#  HTN-Based Tutors: A New Intelligent Tutoring Framework Based on
Hierarchical Task Networks

Momin N. Siddiqui<EMAIL_ADDRESS>0000-0003-1874-7789 Georgia
Institute of TechnologyNorth AvenueAtlantaGAUSA ,  Adit Gupta
<EMAIL_ADDRESS>0000-0002-1218-0630 Drexel University3230 Market
StreetPhiladelphiaPAUSA ,  Jennifer M. Reddig<EMAIL_ADDRESS>0009-0000-6731-9401 Georgia Institute of TechnologyNorth AvenueAtlantaGAUSA
and  Christopher J. MacLellan<EMAIL_ADDRESS>0000-0003-3084-5189
Georgia Institute of TechnologyNorth AvenueAtlantaGAUSA

(2024)

###### Abstract.

Intelligent tutors have shown success in delivering a personalized and
adaptive learning experience. However, there exist challenges regarding the
granularity of knowledge in existing frameworks and the resulting instructions
they can provide. To address these issues, we propose HTN-based tutors, a new
intelligent tutoring framework that represents expert models using
Hierarchical Task Networks (HTNs). Like other tutoring frameworks, it allows
flexible encoding of different problem-solving strategies while providing the
additional benefit of a hierarchical knowledge organization. We leverage the
latter to create tutors that can adapt the granularity of their scaffolding.
This organization also aligns well with the compositional nature of skills.

Human-centered computing, Intelligent tutoring systems, Artificial
Intelligence, Hierarchical Task Network, Scaffolding

††journalyear: 2024††copyright: rightsretained††conference: Proceedings of the
Eleventh ACM Conference on Learning @ Scale; July 18–20, 2024; Atlanta, GA,
USA††booktitle: Proceedings of the Eleventh ACM Conference on Learning @ Scale
(L@S ’24), July 18–20, 2024, Atlanta, GA, USA††doi:
10.1145/3657604.3664702††isbn: 979-8-4007-0633-2/24/07

##  1\. Introduction

Intelligent Tutoring Systems (ITSs) are computer programs that utilize AI
techniques to provide personalized and adaptive learning (Nwana, 1990).
Several randomized controlled trials have demonstrated their effectiveness at
improving student learning gains (Pane et al., 2014; Beal et al., 2007).
Frameworks for intelligent tutoring include constraint-based tutoring, which
uses constraint-based modeling (CBM) to specify domain principles that every
correct solution must follow (Ohlsson, 2016), and example-tracing tutoring,
which uses generalized traces of problem-solving behavior (Aleven et al.,
2016). One of the more well-known frameworks is rule-based tutoring, which
uses production rules to represent the cognitive model, where every rule is
tied to a student skill (Aleven, 2010). Some examples of rule-based tutors
adopted at scale are Cognitive Tutor/MATHia (Anderson et al., 1995; Ritter and
Fancsali, 2016), OATutor (Anastasopoulos et al., 2023), ASSISTments (Heffernan
and Heffernan, 2014) and Apprentice Tutor (AI-ALOE, 2024).

One line of research concerning intelligent tutors is the granularity of
instructions. Granularity, in the context of intelligent tutors, refers to the
amount of reasoning that is handled by the student internally on each step
(VanLehn, 2011). Low granularity means that the system provides detailed,
scaffolded steps to help guide the learner. In contrast, high granularity
means that the system provides much less support—often just a single input box
for the learner to enter the final answer. Research suggests that scaffolding
should ideally be faded, shifting from low granularity to high granularity and
reducing support as learners gain proficiency (McNeill et al., 2006;
Puntambekar and Hubscher, 2005). However, limited research has explored ITS
that dynamically changes the granularity of steps based on student skill
level, functionality that we refer to as adaptive scaffolding. In most
intelligent tutors, granularity is static. It is a feature of the user
interface rather than encoded in the cognitive model, preventing adaption
based on the learner’s skill level. This gap necessitates formalizing a new
expert model for intelligent tutors that can represent knowledge at different
levels of granularity and better support adaptive scaffolding.

Across many existing systems, each skill the user is expected to acquire is
represented independently and discretely (Vanlehn and Niu, 2001). In the case
of rule-based tutors, skills are represented as distinct production rules that
activate when the rule’s conditions are met. We argue that the representation
of skills needs to better capture their compositional and hierarchical nature
(Fischer, 1980).

To address the aforementioned challenges, we present a new framework using
Hierarchical Task Networks (HTNs) that we refer to as HTN-based tutoring. HTNs
are used in automated planning due to their human-like decomposition (Nau et
al., 2003). We leverage this property of HTNs to encode different problem-
solving strategies informed by the student’s knowledge level. Each action is
tied to meaningful abstractions representing knowledge components. HTN-based
tutors capture a key characteristic of skills, they are built upon one another
in compositional hierarchies.

Our contribution is twofold. First, we propose a cognitive modeling framework
that can deliver adaptive scaffolding, aligning with the learner’s progress to
reduce cognitive load for novices and challenge advanced learners. Second, by
modeling the compositional nature of skill development, HTN-based tutors offer
a more realistic and practical framework for learning complex subjects.

