# Continuously Learning to Detect People on the Fly: A Bio-inspired Visual
System for Drones

Ali Safa, , Ilja Ocket, , André Bourdoux, ,
Hichem Sahli, Francky Catthoor, , Georges G.E. Gielen A. Safa, I. Ocket, F.
Catthoor and G. G.E Gielen are with imec and KU Leuven, 3001 Leuven, Belgium
<EMAIL_ADDRESS>Georges.Gielen@kuleuven.be). A.
Bourdoux is with imec, 3001 Leuven, Belgium (Andre.Bourdoux@imec.be). H. Sahli
is with imec and ETRO VUB, 1050 Brussels, Belgium<EMAIL_ADDRESS>
###### Abstract

This paper demonstrates for the first time that a biologically-plausible
spiking neural network (SNN) equipped with Spike-Timing-Dependent Plasticity
(STDP) can continuously learn to detect walking people on the fly using
retina-inspired, event-based cameras. Our pipeline works as follows. First, a
short sequence of event data ($<2$ minutes), capturing a walking human by a
flying drone, is forwarded to a convolutional SNN-STDP system which also
receives teacher spiking signals from a readout (forming a semi-supervised
system). Then, STDP adaptation is stopped and the learned system is assessed
on testing sequences. We conduct several experiments to study the effect of
key parameters in our system and to compare it against conventionally-trained
CNNs. We show that our system reaches a higher peak $F_{1}$ score (+19%)
compared to CNNs with event-based camera frames, while enabling on-line
adaptation.

###### Index Terms:

Bio-inspired vision, continual learning, drones

##  I Introduction

In recent years, the use of micro drones has attracted much attention for
applications ranging from infrastructure inspection to people search and
rescue [1]. In those applications, drones and humans will be moving within the
same environment. Therefore, it is critical to equip micro drones with people
detection pipelines for safety purposes [2].

Following the enormous progress in deep learning, convolutional neural
networks (CNNs) such as You Only Look Once (YOLO) and its variants, constitute
the state of the art in terms of detection and speed performance [3]. However,
the use of such CNNs is not suited for micro drones because of the significant
power budget that conventional CNNs demand ($2$W for an edge TPU, $10$W for a
Jetson Nano) relative to the micro drone power budget ($<1$W). Therefore, a
significant effort has been put in the co-design of CNNs and micro-controller
(MCU) architectures in order to reduce the power and memory consumption budget
of the vision system down to $<100$mW and $<1$MB respectively [4]. Still,
adopting a conventional CNN-hardware design approach leads to rigid systems
that cannot easily adapt themselves to new environments and tasks (unlike the
nervous systems of biological entities in nature).

Figure 2: Continual learning of people detection. Conceptual illustration of a
micro-drone equipped with an event-based camera, an RGB camera and a bio-
inspired SNN-STDP network. The RGB frames are sent to a back-end server where
an oracle CNN returns people detection labels. The labels are used during the
continual learning of the SNN-STDP system.

Indeed, conventional CNNs rely on the standard off-line training procedure
where the CNN is trained on a large dataset capturing the task that needs to
be solved. However, it is not always guaranteed that all scenarios to be
encountered at run time are well captured by the training set (e.g., partially
damaged infrastructures are hard to capture by a dedicated dataset for indoor
search and rescue tasks). In addition, since CNN training requires desktop-
grade computing, the conventional solution for fine-tuning on-board CNNs is to
first re-train the network offline and then send the updated weights back to
the drone. Clearly, this conventional approach has many drawbacks such as high
latency (training takes a long time, prohibiting fast adaptation) and privacy
concerns due to the on-line storage of user data (also causing legal issues
due to e.g., the right to data erasure in the EU111https://gdpr.eu/right-to-
be-forgotten/).

Recently, the use of biologically-inspired spiking neural networks (SNNs) has
gained huge interest for the design of ultra-low-power AI-enabled systems [12,
6]. In contrast to CNNs, SNNs make use of spiking neurons that communicate
through low-complexity binary activations in an event-based manner (vs. frame-
based processing in CNNs), only consuming energy when a spike is emitted [12].
In addition, SNNs can be implemented in massively parallel, non Von Neumann
computing architectures, solving the energy- and latency-expensive memory
bottleneck issues [6]. Finally, the use of a bio-inspired learning rule (vs.
backprop) working in the binary activation domain and local to each neuron
(such as Spike-Timing-Dependent Plasticity or STDP [8]) enables ultra-low-
power learning at the edge (not tackled by embedded CNNs) [9]. Therefore, a
growing number of SNN-STDP computing units have been proposed, achieving a
power consumption as low as $\sim 10$0$\mu$W [9] while enabling on-chip
inference and learning (three orders of magnitude lower power consumption vs.
optimized CNNs running on MCU [4]).

As SNNs require spiking data as input, they are often used in conjunction with
an event-based camera (also called dynamic vision sensor or DVS), inspired by
the inner working of the human eye. DVS cameras are composed of independent
pixels that emit spikes asynchronously whenever the change in light log-
intensity crosses a threshold $C$ [10] (see Fig. 1).

Figure 1: Conceptual illustration of event-based data. The DVS is capturing an
image of ”0”. a) The DVS outputs a spike stream in time and space. b) When the
change $|\Delta L_{ij}|$ in light $L_{ij}$ at pixel $(i,j)$ crosses a
threshold $C$, a spike is emitted. The spike is positive if $\Delta L_{ij}>0$
and negative otherwise.

In this paper, our goal is to investigate the design of a bio-inspired SNN-
STDP system that can continuously learn to perform people detection on drones
from DVS camera data. Fig. 2 illustrates the scenario that we consider in this
work: the drone must explore the environment during $\sim 2$ minutes and
continuously learn to detect a walking human subject using the current data
sample only, without access to the past and the future data. Then, continual
learning (CL) is stopped and we assess the detection performance on a testing
sequence.

As the basis for our people detection network, we use the state-of-the-art
SNN-STDP architecture proposed in [12] and we augment it with three
significant enhancements: i) we motivate and extend the SNN-STDP network to
the use of anti-Hebbian (negative STDP) learning rules; ii) we propose a semi-
supervised method that allows the learning of the logistic regression readout
and the SNN at the same time (vs. disjoint learning in [12]) and iii) we use
our SNN in a more challenging CL setting (vs. offline training in [12]).

This paper is organized as follows. Related works are discussed in Section II.
Background theory is given in Section III. Our methods are presented in
Section IV. Results are discussed in Section V. Conclusions are given in
Section VI.

Figure 4: SNN-STDP architecture. a) The coding layer features $i=1,...,M$
pairs of push-pull LIF neurons ($LIF_{pp}$) that receive the spiking vector
$\bar{s}$ (input) through $\Phi^{T}$ (input weights) and the previous spiking
output vector $\bar{c}$ through $W$ (lateral weights). The output spike trains
$\bar{c}$ are obtained following (7). In addition, the coding layer receives
the re-projection error spike trains $\bar{e}$ through $\Phi^{T}$ and combines
them with the other inputs using (8) to obtain $f_{i}$ locally to each coding
neuron. STDP is applied for learning $\Phi^{T}$ and $W$ with $c_{i}$ and
$f_{i}$ the post-synaptic and $\bar{e}$ the pre-synaptic spike trains. b) The
error layer features $j=1,...,N$ pairs of push-pull LIF neurons $LIF_{pp}$
which receive the entry $s_{j}$ as input and the SNN output $\bar{c}$ through
$\Phi$ (feedback weights, remain identical to the transpose of input weights
during learning). The error layer outputs the spiking vector $\bar{e}$
following (6). STDP is applied for learning $\Phi$ with each $e_{j}(t)$ as the
post-synaptic spike train and $\bar{c}$ as the pre-synaptic spike trains. c)
The pair of push-pull LIF neurons codes both positive and negative values of
the input current $J_{in}$ (output spikes of positive or negative polarity).

##  II Related Works

A growing number of bio-inspired robotic systems have been proposed in the
past decades, mainly focusing on the study of animal-like actuation and motor
control [11]. Complementary to bio-actuation, a number of perception systems
for drones, taking inspiration from the inner workings of the human eye and
the visual cortex, have recently emerged thanks to the advent of neuromorphic
event-based cameras [6, 7, 10]. A line tracking system for drones was proposed
in [6] using a DVS camera processed by an SNN. Similar to our work, the system
of [6] uses an SNN with local bio-inspired learning rules (analogous to STDP)
and demonstrates that such adaptive SNN system can learn to compensate for
external disturbances online. Closer to our work, a DVS-based moving object
tracking system for drones was proposed in [7], using hand-crafted feature
extraction followed by Kalman filtering (processing pipeline not bio-
plausible). Compared to the hand-crafted feature extraction in [7], we
proposed a bio-inspired vision system that can continuously learn to detect
walking people from a drone (vs. detecting any moving object in [7]).

Since sparsely-supervised and unsupervised learning are believed to be key
mechanisms in the brain, a number of bio-inspired architectures have been
proposed for learning spatio-temporal features from event-based data [18, 12].
Among them, the hierarchy of event-based time surfaces (HOTS) was proposed in
[18] by cascading unsupervised layers, trained to cluster event-data
aggregated as time surfaces (exponential decay maps) and by using an output
readout (e.g., logistic regression) to classify the extracted features. In
contrast to HOTS, which uses conventional clustering methods, an unsupervised
SNN-STDP network with better biological plausibility was proposed by us in
[12] for DVS feature learning, achieving state-of-the-art performance on
common DVS benchmarks. Thus, we use the network of [12] as our basis in this
work.

