# An infinite class of exact rotating black hole metrics of modified gravity

Victor I. Afonso<EMAIL_ADDRESS>Unidade Acadêmica de Física,
Universidade Federal de Campina Grande, 58.429-900 Campina Grande-PB, Brazil
Departamento de Física, Universidad del País Vasco UPV/EHU, Leioa-48940, Spain
Gerardo Mora-Pérez<EMAIL_ADDRESS>Departamento de Física Teórica and IFIC,
Centro Mixto Universidad de Valencia - CSIC. Universidad de Valencia,
Burjassot-46100, Valencia, Spain  Gonzalo J. Olmo<EMAIL_ADDRESS>Departamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia
- CSIC. Universidad de Valencia, Burjassot-46100, Valencia, Spain  Emanuele
Orazi<EMAIL_ADDRESS>International Institute of Physics, Federal
University of Rio Grande do Norte, Campus Universitário-Lagoa Nova, Natal-RN
59078-970, Brazil  Escola de Ciencia e Tecnologia, Universidade Federal do Rio
Grande do Norte, Caixa Postal 1524, Natal-RN 59078-970, Brazil  Diego Rubiera-
Garcia<EMAIL_ADDRESS>Departamento de Física Teórica and IPARCOS,
Universidad Complutense de Madrid, E-28040 Madrid, Spain

(August 27, 2024)

###### Abstract

We build an infinite class of exact axisymmetric solutions of a metric-affine
gravity theory, namely, Eddington-inspired Born-Infeld gravity, coupled to an
anisotropic fluid as a matter source. The solution-generating method employed
is not unique of this theory but can be extended to other Ricci-Based Gravity
theories (RBGs), a class of theories built out of contractions of the Ricci
tensor with the metric. This method exploits a correspondence between the
space of solutions of General Relativity and that of RBGs, and is independent
of the symmetries of the problem. For the particular case in which the fluid
is identified with non-linear electromagnetic fields we explicitly derive the
corresponding axisymmetric solutions. Finally, we use this result to work out
the counterpart of the Kerr-Newman black hole when Maxwell electrodynamics is
set on the metric-affine side. Our results open up an exciting new avenue for
testing new gravitational phenomenology in the fields of gravitational waves
and shadows out of rotating black holes.

##  I Introduction

We are currently witnessing the beginning of a new era in gravitational
physics ignited by the consolidation of multimessenger astronomy, namely,
astronomy with different carriers: electromagnetic radiation (now including
shadows), gravitational waves, neutrinos and cosmic rays. The fields of
gravitational waves and shadows have actually reached their adulthood very
recently. Indeed, the detection of gravitational waves out of binary mergers
LIGOScientific:2016aoc ; LIGOScientific:2017vwq , and the outline of the
bright ring of radiation induced by the super-heated plasma surrounding the
supermassive central object of the M87 galaxy EventHorizonTelescope:2019dse ,
are now powerful tools at disposal of the whole community working on the
determination of the nature of compact objects, and on revealing the intimate
nature of the gravitational interaction on its - yet mostly untested - strong
field regime Berti:2015itd .

Within the enormous theoretical pool of compact objects available to run tests
with Cardoso:2019rvt , black holes are undoubtedly still the privileged
candidate. The uniqueness theorems Heusler , our understanding of
gravitational collapse JoshiBook , and the electromagnetic phenomenology
tested so far Bambi:2015kza , singles out the Kerr(-Newman) family of
solutions, described by mass, angular momentum and charge (the latter being
typically neglected in astrophysical environments, see however Zajacek:2019kla
) as the embodiment of the black hole paradigm within General Relativity (GR).
When looking for theoretical alternatives to it, particularly within
gravitational extensions of GR, most attempts typically assume slow-rotation
motion Pani:2011gy ; Yagi:2012ya ; Barausse:2012qh ; Ayzenberg:2014aka ;
Maselli:2015yva ; Adair:2020vso , either in order to decrease the degree of
difficulty in solving the field equations of the theory of gravity under
consideration or to implement numerical recipes, though fully-rotating
solutions are also known Cembranos:2011sr ; Ding:2019mal ; Cano:2019ore ;
Jusufi:2019caq .

The main aim of this work is to use the theoretical weaponry developed in the
last few years, targeted to crack the structure of the field equations of a
certain family of metric-affine gravity theories (i.e. with independent metric
and affine connection), to obtain an infinite class of exact axisymmetric
solutions within it. Such a family is built via scalars out of contractions of
the metric with the Ricci-tensor of the affine connection. These are dubbed as
Ricci-based gravities (RBGs), and the main weapon is the so-called mapping
method Afonso:2018bpv . This method establishes a correspondence between the
spaces of solutions of two RBG theories, in particular allowing to generate
solutions of a given RBG starting from a seed GR solution, via purely
algebraic transformations Afonso:2019fzv ; Olmo:2020fnk ; Guerrero:2021avm .
Critical in this procedure is the identification of the matter Lagrangians
sourcing the RBG/GR sides of the map, a point that has been systematically
implemented for several types of matter fields Afonso:2018hyj ; Delhom:2019zrb
. In the present work we shall consider the case of anisotropic fluids as the
matter source, while our target RBG theory will be given by the so-called
Eddington-inspired Born-Infeld gravity (EiBI), a suitable choice by its good
algebraic properties as well as by its many applications in astrophysics and
cosmology BeltranJimenez:2017doy . By considering the case of non-linear
electrodynamics formulated as anisotropic fluids, our main result will be the
finding of the counterpart of the Kerr-Newman solution in EiBI gravity coupled
to a Maxwell field. The potential applications of this novel result in the
fields of gravitational waves and shadows will also be discussed.

