#  A Little Excitement Across the Horizon

Keith K. Ng  Department of Physics, Nanyang Technological University, 21
Nanyang Link, Singapore 637371  Chen Zhang  Department of Physics and
Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Department of Physics, University of Toronto, 60 St. George St., Toronto,
Ontario, Canada M5S 1A7  Jorma Louko  School of Mathematical Sciences,
University of Nottingham, Nottingham NG7 2RD, UK  Robert B. Mann  Department
of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1,
Canada  Institute for Quantum Computing, University of Waterloo, Waterloo,
Ontario, N2L 3G1, Canada  Perimeter Institute for Theoretical Physics, 31
Caroline St N, Waterloo, ON, N2L 2Y5, Canada

(September 2021)

###### Abstract

We analyse numerically the transitions in an Unruh-DeWitt detector, coupled
linearly to a massless scalar field, in radial infall in (3+1)-dimensional
Schwarzschild spacetime. In the Hartle-Hawking and Unruh states, the
transition probability attains a small local extremum near the horizon-
crossing and is then moderately enhanced on approaching the singularity. In
the Boulware state, the transition probability drops on approaching the
horizon. This is to our knowledge the first analysis of a time-and-space
localised quantum system crossing the horizon of a (3+1)-dimensional black
hole.

_Introduction —_ General expectations of black hole physics – that black hole
evaporation is unitary, well described by local semi-classical quantum field
theory, and respects the equivalence principle – have been shown in recent
years to be quite paradoxical [1, 2]. Indeed, these three reasonable
requirements evidently violate the no-cloning theorem of quantum physics,
suggesting that the horizon must replaced by a high curvature region inducing
a “firewall” that will destroy any observer attempting to fall into the black
hole [3].

While there has been much effort and discussion in attempting to resolve this
conundrum, scant attention has been paid toward understanding in semi-
classical terms the actual behaviour of objects crossing the horizon. This is
of crucial importance, since the only way to determine if a black hole has a
temperature – that is the vacuum of spacetime outside it is in a thermal state
– is to consider the response of some object (a detector) placed in the
vicinity of the black hole [4]. Such detectors are most straightforwardly
modelled as two-level systems known as Unruh-DeWitt (UDW) detectors [5, 6].

When both the detector and the state of the quantum field are stationary, a
number of results about the detector’s response are known [7, 8]; in
particular, a static detector in the exterior of a Schwarzschild black hole
responds thermally to the Hartle-Hawking(-Israel) state [9, 10]. However,
essentially nothing is known about the response of detectors that freely fall
toward a black hole, with the exception of a few studies in lower dimensions.
One such study considered a UDW detector falling radially toward spinless
black hole in (2+1) dimensions [7]. In this case no regions in parameter space
were found where the transition rate was thermal, in accord with the
equivalence principle, but the detector was switched on and off in the region
exterior to the black hole. A study of the transition rate of a UDW detector
infalling toward a $(1+1)$ dimensional Schwarzschild black hole provided
numerical evidence of how its thermal properties are gradually lost during the
infall [11, 12]. A recent study of the same spacetime [13] considered the
entanglement and mutual information acquired by two freely falling detectors
and found that such correlations can be harvested even when the detectors are
causally disconnected by the event horizon.

Here we investigate for the first time the response of a UDW detector
(linearly coupled to a scalar field) in radial free-fall toward a
Schwarzschild black hole and across its horizon. We consider the Hartle-
Hawking(-Israel) [9, 10], Unruh [5], and Boulware [14] states of the field,
where in the last case the detector is switched off before horizon crossing
due to the singularity of the Boulware state at the horizon. Far from the
horizon, we find that the response approaches a Minkowski thermal state
response for the Hartle-Hawking state and the Minkowski vacuum response for
the other two states, as could be expected. Near the horizon, the response in
the Hartle-Hawking and Unruh states attains a small local extremum after
horizon crossing and is then moderately enhanced on approaching the black hole
singularity; in the Boulware state, the response drops as the horizon is
approached. Our work involves pioneering numerical elements, and constitutes,
to our knowledge, the first analysis of a time-and-space localised quantum
system crossing the horizon of a (3+1)-dimensional black hole.

We use throughout units in which $\hbar=c=k_{B}=1$.

_Spacetime and Detector’s Motion —_ The exterior Schwarzschild metric reads

$\displaystyle
ds^{2}=-\left(1-2M/r\right)dt^{2}+\frac{dr^{2}}{1-2M/r}+r^{2}d\Omega^{2},$ (1)

where $M>0$ and $r>2M$, and
$d\Omega^{2}=d\theta^{2}+\sin^{2}\\!\theta\,d\varphi^{2}$ is the metric on the
round unit two-sphere. The Kruskal-Szekeres extension is obtained by
introducing the new coordinates (we follow the conventions of [15])

$\displaystyle\bar{u}$
$\displaystyle=-4M\bigl{(}r/(2M)-1\bigr{)}^{1/2}\,e^{(r-t)/(4M)},$ (2a)
$\displaystyle\bar{v}$
$\displaystyle=4M\bigl{(}r/(2M)-1\bigr{)}^{1/2}\,e^{(r+t)/(4M)},$ (2b)

where initially $\bar{u}<0<\bar{v}$, and then extending the coordinate range
to $\bar{u}\bar{v}<(4M)^{2}$. The metric becomes

$\displaystyle
ds^{2}=-(2M/r)\,e^{-r/(2M)}\,d\bar{u}\,d\bar{v}+r^{2}d\Omega^{2},$ (3)

where $r$ is the unique solution to
$-\bar{u}\bar{v}=(4M)^{2}e^{r/(2M)}\bigl{(}r/(2M)-1\bigr{)}$. The conformal
diagram can be found for example in [15].

We consider the part of the Kruskal-Szekeres extension where $\bar{v}>0$. This
contains the original Schwarzschild exterior at $\bar{u}<0$, the black hole
interior at $0<\bar{u}<(4M)^{2}/\bar{v}$, and the black hole horizon at
$\bar{u}=0$.

