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The scattering coefficients in the above expressions can be calculated through
numerical means; here we use the Teukolsky 0.3.0 package of the Black Hole
Perturbation Toolkit [25]. Since this Mathematica package is only designed to
compute exterior scattering coefficients, we have made adaptations to the code
to extend computations to the spacetime region between the inner and outer
horizons; details on these modifications can be found in Appendix A.

The scalar product of Eq. (18) can then be evaluated along the past null
Cauchy hypersurface where the Unruh state is initialized. The end goal is the
computation of Eq. (17), the vacuum expectation value of the particle number
operator for an observer either at infinity, the event horizon, or the ingoing
or outgoing portions of the inner horizon. These spectral number distributions
will be labeled ${\langle N^{+}_{\text{ext}}\rangle_{\omega\ell m}}$,
${\langle N^{-}_{\text{ext}}\rangle_{\omega\ell m}}$, ${\langle
N^{+}_{\text{int}}\rangle_{\omega\ell m}}$, and ${\langle
N^{-}_{\text{int}}\rangle_{\omega\ell m}}$ for the respective modes of Eqs.
(70)–(73).

The analysis proceeds almost identically to that of the spherical case in Ref.
[6], with one small but crucial difference: the Kerr scattering potential of
Eq. (10) asymptotically approaches a different constant value at infinity
compared to the values at the event horizon and the inner horizon; see Eq.
(65). Thus, observer modes that are initialized as

$\phi_{\text{ob}}\sim\frac{\text{e}^{i\omega r^{*}}}{R\sqrt{4\pi\omega}}$ (74)

at future null infinity will be back-scattered into the form

$\phi_{\text{ob}}\sim\frac{\text{e}^{i\omega_{+}r^{*}}}{R_{+}\sqrt{4\pi\omega}}$
(75)

along the past horizon, and so forth.

The details for the calculation of the resulting number operator vacuum
expectation values from Eq. (17) are given in Appendix B. The result, up to a
normalization factor, is

$\displaystyle\langle N^{+}_{\text{ext}}\rangle_{\omega\ell m}$
$\displaystyle=\left(\frac{\omega_{+}}{\omega}\right)\frac{\left|\mathcal{T}^{+}_{\text{ext},\omega}\right|^{2}}{\text{e}^{\frac{2\pi}{\varkappa_{+}}\omega_{+}}-1},$
(76a) $\displaystyle\langle N^{-}_{\text{ext}}\rangle_{\omega\ell m}$
$\displaystyle=\frac{\left|\mathcal{R}^{-}_{\text{ext},\omega}\right|^{2}}{\text{e}^{\frac{2\pi}{\varkappa_{+}}\omega}-1},$
(76b) $\displaystyle\langle N^{-}_{\text{int}}\rangle_{\omega\ell m}$
$\displaystyle=\left(\frac{\omega_{+}+m\Omega_{-}}{\omega}\right)\frac{\left|\mathcal{T}^{-}_{\text{int},\omega}\mathcal{R}^{-}_{\text{ext},\omega}-\mathcal{R}^{-}_{\text{int},\omega}\text{e}^{\frac{\pi}{\varkappa_{+}}(\omega_{+}+m\Omega_{-})}\right|^{2}}{\text{e}^{\frac{2\pi}{\varkappa_{+}}(\omega_{+}+m\Omega_{-})}-1},$
(76c) $\displaystyle\langle N^{+}_{\text{int}}\rangle_{\omega\ell(-m)}$
$\displaystyle=\left(\frac{\omega_{+}+m\Omega_{-}}{\omega}\right)\frac{\left|\mathcal{T}^{+}_{\text{int},\omega}-\mathcal{R}^{+}_{\text{int},\omega}\mathcal{R}^{-}_{\text{ext},\omega}\text{e}^{\frac{\pi}{\varkappa_{+}}(\omega_{+}+m\Omega_{-})}\right|^{2}}{\text{e}^{\frac{2\pi}{\varkappa_{+}}(\omega_{+}+m\Omega_{-})}-1}+\left(\frac{\omega+m\Omega_{-}}{\omega}\right)\left|\mathcal{R}^{+}_{\text{int},\omega}\mathcal{T}^{-}_{\text{ext},\omega}\right|^{2},$
(76d)

with the surface gravity $\varkappa_{+}$ of Eq. (22), the horizon-limit
frequency $\omega_{+}$ of Eq. (66), and the transmission and reflection
coefficients $\mathcal{T}^{\pm}_{\text{int,ext}}$ and
$\mathcal{R}^{\pm}_{\text{int,ext}}$ of Eqs. (70)–(73). Note that in the limit
${a\to 0}$, the expressions in Eqs. (76) reduce to the same Schwarzschild
expressions obtained in Ref. [6].

The physical interpretation of the vacuum expectation values from Eqs. (76) is
that they measure the spectral emission rate; i.e., the mean number of quanta
in the mode with frequency $\omega$ and angular mode numbers $\ell$ and $m$,
seen by a freely falling observer at each respective location, per the
observer’s proper time $\tau_{\text{ob}}$.

One may at first sight worry that for exterior scattering, when
${\omega<m\Omega_{+}}$ (and similarly, ${\omega<m(\Omega_{+}-\Omega_{-})}$ for
the interior), the frequency prefactors in the above expressions for ${\langle
N_{\text{ext}}^{+}\rangle_{\omega\ell m}}$ and ${\langle
N_{\text{int}}^{\pm}\rangle_{\omega\ell m}}$ become negative. This is
connected to the well-known phenomenon of superradiance, in which the
transmission probability in a rotating system becomes negative and the
absorption probability exceeds unity, so that scattered waves gain amplitude
upon reflection and extract energy from the black hole [26]. However, the
aforementioned negative terms are exactly canceled by the Planckian terms in
the denominator of each expression, which also become negative in the same
superradiant regimes. Therefore, the expected number of particles seen by the
observer will always remain positive.

###  III.2 Scalar modes

First, consider the Hawking radiation from massless scalar mode excitations,
with spin ${s=0}$. The more general bosonic cases (${s=1}$ for photons and
${s=2}$ for gravitons) will be considered in the next section.

Figure 10: Graybody spectra for the Hawking $s$-modes seen by an infalling
observer at infinity (upper left panel), the event horizon (upper right
panel), and the Cauchy horizon (lower left panel for an ingoing observer and
lower right panel for an outgoing observer) in the Kerr spacetime with black
hole spins ${a/M}$ = 0.1, 0.3, 0.5, 0.7, 0.9, 0.96, 0.99, and 0.999 (labeled
with respective colors from dark blue to yellow). Dashed curves show the
corresponding positive-valued blackbody spectra with temperatures
$\varkappa_{+}/(2\pi)$ (upper two panels) and $\varkappa_{-}/(2\pi)$ (lower
two panels) from Eq. (22), while the solid curves are evaluated numerically
from Eqs. (76).

The Hawking spectra for the lowest set of modes (${\ell=0,\ m=0}$) are shown
in Fig. 10. These $s$-wave spectra are computed numerically for a variety of
black hole spin parameters seen by the four observers represented in Eqs.
(76). First, the standard graybody spectrum seen asymptotically far from the
black hole is shown in the upper left panel of Fig. 10. The distribution is
plotted as a spectral intensity, which scales as ${\omega^{3}\langle
N\rangle}$, so that a Planckian blackbody would appear with a quadratic power
law at low frequencies and an exponential drop at high frequencies. Such a
blackbody, with a temperature given by the surface gravity
${\varkappa_{+}/(2\pi)}$, is plotted for each spin parameter with a dashed
curve. All the numerically-evaluated solid curves agree with the blackbody
estimations at high frequencies (i.e., the geometric optics limit). However,
at low frequencies, the graybody spectra differ from their blackbody
counterparts by a power law index of 2, in agreement with the analytic
prediction of Starobinsky in the limit ${\omega\to 0}$ [26].

While the spectrum of Hawking radiation seen by someone looking inward from
asymptotically far away contains entirely straightforward graybody deviations
at low frequencies, the spectrum seen by someone crossing the event horizon
contains graybody deviations at high frequencies, as shown in the upper right
panel of Fig. 10. The spectral intensity is still roughly the same order of
magnitude as that seen at infinity, in accordance with assertion $\\#$1 in
Sec. I. But at higher frequencies, the spectrum drops to zero much more
quickly than one might expect from blackbody predictions. The reason for the
dropoff is that in the geometric optics limit, fewer outgoing modes
originating from an Unruh emitter at the horizon will be reflected and return
to the observer; instead, more will escape as rays to infinity instead of
being back-scattered as waves. Since the Hawking spectrum seen by an observer
looking outward from the event horizon is determined entirely by these
reflected modes (and not from transmitted ingoing modes originating from past
null infinity, which do not exhibit the characteristic exponential peeling),
Hawking radiation detected at the event horizon is suppressed at high
frequencies.