##  III Background theory

###  III-A SNN-STDP fundamentals

In contrast to CNNs, SNNs make use of spiking neurons, often modelled by a
Leaky Integrate-and-Fire (LIF) activation:

$\begin{cases}\frac{dV}{dt}=\frac{1}{\tau_{m}}(J_{in}-V)\\\
\sigma=1,V\xleftarrow{}0\hskip 3.0pt\text{{if}}\hskip 3.0ptV\geq\mu,\hskip
3.0pt\text{{else}}\hskip 3.0pt\sigma=0\end{cases}$ (1)

with $J_{in}$ the input current to the neuron, $\sigma$ the spiking output,
$V$ the membrane potential, $\tau_{m}$ the membrane time constant and $\mu$
the neuron threshold [12]. The scalar input current $J_{in}$ is continuously
integrated in $V$ following (1). When $V$ crosses the firing threshold $\mu$,
the membrane potential is reset back to zero and an output spike is emitted.
The input current $J_{in}$ is obtained by filtering the inner product of the
neural weights and the spiking inputs through a post-synaptic current (PSC)
kernel [12] (estimating the spiking rate):

$J_{in}=\mathcal{PSC}\\{\bar{w}^{T}\bar{s}_{in}(t)\\}$ (2)

with $\bar{s}_{in}(t)$ the input spiking vector (originating from e.g. an
event camera or other spiking neurons), $\bar{w}$ the weight vector and
$\mathcal{P}\mathcal{S}\mathcal{C}\\{\theta(t)\\}=\theta(t)*\frac{1}{\tau_{s}}e^{-t/\tau_{s}}$
the effect of PSC filtering with time constant $\tau_{s}$. Fig. 3 a)
conceptually illustrates the LIF neuron behaviour.

Figure 3: Conceptual illustration of the LIF neuron and STDP learning. a) The
LIF receives as input a pre-synaptic spike train $s_{in}$. When the LIF
membrane potential $V$ rises above the threshold $\mu$, a spike is emitted
($\sigma=1$) and $V$ is set back to zero. b) The synapse strength is modified
following the double exponential STDP rule (3) in function of the difference
between the post- and the pre-synaptic spike times
$\tau_{ij}=t_{post,i}-t_{pre,j}$.

In the SNN architecture used in this work, learning is performed using the
STDP rule, locally modifying the weights $w_{ij}$ of each neuron $i$ as
follows:

$w_{ij}\xleftarrow{}\begin{cases}w_{ij}+A_{+}e^{-\tau_{ij}/\tau_{+}},&\text{if
}\tau_{ij}\geq 0\\\ w_{ij}-A_{-}e^{\tau_{ij}/\tau_{-}},&\text{if
}\tau_{ij}<0\end{cases}$ (3)

with $A_{+},A_{-}$ the long-term potentiation (LTP) and depression (LTD)
weights, $\tau_{+},\tau_{-}$ the potentiation and depression decay constants,
$w_{ij}$ the $j^{th}$ element of the $i^{th}$ neuron weight vector $\bar{w}$
and $\tau_{ij}$ the time difference between the post- and the pre-synaptic
spike times across the $j^{th}$ synapse of neuron $i$ (see Fig. 3 b) [8].
Finally, it can be shown [12] that a good approximation for the long-term
effect of STDP (i.e., its expected value over time $t$) is given by the
product of the post- and pre-synaptic mean spike rates (noted $r_{post,pre}$):

$\Delta
w_{\text{STDP}}\\{s_{post},s_{pre}\\}\approx\eta_{2}(A_{+}\tau_{+}-A_{-}\tau_{-})r_{post}r_{pre}$
(4)

where $\eta_{2}$ is the learning rate.

###  III-B SNN-STDP as Dictionary Learning, Basis Pursuit (DLBP)

As proposed in [12], the SNN-STDP in Fig. 4 is iteratively solving the
unsupervised DLBP problem (inferring $\bar{c}$ using the current $\Phi$ and
learning the next $\Phi$ using the current $\bar{c}$):

$\bar{c},\Phi=\arg\min_{\bar{c},\Phi}\frac{1}{2}||\Phi\bar{c}-\bar{s}||_{2}^{2}+\lambda_{1}||\bar{c}||_{1}+\frac{\lambda_{2}}{2}||\Phi||_{F}^{2}$
(5)

where $\bar{c}$ is the $M$-dimensional output vector containing the mean spike
rates of each output neuron in the SNN, $\Phi$ is the learned network weight
matrix, $\bar{s}$ is the $N$-dimensional input vector, $\lambda_{1}$ is a
sparsity-controlling hyper-parameter and $\lambda_{2}$ is a weight decay
hyper-parameter.

Figure 5: Network architecture for CL and people detection using a DVS camera.
a) Convolutional SNN-STDP followed by a convolutional readout layer. The SNN
output is normalized using (14). The normalized tensor is then fed to a
standard scaling block (15) and the resulting tensor is processed by the
readout. The readout outputs an attention map indicating the position of a
human subject on the image plane. The readout continuously outputs a task-
driven (TD) signal $\bar{u}(t)$ following (23). b) The $i^{th}$ neuron in the
coding layer (see Fig. 4 a) is augmented with an additional synapse to receive
the $i^{th}$ component of the TD signal $u_{i}(t)$. c) The $j^{th}$ neuron in
the error layer (see Fig. 4 b) is augmented with additional inputs receiving
$\bar{u}(t)$.

DLBP (5) is solved iteratively by applying STDP across all neurons as follows
[12]. The $j^{th}$ neuron in the error layer (see Fig. 4 b) receives as input
the spike trains from the coding layer and outputs the reconstruction error in
the spike domain:

$e_{j}=LIF_{pp}\\{\Phi_{j}\bar{c}-s_{j}\\}$ (6)

where $\Phi_{j}$ is locally modified by applying STDP with the $e_{j}$ as the
post- and $\bar{c}(t)$ as the pre-synaptic spike trains. $LIF_{pp}$ denotes
the application of a push-pull pair of LIF neurons (1) (see Fig. 4 c). The
$i^{th}$ neuron in the coding layer (see Fig. 4 a) receives $\bar{s}$ and the
previous output vector $\bar{c}$ as input, and generates the output (where
$\Phi_{i},W_{i}$ are the $i^{th}$ matrix rows):

$c_{i}=LIF_{pp}\\{\Phi_{i}\bar{s}-W_{i}\bar{c}\\}\hskip 3.0pt\textbf{,}\hskip
3.0ptW\equiv\eta_{1}\tilde{W}-I_{M}\hskip 3.0pt\textbf{,}\hskip
3.0pt\tilde{W}\equiv\Phi^{T}\Phi$ (7)

with $\eta_{1}$ the output convergence rate (to be set) and $I_{M}$ the
identity matrix of dimension $M$. Each coding neuron $i$ also receives the
error vector $\bar{e}$ from the error layer and locally combines it with its
inputs $\bar{c}$ and $\bar{s}$ to form the local signal $f_{i}$:

$f_{i}(t)=\underbrace{\eta_{1}^{-1}(W+I_{M})_{i}}_{\tilde{W}_{i}\hskip
3.0pt(\ref{infer})}\bar{c}(t)-(\Phi^{T})_{i}\bar{e}(t)-(\Phi^{T})_{i}\bar{s}(t)\\\\[-15.0pt]
\stackrel{{\scriptstyle\text{(\ref{eerr})}}}{{\approx}}(\tilde{W}-\Phi^{T}\Phi)_{i}\bar{c}(t)$
(8)

Then, STDP is applied across each neuron $i$ using $c_{i}$ as the post- and
$\bar{e}$ as the pre-synaptic spike trains for learning $\Phi^{T}$. STDP is
also applied using $f_{i}$ as the post- and $\bar{e}$ as the pre-synaptic
spike trains for learning $W$ [12]. Finally, it can be remarked that the SNN-
STDP architecture of Fig. 4 can naturally be extended to the convolutional
setting by sweeping the network of Fig. 4 across a $H\times W$ image plane
with kernel size $[l_{x},l_{y}]$ and stride $d$ [12]. A spiking tensor of
dimension:

$[D_{x},D_{y},D_{c}]=[\frac{H-l_{x}}{d}+1,\frac{W-l_{y}}{d}+1,M]$ (9)

is obtained. In this convolutional case, (5) can keep a similar form by
considering $\Phi$ as a circulant matrix and $\bar{s}$, $\bar{c}$, $\bar{e}$
as flattened tensors along the image plane dimension.

##  IV Proposed methods

###  IV-A Extension to anti-Hebbian learning

Anti-Hebbian learning (or negative STDP, see Fig. 3 b) refers to the
application of the opposite of the STDP rule (3) with negative $A_{+}$ and
$A_{-}$ quantities in (3), and has been investigated in a few works [13].
Following the derivations in [12], it can be shown that using negative STDP
for learning the input weights $\Phi^{T}$ and feedback weights $\Phi$ (see
Fig. 4), while using standard STDP for the learning of the lateral weights $W$
(7), the following joint optimisation objective is solved:

$\begin{cases}\bar{c}=\arg\min_{\bar{c}}||\Phi\bar{c}-\bar{s}||_{2}^{2}+\lambda_{1}||\bar{c}||_{1}\\\
\Phi=\arg\min_{\Phi}-\log(||\Phi\bar{c}-\bar{s}||_{2}^{2})+\frac{\lambda_{2}}{2}||\Phi||_{F}^{2}\end{cases}$
(10)

Similar to (5), the SNN iterations infer the next SNN output $\bar{c}$ by
considering the current $\Phi$ constant and learns the next SNN weight matrix
$\Phi$ by considering the current $\bar{c}$ fixed.

Negative STDP solves (10) since:

$\frac{\partial}{\partial\Phi}\\{-\log(||\Phi\bar{c}-\bar{s}||_{2}^{2})\\}=-\frac{(\Phi\bar{c}-\bar{s})\bar{c}^{T}}{||\Phi\bar{c}-\bar{s}||_{2}^{2}}$
(11)

and since it has been shown in [12] that standard STDP approximates the
gradient-based learning:

$\Phi\xleftarrow{}\Phi-\eta_{2}(\Phi\bar{c}-\bar{s})\bar{c}^{T}-\eta_{2}\lambda_{2}\Phi$
(12)

it is now clear that negative STDP approximating:

$\Phi\xleftarrow{}\Phi-\eta_{2}\\{-(\Phi\bar{c}-\bar{s})\bar{c}^{T}\\}-\eta_{2}\lambda_{2}\Phi$
(13)

is using a gradient similar to (11) (up to a dividing scalar oscillating
around $\sim 100$ during our experiments) and is learning $\Phi$ in (10). When
combined with a supervision term (see Section IV-C), the use of the negative
STDP rule can be motivated as a regularization controlling the over-fitting of
the SNN-STDP by preventing the re-projection $||\Phi\bar{c}-\bar{s}||_{2}^{2}$
from becoming too small (see experiments in Section V-B).

