# Mixed Reality: The Interface of the Future

Dipesh Gyawali  Department of Computer Science
Louisiana State University
Baton Rouge, Louisiana, United States
<EMAIL_ADDRESS>
###### Abstract

The world is slowly moving towards everything being simulated digitally and
virtually. Mixed Reality (MR) is the amalgam of the real world with virtual
stimuli. It has great prospects in the future in terms of various applications
additionally with some challenges. This paper focuses on how Mixed Reality
could be used in the future along with the challenges that could arise.
Several application areas along with the potential benefits are studied in
this research. Three research questions are proposed, analyzed, and concluded
through the experiments. While the availability of MR devices could introduce
a lot of potential, specific challenges need to be scrutinized by the
developers and manufacturers. Overall, MR technology has a chance to enhance
personalized, supportive, and interactive experiences for human lives.

###### Index Terms:

Mixed Reality, Hololens, Virtual world, Applications, Challenges

##  I Introduction

Technology is advancing at a rapid pace. The world is slowly moving towards
everything being simulated digitally and virtually. Real-life problems are
being solved by decision-making tools using advanced technology. It is
significant to make the user interface easy to use for proper utilization of
technology. To achieve this goal, many computing technologies have emerged
such as Virtual Reality(VR), Augmented Reality(AR), and Mixed Reality(MR).

Generally, in today’s digital age, there exists a real world, a virtual world,
and a mix of both real and virtual. The real world is the physical world where
everything exists physically and dynamically. The virtual world is a form of
simulation where users can experience different contexts other than the
physical world at the same time. The Hybrid world consists of a mixture of
physical and virtual entities where anything can be manipulated over the
physical world virtually.

Mixed reality (MR) is a hybrid technology where the real world merges with
virtual stimuli. The term ’Mixed Reality’ was coined by Milgram and Kishino[1]
in a paper ”The Taxonomy of Mixed Reality Virtual Devices” in 1994. It
includes AR and VR technologies that overlay virtual objects over physical
entities. MR technology is the outcome of fusing the physical world. MR can be
experienced using special devices e.g. Microsoft Hololens or applications of
mobile phones which introduce the next potential sectors that change the
world.

Mixed Reality is different from AR[4] and VR[2] in the context of how objects
are overlayed and presented. Virtual Reality is sensed through VR devices that
are mounted on the head where users can interact and mimic their natural
behavior such as zooming in and zooming out the object, changing the angle of
projection while moving, etc. There is no physical world in VR where users can
interact with the entities physically. Augmented Reality is different from VR
and can be achieved using AR devices or mobile phones. e.g. Snapchat
application. In AR, there is an overlay of virtual objects over the physical
entities where the interaction between user and object takes place. They can
change different environment settings as well. Mixed Reality[3] is achieved
using a typical MR computing device which establishes new settings of
experiences by the interaction of virtual and physical space. Using MR, users
themselves can manipulate the virtual entities in the physical world to
achieve different configurations and environments.

In this research, various applications of Mixed Reality are studied, and found
how these applications could shape the interface of the future. Additionally,
there are some challenges that need to be considered while using MR
technologies which are also presented in this research. The major
contributions of this paper are:

  * •

Study the various applications of Mixed Reality and its future prospects.

  * •

The way Mixed Reality could be the interface of the future.

  * •

The challenges faced when using Mixed Reality technology.

Figure 1: Microsoft Hololens

##  II Literature Review

Mixed Reality is an emerging technology where big tech companies and academic
institutions are performing research extensively. Speicher et al. presented
basic concepts about mixed reality and characterized MR applications in terms
of the environment number, users, immersion and virtuality level, interaction
degree, input, and output[3]. Similarly, the review of mixed reality, its
current trend, challenges, and prospects have been studied by Rokhsaritalemi
et al. (2020). They have provided a baseline to study MR basic concepts, its
development steps and analytical models, a simulation toolkit, system types,
and architectural types[5]. Egliston et al. propose how MR can be used in the
future and its possibilities[6]. In the same way, Krishna et al. [7] and
Flavian et al.[8] review how MR could be used in Smart Computing Education
Systems and in customer experience. As MR is also the mix-ins of AR and VR,
Cipresso et al., define the past, present, and future of VR and AR technology
research[9]. Azuma et al. from Georgia Tech University, mention the recent
advancement in Augmented Reality[10]. All these studies have focused on the
applications of MR and its possibilities. Milgram [1] defined a ’continuum of
Real to Virtual environments, where AR is one part of the general area of
Mixed Reality’. In VR, the surrounding environment is virtual, while in AR the
surrounding environment is real. There is a lack of sufficient study where MR
could solve real-life problems and the challenges that should be faced which
is presented in this research.

##  III Research Questions

There are a few research questions that this paper aims to address for MR
possibilities. This research addresses the following questions: Q1. Can we see
the increasing trend of applications of Mixed Reality in the future? Q2. Could
MR technologies be the interface of the future? Q3. Are there potential
challenges while exploring the applications of Mixed Reality?

Figure 2: Reality-virtuality Continuum

##  IV Methodology

To address the research question, specific methods are followed with different
settings. Firstly, we reviewed previously published articles in MR-reputed
journals and conferences. These articles were queried using specific keywords
like ”Mixed Reality”, ”Hololens”, and ”Extended Reality” in several databases
and libraries such as IEEEXplore, Springer, ACM, Scopus, Science Direct,
Microsoft Official Site, Google search, etc. The queried articles are related
to Computer Engineering, Electrical Engineering, Computer Science, AR, VR, and
MR domains. The papers that couldn’t be accessed and required to pay are taken
from the Sci-Hub website. The published articles which are reviewed for this
research range from the years 2000-2022. Many Mixed Reality articles were
published within this period. And, within the span of the last five years,
this technology is advancing rapidly. The research methodology is shown in
Fig. 3 and consists of the following steps:

Figure 3: Research Methodology

  * •

Articles were searched and reviewed using specific keywords related to mixed
reality. Multiple databases were queried to find the relevant articles.

  * •

Papers were evaluated qualitatively. The papers talk about different
applications of mixed reality and how it is beneficial in the future.

  * •

The applications areas are seen from medical, education, gaming, etc. that
could solve the existing problem in the future. In addition to that, several
challenges were scrutinized.

  * •

More than 15 articles from general and specific fields were selected based on
their relevance to the proposed research question. They are reviewed in detail
within this research survey.

  * •

The experiments and concepts regarding mixed reality are then presented
sequentially which helps readers to understand easily.

##  V Experiments

###  V-A Overview and Concept

Mixed reality[3] is the technology that fuses the physical with the virtual
world to provide upcoming evolution in human, computer, and environmental
interactions. It’s the next wave in computation. Before it, we had mainframes,
PCs, and smartphones. Now, the concept of AR, VR, and MR are emerging which is
mainstream for consumers and businesses. It liberates people from screen-bound
experiences through interactions with data in the physical world. Mobile
AR[11] nowadays offers mixed reality solutions, especially on social media.
Generally, the applications to mixed reality also include environmental
understanding i.e. anchors and spatial mapping, and human understanding i.e.
eye-tracking, hand-tracking, and speech input, positioning, and locations in
both physical and virtual spaces.

To create true mixed reality interactions and experiences, there are three
essential elements:

  * •

Input methods

  * •

Perceptions from the surrounding

  * •

Cloud-driven computer processing model

The reality-virtuality continuum[1] as shown in Fig. 2 represents the spectrum
of realities as the mixed reality spectrum. At one end, we have the real
physical system in which humans exist. On the other, we have the relative
digital reality.

###  V-B Augmented and Virtual Reality

Nowadays, many big tech companies are doing research on AR and VR. Snapchat
uses AR features on image/video filters. However, most mobile devices and apps
that exist on the market have little environmental perception capabilities.
They cannot fully blend physical and digital realities. Augmented Reality[11]
is achieved by overlaying holograms, graphics, images, and video streams in
the physical world. Virtual reality has one difference in visualizing the real
world. In VR, we cannot see the actual world which makes the experiences fully
immersive digital experience. Mixed Reality can be experienced by
transitioning between AR and VR which places a hologram in the physical world
as if it is physically present. It represents the avatar digitally and
personally present in the physical world in collaboration with people and
objects.

###  V-C Mixed Reality Devices

Mixed Reality perception is achieved by wearing a glass-mounted headset from
which we can see the physical world to interact using digital assets/objects.
The user can manipulate virtual objects to change the environment and interact
using different sensors. For instance, a user can resize an object to fit into
a room. There are several mixed reality devices available in the market in
which hardware devices, chips, sensors, and software are integrated to perform
a task in a coordinated way. MR devices can also act as standalone device, can
be usable with glasses, and allows users to interact with their physical
world. One limitation of MR headsets is it is still in the early stage of
development which limits their production. Additionally, its field of view is
often narrow. These advantages and limitations help to explore more
possibilities in applications and challenges across Mixed Reality. Table I
shows the different mixed reality devices being developed[12]. Fig 4.
represents the decreasing price of VR over the years.

TABLE I: Mixed Reality Devices Brand | Product | Release Date | Country | Price(US)
---|---|---|---|---
Zappar | ZapBox | 2018 | UK | 30
Tesseract | Holoboard | 2018 | India | 349
Occipital | Bridge | 2016 | US | 399
Nreal | Light | 2019 | China | 499
Dim. NXG | AjnaLens | 2018 | India | 1500
Magic Leap | One | 2018 | US | 2295
Microsoft | HoloLens | 2016 | US | 3000
Microsoft | HoloLens 2 | 2019 | US | 3500
Avegant | LightField | 2016 | US | N/A
Figure 4: VR device prices each year

##  VI Applications

Mixed Reality has a big potential to solve problems in various areas in the
future. As this technology is growing, we can expect many possibilities and
applications in different areas which are described below:

###  VI-A Smart Education

The integration of AR and MR technologies in mobile devices has opened up a
lot of opportunities for applications in smart education. It overcomes the
problems in the traditional way of teaching by showing its benefits. According
to a recent study[7], the usage of mixed reality enhances students to learn
complicated logical problems in Chemistry like a synthetic bond. They
mentioned that mixed reality technologies help to picture the basic ideas of
covalent bond to identify the objectives of chemical bonds. In the same way, a
virtual learning environment builds interaction and cooperation among
understudies and teachers. In the normal way of teaching, it’s much more
complicated to visualize higher-dimension data. Teachers can use Mixed Reality
devices to simulate the mathematical analysis and graphs (higher dimensions)
and teach their students so that it is easier for them to understand.

Figure 5: Mixed Reality in Education

Pridhvi et al.[7] conducted the experiment to show the effectiveness of MR in
the education system. They included different application development using
Unity 3D and Unreal Engine to accomplish a practical look for
learners/students. They visualized the Roman Colosseum in VR and also a low-
resolution model of a maze. They also learned about the behavior of animals
according to their adaptation to environments. S.V Viraktamath[14] and Touel
et al.[13] their research mentions that MR devices can be used as a real-time
analytical and interactive tool for instructors to promote the student’s
learning process. They concluded that the environment resulted in better
prospects for education compared to the existing teaching methodology.

###  VI-B Health System

The study was conducted at the Norwegian University of Science and Technology
and Gerup et al.[15] reveals that Mixed Reality is suitable for use in
healthcare sectors because of its 3D capabilities and interactions, with
scalable objects interacting simultaneously by various participants, and its
ability to provide a deeper and better understanding of complex issues. In the
same way, because of COVID-19 and social distancing, this technology has
accelerated a lot.