We consider a detector on an infalling radial timelike geodesic that starts
from infinity with asymptotically vanishing initial velocity. In the Kruskal
coordinates, the trajectory reads

$\displaystyle\bar{u}$
$\displaystyle=-4M\\!\left(z-1\right)\exp\\!\left(z+\tfrac{1}{2}z^{2}+\tfrac{1}{3}z^{3}\right),$
(4a) $\displaystyle\bar{v}$
$\displaystyle=4M\\!\left(z+1\right)\exp\\!\left(-z+\tfrac{1}{2}z^{2}-\tfrac{1}{3}z^{3}\right),$
(4b)

where $z=(\tau/\tau_{H})^{1/3}$, $\tau\in(-\infty,0)$ is the proper time, and
$\tau_{H}=-\tfrac{4}{3}M$. The trajectory starts from past timelike infinity
in the asymptotic past, $\tau\to-\infty$, crosses the horizon at
$\tau=\tau_{H}$, and hits the black hole singularity at $\tau\to 0^{-}$. The
exterior Schwarzschild part of the trajectory can be expressed in the exterior
Schwarzschild coordinates via (2), and similar expressions can be given for
the black hole interior part of the trajectory in terms of interior
Schwarzschild coordinates.

_Detector Preliminaries —_ Our UDW detector [5, 6] is a two-level quantum
system of spatially negligible size, moving through the spacetime on the
timelike worldline $\mathsf{x}(\tau)$, parametrised by the proper time $\tau$.
The detector’s Hilbert space is spanned by two orthonormal states, with the
respective eigenenergies $0$ and $\Omega\in\mathbb{R}\setminus\\{0\\}$,
defined with respect to $\tau$. For $\Omega>0$, the state with eigenenergy $0$
is the ground state and the state with eigenenergy $\Omega$ is the excited
state; for $\Omega<0$, the roles of the states are reversed.

The detector couples linearly to a quantised real scalar field $\phi$. While
simple, this model captures the essential features of light-matter
interactions when angular momentum interchange is negligible [16, 17].

Working to first-order perturbation theory in the coupling constant, the
probability of the detector to make a transition from the eigenenergy $0$
state to the eigenenergy $\Omega$ state is a multiple of the _response
function_ $F(\Omega)$, given by

$\displaystyle F(\Omega)$
$\displaystyle=\int\text{d}\tau\,\text{d}\tau^{\prime}\,\chi(\tau)\chi(\tau^{\prime})\,e^{-i\Omega(\tau-\tau^{\prime})}\,W(\tau,\tau^{\prime}),$
(5)

where
$W(\tau,\tau^{\prime})=\left\langle{\Psi}\left|{\phi(\mathsf{x}(\tau))\phi(\mathsf{x}(\tau^{\prime}))}\right|{\Psi}\right\rangle$
is the pullback of the scalar field’s Wightman function to the detector’s
worldline, $\left|{\Psi}\right\rangle$ denotes the initial state of the field,
and the real-valued switching function $\chi$ specifies how the interaction is
turned on and off. Assuming that $\left|{\Psi}\right\rangle$ has a technical
regularity property known as the Hadamard condition [18],
$W(\tau,\tau^{\prime})$ is a well-defined distribution under mild assumptions
about the detector’s trajectory [19, 20], and then $F(\Omega)$ is well defined
under mild assumptions about $\chi$. As the factor relating $F(\Omega)$ to the
probability depends only on the detector’s internal structure, we (with minor
abuse of terminology) refer to $F(\Omega)$ as the probability.

When the state $\left|{\Psi}\right\rangle$ is a ‘vacuum’ state associated with
a complete orthonormal set of ‘positive frequency’ field modes,
$\\{\Psi_{\beta}(\mathsf{x})\\}$, the Wightman function has the mode sum
expression
$\left\langle{\Psi}\left|{\phi(\mathsf{x})\phi(\mathsf{x}^{\prime})}\right|{\Psi}\right\rangle=\sum_{\beta}\Psi_{\beta}(\mathsf{x})\Psi_{\beta}^{*}(\mathsf{x^{\prime}})$,
and the response function can be written as the mode sum

$\displaystyle
F(\Omega)=\sum_{\beta}\left|\int\text{d}\tau\,\chi(\tau)\,e^{-i\Omega\tau}\,\Psi_{\beta}(\mathsf{x}(\tau))\right|^{2},$
(6)

which we shall employ in what follows.

_Mode Functions and Vacua —_ We take the field $\phi$ to be a real massless
Klein-Gordon field. In the Schwarzchild exterior, we may separate the field
equation, $\Box\phi=0$, by the ansatz

$\displaystyle u(t,r,\theta,\varphi)=\frac{1}{\sqrt{4\pi\omega}}\phi_{\omega
l}(r)Y_{lm}(\theta,\varphi)e^{-i\omega t},$ (7)

where $Y_{lm}$ are the scalar spherical harmonics and the positive separation
parameter $\omega$ is the frequency with respect to Schwarzschild time. The
differential equation for the radial function $\phi_{\omega l}$ may be
conveniently written in terms of the tortoise radial coordinate
$r^{*}=r+2M\ln[r/(2M)-1]\in(-\infty,\infty)$ and the scaling $\phi_{\omega
l}(r)=\rho_{\omega l}(r)/r$, giving the Schrödinger-like equation [21]

$\frac{d^{2}}{dr^{*2}}\rho_{\omega
l}(r)+\left[\omega^{2}-V_{l}(r)\right]\rho_{\omega l}(r)=0,$ (8)

with the effective potential

$V_{l}(r)=\left(1-\frac{2M}{r}\right)\left(\frac{l(l+1)}{r^{2}}+\frac{2M}{r^{3}}\right).$
(9)

As the effective potential approaches zero when $r^{*}\to\infty$ and
$r^{*}\to-\infty$, corresponding respectively to $r\to\infty$ and $r\to 2M$,
there exists a set of mode solutions for $\phi_{\omega l}$, denoted by
$\Phi_{\omega l}^{in}$ and $\Phi_{\omega l}^{up}$, specified uniquely by the
asymptotic behaviour