###  IV-B Proposed bio-inspired SNN-STDP architecture

Fig. 5 a) shows the CL SNN-STDP system used throughout this work to infer
attention maps. We set up a convolutional network by cascading the SNN-STDP
network of Section IV-A (in a convolutional setting with negative STDP
learning) with a convolutional readout in order to learn attention maps. At
the output of the convolutional readout, we use sigmoid activation functions
in order to constrain the dynamic range of the attention map between $0$ and
$1$. The rate of the spiking activity at the SNN output is estimated using a
rolling time window of length $N_{s}=20$. Doing so, a tensor $T_{x,y,c}$
containing the mean spike rates for each spatial coordinate $[x,y]$ and each
channel $c$ is obtained. Then, $T_{x,y,c}$ is normalized along the channel
dimension $c$ to provide invariance to event density [12]:

$\tilde{T}_{x,y,c}=\frac{|T_{x,y,c}|}{\sum_{c}|T_{x,y,c}|}$ (14)

In between the SNN-STDP layer and the readout, we use standard scaling along
the channel dimension $c$ in order to reduce the internal covariate shifts
[14]:

$T^{*}_{x,y,c}=\frac{\tilde{T}_{x,y,c}-\mu_{c}}{\sigma_{c}}$ (15)

where the mean $\mu_{c}$ and standard deviation $\sigma_{c}$ are estimated on-
line. We note the transformations (14)(15) of $\bar{c}$ into $T^{*}$ as
$T^{*}\\{\bar{c}\\}$. Finally, $T^{*}_{x,y,c}$ is zero-padded and fed to the
readout layer with a convolutional kernel size $[r_{x},r_{y},M]$ in order to
obtain an attention map $A_{x,y}$ of dimension $[D_{x},D_{y}]$ following (9).
Finally, we interpolate the attention map $A_{x,y}$ back to the input size
$[H,W]$.

Without the use of the normalization mechanisms (14)(15), we observed the CL
process to fail due to the severe covariate shifts encountered during
learning. In addition, we observed that using a convolutional readout layer
over a standard fully-connected layer is crucial in order to avoid readout
overfitting on the local context encountered by the drone during the CL
process (CL fails when using a fully-connected layer). The architecture of
Fig. 5 a) also shows that the readout generates a vector of task-driven (TD)
spike trains $\bar{u}(t)$ that is fed back to the SNN neurons. The TD spike
trains are used to steer the unsupervised STDP-driven weight dynamics in order
to jointly optimize the SNN weights $\Phi$ and the readout $\Psi$, forming a
task-driven, semi-supervised learning system.

###  IV-C Task-driven semi-supervised learning

During our experiments, we observe that the SNN-STDP system and the readout
should not be optimised independently as in [12] (i.e., unsupervised learning
of SNN and disjoint learning of a readout). Rather, the SNN and the readout
must be learned concurrently. Indeed, if the learning process would be
independent, the way the readout should fit the SNN output in the beginning of
the STDP learning process will not be the same as in the end, since the SNN
weights have independently changed since then, leading to significant
covariate shifts at the input of the readout. Following this discussion, and
the effectiveness of unsupervised DVS learning [12], we opt for a joint semi-
supervised SNN-readout objective:

$\begin{cases}\bar{c}=\arg\min_{\bar{c}}||\Phi\bar{c}-\bar{s}||_{2}^{2}+\lambda_{1}||\bar{c}||_{1}\\\
\Phi=\arg\min_{\Phi}\lambda_{s}\mathcal{L}_{sup}(\bar{\sigma}(\Psi,T^{*}\\{\bar{c}\\},\bar{b}),\bar{y})\\\
\hskip
5.0pt-\lambda_{u}\log(||\Phi\bar{c}-\bar{s}||_{2}^{2})+\frac{\lambda_{2}}{2}||\Phi||_{F}^{2}\\\
\Psi=\arg\min_{\Psi}\lambda_{s}\mathcal{L}_{sup}(\bar{\sigma}(\Psi,T^{*}\\{\bar{c}\\},\bar{b}),\bar{y})\end{cases}$
(16)

where in addition to (10), $\Psi$ is the $K\times M$ weight matrix of the
readout, $\bar{b}$ the learned bias vector, $\sigma(.)$ the sigmoid
activation, $\bar{y}$ the desired output label vector of dimension $K\times
1$, $\mathcal{L}_{sup}$ the supervised loss function and $\lambda_{u,s}$ the
hyper-parameters controlling the strength of the unsupervised and supervised
contributions. As usual, the network infers the next $\bar{c}$ keeping
$\Phi,\Psi$ constant, then learn $\Phi$ keeping $\Psi,\bar{c}$ constant and so
on. We choose $\mathcal{L}_{sup}$ as the focal loss because of its robustness
to imbalanced data compared to the standard cross-entropy [15]:

$\mathcal{L}_{sup}(\sigma_{i}(\Psi_{i},T^{*}\\{\bar{c}\\},\bar{b}),y_{i})=-(1-p_{i})^{\gamma}\log(p_{i})$
(17)

with $\gamma$ setting the robustness to class imbalance and:

$p_{i}=\begin{cases}\sigma_{i}(\Psi_{i},T^{*}\\{\bar{c}\\},\bar{b})\hskip
3.0pt\text{{if}}\hskip 3.0pty_{i}=1\\\
1-\sigma_{i}(\Psi_{i},T^{*}\\{\bar{c}\\},\bar{b})\hskip
3.0pt\text{{otherwise}}\end{cases}$ (18)

In order to optimise $\mathcal{L}_{sup}$ as a function of the SNN weights
$\Phi$, the link between $\bar{c}$ and $\Phi$ must be made explicit. Since
$\bar{c}$ is the result of an iterative process involving the LIF neuron non-
linearity, an analytical function $\bar{c}(\Phi,\bar{s})$ is hard to find and
further relaxations must be done. First, we observe in (16) that for a small
$\lambda_{1}$, $\bar{c}(\Phi,\bar{s})$ is well approximated as:

$\bar{c}(\Phi,\bar{s})\sim(\Phi^{T}\Phi)^{-1}\Phi^{T}\bar{s}$ (19)

Since $\Phi$ is initialized following an i.i.d. zero-mean normal distribution
with standard deviation $\sigma_{w}=0.01$ [12], the following relaxation holds
during the early SNN-STDP learning steps (when most of the learning effect
takes place):

$\bar{c}(\Phi,\bar{s})\sim\frac{1}{N\sigma_{w}^{2}}\Phi^{T}\bar{s}$ (20)

Then, the gradient of $\mathcal{L}_{sup}$ as a function of $\Phi$ is:

$\frac{\partial\mathcal{L}_{sup}}{\partial\Phi}=k\bar{s}(\frac{\partial\mathcal{L}_{sup}}{\partial\bar{c}})^{T}$
(21)

where $k=\frac{1}{N\sigma^{2}}$ is a constant (dropped further on) and the
gradient $\frac{\partial\mathcal{L}_{sup}}{\partial\bar{c}}$ directly follows
from (14)(15)(17)(18).

###  IV-D Task-driven SNN-STDP topology

Thanks to the relaxation (20)(21), the original unsupervised SNN-STDP
architecture of Fig. 4 can now be modified in order to incorporate the effect
of the newly added task-driven term in (16). We observe in (21) that:

$\bar{v}=(\frac{\partial\mathcal{L}_{sup}}{\partial\bar{c}})^{T}$ (22)

is a $1\times M$ vector computed at the readout side (e.g., in an MCU used to
post-process the SNN activity) while the spike train vector $\bar{s}(t)$ in
(21) is already available in both the coding and error layers (see Fig. 4 a).
Therefore, the TD contribution can be injected in the SNN-STDP system by first
converting $\bar{v}$ to a push-pull pair of spike trains as follows:

$\bar{u}(t)=LIF_{pp}\\{\bar{v}\\}$ (23)

Then, each entry $u_{i}$ of $\bar{u}(t)$ can be fed to the $i^{th}$ neuron in
the coding layer via one additional synapse (see Fig. 5 b), and the spiking
vector $\bar{u}(t)$ can be distributed to each error neuron via an additional
set of synapses (see Fig. 5 c).

Finally, in addition to the unsupervised STDP mechanisms already present in
the network of Fig. 4 to solve (5), it follows from (4) that an additional
STDP contribution must be applied to the neurons in the error layer as
follows:

$(\Phi_{j})_{i}\xleftarrow{}(\Phi_{j})_{i}-\lambda_{s}\Delta
w_{\text{STDP}}\\{s_{j}(t),u_{i}(t)\\}$ (24)

with $s_{j}(t)$ the post- and $u_{i}(t)$ the pre-synaptic spike trains. An
additional STDP contribution must also be applied to the neurons in the coding
layer as:

$(\Phi_{i}^{t})_{j}\xleftarrow{}(\Phi_{i}^{t})_{j}-\lambda_{s}\Delta
w_{\text{STDP}}\\{u_{i}(t),s_{j}(t)\\}$ (25)

with $u_{i}(t)$ the post- and $s_{j}(t)$ the pre-synaptic spike trains.
Regarding the lateral weights $W$, it can be shown [12] that the STDP
mechanism adopted in the network of Fig. 4 automatically enforces consistency
between the task-driven learning of $\Phi$ and the convergence of $W$ (since
applying STDP between $f_{i}(t)$ in (8) and $\bar{e}(t)$ in (6) forces $W$ to
converge to $\eta_{1}\Phi^{T}\Phi-I_{M}$ in (7) [12]).

###  IV-E Continual learning strategy

For on-line labelling, we use a pre-trained YOLOv3 network [5] to detect the
presence or absence of a human subject from the RGB frames that are jointly
acquired with the DVS data, and to infer the bounding box coordinates
$[b_{x,1},b_{y,1},b_{x,2},b_{y,2}]$ (normalized to the size of the output map)
indicating the location of the human subject in the image. When a human
subject is present, the corresponding label map $\bar{y}$ (used for the
supervised contribution) is assigned with ones inside the bounding box region
and zeros outside. Since there are significantly more zero-valued entries in
$\bar{y}$, we use median frequency balancing [16] to help mitigate the
learning imbalance in space. Therefore, we weight the readout loss as follows:

$\mathcal{L}_{sup}(\sigma_{i},y_{i})\xleftarrow{}w_{i}\mathcal{L}_{sup}(\sigma_{i},y_{i})\hskip
5.0pt\textbf{with}\hskip 5.0ptw_{i}=\frac{f_{m}}{N_{class,i}}$ (26)

where $N_{class,i}$ is the number of entries in $\bar{y}$ of the same class as
the $i^{th}$ label entry (the class can be either $0$ or $1$ in our detection
case) and $f_{m}$ is the mean of $N_{class=0}$ and $N_{class=1}$.