##  2\. Background

In intelligent tutors, granularity is typically a characteristic of the user
interface and not the cognitive model; granularity reflects the level of
reasoning participants must engage in between interaction points (VanLehn,
2011). A larger grain size implies more reasoning with each interaction.
Systems have been constructed with varying levels of granularity. Answer-based
systems have the largest possible granularity, presenting learners with a
problem and asking them to input the final answer without giving feedback to
learners on intermediate steps. This requires the learner to do extensive
mental reasoning for each response. In contrast, step-based and substep-based
systems offer more frequent, detailed guidance. That is, they break problems
down into multiple steps that the learner needs to perform, reducing the
amount of independent reasoning required before each response. Step-based
tutoring systems (VanLehn, 2006) aim to have users enter the steps they would
naturally write down when solving a problem while showing their work. In
contrast, substep-based tutors give scaffolding and feedback at a level of
detail that is even finer than the steps students would typically exhibit when
showing their work. Therefore, there is a missed opportunity for better
delivery of substep tutoring using instructions, especially when informed by
students’ skill level.

Figure 1. A representation of fraction addition problem-solving knowledge in
rule (left) and HTN formats (right), showing methods (ellipse) and operators
(rectangle) for head tasks, with lettered callouts indicating equivalent steps
in both frameworks.

Hierarchical task networks (HTN) are popular for providing a robust framework
for automated planning, an area of artificial intelligence research (Erol et
al., 1994). HTNs are used in robotics, video games, and military simulations,
among many other applications. Seminal systems such as SHOP and SHOP2 (Nau et
al., 2003) exemplify the effectiveness of HTNs in decomposing abstract tasks
into concrete, manageable tasks through the use of Methods and Operators.
Their hierarchical approach to task decomposition has a natural alignment with
human cognitive skill organization, making it an intuitive framework for
encoding task knowledge. This alignment facilitates a deeper understanding of
planning as a cognitive function. While its potential as a cognitive modeling
tool is evident (Hayes and Scassellati, 2016), research on its use in
intelligent tutors, and in the learning sciences more generally, is limited.
Our proposed system builds upon the domain-independent framework of HTNs to
create expert models for tutoring systems.

##  3\. HTN-Based Tutors

In this section, we will outline the components of our proposed HTN-based
tutoring framework. We will describe the state, the expert model, and how both
support model tracing and tutoring.

###  3.1. State

The system utilizes a short-term working memory that contains an up-to-date
description of the system’s state. The working memory represents the state
using facts, which are logical predicates that describe different elements in
the tutor and how they relate. An example fact describing the value stored in
the “addFraction” field might be:

        Fact(field=’addFraction’, value=’1/2+1/4’)

Facts encode the details of the problem, its expected scaffolding, and the
user’s expertise for each skill as calculated through knowledge tracing
(Corbett and Anderson, 1994). The knowledge of which step should be scaffolded
can be dynamically inferred based on a user’s skill levels using Axioms.
Axioms are described in greater detail in the following section.

###  3.2. Expert Model

The expert model stores problem-solving strategies, including tasks,
operators, methods, and axioms.

####  3.2.1. Tasks

A task represents an activity to perform.111Here we use the HTN definition of
a task (Nau et al., 2003), which is different from typical tutoring system
definition (VanLehn, 2006). Tasks can be performed using an Operator or a
Method. Each task has a definition associated with it, which we call the task
head. The task head is essentially the name or identifier of the task. The
task head corresponds to heads that appears in Operator or Method. In order to
start problem-solving, a task has to be instantiated.

####  3.2.2. Operator

Operators represent primitive behaviors. Each operator has a head,
preconditions, and effects. The preconditions are a partial description of a
state that describes when the operator can be applied and The effects describe
the actions to be take when applied. These usually result in some change to
the state (e.g., updating a tutor field with a particular value). Operators
are similar in many respects to the production rules in rule-based models. The
key difference is that operators are only considered in the context of the
target task that they are known to perform, as specified by their head,
whereas all productions are considered in every state.

####  3.2.3. Method

Methods represent non-primitive behaviors. Like operators, each method has a
head that defines the task it performs and preconditions that describe when it
is applicable. However, unlike operators, methods have subtask decompositions
instead of effects. There can exist multiple methods with the same head to
describe different ways of decomposing a given task under different
conditions. This allows the encoding of multiple strategies for performing a
task. The method is essentially a higher-level abstraction that break down a
task into simpler subtasks.

####  3.2.4. Axiom

Axioms are used to infer facts that are not explicitly asserted in the current
state. Each axiom has a head and preconditions. However, unlike methods and
operators, the head refers to a new fact that can be inferred when its
preconditions are satisfied.

###  3.3. Model Tracing with HTNs

Model tracing is a technique for inferring a student’s mental operations given
their observed behavior on a problem (Anderson et al., 1990), enabling real-
time and contextual feedback on the student’s inferred state.

To support model tracing with HTN-based cognitive models, all problems start
with a single higher-level task that is linked to the problem (e.g., reduce
the logarithmic expression). The HTN model tracing system searches for a way
to recursively decompose this task until it grounds out into operators that
describe actions to be taken in the tutor. For each task, the system retrieves
all the methods or operators that can achieve it and checks to see if their
preconditions are satisfied. If any of the methods and/or operators are
applicable, then it selects one (this choice is represented as an OR
decomposition in Figure 2). If a method is selected, then it decomposes the
current task into a conjunction of subtasks, each of which the system tries to
satisfy via recursive decomposition (the conjunction of subtasks to be
satisfied is represented as an AND decomposition in Figure 2). If an operator
is selected, then it waits for the student to take action corresponding to the
effects described by the operator. If the expected effects are not observed,
then the system backtracks to find an alternative decomposition that matches
the observed student action. When no decomposition matches, the system
identifies the student action as incorrect. However, if the student takes an
expected action, then model tracing continues and the system moves on to the
next subtask that needs to be decomposed. Once all the tasks have been
decomposed into observed student actions, then the system recognizes the
problem as solved.

