# Singular Excitement Beyond the Horizon of a Rotating Black Hole

Sijia Wang<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada  María R. Preciado-
Rivas<EMAIL_ADDRESS>Department of Applied Mathematics,
University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada  Institute for
Quantum Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Massimilano Spadafora<EMAIL_ADDRESS>Department of Physics,
University of Calabria, Rende, Calabria, 87036, Italy  Robert B. Mann
<EMAIL_ADDRESS>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, Waterloo, Ontario, N2L 2Y5, Canada

###### Abstract

Previous studies have shown that an Unruh-DeWitt (UDW) detector, when coupled
linearly to a massless scalar field and permitted to fall radially into
certain black holes, will exhibit non-monotonicity in its transition
properties near the horizon. Specifically, the transition probability of a
detector freely falling into a (3+1)-dimensional Schawrzschild black hole,
when considering the Unruh and Hartle-Hawking vacuum states, was shown to
possess a local extremum at horizon crossing [K.K. Ng et al., New J. Phys. 24
(2022) 103018]. The transition rate of a detector falling into a static
(2+1)-dimensional Bañados-Teitelboim-Zanelli (BTZ) black hole, for the Hartle-
Hawking state, was also found to have multiple local extrema near the horizon
under certain parameter settings [M.R. Preciado-Rivas et al.,
arXiv:2402.14908v1]. These discoveries are of interest, as they suggest that
the event horizon of a black hole may be distinguishable to a local probe when
QFT effects are included. In this paper, we explore the problem of a UDW
detector falling freely into a rotating BTZ black hole. We numerically compute
the detector’s transition rate for different values of black hole mass, black
hole angular momentum, detector energy gap, and field boundary conditions at
infinity. Our results lead to a more generalized description of the behaviour
of particle detectors in BTZ black hole spacetime, from which the previous
non-rotating BTZ case can be retrieved in the limit as angular momentum
vanishes.

##  I Introduction

In the absence of a quantum theory of gravity, relativistic quantum
information provides valuable technical and conceptual insight for
investigating quantum effects in curved spacetime. One common operational
approach in RQI is to use an idealized particle detector to probe quantum
fields, particularly in situations where the notion of a particle is ill-
defined. The simplest such detector is the Unruh-DeWitt (UDW) detector [1, 2],
a two-level quantum system that couples locally to a quantum scalar field. One
quantity of interest in this framework is the transition probability between
the two levels of the detector; another is the derivative of this probability
with respect to total detection time, known as the transition rate. Some well-
known results obtained using the UDW detector model are the Unruh effect [1,
2, 3], wherein an accelerated detector in flat spacetime exhibits a thermal
response proportional to its acceleration, and entanglement degradation [4],
which occurs when two detectors are in relative non-inertial motion.

The particle detector model has also been used to study black hole spacetimes.
Some results obtained in this vein are described in [5], for the static
detector outside of a Schwarzschild black hole, which was found to respond
thermally to the Hartle-Hawking-Israel state [6]; and in [7], for a stationary
detector co-rotating with a Bañados-Teitelboim-Zanelli (BTZ) black hole, which
was found to thermalize at the Hawking temperature.

Much less is known about the response of detectors that freely fall into a
black hole and cross its horizon. The general expectation is that the
transition rate will be smooth and the detector will not thermalize. It has
been shown that a detector freely falling across the horizon of a Bertotti-
Robinson spacetime has a transition rate that linearly decreases as the
horizon is crossed [8]. Several studies have been carried out thus far in
lower-dimensional settings. [9, 10] numerically compute the transition rate of
a UDW detector freely falling into a (1+1)-dimensional Schwarzschild black
hole, finding evidence that the thermal properties of the detector are
gradually lost during infall. In [11], the transition rate of detector
infalling towards a (1+1)-dimensional Cauchy horizon was shown to diverge for
a field in both the Unruh and Hartle-Hawking-Israel vacuum states. Another
study [12] has shown that it is possible for two infalling detectors to
harvest entanglement in (1+1)-dimensional Schwarzschild spacetime, even when
causally separated by the black hole horizon. The response rate of a detector
falling radially in a (2+1)-dimensional static BTZ spacetime was computed in
[7], but the detector had only been switched on outside of the horizon.

Recently, the response function of a detector initially at rest at infinity
and falling freely toward and across the horizon of a (3+1)-dimensional
Schwarzschild black hole was computed [13] for a scalar field in the Unruh and
Hartle-Hawking-Israel states. In both cases, contrary to expectation, a local
extremum was observed in the transition rate as the detector crossed the
horizon. The locus of the extremum was found to depend on the interaction
duration and the detector’s energy gap. This result is of significant
interest, as it suggests that a local probe knows when it is crossing the
Schwarzschild horizon.

The transition rate of a freely falling detector in (2+1)-dimensional non-
rotating BTZ spacetime was computed in [14], exhibiting similar, but richer,
behaviour compared to the Schwarzschild case. In particular, the transition
rate as a function of detector proper time was shown to oscillate slowly
outside of the black hole, but fluctuate more dramatically near the horizon
for certain ranges of the black hole’s mass and the detector’s initial
position. The transition rate then increases and diverges at the singularity.
Moreover, the transition rate was found to be non-differentiable at certain
detector proper times; these cusps in the transition rate were dubbed
“glitches.” Theoretically, one could use these glitches as an early warning
system of whether the observer is about to fall into the horizon. Both the
profile of the detector transition rate and the locations of the glitches are
highly dependent on the black hole mass and the detector’s initial position.

Expanding on the static BTZ black hole research, we investigate the transition
rate of a UDW detector falling into a rotating BTZ black hole in (2+1)
dimensions. The rationale for this investigation is threefold:

  1. 1.

We wish to understand the effect of frame dragging on the response of an
infalling detector.

  2. 2.

We anticipate the possibility of new behaviours in detector transition rate
when rotation is considered, particularly when the black hole approaches
extremality. This intuition is informed by previous research showing that
entanglement harvesting can be amplified by black hole rotation [15],
especially as angular momentum increases to the extremal limit.

  3. 3.

The detector transition rate for a BTZ black hole is efficient to compute.
Rather than the mode sum approach used in the (3+1)-dimensional Schwarzschild
case [13], which is computationally intensive, we can calculate the transition
rate as a much simpler image sum.

We find that the detector transition rate curves for the rotating BTZ
spacetime take on a similar shape as in the non-rotating BTZ case, but the
rotation of the black hole pushes the features in the curves further away from
the singularity. Strikingly, however, the transition rate of the detector does
not diverge at the time the detector reaches the black hole’s singularity when
rotation is present. Additionally, new glitch behaviours arise in the rotating
case. In the non-rotating BTZ case, the detector encounters only one set of
glitches, which are symmetric in the image sum. “Symmetric” in this context
means that, if a glitch occurs at a proper time $\tau_{g}$ in term $n$ of the
image sum, it will also occur at the same time $\tau_{g}$ in term $-n$. In the
rotating BTZ spacetime, by contrast, the detector encounters four types of
glitches—two of which are symmetric in $n$ and two of which are asymmetric.
The sets of glitches present different behaviours for small $|n|$, but all
asymptote monotonically towards the time to the inner horizon (or a reflection
of this about the time to the singularity) as $n\rightarrow\pm\infty$. One can
see how the special case of the non-rotating BTZ spacetime may be recovered
from the rotating BTZ setting: in the limit of vanishing black hole angular
momentum, the time to the inner horizon and its reflection go to the
singularity; thus the glitches are collapsed into the singularity as
$n\to\pm\infty$. The pattern of the glitches in the transition rate provides
us with a predictor for the structure of the spacetime since it depends on the
black hole mass and the detector’s switch-on moment, allowing us to anticipate
if the detector will encounter the black hole horizons.

