# Two-point function of a quantum scalar field in the interior region of a
Kerr black hole

Noa Zilberman<EMAIL_ADDRESS>Department of Physics,
Technion, Haifa 32000, Israel  Marc Casals<EMAIL_ADDRESS><EMAIL_ADDRESS>Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, CEP 22290-180,
Brazil  School of Mathematics and Statistics, University College Dublin,
Belfield, Dublin 4, D04 V1W8, Ireland  Laboratoire Univers et Théories,
Observatoire de Paris, CNRS, Université PSL, Université de Paris, 92190
Meudon, France  Amos Ori<EMAIL_ADDRESS>Department of Physics,
Technion, Haifa 32000, Israel  Adrian C. Ottewill<EMAIL_ADDRESS>School of Mathematics and Statistics, University College Dublin, Belfield,
Dublin 4, D04 V1W8, Ireland

###### Abstract

Quantum field effects on a classical background spacetime may be obtained from
the semiclassical equations of General Relativity with the expectation value
of the stress-energy tensor of the quantum field as a source. This expectation
value can be calculated from Hadamard’s elementary two-point function, which
in practice is given in terms of sums of products of field modes evaluated at
two spacetime points. We derive expressions for the two-point function for a
massless scalar field in the Unruh state on a Kerr black hole spacetime. Our
main result in this paper is a novel expression valid when the two points lie
inside the black hole; we also (re-)derive, using a new method, the known
expression valid when the two points lie outside the black hole. We achieve
these expressions by finding relationships between Unruh modes, defined in
terms of the retarded Kruskal coordinate, and Eddington modes, defined in
terms of the Eddington coordinates. While our starting expression for the two-
point function is written in terms of the Unruh modes, we give our final
expression in terms of the Eddington modes, which have the computational
advantage that they decompose into factors that obey ordinary differential
equations. In an appendix we also derive expressions for the bare mode
contributions to the flux components of the stress-energy tensor for a
minimally-coupled massless scalar field inside the black hole. Our results
thus lay the groundwork for future calculations of quantum effects inside a
Kerr black hole.

##  I Introduction

In the semiclassical framework of quantum field theory on a curved spacetime,
a gravitational field is treated classically whereas matter fields on the
corresponding (background) spacetime are quantized. In practice, the Einstein
field equations of General Relativity are sourced by the renormalized
expectation value of the stress-energy tensor (RSET) for the matter fields in
a certain quantum state. This framework is expected to provide a good
approximation to the Physics when the scales of the system are above the
Planck scales and it has yielded results as important as the emission of
quantum (Hawking) radiation by astrophysical black holes (BHs) [1, 2].

Within quantum field theory on a curved spacetime, one can define Hadamard’s
elementary two-point function (HTPF) as the expectation value of the anti-
commutator of a (say, scalar) field $\hat{\Phi}$ in a certain quantum state
$\left|\Psi\right\rangle$:
$G_{\Psi}^{(1)}(x,x^{\prime})=\left\langle\left\\{\hat{\Phi}\left(x\right),\hat{\Phi}\left(x^{\prime}\right)\right\\}\right\rangle_{\Psi}$,
where $x$ and $x^{\prime}$ are spacetime points, and curly brackets denote
symmetrization with respect to $x$ and $x^{\prime}$ 111More explicitly,
$\left\\{\xi\left(x\right),\zeta\left(x^{\prime}\right)\right\\}\equiv\xi\left(x\right)\zeta\left(x^{\prime}\right)+\xi\left(x^{\prime}\right)\zeta\left(x\right)$,
where $\xi$ and $\zeta$ are two quantities that depend on the spacetime
point.. The HTPF is a solution of the homogeneous wave equation satisfied by
the field $\hat{\Phi}$ and is an important object for various reasons. First,
it is physically relevant in its own right, since it yields the quantum
correlations between different points on the background spacetime. Second,
when subtracting from it an appropriate, purely geometric, renormalization
term (also called the counterterm) [3] and taking the limit
$x^{\prime}\rightarrow x$, the renormalized Wick product
$\left\langle\hat{\Phi}^{2}\left(x\right)\right\rangle_{\text{ren}}^{\Psi}$ is
obtained, which is a manifestation of the quantum vacuum fluctuations. Last
but not least, by applying a certain differential operator on the HTPF minus
the renormalization term [5, 6], and then taking the limit
$x^{\prime}\rightarrow x$, the RSET is obtained, which is a source in the
semiclassical Einstein equations.

In principle, it is possible to define various states for a quantum field on a
BH background spacetime. In the case of a spherically-symmetric (e.g.,
Schwarzschild) BH, the most commonly used states are: (i) the Boulware state
[7, 8], which is meant to model the surrounding of a star-like object, since
this state is empty at (past and future) null infinity and it diverges on the
(past and future) event horizon (EH); (ii) the Unruh state [9], which models
an evaporating BH via the emission of Hawking radiation; and (iii) the Hartle-
Hawking state [10], which models a BH in equilibrium with its own radiation.
When the BH is rotating (Kerr), however, the corresponding Boulware state [11,
12] is no longer empty at future null infinity, as it contains the so-called
Unruh-Starobinskii radiation (essentially, the quantum version of classical
superradiance [14, 13]). Also, for bosons, no state in thermal equilibrium
with the rotating BH can be constructed, i.e., no Hartle-Hawking-like state
exists for bosons in Kerr [15, 12] 222For fermions, on the other hand, a state
in thermal equilibrium can be constructed sufficiently close (specifically,
within the so-called speed-of-light surface) to the rotating BH [16]..
Finally, the Unruh state can be constructed in Kerr ([12] for a scalar field,
[16] for a fermion field and [17] for the electromagnetic field) similarly to
the spherically-symmetric case.

In this paper we shall focus on the Unruh state in Kerr spacetime, which is
the state of relevance for astrophysical BHs.

Our treatment of the Unruh state in Kerr is also significant for the following
reason. Often, in spherical symmetry, one first calculates quantities in the
Hartle-Hawking state since it allows for the Euclideanization technique,
whereby one merely needs to sum over a discrete set of field modes thus
facilitating the calculation; then, if one wishes to calculate the quantity in
another state – such as Unruh – one just calculates the difference between
that quantity in that other state and in the Hartle-Hawking state (such
difference needs no renormalization and so in principle it can be carried out
relatively easily). In Kerr, however, this method of Hartle-Hawking state
subtraction is not applicable, because no Hartle-Hawking-like state exists in
Kerr.

Most calculations for quantum fields on a BH spacetime have focused on
quantities outside the EH of the BH – after all, we are observers located far
away from any BH! Expressions for the HTPF with the two points outside the BH
are known in both the non-rotating [19, 18] and the rotating cases [20, 12]
and they have been used to calculate $\Phi^{2}$ and the RSET. While in the
non-rotating case the RSET outside the BH has been obtained in various
physical settings, in the rotating case the calculation is a lot more
technically involved and in fact it has been achieved only recently and in one
instance: [21].

In its turn, the investigation of effects of quantum fields inside the EH of a
BH may serve to address questions of fundamental conceptual importance. Most
notably, the question of whether the inner horizon (IH) of a rotating and/or
electrically-charged BH is stable under quantum perturbations. Beyond the IH,
the Cauchy initial value problem is not well-posed and so the Einstein field
equations of General Relativity cease to be deterministic. Quantum effects
have been seen to destroy the regularity of the IH of a non-rotating and
electrically-charged (Reissner-Nordström, RN) BH [22]. A similar behavior was
also found in RN-de Sitter BH [23] and, at least for quantum perturbations
approaching the IH from the inside, of a 2+1-dimensional rotating BTZ BH [24,
25]. In all these cases the HTPF was known for the two points inside the EH
and served to calculate the RSET. However, in the most important case of a
Kerr BH, an expression for the HTPF with the two points inside the BH was not
known until the current work and, consequently, no quantitative investigation
of the quantum effects on its IH has yet been carried out 333There have been,
however, qualitative investigations, see [26, 12]..

The main result in this paper is an expression for the HTPF
$G_{U}^{(1)}(x,x^{\prime})=\left\langle\left\\{\hat{\Phi}\left(x\right),\hat{\Phi}\left(x^{\prime}\right)\right\\}\right\rangle_{U}$
for a quantum massless scalar field in the Unruh state
$\left|0\right\rangle_{U}$ with the two points $x$ and $x^{\prime}$ located
inside the Kerr BH between the EH and the IH. One of the main values of this
expression is that it is given in terms of (Eddington) field modes which
decompose into factors that obey ordinary differential equations and so are
relatively easy to calculate, at least numerically. Thus, our expression for
the HTPF is of practical use for potential future calculations of the RSET
inside the EH of a Kerr BH in the Unruh state. (We perform a step in this
direction in Appendix B, where we derive expressions for the bare mode
contribution to the flux components in the BH interior.) Furthermore, once one
achieves renormalization in the Unruh state via the HTPF provided in this
paper, one can use that as the fiducial state with respect to which to
calculate differences and thus easily achieve renormalization in another
state. Prior to obtaining this new expression for the HTPF inside a Kerr BH,
we derive an expression for the HTPF outside a Kerr BH; although this latter
expression was already known, we (re-)derive it by employing a new method
which is the one that we subsequently apply inside the BH. Moreover, in order
to achieve these expressions for the HTPF, we obtain relationships between the
Unruh family of modes (which are defined in terms of the retarded Kruskal
coordinate and serve to define the Unruh state), and Eddington families of
modes (which are defined in terms of the Eddington coordinates and, as
mentioned, decompose by factors). These relationships between families of
field modes are useful in their own right in that they may be readily
applicable to the calculation of two-point functions other than the HTPF, such
as the Wightman function (which is relevant, for example, for the calculation
of the transition probability rate of an Unruh-DeWitt quantum particle
detector [27]). For the reader who is just interested in the new expression
for the HTPF inside the BH, that expression is given in Eq. (134) or,
equivalently, in Eq. (138).

The rest of this paper is organized as follows. Secs. II-IV lay the
foundations for the subject of the paper: the Unruh HTPF for a scalar field on
a Kerr BH interior. In Sec. II we review the Kerr metric and the associated
wave equation satisfied by a massless, uncharged scalar field. Sec. III
introduces the various families of field modes which are relevant to this
paper. The modes allow us to define the Unruh quantum state in Sec. IV and to
derive the (already known) expression for the HTPF outside a Kerr BH in Sec. V
(specifically, Eq. (91)). The paper culminates in Sec. VI, where we obtain the
(new) expression for the HTPF inside a Kerr BH (specifically, Eq. (134), or
(138)). The paper also has two appendices: Appendix A addresses the issue of
IR regularity (i.e. regularity at small frequencies) of our final expressions
for the HTPF; and Appendix B presents a derivation of the bare mode-sum
expressions for the flux components of the RSET, based on the HTPF expression
derived in this paper for the BH interior.

We use units where $c=G=1$ (while $\hbar$ is not taken to be equal to 1) and
metric signature $(-+++)$.

##  II The Kerr metric and the wave equation

###  II.1 The Kerr metric and coordinate systems

The Kerr metric is a vacuum solution to the classical Einstein field
equations, describing a BH of mass $M$ rotating with angular momentum $J$. It
is given by the line element in Boyer-Lindquist coordinates
$(t,r,\theta,\varphi)$:

$\text{d}s^{2}=-\left(1-\frac{2Mr}{\rho^{2}}\right)\text{d}t^{2}+\frac{\rho^{2}}{\Delta}\text{d}r^{2}+\rho^{2}\text{d}\theta^{2}+\left(r^{2}+a^{2}+\frac{2Mra^{2}}{\rho^{2}}\sin^{2}\theta\right)\sin^{2}\theta\text{d}\varphi^{2}-\frac{4Mra}{\rho^{2}}\sin^{2}\theta\text{d}\varphi\text{d}t\,,$
(1)

where $a\equiv J/M$ and

$\displaystyle\rho^{2}$ $\displaystyle\equiv r^{2}+a^{2}\cos^{2}\theta,$
$\displaystyle\Delta$ $\displaystyle\equiv r^{2}-2Mr+a^{2}\,.$

The horizon radii correspond to the roots of the equation $\Delta=0$, yielding
an EH at

$r=r_{+}\equiv M+\sqrt{M^{2}-a^{2}}$

and an IH at

$r=r_{-}\equiv M-\sqrt{M^{2}-a^{2}}.$

Note the resulting restriction on the BH parameters: $\left|a\right|/M\leq 1$.
Throughout this paper we shall only treat the subextremal case, corresponding
to $\left|a\right|/M<1$, and restrict our attention to the region bounded by
$r\geq r_{-}$. We refer to the region $r>r_{+}$ (outside the EH) as the
external Universe or BH _exterior_ , whereas the region bounded by the
horizons $r_{-}<r<r_{+}$ is to be referred to as the BH _interior_. Note that
we might occasionally use the term ”exterior” for $r\geq r_{+}$ (namely,
including $r=r_{+}$), and likewise the term ”interior” for $r_{-}\leq r\leq
r_{+}$, depending on the context. See Fig. 1 for (a portion of) the Penrose
diagram of the analytically-extended sub-extremal Kerr spacetime.

We shall now briefly discuss the behavior of the standard Boyer-Lindquist
coordinates $\left(t,r,\theta,\varphi\right)$ for a free-falling observer
approaching the EH. As in the case of spherical symmetry (e.g., in the
Schwarzschild and RN metrics), the $t$ coordinate diverges at $r=r_{+}$ for an
infalling observer, which motivates the definition of Kruskal coordinates
given below in (9) and (10). However, in Kerr, not only does $t$ diverge on
approaching $r=r_{+}$ but, unlike in spherical symmetry, also the azimuthal
coordinate $\varphi$ diverges there. In other words, a geodesic approaching
the EH undergoes (fictitious) infinite spiraling when presented in the
$\varphi$ coordinate (which is, however, a mere coordinate artifact). One may
compute the (constant) angular velocity with which the EH rotates (or, more
precisely, the geodesic’s limiting value of $\text{d}\varphi/\text{d}t$ at
$r\to r_{+}$) to be

$\Omega_{+}\equiv\frac{a}{2Mr_{+}}=\frac{a}{r_{+}^{2}+a^{2}}\,.$ (2)

This quantity can be used to construct a coordinate that remains regular on
approaching $r\to r_{+}$, defined by:

$\varphi_{+}\equiv\varphi-\Omega_{+}t\,.$ (3)

Similar considerations apply at $r\to r_{-}$, where we analogously define

$\Omega_{-}\equiv\frac{a}{2Mr_{-}},\,\,\,\,\,\,\varphi_{-}\equiv\varphi-\Omega_{-}t\,.$
(4)

For later use, we shall hereby define the tortoise coordinate $r_{*}$ in Kerr
via $\text{d}r/\text{d}r_{*}=\Delta/\left(r^{2}+a^{2}\right)$. We choose the
constant of integration such that 444We note that although this choice of
constant of integration is common in the literature, it differs from other
common choices such as that used in [28].:

$r_{*}=r+\frac{1}{2\kappa_{+}}\log\left(\frac{\left|r-r_{+}\right|}{r_{+}-r_{-}}\right)-\frac{1}{2\kappa_{-}}\log\left(\frac{\left|r-r_{-}\right|}{r_{+}-r_{-}}\right),$
(5)

where $\kappa_{\pm}$ are the two corresponding surface gravity parameters,
given by

$\kappa_{\pm}\equiv\frac{r_{+}-r_{-}}{2\left(r_{\pm}^{2}+a^{2}\right)}\,.$ (6)

Note that $r=r_{+}$ corresponds to $r_{*}\to-\infty$, while $r=r_{-}$ (like
$r\to\infty$ outside the BH) corresponds to $r_{*}\to\infty$.

From here we may define the Eddington coordinates 555While these coordinates
are usually known as “Eddington-Finkelstein coordinates”, we use “Eddington”
for abbreviation., given in the BH exterior by

$u_{\text{ext}}\equiv t-r_{*},\,\,\,v\equiv t+r_{*}\,,$ (7)

and in the BH interior by

$u_{\text{int}}\equiv r_{*}-t,\,\,\,v\equiv r_{*}+t\,.$ (8)

The coordinate $v$ is continuous across the EH (and parameterizes it), whereas
$u_{\text{ext}}$ and $u_{\text{int}}$ diverge there. The regularity of the
metric at the EH may be seen by transforming to a set of Kruskal coordinates,
which we shall denote by $U$ and $V$, given in the BH exterior by:

$U\left(u_{\text{ext}}\right)\equiv-\frac{1}{\kappa_{+}}\exp\left(-\kappa_{+}u_{\text{ext}}\right),\,\,\,V\left(v\right)\equiv\frac{1}{\kappa_{+}}\exp\left(\kappa_{+}v\right)\,,$
(9)

and in the BH interior by

$U\left(u_{\text{int}}\right)\equiv\frac{1}{\kappa_{+}}\exp\left(\kappa_{+}u_{\text{int}}\right),\,\,\,V\left(v\right)\equiv\frac{1}{\kappa_{+}}\exp\left(\kappa_{+}v\right)\,.$
(10)

Note that both Kruskal coordinates $U$ and $V$ are continuous at the EH: The
former vanishes there (from both sides), whereas $V$, just like $v$, regularly
parametrizes the EH. Furthermore, the metric in the $(U,V,\theta,\varphi_{+})$
coordinates is regular and smooth across the EH.

The locus $r=r_{+}$ marks a four-arms cross in the Penrose diagram in Fig. 1.
Out of these four arms, in this paper we only concern with the three included
in the red frame, being the EH (or _right horizon_ , denoted $H_{R}$), the
_past horizon_ $H_{\text{past}}$ (the white hole horizon) and the _left
horizon $H_{L}$_. The other arm, the one at the bottom-left, will not concern
us here as it is located outside the domain of dependence relevant to the
Unruh state (namely, the red frame in Fig. 1).

We note that the IH is also a Cauchy horizon in the sense that it is the
boundary of validity of the Cauchy initial value problem formulated on a
spacelike hypersurface extending from $i^{0}_{R}$ to $i^{0}_{L}$, where
$i^{0}_{R}$ ($i^{0}_{L}$) is spacelike infinity of the external universe at
the right (left) side of Fig. 1. However, in the more physically realistic
case of a BH formed by gravitational collapse, which lacks a past horizon as
well as the entire “left-side” external Universe, it is only the ingoing
section of the IH (see Fig. 1) which retains the causal nature of a Cauchy
horizon.

In Kruskal coordinates, the past horizon $H_{\text{past}}$ is found at $V=0$
and $U<0$, the right horizon $H_{R}$ corresponds to $U=0$ and $V>0$, and the
left horizon $H_{L}$ corresponds to $V=0$ and $U>0$.

In the BH exterior there are, in addition, two null asymptotic boundaries
located at infinity: _past null infinity_ (PNI) is found at $U=-\infty$ and
$V>0$, and _future null infinity_ (FNI) is at $V=\infty$ and $U<0$. See Fig. 1
for locating all the mentioned null surfaces.

