# Rotating black hole solutions for $f(R)$ gravity and Newman Janis Algorithm

Pankaj Chaturvedi<EMAIL_ADDRESS>Department of Physics, Ariel University,
Ariel 40700, Israel  Utkarsh Kumar<EMAIL_ADDRESS>Department of
Physics, Ariel University, Ariel 40700, Israel  Udaykrishna Thattarampilly
<EMAIL_ADDRESS>Department of Physics, Ariel University, Ariel 40700,
Israel  Vishnu Kakkat<EMAIL_ADDRESS>Department of Mathematical Sciences,
Unisa

###### Abstract

We show that the $f(R)$-gravity theories with constant Ricci scalar in the
Jordan/Einstein frame can be described by Einstein or Einstein-Maxwell gravity
with a cosmological term and a modified gravitational constant. We also
propose a modified Newmann-Janis algorithm to obtain the rotating axisymmetric
solutions for the Einstein/Einstein-Maxwell gravity with a cosmological
constant. Using the duality between the two gravity theories we show that the
stationary or static solutions for the Einstein/Einstein-Maxwell gravity with
a cosmological constant will also be the solutions for the dual $f(R)$-gravity
with constant Ricci scalar.

††preprint: APS-XXX

##  I Introduction

The general theory of relativity (GR) is widely accepted as the fundamental
theory of space-time and gravity. Despite predicting numerous observational
tests at large distances and late time scales (Infrared regime) to match the
measurements from solar system tests, GR has gone through several challenges
from the observational and theoretical viewpoints. Cosmological observations
pertaining to cosmic microwave background (CMB), Type Ia supernovae and
several others indicate that the Universe has undergone two phases of cosmic
acceleration namely inflation and dark energy that occurred at early and late
times respectively Riess  _et al._ (1998); Starobinskiǐ (1979); Guth (1981);
Huterer and Turner (1999); Sahni and Starobinsky (2000). GR, in its original
form, is unable to explain these phases of cosmic acceleration. Cosmological
constant used to parametrize the recent accelerated expansion of the Universe
is plagued by hierarchy problems in particle physics Weinberg (2000). Beyond
its inconsistencies with cosmological and astrophysical data, GR also presents
numerous theoretical weaknesses. Most prominently, GR struggles in the
Ultraviolet spectrum, especially when interpreting the physics of black holes
and cosmological singularities at short distances and small time intervals.
This is due to the fact that GR is a 2-derivative action which poses problems
of renormalizability at the quantum level. The non-renormalizability of GR has
spurred interest in higher derivative gravity theories, often referred to as
modified gravities.

A simple yet consequential model for modified gravity is the $f(R)$ gravity
theory in which the Lagrangian density is modified to be an arbitrary analytic
function of the Ricci scalar Bergmann (1968); Ruzmaikina and Ruzmaikin (1969).
The significance of higher-order terms in the $f(R)$ gravity Lagrangian can be
attributed to the fact that they can descend from the low energy limit of
String/M-theory Nojiri and Odintsov (2003a). The simplest $f(R)$-gravity with
$R^{2}$ correction can give rise to a phase of inflation which was first
proposed and explored by StarobinskyStarobinsky (1980). There are numerous
$f(R)$ gravity models to explain the cosmic inflation El Bourakadi  _et al._
(2023); Wang  _et al._ (2023); Ossoulian  _et al._ (2023); Belhaj  _et al._
(2023); Oikonomou  _et al._ (2023); Jeong  _et al._ (2023); Odintsov and Paul
(2023); Oikonomou (2023a); Dioguardi and Racioppi (2023); Cheraghchi  _et al._
(2023); Odintsov  _et al._ (2023); Baffou  _et al._ (2023); Luongo and Mengoni
(2023) and current accelerated expansion of the Universe Capozziello (2002);
Capozziello  _et al._ (2007, 2003); Amendola  _et al._ (2007); Li and Barrow
(2007); Amendola and Tsujikawa (2008); Appleby and Battye (2007); Starobinsky
(2007); Cognola  _et al._ (2008a). In addition to that $f(R)$ gravity theories
also has been investigated in explaining the singularity problem arising in
the strong gravity regime Astashenok  _et al._ (2023); Nava-Callejas  _et al._
(2023); Panotopoulos and Lopes (2018); Yu  _et al._ (2018); Doneva and Pappas
(2018); Cikintoglu (2018); Sussman and Jaime (2017); Gao and Shen (2016);
Cañate  _et al._ (2016); Berti  _et al._ (2015), galaxy rotation curves Mohan
and Goswami (2022); Parbin and Goswami (2023); Shabani and Moraes (2023);
Sharma  _et al._ (2020); Naik  _et al._ (2019, 2018); Matsakos and Diaferio
(2016); Dey  _et al._ (2015); Capozziello  _et al._ (2013), detection of
gravitational waves Dimastrogiovanni  _et al._ (2023); Odintsov and Oikonomou
(2022); Oikonomou (2022); Alves  _et al._ (2023); Oikonomou (2023b); Narang
_et al._ (2023); Inagaki and Taniguchi (2023); Liang  _et al._ (2023); Khlopov
and Chowdhury (2023); Dyadina (2023); Ezquiaga  _et al._ (2023); Alves (2023)
and many more.

The current era of gravitational wave astronomy has presented us with the
possibility of investigating physics of extremely compact objects, such as
black holes and neutron stars Yunes  _et al._ (2016). This has opened new
prospects for observing and testing theoretical models in the strong gravity
regime. As a platform for theories that explain cosmic acceleration and
inflation, it is of paramount importance to explore and test $f(R)$ theories
of modified gravity. The study of the properties of black hole solutions in
these scenarios can provide strong gravity tests for these theories and may
hint toward significant deviations from the GR. This knowledge of black hole
space-time can be obtained from solutions to the field equations, which,
although easy to find analytically in General Relativity (GR), is a non-
trivial task in modified gravity theories. Since most astrophysical objects
are considered to be spinning, there is an interest in finding rotating black
hole solutions for $f(R)$ theories. Several black hole solutions in modified
gravities with or without matter have been studied previously. Outside of
$f(R)$ theories Cvetic  _et al._ (2002); Cai (2002); Moon  _et al._ (2011);
Capozziello  _et al._ (2010); Nzioki  _et al._ (2010); Olmo (2007) such
studies involve finding the black holes solutions for Gauss-Bonnet Gravity,
Lovelock, non-local, and other modified theories studied in Cai (2004);
Matyjasek  _et al._ (2006); Kumar and Panda (2019); Kumar  _et al._ (2018);
Kumar and Panda (2018); Kumar  _et al._ (2020, 2021).