Before discussing the Hawking spectrum seen by someone at the inner horizon,
one feature present in all panels of Fig. 10 is the suppression of Hawking
radiation for faster-spinning black holes. As the spin parameter $a$ is
increased and the curves change color from dark blue to yellow, one may note
that the higher-$a$ curves have overall lower intensities. The faster a black
hole spins, the colder it becomes, regardless of where the observer lies
within the spacetime.

The Hawking spectra seen by an observer at the Cauchy horizon are shown in the
lower two panels of Fig. 10. The lower left panel corresponds to an ingoing
observer looking outward at the sky above (the left portion of $r_{-}$ in Fig.
9), while the lower right panel corresponds to an outgoing observer looking
inward at the past horizon below (the right portion of $r_{-}$ in Fig. 9).
These spectra are plotted alongside the dashed blackbody curves (after taking
the absolute value) for the negative temperature given by the surface gravity
of the inner horizon, $\varkappa_{-}/(2\pi)$. At low frequencies, all the
curves approach the same quadratic power law, but instead of simply falling
off exponentially as the frequency $\omega$ is increased, the curves continue
to climb orders of magnitude higher than any positive-temperature blackbody
would allow. However, the physically measurable spectral intensity does not
contain an ultraviolet divergence for non-zero rotation. Eventually, as
suspected in the Reissner-Nordström case [6], the exponential Wien tail
dominates as ${\omega\to\infty}$ so that the spectral intensity returns to
zero. But as the spin $a$ decreases and the inner horizon approaches the
${r=0}$ singularity, Hawking radiation is able to access higher and higher
frequencies before reaching an ultraviolet cutoff. The Schwarzschild limit
${a/M\to 0}$ is not shown in these lower two panels, since the spectral
intensity in that case becomes infinite at all frequencies.

Figure 11: Hawking spectral intensities for higher $\ell$\- and $m$-modes,
seen by an infalling observer at infinity (upper left panel), the event
horizon (upper right panel), and the Cauchy horizon (lower two panels) in the
Kerr spacetime as a function of the black hole spin parameter ${a/M}$. All
modes are evaluated at a frequency of ${\omega\cdot M=0.1}$. Each individual
$m$-mode for a given $\ell$ are presented for the observer at infinity, while
the $m$-modes are summed for each $\ell$ for the other three observers.

It should be noted that the Cauchy horizon Hawking spectra shown in Fig. 10
are not _a priori_ expected to diverge. The divergent negative Hawking
temperatures seen at the Cauchy horizon correspond to ingoing observers
looking inward ($\kappa^{+}$ for ${P_{r}>0}$) and outgoing observers looking
outward ($\kappa^{-}$ for ${P_{r}<0}$). In contrast, Fig. 10 shows observers
at the Cauchy horizon looking in the direction opposite the Penrose blueshift
singularity—ingoing observers looking outward and outgoing observers looking
inward. Calculations for these latter two scenarios would involve the inner
product of the Unruh emitter’s Kruskal modes with the observer’s Fourier-
decomposed and back-propagated Kruskal modes [27], which is nonetheless
expected to yield an infinite spectral intensity at all frequencies. What Fig.
10 shows is that in addition to the classical (and likely also semiclassical)
blueshift singularity, an observer at the Cauchy horizon will see the entire
sky around them glowing brightly with Hawking radiation.

Beyond the $s$-wave approximation, the Hawking spectral intensities for
higher-$\ell$ modes are shown in Fig. 11. Instead of showing entire spectra as
functions of the frequency $\omega$, these spectral intensities are evaluated
at a specific mid-range frequency (${\omega\cdot M=0.1}$) so that the
dependence on the spin parameter $a$ can be plotted more clearly.

For an observer asymptotically far from the black hole, the higher-$\ell$
spectral intensities are shown in the upper left panel of Fig. 11. As the
black hole spin $a$ is taken to zero, all azimuthal $m$-modes within a given
$\ell$ mode converge to the same value, as expected. These intensities in the
low-$a$ limit drop off as $\ell$ increases, so that the lowest angular mode
dominates the Hawking spectrum. In particular, the $\ell$-modes are spaced
apart by three to five orders of magnitude, in agreement with the approximate
behavior predicted by Starobinsky [26]:

$\left|\mathcal{T}^{+}_{\text{ext}}\right|^{2}\propto\frac{(\omega\cdot
M)^{2\ell+1}(\ell!)^{4}}{[(2\ell)!]^{2}[(2\ell+1)!!]^{2}}.$ (77)

Additionally, note that in the upper left panel of Fig. 11, the green curves
(corresponding to positive $m$, with ${m=\ell}$ located highest in each group)
always lie above the corresponding ${m=0}$ curves, while the magenta curves
(corresponding to negative $m$, with ${m=-\ell}$ located lowest in each group)
always lie below the corresponding ${m=0}$ curves. Physically, Hawking
particles are always being emitted preferentially with the same angular
momentum as the black hole, so that over time, the black hole will tend to
spin down as the Hawking particles carry away excess angular momentum [28].

In the upper right panel of Fig. 11, the Hawking spectral intensity is shown
for an observer crossing the event horizon, yielding quite different behavior
than that of an observer far away. While the ${\ell=0}$ mode dominates for an
asymptotically distant observer, all higher-$\ell$ modes are present when the
observer is in a regime where they are close enough to access more angular
information than $s$-waves. While this panel plots the sum over all $m$-modes
in a given $\ell$-mode, it should be noted that as ${\ell\to\infty}$, all the
modes with ${m=0}$ tend to a constant value, just as quickly as all the modes
${m=0}$ modes for an observer at infinity tend to zero—note the relationship
between $\left|\mathcal{T}^{+}_{\text{ext},\omega}\right|^{2}$ and
$\left|\mathcal{R}^{-}_{\text{ext},\omega}\right|^{2}$ in Eq. (94).

The lower two panels of Fig. 11 similarly show equal contributions from all
higher-$\ell$ modes in the full Hawking spectrum seen by an infalling observer
near the Cauchy horizon. Just as expected, the spectral intensity decreases as
the black hole spin $a$ increases, and the intensity increases or decreases
monotonically with $\ell$, except in the near-extremal case for an outgoing
observer looking inward. The higher-$\ell$ behavior demonstrated in Fig. 11
matches that of the Reissner-Nordström model [6]. Note that as ${a\to 0}$, the
observed Hawking spectral intensity diverges as the inner horizon meets the
${r=0}$ singularity, and as ${a\to M}$, the observed Hawking radiation
vanishes as the inner horizon meets the outer horizon to create an extremal,
zero-temperature spacetime.

###  III.3 Higher-spin modes

Figure 12: Graybody spectra for the electromagnetic [solid curves,
$(s,\ell,m)=(1,1,0)$] and gravitational [dotted curves, $(s,\ell,m)=(2,2,0)$]
components of the Hawking radiation seen by an inertial observer at Boyer-
Lindquist radius $r_{\text{ob}}$. All spectra are evaluated for a Kerr black
hole with angular momentum ${a/M=0.1}$. These higher-spin spectra are
qualitatively similar to their scalar counterparts from Fig. 10, except in the
case of an outgoing observer at the inner horizon, who sees an infrared-
divergent spectrum.

While scalar modes (with spin 0) are commonplace in calculations of
semiclassical effects in curved spacetimes due to the scalar wave equation’s
simplicity and the “physical enough” interpretation of modeling a single
degree of freedom from a photon field, one can obtain more physically
meaningful results by considering the higher-spin generalization of the wave
equation given by Eq. (80).

The Hawking particles that will be considered here are photons from an
electromagnetic field (spin-1) and gravitons from a gravitational field
(spin-2). For all integer spins, the spectra of Eqs. (76) remain Planckian,
with the only modifications arising from the scattering coefficients
calculated from each spin’s corresponding wave equation [29].

The spectra for Hawking radiation from the lowest $\ell$\- and $m$-modes of
spin-1 and spin-2 fields are shown in Fig. 12. In this plot, it can be seen
that the higher-spin fields radiate with roughly the same spectra as in the
scalar case of Fig. 10 for exterior observers. When
${r_{\text{ob}}\to\infty}$, the spin-1 and spin-2 spectra peak at a slightly
higher frequency than the spin-0 spectrum (not shown), but they also trail off
in the infrared regime with a steeper power-law slope than the spin-0
spectrum. Similarly, the spectra for an observer at the event horizon appear
almost identical for different values of spin.

All the spectra in Fig. 12 are shown for a black hole with angular momentum
${a/M=0.1}$. Other values of $a$ yield qualitatively similar results (just as
in Fig. 10), but the low value of $a$ here is chosen since it gives the most
pronounced effects, especially given the proximity of the inner horizon to the
central singularity as ${a\to 0}$.