Figure 1: A portion of the Penrose diagram of the analytically-extended sub-
extremal Kerr spacetime, with 3 systems of coordinates: the Kruskal $U$ and
$V$ (Eqs. (9) and (10)), the outer Eddington $u_{\text{ext}}$ and $v$ (Eq.
(7)), and the inner Eddington $u_{\text{int}}$ and $v$ (Eq. (8)). The
spacetime regions relevant for this paper are within the red frame, consisting
of the BH exterior and interior.

We now make a couple of related observations, relevant to constructing the
families of field modes further on in the paper.

Each of the above mentioned asymptotic null surfaces (the three at $r=r_{+}$
and the two at spacial infinity) can be regularly parameterized by three
coordinates – which may be chosen to be two angular coordinates ($\theta$ and
either $\varphi$ or $\varphi_{+}$) and one Eddington coordinate – as follows:
$H_{\text{past}}$ by $\left(u_{\text{ext}},\theta,\varphi_{+}\right)$, $H_{R}$
by $\left(v,\theta,\varphi_{+}\right)$, $H_{L}$ by
$\left(u_{\text{int}},\theta,\varphi_{+}\right)$, PNI by
$\left(v,\theta,\varphi\right)$ and FNI by
$\left(u_{\text{ext}},\theta,\varphi\right)$. We note that the Eddington
coordinate in each of these null surfaces may also be replaced by the
corresponding Kruskal coordinate.

We also specify here the affine parameters along null geodesics generating
each of these asymptotic null surfaces: $U$ along $H_{\text{past}}$ and
$H_{L}$ (both with fixed $\theta$ and $\varphi_{+}$) and $V$ along $H_{R}$
(again with fixed $\theta$ and $\varphi_{+}$). At PNI and FNI, asymptotic
flatness implies that the affine parameters along these surfaces are simply
the Eddington coordinates $v$ and $u_{\text{ext}}$ respectively (both with
fixed $\theta$ and $\varphi$).

###  II.2 Separation of the wave equation

An uncharged scalar field $\Phi\left(x\right)$ of mass $m$ and coupling $\xi$
to curvature obeys the Klein-Gordon (KG) equation,

$\left(\square-m^{2}-\xi R\right)\Phi=0\,,$ (11)

where $\square$ is the covariant d’Alembertian associated with the background
metric with Ricci scalar $R$. In the case of a massless field which is
minimally-coupled $(\xi=0)$ and/or vanishing Ricci scalar (in particular, a
vacuum spacetime), this equation becomes

$\square\Phi=0\,.$ (12)

Considering Eq. (12) on a Kerr background, one readily obtains the following
explicit form:

$\displaystyle\square\Phi$
$\displaystyle=\left(\frac{\left(r^{2}+a^{2}\right)^{2}}{\Delta}-a^{2}\sin^{2}\theta\right)\frac{\partial^{2}\Phi}{\partial
t^{2}}+\frac{4aMr}{\Delta}\frac{\partial^{2}\Phi}{\partial t\partial\varphi}+$
$\displaystyle+\left(\frac{a^{2}}{\Delta}-\frac{1}{\sin^{2}\theta}\right)\frac{\partial^{2}\Phi}{\partial\varphi^{2}}-\frac{\partial}{\partial
r}\left(\frac{\partial\Phi}{\partial
r}\right)-\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\Phi}{\partial\theta}\right)=0\,.$
(13)

We shall refer to this equation as the scalar Teukolsky equation, after the
general-spin field case in [29]. Utilizing the axial symmetry and time-
translation invariance of the metric and of the master equation, we may
decompose the field into modes

$\Phi_{\omega lm}\left(x\right)=\text{const}\cdot\frac{\psi_{\omega
lm}\left(r\right)}{\sqrt{r^{2}+a^{2}}}e^{-i\omega
t}Z_{lm}^{\omega}\left(\theta,\varphi\right)\,,$ (14)

indexed by the frequency $\omega\in\mathbb{R}$, the azimuthal number
$m\in\mathbb{\mathbb{Z}}$ and the multipolar number
$l\in\mathbb{\mathbb{N}}_{\geq|m|}$, where $x$ is a spacetime point and
$\psi_{\omega lm}\left(r\right)$ is the so-called _radial function_ ; the
$\left(r^{2}+a^{2}\right)^{-1/2}$ factor has been introduced to yield a
convenient one-dimensional scattering-like __ equation for $\psi_{\omega
lm}\left(r\right)$ (see Eq. (18) to follow). The angular functions
$Z_{lm}^{\omega}\left(\theta,\varphi\right)$ are the _spheroidal_ _harmonics_
, given by

$Z_{lm}^{\omega}\left(\theta,\varphi\right)=\left(2\pi\right)^{-1/2}S_{lm}^{\omega}\left(\theta\right)e^{im\varphi}\,,$
(15)

where $S_{lm}^{\omega}\left(\theta\right)$ is the _spheroidal wave function_
[30] solving the eigenvalue problem:

$\frac{1}{\sin\theta}\frac{\text{d}}{\text{d}\theta}\left(\sin\theta\frac{\text{d}S_{lm}^{\omega}\left(\theta\right)}{\text{d}\theta}\right)+\left(a^{2}\omega^{2}\cos^{2}\theta-\frac{m^{2}}{\sin^{2}\theta}+E_{lm}\left(a\omega\right)\right)S_{lm}^{\omega}\left(\theta\right)=0,$
(16)

with $E_{lm}\left(a\omega\right)$ the corresponding eigenvalue, obtained by
imposing regularity at $\theta=0,\pi$. Note that the angular equation is real,
and we shall only be concerned here with real angular functions
$S_{lm}^{\omega}$.

For a given $\omega$, the functions $Z_{lm}^{\omega}$ form a complete basis of
orthonormal functions on the two-sphere, fulfilling

$\displaystyle\int_{0}^{2\pi}\text{d}\varphi\int_{0}^{\pi}\text{d}\theta\sin\theta\,Z_{lm}^{\omega*}\left(\theta,\varphi\right)Z_{l^{\prime}m^{\prime}}^{\omega}\left(\theta,\varphi\right)$
$\displaystyle=\delta_{ll^{\prime}}\delta_{mm^{\prime}}\,.$ (17)

There is no known closed form for the spheroidal harmonics 666In fact,
spheroidal harmonics may be expressed in terms of confluent Heun functions but
only with coefficients which are to be determinable numerically., but in the
spherical case (corresponding to $a\omega=0$) they reduce to the well-known
spherical harmonics $Y_{lm}\left(\theta,\varphi\right)$, whence the spheroidal
wave functions reduce, up to a normalization, to Legendre functions – and the
eigenvalue $E_{lm}\left(a\omega\right)$ simplifies to $l\left(l+1\right)$.

The radial function $\psi_{\omega lm}\left(r\right)$ solves the _radial
equation_

$\frac{\text{d}^{2}\psi_{\omega lm}}{\text{d}r_{*}^{2}}+V_{\omega
lm}\left(r\right)\psi_{\omega lm}=0$ (18)

with the effective potential

$V_{\omega lm}\left(r\right)\equiv\frac{K_{\omega
m}^{2}\left(r\right)-\lambda_{lm}\left(a\omega\right)\Delta}{\left(r^{2}+a^{2}\right)^{2}}-G^{2}\left(r\right)-\frac{\text{d}G\left(r\right)}{\text{d}r_{*}}\,,$
(19)

where

$K_{\omega m}\left(r\right)\equiv\left(r^{2}+a^{2}\right)\omega-
am,\;\;\;\lambda_{lm}\left(a\omega\right)\equiv
E_{lm}\left(a\omega\right)-2am\omega+a^{2}\omega^{2},\;\;\;G\left(r\right)\equiv\frac{r\Delta}{\left(r^{2}+a^{2}\right)^{2}}\,.$
(20)

From Eq. (16) it is evident that flipping the signs of $\omega$ and/or $m$
leaves the angular equation invariant. Since the imposed boundary conditions
(regularity at the two poles) has no explicit reference to either $\omega$ or
$m$, it follows that both the eigenvalue $E_{lm}\left(a\omega\right)$ and the
angular function $S_{lm}^{\omega}$ are invariant (modulo a sign) under such
sign flips. For our purposes, we focus on a simultaneous sign flip of $m$ and
$\omega$:

$S_{l\left(-m\right)}^{-\omega}=(-1)^{m}S_{lm}^{\omega}\quad,\qquad
E_{l(-m)}\left(-a\omega\right)=E_{lm}\left(a\omega\right)\,$ (21)

where we have chosen the sign $(-1)^{m}$ for $S^{\omega}_{lm}$, so that, for
$a\omega=0$, the spheroidal harmonics agree with the standard sign for the
spherical harmonics:
$\left(Z_{l(-m)}^{\omega=0}\right)^{*}=\left(Y_{l}^{-m}\right)^{*}=(-1)^{m}Y_{l}^{m}=(-1)^{m}Z_{lm}^{\omega=0}$.

The situation with the radial equation is slightly more delicate. This
equation, too, is real, as can be seen in Eqs. (18)-(20). Therefore, if
$\psi_{\omega lm}\left(r\right)$ is a solution, its complex conjugate is a
solution too. However, we are physically motivated to choose complex boundary
conditions to the radial solutions (which would correspond to e.g. “ingoing”
or “upgoing” waves) – leading to _complex_ radial functions. The radial
equation (18), too, is invariant under a simultaneous change of signs of
$\omega$ and $m$. This implies that if $\psi_{\omega lm}\left(r\right)$ solves
Eq. (18), it will also be a solution of the radial equation with
$\omega\mapsto-\omega$ and $m\mapsto-m$, and so will be its complex conjugate.
As will become evident in the next section (see Eqs. (29)), from the way the
boundary conditions for the various modes are defined, flipping the signs of
both $\omega$ and $m$ will actually take us from the original mode function
$\psi_{\omega lm}\left(r\right)$ to its complex conjugate.

Examining the effective potential (Eq. (19)) in the asymptotic domains of the
exterior region, $r_{*}\to\infty$ (corresponding to $r\to\infty$) and
$r_{*}\to-\infty$ (corresponding to $r\rightarrow r_{+}$), we find:

$V_{\omega
lm}^{\text{outside}}\rightarrow\begin{cases}\omega^{2},&r\to\infty\,\,\,\,\,\,\,\,\left(r_{*}\to\infty\right)\\\
\omega_{+}^{2},&r\to
r_{+}\,\,\,\,\,\,\,\,\left(r_{*}\to-\infty\right)\end{cases}\,,$ (22)

where we define

$\omega_{+}\equiv\omega-m\Omega_{+}\,.$ (23)

Thus 777To be more precise, this is a consequence of the fact that the
potential is short-range: $V_{\omega lm}=\omega^{2}+O(1/r^{2})$ as
$r\to\infty$ and $V_{\omega
lm}=\omega_{+}^{2}+O\left(e^{2\kappa_{+}r_{*}}\right)$ as $r\to r_{+}$
(corresponding to $r_{*}\to-\infty$)., the asymptotic behavior of solutions to
the radial equation (Eq. (18)) outside the BH is generally of the form $e^{\pm
i\omega r_{*}}$ at $r_{*}\to\infty$ and $e^{\pm i\omega_{+}r_{*}}$ at
$r_{*}\to-\infty$, corresponding to free waves in both these domains.

Similarly, when considering the effective potential in the BH interior, we
obtain

$V_{\omega lm}^{\text{inside}}\rightarrow\begin{cases}\omega_{-}^{2},&r\to
r_{-}\,\,\,\,\,\,\,\,\left(r_{*}\to\infty\right)\\\ \omega_{+}^{2},&r\to
r_{+}\,\,\,\,\,\,\,\,\left(r_{*}\to-\infty\right)\end{cases}\,,$ (24)

where we define

$\omega_{-}\equiv\omega-m\Omega_{-}\,.$ (25)

This is a crucial point for the definition of our modes in Kerr (see Sec.
III), and it differs from the spherically symmetric case (similar to the $m=0$
case here), where the asymptotic behavior of the effective potential leaves
$\omega^{2}$ in all asymptotic domains of the BH exterior and interior.

##  III Families of modes

It is convenient to decompose the field into sets of modes, each providing a
complete set of solutions to Eq. (12) on some spacetime region, which are
orthonormal with respect to the standard KG scalar product, defined by

$\left\langle\psi,\phi\right\rangle\equiv
i\int_{\Sigma}\text{d}\sigma^{\mu}\left(\psi^{*}\phi_{;\mu}-\phi\,\psi_{;\mu}^{*}\right)\,,$
(26)

where $\Sigma$ is the spacelike hypersurface under consideration and
$\text{d}\sigma^{\mu}$ is a future directed normal to $\Sigma$. Note that the
prefactors of the various modes (e.g., the analog of the const. appearing in
Eq. (14)) are chosen such that orthonormality with respect to the KG inner
product is satisfied.

Three families of modes are of particular importance for our purposes: the
outer and inner Eddington modes, and the Unruh modes. Each of these three
families consists of two distinct sets, a “left-moving” and a “right-moving”
one, as specified below. We begin here with a brief general description of the
three families of modes, to be followed by a more detailed presentation. The
various families of modes are illustrated in Fig. 2.

Figure 2: Penrose diagrams for various field modes on Kerr spacetime: the
outer Eddington modes $f_{\omega lm}^{\text{up}}$ (33) and $f_{\omega
lm}^{\text{in}}$ (32); the inner Eddington modes $f_{\omega lm}^{L}$ (44) and
$f_{\omega lm}^{R}$ (43); the Unruh modes
$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}$ (55) and $g_{\omega
lm}^{\text{in}}$ (49). All modes start with unit amplitude on whichever
hypersurface (PNI, $H_{\text{past}}$ or $H_{\text{L}}$) the corresponding dot
lies; part of the wave is reflected and part is transmitted, with the
reflection $\rho$ and transmission $\tau$ coefficients indicated for
$f_{\omega lm}^{\text{in/up}}$. The modes
$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}$ are defined equally throughout
$H_{\text{past}}\cup H_{L}$ in terms of $U$ but they may also be constructed
as the sum of $g_{\hat{\omega}\hat{l}\hat{m}}^{L}$ (Eq. (56), with no support
on $H_{\text{past}}$) and $g_{\hat{\omega}\hat{l}\hat{m}}^{\text{past}}$ (Eq.
(57), with no support on $H_{L}$), respectively coloured in blue and dashed
orange on the diagram for $g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}$.

The _outer Eddington modes_ (see Subsec. III.1) are defined with respect to
the Eddington coordinates on the BH _exterior_ , and consist of two sets of
modes: the outgoing _up_ modes $f_{\omega lm}^{\text{up}}$ which emerge as
free waves from $H_{\text{past}}$, and the ingoing _in_ modes $f_{\omega
lm}^{\text{in}}$ which emerge as free waves from PNI.

The _inner Eddington modes_ (see Sec. III.2) are similarly defined with
respect to the Eddington coordinates on the BH _interior_ , and consist of two
sets of modes: the outgoing _left_ modes $f_{\omega lm}^{L}$ which emerge as
free waves from $H_{L}$, and the ingoing _right_ modes $f_{\omega lm}^{R}$
which emerge as free waves from $H_{R}$.

The _Unruh modes_ (see Sec. III.3) are defined on the entire combined interior
and exterior domain (i.e., the entire red frame in Fig. 1), and consist of two
sets of modes: the outgoing _up_ modes
$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}$ (the indices
$\hat{\omega}\hat{l}\hat{m}$ are to be introduced later on) which emerge from
$H_{\text{past}}\cup H_{L}$ as free waves with respect to the affine parameter
there, the Kruskal coordinate $U$; and the ingoing _in_ modes $g_{\omega
lm}^{\text{in}}$ which emerge from PNI as free waves with respect to the
affine parameter there, the Eddington coordinate $v$ (see the end of Subsec.
II.1).

The Unruh modes are of crucial physical importance for a quantum scalar field
evolving on a BH background. See, in particular, the definition of the Unruh
vacuum in Subsec. IV.3. Our motivation for considering the Eddington modes is
of a different kind: Remarkably, both outer and inner Eddington modes are
entirely decomposable in terms of angular, radial and temporal variables, and
have a general form analogous to Eq. (14). Owing to this decomposition, the
Eddington modes are easily handled numerically – which makes them convenient
as a mathematical tool for the decomposition of the Unruh modes – whereas the
Unruh modes do not generally admit such decomposition.

Throughout the rest of the paper, whenever we have both $\omega_{+}$ and
$\omega$ appearing in the same equation, they should be understood to be
related via the standard relation, $\omega=\omega_{+}+m\Omega_{+}$, or vice
versa, via $\omega_{+}=\omega-m\Omega_{+}$. In all of these cases, there will
always be a well-defined $m$ value, enabling this transformation between
$\omega$ and $\omega_{+}$.

###  III.1 Outer Eddington modes

For the sake of separation of the wave equation (13) in Kerr, we considered a
particular decomposition of the field into modes (Eq. (14)), which was given
in terms of the spheroidal harmonics
$Z_{lm}^{\omega}\left(\theta,\varphi\right)=\frac{1}{\sqrt{2\pi}}S_{lm}^{\omega}\left(\theta\right)e^{im\varphi}$,
the temporal dependence $e^{-i\omega t}$ and the radial function $\psi_{\omega
lm}\left(r\right)$. This decomposition provides the basis for the definition
of the _Eddington_ modes, which we shall generally denote by $f_{\omega lm}$.
In this subsection we introduce the _outer_ Eddington modes, $f_{\omega
lm}^{\text{in}}$ and $f_{\omega lm}^{\text{up}}$ defined exclusively on the BH
exterior.

Based on the asymptotic behavior of the effective potential in the BH exterior
(as given in Eq. (22)), we define two spanning sets of solutions to the radial
equation (18), $\psi_{\omega lm}^{\text{in}}\left(r\right)$ and $\psi_{\omega
lm}^{\text{up}}\left(r\right)$, uniquely determined by their boundary
conditions:

$\psi_{\omega lm}^{\text{in}}\left(r\right)\simeq\begin{cases}\tau_{\omega
lm}^{\text{in}}e^{-i\omega_{+}r_{*}}\,,&r_{*}\to-\infty\\\ e^{-i\omega
r_{*}}+\rho_{\omega lm}^{\text{in}}e^{i\omega
r_{*}}\,,&r_{*}\to\infty\end{cases}\,,$ (27) $\psi_{\omega
lm}^{\text{up}}\left(r\right)\simeq\begin{cases}e^{i\omega_{+}r_{*}}+\rho_{\omega
lm}^{\text{up}}e^{-i\omega_{+}r_{*}}\,,&r_{*}\to-\infty\\\ \tau_{\omega
lm}^{\text{up}}e^{i\omega r_{*}}\,,&r_{*}\to\infty\end{cases}\,,$ (28)

where $\tau_{\omega lm}^{\Lambda}$ and $\rho_{\omega lm}^{\Lambda}$, with
$\Lambda$ denoting either “in” or “up”, are the transmission and reflection
coefficients, respectively. Determination of these coefficients thus requires
numerically solving the one-dimensional scattering of the $\psi_{\omega
lm}^{\Lambda}$ mode off the effective potential (19). We use the symbol
“$\simeq$” to denote asymptotic equivalence.

