# Toward General-Relativistic Magnetohydrodynamics Simulations
in Stationary Non-Vacuum Spacetimes

Prashant Kocherlakota Black Hole Initiative at Harvard University, 20 Garden
Street, Cambridge MA 02138, USA  Harvard-Smithsonian Center for Astrophysics,
60 Garden Street, Cambridge MA 02138, USA  Ramesh Narayan Harvard-Smithsonian
Center for Astrophysics, 60 Garden Street, Cambridge MA 02138, USA  Black Hole
Initiative at Harvard University, 20 Garden Street, Cambridge MA 02138, USA
Koushik Chatterjee Black Hole Initiative at Harvard University, 20 Garden
Street, Cambridge MA 02138, USA  Harvard-Smithsonian Center for Astrophysics,
60 Garden Street, Cambridge MA 02138, USA  Alejandro Cruz-Osorio Institut für
Theoretische Physik, Goethe Universität, Max-von-Laue-Str. 1, 60438 Frankfurt
am Main, Germany  Yosuke Mizuno Tsung-Dao Lee Institute, Shanghai Jiao-Tong
University, Shanghai, 520 Shengrong Road, 201210, China  School of Physics &
Astronomy, Shanghai Jiao-Tong University, Shanghai, 800 Dongchuan Road,
200240, China  Institut für Theoretische Physik, Goethe Universität, Max-von-
Laue-Str. 1, 60438 Frankfurt am Main, Germany

###### Abstract

Accretion of magnetized gas on compact astrophysical objects such as black
holes has been successfully modeled using general relativistic
magnetohydrodynamic (GRMHD) simulations. These simulations have largely been
performed in the Kerr metric, which describes the spacetime of a vacuum and
stationary spinning black hole (BH) in general relativity (GR). The
simulations have revealed important clues on the physics of accretion flows
and jets near the BH event horizon, and have been used to interpret recent
Event Horizon Telescope images of the supermassive BHs, M87∗ and Sgr A∗. GRMHD
simulations require the spacetime metric to be given in horizon-penetrating
coordinates such that all metric coefficients are regular at the event
horizon. Only a few metrics, notably the Kerr metric and its electrically
charged spinning analog, the Kerr-Newman metric, are currently available in
such coordinates. We report here horizon-penetrating forms of a large class of
stationary, axisymmetric, spinning metrics. These can be used to carry out
GRMHD simulations of accretion on spinning, nonvacuum BHs and non-BHs within
GR, as well as accretion on spinning objects described by non-GR metric
theories of gravity.

General Relativity — Accretion — Magnetohydrodynamics — Non-Kerr Spacetimes

##  1 Introduction

General relativistic magneto-hydrodynamic (GRMHD) simulations have emerged as
an indispensable tool in modern astrophysical research, providing a robust
framework for investigating the complex dynamics of magnetized gas within
extreme gravitational environments (e.g., Banyuls et al., 1997; Koide et al.,
1999; Gammie et al., 2003; De Villiers et al., 2003). By virtue of their
ability to simulate diverse accretion regimes, spanning from sub-to-super
Eddington rates, GRMHD simulations offer invaluable insights into the
structure and evolution of a variety of black hole (BH) and neutron star
systems (e.g., Fragile et al., 2007; Narayan et al., 2012; Sadowski & Narayan,
2015; Parfrey & Tchekhovskoy, 2017; Liska et al., 2018; Porth et al., 2019;
Begelman et al., 2022; Chatterjee & Narayan, 2022), as well as astrophysical
phenomena like gamma-ray bursts (e.g., Gottlieb et al., 2022), tidal
disruption events (e.g., Curd & Narayan, 2019; Andalman et al., 2022) and high
energy flares from the centers of galaxies (e.g., Chatterjee et al., 2021;
Porth et al., 2021; Ripperda et al., 2022). Furthermore, GRMHD simulations
self-consistently launch powerful jets that can accelerate to relativistic
speeds and extend to galactic scales, and thus are crucial for understanding
galaxy evolution (e.g., McKinney, 2006; Tchekhovskoy et al., 2011; Chatterjee
et al., 2019; Narayan et al., 2022; Ricarte et al., 2023). Also, the
predictive capabilities of GRMHD simulations facilitate direct comparisons
with observational data, enhancing their applicability in interpreting real
astrophysical systems (e.g., Mościbrodzka et al., 2009; Dexter et al., 2012;
Davelaar et al., 2018; Chael et al., 2019; Chatterjee et al., 2020; Ricarte et
al., 2020; Cruz-Osorio et al., 2022; Ressler et al., 2023), and have helped
motivate experimental tests of gravitational effects such as frame-dragging
(Ricarte et al., 2022) and electromagnetic extraction of energy from spinning
BHs (Chael et al., 2023).

The Event Horizon Telescope (EHT) collaboration recently demonstrated that it
is now possible to “measure” the spacetime metric of astrophysical BHs. More
concretely, we are able to measure the level of agreement of the spacetime
geometry of the supermassive BHs, M87∗ (The EHT Collaboration et al., 2019a,
b; Psaltis et al., 2020) and Sgr A∗ (The EHT Collaboration et al., 2022a, b),
with that of a Kerr BH (Kerr, 1963), which is a vacuum, stationary solution of
general relativity (GR; see, e.g., Wiltshire et al. 2009). Such investigations
allow us, in principle, to test fundamental aspects of GR such as the no-hair
hypothesis (see, e.g., Carter 1971), and whether astrophysical BHs truly have
no surfaces and are devoid of matter (The EHT Collaboration et al., 2022b).

It has also been possible to test the level of agreement of EHT data with non-
BH spacetime geometries (such as boson stars, wormholes, classical naked
singularities, etc.) within GR, as well as both BH and non-BH spacetimes in
non-GR theories (Völkel et al., 2021; Kocherlakota et al., 2021; The EHT
Collaboration et al., 2022b; Vagnozzi et al., 2023). This program has seen
intense activity, and current approaches use a combination of GRMHD
simulations in Kerr BH spacetimes (see, e.g., The EHT Collaboration et al.
2019b, 2022b) as well as non-GR MHD (“xGRMHD”) simulations in non-rotating
spacetimes (Mizuno et al., 2018; Olivares et al., 2020; Fromm et al., 2021;
Röder et al., 2023). In addition, a diverse range of instructive semi-analytic
accretion flow models have been considered (see, e.g., Broderick et al. 2014;
Shaikh et al. 2019; Shaikh & Joshi 2019; Narayan et al. 2019; Paul et al.
2020; Bauer et al. 2022; Özel et al. 2022; Younsi et al. 2023; Kocherlakota &
Rezzolla 2022; Ayzenberg 2022; The EHT Collaboration et al. 2022b) to model
the physics of accretion flows and their emission.

Nearly all the xGRMHD work to date has been restricted to spherically
symmetric “static” spacetime metrics. This is because, apart from the spinning
Kerr metric (Kerr, 1963) and Kerr-Newman metric (Newman et al., 1965), the
only metrics available in the horizon-penetrating “Kerr-Schild” coordinates
required by current GRMHD codes (see, e.g., Font et al. 1998; McKinney &
Gammie 2004; Sadowski et al. 2014; Porth et al. 2017; Liska et al. 2022 for a
discussion of representative GRMHD codes) are static metrics. The one other
exception is Nampalliwar et al. (2022), who use the Kerr-Schild form of the
Johannsen (2013a) stationary metric, which is derived from the well-known
Johannsen & Psaltis (2011) parametrized BH metric.

Astrophysical objects are expected to possess angular momentum. Hence their
spacetimes will in general be axisymmetric (not spherically symmetric), and
their metrics should be “stationary” (not static). For a more realistic
confrontation of the underlying effective theory of gravity and fields in the
vicinity of ultracompact objects to current observations, and in anticipation
of future higher-quality data, it is necessary to be able to model accretion
and emission processes in generic stationary spacetimes. However, other than
the aforementioned Kerr, Kerr-Newman and Johannsen metrics, no other
stationary spacetime metrics have been written down in the horizon-penetrating
coordinates needed for computer simulations. The goal of the present paper is
to eliminate this roadblock.

We introduce a general family of stationary and axisymmetric metrics (Azreg-
Aïnou, 2014a, b) which are derived from a corresponding family of “seed”
spherically symmetric metrics. We then cast these stationary metrics in
horizon-penetrating “Kerr-Schild” form. This is the main contribution of the
present paper. Using the results described in this paper, we will report
elsewhere the first high-resolution, 3D GRMHD simulations in spinning non-Kerr
spacetimes (Chatterjee et al., in prep.a, in prep.b).

We begin in section 2 by introducing the general form of a (spherically-
symmetric) static metric. This form includes the Schwarzschild metric as a
special case but covers a wide range of other models as well. We pick out and
discuss a few well-known examples of the latter.

In section 3 we introduce the Azreg-Aïnou (AA; Azreg-Aïnou 2014b) metric,
which is a stationary, axisymmetric generalization of the static metric
described in section 2. The AA metric was proposed as an ansatz, inspired by
the Newman & Janis (1965) algorithm, to describe generic spinning spacetimes.
This metric has several attractive features, which we touch on in Sec. 1.1 and
consider in greater detail in Sec. 3.

In section 4 we report the form of the AA metric in horizon-penetrating
coordinates, as needed for most GRMHD codes, and in Sec. 5 we write down the
$3+1$ decomposition of the metric required for certain other codes. The
spherical-polar Kerr-Schild coordinates we use to write the horizon-
penetrating AA metric are adapted to the ingoing principal null congruence of
spacetime.

Finally, in section 6 we summarize our findings and present a representative
list of stationary and axisymmetric metrics that are obtained from the AA
metric for specific choices of its metric functions. These metrics may be
directly used in future GRMHD simulations. We also describe the steps needed
to generate other stationary metrics in the future. Throughout we use
geometrized units, $G=c=1$, and the metric signature is $(-,+,+,+)$.

Of necessity, this paper includes a fair amount of technical discussion about
the AA metric and its various forms. A reader who is interested in utilizing
various BH and non-BH metrics to explore with GRMHD simulations, can directly
obtain the main results by reading only sections 2 and 6, along with tables 1
and 2.

###  1.1 Background on the Azreg-Aïnou Metric

While in general the description of an arbitrary stationary and axisymmetric
spacetime requires ten independent metric functions, the AA metric remarkably
makes use of only three functions. Furthermore, when considered as a
generalization of a static metric, two of these functions are fixed by the
original static metric functions. The one remaining free function is a
conformal factor that can be fixed by solving the appropriate field equations.
Moreover, as we discuss, there is a naturally attractive choice for this third
function such that it is no longer a free function. Thus, the AA metric is
both a powerful and surprisingly simple ansatz for spinning spacetimes.

These simplifications are partly a consequence of the AA metric describing
only circular spacetimes, a particularly important subfamily of stationary and
axisymmetric spacetimes which require at most five free functions (see, e.g.,
Sec. 2.2 of Gourgoulhon 2010). The further reduction to just three metric
functions (or indeed only two functions in the most natural from of the AA
metric) is achieved via a novel modification in Azreg-Aïnou (2014b) to the
original Newman-Janis algorithm, where the regularity of a coordinate
transformation is used to eliminate the ad hoc “decomplexification step” that
was required in Newman & Janis (1965) (cf. also Rajan 2016).

Circular spacetimes have been of central interest in explorations of non-Kerr
spacetimes, and several popular parametrized metrics have been constructed to
characterize and study their varied properties. The Johannsen-Psaltis (JP;
Johannsen & Psaltis 2011; Johannsen 2013a) framework has had success, e.g., in
establishing tests of the no-hair conjecture (Johannsen & Psaltis, 2010) and
of the post-Newtonian structure of astrophysical BH spacetimes (Psaltis et
al., 2020). The Konoplya-Rezzolla-Zhidenko (KRZ; Rezzolla & Zhidenko 2014;
Konoplya et al. 2016) framework has been used, e.g., to map well-known static
metrics onto a single low-dimensional space with high accuracy to enable
comparisons between spacetimes (Kocherlakota & Rezzolla, 2020) and to test BH
metrics with gravitational-wave as well as X-ray observations (Völkel &
Barausse, 2020; Cardenas-Avendano et al., 2020). The current (Völkel et al.,
2021; Kocherlakota & Rezzolla, 2022; Younsi et al., 2023) and future
(Ayzenberg, 2022; Kocherlakota et al., 2023) ability to use BH images to test
gravity has also been demonstrated using both these frameworks.