When no human subjects are detected, the label map $\bar{y}$ will only contain
null values for long iteration periods, leading to the over-fitting of the
system on this local context. We alleviate this problem by keeping $\Psi$
fixed when no human subjects are present, and only enabling learning for
$\Phi$. In addition, we prune the task-driven contribution to the learning of
$\Phi$ (22) for values of $\mathcal{L}_{sup}$ smaller than a threshold
$\theta_{th}=0.2$ (empirically found to work well). Doing so, we avoid the
over-fitting of the system by sporadically learning hard examples only.

Regarding the choice of the on-line optimizer for the convolutional readout,
we choose the Adam optimizer [17] because of its adaptive gradient scaling
capability, which, in an on-line learning setting, leads to faster convergence
compared to SGD. In addition, the use of momentum in Adam implicitly
incorporates gradients from the data in hind-side, re-balancing the learning
process over time [17].

###  IV-F Post-processing

A set of discrete detections is obtained from the attention map as follows.
First, a threshold $d_{th}$ is applied to the map and a point cloud is formed.
Then, a set of detected clusters $\\{C_{k}\\}$ is obtained by applying DBSCAN
clustering [19] with min points $=2$ and $\epsilon=5$ (empirically tuned).

##  V Experimental results

In this section, we assess the bio-inspired CL architecture described in
Section IV on the KUL-UAVSAFE dataset [2] for people detection on drones. The
KUL-UAVSAFE dataset features a collection of joint DVS and RGB acquisitions in
an indoor, industrial-like environment, where a human subject is walking
randomly. During our experiments, we choose the three longest acquisitions
($\sim 2$ min.) from the f-wall collection in [2], where the drone and a human
subject are moving in a space surrounded by walls, benches and shelves
(different human subjects are featured to add variability).

We assess the performance of our system via 3-fold cross-validation as
follows. First, one of the acquisitions is used as a learning sequence and is
shown to our CL system only once, in its natural order (the learning sequence
is not shuffled). Then, learning is stopped and a different acquisition is
used to measure the precision-recall curves after post-processing (Section
IV-F), by sweeping $d_{th}$ from $0$ to $1$ and measuring the number of false
alarms, true positives and false negatives in $\\{C_{k}\\}$. This process is
repeated three times using different learning and testing sets and the final
precision-recall curves are obtained as the average over the three learn-test
folds.

Figure 6: Attention maps inferred from the DVS data by our bio-inspired CL
SNN-STDP system. An off-the-shelve YOLOv3 is used for online labelling.

Fig. 6 shows examples of attention maps produced by our system. Table I
reports the SNN-STDP learning parameters (found empirically, starting from the
parameter set used in [12]). Table II reports the parameters used in our
system architecture (tuned empirically). For the readout, we use the default
values of the Adam optimizer [17] with a learning rate $\eta_{r}=5\times
10^{-6}$. Next, we will study the effect of the most important parameters
impacting the CL procedure: the focal loss parameter $\gamma$ (17) and the
semi-supervision hyper-parameters $\lambda_{u,s}$ (16).

$\eta_{1}|\eta_{2}$ | $\mu_{1}|\tau_{m}$ | $\lambda_{2}$ | $A_{+}$ | $A_{-}$ | $\tau_{+}$ | $\tau_{-}$ | $\tau_{s}$
---|---|---|---|---|---|---|---
$0.07|0.05$ | $0.15|0.07$ | $0.002$ | 1 | 0.8 | 0.02 | 0.008 | 0.01
TABLE I: SNN-STDP learning parameters. The simulation time step is $0.005$s. All $\tau$ values are in seconds. $\lambda_{1}$ in (16) is implicitly set by the neuron threshold $\mu$ [12]. $\theta_{th}=0.2$. The negative STDP parameters are $A_{+,-}^{n}=A_{+,-}/100$.  $H$ | $W$ | $l_{x,y}$ | $d$ | $M$ | $r_{x,y}$ | $D_{x}$ | $D_{y}$
---|---|---|---|---|---|---|---
130 | 173 | 30 | 5 | 64 | 12 | 21 | 29
TABLE II: Architectural parameters. $H\times W$: input size. $l_{x}\times
l_{y}$: SNN kernel size, $d$: SNN stride, $M$: number of SNN kernels (neurons
in the coding layer), $r_{x}\times r_{y}$: kernel size of the readout.
$D_{x}\times D_{y}$: shape of the output attention map. The readout has a
stride of 1 and zero-padding.

###  V-A Impact of the focal loss parameter $\gamma$

We explore the impact of $\gamma$ in (17) by keeping all other parameters
constant with $\lambda_{u}=0$, $\lambda_{s}=1$. Fig. 7 shows the measured
precision-recall curves. Fig. 7 also reports the peak $F_{1}$ score as
$\max_{i}\frac{2P_{i}R_{i}}{P_{i}+R_{i}}$ along each precision-recall curve
$(P_{i},R_{i})$ [2]. Fig. 7 shows that the peak $F_{1}$ score is maximised
near $\gamma=0.5$. Therefore, $\gamma=0.5$ in our experiments below.

Figure 7: Impact of the focal loss parameter $\gamma$. The peak $F_{1}$ score
is maximized near $\gamma=0.5$.

###  V-B Impact of the unsupervised and supervised strengths
$\lambda_{u},\lambda_{s}$

Fig. 8 shows the precision-recall curves obtained by varying the unsupervised
(negative STDP) and supervised contributions $\lambda_{u,s}$ in (16). Fig. 8
shows that the peak $F_{1}$ score is maximised near $\lambda_{u}=0.2$,
$\lambda_{s}=0.8$.

Figure 8: Varying the semi-supervised hyper-parameters $\lambda_{u,s}$. The
peak $F_{1}$ score is maximized near $\lambda_{u}=0.2$ and $\lambda_{s}=0.8$.

During our experiments, we also tried the use of the standard positive STDP
rule for the unsupervised contribution, but always reached lower performances
by varying $\lambda_{u}$, due to an increase in false alarms. In contrast, the
use of the negative STDP rule for the unsupervised contribution provides an
opposite weight-steering force to the positive STDP rule used for the
supervised contribution, preventing the over-fitting of the system to
structures in the environment. This leads to less false alarms at higher
recalls and therefore, a higher detection performance (a non-trivial gain of
+$5\%$ on the peak $F_{1}$ score vs. $\lambda_{u}=0$, $\lambda_{s}=1$).

###  V-C Continual SNN-STDP vs. standard CNNs

Since CNNs constitute the standard approach used in the vision pipelines of
drones [4], we compare in Fig. 9 the performance of our CL architecture to two
offline-trained CNNs: a SqueezeNet-based CNN trained with RGB data and a
second SqueezeNet trained with DVS frames following [2] (DVS data averaged
over time into frames at 30 FPS).

Figure 9: Detection precision vs. recall for the presented CL SNN-STDP and the
conventional CNNs with offline training.

Using a SqueezeNet CNN architecture is an appropriate choice for comparison
since SqueezeNet aims to be MCU-friendly, using $\sim 500$KB of weight memory
(comparable to our network that uses $514$KB). In order to fairly compare the
bounding box output of the CNN to the attention map of our SNN-STDP system, we
consider the image region delimited by the bounding box as the active region
of an attention map (region inside the bounding box is filled with $1$ and
outside is filled with $0$). We can then use the same precision-recall
measurement pipeline as used for the SNN. Fig. 9 shows that the CNN-RGB setup
is the best performer (due to less false alarms compared to our SNN), but this
comes at the cost of on-line adaptability. Indeed, the CNN has been trained
offline for a specific environment and needs re-training each time new
situations are encountered. This is in contrast to our CL SNN-STDP system
which can learn and adapt on the fly, as demonstrated in this work.
Interestingly, the CNN-DVS setup is the lowest performer and is significantly
outperformed by the CL SNN-STDP system (+19% on the peak $F_{1}$ score). This
is mainly due to the fact that the SNN introduces time recurrence through its
lateral weights, enabling the network to learn temporal features that are
neglected by the feed-forward CNN-DVS (see [18] for a discussion on temporal
features in DVS data). This clearly shows the effectiveness of the methods
proposed in Section IV for the continual learning of DVS data.

##  VI Conclusion

This paper has presented, to the best of our knowledge, the first continual
learning system for drones that learns to detect people on the fly using a
bio-inspired SNN architecture with STDP learning. After the introduction of
our novel methods, numerous experiments have been described to characterize
the performance of our system and compare it against conventional CNNs. We
have shown that our event-based system reaches a higher peak $F_{1}$ score
(+19%) compared to a same-size CNN processing DVS frames, while enabling on-
line adaptation and learning on the fly. As future work, we plan to increase
the precision of the system at higher recalls by studying how to enhance the
SNN expressivity with more layers and reducing false alarms. We hope that this
work will inspire future research in brain-inspired vision for robotics.

## Acknowledgment

We thank Lars Keuninckx and Tim Verbelen for their precious help. The research
leading to these results has received funding from the Flemish Government (AI
Research Program) and the European Union’s ECSEL Joint Undertaking under grant
agreement n° 826655 - project TEMPO.