Figure 6: Mixed Reality in Health Care

Similarly, Sánchez-Margallo JA[16] performed a study of the application of MR
in medical training and surgical planning that focuses on minimally invasive
surgery. They mentioned MR will reduce the time and effort while performing
the training and surgery as medical training is a long and tiresome process.
They conducted experiments on training in urology that allow visualization of
3D models, perform manipulation in different angles and dimensions, and able
to interact independently with them. They found MR is a significant tool for
studying and visualizing the human pelvic system and anatomy both individually
and in groups.

###  VI-C Engineering and Construction

Construction projects contain a series of interdependent and interrelated
activities. There are many hazardous works that could lead to serious
accidents for the workers. Fei Dai et al. (2018) evaluated the feasibility of
applying an emerging MR technology to enhance communication in the
construction industry[17]. They developed a holographic application in
Microsoft Hololens that helps in real-time collaboration in 3D space where the
application was evaluated through trial and feedback from workers/participants
in the construction industry. The experiment was conducted among 50 workers
where they can communicate without being physically present at the site. Their
experiments demonstrate that 58 percent of total responses agree MR has a
positive impact on risk communication 34 percent are neutral. This helps to
get a sense of how MR could be huge potential for applications in the
construction industry.

Figure 7: Mixed Reality in Engineering and Construction

Similarly, MR can be used for engineering project planning to execution. For
instance, MR can overlay certain elements onto a building plan so that
investors/stakeholders can get a better understanding of the project. MR can
be used to show 3D models, interact with them, provide tours, and track and
document the progress of the project. In addition to this, they can be used
for labeling and smooth collaboration, construction training, etc. These
things seem to create a lot of potential in the construction industry using
MR.

###  VI-D Gaming and Entertainment

Games and Entertainment are one of the major industries where Mixed Reality
has been impacted the most. MR games use collaborative technologies to
entertain, educate, motivate, and inspire people. The design of MR games and
interactive gameplay environments has received a lot of attention across
multiple disciplines. Elizabeth Bonsignore et al.[18] has mentioned that one
example of gamification is the growth of context-aware apps like Foursquare
which uses social and location-based services. Other games include iSpy and
Parallel Kingdom which uses location-based services that offer a playful lens
for visualizing experiences, and relationships of ideas grounded in real
spaces i.e. crossing the boundary between real and virtual.

Likewise, many social media platform that includes interactive mixed-reality
experiences are being developed. Roofie Du et al.[19] from Google presented
Geollery, which is a social media platform to share, create, and explore
geotagged information. It includes a real-time ’rendering of an interactive
and collaborative mirrored world with 3D buildings, user-generated content,
and geo-tagged social media. Using these features, users can view, chat, and
collaborate with other participants remotely in the same spatial context in a
virtual environment. This demonstrates MR features are being developed in the
gaming and entertainment industries.

###  VI-E Government Agency

Mixed Reality also has applications across governmental bodies and agencies. A
recent study conducted shows that AR and MR could be used for security in
customs borders and protections. Similarly, they also mentioned envisioning
transportation security 4.0 which reduces error and risk in security
screening. In the same way, MR features can be used to study the probable
natural calamities region and also have easy management to search and rescue.

###  VI-F Retail and E-commerce

MR can help to fill the lines between brick-and-mortar retail experiences and
e-commerce. The products can be scanned using their tags and leverage
important information from there so that people don’t need to manually fill in
the information. For instance, a retail mobile app can be used to identify all
the products in a store automatically and add them to the shopping cart by
adding computer vision and machine learning features.

Figure 8: Mixed Reality in Retail and E-commerce

##  VII Challenges

Although Mixed Reality induces a lot of potential benefits, there could be
many challenges that need to be considered to operate it. MR system
development has a special software architecture that needs to execute MR
algorithms faster in MR applications. It also has a similar architecture as
others i.e. the server, client, and network. The server manages the data
whereas the client is the user that executes the command. The network acts as
a communicator between the server and the client. Some of the potential
challenges of Mixed Reality lie in the following section:

###  VII-A Architecture

It’s very important to design a suitable architecture to run MR applications
smoothly. The architecture needs to support multiple users, who can
simultaneously manipulate the objects in the environment. There should be non-
noticeable latency in every operation so that user interaction would be
smooth. As its architecture is based on client-server, there is a single point
of failure. So, the server should be up and running every time for execution.
This induces the challenges to be faced by developers in system design.

###  VII-B Network Issue

Networking is a most important part of MR that acts as a communicator between
server and client. By distributing IP, every client can communicate with each
other and access data through the server. If there is any delay in network
communication, latency could be introduced which makes user interaction less
convenient. Thus, the challenges also lie in the networking part.

###  VII-C Data Loss and Security

Data Loss and Security are one of the greatest challenges in Mixed Reality
applications. The MR devices help to overlay the objects in the physical world
where there is continuous data flow across the environment. The data could be
lost on the way to overlaying the objects. In the meantime, anyone can
interrupt the data flowing over the network which could create trouble.
Moreover, the MR applications used in the government sector need to be more
secure in terms of data handling and processing.

###  VII-D Pedagogical Challenges

When MR is used in educational settings, it changes past pedagogical
techniques. The study by Glover and Miller[20] in 2001 mentions that it is a
prerequisite to accepting technology by teachers and students. Both teachers
and students should be prepared for such technologies. Similarly, learning
activities should include well-designed interactions with friends and the
system. There should be minimum information overload and field-of-view should
be accurate. Otherwise, the visualizations are perceived as obtrusive.

###  VII-E Technical Challenges

MR technology and devices are still in their infancy stage. Pascal Knierim et
al.[21] mentioned that it includes a small view field, limited battery life,
uncomfortable wearing devices for a long time, etc. From the fast
technological advances across MR, these challenges could be solved in the
future. Currently, many advanced features like object recognition, hand
gesture recognition, and speech identification are being integrated into MR.
But it’s still a research question of how to design an interactive concept for
MR user experiences.

##  VIII Results

We reviewed several research papers in the field of Mixed Reality and studied
the potential benefits and challenges. From the study we conducted, the
research question that is presented could be answered.

###  VIII-A Can we see the increasing trend of applications of Mixed Reality
in the future?

  * •

The experiments that are performed to analyze the applications of MR in
different fields above show that MR has the potential to be used for solving
the existing problem and a lot of benefits can be leveraged from it. It has
increasing benefits across education, health, construction, gaming, etc. which
could be an interface of the future.

  * •

Despite some negative impacts that MR could bear, we could see a lot of
positive impacts of MR in human lives. This answers our first research
question.

###  VIII-B Could MR technologies be the interface of a future?

  * •

As seen from the increasing number of companies and MR devices year to year in
Table I, a lot of big companies are spending billions to adopt MR technologies
as shown in Table I. Also, the applications seen in various sectors have also
shaped MR to be one of the emerging technologies. This demonstrates MR
technology as the interface of the future.

###  VIII-C Are there potential challenges while exploring the applications of
Mixed Reality?

  * •

Certain challenges to adopting MR technologies in various fields are studied
above. In the same way, there are some uses of MR that can impact human lives
negatively. These all challenges should be cautiously handled in order to
fully utilize the benefits of Mixed Reality. These aspects answer that there
are certain challenges to exploiting the use of Mixed Reality.

##  IX Conclusion

Mixed Reality is an emerging technology with a lot of potential that is being
developed and researched across the globe. Multi-national companies are
spending huge amounts to explore the applications of MR so that it would solve
human problems in the future. This research helps to study the potential
benefits and challenges of MR in the future and the way MR could be an
interface to the future. There are still many areas like sports and
transportation that are not explored in this survey. Also, MR could impact
negatively in some areas which is not described in this research. Overall, MR
technology has a potential role in solving human life problems.

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# The first law of entanglement entropy in AdS black hole backgrounds

Akihiro Ishibashi<EMAIL_ADDRESS>Department of Physics and Research
Institute for Science and Technology, Kindai University, Higashi-Osaka
577-8502, JAPAN  Kengo Maeda<EMAIL_ADDRESS>Faculty of
Engineering, Shibaura Institute of Technology, Saitama 330-8570, JAPAN

###### Abstract

The first law for entanglement entropy in CFT in an odd-dimensional
asymptotically AdS black hole is studied by using the AdS/CFT duality. The
entropy of CFT considered here is due to the entanglement between two
subsystems separated by the horizon of the AdS black hole, which itself is
realized as the conformal boundary of a black droplet in even-dimensional
global AdS bulk spacetime. In $(2+1)$-dimensional CFT, the first law is shown
to be always satisfied by analyzing a class of metric perturbations of the
exact solution of a $4$-dimensional black droplet. In $(4+1)$-dimensions, the
first law for CFT is shown to hold under the Neumann boundary condition at a
certain bulk hypersurface anchored to the conformal boundary of the boundary
AdS black hole. From the boundary view point, this Neumann condition yields
there being no energy flux across the boundary of the boundary AdS black hole.
Furthermore, the asymptotic geometry of a $6$-dimensional small AdS black
droplet is constructed as the gravity dual of our $(4+1)$-dimensional CFT,
which exhibits a negative energy near the spatial infinity, as expected from
vacuum polarization.

##  I Introduction

The entanglement entropy of a quantum field in black hole backgrounds has
attracted much attention as the key concept toward understanding the origin of
the Bekenstein-Hawking entropy (for review, see e.g., [1]). Although it is in
general a Herculean task to calculate an entropy in a curved background, it is
tractable to compute the entropy for strongly coupled conformal fields in the
holographic setting, or the anti-de Sitter (AdS) conformal field theory (CFT)
correspondence, by using the Ryu-Takayanagi formula [2]. In this formula, the
entropy calculation for a subsystem is reduced to a much simpler task of
calculating the area of a minimal or extremal surface anchored to the boundary
of the asymptotically AdS gravity dual.

In the framework of the AdS/CFT duality, the first law of the vacuum
entanglement entropy was shown for any ball-shaped subregions when the AdS
boundary is flat Minkowski spacetime [3, 4]. Conversely, the bulk linearized
Einstein equations around the pure AdS spacetime can be derived when the first
law is satisfied for the subregions in the boundary theory [5], being
consistent with the basic idea [6]. In some sense, the first law of the
entanglement entropy could be a guiding principle for a consistent formulation
of quantum gravity, just like the first law of black hole thermodynamics.
Motivated by this, we holographically explore the first law of the
entanglement entropy for two subsystems of CFT separated by a black hole
horizon in asymptotically AdS spacetime.

In the holographic proof of the first law [5], the Noether charge formula [7]
plays an essential role. This is because the entanglement entropy for any
ball-shaped spatial region in flat AdS boundary corresponds, through a
conformal transformation, to the horizon area of a zero-mass hyperbolic black
hole in the bulk. Therefore it is inferred that the Noether charge formula can
also be applied to the derivation of the first law of the entanglement entropy
of a strongly coupled CFT in black hole backgrounds. In the standard Noether
charge derivation [7] of the first law of the black hole thermodynamics in
$D$-dimensions, one first considers a $(D-1)$-dimensional spacelike
hypersurface which has two $(D-2)$-dimensional boundaries, one at the black
hole horizon (bifurcate surface) and the other at the spatial infinity, and
then computes the Noether charges at these two boundaries. For asymptotically
flat black holes, one can impose a natural boundary condition for physical
fields at the spatial infinity to be consistent with the asymptotic flatness
and black hole no-hair property, but for asymptotically AdS black holes, one
needs to be more careful to consider possible boundary conditions at the AdS
infinity. In the holographic setting, further care may be needed since one
first extends the spacelike hypersurface into the $(D+1)$-dimensional bulk
spacetime to have a $D$-dimensional spacelike hypersurface and, if the
extended hypersurface ends at a bulk boundary (e.g., black hole horizon
isolated from the boundary black hole), then one also has to take into account
the additional contribution to the Noether charge from such a bulk boundary.