The outline of our paper is as follows. In section II, we present the basic
formalism needed to compute the transition rate of a UDW detector freely
falling in a rotating BTZ spacetime. We present in section III the results of
our study, showing how the transition rate depends on the mass of the black
hole, the angular momentum of the black hole, the boundary conditions of the
field at asymptotic infinity, and the energy gap of the detector. We also
present the locations of the four types of glitches in the rotating case, as
well as the transition rate behaviour at the time the detector reaches the
black hole singularity. In section IV, we draw conclusions from the results
and propose directions for future research.

##  II Setup

We consider the Unruh-DeWitt model for a detector coupled to a quantum
massless scalar field. Since much of the formalism has already been derived
[7], we shall only reiterate the ideas relevant to our study.

The UDW detector that we consider is a pointlike qubit, having two states
denoted by $|0\rangle$ and $|E\rangle$. The energy eigenvalues of the states
are $0$ and $E$ respectively, where $E$ may be positive or negative. The
detector starts out in the $\ket{0}$ state, and a positive (negative) $E$
indicates that $\ket{0}$ is the detector’s ground (excited) state. The
detector moves on a timelike worldline $\mathsf{x}(\tau)$, parameterized by
its proper time $\tau$, and interacts with a massless scalar field
$\hat{\phi}$ via the interaction Hamiltonian

$\hat{H}_{\mathrm{int}}=\lambda\chi(\tau)\hat{\mu}(\tau)\otimes\hat{\phi}(\tau)\,,$
(1)

where
$\hat{\mu}(\tau)=\ket{E}\bra{0}e^{\mathsf{i}E\tau}+\ket{0}\bra{E}e^{-\mathsf{i}E\tau}$
is the detector’s monopole moment operator, $\lambda$ is a coupling constant,
and $\chi$ is the switching function, specifying how the detector is switched
on and off.

Perturbatively, the probability that the detector makes a transition from the
state $\ket{0}$ to the state $\ket{E}$ is proportional to the response
function, given by [16]

$\mathcal{F}=\int\mathrm{d}\tau^{\prime}\mathrm{d}\tau^{\prime\prime}\mathrm{e}^{-\mathrm{i}E(\tau^{\prime}-\tau^{\prime\prime})}W(\tau^{\prime},\tau^{\prime\prime}),$
(2)

to leading order, where $W$ is the pullback along the detector’s worldline of
the field’s Wightman function in the state in which the field was initially
prepared [17, 18, 19, 20]. In (2+1) spacetime dimensions, we may take $\chi$
to be the characteristic function of a time interval (i.e., sharp switching)
since $\mathcal{F}$ remains well defined despite the nonsmoothness at the
switch-on and switch-off moments [7]. Moreover, we may consider the detector’s
transition rate, given by (a multiple of) the derivative of $\mathcal{F}$ with
respect to the switch-off moment. The transition rate in the sharp switching
limit and to leading order in the coupling constant is [7]

$\dot{\mathcal{F}}_{\tau}=\frac{1}{4}+2\int_{0}^{\Delta\tau}\mathrm{d}s\operatorname{Re}\left[\mathrm{e}^{-\mathrm{i}Es}W(\tau,\tau-s)\right],$
(3)

where $\tau$ denotes the switch-off moment and $\Delta\tau$ is the total
proper time that the detector operates. $\Delta\tau=\tau-\tau_{0}$, where
$\tau_{0}$ is the switch-on moment. The transition rate can be interpreted in
terms of an ensemble of identical detectors, all following the trajectory
$\mathsf{x}(\tau)$; it tells us the fraction of detectors per unit time that
have undergone a transition when observed at time $\tau$. To measure
$\dot{\mathcal{F}}$ as a function of $\tau$ requires a set of identical
ensembles, each observed at different values of $\tau$ [20, 21].

The line element of the rotating BTZ spacetime is

$ds^{2}=-h(r)dt^{2}+\frac{1}{h(r)}dr^{2}+r^{2}\left[N_{\varphi}(r)dt+d\varphi\right]^{2},$
(4)

where

$\displaystyle h(r)$
$\displaystyle=-M+\frac{r^{2}}{\ell^{2}}+\frac{J^{2}}{4r^{2}},$ (5)
$\displaystyle N_{\varphi}(r)$ $\displaystyle=-\frac{J}{2r^{2}}.$ (6)

$M$ is the (dimensionless) mass of the black hole, and $J$ is its angular
momentum. Additionally, $t\in\mathbb{R}$, $r\in(0,\infty)$, and
$\varphi\in[0,2\pi)$. This metric (4) is a vacuum solution to the Einstein
field equations with cosmological constant $\Lambda=-1/\ell^{2}$, where
$\ell~{}(>0)$ is the AdS length. The outer and inner horizons (where $h(r)=0$)
are given by the expressions

$r^{2}_{\pm}=\frac{1}{2}\left(M\ell^{2}\pm\sqrt{M^{2}\ell^{4}-\ell^{2}J^{2}}\right).$
(7)

By setting $M=1$ (without loss of generality) and $\varphi\rightarrow
y\in(-\infty,\infty)$ in (4), one obtains AdS3-Rindler spacetime. Conversely,
the BTZ spacetime can be obtained from AdS3-Rindler by identifying
$y\rightarrow\phi\in(0,\sqrt{M})$, and then rescaling $\phi$ and the radial
and time coordinates to obtain (4); it is therefore locally equivalent to
anti-de Sitter spacetime. This fact will be important when one considers a
quantum field in BTZ spacetime, for which the correlation functions can be
written as a sum of correlators in AdS3.

We pause to comment on the nature of the BTZ spacetime at $r=0$. In both the
rotating and non-rotating cases, the curvature of the BTZ spacetime is
constant, with a well-defined Kretschmann scalar at every point, including as
$r\to 0$. However the identification that produces the BTZ spacetime is
performed along a spacelike Killing vector, which is simply $\partial_{\phi}$
for the static BTZ black hole and a function of $\partial_{\phi}$ and
$\partial_{t}$ for the rotating BTZ black hole.

These Killing vectors can become null and then timelike in some regions, which
must be excised from the covering anti-de Sitter space to make the
identifications valid; otherwise the spacetime will have closed timelike
curves [22]. This in turn has the consequence that the spacetime becomes
geodesically incomplete and the surface $r=0$ becomes a singularity in the
causal structure [22, 23]. If $J=0$, the manifold is additionally non-
Hausdorff at $r=0$ [22]. If the black hole is formed from collapsing matter
then a genunine curvature singularity appears at $r=0$ [24].

