# Numerical investigation of the dynamics of linear spin $s$ fields on a Kerr
background II:
Superradiant scattering

Károly Csukás111E-mail<EMAIL_ADDRESS>Wigner RCP, H-1121
Budapest, Konkoly Thege Miklós út 29-33, Hungary  István Rácz222E-mail
<EMAIL_ADDRESS>Wigner RCP, H-1121 Budapest, Konkoly Thege
Miklós út 29-33, Hungary

###### Abstract

Superradiant scattering of linear spin $s=0,\pm 1,\pm 2$ fields on Kerr black
hole background is investigated in the time domain by integrating numerically
the homogeneous Teukolsky master equation. The applied numerical setup has
already been used in studying long time evolution and tail behavior of
electromagnetic and metric perturbations on rotating black hole background
[10]. To have a clear setup the initial data is chosen to be of the compact
support, while to optimize superradiance the frequency of the initial data is
fine tuned. Our most important finding is that the rate of superradiance
strongly depends on the relative position of the (compact) support of the
initial data and the ergoregion. When they are well-separated then only a
modest—in case of $s=0$ scalar fields negligible—superradiance occurs, whereas
it can get to be amplified significantly whenever the support of the initial
data and the ergoregion overlap.

##  1 Introduction

Superradiant scattering is one of the most challenging phenomenon in black
hole physics ever since it was pointed out by Misner, Zel’dovich and
Starobinskii [17, 25, 19, 20] that more energy return to infinity than sent in
originally. Superradiance is expected to occur in the scattering of a suitably
frequency tuned scalar, electromagnetic or gravitational wave packet on a
rotating black hole background.

Notably, while studying the superradiant scattering of test fields on black
hole background, most of the works have been done by applying frequency-domain
analyses (see, e.g. [6] and references in it). As opposed to this, time-domain
studies are much rarer (see, e.g., [2, 9, 16, 11, 3]) even though some of the
physically relevant processes cannot be investigated via frequency-domain
analyses. The most natural mathematical framework to investigate scattering of
scalar, electromagnetic and gravitational perturbations in the time-domain is
provided by the homogeneous Teukolsky equation [21]. The conventional argument
rests on the completely plausible assumptions that the “transmitted wave” is
absorbed by the black hole while the “reflected wave” escapes to infinity. It
is also important to be emphasized here that, although the occurrence of
superradiance was found to be completely consistent with the laws of black
hole thermodynamics [4, 22], most of the arguments supporting energy
extraction boil down to the investigation of certain individual, thereby,
preferred eigenstate modes [22].

This paper is to report on our numerical findings on superradiant scattering
of scalar waves and electromagnetic and gravitational perturbations on a
rotating Kerr black hole background. In doing so we shall use the numerical
method we developed to investigate the time evolution of linear fields of spin
$s=0,\pm 1,\pm 2$ on Kerr black hole spacetimes by solving the homogeneous
Teukolsky equation numerically [10]. Accordingly, as in [10], the numerical
setup we shall apply incorporates both the elements of conformal
compactification and those of the hyperbolic initial value problem. Based on
the fact that the fixed Kerr background is smoothly foliated by a two-
parameter family of topological two-spheres—that are the Boyer-Lindquist
$t=const$ and $r=const$ ellipsoids—the evolved fields are expanded in terms of
spin-weighted spherical harmonics. This allows us to evaluate all the angular
derivatives analytically as they can be expressed either in terms of the “eth”
and “ethbar”, $\eth$ and $\overline{\eth}$, operators or by Lie-derivatives
with respect to the axial Killing vector of the Kerr background. The expansion
coefficients are then evolved, in the time-radial section, by applying the
method of lines. The corresponding finite difference part of the code was
verified to be of fourth order accurate [10]. Here it is verified that in the
angular directions the convergence rate is, as expected, indeed exponential.
In principle, the full spectra of nonaxisymmetric configurations can be
investigated, though, in practice, our attention will be restricted to the
values $m=\pm 1,\pm 2$ of the azimuthal parameter. In any numerical
investigation it is critical to verify that the numerical findings are not
simple artifacts of the applied numerical method but they are rooted in true
physical nature of the investigated fields. Therefore—in addition to the
conventional checks on the convergence rates—new type of “energy” and “angular
momentum” balance relations were also used to verify both the proper
implementation of the underlying mathematical model and the reliability and
robustness of the developed numerical schema.

The analysis of individual back-scattered modes—see subsection 2.2 for a brief
discussion on some of the related issues—shows that in order to ensure a
superradiant scattering to occur these modes have to be both frequency tuned
and also guaranteed to submerge deeply into the ergoregion. To demonstrate
that the second assumption is indeed of critical importance three different
types of initial data are applied in our numerical investigations. Each of
these is of compact support. The most significant difference between them is
that for type-I data the compact support is clearly separated from the
ergoregion, for type-II data the support is concentrated to the ergoregion,
whereas for type-III data the support inter-bridges the domains of type-I and
type-II data, i.e. the radial domain of type-III data significantly overlaps
with the ergoregion and also it has a non-negligible part outside of the
ergoregion.

One of our most important findings is that if the initial data is of type-I,
regardless of how well it is frequency-tuned to be superradiant, considerably
large portion of the incident pulse goes through of an almost perfect
reflection before reaching the ergoregion. Most cases this reduces the
transmission coefficient, and, thereby, the effect of superradiance is
suppressed significantly. As opposed to this, in some cases, for type-II and
type-III initial data considerably large superradiance can be observed.

In all circumstances the energy and angular momentum balance relations are
used to give a proper measure of the energy and angular momentum gains. In
doing so the handy conserved currents introduced in [23] are applied which,
however, require the use of a pair of spin $s$ and $-s$ solutions. This, in
particular, means that whenever we intend to characterize the superradiance of
a spin $s$ solution we need to construct an adequate spin $-s$ counterpart
solution to the Teukolsky master equation. Exactly this type of construction
is offered for electromagnetic perturbations by Cohen-Kegeles in [8], and both
for electromagnetic and metric perturbations by Chrzanowski in [7]. All the
related heuristic aspects of the constructions in [8, 7] were explained by
Wald in [24]. As we need to apply these transformations a short review on the
most important elements is provided in the appendix of the present paper.

The rest of this paper is organized as follows. In section 2 some of the basic
notions and notations are recalled including the Teukolsky equation, the mode
analysis of superradiance and our guiding principles applied in selecting the
initial data to get superradiant solutions. Section 3 is to discuss the
regularization of the basic variables, the new type of energy and angular
momentum balance relations, the applied multipole expansions and the actual
initialization of time evolution. Section 4 is to present our numerical
findings. We start, in subsection 4.1, by investigating configurations with
type-I initial data, i.e. the compact support of the data is separated from
the ergoregion. Then, in subsection 4.2, results relevant for configurations
with initial data concentrated to the ergoregion are presented. Subsection 4.3
includes investigations of type-III configurations the radial support of the
initial data of which inter-bridges the domains of types I and II data. The
discussions on our findings are closed by final remarks in section 5. The
paper completed by several appendices providing useful insights on the
analytic backgrounds we used to generate adequate spin $-s$ counterpart to
spin $s$ solutions of the Teukolsky master equation.

##  2 Preliminaries

This section is to recall Teukolsky’s master equation, the mode analyses of
superradiance and also to motivate the choice of the initial data to get
superradiant solutions.

###  2.1 Teukolsky’s master equation

The line element of the Kerr background in Boyer-Lindquist coordinates
$(t,r,\vartheta,\phi)$ is

$(ds)^{2}=\left(1-\frac{2Mr}{\Sigma}\right)(dt)^{2}+\frac{4arM}{\Sigma}\sin^{2}\vartheta\,dtd\phi\\\
-\frac{\Sigma}{\Delta}\,(dr)^{2}-\Sigma\,(d\vartheta)^{2}-\frac{(r^{2}+a^{2})^{2}-a^{2}\Delta\sin^{2}\vartheta}{\Sigma}\sin^{2}\vartheta\,(d\phi)^{2},$
(2.1)

where $\Sigma=r^{2}+a^{2}\cos^{2}\vartheta$ and $\Delta=r^{2}-2Mr+a^{2}$. The
parameters $M$ and $a$ denote the ADM mass and the angular momentum per unit
mass parameters of the Kerr solution, respectively.

A very simple, probably the simplest, covariant form of the homogeneous
Teukolsky master equation was given in [5] and it reads as

$\left[\,\left(\nabla^{a}+s\,\Gamma^{a}\right)\,\left(\nabla_{a}+s\,\Gamma_{a}\right)-4\,s^{2}\,\Psi_{2}\,\right]\,\psi^{(s)}=0\,,$
(2.2)

where $\psi^{(s)}$ is a linear spin $s$ field, and

$\Psi_{2}=-\frac{M}{(r-\mathrm{i}\,a\cos\vartheta)^{3}}$ (2.3)

is the only nonvanishing and gauge invariant Weyl-scalar component on Kerr
background. The components of the “connection vector” $\Gamma^{a}$ in (2.2)
are [5]

$\displaystyle\Gamma^{t}$
$\displaystyle=-\frac{1}{\Sigma}\left[\frac{M(r^{2}-a^{2})}{\Delta}-(r+\mathrm{i}\,a\cos\vartheta)\right],$
(2.4) $\displaystyle\Gamma^{r}$
$\displaystyle=-\frac{1}{\Sigma}\,\big{(}r-M\big{)},$ (2.5)
$\displaystyle\Gamma^{\vartheta}$ $\displaystyle=0,$ (2.6)
$\displaystyle\Gamma^{\phi}$
$\displaystyle=-\frac{1}{\Sigma}\left[\frac{a(r-M)}{\Delta}+\mathrm{i}\,\frac{\cos\vartheta}{\sin^{2}\vartheta}\right]\,.$
(2.7)

###  2.2 Superradiance in mode analysis

Consider a solution $\psi^{(s)}$ to (2.2). It was shown by Teukolsky [21] that
it is advantageous to decompose its temporal Fourier transform,

${\mathscr{F}}\hskip-1.42262pt\psi^{(s)}=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\psi^{(s)}\,e^{-\mathrm{i}\,\omega\,t}dt\,,$
(2.8)

as

${\mathscr{F}}\hskip-1.42262pt\psi^{(s)}(\omega,r,\vartheta,{\phi})=\sum_{\ell=|s|}^{\infty}\sum_{m=-\ell}^{\ell}{}_{s}R_{\ell,\omega}^{m}(r)\cdot{}_{s}S_{\ell,a\omega}^{m}(\vartheta,{\phi})\,,$
(2.9)

where $(t,r,\vartheta,\phi)$ stand again for the Boyer-Lindquist coordinates,
while $\omega$ is the frequency in the time translation direction.

The radial functions ${}_{s}R_{\ell,\omega}^{m}$ in (2.9)—for any specific
value of the angular momentum quantum numbers $\ell,m$ and the frequency
$\omega$—are subject to a one-dimensional Schrödinger-type equation

$\Delta^{-s}\frac{d}{dr}\left(\Delta^{s+1}\,\left(\frac{d}{dr}\,{}_{s}{R}_{\ell,\omega}^{m}\right)\right)+\left[\frac{K^{2}-2\,\mathrm{i}\,s\,(r-M)\,K}{\Delta}+4\,\mathrm{i}\,s\,\omega\,r-{}_{s}\lambda_{\ell,\omega}^{m}\right]\,{}_{s}{R}_{\ell,\omega}^{m}=0\,.$
(2.10)

Note also that the angular factors ${}_{s}S_{\ell,a\omega}^{m}$, in (2.9),
stand for the spin-weighted (oblate) spheroidal harmonic functions, with
oblateness parameter $a\omega$, which are eigenfunctions of a self-adjoint
operator (see, e.g., (2.7) in [22] or (3.74) in [6]), i.e.,
${}_{s}S_{\ell,a\omega}^{m}$ are subject to

$\overline{\eth}\eth{}_{s}S_{\ell,a\omega}^{m}+\left[2\,s\,(a\,\omega\cos\vartheta-1)+a^{2}\,\omega^{2}\cos^{2}\vartheta-{}_{s}E_{\ell,a\omega}^{m}\right]{}_{s}S_{\ell,a\omega}^{m}=0\,.$
(2.11)

The eigenvalues, ${}_{s}\lambda_{\ell,\omega}^{m}$ and
${}_{s}E_{\ell,a\omega}^{m}$, are related as
${}_{s}\lambda_{\ell,\omega}^{m}={}_{s}E_{\ell,a\omega}^{m}+a^{2}\,\omega^{2}-2am\omega-s(s+1)$
and $K=(r^{2}+a^{2})\,\omega-a\,m$. In (2.11) the $\eth$ and $\overline{\eth}$
operators were used which, when acting on a spin-$s$ field $f$, are defined
via the relations

$\displaystyle\eth f$
$\displaystyle=-\sin^{s}\vartheta\left(\partial_{\vartheta}+\frac{\mathrm{i}}{\sin\vartheta}\partial_{\phi}\right)(\sin^{-s}\\!\vartheta\cdot
f)$ (2.12) $\displaystyle\overline{\eth}f$
$\displaystyle=-\sin^{-s}\vartheta\left(\partial_{\vartheta}-\frac{\mathrm{i}}{\sin\vartheta}\partial_{\phi}\right)(\sin^{s}\\!\vartheta\cdot
f)\,.$ (2.13)

In terms of $\eth$ and $\overline{\eth}$ the “generalized” two-dimensional
Laplace-Beltrami operator reads as

$\overline{\eth}\eth
f=\partial_{\vartheta\vartheta}f+\cot\vartheta\,\partial_{\vartheta}f+\frac{1}{\sin^{2}\vartheta}\,\partial_{\phi\phi}f+2\,\mathrm{i}\,s\,\frac{\cot\vartheta}{\sin\vartheta}\,\partial_{\phi}f+s\,(1-s\cot^{2}\vartheta)f\,.$
(2.14)

Returning to the main line of the argument, recall first that, using the
Starobinskii-Teukolsky identity [19, 20, 22], along with the tortoise radial
coordinate $r_{*}$, determined by the relation
$dr/dr_{*}=\Delta/(r^{2}+a^{2})$, the “physical solutions” to (2.10) are
supposed to possess the asymptotic behavior

${R}_{\ell,\omega}^{m}\sim\begin{cases}\mathcal{I}\,e^{-\mathrm{i}\,\omega\,r_{*}}+\mathcal{R}\,e^{+\mathrm{i}\,\omega\,r_{*}}&\mathrm{as}\;\;r\rightarrow\infty\\\
\mathcal{T}\,e^{-\mathrm{i}\,(\omega-m\,\Omega_{+})\,r_{*}}&\mathrm{as}\;\;r\rightarrow
r_{+}\,,\end{cases}$ (2.15)

where $r_{+}=M+\sqrt{M^{2}-a^{2}}$ and $\Omega_{+}=a/(2Mr_{+})$ denote the
(outer) horizon radius and the angular velocity of the Kerr black hole with
respect to the asymptotically stationary observers, respectively. The factors
$\mathcal{I}$, $\mathcal{R}$, and $\mathcal{T}$ denote the incident,
reflection and transmission coefficients at infinity and at the horizon,
respectively. It is important to note and keep in mind that the above boundary
condition at the horizon explicitly assumes the existence of a transmitted
wave submerging deeply into the ergoregion.