##  II The two frames of RBGs and the mapping method

RBGs are built as contractions of the space-time metric $g_{\mu\nu}$ with the
(symmetric) part of the Ricci tensor of an independent connection,
$R_{\mu\nu}(\Gamma)$, via traces of powers of the object
${M^{\mu}}_{\nu}\equiv g^{\mu\alpha}R_{\alpha\nu}$. The symmetric Ricci
requirement is (mostly) imposed in order to guarantee projective invariance,
which safeguards the theory against ghost-like instabilities and trivialises
the role of torsion Afonso:2017bxr ; BeltranJimenez:2019acz ;
BeltranJimenez:2020sqf . The corresponding action is written as

$\displaystyle\mathcal{S}_{RBG}$ $\displaystyle=$
$\displaystyle\frac{1}{2\kappa^{2}}\int
d^{4}x\sqrt{-\text{det}(g)}\mathcal{L}_{G}(g_{\mu\nu},R_{(\mu\nu)}(\Gamma))$
(1) $\displaystyle+$ $\displaystyle\mathcal{S}_{m}(g_{\mu\nu},\psi_{m})\ ,$

where $\kappa^{2}$ is Newton’s constant in suitable units, the gravitational
Lagrangian $\mathcal{L}_{G}$ is built with scalars out of the
${M^{\mu}}_{\nu}$ object above, while the matter sector implements a minimal
coupling with a set of matter fields $\psi_{m}$ via $\mathcal{S}_{m}=\int
d^{4}x\sqrt{-\text{det}(g)}\mathcal{L}_{m}(g_{\mu\nu},\psi_{m})$. The field
equations for the action (1) are obtained by independent variation with
respect to the metric and the connection (metric-affine or Palatini
formalism), and can be conveniently rewritten in an Einstein-frame
representation of the form (see Afonso:2018bpv  for details)

${G^{\mu}}_{\nu}(q)=\tilde{T}{{}^{\mu}}_{\nu}(q)\ ,$ (2)

via the introduction of a new metric $q_{\mu\nu}$, which is compatible with
the independent connection (i.e. $\Gamma$ is Levi-Civita of $q$), and is
related to the space-time metric through the algebraic relation

$q_{\mu\nu}=g_{\mu\alpha}{\Omega^{\alpha}}_{\nu}\ ,$ (3)

where the deformation matrix  ${\Omega^{\mu}}_{\nu}$ depends on the RBG theory
chosen, but can always be written on-shell as a function of the matter fields
(and likely the space-time metric $g_{\mu\nu}$ too). The energy-momentum
tensor appearing in (2) is given by

$\tilde{T}{{}^{\mu}}_{\nu}(q)=\frac{1}{|\Omega|^{1/2}}\left({T^{\mu}}_{\nu}(g)-\delta^{\mu}_{\nu}\left(\mathcal{L}_{G}(g,\psi_{m})+\frac{T(g)}{2}\right)\right)$
(4)

where
$T_{\mu\nu}(g)=\tfrac{-2}{\sqrt{-\text{det}(g)}}\tfrac{\delta\mathcal{S}_{m}}{\delta
g^{\mu\nu}}$ is the usual energy-momentum tensor, while vertical bars denote a
determinant. Provided that all $g$-dependencies in the right-hand side of
Eq.(4) can be systematically replaced by their $q$-counterparts using (3),
this provides not only a relation between the matter and gravitational sectors
of two RBG theories, but also between their spaces of solutions. Since GR
belongs to the RBG family via the trivial choice
$\mathcal{L}_{G}=\text{tr}({M^{\mu}}_{\nu})$, the main attractiveness of this
procedure lies on the possibility of mapping known solutions of GR into any
other RBG of interest.

For the sake of this work let us implement the idea above for the case of
anisotropic fluid matter sources, which covers many cases of physical
interest, such as (obviously) perfect fluids as well as scalar and
electromagnetic fields. Its energy-momentum tensor (in the RBG frame) is
written as

${T^{\mu}}_{\nu}(g)=(\rho+p_{\perp})u^{\mu}u_{\nu}+p_{\perp}\delta^{\mu}_{\nu}+(p_{r}-p_{\perp})\chi^{\mu}\chi_{\nu}\
,$ (5)

with (mutually orthogonal) unit time-like, $g^{\mu\nu}u_{\mu}u_{\nu}=-1$, and
space-like, $g^{\mu\nu}\chi_{\mu}\chi_{\nu}=1$, vectors while
$\\{\rho,p_{r},p_{\perp}\\}$ are the energy density, radial pressure, and
tangential pressure, respectively. Similarly, one can introduce a formally
analogous energy-momentum tensor in the Einstein frame as

