# On well-posedness and algebraic type of the five-dimensional charged
rotating black hole with two equal-magnitude angular momenta

Markus B. Fröb  Institut für Theoretische Physik, Universität Leipzig,
Brüderstraße 16, 04103 Leipzig, Germany  Igor Khavkine  Institute of
Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67 Prague 1,
Czechia  Charles University, Faculty of Mathematics and Physics, Sokolovská
83, 186 75 Prague 8, Czechia  Tomáš Málek  Institute of Mathematics of the
Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czechia  Vojtěch Pravda
Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67
Prague 1, Czechia

###### Abstract

We study various mathematical aspects of the charged rotating black hole with
two equal-magnitude angular momenta in five dimensions. We introduce a
coordinate system that is regular on the horizon and in which Einstein–Maxwell
equations reduce to an autonomous system of ODEs. Employing Bondi and Kruskal-
like coordinates, we analyze the geometric regularity of the black hole metric
at infinity and the horizon, respectively, and the well-posedness of the
corresponding boundary value problem.

We also study the algebraic types of the electromagnetic and curvature
tensors. While outside the horizon the electromagnetic and Ricci tensors are
of type D, the Weyl tensor is algebraically general. The Weyl tensor
simplifies to type II on the horizon and type D on the bifurcation sphere.
These results imply inconsistency of the metric with the Kerr–Schild form with
a geodesic Kerr–Schild vector. This feature is shared by the four-dimensional
Kerr–Newman metric and the vacuum Myers–Perry or charged
Schwarzschild–Tangherlini geometries in arbitrary dimension, but hence not by
the black hole we have considered here.

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##  1 Introduction

Black holes represent the most basic objects of general relativity in four and
higher dimensions. It is relatively straightforward to find static,
spherically symmetric black hole solutions to vacuum Einstein or
Einstein–Maxwell equations. However, it is much more difficult to generalize
such black hole solutions to the rotating case and usually, an additional
simplifying assumption about the metric has to be employed on top of
stationarity and axial symmetry.

In particular, in constructing the Kerr metric in [16], the metric was assumed
to be of Petrov type D. For the construction of a vacuum higher-dimensional
rotating black hole metric [23], the essential assumption was that it can be
cast in the Kerr–Schild form, similarly as in the four-dimensional case:

$g_{\mu\nu}=\eta_{\mu\nu}-2\mathcal{H}k_{\mu}k_{\nu}\,,$ (1)

where $\eta_{\mu\nu}$ is the Minkowski metric, $\mathcal{H}$ a scalar function
and $k$ is a null vector with respect to $\eta_{\mu\nu}$ 111It follows that
$k$ is also null with respect to the full metric $g_{\mu\nu}$..

The above two assumptions—Petrov type D and the Kerr–Schild form—in fact also
hold for the four-dimensional charged rotating black hole described by the
Kerr–Newman metric [26, 25] and for charged Schwarzschild–Tangherlini black
holes [38] in arbitrary dimension. Furthermore, in these cases, the null
eigenvectors of the electromagnetic field $F$ coincide with the principal null
directions of the Weyl tensor [27]. Note also that for Kerr, Kerr–Newman,
Myers–Perry, and Schwarzschild–Tangherlini metrics, the Kerr–Schild vector $k$
is geodesic.

In contrast with the four-dimensional case, where the Kerr–Newman solution was
constructed in less than a year after the publication of the Kerr metric, the
exact solution representing a higher-dimensional charged rotating black hole
is now, three and a half decades after the publication of the Myers–Perry
solution [23], still unknown222Note that various approximate solutions, for
example for small charge [3, 24], slow rotation [2, 1] or large dimensions
[4], are known.. Being a 5-dimensional object, such a black hole is not of
direct relevance to the emerging field of gravitational wave astrophysics,
which has accompanied the development of LIGO, LISA and other gravitational
wave observatories. However, it may be of interest to more detailed future
phenomenological investigations of new physics scenarios like brane world
models, where 5-dimensional gravity interacts with matter confined to a
4-dimensional brane [11]. Independent of their phenomenological significance,
charged rotating black holes are a good theoretical testing ground for methods
of interpolating between known charged static and uncharged rotating
solutions, which besides higher dimensions may become useful also in modified
theories of gravity in 4 dimensions.

In this paper, we focus on the charged rotating black hole in five dimensions
with two equal-magnitude angular momenta, starting with the metric ansatz
previously employed in the numerical studies of this black hole, initiated in
[18, 19] and continued in [24, 9]. These works have demonstrated the existence
(or mathematically speaking, provided very strong evidence thereof) of a
solution to the Einstein–Maxwell equations with this ansatz, by numerically
solving an ordinary boundary value problem in the region between the horizon
and infinity. Their focus was on studying the numerical relationships between
various black hole parameters that can only be determined from the knowledge
of the global solution, rather than from only local approximations. For
instance, while the horizon radius, surface gravity, rotation speed, and
surface electric potential at the horizon are all quantities local to the
horizon, and while the total mass, angular momentum, charge, and gyromagnetic
ratio are local to infinity, definite relations between these two groups of
parameters must be globally determined. However, these previous works have not
discussed or have left implicit certain structural or algebraic properties of
these black holes. It is to these matters that we turn our attention in this
work.

Namely, we focus on two main aspects: the geometric regularity of the black
hole solution at infinity and at the horizon, and on the algebraic type of the
metric in the region between these two extremes. Indeed, we confirm that the
black hole metric has a regular extension across the horizon and that it
possesses a regular null infinity without incoming or outgoing radiation. On
the other hand, we also show that in the bulk the metric is _algebraically
general_ 333Here we refer to the generalization of the standard Petrov
classification of the Weyl tensor in four dimensions to other tensors and to
higher dimensions [8], [22] (see [29] for a review). _Algebraically general_
means that the Weyl tensor does not admit a multiple Weyl aligned null
direction (multiple WAND). In contrast, type II and D spacetime admit one and
(at least) two multiple WANDs respectively. and incompatible with Kerr–Schild
form (1) with geodesic $k$, so it could not have appeared in the lists of
exact solutions obtained under these simplifying assumptions. On the other
hand, employing Kruskal-like coordinates, we show that the metric does become
algebraically special at the horizon, in agreement with the geometric horizon
conjecture of [7] (cf. also [20]).

It is worth noting that while most of our results hold for equal-magnitude
angular momenta, the results on the algebraically general Weyl tensor and non-
compatibility of the metric with the Kerr–Schild form (1) with geodesic $k$
can be obviously extended to the case of generic angular momenta (see [18] for
the corresponding metric ansatz).

The paper is structured as follows: In Section 2, we recall the metric ansatz
from [19] and show why it is not optimal for studying the regularity of the
metric at the horizon by matching it to the known exact uncharged
(Myers–Perry) and non-rotating (charged Schwarzschild–Tangherlini) solutions.
In Section 3, we introduce a different radial coordinate and a modified metric
ansatz, which is better adapted to studying the regularity of the horizon. We
then obtain the corresponding autonomous system of Einstein–Maxwell equations
in the form of a constrained ordinary boundary value problem with singular
end-points. Then a careful comparison of the solution at the singular end-
points with the geometric regularity conditions at infinity and at the horizon
shows that the boundary value problem is indeed well-posed. This analysis also
suggests how to ensure numerical accuracy in future investigations of these
solutions.

In Section 4, we then study the simplifying assumptions holding for Kerr,
Kerr–Newman, Schwarzschild–Tangherlini, and Myers–Perry metrics and we show
that none of these hold for the five-dimensional charged rotating black hole
with two equal-magnitude angular momenta. We show that this black hole is
algebraically general. It also turns out that while the Ricci tensor is of
type D and aligned with a type D Maxwell tensor $F$, the geometric properties
of the common null aligned vector imply that the five-dimensional charged
rotating black hole is not compatible with the Kerr–Schild ansatz (1) with
geodesic $k$. In five dimensions the necessary condition for such
compatibility boils down to $\hat{Q}\hat{J}=0$ (see Equation (60)) which
explains why this ansatz holds only for rotating vacuum or charged non-
rotating five-dimensional black holes.

We should note that the previous work [9] also attempted to put the charged
rotating black hole metric into Kerr–Schild form (1), but they only succeeded
by replacing the reference Minkowski metric $\eta_{ab}$ by a different metric
with unclear algebraic or geometric properties. Moreover, the authors claim
that by using some differential identities, they have simplified the
corresponding Einstein–Maxwell equations to a system on only two unknown
functions, while also giving an asymptotic solution to the resulting
equations. Unfortunately, we have identified a mistake in a key identity that
allowed their simplification and have also found that their asymptotic
solution is not accurate. A more detailed discussion can be found in Appendix
A.

##  2 Metric ansatz

Specializing the metric and vector potential ansatz for equal-magnitude
angular momenta black holes in Einstein–Maxwell theory from [19] to five
dimensions, gives

$\mathop{}\\!\mathrm{d}s^{2}=g_{\mu\nu}\mathop{}\\!\mathrm{d}x^{\mu}\mathop{}\\!\mathrm{d}x^{\nu}=-f\mathop{}\\!\mathrm{d}t^{2}+\frac{m}{f}\left(\mathop{}\\!\mathrm{d}r^{2}+r^{2}\mathop{}\\!\mathrm{d}\theta^{2}\right)\\\
+\frac{n}{f}r^{2}\left[\sin^{2}\theta\left(\mathop{}\\!\mathrm{d}\phi-\frac{\omega}{r}\mathop{}\\!\mathrm{d}t\right)^{2}+\cos^{2}\theta\left(\mathop{}\\!\mathrm{d}\psi-\frac{\omega}{r}\mathop{}\\!\mathrm{d}t\right)^{2}\right]+\frac{m-n}{f}r^{2}\sin^{2}\theta\cos^{2}\theta\left(\mathop{}\\!\mathrm{d}\phi-\mathop{}\\!\mathrm{d}\psi\right)^{2}\,,$
(2)

and

$A=A_{\mu}\mathop{}\\!\mathrm{d}x^{\mu}=a_{0}\mathop{}\\!\mathrm{d}t+a_{\phi}\left(\sin^{2}\theta\mathop{}\\!\mathrm{d}\phi+\cos^{2}\theta\mathop{}\\!\mathrm{d}\psi\right)\,,$
(3)

where the six functions $f$, $m$, $n$, $\omega$, $a_{0}$ and $a_{\phi}$ only
depend on the radial coordinate $r$. Ordering the coordinates as
$t,r,\theta,\phi,\psi$, the metric in matrix form is

$g_{\mu\nu}=\frac{1}{f}\begin{pmatrix}-f^{2}+n\omega^{2}&0&0&-n\omega
r\sin^{2}\theta&-n\omega r\cos^{2}\theta\\\ 0&m&0&0&0\\\ 0&0&mr^{2}&0&0\\\
-n\omega
r\sin^{2}\theta&0&0&\left(m\cos^{2}\theta+n\sin^{2}\theta\right)r^{2}\sin^{2}\theta&(n-m)r^{2}\sin^{2}\theta\cos^{2}\theta\\\
-n\omega
r\cos^{2}\theta&0&0&(n-m)r^{2}\sin^{2}\theta\cos^{2}\theta&\left(m\sin^{2}\theta+n\cos^{2}\theta\right)r^{2}\cos^{2}\theta\end{pmatrix}\,,$

(4)

The outer horizon, located at $r=r_{\text{H}}$, is determined by the condition
$f(r_{\text{H}})=0$, with the value of $r_{\text{H}}$ closest to $r=\infty$.
Two sets of boundary conditions are imposed, one at infinity and one at the
horizon. At infinity ($r\to\infty$) we require asymptotic flatness, namely

$\displaystyle\mathop{}\\!\mathrm{d}s^{2}$
$\displaystyle\sim-\mathop{}\\!\mathrm{d}t^{2}+\mathop{}\\!\mathrm{d}r^{2}+r^{2}\mathop{}\\!\mathrm{d}\theta^{2}+r^{2}\sin^{2}\theta\,\mathop{}\\!\mathrm{d}\phi^{2}+r^{2}\cos^{2}\theta\,\mathop{}\\!\mathrm{d}\phi^{2}\,,$
(5) $\displaystyle A$ $\displaystyle\sim 0\,,$ (6)

where we have followed [19] by not specifying further details about what
higher dimensional asymptotic flatness means (we give a more detailed
definition of asymptotic flatness later in Section 3.1). In [19], the
corresponding boundary condition is stated as the following asymptotic limits:

$\displaystyle f,m,n\sim 1\,,$ (7) $\displaystyle\omega,a_{0},a_{\phi}\sim
0\,.$ (8)

At the horizon ($r=r_{\text{H}}$), we require regularity, which means the
existence of local coordinates in which the metric smoothly extends through
the horizon. In [19], the corresponding boundary condition is stated as
follows:

$\displaystyle f(r_{\text{H}}),m(r_{\text{H}}),n(r_{\text{H}})=0\,,$ (9)
$\displaystyle|\omega(r_{\text{H}})|,|a_{0}|,|a_{\phi}|<\infty\,,$ (10)
$\displaystyle a_{\phi}^{\prime}(r_{\text{H}})=0\,.$ (11)

Following [19, Sec. 3], provided the function $f$, $\omega$ and $a_{0}$ have
asymptotic expansions at spatial infinity ($r\to\infty$) of the form

$f=1-\frac{\hat{M}}{r^{2}}+\cdots\,,\quad\omega=\frac{\hat{J}}{r^{3}}+\cdots\,,\quad
a_{0}=\frac{\hat{Q}}{r^{2}}+\cdots\,,$ (12)

the physical mass, angular momentum and electric charges are respectively
given by

$\displaystyle M$
$\displaystyle=\frac{(d-2)A(S^{d-2})}{16G_{d}}\hat{M}=\frac{3\pi^{2}}{8G_{5}}\hat{M}\,,$
(13) $\displaystyle J$
$\displaystyle=\frac{A(S^{d-2})}{8G_{d}}\hat{J}=\frac{\pi^{2}}{4G_{5}}\hat{J}\,,$
$\displaystyle Q$
$\displaystyle=\frac{(d-3)A(S^{d-2})}{4G_{d}}\hat{Q}=\frac{\pi^{2}}{G_{5}}\hat{Q}\,,$

where $d=5$ is the spacetime dimension, $G_{d}$ is the $d$-dimensional
Newton’s constant, and $A(S^{d-2})=A(S^{3})=2\pi^{2}$ is the area of the unit
$3$-sphere. The physical charges $M$ and $J$ are obtained by integrating the
usual Komar forms at spatial infinity.

###  2.1 Special exact solutions

As we shall see in Section 3, the radial coordinate implied by the ansatz (2)
has non-differentiable behavior (a square root singularity) at the horizon
$r=r_{\text{H}}$. This phenomenon is most efficiently exhibited by matching
the ansatz to the known exact solutions without rotation (5-dimensional
charged Schwarzschild–Tangherlini metric) or without charge (5-dimensional
Myers–Perry metric), which we do next. Since we are interested in making the
comparison in the region exterior to the black hole, we restrict ourselves to
the subextremal ranges of the charge, mass, and angular momentum parameters of
the exact solutions.

Before moving on, let us introduce a new radial coordinate
$\mathsf{r}=\mathsf{r}(r)$, which will play the role of the more familiar
radial coordinate in these exact solutions that regularly extends through the
horizon. It so happens that in all cases considered in this paper, this new
radial coordinate will obey the relationship

$\frac{(\mathop{}\\!\mathrm{d}{r})^{2}}{r^{2}}=\frac{\mathsf{r}^{2}(\mathop{}\\!\mathrm{d}\mathsf{r})^{2}}{(\mathsf{r}^{2}-\mathsf{r}_{+}^{2})(\mathsf{r}^{2}-\mathsf{r}_{-}^{2})}\quad\iff\quad
r=r_{\text{H}}\sqrt{\frac{\sqrt{\mathsf{r}^{2}-\mathsf{r}_{-}^{2}}+\sqrt{\mathsf{r}^{2}-\mathsf{r}_{+}^{2}}}{\sqrt{\mathsf{r}^{2}-\mathsf{r}_{-}^{2}}-\sqrt{\mathsf{r}^{2}-\mathsf{r}_{+}^{2}}}}\,,$
(14)

where $\mathsf{r}_{\pm}$ are some constants and the differential relation has
been solved to match $\mathsf{r}(r_{\text{H}})=\mathsf{r}_{+}$. The square
root singularity is clearly visible at the horizon in the expansion
$r=r_{\text{H}}+\mathcal{O}\\!\left({\sqrt{\mathsf{r}-\mathsf{r}_{+}}}\right)$,
while at infinity $r=\mathcal{O}\\!\left({\mathsf{r}}\right)$ and has a
regular expansion in integer powers of $\mathsf{r}^{-1}$.