$\displaystyle\Phi_{\omega l}^{in}$
$\displaystyle\sim\left\\{\begin{matrix}B^{in}_{\omega l}r^{-1}e^{-i\omega
r^{*}}&r\rightarrow 2M,\\\ r^{-1}e^{-i\omega r^{*}}+A^{in}_{\omega
l}r^{-1}e^{i\omega r^{*}}&r\rightarrow\infty,\end{matrix}\right.$ (10a)
$\displaystyle\Phi_{\omega l}^{up}$
$\displaystyle\sim\left\\{\begin{matrix}r^{-1}e^{i\omega r^{*}}+A^{up}_{\omega
l}r^{-1}e^{-i\omega r^{*}}&r\rightarrow 2M,\\\ B^{up}_{\omega
l}r^{-1}e^{i\omega r^{*}}&r\rightarrow\infty.\end{matrix}\right.$ (10b)

The mode functions defining the Boulware (B) state [14], the Hartle-Hawking
(H) state [9, 10] and the Unruh (U) [5] state in the exterior region can then
be written in terms of $\Phi_{\omega l}^{in}$ and $\Phi_{\omega l}^{up}$ and
their conjugates, as lucidly reviewed in [22], and the H and U modes can be
continued to the black hole interior in the lower half-plane in complexified
$\bar{u}$ [5].

_Response function —_ We are now ready to write out the mode sum expression
(6) for the response function for each of the three vacua.

Consider first a detector that only operates in the Schwarzschild exterior. We
may choose without loss of generality the trajectory to be at $\theta=0$,
which simplifies the spherical harmonics. The pull-back of the Wightman
function to the detector’s worldline then has the mode sum expressions [23, 8]

$\displaystyle W_{B}(\tau,\tau^{\prime})$
$\displaystyle=\sum_{l=0}^{\infty}\int_{0}^{\infty}\text{d}\omega\,\frac{2l+1}{16\pi^{2}\omega}\left(\tilde{\Phi}_{\omega
l}^{up}(r)\tilde{\Phi}_{\omega
l}^{up*}(r^{\prime})\left(\frac{\bar{u}}{\bar{u}^{\prime}}\right)^{i4M\omega}+\tilde{\Phi}_{\omega
l}^{in}(r)\tilde{\Phi}_{\omega
l}^{in*}(r^{\prime})\left(\frac{\bar{v}}{\bar{v}^{\prime}}\right)^{-i4M\omega}\right),$
(11a) $\displaystyle W_{H}(\tau,\tau^{\prime})$
$\displaystyle=\sum_{l=0}^{\infty}\int_{0}^{\infty}\text{d}\omega\,\frac{2l+1}{16\pi^{2}\omega}\left(\frac{e^{4\pi
M\omega}}{2\sinh(4\pi M\omega)}\tilde{\Phi}_{\omega
l}^{up}(r)\tilde{\Phi}_{\omega
l}^{up*}(r^{\prime})\left(\frac{\bar{u}}{\bar{u}^{\prime}}\right)^{i4M\omega}+\frac{e^{-4\pi
M\omega}}{2\sinh(4\pi M\omega)}\tilde{\Phi}_{\omega
l}^{up*}(r)\tilde{\Phi}_{\omega
l}^{up}(r^{\prime})\left(\frac{\bar{u}}{\bar{u}^{\prime}}\right)^{-i4M\omega}\right.$
$\displaystyle\qquad\qquad+\left.\frac{e^{4\pi M\omega}}{2\sinh(4\pi
M\omega)}\tilde{\Phi}_{\omega l}^{in}(r)\tilde{\Phi}_{\omega
l}^{in*}(r^{\prime})\left(\frac{\bar{v}}{\bar{v}^{\prime}}\right)^{-i4M\omega}+\frac{e^{-4\pi
M\omega}}{2\sinh(4\pi M\omega)}\tilde{\Phi}_{\omega
l}^{in*}(r)\tilde{\Phi}_{\omega
l}^{in}(r^{\prime})\left(\frac{\bar{v}}{\bar{v}^{\prime}}\right)^{i4M\omega}\right),$
(11b) $\displaystyle W_{U}(\tau,\tau^{\prime})$
$\displaystyle=\sum_{l=0}^{\infty}\int_{0}^{\infty}\text{d}\omega\,\frac{2l+1}{16\pi^{2}\omega}\left(\frac{e^{4\pi
M\omega}}{2\sinh(4\pi M\omega)}\tilde{\Phi}_{\omega
l}^{up}(r)\tilde{\Phi}_{\omega
l}^{up*}(r^{\prime})\left(\frac{\bar{u}}{\bar{u}^{\prime}}\right)^{i4M\omega}+\frac{e^{-4\pi
M\omega}}{2\sinh(4\pi M\omega)}\tilde{\Phi}_{\omega
l}^{up*}(r)\tilde{\Phi}_{\omega
l}^{up}(r^{\prime})\left(\frac{\bar{u}}{\bar{u}^{\prime}}\right)^{-i4M\omega}\right.$
$\displaystyle\qquad\qquad+\left.\tilde{\Phi}_{\omega
l}^{in}(r)\tilde{\Phi}_{\omega
l}^{in*}(r^{\prime})\left(\frac{\bar{v}}{\bar{v}^{\prime}}\right)^{-i4M\omega}\right),$
(11c)

where

$\displaystyle\tilde{\Phi}_{\omega l}^{up}(r)$ $\displaystyle:=e^{-i\omega
r^{*}}\Phi_{\omega l}^{up}(r),\qquad\tilde{\Phi}_{\omega
l}^{in}(r):=e^{+i\omega r^{*}}\Phi_{\omega l}^{in}(r)\,,$ (12)

and the subscripts B, H and U label the three states. Unprimed and primed
coordinates are evaluated at $\tau$ and $\tau^{\prime}$ respectively. In each
term, the factor depending on the unprimed coordinates comes from the
‘positive frequency’ mode functions defining the corresponding state, and the
factor depending on the primed coordinates comes from the complex conjugate
‘negative frequency’ mode functions.