$\tilde{T}{{}^{\mu}}_{\nu}(q)=(\rho^{q}+p_{\perp}^{q})v^{\mu}v_{\nu}+p^{q}_{\perp}\delta^{\mu}_{\nu}+(p_{r}^{q}-p_{\perp}^{q})\xi^{\mu}\xi_{\nu}\
,$ (6)

with new time-like, $q^{\mu\nu}v_{\mu}v_{\nu}=-1$, and space-like,
$q^{\mu\nu}\xi_{\mu}\xi_{\nu}=1$, vectors and fluid’s functions
$\\{\rho^{q},p^{q}_{r},p^{q}_{\perp}\\}$. Using Eq.(4) and the identifications
$u^{\mu}u_{\nu}=v^{\mu}v_{\nu}$ and $\chi^{\mu}\chi_{\nu}=\xi^{\mu}\xi_{\nu}$,
we find the relations between the functions on each frame as Afonso:2018bpv

$\displaystyle p^{q}_{\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{|\Omega|^{1/2}}\left(\frac{\rho-
p_{r}}{2}-\mathcal{L}_{G}\right)$ (7) $\displaystyle\rho^{q}+p_{\perp}^{q}$
$\displaystyle=$ $\displaystyle\frac{(\rho+p_{\perp})}{|\Omega|^{1/2}}$ (8)
$\displaystyle p^{q}_{r}-p^{q}_{\perp}$ $\displaystyle=$
$\displaystyle\frac{(p_{r}-p_{\perp})}{|\Omega|^{1/2}}\ .$ (9)

On the other hand, the deformation matrix itself can be written as an infinite
power series of the energy-momentum tensor (see Jimenez:2021sqd  for details),
which in the present case can be recast in terms of the fluid’s functions (in
the Einstein frame) as
${\Omega^{\mu}}_{\nu}=\alpha\,\delta^{\mu}_{\nu}+\beta\,v^{\mu}v_{\nu}+\gamma\,\xi^{\mu}\xi_{\nu}$,
with $\\{\alpha,\beta,\gamma\\}$ being model-dependent functions. Therefore,
the mapping equations (7)-(9) provide a relation between the functions
characterizing the fluid on each frame. This allows one to reconstruct the
corresponding matter Lagrangians yielding them, while the fundamental relation
(3) maps the corresponding metric solutions of each frame. Thus, once we
choose a specific RBG theory and some matter field content represented by an
anisotropic fluid, this procedure allows us to map any specific solution of
interest of such an RBG to a solution of any other RBG, in particular, GR
itself. This is so because the mapping holds true irrespective of any
symmetries of the background metric; in other words, it maps the entire spaces
of solutions of GR+$\tilde{\mathcal{L}}_{m}(\psi_{m},q)$ and a given
RBG+$\mathcal{L}_{m}(\psi_{m},g)$.

Among the pool of RBG theories, the so-called Eddington-inspired Born-Infeld
gravity (EiBI) is particularly amiable for calculations. Indeed, in this case
the shape of the deformation matrix is given by the simple algebraic relation
(see BeltranJimenez:2017doy  for details)

$|\Omega|^{1/2}{(\Omega^{-1})^{\mu}}_{\nu}=\lambda\delta^{\mu}_{\nu}-\epsilon\kappa^{2}{T^{\mu}}_{\nu}\
,$ (10)

where $\lambda$ is a constant related to the asymptotic character of the
solutions (from now on we set $\lambda=1$ for asymptotic flatness), while the
length-squared parameter $\epsilon$ encodes the deviations from GR, such that
the EiBI Lagrangian density, which can be expressed as
$\mathcal{L}_{G}=\tfrac{|\Omega|^{1/2}-1}{\epsilon\kappa^{2}}$, recovers the
Einstein-Hilbert one in the limit $|R_{\mu\nu}|\ll\epsilon^{-1}$. Combining
(10) with the set of mapping equations (7)-(9), and using the energy-momentum
tensor (5), the fundamental relation (3) allows us to write the space-time
metric under the compact form

$\displaystyle g_{\mu\nu}$ $\displaystyle=$
$\displaystyle\left(1-\tfrac{\epsilon\kappa^{2}(\rho^{q}-p_{r}^{q})}{2}\right)q_{\mu\nu}$
(11)
$\displaystyle\;-\epsilon\kappa^{2}\left[(\rho^{q}+p_{\perp}^{q})v_{\mu}v_{\nu}+(p_{r}^{q}-p_{\perp}^{q})\xi_{\mu}\xi_{\nu}\right]\
,\quad$

where the right-hand side only involves variables of the Einstein frame
solution. Therefore, given a seed solution obtained within GR, i.e., a
$q_{\mu\nu}$ metric plus a fluid configuration
$\\{\rho^{q},p_{r}^{q},p_{\perp}^{q}\\}$ supporting it, the above equation
provides the corresponding RBG solution. In what follows we shall
particularize this formalism to the case of axially symmetric solutions.