The 5-dimensional _charged Schwarzschild–Tangherlini_ (ST) solution (sometimes
also called the 5-dimensional _Reissner–Nordström_ solution) is given by

$\begin{gathered}\mathop{}\\!\mathrm{d}{s^{2}}=-f\mathop{}\\!\mathrm{d}{t}^{2}+\frac{\mathop{}\\!\mathrm{d}\mathsf{r}^{2}}{f}+\mathsf{r}^{2}(\mathop{}\\!\mathrm{d}\theta^{2}+\sin^{2}\theta\mathop{}\\!\mathrm{d}\phi^{2}+\cos^{2}\theta\mathop{}\\!\mathrm{d}\psi^{2})\,,\\\
A=\left(\frac{\hat{Q}}{\mathsf{r}^{2}}-\hat{C}\right)\mathop{}\\!\mathrm{d}{t}\,,\quad
f=1-\frac{\hat{M}}{\mathsf{r}^{2}}+\frac{4}{3}\frac{\hat{Q}^{2}}{\mathsf{r}^{4}}=\frac{(\mathsf{r}^{2}-\mathsf{r}_{+}^{2})(\mathsf{r}^{2}-\mathsf{r}_{-}^{2})}{\mathsf{r}^{4}}\,,\\\
\hat{M}=\mathsf{r}_{+}^{2}+\mathsf{r}_{-}^{2}\,,\quad\text{with}\quad\hat{Q}=\frac{\sqrt{3}}{2}\mathsf{r}_{+}\mathsf{r}_{-}\,,\quad\end{gathered}$
(15)

where the mass and charge are parametrized by the constants
$(\hat{M},\hat{Q})$ or $(\mathsf{r}_{+},\mathsf{r}_{-})$,
$\mathsf{r}_{+}>\mathsf{r}_{-}$. Matching the ST solution with the ansatz (2)
gives the radial coordinate relation (14) and

$\displaystyle f$
$\displaystyle=\frac{(\mathsf{r}^{2}-\mathsf{r}_{+}^{2})(\mathsf{r}^{2}-\mathsf{r}_{-}^{2})}{\mathsf{r}^{4}}\,,$
(16) $\displaystyle m$ $\displaystyle=\frac{\mathsf{r}^{2}}{r^{2}}f\,,$
$\displaystyle n$ $\displaystyle=\frac{\mathsf{r}^{2}}{r^{2}}f\,,$
$\displaystyle\omega$ $\displaystyle=0\,,$ $\displaystyle a_{0}$
$\displaystyle=\frac{\sqrt{3}}{2}\frac{\mathsf{r}_{+}\mathsf{r}_{-}}{\mathsf{r}^{2}}-\hat{C}\,,$
$\displaystyle a_{\phi}$ $\displaystyle=0\,.$

The 5-dimensional _Myers–Perry_ (MP) solution with equal-magnitude angular
momenta is given by

$\displaystyle\mathop{}\\!\mathrm{d}{s^{2}}$
$\displaystyle=-\mathop{}\\!\mathrm{d}{t}^{2}+\frac{\Sigma\mathsf{r}^{2}\mathop{}\\!\mathrm{d}\mathsf{r}^{2}}{\Sigma^{2}-\hat{M}\mathsf{r}^{2}}+\Sigma\mathsf{r}^{2}(\mathop{}\\!\mathrm{d}\theta^{2}+\sin^{2}\theta\mathop{}\\!\mathrm{d}\phi^{2}+\cos^{2}\theta\mathop{}\\!\mathrm{d}\psi^{2})$
(17)
$\displaystyle\qquad{}+\frac{\hat{M}}{\Sigma}\left(\mathop{}\\!\mathrm{d}{t}+a\sin^{2}\theta\mathop{}\\!\mathrm{d}\phi+a\cos^{2}\theta\mathop{}\\!\mathrm{d}\psi\right)^{2}\,,$
$\displaystyle A$
$\displaystyle=0,\quad\Sigma=\mathsf{r}^{2}+a^{2},\quad\Sigma^{2}-\hat{M}\mathsf{r}^{2}=(\mathsf{r}^{2}-\mathsf{r}_{+}^{2})(\mathsf{r}^{2}-\mathsf{r}_{-}^{2})\,,$
$\displaystyle\qquad\text{with}\quad\hat{M}=(\mathsf{r}_{+}+\mathsf{r}_{-})^{2},\quad
a=\hat{J}/\hat{M}=\sqrt{\mathsf{r}_{+}\mathsf{r}_{-}}\,,\quad$

where the mass and the angular momentum are parametrized by the constants
$(\hat{M},\hat{J})$ or $(\mathsf{r}_{+},\mathsf{r}_{-})$,
$\mathsf{r}_{+}>\mathsf{r}_{-}$. Matching the MP solution with the ansatz (2)
gives

$\displaystyle f$
$\displaystyle=\frac{(\mathsf{r}^{2}-\mathsf{r}_{+}^{2})(\mathsf{r}^{2}-\mathsf{r}_{-}^{2})}{(\mathsf{r}^{2}+\mathsf{r}_{-}\mathsf{r}_{+})^{2}+\mathsf{r}_{+}\mathsf{r}_{-}(\mathsf{r}_{-}+\mathsf{r}_{+})^{2}}=\frac{\Sigma^{2}-\hat{M}\mathsf{r}^{2}}{\Sigma^{2}+a^{2}\hat{M}}\,,$
(18) $\displaystyle m$
$\displaystyle=\frac{(\mathsf{r}^{2}+\mathsf{r}_{-}\mathsf{r}_{+})}{r^{2}}f=\Sigma
f\,,$ $\displaystyle n$
$\displaystyle=\frac{(\mathsf{r}^{2}-\mathsf{r}_{+}^{2})(\mathsf{r}^{2}-\mathsf{r}_{-}^{2})}{r^{2}(\mathsf{r}^{2}+\mathsf{r}_{+}\mathsf{r}_{-})}=\frac{\Sigma^{2}-\hat{M}\mathsf{r}^{2}}{\Sigma}\,,$
$\displaystyle\omega$
$\displaystyle=r\frac{\sqrt{\mathsf{r}_{+}\mathsf{r}_{-}}(\mathsf{r}_{+}+\mathsf{r}_{-})^{2}}{(\mathsf{r}^{2}+\mathsf{r}_{-}\mathsf{r}_{+})^{2}+\mathsf{r}_{+}\mathsf{r}_{-}(\mathsf{r}_{-}+\mathsf{r}_{+})^{2}}=\frac{a\hat{M}}{\Sigma^{2}+a^{2}\hat{M}}\,,$
$\displaystyle a_{0}$ $\displaystyle=0\,,$ $\displaystyle a_{\phi}$
$\displaystyle=0\,.$

##  3 Regularity at infinity and at the horizon

Since the radial coordinate $r$ used in the ansatz (2) has been found to be
non-smooth at the horizons of known exact solutions, we must reparametrize the
ansatz using our new regular coordinate $\mathsf{r}$. In addition, we have
found that removing most of the explicit dependence on $\mathsf{r}$ in the
metric makes the corresponding Einstein equations (given below) autonomous,
which is technically convenient.

Our new _regular_ ansatz for the metric and vector potential is

$\mathop{}\\!\mathrm{d}s^{2}=g_{\mu\nu}\mathop{}\\!\mathrm{d}x^{\mu}\mathop{}\\!\mathrm{d}x^{\nu}=-f\mathop{}\\!\mathrm{d}t^{2}+\frac{\mathsf{m}}{f}\left(\frac{(\mathsf{r}\mathop{}\\!\mathrm{d}\mathsf{r})^{2}}{\mathsf{N}}+\mathop{}\\!\mathrm{d}\theta^{2}\right)\\\
+\frac{\mathsf{N}}{\mathsf{m}}\left[\sin^{2}\theta\left(\mathop{}\\!\mathrm{d}\phi-\varpi\mathop{}\\!\mathrm{d}t\right)^{2}+\cos^{2}\theta\left(\mathop{}\\!\mathrm{d}\psi-\varpi\mathop{}\\!\mathrm{d}t\right)^{2}\right]+\frac{\mathsf{m}^{2}-f\mathsf{N}}{\mathsf{m}f}\sin^{2}\theta\cos^{2}\theta\left(\mathop{}\\!\mathrm{d}\phi-\mathop{}\\!\mathrm{d}\psi\right)^{2}$
(19)

and

$A=A_{\mu}\mathop{}\\!\mathrm{d}x^{\mu}=a_{0}\mathop{}\\!\mathrm{d}t+a_{\phi}\left(\sin^{2}\theta\mathop{}\\!\mathrm{d}\phi+\cos^{2}\theta\mathop{}\\!\mathrm{d}\psi\right)\,.$
(20)

The functions $f$, $\mathsf{m}$, $\mathsf{N}$, $a_{0}$ and $a_{\phi}$ all
depend only on the new radial coordinate $\mathsf{r}$. In matrix form,
ordering the coordinates as $(t,\mathsf{r},\theta,\phi,\psi)$, the metric
reads

$g_{\mu\nu}=\begin{pmatrix}-f+\frac{\mathsf{N}}{\mathsf{m}}\varpi^{2}&0&0&-\frac{\mathsf{N}}{\mathsf{m}}\varpi\sin^{2}\theta&-\frac{\mathsf{N}}{\mathsf{m}}\varpi\cos^{2}\theta\\\
0&\frac{\mathsf{r}^{2}\mathsf{m}}{f\mathsf{N}}&0&0&0\\\
0&0&\frac{\mathsf{m}}{f}&0&0\\\
-\frac{\mathsf{N}}{\mathsf{m}}\varpi\sin^{2}\theta&0&0&\frac{\left(\mathsf{m}^{2}\cos^{2}\theta+f\mathsf{N}\sin^{2}\theta\right)}{f\mathsf{m}}\sin^{2}\theta&\frac{(f\mathsf{N}-\mathsf{m}^{2})}{f\mathsf{m}}\sin^{2}\theta\cos^{2}\theta\\\
-\frac{\mathsf{N}}{\mathsf{m}}\varpi\cos^{2}\theta&0&0&\frac{(f\mathsf{N}-\mathsf{m}^{2})}{f\mathsf{m}}\sin^{2}\theta\cos^{2}\theta&\frac{\left(\mathsf{m}^{2}\sin^{2}\theta+f\mathsf{N}\cos^{2}\theta\right)}{f\mathsf{m}}\cos^{2}\theta\end{pmatrix}\,.$
(21)

Obviously, compared to (2), only the metric part has been reparametrized. We
are using a new radial coordinate $\mathsf{r}$ and the old parametrization is
recovered via the translation formulas

$\displaystyle\mathsf{m}$ $\displaystyle=mr^{2}\,,$ (22a)
$\displaystyle\varpi$ $\displaystyle=\frac{\omega}{r}\,,$ (22b)
$\displaystyle\mathsf{N}$ $\displaystyle=\frac{mnr^{4}}{f}\,,$ (22c)
$\displaystyle\frac{(\mathsf{r}\mathop{}\\!\mathrm{d}\mathsf{r})^{2}}{\mathsf{N}}$
$\displaystyle=\frac{(\mathop{}\\!\mathrm{d}{r})^{2}}{r^{2}}\,.$ (22d)

Note that the radial coordinate $\mathsf{r}$ appears explicitly in the new
ansatz (19) only through the combination
$\mathsf{r}\mathop{}\\!\mathrm{d}\mathsf{r}=\frac{1}{2}\mathop{}\\!\mathrm{d}\mathsf{R}$,
where $\mathsf{R}=\mathsf{r}^{2}$. Hence, it is possible and useful to write
the resulting Einstein equations as an autonomous system of ODEs with respect
to the squared radial coordinate $\mathsf{R}$. In particular, we will see that
the way $\mathsf{N}$ appears in the ansatz gives it a very simple equation of
motion. As long as the location of the horizon satisfies
$\mathsf{r}=\mathsf{r}_{\text{H}}\neq 0$, if $\mathsf{r}$ is a regular
coordinate at the horizon, then so is $\mathsf{R}$. Again, we are interested
in the interval from infinity $\mathsf{r}\sim\infty$ to the horizon
$f(\mathsf{r}=\mathsf{r}_{\text{H}})=0$, the root closest to infinity.

Plugging (19) directly into the Einstein equations gives somewhat complicated
expressions. It is unenlightening to present them directly. Instead, inspired
by the simplifications that have already been noted in [19, 24, 9], we will
give them in equivalent but more structured form. In what follows, we use the
notation
$(-)^{\prime}=\frac{\mathop{}\\!\mathrm{d}}{\mathop{}\\!\mathrm{d}\mathsf{R}}(-)$
and all functions are treated as functions of $\mathsf{R}$. First, we have the
set of conservation laws

$\displaystyle\mathsf{N}^{\prime\prime}$ $\displaystyle=2\implies$
$\displaystyle\mathsf{N}$
$\displaystyle=(\mathsf{R}-\mathsf{r}_{+}^{2})(\mathsf{R}-\mathsf{r}_{-}^{2})\,,$
(23a) $\displaystyle\left(\frac{\mathsf{N}}{f}(a_{0}^{\prime}+\varpi
a_{\phi}^{\prime})\right)^{\prime}$ $\displaystyle=0\implies$
$\displaystyle\frac{\mathsf{N}}{f}(a_{0}^{\prime}+\varpi a_{\phi}^{\prime})$
$\displaystyle=-\hat{Q}\,,$ (23b)
$\displaystyle\left(\frac{\mathsf{N}^{2}}{f\mathsf{m}}\varpi^{\prime}+4\hat{Q}a_{\phi}\right)^{\prime}$
$\displaystyle=0\implies$
$\displaystyle\frac{\mathsf{N}^{2}}{f\mathsf{m}}\varpi^{\prime}+4\hat{Q}a_{\phi}$
$\displaystyle=-2\hat{J}\,,$ (23c)

where $\mathsf{r}_{+},\mathsf{r}_{-},\hat{Q},\hat{J}$ have been introduced as
integration constants. Then we have the boundary value problem (BVP)

$\displaystyle
f^{2}\mathsf{m}\left(\frac{\mathsf{m}}{f^{2}}f^{\prime}\right)^{\prime}-\mathsf{m}\mathsf{m}^{\prime}\left(\frac{\mathsf{m}^{\prime}}{\mathsf{m}}-2\frac{\mathsf{N}^{\prime}}{\mathsf{N}}\right)f-\left(4\frac{\mathsf{m}^{2}}{\mathsf{N}}-f\right)f$
$\displaystyle=\frac{8}{3}\frac{f\mathsf{m}}{\mathsf{N}}\left(2\mathsf{m}^{2}(a_{\phi}^{\prime})^{2}-fa_{\phi}^{2}\right)$
$\displaystyle\quad{}-\frac{4\hat{Q}^{2}}{3}\frac{f^{2}\mathsf{m}^{2}}{\mathsf{N}^{2}}\,,$
(24a) $\displaystyle
f\mathsf{m}\left(\frac{\mathsf{N}}{\mathsf{m}}\mathsf{m}^{\prime}\right)^{\prime}+\left(\frac{f^{2}\mathsf{N}}{\mathsf{m}^{2}}-2f\right)\mathsf{m}$
$\displaystyle=\frac{4}{3}f\left(2\mathsf{m}^{2}(a_{\phi}^{\prime})^{2}-fa_{\phi}^{2}\right)$
$\displaystyle\quad{}+\frac{4\hat{Q}^{2}}{3}\frac{f^{2}\mathsf{m}}{\mathsf{N}}+4\frac{f^{2}\mathsf{m}^{2}}{\mathsf{N}^{2}}(\hat{J}+2\hat{Q}a_{\phi})^{2}\,,$
(24b)
$\displaystyle\mathsf{m}\left(\mathsf{m}a_{\phi}^{\prime}\right)^{\prime}-fa_{\phi}$
$\displaystyle=2\hat{Q}\frac{f\mathsf{m}^{2}}{\mathsf{N}^{2}}(\hat{J}+2\hat{Q}a_{\phi})\,,$
(24c)

and finally we have the constraint

$\mathsf{m}^{2}(a_{\phi}^{\prime})^{2}=fa_{\phi}^{2}-\mathsf{m}+\frac{f\mathsf{N}}{4\mathsf{m}}+\hat{Q}^{2}\frac{f\mathsf{m}}{\mathsf{N}}+\frac{f\mathsf{m}^{2}}{\mathsf{N}^{2}}(\hat{J}+2\hat{Q}a_{\phi})^{2}\\\
+\frac{\mathsf{N}\mathsf{m}^{\prime}}{8}\left(\frac{f^{\prime}}{f}-2\frac{\mathsf{m}^{\prime}}{\mathsf{m}}+4\frac{\mathsf{N}^{\prime}}{\mathsf{N}}\right)-\frac{\mathsf{N}\mathsf{m}f^{\prime}}{8f}\left(2\frac{f^{\prime}}{f}-\frac{\mathsf{m}^{\prime}}{\mathsf{m}}+2\frac{\mathsf{N}^{\prime}}{\mathsf{N}}\right)\,.$
(25)

This last constraint is compatible with the BVP system because, letting
$\mathsf{C}$ be the difference between the left- and right-hand sides of (25),

$(\mathsf{C}/f^{2})^{\prime}=0\mod{\text{\eqref{eq:reg-eom-bvp}}}\,,$ (26)

which means that, if the boundary conditions for the BVP system are chosen
such that $\mathsf{C}/f^{2}\to 0$ either at infinity or at the horizon, then
the constraint $\mathsf{C}=0$ is guaranteed for any BVP solution.

