr\right)\,,$
(42)

where
$H_{4}=\left[\left(r-M\right)\tanh^{-1}\left({\frac{M-r}{\sqrt{H}}}\right)+\sqrt{H}\right]$
and
$H_{5}=\left[\left(r-M\right)\tanh^{-1}\left({\frac{M-r}{\sqrt{H}}}\right)-\sqrt{H}\right]$.
Eq. (III.2) yields the following asymptotic form,

$\displaystyle\varphi^{(1,1)}\approx$ $\displaystyle\,-\frac{12\sigma\xi
a_{\mathrm{KN}}M\cos\theta}{7\sigma_{1}H^{3}r^{9}}\left[\tanh^{-1}\left(\frac{M-r}{\sqrt{H}}\right)\left[r-M\right]-\sqrt{H}\right]$
$\displaystyle\,\quad\times\left(2M^{2}+q^{2}\right)\left(63r\left[M^{2}+q^{2}\right]-8M\left[7q^{2}+9r^{2}\right]\right)\,.$
(43)

Eq. (III.2) yields the general behavior of $\varphi^{(1,1)}$ presented in
Yunes:2007ss  when $q=0$. Eq. (III.2) is $\xi$-times smaller than
$\varphi^{(1,0)}$, therefore presenting the small-coupling approximation self-
consistent. To use this improved $\varphi$ solution in the modified field
equation, we then find a correction to the metric proportional of order
$\xi^{2}\epsilon$, which we neglect in this study. It is important to stress
that to obtain the Reissner-Nordström BH in the static limit, it is enough to
have a constant CS scalar. Indeed, a constant CS scalar is less restrictive
and would allow the topological Pontryagin density in the action which, in
turn, is necessary to set (anti-)self-dual configurations as a ground state
Miskovic:2009bm .

##  IV Properties of the derived BH solution

In the following subsections, we will discuss some physical properties of the
solution derived in the previous section.

###  IV.1 Line-element

Gathering the above data, the line element of the slowly rotating BH can be
written up to $\mathcal{O}\left({a_{\mathrm{KN}}}^{2}\right)$ as

$\displaystyle ds^{2}=$
$\displaystyle\,-g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}-2g_{t\phi}dtd\phi\,,$
(44) $\displaystyle\varphi^{(1,0)}=$
$\displaystyle\,\frac{48\epsilon\sigma{a_{\mathrm{KN}}}}{\sigma_{1}H}\left\\{\sqrt{H}\left(r-M\right)\int\frac{\left(Mr-q^{2}\right)\left(3Mr-2q^{2}\right)\left[\left(r-M\right)\tanh^{-1}\left({\frac{M-r}{\sqrt{H}}}\right)+\sqrt{H}\right]}{r^{7}}dr\right.$
$\displaystyle\,\left.+\left(\sqrt{H}\left(r-M\right)\tanh^{-1}\left({\frac{M-r}{\sqrt{H}}}\right)-H\right)\int\frac{\left(3r^{2}M^{2}-5q^{2}Mr+2q^{4}\right)\left(M-r\right)}{r^{7}}{dr}\right\\}\,,$
(45)

with

$\displaystyle g_{tt}=$
$\displaystyle\,-h-\frac{{a_{\mathrm{KN}}}^{2}\left(2Mr-q^{2}\right)}{r^{4}}\cos^{2}{\theta}\,,$
$\displaystyle g_{t\phi}=$
$\displaystyle\,\,-\frac{\left(2Mr-q^{2}\right)a_{\mathrm{KN}}}{r^{2}}\sin^{2}{\theta}$
$\displaystyle\,-\frac{576a_{\mathrm{KN}}\kappa^{2}\sigma^{2}}{r^{5}H^{3/2}h\sigma_{1}}\int\frac{1}{r^{4}}\left[\int\left\\{r^{2}hH_{1}\int\frac{\left(Mr-q^{2}\right)\left[\left(M-r\right)\tanh^{-1}\left(\frac{M-r}{\sqrt{H}}\right)-\sqrt{H}\right]\left(3Mr-2q^{2}\right)}{r^{7}}dr\right.\right.$
$\displaystyle\,-r^{2}hH_{1}\left[\tanh^{-1}\left({\frac{M-r}{\sqrt{H}}}\right)-\frac{\left(\left(q^{2}+2M^{2}\right)r^{2}-3q^{2}\left[Mr-q^{2}\right]\right)\sqrt{H}}{r^{2}hH}\right]$
$\displaystyle\,\quad\times\left.\left.\int\frac{\left(Mr-q^{2}\right)\left(M-r\right)\left(3Mr-2q^{2}\right)}{r^{7}}dr\right\\}dr\right]dr\,,$
$\displaystyle g_{rr}=$
$\displaystyle\,\frac{1}{h}+\frac{{a_{\mathrm{KN}}}^{2}}{hr^{2}}\left(\cos^{2}\theta-\frac{1}{h}\right)\,,$
$\displaystyle g_{\theta\theta}=$
$\displaystyle\,r^{2}+{a_{\mathrm{KN}}}^{2}\cos^{2}\theta\,,$ $\displaystyle
g_{\phi\phi}=$
$\displaystyle\,r^{2}\sin^{2}\theta+{a_{\mathrm{KN}}}^{2}\sin^{2}\theta\left(1+\frac{2Mr-q^{2}}{r^{2}}\sin^{2}\theta\right)\,,$
$\displaystyle A_{t}=$ $\displaystyle\,-\frac{q}{r}+\frac{q\sin^{2}\theta
a_{\mathrm{KN}}}{r}\,.$ (46)

Eq. (IV.1) is correct in the orders of $\mathcal{O}(2,0)$, $\mathcal{O}(1,1)$,
and $\mathcal{O}(0,2)$. We stress that the terms including $\sigma$ and
$\sigma_{1}$ in $g_{t\phi}$ of Eq. (IV.1) cannot be removed by any coordinate
transformation. We also note that the perturbed parts of the metric are
regular at the horizon, where $r-M+\sqrt{H}$ vanishes. Therefore as long as we
treat the model perturbatively, the scalar invariants are regular at the
horizon.

When $r$ is large i.e., $r\to\infty$, the Pontryagin density
$R_{\nu\mu\alpha\beta}\tilde{R}^{\mu\nu\alpha\beta}$ behaves as:

$\displaystyle R_{\nu\mu\alpha\beta}\tilde{R}^{\mu\nu\alpha\beta}\approx$
$\displaystyle\,-\frac{576\epsilon
a_{\mathrm{KN}}M^{2}\cos\theta}{r^{7}}+\frac{960\epsilon
a_{\mathrm{KN}}Mq^{2}\cos\theta}{r^{8}}-\frac{384\epsilon
a_{\mathrm{KN}}q^{4}\cos\theta}{r^{9}}$
$\displaystyle\,+\frac{432\epsilon\sigma^{2}a_{\mathrm{KN}}M^{3}(2M^{2}+q^{2})\cos\theta}{\sigma_{1}(M^{2}-q^{2})^{3/2}r^{11}}+\mathcal{O}\left(\frac{1}{r^{12}}\right)\,.$
(47)

Eq. (IV.1) shows the corrections coming from the CS term to the order under
consideration in the present study. The Pontryagin density $R\tilde{R}$, which
is proportional to $\square\varphi$, and its deviation from that in the Kerr-
Newman can be calculated by using (III.2). Also, Eq. (IV.1) shows the
correction of the CS scalar field starts in the invariant
$R_{\nu\mu\alpha\beta}\tilde{R}^{\mu\nu\alpha\beta}$ from
$\mathcal{O}\left(\frac{1}{r^{11}}\right)$ and the lower orders appear in this
invariant are due to the contribution coming from the slowly rotating Kerr-
Newman BH. Also, Eq. (IV.1) tells that the cross term of the line element (44)
cannot vanish or be gauged away by any coordinate transformation. This is
because if the cross term in (44) vanishes, $a_{\mathrm{KN}}=0$ and in that
case, the CS scalar field given by Eq. (III.2) also vanishes and the CS theory
reduces to GR of the Einstein theory.

The metric in Eq. (IV.1) involves a true singularity at $r=0$. This can be
investigated by calculating the Kretchmann invariant
$R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$, the squared of Ricci tensor
$R_{\mu\nu}R^{\mu\nu}$, and the Ricci scalar $R$, which ensures the divergence
at $r=0$. Moreover, Eq. (IV.1) shows also that the Pontryagin density and the
CS scalar field diverge at $r=0$.

The location of the event horizon can be found by solving the equation
$g_{tt}g_{\phi\phi}-g_{t\phi}^{2}=0$, which yields,

$\displaystyle
r_{\mathrm{H,KN}}=r_{\mathrm{KN}}=M+\sqrt{M^{2}-{a_{\mathrm{KN}}}^{2}-q^{2}}\,.$
(48)

Moreover, the location of the ergosphere can also be derived by solving the
equation $g_{tt}=0$ for $r$,

$\displaystyle r_{\mathrm{ergo}}=r_{\mathrm{ergo,KN}}\,,$ (49)

with the ergosphere of the Kerr-Newmann solution given by
$r_{\mathrm{ergo,KN}}=M+\sqrt{M^{2}-{a_{\mathrm{KN}}}^{2}\cos^{2}\theta-q^{2}}$.
Eq. (49) tells that the radius of the ergosphere is not changed from that in
the Kerr-Newmann solution.