When the signs of $\omega$ and $m$ are flipped simultaneously, so does the
sign of $\omega_{+}$. This means that the emerging free waves that initiate
the _in_ and _up_ modes ($e^{-i\omega r_{*}}$ or $e^{i\omega_{+}r_{*}}$
respectively) simply undergo complex conjugation under this transformation. We
also recall from the previous section that the radial equation is real, and is
invariant under such a simultaneous sign flip of $\omega$ and $m$. Therefore,
the following symmetry relations are satisfied:

$\displaystyle\psi_{\left(-\omega\right)l\left(-m\right)}^{\Lambda}$
$\displaystyle=\psi_{\omega lm}^{\Lambda*}\,$ (29)

for the radial solutions and

$\displaystyle\rho_{\left(-\omega\right)l\left(-m\right)}^{\Lambda}$
$\displaystyle=\rho_{\omega
lm}^{\Lambda*}\,\,\,\,,\,\,\,\,\,\tau_{\left(-\omega\right)l\left(-m\right)}^{\Lambda}=\tau_{\omega
lm}^{\Lambda*}\,$ (30)

for the corresponding reflection and transmission coefficients, where
$\Lambda$ stands here for either “in” or “up”.

Having introduced the radial functions $\psi_{\omega lm}^{\text{in}}$ and
$\psi_{\omega lm}^{\text{up}}$ and specified their boundary conditions in Eqs.
(27) and (28), the complete orthonormal family of outer Eddington modes are
defined in accordance with Eq. (14):

$\displaystyle f_{\omega lm}^{\text{in}}\left(x\right)$
$\displaystyle\equiv\frac{1}{\sqrt{4\pi\left|\omega\right|\left(r^{2}+a^{2}\right)}}Z_{lm}^{\omega}\left(\theta,\varphi\right)e^{-i\omega
t}\psi_{\omega lm}^{\text{in}}\left(r\right),$ (31) $\displaystyle f_{\omega
lm}^{\text{up}}\left(x\right)$
$\displaystyle\equiv\frac{1}{\sqrt{4\pi\left|\omega_{+}\right|\left(r^{2}+a^{2}\right)}}Z_{lm}^{\omega}\left(\theta,\varphi\right)e^{-i\omega
t}\psi_{\omega lm}^{\text{up}}\left(r\right),$

satisfying the boundary conditions (as emerge from Eqs. (27) and (28)) 888In
the RHS of Eq. (33), being non-zero only at $H_{\text{past}}$, we could have
replaced $r$ in the prefactor by $r_{+}$. However, in Eq. (32) this is not the
case, as well as in similar equations that follow, being non-zero both at
$r=r_{+}$ and at infinity. We thus choose, for the sake of uniformity, to keep
$r$ (rather than $r_{+}$) in the prefactor in the RHS of Eq. (33), as well as
in all similar instances that follow.:

$f_{\omega
lm}^{\text{in}}\left(x\right)\simeq\frac{1}{\sqrt{8\pi^{2}\left|\omega\right|\left(r^{2}+a^{2}\right)}}S_{lm}^{\omega}\left(\theta\right)\begin{cases}e^{-i\omega
v}e^{im\varphi},&\text{PNI}\\\ 0,&H_{\text{past}}\end{cases}\,,$ (32)
$f_{\omega
lm}^{\text{up}}\left(x\right)\simeq\frac{1}{\sqrt{8\pi^{2}\left|\omega_{+}\right|\left(r^{2}+a^{2}\right)}}S_{lm}^{\omega}\left(\theta\right)\begin{cases}0,&\text{PNI}\\\
e^{-i\omega_{+}u_{\text{ext}}}e^{im\varphi_{+}},&H_{\text{past}}\end{cases}\,.$
(33)

Eq. (33) makes use of the relation $e^{-i\omega
t}e^{im\varphi}=e^{-i\omega_{+}t}e^{im\varphi_{+}}$, also useful later in the
paper. Note that, in accordance with the discussion towards the end of Subsec.
II.1, the past asymptotic forms given above (as well as the future asymptotic
forms given below in Eqs. (39)-(40)) are always expressed in terms of the
three regular coordinates on each of the asymptotic null surfaces.

For future use, we also find it beneficial to write Eqs. (32) and (33) in a
slightly different manner by absorbing the $\varphi$\- and
$\varphi_{+}$-dependent factors into the angular functions:

$f_{\omega
lm}^{\text{in}}\left(x\right)\simeq\frac{1}{\sqrt{4\pi\left|\omega\right|\left(r^{2}+a^{2}\right)}}Z_{lm}^{\omega}\left(\theta,\varphi\right)\begin{cases}e^{-i\omega
v},&\text{PNI}\\\ 0,&H_{\text{past}}\end{cases}\,,$ (34) $f_{\omega
lm}^{\text{up}}\left(x\right)\simeq\frac{1}{\sqrt{4\pi\left|\omega_{+}\right|\left(r^{2}+a^{2}\right)}}Z_{lm}^{\omega}\left(\theta,\varphi_{+}\right)\begin{cases}0,&\text{PNI}\\\
e^{-i\omega_{+}u_{\text{ext}}},&H_{\text{past}}\end{cases}\,,$ (35)

where

$Z_{lm}^{\omega}\left(\theta,\varphi_{+}\right)\equiv\frac{1}{\sqrt{2\pi}}S_{lm}^{\omega}\left(\theta\right)e^{im\varphi_{+}}\,.$
(36)

Clearly, for any fixed $\omega\in\mathbb{R}$,
$Z_{lm}^{\omega}\left(\theta,\varphi_{+}\right)$ is a complete family of
functions on the 2-sphere, orthonormal in the sense that

$\displaystyle\int_{0}^{2\pi}\text{d}\varphi_{+}\int_{0}^{\pi}\text{d}\theta\sin\theta\,Z_{lm}^{\omega*}\left(\theta,\varphi_{+}\right)Z_{l^{\prime}m^{\prime}}^{\omega}\left(\theta,\varphi_{+}\right)$
$\displaystyle=\delta_{ll^{\prime}}\delta_{mm^{\prime}}\,.$ (37)

By inspecting Eq. (31) along with Eqs. (29) and (21) we conclude that the
exterior Eddington modes are invariant (modulo a sign introduced by the
angular functions) under simultaneously flipping the signs of $\omega$ and $m$
along with complex conjugation. That is,

$\displaystyle f_{\left(-\omega\right)l\left(-m\right)}^{\Lambda}$
$\displaystyle=(-1)^{m}f_{\omega lm}^{\Lambda*},$ (38)

where $\Lambda$ stands for either “in” or “up”.

For future use, we also provide the asymptotic behavior of the _in_ and _up_
Eddington modes on the future null hypersurfaces:

$\displaystyle f_{\omega
lm}^{\text{in}}\left(x\right)\simeq\frac{1}{\sqrt{8\pi^{2}\left|\omega\right|\left(r^{2}+a^{2}\right)}}S_{lm}^{\omega}\left(\theta\right)\begin{cases}\rho_{\omega
lm}^{\text{in}}e^{-i\omega u_{\text{ext}}}e^{im\varphi},&\text{FNI}\\\
\tau_{\omega
lm}^{\text{in}}e^{-i\omega_{+}v}e^{im\varphi_{+}},&H_{R}\end{cases}\,,$ (39)
$\displaystyle f_{\omega
lm}^{\text{up}}\left(x\right)\simeq\frac{1}{\sqrt{8\pi^{2}\left|\omega_{+}\right|\left(r^{2}+a^{2}\right)}}S_{lm}^{\omega}\left(\theta\right)\begin{cases}\tau_{\omega
lm}^{\text{up}}e^{-i\omega u_{\text{ext}}}e^{im\varphi},&\text{FNI}\\\
\rho_{\omega
lm}^{\text{up}}e^{-i\omega_{+}v}e^{im\varphi_{+}},&H_{R}\end{cases}$
$\displaystyle\,.$ (40)

The _in_ and _up_ outer Eddington modes are illustrated on the top row of Fig.
2. The _in_ mode may be interpreted as a (properly normalized) monochromatic
spherical wave propagating _inwards_ from infinity (PNI) and being partially
reflected back to infinity (FNI) and partially transmitted across the horizon
($H_{R}$), with the relative coefficients of transmission and reflection being
respectively $\tau_{\omega lm}^{\text{in}}$ and $\rho_{\omega
lm}^{\text{in}}$. Similarly, the _up_ mode may be interpreted as a
monochromatic spherical wave propagating _upwards_ from the past horizon
($H_{\text{past}}$) and being partially reflected back to the future horizon
($H_{R}$) and partially transmitted to infinity (FNI), with the relative
coefficients of transmission and reflection being respectively $\tau_{\omega
lm}^{\text{up}}$ and $\rho_{\omega lm}^{\text{up}}$.

###  III.2 Inner Eddington modes

In a similar manner to the definition of the _in_ and _up_ modes outside the
BH, we may additionally define two sets of Eddington modes confined to the BH
interior. Note that, in this spacetime region, the $r_{*}$ coordinate serves
as a temporal coordinate whereas $t$ has a spatial role. This means that a
single initial condition is required for the radial equation (18), which we
simply take as a free wave at $r_{*}\to-\infty$ (corresponding to $r\to
r_{+}$):

$\psi_{\omega lm}^{\text{int}}\simeq e^{-i\omega_{+}r_{*}},\,\,\,\,\,r\to
r_{+}\,.$ (41)

With this radial function we now define the _right_ ($R$) and _left_ ($L$)
sets of orthonormal modes: 999Note that in the treatment of the analogous RN
case in Ref. [31], the _right_ and _left_ modes are defined with an
essentially different temporal dependence (see Eq. (2.16) therein): the
_right_ mode is decomposed with respect to $e^{-i\omega t}$ while the _left_
mode is decomposed with respect to $e^{i\omega t}$, and both share the same
radial function. This was possible since, in RN, the wave equation is
invariant under $\omega\mapsto-\omega$. In Kerr, however, the latter symmetry
does not apply, hence we stick with the canonical decomposition of Eq. (14).
This difference also leads to some differences between several equations below
and their counterparts in Ref. [31].

$\displaystyle f_{\omega lm}^{R}\left(x\right)$
$\displaystyle\equiv\frac{1}{\sqrt{4\pi\left|\omega_{+}\right|\left(r^{2}+a^{2}\right)}}Z_{lm}^{\omega}\left(\theta,\varphi\right)e^{-i\omega
t}\psi_{\omega lm}^{\text{int}}\left(r\right),$ (42) $\displaystyle f_{\omega
lm}^{L}\left(x\right)$
$\displaystyle\equiv\frac{1}{\sqrt{4\pi\left|\omega_{+}\right|\left(r^{2}+a^{2}\right)}}Z_{lm}^{\omega}\left(\theta,\varphi\right)e^{-i\omega
t}\psi_{\omega lm}^{\text{int}*}\left(r\right).$

These modes admit the following asymptotic forms at the right and left
horizons:

$f_{\omega
lm}^{R}\left(x\right)\simeq\frac{1}{\sqrt{8\pi^{2}\left|\omega_{+}\right|\left(r^{2}+a^{2}\right)}}S_{lm}^{\omega}\left(\theta\right)e^{im\varphi_{+}}\begin{cases}0,&H_{L}\\\
e^{-i\omega_{+}v},&H_{R}\end{cases}\,,$ (43) $f_{\omega
lm}^{L}\left(x\right)\simeq\frac{1}{\sqrt{8\pi^{2}\left|\omega_{+}\right|\left(r^{2}+a^{2}\right)}}S_{lm}^{\omega}\left(\theta\right)e^{im\varphi_{+}}\begin{cases}e^{i\omega_{+}u_{\text{int}}},&H_{L}\\\
0,&H_{R}\end{cases}\,.$ (44)

As for the exterior Eddington modes, we shall find it beneficial when
constructing the HTPF to present a slightly different form for Eqs. (43)-(44)
by absorbing the $\varphi$-and $\varphi_{+}$-dependent factors into angular
functions:

$f_{\omega
lm}^{R}\left(x\right)\simeq\frac{1}{\sqrt{4\pi\left|\omega_{+}\right|\left(r^{2}+a^{2}\right)}}Z_{lm}^{\omega}\left(\theta,\varphi_{+}\right)\begin{cases}0,&H_{L}\\\
e^{-i\omega_{+}v},&H_{R}\end{cases}\,,$ (45) $f_{\omega
lm}^{L}\left(x\right)\simeq\frac{1}{\sqrt{4\pi\left|\omega_{+}\right|\left(r^{2}+a^{2}\right)}}Z_{lm}^{\omega}\left(\theta,\varphi_{+}\right)\begin{cases}e^{i\omega_{+}u_{\text{int}}},&H_{L}\\\
0,&H_{R}\end{cases}\,.$ (46)

Analogously to the symmetry in Eq. (38) satisfied by the exterior Eddington
modes, the interior Eddington modes satisfy

$f_{\left(-\omega\right)l\left(-m\right)}^{\Lambda}=(-1)^{m}f_{\omega
lm}^{\Lambda*}\,,$ (47)

where $\Lambda$ stands here for either “ _R_ ” or “ _L_ ”. The _right_ and
_left_ inner Eddington modes are illustrated on the middle row of Fig. 2.

We have readily settled the definition of the inner Eddington family of modes.
However, it is interesting to also inspect the form of the radial function
$\psi_{\omega lm}^{\text{int}}\left(r\right)$ at $r_{*}\to\infty$ (i.e., on
approaching the IH). In correspondence with the asymptotic behavior of the
effective potential (24), the solution of Eq. (18) admits the free asymptotic
form:

$\psi_{\omega lm}^{\text{int}}\simeq A_{\omega
lm}e^{i\omega_{-}r_{*}}+B_{\omega lm}e^{-i\omega_{-}r_{*}},\,\,\,\,\,\,r\to
r_{-},$ (48)

where $A_{\omega lm}$ and $B_{\omega lm}$ are constant complex coefficients.

###  III.3 Unruh modes

The _Unruh modes_ are the basic modes for the field expansion involved in the
definition of the Unruh quantum state (see Subsec. IV.3). There are two
distinct sets of Unruh modes and both inhabit the entire union of the BH
interior and exterior (namely, the red frame in Fig. 1). We shall occasionally
refer to this domain as the “united domain”. This domain has two null
boundaries in its past: The one on the right is PNI. The other boundary, on
the left, is located at $r=r_{+}$; it is the union of $H_{\text{past}}$ and
$H_{L}$. Recall that, as discussed at the end of Subsec. II.1, the
corresponding affine parameters are Eddington $v$ along the past-right
boundary (PNI), and Kruskal $U$ along the past-left boundary
($H_{\text{past}}\cup H_{L}$). We also note that both these affine parameters
span the entire range $\left(-\infty,\infty\right)$ along their respective
past null boundaries (namely $v$ along PNI and $U$ along $H_{\text{past}}\cup
H_{L}$).

The two sets of Unruh modes naturally emerge from these two past null
boundaries, with positive frequencies in each set defined with respect to the
affine parameter along the corresponding boundary. This is unlike the
Eddington modes (introduced above), which are always defined asymptotically
with respect to the corresponding Eddington coordinates $v$ and $u$
101010Since the Unruh modes are introduced here directly for the construction
of a quantum state, we only need to define them with positive frequencies. The
Eddington modes, however, were introduced to be utilized as a mathematical
tool for decomposition. Thus, the latter were defined in Subsecs. III.1-III.2
for negative frequencies as well (hence the absolute value in their
normalization constant).. We now introduce the two sets of Unruh modes, _in_
and _up_ , as outlined above.

####  III.3.1 The _in_ Unruh modes

The _in_ Unruh mode $g_{\omega lm}^{\text{in}}$ originates at PNI as a free
wave with respect to the affine parameter there, the Eddington $v$ coordinate
(i.e., $\propto e^{-i\omega v}$) and vanishes on both $H_{\text{past}}$ and
$H_{L}$. That is, it is endowed with the following past boundary conditions:

$g_{\omega
lm}^{\text{in}}\left(x\right)\simeq\frac{1}{\sqrt{4\pi\omega\left(r^{2}+a^{2}\right)}}Z_{lm}^{\omega}\left(\theta,\varphi\right)\begin{cases}e^{-i\omega
v},&\text{PNI}\\\ 0,&H_{\text{past}}\cup H_{L}\end{cases}\,.$ (49)

Recall that here $\omega$ attains only positive values (see footnote 10). The
_in_ Unruh modes are illustrated at the bottom right diagram of Fig. 2.

Evidently, these boundary conditions are regular. Since the d’Alembertian
operator in Eq. (12) is regular as well, the regularity of the $g_{\omega
lm}^{\text{in}}$ modes is guaranteed throughout the (interior of the) united
domain.

We now restrict our attention to the BH exterior. Notably, the _in_ Unruh mode
$g_{\omega lm}^{\text{in}}$, when constrained to the BH exterior, has the same
boundary conditions (on the past asymptotic null surfaces PNI and
$H_{\text{past}}$) as the _in_ Eddington mode $f_{\omega lm}^{\text{in}}$:
compare Eqs. (34) and (49). That is,

$\displaystyle\left.g_{\omega lm}^{\text{in}}\right|_{\text{PNI}}$
$\displaystyle=\left.f_{\omega lm}^{\text{in}}\right|_{\text{PNI}}$
$\displaystyle\left.g_{\omega lm}^{\text{in}}\right|_{H_{\text{past}}}$
$\displaystyle=0=\left.f_{\omega lm}^{\text{in}}\right|_{H_{\text{past}}}\,.$

Since $g_{\omega lm}^{\text{in}}$ and $f_{\omega lm}^{\text{in}}$ satisfy the
same wave equation (13), it follows that these two quantities are identical
not only on the initial null hypersurfaces $H_{\text{past}}$ and PNI but at
every spacetime point in the BH exterior:

$g_{\omega lm}^{\text{in}}\left(x\right)=f_{\omega
lm}^{\text{in}}\left(x\right)\,,\,\,\,\,\,\,\,\,\,\,r\geq r_{+}\,.$ (50)

In order to find the behavior of $g_{\omega lm}^{\text{in}}$ in the BH
interior, we carry it using Eq. (50) to $H_{R}$, where it fulfills

$\left.g_{\omega lm}^{\text{in}}\right|_{H_{R}}=\left.f_{\omega
lm}^{\text{in}}\right|_{H_{R}}=\sqrt{\frac{\left|\omega_{+}\right|}{\omega}}\tau_{\omega
lm}^{\text{in}}\left.f_{\omega lm}^{R}\right|_{H_{R}}$ (51)

(for the last equality, compare Eq. (39) with Eq. (43)). Again, since
$g_{\omega lm}^{\text{in}}$ and
$\sqrt{\frac{\left|\omega_{+}\right|}{\omega}}\tau_{\omega
lm}^{\text{in}}f_{\omega lm}^{R}$ coincide on $H_{R}$ and on $H_{L}$ as well
(since they both vanish on the latter hypersurface, see Eq. (49) and Eq.
(43)), these solutions are identical everywhere in the BH interior:

$g_{\omega
lm}^{\text{in}}\left(x\right)=\sqrt{\frac{\left|\omega_{+}\right|}{\omega}}\tau_{\omega
lm}^{\text{in}}f_{\omega lm}^{R}\left(x\right)\,,\,\,\,\,\,\,\,\,\,r_{-}\leq
r\leq r_{+}\,.$ (52)

Equations (50) and (52) demonstrate a useful property of the _in_ Unruh modes,
namely that we may match each _in_ Unruh mode, at any given neighborhood, with
a particular Eddington mode: with an _in_ mode at $r\geq r_{+}$ and with a
_right_ mode (up to a specified multiplicative constant) at $r_{-}\leq r\leq
r_{+}$. In particular, this means that, throughout the united domain,
$g_{\omega lm}^{\text{in}}$ decomposes into radial, angular and temporal
terms, as may also be anticipated from its initial conditions given in Eq.
(49) (because $e^{-i\omega v}$ decomposes naturally into a temporal factor
$e^{-i\omega t}$ times a function of $r$, and this separable form of the
$t$-dependence is preserved as $g_{\omega lm}^{\text{in}}$ evolves according
to the $t$-independent wave equation). This situation changes when considering
the _up_ Unruh modes, as we shall do next.