For an observer at the inner horizon, only the spectrum seen by an outgoing
observer looking inward is shown, since the spectra seen by an ingoing
observer there for positive values of the spin weight $s$ are suppressed by
the vanishing of the interior reflection coefficients dictated by the
Heaviside function in Eq. (92). If instead one chooses ${s=-1}$ and ${s=-2}$
(i.e., the outgoing radiative parts of the field; see Footnote 2), as is
common in Kerr perturbation calculations for numerical feasibility, the inner
horizon spectra will appear even more ultraviolet-divergent than in the scalar
case. But even for the positive spin weights shown in Fig. 12, the spectrum of
radiation produced from electromagnetic and gravitational modes is infrared-
divergent, indicating that an outgoing observer looking inward at the
exponentially redshifting and dimming surface of the start that collapsed long
ago will see that surface glow more and more brightly in the infrared as they
approach the Cauchy horizon.

##  IV Discussion

Hawking radiation is an observer-dependent phenomenon. Here we have
generalized the Bogoliubov coefficient calculation of Hawking [1] to determine
the effective temperature of semiclassical radiation seen in the vacuum state
of the locally inertial rest frame of an arbitrary observer within the Kerr
spacetime. Hawking found that if the observer is placed asymptotically far
from the black hole, they will see a small amount of approximately thermal
radiation emerge from the vacuum; here we explore the vast parameter space of
various classes of observers both inside and outside of the black hole, all of
whom will generally see a non-zero amount of Hawking radiation, sometimes
thermal, sometimes not. This radiation appears to originate from the black
hole shadow; i.e., the dimming, redshifting surface of the star that collapsed
long ago to form the black hole.

The main goal of this study has been to extend the results of prior studies of
Hawking radiation in Schwarzschild [5] and Reissner-Nordström [6] black holes
to the more astrophysically relevant case of rotating Kerr black holes. While
the prior studies benefited from spherical symmetry and needed only to examine
freely falling radial observers at different radii $r$, Kerr black holes
possess only azimuthal symmetry, so that the results differ as an observer’s
latitude $\theta$, angular momentum $\mathcal{L}$, and Carter constant
$\mathcal{K}$ are varied.

Here we have examined five classes of observers: those falling along the axis
of rotation (Sec. II.4.1); in a horizostationary orbit (Sec. II.4.2); falling
along the equatorial plane with zero angular momentum (Sec. II.5.1); falling
along the equatorial plane with zero energy (Sec. II.5.2); and crossing
through either the event horizon or the inner Cauchy horizon with arbitrary
finite orbital parameters (Sec. III). In all cases, a non-zero amount of
Hawking radiation is seen regardless of where the observer is. More
importantly, from an analysis of the effective temperature, whenever an
observer reaches the inner horizon, the Hawking radiation will become
blueshift-divergent (i.e., negative-temperature) in at least one direction in
their field of view. Such a divergence was not explicitly calculated in the
spectral analysis of Sec. III; instead, the cases where the Eddington-
Finkelstein modes lead to a 1D scattering problem demonstrate that an inner
horizon observer looking in the _opposite_ direction from the anticipated
semiclassical divergence will still see a high-energy glow of Hawking
radiation.

We additionally conclude here that the amount of Hawking radiation seen by
different Kerr observers remains relatively independent from the spin of the
quantum field, the particular $\ell m$-mode of the quantum state, and the
direction in which the observer looks. Though all of these variables yield
differing results for the Hawking radiation calculations, the key takeaways
concerning the ubiquity of non-zero semiclassical particle creation and an
inner horizon divergence remain unchanged.

One of the biggest outstanding questions that one may ask concerning this
analysis of Hawking radiation in the framework of semiclassical gravity is how
the radiation back-reacts on the spacetime. Here we have kept the spacetime
geometry fixed, but presumably if enough radiation is present, the particles
produced will possess a gravitational field of their own that can change the
underlying metric. Usually, one assumes that these back-reaction effects are
negligibly small and the Kerr metric can still be used as a valid
approximation of the spacetime geometry for astrophysical, rotating black
holes. We find this assumption to be generally true for observers outside of
the event horizon. However, close to the inner horizon, the generally
divergent behavior of observed Hawking radiation suggests that the Kerr metric
is not semiclassically self-consistent there. The back-reaction for
astrophysically relevant perturbations is likely to form a strong, spacelike
singularity just above the inner horizon [13], though other mathematical
predictions also exist in the literature, such as a weak, null singularity
[30] or a rapid implosion toward the formation of a compact horizonless object
[10].

The obvious problem with attempting to analyze the effects of back-reaction
from Hawking radiation in the present framework is that each observer sees a
different amount of Hawking radiation, but they all exist in the same
background metric. Even though the radiation is completely real from the
perspective of any individual observer, the very definition of a particle
depends on an observer’s frame of reference, a concept which seems at first
glance completely at odds with the central claim in semiclassical gravity that
particles arising from quantum fields provide the source of curvature for a
global, classical spacetime metric.

Which observer is the one to see the actual radiation that contributes to the
underlying spacetime geometry? For a full quantum calculation, such a question
appears to be meaningless, for the solution should involve a superposition of
all possible classical metrics, each sourced by the energy-momentum seen by a
different observer. In lieu of this daunting task, the usual approach in
semiclassical gravity is to construct an averaged version of the net energy-
momentum of a quantum field, which can be cast into a Lorentz-covariant form
[31, 32]. This quantity, the renormalized vacuum expectation value ${\langle
T^{\mu\nu}\rangle_{\text{ren}}}$ of the field’s stress-energy tensor (RSET),
is then guaranteed to feed back into the spacetime geometry in the standard
classical way.

However, the RSET is only an averaged quantity and cannot provide all the
information that an observer is able to access about the state of a quantum
field in a curved spacetime. Classically, the stress-energy tensor involves an
integral with each particle’s energy-momentum over a Lorentz-invariant pseudo-
scalar volume element:

$T^{\mu\nu}=\int p^{\mu}p^{\nu}f(\bm{x},\bm{p})\frac{g\
d^{3}p}{p^{0}(2\pi\hbar)^{3}},$ (78)

where $g$ is the number of spin states of the particle and $f$ is the
dimensionless occupation number, which specifies the number $dN$ of particles
with 4-position $x^{\mu}$ and 4-momentum $p^{\mu}$ within the Lorentz-
invariant six-dimensional volume of phase space ${d^{3}x\ d^{3}p}$. The
particle number $N$, directly related to this occupation number $f$, is
precisely the object of study throughout this work. While the RSET is a
difficult object to calculate even for the most symmetric spacetimes, one of
the goals of this work is to show that ${\langle N\rangle}$, the more
fundamental object, is entirely straightforward to calculate in the locally
inertial frame of any observer.

In conclusion, by examining the expectation value of the number operator for a
quantum field placed over a Kerr spacetime, we have analyzed and extended the
same key ideas that were anticipated in Sec. I from prior studies of
spherically symmetric black holes: (1) The Hawking radiation seen by generic
observers passing the vicinity of the event horizon has a negligibly weak
graybody spectrum, (2) the Hawking radiation seen at the Cauchy horizon
possesses a divergent effective temperature for all the classes of observers
examined, (3) the Hawking radiation originating from different directions in
the sky varies considerably for different classes of observers, and (4) the
effective Hawking temperature for certain observers can become negative even
outside of the event horizon, though an ultraviolet divergence in the Hawking
spectrum is not seen for the limited cases considered here. From these results
it is clear that the Kerr metric cannot be trusted in its full global form in
the semiclassical approximation, as a result of the diverging quantum
radiation that would be observed at the inner Cauchy horizon.

##  Appendix A Interior mode scattering in the MST method

The Black Hole Perturbation Toolkit (BHPT) [25] makes use of the Mano-Suzuki-
Takasugi (MST) method [33, 34] to calculate the scattering coefficients for a
Klein-Gordon field with spin weight $s$ in the Kerr spacetime. In this
Appendix, we review the implementation of this method for the scattering of
exterior modes between the event horizon and spatial infinity, and we extend
the analysis to include the scattering of interior modes between the event
horizon and the inner Cauchy horizon.