In this setup by making use of equations (2.10) and (2.11), along with the
asymptotic behavior as fixed in (2.15), one can determine the energy fluxes
carried by the fields from the ergoregion toward infinity. The most striking
implication (see section IV. in [22], and also subsection 3.6.2 in [6] for
details) is that, regardless of the value $s=0,\pm 1,\pm 2$ of the spin of
bosonic fields, the total energy flux per unit solid angle at the horizon is
always proportional to the product
$\omega(\omega-m\,\Omega_{+})\,|\mathcal{T}|^{2}$, i.e.,

$\left.\frac{d^{2}\mathscr{E}}{dtd\Omega}\right|_{r_{+}}\sim\omega(\omega-m\,\Omega_{+})\,|\mathcal{T}|^{2}\,.$
(2.16)

The pertinent positive factors of proportionality—which do not play a primary
role in the present argument—can be read off equations (4.26), (4.32) and
(4.44) in [22]. In accordance to (2.16) whenever $|\mathcal{T}|$ does not
vanish and the inequality

$0<\omega<m\,\Omega_{+}$ (2.17)

holds energy can be extracted by back-scattered individual modes of scalar,
electromagnetic or gravitational perturbations due to their interaction with
the Kerr background.

###  2.3 Initialization of to be superradiant solutions

In preparing frequency-tuned incident wave packets we shall adapt the basic
ideas applied in [9, 16]. Accordingly, we assume that in the asymptotic
region—at least in a sufficiently small neighborhood of the initial data
surface—the solution may be approximated as

$\psi^{(s)}(t,r_{*},\vartheta,\phi)\approx
A\cdot\exp\left[-\mathrm{i}\,\omega_{0}\,(r_{*}-r_{*0}+t)\right]f(r_{*}-r_{*0}+t)\,{}_{s}Y{}_{\ell}^{m}(\vartheta,\phi)\,.$
(2.18)

where $f$ is a smooth function of compact support, and where $A$,
$\omega_{0}$, and $r_{*0}$ are suitable real parameters ($r_{*0}$ is simply to
provide a shift in the tortoise coordinate). Note that though in (2.18) a pure
${}_{s}Y{}_{\ell}^{m}$-mode is specified this does not restrict the generality
of the evolving field as a systematic hierarchy of additional $\ell$-modes
(thought each of them possessing the same $m$-parameter as the initial pure
one) get to be invoked immediately via mode couplings in (2.2). The
corresponding initial data reads as

$\displaystyle\psi^{(s)}(r_{*},\vartheta,\phi)={}$ $\displaystyle
A\cdot\exp\left[-\mathrm{i}\,\omega_{0}\,(r_{*}-r_{*0})\right]f(r_{*}-r_{*0})\,{}_{s}Y{}_{\ell}^{m}(\vartheta,\phi)\,,$
(2.19) $\displaystyle\psi^{(s)}_{t}(r_{*},\vartheta,\phi)={}$ $\displaystyle
A\cdot\exp\left[-\mathrm{i}\,\omega_{0}\,(r_{*}-r_{*0})\right]\,f^{\prime}(r_{*}-r_{*0})\,{}_{s}Y{}_{\ell}^{m}(\vartheta,\phi)-\mathrm{i}\,\omega_{0}\,\psi^{(s)}(r_{*},\vartheta,\phi)\,,$
(2.20)

where $f^{\prime}$ denotes the first derivative of $f$. The Fourier transform,
${\mathscr{F}}\hskip-1.42262pt\psi^{(s)}$, of the approximate solution (2.18)
is of the form

${\mathscr{F}}\hskip-1.42262pt\psi^{(s)}(\omega,r_{*},\vartheta,\phi)\approx
A\cdot\exp\left[-\mathrm{i}\,\omega\,(r_{*}-r_{*0})\right]{\mathscr{F}}\hskip-2.84526ptf(\omega-\omega_{0})\,{}_{s}Y{}_{\ell}{}^{m}(\vartheta,\phi),$
(2.21)

where $\omega$ is the temporal frequency and ${\mathscr{F}}\hskip-2.84526ptf$
stands for the Fourier-transform of $f$. Accordingly,
${\mathscr{F}}\hskip-2.84526ptf$ plays the role of a frequency profile, which
guarantees that whenever ${\mathscr{F}}\hskip-2.84526ptf$ is chosen to be
sufficiently narrow the solution (2.18) in the asymptotic region is well-
approximated by an incident monochromatic wave packet, which by choosing
$\omega_{0}$ from the interval $0<\omega_{0}<m\,\Omega_{+}$ gets to be
superradiant.

##  3 More on the analytic background

This section is to discuss issues related to the regularization of basic
variables, the new type of energy and angular momentum balance relations, the
applied multipole expansion and the initialization of the investigated
solutions.

###  3.1 Regularization

Solutions to the homogeneous Teukolsky master equation (2.2) are known to be
singular either in the $\Delta\rightarrow 0$ or $r\rightarrow\infty$ limit. In
order to regularize the spin $s$ fields $\psi^{(s)}$ new coordinates were
introduced in [10] in two steps. First, the Boyer-Lindquist coordinates,
$(t,r,\vartheta,\phi)$, were replaced by the horizon penetrating ingoing Kerr
coordinates, $(\tau,r,\vartheta,\varphi)$, where $\tau$ and $\varphi$, are
determined via

$\tau=t-r+\int
dr\frac{r^{2}+a^{2}}{\Delta}\qquad\mathrm{and}\qquad\varphi=\phi+\int
dr\frac{a}{\Delta}\,.$ (3.1)

In the second step new coordinates $(T,R)$ were introduced, replacing
$(\tau,r)$, via the relations

$\tau=T+\frac{1+R^{2}}{1-R^{2}}-4M\log(|1-R^{2}|)\qquad\mathrm{and}\qquad
r=\frac{2R}{1-R^{2}}\,.$ (3.2)

Note that in these new coordinates future null infinity, $\mathscr{I}^{+}$, is
represented by the $R=1$ hypersurface, and also that each of the $T=const$
hypersurfaces intersect both the horizon and $\mathscr{I}^{+}$ in regular
spherical cuts. In our numerical investigations the regularized field
variables, $\Phi^{(s)}$, were used which are defined as

$\Phi^{(s)}(T,R,\vartheta,\varphi)=[\,r(R)\cdot\Delta^{s}(R)\,]\cdot\psi^{(s)}(T,R,\vartheta,\varphi)\,.$
(3.3)

In terms of these regularized variables the homogeneous Teukolsky master
equation (2.2) was shown to take the form [10]

$\displaystyle\partial_{TT}\Phi^{(s)}=\frac{1}{\mathscr{A}+\mathscr{B}\cdot
Y_{2}^{0}}\,\Big{[}c_{RR}\cdot\partial_{RR}\Phi^{(s)}+c_{TR}\cdot\partial_{TR}\Phi^{(s)}+c_{T\varphi}\cdot\partial_{T\varphi}\Phi^{(s)}+c_{R\varphi}\cdot\partial_{R\varphi}\Phi^{(s)}$
$\displaystyle+c_{\vartheta\vartheta}\cdot\overline{\eth}\eth\,\Phi^{(s)}+c_{T}\cdot\partial_{T}\Phi^{(s)}+\mathrm{i}\,c_{Ty}\,Y_{1}^{0}\cdot\partial_{T}\Phi^{(s)}+c_{R}\cdot\partial_{R}\Phi^{(s)}+c_{\varphi}\cdot\partial_{\varphi}\Phi^{(s)}+c_{0}\cdot\Phi^{(s)}\Big{]}\,,$
(3.4)

where $Y_{1}^{0}$ and $Y_{2}^{0}$ are the standard (spin $s=0$) spherical
harmonics with $\ell=1,2$ and $m=0$, respectively. The explicit form of the
involved $R$-dependent coefficients can be found in Appendix A of [10].

###  3.2 The conserved currents

In case of scalar perturbations the canonical energy and angular momentum are
completely satisfactory to describe the studied processes. Yet, if $s\not=0$
we have to resort to more complicated formulas which reduce to the scalar case
when s=0. It was shown in [23] that meaningful Lagrangian can be associated
with an arbitrary pair of spin $s$ and $-s$ fields. More importantly, sensible
conserved energy and angular momentum type currents were also introduced in
[23] via the relations

$\displaystyle\mathcal{E}^{a}$
$\displaystyle=(\nabla^{a}-s\Gamma^{a})\psi^{(-s)}T^{b}\partial_{b}\psi^{(s)}+(\nabla^{a}+s\Gamma^{a})\psi^{(s)}T^{b}\partial_{b}\psi^{(-s)}+T^{a}\mathcal{L}\,,$
(3.5) $\displaystyle\mathcal{J}^{a}$
$\displaystyle=(\nabla^{a}-s\Gamma^{a})\psi^{(-s)}\varphi^{b}\partial_{b}\psi^{(s)}+(\nabla^{a}+s\Gamma^{a})\psi^{(s)}\varphi^{b}\partial_{b}\psi^{(-s)}+\varphi^{a}\mathcal{L}\,.$
(3.6)

Here $T^{a}=(\partial_{T})^{a}$ and $\varphi^{a}=(\partial_{\varphi})^{a}$ are
Killing vector fields generating the underlying one-parameter group of
diffeomorphisms. These currents are always conserved when $\psi^{(s)}$ is a
solution to the Teukolsky equation with spin-weight $s$ and $\psi^{(-s)}$ is a
solution with spin-weight $-s$. The two solutions otherwise may be completely
unrelated. In order to attain quantities which can be viewed as if they
characterize a single solution we use a solution $\psi^{(s)}$, along with its
adjoint, to get a proper pair. The motivation and the process of determining
the adjoint solution is explained in details in the Appendix of the present
paper.

Since the covariant divergence of these currents vanish the balance relations

$\int_{\Omega}\nabla_{a}\mathcal{E}^{a}=\int_{\partial\Omega}n_{a}\mathcal{E}^{a}=0\quad\mathrm{and}\quad\int_{\Omega}\nabla_{a}\mathcal{J}^{a}=\int_{\partial\Omega}n_{a}\mathcal{J}^{a}=0$
(3.7)

hold. In the present paper the domain of integration $\Omega$ will always be
the rectangular coordinate domain in $(T,R)$ which is bounded by the initial
and final time slices, $T=T_{i}$ and $T=T_{f}$, and also by the null cylinders
at $R=R_{+}$ and $R=1$, respectively.

The superradiant amplifications can then be characterized by comparing the
energy and angular momentum of the incident wave packet

$E_{0}=\int_{T=T_{i}}n_{a}^{(T)}\mathcal{E}^{a}\sqrt{|h_{T}|}\,\,dR\,\mathrm{d}\vartheta\,\mathrm{d}\varphi\quad\mathrm{and}\quad
J_{0}=\int_{T=T_{i}}n_{a}^{(T)}\mathcal{J}^{a}\sqrt{|h_{T}|}\,\,dR\,\mathrm{d}\vartheta\,\mathrm{d}\varphi$
(3.8)

to the energy and angular momentum radiated toward null infinity

$E_{out}=\int_{R=1}n_{a}^{(R)}\mathcal{E}^{a}\sqrt{|h_{R}|}\,dT\,\mathrm{d}\vartheta\,\mathrm{d}\varphi\quad\mathrm{and}\quad
J_{out}=\int_{R=1}n_{a}^{(R)}\mathcal{J}^{a}\sqrt{|h_{R}|}\,dT\,\mathrm{d}\vartheta\,\mathrm{d}\varphi\,,$
(3.9)

respectively, where $n_{a}^{(T)}=(dT)_{a}/\sqrt{g^{TT}}$ and
$n_{a}^{(R)}=(dR)_{a}/\sqrt{-g^{RR}}$ are the respective normals of the
$T=const$ and $R=const$ hypersurfaces and $h_{T}$ and $h_{R}$ denote the
determinant of the metric induced on these hypersurfaces. Accordingly, if
energy and angular momentum gains occur then $E_{out}>E_{0}$ and
$J_{out}>J_{0}$ and such a situation will always be characterized by the scale
independent ratios

$\delta E=\frac{E_{0}-E_{out}}{E_{0}}\quad\mathrm{and}\quad\delta
J=\frac{J_{0}-J_{out}}{J_{0}}\,,$ (3.10)

respectively. In practice, as $E_{0},E_{out},J_{0},J_{out}$ are all complex
apart from the scalar field case with $s=0$, therefore we always used the
absolute values of $E_{0},E_{out},J_{0},J_{out}$ in evaluating (3.10).

The energy and angular momentum transports do provide important hints on the
histories of the investigated configurations. It has been demonstrated in
earlier studies (see Figs. 12-14 of [13]) that plotting the “volume
normalized” energy and angular momentum current densities is more informative
than the figures showing the energy and angular momentum current densities
themselves. In the present case the use of a slightly different notion of
volume normalized current densities, $\mathscr{E}=\mathscr{E}(T,R)$ and
$\mathscr{J}=\mathscr{J}(T,R)$, turned out to be more appropriate, which are
defined by the integrals

$\mathscr{E}=\int_{0}^{\pi}\int_{0}^{2\pi}n_{a}^{(R)}\mathcal{E}^{a}\sqrt{|h_{R}|}\,\,\mathrm{d}\vartheta\,\mathrm{d}\varphi\,,\quad\mathscr{J}=\int_{0}^{\pi}\int_{0}^{2\pi}n_{a}^{(R)}\mathcal{J}^{a}\sqrt{|h_{R}|}\,\,\mathrm{d}\vartheta\,\mathrm{d}\varphi\,.$
(3.11)

Accordingly, the pure $T$-integrals, $\int_{R}\mathscr{E}(T,R)\,\mathrm{d}T$
and $\int_{R}\mathscr{J}(T,R)\,\mathrm{d}T$, of these current densities, over
specific $R=const$ timelike hypersurfaces give us the energy and angular
momentum fluxes through these $R=const$ hypersurfaces, respectively. Recall
that, in the general case with $s\not=0$, likewise the conserved currents
$\mathcal{E}^{a}$ and $\mathcal{J}^{a}$, the volume normalized energy and
angular momentum current densities have real and imaginary parts both.
Nevertheless, in what follows, we shall plot only the real parts which, in the
considered cases, are always several order larger than the corresponding
imaginary parts.

###  3.3 Multipole expansion

In solving (3.1) the angular derivatives were evaluated analytically by making
use of the operators $\partial_{\varphi}$, $\eth$ and $\overline{\eth}$ by
expanding the dynamical field $\Phi^{(s)}$ in terms of spin-weighted spherical
harmonics as

$\Phi^{(s)}(T,R,\vartheta,\varphi)=\sum_{\ell=|s|}^{\ell_{max}}\sum_{m=-\ell}^{\ell}\phi_{\ell}{}^{m}(T,R)\cdot{}_{s}{Y_{\ell}}{}^{m}(\vartheta,\varphi)\,.$
(3.12)

Then (3.1) can be replaced by a set of coupled $(1+1)$-dimensional linear wave
equations for the expansion coefficients $\phi_{\ell}{}^{m}(T,R)$. The value
of $\ell_{max}$ in (3.12) was chosen to be suitably large in order to keep the
truncation error at the level of numerical precision.

