# Superradiance and quantum states on black hole space-times

Visakan Balakumar1 Rafael Bernar2 and Elizabeth Winstanley1 1 Consortium for
Fundamental Physics, School of Mathematics and Statistics, The University of
Sheffield, Hicks Building, Hounsfield Road, Sheffield. S3 7RH United Kingdom
2 Faculdade de Física, Universidade Federal do Pará, 66075-110, Belém, Pará,
Brazil V.Balakumar@sheffield.ac.ukrbernar<EMAIL_ADDRESS>
###### Abstract

We consider the definition of the Boulware and Hartle-Hawking states for
quantum fields on black hole space-times. The properties of these states on a
Schwarzschild black hole have been understood for many years, but neither of
these states has a direct analogue on a Kerr black hole. We show how
superradiant modes play an important role in the definition of quantum states
on Kerr. Superradiance is also present on static black hole space-times, in
particular for a charged scalar field on a Reissner-Nordström black hole. We
explore whether analogues of the Boulware and Hartle-Hawking states exist in
this situation.

##  1 Quantum field theory on curved space-time

Quantum field theory on curved space-time is a semiclassical approach to
quantum gravity, wherein the space-time is fixed and purely classical. The
matter content in the theory consists of quantum fields, which propagate on
the classical background geometry. One of the most famous results in this
framework is the discovery by Hawking [1] that black holes formed by
gravitational collapse emit thermal quantum radiation with temperature $T_{H}$
given, in natural units by

$T_{H}=\frac{\kappa}{2\pi},$ (1)

where $\kappa$ is the surface gravity of the black hole.

The simplest type of black hole is described by the Schwarzschild metric

$ds^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}+\left(1-\frac{2M}{r}\right)^{-1}dr^{2}+r^{2}\,d\theta^{2}+r^{2}\sin^{2}\theta\,d\varphi^{2},$
(2)

where $M$ is the mass of the black hole (we employ units in which
$G=c=\hbar=k_{{\mathrm{B}}}=1$). On this background geometry, three standard
quantum states have been constructed (see Section 1.2 for more details). The
first is the Unruh state [2], which contains, at future null infinity, an
outgoing flux of Hawking radiation. A key feature of the Unruh state is that
it is regular across the future horizon of the black hole. In contrast, the
Boulware state [3] contains no particles as seen by a static observer far from
the black hole. Finally, the Hartle-Hawking state [4, 5] represents a black
hole surrounded by thermal radiation at the Hawking temperature. This state is
regular everywhere on and outside the event horizon. Our purpose in this work
is to investigate the analogues of these states for quantum scalar fields on
Reissner-Nordström and Kerr black holes. We begin by briefly outlining the
definition of quantum states for a neutral scalar field via canonical
quantization, and how this process is applied on a Schwarzschild black hole.
In Section 2 we consider a neutral scalar field on a Kerr black hole space-
time, and in Section 3 our focus is the Reissner-Nordström black hole, but
with a charged rather than neutral scalar field. Our conclusions are presented
in Section 4.

###  1.1 Canonical quantization of a neutral scalar field

Let $\Phi$ be a neutral scalar field. The process of canonical quantization
begins by expanding the classical field in an orthonormal basis of field
modes, which we write schematically as follows:

$\Phi=\sum_{j}a_{j}\phi_{j}^{+}+a_{j}^{\dagger}{\phi_{j}^{-}},$ (3)

where $j$ represents the quantum numbers characterizing the field modes. Here
the $\phi_{j}^{\pm}$ modes are, respectively, positive and negative frequency
modes. For example, in two-dimensional Minkowski space-time where the modes
are given by plane waves, in terms of time $t$ and a spatial coordinate $x$,
the modes $\phi_{j}^{\pm}$ take the form

$\displaystyle\phi_{j}^{+}$ $\displaystyle\propto e^{-i\omega(t\pm
x)},\qquad\omega>0,$ $\displaystyle\phi_{j}^{-}$ $\displaystyle\propto
e^{-i\omega(t\pm x)},\qquad\omega<0,$ (4)

where $\omega$ is the frequency of the mode. The field is quantized by
promoting the expansion coefficients $a_{j}$, $a_{j}^{\dagger}$ to operators
satisfying the commutation relations

$\left[{\hat{a}}_{j},{\hat{a}}_{k}^{\dagger}\right]=\delta_{jk},\qquad\left[{\hat{a}}_{j},{\hat{a}}_{k}\right]=0,\qquad\left[{\hat{a}}_{j}^{\dagger},{\hat{a}}_{k}^{\dagger}\right]=0.$
(5)

The vacuum state $\left|0\right\rangle$ is then defined as that state
annihilated by the ${\hat{a}}_{j}$ operators, that is:
${\hat{a}}_{j}\left|0\right\rangle=0$ for all $j$. Therefore the definition of
positive and negative frequency modes plays a central role in the construction
of quantum states.

###  1.2 Schwarzschild black hole

We now briefly outline the canonical quantization of a neutral scalar field on
a Schwarzschild black hole with metric (2). Mode solutions of the neutral
scalar field equation on this background take the form

$\phi_{\omega\ell m}(t,r,\theta,\varphi)=\frac{1}{{\mathcal{N}}r}\,e^{-i\omega
t}\,e^{im\varphi}\,Y_{\ell m}(\theta)R_{\omega\ell}(r),$ (6)

where ${\mathcal{N}}$ is a normalization constant, $\omega$ is the frequency
of the modes with respect to Schwarzschild time $t$, the integer
$m\in{\mathbb{Z}}$ is the azimuthal quantum number and $Y_{\ell m}(\theta)$ is
a spherical harmonic with total angular momentum quantum number
$\ell\in{\mathbb{Z}}^{+}$. The radial function $R_{\omega\ell}(r)$ satisfies a
Schrödinger-like equation

$\left[\frac{d^{2}}{dr_{*}^{2}}+V_{\omega\ell}(r)\right]R_{\omega\ell}(r)=0,$
(7)

where $r_{*}$ is the “tortoise” coordinate, defined by the differential
equation

$\frac{dr_{*}}{dr}=\left(1-\frac{2M}{r}\right)^{-1}.$ (8)

As $r\rightarrow r_{h}=2M$, the event horizon radius, the tortoise coordinate
$r_{*}\rightarrow-\infty$, while $r_{*}\rightarrow\infty$ far from the black
hole, as $r\rightarrow\infty$. In these asymptotic regions, the potential
$V_{\omega\ell}(r)$ has the following behaviour:

