# Learn to Compress (LtC): Efficient Learning-based Streaming Video Analytics

Quazi Mishkatul Alam  University of CaliforniaRiversideCAUSA<EMAIL_ADDRESS>,  Israat Haque  Dalhousie UniversityHalifaxCanada<EMAIL_ADDRESS>and  Nael
Abu-Ghazaleh  University of CaliforniaRiversideCAUSA<EMAIL_ADDRESS>
(2023)

###### Abstract.

Video analytics are often performed as cloud services in edge settings, mainly
to offload computation, and also in situations where the results are not
directly consumed at the video sensors. Sending high-quality video data from
the edge devices can be expensive both in terms of bandwidth and power use. In
order to build a streaming video analytics pipeline that makes efficient use
of these resources, it is therefore imperative to reduce the size of the video
stream. Traditional video compression algorithms are unaware of the semantics
of the video, and can be both inefficient and harmful for the analytics
performance. In this paper, we introduce _LtC_ , a collaborative framework
between the video source and the analytics server, that efficiently learns to
reduce the video streams within an analytics pipeline. Specifically, LtC uses
the full-fledged analytics algorithm at the server as a teacher to train a
lightweight student neural network, which is then deployed at the video
source. The student network is trained to comprehend the semantic significance
of various regions within the videos, which is used to differentially preserve
the crucial regions in high quality while the remaining regions undergo
aggressive compression. Furthermore, LtC also incorporates a novel temporal
filtering algorithm based on feature-differencing to omit transmitting frames
that do not contribute new information. Overall, LtC is able to use 28-35%
less bandwidth and has up to 45% shorter response delay compared to recently
published state of the art streaming frameworks while achieving similar
analytics performance.

††copyright: acmcopyright††journalyear: 2023††doi:
XXXXXXX.XXXXXXX††conference: ACM Multimedia; October 28, 2023 – November 3,
2023; Ottawa, Canada††price: 15.00††isbn: 978-1-4503-XXXX-X/18/06

##  1\. Introduction

Video analytics are becoming increasingly important at the edge with the
widespread integration of camera sensors in many scenarios, such as autonomous
systems and IoT devices, to benefit from the visual awareness of the
surroundings. For example, authorities are ramping up the deployment of
surveillance systems to assist in law enforcement and crime prevention (Lin
and Newley Purnell, 2019); transportation systems are moving towards partial
or full automation (Hancock et al., 2019); and unmanned aerial vehicles are
starting to revolutionize a number of industries and remote military
operations (Intelligence, 2021).

In such systems, the videos captured by the cameras are often transferred to
remote cloud servers for analytics (Chen et al., 2015; Chin et al., 2019; Wang
et al., 2019; Zhang et al., 2018; Du et al., 2020; Li et al., 2020), but this
incurs expensive video transportation. Several considerations drive this model
including: (1) the analytics algorithm may require more resources than
available on the end devices; or (2) the analytics consumers may not be
located on the device itself (Qiu et al., 2018) or may require access to
multiple streams of data (Kolar et al., 2016). Transporting high-quality video
to support accurate analytics, consumes significant bandwidth. Thus, a key
research question regarding streaming video analytics is: _how to reduce the
volume of the video data without sacrificing accuracy with respect to the
downstream analytics?_

Conventional video compression algorithms, such as MPEG (Grois et al., 2013)
prioritize maximizing the perceived visual quality of the video over
preserving the essential features for analytics (Pakha et al., 2018; Du et
al., 2020). For example, these algorithms degrade the video quality uniformly
across both regions that are salient for analytics (e.g., object regions) and
those that are not (e.g., background regions). As a result, we are left with
the unsatisfying choice of either losing accuracy by degrading the quality of
the object regions along with the background regions, or losing bandwidth by
keeping everything in sufficient quality to preserve the analytics accuracy
(Pakha et al., 2018).

An alternative approach is semantic compression: such algorithms preserve
necessary features while aggressively degrading unimportant regions, achieving
high compression while preserving analytics accuracy. A challenge in this
approach is how to identify what regions are salient without running the full
analytics at the source. Our insight is that differentiating the relevant
regions from the background regions in a video is easier than the full
analytics task. In the context of object detection and classification,
identifying which areas may have objects is much simpler than the full
classification problem. Another observation that simplifies the problem is
that, given a specific video context, we typically encounter a small subset of
the possible objects – for example, in a highway surveillance context, we
expect different types of vehicles, but not animals, or furniture.

LtC leverages these two observations by using a light-weight student neural
network at the video source, which is trained by a Deep Neural Network (DNN)
based teacher network at the server. The student network identifies the
salient regions within video frames, allowing the source to differentially
compress the video before sending over the network. This training method is
suitable for the analytics tasks where the teacher network infers the
locations of the object regions in addition to the primary task, such as,
object detection and segmentation. We also contribute a novel temporal
filtering algorithm that uses a feature based discriminator to identify
successive video frames where semantic information has not changed
significantly, allowing us to avoid their transmission entirely.

Since LtC operates by specializing the student network to the encountered
scenarios, it is important to detect when these scenarios have changed, as it
is likely to be the case in dynamic environments or with mobile cameras. LtC
also reacts to this occurrence of _concept drift_ when the server discovers
that the information relayed by the source is not effective (e.g., the higher
resolution regions do not yield objects). We observe that the layers of the
student network closer to the input (which we collectively call _the encoder_)
are likely to remain unchanged with the scenario since they embed general
features (i.e., shapes, patterns, etc.). Instead, we train only the layers
closer to the output (which we call _the extension_) mapping features to
posteriors, which significantly reduces the update overhead. The student
network consists of merely 0.5% of the teacher network’s total parameters, and
in the event of concept drift we only need to update as little as 2.2% of the
student’s parameters.

In order to ensure fair comparison, we have implemented LtC and other
baselines by extending an open-source emulator (Du et al., 2020). The video
dataset was obtained from live-streaming public surveillance cameras and
videos from public streaming platforms. Additionally, we regulated the end-to-
end traffic conditions using an open-source network emulator (NetEm, year).
When compared to the closest spatial compression baseline (DDS from SIGCOMM
2020) (Du et al., 2020), LtC uses 23% less bandwidth and has 21-45% shorter
response time. Moreover, it is able to filter 8-14% more frames than the
closes temporal compression baseline (Reducto, also from SIGCOMM 2020) (Li et
al., 2020), and as a result, reduces bandwidth comsumption by 8%. Combining
both spatial and temporal compression, LtC uses 28-35% less bandwidth while
having 14-45% shorter response time than these state-of-the-art alternatives.
Also, LtC substantially outpuerforms commercial video compression standards
including AWStream (Zhang et al., 2018), Cloudseg (Wang et al., 2019) and
Glimpse (Chen et al., 2015) both in terms of F1-score and bandwidth use.

In summary, the contributions of this paper are as follows:

  * •

One-shot semantic spatial compression using lightweight student network. We
use a student network to implement semantic compression without frequent
feedback from the server.

  * •

Temporal filtering through efficient deep feature differencing. We develop a
new approach for filtering redundant frames by measuring changes in the
feature space. We show that this approach is more stable to unimportant
changes in the video (e.g., due to wind, illumination, etc.) and therefore
more effective than manually crafted features because it uses context-specific
knowledge (Li et al., 2020).

  * •

Efficient adaption to concept drift. We introduce an efficient concept drift
detection and update mechanism, significantly lowering the cost of updating
the detector.

##  2\. Motivation and Opportunity

The pertinent question that arises is, _why do we need different compression
algorithms for video analytics?_ Conventional video compression algorithms
(e.g., HVEC or MPEG) are designed to maintain human-perceived visual quality;
these compression algorithms uniformly degrade the quality across the video.
In contrast, the goal of semantic compression is to maintain the accuracy of
downstream analytics, and therefore, by using differential compression
techniques it is able to achieve a better bandwidth-accuracy tradeoffs. The
process of spatial and temporal compression with respect to object
classification is presented in Figure 1.

(a) Spatial compression

(b) Temporal compression

Figure 1. Spatial compression is intra-frame, while temporal compression is
inter-frame

To demonstrate the potential of semantic compression in comparison to
conventional video compression in the context of video analytics, we carry out
a series of experiments. Our results indicate that semantic compression offers
better bandwidth-accuracy tradeoffs than conventional codecs, such as MPEG. To
evaluate the performance of these compression algorithms, we calculate their
F1-scores by comparing the outputs of the full-sized DNN on both the
compressed and the original video data. Alternatively, this approach can be
thought of as the full-sized DNN being placed at the source to act as an
optimal semantic compression, which provides perfect estimates of the
positions and sizes of the required regions for analytics (by the same DNN) at
the server. This optimal algorithm, however, is not feasible in resource-
constrained source devices, and serves only to illustrate the size of the
opportunity.

Figure 2. F1-score of optimal semantic compression vs. MPEG

Figure 2 presents the performance of MPEG against the optimal versions of
spatial, temporal, and spatial-temporal algorithms. Each MPEG bar is
configured such that it consumes either similar or higher bandwidth than the
semantic compression. In all cases, the MPEG suffers a 5-13% F1-score drop for
similar bandwidth use. This is a substantial loss in terms of F1-score: over
the past 5 years, the improvement in the state of the art performance in
ImageNet classification is less than 10% (Beyer et al., 2020; Stock and Cisse,
2018). Also, to reach a 95% F1-score, MPEG is able to compress only 20% of the
highest quality video (e.g., the MPEG bar with 94.7% F1-score), while spatio-
temporal compression (albeit under optimal conditions) is able to compress
over 80%.

(a) Useful frame-regions

(b) Useful frames

Figure 3. Spatial and temporal redundancies in videos

We also use the optimal semantic compression algorithm to characterize the
redundancies present along the spatial and temporal dimensions in real videos
from our dataset listed in Table 3. Generally, in these videos the object
regions constitute a small fraction of the size of the field of view. Across
all the videos, around 80% of the video frames have 33% or fewer object
regions (Figure 3(a)). Also, approximately in 80% of the 1-second batches
sampled from these videos, only 20 frames in 30 frames-per-second (FPS) videos
and 5 frames in 15 FPS videos are necessary for analytics (Figure 3(b)).

##  3\. LtC Design and Implementation

The overall architecture of LtC is presented in Figure 4. The source-side
component is centered around a compact student neural network, which is able
to identify the object regions in order to perform spatial compression. The
student network also provides a tensor based feature vector extracted from its
intermediate layers to be used for temporal filtering. As the camera captures
the video, the source accumulates a fixed-size batch of frames to process at a
time. The student network applies both spatial and temporal compression before
it sends the batch over to the server. As the batch arrives at the server, the
teacher network performs analytics and looks for concept drift. Notably, the
student network is pretrained on historical data allowing itself to quickly
adapt to the present scenario with only a few frames.

Figure 4. Overall architecture of LtC

###  3.1. The student-teacher framework

The student-teacher framework is at the heart of LtC’s system design. This
method of training enables a large neural network to distill its knowledge
into a smaller network (Hinton et al., 2015; Wang and Yoon, 2021). As the
student network specializes only on a subset of the whole data, and tackles a
much simpler task than the actual analytics, its size can be kept small.
Moreover, initial pretraining and subsequent updating allows for fast
retraining with only a small amount of new data.