It is well known that $f(R)$ gravity in Jordan frame can be recast as a non-
minimally coupled scalar-tensor gravity theory in the Einstein frame by means
of a conformal transformation Nojiri and Odintsov (2011); Capozziello and De
Laurentis (2011). Thus for a problem formulated for $f(R)$ gravity, the usual
approach is to first solve the simpler field equations of motion in the
Einstein frame and then use a conformal transformation to revert back to the
Jordan frame. Spherically symmetric black hole solutions of $f(R)$-gravity
theories with or without matter have been studied in the Einstein/Jordan frame
in Mignemi and Wiltshire (1992); Multamaki and Vilja (2006). Although this
conformal transformation of higher order gravity to scalar-tensor gravity is
in general plausible, it does not shed enough information about the physical
relevance of these two theories in different frames. This discrepancy is
related to the fact that using a conformal transformation to go from one frame
to the other the stability of the solutions and their physical meaning can
completely change. This leads to several physical constraints which has to be
imposed on the form of the function $f(R)$ in order to have stable solutions
in both the Einstein and the Jordan frame Bhattacharya and Chakrabarty (2016).
Despite the ambiguities mentioned above, it has been well establish that any
$f(R)$ gravity with constant Ricci scalar is dual to the Einstein gravity with
a cosmological and an effective gravitational constants Tsujikawa and Tatekawa
(2008); De Felice and Tsujikawa (2010).

Motivated by the the above duality of $f(R)$-gravity in the Jordan frame to
scalar-tensor gravity in the Einstein frame, in this work we have explored the
description of constant curvature $f(R)$ gravity in the Einstein frame. We
show that the constant curvature $f(R)$-gravity in the Einstein frame is
indeed described by Einstein Gravity with a cosmological and an effective
gravitational constants at the level of the action and field equations. In
this work, we also examine the possible static or stationary blackhole
solutions for the constant curvature $f(R)$-gravity with Maxwell field in the
Einstein frame.

In general it is not an easy task to obtain the rotating black hole solutions
in the modified or the simple Einstein gravity models with various horizon
topologies because of the nonlinear nature of the field equations. It was
observed by Newman and Janis in the case of the Einstein equations that a
certain algorithm produces the Kerr solution from the corresponding non-
rotating counterpart. This procedure is now known as the Newman-Janis
algorithm (NJA) Newman  _et al._ (1965); Adamo and Newman (2014). This
algorithm has been widely used as a technique to generate the Kerr-like
rotating metrics from the corresponding static metrics, see Shao  _et al._
(2021); Kubiznak  _et al._ (2022); Kamenshchik and Petriakova (2023);
Fernandes (2023); Ghosh  _et al._ (2023). This algorithm provides a way to
generate axisymmetric metrics from a spherically symmetric stationary seed
metric through a particular type of complexification of radial and time
coordinates. Although this algorithm functions effectively within the
framework of classical General Relativity (GR), the reason it yields the same
result for the Kerr metric in classical GR remains somewhat elusive. Moreover,
it has been illustrated in Hansen and Yunes (2013); Ferraro (2014); Ayzenberg
and Yunes (2014) that the NJ algorithm is not suitable for generating
axisymmetric metrics in quadratic gravity models. Furthermore, It is still
unknown about the applicability of the NJ algorithm to the other modified
gravity models. In this paper, we propose a modified version of the NJ
algorithm in order to generate stationary axisymmetric blackhole solutions for
various constant curvature $f(R)$ gravity models in the Einstein frame.

The article is organised as following. In section II, we review the Jordan and
Einstein Frames for modified gravity and establish the duality between the
constant curvature $f(R)$ gravity theories in Jordan frame to Einstein-Maxwell
gravity with a CC. Furthermore, we explicitly show this duality for well-known
examples. Section III deals with the modification in Newmann-Janis Algorithm
to generate the rotating black hole solution for the Einstein gravity with CC.
Finally, we discuss our findings and draw conclusions in section IV.

##  II $f(R)$ gravity in Jordan and Einstein frames

The standard form of the action for $f(R)$ gravity in the so-called Jordan
frame is given by

${\cal S}_{\mathcal{J}}=\frac{1}{2\kappa}\int d^{4}x\sqrt{-g}\;f(R)+{\cal
S}_{M},$ (1)

where $\kappa=8\pi G$ in natural units with $G$ being the four dimensional
gravitational constant, $f(R)$ is a generic function of the Ricci scalar $R$,
and ${\cal S}_{M}$ is the usual matter contribution to the action. Varying the
above action with respect to the metric results in the following Euler-
Lagrange equations of motion

$\left(g_{ab}\Box-\nabla_{a}\nabla_{b}\right)F(R)+F(R)R_{ab}-\frac{1}{2}f(R)g_{ab}=\kappa
T_{ab},$ (2)

where $F(R)=f^{\prime}(R)$ and the matter stress-energy tensor $T_{ab}$ is
given by

$T_{ab}=-\frac{2}{\sqrt{-g}}\frac{\delta{\cal S}_{M}}{\delta g_{ab}}$ (3)

The following action describes the Einstein-Hilbert gravity with a non-
minimally coupled scalar field

$\displaystyle\mathcal{S}_{\mathcal{J}}=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}\left(F(\phi)R-V_{\mathcal{J}}(\phi)\right)+{\cal S}_{M},$ (4)

where

$\displaystyle V_{\mathcal{J}}(\phi)=\phi F(\phi)-f(\phi),$ (5)

is dynamically equivalent to the $f(R)$ gravity action in (1). This may be
seen by considering the variation of the action in (4) with respect to the
metric and to the scalar field-$\phi(x_{\mu})$ which gives us the following
Euler-Lagrange equations of motion

$\displaystyle\left(g_{ab}\Box-\nabla_{a}\nabla_{b}\right)F(\phi)+F(\phi)R_{ab}-\frac{1}{2}f(\phi)g_{ab}=\kappa
T_{ab},$ $\displaystyle F^{\prime}(\phi)(R-\phi)-F(\phi)+f^{\prime}(\phi)=0.$
(6)

Now provided $F^{\prime}(\phi)\neq 0$ and $F(\phi)=f^{\prime}(\phi)$ one gets
the constraint $R=\phi$ from the second equation of motion in (6). Plugging
back the constraint $R=\phi$ in the action (4) and the equations of motion (6)
one recovers the $f(R)$ gravity action and equations of motion. The action in
(4) can now be recast in the Einstein frame by considering the following
conformal transformation

$\displaystyle\widetilde{g}_{ab}=F(\phi)\;g_{ab},$ (7)

where one must impose $F(\phi)>0$ for the regularity of the given
transformation. Given the new metric $\widetilde{g}_{ab}$, the action for the
$f(R)$ gravity can now be written in the Einstein frame as