We note that while the JP and KRZ parametric metrics allow detailed
explorations of black hole spacetimes, the AA ansatz metric can characterize
also non-BH objects such as naked singularities, wormholes, boson stars, etc.
Furthermore, it is clear to see from Johannsen (2013a) that the JP metric uses
a fundamentally different ansatz compared to the AA metric to describe
circular spacetimes, requiring four metric functions (see eq. 51 there), and
motivated by imposing that geodesics be Lioville-separable (eq. 10 there),
i.e., that they possess a Carter constant. This is similarly true for the KRZ
metric, where five free functions are permitted (see eq. 7 there). Konoplya &
Zhidenko (2021) explore a subclass of the KRZ metric that admits a Carter
constant. As we show later, null geodesics in the AA metric always possess a
Carter constant, and in the simplest form of the AA metric (which we favor)
time-like geodesics also have a Carter constant. We note that several other
parametrization frameworks have been constructed and used successfully to
investigate observable effects due to modifications of the spacetime metric in
the strong-field regime (see, e.g., Vigeland et al. 2011; Carson & Yagi 2020).
For the status of parameterizations of non-circular spacetimes, we direct the
reader to Delaporte et al. (2022).

In addition to its simplicity and appealing properties discussed above, our
choice to use the class of AA metrics is further motivated by the knowledge
that nearly all well-known solution metrics across various theories of gravity
can be cast in this form. This metric, therefore, presents an excellent
starting point for a broad forward-modeling study of the effect of the
spacetime metric on various observables of interest. The Liouville-
separability of the null geodesic equations makes computing various
characteristic features of the spacetime such as the location of the photon
shell or the shape of the shadow boundary (see, e.g., Shaikh 2019,
Kocherlakota et al. 2021, Solanki et al. 2022) analytically tractable.
Similarly, the separability of the timelike orbits facilitates a study of,
e.g., equatorial Keplerian orbits, and allows us to go one step further with
semi-analytic techniques. Details regarding geodesic orbits can be found in
appendix C.

##  2 General Class of Static Metrics

Table 1: Metric functions of selected spherically-symmetric and static spacetimes. We have used the coordinate freedom to set $-\hat{g}_{tt}\hat{g}_{rr}=g(r)=1$ in the general metric (1), hence we have only two metric functions, $f(r)$ and $R(r)$. The Schwarzschild and Reissner-Nordström (RN) spacetimes describe the vacuum and the electrovacuum black hole (BH) solutions of general relativity (GR), respectively. The Modified Hayward BH spacetime contains a regular BH and is generated by an anisotropic fluid. The GMGHS BH metric is a solution to the low energy theory of the heterotic string, and contains a scalar field (a dilaton, which vanishes asymptotically) as well as an electromagnetic field. The JNW naked singularity spacetime is generated by a scalar field. The JMN-1 naked singularity spacetime is an interior solution to the Schwarzschild spacetime and is generated by an anisotropic fluid. For easier comparison with the other five models, the JMN-1 metric is presented here with $g(r)=1$ (see Appendix A for details), thus differing from the original form given in eq. (29) of Joshi et al. (2011). The parameter $Q$ denotes an electromagnetic charge, $L$ denotes a (de Sitter) length scale, $R_{b}\geq 2.5M$ denotes a matching radius ($\sigma=M/(R_{\mathrm{b}}-M)$ and $r_{\mathrm{b}}=(1-\sigma)R_{\mathrm{b}}$), and $\hat{\nu}$ denotes a scalar charge ($r_{\star}=2M/(1-\hat{\nu})$ locates the curvature singularity). Object | Theory | Spacetime | $f(r)$ | $R^{2}(r)$
---|---|---|---|---
Black Hole | GR | Schwarzschild | $1-\frac{2M}{r}$ | $r^{2}$
Black Hole | GR | RN | $1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}$ | $r^{2}$
Black Hole | GR | Modified Hayward | $1-\frac{2Mr^{3}}{r^{4}+2L^{4}}$ | $r^{2}$
Black Hole | String | GMGHS | $1-\frac{2M}{r}$ | $r\left(r-\frac{Q^{2}}{M}\right)$
Naked Singularity | GR | JNW | $\left(1-\frac{r_{\star}}{r}\right)^{1-\hat{\nu}}$ | $r^{2}\left(1-\frac{r_{\star}}{r}\right)^{\hat{\nu}}$
Naked Singularity | GR | JMN-1 | $\left(1-\frac{2M}{R_{\mathrm{b}}}\right)\left(\frac{r}{r_{\mathrm{b}}}\right)^{2\sigma}$ | $R_{\mathrm{b}}^{2}\left(\frac{r}{r_{\mathrm{b}}}\right)^{2-2\sigma}$

The line element
$\mathrm{d}s^{2}=\hat{g}_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$ of an
arbitrary static and spherically-symmetric metric $\hat{g}_{\mu\nu}$ can be
written in spherical-polar coordinates, $x^{\mu}=(t,r,\vartheta,\varphi)$, in
the form

$\mathrm{d}s^{2}=\
-f(r)\mathrm{d}t^{2}+\frac{g(r)}{f(r)}~{}\mathrm{d}r^{2}+R^{2}(r)~{}\mathrm{d}\Omega_{2}^{2}\,,$
(1)

where
$\mathrm{d}\Omega_{2}^{2}=\mathrm{d}\vartheta^{2}+\sin^{2}{\vartheta}~{}\mathrm{d}\varphi^{2}$
is the standard line-element on a unit 2-sphere. This form of the metric
involves three functions of the coordinate $r$: $f(r)$, $g(r)$, $R(r)$.
However, an arbitrary static metric can be defined using only two metric
functions, i.e., any one of the metric functions, $f$, $g$, $R$, can be
eliminated by an appropriate change of coordinates (see, e.g., Ch. 14 of
Plebanski & Krasinski 2012). For example, it is possible to find a different
radial coordinate $r$ in which $-\hat{g}_{tt}\hat{g}_{rr}=g(r)=1$ (as we do in
Table 1, and discuss further in Appendix A). However, the required coordinate
transformation does not always lead to analytically convenient metric
functions. We therefore keep an arbitrary $g(r){\color[rgb]{0,0,0}>0}$ in (1).

The function $R(r)$ gives the curvature-radius or areal-radius as a function
of the radial coordinate $r$. In simple models, $R(r)=r$ (see Table 1). When
this is the case, the coordinate-radius $r$ of a 2-sphere is its curvature-
radius, $\kappa_{\mathrm{G}}(r)=\mathscr{R}^{(2)}(r)/2=1/r^{2}$, as well as
its areal-radius, $\mathcal{A}(r)=4\pi r^{2}$. Here we have used
$\kappa_{\mathrm{G}}$, $\mathscr{R}^{(2)}$ and $\mathcal{A}$ to denote the
Gaussian curvature, the Ricci scalar, and the area of a 2-sphere,
respectively. While it is always possible to find a coordinate transformation
to curvature coordinates in which $R(r)=r$, this again may only be tractable
numerically for some models. Finding simple analytic forms for all the metric
functions in such coordinates is not in general possible (see, e.g., Sec. IV D
of Kocherlakota & Rezzolla 2020). Thus, for general convenience, we will leave
$R(r)$ free.

When the metric (1) describes a BH spacetime, its event horizon is a null,
stationary surface. The normal to such a constant-$r$ surface satisfies
$\hat{g}^{\mu\nu}\partial_{\mu}r\cdot\partial_{\nu}r=0$, from which we find
that the horizon is located at the outermost root of $\hat{g}^{rr}=0$, i.e.,
at the outermost root of $f(r)$.

In Table 1 we list the metric functions corresponding to six well-known and
representative spherically-symmetric static solutions in GR and in string
theory. For uniformity, we have written all the metrics in coordinates such
that $g(r)=1$. The first two solutions listed are the canonical nonrotating BH
solutions of GR, namely, the Schwarzschild BH and the charged Reissner-
Nordström (RN) BH, which contain, respectively, a spacelike and a timelike
curvature singularity (see, e.g., Poisson 2004). Several regular BH spacetime
models have been proposed to mimic the desired effect of singularity-resolving
physics (see, e.g., Bardeen 1968; Ayón-Beato & García 1998; Bronnikov 2001;
Dymnikova 2004; Hayward 2006). The Modified Hayward BH (Zhou & Modesto 2023a;
$n=4$) spacetime included in Table 1 is generated by an anisotropic fluid with
a radial equation of state, $\omega_{r}=p_{r}/\rho=-1$, where $\rho$ is the
total energy density and $p_{r}$ is the radial pressure in the fluid rest-
frame. The fluid is dark-energy-like, producing the pressure necessary to
“avoid” a singularity. The charged Gibbons-Maeda-Garfinkle-Horowitz-Strominger
(GMGHS) BH (Gibbons & Maeda, 1988; Garfinkle et al., 1991) is different from
the RN BH of GR because of the presence of a dilaton (a scalar field) that
mediates the interaction between electromagnetism and gravity in the low-
energy effective action of string theory. In particular, the central
singularity of the GMGHS BH remains spacelike.

In addition to the above four BH solutions, Table 1 lists two other non-BH
models. Over the years, non-BH solutions to various theories have been
discussed in the literature and significant progress has been made to
determine ways in which such objects can be distinguished from BHs using
observations (see, e.g., Kocherlakota et al. 2021; The EHT Collaboration et
al. 2022b). The Janis-Newman-Winicour naked singularity (JNW; Janis et al.
1968) spacetime is a static solution generated by a minimally-coupled,
massless scalar field in GR. Indeed, this is also a solution to string theory
when an electromagnetic field is absent (Virbhadra, 1997) and is in addition a
solution to the Brans-Dicke theory with the parameter $\omega=-1$ (Kar, 1997).
Finally, the Joshi-Malafarina-Narayan-1 naked singularity (JMN-1; Joshi et al.
2011) is constructed in GR using an anisotropic fluid with vanishing radial
pressure, $p_{r}=0$.

##  3 Generalization to Stationary Metrics

Inspired by the Newman & Janis (1965) algorithm, Azreg-Aïnou (2014b) proposed
a stationary and axisymmetric generalization of the spherically-symmetric
metric (1) described in section 2. We will refer to this stationary metric as
the AA metric and denote it by $g_{\mu\nu}$ (to distinguish it from
$\hat{g}_{\mu\nu}$ of the spherically symmetric metric). In Boyer-Lindquist
(BL; Boyer & Lindquist 1967) coordinates,
$x^{\mu}=(t,r,\vartheta,\varphi)$,111Note that we do not distinguish between
the labels for the radial coordinates used to write the non-spinning (1) and
the spinning (3) metrics to avoid a proliferation of symbols. Furthermore,
while the metric signature here $(-,+,+,+)$ differs from that in Azreg-Aïnou
(2014b), it can be checked that eq. 3 corresponds to eq. 16 there. the AA
metric is given by

$\displaystyle\mathrm{d}s^{2}=$ $\displaystyle\
\frac{X}{\Sigma}\left[-\left(1-\frac{2F}{\Sigma}\right)\mathrm{d}t^{2}-2\frac{2F}{\Sigma}a\sin^{2}{\vartheta}~{}\mathrm{d}t\mathrm{d}\mathrm{\varphi}\right.$
$\displaystyle\
\left.+\frac{\Sigma}{\Delta}\mathrm{d}r^{2}+\Sigma~{}\mathrm{d}\vartheta^{2}+\frac{\Pi}{\Sigma}\sin^{2}{\vartheta}~{}\mathrm{d}\varphi^{2}\right]\,.$
(2)

Here the parameter $a$ is the spin of the central object and the “stationary”
metric functions $\\{F,\Delta,\Sigma,\Pi\\}$ can be related to the “static”
metric functions $\\{f,g,R\\}$ in (1) as discussed below. This then leaves
only the metric function $X=X(r,\vartheta)$ to be fixed by the field equations
(Azreg-Aïnou, 2014b).

While we focus in this paper on the AA metric, we note that different
generalizations of the static metric (1) are possible, in principle, and can
be achieved by modifying the complex coordinate transformation involved in the
Newman & Janis (1965)-type solution-generating technique (see, e.g., Azreg-
Aïnou 2014a). It is important to note, however, that the particular
transformations used in Azreg-Aïnou (2014a, b) send the Schwarzschild, the
Reissner-Nordström, and the GMGHS metrics, to their appropriate spinning
generalizations, namely to the Kerr, the Kerr-Newman, and the Kerr-Sen
metrics, respectively. We direct the reader to Erbin (2017) for a review of
such solution-generating techniques, and to Sec. 7.1 of Wald (1984) for a
general discussion of stationary, axisymmetric spacetimes, and on the
construction of BL-like coordinates.