## References

  * [1] D. Falanga, K. Kleber, S. Mintchev, D. Floreano and D. Scaramuzza, ”The Foldable Drone: A Morphing Quadrotor That Can Squeeze and Fly,” in IEEE Robotics and Automation Letters, vol. 4, no. 2, pp. 209-216, April 2019, doi: 10.1109/LRA.2018.2885575.
  * [2] A. Safa, T. Verbelen, I. Ocket, A. Bourdoux, F. Catthoor and G. G. E. Gielen, ”Fail-Safe Human Detection for Drones Using a Multi-Modal Curriculum Learning Approach,” in IEEE Robotics and Automation Letters, vol. 7, no. 1, pp. 303-310, Jan. 2022, doi: 10.1109/LRA.2021.3125450.
  * [3] W. Fang, L. Wang and P. Ren, ”Tinier-YOLO: A Real-Time Object Detection Method for Constrained Environments,” in IEEE Access, vol. 8, pp. 1935-1944, 2020, doi: 10.1109/ACCESS.2019.2961959.
  * [4] D. Palossi, A. Loquercio, F. Conti, E. Flamand, D. Scaramuzza and L. Benini, ”A 64-mW DNN-Based Visual Navigation Engine for Autonomous Nano-Drones,” in IEEE Internet of Things Journal, vol. 6, no. 5, pp. 8357-8371, Oct. 2019, doi: 10.1109/JIOT.2019.2917066.
  * [5] J. Redmon, A. Farhadi. (2018). ”YOLOv3: An Incremental Improvement.”
  * [6] A. Vitale, A. Renner, C. Nauer, D. Scaramuzza and Y. Sandamirskaya, ”Event-driven Vision and Control for UAVs on a Neuromorphic Chip,” 2021 IEEE International Conference on Robotics and Automation (ICRA), 2021, pp. 103-109, doi: 10.1109/ICRA48506.2021.9560881.
  * [7] A. Mitrokhin, C. Fermüller, C. Parameshwara and Y. Aloimonos, ”Event-Based Moving Object Detection and Tracking,” 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2018, pp. 1-9, doi: 10.1109/IROS.2018.8593805.
  * [8] Bi, G.q., Poo, M.m. (1998). Synaptic Modifications in Cultured Hippocampal Neurons: Dependence on Spike Timing, Synaptic Strength, and Postsynaptic Cell Type. Journal of Neuroscience, 18(24), 10464–10472.
  * [9] C. Frenkel, M. Lefebvre, J. -D. Legat and D. Bol, ”A 0.086-mm2 12.7-pJ/SOP 64k-Synapse 256-Neuron Online-Learning Digital Spiking Neuromorphic Processor in 28-nm CMOS,” in IEEE Transactions on Biomedical Circuits and Systems, vol. 13, no. 1, pp. 145-158, Feb. 2019, doi: 10.1109/TBCAS.2018.2880425.
  * [10] G. Gallego et al., ”Event-based Vision: A Survey,” in IEEE Transactions on Pattern Analysis and Machine Intelligence, doi: 10.1109/TPAMI.2020.3008413.
  * [11] A. J. Ijspeert (2008). ”Central pattern generators for locomotion control in animals and robots: A review.” Neural Networks, 21(4), 642-653.
  * [12] Ali Safa et al., (2021). ”Learning Event-based Spatio-Temporal Feature Descriptors via Local Synaptic Plasticity: A Biologically-Plausible Perspective of Computer Vision.” (https://arxiv.org/abs/2111.00791), Submitted to IEEE Transaction on Neural Networks and Learning Systems.
  * [13] T. Hu, C. Pehlevan and D. B. Chklovskii, ”A Hebbian/Anti-Hebbian network for online sparse dictionary learning derived from symmetric matrix factorization,” 2014 48th Asilomar Conference on Signals, Systems and Computers, 2014, pp. 613-619, doi: 10.1109/ACSSC.2014.7094519.
  * [14] Ioffe, S. et al., (2015). ”Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift.” In Proceedings of the 32nd International Conference on Machine Learning (pp. 448–456).
  * [15] T. -Y. Lin, P. Goyal, R. Girshick, K. He and P. Dollár, ”Focal Loss for Dense Object Detection,” in IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 42, no. 2, pp. 318-327, 1 Feb. 2020, doi: 10.1109/TPAMI.2018.2858826.
  * [16] D. Eigen and R. Fergus, ”Predicting Depth, Surface Normals and Semantic Labels with a Common Multi-scale Convolutional Architecture,” 2015 IEEE International Conference on Computer Vision (ICCV), 2015, pp. 2650-2658, doi: 10.1109/ICCV.2015.304.
  * [17] Diederik P. Kingma, Jimmy Ba. (2017). ”Adam: A Method for Stochastic Optimization.”
  * [18] X. Lagorce, G. Orchard, F. Galluppi, B. E. Shi and R. B. Benosman, ”HOTS: A Hierarchy of Event-Based Time-Surfaces for Pattern Recognition,” in IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 39, no. 7, pp. 1346-1359, 1 July 2017, doi: 10.1109/TPAMI.2016.2574707.
  * [19] Ester, M., Kriegel, H.P., Sander, J., Xu, X. (1996). ”A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise.” In Proceedings of the Second International Conference on Knowledge Discovery and Data Mining (pp. 226–231). AAAI Press.


# Thermodynamics of AdS planar black holes and holography

Neven Bilić<EMAIL_ADDRESS>Division of Theoretical Physics, Rudjer Bošković
Institute, 10002 Zagreb, Croatia  Júlio C. Fabris<EMAIL_ADDRESS>Departamento de Física, Universidade Federal do Espírito Santo (UFES) Av.
Fernando Ferrari s/n CEP 29.075-910, Vitória, ES, Brazil

###### Abstract

A large AdS Schwarzschild black hole can be approximated by an AdS planar
black hole. However, the temperature of a planar black hole is not well
defined due to the translational invariance of the horizon. We propose to fix
this arbitrariness by imposing the entropy area law. Furthermore, using the
AdS/CFT holography, we propose a relationship between the temperatures of an
AdS planar and large AdS Schwarzschild black holes.

##  1 Introduction

Anti-de Sitter (AdS) Schwarzschild black holes have a curious thermodynamic
property. The horizon temperature of a large AdS Schwarzschild black hole (BH)
increases with increasing horizon radius in contrast to an ordinary
Schwarzschild BH the temperature of which decreases with horizon radius.
Hubeny, Marolf, and Rangamani [1] argue that the thermodynamic behavior of AdS
Schwarzschild BHs is tightly related to the thermodynamic properties of planar
BHs. This argument is based on the fact that a large AdS Schwarzschild BH can
be well approximated by an AdS planar BH [2].

To the best of our knowledge, the AdS planar BHs were first stadied by Lemos
[3] and more recently were shown to appear by scaling a near-extremal brane
solution [4]. Interesting properties of AdS planar BHs were found [5] in the
context of thermodynamics. Besides, planar BHs have recently attracted
considerable attention in non-gravitational contexts. Geometric structures in
the form of a planar BH may have interesting applications in condensed matter
physics [6], 2+1-dimensional superconductor [7, 8, 9, 10], and acoustic
geometry [10, 11, 12].

As we will shortly demonstrate, the horizon of an AdS planar BH is translated
when a planar BH metric undergoes a rescaling transformation, so the horizon
location of an AdS planar BH is not well defined. Following Hubeny, Marolf,
and Rangmani [1] we will refer to this property as ”translational invariance”
of the BH horizon, even though this term could be somewhat misleading in this
context. Based on this property, the authors of Ref. [1] argue that large
Schwarzschild-AdS black holes, although very hot, locally appear at most only
lukewarm.

In this paper, we study the relationship between a large AdS Schwarzschild BH
in $d+1$ dimensions and AdS planar BH in the context of anti-de
Sitter/conformal field theory (AdS/CFT) holography. First, we will show that
the horizon temperature of a planar BH can be fixed by the requirement of the
entropy area law. Then, by invoking the AdS/CFT correspondence conjecture, we
will relate the local temperatures of the planar BH and large AdS
Schwarzschild BH near the AdS boundary.

We divide the remainder of the paper into three sections. In section 2 we
briefly review the thermal properties of AdS Schwarzschild and planar BHs. In
section 3 we derive relationships between the temperatures of the boundary
conformal fluid and of a large AdS BHs by employing AdS/CFT holography.
Concluding remarks are given in section 4.

##  2 AdS Schwarzschild and planar black hole

The metric of an AdS Schwarzschild black hole in $d+1$ dimensions is usually
written in the form [2, 13]

$\displaystyle ds^{2}=f(r)dt^{2}-\frac{1}{f(r)}dr^{2}-r^{2}d\Omega_{d-1}.$ (1)

The function $f$ is defined as

$f(r)=\frac{r^{2}}{\ell^{2}}+1-\mu\left(\frac{\ell}{r}\right)^{d-2},$ (2)

where $\ell$ is the curvature radius of AdSd+1 and the dimensionless parameter
$\mu$ is related to the black-hole mass via [2, 14]

$\mu=\frac{16\pi G_{d+1}M_{\rm bh}}{(d-1)\ell^{d-2}\Omega_{d-1}}.$ (3)

Here $G_{d+1}$ is the $d+1$-dimensional Newton’s constant and

$\Omega_{d-1}=\frac{2\pi^{d/2}}{\Gamma(d/2)}=\left\\{\begin{array}[]{ll}(2\pi)^{d/2}/(d-2)!!,&\mbox{
for even $d$},\\\ 2(2\pi)^{(d-1)/2}/(d-2)!!,&\mbox{ for odd
$d$},\end{array}\right.$ (4)

is the volume of a unit $d-1$-sphere. The parameter $\mu$ can be expressed in
terms of the largest root $r_{+}$ of the equation $f(r)=0$,

$\mu=\left(\frac{r_{+}}{\ell}\right)^{d-2}\left(1+\frac{r_{+}^{2}}{\ell^{2}}\right).$
(5)

Note that the metric (1) with (2) with horizon radius involves two scales: the
AdS curvature $\ell$ and $d+1$-dimensional Newton’s constant $G_{d+1}$

The horizon temperature af an AdS Schwarzschild BH can be derived in the usual
way by approximating the Euclidean metric in the near horizon limit with

$f=(r-r_{+})f_{+},$ (6)

where

$f_{+}=\left.\frac{\partial f}{\partial r}\right|_{r=r_{+}}.$ (7)

Defining the periodicity of the Euclidean time so to avoid the cone
singularity, one finds the inverse horizon temperature

$\beta=\frac{1}{T_{+}}=\frac{4\pi}{f_{+}}=\frac{4\pi\ell^{2}r_{+}}{dr_{+}^{2}+(d-2)\ell^{2}}.$
(8)

Note that, the horizon temperature $T_{+}\rightarrow\infty$ when either
$r_{+}\rightarrow 0$ (small BH) or $r_{+}\rightarrow\infty$ (large BH) unlike
in the Schwarzschild BH in an asymptotically flat background where
$T_{+}\rightarrow 0$ for a large BH. In particular, for a large AdS
Schwarzschild BH we have

$T_{+}=\frac{dr_{+}}{4\pi\ell^{2}}.$ (9)