In this paper, we examine the first law of the entanglement entropy of odd-
dimensional conformal field theories (CFT) in AdS black hole backgrounds by
using the Noether charge formula in the holographic setting. Namely, we
consider a $D$(odd)-dimensional AdS black hole which is realized as a part of
the conformal boundary of the $(D+1)$-asymptotically AdS bulk spacetime. Then,
as briefly mentioned above, we evaluate Noether charges at two
$(D-2)$-boundaries (i.e., the boundary black hole horizon and the boundary AdS
infinity) as well as those at $(D-1)$-bulk boundaries.

For simplicity, we assume that the bulk geometry contains neither horizons in
the IR isolated from the AdS boundary nor black funnels in the Hartle-Hawking
state, which connect two or more boundary black holes [14]. So, the only
possible bulk solution dual to the boundary CFT in an AdS black hole
background is the black droplet solution [14], in which the bulk horizon is
connected to the single boundary black hole. The black droplet solutions with
a boundary black hole in asymptotically flat spacetime were eagerly
constructed with numerics, motivated by the investigation of the Hawking
radiation in strongly coupled field theories [8, 9, 10, 11, 12, 13] (see also
analytic solutions for lower-dimensional black droplets [14] or perturbative
analytic solutions [15, 16]).

Figure 1: A schematic diagram of a time-slice of our black droplet in
asymptotically AdS bulk spacetime of $(D+1)$-dimension. The black circle
describes a $(D-1)$-dimensional sphere as a time-slice of the $D$-dimensional
conformal boundary of the Einstein static cylinder or an asymptotically,
globally AdS spacetime in $(D+1)$-dimension. The conformal boundary of our
black droplet itself is a $D$-dimensional asymptotically AdS black hole
(illustrated in Fig. 2), whose $(D-1)$-dimensional time-slice (dashed blue
half-circle) therefore covers only the half of the $(D-1)$-dimensional sphere.
Note that the boundary $D$-dimensional black hole has two asymptotic regions
(corresponding to A and B of Fig. 2), but the lower half circle (dashed blue
half-circle) shows only one of them, say the region A of Fig. 2. We take our
hypersurface $\Sigma$ as the lower half of this time-slice (shaded region)
enclosed by part of the AdS boundary (part of the dashed blue curve, which
corresponds to the region A), black droplet horizon (black curve) and
$\chi=\pi/2$ (red dotted line). The contribution to the Noether charge arises
from part (red dotted line, $\chi=\pi/2$) of $\partial\Sigma$, as well as from
the bulk horizon and the lower half of the AdS boundary. As we will discuss,
we consider a reflection boundary condition at $\chi=\pi/2$ (red dotted line).

In Fig. 1, we show an example of the black droplet solution in which the bulk
horizon is connected to a boundary black hole in an asymptotically AdS
spacetime. In $(2+1)$-dimensional field theory, we calculate the perturbation
of the black droplet solution with BTZ black hole on the AdS boundary [14].
(For a brane model, the first law of the entanglement entropy was shown [17].)
It is shown that the first law of the entanglement entropy is always
satisfied. In $(4+1)$-dimensional field theory, we show that the first law is
satisfied for the Neumann boundary condition at the equatorial plane in Fig.
1. In the field theory side, the condition corresponds to the no energy flux
condition across the timelike null infinity. Under the boundary condition, we
analytically construct the asymptotic geometry of such a black droplet
solution from the perturbation of the global AdS spacetime. It is shown that
negative energy appears in asymptotic region and it decays with $r^{-3}$ power
in the radial coordinate $r$. This could be understood as a vacuum
polarization effect.

The rest of our paper is organized as follows. In section II, we briefly
review the Noether charge formula [7] and explain our setup. In Sec. III, we
show that the first law of the entanglement entropy is satisfied in
$(2+1)$-dimensional CFT by perturbing the black droplet solution. In Sec. IV,
we derive the Noether charge on the equatorial plane and explore the boundary
condition in which the first law is satisfied in $(4+1)$-dimensional CFT. We
also show that the energy flux across the AdS boundary is zero when the first
law is satisfied. In Sec. V, a black droplet solution with a boundary AdS
black hole is perturbatively constructed from the global AdS spacetime. Sec.
VI is devoted to summary and discussions. In Appendix, we briefly discuss the
Noether charge formula (4) for general perturbations in $(5+1)$-dimensional
bulk metric (25).

##  II Preliminaries: Holographic setting and Noether charge formula

Figure 2: The Penrose diagram of an AdS black hole on the boundary theory. The
Noether charge generically appears on the bulk hypersurface anchored to ${\rm
D}$.

In this section we explain our setup and the strategy before exploring the
first law of the entanglement entropy. Our holographic setup is the following:
We consider a $(D+1)$-dimensional asymptotically (locally) anti-de Sitter
(AlAdS) spacetime $({\cal M},g_{MN})$ which satisfies the vacuum Einstein
equations and whose conformal boundary $(\partial{\cal
M},{\tilde{g}}_{\mu\nu})$ is locally conformally mapped to the $D$-dimensional
static Einstein cylinder. On the conformal boundary $\partial{\cal M}$, we
consider a $D$-dimensional static, asymptotically AdS black hole, whose
spatial section is conformally embedded in the half of the
$(D-1)$-sphere–i.e., a spatial section of the static Einstein cylinder–as
shown by the dashed blue curve in Fig. 1. On such a $D$-dimensional AdS black
hole realized in our holographic setting, we consider the entanglement entropy
for the subregion A on a spacelike hypersurface $S$ separated by the horizon
(see Fig. 2). The entropy is obtained by tracing over the degrees of freedom
located in the subregion B.

From the bulk point of view, there is a static black droplet solution in which
the horizon is connected to the boundary horizon of the boundary AdS black
hole. We assume that there are no bulk black holes disconnected from the AdS
boundaries or black funnels connecting two or more boundary horizons.
According to the Ryu-Takayanagi formula [2], the entanglement entropy between
A and B is obtained by the area of the bulk minimal surface anchored to the
bifurcate surface ${\rm C}$ between ${\rm A}$ and ${\rm B}$. Since the bulk
spacetime is static, there is a bifurcate Killing horizon in the bulk anchored
to the bifurcate surface ${\rm C}$ on the AdS boundary. So, the entanglement
entropy corresponds to the area of the bifurcate surface of the Killing
horizon of the black droplet solution.

We examine the first law of entanglement entropy in the boundary field theory
by applying the Noether charge formula to the bulk Einstein gravity in a
$(D+1)$-dimensional AlAdS spacetime [7]. Let $\xi^{M}$ be a Killing vector
field with respect to the bulk metric $g_{MN}$. The Noether charge
$(D-1)$-form ${\bm{Q}}$ associated with $\xi^{M}$ is given by

$\displaystyle{\bm{Q}}=-\frac{1}{16\pi}\epsilon_{M_{1}M_{2}\cdots
M_{D-1}NL}\nabla^{N}\xi^{L},$ (1)

where here and hereafter we set the $(D+1)$-dimensional gravitational constant
equal to one and we omit the indices $M_{1}M_{2}\cdots$ when writing
differential forms, as in the l. h. s. above. For an arbitrary small deviation
${\delta g}_{MN}$ from the bulk geometry $({\cal M},g_{MN})$, the symplectic
potential $D$-form ${\bm{\Theta}}$ is defined as

$\displaystyle{\bm{\Theta}}(g,\,\delta
g)=\frac{1}{16\pi}\epsilon_{NM_{1}M_{2}\cdots
M_{D}}g^{NJ}g^{LK}(\nabla_{L}\delta g_{JK}-\nabla_{J}\delta g_{LK}).$ (2)

As shown in Ref. [7], the exterior derivative of the variation of the Noether
charge is expressed by the symplectic potential ${\bm{\Theta}}$ as

$\displaystyle d\delta{\bm{Q}}=d(\xi\cdot{\bm{\Theta}}),$ (3)

where the “centered dot” in the r. h. s. of Eq. (3) denotes the contraction of
the Killing vector field $\xi^{M}$ into the first index of the form
${\bm{\Theta}}$. Integrating this over a $D$-dimensional spacelike
hypersurface $\Sigma$, we obtain

$\displaystyle\int_{\partial\Sigma}(\delta{\bm{Q}}-\xi\cdot{\bm{\Theta}}(g,\,\delta
g))=0,$ (4)

where $\partial\Sigma$ is the $(D-1)$-dimensional boundary of the hypersurface
$\Sigma$. Applying this equation to a black hole spacetime with a bifurcate
Killing horizon with respect to $\xi^{M}$, one can obtain the first law.

In the Fefferman-Graham expansion, the bulk metric is represented by

$\displaystyle
ds^{2}=g_{MN}dx^{M}dx^{N}=\frac{L^{2}}{z^{2}}\left(dz^{2}+{\tilde{g}}_{\mu\nu}(z,x)dx^{\mu}dx^{\nu}\right),$
$\displaystyle{\tilde{g}}_{\mu\nu}(z,x)={\tilde{g}}_{(0)\mu\nu}(x)+z^{2}{\tilde{g}}_{(2)\mu\nu}(x)+\cdots+z^{D}{\tilde{g}}_{(D)\mu\nu}(x)+\cdots,\quad$
$\displaystyle x^{M}=(z,x^{\mu})\,\quad(\mu=0,1,\cdots,D-1),$ (5)

where $L$ is the AdS curvature length.

For $D=3$ and $D=5$ cases we are interested in, there is no conformal anomaly,
and hence there is no logarithmic term in (II) [18]. Since the background bulk
solution $g_{MN}$ is static, there is a timelike Killing vector field,
$(\partial_{t})^{M}$ on ${\cal M}$, and let us take
$\xi^{M}=(\partial_{t})^{M}$. We assume that ${\delta g}_{MN}$ is a solution
to the linearized equations of motion satisfying the Dirichlet boundary
condition at the conformal boundary of the bulk AdS ${\cal M}$,

$\displaystyle\delta{\tilde{g}}_{(0)\mu\nu}=0.$ (6)

Since ${\tilde{g}}_{(n)\mu\nu}~{}(0<n<D)$ is the function of
${{\tilde{g}}}_{(0)\mu\nu}$ [18], the leading order of the metric perturbation
is

$\displaystyle{\delta{g}}_{\mu\nu}=O(z^{D-2}).$ (7)

When $\partial\Sigma$ is the bifurcation surface of a bulk black hole, the
second term in Eq. (4) disappears on $\partial\Sigma$ and the first term
yields the variation of the black hole area (entropy). In the black droplet
solutions, the horizon extends to the AdS boundary, and hence the area
diverges towards the boundary. Note however that the variation should be
finite for the Dirichlet boundary condition (6). When $\partial\Sigma$ is
taken as a part of the boundary of $\Sigma$ at the AdS conformal boundary, the
formula (4) provides the variation of the holographic energy $E$ [19], defined
by

$\displaystyle E=\int_{\partial\Sigma}T_{\mu\nu}\xi^{\mu}n^{\nu}\,,$ (8)

where $T_{\mu\nu}$ is the stress-energy tensor of the boundary field theory,
$n^{\mu}$ is the unit normal vector to $\partial\Sigma$, and the volume
element of $\partial\Sigma$ with respect to the conformal metric $g_{(0)}$ is
understood in the integral. Applying the holographic stress-energy formula
[18], $T_{\mu\nu}$ is given in terms of the coefficient
${\tilde{g}}_{(D)\mu\nu}$ as

$\displaystyle T_{\mu\nu}=\frac{DL^{D-1}}{16\pi}{\tilde{g}}_{(D)\mu\nu}\,,$
(9)

in odd dimensional cases.