Expressions for the metric functions $\\{F,\Delta,\Sigma,\Pi\\}$ in the AA
metric (3) are most conveniently written by first defining two auxiliary
functions that are specific combinations of the original static metric
functions, $f(r)$, $g(r)$, $R(r)$,

$\displaystyle A(r)=\ R^{2}/\sqrt{g}\,,\ \ B(r)=\ (f/g)R^{2}\,,$

which can be written even more transparently as

$A=\frac{\hat{g}_{\vartheta\vartheta}}{\sqrt{-\hat{g}_{tt}\hat{g}_{rr}}}\,;\ \
B=\frac{\hat{g}_{\vartheta\vartheta}}{\hat{g}_{rr}}=\hat{g}^{rr}\hat{g}_{\vartheta\vartheta}\,.$
(3)

$A$ and $B$ are each functions of $r$ alone, and in terms of them the
functions $\\{F,\Delta,\Sigma,\Pi\\}$ in the AA metric (3) take the remarkably
simple form

$\displaystyle F(r)=$ $\displaystyle\ (A-B)/2\,,$ (4)
$\displaystyle\Delta(r)=$ $\displaystyle\ B+a^{2}\,,$
$\displaystyle\Sigma(r,\vartheta)=$ $\displaystyle\
A+a^{2}\cos^{2}{\vartheta}\,,$ $\displaystyle\Pi(r,\vartheta)=$
$\displaystyle\ \left(A+a^{2}\right)^{2}-\Delta a^{2}\sin^{2}{\vartheta}\,.$

Note that $F$ and $\Delta$ depend only on $r$, while $\Sigma$ and $\Pi$ are
functions of $r$ and $\vartheta$. The spinning AA metric $g_{\mu\nu}$ (3)
reduces to the nonspinning metric $\hat{g}_{\mu\nu}$ (1) in the limit of
vanishing spin ($a=0$) for appropriate choices of $X$ (see Azreg-Aïnou 2014a
for a discussion on conformal vs. normal fluids).

The determinant $\det{[g_{\mu\nu}]}$ of the AA metric, and of its
$t\varphi-$sector
$\det{[g_{t\varphi}]}:=g_{tt}g_{\varphi\varphi}-g_{t\varphi}^{2}$ are given in
BL coordinates by

$\displaystyle\det{[g_{\mu\nu}]}=$ $\displaystyle\
-(X^{4}/\Sigma^{2})\sin^{2}{\vartheta}\,,$ (5)
$\displaystyle\det{[g_{t\varphi}]}=$ $\displaystyle\
-(X^{2}/\Sigma^{2})\Delta\sin^{2}{\vartheta}\,.$ (6)

The AA metric can also be expressed in the closely-related quasi-isotropic
(QI) coordinates (see, e.g., Sec. 2.3.2 of Gourgoulhon 2010),
$x^{\tilde{\mu}}=(t,\tilde{r},\vartheta,\varphi)$, as

$\displaystyle\mathrm{d}s^{2}=$ $\displaystyle\
\frac{X}{\Sigma}\left[-\frac{X\Delta}{\Pi}\mathrm{d}t^{2}+\frac{\Sigma}{\tilde{r}^{2}}\left(\mathrm{d}\tilde{r}^{2}+\tilde{r}^{2}\mathrm{d}\vartheta^{2}\right)\right.$
$\displaystyle\
\left.+\frac{\Pi}{\Sigma}\sin^{2}{\vartheta}\left(\mathrm{d}\varphi-\Omega_{\mathrm{Z}}\mathrm{d}t\right)^{2}\right]\,,$
(7)

where we have introduced the angular velocity of the zero angular momentum
observer (ZAMO),
$\Omega_{\mathrm{Z}}=-g_{t\varphi}/g_{\varphi\varphi}=2aF/\Pi$ (cf. eqs. 2.4,
2.5 of Bardeen et al. 1972). The form above (3) can be obtained from (3) by
transforming the radial coordinate, $r\mapsto\tilde{r}$, via,
$\tilde{r}(r)=\exp{[\int(\mathrm{d}r/\sqrt{\Delta})]}$. The
$r\vartheta-$sector of the AA metric in these QI coordinates is clearly
conformally flat, as desired. The related cylindrical coordinates,
$x^{\mu\prime}=(t,\rho,z,\varphi)$, with $\rho:=\tilde{r}\sin{\vartheta}$ and
$z:=\tilde{r}\cos{\vartheta}$, are called the Lewis-Papapetrou coordinates
(Lewis, 1932; Papapetrou, 1966).

We briefly illustrate now how the remaining metric function $X$ can be fixed
by considering, as an example, the case when the background matter is an
anisotropic fluid in GR which flows around the spin axis. Since in its own
rest-frame, $\\{e^{\mu}_{(a)}\ (a=0-3)\\}$, the fluid stress-energy-momentum
tensor is diagonal, the Einstein equations imply that the Einstein tensor in
this frame,
$\mathscr{G}_{(a)(b)}=\mathscr{G}_{\mu\nu}e^{\mu}_{(a)}e^{\nu}_{(b)}$, must
also be diagonal. However, it can be shown that two of the off-diagonal
elements of the Einstein tensor, $\mathscr{G}_{(r)(\vartheta)}$ and
$\mathscr{G}_{(t)(\varphi)}$, will in general be non-zero. Demanding that
these two terms vanish yields one nonlinear partial differential equation
(PDE) and one linear PDE, each of which involves the free metric function $X$,
the auxiliary function $A(r)$, and the fluid angular velocity $\Omega$. These
PDEs are given in eqs. 15 and 18 of Azreg-Aïnou (2014a), respectively, where
the fluid angular velocity was chosen to be $\Omega=a/(A+a^{2})$ (see
$e^{\mu}_{t}$ in eq. 16 there). It is interesting to note that this angular
velocity does not correspond to the ZAMO angular velocity introduced above,
$\Omega_{\mathrm{Z}}=2aF/\Pi$. It matches, however, the angular velocity of
the principal null congruences of the spacetime introduced below (see eq. 8).
For non-fluid matter models, additional equations of motion for the matter
fields must be solved. For example, if the matter is a scalar field, the
associated Klein-Gordon equations must additionally be solved. Similarly, if
the matter is an electromagnetic field, the Maxwell equations have to be
accounted for as well. We will not enter into a discussion of this topic but
refer the reader to Erbin (2017).

For the purposes of the present paper, it suffices to note that the subclass
of AA metrics with $X=\Sigma$ is particularly interesting, as it corresponds
to spacetimes that are asymptotically flat (see Appendix B for further
details). Furthermore, it can be checked that this choice for the conformal
factor $X$ always solves the $\mathscr{G}_{(r)(\vartheta)}=0$ equation for
axially-spinning matter. Finally, all of the BH solutions discussed in this
paper have $X=\Sigma$, whether they are vacuum or contain scalar and/or
electromagnetic and/or axion fields. Even though these spacetimes arise as
“solution metrics” to a variety of field equations,
$X=\Sigma=A(r)+a^{2}\cos^{2}\vartheta$ consistently remains a valid choice.
For this subclass of the AA metrics, the possibly divergent behavior of the
Ricci $\mathscr{R}$ and the Kretschmann $\mathscr{K}$ scalars can be seen from
the behavior of their denominators, which go roughly as $\sim\Sigma^{-3}$ and
as $\sim\Sigma^{-6}$ respectively. Thus, when this metric describes a singular
spacetime, the curvature singularity is located at $r=r_{\star}$ such that
$\Sigma(r_{\star},\pi/2)=0$, which corresponds to a ring singularity located
in the equatorial plane. This last equation is equivalent to $R(r_{\star})=0$.

When the general AA metric (3) describes a BH spacetime, as above, a horizon
is present at every location ${\color[rgb]{0,0,0}r=r_{\mathrm{H}}>r_{\star}}$
where $g^{rr}=\Delta/\Sigma=0$. Equivalently, horizons are located at the
real, positive (${\color[rgb]{0,0,0}R(r_{\mathrm{H}})>0}$) roots of
$\Delta(r)$. Conversely, if no such roots exist, then the AA metric does not
correspond to a BH spacetime (e.g., the $a>M$ Kerr metric). The event horizon
in particular is located at the largest such root. Since $\Delta$ is a
function of $r$ alone, it is reassuring to find that the event horizon is
indeed a round sphere in BL coordinates. Furthermore, the horizon angular
velocity, $\Omega_{\mathrm{H}}:=\Omega_{\mathrm{Z}}(r_{\mathrm{H}})$, is
simply given as
$\Omega_{\mathrm{H}}=2aF(r_{\mathrm{H}})/(A(r_{\mathrm{H}})+a^{2})^{2}=a/(A(r_{\mathrm{H}})+a^{2})$.
Clearly, $\Omega_{\mathrm{H}}$ is independent of $\vartheta$, i.e., an AA BH
rotates like a rigid body, as a consequence of the weak rigidity theorem (see,
e.g., Sec. 8.4.4 of Straumann 2013). These desirable properties, which are
well-known for the Kerr and Kerr-Newman metrics, are now seen to be true for
the very wide class of AA stationary BH metrics.

It is interesting to note that the general AA metric (3) can also describe
spacetimes which contain regions that admit closed timelike curves. In such
regions, $g_{\varphi\varphi}<0$ (see, e.g., Sec. V.B of Johannsen 2013b),
i.e., $\Pi<0$ (e.g., the $a>M$ Kerr metric).

We show in Appendix C that the null geodesic equations are Lioville-separable.
Thus, the AA metric always admits a Carter constant for null geodesics.
Furthermore, we also identify a subclass of AA metrics that additionally admit
separable timelike geodesic equations. These require the conformal factor to
take the form $X(r,\vartheta)=X_{r}(r)+X_{\vartheta}(\vartheta)$. For this
subclass, a Carter constant exists for all geodesics. Concomitantly, arbitrary
geodesics of all AA metrics with $X=\Sigma$ possess Carter constants.

The $rr$-component of the AA metric (3) in BL coordinates diverges at the
horizon ($\Delta(r)=0$). The Ricci and Kretschmann scalars, however, reveal
this to be merely a coordinate singularity, an artifact of the choice of
coordinates. The singularity can be eliminated by a change of coordinates, and
this is the topic we turn to next.

##  4 Horizon-Penetrating Coordinates

In this section, we convert the AA metric, which is written in BL coordinates
in equation 3, to a horizon-penetrating form in which no coordinate
singularity is present at the horizon. This is the form that is most useful
for GRMHD and xGRMHD simulations of astrophysical accretion flows.

The ingoing ($-$) and outgoing ($+$) principal null congruences (PNCs) of the
AA spacetime consist of null geodesics $x^{\mu}(\lambda)$ whose tangents
$\ell_{\pm}^{\mu}=\dot{x}^{\mu}=\mathrm{d}x^{\mu}/\mathrm{d}\lambda$ satisfy
$\ell_{\pm}^{\vartheta}=\dot{\vartheta}=0$ and $\ddot{\vartheta}=0$ (see,
e.g., Misner et al. 1973, Hioki & Miyamoto 2008 for further details). The
$1-$form fields $(l_{\pm})_{\mu}$ associated with the PNCs in an AA spacetime
can be expressed elegantly as,222See Appendix C. For completeness, the PNC
vector fields are (cf. eq. 33.39 of Misner et al. 1973 for the Kerr metric),
$\ell_{\pm}^{\mu}=E\frac{\Sigma}{X}\left[\frac{A+a^{2}}{\Delta},\pm
1,0,\frac{a}{\Delta}\right]\,.$ (8)

$\left(\ell_{\pm}\right)_{\mu}=E\left[-1,\pm\frac{\Sigma}{\Delta},0,a\sin^{2}{\vartheta}\right]\,,$
(9)

where $E$ is some constant.