Direct substitution shows that when our system of equations is satisfied, then
so are the unsimplified Einstein equations. Conversely, starting from the
unsimplified Einstein equations and eliminating all variables except
$\mathsf{N}$ gives the first conservation law from (23). Using it to eliminate
$\mathsf{N}^{\prime\prime}$, as well as every variable and its derivatives
that do not appear in the second conservation law, gives that same
conservation law. Repeating this idea once more, gives the third conservation
law. Upon eliminating $\mathsf{N}$, $\varpi$, $a_{0}$ and their derivatives
from the unsimplified equations, solving for
$f^{\prime\prime},\mathsf{m}^{\prime\prime},a_{\phi}^{\prime\prime}$ gives the
BVP system (24). Finally eliminating these and all other second derivatives
leaves the constraint equation (25). The derivation of the unsimplified
equations and the above elimination steps are most easily carried out using
computer algebra; we have relied in part on the xTensor and xCoba packages
from [21].

In the following sections, we will check how the geometric conditions of
asymptotic flatness and regularity at the horizon translate to boundary
conditions for the BVP system (24). Under these conditions, both endpoints
become _regular_ (or _Fuchsian_) _singular points_ , where a solution can be
constructed by the Frobenius method (as a power series in $\mathsf{R}$). At
each endpoint, we perform a Fuchsian analysis (see [17] for a brief exposition
of the standard methods from [40], together with references to the larger
literature), which counts the free parameters in the space of solutions
compatible with the boundary condition. This analysis will help us confirm
that the Einstein equations restricted to our ansatz are indeed well-posed.
Well-posedness is of course a prerequisite property for trying to construct
either numerical or asymptotic approximations to high accuracy. Unfortunately,
the original papers [19, 24] did not report such a well-posedness analysis,
nor specify in what way their numerical methods were compatible with the
expected asymptotic structure of the solution at the horizon and infinity,
which in principle may have led to spurious numerical artifacts near those
points.

###  3.1 Asymptotic flatness

There are different notions of asymptotic flatness both in four and higher
dimensions. We will refer specifically to the existence of a regular null
infinity, where both asymptotic Poincaré symmetries and their charges are well
defined. Specifically in five dimensions, this notion was analyzed in [37] and
translated to precise component asymptotics for the metric in Bondi
coordinates, which is what we will refer to below. Later, the same analysis
was generalized to arbitrary higher dimension in [35]. Earlier work treated
asymptotic flatness at null infinity in even dimensions [15] and at spatial
infinity [36] using conformal methods. A unified treatment of the higher
dimensional asymptotics of scalar, electromagnetic and gravitational fields at
null infinity was given in [34], which is the main reference that we will
follow. Along similar lines, [14] treated higher dimensional asymptotics for
electromagnetic fields at both null and spatial infinities. Independently,
various asymptotic behaviors for electromagnetic fields in higher dimensions
were also analyzed in [31].

Effectively summarizing the results of the above references (most clearly
presented in [34, Sec. II B] for electromagnetic and in [34, Sec. II D–E] for
gravitational fields), the conditions of having an asymptotically flat regular
(past/future) null infinity, with the absence of (incoming/outgoing) radiation
(both gravitational and electromagnetic), but with non-vanishing contributions
to total mass and electric charge, is equivalent to the existence of an
asymptotic coordinate system $(u,r,\theta,\phi,\psi)$ and corresponding
electromagnetic gauge where

$g_{\mu\nu}=\begin{pmatrix}-1&\pm 1&0&0&0\\\ \pm 1&0&0&0&0\\\ 0&0&r^{2}&0&0\\\
0&0&0&r^{2}\sin^{2}\theta&0\\\ 0&0&0&0&r^{2}\cos^{2}\theta\end{pmatrix}\\\
+\begin{pmatrix}\mathcal{O}\\!\left({r^{-2}}\right)&\mathcal{O}\\!\left({r^{-2}}\right)&\mathcal{O}\\!\left({r^{-1}}\right)&\mathcal{O}\\!\left({r^{-1}}\right)&\mathcal{O}\\!\left({r^{-1}}\right)\\\
\mathcal{O}\\!\left({r^{-2}}\right)&0&0&0&0\\\
\mathcal{O}\\!\left({r^{-1}}\right)&0&\mathcal{O}\\!\left({r^{0}}\right)&\mathcal{O}\\!\left({r^{0}}\right)&\mathcal{O}\\!\left({r^{0}}\right)\\\
\mathcal{O}\\!\left({r^{-1}}\right)&0&\mathcal{O}\\!\left({r^{0}}\right)&\mathcal{O}\\!\left({r^{0}}\right)&\mathcal{O}\\!\left({r^{0}}\right)\\\
\mathcal{O}\\!\left({r^{-1}}\right)&0&\mathcal{O}\\!\left({r^{0}}\right)&\mathcal{O}\\!\left({r^{0}}\right)&\mathcal{O}\\!\left({r^{0}}\right)\end{pmatrix}$
(27a) and
$A_{\mu}=\begin{pmatrix}\mathcal{O}\\!\left({r^{-2}}\right)&0&\mathcal{O}\\!\left({r^{-1}}\right)&\mathcal{O}\\!\left({r^{-1}}\right)&\mathcal{O}\\!\left({r^{-1}}\right)\end{pmatrix}\,.$
(27b)

Some explanation is in order. The $\pm$ choice of sign in the metric
distinguishes the past and future null infinities. The exact zero entries in
$g_{\mu\nu}$ and $A_{\mu}$ are part of the gauge fixing conditions. The Bondi
gauge [37] usually also requires that the angular submatrix $g_{IJ}$, where
$I,J$ run through $\theta,\phi,\psi$, has determinant exactly equal to
$r^{6}\sin^{2}\theta\cos^{2}\theta$ (where $\sin^{2}\theta\cos^{2}\theta$ is
the determinant of the unit round $3$-sphere metric in the same coordinates).
However, it is also sufficient that this subdeterminant has this limit only
for $r\to\infty$, with subleading terms given by an asymptotic expansion in
integer powers of $r$, because then the $r$ coordinate can be redefined to
obtain exact equality, without affecting the structure of the asymptotics.
Below, when adapting our ansatz to the Bondi gauge, we will not perform this
extra redefinition, in favor of our more convenient $\mathsf{r}$ coordinate.
However, such a redefinition would of course be possible. Note also that the
Bondi $r$ coordinate is logically distinct from the coordinate denoted by the
same symbol in the metric (2). Since the Bondi radial coordinate is only used
in this section, there should be no confusion between the two.

In dimension $d=5$, in the absence of gravitational radiation, the leading
radial _Coulombic_ asymptotic for both gravitational and electromagnetic
perturbations is
$\mathcal{O}\\!\left({r^{-d+3}}\right)=\mathcal{O}\\!\left({r^{-2}}\right)$ in
an orthonormal basis [34]444Reference [34] uses harmonic instead of Bondi
gauge. But from their discussion, it is clear that one can transform from one
gauge to the other without changing the leading asymptotics., which explains
different decay rates in different coordinate tensor components above.
Coulombic terms are those that can contribute to finite and non-vanishing
mass, angular momentum and electric charges. In the presence of gravitational
and electromagnetic radiation, the perturbations would decay only as
$\mathcal{O}\\!\left({r^{-d/2+1}}\right)=\mathcal{O}\\!\left({r^{-3/2}}\right)$
in an orthonormal basis [34]. It should also be mentioned that the leading
asymptotic terms will be further constrained by the Einstein equations
themselves, which may lead to faster decay for some components, as we shall
see in our case.

We can define the desired Bondi (which could also be called
Eddington–Finkelstein) coordinates as

$\displaystyle\mathop{}\\!\mathrm{d}{u_{\pm}}$
$\displaystyle=\mathop{}\\!\mathrm{d}{t}\pm\sqrt{\frac{\mathsf{m}}{\mathsf{N}}}\frac{\mathop{}\\!\mathrm{d}\mathsf{R}}{2f}\,,$
(28) $\displaystyle r$ $\displaystyle=\mathsf{r}\,,$ $\displaystyle\theta$
$\displaystyle=\theta\,,$ $\displaystyle\mathop{}\\!\mathrm{d}\phi_{\pm}$
$\displaystyle=\mathop{}\\!\mathrm{d}\phi\pm\varpi\sqrt{\frac{\mathsf{m}}{\mathsf{N}}}\frac{\mathop{}\\!\mathrm{d}\mathsf{R}}{2f}\,,$
$\displaystyle\mathop{}\\!\mathrm{d}\psi_{\pm}$
$\displaystyle=\mathop{}\\!\mathrm{d}\psi\pm\varpi\sqrt{\frac{\mathsf{m}}{\mathsf{N}}}\frac{\mathop{}\\!\mathrm{d}\mathsf{R}}{2f}\,.$

In the coordinates $(u_{\pm},\mathsf{r},\theta,\phi_{\pm},\psi_{\pm})$, the
matrix form of the metric is

$g_{\mu\nu}=\begin{pmatrix}-f+\frac{\mathsf{N}}{\mathsf{m}}\varpi^{2}&\pm\sqrt{\frac{\mathsf{r}^{2}\mathsf{m}}{\mathsf{N}}}&0&-\frac{\mathsf{N}}{\mathsf{m}}\varpi\sin^{2}\theta&-\frac{\mathsf{N}}{\mathsf{m}}\varpi\cos^{2}\theta\\\
\pm\sqrt{\frac{\mathsf{r}^{2}\mathsf{m}}{\mathsf{N}}}&0&0&0&0\\\
0&0&\frac{\mathsf{m}}{f}&0&0\\\
-\frac{\mathsf{N}}{\mathsf{m}}\varpi\sin^{2}\theta&0&0&\frac{\left(\mathsf{m}^{2}\cos^{2}\theta+f\mathsf{N}\sin^{2}\theta\right)}{f\mathsf{m}}\sin^{2}\theta&\frac{(f\mathsf{N}-\mathsf{m}^{2})}{f\mathsf{m}}\sin^{2}\theta\cos^{2}\theta\\\
-\frac{\mathsf{N}}{\mathsf{m}}\varpi\cos^{2}\theta&0&0&\frac{(f\mathsf{N}-\mathsf{m}^{2})}{f\mathsf{m}}\sin^{2}\theta\cos^{2}\theta&\frac{\left(\mathsf{m}^{2}\sin^{2}\theta+f\mathsf{N}\cos^{2}\theta\right)}{f\mathsf{m}}\cos^{2}\theta\end{pmatrix}\,,$
(29a) while the vector potential takes on the form
$A_{\mu}-(\mathop{}\\!\mathrm{d}\alpha)_{\mu}=\begin{pmatrix}a_{0}&0&0&a_{\phi}\sin^{2}\theta&a_{\phi}\cos^{2}\theta\end{pmatrix}\,,$
(29b)

with the gauge transformation parameter defined by
$\alpha^{\prime}(\mathsf{r})=-\sqrt{\mathsf{r}^{2}\mathsf{m}/\mathsf{N}}(a_{0}+\varpi
a_{\phi})/f$.

Hence, comparing (29) with (27), both the metric and the vector potential are
in desired Bondi form, provided we require the following leading asymptotics
(recalling that $\mathsf{R}=\mathsf{r}^{2}$)

$\displaystyle f$
$\displaystyle=1+\mathcal{O}\\!\left({\mathsf{R}^{-1}}\right)\,,$ (30)
$\displaystyle\mathsf{m}$
$\displaystyle=\mathsf{R}+\mathcal{O}\\!\left({\mathsf{R}^{0}}\right)\,,$
$\displaystyle a_{\phi}$
$\displaystyle=\mathcal{O}\\!\left({\mathsf{R}^{-1/2}}\right)\,,$
$\displaystyle\mathsf{N}$
$\displaystyle=\mathsf{R}^{2}+\mathcal{O}\\!\left({\mathsf{R}^{1}}\right)\,,$
$\displaystyle\varpi$
$\displaystyle=\mathcal{O}\\!\left({\mathsf{R}^{-3/2}}\right)\,,$
$\displaystyle a_{0}$
$\displaystyle=\mathcal{O}\\!\left({\mathsf{R}^{-1}}\right)\,.$

Note that comparing with (27) has allowed us to determine not only the leading
but also the subleading terms of $f$, $\mathsf{m}$ and $\mathsf{N}$. We have
not yet specified the structure of the full asymptotic expansion in (30),
since in principle we can allow subleading terms with arbitrary fractional
powers of $\mathsf{R}$. However, we are also free to restrict the subleading
terms as we see fit, with the only justification necessary the a posteriori
well-posedness of the BVP part of the Einstein equations, (24) and (23).

So we may assume that the asymptotic expansion is in integer powers of
$\mathsf{R}$, which is consistent because we are solving equations that are
autonomous with respect to $\mathsf{R}$, and perform a Fuchsian analysis of
the BVP system at the singular point $\mathsf{R}=\infty$. For linear
equations, this is the same as applying the method of Frobenius. For a non-
linear equation, the first step is take the leading terms from (30) and check
that they are consistent with the BVP system (24). Namely, the leading order
terms are collected and it is checked that they cancel among themselves. We
have specifically written (24) in a way that all leading terms appear on the
left-hand side. The cancellation, which is verified by direct calculation,
could have imposed some conditions on the undetermined leading coefficient in
$a_{\phi}$, but it turns out that it does not.

The second step is to linearize (24) and apply the method of Frobenius to the
linear equation whose coefficients have asymptotics determined by the leading
terms (30). If $\mathsf{R}=\infty$ turns out to be a regular singular point of
this linear equation, we can extract the corresponding indicial equation and
determine the remaining free coefficients in the expansion. For systems of
ODEs, determining whether a singular point is regular is not entirely trivial.
The procedure must take into account that the leading terms in different
components of the system of unknowns or of the system of equations may be of
different orders, let alone that the components of the equations and of the
unknowns are allowed to mix.

There is a convenient criterion to check when a system of $k$ equations of
order $p$ has a regular singular point (again, see [17] for a brief
exposition). Let $E[v]=0$ be such a system of linear equations, whose
coefficients have asymptotic expansions in powers of $\mathsf{R}$. Let $S$ and
$T$ be $k\times k$ matrices such that all of $S$, $T$, $S^{-1}$, $T^{-1}$ have
components that are Laurent polynomials in $\mathsf{R}$ (for that, it is
necessary and sufficient that $\det S$ and $\det T$ are non-vanishing
monomials). The system has a regular singular point at $\mathsf{R}=\infty$ if
$T^{-1}E[Sv]=E_{0}(\mathsf{R}\,\partial_{\mathsf{R}})[v]+\text{lower order
terms}$, where the components of $E_{0}(x)$ are constant coefficient
polynomials in $x$, $\det E_{0}(x)\neq 0$, and for the purposes of collecting
lower order terms $\mathsf{R}^{n}(\mathsf{R}\,\partial_{\mathsf{R}})^{m}$ has
order $n$ (which will typically be negative). We call matrices $S$ and $T$
_leading multipliers_. The integer solutions of the _indicial equation_ $\det
E_{0}(n)=0$ are the critical exponents (or indices) of $\mathsf{R}^{n}$ whose
coefficients in the expansion of the solution may be a free parameter. The
eigenvectors of $E_{0}(n)$ corresponding to the critical exponents determine
how these free parameters enter the expansion. At a true real regular singular
point, the polynomial degree of $\det E_{0}(n)$ should be $pk$, determined by
the size and order of the system. If the degree is lower, then only a subset
of the solutions exhibit a Frobenius expansion in powers of $\mathsf{R}$ (with
possible logarithmic contributions) and there will exist other solutions that
grow or decay faster than simple powers of $\mathsf{R}$.

The third and last step counts the number of free parameters in the asymptotic
expansion of a solution. If
$v=S\left[\sum_{k>n}v_{k}\mathsf{R}^{n}+v_{n}\mathsf{R}^{n}+O(\mathsf{R})^{n-1}\right]$,
then knowing that the $v_{k>n}$ coefficients solve the ODE system to
appropriate order we can solve for the next coefficient, provided that
$E_{0}(n)$ is invertible, via

$E_{0}(n)v_{n}=e_{n}(v_{k>n})\,,$ (31)

where $e_{n}(-)$ collects all the remaining terms of the original non-linear
ODE system at order $n$. The invertibility of $E_{n}(n)$ fails precisely when
$n$ is a critical exponent. Then two things happen: because $E_{0}(n)$ does
not have full row rank, the consistency of the equation imposes constraints on
$e_{n}(v_{k>n})$ and hence on any free parameters appearing among the
$v_{k>n}$ coefficients; also, because $E_{0}(n)$ does not have full column
rank, new free parameters appear in the solution for the $v_{n}$ coefficients.
In general the number of free parameters gained and lost at a time need not be
the same, so at each critical exponent the total number of free parameters in
the expansion may increase, decrease, or stay the same; it stabilizes after
the last critical exponent.