####  III.3.2 The _up_ Unruh modes

The _up_ Unruh modes are solutions to Eq. (13) emerging from
$H_{\text{past}}\cup H_{L}$ with positive frequency, which we denote by
$\hat{\omega}$ (to distinguish it from the Killing frequency, $\omega$). That
is, the _up_ Unruh modes originate from (the ingoing arms of) $r=r_{+}$ as
$\propto e^{-i\hat{\omega}U}$, with the Kruskal $U$ as defined in Eqs. (9) and
(10).

The desired orthonormal set of _up_ modes is conveniently defined by
specifying the initial value of each of the modes at the initial null
hypersurface $H_{\text{past}}\cup H_{L}$ – as a function of the three regular
coordinates $U,\theta,\varphi_{+}$ which span it (see the end of Subsec.
II.1). This initial-value setup for the _up_ modes is complemented by
requiring these modes to vanish at PNI, see Eq. 54 below. In order for the
_up_ modes to provide (when combined with the _in_ modes defined in the
previous subsection) a complete KG-orthonormal set of solutions to the wave
equation, the aforementioned 3-parameter set of _up_ -modes initial functions
has to be in itself a complete KG-orthonormal set of functions of $U$,
$\theta$ and $\varphi_{+}$ on $H_{\text{past}}\cup H_{L}$. We already chose
the modes’ initial $U$-dependence at $H_{\text{past}}\cup H_{L}$ to be
$\propto e^{-i\hat{\omega}U}$, so all that is left is to specify a complete
orthonormal set of functions of the remaining coordinates $\theta$ and
$\varphi_{+}$ on the 2-sphere. This set can be chosen quite arbitrarily (and
in principle it could also depend on $\hat{\omega}$). It should depend on two
discrete parameters (reflecting the dimensionality of the 2-sphere), which we
here schematically denote by $\hat{l},\hat{m}$; hence we may generally denote
such a set of “initial” angular functions as
$\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}(\theta,\varphi_{+})$ 111111As
mentioned, the Unruh modes introduced here are to be utilized in Subsec. IV.3
for construction of the Unruh quantum state. Generally speaking, what
determines a quantum state is the _frequency_ (and thereby the implied choice
of _positive-frequency modes_), which was here chosen to be the parameter
$\hat{\omega}$ appearing in $e^{-i\hat{\omega}U}$. Then, the remaining choice
of angular functions
$\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}(\theta,\varphi_{+})$ for the _up_
Unruh modes’ initial conditions does not affect the resultant quantum state.
In particular, the final mode-sum structure of the Unruh-state HTPF, as
appears in Eq. (66) or (134), does not depend at all on the choice of
$\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}$ – we show this explicitly in
Subsecs. V.2 and VI.2. . We choose it to be orthonormal in the usual sense,

$\int_{0}^{2\pi}\text{d}\varphi_{+}\int_{0}^{\pi}\text{d}\theta\sin\theta\,\,\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}*}\left(\theta,\varphi_{+}\right)\hat{Z}_{\hat{l}^{\prime}\hat{m}^{\prime}}^{\hat{\omega}}\left(\theta,\varphi_{+}\right)=\delta_{\hat{l}\hat{l}^{\prime}}\delta_{\hat{m}\hat{m}^{\prime}}\,.$
(53)

The set of _up_ modes is then generally defined via its initial conditions at
$H_{\text{past}}\cup H_{L}$ and PNI by

$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}\left(x\right)\simeq\frac{1}{\sqrt{4\pi\hat{\omega}\left(r^{2}+a^{2}\right)}}\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}\left(\theta,\varphi_{+}\right)\begin{cases}0,&\text{PNI}\\\
e^{-i\hat{\omega}U},&H_{\text{past}}\cup H_{L}\end{cases}\,.$ (54)

Recall that $\hat{\omega}$ attains only positive values (see footnote 10). The
_up_ Unruh modes are illustrated at the bottom left diagram of Fig. 2.

We shall choose our arbitrary angular functions
$\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}(\theta,\varphi_{+})$ to be the
simplest complete orthonormal set of angular functions on the two-sphere –
namely the conventional _spherical harmonics_ ,

$Y_{\hat{l}\hat{m}}\left(\theta,\varphi_{+}\right)=\sqrt{\frac{(2\hat{l}+1)}{4\pi}\frac{(\hat{l}-\hat{m})!}{(\hat{l}+\hat{m})!}}P_{\hat{l}}^{\hat{m}}\left(\cos\theta\right)e^{i\hat{m}\varphi_{+}}\,,$

where $P_{\hat{l}}^{\hat{m}}$ are the associated Legendre polynomials.

Thus, we define $g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}$ as a solution to
Eq. (13) with the initial conditions

$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}\left(x\right)\simeq\frac{1}{\sqrt{4\pi\hat{\omega}\left(r^{2}+a^{2}\right)}}Y_{\hat{l}\hat{m}}\left(\theta,\varphi_{+}\right)\begin{cases}0,&\text{PNI}\\\
e^{-i\hat{\omega}U},&H_{\text{past}}\cup H_{L}\end{cases}\,.$ (55)

Conveniently, any given _up_ mode may be written as a sum of two other
solutions to Eq. (13), denoted $g_{\hat{\omega}\hat{l}\hat{m}}^{\text{past}}$
and $g_{\hat{\omega}\hat{l}\hat{m}}^{L}$, which are endowed with the following
initial conditions:

$g_{\hat{\omega}\hat{l}\hat{m}}^{L}\left(x\right)\simeq\frac{1}{\sqrt{4\pi\hat{\omega}\left(r^{2}+a^{2}\right)}}Y_{\hat{l}\hat{m}}\left(\theta,\varphi_{+}\right)\begin{cases}0,&\text{PNI}\\\
0,&H_{\text{past}}\\\ e^{-i\hat{\omega}U},&H_{L}\end{cases}$ (56)
$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{past}}\left(x\right)\simeq\frac{1}{\sqrt{4\pi\hat{\omega}\left(r^{2}+a^{2}\right)}}Y_{\hat{l}\hat{m}}\left(\theta,\varphi_{+}\right)\begin{cases}0,&\text{PNI}\\\
e^{-i\hat{\omega}U},&H_{\text{past}}\\\ 0,&H_{L}\end{cases}\,.$ (57)

That is, while the initial support of
$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}\left(x\right)$ is on the entire
null surface $H_{\text{past}}\cup H_{L}$, the function
$g_{\hat{\omega}\hat{l}\hat{m}}^{L}\left(x\right)$ has its initial support on
$H_{L}$ alone whereas
$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{past}}\left(x\right)$ is initially
supported on $H_{\text{past}}$ only.

One might be slightly confused about our choice of spherical harmonics as the
angular functions here, because it is customary to use the _spheroidal_
harmonics for a Kerr BH. But the only reason for this common use of spheroidal
harmonics in Kerr is to achieve angular separability of the wave equation, as
in Eq. (14). Recall, however, that since the spheroidal harmonics explicitly
depend on $\omega$, this angular separability can only be achieved when the
temporal dependence is precisely of the form $e^{-i\omega t}$ (multiplying
some function of $r$, $\theta$ and $\varphi$). Quite unluckily, an _up_ Unruh
mode admits a more intricate temporal dependence, which does not fit any
single Killing frequency $\omega$. Instead, each such mode is a superposition
of Eddington modes with potentially all possible $\omega$ values; this may be
seen already at the initial hypersurface $H_{\text{past}}\cup H_{L}$, where an
_up_ Unruh mode is $\propto e^{-i\hat{\omega}U}$, recalling that $U$ cannot be
expressed as a sum of $t$-dependent and $r$-dependent pieces, unlike the
Eddington coordinates. Since angular separability cannot be achieved anyway in
this case, there is no advantage in using the spheroidal harmonics for the
_up_ Unruh modes – and we choose the much simpler (and frequency-independent)
spherical harmonics instead.

We point out, however, that despite our choice of spherical harmonics for the
definition of the _up_ Unruh modes, our final mode-sum expressions (in
Eddington modes) for the HTPF are actually given in terms of _spheroidal_
harmonics (as one would naturally expect for Kerr) – as may explicitly be seen
in Eqs. (66) and (134) (along with Eqs. (31) and (42)). In fact, these final
mode-sum expressions for the Unruh-state HTPF are entirely independent of the
choice of angular functions
$\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}\left(\theta,\varphi_{+}\right)$ at
this stage of constructing the _up_ Unruh modes, see footnote 11 above.

It is clear from the discussion above that, unlike what we did for the _in_
Unruh modes (see Eqs. (50) and (52)), it is not possible to match a specific
_up_ Unruh mode with a single Eddington mode (neither in the exterior nor in
the interior of the BH). This reflects our inability to reduce the PDE (13) to
an ODE (such as (18)) for the _up_ Unruh modes. Therefore, in order to allow a
convenient numerical implementation involving the solution of ODEs rather than
PDEs, we shall later Fourier-decompose the
$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}\left(x\right)$ modes in terms of
the (separable) Eddington modes.

###  III.4 Wronskian relations

The absence of a first derivative in the radial equation (18) leads to
$r_{*}$-independence of the Wronskian of any pair of solutions. This Wronskian
conservation yields well-known relations involving the exterior reflection and
transmission coefficients $\rho_{\omega lm}$ and $\tau_{\omega lm}$ (defined
via Eqs. (27) and (28)), as well as relations involving the interior near-IH
coefficients $A_{\omega lm}$ and $B_{\omega lm}$ (defined via Eq. (48)).

##### Relations involving $\rho_{\omega lm}$ and $\tau_{\omega lm}.$

Using the Wronskian conservation on pairs of solutions chosen from
$\psi_{\omega lm}^{\text{in}}$, $\psi_{\omega lm}^{\text{up}}$, $\psi_{\omega
lm}^{\text{in}*}$ and $\psi_{\omega lm}^{\text{up}*}$ (in particular, equating
their Wronskian at $r_{*}\to-\infty$ with their Wronskian at $r_{*}\to\infty$,
using the asymptotic forms given in Eqs. (27) and (28)), yields the following
constraints on the reflection and transmission coefficients:

$\displaystyle\left|\rho_{\omega
lm}^{\text{in}}\right|^{2}+\frac{\omega_{+}}{\omega}\left|\tau_{\omega
lm}^{\text{in}}\right|^{2}=1\,,$ (58) $\displaystyle\left|\rho_{\omega
lm}^{\text{up}}\right|^{2}+\frac{\omega}{\omega_{+}}\left|\tau_{\omega
lm}^{\text{up}}\right|^{2}=1\,,$ $\displaystyle\tau_{\omega
lm}^{\text{up}}=\frac{\omega_{+}}{\omega}\tau_{\omega lm}^{\text{in}}\,,$
$\displaystyle\frac{\rho_{\omega lm}^{\text{up}*}}{\rho_{\omega
lm}^{\text{in}}}=-\frac{\tau_{\omega lm}^{\text{up}*}}{\tau_{\omega
lm}^{\text{up}}}\,.$

The last equation yields, in particular, $\left|\rho_{\omega
lm}^{\text{up}}\right|=\left|\rho_{\omega lm}^{\text{in}}\right|$.

Notably, from the first (second) constraint, modes with $\omega\,\omega_{+}<0$
have $\left|\rho_{\omega lm}^{\text{in}}\right|^{2}>1$
$\left(\left|\rho_{\omega lm}^{\text{up}}\right|^{2}>1\right)$. That is, the
reflected _in_ (_up_) wave has, at FNI ($H_{R}$), an amplitude greater than it
originally had at PNI ($H_{\text{past}}$). This is the classical phenomenon of
_superradiance_ [14, 13]. 121212Note that in RN we have $\omega=\omega_{+}$,
which leads to the simple analogous relation $\left|\rho_{\omega
l}\right|^{2}+\left|\tau_{\omega l}\right|^{2}=1$, implying that there exist
_no_ superradiant modes.

##### Relations involving $A_{\omega lm}$ and $B_{\omega lm}$.

Similarly, Wronskian conservation of the interior radial function
$\psi_{\omega lm}^{\text{int}}$ (see Eq. (41)) and its conjugate $\psi_{\omega
lm}^{\text{int}*}$ relates the internal scattering coefficients $A_{\omega
lm}$ and $B_{\omega lm}$ (defined through Eq. (48)) as follows:

$\left|B_{\omega lm}\right|^{2}-\left|A_{\omega
lm}\right|^{2}=\frac{\omega_{+}}{\omega_{-}}\,.$ (59)

##  IV Quantum states in a Kerr spacetime

All topics outlined in the paper so far were basically purely classical. We
shall now promote our scalar field from a classical field $\Phi$ to a quantum
field operator $\hat{\Phi}$. We first provide a brief review of its
decomposition via annihilation and creation operators and then introduce
various quantum states in Kerr spacetime.

###  IV.1 Generic construction of quantum states

Consider a space of generic positive-frequency mode solutions, $\Phi_{i}$,
with respect to some temporal coordinate (clearly, in curved spacetime, this
choice is not unique). These solutions fulfill the following orthonormality
relations with respect to the KG inner product:

$\left\langle\Phi_{i},\Phi_{j}\right\rangle=\delta_{ij},\,\,\,\,\left\langle\Phi_{i}^{*},\Phi_{j}^{*}\right\rangle=-\delta_{ij},\,\,\,\,\left\langle\Phi_{i},\Phi_{j}^{*}\right\rangle=0\,,$
(60)

so that the union of the set $\Phi_{i}$ (for all $i$) and the set
$\Phi^{*}_{j}$ (for all $j$) is a complete family of orthonormal solutions to
the KG equation (12). We may now expand the field in terms of this basis of
solutions via creation ($\hat{a}_{i}^{\dagger}$) and annihilation
($\hat{a}_{i}$) operators as follows:

$\displaystyle\hat{\Phi}\left(x\right)$
$\displaystyle=\sum_{i}\left(\hat{a}_{i}\Phi_{i}\left(x\right)+\hat{a}_{i}^{\dagger}\Phi_{i}^{*}\left(x\right)\right)\,,$

where the following commutation relations are imposed:

$\left[\hat{a}_{i},\hat{a}_{j}^{\dagger}\right]=\hbar\,\delta_{ij}\hat{1},\,\,\Big{[}\hat{a}_{i},\hat{a}_{j}\Big{]}=0,\,\,\left[\hat{a}_{i}^{\dagger},\hat{a}_{j}^{\dagger}\right]=0\,.$

The vacuum with respect to this family is a state $\left|0\right\rangle$ such
that

$\hat{a}_{i}\left|0\right\rangle=0,\ \text{for all}\ i\,.$

Acting on the vacuum state with the creation operators $\hat{a}_{i}^{\dagger}$
yields the one-particle states

$\left|1_{i}\right\rangle=\hbar^{-1/2}\hat{a}_{i}^{\dagger}\left|0\right\rangle\,,$

and from here one may construct the entire many-particle Fock space.

###  IV.2 The Boulware and (lack of) Hartle-Hawking quantum states

The outlined decomposition scheme is utilized in the construction of quantum
states for a scalar field. As a concrete example, consider the field
decomposition in the Kerr BH exterior in terms of the outer Eddington modes
(Eq. (31))

$\displaystyle\hat{\Phi}\left(x\right)$
$\displaystyle=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\int_{0}^{\infty}\text{d}\omega\left(\hat{b}_{\omega
lm}^{\text{in}}f_{\omega lm}^{\text{in}}\left(x\right)+\hat{b}_{\omega
lm}^{\text{in}\dagger}f_{\omega lm}^{\text{in}*}\left(x\right)\right)$
$\displaystyle+\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\int_{0}^{\infty}\text{d}\omega_{+}\left(\hat{b}_{\omega
lm}^{\text{up}}f_{\omega lm}^{\text{up}}\left(x\right)+\hat{b}_{\omega
lm}^{\text{up}\dagger}f_{\omega lm}^{\text{up}*}\left(x\right)\right)\,,$ (61)

for some operator coefficients $\hat{b}_{\omega lm}^{\text{in}}$ and
$\hat{b}_{\omega lm}^{\text{up}}$. Note that the _up_ modes are defined with
respect to the positive frequency $\omega_{+}$ rather than $\omega$ (this is a
direct result of the asymptotic behavior of the effective potential (22)).
Therefore, the corresponding integration is over positive $\omega_{+}$. The
decomposition in Eq. (61) serves to define the so-called (past) Boulware state
$\left|0\right\rangle_{B}$ (see Refs. [7, 8] for the Schwarzschild case and
Refs. [11, 12] for Kerr) via

$\displaystyle\hat{b}_{\omega lm}^{\text{in}}\left|0\right\rangle_{B}$
$\displaystyle=0,\quad\text{for all $\omega>0$};$
$\displaystyle\hat{b}_{\omega lm}^{\text{up}}\left|0\right\rangle_{B}$
$\displaystyle=0,\quad\text{for all $\omega_{+}>0$}\,.$

In non-rotating BHs (i.e., Schwarzschild and RN), this state is irregular on
both $H_{\text{past}}$ and $H_{\text{R}}$ and is empty on both PNI and FNI
(and so it is said to model a cold star); in the rotating case, it continues
to be empty on PNI but it contains quantum superradiance at FNI (the Unruh-
Starobinskii effect).

The focus of this paper is another state: the Unruh state, which we define in
the next subsection. Before turning to the Unruh state, however, we wish to
give the following remark on another, third state. In non-rotating BHs, one
may consider the _Hartle-Hawking_ (HH) state [10, 32], which corresponds to a
BH in thermal equilibrium, coupled to an infinite bath of radiation. Although
not too realistic, the HH state provides (in the non-rotating case) relative
simplicity due to its time-reversal and time translational invariance, and so
historically it was used to make some progress in the study of the RSET.
However, a state analogous to HH is ill-defined in Kerr (see Ref. [15], as
well as Refs. [12] and [20]). This may be intuitively understood from the
existence of superradiant modes (see Subsec. III.4), for which waves are
reflected back to infinity with increased amplitude, conflicting with the
feasibility of a state of thermal equilibrium. We shall thus consider only the
(highly physically relevant) Unruh state from now on.