From (6), we now obtain

$F_{B}=F_{B}^{up}+F_{B}^{in},\qquad
F_{H}=F_{H}^{up}+F_{H}^{\widetilde{up}}+F_{H}^{in}+F_{H}^{\widetilde{in}},\qquad
F_{U}=F_{H}^{up}+F_{H}^{\widetilde{up}}+F_{B}^{in},$ (13)

for the respective B, H, and U states, where

$\displaystyle F_{B}^{up}(\Omega)$
$\displaystyle=\sum_{l}\int_{0}^{\infty}d\omega\,\frac{2l+1}{16\pi^{2}\omega}\left|\int\text{d}\tau\,\chi(\tau)e^{-i\Omega\tau}\,\tilde{\Phi}^{up}_{\omega
l}(r)\left(\frac{-\bar{u}}{4M}\right)^{i4M\omega}\right|^{2},$ (14a)
$\displaystyle F_{B}^{in}(\Omega)$
$\displaystyle=\sum_{l}\int_{0}^{\infty}d\omega\,\frac{2l+1}{16\pi^{2}\omega}\left|\int\text{d}\tau\,\chi(\tau)e^{-i\Omega\tau}\,\tilde{\Phi}^{in}_{\omega
l}(r)\left(\frac{\bar{v}}{4M}\right)^{-i4M\omega}\right|^{2},$ (14b)
$\displaystyle F_{H}^{up}(\Omega)$
$\displaystyle=\sum_{l}\int_{0}^{\infty}d\omega\,\frac{2l+1}{16\pi^{2}\omega}\times\frac{e^{4\pi
M\omega}}{2\sinh(4\pi
M\omega)}\left|\int\text{d}\tau\,\chi(\tau)e^{-i\Omega\tau}\,\tilde{\Phi}^{up}_{\omega
l}(r)\left(\frac{-\bar{u}}{4M}\right)^{i4M\omega}\right|^{2},$ (14c)
$\displaystyle F_{H}^{\widetilde{up}}(\Omega)$
$\displaystyle=\sum_{l}\int_{0}^{\infty}d\omega\,\frac{2l+1}{16\pi^{2}\omega}\times\frac{e^{-4\pi
M\omega}}{2\sinh(4\pi
M\omega)}\left|\int\text{d}\tau\,\chi(\tau)e^{-i\Omega\tau}\,\tilde{\Phi}^{up*}_{\omega
l}(r)\left(\frac{-\bar{u}}{4M}\right)^{-i4M\omega}\right|^{2},$ (14d)
$\displaystyle F_{H}^{in}(\Omega)$
$\displaystyle=\sum_{l}\int_{0}^{\infty}d\omega\,\frac{2l+1}{16\pi^{2}\omega}\times\frac{e^{4\pi
M\omega}}{2\sinh(4\pi
M\omega)}\left|\int\text{d}\tau\,\chi(\tau)e^{-i\Omega\tau}\,\tilde{\Phi}^{in}_{\omega
l}(r)\left(\frac{\bar{v}}{4M}\right)^{-i4M\omega}\right|^{2},$ (14e)
$\displaystyle F_{H}^{\widetilde{in}}(\Omega)$
$\displaystyle=\sum_{l}\int_{0}^{\infty}d\omega\,\frac{2l+1}{16\pi^{2}\omega}\times\frac{e^{-4\pi
M\omega}}{2\sinh(4\pi
M\omega)}\left|\int\text{d}\tau\,\chi(\tau)e^{-i\Omega\tau}\,\tilde{\Phi}^{in*}_{\omega
l}(r)\left(\frac{\bar{v}}{4M}\right)^{i4M\omega}\right|^{2}.$ (14f)

Consider then a detector that continues to operate behind the horizon. As the
B state is singular on the horizon, we consider only the H and U states. As
shown in the Supplemental Material, following [23], and as discussed in the
context of vacuum polarisation in [24], expressions valid both in the exterior
and interior for $F_{H}^{up}$ and $F_{H}^{\widetilde{up}}$ are

$\displaystyle F_{H}^{up}(\Omega)$
$\displaystyle=\sum_{l}\int_{0}^{\infty}d\omega\,\frac{2l+1}{16\pi^{2}\omega}\times\frac{e^{4\pi
M\omega}}{2\sinh(4\pi
M\omega)}\times\left|\int\text{d}\tau\,\chi(\tau)e^{-i\Omega\tau}\left\\{\Theta(-\bar{u})\tilde{\Phi}^{up}_{\omega
l}(r)\left(\frac{-\bar{u}}{4M}\right)^{i4M\omega}\right.\right.$
$\displaystyle\hskip
25.83325pt\left.\left.+\Theta(\bar{u})\left[\frac{A^{up}\tilde{\Phi}^{in}_{\omega
l}(r)}{B^{in}}\left(\frac{\bar{v}}{4M}\right)^{-i4M\omega}+\frac{e^{-4\pi
M\omega}\tilde{\Phi}^{in*}_{\omega
l}(r)}{B^{in*}}\left(\frac{\bar{u}}{4M}\right)^{i4M\omega}\right]\right\\}\right|^{2},$
(15a) $\displaystyle F_{H}^{\widetilde{up}}(\Omega)$
$\displaystyle=\sum_{l}\int_{0}^{\infty}d\omega\,\frac{2l+1}{16\pi^{2}\omega}\times\frac{e^{-4\pi
M\omega}}{2\sinh(4\pi
M\omega)}\times\left|\int\text{d}\tau\,\chi(\tau)e^{-i\Omega\tau}\left\\{\Theta(-\bar{u})\tilde{\Phi}^{up*}_{\omega
l}(r)\left(\frac{-\bar{u}}{4M}\right)^{-i4M\omega}\right.\right.$
$\displaystyle\hskip
25.83325pt\left.\left.+\Theta(\bar{u})\left[\frac{A^{up*}\tilde{\Phi}^{in*}_{\omega
l}(r)}{B^{in*}}\left(\frac{\bar{v}}{4M}\right)^{i4M\omega}+\frac{e^{4\pi
M\omega}\tilde{\Phi}^{in}_{\omega
l}(r)}{B^{in}}\left(\frac{\bar{u}}{4M}\right)^{-i4M\omega}\right]\right\\}\right|^{2},$
(15b)

where $\Theta$ is the Heaviside theta-function. The responses for the H and U
states are hence given by $F_{H}$ and $F_{U}$ in (13), using (14b), (14e),
(14f) and (15).