The choice of the homogeneous integration constants depends on how to choose
the definition of the mass $M$ and the reduced spin angular momentum
$a_{\mathrm{KN}}$. Therefore, a natural choice is to define such quantities so
that they coincide with those measured by an observer at $r\to\infty$ and we
obtain the metric displayed in (IV.1). With these definitions, the angular
velocity and area of the event horizon are changed from those in the Kerr-
Newmann solution as follows, The new solution given by Eq. (IV.1) amended the
dragging of the inertial frame of the rotation of the BH. This can be
calculated by the angular velocity $\omega_{Z}$ for the zero-angular-momentum
observer, which is defined by,

$\displaystyle\omega_{Z}=-\frac{g_{t\phi}}{g_{\phi\phi}}\,,$ (50)

which yields,

$\displaystyle\omega_{Z}=$ $\displaystyle\,-\frac{2Ma_{\mathrm{KN}}}{r^{3}}$
$\displaystyle\,+\frac{576\kappa^{2}a_{\mathrm{KN}}\sigma^{2}}{r^{5}H^{3/2}h\sigma_{1}}\int\frac{1}{r^{4}}\left[\int\left\\{r^{2}hH_{1}\int\frac{\left(Mr-q^{2}\right)\left[\left(M-r\right)\tanh^{-1}\left({\frac{M-r}{\sqrt{H}}}\right)-\sqrt{H}\right]\left(3Mr-2q^{2}\right)}{r^{7}}{dr}\right.\right.$
$\displaystyle\,-r^{2}hH_{1}\left[\tanh^{-1}\left(\frac{M-r}{\sqrt{H}}\right)-\frac{\left(\left(q^{2}+2M^{2}\right)r^{2}-3q^{2}\left[Mr-q^{2}\right]\right)\sqrt{H}}{r^{2}hH}\right]$
$\displaystyle\quad\times\left.\left.\int{\frac{\left(Mr-q^{2}\right)\left(M-r\right)\left(3Mr-2q^{2}\right)}{r^{7}}}dr\right\\}dr\right]dr\,.$
(51)

Now we are in a position to discuss the conserved charges of the above
solution. We use the following transformation to transform the metric (IV.1)
to the Cartesian coordinates via the following standard transformation
Yunes:2009hc :

$\displaystyle x=$
$\displaystyle\,r\left(1+\frac{a_{\mathrm{KN}}^{2}}{2r^{2}}\right)\cos{\phi}\sin{\theta}\,,$
$\displaystyle y=$
$\displaystyle\,r\left(1+\frac{a_{\mathrm{KN}}^{2}}{2r^{2}}\right)\sin{\phi}\sin{\theta}\,,$
$\displaystyle z=$ $\displaystyle\,r\cos{\theta}\,,$ (52)

We also define $\bar{h}_{\mu\nu}$ by the difference between ${g}_{\mu\nu}$ in
(IV.1) and the flat metric $\eta_{\mu\nu}$,

$\displaystyle\bar{h}_{\mu\nu}\equiv g_{\mu\nu}-\eta_{\mu\nu}\,.$ (53)

We should note that $\bar{h}_{\mu\nu}$ needs not to be small everywhere but we
require that at infinity, away from the black hole, it goes to vanish because
we are considering the asymptotically flat spacetime as a solution. A detailed
discussion on how one can use Eq. (53) to derive the mass formula for the CS-
modified GR in the asymptotically flat spacetime can be found in
(Tekin:2007rn, ). Here we list the mass formula of Chern-Simons-modified GR as
in Tekin:2007rn :

$\displaystyle E=\frac{1}{16\pi
G}\oint_{S^{2}}dS_{i}\,\left(Q^{0i}_{E}\left(\bar{\xi}\right)+\frac{1}{2}Q^{0i}_{E}\left(\bar{\Xi}\right)-\frac{1}{2}Q^{0i}_{C}\left(\bar{\xi}\right)\right)\,,$
(54)

where

$\displaystyle Q^{\mu i}_{E}\left(\bar{\xi}\right)\equiv$
$\displaystyle\,\sqrt{-\bar{g}}\left(\bar{\xi}_{\nu}\bar{\nabla}^{\mu}\bar{h}^{i\nu}-\bar{\xi}_{\nu}\bar{\nabla}^{i}\bar{h}^{\mu\nu}+\bar{\xi}^{\mu}\bar{\nabla}^{i}\,\bar{h}-\bar{\xi}^{i}\bar{\nabla}^{\mu}\bar{h}\right.$
$\displaystyle\,\left.+h^{\mu\nu}\bar{\nabla}^{i}\bar{\xi}_{\nu}-\bar{h}^{i\nu}\bar{\nabla}^{\mu}\bar{\xi}_{\nu}+\bar{\xi}^{i}\bar{\nabla}_{\nu}\bar{h}^{\mu\nu}-\bar{\xi}^{\mu}\bar{\nabla}_{\nu}\,\bar{h}^{i\nu}+\bar{h}\,\bar{\nabla}^{\mu}\,\bar{\xi}^{i}\right)\,,$
$\displaystyle Q^{\mu i}_{C}\left(\bar{\xi}\right)\equiv$
$\displaystyle\,v_{\sigma}\bar{\xi}^{\rho}\epsilon^{\sigma\mu
i\beta}{\mathcal{G}^{L}}_{\rho\beta}+v_{\sigma}\bar{\xi}_{\nu}\epsilon^{\sigma\nu
i\beta}{\mathcal{G}^{L\mu}}_{\beta}+v_{\sigma}\bar{\xi}_{\nu}\epsilon^{\sigma\mu\nu\beta}{\mathcal{G}^{Li}}_{\beta}\,,$
(55)

and the integration is performed on a two-dimensional spacial sphere $S^{2}$
with a large enough radius. In Eq. (IV.1), $\bar{h}$ is the trace of
$\bar{h}^{\mu\nu}$, i.e., $\bar{h}=g_{\mu\nu}\bar{h}^{\mu\nu}$,
${\mathcal{G}^{L}}_{\mu\nu}$ is the linearized form of the Einstein tensor,

$\displaystyle{\mathcal{G}^{L}}_{\mu\nu}=\frac{1}{2}\left(-\bar{\Box}{\bar{h}}_{\mu\nu}-\bar{\nabla}_{\mu}\bar{\nabla}_{\nu}\bar{h}+\bar{\nabla}^{\sigma}\bar{\nabla}_{\nu}{\bar{h}}_{\sigma\mu}+\bar{\nabla}^{\sigma}\bar{\nabla}_{\mu}{\bar{h}}_{\sigma\nu}\right)-\frac{1}{2}{\bar{g}}_{\mu\nu}\left(\bar{\Box}\bar{h}+\bar{\nabla}^{\rho}\bar{\nabla}^{\sigma}{\bar{h}}_{\rho\sigma}\right)\,,$
(56)

and $\bar{\nabla}^{\mu}$ is the covariant derivative with respect to the
background. Because we are considering the flat background (53), we find that
the covariant derivatives reduce to the partial derivatives,
$\bar{\nabla}^{\mu}=\partial^{\mu}$. In Eq. (IV.1), $v_{\sigma}$ is defined in
Eq. (16). Now let us apply the formula of energy given by Eq. (54) to the BH
(IV.1). For the energy, $\bar{\xi}^{\mu}$ is the time-like Killing vector
defined as $\bar{\xi}^{\mu}=(-1,0,0,0)$. An explicit computation of the mass
of this metric using the formula (54) is straightforward. $\bar{\Xi}^{\mu}$
vanishes555 Here $\bar{\Xi}^{\mu}$ is defined as
$\bar{\Xi}^{\mu}\equiv\frac{1}{\sqrt{-g}}v_{\sigma}\epsilon^{\sigma\nu\alpha\mu}\bar{\nabla}_{\alpha}\bar{\xi}_{\nu}\,.$
so $Q_{E}\left(\bar{\Xi}\right)$ term does not contribute. Moreover,
$Q_{C}\left(\bar{\xi}\right)$ term vanishes for various reasons (such as
symmetry and because ${\mathcal{G}^{L\mu}}_{\beta}$ vanishes for this Einstein
space at infinity). From the first part, we obtain $E=M$ at infinity up to
order $a_{\mathrm{KN}}$. Therefore there are no corrections from the CS term.

Now we are ready to write down the formula that enables us to calculate the
angular momentum of the Chern-Simons-modified GR for asymptotically flat space
and generalizes Eq. (54). This formula takes the following form (for more
details, see (Tekin:2007rn, )),

$Q^{0}({\xi}_{\mu})=\frac{1}{16\pi
G}\oint_{S^{2}}dS_{i}\left[\bar{\xi}_{0}(\partial_{j}\bar{h}^{ij}-\partial^{i}\bar{h}^{j}\,_{j})+\bar{\xi}^{i}\partial_{j}\bar{h}^{0j}-\bar{\xi}_{j}\partial^{i}\bar{h}^{0j}+\frac{\sigma_{1}}{2}\epsilon^{ijk}\bar{\xi}_{j}{\mathcal{G}^{L0}}_{k}\right]\,.$
(57)

Eq. (57) for $\bar{\xi}_{0}=(1,0,0,0)$ gives $Q^{0}({\xi}_{0})=E$ and the
formula (57) coincides with the usual ADM one. As Eq. (57) shows, the effect
of the CS term, which appears in the last term, does not contribute.