The notation used throughout this Appendix is self-consistent but may differ
from the notation used in the main body of the paper; instead it is chosen to
match that of the BHPT and its relevant references. The most notable change is
that $R$ here no longer represents the scale length defined below Eq. (II.1)
and instead represents the Teukolsky radial mode function defined via the mode
expansion

$\phi_{\omega\ell m}=\frac{R_{\omega\ell m}(r)\ {}_{s}S^{\omega}_{\ell
m}(\theta)\ \text{e}^{im\varphi-i\omega t}}{\sqrt{4\pi\omega}},$ (79)

vis-à-vis Eq. (7) (therefore ${\psi_{\omega\ell}(r)=R_{\omega\ell
m}(r)\sqrt{r^{2}+a^{2}}}$). The modes ${{}_{s}S^{\omega}_{\ell m}(\theta)}$
represent spin-weighted spheroidal wave functions [18], while the radial modes
${R_{\omega\ell m}(r)}$ satisfy the homogeneous radial Teukolsky equation
[14]:

$\displaystyle\left(\frac{K_{\omega m}^{2}-2is(r-M)K_{\omega
m}}{\Delta}+4is\omega r-\lambda_{\omega\ell m}\right)R_{\omega\ell m}$
$\displaystyle+\Delta^{-s}\frac{d}{dr}\left(\Delta^{s+1}\frac{dR_{\omega\ell
m}}{dr}\right)=0,$ (80)

for a black hole of mass $M$ and spin $a$, vis-à-vis Eq. (9). The horizon
function $\Delta$ is defined in the same way as in the text following Eq.
(II.1), the function ${K_{\omega m}\equiv(r^{2}+a^{2})\omega-ma}$, and the
constant ${\lambda_{\omega\ell m}\equiv\mathscr{E}_{\ell
m}-2ma\omega+a^{2}\omega^{2}-s(s+1)}$, where $\mathscr{E}_{\ell m}$ is the
eigenvalue of the spin-weighted spheroidal wave function
${{}_{s}S^{\omega}_{\ell m}(\theta)}$ [35] and reduces to ${\mathscr{E}_{\ell
m}\to\ell(\ell+1)}$ in the Schwarzschild limit.

The BHPT allows for the following boundary value problem to be solved:
consider two sets of modes $R_{\omega\ell m}^{\text{in}}$ (initialized on past
null infinity) and $R_{\omega\ell m}^{\text{up}}$ (initialized on the past
horizon), which asymptotically approach the boundary values

$R^{\text{in}}_{\omega\ell
m}\to\begin{cases}B^{\text{ref}}_{\text{ext}}r^{-1-2s}\text{e}^{i\omega
r^{*}}+B^{\text{inc}}_{\text{ext}}r^{-1}\text{e}^{-i\omega
r^{*}},&r\to\infty\\\
B^{\text{trans}}_{\text{ext}}|\Delta|^{-s}\text{e}^{-i\omega_{+}r^{*}},&r\to
r_{+}\\\ B^{\text{ref}}_{\text{int}}\
\text{e}^{i\omega_{-}r^{*}}+B^{\text{trans}}_{\text{int}}|\Delta|^{-s}\text{e}^{-i\omega_{-}r^{*}},&r\to
r_{-}\end{cases},$ (81) $R^{\text{up}}_{\omega\ell
m}\to\begin{cases}C^{\text{trans}}_{\text{ext}}r^{-1-2s}\text{e}^{i\omega
r^{*}},&r\to\infty\\\ C^{\text{up}}_{\text{ext}}\
\text{e}^{i\omega_{+}r^{*}}+C^{\text{ref}}_{\text{ext}}|\Delta|^{-s}\text{e}^{-i\omega_{+}r^{*}},&r\to
r_{+}\\\ C^{\text{trans}}_{\text{int}}\
\text{e}^{i\omega_{-}r^{*}}+C^{\text{ref}}_{\text{int}}|\Delta|^{-s}\text{e}^{-i\omega_{-}r^{*}},&r\to
r_{-}\end{cases},$ (82)

where $\omega_{\pm}$ is given by Eq. (66) and the complex constants $B$ and
$C$ are scattering coefficients for either the reflection, incidence, or
transmission of the mode waves. The tortoise coordinate $r^{*}$ is chosen to
be

$r^{*}\equiv
r+\frac{1}{2\varkappa_{+}}\ln\left|\frac{r-r_{+}}{2M}\right|+\frac{1}{2\varkappa_{-}}\ln\left|\frac{r-r_{-}}{2M}\right|,$
(83)

with the surface gravity $\varkappa_{\pm}$ (which is negative at the inner
horizon) given by Eq. (22). Note that the forward-propagated modes
$R_{\omega\ell m}$ and scattering coefficients $B$ and $C$ defined here are
different from the backward-propagated modes $\psi_{\text{ob}}$ and scattering
coefficients $\mathcal{T}$ and $\mathcal{R}$ used in the main text; the
relation between the two will be given at the end of this Appendix.

Conservation of the Wronskian in the scalar case leads to the following
relations between the scattering coefficients:

$\displaystyle\left|B^{\text{ref}}_{\text{ext}}\right|^{2}+\frac{\omega_{+}(r_{+}^{2}+a^{2})}{\omega}\left|B^{\text{trans}}_{\text{ext}}\right|^{2}=\left|B^{\text{inc}}_{\text{ext}}\right|^{2},$
$\displaystyle\left|C^{\text{ref}}_{\text{ext}}\right|^{2}+\frac{\omega}{\omega_{+}(r_{+}^{2}+a^{2})}\left|C^{\text{trans}}_{\text{ext}}\right|^{2}=\left|C^{\text{up}}_{\text{ext}}\right|^{2},$
$\displaystyle\frac{C^{\text{trans}}_{\text{ext}}}{C^{\text{up}}_{\text{ext}}}=\frac{\omega_{+}(r_{+}^{2}+a^{2})}{\omega}\frac{B^{\text{trans}}_{\text{ext}}}{B^{\text{inc}}_{\text{ext}}},$
$\displaystyle\frac{C^{\text{trans}}_{\text{ext}}}{C^{\text{ref}}_{\text{ext}}}=-\frac{\omega_{+}(r_{+}^{2}+a^{2})}{\omega}\frac{B^{\text{trans*}}_{\text{ext}}}{B^{\text{ref*}}_{\text{ext}}},$
$\displaystyle\left|B^{\text{ref}}_{\text{int}}\right|^{2}+\frac{\omega_{+}(r_{+}^{2}+a^{2})}{\omega_{-}(r_{-}^{2}+a^{2})}\left|B^{\text{trans}}_{\text{ext}}\right|^{2}=\left|B^{\text{trans}}_{\text{int}}\right|^{2},$
$\displaystyle\left|C^{\text{up}}_{\text{int}}\right|^{2}-\frac{\omega_{-}(r_{-}^{2}+a^{2})}{\omega_{+}(r_{+}^{2}+a^{2})}\left(\left|C^{\text{trans}}_{\text{int}}\right|^{2}-\left|C^{\text{ref}}_{\text{int}}\right|^{2}\right)=\left|C^{\text{ref}}_{\text{ext}}\right|^{2},$
$\displaystyle
B^{\text{trans}}_{\text{int}}C^{\text{trans}}_{\text{int}}-B^{\text{ref}}_{\text{int}}C^{\text{ref}}_{\text{int}}=\frac{\omega_{+}(r_{+}^{2}+a^{2})}{\omega_{-}(r_{-}^{2}+a^{2})}B^{\text{trans}}_{\text{ext}}C^{\text{up}}_{\text{int}},$
$\displaystyle
B^{\text{trans*}}_{\text{int}}C^{\text{ref}}_{\text{int}}-B^{\text{ref*}}_{\text{int}}C^{\text{trans}}_{\text{int}}=\frac{\omega_{+}(r_{+}^{2}+a^{2})}{\omega_{-}(r_{-}^{2}+a^{2})}B^{\text{trans*}}_{\text{ext}}C^{\text{ref}}_{\text{ext}},$
(84)

where a superscript asterisk (∗) here and elsewhere denotes complex
conjugation, except in the case of the tortoise coordinate $r^{*}$. In what
follows, we will focus on the _in_ modes; the scattering coefficients for the
_up_ modes can be obtained from the _in_ modes through the above Wronskian
conditions. The main results for the application of the MST method in the
exterior portion of the spacetime will be quoted here; for a more complete
review, see Ref. [36].

The solutions to the radial wave Eq. (80) belong to a class of functions known
as confluent Heun functions. However, the mathematical properties of these
functions (especially their asymptotic behavior at each horizon) has been a
mathematical enigma, and to this day, the central two-point connection problem
for these functions still has no explicit solution. However, Svartholm [37]
and Erdélyi [38, 39] early on discovered an integral transform for confluent
Heun functions with a hypergeometric kernel, so that they could be expressed
in a series representation as the sum of (the more tractable) hypergeometric
functions.

Under the MST method, two infinite series expansions of the solutions to Eq.
(80) are found that are valid in different but overlapping regimes. The first
of these expansions, in terms of ordinary hypergeometric functions
${{}_{2}F_{1}}$, is valid for all finite values of $r$ but breaks down as
${r\to\infty}$. Defining the dimensionless parameters444As a reminder, the
parameters defined here and used throughout Appendix A should be treated
independently from the notation of the main body of the paper and instead are
chosen to align with the notation used by MST [33, 34]. In particular,
$\kappa$ does not represent the effective temperature, $\epsilon$ does not
represent the adiabatic control function, and $\tau$ does not represent proper
time.