We consider a freely falling detector that is initially co-rotating with the
black hole. The timelike geodesic followed by the detector is given in BTZ
coordinates by

$\displaystyle r(\tau)$
$\displaystyle=r_{0}\sqrt{1-\left[1-\left(\frac{r_{-}r_{+}}{r_{0}^{2}}\right)^{2}\right]\sin^{2}{\left(\frac{\tau}{\ell}\right)}},$
$\displaystyle\begin{split}t(\tau)&=\frac{\ell^{2}}{r_{+}^{2}-r_{-}^{2}}\left(r_{+}\operatorname{arctanh}{\left[\frac{r_{+}}{r_{0}}\sqrt{\frac{r_{0}^{2}-r_{-}^{2}}{r_{0}^{2}-r_{+}^{2}}}\tan{\left(\frac{\tau}{\ell}\right)}\right]}\right.\\\
&\
-r_{-}\left.\operatorname{arctanh}{\left[\frac{r_{-}}{r_{0}}\sqrt{\frac{r_{0}^{2}-r_{+}^{2}}{r_{0}^{2}-r_{-}^{2}}}\tan{\left(\frac{\tau}{\ell}\right)}\right]}\right),\end{split}$
(8)
$\displaystyle\begin{split}\varphi(\tau)&=\frac{\ell}{r_{+}^{2}-r_{-}^{2}}\left(r_{-}\operatorname{arctanh}{\left[\frac{r_{+}}{r_{0}}\sqrt{\frac{r_{0}^{2}-r_{-}^{2}}{r_{0}^{2}-r_{+}^{2}}}\tan{\left(\frac{\tau}{\ell}\right)}\right]}\right.\\\
&\
-r_{+}\left.\operatorname{arctanh}{\left[\frac{r_{-}}{r_{0}}\sqrt{\frac{r_{0}^{2}-r_{+}^{2}}{r_{0}^{2}-r_{-}^{2}}}\tan{\left(\frac{\tau}{\ell}\right)}\right]}\right),\end{split}$

where $r_{0}$ is the radial position at which the detector is switched on. To
simplify the notation, we will write $\tau/\ell$ as $\tau$.

We consider a massless conformally coupled quantum scalar field
$\hat{\phi}(\mathsf{x})$ satisfying the Klein-Gordon equation,

$(\Box-R/8)\hat{\phi}(\mathsf{x})=0\,,$ (9)

where $\Box$ is the d’Alembert operator, and $R$ is the Ricci scalar. The
Wightman function $W(\mathsf{x},\mathsf{x}^{\prime}):=\langle
0|\hat{\phi}(\mathsf{x})\hat{\phi}(\mathsf{x}^{\prime})|0\rangle$ is the two-
point correlation function in the vacuum state $|0\rangle$ of the field. Since
AdS3 spacetime is locally equivalent to BTZ spacetime, we may express the
field correlations in BTZ as an image sum of the Wightman function in AdS3
spacetime. The specific form is [25, 23]:

$\begin{split}&W_{\text{BTZ}}(\mathsf{x},\mathsf{x}^{\prime})=\sum_{n=-\infty}^{\infty}W_{\text{AdS}}(\mathsf{x},\Gamma^{n}\mathsf{x}^{\prime})\\\
&=\frac{1}{4\pi\sqrt{2}\ell}\sum_{n=-\infty}^{\infty}\left[\frac{1}{\sqrt{\sigma_{\epsilon}(\mathsf{x},\Gamma^{n}\mathsf{x}^{\prime})}}-\frac{\zeta}{\sqrt{\sigma_{\epsilon}(\mathsf{x},\Gamma^{n}\mathsf{x}^{\prime})+2}}\right],\end{split}$
(10)

where $\Gamma:(t,r,\varphi)\mapsto(t,r,\varphi+2\pi)$. The parameter
$\zeta\in\\{-1,0,1\\}$ specifies the boundary conditions of the field at
asymptotic infinity: Neumann, transparent, or Dirichlet, respectively. The
Wightman function with Neumann or Dirichlet boundary conditions corresponds to
a KMS state and is analytic outside of the black hole horizon, thus
characterizing a Hartle-Hawking state [26, 25]. On the other hand, the state
with the transparent boundary condition does not have a clear physical meaning
[26, 25].

$\sigma_{\epsilon}(\mathsf{x},\mathsf{x}^{\prime})$ is the squared geodesic
separation (scaled by $\ell^{2}$) between the two points in the covering AdS3
spacetime, and the $\epsilon$ notation encodes the distributional character of
the Wightman function in the limit as $\epsilon\rightarrow 0_{+}$. Let the
unprimed coordinates correspond to $\mathsf{x}(\tau)$ on the detector
trajectory (II) and the primed coordinates to $\mathsf{x}(\tau-s)$. The
explicit form of
$\sigma_{\epsilon}(\mathsf{x}(\tau),\Gamma^{n}\mathsf{x}(\tau-s))$ is then

$\begin{split}\sigma_{\epsilon}(\mathsf{x}(\tau)&,\Gamma^{n}\mathsf{x}(\tau-s))=\\\
&-1+\cos{(\tau-s)}\left[\cos{(\tau)}K_{n,1}+\sin{(\tau)}K_{n,2}\right]\\\
&+\sin{(\tau-s)}\left[\sin{(\tau)}K_{n,3}+\cos{(\tau)}K_{n,4}\right],\end{split}$
(11)

which we will denote in short as $\sigma_{\epsilon,n}(\tau,\tau-s)$, where

$\displaystyle K_{n,1}=\frac{1}{r_{+}^{2}-r_{-}^{2}}\left[\cosh{\left(2\pi
n\frac{r_{+}}{\ell}\right)}\left(r_{0}^{2}-r_{-}^{2}\right)-\cosh{\left(2\pi
n\frac{r_{-}}{\ell}\right)}\left(r_{0}^{2}-r_{+}^{2}\right)\right]$ (12)
$\displaystyle
K_{n,2}=\frac{1}{r_{0}}\frac{\sqrt{\left(r_{0}^{2}-r_{+}^{2}\right)\left(r_{0}^{2}-r_{-}^{2}\right)}}{r_{+}^{2}-r_{-}^{2}}\left[\sinh{\left(2\pi
n\frac{r_{+}}{\ell}\right)}r_{-}-\sinh{\left(2\pi
n\frac{r_{-}}{\ell}\right)}r_{+}\right]$ (13) $\displaystyle
K_{n,3}=\frac{1}{r_{0}^{2}}\frac{1}{r_{+}^{2}-r_{-}^{2}}\left[\cosh{\left(2\pi
n\frac{r_{-}}{\ell}\right)}r_{+}^{2}\left(r_{0}^{2}-r_{-}^{2}\right)-\cosh{\left(2\pi
n\frac{r_{+}}{\ell}\right)}r_{-}^{2}\left(r_{0}^{2}-r_{+}^{2}\right)\right]$
(14) $\displaystyle
K_{n,4}=\frac{1}{r_{0}}\frac{\sqrt{\left(r_{0}^{2}-r_{+}^{2}\right)\left(r_{0}^{2}-r_{-}^{2}\right)}}{r_{+}^{2}-r_{-}^{2}}\left[\sinh{\left(2\pi
n\frac{r_{-}}{\ell}\right)}r_{+}-\sinh{\left(2\pi
n\frac{r_{+}}{\ell}\right)}r_{-}\right]=-K_{n,2}.$ (15)

The transition rate is then

$\dot{\mathcal{F}}_{\tau}=\frac{1}{4}+\frac{1}{2\pi\sqrt{2}}\sum_{n=-\infty}^{\infty}\int_{0}^{\Delta\tau}\mathrm{d}s\operatorname{Re}\left[\mathrm{e}^{-\mathrm{i}Es}\left(\frac{1}{\sqrt{\sigma_{\epsilon,n}(\tau,\tau-s)}}-\frac{\zeta}{\sqrt{\sigma_{\epsilon,n}(\tau,\tau-s)+2}}\right)\right].$
(16)

Several observations can be made about the rotating case relative to the non-
rotating case described in [14].