In the case of the angular momentum, we consider the case of
$\bar{\xi}_{i}=(0,0,0,1)$ and $Q^{0}\left(\bar{\xi}_{i}\right)=J$. The first
three terms in the integrand of (57) are identical to those in the Einstein
gravity. Because Eq. (IV.1) tells that the corrections coming from the CS term
decrease rapidly compared with the terms appearing in the Kerr geometry in the
Einstein gravity, they do not contribute to the angular momentum. For the last
term $\epsilon^{ijk}\bar{\xi}_{j}{\mathcal{G}^{L0}}_{k}$, which appears due to
the existence of the CS term, the $i$-direction is perpendicular to the two-
dimensional surface $S^{2}$ and therefore the $i$-direction corresponds to the
radial ($r$) direction. On the other hand, ${\bar{\xi}}_{j}$ is a unit vector
corresponding to the $z$-direction. This tells that the $k$-direction
corresponds to the $\phi$-direction and therefore only
${\mathcal{G}^{L}}_{t\phi}$ contributes to $Q^{0}(\bar{\xi}_{i})$. Because Eq.
(IV.1) tells that the metric does not depend on the time $t$ nor angular
$\phi$, all the terms except the first term
$-\frac{1}{2}\bar{\Box}{\bar{h}}_{\mu\nu}$ in the expression (56)
corresponding to ${\mathcal{G}^{L}}_{t\phi}$ vanish trivially. Eq. (IV.1) also
tells that ${\bar{h}}_{t\phi}$ is $\mathcal{O}\left(r^{-1}\right)$ and
therefore $\bar{\Box}{\bar{h}}_{t\phi}\sim\mathcal{O}\left(r^{-3}\right)$.
Because the area of the two-dimensional surface $S^{2}$ is
$\mathcal{O}\left(r^{2}\right)$, the last term in (57) for
$Q^{0}(\bar{\xi}_{i})=J$ does not contribute to $J$ in the limit that the
radius of $S^{2}$ goes to infinity. Therefore there is no correction from the
CS term to the angular momentum and we obtain,

$\displaystyle J=a_{\mathrm{KN}}M\,,$ (58)

which is identical to the angular momentum in the standard Kerr(-Newman) black
hole. Eq. (58) shows that the effect of the CS scalar field on the calculation
of the angular momentum vanishes up to $a_{\mathrm{KN}}$.

Now we are going to discuss the Hawking temperature of the BH solution given
by Eq. (IV.1). The Hawking temperature $T$ is generally defined by the surface
gravity $\kappa$ so that $T=\kappa/\left(2\pi\right)$ and we now obtain
Sheykhi:2012zz ; Sheykhi:2010zz ; Hendi:2010gq ; Sheykhi:2009pf ,

$\displaystyle
T_{\mathrm{H,KN}}=\frac{h^{\prime}\left(r_{\mathrm{H,KN}}\right)}{4\pi}\,.$
(59)

Using Eq. (IV.1) in Eq. (59), we obtain the Hawking temperature as,

$\displaystyle
T_{\mathrm{H,KN}}=\frac{{r_{\mathrm{H,KN}}}^{2}-q^{2}}{4\pi{r_{\mathrm{H,KN}}}^{3}}\,,\quad\mbox{up
to}\quad\mathcal{O}\left(a_{\mathrm{KN}}\right)\,.$ (60)

Eq. (60) does not differ from Reissner-Nordström solution up to
$\mathcal{O}\left(a_{\mathrm{KN}}\right)$.

##  V Geodesic precession in the slowly-rotating charged black hole in the
dynamical Chern-Simons modified gravity

In Harko:2009kj ; Sopuerta:2009iy , the time-like geodesics of the slowly
rotating black hole in dynamical Chern-Simons modified gravity were
considered. Sopuerta et al. Sopuerta:2009iy  investigated the time-like
geodesic equations for the massive particles and discovered that in the Chern-
Simons modified gravity, the location of the innermost stable circular orbit
(ISCO) and the three physical fundamental frequencies associated with the
particle’s time $\tau$ are modified. However, the geodesic precession of
orbits around Chern-Simons black holes is only shown numerically for a few
examples in Ref. Sopuerta:2009iy , with no analytic expression for this
physical quantity. Now let us start using the condition $\theta=\pi/2$, which
puts the orbits on the equatorial plane. In such cases, time-like geodesics
can be found to take the form,

$\displaystyle u^{t}=$
$\displaystyle\,\frac{dt}{d\tau}=\frac{Eg_{\phi\phi}-Lg_{t\phi}}{g^{2}_{t\phi}+g_{tt}g_{\phi\phi}}\,,$
(61) $\displaystyle\mathrm{u}^{\phi}=$
$\displaystyle\,\frac{d\phi}{d\tau}=\frac{Eg_{t\phi}+Lg_{tt}}{g^{2}_{t\phi}+g_{tt}g_{\phi\phi}}\,,$
(62)
$\displaystyle\left(\frac{dr}{d\tau}\right)^{2}+V_{\mathrm{eff}}(r)=E^{2}\,,$
(63)

with the effective potential

$\displaystyle
V_{\mathrm{eff}}(r)=\frac{1}{g_{rr}}\left(1+\frac{E^{2}[g_{rr}(g^{2}_{t\phi}+g_{tt}g_{\phi\phi})-g_{\phi\phi}]+2ELg_{t\phi}+L^{2}g_{tt}}{g^{2}_{t\phi}+g_{tt}g_{\phi\phi}}\right)\,,$
(64)

with $E$ and $L$ being the specific energy and angular momentum of particles
moving in the orbits, respectively. The effective potential $V(r)$ must obey
for a stable circular orbit in the equatorial plane the following equation,

$\displaystyle
V_{\mathrm{eff}}(r)=E^{2}\,,\quad\frac{dV_{\mathrm{eff}}(r)}{dr}=0\,.$ (65)

By solving the above equations, one obtains,

$\displaystyle E=$
$\displaystyle\,\frac{g_{tt}+g_{t\phi}\Omega}{\sqrt{g_{tt}+2g_{t\phi}\Omega-
g_{\phi\phi}\Omega^{2}}}=-{\frac{27a_{\mathrm{KN}}}{5{\xi}\kappa{\beta}r^{3}\left(r^{2}-2\,Mr+q^{2}-r^{4}\Omega^{2}\right)^{3/2}\left({M}^{2}-q^{2}\right)^{3/2}}}$
$\displaystyle\,\times\left({\frac{5}{27}}\kappa\xi\left(q^{2}-M^{2}\right)\beta
r^{2}\left(2M\Omega^{3}\epsilon
r^{5}-\Omega^{3}\epsilon\,q^{2}r^{4}\right)\sqrt{{M}^{2}-q^{2}}\right.$
$\displaystyle\,\left.+\epsilon
M\Omega^{3}{\alpha}^{2}\left[r\left({\frac{40}{21}}\,{M}^{3}+{\frac{20}{21}}Mq^{2}\right)+{M}^{4}+\frac{5}{6}q^{4}+{\frac{25}{6}}{M}^{2}q^{2}\right]\right)$
$\displaystyle\,+\frac{\left[\left(1-\Omega^{2}q^{2}\right)r^{4}-r^{6}\Omega^{2}+2M\Omega^{2}r^{5}+2\left(2{M}^{2}+q^{2}\right)r^{2}-4q^{2}rM+q^{4}-4Mr^{3}\right]}{\left(r^{2}-2Mr+q^{2}-r^{4}\Omega^{2}\right)^{3/2}r}\,,$
$\displaystyle L=$
$\displaystyle\,\frac{-g_{t\phi}+g_{\phi\phi}\Omega}{\sqrt{g_{tt}+2g_{t\phi}\Omega-
g_{\phi\phi}\Omega^{2}}}={\frac{54}{5r^{7}\left(r^{2}-Mr+q^{2}-r^{4}\Omega^{2}\right)^{3/2}\left({M}^{2}-q^{2}\right)^{3/2}{\beta}\kappa{\xi}}}$
$\displaystyle\,\times\left[{\frac{10}{27}}\kappa\xi\,\left(q^{2}-M^{2}\right)\beta\left(M\epsilon\,\Omega^{2}r^{5}-\frac{\epsilon\,\Omega^{2}q^{2}r^{4}}{2}-\frac{Mr^{3}\epsilon}{+}\left(\frac{q^{2}}{4}+{M}^{2}\right)\epsilon
r^{2}-Mr\epsilon q^{2}+\frac{\epsilon
q^{4}}{4}\right)r^{6}\sqrt{{M}^{2}-q^{2}}\right.$
$\displaystyle\,\left.+M{\alpha}^{2}\epsilon\left(Mr+r^{4}\Omega^{2}-\frac{q^{2}}{2}-\frac{r^{2}}{2}\right)\left[\left({\frac{40}{21}}{M}^{3}+{\frac{20}{21}}Mq^{2}\right)r+{M}^{4}+\frac{5q^{4}}{6}+{\frac{25}{6}}{M}^{2}q^{2}\right]\right]a_{\mathrm{KN}}$
$\displaystyle\,-{\frac{\left(r^{6}\Omega-r^{8}\Omega^{3}-2M\Omega
r^{5}+\Omega
q^{2}r^{4}\right)}{\left(r^{2}-2Mr+q^{2}-r^{4}\Omega^{2}\right)^{3/2}r}}\,,$
$\displaystyle\Omega=$
$\displaystyle\,\frac{d\phi}{dt}=\frac{g_{t\phi,r}+\sqrt{(g_{t\phi,r})^{2}+g_{tt,r}g_{\phi\phi,r}}}{g_{\phi\phi,r}}=\frac{\epsilon
a_{\mathrm{KN}}}{\left({M}^{2}-q^{2}\right)^{3/2}{\beta}\kappa{\xi}r^{10}}\left[\left(\left({M}^{2}-q^{2}\right)^{3/2}\beta\kappa\,\xi
r^{7}-18\,{\alpha}^{2}q^{4}\right)M\right.$
$\displaystyle\,\left.-{\frac{108}{5}}{M}^{5}{\alpha}^{2}-36{M}^{4}{\alpha}^{2}r-90{M}^{3}{\alpha}^{2}q^{2}-18{M}^{2}{\alpha}^{2}q^{2}r-q^{2}\left(-q^{2}+{M}^{2}\right)^{3/2}\beta\kappa\xi
r^{6}\right]+\frac{\sqrt{Mr-q^{2}}}{{r^{2}}}\,,$ (66)

with $\Omega$ being the angular velocity of the particle moving in the orbits.