##  III Rotating black holes

Consider the general parametrization of a static, spherically symmetric metric
as

$ds_{q}^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+h(r)d\Omega^{2}\ ,$ (12)

where $d\Omega^{2}=d\theta^{2}+r^{2}\sin^{2}\theta\,d\phi^{2}$. For the sake
of the present work we shall fix the radial function as $h(r)=r^{2}$, so that
the line element above, assumed to arise as a solution of GR coupled to some
matter source, is characterized by a single function $f(r)$. In several works
of recent years, the Janis-Newman algorithm (see Erbin:2016lzq  for a review)
has been employed to arrive at the axisymmetric counterpart of (12). In Boyer-
Lindquist coordinates, its line element is written as (see Azreg-Ainou:2014pra
; Azreg-Ainou:2014aqa ; Toshmatov:2017zpr  for details on this solution)

$\displaystyle ds_{q}^{2}$ $\displaystyle=$ $\displaystyle-\left(1-\frac{2\eta
r}{\Sigma}\right)dt^{2}+\frac{\Sigma}{\Delta}dr^{2}-2a\sin^{2}\theta\frac{2\eta
r}{\Sigma}dtd\phi$ (13) $\displaystyle+$
$\displaystyle\Sigma\,d\theta^{2}+\frac{\sin^{2}\theta}{\Sigma}\left[(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\theta)\right]d\phi^{2}\qquad\
,$

with the following definitions:

$\displaystyle\Sigma$ $\displaystyle=$ $\displaystyle
r^{2}+a^{2}\cos^{2}\theta,$ (14) $\displaystyle 2\eta$ $\displaystyle=$
$\displaystyle r(1-f),$ (15) $\displaystyle\Delta$ $\displaystyle=$
$\displaystyle r^{2}f+a^{2}=r^{2}-2\eta r+a^{2}\ ,$ (16)

where $a=J/M$ is the spin parameter, with $0<a\leq 1$ and bounded from above
by the extremality condition $M=J$. The Kerr-Newman solution belongs to this
family under the choice $f(r)=1-\tfrac{2M}{r}+\tfrac{Q^{2}}{r^{2}}$, with $M$
the ADM mass of the space-time and $Q$ the electric charge.

Due to its rather general character, the line element (13) will be our seed
metric $q_{\mu\nu}$ in order to generate our infinite set of axisymmetric
solutions of EiBI gravity. Assuming motion in the $\phi$ direction, the unit
time-like vector of the fluid sourcing (13) can be written as
$v^{\mu}=(v^{t},0,0,v^{\phi})$, with the angular part typically written as
$v^{\phi}=\omega v^{t}$, where $\omega\equiv d\phi/dt$ is the angular velocity
of the fluid. In order to fix these functions $v^{t}$ and $\omega$, we
introduce an orthornormal basis $e_{\alpha}^{(\mu)}$ given by the so-called
Carter tetrad, whose components take the form Carter

$\displaystyle\quad e_{t}^{(\mu)}$ $\displaystyle=$
$\displaystyle\tfrac{1}{\sqrt{\Sigma\Delta}}(r^{2}+a^{2},0,0,a);\qquad
e_{r}^{(\mu)}=\sqrt{\tfrac{\Delta}{\Sigma}}(0,1,0,0)\quad$ $\displaystyle
e_{\theta}^{(\mu)}$ $\displaystyle=$
$\displaystyle\tfrac{1}{\sqrt{\Sigma}}(0,0,1,0);\quad\;\;e_{\phi}^{(\mu)}=\tfrac{-1}{\sqrt{\Sigma}\sin\theta}(a\sin^{2}\theta,0,0,1)$

From the definition $q^{\mu\nu}v_{\mu}v_{\nu}=-1$ this allows one to identify
$v^{t}=\tfrac{r^{2}+a^{2}}{\sqrt{\Sigma\Delta}}$ and
$\omega=\tfrac{a}{r^{2}+a^{2}}$, which implies that the expression of the
angular velocity is completely general for every fluid able to support the
metric (13). Similarly, the unit space-like vector,
$q^{\mu\nu}\xi_{\mu}\xi_{\nu}=1$, is given by
$\xi^{\mu}=(0,\tfrac{1}{\sqrt{q_{rr}}},0,0)=\tfrac{\Delta}{\Sigma}(0,1,0,0)$.
Next, by lowering the indices with the metric (13) and after some algebra one
gets the expressions

$v_{\mu}=\sqrt{\tfrac{\Delta}{\Sigma}}(-1,0,0,a\sin^{2}\theta)\hskip
2.84544pt;\hskip 8.5359pt\xi_{\mu}=\sqrt{\tfrac{\Sigma}{\Delta}}(0,1,0,0)\ .$
(17)

Therefore, the axially symmetric EiBI metric can be found from the general
expression (11) by writing it formally as

$g_{\mu\nu}=q_{\mu\nu}+\epsilon\kappa^{2}h_{\mu\nu}\ ,$ (18)

where $q_{\mu\nu}$ is the seed GR metric (13), while the EiBI-induced
correction $h_{\mu\nu}$ is written as