Our setup for holographic derivation of the first law for entanglement entropy
in CFT is as follows. As our spatial hypersurface $\Sigma$ for evaluating the
Noether charge formula, we take the lower half of the time-slice enclosed by
part of the AdS boundary, black droplet horizon and $\chi=\pi/2$ (shaded
region in Fig. 1). Our hypersurface $\Sigma$ has the asymptotic boundary
$S\subset\partial\Sigma$, which itself admits its boundary $(D-2)$-sphere
denoted by ${\rm D}$ in Fig. 2. Thus, the hypersurface $\Sigma$ admits the
boundaries and corners. We will impose the reflecting (Neumann) boundary
condition at $\chi=\pi/2$ (red dotted line in Fig. 1). We will discuss in Sec.
IV that this condition is necessary for the purpose of avoiding any additional
contributions to the first law from the upper side halfmoon-shaped region. In
the next section, in more concrete setting, we consider the Noether charge at
the boundaries $\partial\Sigma$ and also study an additional contribution to
the Noether charge from the corner.

##  III The first law of entanglement in the exact droplet solution in
$(3+1)$-dimension

###  III.1 The exact droplet solution in $(3+1)$-dimension

Let us first quickly review the exact solution of a black droplet [14]
constructed by the analytic continuation of the $(3+1)$-Schwarzschild-AdS
solution,

$\displaystyle
ds_{4}^{2}=-F(\rho)d\tau^{2}+\frac{d\rho^{2}}{F(\rho)}+\rho^{2}(d\theta^{2}+\sin^{2}\theta
d\Phi^{2}),$ $\displaystyle
F(\rho)=\frac{\rho^{2}}{L^{2}}+1-\frac{\mu}{\rho},\quad\mu=\frac{\rho_{0}^{3}}{L^{2}}+\rho_{0},$
(10)

where $\rho_{0}$ is the horizon radius. The double Wick rotation

$\displaystyle\tau=i\chi,\qquad\Phi=i\tilde{t}$ (11)

and the coordinate transformation $\tilde{r}:=\cos\theta$ yields the following
warped product type metric including $2$-dimensional de Sitter spacetime,

$\displaystyle
ds^{2}=F(\rho)d\chi^{2}+\frac{d\rho^{2}}{F(\rho)}+\rho^{2}\left(-(1-\tilde{r}^{2})d\tilde{t}^{2}+\frac{d\tilde{r}^{2}}{1-\tilde{r}^{2}}\right).$
(12)

By the rescaling of the coordinates $(\chi,\tilde{r},\tilde{t})$,

$\displaystyle\tilde{t}=\frac{r_{0}}{l^{2}}t,\qquad\tilde{r}=\frac{r_{0}}{r},\qquad\chi=\frac{L}{l}r_{0}\varphi,$
(13)

one obtains the following bubble solution

$\displaystyle
ds^{2}=\frac{d\rho^{2}}{F(\rho)}+\frac{\rho^{2}r_{0}^{2}}{l^{2}r^{2}}\left[-\frac{r^{2}-r_{0}^{2}}{l^{2}}dt^{2}+\frac{l^{2}dr^{2}}{r^{2}-r_{0}^{2}}+\frac{L^{2}r^{2}F(\rho)}{\rho^{2}}d\varphi^{2}\right],$
(14)

where $\varphi$ circle has a period $2\pi$. The $S^{1}$ circle along
$\partial_{\varphi}$ shrinks to zero size at the bubble radius,
$\rho=\rho_{0}$, i. e. , $F(\rho_{0})=0$. By imposing a regularity condition
at the bubble radius, $r_{0}$ is expressed by $\rho_{0}$ as

$\displaystyle r_{0}=\frac{2\rho_{0}lL}{3\rho_{0}^{2}+L^{2}}.$ (15)

Note that the boundary metric (i.e., $3$-dimensional metric in the square
brackets in Eq. (14)) is conformal to the static BTZ black hole with the
horizon radius $r=r_{0}$ [20]. Note also that in the $4$-dimensional bulk,
there is a bulk horizon which extends from the bubble radius $\rho_{0}$, deep
inside the bulk spacetime to the boundary BTZ black hole horizon. In this way,
this bulk metric describes a black droplet as illustrated in Fig. 1.

Let us change the coordinates in Eq. (14) into the Fefferman-Graham coordinate
system (II) by

$\displaystyle\frac{1}{\rho(\eta,\,z)}=z\left(\frac{1}{\eta
L}+\alpha_{2}(\eta)z^{2}+\alpha_{3}(\eta)z^{3}+\cdots\right),$ $\displaystyle
r(\eta,\,z)=r_{0}\left(\eta+\beta_{2}(\eta)z^{2}+\beta_{4}(\eta)z^{4}+\cdots\right),$
$\displaystyle\alpha_{2}=-\frac{\eta^{2}-2}{4L\eta^{3}},\qquad\alpha_{3}=-\frac{\mu}{6L^{2}\eta^{4}},$
$\displaystyle\beta_{2}=\frac{\eta^{2}-1}{2\eta},\qquad\beta_{4}=\frac{\eta^{4}-1}{8\eta^{3}},\cdots.$
(16)

Then, we obtain the stress-energy tensor (9) on the boundary field theory by

$\displaystyle T_{tt}=-\frac{\mu Lr_{0}^{2}(\eta^{2}-1)}{16\pi
l^{4}\eta^{3}},$ $\displaystyle T_{\eta\eta}=\frac{\mu
L}{16\pi\eta^{3}(\eta^{2}-1)},$ $\displaystyle T_{\varphi\varphi}=-\frac{\mu
Lr_{0}^{2}}{8\pi l^{2}\eta}.$ (17)

Since the BTZ black hole is in odd-dimensions, the trace anomaly vanishes,
i.e., ${T_{\mu}^{\mu}}=0$, as seen in Eq. (III.1). So, while even-dimensional
black droplet solutions have infinite energy due to the anomaly [21], the
holographic energy $E$ in Eq. (8) converges to a finite value. When one
applies the formula (4) to the black droplet solution (14), one should note
that the $2$-dimensional surface $\partial\Sigma$ consists of three parts: a
part at the AdS boundary, $z=0$, a part of the bulk bifurcate Killing horizon,
$r=r_{0}$, and a part of the timelike surface at $r=\infty$. In the next
subsection we evaluate Eq. (4) on part of the timelike surface at infinity.

###  III.2 Perturbation of the $(3+1)$-dimensional black droplet solution

We consider metric perturbations $h_{MN}:=\delta g_{MN}$ of the black droplet
solution (14). For convenience we express our coordinate system as
$x^{M}=(y^{a},x^{i})$ with $y^{a}:=(\rho,\varphi),\>x^{i}:=(t,r)$ and denote
by $\tilde{\gamma}_{ij}$ the $2$-dimensional de Sitter metric spanned by
$x^{i}$ in (14). We can impose the following gauge condition:

$\displaystyle h_{ai}=0\,,\quad h_{ij}\propto{\tilde{\gamma}}_{ij}\,.$ (18)

This condition may immediately be justified by, e.g., applying the scalar-type
perturbations of [22] to the $4$-dimensional Schwarzschild-AdS solution
(III.1) with the scalar harmonics ${\mathbb{S}}_{\bf k}(x^{i})$ and taking the
gauge $f_{a}=0,\>H_{T}=0$ in the terminology of [22]. Note however that when
applying the formalism of [22] to the present case, the scalar “harmonics”
${\mathbb{S}}_{\bf k}(x^{i})$ corresponds to the solutions of the Klein-Gordon
equation on the Lorentzian sphere (i..e., de Sitter space with the metric
${\tilde{\gamma}}_{ij}$), rather than the standard eigen-function of the
laplace operator with respect to the unit two-sphere metric.

Now we assume that our perturbations $h_{MN}$ satisfy the symmetry along
$\partial_{\varphi}$. Under the gauge choice (18), the variation of the
Noether charge (1) and the symplectic two form (2) for the Killing vector
$\xi=\partial_{t}$ are calculated for $t=\mbox{const.}$ and $r=\infty$ surface
for the metric (14) as

$\displaystyle\int_{r=\infty}\delta{\bm{Q}}=\frac{1}{32\pi}\int\epsilon_{\rho\varphi
tr}(h_{a}{}^{a}-h_{i}{}^{i})g^{tt}g^{rr}\partial_{r}g_{tt}-\frac{1}{16\pi}\int\epsilon_{\rho\varphi
tr}g^{tt}g^{rr}\left(\partial_{t}{h}_{rt}-\partial_{r}{h}_{tt}\right),$ (19)

and

$\displaystyle\int_{r=\infty}\xi\cdot\Theta=-\frac{1}{16\pi}\int\epsilon_{\rho\varphi
tr}g^{rr}g^{tt}\Biggl{[}\frac{1}{2}h_{i}{}^{i}\partial_{r}g_{tt}-g_{tt}\partial_{r}h_{a}{}^{a}+\partial_{t}{h}_{rt}-\partial_{r}{h}_{tt}\Biggr{]},$
(20)

where we denote the partial traces by
$h_{a}{}^{a}=h_{\rho}{}^{\rho}+h_{\varphi}{}^{\varphi},\>h_{i}{}^{i}=h_{t}{}^{t}+h_{r}{}^{r}$.
Note that as usual, the indices of $h_{\mu\nu}$ are raised and lowered by the
background metric $g_{\mu\nu}$ and its inverse, e.g.,
$h_{\mu}{}^{\nu}:=h_{\mu\lambda}g^{\lambda\nu}$. Then, the combination of Eq.
(19) and (20) yields

$\displaystyle\int_{r=\infty}(\delta{\bm{Q}}-\xi\cdot\Theta)=\frac{1}{16\pi}\int\epsilon_{\rho\varphi
tr}g^{rr}g^{tt}\Biggl{[}\frac{1}{2}h_{a}{}^{a}\partial_{r}g_{tt}-g_{tt}\partial_{r}h_{a}{}^{a}\Biggr{]}.$
(21)

With the condition (18) and the perturbed vacuum Einstein equations, we can
find that the partial trace $h_{a}{}^{a}$ vanishes in $4$-dimensional bulk
[22], i.e.,

$\displaystyle h_{a}{}^{a}={h_{\varphi}}^{\varphi}+{h_{\rho}}^{\rho}=0.$ (22)

Thus, the r.h.s. of Eq. (21) vanishes on $t=\mbox{const.}$ and $r=\infty$
surface. It follows from the holographic interpretation [23] that the first
law of the entanglement entropy $S$ is satisfied:

$\displaystyle\delta E=T\delta S$ (23)

in the black droplet solution, where $T$ denotes the temperature computed from
the surface gravity of the bifurcate Killing horizon. This result is peculiar
to the $4$-dimensional bulk case, because Eq. (22) is satisfied only for $D=3$
case. In the other dimensions, we need to impose boundary conditions for bulk
metric perturbations to derive the first law, as we will see in the next
section.