Using Eq. (66), one obtains Kepler’s third law in the slowly-rotating black-
hole spacetime in the dynamical Chern-Simons modified gravity

$\displaystyle T^{2}=$
$\displaystyle\,4\pi^{2}\left\\{\frac{R^{8}}{\left(\sqrt{q^{4}+MR^{5}-R^{4}q^{2}}+q^{2}\right)^{2}}+\frac{R^{4}M{a_{\mathrm{KN}}}}{35\left(\sqrt{q^{4}+Mr^{5}-R^{4}q^{2}}+q^{2}\right)^{3}{\sigma
1}H^{3/2}\left(q^{4}+MR^{5}-R^{4}q^{2}\right)}\right.$
$\displaystyle\,\times\left(70R^{5}{\sigma_{1}}H^{3/2}q^{4}-54\,H_{3}\,\sigma^{2}\kappa^{2}q^{4}-54H_{3}\sigma^{2}\kappa^{2}MR^{5}+54H_{3}\sigma^{2}\kappa^{2}R^{4}q^{2}+70R^{10}{\sigma_{1}}H^{3/2}M\right.$
$\displaystyle\,-70R^{9}{\sigma_{1}}H^{3/2}q^{2}+45\kappa^{2}\sigma^{2}{H_{2}}Rq^{4}+45\kappa^{2}\sigma^{2}H_{2}\,R^{6}M-45\kappa^{2}\sigma^{2}H_{2}R^{5}q^{2}-54\sqrt{q^{4}+MR^{5}-R^{4}q^{2}}{q}^{2}H_{3}\,\sigma^{2}\kappa^{2}$
$\displaystyle\,\left.\left.+45R\sqrt{q^{4}+MR^{5}-R^{4}q^{2}}q^{2}\kappa^{2}\sigma^{2}H_{2}+70R^{5}\sqrt{q^{4}+MR^{5}-R^{4}q^{2}}q^{2}{\sigma_{1}}\,H^{3/2}\right)+\mathcal{O}\left({a_{\mathrm{KN}}}^{2}\right)\right\\}\,,$
(67)

where $R$ representing the orbital radius and $T=\frac{1}{\Omega}$
representing the orbital period. The subsequent terms in the right-hand side
of Eq. (67) is the correction by the $a_{\mathrm{KN}}$ and the Chern-Simons
term. The correction term disappears as $a$ approaches zero. This is sensible
because as $a_{\mathrm{KN}}$ approaches zero the metric (44) coincides with
one of the Schwarzschild black holes in general relativity. Since the black
hole rotates slowly, the first-order terms in $a_{\mathrm{KN}}$ dominate the
correction. As a result, when the black hole rotates in the same direction as
the particle, i.e., $a_{\mathrm{KN}}>0$, the orbital period $T$ decreases with
the Chern-Simons coupling parameter $\xi$. However, when the black hole
rotates in the opposite direction as the particle, i.e., $a_{\mathrm{KN}}<0$,
the orbital period $T$ tends to increase with the Chern-Simons coupling
parameter $\xi$.

###  V.1 Stability of the BH given by Eq. (IV.1) through the use of geodesic
deviation

The trajectory of a test particle in the BH space-time is prescribed by the
geodesic equations of the following form,

$\displaystyle
0=\frac{d^{2}x^{\alpha}}{d\tau^{2}}+\left\\{\begin{array}[]{c}\alpha\\\
\mu\nu\end{array}\right\\}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\,,$
(70)

where $\tau$ is the affine parameter along the geodesic. The equation of
geodesic deviation has the form dInverno:1992gxs :

$\displaystyle
0=\frac{d^{2}\varepsilon^{\alpha}}{d\tau^{2}}+2\left\\{\begin{array}[]{c}\alpha\\\
\mu\nu\end{array}\right\\}\frac{dx^{\mu}}{d\tau}\frac{d\varepsilon^{\nu}}{d\tau}+\left\\{\begin{array}[]{c}\alpha\\\
\mu\nu\end{array}\right\\}_{,\
\rho}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}\varepsilon^{\rho}\,,$ (75)

with $\varepsilon^{\rho}$ being the deviation 4-vector. Applying (70) and (75)
into (IV.1) , we obtain the geodesic equations in the following form,

$\displaystyle 0=$
$\displaystyle\,\frac{d^{2}t}{d\tau^{2}}+\epsilon\left[Ma_{\mathrm{KN}}-\frac{a_{\mathrm{KN}}q^{2}}{2r}+r^{3}A_{t}\omega\right]\left(\frac{d\phi}{d\tau}\right)^{2}\left\\{1-\frac{h^{\prime}}{2r}\right\\}\,,$
$\displaystyle 0=$
$\displaystyle\,\frac{1}{2}h^{\prime}(r)\left(\frac{dt}{d\tau}\right)^{2}-r\left(\frac{d\phi}{d\tau}\right)^{2}=0\,,\quad
0=\frac{d^{2}\theta}{d\tau^{2}}\,,\quad 0=\frac{d^{2}\phi}{d\tau^{2}}\,.$ (76)

Using the circular orbit

$\displaystyle\theta=\frac{\pi}{2}\,,\quad\frac{d\theta}{d\tau}=0\,,\quad\frac{dr}{d\tau}=0\,,$
(77)

we obtain the geodesic deviation in the following form,

$\displaystyle 0=$
$\displaystyle\,\frac{d^{2}\varepsilon^{0}}{d\tau^{2}}+\frac{h^{\prime}}{h}\frac{dt}{d\tau}\frac{d\varepsilon^{1}}{d\tau}+\epsilon\left[Ma_{\mathrm{KN}}-\frac{a_{\mathrm{KN}}q^{2}}{2r}+r^{3}A_{t}\omega\right]\left(2\frac{d\phi}{d\tau}\frac{d\varepsilon^{3}}{d\tau}-\frac{h^{\prime}}{r}\frac{dt}{d\tau}\frac{d\varepsilon^{0}}{d\tau}\right)$
$\displaystyle\,+\left\\{\epsilon\left[A_{t}r^{2}\left(3\omega+r\omega^{\prime}\right)+\frac{a_{\mathrm{KN}}q^{2}}{2}r^{2}\right]\left(\frac{d\phi}{d\tau}\right)^{2}\right.$
$\displaystyle\,\left.-\frac{\epsilon}{2r^{2}}\left[2r^{3}h^{\prime}\omega+r^{4}h^{\prime}\omega^{\prime}-Ma_{\mathrm{KN}}h^{\prime}+h^{\prime\prime}Ma_{\mathrm{KN}}r+\xi
h^{\prime\prime}r^{4}\omega+\frac{a_{\mathrm{KN}}q^{2}h^{\prime}}{r}-\frac{a_{\mathrm{KN}}q^{2}h^{\prime\prime}}{2}\right]\left(\frac{dt}{d\tau}\right)^{2}\right\\}\varepsilon^{1}\,,$
$\displaystyle 0=$
$\displaystyle\,2\frac{d^{2}\varepsilon^{1}}{d\tau^{2}}-4hr\frac{d\phi}{d\tau}\frac{d\varepsilon^{3}}{d\tau}+2hh^{\prime}\frac{dt}{d\tau}\frac{d\varepsilon^{0}}{d\tau}-\left\\{2\left[h+rh^{\prime}\right]\left(\frac{d\phi}{d\tau}\right)^{2}-\left[hh^{\prime\prime}+h^{\prime
2}\right]\left(\frac{dt}{d\tau}\right)^{2}\right\\}\varepsilon^{1}\,,$
$\displaystyle 0=$
$\displaystyle\,\frac{d^{2}\varepsilon^{2}}{d\tau^{2}}+\left(\frac{d\phi}{d\tau}\right)^{2}\varepsilon^{2}\,,\quad
0=\frac{d^{2}\varepsilon^{3}}{d\tau^{2}}+\frac{2}{r}\frac{d\phi}{d\tau}\frac{d\varepsilon^{1}}{d\tau}\,,$
(78)

where the functions $h=1-\frac{2M}{r}+\frac{q^{2}}{r^{2}}$ and $\omega$ is
defined by the second term of the $g_{t\phi}$ of Eq. (IV.1) via (III.2).

The third equation of (V.1) represents a simple harmonic motion, which ensures
that the motion in the plane $\theta=\frac{\pi}{2}$ is stable. Assuming the
solutions of the remaining equations of (V.1) to be:

$\displaystyle\varepsilon^{0}=k_{1}\mathrm{e}^{i\omega_{1}\phi}\,,\quad\varepsilon^{1}=k_{2}\mathrm{e}^{i\omega_{1}\phi}\,,\quad\mbox{and}\quad\varepsilon^{3}=k_{3}\mathrm{e}^{i\omega_{1}\phi}\,,$
(79)

where $k_{1}$, $k_{2}$, and $k_{3}$ are constants. Substituting (79) into
(V.1), we obtain

$\displaystyle 0=$ $\displaystyle\,h^{\prime}\left\\{2\omega
r^{3}h^{\prime}\sqrt{h^{\prime}r-2h}-\sqrt{2}\epsilon\left[\left(\xi
r^{4}\omega+a_{\mathrm{KN}}\left(Mr-\frac{1}{2}q^{2}\right)\right)rh^{\prime\prime}-\left[\left(r^{7}\xi-\xi
r^{5}\right)\omega^{\prime}-\xi
r^{4}\left(r^{2}+2\right)\omega\right.\right.\right.$
$\displaystyle\,\left.\left.\left.-\frac{a_{\mathrm{KN}}}{2}\left(2q^{2}-5r^{2}q^{2}-2Mr+8r^{3}M\right)\right]h^{\prime}-\left(2\xi
r^{5}\omega^{\prime}-2\xi r^{4}\omega-
a_{\mathrm{KN}}\left(8Mr-5q^{2}\right)\right)hr\right]h\right\\}$
$\displaystyle\,-\left(h^{\prime\prime}h+\left(r^{2}-1\right)h^{\prime
2}+r\left(5h-{\omega_{1}}^{2}\right)h^{\prime}-6h^{2}+2\omega^{2}h\right)$
$\displaystyle\,\quad\times\left[\epsilon h^{\prime}\sqrt{2}\left(\xi
r^{4}\omega+a_{\mathrm{KN}}\left(Mr-\frac{1}{2}q^{2}\right)\right)+r^{2}\omega_{1}\sqrt{h^{\prime}r-2h}\right]r\,.$
(80)

By solving Eq. (V.1), we can derive the form of $\omega_{1}$, but it is rather
tedious. By using the condition $\omega_{1}>0$, we can establish the stability
condition for the space-times (IV.1) Misner:1973prb ; Nashed:2016tbj ;
Nashed:2011fg . This condition is drawn in Figure 1 for different values of
the parameters which characterize the space-time (IV.1).