$h_{\mu\nu}\\!=\\!-\\!\left[\tfrac{1}{2}(\rho^{q}-p_{r}^{q})q_{\mu\nu}+(\rho^{q}+p_{\perp}^{q})v_{\mu}v_{\nu}+(p_{r}^{q}-p_{\perp}^{q})\xi_{\mu}\xi_{\nu}\right]$
(19)

In this expression the unit vectors are those appearing in (17), while the
components of the energy-momentum tensor are found via projection on the
Carter’s tetrad above and comparison with the Einstein equations (2). This
yields the expressions

$\rho^{q}=-p^{q}_{r}=\frac{2\eta^{\prime}r^{2}}{\kappa^{2}\Sigma^{2}}\hskip
2.84544pt;\hskip
11.38092ptp^{q}_{\perp}=p^{q}_{r}-\frac{\eta^{\prime\prime}r+2\eta^{\prime}}{\kappa^{2}\Sigma}\
,$ (20)

where primes denote derivatives with respect to the radial coordinate $r$.
Since both $\rho^{q}$ and $p_{\perp}^{q}$ are functions of such a coordinate,
the above expressions allow, in principle, to write the tangential pressure as
a function of the energy density, i.e. $p_{\perp}^{q}=K(\rho^{q})$. Then, the
EiBI correction (19) simplifies down to

$h_{\mu\nu}=-\left[\rho^{q}q_{\mu\nu}+(\rho^{q}+K(\rho^{q}))(v_{\mu}v_{\nu}-\xi_{\mu}\xi_{\nu})\right]$
(21)

Making explicit the unit vectors obtained in (17), the components of this
correction term can be computed as

$\displaystyle h_{tt}\,$ $\displaystyle=$
$\displaystyle-\tfrac{1}{\Sigma}(\rho^{q}a^{2}\sin^{2}\theta+K(\rho^{q})\Delta)$
(22) $\displaystyle h_{rr}$ $\displaystyle=$
$\displaystyle\tfrac{\Sigma}{\Delta}K(\rho^{q})$ (23) $\displaystyle
h_{\theta\theta}$ $\displaystyle=$ $\displaystyle-\Sigma\,\rho^{q}$ (24)
$\displaystyle h_{t\phi}$ $\displaystyle=$
$\displaystyle\tfrac{a\sin^{2}\theta}{\Sigma}\left(\rho^{q}(r^{2}+a^{2})+K(\rho^{q})\Delta\right)$
(25) $\displaystyle h_{\phi\phi}$ $\displaystyle=$
$\displaystyle-\tfrac{\sin^{2}\theta}{\Sigma}\left({\rho^{q}(r^{2}+a^{2})^{2}+a^{2}\sin^{2}\theta
K(\rho^{q})\Delta}\right)\quad$ (26)

Since the EiBI parameter is heavily constrained by astrophysical
Avelino:2012ge , particle physics Delhom:2019wir ; Jimenez:2021sqd , and
cosmological Benisty:2021laq  observations, the metric (18) will typically
involve mild deviations with respect to the GR solutions at the near-
horizon/photon sphere scale, at least for astrophysical-size black holes (and
even smaller for supermassive ones), therefore posing a challenge for its
observational detectability via gravitational waves and/or shadows. However,
strong deviations will indeed appear as getting closer and closer to the
innermost region of the solutions, where the EiBI-induced corrections can be
the dominant contribution. This fact may give rise to causal and singularity
structures remarkably different from those appearing in GR. Under the presence
of horizons these modifications to the causal structure would be barely
noticeable from the outside, while horizonless compact objects such as
traversable wormholes, which could also arise out of these solutions (see e.g.
Olmo:2013gqa ; Afonso:2019fzv  for examples in the spherically symmetric
case), would represent much better prospects from an observational point of
view.

For further concreteness, we will focus on a specific class of anisotropic
fluid matter sources - non-linear electrodynamics (NED) - for which the
mapping procedure is remarkable simple, while at the same time allows to find
physically relevant solutions. NEDs are given by Lagrangian densities that, in
the Einstein frame, have the form111Here we neglect magnetic fields for
simplicity, though the whole mapping procedure works equally fine should we
have included them. $\tilde{\mathcal{L}}_{m}=\tilde{\varphi}(Z)$, with
$Z=-\tfrac{1}{2}F_{\mu\nu}F^{\mu\nu}$, where
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field strength
tensor of the vector potential $A_{\mu}$, and
$F^{\mu\nu}=q^{\mu\alpha}q^{\nu\beta}F_{\alpha\beta}$. The corresponding field
equations, $\nabla_{\mu}(\sqrt{-q}\tilde{\varphi}_{Z}F^{\mu\nu})=0$, allow to
find $Z=Z(\rho^{q})$, so one can see the relation $p^{q}_{\perp}=K(\rho^{q})$
as characterizing the particular NED model under consideration. Furthermore,
the interest in these models comes from two different niceties of them.