##  IV The first law in $(4+1)$-dimensional field theory in AdS black hole
background

In this section, we examine the first law of the entanglement entropy for CFT
in $D=5$-dimensions by using the holographic principle and applying the
Noether charge formula as in the previous section. For the use of the
holographic ideas, it would be most convenient if there is, as the gravity
dual, any known exact solution of the $(5+1)$-dimensional black droplet that
possesses the boundary $D=5$-dimensional metric

$\displaystyle
ds_{5}^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{3}^{2}\,,\quad
f(r):=1-\frac{M}{r^{2}}+\frac{r^{2}}{L^{2}}\,,$ (24)

where $d\Omega_{3}^{2}:=\sigma_{mn}d\theta^{m}d\theta^{n}$ is the metric of
$3$-dimensional unit sphere with angular coordinates $\theta^{m}~{}(m=1,2,3)$.
Unfortunately there is no such an analytic solution available in
$(5+1)$-dimension. But, fortunately, with the Noether charge formula at our
hand, all we need for our purpose is the asymptotic behavior of the gravity
dual near the boundary rather than the entire structure of the explicitly
written exact solution. In fact, in the $D=3$ case examined in III.2, we have
the exact gravity dual (14) available, but we need not use the explicit
expression of (14) to derive the first law (23) for the boundary CFT.
Therefore, in this section, we simply assume that the desired gravity dual
exists. We will attempt to justify this assumption by perturbatively
constructing a $6$-dimensional black droplet solution in the next section.

Since our $5$-dimensional boundary (24) itself is an asymptotically AdS
spacetime with a $4$-dimensional timelike surface as its conformal boundary
(that is, $r\rightarrow\infty$ in (24): see point D in Fig. 2 and also compare
$\rho=\infty,\chi=\pi/2$ in Fig. 1), by analyzing the asymptotic behavior of
the gravity dual, we can holographically study the asymptotic fall-off
behavior of our $5$-dimensional CFT at large distances $r\rightarrow\infty$.
By doing so we can estimate the energy flux—if it exists—across the
$4$-dimensional boundary $r\rightarrow\infty$ and discuss conditions—if
necessary—for the first law to hold. In Subsec. IV.1, we holographically study
possible energy flux at the boundary $r\rightarrow\infty$ and derive the
condition for having no energy flux across the boundary. For this purpose, it
is sufficient to examine the metric perturbations of the $(5+1)$-dimensional
pure AdS spacetime since the asymptotic behavior of the gravity dual near the
boundary ($\rho\to\infty,\chi\to\pi/2$ of Fig. 1) should be approximately well
described by metric perturbations of the pure AdS background spacetime near
the corresponding boundary (see also Eq. (25) below). Then, in Subsec. IV.2,
by considering metric variations of the $(5+1)$-dimensional static black
droplet—which is assumed to exist as the gravity dual, just mentioned above—we
derive the first law under the no flux condition.

###  IV.1 The energy flux on the perturbed AdS geometry

We consider, as our background, the $(5+1)$-dimensional AdS bulk with the
metric of the form:

$\displaystyle
ds_{6}^{2}=-F_{0}(\rho)dt^{2}+\frac{d\rho^{2}}{F_{0}(\rho)}+\rho^{2}d\Omega_{4}^{2}\,,\quad
F_{0}(\rho):=1+\frac{\rho^{2}}{L^{2}}\,,$ (25)

where $d\Omega_{4}^{2}$ denotes the standard metric of the $4$-dimensional
unit sphere. We use the perturbation formalism of [24], which is most
convenient for the current purpose and is also useful in the next section.
Accordingly we use coordinates $y^{a}:=(t,\rho)$ to express the
$2$-dimensional AdS metric $g_{ab}$ given by the first two terms in (25), and
angular coordinates $x^{i}=(\chi,\theta^{m})$ with $0\leq\chi\leq\pi$ to the
metric $\gamma_{ij}$ of the $4$-dimensional unit sphere $\Omega_{(4)}$:

$d\Omega_{4}^{2}=\gamma_{ij}dx^{i}dx^{j}:=d\chi^{2}+\sin^{2}\\!\chi
d\Omega_{3}^{2}=d\chi^{2}+\sin^{2}\\!\chi\sigma_{mn}d\theta^{m}d\theta^{n}\,.$
(26)

Note that the coordinate $\chi$ is later changed to $r$.

On this background, we consider the metric perturbations, which can in general
be classified into three types: the scalar-, the vector-, and the tensor-type
according to their tensorial behavior on the $4$-dimensional sphere
$\Omega_{(4)}$ as explained in [24]. The tensor-type perturbations are
characterized as the transverse and trace-free part of $\delta g_{ij}$, and in
particular, $\delta g_{t\chi}=0$, and therefore there is no contribution to
the energy flux $T_{tr}$ from the tensor-type perturbations. We discuss the
vector-type perturbations in Appendix. In the following we focus on the
scalar-type metric perturbations.

It is convenient to introduce the harmonic functions ${\mathbb{S}}_{(l)}$ on
the unit four-sphere $\Omega_{(4)}$, which satisfy

$\displaystyle D^{i}D_{i}{{\mathbb{S}}}_{(l)}+l(l+3){\mathbb{S}}_{(l)}=0,\quad
l=0,1,2,\cdots,$ (27)

with $D_{i}$ being the covariant derivative with respect to the metric
$\gamma_{ij}$. Then the scalar-type metric perturbations can be expressed in
terms of ${\mathbb{S}}_{(l)}$ as

$\displaystyle\delta g_{ab}=\sum_{l}H_{(l)ab}{\mathbb{S}}_{(l)},\qquad\delta
g_{ai}=\sum_{l}H_{(l)a}D_{i}{\mathbb{S}}_{(l)},$ $\displaystyle\delta
g_{ij}=\sum_{l}H_{(l)L}\gamma_{ij}{\mathbb{S}}_{(l)}+H_{(l)T}\left(D_{i}D_{j}-\frac{1}{4}\gamma_{ij}D^{k}D_{k}\right){\mathbb{S}}_{(l)},$
(28)

where $H_{(l)ab}$, $H_{(l)a}$, $H_{(l)L}$, and $H_{(l)T}$ are functions of the
coordinates $y^{a}=(t,\rho)$. Hereafter, for brevity we omit the indices $(l)$
labeling the mode, unless otherwise stated. We introduce the following
perturbation variables $Z$ and $Z_{ab}$:

$\displaystyle
Z=4\left(H_{L}+\frac{l(l+3)}{4}H_{T}+2\rho(D^{a}\rho)X_{a}\right),$
$\displaystyle
Z_{ab}=\rho^{2}(H_{ab}+D_{a}X_{b}+D_{b}X_{a})+\frac{3}{4}Zg_{ab},$
$\displaystyle
X_{a}:=-H_{a}+\frac{1}{2}\rho^{2}D_{a}\left(\frac{H_{T}}{\rho^{2}}\right),$
(29)

where $D_{a}$ is the covariant derivative with respect to the $2$-dimensional
unperturbed metric $g_{ab}$. Note that the two variables $Z$ and $Z_{ab}$ are
gauge-invariant. The perturbed Einstein equations can be expressed in terms of
$Z$ and $Z_{ab}$ and furthermore reduced to the following equation for a
single scalar field $\Phi$ on the $2$-dimensional AdS spacetime with the
metric $g_{ab}$:

$\displaystyle D^{a}D_{a}\Phi-\frac{2+l(l+3)}{\rho^{2}}\Phi=0,$ (30)

in terms of which $Z$ and $Z_{ab}$ are expressed as

$\displaystyle
Z_{ab}=\left(D_{a}D_{b}-\frac{1}{L^{2}}g_{ab}\right)(\rho^{2}\Phi),$
$\displaystyle
Z={Z_{a}}^{a}=\left(D^{a}D_{a}-\frac{2}{L^{2}}\right)(\rho^{2}\Phi).$ (31)

Now, using the gauge degrees of freedom, we can choose the following gauge

$\displaystyle H_{\rho}=H_{\rho\rho}=H_{t\rho}=0\,.$ (32)

In this gauge, $X_{\rho}$, $X_{t}$, and $H_{T}$ are determined by the second
and third lines in Eqs. (IV.1) as

$\displaystyle
X_{\rho}=\frac{1}{2\sqrt{F_{0}}}\int\frac{d\rho}{\rho^{2}}\left({\sqrt{F_{0}}}Z_{\rho\rho}-\frac{3Z}{4\sqrt{F_{0}}}\right),$
$\displaystyle
X_{t}=F_{0}\int\frac{d\rho}{\rho^{2}F_{0}}\left({Z_{t\rho}}+i\omega\rho^{2}X_{\rho}\right),$
$\displaystyle H_{T}=2\rho^{2}\int\frac{d\rho}{\rho^{2}}{X_{\rho}},$ (33)

where as previously introduced $F_{0}=1+\rho^{2}/L^{2}$. From the third line
of Eqs. (IV.1), $H_{t}$ is expressed as

$\displaystyle H_{t}=-X_{t}-\frac{1}{2}i\omega H_{T},$ (34)

where we have assumed that all the metric functions depend on $t$ as
$e^{-i\omega t}$.

Let us expand the master variable $\Phi$ near the AdS boundary
$\rho\rightarrow\infty$ as

$\displaystyle\Phi=a_{0}+\frac{a_{1}}{\rho}+\frac{a_{2}}{\rho^{2}}+\cdots.$
(35)

The boundary condition (6) imposes $a_{0}=0$ and sets the integral constants
in Eqs. (IV.1) to zero. Then the stress-energy tensor on the boundary theory
(9) is proportional to the coefficient $a_{1}$.

The perturbed metric is transformed to the Fefferman-Graham coordinate (II) by

$\displaystyle\rho=\frac{\sqrt{1+{r^{2}}/{L^{2}}}}{z}\Biggl{[}1+\frac{L^{2}(L^{2}-r^{2})}{4(L^{2}+r^{2})}z^{2}+\frac{L^{4}(L^{4}-4L^{2}r^{2}+r^{4})}{16(L^{2}+r^{2})^{2}}z^{4}+\cdots\Biggr{]},$
$\displaystyle L\tan\chi=r\frac{4+L^{2}z^{2}}{4-L^{2}z^{2}}.$ (36)

Then, expanding the metric component ${\delta g}_{tr}$ as a series in $z$,
$T_{tr}$ in (9) can be read from the coefficient of $z^{3}$ as

$\displaystyle
T_{tr}\sim\frac{a_{1}}{\left(1+r^{2}/L^{2}\right)^{3/2}}\partial_{r}{\mathbb{S}}.$
(37)

Therefore in the boundary theory side, the energy flux across the AdS boundary
$r=\infty$ (see $\chi\rightarrow\pi/2$ in Fig. 1) becomes

$\displaystyle J\sim\lim_{r\to\infty}\int
T_{t\mu}n^{\mu}\sqrt{1+\frac{r^{2}}{L^{2}}}r^{3}d\Omega_{3}\,dt\sim
a_{1}\lim_{r\to\infty}r^{2}\partial_{r}{\mathbb{S}},$ (38)

where $n^{\mu}$ is the unit normal to the timelike AdS boundary hypersurface
at $r=\infty$. From the harmonic equation (27), ${\mathbb{S}}$ behaves near
the boundary as

$\displaystyle{\mathbb{S}}\simeq A+\frac{B}{r}.$ (39)

By Eq. (38), the condition for no energy flux is

$\displaystyle B=0.$ (40)

In terms of the angle coordinate $\chi$, this condition is equivalent to
imposing

$\displaystyle\lim_{\chi\to{\pi}/{2}}\partial_{\chi}{\mathbb{S}}=0\,,$ (41)

for the harmonic functions. It is easily checked from Eq. (27) that this is
satisfied only for even $l$. In the next subsection, we will check that the
first law of the entanglement entropy is satisfied under the above no-energy
flux conditions.