First, from (16) when calculating $\dot{\mathcal{F}}_{\tau}$, the $n\neq 0$
terms are not invariant under $n\rightarrow-n$. This is different from the
non-rotating case, where there was invariance in $n\rightarrow-n$. Thus, the
$n$ and $-n$ terms must be summed separately in the rotating case.

Second, while the transition rate formula (16) is derived using expression
(11), which is valid only in the black hole exterior, (16) holds over the
detector’s full trajectory, even when the detector operates after entering the
black hole. This follows from the existence of a global analytic chart, by
analytic continuation in (16). This analytic continuation can be performed in
both the rotating and non-rotating cases.

As in the non-rotating case, the transition rate decomposes as

$\dot{\mathcal{F}}_{\tau}=\dot{\mathcal{F}}_{\tau}^{n=0}+\dot{\mathcal{F}}_{\tau}^{n\neq
0},$ (17)

where $\dot{\mathcal{F}}_{\tau}^{n=0}$ consists of the $n=0$ term, and
$\dot{\mathcal{F}}_{\tau}^{n\neq 0}$ consists of the sum over the $n\neq 0$
terms. $\dot{\mathcal{F}}_{\tau}^{n=0}$ is the transition rate of the detector
in AdS3.

The square roots in (16) are positive for positive arguments, and they are
analytically continued to negative arguments by giving $s$ a small negative
imaginary part [7, 17]. This procedure is valid in both the rotating and non-
rotating cases.

For certain values of $s$, the right hand side of (11) vanishes. Specifically,
$\sigma_{\epsilon,n}(\tau,\tau-s)$ vanishes for

$s_{1,2}^{*}(\tau)=\tau-\arctan{\left(\frac{ac\pm
b\sqrt{a^{2}+b^{2}-c^{2}}}{bc\mp a\sqrt{a^{2}+b^{2}-c^{2}}}\right)},$ (18)

where $a=\sin{(\tau)}K_{n,3}+\cos{(\tau)}K_{n,4}$,
$b=\cos{(\tau)}K_{n,1}+\sin{(\tau)}K_{n,2}$ and $c=1$. Similarly, the quantity
$\sigma_{\epsilon,n}(\tau,\tau-s)+2$ in the second term of the integrand
vanishes for

$s_{3,4}^{*}(\tau)=\tau-\arctan{\left(\frac{df\pm
e\sqrt{d^{2}+e^{2}-f^{2}}}{ef\mp d\sqrt{d^{2}+e^{2}-f^{2}}}\right)},$ (19)

with $d=\sin{(\tau)}K_{n,3}+\cos{(\tau)}K_{n,4}$,
$e=\cos{(\tau)}K_{n,1}+\sin{(\tau)}K_{n,2}$ and $f=-1$. These values of $s$
yield singularities in the integrand of (16). However, since they are of the
inverse square root type, they are still integrable.

When $s_{1,2,3,4}^{*}(\tau)$ is equal to one of the bounds of integration in
(16) (i.e., $s_{1,2,3,4}^{*}(\tau)=0$ or $s_{1,2,3,4}^{*}(\tau)=\tau$), the
transition rate function will gain a jump discontinuity in its first
derivative at time $\tau$. Such non-differentiable points in the transition
rate, dubbed “glitches,” were previously observed in the case of a UDW falling
into a non-rotating BTZ black hole [14].

For the static black hole, the glitches encountered by the detector as it
falls towards the singularity originate exclusively from the
$s_{1}^{*}(\tau)=\tau$ case – that is, when $\sigma_{\epsilon,n}(\tau,\tau-s)$
vanishes at the upper bound $\tau$. The singularities $s^{*}_{1}(\tau)$ in the
integrand have a geometric origin, occurring when the detector’s trajectory
crosses the future light cone of the switch-on event. There are many discrete
values of the proper time at which this can happen because the light cones
wrap around the $\varphi$ direction.

For the rotating black hole, all four types of singularities,
$s_{1}^{*}(\tau)$, $s_{2}^{*}(\tau)$, $s_{3}^{*}(\tau)$, and
$s_{4}^{*}(\tau)$, in the integrand contribute in the domain over which the
detector is switched on. We see that we then have three new types of glitches.
The explicit locations of all glitches are respectively

$\displaystyle\tau_{1}^{*}=\arctan{\left(\frac{K_{n,2}+K_{n,1}\sqrt{K_{n,2}^{2}+K_{n,1}^{2}-1}}{K_{n,1}-K_{n,2}\sqrt{K_{n,2}^{2}+K_{n,1}^{2}-1}}\right)},$
(20)
$\displaystyle\tau_{2}^{*}=\arcsin{\left(\sqrt{\frac{1-K_{n,1}}{K_{n,3}-K_{n,1}}}\right)},$
(21)
$\displaystyle\tau_{3}^{*}=\arctan{\left(\frac{-K_{n,2}+K_{n,1}\sqrt{K_{n,2}^{2}+K_{n,1}^{2}-1}}{-K_{n,1}-K_{n,2}\sqrt{K_{n,2}^{2}+K_{n,1}^{2}-1}}\right)},$
(22)
$\displaystyle\tau_{4}^{*}=\arcsin{\left(\sqrt{\frac{1+K_{n,1}}{K_{n,1}-K_{n,3}}}\right)},$
(23)

where the glitches at $\tau_{1}^{*}$ have the same origin as in the static
case, $s^{*}_{1}$ being coincident with the upper bound of integration (i.e.,
$s^{*}_{1}(\tau)=\tau$). The glitches at $\tau_{3}^{*}$ arise from $s^{*}_{3}$
being coincident with the upper bound of integration, which is
$s_{3}^{*}(\tau)=\tau$, but now correspond to the detector’s trajectory
crossing a future-directed timelike curve beginning at the switch-on event.
This is a new feature absent in the static case. The glitches at
$\tau_{2}^{*}$ and $\tau_{4}^{*}$ arise from $s_{2}^{*}$ and $s_{4}^{*}$ being
coincident with the lower bound of integration (i.e., $s_{2,4}^{*}(\tau)=0$) –
another feature that is absent in the static case. Those from $\tau_{2}^{*}$
occur when the detector’s trajectory crosses the future light cone of the
switch-on event, whereas those from $\tau_{4}^{*}$ occur when the detector
intersects a future-directed timelike curve beginning at the switch-on event.
For all four types of glitches, both the light cones and the future-directed
timelike curves can wrap around the $\varphi$ direction, leading to many
possible discrete glitches.

##  III Results

In this section, we present our results for the transition rate of the
detector with different black hole angular momenta, boundary conditions of the
field, masses of the black hole, and energy gaps of the detector. We also
examine the glitches in the rotating case versus the non-rotating case. In all
our results, the initial position of the detector is $r_{0}=100\,r_{+}$, and
we have chosen the detector to be switched-on when the it starts to freely
fall, that is, $\tau_{0}/\ell=0$. Hence, the switch-off time or the total
detection time coincides with the proper time of the detector,
$\Delta\tau/\ell=\tau/\ell$. Consequently, the detector is at its initial
position $r=r_{0}$ when $\Delta\tau/\ell=0$, and it reaches the (geodesic
incompleteness) singularity at $r=0$ when $\Delta\tau/\ell=\pi/2$.

In all of our computations, we truncated the image sums at a term whose
absolute value over the entire detection time interval was on the order of one
part in 10000 of the cumulative total, even in the worst-case scenario. We
only required 1000 terms inside the outer horizon and 200 terms outside of it
in any image sum to obtain curves that did not visually change upon adding
more terms.