$V_{\omega\ell}(r)\rightarrow\omega^{2},\qquad r_{*}\rightarrow\pm\infty.$ (9)

Therefore, as $r_{*}\rightarrow\pm\infty$, we have mode solutions of the form
$e^{-i\omega(t\pm r_{*})}$ with frequency $\omega$, and hence the natural
definition of positive frequency with respect to Schwarzschild time $t$ is
$\omega>0$. A suitable orthonormal basis of field modes is formed by the “in”
and “up” modes, whose radial functions have the following forms in the
asymptotic regions:

$\displaystyle R_{\omega\ell}^{{\mathrm{in}}}(r)$
$\displaystyle\sim\left\\{\begin{array}[]{ll}B_{\omega\ell}^{{\mathrm{in}}}\,e^{-i\omega
r_{*}}&r_{*}\rightarrow-\infty,\\\ e^{-i\omega
r_{*}}+A_{\omega\ell}^{{\mathrm{in}}}\,e^{i\omega
r_{*}}&r_{*}\rightarrow\infty,\end{array}\right.$ (10c) $\displaystyle
R_{\omega\ell}^{{\mathrm{up}}}(r)$
$\displaystyle\sim\left\\{\begin{array}[]{ll}e^{i\omega
r_{*}}+A_{\omega\ell}^{{\mathrm{up}}}\,e^{-i\omega
r_{*}}&r_{*}\rightarrow-\infty,\\\ B_{\omega\ell}^{{\mathrm{up}}}\,e^{i\omega
r_{*}}&r_{*}\rightarrow\infty,\end{array}\right.$ (10f)

where $A_{\omega\ell}^{{\mathrm{in/up}}}$ and
$B_{\omega\ell}^{{\mathrm{in/up}}}$ are complex constants. The “in” modes
represent scalar waves incoming from past null infinity, which are partly
reflected back to future null infinity and partly transmitted down the future
horizon. The “up” modes represent scalar waves which are outgoing near the
past horizon, partly reflected back down the future horizon and partly
transmitted to future null infinity.

Using the “in” and “up” modes, suitably normalized, and defining positive
frequency with respect to Schwarzschild time $t$, the resulting vacuum state
is the Boulware state $|{\mathrm{B}}\rangle$ [3]. However, Schwarzschild time
$t$ is a good time coordinate only in the region outside the event horizon.
Alternatively, we can consider Kruskal time $T$, which is a valid time
coordinate everywhere in the space-time. We can construct an orthonormal basis
of modes having positive frequency with respect to Kruskal time $T$ from the
“in” and “up” modes (see, for example, the construction in [6]). The resulting
vacuum state is then the Hartle-Hawking state $|{\mathrm{H}}\rangle$ [4]. In
the definition of both the Boulware and Hartle-Hawking states, the “in” and
“up” modes are defined to be positive frequency with respect to the same time
coordinate. In contrast, to construct the Unruh state $|{\mathrm{U}}\rangle$
[2], the “in” modes are defined as having positive frequency with respect to
Schwarzschild time $t$, while the “up” modes have positive frequency with
respect to Kruskal time $T$.

To understand the physical properties of these states, in this note we will
consider unrenormalized expectation values of field operators, to avoid the
technical challenges inherent in the renormalization process. Since the
divergent terms which need to be subtracted to give renormalized expectation
values are purely geometric and independent of the quantum state (see, for
example, [7] for a neutral scalar field and [8] for a charged scalar field,
which will be considered in Section 3 below), differences in expectation
values between two quantum states do not require renormalization. For a
general operator ${\hat{O}}$ (for example, the square of the quantum scalar
field or the scalar field stress-energy tensor ${\hat{T}}_{\mu\nu}$), the
unrenormalized expectation values of the operator in the Boulware, Unruh and
Hartle-Hawking states on Schwarzschild space-time can be written as mode sums:

$\displaystyle\langle{\mathrm{B}}|{\hat{O}}|{\mathrm{B}}\rangle$
$\displaystyle=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\int_{0}^{\infty}d{\omega}\left[o_{\omega\ell
m}^{{\mathrm{in}}}+o_{\omega\ell m}^{{\mathrm{up}}}\right],$ (11a)
$\displaystyle\langle{\mathrm{U}}|{\hat{O}}|{\mathrm{U}}\rangle$
$\displaystyle=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\int_{0}^{\infty}d{\omega}\left[o_{\omega\ell
m}^{{\mathrm{in}}}+o_{\omega\ell
m}^{{\mathrm{up}}}\coth\left(\frac{{\omega}}{2T_{H}}\right)\right],$ (11b)
$\displaystyle\langle{\mathrm{H}}|{\hat{O}}|{\mathrm{H}}\rangle$
$\displaystyle=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\int_{0}^{\infty}d{\omega}\left[o_{\omega\ell
m}^{{\mathrm{in}}}+o_{\omega\ell
m}^{{\mathrm{up}}}\right]\coth\left(\frac{{\omega}}{2T_{H}}\right),$ (11c)

where $T_{H}$ is the Hawking temperature of the black hole (1), and
$o^{\rm{in/up}}_{\omega\ell m}$ is the classical value of $O$ when calculated
for a particular scalar field mode. All the unrenormalized expectation values
involve sums over the angular momentum quantum numbers $\ell$ and $m$, and
integrals over positive frequency $\omega>0$.

We interpret the mode sums in (11) as follows. The Boulware state
$|{\mathrm{B}}\rangle$ contains no particles in either the “in” or the “up”
modes, as seen by a static observer far from the black hole. In contrast, the
Hartle-Hawking state $|{\mathrm{H}}\rangle$ contains a thermal distribution of
particles in both the “in” and the “up” modes, at temperature $T_{H}$. Note
that the frequency $\omega$ of each mode appears in the thermal factor in the
mode sum. Finally, the Unruh state $|{\mathrm{U}}\rangle$ is devoid of “in”
mode particles, but contains a thermal flux of particles in the “up” modes,
corresponding to the outgoing Hawking radiation of the black hole.

##  2 Neutral scalar field on Kerr black hole

Having outlined the construction of the standard quantum states for a neutral
scalar field on a static Schwarzschild black hole, we next consider a neutral
scalar field on a rotating Kerr black hole, having metric

$ds^{2}=-\frac{\Delta}{\Sigma}\\!\\!\left[dt-a\sin^{2}\theta\,d\varphi\right]^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma\,d\theta^{2}+\frac{\sin^{2}\theta}{\Sigma}\\!\\!\left[\left(r^{2}+a^{2}\right)d\varphi-a\,dt\right]^{2},$
(12a) where the functions $\Delta(r)$ and $\Sigma(r)$ are given by
$\Delta=r^{2}-2Mr+a^{2},\qquad\Sigma=r^{2}+a^{2}\cos^{2}\theta,$ (12b)

and $M$ is the mass of the black hole, with $a$ the spin parameter.