###  2.1 AdS Planar black hole

As pointed by Witten [2], in the limit of large AdS Schwarzschild BH the
topology goes from ${\bf S}^{1}\times{\bf S}^{d-1}$ to ${\bf S}^{1}\times{\bf
R}^{d-1}$ where a Schwarzschild BH is approximated by a planar BH with a
translationally invariant horizon. To see this, consider a large
Schwarzschild-AdS black hole with $\mu\gg 1$ ($r_{+}\gg\ell$). By rescaling

$r=\mu^{1/d}\frac{\ell^{2}}{\rho},$ (10)

the function $f$ becomes

$f=\mu^{2/d}\left(\frac{\ell^{2}}{\rho^{2}}-\frac{\rho^{d-2}}{\ell^{d-2}}+\mu^{-2/d}\right).$
(11)

Then, by neglecting the last term in brackets and rescaling the time
$t=\mu^{-1/d}\tau$, the line element squared becomes

$ds^{2}=\frac{\ell^{2}}{\rho^{2}}\left[\left(1-\frac{\rho^{d}}{\ell^{d}}\right)d\tau^{2}-\left(1-\frac{\rho^{d}}{\ell^{d}}\right)^{-1}d\rho^{2}-\mu^{2/d}\ell^{2}d\Omega_{d-1}^{2}\right].$
(12)

For $M_{\rm bh}\rightarrow\infty$ (i.e., for $\mu\rightarrow\infty$) the
radius $\mu^{1/d}\ell$ of ${\bf S}^{d-1}$ diverges and hence, ${\bf S}^{d-1}$
becomes flat and looks locally as ${\bf R}^{d-1}$. Now we introduce
coordinates $x_{i}$ near a point $P\in{\bf S}^{d-1}$ such that at $P$,
$d\Omega_{d-1}^{2}=\ell^{-2}\mu^{-2/d}d\mbox{\boldmath$x$}^{2}$. Then the
metric becomes

$ds^{2}=\frac{\ell^{2}}{\rho^{2}}\left[\left(1-\frac{\rho^{d}}{\ell^{d}}\right)d\tau^{2}-\left(1-\frac{\rho^{d}}{\ell^{d}}\right)^{-1}d\rho^{2}-d\mbox{\boldmath$x$}^{2}\right],$
(13)

where

$d\mbox{\boldmath$x$}^{2}=\sum_{i=1}^{d-1}{\rm d}x^{i}{\rm d}x^{i}.$ (14)

The metric (13) describes an asymptotically AdSd+1 geometry with a planar BH
horizon located at $\rho=\ell$. Note that dependence on Newton’s constant
$G_{d+1}$ has disappeared and the metric (13) involves only one scale: the AdS
curvature $\ell$.

Now, it is easy to show by simultaneously rescaling the coordinates that the
metric (13) is equivalent to the whole class of planar BHs represented by the
line element

$ds^{2}=\frac{\ell^{2}}{\rho^{2}}\left[\left(1-\frac{\rho^{d}}{\rho_{\rm
pl}^{d}}\right)d\tau^{2}-\left(1-\frac{\rho^{d}}{\rho_{\rm
pl}^{d}}\right)^{-1}d\rho^{2}-d\mbox{\boldmath$x$}^{2}\right],$ (15)

with horizon located at an arbitrary $\rho_{\rm pl}$.

As before, the horizon temperature of the Planar BH can be calculated by
demanding that the periodicity $\beta_{\rm pl}$ is such that the conical
singularity is removed. One finds

$\beta_{\rm pl}=\frac{1}{T_{\rm pl}}=\frac{4\pi\rho_{\rm pl}}{d},$ (16)

so the temperature of the planar BH increases when the horizon shifts towards
the AdS boundary at $\rho=0$. However, since $\rho_{\rm pl}$ is not fixed, the
temperature $T_{\rm pl}$ is not well defined. Hence, local observations are
independent of $\rho_{\rm pl}$ and, since the AdS scale is the only scale in
the problem, local observers see no excitations above this scale.

The fact that the temperature $T_{\rm pl}$ of the AdS planar BH is not well
defined, is reminiscent of the case for the extreme Reissner-Nordström BH with
AdS near horizon geometry. As argued, e.g., in Refs. [15, 16], the temperature
of the extreme Reissner-Nordström BH is arbitrary since its horizon is
infinitely far away.

Since small $\rho$ corresponds to large $r$, one would naively conclude that
the thermodynamic behavior of the AdS planar BH is consistent with that of the
large AdS Schwarzschild BH. However, this conclusion could be wrong as it is
not obvious that the planar BH temperature is in any way related to the
temperature of the AdS Schwarzschild BH.

###  2.2 Entropy of AdS planar black holes

We closely follow Witten’s procedure for the calculation of the BH entropy.
This procedure applied to the metric (1) with (2) yields the AdS Schwarzschild
BH entropy [2]

$S_{\rm BH}=\frac{r_{+}^{d-1}\Omega_{d-1}}{4G_{d+1}},$ (17)

which demonstrates the area law in agreement with Bekenstein [17] and Hawking
[18]. To calculate the horizon entropy of the AdS planar BH we start from the
metric (15) in which we transform the coordinate $\rho$ to $r=\rho_{\rm
pl}^{2}/\rho$ so that $r$ goes from $r=\rho_{\rm pl}$ to $r=\infty$ as $\rho$
goes from $\rho=\rho_{\rm pl}$ to $\rho=0$. The Euclidean metric is given by

$ds_{\rm E}^{2}=\frac{r^{2}}{\ell^{2}}\left[\left(1-\frac{\rho_{\rm
pl}^{d}}{r^{d}}\right)d\tau^{2}+\left(1-\frac{\rho_{\rm
pl}^{d}}{r^{d}}\right)^{-1}\frac{\ell^{4}}{r^{4}}dr^{2}+d\mbox{\boldmath$x$}^{2}\right],$
(18)

with the associated horizon temperature $T_{\rm pl}$ given by (16).

Following Hawking and Page [19] and Witten [2] we describe the BH free energy
as a renormalized on shell Euclidean bulk action. The on shell AdS BH action
is proportional to an infinite volume of the AdS BH spacetime so it must be be
regularized and renormalized. The regularization is achieved by integrating
the space volume up to a large radius $R$. The renormalized action is then
obtained by subtracting the pure AdS space time volume from the AdS BH
spacetime volume, assuming equality of the local AdS temperature and the
redshifted planar BH temperature at the hypersurface $r=R$. More explicitly

$F_{\rm pl}=-T_{\rm pl}\ln Z=\frac{d\ell^{-2}}{8\pi G_{d+1}\beta_{\rm
pl}}\lim_{R\rightarrow\infty}\left(V_{\rm pl}-V_{\rm AdS}\right),$ (19)

where

$V_{\rm pl}=\int_{0}^{\beta_{\rm
pl}}d\tau\int_{0}^{R}dr\sqrt{g}\left(\int_{-L/2}^{L/2}dx\right)^{d-1},$ (20)
$V_{\rm AdS}=\int_{0}^{\beta_{\rm
AdS}}d\tau\int_{0}^{R}dr\sqrt{g}\left(\int_{-L/2}^{L/2}dx\right)^{d-1}.$ (21)

The length $L$ is the linear size of the hyperplane $r={\rm const.}$. The
inverse AdS temperature $\beta_{\rm AdS}$ is obtained from the equation

$\beta_{\rm AdS}\sqrt{g_{00}^{\rm AdS}}=\beta_{\rm pl}\sqrt{g_{00}^{\rm pl}},$
(22)

where $g_{00}^{\rm AdS}$ and $g_{00}^{\rm pl}$ are the time-time components of
the AdS and-black hole metric tensors, respectively, evaluated at $r=R$. We
find

$\beta_{\rm AdS}=\beta_{\rm pl}\left(1-\frac{\rho_{\rm
pl}^{d}}{R^{d}}\right)^{1/2}.$ (23)

Then, from(19) we obtain the free energy expressed as a function of $\rho_{\rm
pl}$

$F_{\rm pl}=\frac{L^{d-1}}{16\pi G_{d+1}\ell}\left(\frac{\rho_{\rm
pl}}{\ell}\right)^{d},$ (24)

where $L^{d-1}$ is the area of the hyperplane $r={\rm const.}$. From this,
using (16) and the thermodynamic relation between entropy and free energy, we
find

$S_{\rm pl}=-\frac{\partial F_{\rm pl}}{\partial T_{\rm
pl}}=\frac{L^{d-1}}{4G_{d+1}}\left(\frac{\rho_{\rm pl}}{\ell}\right)^{d+1}.$
(25)

Hence, we recover the standard area law for the planar BH entropy if
$\rho_{\rm pl}=\ell$. So, if we demand the validity of the area law, the
metric (12) will be a natural representative of all AdS planar BHs belonging
to the class described by the line element (13) with $0<\rho_{\rm pl}<\infty$.

We shall next invoke the holographic principle to relate the temperature
$T_{\rm pl}$ of the planar BH to the temperature of the conformal fluid at the
boundary of AdS spacetime.

##  3 AdS/CFT holography

AdS/CFT holographic principle relates the bulk geometry to the conformal field
theory at the AdS boundary. To study the CFT at the boundary, it is convenient
to transform both the AdS Schwarzschild metric (1) and the AdS planar BH
metric (15) to the Fefferman-Graham coordinates. In Fefferman-Graham
coordinates the general asymptotically AdS metric can be expressed as

$ds^{2}=\frac{\ell^{2}}{z^{2}}\left(g_{\mu\nu}(z)dx^{\mu}dx^{\nu}-dz^{2}\right),$
(26)

where the $d$-dimensional metric $g_{\mu\nu}$ near the boundary at $z=0$ can
be expanded as [20]

$g_{\mu\nu}=g^{(0)}_{\mu\nu}+z^{2}g^{(2)}_{\mu\nu}+\ldots\,.+z^{d}g^{(d)}_{\mu\nu}+h_{\mu\nu}z^{d}\ln
z^{2}+\mathcal{O}(z^{d+1}).$ (27)

The logarithmic term appears only for $d$ even. Using this expansion one can
express the vacuum expectation value of the boundary CFT stress tensor as [20]

$\langle T_{\mu\nu}\rangle=-\frac{d\ell^{d-1}}{16\pi
G_{d+1}}g^{(d)}_{\mu\nu}+X_{\mu\nu}[g^{(n)}],$ (28)

where $X_{\mu\nu}[g^{(n)}]$ is a function of $g^{(n)}_{\mu\nu}$ with $n<d$.
Its exact form depends on the spacetime dimension and it reflects the
conformal anomalies of the boundary CFT [21] (for $d=4$ and applications in
cosmology see also Refs. [22, 23]). For $d$ odd, there are no gravitational
conformal anomalies and $X_{\mu\nu}$ is equal to zero. For $d$ even, we may
expect that the conformal anonmaly related to the $X$ term will correspond to
the logarithmic correction to the BH entropy [24, 25, 26]. In the following we
will be interested in the dominant area law term in the BH entropy for which
we expect to correspond to the traceless part of the universal term in (28)
proportional to $g^{(d)}_{\mu\nu}$.