We now define the spherical-polar ingoing Kerr-Schild (siKS) coordinates,
$x^{\bar{\mu}}=(\tau,r,\vartheta,\phi)$, as those in which the ingoing PNC
takes the form,

$\left(\ell_{-}\right)_{\bar{\mu}}=E\left[-1,-1,0,a\sin^{2}{\vartheta}\right]\,.$
(10)

To distinguish between the BL ($x^{\mu}$) and the siKS ($x^{\bar{\mu}}$)
coordinate systems, we will use a bar for the indices of the latter. We are
interested in the ingoing PNC since our coordinate system should penetrate the
future horizon $\mathscr{H}^{+}$ when one exists, and cover patches I and II
of the Kruskal (Kruskal, 1960) or Penrose-Carter (Penrose, 1963; Carter, 1966)
diagrams. The Jacobian $\Lambda^{\bar{\mu}}_{\
\mu}=\partial_{\mu}x^{\bar{\mu}}$ for the coordinate transformation from BL
coordinates to siKS coordinates (i.e.,
$\mathrm{d}x^{\bar{\mu}}=\Lambda^{\bar{\mu}}_{\ \mu}\mathrm{d}x^{\mu}$) can be
inferred from equations 9 and 10 to be,

$\Lambda^{\bar{\mu}}_{\ \mu}=\begin{bmatrix}1&2F/\Delta&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&a/\Delta&0&1\end{bmatrix}\,.$ (11)

The time coordinate $\tau$ then is clearly a “tortoise” time coordinate (see,
e.g., Blau 2023),

$\tau=t+\int(2F/\Delta)~{}\mathrm{d}r\,,$ (12)

and we recognize immediately that, in the limit of vanishing spin
($a\rightarrow 0$), the siKS coordinates are analogous to the coordinates
presented in Eddington (1924) and Finkelstein (1958).

Using the above coordinate transformation, we can write the AA metric in siKS
coordinates as,

$g_{\bar{\mu}\bar{\nu}}=\frac{X}{\Sigma}\begin{bmatrix}-\left(1-\frac{2F}{\Sigma}\right)&\frac{2F}{\Sigma}&0&-\left(\frac{2F}{\Sigma}\right)a\sin^{2}{\vartheta}\\\
*&\left(1+\frac{2F}{\Sigma}\right)&0&-\left(1+\frac{2F}{\Sigma}\right)a\sin^{2}{\vartheta}\\\
*&*&\Sigma&0\\\
*&*&*&\left(\frac{\Pi}{\Sigma}\right)\sin^{2}\vartheta\end{bmatrix},$ (13)

where the asterisks denote components that are fixed by symmetry,
$g_{\bar{\mu}\bar{\nu}}=g_{\bar{\nu}\bar{\mu}}$. Notice that the metric
components now diverge only at the spacetime/curvature singularity, present at
$\Sigma=0$. The inverse AA metric in siKS coordinates is

$g^{\bar{\mu}\bar{\nu}}=\frac{\Sigma}{X}\begin{bmatrix}-\left(1+\frac{2F}{\Sigma}\right)&\frac{2F}{\Sigma}&0&0\\\
*&\frac{\Delta}{\Sigma}&0&\frac{a}{\Sigma}\\\ *&*&\frac{1}{\Sigma}&0\\\
*&*&*&\frac{1}{\Sigma\sin^{2}\vartheta}\end{bmatrix}\,.$ (14)

For completeness, we cast the general AA metric into its “classic” Kerr-Schild
form in eq. B2 of Appendix B.

In siKS coordinates, the event horizon is still located at the outermost root
of $g^{\bar{1}\bar{1}}=g^{rr}=0$, i.e., $\Delta(r)=0$ (14), but as we can see
from eq. 13 the metric components $g_{\bar{\mu}\bar{\nu}}$ no longer diverge
at $\Delta=0$. Thus, the coordinate singularity at the horizon has been
removed by changing from BL to siKS coordinates, showing that the latter are
horizon-penetrating coordinates. Furthermore, a comparison of the Kerr metric
in particular, in these coordinates, with that reported in Font et al. (1998),
with $k=1$ there, and in McKinney & Gammie (2004), shows that these are the
horizon-penetrating coordinates canonically used in the context of GRMHD
simulations.

Note that the original set of Kerr-Schild coordinates used to describe the
Kerr spacetime (Kerr, 1963; Kerr & Schild, 2009) are adapted to the outgoing
PNC in which $l_{+}^{\breve{\mu}}\propto\delta^{\breve{\mu}}_{r}$ (see, e.g.,
Wiltshire et al. 2009; Azreg-Aïnou 2014b). The corresponding time coordinate
$u$ is the retarded time as well as a null affine parameter that parametrizes
the outgoing PNC on future null infinity $\mathscr{I}^{+}$. Equivalently, the
ingoing version of the original set of Kerr-Schild coordinates would be
adapted to the ingoing PNC in which
$l_{-}^{\breve{\mu}}\propto\delta^{\breve{\mu}}_{r}$ with the corresponding
time coordinate $v$ being the advanced time as well as a null affine parameter
that parametrizes the ingoing PNC on past null infinity $\mathscr{I}^{-}$.
Clearly, the “original” ingoing Kerr-Schild system is quite different from the
spherical-ingoing Kerr-Schild coordinates which we use in this section and
which are matched to current GRMHD codes. From eq. 10, that the tangent to the
ingoing PNC in our coordinates is given as

$\ell_{-}^{\bar{\mu}}=E\frac{\Sigma}{X}\left[1,-1,0,0\right]\,.$ (15)

##  5 Horizon-Penetrating $3+1$ Form

We can cast the AA metric when written in horizon-penetrating siKS coordinates
into its associated $3+1$ form (Arnowitt et al., 2008) by introducing a
foliation into spacelike hypersurfaces $\Sigma_{\tau}$ that are iso-surfaces
of the scalar time function $\tau$ (see, e.g., Poisson 2004, Gourgoulhon 2007,
Alcubierre 2008, Rezzolla & Zanotti 2013). The unit timelike normal to these
hypersurfaces is

$n_{\bar{\mu}}:=-\alpha\nabla_{\bar{\mu}}\tau{\color[rgb]{0,0,0}=-\alpha\delta_{\bar{\mu}}^{\tau}}\,,$
(16)

where $\alpha:=\sqrt{-1/g^{\tau\tau}}$ is called the lapse function, and from
eq. 14 is given by

$\alpha=[X/(\Sigma+2F)]^{1/2}\,.$ (17)

The four-vector that is dual to the normal, i.e.,
$n^{\bar{\mu}}=g^{\bar{\mu}\bar{\nu}}n_{\bar{\nu}}=-\alpha
g^{\bar{\mu}\tau}=[1/\alpha,-\alpha(2F/X),0,0]$, corresponds to the four-
velocity of the local timelike Eulerian observer. The four-velocity of this
observer in BL coordinates can be obtained as $n^{\mu}=\Lambda^{\mu}_{\
\bar{\mu}}n^{\bar{\mu}}$, where $\Lambda^{\mu}_{\
\bar{\mu}}=\left(\Lambda^{\bar{\mu}}_{\ \mu}\right)^{-1}$. It is
straightforward to then check that the angular velocity of the Eulerian
observer in BL coordinates $\Omega_{\mathrm{E}}:=n^{\varphi}/n^{t}$ matches
the angular velocity of the zero angular momentum observer introduced above
$\Omega_{\mathrm{Z}}=-g_{t\varphi}/g_{\varphi\varphi}=2aF/\Pi$ exactly.
However, the Eulerian observer has nonzero BL radial velocity, i.e.,
$n^{r}=-2\alpha F/X(=n^{\bar{r}})$.

With eq. 16, the induced (Riemannian) metric $\gamma$ on the spacelike
hypersurfaces can now be introduced as

$\gamma_{\bar{\mu}\bar{\nu}}:=g_{\bar{\mu}\bar{\nu}}+n_{\bar{\mu}}n_{\bar{\nu}}\,.$
(18)

The projection tensor, which is used to obtain the spatial components of any
four-vector or tensor, is simply $\gamma^{\bar{\mu}}_{\
\bar{\nu}}=g^{\bar{\mu}\bar{\alpha}}\gamma_{\bar{\alpha}\bar{\nu}}=\delta^{\bar{\mu}}_{\
\bar{\nu}}+n^{\bar{\mu}}n_{\bar{\nu}}$ since $\gamma^{\bar{\mu}}_{\
\bar{\nu}}n^{\bar{\nu}}=0$.

Although the 1-form $n_{\bar{\mu}}\mathrm{d}x^{\bar{\mu}}$ is collinear with
$\mathrm{d}\tau$ (see eq. 16), the dual four-vector
$n^{\bar{\mu}}\partial_{\bar{\mu}}$ is not, in general, collinear with
$\partial_{\tau}$ (since $-\alpha g^{\bar{\mu}\tau}\cancel{\propto}\
\delta^{\bar{\mu}}_{\tau}$). Indeed, this noncollinearity is typically
captured via a four-vector $\beta^{\bar{\mu}}$, which is defined such that
$n_{\bar{\mu}}\beta^{\bar{\mu}}=0$:

$\beta^{\bar{\mu}}:=(\partial_{\tau})^{\bar{\mu}}-\alpha
n^{\bar{\mu}}=\delta^{\bar{\mu}}_{\tau}-\alpha n^{\bar{\mu}}\,.$ (19)

Since by construction $\beta^{\tau}=0$, it is convenient to work instead with
the spacelike three-vector (see, e.g., McKinney & Gammie 2004, Porth et al.
2017)

$\beta^{i}:=\alpha^{2}g^{i\tau}=-g^{i\tau}/g^{\tau\tau}{\color[rgb]{0,0,0}=[2F/(2F+\Sigma)]\delta^{i}_{r}}\,,$
(20)

where the index $i$ takes values $i=1,2,3$. This three-vector measures the
shift of the isolines of the three spatial coordinates with respect to the
normal to the hypersurfaces and is therefore called the shift vector (see,
e.g., Sec. 1.3 of Gourgoulhon 2010).

The desired $3+1$ form of the AA metric can now be written, using the lapse
function $\alpha$ (eq. 17), the shift vector $\beta^{i}$ (eq. 20), and the
components $\gamma_{ij}$ (eq. 18) of the induced metric on the hypersurfaces
$\Sigma_{\tau}$, as

$\mathrm{d}s^{2}=-\alpha^{2}\mathrm{d}\tau^{2}+\gamma_{ij}(\mathrm{d}x^{i}+\beta^{i}\mathrm{d}\tau)(\mathrm{d}x^{j}+\beta^{j}\mathrm{d}\tau)\,.$
(21)

For completeness, we demonstrate how the temporal $u^{\tau}$ and the spatial
$u^{i}$ components of a vector $u^{\bar{\mu}}=(u^{\tau},u^{i})$ can be
obtained using the unit normal and the projection tensor respectively. First,
with $\Gamma=-n_{\bar{\mu}}u^{\bar{\mu}}$, we have $u^{\tau}=\Gamma/\alpha$.
Second, introducing the purely spatial vectors ($v^{0}=0$)
$v^{i}:=(\gamma^{i}_{\ \bar{\mu}}u^{\bar{\mu}})/\Gamma$, we obtain
$u^{i}=\Gamma(v^{i}-\beta^{i}/\alpha)$. One also finds that
$\Gamma=1/\sqrt{1-v^{2}}$, where $v^{2}=v_{i}v^{i}$. When $u$ corresponds to
the four-velocity of a fluid element, $\Gamma$ has the significance of being
the fluid Lorentz factor with respect to the Eulerian observer. An excellent
summary of the 3+1 treatment relevant for GRMHD can be found in Gammie et al.
(2003).

##  6 Summary and Conclusions

We have presented in equation (13), section 4, a horizon-penetrating Kerr-
Schild form of the rather general stationary and axisymmetric Azreg-Aïnou (AA)
metric. This form of the metric can be used as an input when performing
general relativistic (GR) as well as non-GR (xGR) magnetohydrodynamics
simulations of accretion flows onto compact objects in arbitrary metric
theories of gravity. The AA metric can be used to generate a number of popular
metrics that describe the stationary spacetimes in GR corresponding to
electro-vacuum black holes (Kerr, 1963; Newman et al., 1965), regular black
holes such as the Kerr-Hayward models (Hayward, 2006; Bambi & Modesto, 2013;
Zhou & Modesto, 2023a, b), naked singularities such as the Kerr-Janis-Newman-
Winicour spacetime (Solanki et al., 2022), etc. It can also be used to
describe black hole solutions arising in alternative theories of gravity
(e.g., Sen 1992).