###  2.1 Neutral scalar field modes

The neutral scalar field equation on the background (12) has mode solutions of
the form

$\phi_{\omega\ell
m}(t,r,\theta,\varphi)=\frac{1}{{\cal{N}}}\frac{1}{\left(r^{2}+a^{2}\right)^{\frac{1}{2}}}e^{-i\omega
t}e^{im\varphi}S_{\omega\ell m}(\cos\theta)R_{\omega\ell m}(r),$ (13)

where ${\mathcal{N}}$ is a normalization constant, $S_{\omega\ell
m}(\cos\theta)$ is a spheroidal harmonic and the angular quantum numbers
$\ell$, $m$ are as in the Schwarzschild case. The radial function
$R_{\omega\ell m}(r)$ again satisfies a Schrödinger-like equation

$\left[\frac{d^{2}}{dr_{*}^{2}}+V_{\omega\ell m}(r)\right]R_{\omega\ell
m}(r)=0,$ (14)

where the “tortoise” coordinate $r_{*}$ is now defined by

$\frac{dr_{*}}{dr}=\frac{r^{2}+a^{2}}{\Delta}.$ (15)

Close to the event horizon at $r=r_{h}$, $r_{*}\rightarrow-\infty$, and far
from the black hole ($r,r_{*}\rightarrow\infty$), the potential $V_{\omega\ell
m}(r)$ takes the form

$V_{\omega\ell
m}(r)=\left\\{\begin{array}[]{ll}{\widetilde{\omega}}^{2}=\left(\omega-m\Omega_{H}\right)^{2}&{\mbox{as
$r_{*}\rightarrow-\infty$}},\\\ \omega^{2}&{\mbox{as
$r_{*}\rightarrow\infty$}},\end{array}\right.$ (16)

where $\Omega_{H}$ is the angular speed of the event horizon.

At infinity, the potential approaches $\omega^{2}$ as in the Schwarzschild
case, and therefore the natural frequency with respect to the Schwarzschild-
like time coordinate $t$ far from the black hole is simply $\omega$. Close to
the horizon, however, the potential approaches ${\widetilde{\omega}}^{2}$ and
we therefore have a frequency shift, with the natural frequency of field modes
close to the horizon being ${\widetilde{\omega}}$ rather than $\omega$. This
can be seen in the form of the “in” and “up” modes in the asymptotic regions:

$\displaystyle R_{\omega\ell m}^{\rm{in}}(r)$
$\displaystyle\sim\left\\{\begin{array}[]{ll}B_{\omega\ell
m}^{{\mathrm{in}}}\,e^{-i{\widetilde{\omega}}r_{*}}&r_{*}\rightarrow-\infty,\\\
e^{-i\omega r_{*}}+A_{\omega\ell m}^{{\mathrm{in}}}\,e^{i\omega
r_{*}}&r_{*}\rightarrow\infty,\end{array}\right.$ (17c) $\displaystyle
R_{\omega\ell m}^{{\mathrm{up}}}(r)$
$\displaystyle\sim\left\\{\begin{array}[]{ll}e^{i{\widetilde{\omega}}r_{*}}+A_{\omega\ell
m}^{{\mathrm{up}}}\,e^{-i{\widetilde{\omega}}r_{*}}&r_{*}\rightarrow-\infty,\\\
B_{\omega\ell m}^{{\mathrm{up}}}\,e^{i\omega
r_{*}}&r_{*}\rightarrow\infty,\end{array}\right.$ (17f)

where $A_{\omega\ell m}^{{\mathrm{in/up}}}$ and $B_{\omega\ell
m}^{{\mathrm{in/up}}}$ are complex constants. This is our first indication
that the canonical quantization of a neutral scalar field on a Kerr black hole
is more complicated than on Schwarzschild space-time. In particular, the “in”
and “up” modes have different natural frequencies. Since the “in” modes
originate at past null infinity, the natural frequency for these modes is
$\omega$, the frequency far from the black hole. In contrast, since the “up”
modes originate close to the past event horizon, the natural frequency for
these modes is ${\widetilde{\omega}}$.

There is a further complication. The Wronskian of any two linearly independent
solutions of the radial equation (14) is a constant, and this can be used to
derive the following relation between the complex constants appearing in the
asymptotic form of the “in” modes:

$\omega\left[1-\left|A^{\rm{in}}_{\omega\ell
m}\right|^{2}\right]={\widetilde{\omega}}\left|B_{\omega\ell
m}^{\rm{in}}\right|^{2}.$ (18)

From this, it is clear that

$\left|A^{\rm{in}}_{\omega\ell m}\right|^{2}>1{\mbox{ if
}}\omega{\widetilde{\omega}}<0.$ (19)

Therefore, an “in” mode with $\omega{\widetilde{\omega}}<0$ has a reflected
part at future null infinity $A_{\omega\ell m}^{{\mathrm{in}}}\,e^{i\omega
r_{*}}$ which has greater amplitude than the incident part $e^{-i\omega
r_{*}}$. This is the phenomenon of superradiance [9], and results from the
wave extracting rotational energy from the black hole. Similarly, there are
superradiant “up” modes with $\left|A^{\rm{up}}_{\omega\ell m}\right|^{2}>1$
if $\omega{\widetilde{\omega}}<0$.

###  2.2 Unruh state on Kerr

To see the effect of superradiance on quantum states for the neutral scalar
field, one can proceed with canonical quantization as outlined in Section 1.
The Unruh state is constructed by taking “in” modes to have positive frequency
with respect to Schwarzschild-like time $t$ far from the black hole (that is,
$\omega>0$), while the “up” modes have positive frequency with respect to
Kruskal time $T$ near the event horizon. The resulting unrenormalized
expectation values are [10]

$\langle{\mathrm{U}}|{\hat{O}}|{\mathrm{U}}\rangle=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\left[\int_{0}^{\infty}d\omega\,o_{\omega\ell
m}^{{\mathrm{in}}}+\int_{0}^{\infty}d{\widetilde{\omega}}\,o_{\omega\ell
m}^{{\mathrm{up}}}\coth\left(\frac{{\widetilde{\omega}}}{2T_{H}}\right)\right].$
(20)

Note that the integral over the frequency of the “in” modes corresponds to
positive frequency for $\omega>0$ while the integral over the frequency of the
“up” modes is positive frequency for ${\widetilde{\omega}}>0$, so that the two
sets of modes are integrated over their natural frequencies. It can also be
seen that the thermal factor for the “up” modes involves their natural
frequency ${\widetilde{\omega}}$. As in the Schwarzschild case, the Unruh
state contains an outgoing flux of thermal Hawking radiation in the “up”
modes.