_Switching —_ We switch the interaction on and off with

$\displaystyle\chi(\tau)$
$\displaystyle=\begin{cases}{\displaystyle\cos^{4}\\!\left(\frac{\pi(\tau-\tau_{mid})}{2\Delta}\right)}&\text{for}\
\ |\tau-\tau_{mid}|<\Delta,\\\ 0&\textrm{otherwise},\end{cases}$ (16)

where the positive parameter $\Delta$ is half the interaction duration and the
parameter $\tau_{mid}$ is the midpoint of the interaction interval. $\chi$ is
a close approximation to Gaussian switching over its duration [25], and it is
smooth everywhere except at the switch-on and switch-off moments, where it is
still $C^{3}$, which is sufficiently regular to render the response finite.

Note that $\chi$ has peak value $1$, regardless of the total duration, with
the convenient consequence that the response function is dimensionless, but
with the price that the overall magnitudes of the response function for
differing values of $\Delta$ are not directly comparable. Factoring out an
‘overall duration’ effect could be accomplished by including a normalisation
factor proportional to $\Delta^{-1/2}$ [26, 27].

_Results —_ We present here graphs of the response as the detector falls close
to and in the black hole; numerical implementation is given in the
Supplemental Material.

First, we plot in Figure 1 the response function as a function of
$r_{mid}=r(\tau_{mid}),$ for detector frequency $\Omega=5/(2M)$, and switching
time $\Delta=0.3\times 2M$, with respect to all three states. Far from the
horizon, the response function approaches that of a detector in flat space:
for the B and U states, the asymptotic value is the flat space vacuum value
$F_{vac}\approx 0.0334638$, and for the H state, the asymptotic value is the
flat space thermal vacuum value $F_{\beta}\approx 0.0334861$ in the Hawking
temperature $1/(8\pi M)$ (these values are computed in the Supplemental
Material). This is as expected: note that the difference between the H and U
states arises because the H state is in equilibrium with a thermal bath that
extends to infinity, whereas in the U state the black hole is emitting net
radiation whose intensity falls off far from the hole. We likewise see that
the response in the B state falls off as the horizon is approached. The
maximum departure from these asymptotic values is of the order of parts per
thousand. This shows that the major part of the response in our parameter
range is due to the finiteness of the time interval during which the detector
operates.

We also observe (for H and U) a small but discernible peak in the response as
the detector crosses the horizon. Although this region involves some numerical
subtlety in the matching in (15) across the horizon, we have checked (see
Supplemental Material) that the peak is robust on varying the numerical
parameters.

Figure 1: Response for the three states with $\Omega=5/(2M)$ and
$\Delta=0.3\times(2M)$, as a function of $r_{mid}=r(\tau_{mid})$. Note that
the response for the Boulware state is defined only when the full interval in
which the detector operates is outside the horizon. The lower (red) and upper
(blue) dashed horizontal lines indicate respectively the flat space values
$F_{vac}$ and $F_{\beta}$, described in the text.

Next, we plot in Figure 2 the response for selected values of $\Delta$ in the
H state. We again observe a small but discernible peak in the response that is
$\Delta$-dependent. A curious tradeoff becomes visible: if $\Delta$ is too
small, the finer features of the states are overwhelmed by switching effects
[27]; but $\Delta$ cannot be too large since the detector cannot remain
‘switched on’ at the $r=0$ singularity. These facts conspire to limit detector
sensitivity to the state of the field. (In fact, since the B state is defined
only outside the horizon, sensitivity in that case is reduced even further.)

Figure 2: The normalized response $F/F_{asymp}$, where $F_{asymp}$ (inset) is
the response as $r_{mid}\to\infty$ (obtained from the analytic result in the
Supplemental Material), for selected values of the switching duration
parameter $\Delta$, with $\Omega=5/(2M)$, in the Hartle-Hawking state. Lines
end on the left when detector switchoff is at the singularity.

Finally, in Figure 3 we consider the effect of varying the detector’s energy
gap $\Omega$, recalling that $\Omega>0$ corresponds to excitations and
$\Omega<0$ to de-excitations. We specialise to the H state. The response as a
function of $r_{mid}$ exhibits local extrema near and past $r_{mid}=2M$, but
the detailed loci and character of these extrema depend on $\Omega$. When the
detector falls deeper in the black hole, the response exhibits a modest
overall growth (also evident in Figure 1). The magnitude of the
$r_{mid}$-dependent variation is however small, of the order of parts per
thousand, in the parameter range that we have explored.

Figure 3:  The normalized response $F/F_{asymp}$, where $F_{asymp}$ (inset) is
the response as $r_{mid}\to\infty$ (obtained from the analytic result in the
Supplemental Material) for selected values of the energy gap $\Omega$, with
$\Delta=0.3\times 2M$, in the Hartle-Hawking state.

_Conclusions —_ We have obtained for the first time the response of a time-
and-space localised quantum detector falling through the horizon of a
Schwarzschild black hole. When the ambient quantum field is prepared in the
Hartle-Hawking or Unruh states, both of which are regular across the horizon,
we verified that ‘no drama’ occurs when the detector crosses the horizon, as
indeed expected. In the parameter range that we have explored, clear
differences from flat spacetime are discernable by parts per thousand; however
the sensitivity of a detector to the near-horizon geometry is, as we
demonstrated, fundamentally limited by the finite time interval during which
the detector operates. Yet a sufficiently careful measurement can indeed
detect curious features unique to the black hole, features warranting further
study.