Table 2:  Metric functions for the stationary and axisymmetric generalizations of the static and spherically-symmetric spacetimes given in Table 1. The spinning generalizations of a nonspinning “seed” metric (1) is given in Boyer-Lindquist coordinates in eq. 3 and in horizon-penetrating spherical ingoing Kerr-Schild coordinates in eq. 6. These involve five “stationary metric functions” $\\{F,\Delta,\Sigma,\Pi,X\\}$, the first four of which are related to $f$ and $R$ via eq. 24. We show below the spinning generalizations of the specific static solutions listed in Table 1. Since we used coordinates for the static metrics in which $g(r)=1$, $g$ does not appear in the expressions below. For brevity we only show the four metric functions $2F$, $\Delta$, $\Sigma=X$ below. The fifth metric function is given by $\Pi=(R^{2}+a^{2})^{2}-\Delta a^{2}\sin^{2}{\vartheta}$. All the spacetime models listed here are asymptotically flat, and $a$ denotes the usual spin parameter. A variety of physical scenarios are captured by the exemplary set listed here: the vacuum Kerr BH, the electromagnetically charged Kerr-Newman BH, the (modified) Kerr-Hayward spinning BH, the spinning JNW naked singularity, and the spinning JMN-1 naked singularity are solutions of general relativity, whereas the Kerr-Sen spacetime describes electromagnetically-charged BHs of string theory which are also charged under a dilaton field as well as an axion field. Spacetime | $2F=(1-f)R^{2}$ | $\Delta=fR^{2}+a^{2}$ | $\Sigma=X=R^{2}+a^{2}\cos^{2}{\vartheta}$
---|---|---|---
Kerr | $2Mr$ | $r^{2}-2Mr+a^{2}$ | $r^{2}+a^{2}\cos^{2}{\vartheta}$
Kerr-Newman | $2Mr-Q^{2}$ | $r^{2}-2Mr+Q^{2}+a^{2}$ | $r^{2}+a^{2}\cos^{2}\vartheta$
Kerr-Hayward | $\frac{2Mr^{5}}{r^{4}+2L^{4}}$ | $r^{2}-\frac{2Mr^{5}}{r^{4}+2L^{4}}+a^{2}$ | $r^{2}+a^{2}\cos^{2}{\vartheta}$
Kerr-Sen | $2Mr-2Q^{2}$ | $r^{2}-\left(2M+\frac{Q^{2}}{M}\right)r+2Q^{2}+a^{2}$ | $r\left(r-\frac{Q^{2}}{M}\right)+a^{2}\cos^{2}{\vartheta}$
Spinning JNW | $r^{2}\left[\left(1-\frac{r_{\star}}{r}\right)^{\hat{\nu}}-\left(1-\frac{r_{\star}}{r}\right)\right]$ | $r^{2}-\frac{2Mr}{1-\hat{\nu}}+a^{2}$ | $r^{2}\left(1-\frac{r_{\star}}{r}\right)^{\hat{\nu}}+a^{2}\cos^{2}{\vartheta}$
Spinning JMN-1 | $\frac{r^{2}}{(1-\sigma)^{2}}\left[\left(\frac{r}{r_{\mathrm{b}}}\right)^{-2\sigma}-\left(1-\frac{2M}{r_{\mathrm{b}}}\right)\right]$ | $\left(1-\frac{2M}{R_{\mathrm{b}}}\right)\frac{r^{2}}{(1-\sigma)^{2}}+a^{2}$ | $R_{\mathrm{b}}^{2}\left(\frac{r}{r_{\mathrm{b}}}\right)^{2-2\sigma}+a^{2}\cos^{2}{\vartheta}$

To illustrate the ease of generating the Kerr-Schild forms of new stationary
metrics, we now summarize the steps involved. We start with an arbitrary
static and spherically-symmetric “seed” metric, e.g., the ones listed in Table
1, in standard spherical-polar coordinates, $x^{\mu}=(t,r,\vartheta,\varphi)$,

$\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+\frac{g(r)}{f(r)}\mathrm{d}r^{2}+R^{2}(r)\mathrm{d}\Omega_{2}^{2}\,.$
(22)

As we noted in section 2, it is always possible to find a radial coordinate
$r$ in which $g(r)=1$. Since this reduces the number of free functions, we
used such coordinates for the static models listed in Table 1. However, the
required transformation may not always lead to simple analytical forms for
$f(r)$ and $R(r)$. When it does not, it is preferable to use the more general
form (22).

The AA stationary generalization of the metric (22) in spherical-polar ingoing
Kerr-Schild coordinates, $x^{\bar{\mu}}=(\tau,r,\vartheta,\phi)$, takes the
form,

$\displaystyle\mathrm{d}s^{2}=$ $\displaystyle\
\frac{X}{\Sigma}\left[-\left(1-\frac{2F}{\Sigma}\right)\mathrm{d}\tau^{2}+\left(1+\frac{2F}{\Sigma}\right)\mathrm{d}r^{2}\right.$
$\displaystyle\
\left.+\Sigma~{}\mathrm{d}\vartheta^{2}+\frac{\Pi}{\Sigma}\sin^{2}{\vartheta}~{}\mathrm{d}\phi^{2}-2\frac{2F}{\Sigma}a\sin^{2}{\vartheta}~{}\mathrm{d}\tau\mathrm{d}\phi\right.$
$\displaystyle\
\left.+2\frac{2F}{\Sigma}\mathrm{d}\tau\mathrm{d}r-2\left(1+\frac{2F}{\Sigma}\right)a\sin^{2}{\vartheta}~{}\mathrm{d}r\mathrm{d}\phi\right]\,,$
(23)

where the “stationary metric functions” $\\{F,\Delta,\Sigma,\Pi\\}$ are fixed
by the “static metric functions” $\\{f,g,R\\}$ in the seed metric (22) as,

$\displaystyle 2F(r)=$ $\displaystyle\ R^{2}/\sqrt{g}-(f/g)R^{2}\,,$ (24)
$\displaystyle\Delta(r)=$ $\displaystyle\ (f/g)R^{2}+a^{2}\,,$
$\displaystyle\Sigma(r,\vartheta)=$ $\displaystyle\
R^{2}/\sqrt{g}+a^{2}\cos^{2}{\vartheta}\,,$ $\displaystyle\Pi(r,\vartheta)=$
$\displaystyle\ (R^{2}/\sqrt{g}+a^{2})^{2}-\Delta a^{2}\sin^{2}{\vartheta}\,.$

The parameter $a$ describes the spin of the spacetime. The only remaining
freedom is the stationary metric function $X(r,\vartheta)$.

If the background matter in the stationary spacetime is a fluid flowing around
the spin axis ($\vartheta=0$) in general relativity (GR), the unknown function
$X=X(r,\vartheta)$ is fixed by solving two Einstein equations,
$\mathscr{G}_{(r)(\vartheta)}=0$ and $\mathscr{G}_{(t)(\varphi)}=0$, in the
fluid rest frame (the energy-momentum-stress components $T_{(r)(\vartheta)}$
and $T_{(t)(\varphi)}$ of the fluid vanish in this frame). We will not enter
into detailed implications of these equations here but note that $X=\Sigma$ is
always a solution of the first equation, $\mathscr{G}_{(r)(\vartheta)}=0$. The
choice $X=\Sigma$ is additionally appealing since spacetimes with this
property are asymptotically-flat (provided of course that the original seed
static metric is also asymptotically flat). Verifying that $X=\Sigma$ also
solves the second Einstein equation, $\mathscr{G}_{(t)(\varphi)}=0$, in the
fluid frame, and that the resulting energy-momentum-stress tensor is
physically valid, i.e., satisfies various energy conditions, is an involved
calculation and is beyond the scope of this paper. However, if one merely
wants a novel stationary metric in which to carry out GRMHD simulations, and
if one is not too concerned about necessarily satisfying energy conditions,
then the metric (6) with $X=\Sigma$, and a suitable choice of the seed
functions $f(r)$, $g(r)$, $R(r)$, would be a reasonable choice.

We should note that, for non-fluid models, additional equations of motion for
the fields (e.g., the Klein-Gordon equations or the Maxwell equations) must be
solved. The status and successes of such metric-solution-generating techniques
in being able to yield solutions also to the nongravitational field equations
has been discussed in Erbin (2017).

Table 2 presents a compilation of the metric functions of the stationary AA
spacetimes corresponding to the six static “seed” models listed in Table 1.
These are all models for which the choice $X=\Sigma$ is valid. The stationary
Kerr (1963) metric arises naturally as the AA generalization of the static
Schwarzschild metric. It describes vacuum, spinning black holes (BHs) in GR
and has been used extensively in GRMHD simulations. Similarly, the AA
transformation automatically generates the Kerr-Newman metric (Newman et al.,
1965), which describes electromagnetically-charged, spinning BHs in GR, as the
spinning generalization of the non-spinning Reissner-Nordstrom BH. The Kerr-
Hayward model proposed here (the third model in Table 2) is expected to
describe spinning regular (i.e., no singularities) BHs in GR (see, e.g., Bambi
& Modesto 2013; Zhou & Modesto 2023b). The spinning equivalent of the GMGHS BH
solution in string theory is given by the Kerr-Sen metric (Sen, 1992). The
form of the Kerr-Sen metric shown in Table 2 can be put into the form
presented in eq. 20 of Xavier et al. (2020) by replacing $r$ with $r+Q^{2}/M$.
It is worth noting that the spacetime of Kerr-Sen BHs contains not just
electromagnetic fields but also a scalar field (dilaton) as well as an axion
field. All of the BH solutions in Table 2 reduce to the Kerr BH in appropriate
limits ($Q\to 0$, $L\to 0$).

The spinning Janis-Newman-Winicour (JNW) spacetime (the fifth model in Table
2) has been proposed in Solanki et al. (2022) and is expected to describe a
spinning naked singularity spacetime containing scalar field matter. Finally,
the spinning JMN-1 metric proposed here is expected to describe a spinning
naked singularity spacetime containing an anisotropic fluid in GR. Careful
explorations of these spacetimes, especially with a focus on the physicality
of the underlying matter (e.g., how well energy conditions are satisfied), is
left for future work.

Vagnozzi et al. (2023) give an immense compilation of spacetime metrics
including those corresponding to static and spherically-symmetric spacetimes.
These metric functions describe alternative nonspinning black holes which are
of considerable interest and can be studied using simulations. More
importantly, each static model could potentially be used as a seed metric to
obtain the horizon-penetrating form of the stationary metric that it may
belong to via equations 6 and 24. The only open issue, which needs to be
checked in each case, is whether the choice $X=\Sigma$ is physically valid.

We conclude on an optimistic note, envisioning the widespread adoption of
GRMHD simulations, facilitated by our easy-to-use metric formulation. A Kerr
simulation library has recently been used successfully to infer the properties
of plasma present in the close vicinity of Sgr A∗, as well as the spacetime
geometry of this supermassive BH (The EHT Collaboration et al., 2022a, b).
While it might be a challenging endeavor in the near future to construct
similar libraries of fully 3D simulations in the other stationary spacetimes
described in this paper (because of the many additional spacetime parameters
in these models such as $Q$, $L$, $\hat{\nu}$, $\sigma$), it should at least
be possible to build extensive 2D GRMHD simulation libraries for these
spacetimes. This will enable valuable exploration of magnetized relativistic
gas dynamics in diverse spacetimes.

We thank the referee for helpful suggestions and Saurabh for pointing out a
typo. PK, RN, and KC acknowledge support in part from grants from the Gordon
and Betty Moore Foundation and the John Templeton Foundation to the Black Hole
Initiative at Harvard University, and from NSF award OISE-1743747. YM is
supported by the National Natural Science Foundation of China (Grant No.
12273022) and Shanghai target program of basic research for international
scientists (Grant No. 22JC1410600).