###  2.3 Boulware state on Kerr

Considering next the Boulware state, the analogue of (11a) for the neutral
scalar field on Kerr space-time is

$\langle{\mathrm{B}}|{\hat{O}}|{\mathrm{B}}\rangle=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\left[\int_{0}^{\infty}d\omega\,o_{\omega\ell
m}^{{\mathrm{in}}}+\int_{0}^{\infty}d{\widetilde{\omega}}\,o_{\omega\ell
m}^{{\mathrm{up}}}\right],$ (21)

which results from defining positive frequency to be $\omega>0$ for the “in”
modes and ${\widetilde{\omega}}>0$ for the “up” modes. At first glance, it
appears that this state is devoid of particles in the “in” and “up” modes.
However, evaluation of the expectation value
$\langle{\mathrm{B}}|{\hat{T}}_{t}^{r}|{\mathrm{B}}\rangle$ [11] reveals that
the state $|{\mathrm{B}}\rangle$ in fact contains an outgoing flux of
particles at future null infinity, arising from the superradiant modes with
$\omega{\widetilde{\omega}}<0$. This is the Unruh-Starobinskii effect [11,
12], the quantum analogue of classical superradiance.

###  2.4 Hartle-Hawking state on Kerr

Finally in this section, we consider the question of the analogue of the
Hartle-Hawking state $|{\mathrm{H}}\rangle$ on Kerr? For a neutral scalar
field, the Kay-Wald theorem [13, 14] states that there is no thermal
equilibrium state on Kerr which is regular everywhere on and outside the event
horizon and respects the underlying symmetries of the space-time. In other
words, there is no direct analogue of the Hartle-Hawking state for a neutral
quantum scalar field on Kerr space-time. A natural question is whether there
are nonetheless states on Kerr which possess some (but not all) of the
properties we expect for a “Hartle-Hawking”-like state.

There are two such states in the literature. The first [15] (which we call the
$|{\mathrm{CCH}}\rangle$ state), has a thermal distribution of particles in
both the “in” and the “up” modes, with the natural frequencies of each set of
modes appearing in the thermal factors, giving the unrenormalized expectation
values [15]:

$\langle{\mathrm{CCH}}|{\hat{O}}|{\mathrm{CCH}}\rangle=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\left[\int_{0}^{\infty}d\omega\,o_{\omega\ell
m}^{{\mathrm{in}}}\coth\left(\frac{\omega}{2T_{H}}\right)+\int_{0}^{\infty}d{\widetilde{\omega}}\,o_{\omega\ell
m}^{{\mathrm{up}}}\coth\left(\frac{{\widetilde{\omega}}}{2T_{H}}\right)\right].$
(22)

This state is regular everywhere outside the event horizon, but does not
represent an equilibrium state [10]. This may be understood from the fact that
the “in” and “up” modes have different thermal factors.

An alternative state [16] (which we call the $|{\mathrm{FT}}\rangle$ state)
may be defined, in which the “in” and “up” modes have the same thermal
factors. The resulting unrenormalized expectation values are [16]

$\langle{\mathrm{FT}}|{\hat{O}}|{\mathrm{FT}}\rangle=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\left[\int_{0}^{\infty}d\omega\,o_{\omega\ell
m}^{{\mathrm{in}}}\coth\left(\frac{{\widetilde{\omega}}}{2T_{H}}\right)+\int_{0}^{\infty}d{\widetilde{\omega}}\,o_{\omega\ell
m}^{{\mathrm{up}}}\coth\left(\frac{{\widetilde{\omega}}}{2T_{H}}\right)\right].$
(23)

This state does potentially define an equilibrium state. However, closer
inspection [10] reveals that in fact this state is divergent everywhere except
on the axis of rotation of the black hole.

###  2.5 Summary

This brief outline has revealed some subtleties in the definition of quantum
states for a neutral scalar field on a Kerr black hole background. In
particular, there is a frequency shift in the field modes at the horizon
relative to infinity, which results in the “in” and “up” modes having
different natural frequencies and the classical phenomenon of superradiance.
The construction of the Unruh state carries over from the Schwarzschild case
without difficulty, since this treats the “in” and “up” modes differently.
However, there are complications for both the Boulware and Hartle-Hawking
states. The Kerr equivalent of the Boulware state is no longer empty at future
null infinity, but contains an outgoing flux of particles in the superradiant
modes. The Hartle-Hawking state is more problematic. There is no state which
has all the properties we would wish a Hartle-Hawking state to have [13, 14].
In the literature there are two candidate “Hartle-Hawking”-like states [15,
16]. The first [15] has attractive regularity properties but is not an
equilibrium state, while the second [16] is potentially an equilibrium state
but fails to be regular almost everywhere outside the event horizon.

##  3 Charged scalar field on Reissner-Nordström black hole

We now turn to the main focus of this report, the quantization of a charged
scalar field on a Reissner-Nordström (RN) black hole. This section constitutes
an outline of the results in [17, 18] to which the reader is referred for
further details.

###  3.1 Charged scalar field on RN

We consider the RN metric

$ds^{2}=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)\,dt^{2}+\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}\,d\theta^{2}+r^{2}\sin^{2}\theta\,d\varphi^{2},$
(24)

where $M$ is the mass and $Q$ the charge of the black hole. We study a
massless charged scalar field satisfying the field equation

$D_{\mu}D^{\mu}\Phi=0,$ (25)

where $D_{\mu}=\nabla_{\mu}-iqA_{\mu}$ is the covariant derivative, $q$ the
scalar field charge and $A_{\mu}$ the electromagnetic potential, whose only
nonzero component is

$A_{0}=-\frac{Q}{r}.$ (26)

The mode solutions of the charged scalar field equation again take the form
(6), with the radial function $R_{\omega\ell}(r)$ satisfying a Schrödinger-
like equation (7), although the “tortoise” coordinate $r_{*}$ is now given by

$\frac{dr_{*}}{dr}=\frac{1}{1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}}.$ (27)

The potential $V_{\omega\ell}(r)$ is not the same as in the Schwarzschild
case, and depends on the charges of the black hole and scalar field. In the
asymptotic regions $r_{*}\rightarrow\pm\infty$ its behaviour is

$V_{\omega\ell}(r)=\left\\{\begin{array}[]{ll}{\widetilde{\omega}}^{2}=\left(\omega-\frac{qQ}{r_{+}}\right)^{2}&{\mbox{as
$r_{*}\rightarrow-\infty$}},\\\ \omega^{2}&{\mbox{as
$r_{*}\rightarrow\infty$}},\end{array}\right.$ (28)

where $r_{+}$ is the event horizon radius. We see that, as in the Kerr case,
there is a frequency shift at the event horizon relative to infinity.