$\displaystyle
x\equiv\frac{\omega}{\epsilon\kappa}(r_{+}-r),\quad\epsilon\equiv
2M\omega,\quad\kappa\equiv\sqrt{1-\left(\frac{a}{M}\right)^{2}},$
$\displaystyle\epsilon_{\pm}\equiv\frac{\epsilon\pm\tau}{2},\quad\tau\equiv\frac{\epsilon-m\left(\frac{a}{M}\right)}{\kappa},$
(85)

such that the outer ($+$) and inner ($-$) horizon radii are given by
${r_{\pm}\equiv(1\pm\kappa)M}$, this first series is:

$\displaystyle R^{\text{in}}_{\nu}$ $\displaystyle=\text{e}^{i\epsilon\kappa
x}|-x|^{-s-i\epsilon_{+}}|1-x|^{i\epsilon_{-}}\sum_{n=-\infty}^{\infty}a_{n}^{\nu}(s)$
$\displaystyle\times{}_{2}F_{1}(n+\nu+1-i\tau,-n-\nu-i\tau;1-s-2i\epsilon_{+};x),$
(86)

where the parameter $\nu$, called the renormalized angular momentum, is a
generalization of $\ell$ to non-integer values that is fixed by requiring that
the series solution to the Teukolsky equation converges.

Likewise, the second series expansion can be written in terms of confluent
hypergeometric functions and is valid for asymptotically large values of $r$
but fails as ${r\to r_{+}}$. By matching these two expansions, the
coefficients $a_{n}^{\nu}(s)$ in both expansions will satisfy the same three-
term recurrence relation that can be solved numerically to find the minimal
solution.

As ${r\to r_{+}}$ (or equivalently, as ${x\to 0}$), Eq. (A) reduces to

$R^{\text{in}}_{\nu}\to|-x|^{-s-i\epsilon_{+}}\sum_{n=-\infty}^{\infty}a_{n}^{\nu}(s),$
(87)

while Eq. (81) can be written in terms of the parameters of Eq. (85) as

$R^{\text{in}}_{\omega\ell m}\to
B^{\text{trans}}_{\text{ext}}\left(\frac{\epsilon\kappa}{\omega}\right)^{-2s}|-x|^{-s}\text{
e}^{-i\epsilon_{+}\left(\ln|-x|+\kappa+\frac{2\kappa\ln\kappa}{1+\kappa}\right)}.$
(88)

The coefficient $B^{\text{trans}}_{\text{ext}}$ can then be read off by
equating the two expressions from Eqs. (87) and (88). A similar matching
process leads to expressions for the scattering parameters at infinity. The
resulting formulae for $B^{\text{trans}}_{\text{ext}}$,
$B^{\text{inc}}_{\text{ext}}$, and $B^{\text{ref}}_{\text{ext}}$ are given
respectively by Eqs. (167)–(169) of Ref. [36].

Now, consider how the above formalism may be extended to the black hole’s
interior. As ${r\to r_{-}}$ (or equivalently, as ${x\to 1}$), based on the
asymptotic behavior of the hypergeometric function with argument unity, Eqs.
(A) and (81) respectively reduce to

$\displaystyle R^{\text{in}}_{\nu}\to\text{e}^{i\epsilon\kappa}$
$\displaystyle\Bigg{[}|1-x|^{i\epsilon_{-}}\sum_{n=-\infty}^{\infty}a_{n}^{\nu}(s)\frac{\Gamma(1-s-2i\epsilon_{+})\Gamma(-s-2i\epsilon_{-})}{\Gamma(-n-\nu-
s-i\epsilon)\Gamma(n+\nu+1-s-i\epsilon)}H(-s)$
$\displaystyle+|1-x|^{-s-i\epsilon_{-}}\sum_{n=-\infty}^{\infty}a_{n}^{\nu}(s)\frac{\Gamma(1-s-2i\epsilon_{+})\Gamma(s+2i\epsilon_{-})}{\Gamma(-n-\nu-i\tau)\Gamma(n+\nu+1-i\tau)}H(s)\Bigg{]},$
(89) $R^{\text{in}}_{\omega\ell m}\to B^{\text{ref}}_{\text{int}}\text{
exp}\left[i\epsilon_{-}\left(\ln|1-x|-\kappa-\frac{2\kappa\ln\kappa}{1-\kappa}\right)\right]+B^{\text{trans}}_{\text{int}}\left(\frac{\epsilon\kappa}{\omega}\right)^{-2s}|1-x|^{-s}\text{
exp}\left[-i\epsilon_{-}\left(\ln|1-x|-\kappa-\frac{2\kappa\ln\kappa}{1-\kappa}\right)\right],$
(90)

where ${H(s)}$ is the Heaviside step function defined by

$H(s)\equiv\begin{cases}1,&s\geq 0\\\ 0,&s<0\end{cases}.$ (91)

Eq. (89) breaks down for scalar modes when either ${\omega=0}$ or ${a=0}$;
these cases, whose modes asymptotically scale as ${\ln(1-x)}$, must be treated
separately.

Matching Eqs. (89) and (90) leads to expressions for the internal scattering
parameters:

$\displaystyle B^{\text{ref}}_{\text{int}}=H(-s)\text{
e}^{i\kappa\left[\epsilon+\epsilon_{-}\left(1+\frac{2\ln\kappa}{1-\kappa}\right)\right]}$
$\displaystyle\times\sum_{n=-\infty}^{\infty}a_{n}^{\nu}(s)\frac{\Gamma(1-s-2i\epsilon_{+})\Gamma(-s-2i\epsilon_{-})}{\Gamma(-n-\nu-
s-i\epsilon)\Gamma(n+\nu+1-s-i\epsilon)},$ (92) $\displaystyle
B^{\text{trans}}_{\text{int}}$
$\displaystyle=H(s)\left(\frac{\epsilon\kappa}{\omega}\right)^{2s}\text{
e}^{i\kappa\left[\epsilon-\epsilon_{-}\left(1+\frac{2\ln\kappa}{1-\kappa}\right)\right]}$
$\displaystyle\times\sum_{n=-\infty}^{\infty}a_{n}^{\nu}(s)\frac{\Gamma(1-s-2i\epsilon_{+})\Gamma(s+2i\epsilon_{-})}{\Gamma(-n-\nu-i\tau)\Gamma(n+\nu+1-i\tau)}.$
(93)

The step functions in Eqs. (92) and (93) imply that close to the inner
horizon, only the non-negative (non-positive) spin-weighted components of
ingoing (outgoing) waves survive, since these are the radiative (i.e.,
dominant propagating; see Footnote 2) components.

The expressions above for the inner horizon scattering coefficients can then
be implemented in Mathematica alongside the rest of the BHPT’s Teukolsky
package to compute the relevant _in_ scattering coefficients. In order to
connect these results to the calculations in the main body of the paper, a
transformation must be made between the two sets of scattering coefficients
($B$ and $C$ on the one hand and $\mathcal{T}$ and $\mathcal{R}$ on the
other). One of the main problems that gives rise to the need for such a non-
trivial matching is that the main text’s back-propagated modes
${{}^{\text{ext,int}}\psi^{\pm}_{\text{ob}}}$ are initialized on the future
null boundaries, while this Appendix’s forward-propagated modes
$R^{\text{in,up}}_{\omega\ell m}$ are initialized on the past null boundaries.

To find the back-propagated scattering coefficients, first note that through
the conservation of the Wronskian, the transmission and reflection
coefficients $\mathcal{T}^{\pm}_{\text{ext,int}}$ and
$\mathcal{R}^{\pm}_{\text{ext,int}}$ must satisfy the normalization conditions

$\displaystyle\left|\mathcal{R}^{\pm}_{\text{ext},\omega}\right|^{2}+\left|\mathcal{T}^{\pm}_{\text{ext},\omega}\right|^{2}\left(\frac{\omega\mp
m\Omega_{+}}{\omega}\right)$ $\displaystyle=1,$ (94a)
$\displaystyle\left(\left|\mathcal{R}^{\pm}_{\text{int},\omega}\right|^{2}-\left|\mathcal{T}^{\pm}_{\text{int},\omega}\right|^{2}\right)\left(\frac{\omega_{+}+m\Omega_{-}}{\omega}\right)$
$\displaystyle=1.$ (94b)

Consider first the exterior set of modes used to calculate $\langle
N^{\pm}_{\text{ext}}\rangle_{\omega\ell m}$, encoded by a family of observers
asymptotically close to future null infinity and the event horizon. Through
time reversal, these modes ${{}^{\text{ext}}\psi_{\text{ob}}^{\pm}}$ map to
the modes $R^{\text{in,up}}_{\omega\ell m}$ by the transformation
${(\omega,m)\mapsto(-\omega,-m)}$, which corresponds to taking the complex
conjugate of the scattering coefficients, since

$R_{(-\omega)\ell(-m)}=R^{*}_{\omega\ell m},\qquad
B^{\text{trans,ref}}_{(-\omega)\ell(-m)}=B^{\text{trans,ref*}}_{\omega\ell
m}.$ (95)