$\displaystyle\mathcal{S}_{E}=\int
d^{4}x\sqrt{-\widetilde{g}}\left(\frac{\widetilde{R}}{2\kappa}-\frac{1}{2}\widetilde{\nabla}_{a}\widetilde{\phi}\;\widetilde{\nabla}_{b}\widetilde{\phi}-V_{E}(\widetilde{\phi})\right)+\widetilde{\mathcal{S}}_{M},$
(8)
$\displaystyle\widetilde{\phi}=\sqrt{\frac{3}{2\kappa}}\ln{\left[F(\phi)\right]},\quad
V_{E}(\widetilde{\phi})=\frac{V_{\mathcal{J}}(\phi)}{F(\phi)^{2}},$ (9)
$\displaystyle\widetilde{R}=\frac{1}{F(\phi)}\left(R-3\frac{\Box
F(\phi)}{F(\phi)}+\frac{3}{2}\frac{\nabla_{a}F(\phi)\nabla^{a}F(\phi)}{F(\phi)^{2}}\right),$
(10)

where the quantities with the subscript $\widetilde{()}$ are defined with
respect to the new $\widetilde{g}_{ab}$-metric. Moreover, under the
transformation (7) the matter stress-energy tensor transforms as

$\displaystyle\widetilde{T}_{ab}=\frac{T_{ab}}{F(\phi)^{2}}$ (11)

The conformal equivalence of $f(R)$ gravity in the two different frames as
described above, must be accompanied by certain consistency conditions. In
particular, for the case when the form of $f(R)$ is described by a polynomial
function of order greater than two, the correspondence between the two
conformal frames becomes many-to-one Bhattacharya and Chakrabarty (2016). This
implies that there are multiple Einstein frame descriptions of a single
higher-order $f(R)$ theory. As described in the introduction, for a given
higher-order $f(R)$-gravity in the Jordan frame one has to consider
$F^{\prime}(R)>0$ and $F(R)>0$ as the necessary conditions for the existence
of the corresponding Einstein frame. These conditions are also required to
ensure the existence of a matter-dominated era in cosmological evolution in a
high curvature classical regime, as elucidated in Bhattacharya and Chakrabarty
(2016). This motivates us to first consider the possible solutions of the
$f(R)$ gravity in the Einstein frame. We also interpret the equivalence of
such solutions in the Einstein frame with those in the Jordan frame.

###  II.1  Constant curvature black hole solutions for $f(R)$ gravity in
Einstein frame

The action in (8) describes the $f(R)$ gravity in the Einstein frame.
Considering the matter contribution coming solely from the Maxwell field
($A_{a}$) i.e.,

$\displaystyle\widetilde{S}_{M}=-\frac{1}{8\kappa}\int
d^{4}x\sqrt{-\widetilde{g}}\;\widetilde{\mathcal{F}}^{2},$ (12)

where
$\widetilde{\mathcal{F}}^{2}=\widetilde{\mathcal{F}}_{ab}\widetilde{\mathcal{F}}^{ab}$
and
$\widetilde{\mathcal{F}}_{ab}=\widetilde{\nabla}_{a}A_{b}-\widetilde{\nabla}_{b}A_{a}$
is the electromagnetic field strength tensor 111The electromagnetic field
strength tensor in the Einstein frame is related to the one in the Jordan
frame by the transformation,
$\widetilde{\mathcal{F}}_{ab}=\mathcal{F}_{ab}/F(\phi)^{2}$., one can see that
the action in (8) describes an Einstein-Maxwell (EM) gravity non-minimally
coupled to a scalar field. The corresponding Euler-Lagrange equations of the
motion for the field $\widetilde{\phi}$, $\widetilde{g}_{ab}$, and $A_{a}$ can
now be given as

$\displaystyle\widetilde{R}_{ab}-\frac{1}{2}\widetilde{g}_{ab}\widetilde{R}-\left(\widetilde{\nabla}_{a}\widetilde{\phi}\;\widetilde{\nabla}_{b}\widetilde{\phi}-\widetilde{g}_{ab}V_{E}(\widetilde{\phi})\right.$
$\displaystyle\left.+\widetilde{\mathcal{F}}_{ac}\widetilde{\mathcal{F}}^{cb}-\frac{1}{4}\widetilde{g}_{ab}\widetilde{\mathcal{F}}^{2}\right)=0,$
(13)
$\displaystyle\widetilde{\nabla}_{a}\widetilde{\nabla}^{a}\widetilde{\phi}-\frac{\delta
V_{E}(\widetilde{\phi})}{\delta\widetilde{\phi}}=0,$ (14)
$\displaystyle\widetilde{\nabla}_{a}\left(\widetilde{\mathcal{F}}^{ab}\right)=0.$
(15)

Determining a general solution for the above equations of motion is in general
difficult when the scalar field $\widetilde{\phi}$ has a dynamical (coordinate
dependent) solution Cadoni and Pani (2011); Cadoni  _et al._ (2010); Martinez
_et al._ (2004); Gao and Zhang (2004); Cadoni  _et al._ (2011); Serra (2012).
However, in the case when the scalar field $\widetilde{\phi}$ has a constant
profile then assuming,

$\displaystyle\widetilde{\phi}=\mathcal{C},\quad
F(\phi)=e^{\sqrt{\frac{2\kappa}{3}}\mathcal{C}},\quad
V_{E}(\widetilde{\phi})=\frac{\Lambda}{\kappa}e^{-\sqrt{\frac{2\kappa}{3}}\mathcal{C}},$
(16)

where $\mathcal{C}$ and $\Lambda$ are some constants, it may be observed that
the action in (8) reduces to

$\displaystyle\mathcal{S}_{E}=$ $\displaystyle\frac{1}{{2\kappa}}\int
d^{4}x\sqrt{-\widetilde{g}}\left(\widetilde{R}-2\Lambda
e^{-\sqrt{\frac{2\kappa}{3}}\mathcal{C}}-\frac{1}{4}\widetilde{\mathcal{F}}^{2}\right),$
$\displaystyle=$ $\displaystyle\frac{1}{2\widetilde{\kappa}}\int
d^{4}x\sqrt{-g}\left(R-2\Lambda-\frac{1}{4}\mathcal{F}^{2}\right),$ (17)

which describes the Einstein-Maxwell (EM) gravity with a cosmological constant
($\Lambda$) and a modified effective gravitational constant ($G_{eff}$) given
by

$\displaystyle
G_{eff}=\frac{\widetilde{\kappa}}{8\pi}=Ge^{-\sqrt{\frac{2\kappa}{3}}\mathcal{C}}.$
(18)