(a) The plot of $\omega$ w.r.t the radial coordinate $r$. Here we assume the
numerical values of $M$, $\sigma$, $\sigma_{1}$ $q$, and $\epsilon$ as $0.1$,
$0.1$, $0.1$, $q=0.001$, $0.1$.

(b) The plot of $\omega$ w.r.t the radial coordinate $r$. Here we assume the
numerical values of $M$, $\sigma$, $\sigma_{1}$ $q$, and $a_{\mathrm{KN}}$ as
$0.1$, $0.1$, $0.1$, $q=0.001$, $1$.

(c) The plot of $\omega$ w.r.t the radial coordinate $r$. Here we assume the
numerical values of $M$, $\sigma$, $\epsilon$ $q$, and $a_{\mathrm{KN}}$ as
$0.1$, $0.1$, $0.1$, $q=0.001$, $1$.

Figure 1: Schematic plots of the $\omega$ with respect to the radial
coordinate $r$ of the BH solutions Eqs. (IV.1) using different values of
$a_{\mathrm{KN}}$, $\epsilon$ and $\sigma_{1}$.

##  VI Polarization of Photon near BH

We like to consider the possibility to find the BH obtained in this paper by
any observation. In this section, we investigate the propagation of the photon
near the BH in (IV.1) and we focus on the difference in polarization from that
of the Kerr-Newman BH.

In the general background given by $g_{\mu\nu}=g^{(0)}_{\mu\nu}$ and
$A_{\mu}=A^{(0)}_{\mu}$, we may consider the propagation of photon $a_{\mu}$
which is defined by $a_{\mu}\equiv A_{\mu}-A^{(0)}_{\mu}$ by solving the field
equation,

$\displaystyle
0=\frac{1}{\sqrt{-g^{(0)}}}\partial_{\nu}\left\\{\sqrt{-g^{(0)}}g^{(0)\nu\rho}g^{(0)\mu\sigma}\left(\partial_{\rho}a_{\sigma}-\partial_{\sigma}a_{\rho}\right)\right\\}\,.$
(81)

As in (IV.1), we assume

$\displaystyle g^{(0)}_{tt}=-h(r)+\epsilon^{2}h_{tt}(r,\theta)\,,\quad
g^{(0)}_{t\phi}=\epsilon h_{t\phi}\left(r,\theta\right)\,,\quad
g^{(0)}_{rr}=\frac{1}{h(r)}+\epsilon^{2}h_{rr}(r,\theta)\,,\quad
g^{(0)}_{\theta\theta}=r^{2}+\epsilon^{2}h_{\theta\theta}(r,\theta)\,,$
$\displaystyle
g^{(0)}_{\phi\phi}=r^{2}\sin^{2}\theta+\epsilon^{2}h_{\phi\phi}(r,\theta)\,,\quad\mbox{other
components}=0\,,$ (82)

which gives,

$\displaystyle
g^{(0)tt}=-\frac{1}{h(r)}+\epsilon^{2}\frac{-r^{2}\sin^{2}\theta
h_{tt}(r,\theta)+h_{t\phi}\left(r,\theta\right)^{2}}{h(r)^{2}r^{2}\sin^{2}\theta}\,,$
$\displaystyle
g^{(0)t\phi}=\epsilon\frac{h_{t\phi}\left(r,\theta\right)}{h(r)r^{2}\sin^{2}\theta}\,,\quad
g^{(0)rr}=h(r)-\epsilon^{2}{h(r)}^{2}h_{rr}(r,\theta)\,,\quad
g^{(0)\theta\theta}=\frac{1}{r^{2}}-\epsilon^{2}\frac{h_{\theta\theta}(r,\theta)}{r^{4}}\,,$
$\displaystyle
g^{(0)\phi\phi}=\frac{1}{r^{2}\sin^{2}\theta}-\epsilon^{2}\frac{h(r)h_{\phi\phi}(r,\theta)+h_{t\phi}\left(r,\theta\right)^{2}}{h(r)\left(r^{2}\sin^{2}\theta\right)^{2}}\,,\quad\mbox{other
components}=0\,.$ (83)

Here we have used,

$\displaystyle g^{(0)}_{tt}g^{(0)}_{\phi\phi}-{g^{(0)}_{t\phi}}^{2}=$
$\displaystyle\,-h(r)r^{2}\sin^{2}\theta+\epsilon^{2}\left(r^{2}\sin^{2}\theta
h_{tt}(r,\theta)-h(r)h_{\phi\phi}(r,\theta)-h_{t\phi}\left(r,\theta\right)^{2}\right)$
$\displaystyle\frac{1}{g^{(0)}_{tt}g^{(0)}_{\phi\phi}-{g^{(0)}_{t\phi}}^{2}}=$
$\displaystyle\,-\frac{1}{h(r)r^{2}\sin^{2}\theta}-\epsilon^{2}\frac{r^{2}\sin^{2}\theta
h_{tt}(r,\theta)-h(r)h_{\phi\phi}(r,\theta)-h_{t\phi}\left(r,\theta\right)^{2}}{\left(h(r)r^{2}\sin^{2}\theta\right)^{2}}\,.$
(84)

When we choose $a_{t}=0$ gauge condition, Eq. (81) has the following forms,

$\displaystyle 0=$
$\displaystyle\,g^{(0)t\phi}g^{(0)tt}{\partial_{t}}^{2}a_{\phi}+\frac{1}{\sqrt{-g^{(0)}}}\partial_{r}\left\\{\sqrt{-g^{(0)}}g^{(0)rr}g^{(0)tt}\partial_{t}a_{r}\right\\}+\frac{1}{\sqrt{-g^{(0)}}}\partial_{\theta}\left\\{\sqrt{-g^{(0)}}g^{(0)\theta\theta}g^{(0)tt}\partial_{t}a_{\theta}\right\\}$
$\displaystyle\,+g^{(0)\phi\phi}g^{(0)tt}\partial_{\phi}\partial_{t}a_{\phi}\,,$
(85) $\displaystyle 0=$
$\displaystyle\,g^{(0)tt}g^{(0)rr}{\partial_{t}}^{2}a_{r}+g^{(0)t\phi}g^{(0)rr}\partial_{t}\left(\partial_{\phi}a_{r}-\partial_{r}a_{\phi}\right)+g^{(0)\phi
t}g^{(0)rr}\partial_{\phi}\partial_{t}a_{r}$
$\displaystyle\,+\frac{1}{\sqrt{-g^{(0)}}}\partial_{\theta}\left\\{\sqrt{-g^{(0)}}g^{(0)\theta\theta}g^{(0)rr}\left(\partial_{\theta}a_{r}-\partial_{r}a_{\theta}\right)\right\\}+g^{(0)\phi\phi}g^{(0)rr}\partial_{\phi}\left(\partial_{\phi}a_{r}-\partial_{r}a_{\phi}\right)\,,$
(86) $\displaystyle 0=$
$\displaystyle\,g^{(0)tt}g^{(0)\theta\theta}{\partial_{t}}^{2}a_{\theta}+g^{(0)t\phi}g^{(0)\theta\theta}\partial_{t}\left(\partial_{\phi}a_{\theta}-\partial_{\theta}a_{\phi}\right)$
$\displaystyle\,+g^{(0)\phi
t}g^{(0)\theta\theta}\partial_{\phi}\partial_{t}a_{\theta}+\frac{1}{\sqrt{-g^{(0)}}}\partial_{r}\left\\{\sqrt{-g^{(0)}}g^{(0)rr}g^{(0)\theta\theta}\left(\partial_{r}a_{\theta}-\partial_{\theta}a_{r}\right)\right\\}+g^{(0)\phi\phi}g^{(0)\theta\theta}\partial_{\phi}\left(\partial_{\phi}a_{\theta}-\partial_{\theta}a_{\phi}\right)\,,$
(87) $\displaystyle 0=$
$\displaystyle\,g^{(0)tt}g^{(0)\phi\phi}{\partial_{t}}^{2}a_{\phi}+g^{(0)\phi
t}g^{(0)\phi\phi}\partial_{\phi}\partial_{t}a_{\phi}$
$\displaystyle\,+\frac{1}{\sqrt{-g^{(0)}}}\partial_{r}\left\\{\sqrt{-g^{(0)}}g^{(0)rr}g^{(0)\phi\phi}\left(\partial_{r}a_{\phi}-\partial_{\phi}a_{r}\right)\right\\}+\frac{1}{\sqrt{-g^{(0)}}}\partial_{\theta}\left\\{\sqrt{-g^{(0)}}g^{(0)\theta\theta}g^{(0)\phi\phi}\left(\partial_{\theta}a_{\phi}-\partial_{\phi}a_{\theta}\right)\right\\}\,.$
(88)

We consider a small region and the case the wavelength is small enough
compared with the region. Then the background metric
$g_{\mu\nu}=g^{(0)}_{\mu\nu}$ and the background vector field
$A_{\mu}=A^{(0)}_{\mu}$ can be regarded to be adiabatically constant in the
region. Under the assumptions, Eqs. (VI), (VI), (VI), and (VI) in the
$a_{t}=0$ gauge condition have the following forms,