Firstly, the solution for the (electrostatic) spherically symmetric problem
for any NED of the form above within GR is known in exact closed form Diaz-
Alonso:2009xkw :

$f(r)=1-\frac{2M}{r}-\frac{1}{r}\int_{r}^{\infty}R^{2}T_{0}^{0}(R,Q)dR\ ,$
(27)

where $T_{0}^{0}(r,Q)=\tilde{\varphi}(Z)-2Z\tilde{\varphi}_{Z}$ is the
temporal component of the energy-momentum tensor in such a spherically
symmetric scenario, in which the field invariant can be resolved as
$Z\tilde{\varphi}_{Z}^{2}=Q^{2}/r^{4}$. In such a case, the expressions (20)
of the rotating case turn into

$\rho^{q}=-p_{r}^{q}=-\frac{r^{4}T_{0}^{0}}{\Sigma^{2}}\hskip 2.84544pt;\hskip
8.5359ptp_{\perp}^{q}=-\rho^{q}+\frac{4r^{2}T_{0}^{0}+r^{3}(T_{0}^{0})^{\prime}}{2\Sigma}\
,$ (28)

which can be in turn interpreted as the energy density and pressures generated
by an axially symmetric electromagnetic field with components
$A_{\mu}=(A_{t},0,0,A_{\phi})$.

Secondly, the mapping equations (7)-(9) provide the correspondence of NEDs in
the RBG and GR frames, where the former is characterized by a new function
$\mathcal{L}_{m}=\varphi(X)$, with $X=-\tfrac{1}{2}B_{\mu\nu}B^{\mu\nu}$,
where $B_{\mu\nu}=F_{\mu\nu}$ but
$B^{\mu\nu}=g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}$, i.e., its indices are
raised with $g^{\mu\nu}$ rather than with $q^{\mu\nu}$. The explicit shape of
the correspondence was worked out in detail in Delhom:2019zrb , finding the
specific relation between the NEDs on each frame. Therefore, once one defines
the target NED in the RBG (or GR) frame, it is an entertaining algebraic
gymnastics to find its counterpart on the other frame. We can now use this
powerful result in order to work out the counterpart of the Kerr-Newman black
hole within EiBI gravity. This involves the coupling of EiBI to Maxwell
electrodynamics in such a way that, according to the results of Delhom:2019zrb
, the corresponding NED on the GR side is given by (again we restrict
ourselves to purely electric fields):

$\varphi(X)=2\beta^{2}\left(1-\sqrt{1-\frac{X}{\beta^{2}}}\right)\ ,$ (29)

with the identification of constants

$\beta^{2}=-\frac{1}{2\epsilon\kappa^{2}}\ .$ (30)

This is actually the celebrated Born-Infeld (BI) electrodynamics222Should one
have considered magnetic fields, not only $X$ and $Z$ would have picked a term
on them, but also the field invariant associated to the field strength dual
$B_{\mu\nu}^{*}=-\tfrac{1}{2}\epsilon_{\mu\nu\alpha\beta}B^{\alpha\beta}$
would pop up in this expression under the desired BI form in (29). Therefore,
this correspondence is exact for all electromagnetic configurations. See
Delhom:2019zrb  for details., provided that we stick to the identification
between the BI parameter $\beta$ and the EiBI one in the branch $\epsilon<0$,
as forced by (30). As is well known, for electrostatic, spherically symmetric
solutions, the BI electrodynamics yields a field invariant
$X=\tfrac{\beta^{2}Q^{2}}{\beta^{2}r^{4}+Q^{2}}$ allowing to remove the
divergence due to the self-energy of the electron, but not the central black
hole singularity when coupled to GR, see e.g. Diaz-Alonso:2009xkw . However,
when moving to the solutions of its EiBI gravity + Maxwell counterpart (in the
non-rotating regime), which were discussed in detail in Afonso:2018mxn , one
finds the geodesic completeness and singularity-free character of the
corresponding space-times Olmo:2015dba .

In order to build now the rotating version of the EiBI + Maxwell system using
the formalism developed above we first need to compute the following quantity,
which appears in the GR frame via Eq.(27):

$T_{0}^{0}=\frac{2\beta}{r^{2}}\left(\beta
r^{2}-\sqrt{\beta^{2}r^{4}+Q^{2}}\right)\ ,$ (31)

which reduces to the result of Maxwell electrodynamics on such a frame in the
limit $\beta\to\infty$, i.e., $T_{0}^{0}\sim Q^{2}/r^{4}+\mathcal{O}(r^{-8})$.
However, the above limit cannot be directly taken since, via the
identification (30), this would imply taking $\epsilon\to 0$ and EiBI gravity
would boil down to GR, thus simply recovering the usual Kerr-Newman solution.
Incidentally one can note that should one try to map the GR + Maxwell system
(i.e. the usual Kerr-Newman solution), the mapping would provide the
counterpart on the EiBI side as given again by BI electrodynamics, but now the
identification of EiBI/BI constants above holds true only in the $\epsilon>0$
branch. The rotating solutions of such a theory via the mapping were found and
discussed in detail in Ref. Guerrero:2020azx . Multicenter solutions were also
constructed in that way in Olmo:2020fnk .