In each state, the HTN-based system only considers methods that satisfy the
current task/subtask. As a result, it evaluates fewer Methods/Operators than
rule-based tutors, which evaluate all production rules in every state. Figure
1 compares a simplified rule-based expert model, comparable to what is used in
a Cognitive Tutor (Anderson et al., 1995) (on the left) with a simplified HTN-
based expert model (on the right).

##  4\. Example

Having described the framework of HTN-based tutors, it is worthwhile to
showcase an example in the context of a tutor. Figure 2 shows a tutor for
reducing logarithmic expression. The tutor (right) displays different
granularity levels based on the user’s estimated knowledge of each component
(e.g., as estimated via knowledge tracing (Corbett and Anderson, 1994)). A
student who is highly proficient (marked by the green progress bar) sees an
interface similar to an answer-based tutor. A student with intermediate skills
(yellow progress bar) sees an interface similar to step-based, while for a
novice, the interface resembles a sub-step-based tutor, outlining even the
steps that are more traditionally done mentally.

Figure 2 also shows the expected model tracing path corresponding to each user
(left). The numbering at the bottom of nodes represents the ordering sequence
of their respective AND branches. Each path is marked with colors matching
their progression. The green yields no scaffolding, while yellow and red have
provide progressively more scaffolding in each case. The plus button on the
right of the input field expands the scaffolding manually if the student would
like more support. Red implies scaffolding is available, while grey implies
scaffolding has already been expanded for that step.

##  5\. Discussion

Figure 2. An HTN-based tutor implementation for solving logarithmic
expressions is shown. On the left, various paths for model tracing correspond
to different skill levels: green for high-skill, yellow for moderate, and red
for low-skill students.

###  5.1. Adaptive Scaffolding

In intelligent tutors, the focus is typically on providing contextualized
hints and personalized practice sequences rather than adaptive scaffolding
(Anderson et al., 1995; Anastasopoulos et al., 2023). These systems utilize
two primary mechanisms: the outer loop and the inner loop (VanLehn, 2006). The
outer loop selects problems that learners have not yet mastered (Ritter et
al., 2016), focusing their practice where it is needed rather than on skills
they already know. The inner loop offers hints and immediate feedback directly
through the tutor interface, tailoring to the specific context. Sometimes
students with low skills do not have the metacognition to implement the hints
and explanations they are given (Cusi et al., 2020). They require explicit
problem-solving instruction and modeling of their problem-solving strategies
to make progress. Our cognitive models enable alternative methods to
administer adaptive scaffolding.

####  5.1.1. Granularity

Granularity is explicitly represented in HTN’s task hierarchy, where higher-
level tasks are less granular and lower-level tasks represent more
granularity. This allows multiple subtask lists to be authored, catering to
learners of varying skill levels, as seen in figure 2. This approach not only
addresses cognitive load but also enhances learning by dynamically adjusting
the scaffolding based on the HTN model to meet the learner’s evolving needs.
Adaptive granularity enables the tutor to reduce the degrees of freedom (Wood
et al., 1976) for problem-solving while retaining higher-level task context by
means of tailored instructions. Effective scaffolding should gradually fade as
learners gain proficiency (McNeill et al., 2006), allowing them to become more
independent in their problem-solving. Our system progressively reduces the
level of scaffolding provided. As learners advance, the tutor transitions to
an answer-based format, fostering greater learner autonomy and reinforcing
mastery. By giving students practice in a format that more closely resembles
the format of problems on exams and tests, we hypothesize our tutors will
yield better performance gains under testing conditions than typically seen
with ITS (Kulik and Fletcher, 2016).

####  5.1.2. Strategy Recognition

It is valuable for students to receive feedback on the strategies they use, as
this enables them to meta-cognitively assess their problem-solving. Due to
strategy being encoded directly in HTN methods, our model tracing system can
identify the approach taken by students and provide feedback on their
strategic choices. This should help improve student’s current understanding
and enhance their outcomes (Wood et al., 1976).

###  5.2. Representation Power

Skills are compositional in nature (Fischer, 1980), serving as building blocks
for higher-level skills. In contrast to production rule engines, which process
rules individually without a hierarchical knowledge structure, our HTN-based
approach organizes knowledge hierarchically, reflecting the natural
organization of skills. The flexible nature of the HTN-based tutor allows for
the reusability of methods and operators while not compromising the hierarchy
of skills. Moreover, the tasks in the HTN do not have to be strictly
sequential and can have interleaved dependency. This makes it easier to encode
complex problems where tasks do not follow a linear sequence of steps (e.g.,
where the subtasks can be performed in any order).

##  6\. Limitation & Future Work

While HTN-based tutors show promise, they have yet to be tested and integrated
within a large-scale deployment. Integrating our novel cognitive modeling
approach within a platform to support standard inner and outer loop features
would allow for a more holistic evaluation. We plan to investigate the
following questions:

  1. (1)

Does adaptive scaffolding, through adjusting granularity, improve student
learning?

  2. (2)

What is the optimal strategy for adaptive granularity?