By matching the asymptotic relations for the complete set of modes
$R^{\text{in}}_{\omega\ell m}$ and $R^{\text{up}}_{\omega\ell m}$ in the
spin-0 limit of Eqs. (81) and (82) with the appropriately rescaled set of
exterior modes ${{}^{\text{ext}}\psi^{-\text{*}}_{\text{ob}}}$ and
${{}^{\text{ext}}\psi^{+\text{*}}_{\text{ob}}}$ from Eqs. (70) and (71),
respectively, one arrives at the following equalities, after fixing
${B^{\text{inc}}_{\text{ext}}=1}$ and
${C^{\text{up}}_{\text{ext}}=1/\sqrt{r_{+}^{2}+a^{2}}}$:

$\displaystyle\mathcal{T}^{+}_{\text{ext},\omega}=B^{\text{trans*}}_{\text{ext},\omega}\sqrt{r_{+}^{2}+a^{2}},$
(96a)
$\displaystyle\mathcal{R}^{+}_{\text{ext},\omega}=B^{\text{ref*}}_{\text{ext},\omega},$
(96b)
$\displaystyle\mathcal{T}^{-}_{\text{ext},\omega_{+}}=C^{\text{trans*}}_{\text{ext},\omega},$
(96c)
$\displaystyle\mathcal{R}^{-}_{\text{ext},\omega_{+}}=C^{\text{ref*}}_{\text{ext},\omega}\sqrt{r_{+}^{2}+a^{2}}.$
(96d)

Notice the addition of the mode frequency subscripts in the above scattering
coefficients that help highlight a key difference between the ${(B,C)}$
coefficients and the ${(\mathcal{T},\mathcal{R})}$ coefficients—the scattering
process defined by the former is initialized with the frequency eigenmodes of
the wave equation, while the scattering process of the latter is initialized
so that the observer always sees a frequency $\omega$.

Now consider the interior modes used to calculate $\langle
N^{\pm}_{\text{int}}\rangle_{\omega\ell m}$, encoded by a family of observers
reaching the ingoing and outgoing portions of the inner horizon (together with
observers asymptotically close to future null infinity, to form a complete
Cauchy slice). Since the global scattering process now depends on three
singular points and the interior scattering potential is dynamical, the
backward-propagated modes will not map trivially onto the forward-propagated
modes by time reversal. One may instead consider the transformation
${(M,r)\mapsto(-M,-r)}$ as in Ref. [9], which leaves the radial wave Eq. (9)
unchanged aside from a swapping of the asymptotic regimes at the inner and
outer horizons. However, since that transformation leaves the irregular
singular point at ${r\to\infty}$ unchanged, the forward- and backward-
propagated modes must transform into linear combinations of one another.

In order to solve for the interior scattering coefficients, define
coefficients $\alpha^{\pm}_{\text{in,up}}$ that form linear combinations of
the modes $R^{\text{in}}_{\omega\ell m}$ and $R^{\text{up}}_{\omega\ell m}$.
These linear combinations can be asymptotically equated to the modes
${{}^{\text{int}}\psi^{-}_{\text{ob}}}$ initialized along the ingoing portion
of the future Cauchy slice and the modes
${{}^{\text{int}}\psi^{+\text{*}}_{\text{ob}}+{}^{\text{ext}}\psi^{+}_{\text{ob}}}$
initialized along the outgoing portions of the future Cauchy slice:

$\displaystyle\left(\alpha_{\text{in}}^{-}R^{\text{in}}_{\omega\ell
m}+\alpha_{\text{up}}^{-}R^{\text{up}}_{\omega\ell
m}\right)\sqrt{r^{2}+a^{2}}$
$\displaystyle={}^{\text{int}}\psi^{-}_{\text{ob}},$ (97a)
$\displaystyle\left(\alpha_{\text{in}}^{+}R^{\text{in}}_{\omega\ell
m}+\alpha_{\text{up}}^{+}R^{\text{up}}_{\omega\ell
m}\right)\sqrt{r^{2}+a^{2}}$
$\displaystyle={}^{\text{int}}\psi^{+\text{*}}_{\text{ob}}+{}^{\text{ext}}\psi^{+}_{\text{ob}}$
(97b)

Matching the asymptotic relations for the above sets of modes along the inner
horizon leads to the linear coefficient values

$\displaystyle\alpha_{\text{in}}^{+}$
$\displaystyle=\frac{C^{\text{ref}}_{\text{int}}}{D},$
$\displaystyle\alpha_{\text{up}}^{+}=-\frac{B^{\text{trans}}_{\text{int}}}{D},$
(98a) $\displaystyle\alpha_{\text{in}}^{-}$
$\displaystyle=-\frac{C^{\text{trans}}_{\text{int}}}{D},$
$\displaystyle\alpha_{\text{up}}^{-}=\frac{B^{\text{ref}}_{\text{int}}}{D},$
(98b)

where

$D\equiv\left(B^{\text{ref}}_{\text{int}}C^{\text{ref}}_{\text{int}}-B^{\text{trans}}_{\text{int}}C^{\text{trans}}_{\text{int}}\right)\sqrt{r_{-}^{2}+a^{2}}.$
(99)

The scattering coefficients can then be found by matching the asymptotic
relations along the event horizon and at infinity:

$\displaystyle\mathcal{T}^{-}_{\text{int},\omega_{-}}=\left(\alpha^{-}_{\text{in},\omega}B^{\text{trans}}_{\text{ext},\omega}+\alpha^{-}_{\text{up},\omega}C^{\text{ref}}_{\text{ext},\omega}\right)\sqrt{r_{+}^{2}+a^{2}},$
(100a)
$\displaystyle\mathcal{R}^{-}_{\text{int},\omega_{-}}=\alpha^{-}_{\text{up},\omega}C^{\text{up}}_{\text{int},\omega}\sqrt{r_{+}^{2}+a^{2}},$
(100b)
$\displaystyle\mathcal{T}^{+}_{\text{int},\omega_{-}}=\alpha^{+\text{*}}_{\text{up},\omega}C^{\text{up*}}_{\text{int},\omega}\sqrt{r_{+}^{2}+a^{2}},$
(100c)
$\displaystyle\mathcal{R}^{+}_{\text{int},\omega_{-}}=\left(\alpha^{+\text{*}}_{\text{in},\omega}B^{\text{trans*}}_{\text{ext},\omega}+\alpha^{+\text{*}}_{\text{up},\omega}C^{\text{ref*}}_{\text{ext},\omega}\right)\sqrt{r_{+}^{2}+a^{2}},$
(100d)
$\displaystyle\mathcal{T}^{-}_{\text{ext},\omega_{+}}=\frac{\alpha^{-}_{\text{in},\omega}B^{\text{inc}}_{\text{ext},\omega}}{\mathcal{T}^{-}_{\text{int},\omega_{-}}},$
(100e)
$\displaystyle\mathcal{R}^{-}_{\text{ext},\omega_{+}}=\frac{\alpha^{-}_{\text{up},\omega}C^{\text{up}}_{\text{ext},\omega}}{\mathcal{T}^{-}_{\text{int},\omega_{-}}}\sqrt{r_{+}^{2}+a^{2}},$
(100f)
$\displaystyle\mathcal{T}^{+}_{\text{ext},\omega}=\alpha^{+}_{\text{up},\omega}C^{\text{up}}_{\text{ext},\omega}\sqrt{r_{+}^{2}+a^{2}}-\mathcal{R}^{+\text{*}}_{\text{int},\omega_{-}}\mathcal{R}^{-\text{*}}_{\text{ext},\omega_{+}},$
(100g)
$\displaystyle\mathcal{R}^{+}_{\text{ext},\omega}=\alpha^{+}_{\text{in},\omega}B^{\text{inc}}_{\text{ext},\omega}-\mathcal{R}^{+\text{*}}_{\text{int},\omega_{-}}\mathcal{T}^{-\text{*}}_{\text{ext},\omega_{+}}.$
(100h)

Note that, unlike in the exterior case of Eqs. (96), the $B^{\text{inc}}$ and
$C^{\text{up}}$ coefficients from Eqs. (100) do not need to be fixed since all
the $B$ and $C$ coefficients are written in a normalization-free form. Thus,
one may retain the default normalization choice ${B^{\text{trans}}=1}$ used by
the BHPT.

Once the scattering coefficients $B$ and $C$ are computed with the help of the
BHPT, Eqs. (96) can be used to calculate the back-scattering coefficients
$\mathcal{T}$ and $\mathcal{R}$ used in Eqs. (76a)–(76b), while Eqs. (100) can
be used to calculate the back-scattering coefficients used in Eqs.
(76c)–(76d).

##  Appendix B Evaluation of Bogoliubov coefficient scalar products

In this Appendix, details are given for the calculation of the inner products
of Eqs. (17) and (18) leading to the number operators of Eqs. (76). Focus will
be placed on the scalar (spin-0) case, though the final result holds true for
any integer-spin modes.