###  IV.2 The first law of the $6$-dimensional black droplet solution

As mentioned before, throughout this section, we assume that there is a static
black droplet solution whose conformal boundary is the $D=5$ asymptotically
AdS, static $SO(4)$ symmetric black hole with the metric $g_{\mu\nu}(z=0,x)$
given by (24). Such a black droplet solution should admit the Fefferman-Graham
expansion (II) with $D=5$ metric $g_{\mu\nu}(z,x)$ conformal to the boundary
metric (24). In this subsection, as the radial coordinate in the
$D=5$-boundary, we use $\chi$ instead of $r$, so that
$x^{\mu}=(t,\chi,\theta^{m})$.

Now, to derive the first law applying the Noether charge formula, let us
consider metric perturbations $h_{\mu\nu}=\delta g_{\mu\nu}$. For simplicity,
we assume that the perturbed metric satisfies $SO(4)$ symmetry, in which the
only off-diagonal term appeared is $h_{t\chi}$. Since our background boundary
metric is the $5$-dimensional static black hole, such a $SO(4)$ symmetric
perturbation corresponds to monopole or mass perturbation for the boundary
black hole. We briefly comment on possible contributions to the Noether charge
formula from general perturbations without $SO(4)$ symmetry in Appendix.

On our assumptions, the Noether charge (1) and the symplectic form (2) are
evaluated at $t=\mbox{const}.$ and $\chi=\pi/2$ hypersurface as

$\displaystyle\int_{\chi=\pi/2}\delta{\bm{Q}}=-\frac{1}{16\pi}\int\epsilon_{z123t\chi}g^{tt}g^{\chi\chi}\Biggl{[}\partial_{t}h_{\chi
t}-\partial_{\chi}h_{tt}+(\partial_{\chi}g_{tt})\Bigl{(}h_{t}{}^{t}+h_{\chi}{}^{\chi}-\frac{1}{2}h_{\mu}{}^{\mu}\Bigr{)}\Biggr{]},$
(42) $\displaystyle\int_{\chi=\pi/2}[\xi\cdot\Theta]_{z123}$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int\epsilon_{z123t\chi}g^{\chi\chi}\Biggl{[}g^{tt}\partial_{t}h_{\chi
t}+\frac{1}{2}h_{t}{}^{t}g^{tt}\partial_{\chi}g_{tt}-\frac{1}{2}h_{m}{}^{l}g^{mn}\partial_{\chi}g_{nl}$
(43)
$\displaystyle\,+\frac{1}{2}h_{\chi}{}^{\chi}(g^{tt}\partial_{\chi}g_{tt}+g^{mn}\partial_{\chi}g_{mn})-g^{tt}\partial_{\chi}h_{tt}-g^{mn}\partial_{\chi}h_{mn}+h_{m}{}^{n}g^{ml}\partial_{\chi}g_{nl}\Biggr{]},$

where we emphasize that the indices $l,m,n$ represent the angular coordinates
on the unit three-sphere: $\theta^{m}\in\Omega_{3}$. Each term includes the
off-diagonal term $\delta g_{t\chi}$, but this vanishes in the combination of
Eq. (42) and Eq. (43) as follows,

$\displaystyle\int_{\chi=\pi/2}(\delta{\bm{Q}}-\xi\cdot\Theta)$
$\displaystyle=$
$\displaystyle\frac{1}{16\pi}\int\epsilon_{z123t\chi}g^{\chi\chi}\Biggl{[}-\frac{1}{2}g^{tt}h_{m}{}^{m}\partial_{\chi}g_{tt}-\frac{1}{2}g^{mn}h_{m}{}^{l}\partial_{\chi}g_{nl}-\frac{1}{2}h_{\chi}{}^{\chi}g^{mn}\partial_{\chi}g_{mn}+g^{mn}\partial_{\chi}h_{mn}\Biggr{]}.$
(44)

From the expression above (44), we immediately find that if both the
background and perturbed metrics satisfy the Neumann boundary condition

$\displaystyle\lim_{\chi\to{\pi}/{2}}\partial_{\chi}g_{\mu\nu}=\lim_{\chi\to{\pi}/{2}}\partial_{\chi}h_{\mu\nu}=0,$
(45)

then Eq. (44) vanishes. This also shows the first law of the entanglement
entropy (23) in the $D=5$ case, as in the $D=3$ case.

As shown in the previous subsection, the latter condition corresponds to the
no-energy flux condition (41) or (40) for the scalar-type perturbations. As
for the vector- and tensor-type perturbations, it is not obvious from the
above expression alone that one has no contributions to (44). Nevertheless,
one can show, at least for the present $SO(4)$ symmetric case, that the energy
flux caused by the perturbation $h_{\mu\nu}(z,x)$ vanishes on $\chi=\pi/2$ as
follows: The perturbed boundary stress energy tensor $\delta T_{\mu\nu}$
satisfies the conservation law [18]

$\displaystyle\nabla_{\nu}{\delta T_{\mu}}^{\nu}=0,$ (46)

where $\nabla_{\mu}$ is the covariant derivative with respect to the conformal
boundary metric ${\tilde{g}}_{(0)\mu\nu}$ in Eq. (II) under the boundary
condition (6). The $\chi$-component of (46) is written as

$\displaystyle\partial_{\nu}\delta{T_{\chi}}^{\nu}-{\Gamma^{\alpha}}_{\nu\chi}\delta{T_{\alpha}}^{\nu}+{\Gamma^{\nu}}_{\nu\alpha}\delta{T_{\chi}}^{\alpha}=0.$
(47)

Since the only nonzero off-diagonal term of $\delta{T_{\mu}}^{\nu}$ is
$\delta{T_{\chi}}^{t}$ (or $\delta{T_{t}}^{\chi}$), and the background metric
is static diagonal as assumed, the last two terms in Eq. (47) must vanish on
$\chi=\pi/2$ by the boundary condition (45). Thus, Eq. (47) leads to

$\displaystyle\lim_{\chi\to\frac{\pi}{2}}\nabla_{\nu}{\delta
T_{\chi}}^{\nu}=\lim_{\chi\to\frac{\pi}{2}}(\partial_{t}\delta{T_{\chi}}^{t}+\partial_{\chi}\delta{T_{\chi}}^{\chi})=0.$
(48)

The boundary condition (45) implies
$\lim_{\chi\to{\pi}/{2}}\partial_{\chi}{\tilde{g}}_{\mu\nu}=\lim_{\chi\to{\pi}/{2}}\partial_{\chi}{\tilde{h}}_{\mu\nu}=0$.
The fifth-order of the Fefferman-Graham expansion of the perturbed metric
${\tilde{h}}_{(5)\mu\nu}=\delta{\tilde{g}}_{(5)\mu\nu}$ satisfies
$\partial_{\chi}{\tilde{h}}_{(5)\mu\nu}=0$ at $\chi=\pi/2$, and therefore it
follows from the holographic stress-energy tensor formula (9) that
$\lim_{\chi\to{\pi}/{2}}\partial_{\chi}\delta{T_{\chi}}^{\chi}=0$. Then, since
$\partial_{t}{\tilde{h}}_{\mu\nu}=-i\omega{\tilde{h}}_{\mu\nu}$ on the
linearized perturbation, Eq. (48) yields

$\displaystyle\lim_{\chi\to{\pi}/{2}}\delta{T_{\chi}}^{t}=0.$ (49)

This is nothing but the no energy flux condition, as observed in Eq. (40).

##  V The perturbative approach in $(5+1)$-dimensional black droplet solution

In this section, we will construct the asymptotic geometry of the
$6$-dimensional black droplet solution under the boundary condition (45) in
order to justify our analysis based on the holographic method in the last
section. We will also examine the energy density of the boundary field theory.
When the boundary black hole mass parameter $M$ in Eq. (24) is small compared
with the AdS scale, i.e., $M\ll L^{2}$, the asymptotic metric can be
constructed from the static perturbation of the pure AdS spacetime (25). As
shown below, a negative energy appears near the asymptotic region,
inconsistent with the $D=3$ result (III.1). We also show that the holographic
energy (8) is finite. This is because there is no trace anomaly in odd
dimensions, being different from the even dimensional case [21].

We require that the conformal boundary metric of the perturbed bulk metric
coincides with the metric (24) asymptotically in the sense that

$\displaystyle
ds_{5}^{2}=-f(r)dt^{2}+\frac{1}{f(r)}{(1+O(r^{-6}))}dr^{2}+r^{2}\left(1+O(r^{-6})\right)d{\Omega_{3}}^{2}\,,$
(50)

where $f(r)=1-{M}/{r^{2}}+{r^{2}}/{L^{2}}$ as previously introduced by (24).

In the static perturbation with $\omega=0$,
$Z_{t\rho}=H_{t\rho}=H_{t}=X_{t}=0$. Adopting the gauge choice

$\displaystyle H_{\rho}=H_{\rho\rho}=0,$ (51)

$X_{\rho}$, $H_{T}$ are determined by the gauge-invariant variables
$Z_{\rho\rho}$, $Z$ by Eqs. (IV.1). One can always set

$\displaystyle\lim_{\rho\to\infty}\frac{H_{tt}}{\rho^{2}}=0$ (52)

by choosing the integral constant in the first line in Eqs. (IV.1)
appropriately. To construct the boundary metric (50) by perturbing the bulk
metric (25), we find that the angle coordinate $\chi$ in Eq. (25) and the
radial coordinate $r$ of the boundary metric (50) should be related as

$\displaystyle d\chi=\frac{dr}{Lf(r)}\,.$ (53)

This is integrated and approximately expressed at large $r$ as

$\displaystyle\tan\chi=\frac{r}{L}\left(1-\frac{ML^{2}}{5r^{4}}+\frac{3ML^{4}}{35r^{6}}+\cdots\right).$
(54)

Then, $h_{rr}=\delta g_{rr}$ at $O(\rho^{2})$ can be expanded as

$\displaystyle\lim_{\rho\to\infty}\frac{h_{rr}}{\rho^{2}}=\frac{1}{L^{2}f(r)^{2}}\sum_{k=0}\frac{\alpha_{k}}{r^{2k}}.$
(55)

Therefore the leading order of the bulk perturbed metric with $O(\rho^{2})$
becomes

$\displaystyle
ds_{6}^{2}\simeq\frac{d\rho^{2}}{F_{0}(\rho)}+\frac{\rho^{2}}{L^{2}f(r)}\Biggl{[}\left(1+\sum_{k=0}\frac{\alpha_{k}}{r^{2k}}\right)\frac{dr^{2}}{f(r)}+f(r)\left\\{-dt^{2}+L^{2}(\sin^{2}\chi+h_{L})d\Omega_{3}^{2}\right\\}\Biggr{]}+O(\rho^{0}),$
(56)

where $h_{L}$ is the function of $\chi$, arising from the perturbation. If
both the conditions

$\displaystyle\alpha_{0}=\alpha_{1}=\alpha_{2}=0,$ (57)

and

$\displaystyle f(r)L^{2}(\sin^{2}\chi+h_{L})=r^{2}+O(r^{-4}),$ (58)

are satisfied, our bulk metric is conformal to the $5$-dimensional
Schwarzschild-AdS metric (50), up to $O(r^{-4})$. As shown below, both the
conditions (57) and (58) are satisfied for a suitable superposition of the
first three even mode harmonic functions, ${\mathbb{S}}_{(l)}~{}(l=2,4,6)$ in
Eq. (27), which satisfy the boundary condition (45). From now on we show the
indices $(l)$ labeling the mode.