###  III.1 Transition rate for different angular momenta and boundary
conditions

Figure 1: The transition rate $\dot{\mathcal{F}}_{\tau}$ of a UDW detector
freely falling into a rotating BTZ black hole, with $r_{0}/r_{+}=100$,
$M=10^{-4}$, $E\ell=-5$, and (a) $J/M\ell=0$, (b) $J/M\ell=0.5$, and (c)
$J/M\ell=0.999$. We calculated the image sum from $n=-N$ to $n=+N$, where
$N=200$ in the exterior of the black hole and $N=1000$ behind the outer
horizon. The time for the detector to reach the outer horizon is indicated by
the vertical solid line, and the time for the detector to reach the inner
horizon is indicated by the vertical dashed line. The right edge of the plot
is the time to the singularity.

In Fig. 1, we show the transition rate $\dot{\mathcal{F}}_{\tau}$ of the
detector as a function of the total detection time $\Delta\tau/\ell$ for the
three boundary conditions, $\zeta=-1$, 0, and 1, given black hole angular
momentum of (a) $J/M\ell=0$ (static case), (b) $J/M\ell=0.5$, and (c)
$J/M\ell=0.999$. In all three plots, we fix the mass of the black hole to be
$M=10^{-4}$ and the energy gap of the detector to be $E\ell=-5$.

For the values of $M$, $r_{0}$, and $J$ considered in Fig. 1, the behaviour of
the transition rate curve changes over different ranges of $\Delta\tau/\ell$,
and can be analyzed in three cases: (i) the detector is switched off when it
is far away from the black hole, (ii) the detector is switched off when its
radial position is comparable to the outer horizon of the black hole, and
(iii) the detector is switched off after it has crossed the inner horizon of
the black hole.

For short durations of operation, when the detector is far away from the
horizon, the transition rate undergoes slow oscillation as dictated by the
complex exponential function in the integrand of (16). This oscillation, when
computed for the different boundary conditions, will be out of phase for some
values of $\zeta$ compared to the others. For $\Delta\tau/\ell$ corresponding
to a detector position near the outer horizon of the black hole, the
fluctuations in the transition rate become more dramatic. Glitches begin to
appear in this region, and after their occurrence, the transition rate curves
fluctuate with greater frequency and amplitude. The increased fluctuation
arises from the contributions of the $|n|\geq 1$ terms in the image
sum–specifically, when the $n$-th term in the image sum encounters a glitch,
its behaviour will change from slow and weak oscillation to a more rapid and
appreciable oscillation. A detailed discussion on the role of the $|n|\geq 1$
terms is presented in Fig. A1 and Fig. A2 of the appendix.

Outside the inner horizon of the black hole, the transition rate curves are
smooth, except for a finite number of glitches. However, if the detector
remains on after crossing the inner horizon, it observes a transition rate
that is highly irregular. From the insets in Fig. 1 (b) and (c), it can be
seen that, past the inner horizon, the transition rate curves become “jagged”
for the rotating case, with many inflection points at extremely close times to
one another. As the detector continues and approaches the singularity, we see
that the transition rate overall increases. We found, however, that only the
transition rate function in the static case diverges at the singularity.

In the region of proper time where $r(\Delta\tau/\ell)$ is near or behind the
outer horizon, the effect of varying the boundary conditions is overshadowed
by the larger effects noted in the previous paragraphs. Aside from an overall
translation in the vertical direction, there is no appreciable qualitative
difference between the transition rate curves for $\zeta=-1$, 0, and 1 near or
inside of the black hole’s outer horizon. For this reason, in the following
sections, we fix $\zeta=0$.

Comparing the static case in Fig. 1 (a) with the rotating cases in Fig. 1 (b)
and (c), we see that the transition rate curve is similar in all three cases,
except for the addition of new behaviours behind the inner horizon when
angular momentum is present. The main effect of increasing angular momentum is
to push the transition rate curve towards the left (i.e., away from the
singularity). We observe that as the angular momentum increases, the detector
will encounter the features of the transition rate curve sooner.

###  III.2 Transition rate for different black hole masses and angular momenta

Figure 2: The transition rate $\dot{\mathcal{F}}_{\tau}$ of a UDW detector
freely falling into a rotating BTZ black hole, with $r_{0}/r_{+}=100$,
$E\ell=-5$, $\zeta=0$, and (a) $M=10^{-2}$, (b) $M=10^{-4}$, (c) $M=10^{-5}$,
and (d) $M=10^{-6}$. We calculated the image sum from $n=-N$ to $n=+N$, where
$N=200$ in the exterior of the black hole and $N=1000$ behind the outer
horizon. The time for the detector to reach the outer horizon is indicated by
the vertical solid black line, and the times for the detector to reach the
inner horizon are indicated by the vertical dashed lines, where the colours of
the dashed lines match the colours of the black hole angular momenta. The
right edge of the plot is the time to the singularity.

In Fig. 2, we show the transition rate $\dot{\mathcal{F}}_{\tau}$ of the
detector as a function of the detector’s proper time $\Delta\tau/\ell$ for
five black hole angular momenta, $J/M\ell=0$, 0.5, 0.9, 0.99, and 0.999, given
a black hole mass of (a) $M=10^{-2}$, (b) $M=10^{-4}$, (c) $M=10^{-5}$, and
(d) $M=10^{-6}$. In all four plots, we fix the energy gap of the detector to
be $E\ell=-5$ and the boundary condition of the field to be $\zeta=0$.

For the values of $M$, $r_{0}$, and $J$ considered in the previous subsection,
we identified three regions of behaviour for the transition rate function with
respect to the detector operating time. To reiterate, these are (i) the far
region, where the detector is in the distant exterior of the black hole, (ii)
the near outer horizon region, where the detector crosses or approaches close
enough to the outer horizon, and (iii) the inner horizon region, where the
detector has crossed the second horizon of the black hole. Region (i) is
characterized by smooth, slow oscillation of the transition rate function,
region (ii) is characterized by larger, faster oscillations interspersed with
glitches, and region (iii) is characterized by an extremely high concentration
of inflection points, giving the transition rate curve a jagged appearance.

From Fig. 2, it can be seen that detector transition rate is highly sensitive
to black hole mass. In particular, region (ii) expands outward (to the left)
as black hole mass decreases. Region (i) shrinks as a result, while region
(iii) remains confined behind the inner horizon. As region (ii) grows, the
detector encounters glitches further and further away from the horizon.
Moreover, by adjusting the values of $M$ and $J$, the first type of glitch can
be made to appear arbitrarily close to $\Delta\tau/\ell=0$. Thus, for
sufficiently small black hole masses, the detection of glitches in the
transition rate may serve as an early warning for horizon crossing as proposed
in [14].

In the large mass case ($M=10^{-2}$), region (ii) is virtually non-existent.
The smooth oscillation of the transition rate function continues until the
inner horizon, beyond which irregularities begin to appear. In the medium mass
case ($M=10^{-4}$), region (ii) has a relatively small extent, stretching from
$\Delta\tau/\ell\approx 1.5$ to the inner horizon. Thus, we see slow
oscillation for most of the duration of operation, followed by a relatively
brief period of more pronounced oscillations, leading into region (iii). We
then considered a transitional mass, $M=10^{-5}$, for which the features of
the pronounced oscillations in region (ii) develop appreciably and grow
leftward. Finally, in the small mass case ($M=10^{-6}$), the detector
traverses through region (ii) for most of its operating duration, the first
glitch being encountered as early as $\Delta\tau/\ell\approx 5\pi/32$.