For the scalar product along past null infinity ($\mathscr{I}^{-}$), where the
ingoing Eddington-Finkelstein coordinate $v$ runs from ${-\infty}$ to
$\infty$, one may choose

$d\Sigma\ n^{\mu}\sqrt{-g_{\Sigma}}\ \partial_{\mu}=dvd(\cos\theta)d\varphi
R^{2}\partial_{v},$ (101)

while for the scalar product along the past horizon
($\cal{H}_{\text{past}}={\cal{H}_{\text{past}}^{\text{ext}}\cup\cal{H}_{\text{past}}^{\text{int}}}$),
where the outgoing Kruskal-Szekeres coordinate $U$ runs from ${-\infty}$ to 0
in the interior and from $0$ to $\infty$ in the exterior, one has

$d\Sigma\ n^{\mu}\sqrt{-g_{\Sigma}}\
\partial_{\mu}=dUd(\cos\theta)d\varphi_{+}R_{+}^{2}\partial_{U}.$ (102)

The modes to be evaluated along these null hypersurfaces are those of the
emitter:

$\phi_{\text{em}}\to\begin{cases}\displaystyle\frac{\text{e}^{i\bar{m}\varphi}S^{\bar{\omega}}_{\bar{\ell}\bar{m}}(\theta)}{\sqrt{4\pi\bar{\omega}}R}\
\text{e}^{-i\bar{\omega}v},&\mathscr{I}^{-}\\\ \\\
\displaystyle\frac{\text{e}^{i\bar{m}\varphi_{+}}S^{\bar{\omega}}_{\bar{\ell}\bar{m}}(\theta)}{\sqrt{4\pi\bar{\omega}}R_{+}}\
\text{e}^{-i\bar{\omega}U},&\cal{H}_{\text{past}}\end{cases},$ (103)

and those of one of four classes of observers, initialized either at infinity,
at the event horizon, or at the left or right leg of the inner horizon:

$\phi_{\text{ob}}\to\begin{cases}\displaystyle\hat{\mathcal{S}}_{v}\frac{\text{e}^{\pm
im\varphi}S^{\omega}_{\ell m}(\theta)}{\sqrt{4\pi\omega}R}\ \text{e}^{\mp
i\hat{\omega}v},&\mathscr{I}^{-}\\\ \\\
\displaystyle\hat{\mathcal{S}}_{u_{\text{ext}}}\frac{\text{e}^{\pm
im\varphi_{+}}S^{\omega}_{\ell m}(\theta)}{\sqrt{4\pi\omega}R_{+}}\
\text{e}^{\mp
i(\hat{\omega}-m\Omega_{+})u},&\cal{H}_{\text{past}}^{\text{ext}}\\\ \\\
\displaystyle\hat{\mathcal{S}}_{u_{\text{int}}}\frac{\text{e}^{\pm
im\varphi_{+}}S^{\omega}_{\ell m}(\theta)}{\sqrt{4\pi\omega}R_{+}}\
\text{e}^{\mp
i(\hat{\omega}-m\Omega_{+})u},&\cal{H}_{\text{past}}^{\text{int}}\end{cases},$
(104)

where the upper sign corresponds to the observer modes
${}^{\text{ext}}\phi_{\text{ob}}^{+}$, ${}^{\text{ext}}\phi_{\text{ob}}^{-}$,
and ${}^{\text{int}}\phi_{\text{ob}}^{-}$, the lower sign corresponds to the
observer modes ${}^{\text{int}}\phi_{\text{ob}}^{+}$, and where quantities
with hats also take on different forms for each family of observers:

$\displaystyle{}^{\text{ext}}\phi_{\text{ob},\omega}^{+}$ $\displaystyle:$
$\displaystyle\hat{\omega}$ $\displaystyle=\omega,$
$\displaystyle\hat{\mathcal{S}}_{v}=\mathcal{R}^{+}_{\text{ext},\omega},$
$\displaystyle\hat{\mathcal{S}}_{u_{\text{ext}}}$
$\displaystyle=\mathcal{T}^{+}_{\text{ext},\omega},$
$\displaystyle\hat{\mathcal{S}}_{u_{\text{int}}}=0,$
$\displaystyle{}^{\text{ext}}\phi_{\text{ob},\omega}^{-}$ $\displaystyle:$
$\displaystyle\hat{\omega}$ $\displaystyle=\omega+m\Omega_{+},$
$\displaystyle\hat{\mathcal{S}}_{v}=\mathcal{T}^{-}_{\text{ext},\omega},$
$\displaystyle\hat{\mathcal{S}}_{u_{\text{ext}}}$
$\displaystyle=\mathcal{R}^{-}_{\text{ext},\omega},$
$\displaystyle\hat{\mathcal{S}}_{u_{\text{int}}}=0,$
$\displaystyle{}^{\text{int}}\phi_{\text{ob},\omega}^{-}$ $\displaystyle:$
$\displaystyle\hat{\omega}$ $\displaystyle=\omega+m\Omega_{-},$
$\displaystyle\hat{\mathcal{S}}_{v}=\mathcal{T}^{-}_{\text{int},\omega}\mathcal{T}^{-}_{\text{ext},\omega},$
$\displaystyle\hat{\mathcal{S}}_{u_{\text{ext}}}$
$\displaystyle=\mathcal{T}^{-}_{\text{int},\omega}\mathcal{R}^{-}_{\text{ext},\omega},$
$\displaystyle\hat{\mathcal{S}}_{u_{\text{int}}}=\mathcal{R}^{-}_{\text{int},\omega},$
$\displaystyle{}^{\text{int}}\phi_{\text{ob},\omega}^{+}$ $\displaystyle:$
$\displaystyle\hat{\omega}$ $\displaystyle=\omega+m\Omega_{-},$
$\displaystyle\hat{\mathcal{S}}_{v}=\mathcal{R}^{+}_{\text{int},\omega}\mathcal{T}^{-}_{\text{ext},\omega},$
$\displaystyle\hat{\mathcal{S}}_{u_{\text{ext}}}$
$\displaystyle=\mathcal{R}^{+}_{\text{int},\omega}\mathcal{R}^{-}_{\text{ext},\omega},$
$\displaystyle\hat{\mathcal{S}}_{u_{\text{int}}}=\mathcal{T}^{+}_{\text{int},\omega}.$
(105)

Due to the orthogonality of the spheroidal harmonics, the angular pieces can
be integrated out to yield Kronecker $\delta$ functions between the observer’s
and emitter’s mode numbers $\ell$ and $m$. Then one has (where the $\pm$ sign
is once again defined as above)

$\displaystyle\langle\phi_{\text{em}}|\hat{\phi}_{\text{ob}}^{\text{*}}\rangle$
$\displaystyle=\frac{\delta_{\bar{\ell}\ell}\delta_{(\pm\bar{m})m}}{4\pi
i\sqrt{\omega\bar{\omega}}}\bigg{(}\hat{\mathcal{S}}_{v}\int_{-\infty}^{\infty}dv\
\text{e}^{-i\bar{\omega}v}\overset{\leftrightarrow}{\partial}_{v}\text{e}^{\mp
i\hat{\omega}v}$
$\displaystyle+\hat{\mathcal{S}}_{u_{\text{ext}}}\int_{-\infty}^{0}dU\
\text{e}^{-i\bar{\omega}U}\overset{\leftrightarrow}{\partial}_{U}\text{e}^{\mp
i(\hat{\omega}-m\Omega_{+})u}$
$\displaystyle+\hat{\mathcal{S}}_{u_{\text{int}}}\int_{0}^{\infty}dU\
\text{e}^{-i\bar{\omega}U}\overset{\leftrightarrow}{\partial}_{U}\text{e}^{\mp
i(\hat{\omega}-m\Omega_{+})u}\bigg{)}.$ (106)

In evaluating the bidirectional derivative defined by
${\psi\overset{\leftrightarrow}{\partial}_{\mu}\phi\equiv\psi\partial_{\mu}\phi-\phi\partial_{\mu}\psi}$,
the terms in the second and third lines of Eq. (B) that have the form
${\partial_{U}\text{e}^{\mp i(\hat{\omega}-m\Omega_{+})u}}$ can be simplified
through integration by parts, yielding a surface term that can be safely
discarded:

$\displaystyle\langle\phi_{\text{em}}|\hat{\phi}_{\text{ob}}^{\text{*}}\rangle$
$\displaystyle=\frac{\delta_{\bar{\ell}\ell}\delta_{(\pm\bar{m})m}}{4\pi\sqrt{\omega\bar{\omega}}}\bigg{(}(\bar{\omega}\mp\hat{\omega})\hat{\mathcal{S}}_{v}\int_{-\infty}^{\infty}dv\
\text{e}^{-i(\bar{\omega}\pm\hat{\omega})v}$
$\displaystyle+2\bar{\omega}\hat{\mathcal{S}}_{u_{\text{ext}}}\int_{-\infty}^{0}dU\
\text{e}^{-i(\bar{\omega}U\pm(\hat{\omega}-m\Omega_{+})u)}$
$\displaystyle+2\bar{\omega}\hat{\mathcal{S}}_{u_{\text{int}}}\int_{0}^{\infty}dU\
\text{e}^{-i(\bar{\omega}U\pm(\hat{\omega}-m\Omega_{+})u)}\bigg{)}.$ (107)