One way in which these unique features could be further explored is by varying
the geodesic trajectory of the detector. For example, the limitations due to
the detector’s finite operation time could be ameliorated by considering
trajectories that spend more proper time in the near-horizon region.
Alternatively, the region between the closed photon orbit at $r=3M$ and the
innermost stable circular timelike orbit at $r=6M$ can be explored by
nonradial high-eccentricity orbits that come from infinity and return there.
Perhaps Alice the astronaut can gain significant information by just observing
the dance of quantum matter around the hole, without having to dive into the
hole herself.

Our methods are applicable to other quantum states and to other spacetimes.
For instance, the methods can be used to evaluate the response of a detector
to quantum states defined on proposed alternatives to black holes, such as
those discussed in [28]. This may provide a quantum tool for distinguishing a
black hole from its alternatives, and by extension a tool for testing quantum
extensions to general relativity, offering the prospect of taking one more
step towards a unified theory.

_Acknowledgments —_ KN thanks the School of Mathematical Sciences at the
University of Nottingham for hospitality in the early stages of the work. This
work was supported in part by the Natural Science and Engineering Research
Council of Canada and by Asian Office of Aerospace Research and Development
Grant No. FA2386-19-1-4077. JL acknowledges partial support by United Kingdom
Research and Innovation (UKRI) Science and Technology Facilities Council
(STFC) grant ST/S002227/1 “Quantum Sensors for Fundamental Physics” and Theory
Consolidated Grant ST/P000703/1.

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Supplemental Material

_Continuation across the horizon —_ Here we verify the analytic continuation
of $F_{H}^{up}$ and $F_{H}^{\widetilde{up}}$ from the exterior formulas (14c)
and (14d) to the formulas (15) that are valid also inside the horizon, for the
evaluation of the response in the H and U states.

First, note that each of the expressions in (14) is written so that the
quantity within the absolute values comes from the ‘positive frequency’ mode
functions characterising the respective state.

Next, recall that the ‘positive frequency’ mode functions defining the H and U
states have the property that if they are singular on the horizon, where
$\bar{u}$ changes sign, they are continued across the horizon in the lower
half of the complex $\bar{u}$ plane [S1,S2,S3]. As $\bar{v}>0$ in the
spacetime regions in which we are working, and as $\tilde{\Phi}^{in}_{\omega
l}$ (12) is smooth across the black hole horizon, the expressions for
$F_{B}^{in}$, $F_{H}^{in}$ and $F_{H}^{\widetilde{in}}$ in (14) remain well
defined and suitable for numerical evaluation at and behind the horizon. By
contrast, the expressions for $F_{H}^{up}$ and $F_{H}^{\widetilde{up}}$ in
(14) need to be given an appropriate analytic continuation, as described in
[S4], and as we now explain. See [S5] for a discussion of this continuation in
the context of vacuum polarisation.

Consider the exterior formulas (14c) and (14d) for $F_{H}^{up}$ and
$F_{H}^{\widetilde{up}}$. Under each integral, we write (where our convention
for $B^{in}$ differs from [S4])

$\displaystyle\tilde{\Phi}^{up}_{\omega
l}(r)\left(\frac{-\bar{u}}{4M}\right)^{i4M\omega}$
$\displaystyle=\frac{A^{up}\tilde{\Phi}^{in}_{\omega
l}(r)}{B^{in}}\left(\frac{\bar{v}}{4M}\right)^{-i4M\omega}+\frac{\tilde{\Phi}^{in*}_{\omega
l}(r)}{B^{in*}}\left(\frac{-\bar{u}}{4M}\right)^{i4M\omega},$ (S1a)
$\displaystyle\tilde{\Phi}^{up*}_{\omega
l}(r)\left(\frac{-\bar{u}}{4M}\right)^{-i4M\omega}$
$\displaystyle=\frac{A^{up*}\tilde{\Phi}^{in*}_{\omega
l}(r)}{B^{in*}}\left(\frac{\bar{v}}{4M}\right)^{i4M\omega}+\frac{\tilde{\Phi}^{in}_{\omega
l}(r)}{B^{in}}\left(\frac{-\bar{u}}{4M}\right)^{-i4M\omega},$ (S1b)

using (10) and (12), and recalling that $\bar{u}<0$. Continuing now to
positive $\bar{u}$ in the lower half of the complex $\bar{u}$ plane, the
respective first terms on the right-hand sides in (S1) remain well defined as
written, whereas the second terms respectively continue to

$\displaystyle\frac{e^{-4\pi M\omega}\tilde{\Phi}^{in*}_{\omega
l}(r)}{B^{in*}}\left(\frac{\bar{u}}{4M}\right)^{i4M\omega},$ (S2a)
$\displaystyle\frac{e^{4\pi M\omega}\tilde{\Phi}^{in}_{\omega
l}(r)}{B^{in}}\left(\frac{\bar{u}}{4M}\right)^{-i4M\omega}.$ (S2b)

This gives the expressions (15) for $F_{H}^{up}$ and $F_{H}^{\widetilde{up}}$.

_Numerical integration of the radial equation —_ The form of (8) suggests an
obvious strategy: take initial conditions in the “asymptotic free” regime,
then numerically integrate the differential equation. However, things are
somewhat complicated by a few issues. The most obvious of these is that the
modes are defined and normalized on ‘opposite sides’; for instance, while the
in mode is defined by pure ingoing behaviour near the horizon, it is
normalized in the other asymptotic regime, far from the horizon. In this
section, we use $\Phi$ to refer to modes normalized as in (10), and $\phi$ for
an alternate normalization, which will be defined shortly.