In the context of LtC, the teacher DNN trains the student network to determine
the objectness (likelihood of an object being present) in the regions within a
frame. In our approach, the video frames are divided into an array of non-
overlapping equal-size regions. These regions are labeled using the results of
the object detection from the teacher network $\mathcal{T}$, to be used in the
training of the student network $\mathcal{S}$.

The training starts as the camera sends $\mathcal{N}$ video frames
$F^{\mathcal{I}}=[f^{\mathcal{I}}_{1},f^{\mathcal{I}}_{2},...,f^{\mathcal{I}}_{\mathcal{N}}]$
to the server in $\mathcal{I}$th iteration. Upon receiving the frames, the
teacher network $\mathcal{T}$ generates $C_{f}$ bounding boxes
$BB^{f}=[bb^{f}_{1},bb^{f}_{2},...,bb^{f}_{C_{f}}]$ around objects in frame
$f$. Each frame $f$ is split into a $\mathcal{L}\times\mathcal{L}$ array
$X^{f}=[...,x^{f}_{ij},...]$ of non-overlapping same-sized regions, where $i$
and $j$ are 2D indices of a region in the frame. We denote the posterior from
the teacher $\mathcal{T}$ and the student $\mathcal{S}$ networks as
$P_{\mathcal{T}}(y|x^{f}_{ij})$ and $P_{\mathcal{S}}(y|x^{f}_{ij})$,
respectively, where $y$ is the measure of objectness. We use Intersection over
Union (IoU) function to determine $P_{\mathcal{T}}(y|x^{f}_{ij})$ in the
following manner:

(1)
$\mathcal{S}(x^{f}_{ij})=P_{\mathcal{T}}(y|x^{f}_{ij})=\begin{cases}1,&\max^{C_{f}}_{k=1}IoU(x^{f}_{ij},bb^{f}_{k})>0.5\\\
0,&\text{otherwise}\end{cases}$

In order to train a student network that approximates a pretrained teacher
network whose parameters are frozen, we only have to minimize the
Kullback–Leibler (KL)-divergence between the two distributions with respect to
the parameters of the student network, which is equivalent to minimizing the
following loss function in each iteration $\mathcal{I}$:

(2)
$\mathcal{L}(\theta_{S})=-\sum^{\mathcal{N}}_{f=1}\sum^{\mathcal{L}}_{i=1}\sum^{\mathcal{L}}_{j=1}P_{\mathcal{T}}(y|x^{f}_{ij})\log{P_{\mathcal{S}}(y|x^{f}_{ij};\theta_{S})},$

where $\theta_{S}$ are the parameters of the student network.

| Layer type | Output Shape | Param #
---|---|---|---
Encoder | Input Layer | 28x28x3 | 0
2D Convolution Layer | 28x28x16 | 448
2D Max Pooling | 14x14x16 | 0
2D Convolution Layer | 14x14x32 | 4640
2D Max Pooling | 7x7x32 | 0
Flatten Layer | 1568 | 0
Dense Layer | 128 | 200832
Extension | Input Layer | 128 | 0
Dense Layer | 32 | 4128
Dense Layer | 16 | 528
Output Layer | 1 | 17

Table 1. Architecture of the student network

Network type | Param # | Size
---|---|---
Faster R-CNN ResNet-101 (Teacher) | 44M | 196.5 MB
Encoder (Student) | 205920 | 844.2 KB
Extension (Student) | 4673 | 36.5 KB
Student Total | 210593 | 880.7 KB

Table 2. Sizes of the teacher, and the student network (broken into the
encoder, the extension)

We use the Faster R-CNN ResNet-101 (Ren et al., 2015) as our teacher network.
The student network (shown in Table 1) is inspired by the VGG-16 model
(Simonyan and Zisserman, 2014), which also uses alternating convolution and
max pooling layers. The parameters of the full network as well as the student
can be seen in Table 2. Notably, the teacher is over 200x larger than the
student network.

The student network $\mathcal{S}$ is divided into two sub-networks: (1)
Encoder $\mathcal{U}$ and (2) Extension $\mathcal{V}$ (sizes in Figure 2). The
encoder is responsible for converting the input frame-region into a feature
vector. The extension is the later part, which translates the feature vector
into an objectness score. In literature (Zhang et al., 2015; Pan and Yang,
2009), this kind of split is most commonly performed to rapidly train a neural
network for a task by reusing the features of a pretrained model on a similar
task as a starting point. For related tasks, the encoder part of the network
remains relatively unchanged, as it captures low-level features, such as,
shapes, patterns, and other features, which do not drift significantly for
similar objects. High-level features captured by the extension, such as
objectness, tend to be more sensitive to environment changes. In the context
of compression for video analytics, this encoder-extension split is a novel
contribution of LtC. This split has two benefits: (1) it provides a vector of
semantic features that is built implicitly during training, which we leverage
for temporal filtering, and (2) it allows for efficient update of the student
network by updating only the extension.

###  3.2. Temporal filtering using embedded features

Temporal filtering traditionally involves performing a comparison between
frames to see if the new frame includes sufficient new information. This
comparison typically uses manually crafted low-level features that operate
directly on the input image (e.g., SIFT and HOG) (Li et al., 2020). Such
features can be sensitive to small changes to the input resulting from the
presence of wind or similar effects that do not substantially change the
semantic features of the image.

We propose a novel temporal filtering approach that uses directly embedding in
the feature space of the student network to detect salient changes in the
image. Specifically, we use the differences in terms of the feature vector
acquired from the encoder to judge whether a frame should be discarded. The
main advantage using this feature vector is: it is highly contextual and up-
to-date compared to other manually crafted features. Moreover, it is more
robust against noises in the input caused by environmental factors such as
illumination or weather.

For temporal filtering, we calculate the sum of the differences between
feature vectors in consecutive frames. If this difference is below a threshold
value, then that pair is considered identical. We define a difference function
between frames $f_{1}$ and $f_{2}$ as follows:

(3)
$D(f_{1},f_{2})=\sum^{\mathcal{L}}_{i=1}\sum^{\mathcal{L}}_{j=1}{\lVert\mathcal{U}(x^{f_{1}}_{ij})-\mathcal{U}(x^{f_{2}}_{ij})\rVert}^{2}_{2}$

Instead of sending $\mathcal{N}$ frames $F^{\mathcal{I}}$ =
$[f^{\mathcal{I}}_{1},f^{\mathcal{I}}_{2},...,f^{\mathcal{I}}_{\mathcal{N}}]$
the camera uses the difference function to divide $F^{\mathcal{I}}$ into
$\mathcal{M}$ partitions
$[p^{\mathcal{I}}_{1},p^{\mathcal{I}}_{2},...,p^{\mathcal{I}}_{\mathcal{M}}]$
of consecutive frames in the $\mathcal{I}$th iteration, where
$1\leq\mathcal{M}\leq\mathcal{N}$ and all frames in a partition $p$ satisfy:

(4) $\max D(f_{1},f_{2})<th;\forall f_{1},f_{2}\in p$

Lastly, we construct the list of filtered frames $\hat{F}^{\mathcal{I}}$ by
selecting a representative frame $f$ from a partition $p$ as follows:

(5) $f=\operatorname*{argmin}_{x\in p}\sum_{\begin{subarray}{c}x\neq y\\\ y\in
p\end{subarray}}D(x,y)$

###  3.3. Spatial compression using objectness score

Spatial compression is performed after the temporal filtering. Based on the
objectness score of a frame-region given by the student network, that region
is either preserved in high-quality or degraded. LtC uses the _posterior_
$\mathcal{V}(\mathcal{U}(x^{f}_{ij}))$ to define an identity function as
follows:

(6) $I(x_{ij})=\begin{cases}1,&\mathcal{V}(\mathcal{U}(x^{f}_{ij}))>0.5\\\
0,&\text{otherwise}\end{cases}$

We keep a region $x_{ij}$ if $I(x_{ij})$ is 1, or compress it otherwise. Each
frame $f$ is converted into a $\mathcal{L}\times\mathcal{L}$ array
$X^{f}=[...,I(x_{ij})x^{f}_{ij},...]$ of non-overlapping same-sized regions,
where $i$ and $j$ are 2D indices of a region in the frame.

###  3.4. Updating the student network

The student network may become less effective following a significant change
in the environment or scene dynamics, leading to a concept drift. When concept
drift is detected at the server, an update of the student network is
triggered. A copy of the currently operating student network is maintained at
the server, which is evaluated by the teacher network in each iteration to
check if indeed object regions yield high objectness score. If the percentage
falls below a fixed threshold, the server initiates a student-teacher
training, causing the student network to change partially (freezing all other
parameters but the extension) or, in rare cases, wholly. Based on the degree
of change, the server then sends either the extension or the new student
network to the source as an update. We observe that 85% of the updates consist
only of the extension, which is approximately 25x smaller than the encoder as
can be seen in Table 2.

###  3.5. Discussion and General Properties of LtC

A state-of-the-art spatial compression algorithm, DDS (Du et al., 2020), uses
server-side feedback in order to identify the regions that require high-
quality transmission. Specifically, a low-quality baseline transmission of the
video is first sent to the server, which uses the full-sized DNN to identify
the regions of interest that it requires in higher resolution. It sends a
request back to the source, which in turn resends those regions at a higher
quality. Although this approach is able to achieve reasonably high F1-score,
it results in significant delay in server response. This delay includes the
network delay incurred during multiple requests and responses, as well as the
server processing delay after each response. Moreover, the number of feedback
used in the process adversely affects the response delay. These delays are
shown in Figure 5 as a function of the number of feedback rounds used for each
batch of frames. LtC solves this problem by training the student network to
identify the regions of interest, and the compressed video is sent in a single
shot. Although the student network is smaller than the teacher network, LtC
has the advantage of operating on the full resolution video; this is in
contrast to DDS which transmits a low resolution video to detect the areas
with objects.

Figure 5. DDS experiences delays proportional to the number of feedback rounds
used; LtC is one shot

A recently proposed temporal filtering algorithm, Reducto (Li et al., 2020),
uses a profiling based approach, where the server profiles batches of frames
against the optimal actions (what threshold to use for filtering). This
approach is also able to achieve a high F1-score, but has high bandwidth use
as the source sends unfiltered batches to the server during the profiling, as
presented in Figure 6. This profiling happens once at the beginning and every
time the environment changes, and requires a large number of frames for
profiling. LtC solves this problem by using the student network for profiling,
which is both long-lasting due to its generalization capabilities, and is able
to quickly update itself with a few new frames by transferring knowledge from
the previous environments. Additionally, Reducto does not support spatial
compression.

Figure 6. Reducto halts filtering during profiling phases resulting in high
bandwidth use

In the case of moving cameras, such as, dash cam, drone, etc., the background
regions are changing constantly. As a result, use of simple features (e.g.,
pixel values) will result in significant difference between consecutive
frames, even if the objects did not move by much. The feature vector in LtC is
robust to mobility related background changes, and therefore, can recognize
background regions even in the presence of mobility. In Figure 7, we observe
that for a moving camera, difference values calculated using LtC’s features
has the highest Pearson correlation coefficient (0.92) compared to a number of
other commonly used features.

Figure 7. Correlation between the difference values and changes in the
bounding-box results for different features. This experiment uses a video
where the camera is mobile.