The cosmological constant is related to the AdS or dS length ($L$) as
$\Lambda=-\frac{3}{L^{2}}$ or $\Lambda=\frac{3}{L^{2}}$ respectively.
Depending on the sign of the cosmological constant, it is well known that the
EM-gravity with a cosmological constant possesses several solutions namely,
Reissner-Nordstr$\ddot{\text{o}}$m AdS/dS black hole and Kerr-Newmann AdS/dS
black hole. Moreover, for vanishing Maxwell field these solutions reduce to
Scwarzschild AdS/dS black hole and Kerr AdS/dS black hole respectively. These
black holes usually belong to the constant $\widetilde{R}$
($\widetilde{R}=4\Lambda$) solution space of the EM-gravity with a
cosmological constant. Given the constraints (16) together with
$\widetilde{R}=4\Lambda$, (9) and (10), one can see that the curvature $R$ in
the Jordan frame can be fixed to a constant value given by

$\displaystyle R=R_{0}=4\Lambda e^{-\sqrt{\frac{2\kappa}{3}}\mathcal{C}},$
(19)

From the above result, it is straightforward to see that the constant
$\widetilde{R}$ solution space of the EM-gravity with cosmological constant
maps to the constant $R$ solution space of the $f(R)$-gravity in the Jordan
frame. In the next subsection, we discuss the consistency conditions required
to match the constant Ricci scalar solution space in both the EM-gravity with
a cosmological constant and the $f(R)$-gravity in the Jordan frame. We will
also discuss several viable forms of the $f(R)$ function that satisfy the said
conditions.

###  II.2 Constant curvature black hole solutions for $f(R)$ gravity in Jordan
frame

The action describing the $f(R)$ gravity in Jordan frame in (1) where the
matter contribution comes from a Maxwell field ($A_{a}$) can be given as

$\displaystyle\mathcal{S}_{\mathcal{J}}=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}\left(f(R)-\frac{1}{4}\mathcal{F}^{2}\right),$ (20)

where $\mathcal{F}^{2}=\mathcal{F}_{ab}\mathcal{F}^{ab}$ and
$\mathcal{F}_{ab}=\nabla_{a}A_{b}-\nabla_{b}A_{a}$ is the electromagnetic
field strength tensor. The equations of motion for the above action are given
by (2) and the following Maxwell equation

$\displaystyle\nabla_{a}\left(\mathcal{F}^{ab}\right)=0,$ (21)

where the traceless Maxwell stress-energy tensor is given as

$\displaystyle
T_{ab}=\frac{2}{\kappa}\left(\mathcal{F}_{ac}\mathcal{F}^{cb}-\frac{1}{4}g_{ab}\mathcal{F}^{2}\right)$
(22)

Considering the constant curvature scalar $R=R_{0}$, the trace of (2) leads to

$\displaystyle R_{0}=\frac{2f(R_{0})}{F(R_{0})}.$ (23)

which determines the curvature scalar in terms of the function $f(R)$ as long
as $F(R_{0})\neq 0$. The condition that the curvature scalar must assume
constant real values restricts the possible form of the $f(R)$ function. This
also implies the possibility that some theories of $f(R)$-gravity can give
multiple real values of $R_{0}$ while for others one may not have a real
constant value for the curvature scalar. Several models of $f(R)$-gravity,
where one can have a real constant value for the curvature scalar, have been
discussed in Hendi  _et al._ (2012). Thus restricting to such theories of
$f(R)$-gravity with a real constant value for the curvature scalar, one can
use (23) in (2) to obtain

$\displaystyle
R_{ab}-\frac{f(R_{0})}{2F(R_{0})}g_{ab}=\frac{\kappa}{F(R_{0})}T_{ab}.$ (24)

The above equations of motion for the $f(R)$-gravity with constant curvature
scalar are reminiscent of the ones obtained for the usual Einstein gravity
with a cosmological constant

$\displaystyle\Lambda=\frac{f(R_{0})}{2F(R_{0})},$ (25)

and an effective gravitational constant

$\displaystyle G_{eff}=\frac{G}{F(R_{0})},$ (26)

which indicates a duality between the two different theories captured by the
same action in (17). To ensure the positivity of the effective gravitational
constant one has to impose the following conditions

$\displaystyle F(R_{0})>0,\quad F^{\prime}(R_{0})>0,$ (27)

where the second conditions $F^{\prime}(R_{0})>0$ is required for a stable
higher-order $f(R)$-gravity 222Note that the condition $F^{\prime}(R_{0})>0$
comes from the requirement of the positivity of,
$\left.\frac{dG_{eff}}{dR}\right|_{R=R_{0}}$ which ensures the stability of
the $f(R)$-gravity.. We now discuss some known models of $f(R)$ gravity with
constant scalar curvature Hendi  _et al._ (2012) and discuss which of them can
be described by an Einstein gravity with a cosmological constant. We also
study the stability conditions for their constant scalar curvature solutions
described by the following four-dimensional line element

$\displaystyle
ds^{2}=-g(r)\,dt^{2}+\frac{dr^{2}}{g(r)}+r^{2}\,d\Omega^{2}_{k},$ (28)

where

$\displaystyle
d\Omega^{2}_{k}=\begin{cases}d\theta^{2}+\sin^{2}\theta\,d\phi^{2},&k=1\\\
d\theta^{2}+d\phi^{2},&k=0\\\
d\theta^{2}+\sinh^{2}\theta\,d\phi^{2},&k=-1\end{cases}$ (29)

which represents the line element of a 2-sphere for $k=1$, a 2-hyperboloid
$(H_{2})$ for $k=-1$, and flat 2-dimensional line element for $k=0$
respectively.

#### Case (I): $\mathbf{f(R)=R-\mu^{4}/R}$ model

This is one of the earliest models of $f(R)$-gravity proposed in Carroll  _et
al._ (2004) to explain the positive acceleration of the expanding Universe.
Interestingly, this model reduces to the usual Einstein gravity with $f(R)=R$
for very large values of the Ricci scalar. However, for small values of the
Ricci scalar one can not neglect the $1/R$ term implying a modified gravity in
this regime. The field equation for the Maxwell field is given by (21) and for
the metric it is given by

$\displaystyle\left(1+\frac{\mu^{4}}{R^{2}}\right)R_{ab}-\frac{1}{2}\left(1-\frac{\mu^{4}}{R^{2}}\right)Rg_{ab}$
$\displaystyle+\mu^{4}\left(g_{ab}\Box-\nabla_{a}\nabla_{b}\right)R^{-2}=2\left(\mathcal{F}_{ac}\mathcal{F}^{cb}-\frac{1}{4}g_{ab}\mathcal{F}^{2}\right),$
(30)

For the constant-curvature vacuum solutions ($\nabla_{a}R=0$), one has
$R=\pm\sqrt{3}\mu^{2}$. Now on using the metric ansatz (28) in (30), one can
see that the only possible solution is the Schwarzschild-AdS/dS black hole
solution with