$\displaystyle 0=$
$\displaystyle\,g^{(0)t\phi}g^{(0)tt}{\partial_{t}}^{2}a_{\phi}+g^{(0)rr}g^{(0)tt}\partial_{r}\partial_{t}a_{r}+g^{(0)\theta\theta}g^{(0)tt}\partial_{\theta}\partial_{t}a_{\theta}+g^{(0)\phi\phi}g^{(0)tt}\partial_{\phi}\partial_{t}a_{\phi}\,,$
(89) $\displaystyle 0=$
$\displaystyle\,g^{(0)tt}g^{(0)rr}{\partial_{t}}^{2}a_{r}+g^{(0)t\phi}g^{(0)rr}\partial_{t}\left(\partial_{\phi}a_{r}-\partial_{r}a_{\phi}\right)+g^{(0)\phi
t}g^{(0)rr}\partial_{\phi}\partial_{t}a_{r}$
$\displaystyle\,+g^{(0)\theta\theta}g^{(0)rr}\partial_{\theta}\left(\partial_{\theta}a_{r}-\partial_{r}a_{\theta}\right)+g^{(0)\phi\phi}g^{(0)rr}\partial_{\phi}\left(\partial_{\phi}a_{r}-\partial_{r}a_{\phi}\right)\,,$
(90) $\displaystyle 0=$
$\displaystyle\,g^{(0)tt}g^{(0)\theta\theta}{\partial_{t}}^{2}a_{\theta}+g^{(0)t\phi}g^{(0)\theta\theta}\partial_{t}\left(\partial_{\phi}a_{\theta}-\partial_{\theta}a_{\phi}\right)$
$\displaystyle\,+g^{(0)\phi
t}g^{(0)\theta\theta}\partial_{\phi}\partial_{t}a_{\theta}+g^{(0)rr}g^{(0)\theta\theta}\partial_{r}\left(\partial_{r}a_{\theta}-\partial_{\theta}a_{r}\right)+g^{(0)\phi\phi}g^{(0)\theta\theta}\partial_{\phi}\left(\partial_{\phi}a_{\theta}-\partial_{\theta}a_{\phi}\right)\,,$
(91) $\displaystyle 0=$
$\displaystyle\,g^{(0)tt}g^{(0)\phi\phi}{\partial_{t}}^{2}a_{\phi}+g^{(0)\phi
t}g^{(0)\phi\phi}\partial_{\phi}\partial_{t}a_{\phi}$
$\displaystyle\,+g^{(0)rr}g^{(0)\phi\phi}\partial_{r}\left(\partial_{r}a_{\phi}-\partial_{\phi}a_{r}\right)+g^{(0)\theta\theta}g^{(0)\phi\phi}\partial_{\theta}\left(\partial_{\theta}a_{\phi}-\partial_{\phi}a_{\theta}\right)\,.$
(92)

Because $g_{\mu\nu}=g^{(0)}_{\mu\nu}$ can be regarded to be constant, we may
assume

$\displaystyle A_{\mu}=C_{\mu}\mathrm{e}^{-i\omega
t+ip_{r}r+ip_{\theta}\theta+im\phi}\,,$ (93)

with constants, $C_{\mu}$, $\omega$, $p_{r}$, $p_{\theta}$ and an integer $m$.
Then Eqs. (89), (VI), (VI), and (VI) become algebraic equations,

$\displaystyle 0=$
$\displaystyle\,g^{(0)t\phi}\omega^{2}C_{\phi}-g^{(0)rr}p_{r}\omega
C_{r}-g^{(0)\theta\theta}p_{\theta}\omega C_{\theta}-g^{(0)\phi\phi}m\omega
C_{\phi}\,,$ (94) $\displaystyle 0=$
$\displaystyle\,g^{(0)tt}\omega^{2}C_{r}-g^{(0)t\phi}\omega\left(mC_{r}-p_{r}C_{\phi}\right)-g^{(0)\phi
t}m\omega C_{r}$
$\displaystyle\,+g^{(0)\theta\theta}p_{\theta}\left(p_{\theta}C_{r}-p_{r}C_{\theta}\right)+g^{(0)\phi\phi}m\left(mC_{r}-p_{r}C_{\phi}\right)\,,$
(95) $\displaystyle 0=$
$\displaystyle\,g^{(0)tt}\omega^{2}C_{\theta}-g^{(0)t\phi}\omega\left(mC_{\theta}-p_{\theta}C_{\phi}\right)$
$\displaystyle\,-g^{(0)\phi t}m\omega
C_{\theta}+g^{(0)rr}p_{r}\left(p_{r}C_{\theta}-p_{\theta}C_{r}\right)+g^{(0)\phi\phi}m\left(mC_{\theta}-p_{\theta}C_{\phi}\right)\,,$
(96) $\displaystyle 0=$ $\displaystyle\,g^{(0)tt}\omega^{2}C_{\phi}-g^{(0)\phi
t}m\omega
C_{\phi}+g^{(0)rr}p_{r}\left(p_{r}C_{\phi}-mC_{r}\right)+g^{(0)\theta\theta}p_{\theta}\left(p_{\theta}C_{\phi}-mC_{\theta}\right)\,.$
(97)

By defining

$\displaystyle p^{2}\equiv g^{(0)tt}\omega^{2}-2g^{(0)t\phi}\omega
m+g^{(0)rr}{p_{r}}^{2}+g^{(0)\theta\theta}{p_{\theta}}^{2}+g^{(0)\phi\phi}m^{2}\,,$
(98)

we find

$\displaystyle 0=$ $\displaystyle\,\omega\left(g^{(0)t\phi}\omega
C_{\phi}-g^{(0)rr}p_{r}C_{r}-g^{(0)\theta\theta}p_{\theta}C_{\theta}-g^{(0)\phi\phi}mC_{\phi}\right)\,,$
(99) $\displaystyle 0=$
$\displaystyle\,p^{2}C_{r}+p_{r}\left(g^{(0)t\phi}\omega
C_{\phi}-g^{(0)rr}p_{r}C_{r}-g^{(0)\theta\theta}p_{\theta}C_{\theta}-g^{(0)\phi\phi}mC_{\phi}\right)\,,$
(100) $\displaystyle 0=$
$\displaystyle\,p^{2}C_{\theta}+p_{\theta}\left(g^{(0)t\phi}\omega
C_{\phi}-g^{(0)rr}p_{r}C_{r}-g^{(0)\theta\theta}p_{\theta}C_{\theta}-g^{(0)\phi\phi}mC_{\phi}\right)\,,$
(101) $\displaystyle 0=$
$\displaystyle\,p^{2}C_{\phi}+p_{\phi}\left(g^{(0)t\phi}\omega
C_{\phi}-g^{(0)rr}p_{r}C_{r}-g^{(0)\theta\theta}p_{\theta}C_{\theta}-g^{(0)\phi\phi}mC_{\phi}\right)\,.$
(102)

Eq. (99) gives the Gauss-law constraint,

$\displaystyle 0=g^{(0)t\phi}\omega
C_{\phi}-g^{(0)rr}p_{r}C_{r}-g^{(0)\theta\theta}p_{\theta}C_{\theta}-g^{(0)\phi\phi}mC_{\phi}\,,$
(103)

and Eqs. (100), (101), and (102) give the following dispersion relation,

$\displaystyle 0=p^{2}=g^{(0)tt}\omega^{2}-2g^{(0)t\phi}\omega
m+g^{(0)rr}{p_{r}}^{2}+g^{(0)\theta\theta}{p_{\theta}}^{2}+g^{(0)\phi\phi}m^{2}\,,$
(104)

which can be solved with respect to $\omega$

$\displaystyle\omega=\omega_{\pm}\equiv\frac{g^{(0)t\phi}}{g^{(0)tt}}m\pm\sqrt{\left(\frac{g^{(0)t\phi}}{g^{(0)tt}}\right)^{2}m^{2}-\frac{1}{g^{(0)tt}}\left(g^{(0)rr}{p_{r}}^{2}+g^{(0)\theta\theta}{p_{\theta}}^{2}+g^{(0)\phi\phi}m^{2}\right)}\,.$
(105)

For the spherically symmetric case where $g^{(0)t\phi}=0$, $\omega$ is
invariant under the change of the signature of $m$, $m\to-m$. Because Eq.
(103) tells

$\displaystyle C_{\phi}=\frac{1}{g^{(0)\phi\phi}m}\left(g^{(0)t\phi}\omega
C_{\phi}-g^{(0)rr}p_{r}C_{r}-g^{(0)\theta\theta}p_{\theta}C_{\theta}\right)\,,$
(106)

the signature of $m$ expresses the difference between the left rotating
helicity and the right rotating helicity. Although in any axially symmetric
and rotating solution of space-time like the Kerr solution, $g^{(0)t\phi}$
does not vanish, there are several characteristic structures in our model. Eq.
(IV.1) tells the explicit form of $g_{t\phi}$. The first term of $g_{t\phi}$
is common in rotating black hole solutions like the Kerr solution or the Kerr-
Newman solution but the second term is characteristic of our model. The first
term vanishes when $\theta=0$ or $\theta=\pi$, which corresponds to the north
pole or the south pole. This tells that the difference between the dispersion
relations for the left and right rotations vanishes at the poles if there is
not the second term, which does not have the $\theta$ dependence. Furthermore,
the second term increases when $H\equiv M^{2}-q^{2}$ goes to vanish, that is,
in the extremal limit. Of course, the second term appears as a correction
coming from the Chern-Simons term and therefore the expression is valid only
when the Chern-Simons coupling is small but the term can dominate more than
the first term near the limit. The above characteristic structures will give
some effects on the photon which goes through the region near the black hole
although it could be an interesting subject to investigate how the effects on
the photon can be observed.