##  4\. Evaluation

Methodology: We used Tensorflow and Python to implement LtC within an open-
source video streaming emulator for DDS (Du et al., 2020). We extended this
emulator with trace-driven emulation models for CloudSeg (Wang et al., 2019),
Reducto (Li et al., 2020), and Glimpse (Chen et al., 2015). This approach
enables us to carry out fair comparisons within the same environment, and
using similar assumptions for all the baselines. AWStream (Zhang et al., 2018)
was evaluated separately because it was not amenable to trace-driven
emulation.

To the best of our knowledge LtC is the first streaming video analytics
framework that uses both spatial and temporal semantic compression. While
comparing with spatial (or alternatively, temporal) baselines, we turned off
the temporal (and respectively, spatial) part of LtC. Consistent with prior
work (Du et al., 2020; Li et al., 2020) we use object detection as our primary
video analytics task; although it can be specialized to any analytics task
that also infers the location of the object regions, such as segmentation,
tagging, activity recognition, counting, etc. with alternative student
training. We evaluate LtC using a dataset consisting of publicly available
traffic surveillance videos collected from cameras deployed across various
intersection of North America (Cam, 2018; of Auburn Toomer’s Corner Webcam,
2019; Banff Live Cam, 2021; CAM, 2021; La Grange, 2022; Shop, 2022; Patrol,
2023; cam, 2019), drone videos from the VisDrone2021 dataset (Jingdong Wang,
2021), dashcam (through Daytime, 2018, 2021) and parking lot surveillance
videos from public video sharing platforms (lot security camera, 2013).
Details are in Table 3.

Name | # Videos | Total length | FPS | Resolution
---|---|---|---|---
Traffic cam | 8 | live | 15 and 30 | 4K and 1080p
Drone | 8 | 3 mins | 30 | 4K and 1080p
Dash cam | 2 | 3 hours | 30 | 4K
Parking lot | 1 | 5 mins | 15 | 1080p

Table 3. Details of the video dataset

We also evaluate LtC both in a resource-constrained network (1.2Mbps 100ms),
and a more resource rich (100Mbps 20ms) network. We use Linux NetEm (NetEm,
year) to control network bandwidth and delay. Lastly, MPEG is used as a post-
processing step to provide additional compression beyond the spatial and
temporal compression, although any standard compression is equally valid for
this purpose. We add this step to ensure fair comparison of LtC, against a
number of baselines, such as, DDS (Du et al., 2020), AWStream (Zhang et al.,
2018), and Cloudseg (Wang et al., 2019) that integrally use MPEG in their
implementations.

We evaluate LtC on the following performance metrics:

  * •

F1-score: We use F1-score to measure the performance of the downstream
analytics. The results from using an uncompressed image is considered as the
ground truth, which is consistent with the previous literature (Du et al.,
2020; Li et al., 2020).

  * •

Bandwidth use: The bandwidth use of transported video data is normalized with
the size of the MPEG compressed image at the highest resolution (no loss in
quality).

  * •

Response delay: This delay is the time from when a video frame is sent until
the output of the analysis is ready at the server consisting of both network
and processing delays.

###  4.1. Evaluating LtC’s Spatial Compression

We compare LtC with a state-of-the-art spatial compression framework DDS (Du
et al., 2020), and a few other spatial baselines, such as, AWStream (Zhang et
al., 2018) and CloudSeg (Wang et al., 2019). During these comparisons we
disable the temporal filtering module of LtC, since the other baselines do not
support temporal filtering (beyond that provided by MPEG).

We compare the F1-scores and total bandwidth consumption (broken down into
bandwidth used for both low and high resolution phases) of LtC and DDS (Du et
al., 2020) against the normalized resolutions of the low-quality regions. As
presented in Figure 8(a), LtC outperforms DDS across the entire range,
specially in the low resolution range. This is because, DDS sends the low-
quality regions in order to generate feedback at the server, and decreasing
the resolution of these regions results in diminished feedback. Whereas, LtC
has access to high-quality frames at the source, and the low-quality regions
are sent to compensate for any misclassified regions. As a result, at lower
resolutions of the low-quality regions, DDS struggles to find out the
locations of the high-quality regions accurately. Also, as presented in Figure
8(b), the sizes of the low-quality regions are less in LtC than DDS. This is
because, DDS needs to send all the low-quality regions for feedback, however,
LtC can exclude the low-quality regions where it sends them in high-quality,
resulting in less or equivalent bandwidth use at high F1-scores.

(a) F1-score

(b) Bandwidth efficiency

Figure 8. F1-score and bandwidth efficiency of LtC and DDS against the
normalized resolution of the low-quality regions. LtC consistently has higher
F1-score while consuming lower total bandwidth.

Performance against target F1-scores. For a given target F1-score, LtC can
significantly outperform DDS in terms of bandwidth usage even without
considering its temporal filtering. In order to generate accurate feedback
regions to reach an target F1-score, DDS has to send more low quality regions
(almost double) than LtC, as can be seen in Figure 9. In turn, LtC sends more
high quality regions, as it has to send all the regions containing objects in
high resolution. For an increasing F1-score demand, the gain of LtC over DDS
also increases, as the size of the high quality regions remains the same, but
the size of the low quality regions are larger and increases more rapidly for
DDS.

Figure 9. LtC exhibits superior bandwidth efficiency compared to DDS across a
wide range of target F1-scores

Performance against different video types. LtC and DDS perform similarly on
different videos. In general, in a video with sparse frames (in terms of
object presence), the bandwidth savings are higher. However, in denser videos,
LtC sends comparatively more high-quality regions and uses more bandwidth.
Therefore, LtC performs better in sparser videos, and DDS in denser videos.

Figure 10. LtC exhibits superior bandwidth efficiency compared to DDS across
different video types

Performance against other spatial compression baselines. Figure 11(a) presents
the F1-scores against the normalized bandwidth. Overall, LtC compresses 42% of
the highest-quality video and uses 23% less bandwidth than DDS for the same
F1-score. Other spatial compression baselines such as AWStream, and CloudSeg
perform worse because they do not use semantic compression. Additionally,
CloudSeg has a low F1-score because it uses downsampling and super-resolution
based upsampling which appear to harm the semantic features of the video.

The response delay of LtC and DDS is presented in Figure 11(b). LtC has lower
network delay than DDS as it does not require feedback from the server. Also,
LtC has a lower server processing delay then DDS as it only needs to process
the video one time. As a result, LtC achieves an overall 21-45% shorter
response delay than DDS in our two network configurations. This advantage will
likely be higher in networks with higher latency and lower bandwidth.

(a) F1-score and bandwidth efficiency

(b) Response delay

Figure 11. LtC exhibits superior bandwidth-accuracy tradeoffs (upper-left is
better) and lower response delay compared to other spatial compression
baselines. Confidence ellipses contain 1-$\sigma$ (68%) of results.

###  4.2. Evaluating LtC’s Temporal Filtering

In this section, we compare LtC with a state-of-the-art temporal filtering
algorithm Reducto (Du et al., 2020), and an object tracing and filtering based
algorithm Glimpse (Chen et al., 2015). Like before, we disable the spatial
compression module if LtC to enable a fair comparison.

Performance against target F1-score. For a given target F1-score, LtC
outperforms Reducto in terms of number of frames filtered. In order to reach a
high target F1-score, Reducto becomes conservative about using its existing
profiles, and sends out more frames for profiling, as can be seen in Figure
12. In turn, in LtC, the student network learns about object features
implicitly while training. LtC also gains because the filtering based on its
features is more accurate, leading to higher precision in identifying the
correct frames to discard.

Figure 12. LtC sends out less frames compared to Reducto across a wide range
of target F1-score

Performance against different video types. Both LtC and Reducto are sensitive
to the frame-rate and the content of the video, which determines the available
opportunity for temporal filtering. Intuitively, if the frame-rate of a video
is high then we have higher redundancy. In addition, there are more
redundancies in a static or slowly changing video. Among the four kinds of
videos in our dataset, the parking lot dataset is relatively static compared
to the other videos (cars are only going in and out of the parking lot
occasionally). In Figure 13, we can see that for the parking lot videos both
DDS and LtC is able to filter more frames compared to the other videos in the
dataset.

Figure 13. LtC sends out less frames compared to Reducto across different
video types

Performance against other temporal compression baselines. Figure 14 shows that
LtC is able to filter 8-14% more frames, and uses 8% less bandwidth than
Reducto for the same F1-score. The other baseline, Glimpse, falls behind in
F1-score, because it filters more aggressively with a suboptimal policy,
leading to a loss in analytics performance. The response delay of LtC and
Reducto is similar, since hey filter at the source.

Figure 14. LtC exhibits superior bandwidth-accuracy tradeoffs compared to
other temporal filtering baselines

###  4.3. End to end evaluation of LtC

In Figure 15(a), we present the accuracy-bandwidth tradeoff of LtC against all
other baselines. Overall, LtC is uses 28-35% less bandwidth than the closest
baselines DDS and Reducto. Moreover, for both the resource-constrained and the
resource-rich network, LtC has shorter response delay than both DDS and
Reducto. Specifically, LtC achieves 14-45% shorter response delay than these
baselines. Interestingly, for resource-rich network, the server processing
delay of DDS exceeds the network delay, resulting in a significant advantage
for LtC, as can be seen in Figure 15(b).

(a) F1-score and bandwidth efficiency

(b) Response delay

Figure 15. LtC exhibits superior bandwidth-accuracy tradeoffs and lower
response delay compared to all other baselines in an end-to-end setting

Granularity of identifying objects. The student network operates on fixed-size
non-overlapping patches of each frame, to identify if they contain objects. As
the number of patches along a single axis (if this number is 10, then there
are 100 patches in total) increases, the performance of the student network
(measured in Frames Per Second, FPS) decreases rapidly. However, increasing
the number of patches results in little added benefit beyond 20 patches. Even
at a smaller number, such as 10, the bandwidth increases only slightly as the
patches becomes larger. We use 16 patches along a single axis, which can run
at up to 65 FPS on our machine.

Number of layers. The performance of LtC also varies with the number of
convolution layers used in the student network. As the number increases the
FPS value of the student network decreases. However, there is no added benefit
when increasing beyond 2 layers (the configuration we use). The teacher
network runs at 2 FPS, while the student network runs at 65 FPS (32x faster),
due to its much simpler architecture.

Figure 16. Reducing the number of patches has minimal effects on the
bandwidth-accuracy tradeoffs, but increases FPS Figure 17. Reducing the number
of convolutional layers has minimal effects on the bandwidth-accuracy
tradeoffs, but increases FPS

##  5\. Related Work

Spatial compression. A variety of different approaches has been developed to
spatially reduce the size of video data for analytics. Preliminary designs
uses source-side heuristics that are detached from the server. For example,
_AdaScale_ (Chin et al., 2019) and _CloudSeg_ (Wang et al., 2019) reduce the
outgoing video using super-resolution or other heuristics. Recent works tend
to use server-side feedback for real-time adaptation. For example, _AWStream_
(Zhang et al., 2018) collaborates with the server in order to find the optimal
running configuration. The most related approach to LtC is _DDS_ (Du et al.,
2020), which sends high-quality patches to the initially sent low-quality
image in multiple phases per iteration, based on the feedback from the server.
Split-DNNs, such as _Split-brain_ (Emmons et al., 2019), split a DNN into
initial layers on the source and remaining on the server.