$\displaystyle
g(r)=k-\frac{\Lambda}{3}r^{2}-\frac{M}{r},\quad\Lambda=\mp\frac{\sqrt{3}}{4}\mu^{2}$
(31)

which is also a solution to the Einstein gravity with the cosmological
constant ($\Lambda$). Here, it should be noted that for this model of
$f(R)$-gravity it is not possible to have Reisnner-Nordstrom AdS/dS black hole
solutions. Furthermore, the stability conditions (27) in this case become

$\displaystyle\left.\left(1+\frac{\mu^{4}}{R^{2}}\right)\right|_{R=\pm\sqrt{3}\mu^{2}}>0,$
$\displaystyle\left.\left(-\frac{2\mu^{4}}{R^{3}}\right)\right|_{R=\pm\sqrt{3}\mu^{2}}>0$
(32)

which shows that the condition $F^{\prime}(\pm\sqrt{3}\mu^{2})>0$ is violated
for both the Schwarzschild-AdS/dS solutions. In particular, for the
Schwarzschild-dS solution, this violation implies that this model suffers from
the Dolgov-Kawasaki instability Dolgov and Kawasaki (2003). To remove this
instability from the Schwarzschild-dS solution it was proposed in Nojiri and
Odintsov (2003b) to add an additional $R^{2}$ term to the given model.

#### Case (II): $\mathbf{f(R)=R+\alpha R^{n}}$ model

The model of $f(R)$ discussed before suffers from instability problems in the
strong gravity/small R regime but exhibits no problems in the weak
gravity/large R regime. Several viable models of $f(R)$-gravity with no
instability problems in the weak gravity regime have been discussed in Nojiri
and Odintsov (2003b). To resolve such instability problems in a strong gravity
regime, it was proposed in Kobayashi and Maeda (2009) to consider the
corrections proportional to higher orders of curvature such as $R^{n}$ for
$n>1$. This is the motivation for considering the given $f(R)$-gravity model
here. Solving the the field equation (2) for the ansatz (28) gives

$\displaystyle g(r)=k-\frac{\Lambda}{3}r^{2}-\frac{M}{r},$ $\displaystyle
k=\frac{2^{n}(8\Lambda)^{1-n}}{2n-4},~{}n\neq 2,$
$\displaystyle\Lambda=2^{\frac{2}{n-1}}\left(4^{n}(n-2)\,\alpha\right)^{\frac{1}{1-n}}$
(33)

which describes the metric for a Schwarzschild black hole in the presence of a
cosmological constant. Similar to the previous case, one cannot obtain the
charged solution, see Kobayashi and Maeda (2009); Hendi  _et al._ (2012). The
stability conditions (27) in this case now reduce to

$\displaystyle\frac{n-1}{n-2}>0,\quad\frac{n(n-1)}{4\Lambda(n-2)}>0$ (34)

which shows that for $n<0$ the Schwarzschild-AdS solution ($\Lambda<0$) is
stable. However, for $n>2$ the Schwarzschild-dS solution ($\Lambda>0$)
solution is stable and free from the Dolgov-Kawasaki instability.

#### Case (III): $\mathbf{f(R)=R+\lambda\exp{\left(-\xi\;R\right)}}$ model

An interesting and promising model of $f(R)$-gravity can be obtained by adding
an exponential correction term of the form
$\lambda\exp{\left(-\xi\;R\right)},\xi\in\mathcal{R}$ to the usual Einstein
gravity. Notably, this model was shown to agree with cosmological observations
related to the solar system and that of gravitational lensing of galaxies and
clusters Cognola  _et al._ (2008b); Zhang (2007); Elizalde  _et al._ (2011).
Considering this form of $f(R)$-gravity in (2) one gets the corresponding
field equations with $T_{ab}$ as specified in (22). To determine the
solutions, we once again substitute the ansatz (28) for the metric in the
field equations (2) with the given form of the $f(R)$ function. This gives us
the following

$\displaystyle g(r)=k-\frac{\Lambda}{3}r^{2}-\frac{M}{r}+\frac{Q}{r^{2}},$
(35)

indicating that the line element in (28) represents the Reisnner-Nordstrom
AdS/dS black hole solutions. Moreover, one also has the following constraint
relations

$\displaystyle\Lambda=\frac{\lambda
e^{\frac{2}{Q-2}+2}}{2(Q-2)},\quad\xi=\quad\frac{e^{-\frac{2(Q-1)}{Q-2}}(1-Q)}{\lambda},$
(36)

which gives the cosmological constant $(\Lambda)$ and the parameter $\xi$ in
terms of $\lambda$ which we consider as the only free parameter in the theory.
It is straightforward to see that on setting $Q=0$ (i.e., the case of
vanishing Maxwell field) one can also recover the usual Schwarzschild-AdS/dS
solutions for the $f(R)$-gravity. Now to understand the stability of these
solutions, we need to see in what regime the conditions (27) are satisfied. In
this case, the stability conditions now become

$\displaystyle Q>0,\quad\frac{(Q-1)^{2}}{2\Lambda(Q-2)}>0,$ (37)

which shows that Reisnner-Nordstrom AdS or dS are stable for $Q<2$ or $Q>2$
respectively. Whereas, for $Q=0$ only Schwarzschild-AdS solution is stable.

#### Case (IV): $\mathbf{f(R)=R+\eta\left(\log{R}\right)}$ model

In this case, we consider the $f(R)$-gravity model with logarithmic
corrections. Such models involving the logarithm of the Ricci scalar have been
studied in the past to explain the inflationary paradigm in cosmology (see
Amin  _et al._ (2016) and reference therein). Similar to the previous
examples, solving the field equations (2) with the choice of the given $f(R)$
for the metric ansatz (28) one gets the following

$\displaystyle g(r)=k-\frac{\Lambda}{3}r^{2}-\frac{M}{r}+\frac{Q}{r^{2}},$
(38)

which represents the Reisnner-Nordstrom AdS/dS black hole solutions. These
black hole solutions are accompanied by the following constraint relations

$\displaystyle\Lambda=\frac{\eta}{2}\,W\left(\frac{\sqrt{e}}{2\eta}\right),\quad
Q=1+\frac{1}{2W\left(\frac{\sqrt{e}}{2\eta}\right)},$ (39)

where $W$ stands for the Lambert-$W$ or the productlog function. The above
constraints give the cosmological constant and the charge in terms of $\eta$
which is the only free parameter in the theory. For this model of
$f(R)$-gravity one can also obtain the Schwarzschild-AdS/dS black hole
solutions for vanishing Maxwell field. Moreover, in this case, the stability
conditions (27) reduce to the following

$\displaystyle\frac{2-\log{\left(16\Lambda^{2}\right)}}{1-\log{\left(16\Lambda^{2}\right)}}>0,\quad\frac{1}{\log{\left(16\Lambda^{2}\right)}-1}>0$
(40)

which shows that all of the black hole solutions are stable as long as the
condition, $2>\log{\left(16\Lambda^{2}\right)}>1$ is satisfied.