In solving (3.1) standard order reduction techniques were also applied by
introducing $(\phi_{T}){}_{\ell}{}^{m}=\partial_{T}\phi_{\ell}{}^{m}$ and
$(\phi_{R}){}_{\ell}{}^{m}=\partial_{R}\phi_{\ell}{}^{m}$ as our new
variables. The corresponding first order strongly hyperbolic system—relevant
for the vector valued variable
$(\phi_{\ell}{}^{m},(\phi_{R}){}_{\ell}{}^{m},(\phi_{T}){}_{\ell}{}^{m})^{T}$—was
solved numerically by applying the method of lines in a $4^{th}$ order Runge-
Kutta integrator, along with a $6^{th}$ order dissipation term to suppress
high frequency spurious solutions [14].

###  3.4 The choice made for the initial data

In solving (3.1) it is necessary to specify, on a $T=T_{0}\,(\in\mathbb{R})$
initial data surface, a pair of functions $(\phi^{(s)},\phi_{T}^{(s)})$ such
that $\phi^{(s)}=\Phi^{(s)}|_{T=T_{0}}$ and
$\phi_{T}^{(s)}=(\partial_{T}\Phi^{(s)})|_{T=T_{0}}$.

In choosing initial data for the regularized field $\Phi^{(s)}$ we shall
economize the choice we have already discussed in subsection 2.3. In doing so
note first that by combining the coordinate transformations applied in
subsection 3.1 the relations

$t+r_{*}-r_{*0}=\tau+r-r_{*0}=T+\frac{1+R^{2}}{1-R^{2}}-4M\log(|1-R^{2}|)+\frac{2R}{1-R^{2}}-R_{0}$
(3.13)

can be seen to hold, where the $r_{*0}$ shift in the radial tortoise
coordinate, in the last step, is simply replaced by $R_{0}$ that is its
suitable correspondent. Denoting the right hand side of (3.13) by
$H(T,R;R_{0})$ and using the above replacement in (2.18), along with the
regularization (3.3), an approximate incident wave packet is expected to have
the form

$\displaystyle\Phi^{(s)}(T,R,\vartheta,\varphi)={}$
$\displaystyle\mathcal{A}(R)\cdot\psi^{(s)}(T,R,\vartheta,\varphi)$
$\displaystyle\approx{}$
$\displaystyle\mathcal{A}(R)\cdot\exp\left[-\mathrm{i}\,\omega_{0}\,H(T,R;R_{0})\right]f(H(T,R;R_{0}))\,{}_{s}Y{}_{\ell}^{m}(\vartheta,\varphi)\,,$
(3.14)

where $\mathcal{A}$ denotes the product $A\cdot[\,r(R)\cdot\Delta^{s}(R)\,]$
of the amplification factors in (2.18) and (3.3). Accordingly, the
corresponding initial data $(\phi^{(s)},\phi_{T}^{(s)})$ read as

$\displaystyle\phi^{(s)}(R,\vartheta,\varphi)={}$
$\displaystyle\mathcal{A}(R)\cdot\exp\left[-\mathrm{i}\,\omega_{0}\,H_{0}\right]f(H_{0})\,{}_{s}Y{}_{\ell}^{m}(\vartheta,\varphi)$
(3.15) $\displaystyle\phi_{T}^{(s)}(R,\vartheta,\varphi)={}$
$\displaystyle\mathcal{A}(R)\cdot\exp\left[-\mathrm{i}\,\omega_{0}\,H_{0}\right]\,f^{\prime}(H_{0})\,{}_{s}Y{}_{\ell}^{m}(\vartheta,\phi)-\mathrm{i}\,\omega_{0}\,\phi^{(s)}(R,\vartheta,\varphi)\,,$
(3.16)

where $f^{\prime}$ denotes the first order derivative of $f$ and
$H_{0}=H_{0}(R)$ stands for $H(T_{0},R;R_{0})$.

In all of our numerical simulations the radial profile, $f$, was chosen as

$\displaystyle\hskip-8.5359pt{f(x)=\left\\{\begin{array}[]{r
l}\exp\left[4-\left|\frac{w_{1}}{x+\frac{w_{1}}{2}}\right|-\left|\frac{w_{2}}{x-\frac{w_{2}}{2}}\right|\right]\,,&\mathrm{if}\,x\in[-\frac{w_{1}}{2},\frac{w_{2}}{2}]\\\
0\,,&\mathrm{otherwise}\,,\end{array}\right.}$ (3.19)

which is a smooth real function of the real variable $x$ that is vanishing in
the complement of the compact support $[-{w_{1}}/{2},{w_{2}}/{2}]$.

As indicated in the introduction we shall apply three different types of
initial data. The radial profile of each of these are of compact support. The
support of type-I data is clearly separated from the ergoregion, whereas for
type-II data the support is concentrated to the ergoregion. For type-III data
the radial supports inter-bridge the domains of type-I and type-II data.
Accordingly, they start close to the event horizon and cover, across the
ergoregion, larger and larger parts of the domain of outer communication. Some
of the typical $f(H_{0}(R))$ radial profiles are depicted on the panels of
Fig. 1.

(a) Type-I and type-II radial profiles.

(b) Type-III radial profiles.

Figure 1: On the left the radial profiles relevant for type-I and type-II
initial data are plotted, respectively. On the right the profiles pertinent
for type-III data are shown. The two dashed vertical lines are to indicate the
location of the horizon and the maximal radius of the ergoregion.

The compact support of type-I data is visibly separated form the ergoregion
while it is concentrated to the ergoregion for type-II data. On the left panel
the two type-I radial profiles correspond to the specific values
$R_{1}=0.8,R_{0}=0.9,R_{2}=0.95$ and $R_{1}=0.93,R_{0}=0.95,R_{2}=0.98$,
respectively, whereas the corresponding parameters for type-II profile are
$R_{1}=0.454,R_{0}=0.5,R_{2}=0.61$, where $R_{1}$ and $R_{2}$ denote the left
and right edge of the compact support, while $R_{0}$ is to characterize the
asymmetry of the profiles. For symmetric profiles $R_{0}$ also signifies the
location of the maximum value, which, for these symmetric profiles, is at the
mean value of $R_{1}$ and $R_{2}$, i.e. $R_{0}=(R_{1}+R_{2})/2$. The graphs of
type-III radial profiles the common left boundary is at $R_{1}=0.4536$,
whereas the right edges are at $R_{2}=0.7,0.76,0.84,0.95$, respectively.

It is important to emphasize that analogous choices of radial profiles allow
the proper frequency tuning of the solutions, and, in turn, to get the
incident wave packets to be superradiant at least near to the initial data
surface. We emphasize that the $\omega_{0}$ parameter in general differs from
the actual frequency of the solution, however in the asymptotic region it
proves to be a good approximation. This expectation will be verified by
plotting the power spectra

$\mathcal{PS}(\omega,R)=\int_{0}^{\pi}\int_{0}^{2\pi}|\mathscr{F}\Phi^{(s)}|^{2}(\omega,R,\vartheta,\varphi)\,\sin\vartheta\,\mathrm{d}\vartheta\,\mathrm{d}\varphi$
(3.20)

for many of the inspected solutions.

##  4 Numerical results

In proceeding recall that in subsection 2.2 necessary conditions guaranteeing
the occurrence of superradiance were identified. Namely, it was found that
superradiant scatterings cannot occur unless the frequency profile of the
incident wave packet is located in the interval $0<\omega<m\Omega_{+}$, as
well as, it submerges deeply into the ergoregion. We shall see that both of
these conditions are of equal importance. If either of them fails to be
satisfied the product $\omega(\omega-m\,\Omega_{+})\,|\mathcal{T}|^{2}$ on the
rhs of (2.16) may simply be small or, as seen below, in case of type-I spin
$s=0$ fields, practically zero.

###  4.1 Results relevant for type-I initial data

The compact support of type-I initial data is always well-separated from the
ergoregion as indicated on the left panel of Fig. 1. It can be characterized
by three parameters $R_{1},R_{2}$, and $R_{0}$, where $R_{1}$ and $R_{2}$
denote the left and right edge of the compact support, whereas $R_{0}$
signifies the approximate location of the maximum value. In almost all of the
simulations with type-I data these parameters are fixed as
$R_{1}=0.93,R_{0}=0.95,R_{2}=0.98$. Unless otherwise stated, the ADM mass and
specific angular momentum parameters of the Kerr black hole are $M=1$ and
$a=0.99$, respectively. Note also that when the $T-R$\- or
$\omega-R$-dependencies of various quantities are plotted some of the
$R=const$ dark grey dashed lines are always to indicate the equatorial radius
$R_{E}$ of the ergoregion and the location, $R_{+}$, of the black hole
horizon. For $M=1$ and $a=0.99$ the specific values are $R_{E}=0.618$ and
$R_{+}=0.4533$. The mean value, $\widetilde{\omega}$, of the frequency profile
of the applied initial data, (3.15) and (3.16), is expected to be below but
close to the upper bound $m\,\Omega_{+}$. The specific numerical values of
$\omega_{0}$ which is used in producing the specific spectra will always be
given in each of the reported cases.

####  4.1.1 Scalar field

Beside $R_{1},R_{2}$, and $R_{0}$ determining the radial profile the two
solutions inspected in details are characterized by the physical parameters
$s=0,\omega_{0}=0.3,\ell=1,m=\pm 1$. Note that the sign of the “magnetic
quantum number” $m$ has a simple interpretation. Configurations with $m>0$ are
corotating, whereas with $m<0$ they are counterrotating with respect to the
background Kerr black hole. Before inspecting the pertinent energy and angular
momentum transports, it is rewarding to have a glance of the power spectra,
$\mathcal{PS}=\mathcal{PS}(\omega,R)$, of the corresponding solutions depicted
by the panels in Fig. 2.

(a) $m=+1$

(b) $m=-1$

Figure 2: The power spectra, $\mathcal{PS}(\omega,R)$, of solutions with
$m=+1$ and $m=-1$ are shown on the left and right panel, respectively. The
other characteristic parameters are $s=0$, $\ell=1,\omega_{0}=0.3$.

Notice that on the left panel with $m=+1$ the power spectra of the incident
wave packet is located—for all values of $R$ not only in the vicinity of the
initial data surface where the frequency was tuned—in the desired superradiant
interval, $0<\omega<m\,\Omega_{+}$. It is also remarkable that instead of
going down to the black hole horizon the power spectra starts to be fading
away as the ergoregion is approached. This behavior is confirmed by the
corresponding energy and angular momentum transports plotted on the left panel
of Fig. 3.

The power spectra of the counterrotating solution, with $m=-1$, and with the
same physical parameters, $s=0$, $\ell=1$ $\omega_{0}=0.3$, is in the interval
$m\,\Omega_{+}<\omega<0$ which is the reflection of the superradiant range to
the $\omega=0$ axis. As verified by the corresponding energy and angular
momentum transports plotted on the right panel of Fig. 3 considerably large
part of the energy and angular momentum of the solution falls into the black
hole directly.

(a) Solution with $s=0,\ell=1,m=+1$.

(b) Solution with $s=0,\ell=1,m=-1$.

Figure 3: The volume normalized energy current densities,
$\mathscr{E}=\mathscr{E}(T,R)$, are plotted for solutions with parameters
$s=0,\omega_{0}=0.3,\ell=1,m=\pm 1$. The corresponding values of the angular
momentum current densities, $\mathscr{J}=\mathscr{J}(T,R)$, are indicated at
the colored contours.

On Fig. 3 the volume normalized energy and angular momentum current densities,
$\mathscr{E}=\mathscr{E}(T,R)$ and $\mathscr{J}=\mathscr{J}(T,R)$, are
depicted. As the angular momentum current densities are almost identical in
their functional behavior to that of the energy current densities they will
always be indicated by using colored contours and numbers signify their values
at the pertinent level sets. The left panel of Fig. 3, corresponding to the
corotating solution with $m=+1$, indicates that essentially all the energy and
angular momentum content of the incident wave packet bounces back before
entering the ergoregion. As opposed to this on the right panels, corresponding
to the counterrotating solution with $m=-1$, apparently almost all the energy
and angular momentum content of the incident wave packet falls directly into
the black hole.

(a) The energy gain.

(b) The angular momentum gain.

Figure 4: The time dependence of the energy and angular momentum gains,
$\delta E=(E_{0}-E_{out})/E_{0}$ and $\delta J=(J_{0}-J_{out})/J_{0}$, are
plotted for the investigated solutions with $m=+1$ and $m=-1$.

All the foregoing observations are also confirmed by the panels of Fig. 4
depicting the respective time dependence of the energy and angular momentum
gains, $\delta E=(E_{0}-E_{out})/E_{0}$ and $\delta J=(J_{0}-J_{out})/J_{0}$.
In virtue of the graphs on the panels of Fig. 4 for the solution with
$m=+1$—in accordance with the left panel of Fig. 3—the incident wave packet go
through (in accordance with the claims in [9, 16]) an almost perfect
reflection. Accordingly, no energy or angular momentum fall into the black
hole, i.e. the energy and angular momentum content leave the domain through
null infinity. As opposed to this, the counterrotating solution with $m=-1$
about $16\%$ of the energy and about $17\%$ of the angular momentum contained
by the incident wave packet leaves the domain through ${\mathscr{I}}^{+}$
while the rest is falling directly into the black hole as it is visible on the
right panel of Fig. 3.

It is of interest to know what happens if for a type-I counterrotating
solution $\omega_{0}$ is replaced by $-\omega_{0}$. One may suspect that this
will shift the power spectra of the pertinent solution into the superradiant
range. To see that indeed this happens follows from the choice we made for the
initial data $(\phi^{(s)},\phi_{T}^{(s)})$, in (3.15) and (3.16), and by
applying the relation
$\overline{{}_{s}Y{}_{\ell}{}^{m}}=(-1)^{s+m}\cdot{}_{-s}Y{}_{\ell}{}^{-m}$.
It is straightforward to see then that the initial data of a counterrotating
scalar configuration with $-\omega_{0}$ is $(-1)^{m}$ times of the complex
conjugate of the initial data of a corotating solution with $\omega_{0}$. For
this reason the insight provided by the inspection of time evolution of scalar
initial data yielded by the simultaneous replacements $m\rightarrow-m$,
$\omega_{0}\rightarrow-\omega_{0}$ is very limited. Note also that for type-I
counterrotating solutions with positive $\omega_{0}$ the power spectra is
always out of the superradiant range. Accordingly, their behavior does not
differ significantly from those of the corresponding counter-rotating
solutions. For this reason, in the rest of subsection 4.1, we shall not
investigate these type of configurations either. As seen later counterrotating
solutions do not play significant role in case of type-II configurations
either, nevertheless, they will be investigated in some extent in subsection
4.2.

####  4.1.2 Outgoing electromagnetic and gravitational perturbations

The spin $s=+1$ electromagnetic and spin $s=+2$ gravitational perturbations
are usually referred to as outgoing modes. In this subsection, for both cases,
the applied radial profile is fixed by the values
$R_{1}=0.8,R_{0}=0.9,R_{2}=0.95$, where $R_{1}$ and $R_{2}$ denote the left
and right edge of the compact support (see Fig. 1). Since for the positive
spin value there is no significant difference between the functional behavior
of the electromagnetic and gravitational perturbations they are treated
together in this subsection. The wave packets were frequency tuned as it is
depicted by the power spectras, $\mathcal{P}S$, plotted on Fig. 5. It is
clearly visible on both panels of Fig. 5 that the incident wave packets are
well-tuned to be superradiant.

(a) $\mathcal{PS}$ for solution with $s=+1,m=+1$.