Temporal filtering. There has been a number of systems that use source-side,
or edge-based heuristics to filter frames. For example, _Vigil_ (Zhang et al.,
2015) uses a light-weight object counter model on the edge to filter out
frames based on number of objects present. _FilterForward_ (Canel et al.,
2019) uses light-weight application specific binary classifiers to assist in
filtering. _Glimpse_ (Chen et al., 2015) uses a caching and object-tracking
technique to fill out results in the omitted frames. Finally, similar to LtC,
_Reducto_ is a feature-differencing approach (Li et al., 2020), which finds
out the optimal action for a batch of frames by building a profile. Reducto
uses shallower features than LtC, and needs expensive retraining to identify
optimal actions in the presence of scene dynamics.

Tiny models. Recently, there has been developments of tiny neural network
models (Adarsh et al., 2020; Redmon and Farhadi, 2018) (e.g. Tiny YOLO
versions) that can run in resource-constrained end devices at high FPS, and
with reasonable accuracy. However, compression using these tiny models is
suboptimal as they perform poorly in videos with smaller objects.

Optimizing data transport in sensor networks. There is a rich body of work in
the sensor network community on reducing the dimensionality of transported
data (e.g., (Deshpande et al., 2004)). Network based video streaming
frameworks such as LtC and others (Du et al., 2020; Li et al., 2020) may be
viewed as an instance of this line of work for video sensors. In the context
of video sensor networks, a number of platforms also use compression. However,
they are targeted towards deeply embedded platforms and use traditional
compression (Seema and Reisslein, 2011), or even just still images, for
example, using camera traps (Kays et al., 2009).

##  6\. Conclusion

In this work, we present a compression algorithm that can work in real-time on
video streams targeted for analytics. The proposed scheme leverages a novel
student-teacher framework, and integrates both spatial and temporal
compression in a complementary way. In addition to that, LtC significantly
outperforms the state of the art proposals in terms of accuracy and bandwidth
use.

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aainstitutetext: Department of Physics, Kyushu University, 744 Motooka, Nishi-
Ku, Fukuoka 819-0395, Japanbbinstitutetext: RIKEN iTHEMS, Wako, Saitama
351-0198, Japan

# Global study of the scalar quasi-normal modes of Kerr-AdS5 black holes:
Stability, thermality, and horizon area quantization

Issei Koga b Naritaka Oshita a Kazushige Ueda

###### Abstract

We numerically explore the structure of quasi-normal (QN) frequencies of the
five-dimensional small and large Kerr-anti de Sitter (Kerr-AdS5) black hole
with equal and unequal rotations. Our investigation also covers low and high
Hawking temperatures. We then study the stability of the Kerr-AdS5 black hole
and the structure of highly damped QN modes, which would reflect the
thermodynamic property of the Kerr-AdS5 black hole. We find that the highly
damped complex QN frequencies of a nearly maximally spinning Kerr-AdS5 black
hole have the periodic separation of the surface gravity at the horizon in the
imaginary part while the real part converges to the superradiant frequency,
which may be relevant to the pole structure of the thermal Green’s function in
the corresponding conformal field theory on the Kerr-AdS5 boundary. Finally,
we discuss a relation between the QN modes of the Kerr-AdS5 black hole and the
Hod’s conjecture on the horizon area quantization along with the analysis of
the horizon topology of the Kerr-AdS5 black hole. We show that in general, an
ultra-spinning Kerr-AdS5 black hole, whose spin parameter is infinitesimally
close to the AdS curvature radius, has its non-compact horizon, and based on
the Hod’s conjecture, we argue that the horizon area may be continuous, that
is, the unit area of the horizon vanishes in the ultra-spinning regime.

††preprint: RIKEN-iTHEMS-Report-22

##  1 Introduction

Higher-dimensional gravity is important in the holographic principle ’t Hooft
(1993); Susskind, Thorlacius, and Uglum (1993); Susskind (1995); Maldacena
(1998), the brane-world scenario Horava and Witten (1996); Arkani-Hamed,
Dimopoulos, and Dvali (1998); Randall and Sundrum (1999a, b); Arkani-Hamed
_et al._ (2000), and the landscape Susskind (2003) of the string theory. In
these contexts, black holes in higher dimensions play pivotal roles, e.g.,
thermality of the conformal field theory (CFT) on the anti-de Sitter (AdS)
boundary Witten (1998); Banks  _et al._ (1998), the source of dark radiation
in brane-world models Kraus (1999); Ida (2000); Mukohyama (2000), and so on.
Another interesting nature of higher-dimensional black holes is that there is
no corresponding theorem to a uniqueness theorem in higher dimensions,
although it exists in the four-dimensional spacetime. Thus the stability of
black holes in higher dimensions is non-trivial. The analysis of quasi-normal
(QN) modes of black holes is a good probe to see its stability. The positivity
of the imaginary part of complex QN frequencies leads to the exponential
growth of the perturbation of black holes. The structure of QN modes is also
relevant to the context of the holography as the QN modes of black holes in
AdS space are conjectured to be dual with the poles of Green’s function of the
CFT on the AdS boundary (see Ref. Birmingham, Sachs, and Solodukhin (2002) for
the simplest case of the duality in the BTZ black hole). The structure of QN
modes of black holes could shed light on the quantum nature of black hole
horizons as well. According to the Hod’s conjecture Hod (1998), the real part
of highly damped QN frequencies determines the unit area of a horizon. It
would be an interesting question if the concept of the horizon area
quantization can be extended to higher-dimensional gravity for which there are
rich structures such as black holes, black rings, black strings, and so on.

In this work, we consider the QN frequencies of a scalar field in the Kerr-
AdS5 spacetime Myers and Perry (1986) to understand the stability, thermality,
and area quantization of the black hole horizon. The scalar perturbations of
the Kerr-AdS5 black hole are governed by the Klein-Gordon equation that
reduces to the Heun’s differential equation by performing the coordinate
transformations and the redefinition of the perturbation variables Aliev and
Delice (2009). Therefore, the solution of the equation is represented by the
Heun function.

The analysis of AdS5 spacetime with a black hole and/or compactified extra-
dimensional space has been done mainly for holographic applications Murata
(2009); Murata, Nishioka, and Tanahashi (2009); Cardoso  _et al._ (2014);
Aref’eva, Golubtsova, and Gourgoulhon (2021); Nian and Pando Zayas (2020).
Analytical and numerical studies of the QN modes of the Kerr-AdS5 spacetime
have been recently done in Ref. Barragán Amado, Carneiro Da Cunha, and
Pallante (2019); Amado, da Cunha, and Pallante (2021); da Cunha and Cavalcante
(2021) by performing the expansion of the $\tau$-function for the Painlevé
transcendents. In special cases, e.g., nearly equal spins, small spins, near
extremal, or small mass regimes, the expansion of the $\tau$-function
converges fast and makes the computation of QN frequencies tractable. On the
other hand, in our computation, we use the solution of the Teukolsky equations
represented by the Heun functions, which makes the computation of the QN modes
substantially fast, and allows us to investigate broader parameter regions of
the Kerr-AdS5 spacetime. Also, our computation can be applied to search for
not only the fundamental QN mode but also other overtones, including highly
damped modes. Our computation with those advantages allows us to make a
“global map” of the locations of QN modes of Kerr-AdS5 spacetime, which would
be useful to understand the stability of the background spacetime and even the
Kerr-AdS/CFT correspondence Amado, Carneiro da Cunha, and Pallante (2017).

In the next section, we provide a review of the scalar perturbation of the
Kerr-AdS5 black hole. In Sec. 3, we study the QN modes of the Kerr-AdS5 black
hole with equal spins in the small and large mass regimes. In Sec. 4, we
investigate the QN modes for the unequal spins in the small and large mass
regimes. In Sec. 5, we discuss some implications to the area quantization of
the Kerr-AdS5 black hole by applying the Hod’s conjecture to our results.
Also, we analyze the topology of the horizon of the Kerr-AdS5 black hole in
the ultra-spinning limit for which its spin parameter is infinitesimally close
to the AdS curvature radius $\ell$. Finally, we summarize our results and
conclusions in Sec. 6.

##  2 Formalism

The Kerr-AdS5 spacetime has the following metric

$\displaystyle\begin{split}ds^{2}&=-\frac{\Delta_{r}}{\rho^{2}}\left(dt-\frac{a_{1}\sin^{2}\theta}{1-a_{1}^{2}}d\phi-\frac{a_{2}\cos^{2}\theta}{1-a_{2}^{2}}d\psi\right)^{2}+\frac{\Delta_{\theta}\sin^{2}\theta}{\rho^{2}}\left(a_{1}dt-\frac{r^{2}+a_{1}^{2}}{1-a_{1}^{2}}d\phi\right)^{2}\\\
&+\frac{1+r^{2}}{r^{2}\rho^{2}}\left(a_{1}a_{2}dt-\frac{a_{2}(r^{2}+a_{1}^{2})\sin^{2}\theta}{1-a_{1}^{2}}d\phi-\frac{a_{1}(r^{2}+a_{2}^{2})\cos^{2}\theta}{1-a_{2}^{2}}d\psi\right)^{2}\\\
&+\frac{\Delta_{\theta}\cos^{2}\theta}{\rho^{2}}\left(a_{2}dt-\frac{r^{2}+a_{2}^{2}}{1-a_{2}^{2}}d\psi\right)^{2}+\frac{\rho^{2}}{\Delta_{r}}dr^{2}+\frac{\rho^{2}}{\Delta_{\theta}}d\theta^{2},\end{split}$
(1)

where $M$ is the mass parameter, the AdS curvature radius is set to $\ell=1$,
$a_{1}$ and $a_{2}$ are the spin parameters for the two rotations of the Kerr-
AdS5 black hole, and

$\displaystyle\Delta_{r}$
$\displaystyle\equiv\frac{1}{r^{2}}(r^{2}+a_{1}^{2})(r^{2}+a_{2}^{2})(1+r^{2})-2M,$
(2) $\displaystyle\Delta_{\theta}$ $\displaystyle\equiv
1-a_{1}^{2}\cos^{2}\theta-a_{2}^{2}\sin^{2}\theta,$ (3)
$\displaystyle\rho^{2}$ $\displaystyle\equiv
r^{2}+a_{1}^{2}\cos^{2}\theta+a_{2}^{2}\sin^{2}\theta.$ (4)

Based on the thermodynamic description of the Kerr-AdS5 spacetime, the
Arnowitt-Deser-Misner (ADM) mass and angular momentum are given by Gibbons,
Perry, and Pope (2005); Hollands, Ishibashi, and Marolf (2005); Olea (2007)

${\cal M}\equiv\frac{\pi
M(2\Xi_{1}+2\Xi_{2}-\Xi_{1}\Xi_{2})}{4\Xi_{1}^{2}\Xi_{2}^{2}},\ {\cal
J}_{\phi}\equiv\frac{\pi Ma_{1}}{2\Xi_{1}^{2}\Xi_{2}},\ {\cal
J}_{\psi}\equiv\frac{\pi Ma_{2}}{2\Xi_{1}\Xi_{2}^{2}},$ (5)

where $\Xi_{i}\equiv 1-a_{i}^{2}$ ($i=1,2$). The spin parameters are
restricted to $a_{i}\leq 1$, for which all the physical quantities in (5) are
well-defined. We here compute the QN modes of the Kerr-AdS5 black hole and
investigate the instability of a scalar field $\Phi(t,r,\theta,\phi,\psi)$
with mass $\mu$. Let us start with the Klein-Gordon equation