###  IV.3 The Unruh quantum state

The _Unruh_ state [33] is widely recognized as a physically realistic vacuum
quantum state, describing an evaporating BH (and thus, by definition, is not
time-reversal invariant). The Unruh state is constructed to resemble the
quantum state arising at late times for a BH formed by gravitational collapse.
It is defined by taking positive frequencies with respect to the affine
parameters along both initial null hypersurfaces (see the formulation of the
Unruh modes in Subsec. III.3). I.e., positive frequencies are defined with
respect to $v$ (the affine coordinate on PNI) for incoming modes, and with
respect to $U$ (the affine coordinate on $H_{\text{past}}$ and $H_{L}$) for
outgoing modes.

For a straight-forward definition of the Unruh vacuum state, we decompose the
metric in terms of the Unruh modes (Eqs. (49) and (55)):

$\displaystyle\hat{\Phi}\left(x\right)=$
$\displaystyle\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\int_{0}^{\infty}\text{d}\omega\left(\hat{a}_{\omega
lm}^{\text{in}}g_{\omega lm}^{\text{in}}\left(x\right)+\hat{a}_{\omega
lm}^{\text{in}\dagger}g_{\omega lm}^{\text{in}*}\left(x\right)\right)$ (62)
$\displaystyle+\sum_{\hat{l}=0}^{\infty}\sum_{\hat{m}=-\hat{l}}^{\hat{l}}\int_{0}^{\infty}\text{d}\hat{\omega}\left(\hat{a}_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}\left(x\right)+\hat{a}_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}\dagger}g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}*}\left(x\right)\right)\,,$

where $\hat{a}_{\omega lm}^{\text{in}}$ and $\hat{a}_{\omega
lm}^{\text{in$\dagger$}}$ ($\hat{a}_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}$
and $\hat{a}_{\hat{\omega}\hat{l}\hat{m}}^{\text{up$\dagger$}}$) are the
creation and annihilation operators corresponding to the _in_ (_up_) Unruh
modes, and the set of quantum numbers $\hat{\omega}\hat{l}\hat{m}$ classifying
the _up_ Unruh modes are as discussed in Subsec. III.3.2.

The Unruh state $\left|0\right\rangle_{U}$ is then defined as the vacuum state
with respect to the Unruh decomposition (62), namely, it is the state
annihilated by all Unruh-modes annihilation operators:

$\displaystyle\hat{a}_{\omega lm}^{\text{in}}\left|0\right\rangle_{U}$
$\displaystyle=0,\quad\text{for all $\omega>0$};$
$\displaystyle\hat{a}_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}\left|0\right\rangle_{U}$
$\displaystyle=0,\quad\text{for all $\hat{\omega}>0$}\,.$

The Unruh state involves no incoming flux at PNI, and an outgoing flux of
thermal radiation at FNI, in correspondence with Hawking radiation of an
evaporating BH. The corresponding RSET is expected to be regular across the
interior of the united domain (since the united domain is the future domain of
dependence of the two Unruh-state initial null hypersurfaces, PNI and
$H_{\text{past}}\cup H_{L}$). In particular, this expectation for regularity
applies at $H_{R}$, but not at $H_{\text{past}}\cup H_{L}$.

##  V Constructing the Unruh state HTPF in the exterior of a Kerr BH

From the decomposition of the field $\hat{\Phi}$ in terms of Unruh modes (Eq.
(62)), applying the commutation relations
$\left[\hat{a}_{I}^{\Lambda},\hat{a}_{I^{\prime}}^{\Lambda^{\prime}\dagger}\right]=\hbar\,\delta_{II^{\prime}}\delta_{\Lambda\Lambda^{\prime}}\hat{1}$
(where $I$ denotes the set of all three quantum numbers and $\Lambda$ is
either “up” or “in”), we obtain the mode-sum expression of the Unruh state
HTPF in terms of the Unruh modes $g_{\omega lm}^{\text{in}}$ and
$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}$ (defined in Eqs. (49) and (55)):

$G_{U}^{(1)}(x,x^{\prime})\equiv\left\langle\left\\{\hat{\Phi}\left(x\right),\hat{\Phi}\left(x^{\prime}\right)\right\\}\right\rangle_{U}=\left\langle\hat{\Phi}\left(x\right)\hat{\Phi}\left(x^{\prime}\right)+\hat{\Phi}\left(x^{\prime}\right)\hat{\Phi}\left(x\right)\right\rangle_{U}=G_{U}^{\text{in}}\left(x,x^{\prime}\right)+G_{U}^{\text{up}}\left(x,x^{\prime}\right),$
(63)

where $x$ and $x^{\prime}$ are spacetime points and we denote

$\displaystyle G_{U}^{\text{in}}\left(x,x^{\prime}\right)$
$\displaystyle\equiv\hbar\sum_{l,m}\int_{0}^{\infty}\text{d}\omega\left\\{g_{\omega
lm}^{\text{in}}\left(x\right),g_{\omega
lm}^{\text{in}*}\left(x^{\prime}\right)\right\\},$ (64) $\displaystyle
G_{U}^{\text{up}}\left(x,x^{\prime}\right)$
$\displaystyle\equiv\hbar\sum_{\hat{l},\hat{m}}\int_{0}^{\infty}\text{d}\hat{\omega}\left\\{g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}\left(x\right),g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}*}\left(x^{\prime}\right)\right\\}\,,$
(65)

and, recall,
$\left\\{\xi\left(x\right),\zeta\left(x^{\prime}\right)\right\\}\equiv\xi\left(x\right)\zeta\left(x^{\prime}\right)+\xi\left(x^{\prime}\right)\zeta\left(x\right)$.

Throughout the paper, we shall use the shorthand notation
$\sum_{l,m}\equiv\sum_{l=0}^{\infty}\sum_{m=-l}^{l}$ (and likewise for
$\sum_{\hat{l},\hat{m}}$). Remarkably, it has been found (see Eq. (3.22) in
Ref. [20], as well as Eq. (3.18c) in Ref. [12] 131313Note that Ref. [12]
considers a slightly different two-point function (TPF) from our
$G_{U}^{(1)}\left(x,x^{\prime}\right)$, namely the _non-symmetrized_ TPF,
usually called the Wightman function
$G^{+}(x,x^{\prime})=\langle\hat{\Phi}(x)\hat{\Phi}(x^{\prime})\rangle$ rather
than the Hadamard two-point function
$G^{(1)}(x,x^{\prime})=\langle\bigl{\\{}\hat{\Phi}(x),\hat{\Phi}(x^{\prime})\bigr{\\}}\rangle$.
In particular, it is trivial to derive the symmetrized TPF from the non-
symmetrized one provided in Ref. [12], thereby obtaining Eq. (66). , with the
analogous Schwarzschild case in Refs. [18, 19])), that the mode-sum expression
in Eq. (63) may be decomposed in terms of the more manageable outer Eddington
modes (defined and discussed in Subsec. III.1), yielding the expression:

$\displaystyle G_{U}^{(1)}\left(x,x^{\prime}\right)=$
$\displaystyle\hbar\sum_{l,m}\left[\int_{0}^{\infty}\text{d}\omega\left\\{f_{\omega
lm}^{\text{in}}\left(x\right),f_{\omega
lm}^{\text{in}*}\left(x^{\prime}\right)\right\\}+\int_{0}^{\infty}\text{d}\omega_{+}\coth\left(\frac{\pi\omega_{+}}{\kappa_{+}}\right)\left\\{f_{\omega
lm}^{\text{up}}\left(x\right),f_{\omega
lm}^{\text{up}*}\left(x^{\prime}\right)\right\\}\right].$ (66)

In what follows in this section, we shall present the derivation of Eq. (66)
in the Kerr BH exterior via a procedure different from the methods previously
used in the mentioned references. This procedure is the same as that which we
use later on to derive the HTPF in the Kerr interior and is an extension to
Kerr of the procedure used in Ref. [31] for the interior of a RN BH. In doing
so, we recover the known result (Eq. (66)). Proceeding in this way will allow
us to demonstrate our method on a Kerr BH background in a simpler case (i.e.
outside the BH) before delving into the BH interior, and to handle various
issues special to Kerr that arise already in the BH exterior.

###  V.1 Mode decomposition of the exterior Unruh HTPF

Concentrating on the BH exterior, we wish to express Eqs. (64) and (65) in
terms of the exterior Eddington modes, $f_{\omega lm}^{\text{in}}$ and
$f_{\omega lm}^{\text{up}}$, defined in Subsec. III.1.

In various stages of the computation to be carried out, it turns out to be
very useful to define a new version of $f_{\omega lm}^{\text{up}}$, which
carries an index $\omega_{+}$ rather than $\omega$. We shall use the notation
$f_{\omega_{+}lm}^{\text{up}\left(+\right)}$ as the $\omega_{+}$-indexed
version of $f_{\omega lm}^{\text{up}}$. That is, for a certain set $\omega
lm$, we define

$f_{\omega_{+}lm}^{\text{up}\left(+\right)}\equiv
f_{\omega\left(\omega_{+},m\right)lm}^{\text{up}}$ (67)

where $\omega$ on the RHS is related to $\omega_{+}$ and $m$ on the LHS by the
standard relation, $\omega\left(\omega_{+},m\right)=\omega_{+}+m\Omega_{+}$.
All relations and equations from the previous sections that include $f_{\omega
lm}^{\text{up}}$ may now be carried to this section with the simple
replacement $f_{\omega lm}^{\text{up}}\mapsto
f_{\omega_{+}lm}^{\text{up}\left(+\right)}$ 141414A clarification regarding
our notation may be in order here (particularly related to the
interchangeability of $f_{\omega lm}^{\text{up}}$ and
$f_{\omega_{+}lm}^{\text{up}\left(+\right)}$): Once the object
$f_{\omega_{+}lm}^{\text{up}\left(+\right)}$ has been defined here (in Eq.
(67)), our notational rules allow us to write equations of the form, e.g.,
$f_{\omega_{+}lm}^{\text{up}\left(+\right)}=f_{\omega lm}^{\text{up}}$ (just
to give a simple illustrative example). The exact meaning of an equality of
this type has been clarified in Sec. (III), and we repeat it here for clarity:
Whenever a part of a given equation depends on $\omega$ and another part
depends on $\omega_{+}$, the latter is to be viewed as given by
$\omega-m\Omega_{+}$. (Or, if one prefers, the other way around: $\omega$ may
be viewed to be given by $\omega_{+}+m\Omega_{+}$.) . In what follows, we
shall use the object $f_{\omega_{+}lm}^{\text{up}\left(+\right)}$ (rather than
$f_{\omega lm}^{\text{up}}$) as a tool until we reach the final expression
(Eq. (90)), which will then be re-expressed in terms of the usual $f_{\omega
lm}^{\text{up}}$.

We begin with the _in_ contribution. We may readily use the equality presented
in Eq. (50) between $g_{\omega lm}^{\text{in}}$ and $f_{\omega
lm}^{\text{in}}$ in the BH exterior to express Eq. (64) as the mode sum

$G_{U}^{\text{in}}\left(x,x^{\prime}\right)=\hbar\sum_{l,m}\int_{0}^{\infty}\text{d}\omega\left\\{f_{\omega
lm}^{\text{in}}\left(x\right),f_{\omega
lm}^{\text{in}*}\left(x^{\prime}\right)\right\\}\,.$ (68)

Likewise, the _up_ counterpart $G_{U}^{\text{up}}\left(x,x^{\prime}\right)$
requires establishing a relation between the _up_ Unruh modes
$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}$ and the exterior Eddington modes.
However, as discussed at the end of Subsec. III.3.2, that task is a more
complicated one – compared to the _in_ Unruh modes, but also compared to the
spherical symmetry counterpart.

We shall now introduce a notation to be used occasionally throughout the rest
of the paper. In determining a quantum state, the frequency has a special role
over the other quantum numbers. Let us then denote collectively all other
quantum numbers by $J$ (or sometimes $\hat{J}$). Here, more specifically,
$J\equiv\left(l,m\right)$ and $\hat{J}\equiv\left(\hat{l},\hat{m}\right)$
(where the latter indices were introduced in Subsec. III.3.2). Then, we shall
also use the shorthand notation
$\sum_{J}\equiv\sum_{l,m}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}$, and likewise
for $\sum_{\hat{J}}$.

To relate the _up_ Unruh modes $g_{\hat{\omega}\hat{J}}^{\text{up}}$ and the
Eddington modes on the BH exterior, we turn our attention to the relevant
asymptotic surfaces. Recall that on the past horizon, we have (see Eqs. (55)
and (35))

$\left.g_{\hat{\omega}\hat{J}}^{\text{up}}\right|_{H_{\text{past}}}=\frac{1}{\sqrt{4\pi\hat{\omega}\left(r_{+}^{2}+a^{2}\right)}}Y_{\hat{J}}\left(\theta,\varphi_{+}\right)e^{-i\hat{\omega}U\left(u_{\text{ext}}\right)}$
(69)

and

$\displaystyle\left.f_{\omega_{+}J}^{\text{up}\left(+\right)}\right|_{H_{\text{past}}}$
$\displaystyle=\frac{1}{\sqrt{4\pi\left|\omega_{+}\right|\left(r_{+}^{2}+a^{2}\right)}}Z_{J}^{\omega}\left(\theta,\varphi_{+}\right)e^{-i\omega_{+}u_{\text{ext}}}\,.$
(70)

As was already spelled out in Eq. (37), the set of spheroidal harmonics
$Z_{J}^{\omega}\left(\theta,\varphi_{+}\right)$ is a complete orthonormal
family on the 2-sphere for any fixed $\omega$. For the analysis below, it is
important to note that this set also forms a complete orthonormal family for
any fixed $\omega_{+}$. It is orthonormal in the sense

$\displaystyle\int_{0}^{2\pi}\text{d}\varphi_{+}\int_{0}^{\pi}\text{d}\theta\,\sin\theta\,\left[Z_{lm}^{\omega\left(\omega_{+},m\right)}\left(\theta,\varphi_{+}\right)\right]^{*}Z_{l^{\prime}m^{\prime}}^{\omega\left(\omega_{+},m^{\prime}\right)}\left(\theta,\varphi_{+}\right)$
$\displaystyle=\delta_{ll^{\prime}}\delta_{mm^{\prime}}\,,$ (71)

where, as usual,
$\omega\left(\omega_{+},m\right)\equiv\omega_{+}+m\Omega_{+}$. This is so
because
$\int_{0}^{2\pi}e^{i\left(m^{\prime}-m\right)\varphi_{+}}\text{d}\varphi_{+}=2\pi\delta_{mm^{\prime}}$,
hence the $\theta$-integral in Eq. (71) needs only to be carried out for
$m^{\prime}=m$ – in which case the equality of $\omega_{+}$ (in the two $Z$
functions) is fully equivalent to the equality of $\omega$. A fairly similar
argument may be used to show that the completeness of the family
$Z_{J}^{\omega}\left(\theta,\varphi_{+}\right)$ with fixed $\omega$ also
implies its completeness with fixed $\omega_{+}$.

Now, in order to decompose $g_{\hat{\omega}\hat{J}}^{\text{up}}$ in terms of
$f_{\omega_{+}J}^{\text{up}\left(+\right)}$ on $H_{\text{past}}$, we introduce
two sets of coefficients, $\alpha_{\hat{\omega}\omega_{+}}$ and
$C_{\hat{J}J}^{\omega_{+}}$, aimed at handling the frequential and angular
factors respectively. The Fourier coefficients
$\alpha_{\hat{\omega}\omega_{+}}$ are given by the inverse Fourier transform

$\alpha_{\hat{\omega}\omega_{+}}\equiv\int_{-\infty}^{\infty}\text{d}u_{\text{ext}}\,e^{-i\hat{\omega}U\left(u_{\text{ext}}\right)}e^{i\omega_{+}u_{\text{ext}}}\,.$
(72)

This integral may be evaluated as described in Ref. [31] 151515See Eq. (3.3)
therein, and apply the notation change
$\tilde{\omega}\mapsto\omega_{+},\omega\mapsto\hat{\omega}$., yielding

$\alpha_{\hat{\omega}\omega_{+}}=\frac{1}{\kappa_{+}}\left(\frac{\hat{\omega}}{\kappa_{+}}\right)^{i\omega_{+}/\kappa_{+}}e^{\pi\omega_{+}/2\kappa_{+}}\Gamma\left(-i\frac{\omega_{+}}{\kappa_{+}}\right)\,.$
(73)

Similarly, $C_{\hat{J}J}^{\omega_{+}}$, the coefficients translating between
the two bases of orthonormal functions on the two-sphere, $Z_{J}^{\omega}$ and
$Y_{\hat{J}}$, are defined by

$C_{\hat{J}J}^{\omega_{+}}\equiv\int_{0}^{2\pi}\text{d}\varphi_{+}\int_{0}^{\pi}\text{d}\theta\sin\theta\,\,\left[Z_{J}^{\omega\left(\omega_{+},m\right)}\left(\theta,\varphi_{+}\right)\right]^{*}Y_{\hat{J}}\left(\theta,\varphi_{+}\right)$
(74)

(recall that in the square brackets the information about the value of $m$ in
$\omega\left(\omega_{+},m\right)$ is encoded in $J$).