(b) $\mathcal{PS}$ for solution with $s=+2,m=+2$

Figure 5: The power spectra of the solutions with parameters
$s=+1,\omega_{0}=0.21339,\ell=1,m=+1$ and
$s=+2,\omega_{0}=0.21339,\ell=2,m=+2$ are shown on the left and right panel,
respectively.

The corresponding energy and angular momentum transports (see the panels of
Fig. 6)

(a) $\mathscr{E}$ and $\mathscr{J}$ with $s=+1$ and $m=+1$.

(b) $\mathscr{E}$ and $\mathscr{J}$ with $s=+2$ and $m=+2$.

Figure 6: The normalized energy and angular momentum current densities,
$\mathscr{E}(T,R)$ and $\mathscr{J}(T,R)$, are plotted for outgoing
electromagnetic and gravitational perturbations with physical parameters
$s=+1,\ell=1,m=+1,\omega_{0}=0.21339$ and
$s=+2,\ell=2,m=+2,\omega_{0}=0.21339$, respectively.

do also indicate that the $s=+1$electromagnetic and $s=+2$ gravitational
perturbations behave analogously. In accordance with their reputation, for
these type-I configurations the wave packets start to move directly toward
null infinity and they quickly leave the domain of outer communication. Note
also that this does not allow us to have energy and angular-momentum gains in
either case. It is also worth emphasizing that all the foregoing observations
are unaffected by the concrete value of the spin as only the outgoing
character of considered electromagnetic and gravitational perturbations what
matters. The above conclusions are insensitive to the sign of $m$, i.e. to the
co- or counterrotating character of these perturbations.

In closing this section it is informative to have a glance of the energy and
angular momentum balances. The right panel of Fig. 7 is to verify the fourth
order convergence, in the $T-R$ section, of the real part of the energy
balance (see also the analogous figures, e.g. Fig. 4, in [10]).

(a) The convergence rate in the $T-R$ section is verified by plotting the real
part of the energy balance for the indicated two radial resolutions.

(b) The relative error $E_{\ell_{max},\Delta\ell_{max}}(\Phi^{(-1)})$ is
plotted, for the indicated $\ell_{max}$ values, at the location $R=0.8$ and on
the interval $0\leq T\leq 100$.

Figure 7: The convergence rates in the $T-R$ and in the angular sections are
verified for the co-rotating solution—with parameters
$s=+1,\omega_{0}=0.21339,\ell=1$ and $m=+1$, respectively. The convergence
rate in the $T-R$ section is of $4^{th}$ order whereas in the angular section
it is exponential.

The convergence rate in the angular section is also of obvious interest.
This—in applying the multipole expansions, as described in section 3—can be
monitored by varying the value of $\ell_{max}$, i.e., the value of $\ell$
where the infinite series is truncated in (3.12). To characterize the
$\ell_{max}$ dependence the relative error of the basic variable $\Phi^{(s)}$,
defined via the ratio,

$E_{\ell_{max},\Delta\ell_{max}}(\Phi^{(s)})=\frac{||\Phi^{(s)}_{\ell_{max}}-\Phi^{(s)}_{\ell_{max}+\Delta\ell_{max}}||}{||\Phi^{(s)}_{\ell_{max}+\Delta\ell_{max}}||}$
(4.21)

is monitored, where $||\cdot||$ denotes the $C^{0}$ norm. On the right panel
of Fig. 7 the relative error, relevant for the sequence of values
$\ell_{max}=4,6,8,10$, is plotted for an outgoing electromagnetic perturbation
with physical parameters $s=+1,\omega_{0}=0.21339,\ell=1,m=+1$ for the time
interval $0\leq T\leq 100$. The apparent linear shift in the succeeding graphs
indicates that the rate of convergence in the angular section is, as expected,
exponential.

####  4.1.3 Ingoing electromagnetic and gravitational perturbations

The applied radial profile is determined here again by the parameters
$R_{1}=0.93,R_{0}=0.95,R_{2}=0.98$ (see Fig. 1). The other physical parameters
are $s=-1,\omega_{0}=0.28,\ell=1,m=+1$ and $s=-2,\omega_{0}=0.65,\ell=2,m=+2$,
respectively. The story is very similar to the one reported in case of ingoing
corotating scalar fields. The power spectra
$\mathcal{PS}=\mathcal{PS}(\omega,R)$ of the ingoing electromagnetic and
gravitational perturbations are shown on the two panels of Fig. 8.

(a) $\mathcal{PS}$ for solution with $s=-1,m=+1$.

(b) $\mathcal{PS}$ for solution with $s=-2,m=+2$.

Figure 8: The power spectra of ingoing and corotating electromagnetic and
gravitational perturbations—with physical parameters
$s=-1,\omega_{0}=0.28,\ell=1,m=+1$ and $s=-2,\omega_{0}=0.65,\ell=2,m=+2$, are
plotted, respectively. Notice that considerably large fraction of both of the
incident wave packets, are in the superradiant regime and remain there for the
entire evolution.

The power spectra for the corotating electromagnetic and gravitational
perturbations are located entirely in the superradiant interval
$0<\omega<m\,\Omega_{+}$ not only in the vicinity of the compact support of
the initial data surface.

(a) $\mathscr{E}$ and $\mathscr{J}$ for $s=-1$ and $m=+1$.

(b) $\mathscr{E}$ and $\mathscr{J}$ for $s=-2$ and $m=+2$.

Figure 9: The volume normalized energy and angular momentum current densities,
$\mathscr{E}(T,R)$ and $\mathscr{J}(T,R)$ are plotted for ingoing and
corotating electromagnetic and gravitational perturbations, respectively. The
physical parameters are exactly the same as those applied in producing Fig. 8.

Accordingly, only a small part of the energy and angular momentum of the
incident wave falls into the black hole. There are also small fractions of the
power spectra with slightly larger frequency than $m\,\Omega_{+}$. These, by
determining the power spectra relevant only for the late time period, can be
seen to correspond to the quasinormal oscillations of the electromagnetic and
gravitational perturbations, respectively.

The energy and angular momentum transports are in accordance with the comments
have just made by inspecting the power spectras. On the left and right panels
of Fig. 9 the volume normalized energy and angular momentum current densities,
$\mathscr{E}=\mathscr{E}(T,R)$ and $\mathscr{J}=\mathscr{J}(T,R)$ are shown
for the inspected ingoing and corotating electromagnetic and gravitational
perturbations, respectively.

As before, the angular momentum current densities are indicated only by
colored contours such that the pertinent values are marked by the numbers next
to these contours.

(a) The energy gains.

(b) The angular momentum gains.

Figure 10: The time dependence of the energy and angular momentum gains,
$\delta E=(E_{0}-E_{out})/E_{0}$ and $\delta J=(J_{0}-J_{out})/J_{0}$, are
plotted for the investigated ingoing and corotating electromagnetic and
gravitational perturbations, respectively. As it is indicated by these panels
the gains are still modest as they are about $2\%$ and $32.5\%$ for
electromagnetic and gravitational perturbations, respectively.

Both of the panels verify that considerably large fraction of the energy and
angular momentum of the ingoing, corotating configurations bounced back before
entering to the ergoregion and leave the domain of outer communication via
null infinity $\mathscr{I}^{+}$, that is represented by the $R=1$ surface.
Nevertheless, it is also clearly visible that dynamical processes occur also
in the ergoregion which indicates that superradiance might also occur in these
cases.

Recall that to quantify superradiance the energy and angular momentum gains,
$\delta E=(E_{0}-E_{out})/E_{0}$ and $\delta J=(J_{0}-J_{out})/J_{0}$, are
used. On the two panels of Fig. 10 the time dependence of these quantities are
plotted for the investigated ingoing and co-rotating electromagnetic and
gravitational perturbations, respectively. As noted earlier, if either $\delta
E$ or $\delta J$ gets negative it means that superradiance does indeed occur.
Although the energy and angular momentum gains, $\delta E$ and $\delta J$, are
modest $\sim 2\%$ for electromagnetic perturbations, whereas the scale of
superradiance is about $~{}32.5\%$ for gravitational perturbations.

###  4.2 Results relevant for type-II initial data

In this section type-II configurations will be considered. This means that the
compact support of the initial data and the ergoregion overlap significantly.
Actually, in all the reported cases we shall choose the outer radius of the
compact support to be slightly smaller than the pertinent equatorial radius of
the ergoregion. In practice, the characteristic parameters of the radial
profile of the compact support will be set as $R_{1}=0.4536,R_{2}=0.615$, and
$R_{0}=0.5$, where $R_{1}$ and $R_{2}$ denote the left and right edges of the
compact support (see Fig. 1).

(a) $\mathscr{E}$ and $\mathscr{J}$ with $s=-2$ and $m=+2$.

(b) $\mathscr{E}$ and $\mathscr{J}$ with $s=-2$ and $m=-2$.

Figure 11: The volume normalized energy and angular momentum current
densities, $\mathscr{E}(T,R)$ and $\mathscr{J}(T,R)$ are plotted for ingoing
electromagnetic and gravitational perturbations, respectively. The physical
parameters are $s=-2,\omega_{0}=0.725,\ell=2,m=\pm 2$.

Note also that, as in the previous section, the mass and specific angular
momentum parameters of the Kerr black hole are $M=1$ and $a=0.99$,
respectively. These physical parameters set the horizon radius as
$R_{+}=0.4533$ while the equatorial radius of the ergoregion is $R_{E}=0.618$.

####  4.2.1 Ingoing scalar, electromagnetic and gravitational perturbations

(a) Energy gains: $s=-2,m=\pm 2$.

(b) Angular momentum gains: $s=-2,m=\pm 2$.

Figure 12: The time dependence of the energy and angular momentum gains,
$\delta E=(E_{0}-E_{out})/E_{0}$ and $\delta J=(J_{0}-J_{out})/J_{0}$, are
plotted for the investigated ingoing electromagnetic and gravitational
perturbations, respectively. These panels verify that no gains thereby no
superradiance happens.

The most important difference between the type-I and type-II configurations is
that while the ingoing type-I electromagnetic or gravitational perturbations
were superradiant they do not show superradiance at all when type-II initial
data is used. Instead, the incident wave packets fall directly toward the
black hole region and leave the computational domain through the black hole
horizon.

Accordingly, neither of the power spectra of these type-II ingoing spin $s=0$,
$s=-1$ and $s=-2$ configurations is informative because in each of these cases
the truly dynamical part of the evolution is rather short.

This interpretation is verified by the panels on Fig. 11 where the energy and
angular momentum transports of a gravitational perturbation with physical
parameters $s=-2,\omega_{0}=0.725,\ell=2,m=\pm 2$ are plotted.

These figures are typical in the sense that completely analogous processes
occur in case of ingoing type-II spin $s=0$ and $s=-1$ configurations.

(a) $\mathscr{E}$ and $\mathscr{J}$ with $s=0,m=+1$.

(b) $\mathscr{E}$ and $\mathscr{J}$ with $s=0,m=-1$.

Figure 13: The volume normalized energy and angular momentum current
densities, $\mathscr{E}(T,R)$ and $\mathscr{J}(T,R)$ are plotted for ingoing
scalar perturbations with reversed time derivative and with physical
parameters $s=0,\omega_{0}=0.3,\ell=1,m=\pm 1$.

Exactly the same conclusion can be drawn by inspecting the energy and angular
momentum gains, $\delta E=(E_{0}-E_{out})/E_{0}$ and $\delta
J=(J_{0}-J_{out})/J_{0}$, plotted on the panels of Fig. 12. It is clearly
visible that essentially the entire energy and angular momentum stored by the
initial data leave, in a very short period, the domain of outer communication
via the black hole event horizon.

(a) $\mathcal{PS}$ for solution with $s=0,m=+1$.

(b) $\mathcal{PS}$ for solution with $s=0,m=-1$.

Figure 14: The power spectra of the ingoing scalar perturbations—with reversed
time derivative and with physical parameters $s=0,\omega_{0}=0.3,\ell=1,m=\pm
1$—are plotted, respectively.

(a) $\mathcal{PS}$ for solution with $s=+1,m=+1$.

(b) $\mathcal{PS}$ for solution with $s=+1,m=-1$.

(c) $\mathcal{PS}$ for solution with $s=+2,m=+2$.

(d) $\mathcal{PS}$ for solution with $s=+2,m=-2$.

Figure 15: The power spectra of the outgoing electromagnetic and gravitational
perturbations—with physical parameters $s=-1,\omega_{0}=0.25,\ell=1,m=\pm 1$
and $s=-2,\omega_{0}=0.65,\ell=2,m=\pm 2$—are plotted, respectively.

As nothing interesting happened in these ingoing type-II configurations it was
tempting to ask what happens if in the initial data
$(\phi^{(s)},\phi_{T}^{(s)})$ the time derivative is reversed, i.e. the
initial data $(\phi^{(s)},-\phi_{T}^{(s)})$ is used. One might naively expect
that this also reverses the direction of energy and angular momentum
transports of the pertinent configurations. On contrary to this
expectation—indicated by the panels on Fig. 13—only tiny fragments of the wave
packets start to move outwards, in spite of the fact that the power spectra
shown on the panels of Fig. 14 could allow more interesting dynamics. It is
also somewhat counterintuitive that no energy gain occurs while the angular
momentum gains are significant for both co- and counterrotating
configurations. It is also counterintuitive that, in the considered ingoing
time derivative reversed cases, the angular momentum gain is the strongest for
scalar configurations it is much modest for electromagnetic perturbations and
even this type of gain is missing for linear $s=-2$ metric perturbations. The
angular momentum gain for scalar field—with physical parameters
$s=0,\omega_{0}=0.3,\ell=1,m=\pm 1$ and with the replacement
$\phi_{T}^{(s)}\rightarrow-\phi_{T}^{(s)}$—is about $40\%$ for the corotating
while it is about $315\%$ for counterrotating configuration.

####  4.2.2 Outgoing electromagnetic and gravitational perturbations

The type-II outgoing scalar configurations were already considered in the
previous subsection where time evolution with reversed time derivative was
discussed. Whence, here only the electromagnetic and gravitational
perturbations will be considered. As noted before the main difference between
the type-I and type-II configurations is that the role of the ingoing and
outgoing configurations gets reversed.

(a) $\mathscr{E}$ and $\mathscr{J}$ with $s=+1$ and $m=+1$.

(b) $\mathscr{E}$ and $\mathscr{J}$ with $s=+1$ and $m=-1$.

(c) $\mathscr{E}$ and $\mathscr{J}$ with $s=+2$ and $m=+2$.

(d) $\mathscr{E}$ and $\mathscr{J}$ with $s=+2$ and $m=-2$.

Figure 16: The volume normalized energy and angular momentum current
densities, $\mathscr{E}(T,R)$ and $\mathscr{J}(T,R)$ are plotted for type-II
outgoing electromagnetic and gravitational perturbations, respectively. The
physical parameters are exactly the same as those applied in producing Fig. 15
.

More specifically, while the outgoing configurations did not show any sort of
superradiance for type-I electromagnetic or gravitational perturbations they
step forward to be the main actors in superradiant processes of type-II
configurations. It is not surprising that the strength of superradiance of
outgoing type-II configurations does also depend on spin. It is rather modest
for $s=1$ electromagnetic fields, whereas it is getting considerable for $s=2$
gravitational perturbations.