$\left[\Box-\mu^{2}\right]\Phi=0,$ (6)

and decomposing $\Psi$ as $\Psi=e^{-i\omega
t+im_{1}\phi+im_{2}\psi}\Theta(\theta)\Pi(r)$, one has the radial and angular
equations:

$\displaystyle\begin{split}&\frac{1}{r}\frac{d}{dr}\left(r\Delta_{r}\frac{d\Pi(r)}{dr}\right)-\left[\lambda+\mu^{2}r^{2}+\frac{1}{r^{2}}\left(a_{1}a_{2}\omega-
a_{2}(1-a_{1}^{2})m_{1}-a_{1}(1-a_{2}^{2})m_{2}\right)^{2}\right]\Pi(r)\\\
&+\frac{(r^{2}+a_{1}^{2})^{2}(r^{2}+a_{2}^{2})^{2}}{r^{4}\Delta_{r}}\left(\omega-\frac{m_{1}a_{1}(1-a_{1}^{2})}{r^{2}+a_{1}^{2}}-\frac{m_{2}a_{2}(1-a_{2}^{2})}{r^{2}+a_{2}^{2}}\right)^{2}\Pi(r)=0,\end{split}$
(7)
$\displaystyle\begin{split}&\frac{1}{\sin\theta\cos\theta}\frac{d}{d\theta}\left(\sin\theta\cos\theta\Delta_{\theta}\frac{d\Theta(\theta)}{d\theta}\right)-\left[-\lambda+\omega^{2}+\frac{(1-a_{1}^{2})m_{1}^{2}}{\sin^{2}\theta}+\frac{(1-a_{2}^{2})m_{2}^{2}}{\cos^{2}\theta}\right.\\\
&\left.-\frac{(1-a_{1}^{2})(1-a_{2}^{2})}{\Delta_{\theta}}(\omega+m_{1}a_{1}+m_{2}a_{2})^{2}+\mu^{2}(a_{1}^{2}\cos^{2}\theta+a_{2}^{2}\sin^{2}\theta)\right]\Theta(\theta)=0,\end{split}$
(8)

where $\lambda$ is the separation constant to be determined so that
$\Theta(\theta)$ is regular at $\theta=0$ and $\theta=\pi/2$ Aliev and Delice
(2009). Performing the following transformations:

$\displaystyle r$ $\displaystyle\to
z\equiv\frac{r^{2}-r_{-}^{2}}{r^{2}-r_{0}^{2}},$ (9) $\displaystyle\Pi(r)$
$\displaystyle\to R(z)\equiv
z^{\theta_{-}/2}(z-z_{0})^{\theta_{+}/2}(z-1)^{-\Delta/2}\Pi(z),$ (10)
$\displaystyle\sin^{2}\theta$ $\displaystyle\to
u\equiv\frac{\sin^{2}\theta}{\sin^{2}\theta-\chi_{0}},\ \text{with}\
\chi_{0}\equiv\frac{1-a_{1}^{2}}{a_{2}^{2}-a_{1}^{2}},$ (11)
$\displaystyle\Theta(\theta)$ $\displaystyle\to S(u)\equiv
u^{-m_{1}/2}(u-1)^{-\Delta/2}(u-u_{0})^{-m_{2}/2}\Theta(u),$ (12)

with $r_{-}$ and $r_{+}$ being the inner and outer horizon radii,
respectively, $r_{0}$ being the imaginary root of $\Delta_{r}$, $\Delta\equiv
2+\sqrt{4+\mu^{2}}$, and $u_{0}\equiv(a_{2}^{2}-a_{1}^{2})/(a_{2}^{2}-1)$, the
radial and angular equations reduce to the Heun’s differential equations:

$\displaystyle\frac{d^{2}R}{dz^{2}}+\left[\frac{1-\theta_{-}}{z}+\frac{-1+\Delta}{z-1}+\frac{1-\theta_{+}}{z-z_{0}}\right]\frac{dR}{dz}+\left(\frac{\kappa_{1}\kappa_{2}}{z(z-1)}-\frac{K}{z(z-1)(z-z_{0})}\right)R=0,$
(13)
$\displaystyle\frac{d^{2}S}{du^{2}}+\left[\frac{1+m_{1}}{u}+\frac{-1+\Delta}{u-1}+\frac{1+m_{2}}{u-u_{0}}\right]\frac{dS}{du}+\left(\frac{q_{1}q_{2}}{u(u-1)}-\frac{Q}{u(u-1)(u-u_{0})}\right)S=0,$
(14)

where

$\displaystyle\theta_{i}$ $\displaystyle\equiv\frac{i}{2\pi}\frac{\omega-
m_{1}\Omega_{i,1}-m_{2}\Omega_{i,2}}{T_{i}},$ (15) $\displaystyle T_{i}$
$\displaystyle\equiv\frac{r_{i}^{2}\Delta_{r}^{\prime}(r_{i})}{4\pi(r_{i}^{2}+a_{1}^{2})(r_{i}^{2}+a_{2}^{2})},\
\Omega_{i,1}\equiv\frac{a_{1}\Xi_{1}}{r_{i}^{2}+a_{1}^{2}},\
\Omega_{i,2}\equiv\frac{a_{2}\Xi_{2}}{r_{i}^{2}+a_{2}^{2}},$ (16)
$\displaystyle\kappa_{1}$
$\displaystyle\equiv-\frac{1}{2}\left(\theta_{-}+\theta_{+}-\Delta-\theta_{0}\right),\
\kappa_{2}\equiv-\frac{1}{2}\left(\theta_{-}+\theta_{+}-\Delta+\theta_{0}\right),$
(17) $\displaystyle q_{1}$
$\displaystyle\equiv\frac{1}{2}\left(m_{1}+m_{2}+\Delta-\zeta\right),\
q_{2}\equiv\frac{1}{2}\left(m_{1}+m_{2}+\Delta+\zeta\right),$ (18)
$\displaystyle\zeta$ $\displaystyle\equiv\omega+a_{1}m_{1}+a_{2}m_{2},$ (19)
$\displaystyle\begin{split}K&\equiv-\frac{1}{4}\left\\{\frac{\lambda+\mu^{2}r_{-}^{2}-\omega^{2}}{r_{+}^{2}-r_{0}^{2}}+(z_{0}-1)[(\theta_{+}+\theta_{-}-1)^{2}-\theta_{0}^{2}-1]\right.\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+z_{0}\left[2(\theta_{+}-1)(1-\Delta)+(2-\Delta)^{2}-2\right]\right.\biggr{\\}},\end{split}$
(20)
$\displaystyle\begin{split}Q&\equiv-\frac{1}{4}\left\\{\frac{\omega^{2}+a_{1}^{2}\mu^{2}-\lambda}{a_{2}^{2}-1}+u_{0}\left[(m_{2}+\Delta-1)^{2}-m_{2}^{2}-1\right]\right.\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+(u_{0}-1)\left[(m_{1}+m_{2}+1)^{2}-\zeta^{2}-1\right]\right.\biggr{\\}}.\end{split}$
(21)

The general solution of the Heun’s differential equation, (13), is

$\displaystyle\begin{split}R&=c_{0}R_{\rm in}(\omega,z)+d_{0}R_{\rm
out}(\omega,z)\\\ &\equiv
c_{0}H\ell\left(\frac{z_{0}}{z_{0}-1},\frac{-K}{z_{0}-1};\kappa_{1},\kappa_{2},1-\theta_{+},\Delta-1;\frac{z_{0}-z}{z_{0}-1}\right)\\\
&+d_{0}\left(\frac{z_{0}-z}{z_{0}-1}\right)^{1-\epsilon}H\ell\left(\frac{z_{0}}{z_{0}-1},\frac{\theta_{+}[z_{0}(\Delta-\theta_{-})-1+\theta_{-}]}{z_{0}-1}-\frac{K}{z_{0}-1};\right.\\\
&\left.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\kappa_{1}+\theta_{+},\kappa_{2}+\theta_{+},1+\theta_{+},\Delta-1;\frac{z_{0}-z}{z_{0}-1}\right),\end{split}$
(22)

for $z\sim z_{0}(r\sim r_{+})$, and

$\displaystyle\begin{split}R&=c_{1}R_{\rm AdS}(\omega,z)+d_{1}R_{\rm
Div}(\omega,z)\\\ &\equiv
c_{1}H\ell\left(1-z_{0},\kappa_{1}\kappa_{2}-\tilde{K};\kappa_{1},\kappa_{2},\delta-1,1-\theta_{-};1-z\right)\\\
&+d_{1}(1-z)^{\theta_{-}}H\ell\left(1-z_{0},[(1-z_{0})(1-\theta_{-})+1-\theta_{+}](2-\Delta)+\kappa_{1}\kappa_{2}-\tilde{K};\right.\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\kappa_{1}+2-\Delta,\kappa_{2}+2-\Delta,3-\Delta,1-\theta_{-};1-z\Bigr{)},\end{split}$
(23)

for $z\sim 1(r\sim\infty)$. Here $c_{i}$ and $d_{i}$ ($i=0,1$) are arbitrary
constants and we define $\tilde{K}\equiv K+\kappa_{1}\kappa_{2}z_{0}$. For the
angular equation (14), its general solution is

$\displaystyle\begin{split}S&=\tilde{c}_{0}H\ell\left(u_{0},\tilde{Q};q_{1},q_{2},1+m_{1},\Delta-1;u\right)\\\
&+\tilde{d}_{0}z^{m_{1}}H\ell\left(u_{0},\tilde{Q}-m_{1}[u_{0}(\Delta-1)+m_{2}+1];\right.\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}q_{1}-m_{1},q_{2}-m_{1},1-m_{1},\Delta-1;u\Bigr{)}\end{split}$
(24)

around $u\sim 0$ ($\theta\sim 0$), and

$\displaystyle\begin{split}S&=\tilde{c}_{1}H\ell\left(\frac{u_{0}}{u_{0}-1},\frac{-Q}{u_{0}-1};q_{1},q_{2},1+m_{2},\Delta-1;\frac{u_{0}-u}{u_{0}-1}\right)\\\
&+\tilde{d}_{1}\left(\frac{u_{0}-u}{u_{0}-1}\right)^{-m_{2}}H\ell\left(\frac{u_{0}}{u_{0}-1},\frac{-Q-m_{2}[u_{0}(\Delta+m_{1})-(1+m_{1})]}{u_{0}-1};\right.\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}q_{1}-m_{2},q_{2}-m_{2},1-m_{2},\Delta-1;\frac{u_{0}-u}{u_{0}-1}\biggr{)}\end{split}$
(25)

around $u\sim u_{0}$ ($\theta\sim\pi/2$), where $\tilde{c}_{i}$ and
$\tilde{d}_{i}$ ($i=0,1$) are arbitrary constants and $\tilde{Q}\equiv
Q+q_{1}q_{2}u_{0}$. To ensure the ingoing and Dirichlet boundary conditions at
$r=r_{+}$ and $r=\infty$, respectively, one has to impose the following
boundary condition for $\Pi(z)$

$\displaystyle\Pi(z)\sim\begin{cases}(z-z_{0})^{-\theta_{+}/2}&\text{for}\
z\to z_{0}\ (r\to r_{+}),\\\ (z-1)^{\Delta/2}&\text{for}\ z\to 1\
(r\to\infty),\end{cases}$ (26)

and for $\Theta(u)$, the regular condition at $u=0$ ($\theta=0$) and $u=u_{0}$
($\theta=\pi/2$) is