The $\alpha_{\hat{\omega}\omega_{+}}$ and $C_{\hat{J}J}^{\omega_{+}}$
coefficients, as defined in Eqs. (72) and (74), allow a translation from the
Unruh to the Eddington frequential and angular factors via the relations

$e^{-i\hat{\omega}U\left(u_{\text{ext}}\right)}=\frac{1}{2\pi}\int_{-\infty}^{\infty}\text{d}\omega_{+}\,\alpha_{\hat{\omega}\omega_{+}}e^{-i\omega_{+}u_{\text{ext}}}\,$
(75)

and

$Y_{\hat{J}}\left(\theta,\varphi_{+}\right)=\sum_{J}C_{\hat{J}J}^{\omega_{+}}Z_{J}^{\omega(\omega_{+},m)}\left(\theta,\varphi_{+}\right)\,,$
(76)

where, recall, in the last equality the sum over $J$ is carried out with fixed
$\omega_{+}$. The first of these relations is just the inversion of the
(inverse) Fourier transfom in Eq. (72). To derive the second relation, we use
the (fixed-$\omega_{+}$) completeness of the spheroidal harmonics to decompose
the spherical harmonics as

$Y_{\hat{J}}\left(\theta,\varphi_{+}\right)=\sum_{J}P_{\hat{J}J}^{\omega_{+}}Z_{J}^{\omega(\omega_{+},m)}\left(\theta,\varphi_{+}\right)\,,$
(77)

where $P_{\hat{J}J}^{\omega_{+}}$ denote the coefficients of the
decomposition. Substituting this decomposition into the RHS of Eq. (74), and
recalling the spheroidal harmonics orthonormality, one readily sees that
$C_{\hat{J}J}^{\omega_{+}}=P_{\hat{J}J}^{\omega_{+}}$, hence Eq. (77) reduces
to the desired relation (76). In a similar manner (this time employing the
orthonormality and completeness of the _spherical_ harmonics), one can
decompose the spheroidal harmonics and show that

$Z_{J}^{\omega(\omega_{+},m)}\left(\theta,\varphi_{+}\right)=\sum_{\hat{J}}C_{\hat{J}J}^{\omega_{+}*}Y_{\hat{J}}\left(\theta,\varphi_{+}\right)\,.$
(78)

For future use, we also note that the following relation holds:

$\sum_{\hat{J}}C_{\hat{J}J}^{\omega_{+}}C_{\hat{J}J^{\prime}}^{\omega_{+}*}=\delta_{JJ^{\prime}}\,.$
(79)

To see this, we substitute Eq. (76) into the RHS of Eq. (78) (with index
renaming $J\mapsto J^{\prime}$ in the latter) and obtain

$Z_{J^{\prime}}^{\omega(\omega_{+},m^{\prime})}\left(\theta,\varphi_{+}\right)=\sum_{J}\left[\sum_{\hat{J}}C_{\hat{J}J}^{\omega_{+}}C_{\hat{J}J^{\prime}}^{\omega_{+}*}\right]Z_{J}^{\omega(\omega_{+},m)}\left(\theta,\varphi_{+}\right)\,.$
(80)

Recalling the orthonormality of the spheroidal harmonics, the term in square
brackets must be the identity matrix with components $\delta_{JJ^{\prime}}$,
yielding Eq. (79).

Substituting Eqs. (75) and (76) in Eq. (69) and comparing to Eq. (70), the
decomposition of $g_{\hat{\omega}\hat{J}}^{\text{up}}$ in terms of
$f_{\omega_{+}J}^{\text{up}\left(+\right)}$ on the null hypersurface
$H_{\text{past}}$ may now be written as follows:

$\displaystyle\sqrt{\hat{\omega}}\left.g_{\hat{\omega}\hat{J}}^{\text{up}}\right|_{H_{\text{past}}}$
$\displaystyle=\frac{1}{2\pi}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\,\alpha_{\hat{\omega}\omega_{+}}C_{\hat{J}J}^{\omega_{+}}\left.f_{\omega_{+}J}^{\text{up}\left(+\right)}\right|_{H_{\text{past}}}\,.$
(81)

In addition, recall that both $g_{\hat{\omega}\hat{J}}^{\text{up}}$ and
$f_{\omega_{+}J}^{\text{up}\left(+\right)}$ vanish on PNI: see Eqs. (35) and
(55). Thus,

$\sqrt{\hat{\omega}}\left.g_{\hat{\omega}\hat{J}}^{\text{up}}\right|_{\text{PNI}}=0=\frac{1}{2\pi}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\,\alpha_{\hat{\omega}\omega_{+}}C_{\hat{J}J}^{\omega_{+}}\left.f_{\omega_{+}J}^{\text{up}\left(+\right)}\right|_{\text{PNI}}\,.$
(82)

Since $g_{\hat{\omega}\hat{J}}^{\text{up}}$ and
$f_{\omega_{+}J}^{\text{up}\left(+\right)}$ satisfy the same wave equation
(13), it follows that these two quantities are related as prescribed in Eqs.
(81) and (82) not only on the initial null hypersurfaces $H_{\text{past}}$ and
PNI but throughout the BH exterior. That is,

$\sqrt{\hat{\omega}}\,g_{\hat{\omega}\hat{J}}^{\text{up}}(x)=\frac{1}{2\pi}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\,\alpha_{\hat{\omega}\omega_{+}}C_{\hat{J}J}^{\omega_{+}}f_{\omega_{+}J}^{\text{up}\left(+\right)}(x)\,\,\,,\,\,\,\,\,\,\,\,\,\,r\geq
r_{+}\,.$ (83)

Next, the HTPF _up_ mode contribution
$G_{U}^{\text{up}}\left(x,x^{\prime}\right)$ in terms of Eddington modes is
achieved by substituting Eq. (83) in Eq. (65):

$\displaystyle G_{U}^{\text{up}}\left(x,x^{\prime}\right)=$ (84)
$\displaystyle\frac{\hbar}{4\pi^{2}}\sum_{\hat{J}}\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{\hat{\omega}}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\alpha_{\hat{\omega}\omega_{+}}C_{\hat{J}J}^{\omega_{+}}\sum_{J^{\prime}}\int_{-\infty}^{\infty}\text{d}\omega^{\prime}_{+}\sqrt{\left|\omega^{\prime}_{+}\right|}\,\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{*}C_{\hat{J}J^{\prime}}^{\omega^{\prime}_{+}*}\left\\{f_{\omega_{+}J}^{\text{up}\left(+\right)}\left(x\right),f_{\omega^{\prime}_{+}J^{\prime}}^{\text{up}\left(+\right)*}\left(x^{\prime}\right)\right\\}\,.$

We conveniently rearrange Eq. (84) as follows 161616This re-arrangement
involves interchanges of the summation over $\hat{J}$ and the integration over
$\hat{\omega}$ with all subsequent operations (summations and integrations).
We do not attempt to rigorously justify this manipulation (or similar ones
that appear later on). Nevertheless, after implementing this re-arrangement,
we do recover the correct, well known, result quoted in Eq. (66) above. This
may be considered a justification for the manipulations entailed.:

$\displaystyle G_{U}^{\text{up}}\left(x,x^{\prime}\right)=$ (85)
$\displaystyle\frac{\hbar}{4\pi^{2}}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\sum_{J^{\prime}}\int_{-\infty}^{\infty}\text{d}\omega^{\prime}_{+}\sqrt{\left|\omega^{\prime}_{+}\right|}\left\\{f_{\omega_{+}J}^{\text{up}\left(+\right)}\left(x\right),f_{\omega^{\prime}_{+}J^{\prime}}^{\text{up}\left(+\right)*}\left(x^{\prime}\right)\right\\}\sum_{\hat{J}}C_{\hat{J}J}^{\omega_{+}}C_{\hat{J}J^{\prime}}^{\omega^{\prime}_{+}*}\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{\hat{\omega}}\alpha_{\hat{\omega}\omega_{+}}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{*}\,.$

We can now perform the $\hat{\omega}$-integral appearing in the above
equation, using Eq. (73): 171717To obtain this integral, one may rewrite it as
$\frac{1}{\kappa_{+}}e^{\left(\omega_{+}+\omega^{\prime}_{+}\right)\pi/2\kappa_{+}}\Gamma\left(-i\omega_{+}/\kappa_{+}\right)\Gamma\left(i\omega^{\prime}_{+}/\kappa_{+}\right)\int_{-\infty}^{\infty}\text{d}s\,e^{is\left(\omega_{+}-\omega^{\prime}_{+}\right)}$,
where $s\equiv\frac{1}{\kappa_{+}}\ln\left(\hat{\omega}/\kappa_{+}\right)$,
and use the relation
$\left|\Gamma\left(i\omega_{+}/\kappa_{+}\right)\right|^{2}=\pi\left(\kappa_{+}/\omega_{+}\right)/\sinh\left(\pi\omega_{+}/\kappa_{+}\right)$.

$\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{\hat{\omega}}\alpha_{\hat{\omega}\omega_{+}}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{*}=\frac{4\pi^{2}}{\omega_{+}}\frac{1}{1-e^{-2\pi\omega_{+}/\kappa_{+}}}\delta\left(\omega_{+}-\omega^{\prime}_{+}\right)\,.$
(86)

With this identity, Eq. (85) reduces, after performing the trivial integration
of the $\delta$-function over $\omega^{\prime}_{+}$, to

$G_{U}^{\text{up}}\left(x,x^{\prime}\right)=\hbar\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\frac{\left|\omega_{+}\right|}{\omega_{+}}\frac{1}{1-e^{-2\pi\omega_{+}/\kappa_{+}}}\sum_{J^{\prime}}\left\\{f_{\omega_{+}J}^{\text{up}\left(+\right)}\left(x\right),f_{\omega{}_{+}J^{\prime}}^{\text{up}\left(+\right)*}\left(x^{\prime}\right)\right\\}\sum_{\hat{J}}C_{\hat{J}J}^{\omega_{+}}C_{\hat{J}J^{\prime}}^{\omega{}_{+}*}\,.$

Next we use Eq. (79) to obtain

$G_{U}^{\text{up}}\left(x,x^{\prime}\right)=\hbar\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\text{\,sign}\left(\omega_{+}\right)\frac{1}{1-e^{-2\pi\omega_{+}/\kappa_{+}}}\left\\{f_{\omega_{+}J}^{\text{up}\left(+\right)}\left(x\right),f_{\omega_{+}J}^{\text{up}\left(+\right)*}\left(x^{\prime}\right)\right\\}\,.$
(87)

Finally, we would like to “fold” the $\omega_{+}$-integral in this equation so
that only modes with $\omega_{+}>0$ show up. To this end, we note that since
the mapping $\left(\omega,m\right)\mapsto\left(-\omega,-m\right)$ is
equivalent to $\left(\omega_{+},m\right)\mapsto\left(-\omega_{+},-m\right)$,
we may rewrite Eq. (38) (with $\Lambda$ taken as ”up”) as

$f_{\left(-\omega_{+}\right)l\left(-m\right)}^{\text{up}\left(+\right)}=(-1)^{m}f_{\omega_{+}lm}^{\text{up}\left(+\right)*}\,.$
(88)

We now recall that $\sum_{J}\equiv\sum_{l=0}^{\infty}\sum_{m=-l}^{l}$ and
concentrate on the summation over $m$. For each given $l$ we have

$\sum_{m=-l}^{l}\left\\{f_{\left(-\omega_{+}\right)lm}^{\text{up}\left(+\right)}\left(x\right),f_{\left(-\omega_{+}\right)lm}^{\text{up}\left(+\right)*}\left(x^{\prime}\right)\right\\}=\sum_{m=-l}^{l}\left\\{f_{\omega_{+}l\left(-m\right)}^{\text{up}\left(+\right)}\left(x\right),f_{\omega_{+}l\left(-m\right)}^{\text{up}\left(+\right)*}\left(x^{\prime}\right)\right\\}=\sum_{m=-l}^{l}\left\\{f_{\omega_{+}lm}^{\text{up}\left(+\right)}\left(x\right),f_{\omega_{+}lm}^{\text{up}\left(+\right)*}\left(x^{\prime}\right)\right\\}\,,$

where the first equality follows from Eq. (88), and the last equality simply
involves a renaming of the summation index $m\mapsto-m$. Using the group index
$J$, this may be expressed as

$\sum_{J}\left\\{f_{\left(-\omega_{+}\right)J}^{\text{up}\left(+\right)}\left(x\right),f_{\left(-\omega_{+}\right)J}^{\text{up}\left(+\right)*}\left(x^{\prime}\right)\right\\}=\sum_{J}\left\\{f_{\omega_{+}J}^{\text{up}\left(+\right)}\left(x\right),f_{\omega_{+}J}^{\text{up}\left(+\right)*}\left(x^{\prime}\right)\right\\}\,.$

Eq. (87) can thus be rewritten as

$G_{U}^{\text{up}}\left(x,x^{\prime}\right)=\hbar\sum_{J}\int_{0}^{\infty}\text{d}\omega_{+}\left[\frac{1}{1-e^{-2\pi\omega_{+}/\kappa_{+}}}-\frac{1}{1-e^{2\pi\omega_{+}/\kappa_{+}}}\right]\left\\{f_{\omega_{+}J}^{\text{up}\left(+\right)}\left(x\right),f_{\omega{}_{+}J}^{\text{up}\left(+\right)*}\left(x^{\prime}\right)\right\\}\,.$

As one can easily see, the term in square brackets is
$\coth\left(\pi\omega_{+}/\kappa_{+}\right)$. Our final result for
$G_{U}^{\text{up}}$ is, therefore,

$G_{U}^{\text{up}}\left(x,x^{\prime}\right)=\hbar\sum_{J}\int_{0}^{\infty}\text{d}\omega_{+}\coth\left(\frac{\pi\omega_{+}}{\kappa_{+}}\right)\left\\{f_{\omega_{+}J}^{\text{up}\left(+\right)}\left(x\right),f_{\omega{}_{+}J}^{\text{up}\left(+\right)*}\left(x^{\prime}\right)\right\\}\,.$
(89)

As prescribed in Eq. (63), we may now put together the _up_ and _in_ mode
contributions, as given in Eqs. (68) and (89), to yield
$G_{U}^{\left(1\right)}\left(x,x^{\prime}\right)$:

$G_{U}^{(1)}(x,x^{\prime})=\hbar\sum_{l,m}\left[\int_{0}^{\infty}\text{d}\omega\left\\{f_{\omega
lm}^{\text{in}}\left(x\right),f_{\omega
lm}^{\text{in}*}\left(x^{\prime}\right)\right\\}+\int_{0}^{\infty}\text{d}\omega_{+}\coth\left(\frac{\pi\omega_{+}}{\kappa_{+}}\right)\left\\{f_{\omega_{+}lm}^{\text{up}\left(+\right)}\left(x\right),f_{\omega_{+}lm}^{\text{up}\left(+\right)*}\left(x^{\prime}\right)\right\\}\right]\,.$
(90)

Finally, we may retrieve the standard $\omega$-indexed notation, replacing
$f_{\omega_{+}lm}^{\text{up}\left(+\right)}$ by $f_{\omega lm}^{\text{up}}$
(since, as mentioned above, they represent the same object). Then, the mode-
sum expression of the Unruh-state HTPF outside a Kerr BH, in terms of
Eddington modes, is as previously quoted (in Eq. (66)):

$G_{U}^{(1)}(x,x^{\prime})=\hbar\sum_{l,m}\left[\int_{0}^{\infty}\text{d}\omega\left\\{f_{\omega
lm}^{\text{in}}\left(x\right),f_{\omega
lm}^{\text{in}*}\left(x^{\prime}\right)\right\\}+\int_{0}^{\infty}\text{d}\omega_{+}\coth\left(\frac{\pi\omega_{+}}{\kappa_{+}}\right)\left\\{f_{\omega
lm}^{\text{up}}\left(x\right),f_{\omega
lm}^{\text{up}*}\left(x^{\prime}\right)\right\\}\right]\,.$ (91)

It may be shown (however, outside the scope of this paper) that at small
$\omega_{+}$, the _up_ radial function $\psi_{\omega lm}^{\text{up}}$ behaves
(to leading order) as $\omega_{+}$, which ensures regularity of the
$\int_{0}^{\infty}\text{d}\omega_{+}$ integral at small $\omega_{+}$.
(Regularity of the $\int_{0}^{\infty}\text{d}\omega$ integral at small
$\omega$ is similarly ensured.)

###  V.2 Invariance to the choice of the angular functions
$\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}\left(\theta,\varphi_{+}\right)$

Finally, we comment on the invariance of our final result (91) in the exterior
of a Kerr BH with respect to the choice of the angular functions
$\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}\left(\theta,\varphi_{+}\right)$, used
for prescribing the _up_ Unruh-modes initial data at $H_{\text{past}}\cup
H_{L}$. As discussed in Subsec. III.3.2,
$\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}\left(\theta,\varphi_{+}\right)$ can
be any set of angular functions which is orthonormal and complete on the
2-sphere (and in particular it may depend on the mode’s Kruskal frequency
$\hat{\omega}$). However, for the sake of concreteness (as well as simplicity
and brevity), in the analysis above we made the specific choice
$\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}\left(\theta,\varphi_{+}\right)=Y_{\hat{l}\hat{m}}\left(\theta,\varphi_{+}\right)$.
Here we shall briefly consider how the analysis would proceed, and ultimately
what would be the final resultant mode structure of the HTPF, if one chose to
work with generic angular functions $\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}$
rather than the specific functions $Y_{\hat{l}\hat{m}}$.

Let us examine the consequence of replacing $Y_{\hat{l}\hat{m}}$ everywhere by
$\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}$ (and likewise $Y_{\hat{J}}$ by
$\hat{Z}_{\hat{J}}^{\hat{\omega}}$), starting at Eq. (69). The
$C$-coefficients, relating the spheroidal harmonics with the _up_ Unruh-modes
initial angular functions (which now depend on $\hat{\omega}$), accordingly
acquire an extra index $\hat{\omega}$. That is, Eq. (74) is replaced by

$C_{\hat{J}J}^{\hat{\omega}\omega_{+}}\equiv\int_{0}^{2\pi}\text{d}\varphi_{+}\int_{0}^{\pi}\text{d}\theta\sin\theta\,\,\left[Z_{J}^{\omega(\omega_{+},m)}\left(\theta,\varphi_{+}\right)\right]^{*}\hat{Z}_{\hat{J}}^{\hat{\omega}}\left(\theta,\varphi_{+}\right)\,,$
(92)

and every instance of $C_{\hat{J}J}^{\omega_{+}}$ is replaced by
$C_{\hat{J}J}^{\hat{\omega}\omega_{+}}$. Under these replacements, all
equations up to Eq. (84) (inclusive) hold in their new analogous form. For
later use, we quote in particular the new form of Eq. (79), simply adding the
index $\hat{\omega}$ to both coefficients:

$\sum_{\hat{J}}C_{\hat{J}J}^{\hat{\omega}\omega_{+}}C_{\hat{J}J^{\prime}}^{\hat{\omega}\omega{}_{+}*}=\delta_{JJ^{\prime}}\,.$
(93)

Things become slightly more delicate when arriving at (the
$Y_{\hat{J}}\mapsto\hat{Z}_{\hat{J}}^{\hat{\omega}}$ counterpart of) Eq. (85):
Here, the term
$\sum_{\hat{J}}C_{\hat{J}J}^{\omega_{+}}C_{\hat{J}J^{\prime}}^{\omega^{\prime}_{+}*}$
should of course be replaced by
$\sum_{\hat{J}}C_{\hat{J}J}^{\hat{\omega}\omega_{+}}C_{\hat{J}J^{\prime}}^{\hat{\omega}\omega^{\prime}_{+}*}$.
Naively one might be concerned that since this term exhibits explicit
dependence on $\hat{\omega}$, it would be necessary to keep it inside the
$\hat{\omega}$ integral (i.e. the integral at the very end of the RHS of Eq.
(85), which is then evaluated in Eq. (86)). It is therefore crucial to note
that (as we shall shortly justify)

$\sum_{\hat{J}}C_{\hat{J}J}^{\hat{\omega}\omega_{+}}C_{\hat{J}J^{\prime}}^{\hat{\omega}\omega^{\prime}_{+}*}=\tilde{C}_{JJ^{\prime}}^{\omega_{+}\omega^{\prime}_{+}*}\,,$
(94)

where $\tilde{C}_{JJ^{\prime}}^{\omega_{+}\omega^{\prime}_{+}}$ is defined by

$\tilde{C}_{JJ^{\prime}}^{\omega_{+}\omega^{\prime}_{+}}\equiv\int_{0}^{2\pi}\text{d}\varphi_{+}\int_{0}^{\pi}\text{d}\theta\sin\theta\,\,\left[Z_{J^{\prime}}^{\omega\left(\omega^{\prime}_{+},m^{\prime}\right)}\left(\theta,\varphi_{+}\right)\right]^{*}Z_{J}^{\omega\left(\omega_{+},m\right)}\left(\theta,\varphi_{+}\right),$
(95)

which by construction is independent of $\hat{\omega}$. 181818As one can
easily see by performing the integration over $\varphi_{+}$,
$\tilde{C}_{JJ^{\prime}}^{\omega_{+}\omega^{\prime}_{+}}$ vanishes for any
$m^{\prime}\neq m$; but this specific property is not needed here.