In general, leading multiplier matrices may be quite complicated, but we have
specifically written the BVP system (24) to make them simple. Namely, the
resulting $E_{0}(\mathsf{R}\,\partial_{\mathsf{R}})[v]$ operator is

$\begin{bmatrix}(\mathsf{R}\,\partial_{\mathsf{R}})^{2}+1&2(\mathsf{R}\,\partial_{\mathsf{R}}-1)&0\\\
1&(\mathsf{R}\,\partial_{\mathsf{R}}+2)(\mathsf{R}\,\partial_{\mathsf{R}}-1)&0\\\
0&0&(\mathsf{R}\,\partial_{\mathsf{R}}+1)(\mathsf{R}\,\partial_{\mathsf{R}}-1)\end{bmatrix}\begin{bmatrix}\delta
f\\\ \frac{\delta\mathsf{m}}{\mathsf{R}}\\\ \delta a_{\phi}\end{bmatrix}\,,$
(32)

where $S$ multiplies $\delta\mathsf{m}$ by $\mathsf{R}^{-1}$ and $T^{-1}$
multiplies the middle equation by $\mathsf{R}$, while acting as the identity
on other components. The critical exponents are $k=1~{}(2),0~{}(1),-1~{}(3)$,
with multiplicities indicated in parentheses. Because of the constraints on
the leading terms from (30), the coefficients corresponding to the $k=1,0$
exponents must vanish. At $k=-1$, one free parameter is allowed in front of
the leading term of $a_{\phi}$ and two more free parameters appear in the
subleading terms of $f$ and $\mathsf{m}$, with algebraic multiplicity $2$. As
is usual in applying the Frobenius method, the algebraic multiplicity $2$
implies that the two independent coefficients appear in front of
$\mathsf{R}^{-1}$ and $\mathsf{R}^{-1}\log\mathsf{R}$ in the expansions of $f$
and $\mathsf{m}/\mathsf{R}$. However, the logarithmic terms are incompatible
with the asymptotics (30) corresponding to a regular null infinity. Thus, only
one free coefficient at order $\mathsf{R}^{-1}$ remains. The remaining two
free parameters remain free at all further orders of the expansion.

Once the asymptotic expansions for $f,\mathsf{m},a_{\phi}$ have been
determined, we can also asymptotically integrate the first order conservation
laws (23) for $\varpi$ and $a_{0}$. For each of them, there is a free
parameter that appears as an additive constant.

In summary, the following leading terms uniquely fix the most general
asymptotic expansion of a solution in powers of $\mathsf{R}$ that is
compatible with the asymptotic form (30):

$\displaystyle f$
$\displaystyle=1+\frac{f^{(-1)}}{\mathsf{R}}+\mathcal{O}\\!\left({\mathsf{R}^{-2}}\right)\,,$
(33) $\displaystyle\mathsf{m}$
$\displaystyle=\mathsf{R}+\mathsf{m}^{(0)}+\mathcal{O}\\!\left({\mathsf{R}^{-1}}\right)\,,$
$\displaystyle a_{\phi}$
$\displaystyle=\frac{a_{\phi}^{(-1)}}{\mathsf{R}}+\mathcal{O}\\!\left({\mathsf{R}^{-2}}\right)\,,$
$\displaystyle\mathsf{N}$
$\displaystyle=\mathsf{R}^{2}-(\mathsf{r}_{+}^{2}+\mathsf{r}_{-}^{2})\mathsf{R}+\mathsf{r}_{+}^{2}\mathsf{r}_{-}^{2}\,,$
$\displaystyle\varpi$
$\displaystyle=\varpi^{(0)}+\frac{\hat{J}}{\mathsf{R}^{2}}+\mathcal{O}\\!\left({\mathsf{R}^{-3}}\right)\,,$
$\displaystyle a_{0}$
$\displaystyle=a_{0}^{(0)}+\frac{\hat{Q}}{\mathsf{R}}+\mathcal{O}\\!\left({\mathsf{R}^{-2}}\right)\,,$

which has exactly two free parameters ($a_{\phi}^{(-1)}$ and a linear
combination of $f^{(-1)}$ and $\mathsf{m}^{(0)}$), since the yet undetermined
coefficients are constrained by

$\displaystyle
f^{(-1)}-2\mathsf{m}^{(0)}-(\mathsf{r}_{+}^{2}+\mathsf{r}_{-}^{2})$
$\displaystyle=0\,,$ (34) $\displaystyle\varpi^{(0)}$ $\displaystyle=0\,,$
$\displaystyle a_{0}^{(0)}$ $\displaystyle=0\,,$

after enforcing compatibility with the prescribed asymptotic limits of
$f,\mathsf{m},\varpi$ and $a_{0}$. Comparing with (12) justifies the notation
$\hat{J}$ and $\hat{Q}$ for the coefficients in the expansion of $\omega$ and
$a_{0}$ and also allows us to identify

$f^{(-1)}=-\hat{M}\,,$ (35)

with $\hat{M}$, $\hat{J}$ and $\hat{Q}$ being proportional to the physical
mass, angular momentum and electric charges, respectively, as specified in
(13). Finally, there are no further constraints coming from enforcing the
limit $\mathsf{C}/f^{2}\to 0$, which is needed to ensure that the constraint
$\mathsf{C}=0$ (23) is enforced by the compatibility equation (26). Recall
that $\mathsf{r}_{+}^{2},\mathsf{r}_{-}^{2}$ are integration constants that
appear in (23). Note that, after we have carefully taken into account the
Einstein–Maxwell equations, the final leading asymptotics
$a_{\phi}=\mathcal{O}\\!\left({\mathsf{R}^{-1}}\right)$ and
$\varpi=\mathcal{O}\\!\left({\mathsf{R}^{-2}}\right)$ decay faster than
strictly required by asymptotic flatness in (30).

Having shown that at infinity the series expansion for the solution is
uniquely fixed as a function of $\mathsf{r}_{\pm}$, $\hat{J}$, $\hat{Q}$,
$a_{\phi}^{(-1)}$ and say $\hat{M}$, we report here the first few terms of
$f$, $\mathsf{m}$, and $a_{\phi}$ extending (33) (defining
$\mathsf{r}_{\text{H}}^{2}=\mathsf{r}_{+}^{2}-\mathsf{r}_{-}^{2})$:

$\displaystyle f$
$\displaystyle=1-\frac{\hat{M}}{(\mathsf{R}-\mathsf{r}_{-}^{2})}+\frac{\frac{1}{2}\hat{M}\left(\hat{M}-\mathsf{r}_{\text{H}}^{2}\right)+\frac{4}{3}\hat{Q}^{2}}{(\mathsf{R}-\mathsf{r}_{-}^{2})^{2}}$
(36)
$\displaystyle\quad{}+\frac{\frac{4}{9}\left(a_{\phi}^{(-1)}\right)^{2}-\frac{1}{6}\hat{M}\left(\hat{M}^{2}-3\hat{M}\mathsf{r}_{\text{H}}^{2}+2\mathsf{r}_{\text{H}}^{4}\right)+\frac{4}{9}\hat{Q}^{2}\left(-4\hat{M}+3\mathsf{r}_{\text{H}}^{2}\right)+\frac{2}{3}\hat{J}^{2}}{(\mathsf{R}-\mathsf{r}_{-}^{2})^{3}}$
$\displaystyle\quad{}+\frac{\frac{2}{9}\left(a_{\phi}^{(-1)}\right)^{2}\left(-2\hat{M}+3\mathsf{r}_{\text{H}}^{2}\right)+\frac{16}{9}a_{\phi}^{(-1)}\hat{J}\hat{Q}+\frac{1}{9}\hat{Q}^{2}\left(11(\hat{M}-\mathsf{r}_{\text{H}}^{2})^{2}-2\hat{M}\mathsf{r}_{\text{H}}^{2}\right)}{(\mathsf{R}-\mathsf{r}_{-}^{2})^{4}}$
$\displaystyle\qquad{}+\frac{\frac{1}{24}\hat{M}(\hat{M}-\mathsf{r}_{\text{H}}^{2})(\hat{M}-2\mathsf{r}_{\text{H}}^{2})(\hat{M}-3\mathsf{r}_{\text{H}}^{2})+\hat{J}^{2}\left(-\frac{7}{6}\hat{M}+\mathsf{r}_{\text{H}}^{2}\right)+\frac{32}{27}\hat{Q}^{4}}{(\mathsf{R}-\mathsf{r}_{-}^{2})^{4}}$
$\displaystyle\quad{}+\mathcal{O}\\!\left({\mathsf{R}^{-5}}\right)\,,$
$\displaystyle\mathsf{m}$
$\displaystyle=(\mathsf{R}-\mathsf{r}_{-}^{2})-\frac{1}{2}\left(\hat{M}+\mathsf{r}_{\text{H}}^{2}\right)+\frac{\frac{4}{3}\hat{Q}^{2}}{(\mathsf{R}-\mathsf{r}_{-}^{2})}$
$\displaystyle\quad{}+\frac{\frac{1}{72}\left(16\left(a_{\phi}^{(-1)}\right)^{2}+3\hat{M}^{3}-3\hat{M}\mathsf{r}_{\text{H}}^{4}\right)+\frac{2}{9}\hat{Q}^{2}\left(-4\hat{M}+3\mathsf{r}_{\text{H}}^{2}\right)+\frac{5}{6}\hat{J}^{2}}{(\mathsf{R}-\mathsf{r}_{-}^{2})^{2}}$
$\displaystyle\quad{}+\frac{\frac{1}{360}\left(-\hat{M}+5\mathsf{r}_{\text{H}}^{2}\right)\left(16\left(a_{\phi}^{(-1)}\right)^{2}+3\hat{M}^{3}-3\hat{M}\mathsf{r}_{\text{H}}^{4}\right)+\frac{88}{45}a_{\phi}^{(-1)}\hat{J}\hat{Q}}{(\mathsf{R}-\mathsf{r}_{-}^{2})^{3}}$
$\displaystyle\qquad{}+\frac{\frac{1}{45}\hat{Q}^{2}\left(9\hat{M}^{2}-40\hat{M}\mathsf{r}_{\text{H}}^{2}+23\mathsf{r}_{\text{H}}^{4}\right)+\hat{J}^{2}\left(-\frac{29}{30}\hat{M}+\frac{5}{6}\mathsf{r}_{\text{H}}^{2}\right)+\frac{128}{135}\hat{Q}^{4}}{(\mathsf{R}-\mathsf{r}_{-}^{2})^{3}}$
$\displaystyle\quad{}+\mathcal{O}\\!\left({\mathsf{R}^{-4}}\right)\,,$
$\displaystyle a_{\phi}$
$\displaystyle=\frac{a_{\phi}^{(-1)}}{(\mathsf{R}-\mathsf{r}_{-}^{2})}+\frac{\frac{2}{3}\hat{J}\hat{Q}+\frac{1}{6}a_{\phi}^{(-1)}\left(\hat{M}+3\mathsf{r}_{\text{H}}^{2}\right)}{(\mathsf{R}-\mathsf{r}_{-}^{2})^{2}}$
$\displaystyle\quad{}+\frac{\frac{1}{12}a_{\phi}^{(-1)}\left(\hat{M}^{2}+2\hat{M}\mathsf{r}_{\text{H}}^{2}+3\mathsf{r}_{\text{H}}^{4}\right)+\frac{1}{6}\hat{J}\hat{Q}\left(-\hat{M}+4\mathsf{r}_{\text{H}}^{2}\right)}{(\mathsf{R}-\mathsf{r}_{-}^{2})^{3}}$
$\displaystyle\quad{}+\frac{-\frac{1}{360}a_{\phi}^{(-1)}\left(16\left(a_{\phi}^{(-1)}\right)^{2}-6\hat{M}^{3}-45\hat{M}^{2}\mathsf{r}_{\text{H}}^{2}-48\hat{M}\mathsf{r}_{\text{H}}^{4}-45\mathsf{r}_{\text{H}}^{6}\right)}{(\mathsf{R}-\mathsf{r}_{-}^{2})^{4}}$
$\displaystyle\qquad{}+\frac{-\frac{34}{135}a_{\phi}^{(-1)}\hat{M}\hat{Q}^{2}-\frac{7}{30}a_{\phi}^{(-1)}\hat{J}^{2}+\frac{1}{60}\hat{J}\hat{Q}\left(5\hat{M}^{2}-15\hat{M}\mathsf{r}_{\text{H}}^{2}+34\mathsf{r}_{\text{H}}^{4}\right)+\frac{8}{135}\hat{J}\hat{Q}^{3}}{(\mathsf{R}-\mathsf{r}_{-}^{2})^{4}}$
$\displaystyle\quad{}+\mathcal{O}\\!\left({\mathsf{R}^{-5}}\right)\,.$

As discussed in more detail in Section 3.3, our reduced Einstein–Maxwell
equations possess the shift symmetry
$\mathsf{R}\mapsto\mathsf{R}+\mathsf{R}_{0}$, which also shifts the constants
$\mathsf{r}_{\pm}^{2}\to\mathsf{r}_{\pm}^{2}+\mathsf{R}_{0}$. The above
expansion was obtained under the simplifying assumption $\mathsf{r}_{-}^{2}=0$
and then converted to a form that is manifestly invariant under this shift
symmetry. Elsewhere in the text, we will use $\mathsf{r}_{\text{H}}$ to denote
the location of the outer horizon in general. So, strictly speaking, the
numerical value of $\mathsf{r}_{\text{H}}$ in the above expansion should only
be interpreted as the location of the horizon under the assumption
$\mathsf{r}_{-}^{2}=0$.

This expansion is used several times in Section 4 to test whether some
differential constraints forced by special algebraic types would be satisfied
on-shell, that is, satisfying the Einstein–Maxwell equations and our boundary
conditions. For instance, deducing the condition (60) required an expansion at
least to the order given above.

###  3.2 Regularity at the horizon

The vanishing $f(\mathsf{r}_{\text{H}})$ on the horizon signals a coordinate
singularity there, as is expected for coordinates adapted to a timelike
Killing vector when it becomes null. The regularity of the metric (2) and the
corresponding vector potential at the horizon must then be checked in other
coordinates that penetrate the horizon, where the only requirement for a
tensor field to be regular is for it to have smooth components in that
coordinate system.

Traditionally, the simplest horizon penetrating coordinates on a black hole
are of Eddington–Finkelstein type. The coordinates (28) that we have
introduced to play the role of the Bondi frame at null infinity are actually
also of Eddington–Finkelstein type at the horizon. The
$(u_{+},\mathsf{R},\theta,\phi_{+},\psi_{+})$ coordinates regular at the
future horizon and $(u_{-},\mathsf{R},\theta,\phi_{-},\psi_{-})$ ones regular
at the past horizon. We will show below the horizon regularity of our improved
ansatz (19) with respect to these coordinates. Unfortunately, neither of these
coordinate systems is regular at the bifurcation sphere where the two horizons
intersect. Thus we will also introduce Kruskal-like coordinates that cover a
neighborhood of the bifurcation sphere and show that our improved ansatz is
regular there as well.

To reproduce the horizon behavior (9) imposed in the original work [19], we
must set $f=\mathsf{m}=\mathsf{N}=0$ at
$\mathsf{R}=\mathsf{r}_{\text{H}}^{2}$. As a consequence of the integration of
the equation of motion (23) for $\mathsf{N}$, we must set
$\mathsf{r}_{\text{H}}=\mathsf{r}_{+}^{2}$, where we have required
$\mathsf{r}_{+}\neq 0$ and ordered the integration constants as
$\mathsf{r}_{+}^{2}>\mathsf{r}_{-}^{2}$. Combining these requirements with
regularity at the horizon, via smoothness of the tensor components (29) in
Eddington–Finkelstein coordinates, gives the horizon asymptotics

$\displaystyle f$
$\displaystyle=\mathcal{O}\\!\left({\mathsf{R}-\mathsf{r}_{\text{H}}^{2}}\right)\,,$
(37a) $\displaystyle\mathsf{m}$
$\displaystyle=\mathcal{O}\\!\left({\mathsf{R}-\mathsf{r}_{\text{H}}^{2}}\right)\,,$
(37b) $\displaystyle a_{\phi}$
$\displaystyle=\mathcal{O}\\!\left({1}\right)\,,$ (37c)
$\displaystyle\mathsf{N}$
$\displaystyle=(\mathsf{r}_{+}^{2}-\mathsf{r}_{-}^{2})(\mathsf{R}-\mathsf{r}_{\text{H}}^{2})+\mathcal{O}\\!\left({(\mathsf{R}-\mathsf{r}_{\text{H}}^{2})^{2}}\right)\,,$
(37d) $\displaystyle\varpi$ $\displaystyle=\mathcal{O}\\!\left({1}\right)\,,$
(37e) $\displaystyle a_{0}$ $\displaystyle=\mathcal{O}\\!\left({1}\right)\,,$
(37f)

with each function also smooth in $\mathsf{R}$ at the horizon and where the
leading coefficients of $f$ and $\mathsf{m}$ are non-vanishing. Again, the
asymptotics of $\mathsf{N}$, $\varpi$ and $a_{0}$ are obtained by integrating
the conservation laws (23), which remain consistent with the regularity of the
tensor coefficients in (29).

Kruskal (or Kruskal–Szekeres) coordinates for Schwarzschild spacetime are
well-known. It is not as easy to find analogous coordinates constructed for
the Kerr and related black holes. The original construction by Carter [6] was
restricted to the $(t,r)$ plane along the rotation axis. Pretorius and Israel
[33] seem to have been the first to construct global double-null coordinates
regular on the Kerr bifurcation sphere. Motivated by applications in the
global non-linear stability of Kerr and related black holes, analogous double-
null coordinates have been constructed also for a wider class of geometries,
see [10, 5] and references therein.