In principle, one can consider a more general form of the $f(R)$ function
combining the models discussed here resulting in more exotic forms of
$f(R)$-gravity Hendi  _et al._ (2012). It may also be seen that for some
models of constant curvature $f(R)$-gravity only Schwarzschild-AdS/dS black
hole solutions are possible whereas for others it is possible to obtain
Reisnner-Nordstrom AdS/dS black hole solutions also. This observation clearly
implies a duality between the constant curvature $f(R)$-gravity theories and
the Einstein-Maxwell gravity with a cosmological constant. Having obtained the
static spherically symmetric solutions to the cases of $f(R)$-gravity one can
now look for their rotating axisymmetric stationary solutions. In general, the
way to obtain such rotating solutions is to use the Newmann-Janis algorithm.
However, as described in the introduction using the NJ algorithm for modified
gravity theories introduces pathologies in the resulting axially-symmetric
metric Hansen and Yunes (2013). In the next section, we propose a modified NJ
algorithm for obtaining rotating solutions to Einstein-Maxwell gravity with a
cosmological constant. Then by exploiting the duality described here, one can
show that these rotating solutions for Einstein-Maxwell gravity with a
cosmological constant will also be the solutions for the constant curvature
$f(R)$-gravity theories.

##  III Newmann Janis Algorithm: Einstein gravity with cosmological constant

The original Newman-Janis (NJ) algorithm was proposed as a five-step procedure
for generating new rotating axisymmetric solutions from known static
spherically symmetric solutions (also known as seed metric) of Einstein
gravity in flat space-time Newman and Janis (1965); Newman  _et al._ (1965);
Adamo and Newman (2014). In this section, we describe a modified Newman-Janis
algorithm that generates rotating solutions for Einstein gravity with a
cosmological constant from its known static spherically symmetric solutions.
Here the Schwarzschild or the Reissner-Nordstr$\ddot{\text{o}}$m AdS/dS
solutions can be considered as the seed metrices for the Einstein gravity with
a cosmological constant. To describe the five steps of the said algorithm for
the present case, we start with the following general spherically symmetric
static seed metric

$ds^{2}=-F(r)\,dt^{2}+G(r)^{-1}\,dr^{2}+H(r)\,\left(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}\right).$
(41)

which is used to generate the rotating solutions. Given the above seed metric,
the first step of the NJ algorithm is to write it in terms of the Eddington-
Finkelstein coordinates ($x_{\mu}=\left\\{u,r,\theta,\phi\right\\}$) using the
following transformation

$du=dt-\frac{dr}{\sqrt{F\,G}}.$ (42)

The second step of the algorithm involves expressing the contravariant form of
the seed metric in terms of a null tetrad,
$e^{\mu}_{a}=\left\\{l^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu}\right\\}$ as:

$\displaystyle
g^{\mu\nu}=l^{\mu}n^{\nu}+l^{\nu}n^{\mu}-m^{\mu}\bar{m}^{\nu}-m^{\nu}\bar{m}^{\mu}\,,$
(43)

where

$\displaystyle l_{\mu}l^{\mu}$ $\displaystyle=$ $\displaystyle
m_{\mu}m^{\mu}=n_{\mu}n^{\mu}=l_{\mu}m^{\mu}=n_{\mu}m^{\mu}=0,$ $\displaystyle
l_{\mu}n^{\mu}$ $\displaystyle=$ $\displaystyle-m_{\mu}\bar{m}^{\mu}=1$ (44)

with $\bar{m}_{\mu}$ being the complex conjugate of the $m_{\mu}$ vector. For
the seed metric 41, the form of the null tetrad can be obtained as

$\displaystyle l^{\mu}$ $\displaystyle=$ $\displaystyle\delta_{r}^{\mu},$
$\displaystyle n^{\mu}$ $\displaystyle=$
$\displaystyle\sqrt{F/G}\,\delta_{u}^{\mu}-\left(F/2\right)\,\delta_{r}^{\mu},$
$\displaystyle m^{\mu}$ $\displaystyle=$
$\displaystyle\left(\delta_{\theta}^{\mu}+\frac{i}{\sin{\theta}}\,\delta_{\phi}^{\mu}\right)/\sqrt{2H}.$
(45)

Having obtained the null tetrad, the third step is to extend the Eddington-
Finkelstein coordinates ($x_{\mu}$) to a new set of complex coordinates using
the following transformation

$\displaystyle d\widetilde{u}$ $\displaystyle\to$ $\displaystyle
du+i\,a\,P(\theta),\quad d\widetilde{r}\to dr-i\,a\,\sin{\theta},$
$\displaystyle d\widetilde{\phi}$ $\displaystyle\to$ $\displaystyle
d\phi+i\,a\,Q(\theta),\quad\theta\to\theta.$ (46)

where $a$ is some constant and the old tetrad and metric are recovered when
one imposes the constraint, $x_{\mu}=\overline{x}_{\mu}$ to the above
coordinate transformation. Here it is to be noted that the usual NJ algorithm
for the Einstein gravity in flat spacetime involves the complexification of
only $\left\\{u,r\right\\}$-coordinates. However, in the present case of
Einstein gravity with a cosmological constant, one requires an additional
complexification of $\phi$-coordinate as well. Thus to summarize, the effect
of this transformation is to create a new metric whose components are (real)
functions of the complex coordinates. Thus on using the transformation given
in eq.(46), the components $F(r)$, $G(r)$ and $H(r)$ of the metric (41)
transform in to the new functions $A(\overline{r},a)$, $B(\overline{r},a)$ and
$C(\overline{r},a)$ respectively. We now consider the following ansatz for the
functions $A$, $B$ and $C$

$\displaystyle A(\overline{r})=A(r,\theta)$ $\displaystyle=$
$\displaystyle\frac{\Delta_{r}(r,)-a^{2}\,\sin^{2}{\theta}\,\Delta_{\theta}(\theta)}{r^{2}+a^{2}\,\cos^{2}\theta},$
$\displaystyle B(\overline{r})=B(r,\theta)$ $\displaystyle=$
$\displaystyle\frac{1}{A(r,\theta)},$ $\displaystyle
C(\overline{r})=C(r,\theta)$ $\displaystyle=$
$\displaystyle\frac{\left(r^{2}+a^{2}\,\cos^{2}{\theta}\right)}{\Delta_{\theta}(\theta)}$
(47)

which is inspired by the Kerr-AdS metric where we match its
$\left\\{uu\right\\}$ and $\left\\{\theta\theta\right\\}$ components with the
functions $A(r,\theta)$ and $C(r,\theta)$ respectively.