To address the first question, we will compare the performance of intelligent
tutors with static granularity to HTN-based tutors with adaptive granularity
through controlled experiments. Students will be randomly assigned to one of
three groups: an HTN-based tutor with static scaffolding, one with adaptive
scaffolding, or a control group without tutoring. We will evaluate
effectiveness using learning gains measured by pre- and post-tests. For the
second question, we will use a similar design with two treatment groups: one
where granularity follows a U-shaped curve (starting large, decreasing, then
increasing) and another where granularity follows a sigmoid curve (starting
small and increasing). Outcomes will be assessed using pre- and post-tests.

##  7\. Conclusion

To address the limitations of existing tutoring systems, we present an
alternative framework for tutor cognitive models that uses HTNs. HTN-based
tutors provide better personalization by enabling adaptive scaffolding
functionalities that have thus far been under explored. The hierarchical
nature of the system enables reusability of skills across tutors while
maintaining their hierarchical association. It allows the composition of
skills to formulate new higher-level skills, enabling better knowledge
compilation (Fischer, 1980). The human-like encoding of skills opens the
opportunity to incorporate several features that will help us deliver a better
experience for learners.

##  8\. Acknowledgement

This project is supported by National Science Foundation under Grant No.
2247790 and Grant No. 2112532. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the author(s) and do
not necessarily reflect the views of the National Science Foundation.

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# Asymptotic flatness at spatial infinity in higher dimensions

Kentaro Tanabe  Yukawa Institute for Theoretical Physics, Kyoto University,
Kyoto 606-8502, Japan  Norihiro Tanahashi  Tetsuya Shiromizu  Department of
Physics, Kyoto University, Kyoto 606-8502, Japan

###### Abstract

A definition of asymptotic flatness at spatial infinity in $d$ dimensions
($d\geq 4$) is given using the conformal completion approach. Then we discuss
asymptotic symmetry and conserved quantities. As in four dimensions, in $d$
dimensions we should impose a condition at spatial infinity that the
“magnetic” part of the $d$-dimensional Weyl tensor vanishes at faster rate
than the “electric” part does, in order to realize the Poincare symmetry as
asymptotic symmetry and construct the conserved angular momentum. However, we
found that an additional condition should be imposed in $d>4$ dimensions.

###### pacs:

04.20.Ha

##  I Introduction

If one considers an “isolated” system in general relativity, one should impose
some asymptotic boundary conditions on gravitational fields. As one of such
conditions, there is the asymptotically flat condition, which states that the
metric should approach to Minkowski metric at “far away” place from
gravitational sources. In order to define the notion of this “far away”
covariantly, one often uses the conformal completion method introduced by
Penrose Penrose . In this method, physical space-time $M$ is conformally
embedded to unphysical space-time $\hat{M}$ with boundary, and this boundary
is constituted of spatial infinity and null infinity. Hence, one can define
asymptotic flatness, imposing some proper boundary conditions at this spatial
infinity or null infinity.

In four dimensions, asymptotic flatness at spatial infinity was investigated
using the conformal completion method by Ashtekar and Hansen AH . They
revealed that asymptotic symmetry at spatial infinity can be reduced to the
Poincare symmetry which is a symmetry associated with “background” flat
metric, and constructed $4$-momentum and angular momentum. On the other hand,
in higher dimensions, there is only a few works about asymptotic structure at
spatial infinity Shiromizu:2004jt  or null infinity Hollands:2003ie  though
recently the importance of higher dimensional black holes is increasing in
string theory and TeV gravity scenario HDBH ; Review .

While in four dimensions, uniqueness theorem was obtained Israel , we cannot
prove the uniqueness for stationary black holes (counterexamples are Myers-
Perry black hole MP  and black ring BR  with the same mass and angular
momentum) in higher dimensions (although uniqueness was shown in GIS  for
static black holes). If one would like to classify these higher dimensional
black holes using some parameters, the investigation on asymptotic structure
at spatial infinity could play a key role.

The purpose of this paper is to define asymptotic flatness and investigate
asymptotic structure at spatial infinity in higher dimensions, following
Ashtekar and Hansen AH . (The reference Shiromizu:2004jt  investigates into
asymptotic flatness in higher dimensions following Ashtekar and Romano
Ashtekar:1991vb . This analysis is useful when one is interested only in
spatial infinity. For full understanding of asymptotic structures, however,
Ashtekar and Hansen’s work is appropriate.)

The rest of this paper is organized as follows. In the section II, we define
asymptotic flatness at spatial infinity following Ashtekar and Hansen AH . In
the section III, we investigate asymptotic structure: asymptotic symmetry and
conserved quantities. Finally, we give a summary and discussion in the section
IV. In the appendix A we introduce some important concepts in this literature
such as directional dependence, and in the appendix B we summarize basic
features of conformal completion taking Minkowski space-time for an example.
Some important equations in this literature are derived in the appendix C, and
in the appendix D we prove the equivalence of our expressions for conserved
quantities with the ADM formulae.

##  II Definition

We define asymptotic flatness at spatial infinity ($i^{0}$) in $d$ dimensions
using the conformal completion method developed by Ashtekar and Hansen in four
dimensions AH . In this paper, for simplicity we assume physical space-time
$(M,g_{ab})$ satisfies the vacuum Einstein equation $R_{ab}=0$. It is easy to
extend our current work to more general non-vacuum cases as long as one
focuses on the asymptotically flat space-time.

Definition: $d$-dimensional physical space-time $(M,g_{ab})$ will be said to
be asymptotically flat at spatial infinity $i^{0}$ if there exists
$(\hat{M},\hat{g}_{ab})$, where $\hat{g}_{ab}$ is $C^{>d-4}$ at $i^{0}$ (see
Appendix A for the definition of $C^{>n}$), and embedding of $M$ into
$\hat{M}$ satisfying the following conditions:

  1. 1.