##  Appendix A Alternative Form for Arbitrary Static and Spherically-Symmetric
Metrics

Given an arbitrary spherically-symmetric and static metric $\hat{g}_{\mu\nu}$
in arbitrary spherical-polar coordinates of the form (1),

$\mathrm{d}s^{2}=\
-f(r)\mathrm{d}t^{2}+\frac{g(r)}{f(r)}\mathrm{d}r^{2}+R^{2}(r)\mathrm{d}\Omega_{2}^{2}\,,$
(A1)

we can put it into the form where $g(r)=1$ as,

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=\
-f(r)\mathrm{d}t^{2}+\frac{g(r)}{f(r)}\left(\partial_{\rho}r\right)^{2}\mathrm{d}\rho^{2}+R^{2}(r)\mathrm{d}\Omega_{2}^{2}\,,$
$\displaystyle=\
-f(\rho)\mathrm{d}t^{2}+\frac{\mathrm{d}\rho^{2}}{f(\rho)}+R^{2}(\rho)\mathrm{d}\Omega_{2}^{2}\,,$
(A2)

where in writing the last equality we have imposed the condition that
$r(\rho)$ satisfy the ordinary differential equation,

$\sqrt{g(r)}~{}\partial_{\rho}r=1\,,$ (A3)

and $f(\rho)=f(r(\rho))$. Note that the above equation can be rewritten as

$\sqrt{-\hat{g}_{tt}(r)\hat{g}_{rr}(r)}~{}\partial_{\rho}r=1\,.$ (A4)

Relabeling $\rho$ by $r$ in eq. A, we have,

$\mathrm{d}s^{2}=\
-f(r)\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{f(r)}+R^{2}(r)\mathrm{d}\Omega_{2}^{2}\,.$
(A5)

We now demonstrate how we can straightforwardly obtain the JMN-1 naked
singularity metric in the form given above (A5), where
$-\hat{g}_{tt}\hat{g}_{rr}=1$. Its original form, as given in eq. 29 of Joshi
et al. (2011), uses areal-radial coordinates,
$x^{\mu}=(t,R,\vartheta,\varphi)$, in which $-\hat{g}_{tt}\hat{g}_{RR}\neq 1$.
From the original form,

$\mathrm{d}s^{2}=\
-\left(1-M_{0}\right)\left(\frac{R}{R_{\mathrm{b}}}\right)^{\frac{M_{0}}{1-M_{0}}}\mathrm{d}t^{2}+\frac{\mathrm{d}R^{2}}{\left(1-M_{0}\right)}+R^{2}~{}\mathrm{d}\Omega_{2}^{2}\,,$

where the compactness parameter $M_{0}$ is given in terms of the ADM mass $M$
and the physical boundary or matching areal-radius $R_{\mathrm{b}}$ as
$M_{0}=2M/R_{\mathrm{b}}$, we can write the desired equation (A4) to put it in
the “$g(r)=1$” coordinates (A5) used in Table 1 as,

$\displaystyle\left(\frac{R}{R_{\mathrm{b}}}\right)^{\alpha}\mathrm{d}R=\mathrm{d}r\,,$
(A6)

where $\alpha=M_{0}/(2-2M_{0})$, not to be confused with the lapse function,
which yields a solution,

$\frac{R_{\mathrm{b}}}{\alpha+1}\left(\frac{R}{R_{\mathrm{b}}}\right)^{\alpha+1}=r+k\,.$
(A7)

The integration constant $k$ above can be set to zero so that at $R=0$ we also
have $r=0$. The above equation is easily invertible as,

$R(r)=R_{\mathrm{b}}\left[\left(\frac{\alpha+1}{R_{\mathrm{b}}}\right)r\right]^{\frac{1}{\alpha+1}}\,.$
(A8)

Now, with $M_{0}=2\alpha/(1+2\alpha)$, and

$\partial_{r}R(r)=\left[\left(\frac{\alpha+1}{R_{\mathrm{b}}}\right)r\right]^{\frac{-\alpha}{\alpha+1}}\,,$
(A9)

we can rewrite the metric given in eq. A as,

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=\
-\left(\frac{1}{1+2\alpha}\right)\left[\left(\frac{\alpha+1}{R_{\mathrm{b}}}\right)r\right]^{\frac{2\alpha}{\alpha+1}}\mathrm{d}t^{2}$
(A10) $\displaystyle\
+\left(1+2\alpha\right)\left[\left(\frac{\alpha+1}{R_{\mathrm{b}}}\right)r\right]^{\frac{-2\alpha}{\alpha+1}}\mathrm{d}r^{2}$
$\displaystyle\
+R_{\mathrm{b}}^{2}\left[\left(\frac{\alpha+1}{R_{\mathrm{b}}}\right)r\right]^{\frac{2}{\alpha+1}}~{}\mathrm{d}\Omega_{2}^{2}\,.$

Introducing $\sigma$ and $r_{\mathrm{b}}$,

$\displaystyle\sigma$ $\displaystyle=\
\frac{\alpha}{\alpha+1}=\frac{M}{R_{\mathrm{b}}-M}\,,$ (A11) $\displaystyle
r_{\mathrm{b}}$ $\displaystyle=\
(1-\sigma)R_{\mathrm{b}}=\left(\frac{R_{\mathrm{b}}-2M}{R_{\mathrm{b}}-M}\right)R_{\mathrm{b}}\,,$

so that, $\alpha=\sigma/(1-\sigma)$, we can simplify eq. A10 and write
instead,

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=\
-\left(1-\frac{2M}{R_{\mathrm{b}}}\right)\left(\frac{r}{r_{\mathrm{b}}}\right)^{2\sigma}\mathrm{d}t^{2}$
(A12) $\displaystyle\
+\left(1-\frac{2M}{R_{\mathrm{b}}}\right)^{-1}\left(\frac{r}{r_{\mathrm{b}}}\right)^{-2\sigma}\mathrm{d}r^{2}$
$\displaystyle\
+R_{\mathrm{b}}^{2}\left(\frac{r}{r_{\mathrm{b}}}\right)^{2-2\sigma}\mathrm{d}\Omega_{2}^{2}\,.$

This is the form of the JMN-1 metric reported in Table 1. We note that while
inverting $r(R)$, given in eq. A7, to $R(r)$, given in eq. A8, was trivial in
this case, this step might not be analytically feasible in general (assuming
eq. A4 admits a closed-form solution in the first place). Therefore, it is
convenient to use three metric functions, $f,g,R$, in general, as in eq. 1.

##  Appendix B A Subclass of Asymptotically-Flat Stationary Metrics

Since we are focused here on asymptotically-flat spacetimes (see, e.g., Adamo
et al. 2009), let us note first that for the seed metric (1) to be
asymptotically-flat, its metric functions must become $R(r)=r$, $f(r)=1$, and
$g(r)=1$ in the limit $r\rightarrow\infty$. Then, if we define
$\tilde{\eta}_{\bar{\mu}\bar{\nu}}$ as,

$\tilde{\eta}_{\bar{\mu}\bar{\nu}}=\begin{bmatrix}-1&0&0&0\\\
0&1&0&-a\sin^{2}{\vartheta}\\\ 0&0&A+a^{2}\cos^{2}{\vartheta}&0\\\
0&-a\sin^{2}{\vartheta}&0&(A+a^{2})\sin^{2}{\vartheta}\end{bmatrix}\,,$ (B1)

we will find that as $r\rightarrow\infty$, the Riemann tensor associated with
$\tilde{\eta}$ vanishes identically for such asymptotically-flat seeds. Thus,
asymptotically, $\tilde{\eta}$ is the flat Minkowski metric $\eta$, in
disguise, i.e., $\lim_{r\rightarrow\infty}\tilde{\eta}=\eta$. We stress that
the Riemann tensor does not vanish in general at finite coordinate radii $r$
for the metric tensor $\tilde{\eta}$.

Now, the AA metric in the siKS coordinates (13) can be expressed in terms of
$\tilde{\eta}$ and the tangent to the ingoing PNC $\ell_{-}$ (10) everywhere
as,

$g_{\bar{\mu}\bar{\nu}}=\frac{X}{\Sigma}\left[\tilde{\eta}_{\bar{\mu}\bar{\nu}}+\frac{2F}{\Sigma}(\ell_{-})_{\bar{\mu}}(\ell_{-})_{\bar{\nu}}\right]\,.$
(B2)

We note further that $\ell_{-}$ is null with respect to both $g$ and
$\tilde{\eta}$. We refer to the form of the metric above as the generalized
(spherical ingoing) Kerr-Schild form of the AA metric (see Sec. 32.5 of
Stephani et al. 2009 for related discussion). This can be seen to be
conformally related to the “classic” Kerr-Schild form, as given in eq. 1.1 of
Kerr & Schild (2009).

Therefore, for the AA metric itself to be asymptotically-flat, we require (a)
$\lim_{r\rightarrow\infty}(2F/\Sigma)=0$, and (b)
$\lim_{r\rightarrow\infty}(X/\Sigma)=1$. The first of these conditions is met
due to the properties of the asymptotically-flat seed metric functions

$\displaystyle\lim_{r\rightarrow\infty}\frac{2F}{\Sigma}=\lim_{r\rightarrow\infty}\frac{(1/\sqrt{g}-f/g)R^{2}}{(f/g)R^{2}+a^{2}\cos^{2}{\vartheta}}=0\,,$
(B3)

whereas the second is trivially met when $X=\Sigma$. Thus, asymptotically-flat
seed metrics admit asymptotically-flat stationary generalizations when
$X=\Sigma$. Note that this latter condition ($X=\Sigma$) is a sufficient but
not a necessary condition. It is instructive to compare the asymptotically
Minkowski metric in our siKS coordinates (B1) with the analogous form of the
Minkowski metric in the original (outgoing) Kerr-Schild coordinates for the
Kerr metric, as given, e.g., in equations 1.7 and 1.13 of Wiltshire et al.
(2009).

##  Appendix C Separability of the geodesic equation for the Stationary metric

The Lagrangian $\mathscr{L}$ describing a geodesic orbit $x^{\mu}(\lambda)$ is
given as $2\mathscr{L}=u_{\mu}u^{\mu}$, where
$u^{\mu}=\mathrm{d}x^{\mu}/\mathrm{d}\lambda=\dot{x}^{\mu}$ is the four-
velocity along the geodesic and $\lambda$ is an affine parameter along it.
Working now in Boyer-Lindquist coordinates (3), due to the two Killing
symmetries of the spacetime that are generated by $T=\partial_{t}$ and
$\Phi=\partial_{\varphi}$, we can find momenta,
$p_{\mu}=\partial_{\dot{x}^{\mu}}\mathscr{L}=\partial_{u^{\mu}}\mathscr{L}=u_{\mu}$,
that are conserved, corresponding to the two cyclic variables. These are
identified as being the energy $E=-u_{\mu}T^{\mu}$ and the azimuthal angular
momentum $L=u_{\mu}\Phi^{\mu}$ of the orbit respectively,333For $K$ a Killing
vector,
$u^{\mu}\nabla_{\mu}(u^{\nu}K_{\nu})=(u^{\mu}\nabla_{\mu}u^{\nu})K_{\nu}+u^{\mu}u^{\nu}(\nabla_{\mu}K_{\nu})=0$.
The first and the second terms vanish due to the geodesic equation and the
Killing equation respectively. which can be used to obtain

$\displaystyle\frac{\dot{t}}{E}$ $\displaystyle=\
-\frac{X}{\Sigma^{2}\det{[g_{t\varphi}]}}\left[\Pi-2Fa\xi\right]\sin^{2}{\vartheta}\,,$
(C1) $\displaystyle\frac{\dot{\varphi}}{E}$ $\displaystyle=\
-\frac{X}{\Sigma^{2}\det{[g_{t\varphi}]}}\left[2Fa\sin^{2}{\vartheta}+(\Sigma-2F)\xi\right]\,,$

where we have introduced the first impact parameter, $\xi:=L/E$, and the
determinant of the $t\varphi-$sector of the AA metric tensor in BL coordinates
(3),
$\det{[g_{t\varphi}]}:=g_{tt}g_{\varphi\varphi}-g_{t\varphi}^{2}{\color[rgb]{0,0,0}=-(X^{2}/\Sigma^{2})\Delta\sin^{2}{\vartheta}}$.

Now with eq. C1 we can write,

$\displaystyle\frac{2\mathscr{L}X}{E^{2}}\Delta=$ $\displaystyle\
\left[X^{2}\frac{\dot{r}^{2}}{E^{2}}-\left(\Delta+2F-a\xi\right)^{2}\right]$
(C2) $\displaystyle\
+\Delta\left[X^{2}\frac{\dot{\vartheta}^{2}}{E^{2}}+\left(a\sin{\vartheta}-\xi\csc{\vartheta}\right)^{2}\right]\,.$

It is easy to see then that the geodesic equation for null geodesics
($2\mathscr{L}=0$) is fully separable,

$\displaystyle X^{2}\frac{\dot{\vartheta}^{2}}{E^{2}}=$ $\displaystyle\
\eta^{2}-\left(a\sin{\vartheta}-\xi\csc{\vartheta}\right)^{2}=:\Theta_{0}(\vartheta)\,,$
(C3) $\displaystyle X^{2}\frac{\dot{r}^{2}}{E^{2}}=$ $\displaystyle\
\left(\Delta+2F-a\xi\right)^{2}-\Delta\eta^{2}=:\mathscr{R}_{0}(r)\,,$ (C4)

where we have introduced a separation constant $\eta^{2}$ which is related to
the Carter constant $C$ through $\eta^{2}=C/E^{2}$. The Carter constant in
turn can be used to demonstrate the existence of a Killing-Yano tensor and an
associated hidden symmetry of the motion.