For our purposes, since we are interested only in a neighborhood of the
bifurcation sphere, it is easiest to follow the idea from the much simpler
construction by Hayward [13]. Namely, consider a constant $\kappa$, which we
will choose appropriately, and the coordinates $(U,V,\theta,\Phi,\Psi)$
defined by

$\displaystyle\frac{\mathop{}\\!\mathrm{d}{U}}{U}-\frac{\mathop{}\\!\mathrm{d}{V}}{V}$
$\displaystyle=\kappa\mathop{}\\!\mathrm{d}{t}\,,$
$\displaystyle\mathop{}\\!\mathrm{d}{t}$
$\displaystyle=\frac{1}{\kappa}\left(\frac{\mathop{}\\!\mathrm{d}{U}}{U}-\frac{\mathop{}\\!\mathrm{d}{V}}{V}\right)\,,$
(38)
$\displaystyle\frac{\mathop{}\\!\mathrm{d}{U}}{U}+\frac{\mathop{}\\!\mathrm{d}{V}}{V}$
$\displaystyle=\kappa\sqrt{\frac{\mathsf{m}}{\mathsf{N}}}\frac{\mathop{}\\!\mathrm{d}\mathsf{R}}{2f}\,,$
$\displaystyle\sqrt{\frac{\mathsf{m}}{\mathsf{N}}}\frac{\mathop{}\\!\mathrm{d}\mathsf{R}}{2f}$
$\displaystyle=\frac{1}{\kappa}\left(\frac{\mathop{}\\!\mathrm{d}{U}}{U}+\frac{\mathop{}\\!\mathrm{d}{V}}{V}\right)\,,$
$\displaystyle\mathop{}\\!\mathrm{d}\Phi$
$\displaystyle=\mathop{}\\!\mathrm{d}\phi-\varpi(\mathsf{r}_{\text{H}}^{2})\mathop{}\\!\mathrm{d}{t}\,,$
$\displaystyle\mathop{}\\!\mathrm{d}\phi$
$\displaystyle=\mathop{}\\!\mathrm{d}\Phi+\frac{\varpi(\mathsf{r}_{\text{H}}^{2})}{\kappa}\left(\frac{\mathop{}\\!\mathrm{d}{U}}{U}-\frac{\mathop{}\\!\mathrm{d}{V}}{V}\right)\,,$
$\displaystyle\mathop{}\\!\mathrm{d}\Psi$
$\displaystyle=\mathop{}\\!\mathrm{d}\psi-\varpi(\mathsf{r}_{\text{H}}^{2})\mathop{}\\!\mathrm{d}{t}\,,$
$\displaystyle\mathop{}\\!\mathrm{d}\psi$
$\displaystyle=\mathop{}\\!\mathrm{d}\Psi+\frac{\varpi(\mathsf{r}_{\text{H}}^{2})}{\kappa}\left(\frac{\mathop{}\\!\mathrm{d}{U}}{U}-\frac{\mathop{}\\!\mathrm{d}{V}}{V}\right)\,.$

Integrating the second equation, we find

$\mathop{}\\!\mathrm{d}\ln(UV)=\frac{\kappa}{2f^{\prime}(\mathsf{r}_{\text{H}}^{2})}\sqrt{\frac{\mathsf{m}^{\prime}(\mathsf{r}_{\text{H}}^{2})}{\mathsf{N}^{\prime}(\mathsf{r}_{\text{H}}^{2})}}\frac{\mathop{}\\!\mathrm{d}\mathsf{R}}{\mathsf{R}-\mathsf{r}_{\text{H}}^{2}}+\mathcal{O}\\!\left({(\mathsf{R}-\mathsf{r}_{\text{H}}^{2})^{0}}\right)\\\
\iff\quad UV=(\mathsf{R}/\mathsf{r}_{\text{H}}^{2}-1)h(\mathsf{R})\,,\quad
h(\mathsf{R})=1+\mathcal{O}\\!\left({\mathsf{R}-\mathsf{r}_{\text{H}}^{2}}\right)\,,$
(39)

provided we choose the overall integration constant appropriately and set

$\kappa=2f^{\prime}(\mathsf{r}_{\text{H}}^{2})\sqrt{\frac{\mathsf{N}^{\prime}(\mathsf{r}_{\text{H}}^{2})}{\mathsf{m}^{\prime}(\mathsf{r}_{\text{H}}^{2})}}\,.$
(40)

In these new Kruskal coordinates, denoting
$\Delta\varpi=\varpi-\varpi(\mathsf{r}_{\text{H}}^{2})$, the metric takes the
form

$g_{\mu\nu}=\begin{pmatrix}V^{2}\frac{(\Delta\varpi)^{2}\mathsf{N}}{\kappa^{2}U^{2}V^{2}\mathsf{m}}&\frac{2f}{\kappa^{2}UV}-UV\frac{(\Delta\varpi)^{2}\mathsf{N}}{\kappa^{2}U^{2}V^{2}\mathsf{m}}&0&-V\frac{(\Delta\varpi)\mathsf{N}}{\kappa
UV\mathsf{m}}\sin^{2}\theta&-V\frac{(\Delta\varpi)\mathsf{N}}{\kappa
UV\mathsf{m}}\cos^{2}\theta\\\
\frac{2f}{\kappa^{2}UV}-UV\frac{(\Delta\varpi)^{2}\mathsf{N}}{\kappa^{2}U^{2}V^{2}\mathsf{m}}&U^{2}\frac{(\Delta\varpi)^{2}\mathsf{N}}{\kappa^{2}U^{2}V^{2}\mathsf{m}}&0&U\frac{(\Delta\varpi)\mathsf{N}}{\kappa
UV\mathsf{m}}\sin^{2}\theta&U\frac{(\Delta\varpi)\mathsf{N}}{\kappa
UV\mathsf{m}}\cos^{2}\theta\\\ 0&0&\frac{\mathsf{m}}{f}&0&0\\\
-V\frac{(\Delta\varpi)\mathsf{N}}{\kappa
UV\mathsf{m}}\sin^{2}\theta&U\frac{(\Delta\varpi)\mathsf{N}}{\kappa
UV\mathsf{m}}\sin^{2}\theta&0&\frac{\left(\mathsf{m}^{2}\cos^{2}\theta+f\mathsf{N}\sin^{2}\theta\right)}{f\mathsf{m}}\sin^{2}\theta&\frac{(f\mathsf{N}-\mathsf{m}^{2})}{f\mathsf{m}}\sin^{2}\theta\cos^{2}\theta\\\
-V\frac{(\Delta\varpi)\mathsf{N}}{\kappa
UV\mathsf{m}}\cos^{2}\theta&U\frac{(\Delta\varpi)\mathsf{N}}{\kappa
UV\mathsf{m}}\cos^{2}\theta&0&\frac{(f\mathsf{N}-\mathsf{m}^{2})}{f\mathsf{m}}\sin^{2}\theta\cos^{2}\theta&\frac{\left(\mathsf{m}^{2}\sin^{2}\theta+f\mathsf{N}\cos^{2}\theta\right)}{f\mathsf{m}}\cos^{2}\theta\end{pmatrix}\,.$

(41)

Recalling that
$\Delta\varpi,f,\mathsf{m},\mathsf{N}=\mathcal{O}\\!\left({\mathsf{R}-\mathsf{r}_{\text{H}}^{2}}\right)=\mathcal{O}\\!\left({UV}\right)$,
with the leading terms of $f$ and $\mathsf{m}$ non-vanishing, all the
components and also

$\det
g=-\frac{4\mathsf{m}\mathsf{N}}{\kappa^{4}U^{2}V^{2}}\cos^{2}\theta\to-\frac{4\mathsf{r}_{\text{H}}^{4}\mathsf{m}^{\prime}(\mathsf{r}_{\text{H}}^{2})\mathsf{N}^{\prime}(\mathsf{r}_{\text{H}}^{2})}{\kappa^{4}}\cos^{2}\theta$
(42)

remain finite as $\mathsf{R}\to\mathsf{r}_{\text{H}}^{2}$. Hence, under the
conditions already imposed by (37), the metric is regular in the Kruskal
coordinates $(U,V,\theta,\Phi,\Psi)$ in a neighborhood of the bifurcation
sphere $U=V=0$, where the future and past horizons
$\mathsf{R}=\mathsf{r}_{\text{H}}^{2}$ intersect. The vector potential
transforms to

$A=\frac{\Delta(a_{0}+\varpi a_{\phi})}{\kappa
UV}\left(V\mathop{}\\!\mathrm{d}{U}-U\mathop{}\\!\mathrm{d}{V}\right)+a_{\phi}(\sin^{2}\theta\mathop{}\\!\mathrm{d}\Phi+\cos^{2}\theta\mathop{}\\!\mathrm{d}\Psi)\\\
+\mathop{}\\!\mathrm{d}\left[\frac{a_{0}(\mathsf{r}_{\text{H}}^{2})+\varpi(\mathsf{r}_{\text{H}}^{2})a_{\phi}(\mathsf{r}_{\text{H}}^{2})}{\kappa}\ln\frac{U}{V}\right]\,,$
(43)

where we have used the shortcut notation

$\Delta(a_{0}+\varpi
a_{\phi})=\left(a_{0}+\varpi(\mathsf{r}_{\text{H}}^{2})a_{\phi}\right)-\left(a_{0}(\mathsf{r}_{\text{H}}^{2})+\varpi(\mathsf{r}_{\text{H}}^{2})a_{\phi}(\mathsf{r}_{\text{H}}^{2})\right)\,.$
(44)

As discussed in the case of Eddington–Finkelstein-like coordinates, the last
term of the vector potential is singular but pure gauge (it is exact), while
the rest of the terms are manifestly regular.

Explicitly solving the coordinate transformation, we find

$\displaystyle U$
$\displaystyle=\sqrt{\mathsf{R}/\mathsf{r}_{\text{H}}^{2}-1}\,\mathrm{e}^{\kappa
t/2}\sqrt{h}\,,$ $\displaystyle\mathop{}\\!\mathrm{d}{U}$
$\displaystyle=\frac{\kappa}{2}\sqrt{\mathsf{R}/\mathsf{r}_{\text{H}}^{2}-1}\,\mathrm{e}^{\kappa
t/2}\sqrt{h}\left(\sqrt{\frac{\mathsf{m}}{\mathsf{N}}}\frac{\mathop{}\\!\mathrm{d}\mathsf{R}}{2f}+\mathop{}\\!\mathrm{d}{t}\right)\,,$
(45) $\displaystyle V$
$\displaystyle=\sqrt{\mathsf{R}/\mathsf{r}_{\text{H}}^{2}-1}\,\mathrm{e}^{-\kappa
t/2}\sqrt{h}\,,$ $\displaystyle\mathop{}\\!\mathrm{d}{V}$
$\displaystyle=\frac{\kappa}{2}\sqrt{\mathsf{R}/\mathsf{r}_{\text{H}}^{2}-1}\,\mathrm{e}^{-\kappa
t/2}\sqrt{h}\left(\sqrt{\frac{\mathsf{m}}{\mathsf{N}}}\frac{\mathop{}\\!\mathrm{d}\mathsf{R}}{2f}-\mathop{}\\!\mathrm{d}{t}\right)\,,$
$\displaystyle\Phi$ $\displaystyle=\phi-\varpi(\mathsf{r}_{\text{H}}^{2})t\,,$
$\displaystyle\mathop{}\\!\mathrm{d}\Phi$
$\displaystyle=\mathop{}\\!\mathrm{d}\phi-\varpi(\mathsf{r}_{\text{H}}^{2})\mathop{}\\!\mathrm{d}{t}$
$\displaystyle\Psi$ $\displaystyle=\psi-\varpi(\mathsf{r}_{\text{H}}^{2})t\,,$
$\displaystyle\mathop{}\\!\mathrm{d}\Psi$
$\displaystyle=\mathop{}\\!\mathrm{d}\psi-\varpi(\mathsf{r}_{\text{H}}^{2})\mathop{}\\!\mathrm{d}{t}\,.$

Next, we repeat the Fuchsian analysis of
$\mathsf{R}=\mathsf{r}_{\text{H}}^{2}$ as a regular singular point of the BVP
part (24) of the Einstein–Maxwell equations restricted to our ansatz, to
determine the remaining free parameters in the Taylor expansion of the
solution. Since the steps of the procedure were already explained in Section
3.1, we merely summarize the results. For convenience, we will denote
$\rho\equiv\mathsf{R}-\mathsf{r}_{\text{H}}^{2}$.

Plugging the leading terms (37) into the BVP system (24), a direct calculation
shows that they are compatible, without any new restrictions on the
undetermined coefficients. Again, we have written the BVP system in such a way
that all cancelling leading terms appear on the left-hand side. Upon
linearization, we find that the leading multiplier matrices $S$ and $T$ may
just be taken to be identity. The resulting $E_{0}(\rho\partial_{\rho})[v]$
operator is

$\begin{bmatrix}(\rho\partial_{\rho}-3)(\rho\partial_{\rho}-1)&\frac{f^{(1)}}{\mathsf{m}^{(1)}}(\rho\partial_{\rho}-1)&0\\\
0&(\rho\partial_{\rho}-1)^{2}&0\\\
-\frac{a_{\phi}^{(0)}}{(\mathsf{m}^{(1)})^{2}}-\frac{2\hat{Q}(\hat{J}+2\hat{Q}a_{\phi}^{(0)})}{(\mathsf{r}_{+}^{2}-\mathsf{r}_{-}^{2})^{2}}&-\frac{4f^{(1)}\hat{Q}(\hat{J}+2\hat{Q}a_{\phi}^{(0)})}{\mathsf{m}^{(1)}(\mathsf{r}_{+}^{2}-\mathsf{r}_{-}^{2})^{2}}&(\rho\partial_{\rho})^{2}\end{bmatrix}\begin{bmatrix}\delta
f\\\ \delta\mathsf{m}\\\ \delta a_{\phi}\end{bmatrix}\,.$ (46)

Exchanging the first two rows and columns, we see that the matrix is lower
triangular, so that we can immediately read off its eigenvalues and
determinant (the indicial equation). The critical $\rho^{k}$ exponents are
$k=0~{}(2),1~{}(3),3~{}(1)$, with multiplicities indicated in parentheses.
Note that regularity at the horizon excludes any logarithmic terms like
$\rho^{k}(\log\rho)^{l}$ in the expansion. Multiplicities in the critical
exponents that generate the same eigen-vectors do not generate more free
parameters. At $k=0$ we already have the free leading coefficient of
$a_{\phi}$. At $k=1$, we have the free leading coefficients of $f$ and
$\mathsf{m}$. At $k=3$ we have one new free coefficient in the expansion of
$f$ and no restrictions on the other free coefficients. Finally, enforcing the
limit $\mathsf{C}/f^{2}\to 0$ needed to ensure that the constraint
$\mathsf{C}=0$ (25) is enforced by the compatibility equation (26) does not
bring in any new restrictions on the free parameters. Once the Taylor series
for $f,\mathsf{m},a_{\phi}$ have been determined, we can also integrate the
first order conservation laws (23) for $\varpi$ and $a_{0}$. For each of them,
there is a free parameter that appears as an additive constant. In summary,
the following leading terms uniquely fix the Taylor series of a solution in
powers of $\rho=\mathsf{R}-\mathsf{r}_{\text{H}}^{2}$, with all free
parameters explicitly shown:

$\displaystyle f$
$\displaystyle=f^{(1)}\rho+\cdots+f^{(3)}\rho^{3}+\mathcal{O}\\!\left({\rho^{4}}\right)\,,$
(47) $\displaystyle\mathsf{m}$
$\displaystyle=\mathsf{m}^{(1)}\rho+\mathcal{O}\\!\left({\rho^{2}}\right)\,,$
$\displaystyle a_{\phi}$
$\displaystyle=a_{\phi}^{(0)}+\mathcal{O}\\!\left({\rho}\right)\,,$
$\displaystyle\mathsf{N}$
$\displaystyle=(\mathsf{r}_{+}^{2}-\mathsf{r}_{-}^{2})\rho+\rho^{2}\,,$
$\displaystyle\varpi$
$\displaystyle=\varpi^{(0)}+\mathcal{O}\\!\left({\rho}\right)\,,$
$\displaystyle a_{0}$
$\displaystyle=a_{0}^{(0)}+\mathcal{O}\\!\left({\rho}\right)\,.$

The additive constants $\varpi^{(0)}$ and $a_{0}^{(0)}$ are not fixed by the
requirements of regularity at the horizon (37). Instead, they must be fixed by
enforcing compatibility with the prescribed limits (30) at infinity, as was
discussed at the end of Section 3.1. Thus, the parameters $\varpi^{(0)}$ and
$a_{0}^{(0)}$ are uniquely fixed, but their values cannot be obtained locally
at the horizon.

###  3.3 Well-posedness

In this section we discuss the well-posedness of the BVP system (24) with the
boundary conditions prescribed by asymptotic flatness (Section 3.1) and
regularity at the horizon (Section 3.2), as well as the overall uniqueness of
the corresponding charged rotating black hole solution.