$\bar{J}(i^{0})=\hat{M}-M$, where $\bar{J}(i^{0})$ is the closure of the union
of chronological future and past of $i^{0}$.

  2. 2.

There exists a function $\Omega$ on $\hat{M}$ that is $C^{2}$ at $i^{0}$ such
that $\hat{g}_{ab}=\Omega^{2}g_{ab}$ on $M$ and
$\hat{\nabla}_{a}\hat{\nabla}_{b}\Omega\hat{=}2\hat{g}_{ab}$, $\Omega\hat{=}0$
and $\hat{\nabla}_{a}\Omega\hat{=}0$ at $i^{0}$ on $\hat{M}$.

Here, and $\hat{\nabla}_{a}$ is the connection for $\hat{g}_{ab}$, and
$\hat{=}$ implies the evaluation on $i^{0}$ (i.e. “$=\lim_{\rightarrow
i^{0}}$” is equivalent to “$\hat{=}$”). The first condition requires that, in
$\hat{M}$, $i^{0}$ is connected to the points on $M$ only via spacelike
curves. The second condition says that $\Omega$ behaves $\sim 1/r^{2}$ near
$i^{0}$. This is the same asymptotic behavior as in the Minkowski space-time
(see Appendix B).

Since we assume $\hat{g}_{ab}$ is $C^{>d-4}$ at $i^{0}$,
$\hat{\partial}_{a_{1}}\cdots\hat{\partial}_{a_{(d-3)}}\hat{g}_{bc}$ has
directional dependent limit at $i^{0}$ (where $\hat{\partial}$ is flat
connection on $i^{0}$). This condition is equivalent to one such that
$\Omega^{(5-d)/2}\hat{R}_{abcd}$ has directional dependent limit at $i^{0}$.
When we discuss asymptotic structure, we often use the Weyl tensor
$\hat{C}_{abcd}$ as asymptotic gravitational fields. Thus, it is convenient to
use the latter condition on $\hat{R}_{abcd}$ for the discussions hereafter.

##  III asymptotic structure

In this section, we show how to derive the asymptotic structure from the
asymptotic flatness definition. Firstly, we discuss asymptotic symmetry in the
section III.1. We show that the asymptotic symmetry is constituted of the
Lorentz group and supertranslation group (infinite group of angular-dependent
translation) in higher dimensions. In the section III.2, we define asymptotic
fields and study their transformation behavior under supertranslation. We find
that supertranslation group reduces to the Poincare group if we impose an
additional asymptotic condition $B_{a_{1}a_{2}\cdots a_{d-2}}\hat{=}0$ in the
definition of asymptotic flatness. We define conserved quantities
($d$-momentum and angular momentum) associated to this Poincare symmetry in
the section III.3. We confirm that the conserved quantities we define agree
with the ADM formulae in this section and the appendix D.

###  III.1 Asymptotic symmetry

The asymptotic symmetry is a group of mappings which conserve asymptotic
structure. Here, by asymptotic structure we mean
$(\hat{g}_{ab},\Omega^{(4-d)/2}\hat{\partial}\hat{g}_{bc})$ at $i^{0}$, since
we impose $C^{>d-4}$ condition on the behavior of $\hat{g}_{ab}$ at $i^{0}$.
In order to investigate this asymptotic symmetry, we consider the generator
$\hat{\xi}$ of the asymptotic symmetry on $\hat{M}$. This generator
$\hat{\xi}$ should be an extension of $\xi$, which is a generator of
diffeomorphism on $M$. This extension $\hat{\xi}$ of $\xi$ to $i^{0}$ should
satisfy

  1. 1.

$\hat{\xi}\hat{=}0$ ,

  2. 2.

$\hat{\nabla}_{(a}\hat{\xi}_{b)}\hat{=}0$ ,

  3. 3.

$\hat{\nabla}_{(a}\hat{\xi}_{b)}$ is a $C^{>d-4}$ tensor at $i^{0}$.

Roughly speaking, these conditions set the behavior of components of
$\hat{\xi}$ near $i^{0}$ as

$\hat{\xi}^{a}\,\sim\,\frac{1}{r}+\frac{1}{r^{d-2}}.$ (1)

The first condition says that a generator $\hat{\xi}$ does not touch $i^{0}$.
The second condition implies that $\hat{\xi}$ is asymptotically a Killing
vector, i.e. $\hat{g}_{ab}$ at $i^{0}$ is not changed. Before explaining the
meaning of the third condition, let us consider the gauge freedom of the
conformal completion. First, let $\omega$ be a function on $\hat{M}$,
$C^{>d-4}$ at $i^{0}$ and $\omega\hat{=}1$. Then, a conformal completion such
that $\hat{g}^{\prime}_{ab}=(\omega\Omega)^{2}g_{ab}$ is equivalent to
$\hat{g}_{ab}=\Omega^{2}g_{ab}$, because $\omega\Omega$ satisfies

$\omega\Omega\hat{=}0\,,\,\hat{\nabla}_{a}(\omega\Omega)\hat{=}0\,,\,\hat{\nabla}_{a}\hat{\nabla}_{b}(\omega\Omega)\hat{=}2\hat{g}_{ab}.$
(2)

Then, we cannot distinguish these two conformal completions under the
asymptotic flatness definition in section II. This gauge freedom $\omega$ of
the conformal completion reshuffles the value
$\Omega^{(4-d)/2}\hat{\partial}\hat{g}_{bc}$ in the asymptotic structure as