The “in” and “up” modes take the form (17) in the asymptotic regions, but with
${\widetilde{\omega}}$ defined by (28) rather than (16). For modes with
$\omega{\widetilde{\omega}}<0$ there is charge superradiance [19], with
classical waves being amplified upon scattering by the black hole. In this
situation the scattered wave is extracting electrostatic rather than
rotational energy from the black hole.

Following the discussion in the previous section, we now seek to understand
the effect of charge superradiance on the construction of the standard quantum
states for a charged scalar field on an RN black hole, again considering
unrenormalized expectation values. The observables of interest are the scalar
condensate ${\widehat{SC}}$, current ${\hat{J}}^{\mu}$ and stress-energy
tensor ${\hat{T}}_{\mu\nu}$, given in terms of the scalar field operator
${\hat{\Phi}}$ as follows:

$\displaystyle{\widehat{SC}}$
$\displaystyle=\frac{1}{2}\left[{\hat{\Phi}}{\hat{\Phi}}^{\dagger}+{\hat{\Phi}}^{\dagger}{\hat{\Phi}}\right],$
(29a) $\displaystyle{\hat{J}}^{\mu}$
$\displaystyle=\frac{iq}{16\pi}\left[{\hat{\Phi}}^{\dagger}\left(D^{\mu}{\hat{\Phi}}\right)+\left(D^{\mu}{\hat{\Phi}}\right){\hat{\Phi}}^{\dagger}-{\hat{\Phi}}\left(D^{\mu}{\hat{\Phi}}\right)^{\dagger}-\left(D^{\mu}{\hat{\Phi}}\right)^{\dagger}{\hat{\Phi}}\right],$
(29b) $\displaystyle{\hat{T}}_{\mu\nu}$
$\displaystyle=\frac{1}{4}\left\\{\left(D_{\mu}{\hat{\Phi}}\right)^{\dagger}D_{\nu}{\hat{\Phi}}+D_{\nu}{\hat{\Phi}}\left(D_{\mu}{\hat{\Phi}}\right)^{\dagger}+\left(D_{\nu}{\hat{\Phi}}\right)^{\dagger}D_{\mu}{\hat{\Phi}}+D_{\mu}{\hat{\Phi}}\left(D_{\nu}{\hat{\Phi}}\right)^{\dagger}\right.$
$\displaystyle~{}\left.\qquad-\frac{1}{2}g_{\mu\nu}g^{\rho\sigma}\left[\left(D_{\rho}{\hat{\Phi}}\right)^{\dagger}D_{\sigma}{\hat{\Phi}}+D_{\sigma}{\hat{\Phi}}\left(D_{\rho}{\hat{\Phi}}\right)^{\dagger}+\left(D_{\sigma}{\hat{\Phi}}\right)^{\dagger}D_{\rho}{\hat{\Phi}}+D_{\rho}{\hat{\Phi}}\left(D_{\sigma}{\hat{\Phi}}\right)^{\dagger}\right]\right\\}.$
(29c)

We particularly focus on the fluxes of charge ${\mathcal{K}}$ and energy
${\mathcal{L}}$, which are given by the following expectation values of
components of the current and stress-energy tensor:

$\langle{\hat{{{J}}}}^{r}\rangle=-\frac{{\mathcal{K}}}{r^{2}},\qquad\langle{\hat{{{T}}}}^{r}_{t}\rangle=-\frac{{\mathcal{L}}}{r^{2}}+\frac{4\pi
Q{\mathcal{K}}}{r^{3}}.$ (30)

###  3.2 Unruh state on RN

We begin our study of quantum states for a charged scalar field on an RN black
hole with an uncontroversial state, namely the Unruh state. The construction
of this state follows that in the Kerr case, resulting in the unrenormalized
expectation values [18, 20]

$\langle{\rm{U}}|{\hat{{O}}}|{\rm{U}}\rangle=\frac{1}{2}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\left[\int_{-\infty}^{\infty}d\omega\,o_{\omega\ell
m}^{\rm{in}}+\int_{-\infty}^{\infty}d{\widetilde{\omega}}\,o_{\omega\ell
m}^{\rm{up}}\coth\left|\frac{{\widetilde{\omega}}}{2T_{H}}\right|\,\right].$
(31)

The mode sums involve contributions from both positive and negative frequency
modes, due to the charge of the scalar field. As in the Kerr case, there are
no particles in the “in” modes and a thermal distribution of particles in the
“up” modes, with the frequency ${\widetilde{\omega}}$ (the natural frequency
for the “up” modes) in the thermal factor. The thermal nature is clear from
the expressions for the fluxes of charge and energy in this state [18, 20]:

$\displaystyle{\mathcal{K}}_{{\mathrm{U}}}$
$\displaystyle=\frac{q}{64\pi^{3}}\sum_{\ell=0}^{\ell}\int_{0}^{\infty}d\omega\left(2\ell+1\right)\omega\left[\frac{{\left|{B}^{\mathrm{up}}_{\omega\ell}\right|}^{2}}{{\widetilde{\omega}}\left(\exp{\left[\frac{{{\widetilde{\omega}}}}{T_{H}}\right]}-1\right)}-\frac{{\left|{B}^{\mathrm{up}}_{-\omega\ell}\right|}^{2}}{{\overline{\omega}}\left(\exp{\left[\frac{{\overline{\omega}}}{T_{H}}\right]}-1\right)}\right],$
(32a) $\displaystyle{\mathcal{L}}_{{\mathrm{U}}}$
$\displaystyle=\frac{1}{16\pi^{2}}\sum_{\ell=0}^{\ell}\int_{0}^{\infty}d\omega\left(2\ell+1\right)\omega^{2}\left[\frac{{\left|{B}^{\mathrm{up}}_{\omega\ell}\right|}^{2}}{{\widetilde{\omega}}\left(\exp\left[\frac{{\widetilde{\omega}}}{T_{H}}\right]-1\right)}+\frac{{\left|{B}^{\mathrm{up}}_{-\omega\ell}\right|}^{2}}{{\overline{\omega}}\left(\exp\left[\frac{{\overline{\omega}}}{T_{H}}\right]-1\right)}\right],$
(32b)

where for negative frequency modes the term in the thermal factor is

${\overline{\omega}}=\omega+\frac{qQ}{r_{+}}.$ (33)

The fluxes (32) contain contributions from both positive and negative
frequency modes. For positive frequency modes, there is thermal emission with
an effective chemical potential $qQ/r_{+}$ [1, 20]. The sign of the effective
chemical potential is reversed for negative frequency modes.