In the present EiBI + Maxwell system under consideration, one just needs to
insert the expression (31) into those of the energy density and pressures of
the rotating case, Eq.(28), to get

$\displaystyle\rho^{q}$ $\displaystyle=$ $\displaystyle\frac{2\beta
r^{2}}{\Sigma^{2}}\left(\sqrt{\beta^{2}r^{4}+Q^{2}}-\beta r^{2}\right)$ (32)
$\displaystyle K$ $\displaystyle=$
$\displaystyle-\rho^{q}+\frac{2\beta}{\Sigma}\left(2\beta
r^{2}-\frac{2\beta^{2}r^{4}+Q^{2}}{\sqrt{\beta^{2}r^{4}+Q^{2}}}\right)\ ,$
(33)

which, as expected, reduce to their non-rotating EiBI-Maxwell counterparts in
the limit $a\to 0$ (where $\Sigma\to r^{2})$.

We now have everything needed in order to generate the counterpart of the
Kerr-Newman solution in the EiBI+Maxwell system via Eq.(18). The background
metric $q_{\mu\nu}$ is given by the line element (13), where the spherically
symmetric function $f(r)$ feeding it via (15) is defined by Eq.(27), after
inserting there the expression (31) for the Born-Infeld electrostatic field.
The corresponding expression admits an analytic integration to yield the
result

$\displaystyle f(r)=1-\frac{2M}{r}$ (34)
$\displaystyle+\frac{2\beta^{2}}{3}\left[r^{2}-\sqrt{r^{4}+\frac{Q^{2}}{\beta^{2}}}+\frac{2Q^{2}}{\beta^{2}r^{2}}\
_{2}F_{1}\left[\frac{1}{4},\frac{1}{2},\frac{5}{4},\frac{-Q^{2}}{\beta^{2}r^{4}}\right]\right]$

where ${}_{2}F_{1}[a,b,c;z]$ is a hypergeometric function. This unambiguously
defines the shape of the seed (GR-based) rotating solution, parameterized by
mass $M$, charge $Q$, spin $a$, and Born-Infeld constant $\beta$. As for the
(EiBI-driven) correcting term $h_{\mu\nu}$, its components are given in
Eqs.(22)-(26), after inserting the expressions (32) and (33) for the energy
density and pressure of the rotating fluid, respectively. This completes our
construction of the axially symmetric metric of the EiBI + Maxwell system,
i.e., the counterpart of the Kerr-Newman metric.

Note that in this Kerr-Newman-like metric the BI-dependencies in $\beta^{2}$
are replaced by EiBI dependencies in $\epsilon$ via the identification (30),
which therefore becomes the only parameter encoding the deviations of this
rotating metric with respect to the Kerr-Newman solution of GR.

It should also be noted that far from the sources, $r\to\infty$, where the
energy density and pressure approximate their vacuum values, $\rho^{q}\approx
p_{\perp}^{q}\approx 0$, one has $g_{\mu\nu}\approx q_{\mu\nu}$, and the EiBI
Kerr-Newman line element boils down to

$\displaystyle ds_{g}^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M}{r}+\mathcal{O}(r^{-2})\right)dt^{2}$
$\displaystyle+$
$\displaystyle\left(1+\mathcal{O}(r^{-1})\right)(dr^{2}+r^{2}d\Omega^{2})$
$\displaystyle+$
$\displaystyle\left(\frac{4aM\sin^{2}\theta}{r}+\mathcal{O}(r^{-2})\right)dtd\phi\
,$

in agreement with the expectations of the weak-field limit of the Kerr-Newman
solution of GR. The EiBI corrections appear at order $\mathcal{O}(r^{-6})$ and
therefore they are strongly suppressed in this limit. As a consequence, this
EiBI Kerr-Newman metric naturally passes the same weak-field limit tests as
the GR Kerr-Newman does, such as the consistence with star’s orbital motions
around them. However, in the strong-field regime the EiBI corrections in the
parameter $\epsilon$, fueled by the growth of the energy density and pressure
appearing in the $h_{\mu\nu}$ term of Eq.(18), become non-negligible, to the
point that in the innermost region of the solution it may constitute the
dominant contribution.

Due to the fact that, in particular, the $g_{tt}$ and $g_{rr}$ components in
the EiBI Kerr-Newman solution get corrections in the energy density and
pressure (suppressed by the EiBI length-squared scale), modifications to the
astrophysically relevant surfaces of their GR counterparts do occur. Indeed,
it affects both the (ergo-)horizons and the three-dimensional shell of
unstable null geodesics Johnson:2019ljv . Since these are the main geometrical
features needed for the analysis of gravitational wave radiation
Ezquiaga:2020dao  as well as shadows and light rings from accretion disks
Gralla:2019xty ; Chael:2021rjo  their study is of uttermost interest. On the
other hand, the singularity structure in terms of geodesic completeness and
behaviour of curvature scalars is also modified. For instance, the ring
singularity of the GR Kerr-Newman solution, which is revealed in the zeros of
the $g_{\theta\theta}=\Sigma$ component, and amounts to
$\\{r=0,\theta=\pi/2\\}$, is modified in the present EiBI Kerr-Newman geometry
to $\Sigma(1-\epsilon\kappa^{2}\rho^{q})=0$, where the energy-density
corrections appearing via (32) are again the relevant protagonist. An in-depth
analysis of these features will be carried out elsewhere.