$\displaystyle\Omega^{(4-d)/2}\left(\hat{\nabla}_{a}^{\prime}-\hat{\nabla}_{a}\right)\hat{v}_{b}\quad$
$\displaystyle\;\hat{=}\;\frac{1}{\omega}\Bigl{[}\;\;\delta^{c}_{a}\Omega^{(4-d)/2}\hat{\nabla}_{b}\omega$
$\displaystyle+\delta^{c}_{b}\Omega^{(4-d)/2}\hat{\nabla}_{a}\omega$
$\displaystyle-
g_{ab}\Omega^{(4-d)/2}\hat{\nabla}^{c}\omega\;\;\Bigr{]}\hat{v}_{c}\;,$ (3)

where $\hat{\nabla}_{a}^{\prime}$ is the connection for
$\hat{g}_{ab}^{\prime}$ and $\hat{v}_{a}$ is any vector. This equation can
also be written as

$\Omega^{(4-d)/2}\hat{\nabla}_{a}\hat{\nabla}_{(b}\hat{\xi}_{c)}\;\hat{=}\;2\Omega^{(4-d)/2}(\hat{\nabla}_{a}\omega)\hat{g}_{bc}.$
(4)

Thus, asymptotic structure $\Omega^{(4-d)/2}\hat{\partial}\hat{g}_{bc}$ has an
ambiguity coming from gauge freedom $\omega$, and this ambiguity is reshuffled
by order $1/r^{d-2}$ part of $\hat{\xi}$. Hence, asymptotic symmetry is the
group of transformations which does not change the asymptotic structure except
for this gauge ambiguity. Then, we call this asymptotic symmetry
transformation, which is induced by order $1/r^{d-2}$ component of
$\hat{\xi}$, supertranslation group. As any two generators $\hat{\xi}^{1}$,
$\hat{\xi}^{2}$ of supertranslation group commute:

$\displaystyle\left[\hat{\xi}^{1},\hat{\xi}^{2}\right]^{a}$
$\displaystyle\sim\frac{1}{r^{d-2}}\frac{\partial}{\partial
U}r^{2-d}\sim\mathcal{O}\left(\frac{1}{r^{2d-5}}\right)\;,$ (5)

supertranslation group is abelian (where we use the fact that the contribution
to $\Omega^{(4-d)/2}\hat{\partial}\hat{g}_{bc}$ from $\mathcal{O}(1/r^{2d-5})$
part of $\hat{\xi}$ is only $\mathcal{O}(1/r^{d-3})$, which is regarded as
zero at $i^{0}$, and thus that part cannot transform the asymptotic
structure). Because of angular dependence of $\omega$, however,
supertranslation group has infinite translational directions. In this stage,
asymptotic symmetry is not expected to be the Poincare symmetry.

###  III.2 Asymptotic fields

In order to construct conserved quantities associated with the asymptotic
symmetry, we define asymptotic gravitational fields using the Weyl tensor
$\hat{C}_{ambn}$ as111 In the definition of the magnetic part of the Weyl
tensor (7), the power of $\Omega$ is determined by the following evaluation.
Since $a_{1},\cdots,a_{d-3}$ are indices for angular coordinates and $m$ is
for the radial coordinate in polar coordinates, one of $p$ and $q$ has to be
for the time coordinate $t$ and the other one has to be for an angular
coordinate $\varphi$. Each parts in the magnetic part behaves near $i^{0}$ as
$\hat{\epsilon}_{a_{1}\cdots
a_{d-3}mpq}=\mathcal{O}(\sqrt{-\hat{g}})=\mathcal{O}(r^{2-d})$,
$\hat{C}^{p}{}_{qbn}=\mathcal{O}(r^{5-d})$, $g_{tt}=\mathcal{O}(1)$,
$g_{\varphi\varphi}=\mathcal{O}(r^{-2})$ and $\hat{\eta}^{a}=\mathcal{O}(1)$.
Thus, $\hat{\epsilon}_{a_{1}\cdots
a_{d-3}mpq}\hat{C}^{pq}{}_{bn}\hat{\eta}^{m}\hat{\eta}^{n}=\mathcal{O}(r^{9-2d})\sim\Omega^{(2d-9)/2}$,
and we have to multiply an inverse of this factor to define a regular
quantity.
$\hat{E}_{ab}$ is a symmetric traceless tensor since the Weyl tensor is
traceless. $\hat{B}_{a_{1}\cdots a_{d-3}b}$ is also a traceless tensor;
$\hat{B}_{a_{1}\cdots a_{d-3}b}\hat{g}^{a_{i}a_{j}}=0$ due to antisymmetry of
$\hat{\epsilon}$ in Eq. (7); $\hat{B}_{a_{1}\cdots
a_{d-3}b}\hat{g}^{a_{i}b}=0$ since it contains $\hat{C}^{[pqb]n}=0$. This
$\hat{B}_{a_{1}\cdots a_{d-3}b}$ is antisymmetric on the first $d-3$ indices
$a_{i}$ ($i=1,\cdots,d-3$). There are no symmetry between the last index $b$
and the other indices $a_{i}$ in general, though in the four-dimensional case
the magnetic part $\hat{B}_{ab}$ is symmetric.