###  3.3 Boulware state on RN

We now turn to the construction of a “Boulware”-like state for a charged
scalar field on RN space-time. We quantize the “in” and “up” modes with
respect to their natural frequencies, defining positive frequency to be
$\omega>0$ for the “in” modes and ${\widetilde{\omega}}>0$ for the “up” modes.
We denote the resulting vacuum state $|{\rm{B}}_{1}\rangle$. The construction
gives the following unrenormalized expectation values [18]

$\langle{\rm{B}}_{1}|{\hat{{O}}}|{\rm{B}}_{1}\rangle=\frac{1}{2}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\left[\int_{-\infty}^{\infty}d\omega\,o_{\omega\ell
m}^{{\rm{in}}}+\int_{-\infty}^{\infty}d{\widetilde{\omega}}\,o_{\omega\ell
m}^{{\rm{up}}}\right].$ (34)

As in the Kerr case, this state is not empty at future null infinity. There is
an outgoing flux of charge and energy in the superradiant “up” modes [17]:

$\displaystyle{\mathcal{K}}_{{\mathrm{B}}_{1}}$
$\displaystyle=\frac{q}{64\pi^{3}}\sum_{\ell=0}^{\infty}\int_{\min\\{\frac{qQ}{r_{+}},0\\}}^{\max\\{\frac{qQ}{r^{+}},0\\}}d\omega\frac{\omega}{\left|{\widetilde{\omega}}\right|}\left(2\ell+1\right)\left|B^{\rm{up}}_{\omega\ell}\right|^{2},$
(35a) $\displaystyle{\mathcal{L}}_{{\mathrm{B}}_{1}}$
$\displaystyle=\frac{1}{16\pi^{2}}\sum_{\ell=0}^{\infty}\int_{\min\\{\frac{qQ}{r_{+}},0\\}}^{\max\\{\frac{qQ}{r^{+}},0\\}}d\omega\frac{\omega^{2}}{\left|{\widetilde{\omega}}\right|}\left(2\ell+1\right)\left|B^{\rm{up}}_{\omega\ell}\right|^{2}.$
(35b)

The question is then whether it is possible to construct a state which is as
empty as possible at both past and future null infinity. Such a state, denoted
$|{\rm{B}}\rangle$, is constructed in [18], using creation and annihilation
operators which satisfy nonstandard commutation relations (see [18] for
details). As far as unrenormalized expectation values are concerned, the
superradiant “up” modes make a contribution which has the opposite sign to
that in (34) [18]:

$\langle{\rm{B}}|{\hat{{O}}}|{\rm{B}}\rangle=~{}\frac{1}{2}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\left[\int_{-\infty}^{\infty}d\omega\left[o_{\omega\ell
m}^{\rm{in}}+o_{\omega\ell
m}^{\rm{up}}\right]-2\int_{\min\\{0,\frac{qQ}{r_{+}}\\}}^{\max\\{0,\frac{qQ}{r_{+}}\\}}d\omega\,o_{\omega\ell
m}^{\rm{up}}\right].$ (36)

The fluxes of charge and energy in this state vanish:

${\mathcal{K}}_{\mathrm{B}}=0,\qquad{\mathcal{L}}_{\mathrm{B}}=0,$ (37)

and therefore the state $|{\rm{B}}\rangle$ is as empty as possible at both
past and future null infinity.

As with the Boulware state on Schwarzschild space-time, the state
$|{\rm{B}}\rangle$ diverges on the event horizon of the RN black hole. It is
argued in [10] that there is no vacuum state for a neutral quantum scalar
field on Kerr space-time which is as empty as possible at both past and future
null infinity. This appears to also be the case for a charged quantum scalar
field on RN space-time, since the “Boulware”-like state $|{\rm{B}}\rangle$ is
not a conventional vacuum state, as its construction relies on nonstandard
commutation relations [18].

###  3.4 Hartle-Hawking state on RN

Finally we consider the construction of an analogue of the Hartle-Hawking
state for a charged quantum scalar field on an RN black hole. Following our
discussion of “Hartle-Hawking”-like states on Kerr space-time, we begin with a
state $|{\mathrm{H}}_{1}\rangle$, which is the analogue of the
$|{\mathrm{CCH}}\rangle$ state on Kerr. Unrenormalized expectation values in
this state take the form [18]

$\langle{\mathrm{H}}_{1}|{\hat{O}}|{\mathrm{H}}_{1}\rangle=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\left[\int_{-\infty}^{\infty}d\omega\,o_{\omega\ell
m}^{{\mathrm{in}}}\coth\left|\frac{\omega}{2T_{H}}\right|+\int_{-\infty}^{\infty}d{\widetilde{\omega}}\,o_{\omega\ell
m}^{{\mathrm{up}}}\coth\left|\frac{{\widetilde{\omega}}}{2T_{H}}\right|\,\right].$
(38)

Both the “in” and “up” modes contain a thermal distribution of particles at
the Hawking temperature $T_{H}$, however the frequencies in the thermal
factors are different. The “in” modes are thermalized with respect to their
natural frequency $\omega$, while the “up” modes are thermalized with respect
to their natural frequency ${\widetilde{\omega}}$.