The magnitude of the transition rate increases more quickly beyond the horizon
when the black hole mass is small. Far away from the horizon,
$\dot{\mathcal{F}}_{\tau}$ oscillates about the value of approximately 0.5,
which is true for all four plots in Fig. 2. In the large mass case, the
transition rate remains on the order of 1 up to $\Delta\tau/\ell=\pi/2$.
However, in the medium mass case, the magnitude of the transition rate grows
to roughly 10 in the interval up to $\Delta\tau/\ell=\pi/2$, while in the
small mass case, the magnitude grows over 100. Strikingly, we observe again
that the transition rate function does not diverge at $\Delta\tau/\ell=\pi/2$
except when $J/M\ell=0$. We note that the smooth appearance of the transition
rate curves behind the inner horizon, for the smaller masses, is purely due to
the scaling of the plots. When zoomed in sufficiently, one would observe the
same jagged shape of the transition rate curve as in the large mass plot.

Finally, we comment on the effect of angular momentum. As was observed in Fig.
1, increasing the angular momentum of the black hole stretches the transition
rate curves away from the singularity. Region (iii) grows outward as the
angular momentum pushes the inner horizon towards the outer horizon, and
region (ii) also stretches outward to some extent. Consequently, keeping all
other parameters constant, the same features of the transition rate curve are
encountered by the detector at an earlier proper time when the angular
momentum is larger.

###  III.3 Transition rate for different detector energy gaps

Figure 3: The transition rate $\dot{\mathcal{F}}_{\tau}$ of a UDW detector
freely falling into a rotating BTZ black hole, with $r_{0}/r_{+}=100$,
$J/M\ell=0.999$, $\zeta=0$, and (a) $M=10^{-2}$, (b) $M=10^{-4}$, and (c)
$M=10^{-6}$. We calculated the image sum from $n=-N$ to $n=+N$, where $N=200$
in the exterior of the black hole and $N=1000$ behind the outer horizon. The
time for the detector to reach the outer horizon is indicated by the vertical
solid line, and the time for the detector to reach the inner horizon is
indicated by the vertical dashed line. The right edge of the plot is the time
to the singularity.

In Fig. 3, we show the transition rate $\dot{\mathcal{F}}_{\tau}$ of the
detector as a function of the detector’s proper time $\Delta\tau/\ell$ for
four energy gaps, $E\ell=-20$, -5, 5, and 20, given a black hole mass of (a)
$M=10^{-2}$, (b) $M=10^{-4}$, and (c) $M=10^{-6}$. In all three plots, we fix
the angular momentum of the black hole to be $J/M\ell=0.999$ and the boundary
condition of the field to be $\zeta=0$.

We find that the transition rate curves for all of the energy gaps considered
respect the observations regarding black hole mass variation in the previous
subsection. Namely, the smaller the mass of the black hole, the further
outward region (ii) grows, which holds for all detector energy gaps, as
expected since the energy gap does not affect the location of the glitches.
Additionally, the magnitude of the transition rate grows more quickly near the
singularity when the black hole mass is small, which is observed for any
energy gap considered.

In region (i), which is the weak oscillation portion of
$\dot{\mathcal{F}}_{\tau}$, we find that larger $|E\ell|$ corresponds to
faster oscillations. This result is as expected, as $E$ appears in the
argument of the complex exponential function in the integrand of Eq. (16), and
thus directly adjusts the oscillation frequency. When the detector is far away
from the black hole, the transition rate curves corresponding to
$E\ell=+\Omega$ and $E\ell=-\Omega$ are roughly reflections of one another
with respect to the horizontal line $\dot{\mathcal{F}}_{\tau}=0.25$.

As the detector approaches the black hole, the behaviour of the transition
rate curves becomes significantly different for different energy gaps,
especially when the distance between the detector and the black hole’s outer
horizon is small. Although the energy gap does not affect the location of the
glitches, its magnitude affects the frequency of the oscillations after each
glitch. The insight to be gained is that, when constructing a UDW detector,
the value of the energy gap plays a large role in defining what one expects to
observe as the detector falls into the black hole.

Figure 4: The transition rate $\dot{\mathcal{F}}_{\tau}$ of a UDW detector
freely falling into a rotating BTZ black hole, with $r_{0}/r_{+}=100$,
$E\ell=-20$, $\zeta=0$, and $M=10^{-6}$. We calculated the image sum from
$n=-N$ to $n=+N$, where $N=200$ in the exterior of the black hole and $N=1000$
behind the outer horizon. The time for the detector to reach the outer horizon
is indicated by the vertical solid black line, and the times for the detector
to reach the inner horizon are indicated by the vertical dashed lines, where
the colours of the dashed lines match the colours of the black hole angular
momenta. The right edge of the plot is the time to the singularity.

To further connect these results to our observations in the previous
subsection, we show, in Fig. 4, the transition rate $\dot{\mathcal{F}}_{\tau}$
of the detector as a function of the detector’s proper time $\Delta\tau/\ell$
for $E\ell=-20$ (as opposed to $E\ell=-5$ in Fig. 2), given a black hole mass
of $M=10^{-6}$. We then compare the transition rates for different black hole
angular momenta–specifically, $J/M\ell=0$, 0.5, 0.9, 0.99, and 0.999, as
before.

We find that, when the detector is near the outer horizon, the property that
increasing angular momentum stretches the transition rate curve to the left is
preserved for this new energy gap. However, the behaviour in the small mass
case ($M=10^{-6}$) is the most complex, due to the far extent of region (ii).
Thus, for example, we see significant discrepancy between the transition rate
curves in Fig. 4 at $\Delta\tau/\ell\approx 3\pi/8$, which arises from the
highly non-trivial interaction between the oscillatory properties of the
transition rate and the effects of glitches.

###  III.4 Glitches in the rotating BTZ problem

Figure 5: The distribution of glitches in the (a) static case compared to the
(b) rotating case with $J/M\ell=0.9$, where we have fixed $M=10^{-4}$ and
$r_{0}/r_{+}=100$. We plot detector proper time $\Delta\tau/\ell$ on the
vertical axis and term number $n$ on the horizontal axis. The four types of
glitches are denoted by different markers. The circle and square markers
correspond to glitches that arise from singularities in the terms independent
of the field’s boundary conditions in (16), whereas the diamond and triangle
markers correspond to glitches that arise from the terms that depend on the
boundary conditions. We have extended the vertical axis past the proper time
to the singularity for purposes of illustration. The lower solid (dashed)
horizontal line indicates the time to the outer (inner) horizon. The upper
lines are reflections of the time to the horizons with respect to
$\Delta\tau/\ell=\pi/2$.

In Fig. 5, we show the distribution of glitches in the static BTZ case
compared to the rotating BTZ case. We set the parameters to be $M=10^{-4}$ and
$r_{0}/r_{+}=100$, with $J/M\ell=0.9$ for the angular momentum of the black
hole in the rotating case. The detector proper time $\Delta\tau/\ell$ is
plotted on the vertical axis against $n$ on the horizontal axis, where $n$ is
the term number in the image sum. Thus, each point on the plot indicates a
glitch at operating time $\Delta\tau/\ell$ arising from the $n$-th term of the
summation.