The power spectra of the outgoing electromagnetic and gravitational
perturbations—with physical parameters $s=1,\omega_{0}=0.28,\ell=1,m=\pm 1$
and $s=2,\omega_{0}=0.725,\ell=2,m=\pm 2$—, depicted on Fig. 15, are not
extremely informative. There are some not too intensive spots in the critical
$0<\omega<m\,\Omega_{+}$ interval, while the intense ones outside of this
interval are associated with lasting quasinormal ringings.

The energy and angular momentum transfers, depicted on Fig. 16, do indicate a
long lasting breathing structure that takes place mainly in the ergoregion and
that is supposedly a manifestation of superradiant processes.

(a) Energy gains: $s=+1,m=\pm 1$.

(b) Angular momentum gains: $s=+1,m=\pm 1$.

(c) Energy gains: $s=+2,m=\pm 2$.

(d) Angular momentum gains: $s=+2,m=\pm 2$.

Figure 17: The time dependence of the energy and angular momentum gains,
$\delta E=(E_{0}-E_{out})/E_{0}$ and $\delta J=(J_{0}-J_{out})/J_{0}$, are
plotted for the investigated outgoing and corotating electromagnetic and
gravitational perturbations, respectively. As it is indicated by these panels
there is only angular momentum gain for electromagnetic perturbations whereas
there are both energy and angular momentum gains for gravitational
perturbations.

(a) Energy gains: $s=0,m=\pm 1$.

(b) Angular momentum gains: $s=0,m=\pm 1$.

Figure 18: The time dependence of the energy and angular momentum gains,
$\delta E=(E_{0}-E_{out})/E_{0}$ and $\delta J=(J_{0}-J_{out})/J_{0}$, are
plotted for ingoing scalar perturbations (with reversed time derivative),
respectively. The physical parameters are as those are on Figs. 14 and 13.

Recall that the ratios $\delta E=(E_{0}-E_{out})/E_{0}$ and $\delta
J=(J_{0}-J_{out})/J_{0}$ were designed to scale superradiance. If energy
superradiance and angular momentum gain occurs in scattering processes then
the energy and angular momentum, $E_{out}$ and $J_{out}$, received at null
infinity, ${\mathscr{I}}^{+}$, have to be larger than the energy and angular
momentum, $E_{0}$ and $J_{0}$, of the initial wave packet, which means that
$\delta E$ and $\delta J$ get to be negative. Notably, on the top panels of
Fig. 17 depicting the electromagnetic case, (likewise on the analogous panels
of Fig. 18) only $\delta J$ gets negative. The angular momentum gain is $\sim
200\%$ for corotating, while it is $\sim 350\%$ for counterrotating
electromagnetic perturbations. Both energy superradiance and angular momentum
gain occur in case of gravitational perturbations. The energy gain is $\sim
116.7\%$ for corotating and $\sim 6\%$ for counterrotating configurations,
whereas the corresponding angular momentum gains are $\sim 100\%$ and $\sim
5\%$, respectively. Remarkably, the angular momentum gain (again likewise in
case of type-II outgoing scalar configurations with reversed time derivative)
is larger for counterrotating configurations while in case of gravitational
perturbations, the situation is quite the opposite, the corotating
configurations are found to be more superradiant.

###  4.3 Results relevant for type-III initial data

So far in using type-I and type-II initial data only the $\omega_{0}$
characteristic frequency was fine tuned to amplify the effect of
superradiance. The type-III initial data, considered in this section, is to
probe the strength of superradiance with respect to the variation of the width
of the radial profile. In doing so we shall modify the type-II radial profile
by keeping the left boundary at $R_{1}=0.4536$ fixed but moving the right edge
outwards such that $R_{2}$ takes the values $R_{2}=0.7,0.76,0.84,0.95$. The
maximum of the radial profile is located approximately at the mean value of
$R_{1}$ and $R_{2}$, i.e. $R_{0}=(R_{1}+R_{2})/2$ for each of the considered
configurations as shown on the right panel of Fig. 1.

####  4.3.1 Ingoing and outgoing scalar wave packets

It is somewhat unexpected but very little change occurs when type-III initial
data is replacing the type-II ones in the scalar field case. More
specifically, no superradiance occurred when either of the four type-III co-
or counterrotating incident wave packets were applied in the “ingoing” case
with initial data as specified in (3.15) and (3.16).

As seen earlier somewhat more interesting dynamics may occur when time
derivative reversed type-II scalar wave packets were investigated.

Figure 19: The volume normalized energy and angular momentum current
densities, $\mathscr{E}(T,R)$ and $\mathscr{J}(T,R)$ are plotted for in- and
outgoing type-III scalar wave packets, respectively.

These time derivative reversed scalar field configurations may be considered
as being “outgoing” ones.

(a) Energy gains: $s=0,m=-1$.

(b) Angular momentum gains: $s=0,m=-1$.

Figure 20: No energy but notable angular momentum gains occur for
counterrotating type-III outgoing scalar perturbations.

To verify the rightfulness of the use of the terminology ingoing and outgoing
one it is rewarding to inspect the energy transports of scalar wave packets
shearing the compact support and the value of $\omega_{0}$, with $R_{2}=0.84$
and $\omega_{0}=0.3$, shown by the two panels of Fig. 19. While on the left
panel the energy and angular momentum transfer, with initial data given as in
(3.15) and (3.16), is visibly ingoing for the time derivative reversed case,
shown on the right panel, the energy and angular momentum move to the outer
direction.

Notably, what we have experienced in case of counterrotating outgoing type-II
scalar field configurations manifest itself for the type-III wave packets too.
More specifically, for time derivative reversed scalar fields—with physical
parameters $s=0,\omega_{0}=0.3,\ell=1,m=-1$ —while no energy superradiance
occurs the angular momentum gain ranges between $49\%$ and $135\%$ as shown on
the right panel of Fig. 20.

####  4.3.2 Ingoing electromagnetic and gravitational perturbations

It was a generic feature of type-II ingoing, $s=-1$ and $s=-2$,
electromagnetic and gravitational perturbations that neither of them shows
superradiance. Instead considerable large part of the initial pulse was
leaving the domain of outer communication in a short period via the event
horizon. Exactly the same phenomenon shows up in case of type-III ingoing
configurations. The only notable difference is that the evolution gets
somewhat longer. Neither energy nor angular momentum gains show up regardless
whether co- or counterrotating ingoing $s=-1$ electromagnetic or $s=-2$
gravitational perturbations were investigated.

####  4.3.3 Outgoing electromagnetic and gravitational perturbations

The corotating outgoing electromagnetic and gravitational perturbations are
not too much exciting either. There are no energy or angular momentum gains
for $s=+1,\ell=1,m=+1$ electromagnetic perturbations. The evolution relevant
for the four radial profiles differ only by the ratio of the energy or angular
momentum that leave the domain of outer communication either through the event
horizon or via null infinity.

As opposed to this the evolution of counterrotating type-III outgoing $s=+1$
electromagnetic perturbations show somewhat more exciting behavior.

(a) Energy gains: $s=+1,m=-1$.

(b) Angular momentum gains: $s=+1,m=-1$.

Figure 21: The time dependence of the energy and angular momentum gains,
$\delta E$ and $\delta J$, are plotted for counterrotating type-III outgoing
electromagnetic perturbations.

The energy and angular momentum gains of counterrotating type-III outgoing
electromagnetic perturbations—with physical parameters
$s=+1,\omega_{0}=0.28,\ell=1,m=-1$ —are shown on the panels of Fig. 21. There
is a slight $\sim 8\%$ energy superradiance for the widest radial profile,
with $R_{2}=0.95$. As opposed to this, no angular momentum gains happens for
this profile but stronger and stronger angular momentum gains, $\sim 10\%,\sim
60\%,\sim 120\%$, can be seen to happen for the other three radial profiles
while the domain of the radial profiles is getting narrower and narrower.

The type-III outgoing gravitational perturbations behave significantly
differently. The energy and angular momentum transfers of the co- or
counterrotating type-III outgoing gravitational perturbations—depicted on Fig.
22 with physical parameters $s=+2,\omega_{0}=0.65,\ell=2,m=+2,R_{2}=0.84$—are
very close to the ones we saw in case of analogous type-II configurations.

(a) $\mathscr{E}$ and $\mathscr{J}$ with $s=+2$ and $m=+2$.

(b) $\mathscr{E}$ and $\mathscr{J}$ with $s=+2$ and $m=-2$.

Figure 22: The volume normalized energy and angular momentum current
densities, $\mathscr{E}(T,R)$ and $\mathscr{J}(T,R)$ are plotted for outgoing
co- and counterrotating gravitational perturbations—with physical parameters
$s=+2,\omega_{0}=0.65,\ell=2,m=\pm 2,R_{2}=0.84$—, respectively.

Nevertheless, as it is depicted on the panels of Fig. 23 the outgoing
corotating type-III gravitational perturbations—with physical parameters
$s=+2,\omega_{0}=0.65,\ell=2,m=+2$—produced so far the strongest of energy
gains, $\sim 100\%,\sim 320\%,\sim 390\%$, though the very same configuration
did not show the slightest angular momentum gain.

(a) Energy gains: $s=+2,m=+2$.

(b) Angular momentum gains: $s=+2,m=+2$.

Figure 23: The time dependence of the energy and angular momentum gains,
$\delta E=(E_{0}-E_{out})/E_{0}$ and $\delta J=(J_{0}-J_{out})/J_{0}$, are
plotted for outgoing corotating gravitational perturbations. Note that on the
left panel the graph of the strongest energy gain $\sim 390\%$, we have found,
can also be seen.

Notably, even the counterrotating outgoing gravitational perturbations—with
physical parameters $s=+2,\omega_{0}=0.65,\ell=2,m=-2$, for either of the
type-III radial profiles,—produced some energy and angular momentum gains as
it is depicted on the panels of Fig. 24.

(a) Energy gains: $s=+2,m=-2$.

(b) Angular momentum gains: $s=+2,m=-2$.

Figure 24: The time dependence of the energy and angular momentum gains,
$\delta E=(E_{0}-E_{out})/E_{0}$ and $\delta J=(J_{0}-J_{out})/J_{0}$, are
plotted for outgoing counterrotating gravitational perturbations.

Nevertheless, the scale of superradiance, produced by counterrotating type-III
outgoing gravitational perturbations with $R_{2}=0.7,0.76,0.84,0.95$, is
considerably weaker as $\sim 0.01\%,\sim 0.9\%,\sim 1.8\%,\sim 2.5\%$ energy
gains and $\sim 1.0\%,\sim 3.3\%,\sim 5.1\%,\sim 7\%$ angular momentum gains
showed up in the respective investigations.

##  5 Final remarks

In this paper various aspects of superradiance, on rotating Kerr black hole
background, were investigated in the time domain by solving the source free
Teukolsky master equation relevant for spin $-2,-1,0,+1,+2$ fields
numerically. In doing so three different types of initial data were applied.
The radial profile of each of these were of compact support. For type-I
configurations the support of the radial profile is clearly separated from the
ergoregion located in the domain of outer communications. By contrast, the
support type-II data is concentrated onto the ergoregion. (Some aspects of the
angular dependence of type-II data is depicted on the left panel of Fig. 25.
See the explanation in the caption of this figure.) For type-III data the
corresponding compact support was chosen to start next to the event horizon
and to cover, across the ergoregion, larger and larger portions of the domain
of outer communication. In addition the initial data—by frequency tuning the
incident wave packets via specifying the $\omega_{0}$ parameter in (3.16)
properly—was chosen to be optimally superradiant.

One of the main lessons drawn from our investigations is that the scale and
various characters of superradiance primarily depends on the type or on the
location of the compact support of the radial profile of the initial data. For
type-I corotating configurations, regardless how well they are frequency-
tuned, considerably large portion of the incident wave packets endure an
almost perfect reflection while approaching the ergoregion. This, in turn,
reduces the transmission coefficient, and, thereby, the effect of
superradiance is suppressed significantly in the corresponding cases. For
instance, regardless of the type of the initial data for the scalar field case
no energy superradiance occur. For this reason it is surprising that angular
momentum gain happened in case of time derivative reversed type-II and type-
III outgoing scalar fields. The scale of superradiance remained modest
regardless of the type of the initial data for electromagnetic perturbations.
Impressively strong energy superradiance with notable $\sim 116.7\%$ and $\sim
320\%,\sim 390\%$ energy gains occurred only for co-rotating type-II and type-
III gravitational perturbations, respectively.

In [9, 16], studying the evolution of scalar perturbations, we experienced an
analogous suppression of superradiance while initial data well-separated from
the ergoregion was used. Notably, in our present studies, we found a
significant superradiance deficit for well-tuned type-I initial data even for
spin $s=-1$ and $s=-2$ perturbations. In other words, instead of experiencing
superradiance, most of these type-I inbound wave packets get reflected before
reaching the ergoregion. This phenomenon seems to be explained by the first
and second laws of black hole thermodynamics as follows. On the one hand,
according to the first law, the change in the energy, area, and angular
momentum of a black hole must be related as

$dE=\tfrac{\kappa}{8\,\pi}dA+\Omega_{+}\,dJ\,.$ (5.22)

On the other hand, according to the second law, in the examined completely
classical scattering processes, the area of the black hole cannot decrease,
i.e.

$dA\geq 0\,.$ (5.23)

Using these two relations, we get

$0\leq dE-\Omega_{+}\,dJ\,,$ (5.24)

i.e., any inbound wave packet whose energy and momentum are such that the
expression on the right-hand side becomes negative goes against the laws of
black hole thermodynamics. Therefore, such a wave packet must undergo a strong
reflection. Interestingly, regardless of the spin, the term on the right is
negative for all type-I initial data tuned to be superradiant. It is not
surprising then that if the support of the initial data is well-separated from
the ergoregion, strong reflections occur, and superradiance gets to be
suppressed even for spin $s=-1$ and $s=-2$ perturbations.

It is worth noting that the observed strong reflection could also be viewed as
a highly indirect justification of Penrose’s weak censorship hypothesis.
Assume that a type-I wave packet tuned to the superradiant falls unhindered
into a highly spinning Kerr black hole. Provided that backreaction could
occur, the wave packet’s oversized angular momentum would turn the singularity
of the black hole into a naked one.

To provide a proper measure of the scale of energy superradiance and angular
momentum gain we applied the energy and angular momentum balance relations
tested previously in [10]. As it was pointed out in [23], conserved currents
can be associated with essentially arbitrary pairs of spin $s$ and $-s$
solutions to the Teukolsky master equation. In our investigations the aim was
to characterize the superradiance of specific spin $s$ configurations. In
order to keep the distinguished position of these specific solutions we
decided to use the spin $-s$ adjoint solutions to the Teukolsky master
equation. This required the use of the constructions offered for
electromagnetic perturbations by Cohen-Kegeles in [8, 7], and for metric
perturbations by Chrzanowski in [7] as it is outlined in the appendix of the
present paper. It is important to emphasize that when energy and angular
momentum gains were reported by referring to via the quantities $\delta E$ and
$\delta J$ it is always guaranteed that the energy and angular momentum
balances hold at least up to the order $10^{-4}\cdot\delta E$ and
$10^{-4}\cdot\delta J$, respectively.