##  VII Conclusion and discussions

A new non-trivial natural slowly rotating BH solution using the dCS
gravitational theory has been derived Yunes:2007ss . The charge in the study
of BH solutions is important because BH generated in a collider could have an
electric field. Therefore, we included the effect of electric charge in the
dCS field equations by including the effect of Maxwell’s electromagnetic
field. We are interested in the effects of the dCS term and therefore we did
not study the non-dynamical case because the result of this theory is not
changed from the result of the Reissner-Nordström black hole solution.

If we include the electric charge and the electromagnetic field, the scalar of
the SC field in (13) is modified, and therefore, the expression of the scalar
field $\varphi^{(1,0)}$ and $\varphi^{(1,1)}$ are not equivalent to the forms
given in Yunes:2007ss . Moreover, when we apply the expression of the CS
scalar field $\varphi^{(1,0)}$ in the field equation (12), we obtain the
rectification of the metric up to order $\epsilon$. This rectification
$\omega^{(1,1)}$ gives an asymptotic form of order
$\mathcal{O}\left(\frac{1}{r^{7}}\right)$ for large $r$, which is much weaker
than the expression given in Yunes:2007ss  where the leading order behavior is
$\mathcal{O}\left(\frac{1}{r^{6}}\right)$. Moreover, we found that the leading
form of the Pontryagin density
$R_{\nu\mu\alpha\beta}\tilde{R}^{\mu\nu\alpha\beta}$ has the form
$\mathcal{O}\left(\frac{1}{r^{7}}\right)$ when $r$ is large, which agrees with
the results given in the literature, however, the next-to-leading term is of
order $\mathcal{O}\left(\frac{1}{r^{8}}\right)$, which depends on the charge
and is stronger than the one given in Yagi:2012ya . Moreover, we studied the
stability of the charged BH solution using the geodesic deviations. We obtain
the condition of the stability and discuss its behavior graphically.

We also investigate the possibility that the BH in this paper could be found
by any observation. We focused on the polarization of the photon which
propagates near the black hole. In the case of the Kerr solution or the Kerr-
Newman solution, the difference between the dispersion relations for the left
and right rotations vanishes at the north and south poles but in the BH
solution in this paper, the difference does not vanish there. Furthermore, in
the extremal limit. where $H\equiv M^{2}-q^{2}$ goes to vanish, the correction
coming from the Chern-Simons term becomes dominant. Therefore it could be an
interesting subject to investigate how these effects on the photon can be
observed.

Finally, we close our study with the following: In the present paper, we
derive the charged electric BH solution in dCS by using a linear Maxwell
field. Other forms of the Maxwell field, i.e., non-linear forms of the Maxwell
field are not beneficial because the asymptotic form of the non-linear Maxwell
field will be $\mathcal{O}(\epsilon^{2})$. Another case that could be
interesting to study is to assume the magnetic field in addition to the
electric field. This study will be carried out elsewhere. Also, another
interesting case is to take into account the effect of the electric charge and
the potential $V(\varphi)$ for the scalar field $\varphi$ because we only
considered the case $V(\varphi)=0$ in this paper. Also, this case will be our
future work.

## Acknowledgment

The authors would like to thank the anonymous referee for her/his critical
comments which put the paper in clear formalism.