###  3.1 AdS Schwarzschild BH

Consider first the AdS Schwarzschild BH. By transforming the radial coordinate
$r=\ell^{2}/\rho$, we rewrite the metric (1) in the form similar to (12),

$ds^{2}=\frac{\ell^{2}}{\rho^{2}}\left[\left(1+\frac{\rho^{2}}{\ell^{2}}-\mu\frac{\rho^{d}}{\ell^{d}}\right)d\tau^{2}-\left(1+\frac{\rho^{2}}{\ell^{2}}-\mu\frac{\rho^{d}}{\ell^{d}}\right)^{-1}d\rho^{2}-\ell^{2}d\Omega_{d-1}^{2}\right].$
(29)

Then, the transformation

$\frac{dz}{z}=\frac{d\rho}{\rho}\left(1+\frac{\rho^{2}}{\ell^{2}}-\mu\frac{\rho^{d}}{\ell^{d}}\right)^{-1/2}$
(30)

will take us to the Fefferman-Graham form. The integration of (30) cannot be
performed in terms of elementary functions, with the exception of $d=2$ and
$d=4$ [22]. However, as we are interested in the expansion (27) near $z=0$ up
to the order $z^{d}$, we can apply an ansatz to the metric in Fefferman-Graham
coordinates:

$ds^{2}=\frac{\ell^{2}}{z^{2}}\left[H(z)d\tau^{2}-F(z)\ell^{2}d\Omega_{d-1}^{2}-dz^{2}\right].$
(31)

Here,

$F(z)=1+\sum_{n=2}^{\infty}f_{n}\frac{z^{n}}{\ell^{n}},\quad
H(z)=1+\sum_{n=2}^{\infty}h_{n}\frac{z^{n}}{\ell^{n}},$ (32)

where the coefficients $f_{n}$ and $h_{n}$ are to be determined by equating
the line element (31) with (29) and making use of (30). We find

$F(z)=1-\frac{1}{2}\frac{z^{2}}{\ell^{2}}+\frac{1}{16}\frac{z^{4}}{\ell^{4}}+\frac{1}{d}\mu\frac{z^{d}}{\ell^{d}}+\dots,$
(33)
$H(z)=1+\frac{1}{2}\frac{z^{2}}{\ell^{2}}+\frac{1}{16}\frac{z^{4}}{\ell^{4}}-\frac{d-1}{d}\mu\frac{z^{d}}{\ell^{d}}+\dots,$
(34)

so the background is the static Einstein universe ${\bf R}\times{\bf S}^{d-1}$
with line element

$g^{(0)}_{\mu\nu}dx^{\mu}dx^{\nu}=d\tau^{2}-\ell^{2}d\Omega_{d-1}^{2}$ (35)

For $d\neq 4$ we find

$g^{(d)}_{\mu\nu}dx^{\mu}dx^{\nu}=-\frac{\mu}{d\ell^{d}}[(d-1)d\tau^{2}+\ell^{2}d\Omega_{d-1}^{2}],$
(36)

with trace ${\rm Tr}g^{(d)}=0$. For $d=4$ we have

$g^{(4)}_{\mu\nu}dx^{\mu}dx^{\nu}=-\frac{1}{4\ell^{4}}\left[\left(3\mu-\frac{1}{4}\right)d\tau^{2}+\left(\mu+\frac{1}{4}\right)\ell^{2}d\Omega_{3}^{2}\right],$
(37)

with ${\rm Tr}g^{(4)}=1/(4\ell^{4})$.

To find the traceless part of the holographic stress tensor (28) we just
subtract the trace, so for any $d\geq 2$ we have

$\langle T^{(d)}_{\mu\nu}\rangle=-\frac{d\ell^{d-1}}{16\pi
G_{d+1}}\left(g^{(d)}_{\mu\nu}-\frac{1}{d}{\rm
Tr}g^{(d)}g^{(0)}_{\mu\nu}\right).$ (38)

By writing the BH mass parameter $\mu$ in terms of $r_{+}$ by Eq. (5), we
obtain

$\langle{{T^{(d)}}^{\mu}}_{\nu}\rangle=\frac{\ell^{d-1}}{16\pi
G_{d+1}}\left(\frac{r_{+}}{\ell^{2}}\right)^{d}\left(1+\frac{\ell^{2}}{r_{+}^{2}}\right)\;{\rm
diag}\left(d-1,-1,-1,...\right).$ (39)

This stress tensor is of the form of the stress tensor of a $d$-dimensional
conformal fluid (CF) with the equation of state

$p_{\rm CF}=\varepsilon_{\rm CF}/(d-1),$ (40)

where $p_{\rm CF}$ and $\varepsilon_{\rm CF}$ are the fluid pressure and
energy density, respectively, and

$\varepsilon_{\rm CF}=\frac{(d-1)\ell^{d-1}}{16\pi
G_{d+1}}\left(\frac{r_{+}}{\ell^{2}}\right)^{d}\left(1+\frac{\ell^{2}}{r_{+}^{2}}\right).$
(41)

We now assume that the conformal fluid at the boundary is equivalent to the
black-body radiation at a temperature $T$. For a theory with $n_{\rm B}$
massless bosons and $n_{\rm F}$ massless fermions in $d-1$ spatial dimensions,
the energy density of a gas at a temperature $T$ is given by

$\varepsilon=\int\frac{d^{d-1}k}{(2\pi)^{d-1}}\left(\frac{n_{\rm
B}k}{e^{k/T}-1}+\frac{gn_{\rm F}k}{e^{k/T}+1}\right)=a_{d}T^{d},$ (42)

the pressure by $p=\varepsilon/(d-1)$, and the entropy density by

$s=T^{-1}(p+\varepsilon)=\frac{da_{d}}{d-1}T^{d-1},$ (43)

where

$a_{d}=\frac{\Omega_{d-2}\Gamma(d)\zeta(d)}{(2\pi)^{d-1}}\left(n_{\rm
B}+(1-2^{1-d})gn_{\rm F}\right).$ (44)

The factor $g$ is included to account for degeneracy due to internal degrees
of freedom of the fermions. For example, for Weyl fermions in $4$-dimensional
spacetime, $g=2$.

For a large BH ($r_{+}\gg\ell$), the energy density $\varepsilon_{\rm CF}$
takes the form (42) with the black-body temperature $T\propto r_{+}$. Then,
according to (9), the temperature $T$ of the conformal fluid at the AdS
boundary is proportional to the horizon temperature $T_{+}$ of the large AdS
Schwarzschild BH. Hence, the entropy density (43) of the conformal fluid at
the boundary scales as $T_{+}^{d-1}$ as expected [2].

###  3.2 AdS Planar BH

Next, we apply the above procedure to a planar BH. Starting from (15) we
transform the coordinate $\rho$ into $z$ by

$\frac{dz}{z}=\frac{d\rho}{\rho}\left(1-\frac{\rho^{d}}{\rho_{\rm
pl}^{d}}\right)^{-1/2}.$ (45)

integrating this we find

$\left(\frac{z}{z_{\rm pl}}\right)^{d}=\frac{1-\sqrt{1-(\rho/\rho_{\rm
pl})^{d}}}{1+\sqrt{1-(\rho/\rho_{\rm pl})^{d}}},$ (46)

where $z_{\rm pl}$ is an arbitrary positive integration constant which can be
interpreted as the location of the planar BH horizon in Fefferman-Graham
coordinates. In the limit $\rho\rightarrow 0$ we have

$(z/z_{\rm pl})^{d}=\frac{1}{4}(\rho/\rho_{\rm
pl})^{d}+\mathcal{O}((\rho/\rho_{\rm pl})^{2d}),$ (47)

so it is natural to identify $z_{\rm pl}^{d}\equiv 4\rho_{\rm pl}^{d}$. The
inverse to (46) is easily found:

$\rho=z\left(1+\frac{1}{4}\left(\frac{z}{\rho_{\rm
pl}}\right)^{d}\right)^{-2/d}.$ (48)

Using this we can recast the metric (15) in the Fefferman-Graham form

$ds^{2}=\frac{\ell^{2}}{z^{2}}\left[\left(1-\frac{1}{4}\frac{z^{d}}{\rho_{\rm
pl}^{d}}\right)^{2}\left(1+\frac{1}{4}\frac{z^{d}}{\rho_{\rm
pl}^{d}}\right)^{4/d-2}d\tau^{2}-\left(1+\frac{1}{4}\frac{z^{d}}{\rho_{\rm
pl}^{d}}\right)^{4/d}d\mbox{\boldmath$x$}^{2}-dz^{2}\right],$ (49)

We now compare this with the general expression for the asymptotically AdS
metric (26). Expanding the functions in (49) and comparing with (26) and (27)
we find

$g^{(0)}_{\mu\nu}=\eta_{\mu\nu},\quad\quad
g^{(d)}_{\mu\nu}=-\frac{1}{d\rho_{\rm pl}^{d}}\;{\rm
diag}\left(d-1,1,1,...\right),$ (50)

so the traceless part of the stress tensor is given by

$\langle{{T^{(d)}}^{\mu}}_{\nu}\rangle=\frac{\ell^{d-1}\rho_{\rm
pl}^{-d}}{16\pi G_{d+1}}\;{\rm diag}\left(d-1,-1,-1,...\right).$ (51)

As in the case of the AdS Schwarzschild BH, this form is identical to the
stress tensor of the $d$-dimensional conformal fluid with the equation of
state $p_{\rm CF}=\varepsilon_{\rm CF}/(d-1)$, where the energy density is
given by

$\varepsilon_{\rm
CF}=\langle{{T^{(d)}}^{0}}_{0}\rangle=\frac{(d-1)\ell^{d-1}\rho_{\rm
pl}^{-d}}{16\pi G_{d+1}}.$ (52)

We can compare this with the energy density (42) of black-body radiation and
relate $\rho_{\rm pl}$ to the black-body temperature $T\propto\rho_{\rm
pl}^{-1}$. As before, we find that the entropy of the conformal fluid at the
boundary scales as $T_{\rm pl}^{d-1}$.