##  IV Conclusion

In this work we have combined several theoretical results found in the last
few years in order to produce an infinite class of exact axisymmetric
solutions of modified gravity, which are suitable frameworks for representing
alternative black holes beyond the Kerr-Newman solution. Using anisotropic
fluids as the matter source, this has been achieved by employing a mapping
between the spaces of solutions of certain families of metric-affine
gravities, dubbed as Ricci-based gravities, and GR itself. This mapping method
allows to obtain a solution of the former using a seed solution of the latter
via purely algebraic transformations, once the identification between the
matter Lagrangian densities on each side of the mapping is made.

In order to obtain this class of axisymmetric solutions we have first
considered the Newman-Janis technique which, starting from a general
spherically symmetric metric, allows to find its rotating counterpart,
assuming GR as the theory of gravity supporting it. At the same time, the use
of the Carter’s tetrad allows to find the energy density, pressure, and
angular velocity of the fluid generating such an axisymmetric solution by
demanding the fulfillment of Einstein equations. This general family of GR
rotating solutions thus becomes the seed on which to grow the map. In the case
of EiBI gravity, a particularly agreeable member of the RBG class, this allows
to find its rotating counterpart in exact closed form, which can be decomposed
as the original GR one plus a correction term(s) suppressed by the EiBI
length-squared scale. Next we turned to (non-linear) electromagnetic fields,
which can be described as anisotropic fluids where the choice of the relation
$p_{\perp}^{q}=K(\rho^{q})$ characterizes a particular model. Using the fact
that the corresponding spherically symmetric solutions (to be Newman-Janis-
transformed) are known in exact form for any such model, we first found the
correspondence between EiBI gravity coupled to a Maxwell field and GR coupled
to a Born-Infeld-type electrodynamics, and next combined both results to
determine the counterpart of the Kerr-Newman solution in this framework. This
identification requires the EiBI constant to be negative, which is precisely
the branch of solutions that, in the spherically symmetric case, removes the
singularity issue via a restoration of geodesic completeness for every null
and time-like curve.

In our opinion, this modified Kerr-Newman solution of the EiBI + Maxwell
system, which illustrates the capabilities of the mapping method, is the most
succulent result of the present work. While it reduces in the weak-field limit
to the GR Kerr-Newman one, therefore being automatically compatible with the
observed orbital motions in such a regime, it encapsulates the departures from
it arising in the strong-field regime via a single additional (EiBI)
parameter. For non-vanishing values of the energy density of the matter
(electromagnetic) field, the modified gravitational physics yields departures
from GR predictions in terms of (ergo-)horizons, critical (null) surfaces, and
the innermost structure of the black hole, the latter affecting the ring-like
singularity issue. These aspects are of great interest not only from a purely
theoretical point of view, but also for the sake of multimessenger astronomy.
This is so because EiBI-induced effects happening outside the event horizon
are expected to yield modifications to the waveforms of gravitational waves
out of binary mergers, and to the shape of their optical appearance and
shadows when illuminated by accretion disks. Besides, should the successful
removal of space-time singularities of the spherically symmetric EiBI+Maxwell
system be extended to the axisymmetric case, this would imply that horizonless
Kerr-Newman solutions of this kind (supplemented with a suitable gravitational
collapse mechanism) would also be acceptable as compact objects alternative to
GR black holes, i.e., black hole mimickers Shao:2020weq . Combined, these
results would allow to test the existence of new gravitational dynamics beyond
the present weak-field limit tests EventHorizonTelescope:2020qrl . We are
currently working to extract and analyze all these features from the above
solutions and we hope to report on this issue very soon.

## Acknowledgments

DRG is funded by the _Atracción de Talento Investigador_ programme of the
Comunidad de Madrid (Spain) No. 2018-T1/TIC-10431. This work is supported by
the Spanish Grants FIS2017-84440-C2-1-P, PID2019-108485GB-I00 and
PID2020-116567GB-C21 funded by MCIN/AEI/10.13039/501100011033 (“ERDF A way of
making Europe” and “PGC Generación de Conocimiento”), the project
PROMETEO/2020/079 (Generalitat Valenciana), the project H2020-MSCA-RISE-2017
Grant FunFiCO- 777740, the project i-COOPB20462 (CSIC), the FCT projects No.
PTDC/FIS-PAR/31938/2017 and PTDC/FIS-OUT/29048/2017, and the Edital 006/2018
PRONEX (FAPESQ-PB/CNPQ, Brazil, Grant 0015/2019). VIA thanks the Theoretical
Physics Department of the Complutense University of Madrid for kind
hospitality while doing this work. This article is based upon work from COST
Action CA18108, supported by COST (European Cooperation in Science and
Technology).

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