When we enter initial values into the differential equation, we must use the
asymptotic regime where the mode is expressible as a single exponential; more
specifically, we use an initial value such that

$\displaystyle\phi_{\omega l}^{in}$ $\displaystyle\sim r^{-1}e^{-i\omega
r^{*}},$ $\displaystyle r\rightarrow 2M,$ (S3a) $\displaystyle\phi_{\omega
l}^{up}$ $\displaystyle\sim r^{-1}e^{i\omega r^{*}},$ $\displaystyle
r\rightarrow\infty.$ (S3b)

However, this normalization is clearly at odds with the normalization required
for $\Phi$.

In order to resolve this problem, we can introduce two further sets of
solutions: the downward and outward modes, which we define by

$\displaystyle\phi_{\omega l}^{out}$ $\displaystyle\sim r^{-1}e^{i\omega
r^{*}},$ $\displaystyle r\rightarrow 2M,$ (S4a) $\displaystyle\phi_{\omega
l}^{down}$ $\displaystyle\sim r^{-1}e^{-i\omega r^{*}},$ $\displaystyle
r\rightarrow\infty.$ (S4b)

Normalization is then a matter of expressing each physical mode in terms of a
superposition of two other modes; this allows us to deduce the correct factor
by which we must multiply $\phi$ to yield $\Phi$. In essence, we are solving
for the transmission and reflection coefficients in (10).

While we could, in principle, simply take the boundary conditions at face
value for some suitably asymptotic finite values of $r^{*}$, there are more
precise estimates of the modes in asymptotic regimes available. Following
Leaver [S6], we can make estimates for the up and down modes far from the
horizon as follows:

$\displaystyle\phi_{\omega l}^{up}$
$\displaystyle\sim\frac{1}{\sqrt{r(r-{2M})}}H^{+}_{\nu_{a}}(-2M\omega,\omega
r)\exp\\!\left[\mathrm{i}\left(-2M\omega\log(4M\omega)+\tfrac{\pi}{2}\nu_{a}-\sigma_{\nu_{a}}(-{2M}\omega)\right)\right],$
(S5a) $\displaystyle\phi_{\omega l}^{down}$
$\displaystyle\sim\frac{1}{\sqrt{r(r-{2M})}}H^{-}_{\nu_{a}}(-2M\omega,\omega
r)\exp\\!\left[-\mathrm{i}\left(-2M\omega\log(4M\omega)+\tfrac{\pi}{2}\nu_{a}-\sigma_{\nu_{a}}(-{2M}\omega)\right)\right],$
(S5b)

where $H^{\pm}$ are Coulomb wave functions, the Coulomb phase shift is

$\sigma_{\nu}(\eta)=-\frac{\mathrm{i}}{2}\log\frac{\Gamma(\nu+1+\mathrm{i}\eta)}{\Gamma(\nu+1-\mathrm{i}\eta)},$
(S6)

and $\nu_{a}=\frac{1}{2}(-1+\sqrt{(2l+1)^{2}-12(2M\omega)^{2}})$. The final
factor ensures that the phase (i.e. complex argument) of the mode matches the
asymptotic solution $e^{\pm i\omega r}$. Note that some other references use a
different expression for $\sigma$ involving arg; such expressions are only
valid for real $\omega$.

On the other hand, we estimate the in and out modes near the horizon using the
Jaffé power series solution given in Eqs. (49)–(51) of [S6], For the in modes,

$\phi_{\omega l}^{in}\sim\frac{1}{r}e^{-i\omega
r}\left(\frac{r}{2M}-1\right)^{-\mathrm{i}2M\omega}\sum_{n=0}^{\infty}a_{n}\left(1-\frac{2M}{r}\right)^{n},$
(S7)

where

$\displaystyle a_{-1}$ $\displaystyle=0,\qquad a_{0}=1,$
$\displaystyle(1+n)(1+n-2\mathrm{i}(2M\omega))a_{n+1}$ $\displaystyle+$
$\displaystyle(-1-l(l+1)-2n(1+n))a_{n}+n^{2}a_{n-1}=0,$ (S8)

and for the out modes we replace $\omega$ by $-\omega$ in the equations above.
While these solutions are, in principle, analytic, in practice the convergence
becomes extremely slow far from the horizon; as a result, it is more practical
to use them to find the asymptotic value of the modes. Specifically, we
evaluate the Jaffé asymptotic expressions at fixed boundary values of $r$ just
outside and inside the horizon, then use the differential equation to
extrapolate further away. Our results are robust to the choice of values for
these two parameters.

Unfortunately, this numerical integration method is not without issues. The
most serious problems occur when the energy of the mode is very small compared
to the angular momentum $l$ – or rather, the effective potential $V_{l}(r)$.
In that case, continuing the analogy with one-dimensional scattering, the
incident and reflected waves will have similar magnitudes, while the
transmitted wave will be very, very small; the relative disparity in
magnitudes amplifies numerical imprecision. Some care is therefore needed.

_Convergence of the mode sums and integrals —_ The evaluation of the response
function is done mode by mode, and each ‘mode’ makes a positive contribution
to the response function. Therefore, calculating the response function is a
simple matter of summing over the contributions of all modes, questions of
convergence notwithstanding. It thus remains to show, analytically or
otherwise, that the sum over $l$ and $\omega$ remains finite; while a suitably
chosen switching function will accelerate convergence over $\omega$, we must
carefully consider the sum over $l$. Outside, of course, the sum over $l$
behaves similarly to the Minkowski case, but the sum inside the black hole
needs more care.

In practice, our program proceeds as follows: we evaluate the mode functions
for each $l$ on a discretized $\omega$ grid, which is then interpolated and
integrated over, taking care to evaluate at enough $\omega$ points to ensure
smoothness of the interpolating function. (The low $\omega$ regime, in
particular, requires evaluation to high resolution in order to capture the
features of the sharp peaks found there.) These $\omega$ integrals are then
summed over to a suitable maximum $l$.