The fundamental principal null congruences (PNCs) of the AA spacetime consist
of null geodesics that satisfy $\dot{\vartheta}=0$ and $\ddot{\vartheta}=0$
(see, e.g., Misner et al. 1973, Hioki & Miyamoto 2008; see also Sec. 2.3 of
Adamo et al. 2009). This is equivalent to requiring that $\Theta_{0}=0$ and
$\partial_{\vartheta}\Theta_{0}=0$, which yields a solution $\eta=0$ and
$\xi=a\sin^{2}{\vartheta}$. On a related note, the Newman & Penrose (1962)
complex null tetrad adapted to the outgoing ($\dot{r}>0$) PNC for the AA
metric can be found in Azreg-Aïnou (2014a).

If $X(r,\vartheta)$ is of the form
$X(r,\vartheta)=X_{r}(r)+X_{\vartheta}(\vartheta)$, then the geodesic equation
is separable even for non-null orbits,

$\displaystyle X^{2}\frac{\dot{\vartheta}^{2}}{E^{2}}=$ $\displaystyle\
\eta^{2}-\left(a\sin{\vartheta}-\xi\csc{\vartheta}\right)^{2}+\frac{2\mathscr{L}}{E^{2}}X_{\vartheta}=:\Theta(\vartheta)\,,$
$\displaystyle X^{2}\frac{\dot{r}^{2}}{E^{2}}=$ $\displaystyle\
\left(\Delta+2F-a\xi\right)^{2}-\Delta\eta^{2}+\frac{2\mathscr{L}}{E^{2}}\Delta
X_{r}=:\mathscr{R}(r)\,.$ (C5)

For the special case when $X=\Sigma$ these are given simply as $X_{r}(r)=A(r)$
and $X_{\vartheta}(\vartheta)=a^{2}\cos^{2}{\vartheta}$.