The observed number operator ${\langle\hat{N}\rangle_{\omega\ell m}}$ can now
be evaluated via Eq. (17) by taking the square of the complex conjugate of Eq.
(B) and summing over all emitter modes. The integral in the first line of Eq.
(B) reduces to a Dirac $\delta$ function that either vanishes (upper sign) or
leaves a small additive factor (lower sign)—these values can be ascertained by
noting that the emitter’s modes $\phi_{\text{em}}$ are normalized along past
null infinity as

$\displaystyle\langle\phi_{\text{em}}^{\omega\ell
m}|\phi_{\text{em}}^{\bar{\omega}\bar{\ell}\bar{m}}\rangle=-\langle\phi_{\text{em}}^{\omega\ell
m\text{*}}|\phi_{\text{em}}^{\bar{\omega}\bar{\ell}\bar{m}\text{*}}\rangle=\delta(\omega-\bar{\omega})\delta_{\ell\bar{\ell}}\delta_{m\bar{m}},$
$\displaystyle\langle\phi_{\text{em}}^{\omega\ell
m}|\phi_{\text{em}}^{\bar{\omega}\bar{\ell}\bar{m}\text{*}}\rangle=0.$ (108)

The integrals in the second and third lines of Eq. (B), on the other hand, are
the origin of the Planckian distribution. Using the definition of $U$ from Eq.
(21), these can be evaluated in terms of $\Gamma$ functions [2]. The resulting
number operator can then be found after taking the modulus squared:

$\displaystyle\langle\hat{N}\rangle_{\omega\ell\\{\pm
m\\}}=\frac{1}{4\pi^{2}\omega}\int_{0}^{\infty}d\bar{\omega}\ \bar{\omega}\
\bigg{|}\left\\{\genfrac{}{}{0.0pt}{}{0}{2\pi\hat{\mathcal{S}}_{v}\delta(\bar{\omega}-\hat{\omega})}\right\\}$
$\displaystyle+\left(\hat{\mathcal{S}}_{u_{\text{ext}}}+(-1)^{-z}\hat{\mathcal{S}}_{u_{\text{int}}}\right)\Gamma(z)(-i\bar{\omega})^{-z}\varkappa_{+}^{z-1}\bigg{|}^{2},$
(109)

where we have defined the quantity

$z\equiv 1\pm\frac{i(\hat{\omega}-m\Omega_{+})}{\varkappa_{+}}.$ (110)

The squared modulus of Eq. (B) can be simplified by the property

$\left|\Gamma\left(1+bi\right)\right|^{2}=\frac{\pi b}{\sinh(\pi b)}.$ (111)

First, consider the upper sign of Eq. (B), corresponding to the observer modes
${}^{\text{ext}}\phi_{\text{ob}}^{+}$, ${}^{\text{ext}}\phi_{\text{ob}}^{-}$,
and ${}^{\text{int}}\phi_{\text{ob}}^{-}$ (i.e., all modes except those
originating from an outgoing observer at the inner horizon, who has
exp(${i\omega t}$) instead of exp(${-i\omega t}$)). Then, assuming the
frequencies $\hat{\omega}$ and $\bar{\omega}$ are strictly positive, the first
term in the integrand of Eq. (B) will vanish, and the observed number operator
will simplify to

$\displaystyle\langle\hat{N}\rangle_{\omega\ell m}$
$\displaystyle=\frac{\hat{\omega}-m\Omega_{+}}{\omega}\frac{\left|\hat{\mathcal{S}}_{u_{\text{ext}}}-\text{e}^{\frac{\pi}{\varkappa_{+}}(\hat{\omega}-m\Omega_{+})}\hat{\mathcal{S}}_{u_{\text{int}}}\right|^{2}}{\text{e}^{\frac{2\pi}{\varkappa_{+}}(\hat{\omega}-m\Omega_{+})}-1}$
$\displaystyle\times\frac{1}{2\pi\varkappa_{+}}\int_{0}^{\infty}\frac{d\bar{\omega}}{\bar{\omega}}.$
(112)

The integral over the emitter’s frequency modes in the second line of Eq. (B)
diverges, but such behavior is not a problem and is actually to be expected.
The divergence originates from the use of continuum-normalized plane waves to
initialize the modes detected over an entire Cauchy hypersurface, which
inevitably leads to an infinite amount of Hawking radiation reaching an
observer throughout the infinite amount of time left in the future. If instead
of plane waves, a more physically realistic choice is used to represent the
observer’s modes, such as a finite wave packet distribution, then the
resulting integral will be regularized and the second line in Eq. (B) will
reduce to unity [40]. In particular, one may model the observer as a particle
detector sensitive only to frequencies within a small $\epsilon$ of
${\omega\sim j\epsilon\sim(j+1)\epsilon}$, which is turned on at a time
${u=2\pi n/\epsilon}$ for a duration ${2\pi/\epsilon}$ (for integers $j$ and
$n$). Then the observer’s modes will appear as

$\phi_{\text{ob}}^{\text{reg}}=\frac{1}{\sqrt{\epsilon}}\int_{j\epsilon}^{(j+1)\epsilon}d\omega\
\text{e}^{2\pi i\omega n/\epsilon}\phi_{\text{ob}}.$ (113)

At late times (large $n$), the above expression will yield exactly the first
line of Eq. (B), with the remaining terms on the second line reducing to
unity. This expression for the expectation value of the number operator
${\langle\hat{N}\rangle_{\omega\ell m}}$ reproduces Eqs. (76a)–(76c) after
substituting the respective observer quantities from Eqs. (B).

Finally, consider the lower sign of Eq. (B), corresponding to the observer
modes ${{}^{\text{int}}\phi_{\text{ob}}^{+}}$. As in the previous case, it
will be helpful to consider a wave-packet version of the observer’s modes,
since the integral over the complex square modulus of a Dirac delta
distribution must be regularized in some meaningful way.

For the portion of the observer’s modes along past null infinity, performing
the integral of Eq. (113) gives

$\phi_{\text{ob}}^{\text{reg}}\propto\hat{\mathcal{S}}_{v}\frac{\sqrt{\epsilon}}{\frac{v\epsilon}{2}+n\pi}\
\text{e}^{i\frac{v\epsilon}{2}(2j-1)}\sin\left(\frac{v\epsilon}{2}\right).$
(114)

The Fourier transform of these modes over the emitter’s frequency yields the
inner product

$\langle\phi_{\text{em}}|\hat{\phi}_{\text{ob}}^{\text{reg*}}\rangle_{\mathscr{I}^{-}}=\delta_{\bar{\ell}\ell}\delta_{(-\bar{m})m}\hat{\mathcal{S}}_{v}\sqrt{\frac{\bar{\omega}}{\omega}}\
\frac{\text{e}^{2\pi i\bar{\omega}n/\epsilon}}{\sqrt{\epsilon}}$ (115)

in the frequency range ${(j-1)\epsilon<\bar{\omega}<j\epsilon}$, and 0
everywhere else. Once this quantity’s complex modulus is squared and summed
over the emitter’s modes, a constant term will remain of the form

$\frac{(2j-1)\epsilon}{2\omega}\hat{\mathcal{S}}_{v}\sim\frac{\hat{\omega}}{\omega}\hat{\mathcal{S}}_{v},$
(116)

since for small $\epsilon$ the quantity $j\epsilon$ is precisely the
observer’s frequency ${\hat{\omega}}$ back-propagated to past null infinity.
Thus, the number operator of Eq. (17) for an outgoing observer at the inner
horizon reduces to

$\displaystyle\langle\hat{N}\rangle_{\omega\ell(-m)}=$
$\displaystyle-\frac{\hat{\omega}-m\Omega_{+}}{\omega}\frac{\left|\hat{\mathcal{S}}_{u_{\text{ext}}}-\text{e}^{-\frac{\pi}{\varkappa_{+}}(\hat{\omega}-m\Omega_{+})}\hat{\mathcal{S}}_{u_{\text{int}}}\right|^{2}}{\text{e}^{-\frac{2\pi}{\varkappa_{+}}(\hat{\omega}-m\Omega_{+})}-1}+\frac{\hat{\omega}}{\omega}\hat{\mathcal{S}}_{v}.$
(117)

Note that in principle Eq. (B) will contain an additional cross term when the
complex modulus of Eq. (B) is squared, which can be expressed in terms of
incomplete gamma functions. However, in the late-time limit of large $n$, this
term and its complex conjugate become negligibly small and thus are not
included here.

The expression in Eq. (B) for the expectation value of the number operator
${\langle\hat{N}\rangle_{\omega\ell m}}$ reproduces Eq. (76d) after
substituting the respective observer quantities from Eqs. (B).

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