This approach raises a technical issue: by summing over modes in $\omega$ and
$l$, we are effectively summing in two infinite dimensions. We deal with this
by putting in cut-offs on $\omega$ and $l$ such that any term beyond the
cutoff values have contributions smaller than $0.0001\%$ of the total sum. We
find that in general the cut-off $l_{\rm max}$ grows with larger distances,
smaller switching widths or decreasing energy gaps. We have also checked the
robustness of our results by refining the grid of the $\omega$ summation.

_Comparison with a static detector in Minkowski spacetime —_ Here we compare
our infalling detector results to the response of a static detector in a
thermal state in Minkowski spacetime, and, as the zero temperature limit, to
the response of an inertial detector in Minkowski vacuum.

Starting with the general expression (5) for the response function, and
assuming that $W(\tau,\tau^{\prime})$ depends on its two arguments only
through their difference, $W(\tau,\tau^{\prime})=W(\tau-\tau^{\prime})$, the
response function takes the form [S7]

$\displaystyle
F(\Omega)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\textrm{d}\omega\left|\widehat{\chi}(\omega)\right|^{2}\widehat{W}(\Omega-\omega),$
(S9)

where the hat stands for the Fourier transform, in the convention

$\displaystyle\widehat{\chi}(\omega)=\int_{-\infty}^{\infty}\textrm{d}\tau\,e^{-i\omega\tau}\chi(\tau).$
(S10)

We now specialise to four-dimensional Minkowski spacetime. For a detector that
is static in the rest frame of a thermal state of temperature $\beta^{-1}>0$,
we have [S8]

$\displaystyle\widehat{W}_{\beta}(\omega)=\frac{\omega}{2\pi\\!\left(e^{\beta\omega}-1\right)}.$
(S11)

We write the switching function (16) as

$\displaystyle\chi(\tau)=\begin{cases}\cos^{4}\bigl{(}\tau/(2\alpha)\bigr{)}&\text{for}\
\ |\tau|<\pi\alpha,\\\ 0&\textrm{otherwise},\end{cases}$ (S12)

where the positive parameter $\alpha$ is related to $\Delta$ in (16) by
$\alpha=\Delta/\pi$, and the parameter $\tau_{mid}$ in (16) has been dropped,
by time translation invariance of the Minkowski thermal state. It follows that

$\displaystyle\widehat{\chi}(\omega)$ $\displaystyle=\alpha H(\alpha\omega),$
(S13)

where

$\displaystyle H(z)$ $\displaystyle=\frac{3\sin(\pi z)}{z(1-z^{2})(4-z^{2})}.$
(S14)

Combining these observations, the response function takes the form

$\displaystyle
F_{\beta}(\Omega)=\frac{1}{4\pi^{2}}\int_{-\infty}^{\infty}\frac{\textrm{d}z\,z}{1-e^{-(\beta/\alpha)z}}\,H^{2}(z+\alpha\Omega).$
(S15)

A plot of $F_{\beta}$ is given in Figure S1. In the zero temperature limit,
$\beta\to\infty$, $F_{\beta}$ reduces to the Minkowski vacuum response,

$\displaystyle
F_{vac}(\Omega)=\frac{1}{4\pi^{2}}\int_{0}^{\infty}\textrm{d}z\,zH^{2}(z+\alpha\Omega),$
(S16)

which can be expressed as a sum of sine and cosine integrals [S9].

Figure S1: Plot of the Minkowski thermal state response $F_{\beta}$ (S15) as a
function of $\alpha\Omega$ and $\beta/\alpha$.

We wish to compare $F_{\beta}$ and $F_{vac}$ with the Schwarzschild response
shown in Figure 1, in which $\Delta=0.3\times 2M$ and $\Omega=5/(2M)$. In
terms of our parameters, this means $\alpha\Omega=1.5/\pi$, and, using for
$\beta$ the Schwarzschild infinity value $8\pi M$, $\beta/\alpha=40\pi^{2}/3$.
With these values, we have $F_{vac}\approx 0.0334638$ and $F_{\beta}\approx
0.0334861$, shown as horizontal dashed lines in Figure 1.

In the Schwarzschild responses shown in Figures 2 and 3, the normalisation at
infinity is obtained by setting $F_{\text{asymp}}$ to $F_{\beta}$ with
$\beta=8\pi M$ and $\alpha=\Delta/\pi$.

References

  * [S1]

W. G. Unruh, “Notes on black hole evaporation,” Phys. Rev. D 14, 870 (1976).

  * [S2]

N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge
University Press, Cambridge, UK, 1982).

  * [S3]

M. Casals, S. R. Dolan, B. C. Nolan, A. C. Ottewill and E. Winstanley,
“Quantization of fermions on Kerr space-time,” Phys. Rev. D 87, 064027 (2013)
[arXiv:1207.7089 [gr-qc]].

  * [S4]

L. Hodgkinson, Particle detectors in curved spacetime quantum field theory,
Ph.D. thesis, University of Nottingham (2013), arXiv:1309.7281 [gr-qc].

  * [S5]

A. Lanir, A. Levi and A. Ori, “Mode-sum renormalization of
$\langle\Phi^{2}\rangle$ for a quantum scalar field inside a Schwarzschild
black hole,” Phys. Rev. D 98, 084017 (2018) [arXiv:1808.06195 [gr-qc]].

  * [S6]

E. W. Leaver, “Solutions to a generalized spheroidal wave equation:
Teukolsky’s equations in general relativity, and the two-center problem in
molecular quantum mechanics,” J. Math. Phys. 27, 1238 (1986).

  * [S7]

C. J. Fewster, B. A. Juárez-Aubry and J. Louko, “Waiting for Unruh,” Class.
Quant. Grav. 33, 165003 (2016) [arXiv:1605.01316 [gr-qc]].

  * [S8]

S. Takagi, “Vacuum Noise and Stress Induced by Uniform Acceleration: Hawking-
Unruh Effect in Rindler Manifold of Arbitrary Dimension,” Prog. Theor. Phys.
Suppl. 88, 1 (1986).

  * [S9]

NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release
1.1.2 of 2021-06-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I.
Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S.
Cohl, and M. A. McClain, eds.