The fourth step in the algorithm is to write the transformed null tetrad using
the complex coordinate transformation introduced in (46) as

$\displaystyle l^{\mu}$ $\displaystyle=\delta_{r}^{\mu},$ $\displaystyle
n^{\mu}$ $\displaystyle=\delta_{u}^{\mu}-\left(A/2\right)\,\delta_{r}^{\mu},$
$\displaystyle m^{\mu}$
$\displaystyle=\frac{1}{\sqrt{2C}}\left(\delta_{\theta}^{\mu}+ia\left(\delta_{u}^{\mu}P-\delta_{r}^{\mu}\sin{\theta}\right)+i\left(\csc{\theta}+Q\right)\delta_{\phi}^{\mu}\right),$
(48)

which on using the eq. (43) yields the contravariant form of the transformed
seed metric with the following non vanishing elements

$\displaystyle g^{uu}$
$\displaystyle=\frac{2\,a^{2}\,P^{2}(\theta)\,\Delta_{\theta}(\theta)}{\Sigma(r,\theta)},$
$\displaystyle g^{ur}$
$\displaystyle=g^{ru}=-1-\frac{2\,a^{2}\,P(\theta)\,\sin{\theta}\,\Delta_{\theta}(\theta)}{\Sigma(r,\theta)},$
$\displaystyle g^{u\phi}$
$\displaystyle=g^{u\phi}=\frac{2\,a\,P(\theta)\,\left(\csc{\theta}+Q(\theta)\right)\,\Delta_{\theta}(\theta)}{\Sigma(r,\theta)},$
$\displaystyle g^{rr}$
$\displaystyle=\frac{2\,\Delta_{r}(r)}{\Sigma(r,\theta)},\quad
g^{\theta\theta}=\frac{2\,\Delta_{\theta}(\theta)}{\Sigma(r,\theta)}$
$\displaystyle g^{r\phi}$ $\displaystyle=g^{\phi
r}=-\frac{2\,a\,\left(1+Q(\theta)\,\sin{\theta}\right)\,\Delta_{\theta}(\theta)}{\Sigma(r,\theta)},$
$\displaystyle g^{\phi\phi}$
$\displaystyle=\frac{2\,\left(\csc{\theta}+Q(\theta)\right)^{2}\,\Delta_{\theta}(\theta)}{\Sigma(\theta)}\,,$
(49)

where for brevity we have introduced the function
$\Sigma(r,\theta)=\left(a^{2}+2\,r^{2}+a^{2}\,\cos{2\theta}\right)$, in the
above expressions. The line element corresponding to the above-transformed
metric can now be given as

$ds^{2}=\frac{2a^{2}\Delta_{\theta}(\theta)\sin^{2}\theta-\Delta_{r}(r)}{\Sigma(r,\theta)}\,du^{2}-2\,du\,dr\\\
+\frac{\Sigma(r,\theta)}{2\,\Delta_{\theta}(\theta)}\,d\theta^{2}+2\,a\,\frac{P(\theta)}{\csc{\theta}+Q(\theta)}\,drd\phi\\\
+\,2\,a\,\frac{2P(\theta)\left(\Delta_{r}(r)-a^{2}\Delta_{\theta}(\theta)\sin^{2}{\theta}\right)-\sin{\theta}\Sigma(r,\theta)}{\Sigma(r,\theta)(Q(\theta)+\csc{\theta})}\,dud\phi\\\
+\frac{4a^{2}\sin\theta
P(\theta)+\frac{\Sigma(r,\theta)}{\Delta_{\theta}(\theta)}+\frac{4P(\theta)^{2}\left(a^{4}\Delta_{\theta}(\theta)\sin^{2}{\theta}-a^{2}\Delta_{r}(r)\right)}{\Sigma(r,\theta)}}{2(Q(\theta)+\csc{\theta})^{2}}\,d\phi^{2}$
(50)

which is nothing but the line element of a rotating-AdS black hole solution in
the Eddington-Finkelstein coordinates.

The fifth and final step of the algorithm is to go back to Boyer-Lindquist
coordinates (BLC) using the following global coordinates transformations

$du=dt-\frac{a^{2}+r^{2}}{\Delta_{r}(r)},\,d\phi=d\phi-\frac{a\,S}{\Delta_{r}(r)}\,.$
(51)

where $S$ is a constant to be determined later on. The line element of the
rotating-AdS metric in BLC can now be given as follows

$ds^{2}=\frac{2\left(a^{2}\Delta_{\theta}(\theta)\sin^{2}\theta-\Delta_{r}(r)\right)}{\Sigma(r,\theta)}\,dt^{2}\\\
+\frac{\Sigma(r,\theta)}{2\Delta_{r}(r)}\,dr^{2}+\frac{\Sigma(r,\theta,a)}{2\Delta_{\theta}(\theta)}\,d\theta^{2}\\\
-4\,a\,\frac{\sin^{2}\theta\left(\Delta_{\theta}(\theta)\left(a^{2}+r^{2}\right)-\Delta_{r}(r)\right)}{S\,\Sigma(r,\theta)}\,dt\,d\phi\\\
+\frac{\sin^{2}\theta\left(2\Delta_{\theta}(\theta)\left(a^{2}+r^{2}\right)^{2}-2a^{2}\Delta_{r}(r)\sin^{2}\theta\right)}{S^{2}\,\Sigma(r,\theta)}\,d\phi^{2}$
(52)

where in deriving the above line element, we have considered that all the off-
diagonal elements of the corresponding metric should vanish except its
$\left\\{t\phi\right\\}$ component. This further gives us two constraint
relations that determine the unknown functions $P(\theta)$ and $Q(\theta)$ in
terms of the functions $\Delta_{r}$ and $\Delta_{\theta}$ as

$\displaystyle P(\theta)$ $\displaystyle=$
$\displaystyle\frac{\sin\theta}{\Delta_{\theta}(\theta)}$ (53) $\displaystyle
Q(\theta)$ $\displaystyle=$
$\displaystyle\csc\theta\,\left(-1+\frac{S}{\Delta_{\theta}(\theta)}\right),$
(54)

To this end, it is to be noted that the line element in (52), derived from the
modified Newman Janis algorithm discussed here, has two unknown functions
$\Delta_{r}$, $\Delta_{\theta}$ and a constant $S$. One can fix these unknowns
using the equations of motion of Einstein-Maxwell gravity with a cosmological
constant whose action can be given as