As might be expected, the state $|{\mathrm{H}}_{1}\rangle$ is not an
equilibrium state. The fluxes of charge and energy in this state are [18]:

$\displaystyle{\mathcal{K}}_{{\mathrm{H}}_{1}}$
$\displaystyle={\mathcal{K}}_{{\mathrm{U}}}-\frac{q}{64\pi^{3}}\sum_{\ell=0}^{\infty}\int_{0}^{\infty}d\omega\,\frac{\left(2\ell+1\right){\widetilde{\omega}}^{2}}{\omega\\!\left(\exp\left[\frac{\omega}{T_{H}}\right]-1\right)}\left[\frac{1}{{\widetilde{\omega}}}\left|B_{\omega\ell}^{\rm{in}}\right|^{2}-\frac{1}{{\overline{\omega}}}\left|B_{-\omega\ell}^{\rm{in}}\right|^{2}\right],$
(39a) $\displaystyle{\mathcal{L}}_{{\mathrm{H}}_{1}}$
$\displaystyle={\mathcal{L}}_{{\mathrm{U}}}-\frac{1}{16\pi^{2}}\sum_{\ell=0}^{\infty}\int_{0}^{\infty}d\omega\,\frac{\left(2\ell+1\right){\widetilde{\omega}}^{2}}{\left(\exp\left[\frac{\omega}{T_{H}}\right]-1\right)}\left[\frac{1}{{\widetilde{\omega}}}\left|B_{\omega\ell}^{\rm{in}}\right|^{2}+\frac{1}{{\overline{\omega}}}\left|B_{-\omega\ell}^{\rm{in}}\right|^{2}\right],$
(39b)

where ${\mathcal{K}}_{{\mathrm{U}}}$ and ${\mathcal{L}}_{\mathrm{U}}$ are,
respectively, the fluxes of charge and energy in the Unruh state (32). In the
“up” modes, the fluxes are the same as those for the Unruh state
$|{\mathrm{U}}\rangle$, since the thermalization of the “up” modes is the same
in these two states. The state $|{\mathrm{H}}_{1}\rangle$ has additional
thermal fluxes for the “in” modes, with the thermal factor containing the
natural frequency $\omega$ for these modes.

Again following the constructions in the Kerr case, our second “Hartle-
Hawking”-like state, $|{\mathrm{H}}_{2}\rangle$, has identical thermal factors
for both the “in” and “up” modes, with unrenormalized expectation values given
by [18]

$\langle{\mathrm{H}}_{2}|{\hat{O}}|{\mathrm{H}}_{2}\rangle=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\left[\int_{-\infty}^{\infty}d{\widetilde{\omega}}\,o_{\omega\ell
m}^{{\mathrm{in}}}\coth\left|\frac{{\widetilde{\omega}}}{2T_{H}}\right|+\int_{-\infty}^{\infty}d{\widetilde{\omega}}\,o_{\omega\ell
m}^{{\mathrm{up}}}\coth\left|\frac{{\widetilde{\omega}}}{2T_{H}}\right|\,\right].$
(40)

The state $|{\mathrm{H}}_{2}\rangle$ is the analogue, for a charged quantum
scalar field on an RN black hole, of the state $|{\mathrm{FT}}\rangle$ for a
neutral quantum scalar field on a Kerr black hole.

Unlike the state $|{\mathrm{FT}}\rangle$ on Kerr, the state
$|{\mathrm{H}}_{2}\rangle$ is not an equilibrium state. It has nonzero fluxes
of charge and energy in the superradiant “in” modes:

$\displaystyle{\mathcal{K}}_{{\mathrm{H}}_{2}}$
$\displaystyle=\frac{q}{64\pi^{3}}\sum_{\ell=0}^{\infty}\int_{\min\left\\{\frac{qQ}{r_{+}},0\right\\}}^{\max\left\\{\frac{qQ}{r_{+}},0\right\\}}d\omega\,\frac{\left|{\widetilde{\omega}}\right|}{\omega}\left(2\ell+1\right)\left|{{B}}_{\omega\ell}^{\mathrm{in}}\right|^{2}\coth\left|\frac{{\widetilde{\omega}}}{2T_{H}}\right|,$
(41a) $\displaystyle{\mathcal{L}}_{{\mathrm{H}}_{2}}$
$\displaystyle=\frac{1}{16\pi^{2}}\sum_{\ell=0}^{\infty}\int_{\min\left\\{\frac{qQ}{r_{+}},0\right\\}}^{\max\left\\{\frac{qQ}{r_{+}},0\right\\}}d\omega\,\left|{\widetilde{\omega}}\right|\left(2\ell+1\right)\left|{{B}}_{\omega\ell}^{\mathrm{in}}\right|^{2}\coth\left|\frac{{\widetilde{\omega}}}{2T_{H}}\right|.$
(41b)

In [18], the properties of the state $|{\mathrm{H}}_{2}\rangle$ are studied by
considering the differences in expectation values between this state and the
uncontroversial Unruh state $|{\mathrm{U}}\rangle$. The differences in the
expectation values of the current and stress-energy tensor are regular
everywhere outside the event horizon. Since it is anticipated that the Unruh
state $|{\mathrm{U}}\rangle$ is also regular everywhere outside the event
horizon, this may lead us to deduce that the state $|{\mathrm{H}}_{2}\rangle$
is also regular everywhere outside the event horizon. However, there is a
problem when we consider the difference in expectation values of the scalar
condensate:

$\langle{\mathrm{H}}_{2}|{\widehat{{SC}}}|{\mathrm{H}}_{2}\rangle-\langle{\mathrm{U}}|{\widehat{{SC}}}|{\mathrm{U}}\rangle=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\int_{-\infty}^{\infty}d\omega\,\frac{1}{\exp\left|\frac{{\widetilde{\omega}}}{T_{H}}\right|-1}\left|\phi^{\rm{in}}_{\omega\ell
m}\right|^{2}.$ (42)

The integrand in (42) has a pole when ${\widetilde{\omega}}=0$, and the
integral is not convergent. Given the expected regularity of the Unruh state
$|{\mathrm{U}}\rangle$, we infer that the state $|{\mathrm{H}}_{2}\rangle$ is
divergent everywhere. This situation is similar to that for the state
$|{\mathrm{FT}}\rangle$ on Kerr, whose divergent nature is revealed by
considering the vacuum polarization (the scalar condensate for a neutral
scalar field) [10].