The BVP system (24) is a second order ODE system for three unknowns, with
external parameters $\mathsf{r}_{+}^{2}$, $\mathsf{r}_{-}^{2}$, $\hat{J}$ and
$\hat{Q}$. With the external parameters fixed, its solution space is hence
6-dimensional (counting the free initial data at any regular point, for
instance). The conclusion of Section 3.1 is that there is a 2-dimensional
space of solutions compatible with asymptotic flatness (parametrized by the
constants $f^{(-1)},\mathsf{m}^{(0)},a_{\phi}^{(-1)}$ (33), with one
constraint (34) among them). The conclusion of Section 3.2 is that there is a
4-dimensional space of solutions compatible with regularity at the horizon,
parametrized by the constants
$f^{(1)},f^{(3)},\mathsf{m}^{(1)},a_{\phi}^{(0)}$ in Equation (47). Hence,
within the overall 6-dimensional solution space, the intersection of these 2-
and 4-dimensional families will generically be 0-dimensional, meaning that the
solution if it exists is expected to be locally unique (under small
perturbations of free parameters at either boundary). In other words, the
boundary conditions prescribed by Sections 3.1 and Section 3.2 define a well-
posed BVP for the non-linear system (24).

The BVP system (24) itself depends on 4 external parameters, the integration
constants $\mathsf{r}_{+}^{2},\mathsf{r}_{-}^{2},\hat{J},\hat{Q}$ from the
conservation laws (23). Two of them, $\hat{J}$ and $\hat{Q}$, have direct
physical interpretations as being proportional to the total angular momentum
and total charge, respectively (13). Because the total Einstein–Maxwell system
of equations restricted to the ansatz (19) is autonomous with respect to
$\mathsf{R}$, the coordinate shift $\mathsf{R}\to\mathsf{R}+\mathsf{R}_{0}$ is
a symmetry, since it is also compatible with both the horizon and asymptotic
boundary conditions. However, being a simple coordinate transformation it does
not change the isometry class of the solution. Note that this shift affects
the integration constants as
$\mathsf{r}_{\pm}^{2}\to\mathsf{r}_{\pm}^{2}+\mathsf{R}_{0}$. We can use the
shift symmetry to place the horizon at any convenient location, for example by
setting $\mathsf{r}_{-}^{2}=0$, which we will assume from now on. This choice
places the horizon at $\mathsf{r}_{\text{H}}=\mathsf{r}_{+}$. Having decided
the location of the horizon, by the well-posedness of the BVP discussed in the
previous paragraph, the solution is (locally) uniquely fixed by the three
remaining parameters. In particular, the total mass can be written as
$\hat{M}=\hat{M}(\mathsf{r}_{\text{H}}^{2},\hat{J},\hat{Q})$, where we can use
the relation (35) between $f^{(-1)}$ and $\hat{M}$. Physically, it is more
convenient to dynamically invert this relationship to
$\mathsf{r}_{\text{H}}^{2}=\mathsf{r}_{\text{H}}^{2}(\hat{M},\hat{J},\hat{Q})$.

In summary, we see that the isometry class of a charged rotating black hole
with ansatz (19) and satisfying the Einstein–Maxwell equations is (locally)
uniquely determined by its total mass $\hat{M}$, total angular momentum
$\hat{J}$, and total charge $\hat{Q}$.

As proof of concept, taking the above well-posedness discussion into account,
we have implemented a numerical solver for the BVP system (24) that takes the
external $\hat{M}$, $\hat{J}$ and $\hat{Q}$ parameters as input and constructs
the corresponding solution of the Einstein–Maxwell equations both outside and
inside the horizon. On the exterior region, we use BVP solver in a basis of
Chebyshev polynomials that are weighted at infinity and the horizon by the
expected asymptotics and boundary conditions. Once the solution on the
exterior region has converged, its values and derivatives near the horizon are
used to feed into a standard ODE solver for the interior region. Figure 1
illustrates the resulting solution for generic values of the external
parameters. In particular, we see that even a naive extrapolation of the
exterior solution inside the horizon shows rather good agreement with the
interior solution on a sizable neighborhood of the horizon.

Figure 1: Numerical solution for the black hole with external parameters
$\hat{M}=5$, $\hat{J}=1$, $\hat{Q}=2$. The horizon radius is dynamically
determined to be $\mathsf{r}_{\text{H}}\approx 1.32464$. The horizontal
variable is $\bar{\mathsf{R}}=\mathsf{r}_{\text{H}}^{2}/\mathsf{R}$, with
$\bar{\mathsf{R}}=0$ corresponding to infinity and $\bar{\mathsf{R}}=1$ to the
horizon, indicated by the dashed vertical line. The barred functions have been
rescaled as $\bar{f}=f/(1-\bar{R})$,
$\bar{\mathsf{m}}=\mathsf{m}\bar{\mathsf{R}}/\mathsf{r}_{\text{H}}^{2}$ and
$\bar{a}_{\phi}=a(1-\bar{\mathsf{R}})$. The thick curves show the exterior
solution and its extrapolation some distance inside the horizon. The thin
curves show the interior solution.

##  4 Algebraic classification

In this section, we study algebraic types of curvature tensors and of the
electromagnetic tensor in the charged Myers–Perry metric.

Recall that four-dimensional Kerr and Kerr–Newman black holes are of Petrov
type D and both metrics admit a Kerr–Schild form (1) with geodesic $k$.
Furthermore, for the Kerr–Newman black hole, the null eigenvectors of the
electromagnetic tensor $F$ coincide with the principal null directions of the
Weyl tensor [27] (_i.e._ , Weyl tensor and $F$ are aligned).

Although the vacuum Myers–Perry black hole is of type D [32, 12] and admits
Kerr–Schild form with geodesic $k$, it has been pointed out already in [23],
that its charged generalization does not admit the Kerr–Schild form assuming
that the vector potential $A$ coincides (up to normalization) with $k$. As a
consequence of the algebraically special form of the Ricci tensor in the
charged Myers–Perry black hole, we will also arrive at the incompatibility of
the charged Myers–Perry black hole metric with Kerr–Schild ansatz (1) with
geodesic $k$, but without employing the strong constraint
$\mbox{\boldmath{$A$}}\propto\mbox{\boldmath{$k$}}$.

For determining the compatibility of the charged Myers–Perry metric with the
Kerr–Schild form with geodesic $k$, we will employ the following proposition
from [28]:

######  Proposition 1 (Geodesicity of the Kerr–Schild vector $k$ [28]).

The null vector $k$ in the Kerr–Schild metric (1) is geodesic if, and only if,
it is an aligned null direction of the Ricci tensor, _i.e._ , if
$R_{ab}k^{a}k^{b}=0$.

Note (see [28]) that if $k$ in the Kerr–Schild metric (1) is geodesic, then
$k$ is also a multiple aligned Weyl direction (WAND) of the spacetime, and
thus the spacetime is algebraically special. Since we also show in this
section that the charged Myers–Perry metric is not algebraically special, this
could be used as another argument against compatibility with the Kerr–Schild
form with geodesic $k$.

We also discuss the algebraic properties of the test electromagnetic field in
the Myers–Perry background and compare this field with the weak field limit of
the charged Myers–Perry metric.

###  4.1 Algebraic type of Maxwell and Ricci tensors

The Maxwell tensor obtained from the potential (20) reads

$F_{\mu\nu}=\begin{pmatrix}0&-a_{0}^{\prime}&0&0&0\\\
a_{0}^{\prime}&0&0&a_{\phi}^{\prime}\sin^{2}\theta&a_{\phi}^{\prime}\cos^{2}\theta\\\
0&0&0&a_{\phi}\sin(2\theta)&-a_{\phi}\sin(2\theta)\\\
0&-a_{\phi}^{\prime}\sin^{2}\theta&-a_{\phi}\sin(2\theta)&0&0\\\
0&-a_{\phi}^{\prime}\cos^{2}\theta&a_{\phi}\sin(2\theta)&0&0\end{pmatrix}\,.$
(48)

The eigenvalues and eigenvectors of $F^{\mu}{}_{\nu}$ are given by

  1. 1.

Eigenvalue 0, eigenvector
$k_{(0)}^{\mu}=\frac{2}{\alpha}(a_{\phi}^{\prime},0,0,-a_{0}^{\prime},-a_{0}^{\prime})$.

  2. 2.

Eigenvalues $\pm\alpha$, where

$\alpha=2\sqrt{-fa_{\phi}^{\prime 2}+\frac{\mathsf{N}}{\mathsf{m}}a^{\prime
2}}\,,$ (49)

and $a^{\prime}=a_{0}^{\prime}+\varpi a_{\phi}^{\prime}$, eigenvectors

$k_{(\pm)}^{\mu}=\sqrt{\frac{2}{f\mathsf{m}\mathsf{N}}}\alpha\left(-\mathsf{N}a^{\prime},\mp
f\mathsf{N}\alpha,0,f\mathsf{m}a_{\phi}^{\prime}-\mathsf{N}\omega
a^{\prime},f\mathsf{m}a_{\phi}^{\prime}-\mathsf{N}\omega a^{\prime}\right)\,.$
(50)

  3. 3.

Eigenvalues $\pm\mathrm{i}\beta$, where $\beta=\dfrac{2fa_{\phi}}{\mathsf{m}}$
with eigenvectors

$\ell_{(\pm)}^{\mu}=\sqrt{f/{2\mathsf{m}}}\left(0,0,\pm\mathrm{i},-\cot\theta,\tan\theta\right)\,.$
(51)

The above eigenvectors can be used to construct a null basis

$\begin{gathered}\mbox{\boldmath{$\ell$}}=\mbox{\boldmath{$m$}}_{(0)}=\mbox{\boldmath{$k$}}^{(+)}\,,\qquad\mbox{\boldmath{$n$}}=\mbox{\boldmath{$m$}}_{(1)}=-\mbox{\boldmath{$k$}}^{(-)}\,,\qquad\mbox{\boldmath{$m$}}_{(2)}=\mbox{\boldmath{$k$}}^{(0)}\,,\\\
\mbox{\boldmath{$m$}}_{(3)}=\sqrt{2}\,\Re\mbox{\boldmath{$\ell$}}^{(+)}\,,\qquad\mbox{\boldmath{$m$}}_{(4)}=\sqrt{2}\,\Im\mbox{\boldmath{$\ell$}}^{(+)}\,.\end{gathered}$
(52)

The only non-vanishing scalar products are then
$\mbox{\boldmath{$\ell$}}\cdot\mbox{\boldmath{$n$}}=1$ and
$\mbox{\boldmath{$m$}}_{(i)}\cdot\mbox{\boldmath{$m$}}_{(j)}=\delta_{ij}$ for
$i,j=2,3,4$ and the metric reads

$g_{\mu\nu}=2\ell_{(\mu}n_{\nu)}+\delta_{ij}m^{(i)}_{\mu}m^{(j)}_{\nu}\,.$
(53)

In this basis, the Maxwell tensor reads

$F_{\mu\nu}=2\alpha\ell_{[\mu}n_{\nu]}-2\beta m^{(3)}_{[\mu}m^{(4)}_{\nu]}\,,$
(54)

and the stress-energy tensor has the form

$\begin{split}T_{\mu\nu}&=F_{\mu\rho}F_{\nu}{}^{\rho}-\frac{1}{4}g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}\\\
&=\frac{1}{2}\left(\alpha^{2}-\beta^{2}\right)m^{(2)}_{\mu}m^{(2)}_{\nu}+\left(\alpha^{2}+\beta^{2}\right)\left(m^{(3)}_{\mu}m^{(3)}_{\nu}+m^{(4)}_{\mu}m^{(4)}_{\nu}-2\ell_{(\mu}n_{\nu)}\right)\,.\end{split}$
(55)

Since both tensors $F$ and $T$ in (54) and (55) contain only boost weight zero
terms, they are both algebraically special, more precisely of type D in the
algebraic classification of tensors [22]. Note that this property holds off-
shell (_i.e._ , without imposing the field equations).

The Einstein equations (with $4\pi G_{5}=1$)

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=2T_{\mu\nu}\,,\qquad
R_{\mu\nu}=2T_{\mu\nu}-\frac{2}{3}g^{\rho\sigma}T_{\rho\sigma}g_{\mu\nu}\,,$
(56)

take the form

$\begin{split}R_{\mu\nu}&=-\frac{4}{3}\left(2\alpha^{2}+\beta^{2}\right)\ell_{(\mu}n_{\nu)}+\frac{2}{3}\left(\alpha^{2}-\beta^{2}\right)m^{(2)}_{\mu}m^{(2)}_{\nu}\\\
&\quad+\frac{2}{3}\left(\alpha^{2}+2\beta^{2}\right)m^{(3)}_{\mu}m^{(3)}_{\nu}+\frac{2}{3}\left(\alpha^{2}+2\beta^{2}\right)m^{(4)}_{\mu}m^{(4)}_{\nu}\,.\end{split}$
(57)

The Ricci tensor is thus of type D on-shell (_i.e._ , assuming that the
Einstein equations hold).

Consider a general vector
$\mbox{\boldmath{$v$}}=v_{0}\mbox{\boldmath{$n$}}+v_{1}\mbox{\boldmath{$\ell$}}+v_{2}\mbox{\boldmath{$m$}}_{(2)}+v_{3}\mbox{\boldmath{$m$}}_{(3)}+v_{4}\mbox{\boldmath{$m$}}_{(4)}$
in the above null basis. We want to identify all null vectors $v$ obeying
$R_{ab}v^{a}v^{b}=0.$ Provided $\alpha\neq 0$, from the null condition and
Equation (57) it follows that $v_{2}=v_{3}=v_{4}=0$ and $v_{0}v_{1}=0$. Up to
a factor, $\ell$ and $n$ are thus the only null vectors obeying
$R_{ab}v^{a}v^{b}=0$. The exceptional case $\alpha=0$, after using the
conservation laws (23), corresponds to

$-fa_{\phi}^{\prime 2}+\frac{\hat{Q}^{2}f^{2}}{\mathsf{N}\mathsf{m}}=0.$ (58)

This condition looks independent from the reduction of the equations of motion
to the constrained BVP (24) and (25), so it is likely inconsistent with them.
Formally, this can be checked by substituting the leading order asymptotic
expansion expansion (33). Setting to zero the resulting leading
$\mathcal{O}\\!\left({\mathsf{R}^{-3}}\right)$ coefficient gives $\hat{Q}=0$,
meaning that only the uncharged case allows $\alpha=0$.

The null congruence $\ell$ (and $n$) is geodesic if and only if

$a_{\phi}^{\prime}a_{0}^{\prime\prime}-a_{0}^{\prime}a_{\phi}^{\prime\prime}=0\,.$
(59)

This condition also looks independent from the reduction of the equations of
motion to the constrained BVP (24) and (25), so it is likely inconsistent with
them. Formally, this can be checked by continuing the expansion (33) by two
more subleading orders (as we have done in (36)) and plugging it into the
desired geodesic condition (59). Setting to zero any non-vanishing terms in
the resulting asymptotic series then imposes non-trivial constraints on the
parameters which determine the general solution (namely $\hat{M}$,
$a_{\phi}^{(-1)}$ and $\mathsf{r}_{\pm},\hat{J},\hat{Q}$, as discussed in
Section 3.1). The first non-trivial term in the expansion of (59) appears at
order $\mathcal{O}\\!\left({\mathsf{R}^{-6}}\right)$. Setting to zero the
coefficients of $\mathsf{R}^{-6}$ and $\mathsf{R}^{-7}$ determines the
$\hat{M}$ and $a_{\phi}^{(-1)}$ parameters, while plugging these values into
the coefficient of $\mathsf{R}^{-8}$ and setting it to zero results in the
simple constraint

$\hat{J}^{3}\hat{Q}^{5}=0\,.$ (60)

In other words, the geodesic condition (59) on $\ell$ cannot hold whenever
both $\hat{J}\neq 0$ and $\hat{Q}\neq 0$.

This result, together with Proposition 1, clearly explains why the metric for
a charged rotating black hole in five dimensions cannot be written in the
Kerr–Schild form (1) with a geodesic $k$, while the limiting cases with either
$\hat{Q}=0$ or $\hat{J}=0$ are compatible with this form.

####  4.1.1 Electromagnetic test field

It is of also of interest to compare the results of the previous section with
the cases of a test field and the weak-field limit. It is well known that in a
vacuum spacetime, Killing vectors can be used to construct electromagnetic
test fields obeying vacuum Maxwell equations in the background spacetime [39].
The contribution of the test field to the energy-momentum tensor is neglected
and thus test fields have a physical meaning only in the weak-field limit.
Electromagnetic test fields for the five-dimensional Myers–Perry black hole
were analyzed in [3]. Additional properties of test fields in various five-
dimensional black hole/ring spacetimes were studied in [27]. In particular, it
has been shown in [27] that for the Myers–Perry black hole, null eigenvectors
of the test field are aligned with the WANDs of the background vacuum
Myers–Perry metric. The Weyl tensor and the Maxwell tensor are in this case
both of type D and aligned.

The Maxwell tensor corresponding to the test field potential
$A^{\mu}=\hat{\alpha}\delta^{\mu}_{t}$ reads

$F_{\mu\nu}=\dfrac{\hat{\alpha}\hat{M}}{(\mathsf{R}+a^{2})^{2}}\begin{pmatrix}0&1&0&0&0\\\
-1&0&0&a\sin^{2}\theta&a\cos^{2}\theta\\\
0&0&0&-a(\mathsf{R}+a^{2})\sin(2\theta)&a(\mathsf{R}+a^{2})\sin(2\theta)\\\
0&-a\sin^{2}\theta&a(\mathsf{R}+a^{2})\sin(2\theta)&0&0\\\
0&-a\cos^{2}\theta&-a(\mathsf{R}+a^{2})\sin(2\theta)&0\end{pmatrix}\,,$

(61)

and the eigenvalues and corresponding null eigenvectors of $F^{\mu}{}_{\nu}$
are given by

$\pm\alpha_{\text{T}}=\pm\frac{2\hat{\alpha}\hat{M}\sqrt{\mathsf{R}}}{(\mathsf{R}+a^{2})^{2}}\,,$
(62)
$k_{\text{T}(\pm)}^{\mu}=\left(a+\frac{\mathsf{R}}{a},\mp\frac{2\sqrt{\mathsf{R}}}{a(a^{2}+\mathsf{R})}\left(a^{4}+2a^{2}\mathsf{R}+\mathsf{R}(\mathsf{R}-\hat{M})\right),0,1,1\right)\,,$
(63)

where the subscript T stands for “test field”.