Recall from section II that the four types of glitches arise from setting
$s_{i}^{*}(\tau)=0$ or $s_{i}^{*}(\tau)=\tau$, with the explicit proper times
of the glitches given in Eqs (20-23). In the non-rotating case, the glitches
are symmetric under $n\rightarrow-n$. In the rotating case, two of the four
types of glitches become asymmetric under $n\rightarrow-n$, though they remain
symmetric to one another by a reflection through the “origin,” which we take
to be ($n=0$, $\Delta\tau/\ell=\pi/2)$. The fact that the glitches are not
symmetric as $n\rightarrow-n$ in the rotating case is consistent with the fact
that $\dot{\mathcal{F}}_{\tau}$ is in general not symmetric as
$n\rightarrow-n$. It is important to note that glitches beyond
$\Delta\tau/\ell=\pi/2$ are unphysical, as the detector will have already
reached the singularity by then, but we include them in Fig. 5 to illustrate
the mathematical structure of the non-differentiable points of the transition
rate.

In the rotating case, we also observe that three of the four types of glitches
are confined behind the inner horizon. Asymptotically, all glitches (and their
reflection with respect to $\Delta\tau/\ell=\pi/2$) accumulate at the inner
horizon (or its reflection with respect to $\Delta\tau/\ell=\pi/2$) as
$n\rightarrow\pm\infty$. Only one type of glitch (which arises from the first
square root for $s_{1}^{*}(\tau)=\tau$) is able to escape from the inner
horizon for small $|n|$. Thus, in the limit as $J\rightarrow 0$ for the black
hole, and the inner horizon goes to the singularity, we are left with only one
observable type of glitch. We find that, despite the richer behaviour of
glitches in the rotating BTZ case, we are able to retrieve the non-rotating
case in a continuous limit.

Finally, we reiterate that the proper times at which the detector encounters
the glitches depends on the initial position of the detector as well as the
mass and angular momentum of the black hole. As mass decreases or angular
momentum increases, the glitches will stretch downwards in the plots in Fig.
5, meaning that they will be encountered by the detector at earlier proper
times. However, the overall shapes of the curves formed by the four types of
glitches will remain the same.

###  III.5 Detector transition rate at the singularity

Figure 6: The transition rate at the singularity
($\dot{\mathcal{F}}_{\pi\ell/2}$) as a function of black hole angular
momentum, for different black hole masses. Plots (a) and (b) correspond to a
detector energy gap of $E\ell=-5$, while plots (c) and (d) correspond to a gap
of $E\ell=+5$. For all plots, $\zeta=0$ and $r_{0}/r_{+}=100$. When the black
hole mass is large, the transition rate at the singularity is roughly constant
for all angular momenta. When black hole mass is small, the transition rate at
the singularity decays as an inverse power of $J/M\ell$, as shown by parallel
linear traces on the double-log plot (a), (c). However, at intermediate
masses, the relationship between $\dot{\mathcal{F}}_{\pi\ell/2}$ and $J/M\ell$
is highly non-trivial, with fluctuations and glitches. Plots (b) and (d)
highlight the non-trivial glitch structure within this intermediary band of
masses.

As observed in previous sections, the transition rate of the detector does not
diverge at the singularity when the black hole is rotating. This absence of a
divergence is an important qualitative difference between the static and
rotating BTZ scenarios. We conjecture that the divergence in the transition
rate for vanishing angular momentum at the time the detector reaches the
singularity is a consequence of the manifold breaking down at the singularity.
We find for the rotating black hole that the transition rate is everywhere
finite, commensurate with the manifold being well defined at the singularity.

In Fig. 6, we show the transition rate $\dot{\mathcal{F}}_{\tau}$ of the
detector at the singularity (i.e., when $\tau/\ell=\pi/2$) as a function of
black hole angular momentum $J/M\ell\in(0,1)$ for black hole mass ranging from
$M=10^{-6}$ to $M=10$. We compare the $E\ell=-5$ and $E\ell=+5$ cases.

When black hole mass is large, the detector’s transition rate at the
singularity is essentially independent of the angular momentum $J$. This
occurs because the image sum converges more quickly for larger $M$. When $M$
is sufficiently large, the sum is dominated by the $n=0$ term, and thus, in
the large mass limit, the transition rate at the singularity tends to the
transition rate in pure AdS3 spacetime.

When black hole mass is small, we observe the detector’s transition rate at
the singularity to decay as an inverse power of $J/M\ell$, as seen from the
straight lines in the double-log plot in Fig. 6 (a), (c). As long as the mass
is non-zero, the transition rate at the singularity will be finite.
Numerically we find that the transition rate continues to increase as
$M\rightarrow 0$ for all $J$.

Between the small and large mass extremes is a range of masses for which the
relationship between the transition rate at the singularity and the angular
momentum is highly non-trivial. For these masses the transition rate at the
singularity is a non-monotonic function of $J/M\ell$. Moreover, there are
glitches in the transition rate, which occur when the geodesic distance under
one of the square roots in (16) vanishes at $\tau/\ell=\pi/2$. In Fig. 6, we
considered the transparent boundary condition $\zeta=0$, and thus the glitches
in the plots correspond only to $\sigma_{n}=0$. However, for the other
boundary conditions, one obtains additional glitches when $\sigma_{n}+2=0$.

The behaviour of the transition rate as a function of $J/M\ell$ does not
depend qualitatively on the gap of the detector. We obtain the same large,
small, and intermediary response structure at the singularity for positive
detector energy gaps ($E\ell=+5$) and negative ones ($E\ell=-5$). This occurs
because the Wightman function depends on $M$ but not on $E$, whereas $E$
controls the phase in (16). Thus, the small and large mass limiting behaviours
are independent of $E$.

##  IV Conclusion

We numerically calculated the transition rate of an Unruh-DeWitt detector
coupled to a massless conformal scalar field as the detector falls freely into
a rotating BTZ black hole along a co-rotating trajectory. We explored the
effects of different black hole masses, black hole angular momenta, detector
energy gaps, and field boundary conditions at infinity. To calculate the
transition rate, we expressed the Wightman function of the BTZ black hole as
an image sum of Wightman functions in AdS3, and then computed the transition
rate as a sum of the contributions arising from each image. Unlike in the non-
rotating case, the terms in the transition rate when rotation is present are
not invariant under $n\rightarrow-n$. We found that selecting different
boundary condition for the field at infinity did not change the qualitative
behaviour of the transition rate curves. However, black hole mass and detector
energy gap did significantly impact the transition rate, while black hole
angular momentum also played a role. Specifically, the presence of angular
momentum caused a stretching of the transition rate curve away from the
singularity.

As with the non-rotating BTZ case, we discovered that the detector’s
transition rate is non-differentiable at certain discrete values of the
detector’s proper time. Following the nomenclature in [14], we refer to these
points of non-differentiability as glitches. There are four types of glitches
in the rotating BTZ case, which are asymmetric for $n\rightarrow-n$. The
exotic glitches (i.e., those that were not seen in the static case) live
entirely behind the inner horizon, and thus vanish as expected as angular
momentum of the black hole goes to zero. The locations of the glitches depend
on the black hole mass and angular momentum, and also (though we did not
consider it in this study) the initial radial position of the detector.

We observed that, in the presence of rotation, the detector’s transition rate
at the singularity does not diverge. Furthermore, we identified a range of
masses for which the transition rate at the singularity exhibits glitches when
plotted as a function of black hole angular momentum. In the small mass limit,
the transition rate at singularity decays as an inverse power of angular
momentum, while in the large mass limit, the transition rate at singularity
tends to the transition rate in pure AdS3 spacetime.