Note that taking $\omega^{\prime}_{+}=\omega_{+}$ in the RHS of Eq. (95)
reduces to the LHS of the orthonormality relation (37). Hence

$\tilde{C}_{JJ^{\prime}}^{\omega_{+}\omega{}_{+}}=\delta_{JJ^{\prime}},$ (96)

and Eq. (94) thus reduces appropriately to Eq. (93).

In order to establish Eq. (94) we first note (using an argument similar to the
one employed above for the justification of Eq. (76)) that
$\tilde{C}_{JJ^{\prime}}^{\omega_{+}\omega^{\prime}_{+}}$ are actually the
coefficients relating the two sets of spheroidal harmonics
$Z_{J}^{\omega\left(\omega_{+},m\right)}$ and
$Z_{J^{\prime}}^{\omega\left(\omega^{\prime}_{+},m^{\prime}\right)}$, via
191919Explicitly, the derivation is as follows: From the (fixed-$\omega_{+}$)
completeness of the spheroidal harmonics, one can write (in analogy to Eq.
(77))
$Z_{J}^{\omega\left(\omega_{+},m\right)}\left(\theta,\varphi_{+}\right)=\sum_{J^{\prime}}P_{JJ^{\prime}}^{\omega_{+}\omega^{\prime}_{+}}Z_{J^{\prime}}^{\omega(\omega^{\prime}_{+},m^{\prime})}\left(\theta,\varphi_{+}\right),$
where $P_{JJ^{\prime}}^{\omega_{+}\omega^{\prime}_{+}}$ are the coefficients
of the decomposition. Substituting this decomposition into Eq. (95), and using
the orthonormality of the spheroidal harmonics, one obtains
$P_{JJ^{\prime}}^{\omega_{+}\omega^{\prime}_{+}}=\tilde{C}_{JJ^{\prime}}^{\omega_{+}\omega^{\prime}_{+}}$.

$Z_{J}^{\omega\left(\omega_{+},m\right)}\left(\theta,\varphi_{+}\right)=\sum_{J^{\prime}}\tilde{C}_{JJ^{\prime}}^{\omega_{+}\omega^{\prime}_{+}}Z_{J^{\prime}}^{\omega\left(\omega^{\prime}_{+},m^{\prime}\right)}\left(\theta,\varphi_{+}\right)\,.$
(97)

Then, Eq. (94) naturally follows from the completeness and orthonormality of
each of the three involved families of angular functions, namely
$Z_{J}^{\omega\left(\omega_{+},m\right)}$,
$Z_{J^{\prime}}^{\omega\left(\omega^{\prime}_{+},m^{\prime}\right)}$ and
$\hat{Z}_{\hat{J}}^{\hat{\omega}}$, by a slight generalization of the argument
described right after Eq. (79) 202020More precisely, one would need to combine
the $Y_{\hat{J}}\mapsto\hat{Z}_{\hat{J}}^{\hat{\omega}}$ counterparts of Eqs.
(76) and (78), taking $J\mapsto J^{\prime}$ and
$\omega_{+}\mapsto\omega^{\prime}_{+}$ in the former, to yield the (slightly
generalized) counterpart of Eq. (80):
$Z_{J}^{\omega(\omega_{+},m)}\left(\theta,\varphi_{+}\right)=\sum_{J^{\prime}}\left[\sum_{\hat{J}}C_{\hat{J}J}^{\hat{\omega}\omega_{+}*}C_{\hat{J}J^{\prime}}^{\hat{\omega}\omega^{\prime}_{+}}\right]Z_{J^{\prime}}^{\omega(\omega^{\prime}_{+},m^{\prime})}\left(\theta,\varphi_{+}\right)\,.$
Recalling the coefficients of the decomposition in Eq. (97) are unique, one
obtains
$\sum_{\hat{J}}C_{\hat{J}J}^{\hat{\omega}\omega_{+}*}C_{\hat{J}J^{\prime}}^{\hat{\omega}\omega^{\prime}_{+}}=\tilde{C}_{JJ^{\prime}}^{\omega_{+}\omega^{\prime}_{+}}\,.$
The desired result, Eq. (94), is then achieved by complex conjugation..

Now, owing to Eq. (94), we are allowed to place the term
$\sum_{\hat{J}}C_{\hat{J}J}^{\hat{\omega}\omega_{+}}C_{\hat{J}J^{\prime}}^{\hat{\omega}\omega^{\prime}_{+}*}$
out of the $\hat{\omega}$ integral, just as in Eq. (85). From this point on
the analysis proceeds in a completely analogous manner (recalling Eq. (93)) to
the previous subsection, and the final result (91) is again obtained, this
time using the generic angular functions
$\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}$ rather than the spherical harmonics
$Y_{\hat{l}\hat{m}}$.

To avoid confusion, we also emphasize that in this final expression for the
mode structure of the Unruh-state HTPF, the modes that appear are the
Eddington modes $f_{\omega lm}^{\text{in}}$ and $f_{\omega lm}^{\text{up}}$,
which are of course separable in terms of _spheroidal harmonics_ (this is
regardless of the nature of the angular functions
$\hat{Z}_{\hat{l}\hat{m}}^{\hat{\omega}}\left(\theta,\varphi_{+}\right)$ that
were chosen earlier in the process).

##  VI Constructing the Unruh state HTPF in the interior of a Kerr BH

In this section we shall finally construct the mode-sum expression for the
Unruh-state HTPF inside a Kerr BH in terms of Eddington modes. We shall follow
here an analogous procedure to the one carried out in Ref. [31] in the RN
case, while noting that the presence of rotation induces some essential
differences. Basically it is the procedure demonstrated already in Sec. V for
the exterior of a Kerr BH, although there are some notable technical
differences. In the first subsection we shall carry out the actual derivation
of the expression for the HTPF and in the second one we shall prove that the
expression is invariant with respect to the choice of the initial angular
functions.

###  VI.1 Mode decomposition of the interior Unruh HTPF

In the analysis to follow, just like in the exterior counterpart presented in
Sec. V, we shall use the $\omega_{+}$-indexed versions (that is, objects
carrying an index $\omega_{+}$ rather than $\omega$) of the Eddington modes
$f_{\omega lm}^{R}$, $f_{\omega lm}^{L}$ and $f_{\omega lm}^{\text{up}}$. The
$\omega_{+}$ version of $f_{\omega lm}^{\text{up}}$ has been introduced in Eq.
(67), and for the interior modes we define in a similar manner:

$f_{\omega_{+}lm}^{\Lambda\left(+\right)}\equiv
f_{\omega\left(\omega_{+},m\right)lm}^{\Lambda},$

where $\Lambda$ is either “ _R_ ” or “ _L_ ”, and
$\omega\left(\omega_{+},m\right)\equiv\omega_{+}+m\Omega_{+}$. In addition, we
shall use the notation

$\rho_{\omega_{+}lm}^{\text{up}\left(+\right)}\equiv\rho_{\omega\left(\omega_{+},m\right)lm}^{\text{up}}$

as the $\omega_{+}$-indexed version of $\rho_{\omega lm}^{\text{up}}$. All
relations and equations containing $f_{\omega lm}^{R}$, $f_{\omega lm}^{L}$,
$f_{\omega lm}^{\text{up}}$ and $\rho_{\omega lm}^{\text{up}}$ are to be
carried from previous sections to this section along with a simple replacement
of the $\omega$-indexed objects with their $\omega_{+}$-indexed counterparts,
as defined above (see also footnote (14)).

We begin with Eqs. (63)-(65), which are valid in the interior as well as the
exterior of the BH, aiming for a mode-sum decomposition of both the _in_ and
_up_ contributions in terms of the interior Eddington modes. Starting with
$G_{U}^{\text{in}}\left(x,x^{\prime}\right)$, we may readily use the relation
between the _in_ Unruh modes and the _right_ Eddington modes, as given in Eq.
(52), which holds throughout the BH interior. Then, Eq. (64) may be written as

$G_{U}^{\text{in}}\left(x,x^{\prime}\right)=\hbar\sum_{l,m}\int_{0}^{\infty}\text{d}\omega\frac{\left|\omega_{+}\right|}{\omega}\left|\tau_{\omega
lm}^{\text{in}}\right|^{2}\left\\{f_{\omega_{+}lm}^{R\left(+\right)}\left(x\right),f_{\omega_{+}lm}^{R\left(+\right)*}\left(x^{\prime}\right)\right\\}\,.$
(98)

The contribution from the _up_ Unruh modes is, as expected (see discussion in
Subsec. III.3), less straightforward to decompose in terms of Eddington modes.
We find it convenient to start by writing
$G_{U}^{\text{up}}\left(x,x^{\prime}\right)$ in Eq. (65) in terms of the two
$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}$ components:
$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{past}}$ and
$g_{\hat{\omega}\hat{l}\hat{m}}^{L}$ (see Eqs. (56)-(57)). Recalling that

$g_{\hat{\omega}\hat{l}\hat{m}}^{\text{up}}\left(x\right)=g_{\hat{\omega}\hat{l}\hat{m}}^{\text{past}}\left(x\right)+g_{\hat{\omega}\hat{l}\hat{m}}^{L}\left(x\right)\,,$

which is valid throughout the united domain, and substituting this relation
into Eq. (65), we readily obtain

$\displaystyle G_{U}^{\text{up}}\left(x,x^{\prime}\right)$
$\displaystyle=\hbar\sum_{\hat{J}}\int_{0}^{\infty}\text{d}\hat{\omega}\left[\left\\{g_{\hat{\omega}\hat{J}}^{\text{past}}\left(x\right),g_{\hat{\omega}\hat{J}}^{\text{past}*}\left(x^{\prime}\right)\right\\}+\left\\{g_{\hat{\omega}\hat{J}}^{L}\left(x\right),g_{\hat{\omega}\hat{J}}^{L*}\left(x^{\prime}\right)\right\\}\right.$
(99)
$\displaystyle\left.+\left\\{g_{\hat{\omega}\hat{J}}^{\text{past}}\left(x\right),g_{\hat{\omega}\hat{J}}^{L*}\left(x^{\prime}\right)\right\\}+\left\\{g_{\hat{\omega}\hat{J}}^{L}\left(x\right),g_{\hat{\omega}\hat{J}}^{\text{past}*}\left(x^{\prime}\right)\right\\}\right]\,,$

where the integration is over the Kruskal frequency $\hat{\omega}$ and we use
the notation previously introduced, $\hat{J}=\left(\hat{l},\hat{m}\right)$.

Aiming for a decomposition of both $g_{\hat{\omega}\hat{J}}^{\text{past}}$ and
$g_{\hat{\omega}\hat{J}}^{L}$ in terms of Eddington modes, we shall follow the
same reasoning as in Sec. V, where we defined coefficients relating the
frequential and angular factors of the Unruh and Eddington modes under
consideration, constrained to the relevant asymptotic null surfaces. The
angular coefficients $C_{\hat{J}J}^{\omega_{+}}$, defined in Eq. (74), will be
utilized in the BH interior exactly as they were in the BH exterior. However,
as we shall see, adjusting the various frequential factors will require
defining an additional set of Fourier coefficients, along with the ones
already defined in the BH exterior. For future use, we rename the
$\alpha_{\hat{\omega}\omega_{+}}$ coefficients, as defined in Eq. (72), by
$\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}$ (adding a superscript “past”).
That is,

$\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}\equiv\int_{-\infty}^{\infty}\text{d}u_{\text{ext}}\,e^{-i\hat{\omega}U\left(u_{\text{ext}}\right)}e^{i\omega_{+}u_{\text{ext}}}\,,$
(100)

and it is explicitly given by (see Eq. (73))

$\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}=\frac{1}{\kappa_{+}}\left(\frac{\hat{\omega}}{\kappa_{+}}\right)^{i\omega_{+}/\kappa_{+}}e^{\pi\omega_{+}/2\kappa_{+}}\Gamma\left(-i\frac{\omega_{+}}{\kappa_{+}}\right)\,.$
(101)

We shall begin with $g_{\hat{\omega}\hat{J}}^{\text{past}}$, emerging from
$H_{\text{past}}$ and vanishing on $H_{L}$ and PNI, hence identical to
$g_{\hat{\omega}\hat{J}}^{\text{up}}$ when restricted to the BH exterior
(compare Eqs. (55) and (57)). This allows us to relate to the analysis carried
out in Sec. V, and replace $g_{\hat{\omega}\hat{J}}^{\text{up}}$ by
$g_{\hat{\omega}\hat{J}}^{\text{past}}$ in the LHS of Eq. (83). Explicitly,
this relates $g_{\hat{\omega}\hat{J}}^{\text{past}}$ and
$f_{\omega_{+}J}^{\text{up}\left(+\right)}$ throughout the BH exterior as
follows:

$\sqrt{\hat{\omega}}\,g_{\hat{\omega}\hat{J}}^{\text{past}}=\frac{1}{2\pi}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\,\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}C_{\hat{J}J}^{\omega_{+}}f_{\omega_{+}J}^{\text{up}\left(+\right)}\,\,,\,\,\,\,\,\,\,\,\,\,r\geq
r_{+}\,.$

Aiming at the BH interior, we carry the above relation over to the common
boundary of the BH exterior and interior – namely the hypersurface $H_{R}$
(the EH). This yields

$\sqrt{\hat{\omega}}\left.g_{\hat{\omega}\hat{J}}^{\text{past}}\right|_{H_{R}}=\frac{1}{2\pi}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\,\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}C_{\hat{J}J}^{\omega_{+}}\left.f_{\omega_{+}J}^{\text{up}\left(+\right)}\right|_{H_{R}}\,.$
(102)

Now, we wish to re-express this in terms of the _interior_ Eddington modes
instead of the exterior _up_ modes. By comparing Eq. (40) with Eq. (43), we
register their relation on $H_{R}$:

$\left.f_{\omega_{+}J}^{\text{up}\left(+\right)}\right|_{H_{R}}=\rho_{\omega_{+}J}^{\text{up}\left(+\right)}\left.f_{\omega_{+}J}^{R\left(+\right)}\right|_{H_{R}}\,.$
(103)

Substituting this into Eq. (102) we find

$\sqrt{\hat{\omega}}\left.g_{\hat{\omega}\hat{J}}^{\text{past}}\right|_{H_{R}}=\frac{1}{2\pi}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\,\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}C_{\hat{J}J}^{\omega_{+}}\rho_{\omega_{+}J}^{\text{up}\left(+\right)}\left.f_{\omega_{+}J}^{R\left(+\right)}\right|_{H_{R}}\,.$
(104)

In addition, both $g_{\hat{\omega}\hat{J}}^{\text{past}}$ and
$f_{\omega_{+}J}^{R\left(+\right)}$ vanish on $H_{L}$ (see Eqs. (57) and
(43)), implying that this relation actually holds throughout the BH interior:

$\sqrt{\hat{\omega}}g_{\hat{\omega}\hat{J}}^{\text{past}}(x)=\frac{1}{2\pi}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\,\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}C_{\hat{J}J}^{\omega_{+}}\rho_{\omega_{+}J}^{\text{up}\left(+\right)}f_{\omega_{+}J}^{R\left(+\right)}(x)\,\,,\,\,\,\,\,\,\,\,\,\,r_{-}\leq
r\leq r_{+}\,.$ (105)

We now proceed similarly with $g_{\hat{\omega}\hat{J}}^{L}$, whose form on
$H_{L}$ is (see Eq. (56))

$\displaystyle\left.g_{\hat{\omega}\hat{J}}^{L}\left(x\right)\right|_{H_{L}}$
$\displaystyle=\frac{Y_{\hat{J}}\left(\theta,\varphi_{+}\right)}{\sqrt{4\pi\hat{\omega}\left(r_{+}^{2}+a^{2}\right)}}e^{-i\hat{\omega}U\left(u_{\text{int}}\right)}\,,$

aiming to relate it to the _left_ Eddington mode, whose form on $H_{L}$ is
(see Eq. (46))

$\left.f_{\omega_{+}J}^{L\left(+\right)}\left(x\right)\right|_{H_{L}}=\frac{Z_{J}^{\omega}\left(\theta,\varphi_{+}\right)}{\sqrt{4\pi\left|\omega_{+}\right|\left(r_{+}^{2}+a^{2}\right)}}e^{i\omega_{+}u_{\text{int}}}\,.$

This resembles the case of decomposing $g_{\hat{\omega}\hat{J}}^{\text{up}}$
in terms of $f_{\omega_{+}J}^{\text{up}\left(+\right)}$ on $H_{\text{past}}$,
as carried out in Sec. V, with a modification in the frequential factors: here
we have $e^{-i\hat{\omega}U\left(u_{\text{int}}\right)}$ and
$e^{i\omega_{+}u_{\text{int}}}$ as the Unruh and Eddington frequential
factors, respectively, instead of
$e^{-i\hat{\omega}U\left(u_{\text{ext}}\right)}$ and
$e^{-i\omega_{+}u_{\text{ext}}}$, respectively.