###  3.3 Temperature relationships

In this section, based on the correspondence between the CFT entropy at the
boundary and the BH entropies, we establish relationships between the
temperatures $T$, $T_{+}$, and $T_{\rm pl}$, of the boundary conformal fluid,
the large Schwarzscild BH, and planar BH, respectively.

First, we equate the Schwarzschild BH entropy $S_{\rm BH}$ given by (17) with
the total entropy of the black-body radiation on the boundary

$S=\lim_{R\rightarrow\infty}\frac{da_{d}}{d-1}T^{d-1}R^{d-1}\Omega_{d-1},$
(53)

where $R^{d-1}\Omega_{d-1}$ is the area of the large hypersphere of radius
$R$, which approaches the boundary. From the equation $S=S_{\rm BH}$ we find

$T^{d-1}=\frac{d-1}{4da_{d}G_{d+1}}\frac{r_{+}^{d-1}}{R^{d-1}}.$ (54)

Then, for a large BH ($r_{+}\gg\ell$), using (9) we find a relationship
between the temperatures $T$ and $T_{+}$

$T=\frac{4\pi\ell}{d}\left(\frac{d-1}{4da_{d}G_{d+1}}\right)^{1/(d-1)}\frac{T_{+}}{R/\ell}.$
(55)

Note that, according to Tolman’s relation applied to the asymptotic AdS
spacetime [29], the local temperature near the boundary is given by

$T_{+,\rm loc}\equiv\frac{T_{+}}{\sqrt{g_{00}}}=\frac{T_{+}}{R/\ell},$ (56)

and hence, the expression (55) relates the temperature of the holographic
conformal fluid to the local Schwarzschild BH temperature near the AdS
boundary at $r=R$.

Consider next the planar BH geometry. In this case, the total entropy of the
black-body radiation on the boundary is given by

$S=\lim_{\epsilon\rightarrow
0}\frac{da_{d}}{d-1}T^{d-1}\frac{\ell^{d-1}}{\epsilon^{d-1}}L^{d-1},$ (57)

where $(\ell^{d-1}/\epsilon^{d-1})L^{d-1}$ is the area of the hypersurface at
$\rho=\epsilon$, which approaches the boundary as $\epsilon\rightarrow 0$.
Then, from the equation $S=S_{\rm pl}$ we find

$T^{d-1}=\frac{d-1}{4da_{d}G_{d+1}}\frac{\epsilon^{d-1}}{\ell^{d-1}}\frac{\rho_{\rm
pl}^{d+1}}{\ell^{d+1}}.$ (58)

As mentioned in Sec. 2.2, to recover the area law for the planar BH we set
$\rho_{\rm pl}=\ell$. In this case, according to (16), the temperature of the
planar BH is $T_{\rm pl}=d/(4\pi\ell)$. Using this and Eq. (58), the
temperature $T$ may be expressed as

$T=\frac{4\pi\ell}{d}\left(\frac{d-1}{4da_{d}G_{d+1}}\right)^{1/(d-1)}\frac{T_{\rm
pl}}{\ell/\epsilon}.$ (59)

Here, similar to case of Schwarzschild BH geometry, the quantity $T_{\rm
pl}/(\ell/\epsilon)$ can be regarded as the local temperature near the AdS
boundary at $\rho=\epsilon$,

$T_{\rm pl,loc}\equiv\frac{T_{\rm pl}}{\sqrt{g_{00}}}=\frac{T_{\rm
pl}}{\ell/\epsilon}.$ (60)

Hence, the expression (59) relates the temperature of the holographic
conformal fluid to the planar BH local temperature near the AdS boundary.

Now, we identify the temperature $T$ of the boundary conformal fluid defined
by (59) with that defined by (55). As a consequence, we obtain a relation
between the redshifted local temperatures of the planar BH and the large
Schwarzschild BH measured near the AdS boundary

$\frac{T_{+}}{R/\ell}=\frac{T_{\rm pl}}{\ell/\epsilon}.$ (61)

Hence, the large AdS Schwarzschild BH appears as hot as the corresponding
planar BH. Using (61) and (16) with $\rho_{\rm pl}=\ell$ we can express the
horizon temperature of the large AdS Schwarzschild BH in terms of the a planar
BH temperature

$T_{+}\sim T_{\rm pl}=\frac{d}{4\pi\ell}.$ (62)

Equations (61) and (62), as our main result, are obtained by invoking the
AdS/CFT conjecture and assuming that the temperature of an AdS planar BH can
be fixed by imposing the entropy area law. In this way, we have found that the
temperature of a large Schwarzschild BH and that of the corresponding planar
BH as seen by an observer near the AdS boundary is of the same order of
magnitude.

##  4 Conclusions

By making use of the AdS/CFT correspondence we have calculated the traceless
part of the vacuum expectation value of the stress tensor of the boundary CFT
corresponding to the bulk geometry of AdS Schwarzschild and planar BHs. Based
on the correspondence between the CFT entropy at the boundary and the BH
entropies, we have established relationships between the temperatures $T$,
$T_{+}$, and $T_{\rm pl}$ of the conformal fluid at the boundary, large
Schwarzschild BH, and planar BH, respectively.

In this way, we predict that the temperatures of large AdS Schwarzschild and
planar BHs are of the same order of magnitude as seen by an observer near the
AdS boundary. Because of the diverging redshift near the boundary, local
observers measure a large redshifted temperature of the AdS Schwarzschild BH
and the actual horizon temperature $T_{+}$ is unobservable. This contrasts
with the case of ordinary Schwarzschild BH where asymptotic observers see the
actual horizon temperature.

Our result confirms the argument of Ref. [1] based on the translational
invariance of the planar BH horizon and previous assertion [27, 28] that the
high temperatures associated with large values of $T_{+}$ are not locally
observable.

## Acknowledgments

N. Bilić has been partially supported by the ICTP - SEENET-MTP project NT-03
Cosmology - Classical and Quantum Challenges. J. C. Fabris thanks CNPq
(Brazil) and FAPES (Brazil) for partial support.

## References

  * [1] V. E. Hubeny, D. Marolf and M. Rangamani, Class. Quant. Grav. 27, 095018 (2010) [arXiv:0911.4144 [hep-th]].
  * [2] E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998) [hep-th/9803131].
  * [3] J. P. S. Lemos, Class. Quant. Grav. 12, 1081-1086 (1995) [arXiv:gr-qc/9407024 [gr-qc]].
  * [4] G. T. Horowitz and S. F. Ross, JHEP 9804, 015 (1998) [hep-th/9803085].
  * [5] C. S. Peca and J. P. S. Lemos, J. Math. Phys. 41, 4783-4789 (2000) [arXiv:gr-qc/9809029 [gr-qc]].
  * [6] S. A. Hartnoll, Class. Quant. Grav. 26, 224002 (2009) [arXiv:0903.3246 [hep-th]].
  * [7] N. Bobev, A. Kundu, K. Pilch and N. P. Warner, JHEP 1203, 064 (2012) [arXiv:1110.3454 [hep-th]].
  * [8] T. Albash and C. V. Johnson, JHEP 1205, 079 (2012) [arXiv:1202.2605 [hep-th]].
  * [9] A. Chakraborty, Class. Quant. Grav. 37, no.6, 065021 (2020) [arXiv:1903.00613 [hep-th]].
  * [10] N. Bilić and J. C. Fabris, Class. Quant. Grav. 38, no.16, 165007 (2021) [arXiv:2101.12494 [hep-th]].
  * [11] S. Hossenfelder, Phys. Rev. D 91, no. 12, 124064 (2015) [arXiv:1412.4220 [gr-qc]].
  * [12] N. Bilic and H. Nikolic, Class. Quant. Grav. 35, no. 13, 135008 (2018) [arXiv:1802.03267 [gr-qc]].
  * [13] G. T. Horowitz and R. C. Myers, Phys. Rev. D 59, 026005 (1998) [hep-th/9808079].
  * [14] R. C. Myers, Phys. Rev. D 60, 046002 (1999) [hep-th/9903203].
  * [15] S. W. Hawking and G. T. Horowitz, Class. Quant. Grav. 13, 1487-1498 (1996) [arXiv:gr-qc/9501014].
  * [16] F. G. Alvarenga, A. B. Batista, J. C. Fabris and G. T. Marques, Phys. Lett. A 320, 83-94 (2003) [arXiv:gr-qc/0306030]; arXiv:gr-qc/0309022.
  * [17] J. Bekenstein, Phys. Rev. D 7, 2333 (1973).
  * [18] S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).
  * [19] S. W. Hawking and D. N. Page, Commun. Math. Phys. 87, 577 (1983).
  * [20] S. de Haro, S. N. Solodukhin and K. Skenderis, Commun. Math. Phys. 217, 595 (2001) [hep-th/0002230].
  * [21] M. Henningson and K. Skenderis, JHEP 07, 023 (1998) [arXiv:hep-th/9806087 [hep-th]].
  * [22] P. S. Apostolopoulos, G. Siopsis and N. Tetradis, Phys. Rev. Lett. 102, 151301 (2009) [arXiv:0809.3505 [hep-th]],
  * [23] N. Bilić, Phys. Rev. D 93, no.6, 066010 (2016) [arXiv:1511.07323 [gr-qc]].
  * [24] D. V. Fursaev, Phys. Rev. D 51, 5352-5355 (1995) [arXiv:hep-th/9412161 [hep-th]].
  * [25] R. G. Cai, L. M. Cao and N. Ohta, JHEP 04, 082 (2010) [arXiv:0911.4379 [hep-th]].
  * [26] E. J. Chang, C. J. Chou and Y. Yang, Phys. Rev. D 98, no.10, 106016 (2018) [arXiv:1805.06117 [hep-th]].
  * [27] S. Hemming and L. Thorlacius, JHEP 0711, 086 (2007) [arXiv:0709.3738 [hep-th]].
  * [28] R. Gregory, S. F. Ross and R. Zegers, JHEP 0809, 029 (2008) [arXiv:0802.2037 [hep-th]].
  * [29] D. N. Page and K. C. Phillips, Gen. Rel. Grav. 17, 1029-1042 (1985)