####  4.1.2 Behaviour for small charge and comparison with the test field

Let us introduce the function $b$ by

$a_{\phi}=\hat{Q}b\,.$ (64)

Then the field equations (24) and constraint (25) contain only even powers of
$\hat{Q}$, and for small charge we can expand the metric and potential
functions in terms of $\hat{Q}^{2}$:

$f=\sum_{k=0}^{\infty}f_{(k)}\frac{\hat{Q}^{2k}}{k!}\,,\quad\mathsf{m}=\sum_{k=0}^{\infty}\mathsf{m}_{(k)}\frac{\hat{Q}^{2k}}{k!}\,,\quad
b=\sum_{k=0}^{\infty}b_{(k)}\frac{\hat{Q}^{2k}}{k!}\,.$ (65)

By neglecting second and higher powers of $\hat{Q}$, the field equations (24)
and constraint (25) for the leading terms $f_{(0)}$, $\mathsf{m}_{(0)}$ and
$b_{(0)}$ reduce to the vacuum field equations for the uncharged Myers–Perry
black hole, plus one additional equation for $b_{(0)}$:

$\mathsf{m}_{(0)}(b_{(0)}^{\prime}\mathsf{m}_{(0)})^{\prime}-b_{(0)}f_{(0)}=2\hat{J}\frac{f_{(0)}\mathsf{m}_{(0)}^{2}}{\mathsf{N}^{2}}\,.$
(66)

The Functions $f_{(0)}$ and $\mathsf{m}_{(0)}$ thus correspond to the vacuum
Myers–Perry metric (18), and Equation (66) is solved by

$b_{(0)}=\frac{\hat{J}}{\hat{M}(a^{2}+R)}\,.$ (67)

Setting $\hat{\alpha}=\hat{Q}/\hat{M}$, the eigenvalue $\alpha$ (49) of the
Maxwell tensor reduces to the eigenvalue $\alpha_{\text{T}}$ (62) of the test
field and the eigenvectors (50) reduce to the test field eigenvectors (63).

Therefore, in the weak field limit the Maxwell tensor (48) reduces to the test
field tensor (61) and is aligned with the (vacuum) background Weyl tensor. As
we will see in the next section, this alignment of the Weyl and
electromagnetic tensors does not survive beyond the weak field approximation.

###  4.2 Algebraic type of the Weyl tensor

####  4.2.1 Weyl type of the metric ansatz

It is well-known that the Kerr–Newman black hole in four dimensions is an
algebraically special solution of Weyl type D. As we will point out below, for
the five-dimensional charged rotating black hole this does not hold anymore.
The algebraic classification of the Weyl tensor in higher dimensions is based
on the alignment of a null vector, the so-called Weyl aligned null direction
(WAND), where the definition of the Weyl types is linked with the multiplicity
of such a WAND (see for example [29] for a review). The standard approach to
determine the algebraic type of a given Weyl tensor is to find an aligned null
frame in which the vanishing components of the Weyl tensor then indicate its
type. Here, we employ the higher-dimensional extension of the Bel–Debever
criteria [30], another simpler and frame-independent method to answer the
question whether the five-dimensional charged rotating black hole solution is
algebraically special. The Bel–Debever criterion for a specific algebraic type
is a polynomial equation involving the Weyl tensor and an unknown null vector
$\ell$, which is the respective WAND of given multiplicity. For instance, the
Weyl tensor is of algebraic type II if and only if there exists a null vector
$\ell$ satisfying

$\mathcal{C}^{\text{II}}_{\mu\nu\rho\sigma\tau}\equiv\ell_{[\sigma}C_{\mu]\lambda[\nu\rho}\ell_{\tau]}\ell^{\lambda}=0\,.$
(68)

Taking the regular metric ansatz (19) in terms of the radial coordinate
$\mathsf{R}$ into account, it is convenient to start with a general $\ell$ of
the form

$\ell^{\mu}=\left(-\frac{\hat{\ell}^{t}}{f},2\sqrt{\frac{\mathsf{N}}{\mathsf{m}}}\hat{\ell}^{\mathsf{R}},\frac{\hat{\ell}^{\theta}}{\sqrt{\mathsf{m}}},\sqrt{\frac{\mathsf{m}}{f\mathsf{N}}}\hat{\ell}^{\phi}-\frac{\hat{\ell}^{t}\varpi}{f},\sqrt{\frac{\mathsf{m}}{f\mathsf{N}}}\hat{\ell}^{\psi}-\frac{\hat{\ell}^{t}\varpi}{f}\right)\,,$
(69)

which simplifies the condition for $\ell$ being null:

$\begin{split}&8\left[(\hat{\ell}^{t})^{2}-(\hat{\ell}^{\mathsf{R}})^{2}-(\hat{\ell}^{\theta})^{2}\right]-\left[3(\hat{\ell}^{\phi})^{2}+2\hat{\ell}^{\phi}\hat{\ell}^{\psi}+3(\hat{\ell}^{\psi})^{2}\right]\\\
&\quad+4\left[(\hat{\ell}^{\phi})^{2}-(\hat{\ell}^{\psi})^{2}\right]\cos(2\theta)-(\hat{\ell}^{\phi}-\hat{\ell}^{\psi})^{2}\left[\cos(4\theta)+\frac{2\mathsf{m}^{2}}{f\mathsf{N}}\sin^{2}(2\theta)\right]=0\,.\end{split}$
(70)

The Bel–Debever criterion (68) for the Weyl tensor of our ansatz (19) is
considerably complicated. It turns out that the simplest component of the
rank-5 tensor $\mathcal{C}^{\text{II}}_{\mathsf{R}t\phi\theta\psi}$ factorizes
and yields the condition

$\left[(\hat{\ell}^{\theta})^{2}+(\hat{\ell}^{\mathsf{R}})^{2}\right]\left[\hat{\ell}^{t}\left(\mathsf{N}f^{\prime}+f\mathsf{N}^{\prime}-2\frac{f\mathsf{N}}{\mathsf{m}}\mathsf{m}^{\prime}\right)+\sqrt{\frac{f\mathsf{N}^{3}}{\mathsf{m}}}(\hat{\ell}^{\phi}\sin^{2}\theta+\hat{\ell}^{\psi}\cos^{2}\theta)\varpi^{\prime}\right]=0\,,$
(71)

where a prime denotes a derivative with respect to $\mathsf{R}$. We thus
obtain two branches: either the first bracket vanishes, which entails
$\ell^{\theta}=\ell^{\mathsf{R}}=0$, or the second bracket vanishes. Both
conditions ensure that the component
$\mathcal{C}^{\text{II}}_{\mathsf{R}t\phi\theta\psi}$ vanishes, and for each
branch we can simplify the criterion (68) by substituting back the
corresponding condition. We then repeat this procedure: picking up the
remaining simplest component which factorizes, and plugging the condition in
each new subbranch into the criterion (68), until all the components of the
criterion are satisfied. In this way, we obtain at the end 109 sets of
conditions on the metric functions and the components of the WAND $\ell$ under
which the Bel–Debever criterion for type II (68) is satisfied. However, all
these sets are contained in the following 9 disjunct cases:

$\displaystyle\begin{split}1:\quad&(\hat{\ell}^{t})^{2}=(\hat{\ell}^{\mathsf{R}})^{2}+(\hat{\ell}^{\theta})^{2}+(\hat{\ell}^{\phi})^{2}\sin^{2}\theta+(\hat{\ell}^{\psi})^{2}\cos^{2}\theta\,,\\\
&f\mathsf{N}=\mathsf{m}^{2},\quad\varpi^{\prime}=0\,,\quad
f^{\prime\prime}=\frac{(f^{\prime})^{2}}{4f}+f\frac{4\mathsf{N}(\mathsf{N}^{\prime\prime}+1)-3(\mathsf{N}^{\prime})^{2}}{12\mathsf{N}^{2}}\,,\end{split}$
(72a)
$\displaystyle\begin{split}2:\quad&\hat{\ell}^{\theta}=0\,,\quad\hat{\ell}^{\psi}=\hat{\ell}^{\phi}\,,\quad(\hat{\ell}^{t})^{2}=(\hat{\ell}^{\mathsf{R}})^{2}+(\hat{\ell}^{\phi})^{2}\,,\\\
&\left(\frac{f\mathsf{N}}{\mathsf{m}^{2}}\right)^{\prime}=\varpi^{\prime}=0\,,\quad
f^{\prime\prime}=\frac{f^{2}}{3\mathsf{m}^{2}}+\frac{(f^{\prime})^{2}}{4f}+f\frac{4\mathsf{N}\mathsf{N}^{\prime\prime}-3(\mathsf{N}^{\prime})^{2}}{12\mathsf{N}^{2}}\,,\end{split}$
(72b)
$\displaystyle\begin{split}3:\quad&\hat{\ell}^{\theta}=0\,,\quad\hat{\ell}^{\phi}=0\,,\quad\hat{\ell}^{\psi}=0\,,\quad(\hat{\ell}^{t})^{2}=(\hat{\ell}^{\mathsf{R}})^{2}\,,\\\
&\left(\frac{f\mathsf{N}}{\mathsf{m}^{2}}\right)^{\prime}=\varpi^{\prime}=0\,,\quad
f^{\prime\prime}=\frac{5f^{2}}{3\mathsf{m}^{2}}+\frac{(f^{\prime})^{2}}{4f}+f\frac{4\mathsf{N}(\mathsf{N}^{\prime\prime}-4)-3(\mathsf{N}^{\prime})^{2}}{12\mathsf{N}^{2}}\,,\end{split}$
(72c)
$\displaystyle\begin{split}4:\quad&\hat{\ell}^{\theta}=0\,,\quad\hat{\ell}^{\phi}=0\,,\quad\hat{\ell}^{\psi}=0\,,\quad(\hat{\ell}^{t})^{2}=(\hat{\ell}^{\mathsf{R}})^{2}\,,\\\
&\left(\frac{f\mathsf{N}}{\mathsf{m}^{2}}\right)^{\prime}=\varpi^{\prime}=0\,,\quad
f^{\prime\prime}=-\frac{f^{2}}{3\mathsf{m}^{2}}+\frac{(f^{\prime})^{2}}{4f}+f\frac{4\mathsf{N}(\mathsf{N}^{\prime\prime}+2)-3(\mathsf{N}^{\prime})^{2}}{12\mathsf{N}^{2}}\,,\end{split}$
(72d)
$\displaystyle\begin{split}5:\quad&\hat{\ell}^{\theta}=0\,,\quad\hat{\ell}^{\phi}=0\,,\quad\hat{\ell}^{\psi}=0\,,\quad(\hat{\ell}^{t})^{2}=(\hat{\ell}^{\mathsf{R}})^{2}\,,\\\
&\left(\frac{f\mathsf{N}}{\mathsf{m}^{2}}\right)^{\prime}=0\,,\quad\varpi^{\prime\prime}=\frac{3(\mathsf{N}f^{\prime}-f\mathsf{N}^{\prime})\varpi^{\prime}}{4f\mathsf{N}}\,,\end{split}$
(72e)
$\displaystyle\begin{split}6:\quad&\hat{\ell}^{t}=\hat{\ell}^{\phi}\sqrt{\frac{\mathsf{N}}{f\mathsf{m}}}\varpi\,,\quad\hat{\ell}^{\theta}=0\,,\quad\hat{\ell}^{\psi}=\hat{\ell}^{\phi}\,,\quad(\hat{\ell}^{\mathsf{R}})^{2}=(\hat{\ell}^{\phi})^{2}\left(\frac{\mathsf{N}}{f\mathsf{m}}\varpi^{2}-1\right)\,,\\\
&\left(\frac{f\mathsf{N}\varpi}{\mathsf{m}^{2}}\right)^{\prime}=0\,,\quad
4\mathsf{m}^{2}f^{\prime\prime}=f^{2}+\frac{\mathsf{m}f^{\prime}(2\mathsf{m}f^{\prime}-3\mathsf{N}\varpi\varpi^{\prime})}{f}-f((\mathsf{m}^{\prime})^{2}-2\mathsf{m}\mathsf{m}^{\prime\prime})+3\mathsf{N}\varpi\mathsf{m}^{\prime}\varpi^{\prime}\\\
&\quad-\mathsf{m}(f^{\prime}\mathsf{m}^{\prime}-2\mathsf{N}\varpi\varpi^{\prime\prime})\,,\end{split}$
(72f)
$\displaystyle\begin{split}7:\quad&\hat{\ell}^{t}=\hat{\ell}^{\phi}\sqrt{\frac{\mathsf{N}}{f\mathsf{m}}}\varpi\,,\quad\hat{\ell}^{\theta}=0\,,\quad\hat{\ell}^{\psi}=\hat{\ell}^{\phi}\,,\quad(\hat{\ell}^{\mathsf{R}})^{2}=(\hat{\ell}^{\phi})^{2}\left(\frac{\mathsf{N}}{f\mathsf{m}}\varpi^{2}-1\right)\,,\\\
&\left(\frac{f\mathsf{N}\varpi}{\mathsf{m}^{2}}\right)^{\prime}=0\,,\quad(\mathsf{N}\varpi^{2}-f\mathsf{m})\varpi^{\prime\prime}=\frac{8f^{3}\varpi+3\mathsf{m}\mathsf{N}\varpi^{2}f^{\prime}\varpi^{\prime}}{4f\mathsf{m}}-f\mathsf{m}\frac{2\varpi^{2}-\varpi\mathsf{N}^{\prime}\varpi^{\prime}+\mathsf{N}(\varpi^{\prime})^{2}}{\mathsf{N}\varpi}\\\
&\quad-\varpi^{\prime}\frac{6\mathsf{m}f^{\prime}+\varpi(3\varpi\mathsf{N}^{\prime}-7\mathsf{N}\varpi^{\prime})}{4}\,,\quad
12f^{\prime\prime}=\frac{20f^{2}}{\mathsf{m}^{2}}+\frac{3(f^{\prime})^{2}}{f}-4\varpi^{\prime}\left(\frac{3f^{\prime}}{\varpi}-\frac{5\mathsf{N}\varpi^{\prime}}{\mathsf{m}}\right)\\\
&\quad-\frac{3f(\mathsf{N}^{\prime})^{2}}{\mathsf{N}^{2}}+2f\frac{5\mathsf{N}^{\prime}\varpi^{\prime}-2\varpi(4-\mathsf{N}^{\prime\prime})}{\mathsf{N}\varpi}-f\frac{11(\varpi^{\prime})^{2}-12\varpi\varpi^{\prime\prime}}{\varpi^{2}}\,,\end{split}$
(72g)
$\displaystyle\begin{split}8:\quad&\hat{\ell}^{\mathsf{R}}=0\,,\quad\hat{\ell}^{\theta}=0\,,\quad\hat{\ell}^{\psi}=\hat{\ell}^{\phi}\,,\quad(\hat{\ell}^{t})^{2}=(\hat{\ell}^{\phi})^{2}\,,\quad\varpi^{\prime}=\frac{\hat{\ell}^{\phi}}{\hat{\ell}^{t}}\sqrt{\frac{f^{3}\mathsf{N}}{\mathsf{m}^{3}}}\left(\frac{\mathsf{m}^{2}}{f\mathsf{N}}\right)^{\prime}\,,\\\
&6f^{\prime\prime}=\frac{f^{2}}{\mathsf{m}^{2}}+3\frac{(f^{\prime})^{2}}{f}-\frac{f^{\prime}\mathsf{N}^{\prime}}{N}+\frac{2f(\mathsf{N}^{\prime})^{2}}{\mathsf{N}^{2}}-3f\frac{(\mathsf{m}^{\prime})^{2}-2\mathsf{m}\mathsf{m}^{\prime\prime}}{\mathsf{m}^{2}}-f\frac{\mathsf{m}^{\prime}\mathsf{N}^{\prime}+2\mathsf{m}\mathsf{N}^{\prime\prime}}{\mathsf{m}\mathsf{N}}\,,\end{split}$
(72h)
$\displaystyle\begin{split}9:\quad&\hat{\ell}^{t}=-\hat{\ell}^{\phi}\sqrt{\frac{f\mathsf{N}^{3}}{\mathsf{m}}}\frac{\varpi^{\prime}}{\mathsf{N}f^{\prime}+f\mathsf{N}^{\prime}-2\frac{f\mathsf{N}}{\mathsf{m}}\mathsf{m}^{\prime}}\,,\quad\hat{\ell}^{\theta}=0\,,\quad\hat{\ell}^{\psi}=\hat{\ell}^{\phi}\,,\quad(\hat{\ell}^{\mathsf{R}})^{2}+(\hat{\ell}^{\phi})^{2}=(\hat{\ell}^{t})^{2}\,,\\\
&\frac{f^{\prime\prime}}{2f}=\frac{(\mathsf{m}^{\prime})^{2}}{\mathsf{m}^{2}}-\frac{2\mathsf{m}^{\prime}\mathsf{N}^{\prime}}{\mathsf{m}\mathsf{N}}+\frac{\mathsf{m}^{\prime\prime}}{\mathsf{m}}+\left(\frac{2f^{\prime}}{f}+\frac{\varpi^{\prime\prime}}{\varpi^{\prime}}\right)\left(\frac{f^{\prime}}{2f}+\frac{\mathsf{N}^{\prime}}{2\mathsf{N}}-\frac{\mathsf{m}^{\prime}}{\mathsf{m}}\right)+\frac{(\mathsf{N}^{\prime})^{2}}{\mathsf{N}^{2}}-\frac{\mathsf{N}^{\prime\prime}}{2\mathsf{N}}\,,\\\
&2(2f\mathsf{N}\mathsf{m}^{\prime}-\mathsf{m}(\mathsf{N}f^{\prime}+f\mathsf{N}^{\prime}))\mathsf{m}^{\prime\prime}=\frac{2\mathsf{m}^{2}\mathsf{N}(f^{\prime})^{3}}{f^{2}}+\frac{2f\mathsf{N}(\mathsf{m}^{\prime})^{3}}{\mathsf{m}}+f(\mathsf{N}f^{\prime}-(\mathsf{m}^{\prime})^{2}\mathsf{N}^{\prime})\\\
&\quad-f^{2}\frac{2\mathsf{N}\mathsf{m}^{\prime}-\mathsf{m}\mathsf{N}^{\prime}}{\mathsf{m}}-\mathsf{m}f^{\prime}\frac{-2\mathsf{m}f^{\prime}\mathsf{N}^{\prime}-3\mathsf{N}^{2}(\varpi^{\prime})^{2}+\mathsf{N}(5f^{\prime}\mathsf{m}^{\prime}+4\mathsf{m}f^{\prime\prime})}{f}-\mathsf{m}\mathsf{N}^{\prime}(f^{\prime}\mathsf{m}^{\prime}+4\mathsf{m}f^{\prime\prime})\\\
&\quad+\mathsf{N}\mathsf{m}^{\prime}(f^{\prime}\mathsf{m}^{\prime}+8\mathsf{m}f^{\prime\prime})-\mathsf{N}^{2}\varpi^{\prime}(3\mathsf{m}^{\prime}\varpi^{\prime}+2\mathsf{m}\varpi^{\prime\prime})\,.\end{split}$
(72i)