Thus far we have postulated two “Hartle-Hawking”-like states: the first,
$|{\mathrm{H}}_{1}\rangle$, is regular everywhere outside the event horizon
but is not an equilibrium state; the second, $|{\mathrm{H}}_{2}\rangle$, is
also not an equilibrium state but is divergent everywhere outside the event
horizon. In [18], a third potential “Hartle-Hawking”-like state,
$|{\mathrm{H}}\rangle$, is constructed. Like the state $|{\mathrm{B}}\rangle$,
the construction of the state $|{\mathrm{H}}\rangle$ relies on the use of
creation and annihilation operators satisfying nonstandard commutation
relations. This has the effect of reversing the sign of the contribution of
the superradiant “in” modes to unrenormalized expectation values, compared to
those in the state $|{\mathrm{H}}_{2}\rangle$, yielding:

$\langle{\rm{H}}|{\hat{{O}}}|{\rm{H}}\rangle=\frac{1}{2}\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\left[\int_{-\infty}^{\infty}d{\widetilde{\omega}}\,\left[o_{\omega\ell
m}^{\rm{in}}+o_{\omega\ell
m}^{\rm{up}}\right]\coth\left|\frac{{\widetilde{\omega}}}{2T_{H}}\right|\right.\\\
\left.-2\int_{\min\\{0,\frac{qQ}{r_{+}}\\}}^{\max\\{0,\frac{qQ}{r_{+}}\\}}d\omega\,o_{\omega\ell
m}^{\rm{in}}\coth\left|\frac{{\widetilde{\omega}}}{2T_{H}}\right|\,\right].$
(43)

Unlike the states $|{\mathrm{H}}_{1}\rangle$ and $|{\mathrm{H}}_{2}\rangle$,
our new state $|{\mathrm{H}}\rangle$ is an equilibrium state, with vanishing
fluxes of both charge and energy [18]:

${\mathcal{K}}_{\mathrm{H}}=0,\qquad{\mathcal{L}}_{\mathrm{H}}=0.$ (44)

This is one of the properties we were seeking. The other is regularity
everywhere outside the event horizon.

Based on our experience with the states $|{\mathrm{H}}_{1}\rangle$ and
$|{\mathrm{H}}_{2}\rangle$, the key quantity to examine is the scalar
condensate. We find the following expression for the difference in expectation
values of the scalar condensate in the state $|{\mathrm{H}}\rangle$ and the
Unruh state:

$\langle{\rm{H}}|{\widehat{{SC}}}|{\rm{H}}\rangle-\langle{\rm{U}}|{\widehat{{SC}}}|{\rm{U}}\rangle=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}\int_{0}^{\infty}d\omega\left[\frac{\left|\phi_{\omega\ell
m}^{\rm{in}}\right|^{2}}{\exp\left(\frac{{\widetilde{\omega}}}{T_{H}}\right)-1}+\frac{\left|\phi_{-\omega\ell
m}^{\rm{in}}\right|^{2}}{\exp\left(\frac{{\overline{\omega}}}{T_{H}}\right)-1}\right],$
(45)

where ${\widetilde{\omega}}$ is given by (28) and ${\overline{\omega}}$ by
(33). This clearly has a thermal distribution of particles in the “in” modes,
with the chemical potential changing signs for the negative frequency modes
relative to the positive frequency modes (compare (32) for the fluxes in the
“up” modes in the Unruh state). Each integral in (45) has a pole in the
denominator when either ${\widetilde{\omega}}=0$ or ${\overline{\omega}}=0$.
However, unlike the poles arising in the integrand in (42), the Cauchy
principal values of the integrals in (45) exist, leading to a regular scalar
condensate expectation value in the $|{\mathrm{H}}\rangle$ state [18]. Further
numerical investigations [18] reveal that the differences in expectation
values for the scalar condensate, current and stress-energy tensor between the
state $|{\mathrm{H}}\rangle$ and the Unruh state are all regular everywhere
outside the event horizon. We therefore deduce that the state
$|{\mathrm{H}}\rangle$ is, like the Unruh state, regular everywhere outside
the event horizon.

##  4 Conclusions

The standard Unruh [2], Boulware [3] and Hartle-Hawking [4] states for quantum
fields on a Schwarzschild black hole were constructed in the 1970s and since
then have been extensively studied. Extending these definitions to more
general black holes has proven to be nontrivial. On a rotating Kerr black
hole, for a neutral scalar field there is no Hartle-Hawking state [13, 14],
and attempts to define a state with some (but not all) of the properties of a
Hartle-Hawking state result in either a state [15] which is regular outside
the event horizon but not an equilibrium state, or an equilibrium state [16]
which is divergent almost everywhere [10]. A key property of bosonic quantum
fields on Kerr space-time is the classical phenomenon of superradiance [9], by
which an incident wave is amplified upon scattering by the black hole. In this
note we have described superradiance as resulting from a frequency shift in
scalar field modes at the event horizon relative to infinity. This frequency
shift complicates the construction of quantum states via canonical
quantization, since the first step is to split the field modes into “positive”
and “negative” frequencies.

A frequency shift, and consequent superradiance, also occurs for a charged
scalar field on a Reissner-Nordström black hole. In analogy with the Kerr
case, we have found that the definition of the standard quantum states in this
set-up is also complicated by the superradiant modes. As well as analogues of
the “Hartle-Hawking”-like states [15, 16] discussed above for a neutral scalar
field on Kerr, we have postulated two new states, $|{\mathrm{B}}\rangle$ and
$|{\mathrm{H}}\rangle$. The state $|{\mathrm{B}}\rangle$ is as empty as
possible at both past and future null infinity, and is analogous to the
Boulware state on Schwarzschild. The state $|{\mathrm{H}}\rangle$ is the
analogue of the Hartle-Hawking state on Schwarzschild — it is a thermal
equilibrium state and appears to be regular everywhere outside the event
horizon. However, we anticipate that a Kay-Wald-like theorem [13, 14] will
still hold for a charged scalar field on Reissner-Nordström space-time, since
the construction of both the states $|{\mathrm{B}}\rangle$ and
$|{\mathrm{H}}\rangle$ relies on using creation and annihilation operators
which do not satisfy the standard commutation relations (see [18] for
details). Therefore, it is likely that these states do not satisfy the
assumptions of a Kay-Wald-like theorem [21].

## Acknowledgments

We thank the organizers of the workshop “Avenues of Quantum Field Theory in
Curved Space-Time” for a very interesting and stimulating meeting, at which
this work was presented. We thank Luís C. B. Crispino and Bernard Kay for
invaluable discussions on this project. V.B. thanks STFC for the provision of
a studentship supporting this work, and the Faculdade de Física, Universidade
Federal do Pará, for hospitality while this work was in progress. The work of
V.B. is also supported by an EPSRC Mathematical Sciences Research
Associateship. The work of R.P.B. is financed in part by Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES, Brazil) - Finance Code
001 and by Conselho Nacional de Desenvolvimento Científico e Tecnológico
(CNPq, Brazil). The work of E.W. is supported by the Lancaster-Manchester-
Sheffield Consortium for Fundamental Physics under STFC grant ST/T001038/1.
This research has also received funding from the European Union’s Horizon 2020
research and innovation program under the H2020-MSCA-RISE-2017 Grant No.
FunFiCO-777740.

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