## References

  * (1) C. M. Will, Living Rev. Rel. 17, 4 (2014) doi:10.12942/lrr-2014-4 [arXiv:1403.7377 [gr-qc]].
  * (2) B. Bertotti, L. Iess and P. Tortora, Nature 425, 374-376 (2003) doi:10.1038/nature01997
  * (3) I. H. Stairs, Living Rev. Rel. 6, 5 (2003) doi:10.12942/lrr-2003-5 [arXiv:astro-ph/0307536 [astro-ph]].
  * (4) A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gursel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero and K. S. Thorne, et al. Science 256, 325-333 (1992) doi:10.1126/science.256.5055.325
  * (5) B. P. Abbott et al. [LIGO Scientific], Rept. Prog. Phys. 72, 076901 (2009) doi:10.1088/0034-4885/72/7/076901 [arXiv:0711.3041 [gr-qc]].
  * (6) G. M. Harry [LIGO Scientific], Class. Quant. Grav. 27, 084006 (2010) doi:10.1088/0264-9381/27/8/084006
  * (7) F. Acernese et al. [VIRGO], Class. Quant. Grav. 25, 225001 (2008) doi:10.1088/0264-9381/25/22/225001 [arXiv:0803.0376 [gr-qc]].
  * (8) B. Caron, A. Dominjon, C. Drezen, R. Flaminio, X. Grave, F. Marion, L. Massonnet, C. Mehmel, R. Morand and B. Mours, et al. Class. Quant. Grav. 14, 1461-1469 (1997) doi:10.1088/0264-9381/14/6/011
  * (9) A. Giazotto, Nucl. Instrum. Meth. A 289, 518-525 (1990) doi:10.1016/0168-9002(90)91525-G
  * (10) F. Acernese et al. [VIRGO], Class. Quant. Grav. 32, no.2, 024001 (2015) doi:10.1088/0264-9381/32/2/024001 [arXiv:1408.3978 [gr-qc]].
  * (11) E. Berti, K. Yagi and N. Yunes, Gen. Rel. Grav. 50, no.4, 46 (2018) doi:10.1007/s10714-018-2362-8 [arXiv:1801.03208 [gr-qc]].
  * (12) E. Berti, K. Yagi, H. Yang and N. Yunes, Gen. Rel. Grav. 50, no.5, 49 (2018) doi:10.1007/s10714-018-2372-6 [arXiv:1801.03587 [gr-qc]].
  * (13) A. G. Riess et al. [Supernova Search Team], Astrophys. J. 607, 665-687 (2004) doi:10.1086/383612 [arXiv:astro-ph/0402512 [astro-ph]].
  * (14) V. C. Rubin and W. K. Ford, Jr., Astrophys. J. 159, 379-403 (1970) doi:10.1086/150317
  * (15) P. Helbig, Astron. Astrophys. 350, 1-8 (1999) [arXiv:astro-ph/9904311 [astro-ph]].
  * (16) D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones and D. Zaritsky, Astrophys. J. Lett. 648, L109-L113 (2006) doi:10.1086/508162 [arXiv:astro-ph/0608407 [astro-ph]].
  * (17) M. Tegmark et al. [SDSS], Phys. Rev. D 74, 123507 (2006) doi:10.1103/PhysRevD.74.123507 [arXiv:astro-ph/0608632 [astro-ph]].
  * (18) N. Aghanim et al. [Planck], Astron. Astrophys. 641, A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)] doi:10.1051/0004-6361/201833910 [arXiv:1807.06209 [astro-ph.CO]].
  * (19) D. Huterer and M. S. Turner, Phys. Rev. D 60, 081301 (1999) doi:10.1103/PhysRevD.60.081301 [arXiv:astro-ph/9808133 [astro-ph]].
  * (20) B. J. Carr, Ann. Rev. Astron. Astrophys. 32, 531-590 (1994) doi:10.1146/annurev.aa.32.090194.002531
  * (21) R. J. Scherrer, Phys. Rev. Lett. 93, 011301 (2004) doi:10.1103/PhysRevLett.93.011301 [arXiv:astro-ph/0402316 [astro-ph]].
  * (22) A. Ashtekar and J. Lewandowski, Class. Quant. Grav. 21, R53 (2004) doi:10.1088/0264-9381/21/15/R01 [arXiv:gr-qc/0404018 [gr-qc]].
  * (23) C. Rovelli, Univ. Pr., 2004, doi:10.1017/CBO9780511755804
  * (24) R. Jackiw and S. Y. Pi, Phys. Rev. D 68, 104012 (2003) doi:10.1103/PhysRevD.68.104012 [arXiv:gr-qc/0308071 [gr-qc]].
  * (25) S. Alexander and N. Yunes, Phys. Rept. 480, 1-55 (2009) doi:10.1016/j.physrep.2009.07.002 [arXiv:0907.2562 [hep-th]].
  * (26) S. H. S. Alexander and S. J. Gates, Jr., JCAP 06, 018 (2006) doi:10.1088/1475-7516/2006/06/018 [arXiv:hep-th/0409014 [hep-th]].
  * (27) S. Weinberg, Phys. Rev. D 77, 123541 (2008) doi:10.1103/PhysRevD.77.123541 [arXiv:0804.4291 [hep-th]].
  * (28) V. Taveras and N. Yunes, Phys. Rev. D 78, 064070 (2008) doi:10.1103/PhysRevD.78.064070 [arXiv:0807.2652 [gr-qc]].
  * (29) G. Calcagni and S. Mercuri, Phys. Rev. D 79, 084004 (2009) doi:10.1103/PhysRevD.79.084004 [arXiv:0902.0957 [gr-qc]].
  * (30) N. Yunes and D. N. Spergel, Phys. Rev. D 80, 042004 (2009) doi:10.1103/PhysRevD.80.042004 [arXiv:0810.5541 [gr-qc]].
  * (31) S. Alexander and N. Yunes, Phys. Rev. Lett. 99, 241101 (2007) doi:10.1103/PhysRevLett.99.241101 [arXiv:hep-th/0703265 [hep-th]].
  * (32) T. L. Smith, A. L. Erickcek, R. R. Caldwell and M. Kamionkowski, Phys. Rev. D 77, 024015 (2008) doi:10.1103/PhysRevD.77.024015 [arXiv:0708.0001 [astro-ph]].
  * (33) S. Alexander and N. Yunes, Phys. Rev. D 75, 124022 (2007) doi:10.1103/PhysRevD.75.124022 [arXiv:0704.0299 [hep-th]].
  * (34) Y. Nakamura, D. Kikuchi, K. Yamada, H. Asada and N. Yunes, Class. Quant. Grav. 36, no.10, 105006 (2019) doi:10.1088/1361-6382/ab04c5 [arXiv:1810.13313 [gr-qc]].
  * (35) K. Yagi, L. C. Stein, N. Yunes and T. Tanaka, Phys. Rev. D 87, 084058 (2013) [erratum: Phys. Rev. D 93, no.8, 089909 (2016)] doi:10.1103/PhysRevD.87.084058 [arXiv:1302.1918 [gr-qc]].
  * (36) N. Yunes and F. Pretorius, Phys. Rev. D 79, 084043 (2009) doi:10.1103/PhysRevD.79.084043 [arXiv:0902.4669 [gr-qc]].
  * (37) K. Yagi, N. Yunes and T. Tanaka, Phys. Rev. D 86, 044037 (2012) [erratum: Phys. Rev. D 89, 049902 (2014)] doi:10.1103/PhysRevD.86.044037 [arXiv:1206.6130 [gr-qc]].
  * (38) A. Maselli, P. Pani, R. Cotesta, L. Gualtieri, V. Ferrari and L. Stella, Astrophys. J. 843, no.1, 25 (2017) doi:10.3847/1538-4357/aa72e2 [arXiv:1703.01472 [astro-ph.HE]].
  * (39) R. McNees, L. C. Stein and N. Yunes, Class. Quant. Grav. 33, no.23, 235013 (2016) doi:10.1088/0264-9381/33/23/235013 [arXiv:1512.05453 [gr-qc]].
  * (40) B. Chen and L. C. Stein, Phys. Rev. D 97, no.8, 084012 (2018) doi:10.1103/PhysRevD.97.084012 [arXiv:1802.02159 [gr-qc]].
  * (41) N. Yunes and C. F. Sopuerta, Phys. Rev. D 77, 064007 (2008) doi:10.1103/PhysRevD.77.064007 [arXiv:0712.1028 [gr-qc]].
  * (42) C. W. F. Everitt, D. B. DeBra, B. W. Parkinson, J. P. Turneaure, J. W. Conklin, M. I. Heifetz, G. M. Keiser, A. S. Silbergleit, T. Holmes and J. Kolodziejczak, et al. Phys. Rev. Lett. 106, 221101 (2011) doi:10.1103/PhysRevLett.106.221101 [arXiv:1105.3456 [gr-qc]].
  * (43) Y. Ali-Haimoud and Y. Chen, Phys. Rev. D 84, 124033 (2011) doi:10.1103/PhysRevD.84.124033 [arXiv:1110.5329 [astro-ph.HE]].
  * (44) H. Okawara, K. Yamada and H. Asada, Phys. Rev. Lett. 109, 231101 (2012) doi:10.1103/PhysRevLett.109.231101 [arXiv:1210.4628 [gr-qc]].
  * (45) H. Okawara, K. Yamada and H. Asada, Phys. Rev. D 87, no.8, 084038 (2013) doi:10.1103/PhysRevD.87.084038 [arXiv:1302.0002 [gr-qc]].
  * (46) D. Kikuchi, N. Omoto, K. Yamada and H. Asada, Phys. Rev. D 90, no.6, 064036 (2014) doi:10.1103/PhysRevD.90.064036 [arXiv:1405.7472 [gr-qc]].
  * (47) N. Loutrel, T. Tanaka and N. Yunes, Class. Quant. Grav. 36, no.10, 10LT02 (2019) doi:10.1088/1361-6382/ab15fa [arXiv:1806.07425 [gr-qc]].
  * (48) K. Konno, T. Matsuyama and S. Tanda, Phys. Rev. D 76, 024009 (2007) doi:10.1103/PhysRevD.76.024009 [arXiv:0706.3080 [gr-qc]].
  * (49) D. Grumiller and N. Yunes, Phys. Rev. D 77, 044015 (2008) doi:10.1103/PhysRevD.77.044015 [arXiv:0711.1868 [gr-qc]].
  * (50) K. Konno, T. Matsuyama and S. Tanda, Prog. Theor. Phys. 122, 561-568 (2009) doi:10.1143/PTP.122.561 [arXiv:0902.4767 [gr-qc]].
  * (51) L. Amarilla, E. F. Eiroa and G. Giribet, Phys. Rev. D 81, 124045 (2010) doi:10.1103/PhysRevD.81.124045 [arXiv:1005.0607 [gr-qc]].
  * (52) Y. Brihaye and E. Radu, Phys. Lett. B 764, 300-305 (2017) doi:10.1016/j.physletb.2016.11.055 [arXiv:1610.09952 [gr-qc]].
  * (53) A. Cisterna, C. Corral and S. del Pino, Eur. Phys. J. C 79, no.5, 400 (2019) doi:10.1140/epjc/s10052-019-6910-5 [arXiv:1809.02903 [gr-qc]].
  * (54) C. W. Misner, K. S. Thorne and J. A. Wheeler, W. H. Freeman, 1973, ISBN 978-0-7167-0344-0, 978-0-691-17779-3
  * (55) B. Tekin, Phys. Rev. D 77, 024005 (2008) doi:10.1103/PhysRevD.77.024005 [arXiv:0710.2528 [gr-qc]].
  * (56) S. Deser and B. Tekin, Phys. Rev. Lett. 89, 101101 (2002) doi:10.1103/PhysRevLett.89.101101 [arXiv:hep-th/0205318 [hep-th]].
  * (57) S. Deser and B. Tekin, Phys. Rev. D 67, 084009 (2003) doi:10.1103/PhysRevD.67.084009 [arXiv:hep-th/0212292 [hep-th]].
  * (58) S. Deser and B. Tekin, Class. Quant. Grav. 20, L259 (2003) doi:10.1088/0264-9381/20/21/L01 [arXiv:gr-qc/0307073 [gr-qc]].
  * (59) D. Grumiller, R. B. Mann and R. McNees, Phys. Rev. D 78, 081502 (2008) doi:10.1103/PhysRevD.78.081502 [arXiv:0803.1485 [gr-qc]].
  * (60) K. S. Thorne and J. B. Hartle, Phys. Rev. D 31, 1815-1837 (1984) doi:10.1103/PhysRevD.31.1815
  * (61) J. B. Hartle and K. S. Thorne, Astrophys. J. 153, 807 (1968) doi:10.1086/149707
  * (62) O. Miskovic and R. Olea, Phys. Rev. D 79, 124020 (2009) doi:10.1103/PhysRevD.79.124020 [arXiv:0902.2082 [hep-th]].
  * (63) C. Y. Liu, D. S. Lee and C. Y. Lin, Class. Quant. Grav. 34, no.23, 235008 (2017) doi:10.1088/1361-6382/aa903b [arXiv:1706.05466 [gr-qc]].
  * (64) A. M. A. Zahrani, V. P. Frolov and A. A. Shoom, Phys. Rev. D 87, no.8, 084043 (2013) doi:10.1103/PhysRevD.87.084043 [arXiv:1301.4633 [gr-qc]].
  * (65) J. B. Hartle, Cambridge University Press, 2021, ISBN 978-1-00-904260-4 doi:10.1017/9781009042604
  * (66) J. M. Bardeen, W. H. Press and S. A. Teukolsky, Astrophys. J. 178, 347 (1972) doi:10.1086/151796
  * (67) A. Ori and K. S. Thorne, Phys. Rev. D 62, 124022 (2000) doi:10.1103/PhysRevD.62.124022 [arXiv:gr-qc/0003032 [gr-qc]].
  * (68) M. Blaschke and Z. Stuchlík, Phys. Rev. D 94, no.8, 086006 (2016) doi:10.1103/PhysRevD.94.086006 [arXiv:1610.07462 [gr-qc]].
  * (69) D. Pugliese, H. Quevedo and R. Ruffini, Phys. Rev. D 88, no.2, 024042 (2013) doi:10.1103/PhysRevD.88.024042 [arXiv:1303.6250 [gr-qc]].
  * (70) A. N. Aliev and A. E. Gumrukcuoglu, Phys. Rev. D 71, 104027 (2005) doi:10.1103/PhysRevD.71.104027 [arXiv:hep-th/0502223 [hep-th]].
  * (71) R. d’Inverno, “Introducing Einstein’s relativity,” isbn = ”978-0-19-859686-8”, year = ”1992”
  * (72) T. Harko, Z. Kovacs and F. S. N. Lobo, Class. Quant. Grav. 27, 105010 (2010) doi:10.1088/0264-9381/27/10/105010 [arXiv:0909.1267 [gr-qc]].
  * (73) C. F. Sopuerta and N. Yunes, Phys. Rev. D 80, 064006 (2009) doi:10.1103/PhysRevD.80.064006 [arXiv:0904.4501 [gr-qc]].
  * (74) A. Sheykhi, Phys. Rev. D 86, 024013 (2012) doi:10.1103/PhysRevD.86.024013 [arXiv:1209.2960 [hep-th]].
  * (75) A. Sheykhi, Eur. Phys. J. C 69, 265-269 (2010) doi:10.1140/epjc/s10052-010-1372-9 [arXiv:1012.0383 [hep-th]].
  * (76) S. H. Hendi, A. Sheykhi and M. H. Dehghani, Eur. Phys. J. C 70, 703-712 (2010) doi:10.1140/epjc/s10052-010-1483-3 [arXiv:1002.0202 [hep-th]].
  * (77) A. Sheykhi, M. H. Dehghani and S. H. Hendi, Phys. Rev. D 81, 084040 (2010) doi:10.1103/PhysRevD.81.084040 [arXiv:0912.4199 [hep-th]].
  * (78) G. G. L. Nashed and W. El Hanafy, Eur. Phys. J. C 77, no.2, 90 (2017) doi:10.1140/epjc/s10052-017-4663-6 [arXiv:1612.05106 [gr-qc]].
  * (79) G. G. L. Nashed, Annalen Phys. 523, 450-458 (2011) doi:10.1002/andp.201100030 [arXiv:1105.0328 [gr-qc]].