$\displaystyle\Theta(u)\sim\begin{cases}u^{|m_{1}|/2}&\text{for}\ u\to 0,\\\
(u-u_{0})^{|m_{2}|/2}&\text{for}\ u\to u_{0}.\end{cases}$ (27)

To satisfy the boundary condition (26), $R(z)$ should take the following form
at the boundaries

$\displaystyle R$ $\displaystyle\sim R_{\rm in}(\omega,z)\ \text{for}\ z\sim
z_{0},$ (28) $\displaystyle R$ $\displaystyle\sim R_{\rm AdS}(\omega,z)\
\text{for}\ z\sim 1.$ (29)

For $u\sim 0$, on the other hand, $S(u)$ takes the following form

$\displaystyle
S=\begin{cases}&H\ell(u_{0},\tilde{Q};q_{1},q_{2},1+m_{1},\Delta-1;u)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{for}\
m_{1}\geq 0,\\\
&z^{-m_{1}}H\ell\Bigl{(}u_{0},-m_{1}[u_{0}(\Delta-1)+1+m_{2}]+\tilde{Q};\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}q_{1}-m_{1},q_{2}-m_{1},1-m_{1},\Delta-1;u\Bigr{)}~{}~{}~{}~{}~{}~{}~{}\text{for}\
m_{1}\leq 0,\end{cases}$ (30)

and for $u\sim u_{0}$, $S(u)$ is

$\displaystyle S=\begin{cases}&\displaystyle
H\ell\left(\frac{u_{0}}{u_{0}-1},\frac{q_{1}q_{2}u_{0}-\tilde{Q}}{u_{0}-1};q_{1},q_{2},1+m_{2},\Delta-1;\frac{u_{0}-u}{u_{0}-1}\right)~{}~{}~{}\text{for}\
m_{2}\geq 0\\\
&\displaystyle\left(\frac{u_{0}-u}{u_{0}-1}\right)^{-m_{2}}H\ell\biggl{(}\frac{u_{0}}{u_{0}-1},\frac{-m_{2}[u_{0}(m_{1}+\Delta)-1-m_{1}]}{u_{0}-1}+\frac{q_{1}q_{2}u_{0}-\tilde{Q}}{u_{0}-1};\\\
&\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}q_{1}-m_{2},q_{2}-m_{2},1-m_{2},\Delta-1;\frac{u_{0}-u}{u_{0}-1}\biggr{)}~{}~{}~{}~{}~{}\text{for}\
m_{2}\leq 0.\end{cases}$ (31)

In the following sections, we use the Wolfram Mathematica to search for the
eigenvalues $\lambda_{lm_{1}m_{2}n}$ and QN frequencies
$\omega_{lm_{1}m_{2}n}$ for which the boundary conditions (28) - (31) are
satisfied.

Figure 1: Plot of QN frequencies for $l=1,2,3$ modes. We set
$a_{1}=a_{2}=4\times 10^{-4}$, $\mathcal{M}=10^{-5}$ and $\mu=10^{-2}$.

Figure 2: Plot of QN frequencies for various spin parameters. In the left and
right panels, the imaginary part is shown in log and linear scale,
respectively. Each marker indicates the complex value of each QN frequency. We
set $\mathcal{M}=10^{-5}$ and $\mu=10^{-2}$, and the spin parameters are in
the range of $10^{-4}\leq a(=a_{1}=a_{2})\leq 10^{-3}$. The angular modes are
fixed as $(l,m_{1},m_{2})=(1,1,0)$ that is equivalent to $(1,0,1)$.

##  3 Stability analysis for equal spins ($a_{1}=a_{2}$)

In this section, we numerically investigate the QN frequencies for a Kerr-AdS5
black hole with $a_{1}=a_{2}\equiv a$. For equal spins, the angular equation
reduces to the hypergeometric differential equation, and one can obtain the
analytic expression of the eigenvalue $\lambda$ Aliev and Delice (2009)

$\lambda=(1-a^{2})\left[l(l+2)-2\omega
a(m_{1}+m_{2})-a^{2}(m_{1}+m_{2})^{2}\right]+a^{2}\omega^{2}+a^{2}\Delta(\Delta-4),$
(32)

where $l=0,1,2,...$ is the angular mode. We can obtain QN frequencies by
solving the radial equation (13) with $\lambda=\lambda(\omega)$ given in (32)
and by searching for $\omega=\omega_{lm_{1}m_{2}n}$ at which the obtained
solution satisfies the boundary condition (26).

###  3.1 Small black holes ${\cal M}\ll 1$

For small black holes, QN frequencies are localized near the real axis of the
complex frequency plane,
$|\text{Re}(\omega_{lm_{1}m_{2}n})|\gg|\text{Im}(\omega_{lm_{1}m_{2}n})|$, due
to trapped modes in the AdS boundary. Also, the superradiant instability is
caused by the resonance between the ergoregion and AdS boundary. The stability
of the black hole can be read from the sign of the imaginary part of QN
frequencies, and $\text{Im}(\omega_{lm_{1}m_{2}n})>0$ means that the
background spacetime is unstable against linear perturbations. For small Kerr-
AdS5 black holes, the unstable QN modes satisfy the following condition

$\text{Re}({\omega_{lm_{1}m_{2}n}})<m_{1}\Omega_{+,1}+m_{2}\Omega_{+,2}\equiv\Omega.$
(33)

Here it is natural to ask which overtone leads to the most significant
instability when multiple overtones satisfy the above condition.

Figure 3: Plots of the coefficient $D_{1}(\omega)$ defined in (34). The
parameters are set to $a=\pm 0.07$, $M=0.01$, and $(l,m_{1},m_{2})=(2,1,1)$.
The type-II modes are numbered by non-negative integers, $n$, in ascending
order of the damping rates.

To see this, we first investigate QN modes for $l=1,2,3$ in Figure 1, and it
is found that $l=1$ mode leads to the most significant instability. We then
study the spin-dependence of QN modes with $(l,m_{1},m_{2})=(1,1,0)$, and the
result is shown in Figure 2. It is shown that rapid rotations destabilize the
Kerr-AdS5 black holes, and the peak of $\text{Im}(\omega_{lm_{1}m_{2}n})$ is
slightly below the superradiant frequency of $\Omega$. We also confirm that
our result shown in Figure 2 is consistent with the condition of the
superradiant instability (33) as $\text{Re}({\omega_{lm_{1}m_{2}n}})<136.422$
and $\text{Re}({\omega_{lm_{1}m_{2}n}})<11.795$ for $a=10^{-3}$ and
$a=10^{-4}$, respectively. In the following, we will omit the subscript $+$
from $\Omega_{+,i}$. Besides the QN modes localized near the real axis of
$\omega$, highly damped QN modes are localized near the line of
$\text{Re}(\omega)=\Omega$ in the complex frequency plane, as is shown in
Figure 3. The coefficient $D_{1}(\omega)$, defined as

$R_{\rm BH}(\omega,z)=C_{1}(\omega)R_{\rm AdS}(\omega,z)+D_{1}(\omega)R_{\rm
Div}(\omega,z),$ (34)

is plotted in Figure 3, and it vanishes at $\omega=\omega_{lm_{1}m_{2}n}$.
Hereinafter, we call QN modes localized near the real axis of $\omega$ and
those localized near the superradiant frequency type I and type II,
respectively.

Figure 4: Trajectories of QN frequencies for $M=0.01$,
$(l,m_{1},m_{2})=(2,1,1)$, and $\mu=0.01$. The solid and dashed lines indicate
the type-I and type-II QN modes, respectively. The spin parameter $a$ runs
from $a=10^{-4}$ (brown triangles) to $a\simeq 0.999996a_{\rm max}$ (blue
dots) in (b) and (c). The spin parameter runs up to $a=0.99999a_{\rm max}$
(blue dots) in (d). The maximum spin parameter is $a_{\rm max}=0.070536282...$
and the red squares in (d) indicate the QN frequencies at $a=0.07$. The arrows
indicate the direction in which the spin parameter, $a$, increases.

The type-I modes are caused by the resonance between the AdS barrier and the
angular momentum barrier. On the other hand, type-II modes may be relevant to
the thermality of the Kerr-AdS5 black hole as their separation in frequency
space is nearly equal to the surface gravity of Kerr-AdS5 black hole $2\pi
T_{\rm H}$, and the real part of the type-II modes can be interpreted as the
chemical potential. Figure 4 shows the trajectories of the type-I and type-II
modes with respect to the change of the Hawking temperature. One can see that
the lower the Hawking temperature of the Kerr-AdS5 black hole is, the stronger
the localization of type-II modes at the superradiant frequency is (see Figure
5). Also, the separation of the imaginary part of QN frequencies, defined as
$\Delta\text{Im}(\omega_{lm_{1}m_{2}n})\equiv\text{Im}(\omega_{lm_{1}m_{2}n}-\omega_{lm_{1}m_{2}(n+1)})$,
approaches $2\pi T_{\rm H}$ in the zero-temperature limit. It looks like the
behaviors of highly damped QN modes are insensitive to $\mu$ at least for
$0.01\leq\mu\leq 10$.

In the next subsection, we will see the massive case, ${\cal M}\gtrsim 1$, for
which the lowest Hawking temperature is nonzero and finite due to the upper
bound on the two spins ($a_{i}\leq 1$ with $i=1,2$). We will show the
structure of the type-II modes in the complex frequency plane for the lowest
Hawking temperature.

Figure 5:  The real part and the separation in the imaginary part of QN
frequencies are shown with respect to $(a_{\rm max}-a)/a_{\max}$ in the upper
left and upper right panels, respectively. We change the opacity of each line
to distinguish the case of $\mu=0.01$ (transparent lines) and $\mu=10$ (opaque
lines). The other parameters are set to $M=0.01$ and
$(l,m_{1},m_{2})=(2,1,1)$. In the left panel, the black solid line indicates
the superradiant frequency, $\Omega$, that depends on $a$, and the black
dashed line shows the value of $\Omega$ at the extremal case ($a=a_{\rm
max}$). The blue dashed line in the right panel is $2\pi T_{\rm H}(a)$. As a
reference, the region of $10^{-3.3}\leq(a_{\rm max}-a)/a_{\rm max}\leq
10^{-3.26}$ is zoomed in and displayed in the lower panels.

###  3.2 Large black holes ${\cal M}\gtrsim 1$

Massive Kerr-AdS5 black holes (${\cal M}\gtrsim 1$) are stable against linear
perturbations, which is equivalent to $\text{Im}(\omega_{lm_{1}m_{2}n})<0$ for
all modes, since the cavity between the angular momentum barrier of the black
hole and the AdS boundary to cause the resonant instability does not exist for
massive black holes. The trajectories of QN modes for $M=5$ is shown in Figure
6. Note that type-I modes (dashed lines) correspond to the QN modes that
localize near the real axis of $\omega$ in the small mass regime (c.f. Sec.
3.1).

Figure 6: Trajectories of QN frequencies for $M=5$ and
$(l,m_{1},m_{2})=(2,1,1)$. The spin parameter $a$ runs from $a=0.01$ to
$a=0.99$. The dashed and solid lines indicate the type-I and type-II QN modes,
respectively. The arrows indicate the direction in which the spin parameter,
$a$, increases.