## References

  * Adamo et al. (2009) Adamo, T. M., Kozameh, C., & Newman, E. T. 2009, Living Reviews in Relativity, 12, 6, doi: 10.12942/lrr-2009-6
  * Alcubierre (2008) Alcubierre, M. 2008, Introduction to 3+1 Numerical Relativity
  * Andalman et al. (2022) Andalman, Z. L., Liska, M. T. P., Tchekhovskoy, A., Coughlin, E. R., & Stone, N. 2022, MNRAS, 510, 1627, doi: 10.1093/mnras/stab3444
  * Arnowitt et al. (2008) Arnowitt, R., Deser, S., & Misner, C. W. 2008, Gen. Rel. Grav., 40, 1997, doi: 10.1007/s10714-008-0661-1
  * Ayón-Beato & García (1998) Ayón-Beato, E., & García, A. 1998, Phys. Rev. Lett., 80, 5056, doi: 10.1103/PhysRevLett.80.5056
  * Ayzenberg (2022) Ayzenberg, D. 2022, Class. Quantum Grav., 39, 105009, doi: 10.1088/1361-6382/ac655d
  * Azreg-Aïnou (2014a) Azreg-Aïnou, M. 2014a, Eur. Phys. J. C, 74, 2865, doi: 10.1140/epjc/s10052-014-2865-8
  * Azreg-Aïnou (2014b) —. 2014b, Phys. Rev. D, 90, 064041, doi: 10.1103/PhysRevD.90.064041
  * Bambi & Modesto (2013) Bambi, C., & Modesto, L. 2013, Phys. Lett. B, 721, 329, doi: 10.1016/j.physletb.2013.03.025
  * Banyuls et al. (1997) Banyuls, F., Font, J. A., Ibáñez, J. M., Martí, J. M., & Miralles, J. A. 1997, ApJ, 476, 221, doi: 10.1086/303604
  * Bardeen (1968) Bardeen, J. M. 1968, Proc. Int. Conf. GR5, Tbilisi, U.S.S.R.
  * Bardeen et al. (1972) Bardeen, J. M., Press, W. H., & Teukolsky, S. A. 1972, ApJ, 178, 347, doi: 10.1086/151796
  * Bauer et al. (2022) Bauer, A. M., Cárdenas-Avendaño, A., Gammie, C. F., & Yunes, N. 2022, ApJ, 925, 119, doi: 10.3847/1538-4357/ac3a03
  * Begelman et al. (2022) Begelman, M. C., Scepi, N., & Dexter, J. 2022, MNRAS, 511, 2040, doi: 10.1093/mnras/stab3790
  * Blau (2023) Blau, M. 2023, Lecture Notes on General Relativity.  http://www.blau.itp.unibe.ch/newlecturesGR.pdf
  * Boyer & Lindquist (1967) Boyer, R. H., & Lindquist, R. W. 1967, J. Math. Phys., 8, 265, doi: 10.1063/1.1705193
  * Broderick et al. (2014) Broderick, A. E., Johannsen, T., Loeb, A., & Psaltis, D. 2014, ApJ, 784, 7, doi: 10.1088/0004-637X/784/1/7
  * Bronnikov (2001) Bronnikov, K. A. 2001, Phys. Rev. D, 63, 044005, doi: 10.1103/PhysRevD.63.044005
  * Cardenas-Avendano et al. (2020) Cardenas-Avendano, A., Nampalliwar, S., & Yunes, N. 2020, Class. Quant. Grav., 37, 135008, doi: 10.1088/1361-6382/ab8f64
  * Carson & Yagi (2020) Carson, Z., & Yagi, K. 2020, Phys. Rev. D, 101, 084030, doi: 10.1103/PhysRevD.101.084030
  * Carter (1966) Carter, B. 1966, Phys. Rev., 141, 1242, doi: 10.1103/PhysRev.141.1242
  * Carter (1971) Carter, B. 1971, Phys. Rev. Lett., 26, 331, doi: 10.1103/PhysRevLett.26.331
  * Chael et al. (2023) Chael, A., Lupsasca, A., Wong, G. N., & Quataert, E. 2023, arXiv e-prints, arXiv:2307.06372, doi: 10.48550/arXiv.2307.06372
  * Chael et al. (2019) Chael, A., Narayan, R., & Johnson, M. D. 2019, MNRAS, 486, 2873, doi: 10.1093/mnras/stz988
  * Chatterjee et al. (in prep.a) Chatterjee, K., Kocherlakota, P., Younsi, Z., & Narayan, R. in prep.a
  * Chatterjee et al. (2019) Chatterjee, K., Liska, M., Tchekhovskoy, A., & Markoff, S. B. 2019, MNRAS, 490, 2200, doi: 10.1093/mnras/stz2626
  * Chatterjee & Narayan (2022) Chatterjee, K., & Narayan, R. 2022, ApJ, 941, 30, doi: 10.3847/1538-4357/ac9d97
  * Chatterjee et al. (in prep.b) Chatterjee, K., Younsi, Z., Kocherlakota, P., & Narayan, R. in prep.b
  * Chatterjee et al. (2020) Chatterjee, K., Younsi, Z., Liska, M., et al. 2020, MNRAS, 499, 362, doi: 10.1093/mnras/staa2718
  * Chatterjee et al. (2021) Chatterjee, K., Markoff, S., Neilsen, J., et al. 2021, MNRAS, 507, 5281, doi: 10.1093/mnras/stab2466
  * Cruz-Osorio et al. (2022) Cruz-Osorio, A., Fromm, C. M., Mizuno, Y., et al. 2022, Nature Astronomy, 6, 103, doi: 10.1038/s41550-021-01506-w
  * Curd & Narayan (2019) Curd, B., & Narayan, R. 2019, MNRAS, 483, 565, doi: 10.1093/mnras/sty3134
  * Davelaar et al. (2018) Davelaar, J., Mościbrodzka, M., Bronzwaer, T., & Falcke, H. 2018, A&A, 612, A34, doi: 10.1051/0004-6361/201732025
  * De Villiers et al. (2003) De Villiers, J.-P., Hawley, J. F., & Krolik, J. H. 2003, ApJ, 599, 1238, doi: 10.1086/379509
  * Delaporte et al. (2022) Delaporte, H., Eichhorn, A., & Held, A. 2022, Class. Quantum Grav., 39, 134002, doi: 10.1088/1361-6382/ac7027
  * Dexter et al. (2012) Dexter, J., McKinney, J. C., & Agol, E. 2012, MNRAS, 421, 1517, doi: 10.1111/j.1365-2966.2012.20409.x
  * Dymnikova (2004) Dymnikova, I. 2004, Class. Quantum Grav., 21, 4417, doi: 10.1088/0264-9381/21/18/009
  * Eddington (1924) Eddington, A. S. 1924, Nature, 113, 192, doi: 10.1038/113192a0
  * Erbin (2017) Erbin, H. 2017, Universe, 3, 19, doi: 10.3390/universe3010019
  * Finkelstein (1958) Finkelstein, D. 1958, Phys. Rev., 110, 965, doi: 10.1103/PhysRev.110.965
  * Font et al. (1998) Font, J. A., Ibáñez, J. M., & Papadopoulos, P. 1998, ApJL, 507, L67, doi: 10.1086/311666
  * Fragile et al. (2007) Fragile, P. C., Blaes, O. M., Anninos, P., & Salmonson, J. D. 2007, ApJ, 668, 417, doi: 10.1086/521092
  * Fromm et al. (2021) Fromm, C. M., Mizuno, Y., Younsi, Z., et al. 2021, A&A, 649, A116, doi: 10.1051/0004-6361/201937335
  * Gammie et al. (2003) Gammie, C. F., McKinney, J. C., & Tóth, G. 2003, ApJ, 589, 444, doi: 10.1086/374594
  * Garfinkle et al. (1991) Garfinkle, D., Horowitz, G. T., & Strominger, A. 1991, PhRvD, 43, 3140, doi: 10.1103/PhysRevD.43.3140
  * Gibbons & Maeda (1988) Gibbons, G. W., & Maeda, K.-i. 1988, Nucl. Phys. B, 298, 741, doi: 10.1016/0550-3213(88)90006-5
  * Gottlieb et al. (2022) Gottlieb, O., Lalakos, A., Bromberg, O., Liska, M., & Tchekhovskoy, A. 2022, MNRAS, 510, 4962, doi: 10.1093/mnras/stab3784
  * Gourgoulhon (2007) Gourgoulhon, E. 2007, arXiv e-prints, gr, doi: 10.48550/arXiv.gr-qc/0703035
  * Gourgoulhon (2010) —. 2010, arXiv e-prints, arXiv:1003.5015, doi: 10.48550/arXiv.1003.5015
  * Hayward (2006) Hayward, S. A. 2006, Phys. Rev. Lett., 96, 031103, doi: 10.1103/PhysRevLett.96.031103
  * Hioki & Miyamoto (2008) Hioki, K., & Miyamoto, U. 2008, Phys. Rev. D, 78, 044007, doi: 10.1103/PhysRevD.78.044007
  * Janis et al. (1968) Janis, A. I., Newman, E. T., & Winicour, J. 1968, Phys. Rev. Lett., 20, 878, doi: 10.1103/PhysRevLett.20.878
  * Johannsen (2013a) Johannsen, T. 2013a, Phys. Rev. D, 88, 044002, doi: 10.1103/PhysRevD.88.044002
  * Johannsen (2013b) —. 2013b, Phys. Rev. D, 87, 124017, doi: 10.1103/PhysRevD.87.124017
  * Johannsen & Psaltis (2010) Johannsen, T., & Psaltis, D. 2010, ApJ, 716, 187, doi: 10.1088/0004-637X/716/1/187
  * Johannsen & Psaltis (2011) —. 2011, Phys. Rev. D, 83, 124015, doi: 10.1103/PhysRevD.83.124015
  * Joshi et al. (2011) Joshi, P. S., Malafarina, D., & Narayan, R. 2011, Class. Quantum Grav., 28, 235018, doi: 10.1088/0264-9381/28/23/235018
  * Kar (1997) Kar, S. 1997, Phys. Rev. D, 55, 4872, doi: 10.1103/PhysRevD.55.4872
  * Kerr (1963) Kerr, R. P. 1963, PhRvL, 11, 237, doi: 10.1103/PhysRevLett.11.237
  * Kerr & Schild (2009) Kerr, R. P., & Schild, A. 2009, Gen. Rel. Grav., 41, 2485, doi: 10.1007/s10714-009-0857-z
  * Kocherlakota & Rezzolla (2020) Kocherlakota, P., & Rezzolla, L. 2020, Phys. Rev. D, 102, 064058, doi: 10.1103/PhysRevD.102.064058
  * Kocherlakota & Rezzolla (2022) —. 2022, MNRAS, 513, 1229, doi: 10.1093/mnras/stac891
  * Kocherlakota et al. (2023) Kocherlakota, P., Rezzolla, L., Roy, R., & Wielgus, M. 2023, arXiv e-prints, arXiv:2307.16841, doi: 10.48550/arXiv.2307.16841
  * Kocherlakota et al. (2021) Kocherlakota, P., Rezzolla, L., et al. 2021, PhRvD, 103, 104047, doi: 10.1103/PhysRevD.103.104047
  * Koide et al. (1999) Koide, S., Shibata, K., & Kudoh, T. 1999, ApJ, 522, 727, doi: 10.1086/307667
  * Konoplya et al. (2016) Konoplya, R., Rezzolla, L., & Zhidenko, A. 2016, Phys. Rev. D, 93, 064015, doi: 10.1103/PhysRevD.93.064015
  * Konoplya & Zhidenko (2021) Konoplya, R. A., & Zhidenko, A. 2021, Phys. Rev. D, 103, 104033, doi: 10.1103/PhysRevD.103.104033
  * Kruskal (1960) Kruskal, M. D. 1960, Phys. Rev., 119, 1743, doi: 10.1103/PhysRev.119.1743
  * Lewis (1932) Lewis, T. 1932, Proceedings of the Royal Society of London Series A, 136, 176, doi: 10.1098/rspa.1932.0073
  * Liska et al. (2018) Liska, M., Hesp, C., Tchekhovskoy, A., et al. 2018, MNRAS, 474, L81, doi: 10.1093/mnrasl/slx174
  * Liska et al. (2022) Liska, M. T. P., Chatterjee, K., Issa, D., et al. 2022, ApJS, 263, 26, doi: 10.3847/1538-4365/ac9966
  * McKinney (2006) McKinney, J. C. 2006, MNRAS, 368, 1561, doi: 10.1111/j.1365-2966.2006.10256.x
  * McKinney & Gammie (2004) McKinney, J. C., & Gammie, C. F. 2004, ApJ, 611, 977, doi: 10.1086/422244
  * Misner et al. (1973) Misner, C. W., Thorne, K. S., & Wheeler, J. A. 1973, Gravitation (Princeton Univ. Pr.)
  * Mizuno et al. (2018) Mizuno, Y., Younsi, Z., Fromm, C. M., et al. 2018, Nature Astron., 2, 585, doi: 10.1038/s41550-018-0449-5
  * Mościbrodzka et al. (2009) Mościbrodzka, M., Gammie, C. F., Dolence, J. C., Shiokawa, H., & Leung, P. K. 2009, ApJ, 706, 497, doi: 10.1088/0004-637X/706/1/497
  * Nampalliwar et al. (2022) Nampalliwar, S., Yfantis, A. I., & Kokkotas, K. D. 2022, Phys. Rev. D, 106, 063009, doi: 10.1103/PhysRevD.106.063009
  * Narayan et al. (2022) Narayan, R., Chael, A., Chatterjee, K., Ricarte, A., & Curd, B. 2022, MNRAS, 511, 3795, doi: 10.1093/mnras/stac285
  * Narayan et al. (2019) Narayan, R., Johnson, M. D., & Gammie, C. F. 2019, ApJ, 885, L33, doi: 10.3847/2041-8213/ab518c
  * Narayan et al. (2012) Narayan, R., Sadowski, A., Penna, R. F., & Kulkarni, A. K. 2012, MNRAS, 426, 3241, doi: 10.1111/j.1365-2966.2012.22002.x
  * Newman & Penrose (1962) Newman, E., & Penrose, R. 1962, J. Math. Phys., 3, 566, doi: 10.1063/1.1724257
  * Newman et al. (1965) Newman, E. T., Couch, E., Chinnapared, K., et al. 1965, J. Math. Phys., 6, 918, doi: 10.1063/1.1704351
  * Newman & Janis (1965) Newman, E. T., & Janis, A. I. 1965, J. Math. Phys., 6, 915, doi: 10.1063/1.1704350
  * Olivares et al. (2020) Olivares, H., Younsi, Z., Fromm, C. M., et al. 2020, MNRAS, 497, 521, doi: 10.1093/mnras/staa1878
  * Özel et al. (2022) Özel, F., Psaltis, D., & Younsi, Z. 2022, ApJ, 941, 88, doi: 10.3847/1538-4357/ac9fcb
  * Papapetrou (1966) Papapetrou, A. 1966, Annales de L’Institut Henri Poincare Section (A) Physique Theorique, 4, 83
  * Parfrey & Tchekhovskoy (2017) Parfrey, K., & Tchekhovskoy, A. 2017, ApJ, 851, L34, doi: 10.3847/2041-8213/aa9c85
  * Paul et al. (2020) Paul, S., Shaikh, R., Banerjee, P., & Sarkar, T. 2020, J. Cosmology Astropart. Phys, 2020, 055, doi: 10.1088/1475-7516/2020/03/055
  * Penrose (1963) Penrose, R. 1963, Phys. Rev. Lett., 10, 66, doi: 10.1103/PhysRevLett.10.66
  * Plebanski & Krasinski (2012) Plebanski, J., & Krasinski, A. 2012, An Introduction to General Relativity and Cosmology (Cambridge Univ. Pr.)
  * Poisson (2004) Poisson, E. 2004, A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics (Cambridge Univ. Pr.)
  * Porth et al. (2021) Porth, O., Mizuno, Y., Younsi, Z., & Fromm, C. M. 2021, MNRAS, 502, 2023, doi: 10.1093/mnras/stab163
  * Porth et al. (2017) Porth, O., Olivares, H., Mizuno, Y., et al. 2017, Computational Astrophysics and Cosmology, 4, 1, doi: 10.1186/s40668-017-0020-2
  * Porth et al. (2019) Porth, O., Chatterjee, K., Narayan, R., et al. 2019, ApJS, 243, 26, doi: 10.3847/1538-4365/ab29fd
  * Psaltis et al. (2020) Psaltis, D., Medeiros, L., Christian, P., Özel, F., et al. 2020, PhRvL, 125, 141104, doi: 10.1103/PhysRevLett.125.141104
  * Rajan (2016) Rajan, D. 2016, arXiv e-prints, arXiv:1601.03862.  https://arxiv.org/abs/1601.03862
  * Ressler et al. (2023) Ressler, S. M., White, C. J., & Quataert, E. 2023, MNRAS, 521, 4277, doi: 10.1093/mnras/stad837
  * Rezzolla & Zanotti (2013) Rezzolla, L., & Zanotti, O. 2013, Relativistic Hydrodynamics (Oxford Univ. Pr.)
  * Rezzolla & Zhidenko (2014) Rezzolla, L., & Zhidenko, A. 2014, Phys. Rev. D, 90, 084009, doi: 10.1103/PhysRevD.90.084009
  * Ricarte et al. (2023) Ricarte, A., Narayan, R., & Curd, B. 2023, arXiv e-prints, arXiv:2307.04621, doi: 10.48550/arXiv.2307.04621
  * Ricarte et al. (2022) Ricarte, A., Palumbo, D. C. M., Narayan, R., Roelofs, F., & Emami, R. 2022, ApJ, 941, L12, doi: 10.3847/2041-8213/aca087
  * Ricarte et al. (2020) Ricarte, A., Prather, B. S., Wong, G. N., et al. 2020, MNRAS, 498, 5468, doi: 10.1093/mnras/staa2692
  * Ripperda et al. (2022) Ripperda, B., Liska, M., Chatterjee, K., et al. 2022, ApJ, 924, L32, doi: 10.3847/2041-8213/ac46a1
  * Röder et al. (2023) Röder, J., Cruz-Osorio, A., Fromm, C. M., et al. 2023, A&A, 671, A143, doi: 10.1051/0004-6361/202244866
  * Sadowski & Narayan (2015) Sadowski, A., & Narayan, R. 2015, MNRAS, 453, 3213, doi: 10.1093/mnras/stv1802
  * Sadowski et al. (2014) Sadowski, A., Narayan, R., McKinney, J. C., & Tchekhovskoy, A. 2014, MNRAS, 439, 503, doi: 10.1093/mnras/stt2479
  * Sen (1992) Sen, A. 1992, Phys. Rev. Lett., 69, 1006, doi: 10.1103/PhysRevLett.69.1006
  * Shaikh (2019) Shaikh, R. 2019, Phys. Rev. D, 100, 024028, doi: 10.1103/PhysRevD.100.024028
  * Shaikh & Joshi (2019) Shaikh, R., & Joshi, P. S. 2019, J. Cosmology Astropart. Phys, 2019, 064, doi: 10.1088/1475-7516/2019/10/064
  * Shaikh et al. (2019) Shaikh, R., Kocherlakota, P., Narayan, R., & Joshi, P. S. 2019, MNRAS, 482, 52, doi: 10.1093/mnras/sty2624
  * Solanki et al. (2022) Solanki, D. N., Bambhaniya, P., Dey, D., Joshi, P. S., & Pathak, K. N. 2022, Eur. Phys. J. C, 82, 77, doi: 10.1140/epjc/s10052-022-10045-1
  * Stephani et al. (2009) Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., & Herlt, E. 2009, Exact Solutions of Einstein’s Field Equations (Cambridge Univ. Pr.), doi: 10.1017/CBO9780511535185
  * Straumann (2013) Straumann, N. 2013, General Relativity, doi: 10.1007/978-94-007-5410-2
  * Tchekhovskoy et al. (2011) Tchekhovskoy, A., Narayan, R., & McKinney, J. C. 2011, MNRAS, 418, L79, doi: 10.1111/j.1745-3933.2011.01147.x
  * The EHT Collaboration et al. (2019a) The EHT Collaboration, et al. 2019a, ApJ, 875, L5, doi: 10.3847/2041-8213/ab0f43
  * The EHT Collaboration et al. (2019b) —. 2019b, ApJL, 875, L6, doi: 10.3847/2041-8213/ab1141
  * The EHT Collaboration et al. (2022a) —. 2022a, ApJ, 930, L16, doi: 10.3847/2041-8213/ac6672
  * The EHT Collaboration et al. (2022b) —. 2022b, ApJL, 930, L17, doi: 10.3847/2041-8213/ac6756
  * Vagnozzi et al. (2023) Vagnozzi, S., Roy, R., Tsai, Y.-D., et al. 2023, Class. Quant. Grav., 40, 165007, doi: 10.1088/1361-6382/acd97b
  * Vigeland et al. (2011) Vigeland, S., Yunes, N., & Stein, L. C. 2011, Phys. Rev. D, 83, 104027, doi: 10.1103/PhysRevD.83.104027
  * Virbhadra (1997) Virbhadra, K. S. 1997, Int. J. of Mod. Phys. A, 12, 4831, doi: 10.1142/S0217751X97002577
  * Völkel & Barausse (2020) Völkel, S. H., & Barausse, E. 2020, Phys. Rev. D, 102, 084025, doi: 10.1103/PhysRevD.102.084025
  * Völkel et al. (2021) Völkel, S. H., Barausse, E., Franchini, N., & Broderick, A. E. 2021, Class. Quant. Grav., 38, 21LT01, doi: 10.1088/1361-6382/ac27ed
  * Wald (1984) Wald, R. M. 1984, General Relativity (Chicago Univ. Pr.), doi: 10.7208/chicago/9780226870373.001.0001
  * Wiltshire et al. (2009) Wiltshire, D. L., Visser, M., & Scott, S. M. 2009, The Kerr spacetime: Rotating Black Holes in General Relativity (Cambridge Univ. Pr.)
  * Xavier et al. (2020) Xavier, S. V. M. C. B., Cunha, P. V. P., Crispino, L. C. B., & Herdeiro, C. A. R. 2020, Int. J. Mod. Phys. D, 29, 2041005, doi: 10.1142/S0218271820410059
  * Younsi et al. (2023) Younsi, Z., Psaltis, D., & Özel, F. 2023, ApJ, 942, 47, doi: 10.3847/1538-4357/aca58a
  * Zhou & Modesto (2023a) Zhou, T., & Modesto, L. 2023a, Phys. Rev. D, 107, 044016, doi: 10.1103/PhysRevD.107.044016
  * Zhou & Modesto (2023b) —. 2023b, arXiv e-prints, arXiv:2303.11322, doi: 10.48550/arXiv.2303.11322