We thus define new Fourier coefficients relating
$e^{-i\hat{\omega}U\left(u_{\text{int}}\right)}$ and
$e^{i\omega_{+}u_{\text{int}}}$, which we shall denote by
$\alpha_{\hat{\omega}\omega_{+}}^{L}$, given by the inverse Fourier transform:

$\alpha_{\hat{\omega}\omega_{+}}^{L}\equiv\int_{-\infty}^{\infty}\text{d}u_{\text{int}}e^{-i\hat{\omega}U\left(u_{\text{int}}\right)}e^{-i\omega_{+}u_{\text{int}}}\,.$
(106)

This integral may be found by inspecting Eq. (106) alongside Eq. (100),
changing the integration variable from $u_{\text{int}}$ to $-u_{\text{ext}}$
(which also implies $U(u_{\text{int}})\mapsto-U(u_{\text{ext}}$), see Eqs.
(9)-(10)), which results in
$\alpha_{\hat{\omega}\omega_{+}}^{L}=\alpha_{\hat{\omega}\left(-\omega_{+}\right)}^{\text{past}*}$.
Applying this relation to Eq. (101) then yields 212121This expression is also
given in Eq. (3.5) in Ref. [31], however, there are some notational
differences in this case that need to be bridged: Basically, the notation in
Ref. [31] is related to ours as prescribed in footnote 15, namely
$\tilde{\omega}\mapsto\omega_{+},\omega\mapsto\hat{\omega}$. However, in the
RHS of Eq. (107) our $\omega_{+}$ should actually be mapped to
$-\tilde{\omega}$ (this extra change of sign has to do with the issue
discussed earlier in footnote 9).:

$\alpha_{\hat{\omega}\omega_{+}}^{L}=\frac{1}{\kappa_{+}}\left(\frac{\hat{\omega}}{\kappa_{+}}\right)^{i\omega_{+}/\kappa_{+}}e^{-\pi\omega_{+}/2\kappa_{+}}\Gamma\left(-i\frac{\omega_{+}}{\kappa_{+}}\right)\,.$
(107)

Comparing Eq. (107) with Eq. (101), we find
$\alpha_{\hat{\omega}\omega_{+}}^{L}$ in terms of
$\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}$:

$\alpha_{\hat{\omega}\omega_{+}}^{L}=\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}e^{-\pi\omega_{+}/\kappa_{+}}$
(108) 222222One might be concerned about the notable difference between our
Eq. (108), describing the relation between
$\alpha_{\hat{\omega}\omega_{+}}^{L}$ and
$\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}$, and the rightmost side of Eq.
(3.5) in Ref. [31] (which states that
$\alpha_{\omega\tilde{\omega}}^{L}=\alpha_{\omega\tilde{\omega}}^{\text{past}*}$).
To reconcile this difference, note that when transforming our
$\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}$ and
$\alpha_{\hat{\omega}\omega_{+}}^{L}$ to the notation of Ref. [31], the former
becomes $\alpha_{\omega\tilde{\omega}}^{\text{past}}$, but
$\alpha_{\hat{\omega}\omega_{+}}^{L}$ is translated to
$\alpha_{\omega\left(-\tilde{\omega}\right)}^{L}$ (see footnotes 15 and 21, as
well as 9). Indeed, considering the relation between
$\alpha_{\omega\left(-\tilde{\omega}\right)}^{L}$ and
$\alpha_{\omega\tilde{\omega}}^{\text{past}}$ therein would yield the analog
of Eq. (108) (namely,
$\alpha_{\omega\left(-\tilde{\omega}\right)}^{L}=\alpha_{\omega\tilde{\omega}}^{\text{past}}e^{-\pi\tilde{\omega}/\kappa_{+}}$).

.

The two $H_{L}$-projected functions
$\left.g_{\hat{\omega}\hat{J}}^{L}\left(x\right)\right|_{H_{L}}$ and
$\left.f_{\omega_{+}J}^{L\left(+\right)}\left(x\right)\right|_{H_{L}}$ may now
be related using the conversion coefficients
$\alpha_{\hat{\omega}\omega_{+}}^{L}$ and $C_{\hat{J}J}^{\omega_{+}}$ (in full
analogy with Eq. (83), replacing
$\alpha_{\hat{\omega}\omega_{+}}=\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}$
by $\alpha_{\hat{\omega}\omega_{+}}^{L}$):

$\sqrt{\hat{\omega}}\left.g_{\hat{\omega}\hat{J}}^{L}\right|_{H_{L}}=\frac{1}{2\pi}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\alpha_{\hat{\omega}\omega_{+}}^{L}C_{\hat{J}J}^{\omega_{+}}\left.f_{\omega_{+}J}^{L\left(+\right)}\right|_{H_{L}}\,.$
(109)

Since both the _left_ Unruh and Eddington modes vanish on $H_{R}$, this
relation between $g_{\hat{\omega}\hat{J}}^{L}$ and
$f_{\omega_{+}J}^{L\left(+\right)}$ actually holds throughout the BH interior:

$\sqrt{\hat{\omega}}g_{\hat{\omega}\hat{J}}^{L}\left(x\right)=\frac{1}{2\pi}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\alpha_{\hat{\omega}\omega_{+}}^{L}C_{\hat{J}J}^{\omega_{+}}f_{\omega_{+}J}^{L\left(+\right)}(x)\,\,,\,\,\,\,\,\,\,\,r_{-}\leq
r\leq r_{+}\,.$ (110)

Now, substituting Eqs. (105) and (110) into Eq. (99),
$G_{U}^{\text{up}}\left(x,x^{\prime}\right)$ may be written as

$G_{U}^{\text{up}}\left(x,x^{\prime}\right)=\hbar\left(I_{RR}+I_{LL}+I_{RL}+I_{LR}\right)\,,$
(111)

where

$\displaystyle I_{RR}\equiv$ (112)
$\displaystyle\sum_{\hat{J}}\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{4\pi^{2}\hat{\omega}}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}C_{\hat{J}J}^{\omega_{+}}\rho_{\omega_{+}J}^{\text{up}\left(+\right)}\sum_{J^{\prime}}\int_{-\infty}^{\infty}\text{d}\omega^{\prime}_{+}\sqrt{\left|\omega^{\prime}_{+}\right|}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{\text{past}*}C_{\hat{J}J^{\prime}}^{\omega^{\prime}_{+}*}\rho_{\omega^{\prime}_{+}J^{\prime}}^{\text{up}\left(+\right)*}\left\\{f_{\omega_{+}J}^{R\left(+\right)}\left(x\right),f_{\omega^{\prime}_{+}J^{\prime}}^{R\left(+\right)*}\left(x^{\prime}\right)\right\\}\,,$
$\displaystyle I_{LL}$
$\displaystyle\equiv\sum_{\hat{J}}\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{4\pi^{2}\hat{\omega}}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\alpha_{\hat{\omega}\omega_{+}}^{L}C_{\hat{J}J}^{\omega_{+}}\sum_{J^{\prime}}\int_{-\infty}^{\infty}\text{d}\omega^{\prime}_{+}\sqrt{\left|\omega^{\prime}_{+}\right|}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{L*}C_{\hat{J}J^{\prime}}^{\omega^{\prime}_{+}*}\left\\{f_{\omega_{+}J}^{L\left(+\right)}\left(x\right),f_{\omega^{\prime}_{+}J^{\prime}}^{L\left(+\right)*}\left(x^{\prime}\right)\right\\}\,,$
(113)

$\displaystyle I_{RL}$
$\displaystyle\equiv\sum_{\hat{J}}\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{4\pi^{2}\hat{\omega}}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}C_{\hat{J}J}^{\omega_{+}}\rho_{\omega_{+}J}^{\text{up}\left(+\right)}\sum_{J^{\prime}}\int_{-\infty}^{\infty}\text{d}\omega^{\prime}_{+}\sqrt{\left|\omega^{\prime}_{+}\right|}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{L*}C_{\hat{J}J^{\prime}}^{\omega^{\prime}_{+}*}\left\\{f_{\omega_{+}J}^{R\left(+\right)}\left(x\right),f_{\omega^{\prime}_{+}J^{\prime}}^{L\left(+\right)*}\left(x^{\prime}\right)\right\\}\,,$
(114)

$\displaystyle I_{LR}$
$\displaystyle\equiv\sum_{\hat{J}}\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{4\pi^{2}\hat{\omega}}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\alpha_{\hat{\omega}\omega_{+}}^{L}C_{\hat{J}J}^{\omega_{+}}\sum_{J^{\prime}}\int_{-\infty}^{\infty}\text{d}\omega^{\prime}_{+}\sqrt{\left|\omega^{\prime}_{+}\right|}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{\text{past}*}C_{\hat{J}J^{\prime}}^{\omega^{\prime}_{+}*}\rho_{\omega^{\prime}_{+}J^{\prime}}^{\text{up}\left(+\right)*}\left\\{f_{\omega_{+}J}^{L\left(+\right)}\left(x\right),f_{\omega^{\prime}_{+}J^{\prime}}^{R\left(+\right)*}\left(x^{\prime}\right)\right\\}\,.$
(115)

We rearrange Eqs. (112)-(115) into a form similar to that of Eq. (85):

$\displaystyle I_{RR}$
$\displaystyle=\frac{1}{4\pi^{2}}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\sum_{J^{\prime}}\int_{-\infty}^{\infty}\text{d}\omega^{\prime}_{+}\sqrt{\left|\omega^{\prime}_{+}\right|}\rho_{\omega_{+}J}^{\text{up}\left(+\right)}\rho_{\omega^{\prime}_{+}J^{\prime}}^{\text{up}\left(+\right)*}\cdot$
$\displaystyle\left\\{f_{\omega_{+}J}^{R\left(+\right)}\left(x\right),f_{\omega^{\prime}_{+}J^{\prime}}^{R\left(+\right)*}\left(x^{\prime}\right)\right\\}\sum_{\hat{J}}C_{\hat{J}J}^{\omega_{+}}C_{\hat{J}J^{\prime}}^{\omega^{\prime}_{+}*}\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{\hat{\omega}}\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{\text{past}*}\,,$
(116) $\displaystyle I_{LL}$
$\displaystyle=\frac{1}{4\pi^{2}}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\sum_{J^{\prime}}\int_{-\infty}^{\infty}\text{d}\omega^{\prime}_{+}\sqrt{\left|\omega^{\prime}_{+}\right|}\cdot$
$\displaystyle\left\\{f_{\omega_{+}J}^{L\left(+\right)}\left(x\right),f_{\omega^{\prime}_{+}J^{\prime}}^{L\left(+\right)*}\left(x^{\prime}\right)\right\\}\sum_{\hat{J}}C_{\hat{J}J}^{\omega_{+}}C_{\hat{J}J^{\prime}}^{\omega^{\prime}_{+}*}\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{\hat{\omega}}\alpha_{\hat{\omega}\omega_{+}}^{L}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{L*}\,,$
(117) $\displaystyle I_{RL}$
$\displaystyle=\frac{1}{4\pi^{2}}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\sum_{J^{\prime}}\int_{-\infty}^{\infty}\text{d}\omega^{\prime}_{+}\sqrt{\left|\omega^{\prime}_{+}\right|}\rho_{\omega_{+}J}^{\text{up}\left(+\right)}\cdot$
$\displaystyle\left\\{f_{\omega_{+}J}^{R\left(+\right)}\left(x\right),f_{\omega^{\prime}_{+}J^{\prime}}^{L\left(+\right)*}\left(x^{\prime}\right)\right\\}\sum_{\hat{J}}C_{\hat{J}J}^{\omega_{+}}C_{\hat{J}J^{\prime}}^{\omega^{\prime}_{+}*}\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{\hat{\omega}}\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{L*}\,,$
(118) $\displaystyle I_{LR}$
$\displaystyle=\frac{1}{4\pi^{2}}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\sqrt{\left|\omega_{+}\right|}\sum_{J^{\prime}}\int_{-\infty}^{\infty}\text{d}\omega^{\prime}_{+}\sqrt{\left|\omega^{\prime}_{+}\right|}\rho_{\omega^{\prime}_{+}J^{\prime}}^{\text{up}\left(+\right)*}\cdot$
$\displaystyle\left\\{f_{\omega_{+}J}^{L\left(+\right)}\left(x\right),f_{\omega^{\prime}_{+}J^{\prime}}^{R\left(+\right)*}\left(x^{\prime}\right)\right\\}\sum_{\hat{J}}C_{\hat{J}J}^{\omega_{+}}C_{\hat{J}J^{\prime}}^{\omega^{\prime}_{+}*}\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{\hat{\omega}}\alpha_{\hat{\omega}\omega_{+}}^{L}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{\text{past}*}\,.$
(119)

At this point it becomes clear (after renaming the indices
$\omega_{+}\leftrightarrow\omega^{\prime}_{+}$ and $J\leftrightarrow
J^{\prime}$ in the last equation) that $I_{LR}=I_{RL}^{*}$.

In order to proceed as we did in the BH exterior, we need to perform the
integrals on the rightmost side of each equality, of the form
$\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{\hat{\omega}}\alpha_{\hat{\omega}\omega_{+}}^{\Lambda_{1}}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{\Lambda_{2}*}$
with $\Lambda_{1,2}$ either “past” or “ _L_ ”. The result of the integral
$\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{\hat{\omega}}\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{\text{past}*}$
has already been given in Eq. (86). Next, we use the relation between
$\alpha_{\hat{\omega}\omega_{+}}^{L}$ and
$\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}$ given in Eq. (108), along with
the previously mentioned integral (Eq. (86)), to find

$\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{\hat{\omega}}\alpha_{\hat{\omega}\omega_{+}}^{L}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{L*}=e^{-\pi\left(\omega_{+}+\omega^{\prime}_{+}\right)/\kappa_{+}}\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{\hat{\omega}}\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{\text{past}*}=\frac{4\pi^{2}}{\omega_{+}}\frac{1}{e^{2\pi\omega_{+}/\kappa_{+}}-1}\delta\left(\omega_{+}-\omega^{\prime}_{+}\right)$
(120)

and

$\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{\hat{\omega}}\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{L*}=e^{-\pi\omega^{\prime}_{+}/\kappa_{+}}\int_{0}^{\infty}\frac{\text{d}\hat{\omega}}{\hat{\omega}}\alpha_{\hat{\omega}\omega_{+}}^{\text{past}}\alpha_{\hat{\omega}\omega^{\prime}_{+}}^{\text{past}*}=\frac{4\pi^{2}}{\omega_{+}}\frac{1}{e^{\pi\omega_{+}/\kappa_{+}}-e^{-\pi\omega_{+}/\kappa_{+}}}\delta\left(\omega_{+}-\omega^{\prime}_{+}\right)\,.$
(121)

Performing a computation very similar to the one carried out in Eq. (87) in
the BH exterior, substituting the integrals in Eqs. (86), (120) and (121) into
Eqs. (116), (117) and (118) respectively, and making use of Eq. (79), we
obtain, after translating the exponential factors into corresponding
hyperbolic-geometric functions:

$\displaystyle I_{RR}$
$\displaystyle=\frac{1}{2}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\frac{\left|\omega_{+}\right|}{\omega_{+}}\left[\coth\left(\frac{\pi\omega_{+}}{\kappa_{+}}\right)+1\right]\left[\left|\rho_{\omega_{+}J}^{\text{up}\left(+\right)}\right|^{2}\left\\{f_{\omega_{+}J}^{R\left(+\right)}\left(x\right),f_{\omega{}_{+}J}^{R\left(+\right)*}\left(x^{\prime}\right)\right\\}\right]\,,$
(122) $\displaystyle I_{LL}$
$\displaystyle=\frac{1}{2}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\frac{\left|\omega_{+}\right|}{\omega_{+}}\left[\coth\left(\frac{\pi\omega_{+}}{\kappa_{+}}\right)-1\right]\left[\left\\{f_{\omega_{+}J}^{L\left(+\right)}\left(x\right),f_{\omega{}_{+}J}^{L\left(+\right)*}\left(x^{\prime}\right)\right\\}\right]\,,$
(123)

and

$I_{RL}=I_{LR}^{*}=\frac{1}{2}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\frac{\left|\omega_{+}\right|}{\omega_{+}}\left[\text{cosech}\left(\frac{\pi\omega_{+}}{\kappa_{+}}\right)\right]\left[\rho_{\omega_{+}J}^{\text{up}\left(+\right)}\left\\{f_{\omega_{+}J}^{R\left(+\right)}\left(x\right),f_{\omega{}_{+}J}^{L\left(+\right)*}\left(x^{\prime}\right)\right\\}\right]\,,$
(124)

where $\text{cosech}\equiv 1/\sinh$.

Next, we would like to fold these three integrals through $\omega_{+}=0$, just
as we did in Sec. V for the BH exterior. To this end, we first note that in
all three equations (122)-(124), the RHS is of the general form

$I\equiv\frac{1}{2}\sum_{J}\int_{-\infty}^{\infty}\text{d}\omega_{+}\frac{\left|\omega_{+}\right|}{\omega_{+}}H\left(\omega_{+}\right)F_{\omega_{+}J}\left(x,x^{\prime}\right)=\frac{1}{2}\sum_{l,m}\int_{-\infty}^{\infty}\text{d}\omega_{+}\,\text{sign}\left(\omega_{+}\right)H\left(\omega_{+}\right)F_{\omega_{+}lm}\left(x,x^{\prime}\right)\,,$
(125)

where $H\left(\omega_{+}\right)$ and
$F_{\omega_{+}lm}\left(x,x^{\prime}\right)$ respectively stand for the first
and second terms in square brackets in each of these three equations.
Furthermore, we once again note that since
$\left(\omega,m\right)\mapsto\left(-\omega,-m\right)$ is identical to
$\left(\omega_{+},m\right)\mapsto\left(-\omega_{+},-m\right)$, we may rewrite
Eq. (47) as

$f_{\left(-\omega_{+}\right)l\left(-m\right)}^{\Lambda\left(+\right)}=\left(-1\right){}^{m}f_{\omega_{+}lm}^{\Lambda\left(+\right)*}$
(126)

(with $\Lambda$ either “ _R_ ” or “ _L_ ”) and the invariance relation of
$\rho^{\text{up}}$ included in Eq. (30) as

$\rho_{\left(-\omega_{+}\right)l\left(-m\right)}^{\text{up}\left(+\right)}=\rho_{\omega_{+}lm}^{\text{up}\left(+\right)*}\,.$
(127)

Eqs. (126) and (127) may now be used to show that the function
$F_{\omega_{+}lm}\left(x,x^{\prime}\right)$ in all three cases (122)-(124)
satisfies

$F_{\left(-\omega_{+}\right)lm}\left(x,x^{\prime}\right)=F_{\omega_{+}l\left(-m\right)}^{*}\left(x,x^{\prime}\right)\,.$

Then, summing over $l$ and $m$ (recalling
$\sum_{l,m}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}$), we obtain

$\sum_{l,m}F_{\left(-\omega_{+}\right)lm}\left(x,x^{\prime}\right)=\sum_{l,m}F_{\omega_{+}lm}^{*}\left(x,x^{\prime}\right)=\sum_{J}F_{\omega_{+}J}^{*}\left(x,x^{\prime}\right)\,.$

Therefore, by folding the $\omega_{+}$-integral in Eq. (125), we obtain the
following explicit form:

$I=\frac{1}{2}\sum_{J}\int_{0}^{\infty}\text{d}\omega_{+}\left[H\left(\omega_{+}\right)F_{\omega_{+}J}\left(x,x^{\prime}\right)-H\left(-\omega_{+}\right)F_{\omega_{+}J}^{*}\left(x,x^{\prime}\right)\right]\,.$
(128)

Note that the function $H\left(\omega_{+}\right)$ potentially has both a part
that is an even function of $\omega_{+}$and a part that is an odd function.
Then, from Eq. (128) it is clear that the even part of
$H\left(\omega_{+}\right)$ leaves out the imaginary part of
$F_{\omega_{+}J}\left(x,x^{\prime}\right)$, while the odd part of
$H\left(\omega_{+}\right)$ leaves out the real part of
$F_{\omega_{+}J}\left(x,x^{\prime}\right)$. Applying this general folding
structure to Eqs. (122)-(124) and noticing that
$F_{\omega_{+}J}\left(x,x^{\prime}\right)$ is actually real for $I_{RR}$ and