In that case, the type-I modes are caused by the resonance in the AdS
boundary, and thus, the type-I modes periodically appear near the real axis of
$\omega$, which is similar to the normal modes of a vibrating string. In the
massive case, on the other hand, type-I modes are highly suppressed especially
for $a\to 1$, while type-II modes localize at $\text{Re}(\omega)=\Omega$ (see
Figure 6). Note that in the limit of $a\to 1$, the superradiant frequency also
vanishes $\Omega\propto(1-a^{2})\to 0$. We also find that the type-II modes
appear from the region of $\text{Re}(\omega)>\Omega$ for $a>0$ and they do
from the region of $\text{Re}(\omega)<\Omega$ for the counter rotations
($a<0$), as is shown in Figure 7, where the absolute value of
$|D_{1}(\omega)|$ is shown in the log-scale.

Figure 7: 3D plots of the coefficient $D_{1}(\omega)$ in the complex frequency
plane. The zeros of $D_{1}$ correspond to the complex QN frequencies,
$\omega=\omega_{lm_{1}m_{2}n}$, and the pink and black arrows indicate the
type-I and type-II modes, respectively. Here we set $(l,m_{1},m_{2})=(2,1,1)$,
$M=5$, and $\mu=0.01$. The spins are set to $a=0.4$ (left) and $a=-0.4$
(right). Figure 8: Plots of $\Delta\text{Im}(\omega_{211n})$ for $\mu=10$
(left) and $\mu=0.01$ (right). The black line indicates 2$\pi T_{\rm H}(a)$,
and black dashed line indicates 2$\pi T_{\rm H}(a\rightarrow 1)$. The other
parameters are set to $M=10$ and $(l,m_{1},m_{2})=(2,1,1)$. Figure 9: Plots of
$\text{Re}(\omega_{211n})$ for $\mu=10$ (upper left) and $\mu=0.01$ (upper
right). The other parameters are set to $M=10$ and $(l,m_{1},m_{2})=(2,1,1)$.
We change the opacity of each line to distinguish the case of $\mu=0.01$
(transparent lines) and $\mu=10$ (opaque lines). The region of $1.0\times
10^{-3}\leq 1-a\leq 1.1\times 10^{-3}$ is zoomed in and displayed in the lower
panel for comparison.

We confirm that the separation $\Delta\text{Im}(\omega_{lm_{1}m_{2}n})$
approaches $2\pi T_{\rm H}$ for $a\to 1$ as is shown in Figure 8. It looks
like the separation converges to $2\pi T_{\rm H}$ for highly damped modes
(higher overtone number $n$) though it is dispersive, and its convergence is
weaker when the mass of the scalar field, $\mu$, is massive. This implies that
the convergence of $\Delta\text{Im}(\omega_{lm_{1}m_{2}n})$ to $2\pi T_{\rm
H}$ for $n\gg 1$ is delayed or that the thermality of the Kerr-AdS5 black hole
would be disturbed by the mass of surrounding fields. Figure 9 shows that
nevertheless the real part of type-II modes strongly converges to the
superradiant frequency $\Omega$ in the ultra-spinning limit for both small and
large $\mu$. As a final remark of this section, we have numerically computed
the type-II QN frequencies up to finite overtone numbers. There is still a
possibility that even high-temperature Kerr-AdS5 black holes have the thermal
structure of their QN modes at $n\to\infty$, i.e., the real part and the
separation of the imaginary part of the type-II modes match $\Omega$ and $2\pi
T_{\rm H}$, respectively, in the highly damped limit.

##  4 Stability analysis for unequal spins ($a_{1}>a_{2}$)

In this section, we numerically investigate the configuration of QN modes in
the complex frequency plane for Kerr-AdS5 black holes with unequal-spin
parameters $a_{1}>a_{2}$. In this case, the symmetry of spacetime reduces to
$U(1)\times U(1)$ while it is enhanced to $U(2)$ for $a_{1}=a_{2}$. We
investigate how the reduction of the symmetry affects the superradiant
instability in Sec. 4.1 and study the structure of highly damped QN modes for
unequal spins in Sec. 4.2.

###  4.1 Small black holes ${\cal M}\ll 1$

To see which harmonic mode is the most significant to destabilize the system,
we numerically compute the QN frequencies for various values of the spin
ratio, $a_{2}/a_{1}$. We then numerically confirm that the most dominant
instability is caused by the mode of $(l,m_{1},m_{2})=(1,1,0)$ for
$a_{1}>a_{2}$ as is shown in Figure 10. Therefore, we consider the instability
of $(1,1,0)$ only and investigate how the reduction of the symmetry of
spacetime affects the instability111The opposite hierarchy between $a_{1}$ and
$a_{2}$ $(a_{2}/a_{1}>1)$ results in the suppression of the instability of
$(1,1,0)$, and then the instability caused by $(1,0,1)$ mode becomes
dominant.. We compute the QN frequencies by fixing the ADM mass, the mass of
the scalar field, and the superradiant frequency in Figure 11. As a result, we
find that the instability is more significant when the spin ratio
$a_{2}/a_{1}$ is smaller. Our result implies that the symmetry reduction of
the Kerr-AdS5 black hole leads to the enhancement of the superradiant
instability when ${\cal M}$, $\mu$, and $\Omega_{1}$ are fixed.

Figure 10: QN frequencies of $l=1,2,3$ modes for various ratio of spin
parameters ($a_{2}/a_{1}=0.1,$ $0.5$, $0.9$, and $1$). We set the background
metric as $M=10^{-5}$, $\mu=10^{-2}$, and $a_{1}=10^{-4}$. Figure 11: Plot of
QN frequencies for various values of the spin ratio. Each marker indicates the
complex value of each QN frequency. We set $\mathcal{M}=10^{-5}$,
$\mu=10^{-2}$, and superradiant frequency $\Omega=116$, and the spin ratio is
set to $a_{2}/a_{1}=1$, $0.9$, $0.5$, and $0.1$. The angular modes are fixed
as $(l,m_{1},m_{2})=(1,1,0)$. The black dashed line indicates the superradiant
frequency.

We found that the structure of type-II modes exhibits the thermodynamic nature
of the Kerr-AdS5 black hole in the low-temperature or ultra-spinning limits
with $a_{1}=a_{2}$ (see Sec. 3.1 and 3.2). We here investigate if the thermal
interpretation of the type-II modes holds even for unequal spins ($a_{1}\neq
a_{2}$). The left panel in Figure 12 shows that the real part of type-II modes
approaches the superradiant frequency for lower $T_{\rm H}$, which is similar
to the case of small Kerr-AdS5 black holes with equal spins (see Figure 5).
From the right panel in Figure 12, one can also see that
$\Delta\text{Im}(\omega_{211n})$ matches $2\pi T_{\rm H}$ in the low-
temperature limit. From those results, we confirm that the thermodynamic
nature of the Kerr-AdS5 black hole we observed for the equal spin case still
holds even for unequal spins.

Figure 12: The real part and the separation of the imaginary part of QN
frequencies are shown with respect to $(a_{\rm max}-a_{1})/a_{\max}$ in the
left and right panels, respectively. We set $a_{2}/a_{1}=0.9$, $M=0.01$, and
$\mu=0.01$. In the left panel, the black solid line indicates the superradiant
frequency, $\Omega$, that depends on $a_{1}$, and the black dashed line shows
the value of $\Omega$ at the extremal case ($a_{1}=a_{\rm max}$). The blue
dashed line in the right panel shows $2\pi T_{\rm H}(a_{1})$.

###  4.2 Large black holes ${\cal M}\gtrsim 1$

We here investigate the type-II QN frequencies of a large black hole ${\cal
M}\gtrsim 1$ with unequal spins. The Kerr-AdS5 spacetime is stable for ${\cal
M}\gtrsim 1$ in the case of unequal spins, $a_{1}>a_{2}$, as well. In Figure
13, the QN frequencies are shown up to the 4th overtone. We can see that each
QN frequency has a bent path when the Hawking temperature changes while the
other quantities are fixed (left panel in Figure 13). As a Kerr-AdS5 black
hole becomes massive, the background spacetime is getting stabilized, which
can be seen in the right panel of Figure 13 and is similar to the equal spin
case.

Figure 13: Trajectories of QN frequencies in the complex frequency plane. In
the left panel, we set ${\cal M}=1$ and the Hawking temperature changes from
the lower value of $T_{\rm H}=0.01$ to the maximum temperature while fixing
the spin ratio $a_{2}/a_{1}$. The red open circles indicate the QN frequencies
for which $T_{\rm H}$ takes the maximum value. In the right panel, the Hawking
temperature is fixed at $T_{\rm H}=0.01$ and the ADM mass changes from ${\cal
M}=0.01$ (black open circles) to ${\cal M}=1$. The arrows indicate the
direction in which the ADM mass increases. The harmonic mode is set to
$(l,m_{1},m_{2})=(2,1,1)$. Figure 14: The separation of the imaginary part of
the type-II QN frequencies is shown as a function of $1-a_{1}$ in the ultra-
spinning limit. The harmonic mode is set to $(l,m_{1},m_{2})=(2,1,1)$ and
$M=10$. The black solid lines show the values of $2\pi T_{\rm H}(a_{1})$.
Figure 15: Real part of the QN frequencies in the ultra-spinning limit. The
spin ratio is fixed as $a_{2}/a_{1}=0.1$, $0.5$, $0.8$, and $1$, the mass
parameter is $M=10$, and the harmonic mode is $(l,m_{1},m_{2})=(2,1,1)$. The
QN frequencies are shown up to the 4th overtone. For the equal spins, the real
part of QN frequencies approaches zero as $\Omega\to 0$ in the ultra-spinning
limit. For unequal spin case, $\text{Re}(\omega_{211n})$ saturates at
$\Omega>0$. The black lines indicate the value of $\Omega$.

We have numerically shown that the damping rates of the type-II QN modes of a
small Kerr-AdS5 black hole with unequal spins also have the periodicity of
$2\pi T_{\rm H}$. We here numerically check that the intriguing thermodynamic
property of highly damped modes holds even for Kerr-AdS5 black holes with
large mass and unequal spins. We show the separation
$\Delta\text{Im}(\omega_{lm_{1}m_{2}n})$ as a function of $1-a_{1}$ in Figure
14. For highly damped modes, the separation approaches $2\pi T_{\rm H}$ in the
ultra-spinning limit. Also, Figure 15 shows the real part of the type-II QN
frequencies for up to the 4th overtone, and one can see that
$\text{Re}(\omega_{lm_{1}m_{2}n})$ approaches $\text{Re}(\omega)=\Omega$ as
$a_{1}\to 1$. In the next section, we discuss an implication we can obtain by
combining our result and the Hod’s conjecture on the black hole area
quantization Hod (1998).

##  5 Area quantization of the Kerr-AdS5 black hole

In the previous section, we found that the real part of the highly damped QN
frequencies approaches the superradiant frequency. The strong localization at
$\text{Re}(\omega)=\Omega$ can be seen in the low-temperature or ultra-
spinning limits. According to the Hod’s conjecture Hod (1998), the asymptotic
value of the real part of QN frequencies of massless fields determines the
smallest size of a quantized horizon area, $\Delta A$, and the horizon area is
given by $A=N_{\rm max}\Delta A$. The integer $N_{\rm max}$ is the total
number of the unit area on the horizon. In our numerical code, it is
technically difficult to handle the exactly massless scalar perturbation. On
the other hand, at least in the range of $10^{-12}\leq\mu\leq 0.01$, we
confirm that the values of QN frequencies converge well (see Figure 16). Hence
we assume that the scalar field with $\mu=0.01$ is effectively massless in our
situation. In this section, we discuss the relation between the QN modes and
the horizon area quantization for the Kerr-AdS5 black holes.