When no energy gain but only angular momentum gain occurs—at least in the
scalar field case where the conserved energy and angular momentum currents,
(3.5) and (3.6), possess immediate physical interpretations—one may suspect
that frame dragging enhances the corresponding superradiant behavior. This
expectation is supported by the fact that the signs of the angular momentum
fluxes through the black hole horizon $J_{in}$ and through null infinity
$J_{out}$ were opposite of each other such that, apart from a short initial
dynamical period, $J_{in}$ is negative. Notably, as indicated by the $(b)$
panels of Figs. 18, 17, 20, and 21, analogous phenomenon can be seen to apply
not only to type-II but to type-III outgoing (i.e. time derivative reversed)
scalar configurations, as well as, to type-II and type-III counterrotating
electromagnetic configurations. On the two panels of Fig. 26 the normalized
and sign correct angular momentum fluxes through the black hole horizon
$\mathcal{J}_{in}={J}_{in}/|J_{0}|$ and through null infinity
$\mathcal{J}_{out}={J}_{out}/|J_{0}|$, along with the “normalized” and “sign
correct” angular momentum content $\mathcal{J}(T)={J}(T)/|J_{0}|$ of the
$T=const$ time slices are shown separately for the type-II outgoing scalar
configurations which were used on Fig. 18. Each one of these is normalized by
the absolute value $|J_{0}|=|J(T=0)|$ of the angular momentum, $J_{0}$, on the
initial slice. Note that the sign correctness, as expected, means here that
$J_{0}$ is positive for co-rotating, $m=+1$, configurations, whereas it is
negative for counterrotating, $m=-1$, configurations.

Figure 25: On the left panel the $\varphi=0$ section of a type-II initial data
configuration proportional to ${}_{1}Y_{2}^{2}$, spin $1$ $\ell=m=+2$, spin-
weighted spherical harmonic is depicted. The inner colored contour signify the
location of the event horizon, the other one does it for the outer boundary
ergoregion. Null infinity is located at the $R=1$ semicircle bounding the
homogeneously colored region. Note that left vertical boundary denotes the
axis of rotation of the Kerr geometry which is axially symmetric,
nevertheless, one should keep in mind that the spin-weighted spherical
harmonic ${}_{1}Y_{2}^{2}$ is not axially symmetric.
On the right panel the $\varphi=const,\vartheta=\pi/2$ slice of the Carter-
Penrose diagram of domain of outer communication of the Kerr spacetime is
shown. The dotted line is to indicate the location of the outer boundary of
the ergoregion. This figure is to demonstrate that the intersection of the
causal past, $J^{-}[\mathcal{D}]$, of the compact support $\mathcal{D}$ of the
initial data at $T=0$ and the progenitor time slices will always have a non-
negligible part overlapping with the ergoregion if the inner edge of the
support is chosen to be close to the black hole horizon.

Accordingly, $\mathcal{J}_{out}=J_{out}/|J_{0}|$ and the angular momentum gain
$\delta J=(|J_{0}|-|J_{out}|)/|J_{0}|$ are, regardless the sign of the $m$
azimuthal parameter, related via the relation $\delta
J=1-|\mathcal{J}_{out}|$. It is clearly visible on the panels of Fig. 26 that
regardless of the sign of $J_{0}$, apart from a short initial dynamical
period, the flux through the event horizon is negative, whereas the flux
through ${\mathscr{I}}^{+}$ is positive. In addition, as it is anticipated,
the frame dragging effect can also be seen (compare the panels of Fig. 26) to
be stronger for the counterrotating configuration. Note finally that this
angular momentum gain, caused by frame dragging, is in entire accordance with
the picture we have on mind concerning energy gain in the Penrose process. To
see this note that an enhanced negative angular momentum flux through the
black hole event horizon ensures the appearance of a scaled up positive flux
at null infinity.

It is important to mention that in spite of the detailed investigations
carried out in the present paper there remained a lot of interesting
configurations to be studied. As an immediate example one may think of initial
data with radial profile of noncompact support which were left out of the
present investigations. This is mainly because in evaluating the energy and
angular momentum balance relations we treated the spin $-s$ adjoint solutions,
$\Phi^{(-s)}_{adj}$, as suitable counterparts of the spin $s$ solutions,
$\Phi^{(s)}$, to the Teukolsky master equation. There is, however, a
substantial technical difficulty coming with this choice. Namely, the use of
adjoint solutions in case of gravitational perturbations requires the
evaluation of the fourth order derivatives of the $\Phi^{(s)}$ solutions. With
the choice of radial profile of compact support we could do this by keeping
sufficiently good level of accuracy but this was not possible to be done with
initial data the radial support of which extended to null infinity. In this
respect one could also carry out interesting follow up studies, for instance,
by applying in the evaluation of the energy and angular momentum balance
relation spin $-1,0,+1$ projections of the same electromagnetic perturbations,
e.g. based on the Fackerell-Ipser type approach [12], or spin $-2,-1,0,+1,+2$
projections of gravitational perturbations, after applying suitable gauge
choice, as suggested in [1]. This technically could exclude the use of second
or fourth order derivatives of spin $s$ solutions to the Teukolsky master
equation and, in addition, possibly physically more adequate energy and
angular momentum currents could be defined this way in the $s=\pm 1,\pm 2$
field cases.

(a) $s=0,m=+1$.

(b) $s=0,m=-1$

Figure 26: The time dependence of the normalized and sign correct angular
momentum fluxes $\mathcal{J}_{in}={J}_{in}/|J_{0}|$ and
$\mathcal{J}_{out}={J}_{out}/|J_{0}|$ at the horizon and at
${\mathscr{I}}^{+}$, along with the normalized and sign correct angular
momentum content $\mathcal{J}={J}{(T)}/|J_{0}|$ of the $T=const$ time slices,
are plotted for ingoing co- and counterrotating (time derivative reversed)
scalar perturbations, respectively.

Let us finish by recalling again that in order to have a sizable superradiance
one has to use type-II or type-III initial data. As described above these are
configurations where the relevant radial profile of the initial data overlaps
considerably with the ergoregion. It is natural to ask then if on the
“progenitor time slices” of the corresponding configurations could get to be
separated from the ergoregion or not. Since the causal past of the compact
support of the original initial data intersects all the earlier hyperbolic
time slices near to the black hole horizon the pertinent compact supports of
the progenitor time slices will always remain partly within the ergoregion (as
indicated on the right panel of Fig. 25). Our numerical experiments—performed
by evolving the corresponding initial data configurations backward in
time—confirm this expectation. All these simple observations do also indicate
that sizable superradiant scattering cannot occur unless the initial data is
arranged to have considerable overlap with the ergoregion. In other words, if
an advanced civilization would try to extract sizable energy or angular
momentum from a rotating black hole that cannot be done without being able to
engineer, say by some astrophysical processes, suitable scalar,
electromagnetic or gravitational perturbations such that nontrivial parts of
these perturbation are prepared to have a domain of support ranging
essentially from the black hole event horizon, across the ergoregion, to
certain location in the domain of outer communication.

## Acknowledgments

The authors are grateful to András László and Gábor Tóth for helpful comments
and suggestions. This research was supported in part by the NKFIH grants
K-115434.

## Appendix: Generating adjoint solutions

This appendix is to recall basics of the methods we used to generate adjoint
solutions to Teukolsky’s master equation. In doing so the proposals made by
Cohen-Kegeles [8], Chrzanowski [7], explained later by Wald in [24], were
followed. In the following subsections the basic ideas and elements of the
aforementioned constructions will be outlined first for the generic case,
while in the succeeding sections the explicit form of the transformations
applicable to electromagnetic perturbations and linear metric perturbations,
respectively, are recalled. For the sake of simplicity in the outlined
arguments considerations will be restricted to the case of homogeneous TMEs.
Note, however, that all the details relevant for the presence of sources can
be found in [8, 7, 24].

##  Appendix A Outline of the generic argument

Consider a physical field f that is subject to a linear partial differential
equation (PDE)

$\mathcal{E}\big{(}\textit{{f}}\big{)}=0\,,$ (GA.1)

where, in general, $\mathcal{E}$ is a tensor valued linear partial derivative
operator and the dependent variable f is a tensor field. 333Since the metric
$g_{ab}$ is assumed to be defined everywhere on $M$ we may, without loss of
generality, restrict considerations to tensor fields of type $(0,m)$, where
$m$ is a positive integer.

###  A.1 Reformulations

It was assumed in [24] that there exist linear partial derivative operators
${{}^{(s)\hskip-1.42262pt}}\mathcal{O}$,
${{}^{(s)\hskip-1.42262pt}}\mathcal{T}$ and
${{}^{(s)\hskip-1.42262pt}}\mathcal{S}$, in addition to $\mathcal{E}$, such
that the followings hold 444Note that there is a slight notational difference
with respect to the one applied in [24]. Namely the role of
${{}^{(s)\hskip-1.42262pt}}\mathcal{O}$ and
${{}^{(s)\hskip-1.42262pt}}\mathcal{T}$ are changed as it appears to be more
adequate to denote by ${{}^{(s)\hskip-1.42262pt}}\mathcal{T}$ the Teukolsky
operator, defined by (GA.3).

  * •

the tensorial variable f is mapped by ${{}^{(s)\hskip-1.42262pt}}\mathcal{O}$
to a spin-weight-$s$ variable, denoted by $\psi{{}^{(s)\hskip-1.42262pt}}$,
such that the relation

$\psi{{}^{(s)\hskip-1.42262pt}}={{}^{(s)\hskip-1.42262pt}}\mathcal{O}(\textit{{f}})$
(GA.2)

holds.

  * •

${{}^{(s)\hskip-1.42262pt}}\mathcal{T}$ is Teukolsky’s linear operator acting
on a spin-$s$ variable generated by ${{}^{(s)\hskip-1.42262pt}}\mathcal{O}$
such that

${{}^{(s)\hskip-1.42262pt}}\mathcal{T}{{}^{(s)\hskip-1.42262pt}}\mathcal{O}(\textit{{f}})={{}^{(s)\hskip-1.42262pt}}\mathcal{T}\psi{{}^{(s)\hskip-1.42262pt}}$
(GA.3)

holds.

  * •

finally, the linear operator ${{}^{(s)\hskip-1.42262pt}}\mathcal{S}$ is
supposed to represent all the manipulations that must be performed to get, by
combining it with $\mathcal{E}$, the Teukolsky operator
${{}^{(s)\hskip-1.42262pt}}\mathcal{T}$, i.e.

${{}^{(s)\hskip-1.42262pt}}\mathcal{S}\mathcal{E}\big{(}\textit{{f}}\big{)}={{}^{(s)\hskip-1.42262pt}}\mathcal{T}{{}^{(s)\hskip-1.42262pt}}\mathcal{O}\big{(}\textit{{f}}\big{)}={{}^{(s)\hskip-1.42262pt}}\mathcal{T}\psi{{}^{(s)\hskip-1.42262pt}}\,.$
(GA.4)

This, in particular means that

    * –

for the linear partial derivative operators $\mathcal{E}$,
${{}^{(s)\hskip-1.42262pt}}\mathcal{O}$,
${{}^{(s)\hskip-1.42262pt}}\mathcal{T}$ and
${{}^{(s)\hskip-1.42262pt}}\mathcal{S}$ the relation

${{}^{(s)\hskip-1.42262pt}}\mathcal{S}\mathcal{E}={{}^{(s)\hskip-1.42262pt}}\mathcal{T}{{}^{(s)\hskip-1.42262pt}}\mathcal{O}$
(GA.5)

holds, and

    * –

whenever f is a solution to $\mathcal{E}(\textit{{f}})=0$ then
$\psi{{}^{(s)\hskip-1.42262pt}}={{}^{(s)\hskip-1.42262pt}}\mathcal{O}(\textit{{f}})$
is a solution to
${{}^{(s)\hskip-1.42262pt}}\mathcal{T}\psi{{}^{(s)\hskip-1.42262pt}}=0$.

###  A.2 The adjoint operator

Consider a linear operator $\mathcal{P}$ that maps $n$-index tensor fields to
$m$-index tensor fields. The linear operator $\mathcal{P}^{\dagger}$ is called
the adjoint of $\mathcal{P}$ if it maps $m$-index tensor fields to $n$-index
tensor fields, and it is determined, up to the total divergence
$\nabla_{a}\sigma^{a}$, where $\sigma^{a}$ is a smooth vector field on
$\mathbb{R}^{4}$, via the relation

$\psi^{a_{1}\ldots a_{m}}(\mathcal{P}\textit{{f}})_{a_{1}\ldots
a_{m}}-(\mathcal{P}^{\dagger}\textit{{f}})^{b_{1}\ldots
b_{n}}\phi_{b_{1}\ldots b_{n}}=\nabla_{a}\sigma^{a}\,.$ (GA.6)

For the composition of the linear operators $\mathcal{P}$ and $\mathcal{Q}$
the familiar relation
$(\mathcal{P}\mathcal{Q})^{\dagger}=\mathcal{Q}^{\dagger}\mathcal{P}^{\dagger}$
can be seen to hold. A linear operator $\mathcal{P}$ is called to be self-
adjoint if $\mathcal{P}^{\dagger}=\mathcal{P}$ holds, which can happen only if
$m=n$.

###  A.3 Wald’s $1^{st}$ theorem

The first claim in [24] was that if $\psi{{}^{(-s)\hskip-1.42262pt}}$ is a
solution to the adjoint of Teukolsky’s equation then
$\textit{{f}}={{}^{(-s)\hskip-1.42262pt}}\mathcal{S}^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}}$
is a solution to the original linear equation (GA.1). More precisely, we have
the following.

###### Theorem A.1

Assume that $\psi{{}^{(-s)\hskip-1.42262pt}}$ is the solution to the adjoint
of the Teukolsky equation

${{}^{(-s)\hskip-1.42262pt}}\mathcal{T}^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}}=0\,,$
(GA.7)

where
${{}^{(-s)\hskip-1.42262pt}}\mathcal{T}^{\dagger}=\big{[}{{}^{(s)\hskip-1.42262pt}}\mathcal{T}\big{]}^{\dagger}$,
and assume that $\mathcal{E}$ is self-adjoint, i.e.
$\mathcal{E}^{\dagger}=\mathcal{E}$. Then, in virtue of (GA.5), the tensor
field
$\textit{{f}}={{}^{(-s)\hskip-1.42262pt}}\mathcal{S}^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}}$
is a solution to (GA.1) as the following relations hold

$\displaystyle\mathcal{E}({{}^{(-s)\hskip-1.42262pt}}\mathcal{S}^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}})$
$\displaystyle=\mathcal{E}^{\dagger}\big{(}{{}^{(-s)\hskip-1.42262pt}}\mathcal{S}^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}}\big{)}=({{}^{(s)\hskip-1.42262pt}}\mathcal{S}\mathcal{E})^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}}=({{}^{(s)\hskip-1.42262pt}}\mathcal{T}{{}^{(s)\hskip-1.42262pt}}\mathcal{O})^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}}$
$\displaystyle={{}^{(-s)\hskip-1.42262pt}}\mathcal{O}^{\dagger}\big{(}{{}^{(-s)\hskip-1.42262pt}}\mathcal{T}^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}}\big{)}=0\,.$
(GA.8)

###  A.4 Wald’s $2^{nd}$ theorem

In [24] it was also shown that whenever $\psi{{}^{(-s)\hskip-1.42262pt}}$ is a
solution to the adjoint of the Teukolsky equation then
$\psi{{}^{(s)\hskip-1.42262pt}}={{}^{(s)\hskip-1.42262pt}}\mathcal{O}\big{(}{{}^{(-s)\hskip-1.42262pt}}\mathcal{S}^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}}\,\big{)}$
is a solution to the opposite spin Teukolsky equation, i.e.,
${{}^{(s)\hskip-1.42262pt}}\mathcal{T}\psi{{}^{(s)\hskip-1.42262pt}}=0$.