Our study was motivated firstly by the observation that the response function
of a detector interacting with the Hartle-Hawking(-Israel) state of a massless
scalar field and freely falling into a (3+1)-dimensional Schwarzschild black
hole exhibits a local extremum at the horizon crossing [13]. Following this
work with the Schwarzschild black hole was a further study [14], wherein a UDW
detector was permitted to fall radially into a (2+1)-dimensional non-rotating
BTZ black hole. In this latter study, in addition to the discovery of
glitches, it was found that more than one local extremum could exist near the
horizon under certain parameters. Our results for the (2+1)-dimensional
rotating BTZ black hole are commensurate with both of these previous studies,
and moreover provides a more general description of UDW detectors in BTZ
spacetime (from which the previous non-rotating BTZ case can be derived). The
possibility of using glitches as an early warning for horizon crossing, so
long as black hole mass is sufficiently small, remains valid in the presence
of rotation.

One limitation of our study is that we calculated the transition rate of the
detector during infall, as opposed to the response function, which is
proportional to the transition probability of the detector and thus easier to
interpret. This constraint makes the comparison with [13] somewhat cumbersome,
as the (3+1)-dimensional Schwarzschild study calculated the response function
directly. However, we may still observe a number of qualitative similarities
between the two sets of results. Another limitation is that we have considered
a (2+1)-dimensional spacetime with constant curvature, whereas physical black
holes are (3+1)-dimensional and have varying spacetime curvature.
Nevertheless, we expect the results of our study to provide useful insight on
the nature of quantum fields in rotating black hole spacetime backgrounds.

A future avenue of research would be to study the extanglement dynamics of
multiple detectors in rotating BTZ spacetime, specifically when one or more of
the detectors falls behind the horizon. Additionally, the work on infalling
UDW detectors can be extended to other black hole spacetimes where the
curvature may not be constant. The role of hidden topologies may also be a
topic of interest–for instance, comparing the transition rates of detector
falling into a BTZ black hole versus its geon counterpart. Finally, one could
explore the transition probability of a detector falling into a black hole
while coupled to other fields–for example, a fermionic field.

## Acknowledgments

We thank Jorma Louko for helpful discussions on aspects of this project. This
work was supported in part by the Natural Sciences and Engineering Research
Council of Canada. M. R. Preciado-Rivas gratefully acknowledges the support
from the Mike and Ophelia Lazaridis Graduate Fellowship.

*

##  Appendix A Individual terms

In this appendix, we plot selected individual contributions from terms in the
image sum using $M=10^{-4}$ and $J/M\ell=0.9$, shown in Fig. A1 for $\zeta=0$
and in Fig. A2 for $\zeta=\pm 1$. Following the notation used in Fig. 5, the
circle and square markers indicate glitches that correspond to the first
($\sigma_{n}(\tau,\tau-s)=0$, $s(\tau)=\tau$) and second type
($\sigma_{n}(\tau,\tau-s)=0$, $s(\tau)=0$) of glitches, which arise from the
terms independent of the boundary conditions. The diamond and triangle markers
correspond to the third ($\sigma_{n}(\tau,\tau-s)+2=0$, $s(\tau)=\tau$) and
fourth type ($\sigma_{n}(\tau,\tau-s)+2=0$, $s(\tau)=0$) of glitches, which
arise from terms that depend on the boundary conditions.

Figure A1: A few individual contributions to the transition rate calculated
with $\zeta=0$ and the same parameters as in Fig. 5. (a), (c) and (e)
correspond to terms of negative $n$, meanwhile (b), (d) and (f) correspond to
terms of positive $n$. The time at the outer (inner) horizon is indicated with
a solid (dashed) vertical line. The glitches are shown with markers following
the notation in Fig. 5. The circle and square markers correspond to the first
and second type of glitches which arise from the terms independent of the
boundary condition.

Figure A2: Individual contributions to the transition rate calculated with
$\zeta=-1$ or $\zeta=1$ and the same parameters as in Fig. 5. We show values
of $n$ that differ considerably with respect to the terms calculated with
$\zeta=0$. These correspond to (c), (e) and (f) of Fig. A1. The glitches are
shown with markers following the notation in Fig. 5. The diamond and triangle
markers correspond to the third and fourth type of glitches, which arise from
singularities in the term that depend on the field’s boundary conditions.

As seen in Fig. A1 (a) and (b), the shape of the first few terms is mostly
determined by the oscillations following the first type of glitches. On the
other hand, there are no oscillations following the second type of glitches,
which are located very close to the time to the singularity. For these first
terms, the glitches for negative and positive terms occur at times that are
relatively close to each other. As a result, the shape of the individual
contributions to the transitions rate are essentially the same for positive
and negative $n$ terms.

As observed in Fig. A1 (c) and (d), it is for $n=-5$ and $n=5$ that
qualitative differences in the contributions become apparent: while the
negative $n$ terms attain three local extrema, the positive $n$ terms attain
only two or none. This symmetry loss happens because the glitches of the first
type are located in the band between the time to the horizon and the time to
the singularity for negative $n$ but are located past the time to the
singularity when $n$ is positive. Moreover, these glitches tend to the inner
horizon crossing as $n\to-\infty$, which is observed in Fig. A1 (e), but tend
to a reflection of the inner horizon crossing about $\pi/2$ for $n\to\infty$.

In Fig. A1 (e) and (f), we observe that the amplitude of the contributions
decreases as $|n|$ increases. The terms decaying fast enough is what ensures
that the transition rate remains finite over all the domain of the total
detection time, including the time to the singularity and the horizons. We
also observe that the overall shape is maintained but shrunken along the total
detection time axis for bigger $|n|$. This behaviour is what produces the
jagged region in the transition rate behind the inner horizon crossing.

The role of the third (diamond) and fourth (triangle) types of glitches is
illustrated in Fig. A2, where we have chosen to show the plots analogous to
Fig. A1 (c), (e), and (f) because they differ the most when compared to the
contributions calculated with $\zeta=0$. For small $|n|$ or for positive $n$,
glitches of the third type only occur after the time to the singularity.
Hence, these contributions do not differ substantially from $\zeta=0$.
However, as the magnitude of $n$ increases, the third type of glitches starts
to appear in the contributions for negative $n$ terms, as observed in Fig. A2
(a) and (b). As expected, the oscillation behind this type of glitches has an
opposite phase for $\zeta=\pm 1$.

We observe in Fig. A2 (c) and (d) that glitches of both the third and fourth
types are present in the terms for negative and large magnitude $n$, but the
amplitude of the curves after the fourth type of glitches is negligible
compared to the rest of the contribution. Since the glitches of the third type
tend to a reflection of the inner horizon crossing with respect to $\pi/2$ as
$n\to\infty$, they are not present in the plots of Fig. A2 (e) and (f).
Although the behaviour is qualitatively different for $\zeta=\pm 1$, the
amplitude of the contributions after the fourth type of glitches is again
negligible when compared to other terms.

Overall, we note that the terms that are mostly determined by the first type
of glitches, which are the ones with small $|n|$, resemble those that were
present in the non-rotating case of [14]. We also note that when more terms
are included in the image sum, the difference between the static and the
rotating black holes becomes apparent because the exotic types of glitches are
present. These new types of glitches, that were absent in the static case,
occur close to one another behind the inner horizon crossing and do not pile
up at the time to the singularity, which results in transition rate curves
that are jagged behind the inner horizon crossing and finite at the time to
the singularity.

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