Let us briefly review the Hod’s conjecture and its physical interpretation.
Hod proposed that the overtone number $n$ may be interpreted as a quantum
number characterizing the energy levels of a black hole as an analogy of a
hydrogen atom, as in the Bohr’s corresponding principle. Based on this idea,
the discretized values of the real part of QN frequencies might be associated
with the energy of quanta which the black hole can emit or absorb. From this
point of view, the mass of a Schwarzschild black hole may be quantized as

$\Delta M=\lim_{n\to\infty}\text{Re}(\omega_{lm_{1}m_{2}n}),$ (35)

which leads to the horizon area quantization as $\Delta A/4G=\Delta M/T_{\rm
H}$.

Figure 16: Real part of QN frequencies of $(l,m_{1},m_{2})=(1,1,0)$ and $n=0$
and $4$. The mass of the scalar field is set to $\mu=0.01$, $10^{-7}$, and
$10^{-12}$. The parameters are set to $M=10$ and $a_{1}=a_{2}$.

At least, for the Schwarzschild Motl (2003) and Kerr Berti  _et al._ (2003);
Berti, Cardoso, and Yoshida (2004) black holes, the asymptotic value of the
real part of QN frequencies is independent of the angular index $l$ and is
universal. The quantization of the Kerr-AdS5 black hole mass may also lead to
the horizon area quantization222However, the Hod’s conjecture may be subtle
for the Kerr spacetime if the asymptotic value of the real part of QN
frequencies is $m\Omega$ for a Kerr black hole with ($a>0$). The naive
application of the Hod’s conjecture to that case leads to $\Delta A=0$. See
Ref. Berti  _et al._ (2003); Berti, Cardoso, and Yoshida (2004) for more
details.. The first law of black hole thermodynamics in the Kerr-AdS5
spacetime is Gibbons, Perry, and Pope (2005)

$\displaystyle\Delta{\cal M}=T_{\rm H}\frac{\Delta
A}{4G}+\sum_{i=1}^{2}\Omega_{i}\Delta J_{i}.$ (36)

Assuming (35), $\Delta J_{i}=m_{i}$, and

$\lim_{n\to\infty}\text{Re}(\omega_{lm_{1}m_{2}n})\to\Omega=\sum_{i}m_{i}\Omega_{i},$
(37)

the first law reduces to

$T_{\rm H}\frac{\Delta A}{4G}=0.$ (38)

It means that when $T_{\rm H}\neq 0$, the horizon is no longer quantized and
becomes a continuum, at least based on the Hod’s conjecture. Note that our
numerical results just imply that the real part of QN frequencies for the
first several overtones approaches $\Omega$, in the low-temperature limit.
Therefore, the strong convergence of $\text{Re}(\omega_{lm_{1}m_{2}n})$ to
$\Omega$ for $n\to\infty$ is still non-trivial from our investigation. Also,
for small black holes ${\cal M}\ll 1$, the lowest Hawking temperature is zero,
and so it is non-trivial if $\Delta A=0$ at $T_{\rm H}=0$. On the other hand,
for large black holes, the temperature is non-zero and finite even at the
ultra-spinning limit, and therefore, the assumption of (37) leads to $\Delta
A=0$ for ${\cal M}\gtrsim 1$.

We can show that in the ultra-spinning limit, the horizon area of the Kerr-
AdS5 black hole becomes singular. To demonstrate this, let us perform the
following coordinate transformations in the metric (1):

$\phi\to\sqrt{1-a_{1}^{2}}\phi,\ \psi\to\sqrt{1-a_{2}^{2}}\psi.$ (39)

Considering the geometry of constant $(t,r)$ surface in the near-horizon limit
($r\to r_{+}$), the line element for the ultra-spinning limit reduces to

$ds^{2}=\lim_{a\to
1}\frac{r_{+}^{2}+1}{1-a^{2}}\left[d\theta^{2}+\frac{(r_{+}^{2}+1)}{r_{+}^{2}}\left(\sin^{2}\theta
d\phi+\cos^{2}\theta d\psi\right)^{2}\right],$ (40)

where we take $a_{1}=a_{2}=a$. This shows that in the limit, the horizon area
diverges in the coordinates. On the other hand, when we take the limit of
$a_{1}\to 1$ while keeping $a_{2}<1$, the topology of the horizon becomes non-
compact. As a simple example, let us look into the case of $a_{1}\to 1$ and
$a_{2}=0$. In that case, the metric reduces to

$\displaystyle ds^{2}=\frac{(r_{+}^{2}+1)^{2}}{\rho^{2}}\sin^{4}\theta
d\phi^{2}+\frac{r_{+}^{4}}{\rho^{2}}\sin^{2}\theta\cos^{2}\theta
d\psi^{2}+\frac{r_{+}^{2}(r_{+}^{2}+1)}{\rho^{2}}\cos^{4}\theta
d\psi^{2}+\frac{\rho^{2}}{\sin^{2}\theta}d\theta^{2},$ (41)

where we performed the following transformations for the original metric

$\phi\to(1-a_{1}^{2})\phi,\ \psi\to(1-a_{2}^{2})\psi.$ (42)

The metric in (41) appears to be ill-defined at $\theta=0$. However, by
performing a further transformation of

$\xi=1-\cos\theta,$ (43)

the metric near the pole of $\theta=0$ is

$ds^{2}\simeq(r_{+}^{2}+1)\left(4\xi^{2}d\phi^{2}+\frac{d\xi^{2}}{4\xi^{2}}\right)+r_{+}^{2}d\psi^{2},$
(44)

which has the submanifold of $\mathbb{H}^{2}$ on $\phi-\xi$ surface, and we
find that the constant $(t,r)$ surface is non-compact in the ultra-spinning
limit. One can easily find that for $a_{1}\to 1$ and $a_{2}\neq 0$, the
$\phi-\xi$ surface near $r=r_{+}$ and $\theta=0$ is also non-compact as its
metric is

$ds^{2}\simeq(r_{+}^{2}+1)\left(4\xi^{2}\frac{r_{+}^{2}+a_{2}^{2}}{r_{+}^{2}}d\phi^{2}+\frac{1}{1-a_{2}^{2}}\frac{d\xi^{2}}{4\xi^{2}}\right).$
(45)

Indeed, a similar non-compact horizon appears even in the Kerr-AdS4 black hole
Klemm (2014); Hennigar, Kubizňák, and Mann (2015); Hennigar  _et al._ (2015).

In summary, we have applied the Hod’s proposal to our result that the real
part of scalar QN frequencies approaches the superradiant frequency in the
low-temperature or ultra-spinning limits. Then we have concluded that it leads
to a continuous horizon area, i.e. unquantized horizon area, when naively
applying the Hod’s conjecture to Kerr-AdS5 black holes. Also, in the special
cases involving the ultra-spinning regime, the total horizon area diverges, or
the topology of the Kerr-AdS5 black hole horizon is non-compact.

##  6 Discussion and Conclusion

In this paper, we have numerically investigated the structure of scalar quasi-
normal (QN) frequencies of the five-dimensional Kerr anti-de Sitter (Kerr-
AdS5) black hole. Our numerical investigation covers a broad range of
parameter regions for the Kerr-AdS5 black hole.

In Sec. 3.1, we have studied the instability of small Kerr-AdS5 black holes
with equal spins. We have confirmed that the strong instability is caused by
$l=1$, and the most unstable QN mode has its frequency close to the
superradiant frequency defined by $\Omega\equiv
m_{1}\Omega_{+,1}+m_{2}\Omega_{+,2}$. This is the case even for unequal spins,
as shown in Sec. 4.1. We have also investigated if the superradiant
instability is amplified when the Kerr-AdS5 black hole has a hierarchy between
the two spins, i.e., $a_{2}/a_{1}<1$. As we have checked that the most
unstable harmonics is $(l,m_{1},m_{2})=(1,1,0)$ for $a_{2}<a_{1}$ (Figure 10),
we computed the unstable QN frequencies of $(1,1,0)$ mode for different spin
ratios while fixing the angular velocity $\Omega_{1}$ which is equivalent to
the superradiant frequency for $(1,1,0)$ mode. Then, we have found that the
stronger the hierarchy of the two spins is, the more enhanced the instability
is (Figure 11). We conclude that the superradiant instability of the Kerr-AdS5
black holes is enhanced when the symmetry of the spacetime is reduced. The QN
modes that could induce the instability of small Kerr-AdS5 black holes
localize near the real axis in the complex frequency plane. We call those
modes type-I modes. On the other hand, we have investigated highly damped QN
modes (type-II modes), which differ from the type-I modes and might be
relevant to the thermodynamic nature of the Kerr-AdS5 black hole as the
separation of their imaginary part is $\sim 2\pi T_{\rm H}$. We confirmed that
the separation matches $2\pi T_{\rm H}$ in high accuracy in the low-
temperature limit.

In Sec. 3.2 and Sec. 4.2, we have investigated the scalar QN modes for large
Kerr-AdS5 black holes with equal and unequal spins, respectively. As spin
parameters approach the AdS curvature radius, that is the ultra-spinning limit
($a_{i}\to 1$), type-I modes are suppressed and type-II modes are excited
along $\text{Re}(\omega)=\Omega$. The type-II modes with higher damping rates
have the periodic separation in their imaginary parts, and the separation for
the first several tones matches the surface gravity of the horizon when
$a_{i}\to 1$ and $\mu\ll 1$ (see Figure 8). The thermal nature holds even for
large Kerr-AdS5 black holes with unequal spins (Figure 14 and 15). It would be
interesting to study how this property can be relevant to the pole structure
of the thermal Green’s function in the corresponding conformal field theory
(CFT) on the AdS boundary.

Based on the Hod’s conjecture regarding the horizon area quantization Hod
(1998), the asymptotic value of $\text{Re}(\omega_{lm_{1}m_{2}n})$ plays an
important role in determining the one-bit size of a black hole area $\Delta A$
as $T_{\rm H}\Delta
A/4G=\lim_{n\to\infty}\text{Re}(\omega_{lm_{1}m_{2}n})-\Omega$. Therefore, as
we discussed in Sec. 5, the convergence of the real part of type-II QN
frequencies to $\Omega$ for large $n$ implies that the horizon area is no
longer quantized but is continuous when naively applying the Hod’s conjecture
to the Kerr-AdS5 black holes. From our numerical result, the convergence is
likely at least for the low-temperature and ultra-spinning regimes. We leave
the further study of highly damped QN modes in an analytical way for future
work. We also analyzed the horizon topology of the Kerr-AdS5 black hole and
found that in general, an ultra-spinning Kerr-AdS5 black hole has its non-
compact horizon.

## Acknowledgement

This work was supported in part by the Special Postdoctoral Researcher (SPDR)
Program at RIKEN (NO), Incentive Research Project at RIKEN (NO), Grant-in-Aid
for Scientific Research (KAKENHI) project (21K20371) (NO), and Japan Society
for the Promotion of Science(JSPS) Grant No. 20J22946(KU).

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