###### Theorem A.2

Assume that $\psi{{}^{(-s)\hskip-1.42262pt}}$ is the solution to the adjoint
of the Teukolsky equation

${{}^{(-s)\hskip-1.42262pt}}\mathcal{T}^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}}=0\,.$
(GA.9)

Then, in virtue of (A.1),
$\mathcal{E}\big{(}{{}^{(-s)}\hskip-1.42262pt}\mathcal{S}^{\dagger}{{}^{(-s)\\!}}\psi\big{)}=0$,
which, along with (GA.5) and the self-adjointness of $\mathcal{E}$, implies
that
$\psi{{}^{(s)\hskip-1.42262pt}}={{}^{(s)}\hskip-1.42262pt}\mathcal{O}\big{(}{{}^{(-s)}\hskip-1.42262pt}\mathcal{S}^{\dagger}\psi{{}^{(-s)}\hskip-1.42262pt}\big{)}$
is indeed a solution to the Teukolsky equation as the following relations hold

$\displaystyle{{}^{(s)\hskip-1.42262pt}}\mathcal{T}\big{(}{{}^{(s)\hskip-1.42262pt}}\mathcal{O}\big{(}{{}^{(-s)\hskip-1.42262pt}}\mathcal{S}^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}}\big{)}\big{)}$
$\displaystyle=\big{(}{{}^{(s)\hskip-1.42262pt}}\mathcal{T}{{}^{(s)\hskip-1.42262pt}}\mathcal{O}\big{)}\big{(}{{}^{(-s)\hskip-1.42262pt}}\mathcal{S}^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}}\big{)}=\big{(}{{}^{(s)\hskip-1.42262pt}}\mathcal{S}\mathcal{E}^{\dagger}\big{)}\big{(}{{}^{(-s)\hskip-1.42262pt}}\mathcal{S}^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}}\big{)}$
$\displaystyle={{}^{(s)\hskip-1.42262pt}}\mathcal{S}\big{(}\mathcal{E}\big{(}{{}^{(-s)\hskip-1.42262pt}}\mathcal{S}^{\dagger}\psi{{}^{(-s)\hskip-1.42262pt}}\big{)}\big{)}=0\,.$
(GA.10)

##  Appendix B The electromagnetic and metric perturbations

As it was observed first by Teukolsky [21] the electromagnetic and metric
perturbations of a fixed Kerr background, respectively, whenever they are
written out by making use of the Newman-Penrose formalism [18], the equations
relevant for the extreme spin values decouple from the other equations. What
makes these extreme spin modes to be distinguished is that essential physical
information—such as ingoing and outgoing radiation—is carried by them. 555The
basic notions and notation of the Newman-Penrose formalism [18], along with
the gauge fixing applied by Kinnersley [15], is assumed to be known and they
will be cited without explanations.

###  B.1 Electromagnetic perturbations

The symbolic form of (GA.1) describing the electromagnetic perturbations of
algebraically special background reads as

$\mathcal{E}_{E}(A_{b})=0\,,$ (ED.1)

where now the vector potential $A_{a}$ plays the role of the $(0,1)$ type
tensor field f, and the linear operator $\mathcal{E}_{E}$ acts on the vector
potential as

$\big{[}\mathcal{E}_{E}(A_{b})\big{]}_{a}=\big{[}\,g^{b}{}_{a}\,\Box-g^{bc}\nabla_{c}\nabla_{a}\,\big{]}\,A_{b}\,.$
(ED.2)

It is straightforward to check that the linear operator $\mathcal{E}_{E}$ is
self-adjoint.

The additional linear operators ${{}^{(s)\hskip-1.42262pt}}\mathcal{O}$,
${{}^{(s)\hskip-1.42262pt}}\mathcal{T}$ and
${{}^{(s)\hskip-1.42262pt}}\mathcal{S}$, relevant for the extreme spin values
$s=+1$ and $s=-1$, along with the pertinent form of the two theorems by Wald,
are given in the following subsections.

####  B.1.1 The ${{}^{(\pm 1)\hskip-1.42262pt}}\mathcal{O}$ operators

Recall first that the Maxwell spinor components $\phi_{0}$ and $\phi_{1}$ are
defined by the contractions

$\displaystyle\phi_{0}$
$\displaystyle=\ell^{a}m^{b}F_{ab}=[m^{a}D-\ell^{a}\delta]\,A_{a}\,,$ (ED.3)
$\displaystyle\phi_{2}$
$\displaystyle=\overline{m}^{a}n^{b}F_{ab}=[n^{a}\overline{\delta}-\overline{m}^{a}\Delta]\,A_{a}\,.$
(ED.4)

In terms of these the Teukolsky variables read as [21]

$\displaystyle\psi^{(+1)}$ $\displaystyle=\phi_{0}\,,$ (ED.5)
$\displaystyle\psi^{(-1)}$ $\displaystyle=(\Psi_{2})^{-2/3}\,\phi_{2}\,,$
(ED.6)

where the only nonvanishing and gauge invariant Weyl-scalar component on Kerr
background is given by the contraction of the Weyl tensor $C_{abcd}$ and the
tetrad vectors as [21]

$\Psi_{2}=-C_{abcd}\,l^{a}{m}{}^{\,b}n^{c}\overline{m}{}^{\,d}\,.$ (ED.7)

The operators ${{}^{(\pm 1)\hskip-1.42262pt}}\mathcal{O}$, mapping the field
$A_{a}$ to the extreme spin Teukolsky variables $\psi^{(\pm 1)}$, can be read
off the above relations. They are

$\displaystyle\psi^{(+1)}=[{{}^{(+1)\hskip-1.42262pt}}\mathcal{O}]^{a}A_{a}$
$\displaystyle=\big{[}m^{a}\,D-\ell^{a}\,\delta\big{]}\,A_{a}$
$\displaystyle=\big{[}(D-\epsilon+\overline{\epsilon}-\overline{\rho})\,m^{a}-(\delta+\overline{\varpi}-\beta-\overline{\alpha})\,\ell^{a}\big{]}\,A_{a},$
(ED.8)
$\displaystyle\psi^{(-1)}=[{{}^{(-1)\hskip-1.42262pt}}\mathcal{O}]^{a}A_{a}$
$\displaystyle=(\Psi_{2})^{-2/3}\big{[}n^{a}\,\overline{\delta}-\overline{m}^{a}\,\Delta\big{]}\,A_{a}$
$\displaystyle=(\Psi_{2})^{-2/3}\big{[}(\overline{\delta}+\alpha+\overline{\beta}-\overline{\tau})\,n^{a}-(\Delta+\overline{\mu}-\overline{\gamma}+\gamma)\,\overline{m}^{a}\big{]}\,A_{a}\,.$
(ED.9)

####  B.1.2 The ${{}^{(\pm 1)\hskip-1.42262pt}}\mathcal{T}$ and ${{}^{(\pm
1)\hskip-1.42262pt}}\mathcal{T}^{\dagger}$ operators

Recall that by applying the operators ${{}^{(\pm
1)\hskip-1.42262pt}}\mathcal{T}$ to the fields $\psi^{(\pm 1)}$ we get the
homogeneous TME. The relevant form of the operators ${{}^{(\pm
1)\hskip-1.42262pt}}\mathcal{T}$ are as [7, 24]

$\displaystyle{{}^{(+1)\hskip-1.42262pt}}\mathcal{T}=\big{[}\,(D-\epsilon{}$
$\displaystyle+\overline{\epsilon}-2\rho-\overline{\rho})\,(\Delta+\mu-2\gamma)$
$\displaystyle-(\delta-\beta-\overline{\alpha}-2\tau+\overline{\varpi})\,(\overline{\delta}+\varpi-2\alpha)\,\big{]}\,,$
(ED.10)
$\displaystyle{{}^{(-1)\hskip-1.42262pt}}\mathcal{T}=\big{[}\,(\Delta+\rho{}$
$\displaystyle-\overline{\rho}+\overline{\mu})\,(D+\rho+2\epsilon)$
$\displaystyle-(\overline{\delta}+\alpha+\overline{\beta}-\overline{\tau})\,(\delta+\tau+2\beta)\,\big{]}\,.$
(ED.11)

Since the following relations hold for the adjoint of the Newman–Penrose
derivative operators

$\displaystyle D^{\dagger}$
$\displaystyle=-(D+\epsilon+\overline{\epsilon}-\rho-\overline{\rho}),$
(ED.12) $\displaystyle\Delta^{\dagger}$
$\displaystyle=-(\Delta-\gamma-\overline{\gamma}+\mu+\overline{\mu}),$ (ED.13)
$\displaystyle\delta^{\dagger}$
$\displaystyle=-(\delta+\beta-\overline{\alpha}-\tau+\overline{\varpi}).$
(ED.14)

the relations

$\displaystyle{{}^{(+1)\hskip-1.42262pt}}\mathcal{T}^{\dagger}=\big{[}{{}^{(-1)\hskip-1.42262pt}}\mathcal{T}\big{]}^{\dagger}\,,$
(ED.15)
$\displaystyle{{}^{(-1)\hskip-1.42262pt}}\mathcal{T}^{\dagger}=\big{[}{{}^{(+1)\hskip-1.42262pt}}\mathcal{T}\big{]}^{\dagger}\,,$
(ED.16)

applied in Theorem A.1 also follow.

####  B.1.3 The ${{}^{(\pm 1)\hskip-1.42262pt}}\mathcal{S}$ and ${{}^{(\mp
1)\hskip-1.42262pt}}\mathcal{S}^{\dagger}$ operators

The operators ${{}^{(\pm 1)\hskip-1.42262pt}}\mathcal{S}$ can be seen to take
the form [8, 7]

$\displaystyle{{}^{(+1)\hskip-1.42262pt}}\mathcal{S}{}^{a}$
$\displaystyle=(\delta-\beta-\overline{\alpha}-2\tau+\overline{\varpi})\,\ell^{a}-(D-\epsilon+\overline{\epsilon}-2\rho-\overline{\rho})\,m^{a}\,,$
(ED.17) $\displaystyle{{}^{(-1)\hskip-1.42262pt}}\mathcal{S}{}^{a}$
$\displaystyle=(\Psi_{2})^{-2/3}[(\Delta+\rho-\overline{\rho}+2\mu+\overline{\mu})\,\overline{m}^{a}-(\overline{\delta}+\alpha+\overline{\beta}+2\varpi-\overline{\tau})\,n^{a}]\,,$
(ED.18)

whereas their adjoints are

$\displaystyle\big{[}{{}^{(+1)\hskip-1.42262pt}}\mathcal{S}{}^{\dagger}\big{]}^{a}$
$\displaystyle=[\,m^{a}\,(D+2\epsilon+\rho)-\ell^{a}\,(\delta+2\beta+\tau)\,]\,,$
(ED.19)
$\displaystyle\big{[}{{}^{(-1)\hskip-1.42262pt}}\mathcal{S}{}^{\dagger}\big{]}^{a}$
$\displaystyle=[\,n^{a}\,(\overline{\delta}-2\alpha-\varpi)-\overline{m}^{a}\,(\Delta-2\gamma-\mu)\,]\,(\Psi_{2})^{-2/3}\,.$
(ED.20)

###  B.2 Wald’s $1^{st}$ theorem in case of electrodynamics

By applying these adjoint operators ${{}^{(\pm
1)\hskip-1.42262pt}}\mathcal{S}{}^{\dagger}$ to solutions $\psi^{(+1)}$ and
$\psi^{(-1)}$ to the Teukolsky equations we can generate solutions
${{}^{(+1)\hskip-1.42262pt}}{\widehat{A}}_{a}$ and
${{}^{(-1)\hskip-1.42262pt}}{\widehat{A}}_{a}$ to (ED.1), respectively, given
as

$\displaystyle{{}^{(-1)\hskip-1.42262pt}}{\widehat{A}}_{a}={}$
$\displaystyle\mathfrak{Re}\,\big{[}\,{{}^{(-1)\hskip-1.42262pt}}\mathcal{S}{}^{\dagger}\psi^{(-1)}\,\big{]}$
$\displaystyle={}$
$\displaystyle\tfrac{1}{2}\big{[}\,m_{a}\,(D+2\epsilon+\rho)\,\psi^{(-1)}-\ell_{a}\,(\delta+2\beta+\tau)\,\psi^{(-1)}$
$\displaystyle+\overline{m}_{a}\,(D+2\overline{\epsilon}+\overline{\rho})\,\overline{\psi^{(-1)}}-\ell_{a}\,(\overline{\delta}+2\overline{\beta}+\overline{\tau})\,\overline{\psi^{(-1)}}\,\big{]},$
(ED.21)

and

$\displaystyle{{}^{(+1)\hskip-1.42262pt}}{\widehat{A}}_{a}={}$
$\displaystyle\mathfrak{Re}\,\big{[}{{}^{(+1)\hskip-1.42262pt}}\mathcal{S}{}^{\dagger}\psi^{(+1)}\,\big{]}$
$\displaystyle=$
$\displaystyle\tfrac{1}{2}\,\big{[}n_{a}\,(\overline{\delta}-2\alpha-\varpi)\,(\Psi_{2})^{-2/3}\psi^{(+1)}-\overline{m}_{a}\,(\Delta-2\gamma-\mu)\,(\Psi_{2})^{-2/3}\psi^{(+1)}$
$\displaystyle+n_{a}\,(\delta-2\overline{\alpha}-\overline{\varpi})\,\overline{(\Psi_{2})^{-2/3}\psi^{(+1)}}-m_{a}\,(\Delta-2\overline{\gamma}-\overline{\mu})\,\overline{(\Psi_{2})^{-2/3}\psi^{(+1)}}\,\big{]}\,,$
(ED.22)

where $\mathfrak{Re}$ denotes the operation taking the real value of the
pertinent complex variable. Note that neither
${{}^{(+1)\hskip-1.42262pt}}{\widehat{A}}_{a}$ nor
${{}^{(-1)\hskip-1.42262pt}}{\widehat{A}}_{a}$ coincides with the vector
potential $A_{a}$ from which the fields $\psi^{(+1)}$ and $\psi^{(-1)}$ are
generated by the relation $\psi^{(\pm 1)}=[{{}^{(\pm
1)\hskip-1.42262pt}}\mathcal{O}]^{a}A_{a}$.