$\displaystyle{\cal S}=\frac{1}{16\pi G}\int
d^{4}x\sqrt{-g}\left(R-2\Lambda-\frac{1}{4}\mathcal{F}^{2}\right),$ (55)

where $G$ is the four-dimensional gravitational constant, $\Lambda$ is the
cosmological constant. On solving the Einstein field equations for the
rotating metric given in eq. (52), one can determine the unknown functions
$\Delta_{r}(r)$, $\Delta_{\theta}(\theta)$, and the constant $S$ as

$\displaystyle\Delta_{r}(r)$ $\displaystyle=$
$\displaystyle\left(a^{2}+r^{2}\right)\left(1-\frac{\Lambda
r^{2}}{3}\right)-2Gmr+Q^{2}$ $\displaystyle\Delta_{\theta}(\theta)$
$\displaystyle=$ $\displaystyle 1+\frac{\Lambda a^{2}}{3}\cos^{2}\theta,\quad
S=1+\frac{\Lambda a^{2}}{3},$ (56)

where $Q$ is the charge of the black hole and the cosmological constant is
related to the AdS or dS length ($L$) as $\Lambda=-\frac{3}{L^{2}}$ or
$\Lambda=\frac{3}{L^{2}}$ respectively. Plugging the above form of the
functions $\Delta_{r}(r)$, $\Delta_{\theta}(\theta)$, and the constant $S$ in
eq.(52) it is obvious to see that one gets the line element corresponding to a
Kerr-Newmann-AdS/dS black hole. The mass M and angular momentum J of the Kerr-
Newmann-AdS/dS black hole are related to the parameters $m$ and $a$ through
the relations $M=m/\Sigma^{2},~{}J=am/\Sigma^{2}$ respectively. One can also
obtain the Kerr-AdS/dS black hole solution described by (52) and (56) for
$Q=0$, to the Einstein gravity with a cosmological constant by using the same
algorithm.

$\mathbf{f(R)}$gravity | Dual | Black hole solutions (AdS/dS ) |  Dictionary with $\mathbf{G_{eff}=G/F(4\Lambda)}$
---|---|---|---
$\mathbf{R-\mu^{4}/R}$ | $\mathbf{R-2\Lambda}$ | Schwarzschild and Kerr | $\Lambda=\mp\frac{\sqrt{3}}{4}\mu^{2}$
$\mathbf{R+\alpha R^{n}}$ | $\mathbf{R-2\Lambda}$ | Schwarzschild and Kerr | $\Lambda=2^{\frac{2}{n-1}}\left(4^{n}(n-2)\,\alpha\right)^{\frac{1}{1-n}}$
$\mathbf{R+\lambda\exp{\left(-\xi\;R\right)}}$ | $\mathbf{R-2\Lambda}$ | Schwarzschild and Kerr | $\Lambda=-\frac{\lambda e}{4},\quad\xi=\frac{1}{e\lambda}$
| $\mathbf{R-2\Lambda-\frac{1}{4}\mathcal{F}^{2}}$ | Reisnner-Nordstrom and Kerr-Newmann | $\Lambda=\frac{\lambda e^{\frac{2}{Q-2}+2}}{2(Q-2)},\quad\xi=\quad\frac{e^{-\frac{2(Q-1)}{Q-2}}(1-Q)}{\lambda}$
$\mathbf{R+\eta\log{\left(R\right)}}$ | $\mathbf{R-2\Lambda}$ | Schwarzschild and Kerr | $\Lambda=\frac{\eta}{2}\,W\left(\frac{\sqrt{e}}{2\eta}\right)$
| $\mathbf{R-2\Lambda-\frac{1}{4}\mathcal{F}^{2}}$ | Reisnner-Nordstrom and Kerr-Newmann | $\Lambda=\frac{\eta}{2}\,W\left(\frac{\sqrt{e}}{2\eta}\right),\quad Q=1+\frac{1}{2W\left(\frac{\sqrt{e}}{2\eta}\right)}$
Table 1: A table showing the dictionary between the $f(R)$-gravities and their
duals. The parameters $G$ and $G_{eff}$ denote the gravitational constant of
$f(R)$-gravities and their duals respectively.

Now going back to the cases of $f(R)$-gravity discussed before, for
$f(R)=R-\mu^{4}/R$ one can see that the Kerr-AdS/dS black hole solution
described by (52) and (56) for $Q=0$, is also a solution to the field
equations (30) for vanishing Maxwell field with the identification of the
cosmological constant ($\Lambda$) with the parameter $\mu$ given in (31) as
before. In the Table (1) we summarize the dualities of different
$f(R)$-gravities with constant Ricci scalar and their solutions with Einstein
gravity with a cosmological constant in the presence or absence of Maxwell
field.

##  IV Discussion and Conclusion

In this Paper, we use the conformal transformation to express the $f(R)$
gravity in the Jordan to Einstein frame. We find that constant curvature
$f(R)$ gravity theories in the Jordan frame are dual to Einstein-Maxwell
gravity with a cosmological constant or modification in effective
gravitational constant. We show the existence of the aforementioned duality by
giving specific examples of well-known $f(R)$ gravity theories. Table 1 shows
the several $f(R)$ gravity theories, their dual and the effective
gravitational constant ($G_{eff}$). We further use this fact to derive the
rotating blackhole solutions for generalized $f(R)$ gravity with constant
curvature. Previously, the Newman-Janis algorithm was used to generate the
rotating black hole solution for Einstein and modified gravity. However, the
NJ algorithm is only known to give accurate rotating BHs for the Einstein
Gravity. We then present the modified NJ algorithm to generate the
axisymmetric rotating Einstein-Maxwell gravity with CC. Our modified NJ
algorithm involves an additional complexification of $\phi$-coordinate to
obtain the rotating space-time for Einstein gravity with $\Lambda$. This
additional complexification gives the tractable form of transformed rotating
metric with two unknown functions which are determined from the field
equations of the gravity theory under consideration. The determination of
these unknown functions in transformed metric ensures that the resulting
rotating solution is indeed a solution of that particular theory.

We have presented our results for the modified gravity theories assuming the
constant curvature solutions. However, we believe that one can also map the
solutions of the solutions of $f(R)$-gravity with dynamical Ricci scalar to
those of the EM-gravity non-minimally coupled to a scalar field in the
Einstein/Jordan frame. We also plan to explore the implications of such
duality between two different gravitational theories in the context of
gauge/gravity duality Maldacena (1998). We leave these interesting avenues for
future works.

###### Acknowledgements.

We acknowledge Ido Ben-Dayan for offering suggestions and encouragement. PC is
supported by the postdoctoral program at the Ariel University. VK acknowledges
the postdoctoral grant of Unisa.

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