The analytic solution of Eq. (30) that satisfies the regularity at $\rho=0$ is
given by the hypergeometric function $F$ with the argument
$F_{0}(\rho)=1+\rho^{2}/L^{2}$ (see [24]):

$\displaystyle\Phi_{(l)}$ $\displaystyle=$ $\displaystyle
a_{0(l)}\left[\frac{F_{0}(\rho)-1}{F_{0}(\rho)}\right]^{1+{l}/{2}}\cdot\Biggl{\\{}F\left(\frac{l+2}{2},\frac{l+2}{2},\frac{1}{2},\frac{1}{F_{0}(\rho)}\right)$
(59)
$\displaystyle\qquad\qquad\qquad\qquad\qquad-2\left[\frac{\Gamma\left(\frac{l+3}{2}\right)}{\Gamma\left(\frac{l+2}{2}\right)}\right]^{2}\frac{1}{\sqrt{F_{0}(\rho)}}\cdot
F\left(\frac{l+3}{2},\frac{l+3}{2},\frac{3}{2},\frac{1}{F_{0}(\rho)}\right)\Biggr{\\}}.$

This equation enables us to describe $a_{1(l)}$ by $a_{0(l)}$ as follows,

$\displaystyle
a_{1(l)}=-2L\left[\frac{\Gamma\left(\frac{l+3}{2}\right)}{\Gamma\left(\frac{l+2}{2}\right)}\right]^{2}a_{0(l)}.$
(60)

The first three even mode harmonic functions ${\mathbb{S}}_{(l)}$ are given by

$\displaystyle{\mathbb{S}}_{(2)}(\chi)=\frac{3}{4}(5\cos^{2}\chi-1),$
$\displaystyle{\mathbb{S}}_{(4)}(\chi)=\frac{5}{8}(1-14\cos^{2}\chi+21\cos^{4}\chi),$
$\displaystyle{\mathbb{S}}_{(6)}(\chi)=\frac{7}{64}(-5+135\cos^{2}\chi-495\cos^{4}\chi+429\cos^{6}\chi).$
(61)

By Eqs. (IV.1), the leading term of $H_{(l)T}$ becomes

$\displaystyle H_{(l)T}=c_{(l)}\rho^{2}+O(1),$ (62)

where $c_{(l)}$ is the integral constant for $H_{(l)T}$ in Eqs. (IV.1). Then,
from (IV.1), we find that $\delta g_{\chi\chi}$ is asymptotically,
$\rho\rightarrow\infty$, described by

$\displaystyle h_{\chi\chi}$
$\displaystyle=\sum_{l=2,4,6}\Biggl{[}\left(H_{(l)L}+\frac{l(l+3)}{4}H_{(l)T}\right){\mathbb{S}}_{(l)}(\chi)+H_{(l)T}\partial_{\chi}^{2}{\mathbb{S}}_{(l)}(\chi)\Biggr{]}$
$\displaystyle=\sum_{l=2,4,6}\left(\frac{3a_{0(l)}}{L^{2}}{\mathbb{S}}_{(l)}+c_{(l)}\partial_{\chi}^{2}{\mathbb{S}}_{(l)}\right)\rho^{2}+O(1),$
(63)

where Eq. (27) is used in the first line. As for the second line, we first
write $Z_{ab},Z$ in Eqs. (IV.1) and $X_{\rho}$ in Eq. (IV.1) using Eq. (59)
and obtain the expression of $H_{L},H_{T}$, which we then insert in the first
line. Note that when evaluating $X_{\rho}$, one has to integrate the r.h.s. of
the first line of Eq. (IV.1) and choose the integration constant so that the
condition (52) is satisfied, as mentioned before. Then, one finds in the
leading order,

$X_{(l)\rho}\sim-\frac{a_{0(l)}}{\rho}+O(1/\rho^{3}).$ (64)

Together with

$\displaystyle Z_{(l)}\sim 4a_{0(l)}\frac{\rho^{2}}{L^{2}}+O(1)\,,\quad$ (65)

one obtains

$H_{L}+\frac{l(l+3)}{4}H_{T}=\frac{Z}{4}-2\rho(D^{a}\rho)X_{a}\sim
3a_{0(l)}\frac{\rho^{2}}{L^{2}}+O(1)\,,$ (66)

which provides the expression of (V). This $h_{\chi\chi}$ can be expanded as a
series in $1/r$ by using Eq. (54). The result is

$\displaystyle\lim_{\rho\to\infty}\displaystyle\frac{h_{\chi\chi}}{\rho^{2}}$
$\displaystyle=$
$\displaystyle\frac{5}{32}(48c_{(2)}-112c_{(4)}+189c_{(6)})-\displaystyle\frac{3}{64L^{2}}(48a_{0(2)}-40a_{0(4)}+35a_{0(6)})$
(67) $\displaystyle+$
$\displaystyle\displaystyle\frac{5}{64r^{2}}\Bigl{\\{}144a_{0(2)}-336a_{0(4)}+567a_{0(6)}-16L^{2}(12c_{(2)}-154c_{(4)}+567c_{(6)})\Bigr{\\}}$
$\displaystyle+$
$\displaystyle\displaystyle\frac{5}{32r^{4}}\Bigl{\\{}-72a_{0(2)}+420a_{0(4)}-1323a_{0(6)}+L^{2}(96c_{(2)}-2576c_{(4)}+19089c_{(6)})\Bigr{\\}}+O\left(\frac{1}{r^{6}}\right).$

From this expansion, the coefficients $\alpha_{k}$ in Eq. (55) can be
determined. Then, under the condition (57), each coefficient $c_{(l)}$ can be
obtained as

$\displaystyle c_{(2)}=\frac{864a_{0(2)}+240a_{0(4)}+245a_{0(6)}}{5280L^{2}},$
$\displaystyle
c_{(4)}=\frac{-144a_{0(2)}+240a_{0(4)}+115a_{0(6)}}{2080L^{2}},$
$\displaystyle
c_{(6)}=-\frac{576a_{0(2)}+600a_{0(4)}-6895a_{0(6)}}{90090L^{2}}.$ (68)

Similarly, $h_{L}$ in Eq. (56) is expanded in terms of the harmonic functions
$S_{(l)}$ as

$\displaystyle
h_{L}=\sum_{l=2,4,6}\left(\frac{3a_{0(l)}}{L^{2}}\sin^{2}\chi{\mathbb{S}}_{(l)}+c_{(l)}\sin\chi\cos\chi{\mathbb{S}}_{(l),\chi}\right).$
(69)

Substituting Eq. (54) into Eq. (58), we determine each coefficient
$a_{0(l)}~{}(l=2,4,6)$ in terms of the small parameter $M$ as

$\displaystyle a_{0(2)}=-\frac{4M}{297},\quad a_{0(4)}=-\frac{16M}{585},\quad
a_{0(6)}=-\frac{64M}{5005}.$ (70)

Therefore the asymptotic bulk geometry is completely determined by the
coefficients.

Due to the gauge choice (51), the coordinate transformation

$\displaystyle\rho=\frac{L}{2}\left(\frac{1}{z}-z\right)$ (71)

leads to the Fefferman-Graham coordinate (II) in the original conformally
static Einstein frame, $x^{\mu}=(t,\chi,\theta^{m})$, $m=1,2,3$. Then, under
the boundary condition (52), ${\tilde{h}}_{tt}$ in Eqs. (IV.1) can be expanded
as a series in $z$,

$\displaystyle{\tilde{h}}_{tt}=\sum_{l=2,4,6}\frac{4-l(l+3)}{L^{2}}\Biggl{\\{}\frac{3}{8}+\frac{3}{8}[1-l(l+3)]z^{2}+\frac{4l(l+3)}{5}\left[\frac{\Gamma\left(\frac{3+l}{2}\right)}{\Gamma\left(\frac{l+2}{2}\right)}\right]^{2}z^{3}\Biggr{\\}}a_{0(l)}{\mathbb{S}}_{(l)}+O(z^{4}),$
(72)

where we used Eq. (60). So, according to the AdS/CFT dictionary, the energy
density $T_{tt}$ in the static Einstein universe can be read off from the
$z^{3}$ coefficient as

$\displaystyle
T_{tt}\simeq-\frac{3285L^{2}M}{1024}+O\left(\chi-\frac{\pi}{2}\right)^{2}$
(73)

combined with the result (70). Note that the Schwarzschild-AdS metric on the
boundary (50) is obtained by a conformal transformation from the static
Einstein frame by

$\displaystyle g_{\mu\nu}\to e^{2\sigma}g_{\mu\nu},\quad
e^{\sigma}=\sqrt{f(r)}\,.$ (74)

Then, the stress-energy tensor is obtained from the transformation

$\displaystyle T_{\mu\nu}\to e^{-3\sigma}T_{\mu\nu}.$ (75)

The asymptotic energy density in the $5$-dimensional Schwarzschild-AdS metric
is obtained by

$\displaystyle T_{tt}\to{T_{tt}}f(r)^{-3/2}\sim r^{-3}\,.$ (76)

Thus, the holographic energy (8) becomes finite.

Since the negative energy density (73) in the asymptotic region is
proportional to the mass parameter $M$ in the Schwarzschild AdS black hole
(24), this reflects the curvature of the black hole. So, one may say that the
negative energy density is caused by the vacuum polarization effect, as
explained in [9]. This behavior is qualitatively the same as the case of
$5$-dimensional small black droplet numerical solutions [21] with an AdS black
hole on the boundary.

##  VI Summary and discussions

In this paper, we have derived the first law of the entanglement entropy for
two subsystems separated by an AdS black hole for odd-dimensional CFTs by
using the holographic method and applying the Noether charge formula. We have
seen that the nonvanishing contribution to our Noether charge generically
arises not only at the bifurcate horizon and the asymptotic infinity but also
at a spacelike hypersurface which terminates at the“equatorial plane” on the
boundary static Einstein universe (see the dotted red line in Fig. 1). This is
because the boundary AdS black hole covers only one-half of the global AdS
boundary.

For $(2+1)$-dimensional CFT, we have, as its gravity dual, the exact black
droplet solution [14], and therefore by perturbing it, we have holographically
shown that the first law is satisfied without imposing any additional
conditions. As for $(4+1)$-dimensional CFT, restricted to the $SO(4)$
symmetric perturbations, we have shown in Sec. IV that the first law is
satisfied when there is no energy flux across the timelike conformal boundary,
which corresponds to the Neumann boundary condition on the spacelike bulk
hypersurface at $\chi=\pi/2$. We give a brief discussion on the analysis of
generic perturbations in Appendix. We believe that an analysis for general
perturbations without $SO(4)$ symmetry would also show the first law, but in
the present paper, we have not been able to fully clarify this issue.