$\displaystyle\hat{E}_{ab}\;\hat{=}\;\Omega^{(5-d)/2}\hat{C}_{ambn}\hat{\eta}^{m}\hat{\eta}^{n},$
(6) $\displaystyle\hat{B}_{a_{1}\cdots
a_{d-3}b}\;\hat{=}\;\Omega^{(9-2d)/2}\hat{\epsilon}_{a_{1}\cdots
a_{d-3}mpq}\hat{C}^{pq}{}_{bn}\hat{\eta}^{m}\hat{\eta}^{n},$ (7)

where $\hat{\epsilon}_{a_{1}\cdots
a_{d-3}mpq}\equiv\sqrt{-\hat{g}}E_{a_{1}\cdots a_{d-3}mpq}$ is a totally
antisymmetric tensor in $\hat{M}$, and we take the convention that
$E_{012\cdots d-1}=1$. $\hat{\eta}_{a}\hat{=}\hat{\nabla}_{a}\Omega^{1/2}$ is
a normal vector to $\Omega=\text{constant}$ surface which becomes a unit
vector at $i^{0}$. We call these asymptotic fields (6) and (7) electric and
magnetic parts of the Weyl tensor respectively. As these fields do not have
components parallel to $\hat{\eta}^{a}$, we can regard them as fields on a
timelike hypersurface $\mathcal{S}$ normal to $\hat{\eta}^{a}$.

Firstly, let us derive asymptotic field equations. Using the Bianchi identity
in the physical vacuum space-time $\nabla_{[m}C_{ab]cd}=0$, we obtain the
following equation in terms of the unphysical space-time quantities:

$\hat{\nabla}_{[m}\hat{C}_{ab]cd}=\Omega^{-1}\left(\hat{g}_{c[m}\hat{C}_{ab]pd}\hat{\nabla}^{p}\Omega+\hat{g}_{d[m}\hat{C}_{ab]cp}\hat{\nabla}^{p}\Omega\right)\;.$
(8)

It is better to rewrite the left-hand side as

$\displaystyle\hat{\nabla}_{[m}\hat{C}_{ab]cd}=\Omega^{-1}\biggl{[}\Omega^{(d-3)/2}\hat{\nabla}_{[m}\\!\Bigl{(}\Omega^{(5-d)/2}\hat{C}_{ab]cd}\Bigr{)}\;\;$
$\displaystyle-\frac{5-d}{2}(\hat{\nabla}_{[m}\Omega)\hat{C}_{ab]cd}$
$\displaystyle\biggr{]},$ (9)

since $\Omega^{(5-d)/2}\hat{C}_{abcd}$ have directional dependent limit at
$i^{0}$. We project these equations into the timelike hypersurface
$\mathcal{S}$, and contract with $\hat{\eta}^{a}$. Then, we get the equations
for the electric part

$\hat{D}_{a}\hat{E}_{bc}-\hat{D}_{b}\hat{E}_{ac}\;\hat{=}\;(4-d)\hat{h}_{a}^{~{}p}\hat{h}_{b}^{~{}q}\hat{h}_{c}^{~{}r}\Omega^{(5-d)/2}\hat{C}_{pqrm}\hat{\eta}^{m}$
(10)

and for the magnetic part

$\displaystyle\hat{D}_{b}\hat{B}_{a_{1}\cdots a_{d-3}c}-\hat{D}_{c}$
$\displaystyle\hat{B}_{a_{1}\cdots a_{d-3}b}$ (11)
$\displaystyle\hat{=}-(d-3)$
$\displaystyle\Omega^{(9-2d)/2}\hat{\epsilon}_{a_{1}\cdots
a_{d-3}}{}^{fpq}\,{}^{(d-1)}\hat{C}_{bcpq}\hat{\eta}_{f}\,,$

where $\hat{h}_{ab}$ is the induced metric on $\mathcal{S}$, and

$\hat{D}_{a}\hat{v}_{b}\equiv\Omega^{1/2}\hat{h}_{a}^{~{}p}\hat{h}_{b}^{~{}q}\hat{\nabla}_{p}\hat{v}_{q}$
(12)

is a regular differentiation with respect to $\hat{h}_{ab}$ on $\mathcal{S}$.
${}^{(d-1)}\hat{C}_{abcd}$ is the $(d-1)$-dimensional Weyl tensor with respect
to $\hat{h}_{ab}$, and $\Omega^{(5-d)/2}\,{}^{(d-1)}\hat{C}^{a}{}_{bcd}$ have
a directional dependent limit at $i^{0}$. (For detailed derivations of Eqs.
(10) and (11), see Appendix C.1.)

Next, in order to see how these fields transform under the supertranslation,
we introduce potentials of the Weyl tensor. To do so, we will use the Bianchi
identity in the unphysical space-time

$\hat{\nabla}_{m}\hat{C}_{abc}{}^{m}+\frac{2(d-3)}{d-2}\hat{\nabla}_{[a}\hat{S}_{b]c}=0\;\;,$
(13)

where

$\hat{S}_{ab}\equiv\hat{R}_{ab}-\frac{\hat{R}}{2(d-1)}\hat{g}_{ab}\;\;.$ (14)

Since we assume $\hat{g}_{ab}$ to be $C^{>d-4}$,
$\Omega^{(5-d)/2}\hat{S}_{ab}$ admits directional dependent limit at $i^{0}$.
Then, we define potentials as