In other words, the metric ansatz (19) is of Weyl type II if and only if the
conditions of any of these 9 cases are met. Nevertheless, it is not clear so
far whether these conditions could be satisfied on-shell. First, we recall
that the uncharged and non-rotating limits of the charged rotating solution
lead to known exact solutions, namely the 5-dimensional Myers–Perry black hole
and the 5-dimensional charged Schwarzschild–Tangherlini black hole,
respectively, which are given in Section 2.1. Although both special solutions
are of Weyl type D, they satisfy the type II conditions from _distinct_ cases
which suggests that the charged rotating solution is of more general algebraic
type. Specifically, the Myers–Perry black hole fulfills the conditions of case
9, whereas the charged Schwarzschild–Tangherlini black hole meets the
conditions of case 5.

Now, let us discuss the compatibility of the conditions (72) with a general
charged rotating solution of the Einstein–Maxwell equations. The conditions of
cases 1, 2, 3 and 4 are satisfied only in the non-rotating limit, since for
$\varpi^{\prime}=0$ we obtain $\hat{J}=0$ from the conservation law (23c)
together with the BVP (24c). To analyze the remaining cases, we make use of
the on-shell asymptotic expansions (36) of the functions $f$, $\mathsf{m}$ and
$a_{\phi}$, where without loss of generality we set $\mathsf{r}_{-}^{2}=0$. In
the case 5, plugging the asymptotic expansions into
$(\mathsf{N}f/\mathsf{m}^{2})^{\prime}=0$, the numerator given by
$\mathsf{N}f=\mathsf{R}^{2}-(\hat{M}+\mathsf{r}_{\text{H}}^{2})\mathsf{R}+\frac{1}{2}\hat{M}(\hat{M}+\mathsf{r}_{\text{H}}^{2})+\frac{4}{3}\hat{Q}^{2}+\mathcal{O}\\!\left({\mathsf{R}^{-1}}\right)$
and the denominator given by
$\mathsf{m}^{2}=\mathsf{R}^{2}-(\hat{M}+\mathsf{r}_{\text{H}}^{2})\mathsf{R}+\frac{1}{4}(\hat{M}+\mathsf{r}_{\text{H}}^{2})^{2}+\frac{8}{3}\hat{Q}^{2}+\mathcal{O}\\!\left({\mathsf{R}^{-1}}\right)$
differ at order $\mathsf{R}^{0}$. Therefore, $\mathsf{N}f/\mathsf{m}^{2}$ in
general depends on $\mathsf{R}$ and the conditions of case 5 are not
fulfilled. Analogously, in cases 6 and 7, from the first terms of the
asymptotic expansion of $(\mathsf{N}f\varpi/\mathsf{m}^{2})^{\prime}=0$ it is
obvious that the fraction of
$\mathsf{N}f\varpi=\hat{J}+\mathcal{O}\\!\left({\mathsf{R}^{-1}}\right)$ and
$\mathsf{m}^{2}=\mathsf{R}^{2}+\mathcal{O}\\!\left({\mathsf{R}}\right)$ is not
independent of $\mathsf{R}$, such that also this condition is not met for a
general solution. Substituting the asymptotic expansion (36) in the condition
$\varpi^{\prime}=\pm\sqrt{\mathsf{N}f^{3}/\mathsf{m}^{3}}[\mathsf{m}^{2}/(\mathsf{N}f)]^{\prime}$
of case 8 we obtain
$2\hat{J}/\mathsf{R}^{3}=\pm(16\hat{Q}^{2}-3\hat{M}^{2}+3\mathsf{r}_{\text{H}}^{4})/(6\mathsf{R}^{7/2})+\mathcal{O}\\!\left({\mathsf{R}^{-4}}\right)$,
requiring $\hat{J}=0$. Lastly, in case 9 the leading order term in its
condition reads
$-4\hat{J}\mathsf{R}+\mathcal{O}\\!\left({\mathsf{R}^{0}}\right)=0$, and again
requires that $\hat{J}=0$, such that none of the conditions can be fulfilled
for a general charged rotating solution.

####  4.2.2 Weyl type on the horizon

We have shown above that in the bulk the charged Myers–Perry metric is
algebraically general. In this section, we determine the algebraic type of the
metric on the horizon. The form (41) is manifestly regular at the horizons and
the bifurcation sphere, but it is more cumbersome than necessary. In Kruskal-
like coordinates, an a priori regular ansatz may be parameterized as follows,
where any function with implicit coordinate dependence depends only on the
product $UV$ and have non-vanishing leading terms of order zero, _e.g._ ,
$F=F(UV)=F(0)+\mathcal{O}\\!\left({UV}\right)$:

$\mathop{}\\!\mathrm{d}{s}^{2}=2F\mathop{}\\!\mathrm{d}{U}\mathop{}\\!\mathrm{d}{V}+N[W(U\mathop{}\\!\mathrm{d}{V}-V\mathop{}\\!\mathrm{d}{U})-\sin^{2}\theta\mathop{}\\!\mathrm{d}\phi-\cos^{2}\theta\mathop{}\\!\mathrm{d}\psi]^{2}\\\
+M\mathop{}\\!\mathrm{d}\theta^{2}+M\sin^{2}\theta\cos^{2}\theta(\mathop{}\\!\mathrm{d}\phi-\mathop{}\\!\mathrm{d}\psi)^{2}\,,$
(73)

$g_{\mu\nu}=\begin{pmatrix}V^{2}NW^{2}&F-UVNW^{2}&0&VNW\sin^{2}\theta&VNW\cos^{2}\theta\\\
F-UVNW^{2}&U^{2}NW^{2}&0&-UNW\sin^{2}\theta&-UNW\cos^{2}\theta\\\ 0&0&M&0&0\\\
VNW\sin^{2}\theta&-UNW\sin^{2}\theta&0&(N\sin^{2}\theta+M\cos^{2}\theta)\sin^{2}\theta&(N-M)\sin^{2}\theta\cos^{2}\theta\\\
VNW\cos^{2}\theta&-UNW\cos^{2}\theta&0&(N-M)\sin^{2}\theta\cos^{2}\theta&(N\cos^{2}\theta+M\sin^{2}\theta)\cos^{2}\theta\\\
\end{pmatrix}\,.$

(74)

To determine the type of the Weyl tensor on the horizon, we employ the null
coframe $\mbox{\boldmath{$\ell$}}=\mathop{}\\!\mathrm{d}{U}$,
$\mbox{\boldmath{$n$}}=\mathop{}\\!\mathrm{d}{V}$,
$\mbox{\boldmath{$m$}}^{(2)}=\mathop{}\\!\mathrm{d}{\theta}$,
$\mbox{\boldmath{$m$}}^{(3)}=\mathop{}\\!\mathrm{d}\phi-\mathop{}\\!\mathrm{d}\psi$,
$\mbox{\boldmath{$m$}}^{(4)}=W(V\mathop{}\\!\mathrm{d}{U}-U\mathop{}\\!\mathrm{d}{V})+\mathop{}\\!\mathrm{d}\phi$
(up to normalization).

Calculating all boost-weight +2 components of the Weyl tensor
$C^{\alpha\beta\gamma\delta}\ell_{\alpha}m^{(i)}{}_{\beta}\ell_{\gamma}m^{(j)}{}_{\delta}$,
we find that they all vanish for $U=0$. Furthermore, all boost-weight +1
components vanish there as well. It follows that $\ell$ is a multiple WAND at
$U=0$. Similarly, $n$ is a multiple WAND at $V=0$. Therefore, the spacetime is
at least of Weyl type II at either of the horizons ($V=0$ or $U=0$) and type D
at the bifurcation sphere ($U=V=0$). The same results also hold for the
Riemann tensor. Note that these observations hold even off-shell, without
imposing the Einstein–Maxwell equations.

##  5 Discussion

We have studied various mathematical aspects of a charged rotating black hole
with two equal-magnitude angular momenta in five dimensions.

In Section 3, we proposed a metric ansatz (19) which is regular on the
horizon. Furthermore, for this metric ansatz the Einstein–Maxwell system (24)
and (25) is autonomous, which is technically convenient. We studied geometric
regularity conditions for the Einstein–Maxwell system (24) following from
asymptotic flatness with the absence of incoming and outgoing radiation, as
well as regularity at the horizon. By comparing the solution at the singular
end-points with these conditions, we showed that the boundary value problem is
well-posed.

In Section 4, we performed the algebraic classification of the curvature
tensors and the electromagnetic tensor $F$. We found that the Weyl tensor is
algebraically general in the bulk, type II on the horizon, and type D on the
bifurcation sphere. The electromagnetic field strength tensor $F$ and the
stress-energy tensor $T$ are both of type D off-shell, while the Ricci tensor
is type D on-shell. These algebraic properties are inconsistent with the
Kerr–Schild form (1) of the metric with geodesic $k$, such that the charged
rotating black hole with two equal-magnitude angular momenta in five
dimensions cannot be described by such a Kerr–Schild metric. On the other
hand, changing to coordinates appropriate for studying the on-horizon
properties of the spacetime, the metric (73) takes a form somewhat reminiscent
of the Kerr–Schild form. In this case, however, the background metric is a
four-dimensional type D warped product metric, and the “Kerr–Schild” vector is
not null.

## Acknowledgments

This work has been supported by the Research Plan RVO 67985840 and Czech
Science Foundation Grant No. 19-09659S (I.K., T.M., and V.P.). M.B.F. is
supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) — project no. 396692871 within the Emmy Noether grant CA1850/1-1
and project no. 406116891 within the Research Training Group RTG 2522/1. In
addition, I.K. was partially supported by the Praemium Academiae of M. Markl.

##  Appendix A Comparison with the work of Fan, Liang, Mei

In [9], Fan _et al._ looked at the same metric ansatz, introduced by Kunz _et
al._ [19, 24], for a 5-dimensional charged rotating black hole that we are
considering in this work. They claim to have reduced the corresponding
Einstein equations to only two scalar variables (their $Z$ and $A_{\phi}$). In
contrast, in Section 3, the most we could reduce our system to is the three
scalar variables $f$, $\mathsf{m}$, $a_{\phi}$, in Equations (24) and (25).
Unfortunately, a closer comparison reveals some inconsistencies in their
formulas.

In more detail, the parametrization of the ansatz used in [9] is

$\mathop{}\\!\mathrm{d}s^{2}=-F\mathop{}\\!\mathrm{d}t^{2}+\frac{\mathop{}\\!\mathrm{d}R^{2}}{W}+R^{2}\left(\frac{\mathop{}\\!\mathrm{d}x^{2}}{1-x^{2}}+(1-x^{2})\mathop{}\\!\mathrm{d}\phi^{2}+x^{2}\mathop{}\\!\mathrm{d}\psi^{2}\right)\\\
+N\left[(1-x^{2})\mathop{}\\!\mathrm{d}\phi+x^{2}\mathop{}\\!\mathrm{d}\psi\right]^{2}-2B\left[(1-x^{2})\mathop{}\\!\mathrm{d}\phi+x^{2}\mathop{}\\!\mathrm{d}\psi\right]\mathop{}\\!\mathrm{d}t\,,$
(75a)
$A=A_{t}\mathop{}\\!\mathrm{d}t+A_{\phi}\left[(1-x^{2})\mathop{}\\!\mathrm{d}\phi+x^{2}\mathop{}\\!\mathrm{d}\psi\right]\,.$
(75b)

where a comparison with our parametrization (19) reveals the dictionary

$\displaystyle x$ $\displaystyle=\cos\theta\,,$ (76a) $\displaystyle r$
$\displaystyle=\sqrt{\mathsf{m}/f}\,,$ (76b) $\displaystyle F$
$\displaystyle=f-\frac{\mathsf{N}}{\mathsf{m}}\varpi^{2}\,,$ (76c)
$\displaystyle W$
$\displaystyle=\mathsf{N}\left(\frac{\mathsf{m}^{\prime}}{\mathsf{m}}-\frac{f^{\prime}}{f}\right)^{2}\,,$
(76d) $\displaystyle N$
$\displaystyle=\frac{\mathsf{N}}{\mathsf{m}}-\frac{\mathsf{m}}{f}\,,$ (76e)
$\displaystyle B$ $\displaystyle=\frac{\mathsf{N}}{\mathsf{m}}\varpi\,,$ (76f)
$\displaystyle A_{t}$ $\displaystyle=a_{0}\,,$ (76g) $\displaystyle A_{\phi}$
$\displaystyle=a_{\phi}\,,$ (76h)

where all derivatives on the right-hand side are with respect to our
coordinate $\mathsf{R}=\mathsf{r}^{2}$. In the two equations,

$\displaystyle W$
$\displaystyle=\frac{16Z(c_{0}+Z)}{r^{2}(\partial_{r}Z)^{2}}\,,$ Eq. (3) of
[9] $\displaystyle N$
$\displaystyle=\frac{Z}{r^{2}F}-\frac{B^{2}}{F}-r^{2}\,,$ Eq. (4) of [9]

the first one introduces a new variable $Z=Z(r)$ and a constant $c_{0}$, while
the second one follows from this definition in conjunction with the Einstein
equations. Using our dictionary and simple algebraic manipulation, we find
that $Z$ and $c_{0}$ are uniquely determined by the identities

$Z=\mathsf{N}=(\mathsf{R}-\mathsf{r}_{+}^{2})(\mathsf{R}-\mathsf{r}_{-}^{2})\,,\quad
c_{0}+Z=(\mathsf{N}^{\prime}/2)^{2}=\left(\mathsf{R}-\frac{\mathsf{r}_{+}^{2}+\mathsf{r}_{-}^{2}}{2}\right)^{2}\,,$
(77)

where we have used the form of $\mathsf{N}$ (23) imposed by the Einstein
equations. Next, after introducing the variable $\tilde{A}_{\phi}$ in their
Equation (7), their formula (6) for $F$ is equivalent to a combination of our
(23c) and (24c). However, their final complicated formula (8) for $B$ turns
out to be inconsistent. We have verified the inconsistency by plugging in our
general asymptotic solution (36), for any non-trivial values of the free
parameters $\hat{M}$ and $a_{\phi}^{(-1)}$. Unfortunately, the claimed
reduction by [9] of the Einstein equations to just the two variables $Z$ and
$A_{\phi}$ cannot be correct without the validity of their Equation (8), which
we have also verified cannot be saved by undoing possible sign errors or other
minor typos. Finally, the asymptotic solution at infinity (9) for $Z$ and
$A_{\phi}$ in [9] cannot be matched up to our asymptotic solution (36) by any
correspondence between their free parameters ($m,a,s$) and ours
($\hat{M},a_{\phi}^{(-1)}$), with an inconsistency appearing already at the
next subleading order.

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