We would like to emphasize again that in our holographic setup, in order to
compute the Noether charge, we have taken $\Sigma$ as the lower half shaded
region in Fig. 1 and imposed, in particular, the reflection (Neumann) boundary
condition at $\chi=\pi/2$. The reflection boundary condition at $\chi=\pi/2$
is necessary to avoid additional contributions to the first law from the upper
side of Fig. 1. One may think of what happens if one takes $\Sigma$ as the
entire time-slice. For example, if $\Sigma$ is taken as the union of the
single black droplet in the lower half and the regular (no black droplet)
region in the upper half of Fig. 1, then the reflection condition at
$\chi=\pi/2$ is no longer satisifed, and hence there is no guarantee that the
first law can be derived. First of all, if $\Sigma$ is taken as the entire
time-slice, then the corresponding boundary CFT would live in a spatially
compact universe, rather than in an asymptotically AdS boundary spacetime, and
the notion of a boundary black hole itself is no longer clearly defined.

In Sec. V, under the Neumann boundary condition for no-energy flux, we have
constructed the asymptotic geometry of a small black droplet solution from
scalar-type of the linear metric perturbations of the pure AdS spacetime. For
this purpose, we have expanded the metric perturbations by scalar harmonics on
the sphere and also set the boundary black hole mass $M$ as the small
parameter. To satisfy the Neumann boundary condition, only even modes of the
scalar harmonics are permitted.

In Ref. [25], quantization of conformally coupled scalar fields in anti-de
Sitter spacetime was considered in various schemes. One is the “transparent”
boundary condition in which each positive frequency classical solution
propagates beyond the timelike boundary at spatial infinity, and hence the
Klein-Gordon inner product is defined on the whole Einstein static cylinder,
by adding another “virtual” AdS spacetime. In this case, observers living in
an asymptotically AdS spacetime would recognize that some information or
energy is lost through the timelike boundary. Another boundary condition is
the “reflective” boundary condition in which each positive frequency classical
solution is reflected at the timelike boundary at infinity, and therefore
information or energy loss does not occur on the boundary. In this paper, by
adapting the latter “reflective” boundary condition, we have shown the first
law of the entanglement entropy for $D=5$ CFT in asymptotically AdS spacetime.
Conversely, if the first law is satisfied in asymptotically AdS spacetime, the
“reflective” boundary condition is derived for the holographic stress-energy
tensor. On the other hand, in the $D=3$ case, the Noether charge at the
infinity is always zero, being irrespective of the boundary conditions in the
holographic model we consider. It would be interesting to explore whether the
first law of the entanglement entropy is satisfied for various quantizations
of a free scalar field in BTZ background.

## Acknowledgements

We would like to thank Takashi Okamura for useful discussions. This work was
supported in part by JSPS KAKENHI Grant Numbers 17K05451, 20K03975 (KM) and
15K05092, 20K03938 (AI).

##  VII Appendix

In this appendix we consider the Noether charge formula (4) for general
perturbations in $(5+1)$-dimensional pure AdS bulk background metric (25).
Note that although the perturbation analysis in Sec. IV.2 is restricted to
$SO(4)$ symmetric case, the background geometry treated in Sec. IV.2 is the
black droplet solution, which we assume to exist. In this sense the analysis
in Sec. IV.2 is more relevant to our purpose for obtaining the first law than
the analysis below.

As introduced in sec. IV.1, we use the coordinates $y^{a}:=(t,\rho)$ in the
$2$-dimensional AdS spacetime and the angular coordinates
$x^{i}=(\chi,\theta^{m})$. We denote our Killing vector field by
$\xi^{M}=(\xi^{a},\xi^{i})$. The integrand of the Noether charge formula (4)
is in the present case a $4$-form on the $4$-sphere
$(\Omega_{(4)},\gamma_{ij})$. For generic metric perturbations $\delta
g_{MN}=h_{MN}=(h_{ab},h_{ai},h_{ij})$, we have

$\displaystyle\delta{\bm{Q}}-\xi\cdot{\bm{\Theta}}(g,\,\delta g)$
$\displaystyle=$ $\displaystyle-\frac{1}{16\pi}\epsilon_{i_{1}\cdots
i_{4}}{}^{ab}\Bigg{[}\left(\frac{1}{2}hg^{ac}-h^{ac}\right)D_{c}\xi^{b}-h^{aj}D_{j}\xi^{b}+\frac{D^{b}\rho}{\rho}h^{a}_{j}\xi^{j}$
(77)
$\displaystyle\qquad\qquad\qquad+\frac{1}{2}(D^{a}h^{b}_{c}-D^{b}h^{a}_{c})\xi^{c}+\frac{\rho}{2}\left\\{D^{a}\left(\frac{h^{b}_{j}}{\rho}\right)-D^{b}\left(\frac{h^{a}_{j}}{\rho}\right)\right\\}\xi^{j}$
$\displaystyle\qquad\qquad\qquad-\xi^{a}\left\\{D_{c}h^{bc}+D_{j}h^{bj}+n\frac{D_{c}\rho}{\rho}h^{bc}-\frac{D^{b}\rho}{\rho}h^{k}_{k}-D^{b}h\right\\}\Bigg{]}.$

The tensor-type metric perturbations $h_{MN}$ are tangential to
$(\Omega_{(4)},\gamma_{ij})$ with no components along $y^{a}$ direction, and
behave as a transverse-traceless tensor on $(\Omega_{(4)},\gamma_{ij})$. It is
straightforward to see that the $4$-form (77) vanishes for tensor-type
perturbations.

Now let us consider the vector-type perturbations. We introduce the vector
harmonics on $\Omega_{(4)}$, which satisfy

$\displaystyle
D^{i}D_{i}{{\mathbb{V}}}_{(l_{v})j}+[l_{v}(l_{v}+3)-1]{\mathbb{V}}_{(l_{v})j}=0,\quad
D^{i}{\mathbb{V}}_{(l_{v})i}=0,\quad l_{v}=1,2,3,\cdots.$ (78)

The vector-type metric perturbations can be expanded in terms of
${\mathbb{V}}_{(l_{v})i}$ as

$\displaystyle\delta g_{ab}=0,\qquad\delta
g_{ai}=\sum_{l_{v}}H^{(1)}_{(l_{v})a}{\mathbb{V}}_{(l_{v})i},$
$\displaystyle\delta
g_{ij}=\sum_{l_{v}}H^{(1)}_{T(l_{v})}\left(D_{i}{{\mathbb{V}}}_{(l_{v})j}-D_{j}{{\mathbb{V}}}_{(l_{v})i}\right).$
(79)

Hereafter we omit the mode indices $(l_{v})$ for notational simplicity. Note
that since the gauge transformation of the vector-type perturbations is
generated by a vector field $X_{i}=X^{(1)}{\mathbb{V}}_{i}$ with a scalar
$X^{(1)}$ on the $2$-dimensional AdS spacetime, one can impose, for instance,
the gauge condition $H^{(1)}_{T}=0$.

The gauge invariant variables for the vector-type perturbations are given by

$\displaystyle
Z_{a}=H_{a}^{(1)}-\rho^{2}D_{a}\left(\frac{H^{(1)}_{T}}{\rho^{2}}\right).$
(80)

This variable can be expressed by a single master variable $\Phi_{V}$ on the
$2$-dimensional AdS spacetime spanned by $(t,\rho)$ as

$\displaystyle Z_{a}=\frac{1}{\rho^{2}}\epsilon_{ab}D^{b}(\rho^{2}\Phi_{V}),$
(81)

where $\epsilon_{ab}$ is the metric compatible volume element on the
$2$-dimensional AdS spacetime $g_{ab}$. The master variable $\Phi_{V}$
satisfies

$\displaystyle
D^{a}D_{a}\Phi_{V}-\left\\{\frac{2}{L^{2}}+\frac{2+l_{v}(l_{v}+3)}{\rho^{2}}\right\\}\Phi_{V}=0.$
(82)

The normalizable solution $\Phi_{V}$ which also satisfies the regularity
condition in the bulk is given by eq. (154) of [24] with
$\nu=3/2,\sigma=\sqrt{l_{v}(l_{v}+3)+9/4}$. (Note that for the normalizable
solution, the frequency $\omega$ is quantized to be
$\omega=\mp(2m+1+\nu+\sigma)$, $m=0,1,2,\dots$.) Then, from eq. (154) of [24],
one can see the asymptotic behavior of $\Phi_{V}$ as

$\displaystyle\Phi_{V}=\frac{b_{1}}{\rho^{2}}+\frac{b_{2}}{\rho^{4}}+\cdots.$
(83)

It then follows that at large $\rho$,

$Z_{t}\sim-\frac{2b_{2}/L^{2}}{\rho^{3}}+O(1/\rho^{5}),\qquad
Z_{\rho}\sim-i\omega
L^{2}\left(\frac{b_{1}}{\rho^{4}}+\frac{b_{2}}{\rho^{6}}+\cdots\right).$ (84)

For the vector-type perturbations, the Noether charge formula (77) is
expressed in terms of the gauge-invariant variable (81) as

$\displaystyle\delta{\bm{Q}}-\xi\cdot{\bm{\Theta}}(g,\,\delta
g)=-\frac{1}{16\pi}\epsilon_{i_{1}\cdots
i_{4}}{}^{ab}\left[-\frac{Z_{a}}{\rho^{2}}{\mathbb{V}}^{i}D_{i}\xi_{b}+D_{[a}Z_{b]}\xi^{i}{\mathbb{V}}_{i}\right].$
(85)

So far, we have not yet used the property that $\xi^{M}$ is the Killing
vector. In Sec. IV.1, we are concerned with the static black hole background
and our Killing vector is $\xi^{a}\partial_{a}=\partial/\partial t$,
$\xi^{i}=0$. Thus, in this case, one can immediately see that the above
formula (85) vanishes.

Now suppose that a rotating black droplet solution be available, and let us
consider $\xi^{i}\neq 0$. In that case, one should be able to choose $\xi^{i}$
as a linear combination of the rotational symmetry generators of
$\Omega_{(4)}$. It is known that for the dipole moment $l_{v}=1$,
${\mathbb{V}}^{i}$ itself is a Killing vector field on $\Omega_{(4)}$, and
therefore one can take $\xi^{i}$ one of such ${\mathbb{V}}^{i}$ so that
$\xi^{i}{\mathbb{V}}_{i}\neq 0$. Then, one should, in principle, be able to
find a Killing vector field $\xi^{M}\partial_{M}=\partial/\partial
t+\Omega_{(A)}\partial/\partial\psi_{(A)}$ which becomes tangent to the null
generators of the Killing horizon of the black droplet solution, where
$\Omega_{(A)}$ are certain constants corresponding to the angular velocity of
the black hole, and $\psi_{(A)}$ angular Killing parameters for $\xi^{i}$. For
such a Killing vector field $\xi^{M}$, the second-term of r.h.s. of the above
formula (85) should provide the term proportional to the variation of the
angular momenta. In fact, substituting (84) into (85) gives a nonvanishing
term proportional to $\Omega_{(A)}(\omega^{2}L^{2}b_{1}+6b_{2}/L^{2})$. For
the stationary case $\omega=0$, this can be viewed as adding an angular
momentum to the background black hole by the dipole perturbation.

The analysis of generic perturbations of the scalar-type appears to be much
more involved with more terms in the Noether charge formula (77). If we can
evaluate the integration of the above formula (77) for both the vector-type
and scalar-type perturbations, not only at
$\rho\rightarrow\infty,\,\chi\rightarrow\pi/2$ but also at the Killing horizon
of the (hypothetical) rotating black droplet as well as other relevant
boundaries, then we expect that the first law with work term should be
obtained. To fully verify this argument is, however, beyond the scope of the
present paper where we consider only linear perturbations and their asymptotic
behavior.

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