###  B.3 Wald’s $2^{nd}$ theorem in case of electrodynamics

In virtue of Theorem A.2 while acting on the above yielded vector potentials
${{}^{(\pm 1)\hskip-1.42262pt}}{\widehat{A}}_{a}$ by the operators ${{}^{(\pm
1)\hskip-1.42262pt}}\mathcal{O}$ from the $\psi^{(\pm 1)}$ fields new spin
$\pm 1$ solutions $\widehat{\psi}^{(\pm 1)}$ can be generated which can be
given as

$\displaystyle\widehat{\psi}^{(+1)}$
$\displaystyle=\tfrac{1}{2}\,(D-\epsilon+\overline{\epsilon}-\overline{\rho})\,(D+2\overline{\epsilon}+\overline{\rho})\,\overline{\psi^{(-1)}}\,,$
(ED.23) $\displaystyle\widehat{\psi}^{(-1)}$
$\displaystyle=\tfrac{1}{2}\,(\Psi_{2})^{-2/3}\,(\Delta+\overline{\mu}-\overline{\gamma}+\gamma)(\Delta-2\overline{\gamma}-\overline{\mu})[\,(\overline{\Psi_{2}})^{-2/3}\overline{\psi^{(+1)}}\,]\,.$
(ED.24)

By applying the gauge fixing of Kinnersley [15], implying in particular
$\epsilon=0$, along with the equation $D\rho=\rho^{2}$ [which is guaranteed by
the Petrov D type of the background], (ED.23) simplifies to

$\displaystyle\widehat{\psi}^{(+1)}$
$\displaystyle=\tfrac{1}{2}\,DD\,\overline{\psi^{(-1)}}\,.$ (ED.25)

###  B.4 Linear metric perturbations

The linearized Einstein’s equations

$\mathcal{E}_{G}(h_{ef})=0\,,$ (MP.1)

where the metric perturbation $h_{ab}\,(=g_{ab}-\eta_{ab})$ plays the role of
the $(0,2)$ type tensor field $f$, whereas the linear operator
$\mathcal{E}_{G}$ acts on the $h_{ab}$ perturbations as

$[\mathcal{E}_{G}(h_{ef})]_{ab}=\tfrac{1}{2}\Big{[}g^{f}{}_{a}g^{eg}\nabla_{g}\nabla_{b}+g^{f}{}_{b}g^{eg}\nabla_{g}\nabla_{a}\\\
-g^{e}{}_{a}g^{f}{}_{b}\Box-g^{ef}\nabla_{a}\nabla_{b}-g_{ab}g^{eg}g^{fh}\nabla_{g}\nabla_{h}+g_{ab}g^{ef}\Box\,\Big{]}\,h_{ef}\,.$
(MP.2)

The operator $\mathcal{E}_{G}$ is self-adjoint and the additional linear
operators ${{}^{(s)\hskip-1.42262pt}}\mathcal{O}$,
${{}^{(s)\hskip-1.42262pt}}\mathcal{T}$ and
${{}^{(s)\hskip-1.42262pt}}\mathcal{S}$, relevant for the spin $s=+2$ and
$s=-2$ fields, along with the pertinent form of the theorems by Wald, are
given in the succeeding subsections.

####  B.4.1 The ${{}^{(\pm 2)\hskip-1.42262pt}}\mathcal{O}$ operators

Recall first that the Weyl spinor components $\Psi_{0}$ and $\Psi_{4}$ are
defined by the contractions

$\displaystyle\Psi_{0}$ $\displaystyle=-C_{abcd}\,l^{a}m^{b}l^{c}m^{d}\,,$
(MP.3) $\displaystyle\Psi_{4}$
$\displaystyle=-C_{abcd}\,n^{a}\overline{m}{}^{\,b}n^{c}\overline{m}{}^{\,d}\,,$
(MP.4)

whereas the Teukolsky variables, satisfying the homogeneous Teukolsky master
equations, respectively, read as [21]

$\displaystyle\psi^{(+2)}=$ $\displaystyle\
\Psi_{0}=[{{}^{(+2)\hskip-1.42262pt}}\mathcal{O}]^{ab}h_{ab}\,,$ (MP.5)
$\displaystyle\psi^{(-2)}=$ $\displaystyle\
(\Psi_{2})^{-4/3}\cdot\Psi_{4}=[{{}^{(-2)\hskip-1.42262pt}}\mathcal{O}]^{ab}h_{ab}\,,$
(MP.6)

where the explicit form of the operators ${{}^{(\pm
2)\hskip-1.42262pt}}\mathcal{O}$, relating the metric perturbations $h_{ab}$
to the extreme spin value Teukolsky variables $\psi^{(\pm 2)}$, can be deduced
from equations (B11) and (B12) of [7] and they read as

$\displaystyle[{{}^{(+2)\hskip-1.42262pt}}\mathcal{O}]^{ab}h_{ab}={}$
$\displaystyle\Psi_{0}=-\tfrac{1}{2}\\{(\delta+\overline{\varpi}-3\beta-\overline{\alpha})(\delta+\overline{\varpi}-2\beta-2\overline{\alpha})\,\ell^{a}\ell^{b}$
$\displaystyle+(D-\overline{\rho}-3\epsilon+\overline{\epsilon})(D-\overline{\rho}-2\epsilon+2\overline{\epsilon})\,m^{a}m^{b}$
$\displaystyle-[(D-\overline{\rho}-3\epsilon+\overline{\epsilon})(\delta+2\overline{\varpi}-2\beta)$
$\displaystyle+(\delta+\overline{\varpi}-3\beta-\overline{\rho})(D-2\overline{\rho}-2\epsilon)]\,n^{(a}\overline{m}^{b)}\\}\,h_{ab}\,,$
(MP.7)
$\displaystyle\hskip-11.38092pt[{{}^{(-2)\hskip-1.42262pt}}\mathcal{O}]^{ab}h_{ab}={}$
$\displaystyle(\Psi_{2})^{-4/3}\cdot\Psi_{4}$ $\displaystyle={}$
$\displaystyle-\tfrac{1}{2}(\Psi_{2})^{-4/3}\\{(\overline{\delta}-\overline{\tau}+3\alpha+\overline{\beta})(\overline{\delta}-\overline{\tau}+2\alpha+2\overline{\beta})\,n^{a}n^{b}$
$\displaystyle+(\Delta+\overline{\mu}+3\rho-\overline{\rho})(\Delta+\overline{\mu}+2\rho-2\overline{\rho})\,\overline{m}^{a}\overline{m}^{b}$
$\displaystyle-[(\Delta+\overline{\mu}+3\rho-\overline{\rho})(\overline{\delta}-2\overline{\tau}+2\alpha)$
$\displaystyle+(\overline{\delta}-\overline{\tau}+3\alpha+\overline{\beta})(\Delta+2\overline{\mu}+2\rho)]\,n^{(a}\overline{m}^{b)}\\}\,h_{ab}.$
(MP.8)

####  B.4.2 The ${{}^{(\pm 2)\hskip-1.42262pt}}\mathcal{T}$ operators

By applying the operators ${{}^{(\pm 2)\hskip-1.42262pt}}\mathcal{T}$ to the
fields $\psi^{(\pm 2)}$ we get the homogeneous TME. These operators ${{}^{(\pm
2)\hskip-1.42262pt}}\mathcal{T}$ read as [7]

$\displaystyle{{}^{(+2)\hskip-1.42262pt}}\mathcal{T}={}$
$\displaystyle[(D-3\epsilon+\overline{\epsilon}-4\rho-\overline{\rho})(\Delta-4\rho+\mu)$
$\displaystyle-(\delta+\overline{\varpi}-\overline{\alpha}-3\beta-4\tau)(\overline{\delta}-4\alpha+\varpi)-3\Psi_{2}]\,,$
(MP.9) $\displaystyle{{}^{(-2)\hskip-1.42262pt}}\mathcal{T}={}$
$\displaystyle[(\Delta+3\rho-\overline{\rho}+\overline{\mu})(D+4\epsilon+3\rho)$
$\displaystyle-(\overline{\delta}-\overline{\tau}+\overline{\beta}+3\alpha)(\delta+4\beta+3\tau)-3\Psi_{2}]\,.$
(MP.10)

A straightforward but heavy calculation verifies then that the relations

$\displaystyle{{}^{(+2)\hskip-1.42262pt}}\mathcal{T}^{\dagger}=\big{[}{{}^{(-2)\hskip-1.42262pt}}\mathcal{T}\big{]}^{\dagger}\,,$
(MP.11)
$\displaystyle{{}^{(-2)\hskip-1.42262pt}}\mathcal{T}^{\dagger}=\big{[}{{}^{(+2)\hskip-1.42262pt}}\mathcal{T}\big{]}^{\dagger}\,,$
(MP.12)

applied in Theorem A.1, also hold.

####  B.4.3 The ${{}^{(\pm 2)\hskip-1.42262pt}}\mathcal{S}$ operators

The operators ${{}^{(\pm 2)\hskip-1.42262pt}}\mathcal{S}$ can be seen to take
the form [7]

$\displaystyle{{}^{(+2)\hskip-1.42262pt}}\mathcal{S}{}^{ab}$
$\displaystyle\hskip
11.38092pt=(\delta+\overline{\varpi}-\overline{\alpha}-3\beta-4\tau)\big{[}(D-2\epsilon-2\overline{\rho})\,\ell^{(a}m^{b)}-(\delta+\overline{\varpi}-2\overline{\alpha}-2\beta)\,\ell^{a}\ell^{b}\big{]}$
$\displaystyle\hskip
22.76228pt+(D-3\epsilon+\overline{\epsilon}-4\rho-\overline{\rho})\big{[}(\delta+2\overline{\varpi}-2\beta)\,\ell^{(a}m^{b)}-(D-2\epsilon+2\overline{\epsilon}-\overline{\rho})\,m^{a}m^{b}\big{]}\,,$
(MP.13)
$\displaystyle{{}^{(-2)\hskip-1.42262pt}}\mathcal{S}{}^{ab}=(\Psi_{2})^{-4/3}$
$\displaystyle\hskip
11.38092pt\cdot\Big{\\{}(\Delta+3\rho-\overline{\rho}+4\mu+\overline{\mu})\big{[}(\overline{\delta}-2\overline{\tau}+2\alpha)\,n^{(a}\overline{m}^{b)}-(\Delta+2\rho-2\overline{\rho}+\overline{\mu})\,\overline{m}^{a}\overline{m}^{b}\big{]}$
$\displaystyle\hskip
17.07182pt+(\overline{\delta}-\overline{\tau}+\overline{\beta}+3\alpha+4\varpi)\big{[}(\Delta+2\rho+2\overline{\mu})\,n^{(a}\overline{m}^{b)}-(\overline{\delta}-\overline{\tau}+2\overline{\beta}+2\alpha)\,n^{a}n^{b}\big{]}\Big{\\}}\,.$
(MP.14)

The adjoints of operators ${{}^{(\pm 2)\hskip-1.42262pt}}\mathcal{S}$ can be
given as [7]

$\displaystyle[{{}^{(+2)\hskip-1.42262pt}}\mathcal{S}{}^{\dagger}]_{ab}$
$\displaystyle=\big{[}\ell_{(a}m_{b)}\,(D+3\epsilon+\overline{\epsilon}-\rho+\overline{\rho})-\ell_{a}\ell_{b}\,(\delta+3\beta+\overline{\alpha}-\tau)\big{]}(\delta+4\beta+3\tau)$
$\displaystyle+\big{[}\ell_{(a}m_{b)}\,(\delta+3\beta-\overline{\alpha}-\tau-\overline{\varpi})-m_{a}m_{b}\,(D+3\epsilon-\overline{\epsilon}-\rho)\big{]}\,(D+4\epsilon+3\rho)$
(MP.15)
$\displaystyle[{{}^{(-2)\hskip-1.42262pt}}\mathcal{S}{}^{\dagger}]_{ab}$
$\displaystyle=\Big{\\{}\big{[}n_{(a}\overline{m}_{b)}\,(\overline{\delta}-3\alpha+\overline{\beta}+\varpi+\overline{\tau})-\overline{m}_{a}\overline{m}_{b}\,(\Delta-3\rho+\overline{\rho}+\mu)\big{]}(\Delta-4\rho-3\mu)$
$\displaystyle+\big{[}n_{(a}\overline{m}_{b)}\,(\Delta-3\rho-\overline{\rho}+\mu-\overline{\mu})-n_{a}n_{b}\,(\overline{\delta}-3\alpha-\overline{\beta}+\varpi)\big{]}(\overline{\delta}-4\alpha-3\varpi)\Big{\\}}$
$\displaystyle\hskip 304.4444pt\cdot(\Psi_{2})^{-4/3}$ (MP.16)

###  B.5 Transformations of linear metric perturbations

In virtue of Theorem A.1 by applying these adjoint operators ${{}^{(\pm
2)\hskip-1.42262pt}}\mathcal{S}{}^{\dagger}$ to solutions $\psi^{(+2)}$ and
$\psi^{(-2)}$ to the Teukolsky equations solutions
${{}^{(+2)\hskip-1.42262pt}}{\widehat{h}}_{ab}$ and
${{}^{(-2)\hskip-1.42262pt}}{\widehat{h}}_{ab}$ to (MP.1), respectively, can
be generated, given formally as

$\displaystyle{{}^{(+2)}}{\widehat{h}}_{ab}$
$\displaystyle=\mathfrak{Re}\big{[}[{{}^{(+2)\hskip-1.42262pt}}\mathcal{S}{}^{\dagger}]_{ab}\,\psi^{(+2)}\big{]}$
(MP.17) $\displaystyle{{}^{(-2)}}{\widehat{h}}_{ab}$
$\displaystyle=\mathfrak{Re}\big{[}[{{}^{(-2)\hskip-1.42262pt}}\mathcal{S}{}^{\dagger}]_{ab}\,\psi^{(-2)}\big{]}$
(MP.18)

In analogy to the electromagnetic case neither ${{}^{(+2)}}{\widehat{h}}_{ab}$
nor ${{}^{(-2)}}{\widehat{h}}_{ab}$ has anything to do with the metric
perturbation $h_{ab}$ from which the fields $\psi^{(+2)}$ and $\psi^{(-2)}$
are generated as $\psi^{(\pm 2)}=[{{}^{(\pm
2)\hskip-1.42262pt}}\mathcal{O}]^{ab}h_{ab}$.

By appealing now to Theorem A.1 we get that by acting on the metric
perturbations ${{}^{(+2)}}{\widehat{h}}_{ab}$ and
${{}^{(-2)}}{\widehat{h}}_{ab}$ by the operators ${{}^{(\pm
2)\hskip-1.42262pt}}\mathcal{O}$ we can generate, from the $\psi^{(\pm 2)}$
new fields spin $\pm 2$ solutions $\widehat{\psi}^{(\pm 2)}$ which can be
given as [7]

$\widehat{\psi}^{(+2)}=\tfrac{1}{4}(D-\overline{\rho}-3\epsilon+\overline{\epsilon})(D-\overline{\rho}-2\epsilon+2\overline{\epsilon})(D-3\overline{\epsilon}+\epsilon-4\overline{\rho}-\rho)(D-2\overline{\epsilon}+2\epsilon-\rho)\,\overline{\psi^{(-2)}}.$
(MP.19) $\displaystyle\widehat{\psi}^{(-2)}={}$
$\displaystyle\tfrac{1}{4}(\Psi_{2})^{-4/3}$
$\displaystyle\cdot(\Delta+\overline{\mu}+3\rho-\overline{\rho})(\Delta+\overline{\mu}+2\rho-2\overline{\rho})(\Delta-3\overline{\rho}+\rho+\overline{\mu})(\Delta-4\overline{\rho}-3\overline{\mu})$
$\displaystyle\cdot(\overline{\Psi_{2}})^{-4/3}\,\overline{\psi^{(2)}}$
(MP.20)

As in the electromagnetic case, by applying the gauge fixing of Kinnersley
[15], with $\epsilon=0$, along with the equation $D\rho=\rho^{2}$ (MP.19)
simplifies to

$\displaystyle\widehat{\psi}^{(2)}=$
$\displaystyle\tfrac{1}{4}DDDD\overline{\psi^{(-2)}}.$ (MP.21)

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