$\displaystyle\hat{E}\;$
$\displaystyle\hat{=}\;\Omega^{(5-d)/2}\hat{S}_{pq}\hat{\eta}^{p}\hat{\eta}^{q}\;\;\;,$
(15) $\displaystyle\hat{Q}_{a}\;$
$\displaystyle\hat{=}\;\Omega^{(5-d)/2}\hat{S}_{pq}\hat{h}_{a}^{~{}p}\hat{\eta}^{q}\;\;,$
(16) $\displaystyle\hat{U}_{ab}\;$
$\displaystyle\hat{=}\;\Omega^{(5-d)/2}\hat{S}_{pq}\hat{h}_{a}^{~{}p}\hat{h}_{b}^{~{}q}\;.$
(17)

Using Eqs. (8) and (13), we can write down the electric and the magnetic part
in terms of potentials as

$\hat{E}_{ab}\,\hat{=}\,\frac{-1}{2(d-2)}\left[\frac{1}{d-3}\hat{D}_{a}\hat{D}_{b}\hat{E}+\hat{E}\hat{h}_{ab}+(4-d)\hat{U}_{ab}\right],$
(18) $\displaystyle\\!\hat{B}_{a_{1}\cdots a_{d-3}b}\,\hat{=}\,\frac{-1}{d-2}$
$\displaystyle\hat{\epsilon}_{a_{1}\cdots
a_{d-3}mpq}\hat{\eta}^{m}\Omega^{(4-d)/2}$
$\displaystyle\qquad\times\hat{D}^{p}\left(\hat{U}^{q}_{~{}b}-\frac{1}{d-3}\hat{E}\hat{h}^{q}_{~{}b}\right)$
$\displaystyle\equiv\frac{-1}{d-2}$ $\displaystyle\hat{\epsilon}_{a_{1}\cdots
a_{d-3}mpq}\Omega^{(4-d)/2}\hat{\eta}^{m}\hat{D}^{p}\hat{\mathcal{K}}^{q}_{~{}b}\,\,,$
(19)

where we define a tensor $\hat{\mathcal{K}}_{ab}$ by Eq. (19). (Eqs. (18) and
(19) are derived in Appendix C.2.)

Now, we observe transformation behaviors of the asymptotic fields under the
supertranslation. In a supertranslational transformation
$\hat{g}_{ab}\rightarrow\hat{g}_{ab}^{\prime}=\omega^{2}\hat{g}_{ab}$, where
$\omega$ is a $C^{>d-4}$ function ($\omega\hat{=}1$), $\hat{S}_{ab}$
transforms as

$\displaystyle\hat{S}_{ab}^{\prime}=\hat{S}_{ab}$
$\displaystyle-(d-2)\omega^{-1}\hat{\nabla}_{a}\hat{\nabla}_{b}\omega$
$\displaystyle+2(d-2)\omega^{-2}(\hat{\nabla}_{a}\omega)(\hat{\nabla}_{b}\omega)$
$\displaystyle+\frac{2-d}{2}\omega^{-2}\hat{g}_{ab}(\hat{\nabla}_{m}\omega)(\hat{\nabla}^{m}\omega)\;.$
(20)

Since $\omega$ is $C^{>d-4}$ and $\omega\hat{=}1$, it can be written as

$\omega=1+\Omega^{(d-3)/2}\alpha\;,$ (21)

where $\alpha$ is a function which has directional dependent limit at $i^{0}$.
Then, the potentials $\hat{E}$ and $\hat{U}_{ab}$ transform under the
supertranslational transformation as

$\displaystyle\hat{E}^{\prime}\;\hat{=}\;\hat{E}-(d-2)(d-3)(d-4)\alpha\;,$
(22)
$\displaystyle\hat{U}_{ab}^{\prime}\;\hat{=}\;\hat{U}_{ab}-(d-2)\left(\hat{D}_{a}\hat{D}_{b}\alpha+(d-3)\alpha\hat{h}_{ab}\right)\;.$
(23)

To show these equations, we use a relation

$\displaystyle\Omega^{(4-d)/2}\hat{\eta}^{a}\hat{\nabla}_{a}\omega$
$\displaystyle\;\hat{=}\;\Omega^{1/2}\hat{\eta}^{a}\hat{\nabla}_{a}\alpha+(d-3)\alpha$
$\displaystyle\;\hat{=}\;(d-3)\alpha\;.$ (24)

The second equality in this relation holds since $\alpha$ has directional
dependent limit at $i^{0}$ and $\hat{\eta}^{a}\hat{\nabla}_{a}\alpha\hat{=}0$.
We note that only $\hat{\nabla}_{a}\hat{\nabla}_{b}\omega$ term of Eq. (20)
contributes to the variation of $\hat{E}$ and $\hat{U}_{ab}$.

It is easy to check that the electric part does not change in this
transformation. On the other hand, the potential of the magnetic part
$\hat{\mathcal{K}}_{ab}$ transforms as

$\hat{\mathcal{K}}_{ab}^{\prime}\;\hat{=}\;\hat{\mathcal{K}}_{ab}-(d-2)(\hat{D}_{a}\hat{D}_{b}\alpha+\alpha\hat{h}_{ab})\;.$
(25)

Hence, the magnetic part $\hat{B}_{a_{1}\cdots a_{d-3}b}$ does change under
the supertranslational transformation.

###  III.3 Conserved quantities and Poincare symmetry

Let us construct conserved quantities and the asymptotic symmetry in this
section. First, as in four dimensions, we impose an additional condition

$\hat{B}_{a_{1}\cdots a_{d-3}b}\;\hat{=}\;0\;.$ (26)

This condition implies that the Taub-NUT charge is zero. Although it is of
course possible to consider asymptotically locally Minkowski space-time with
$\hat{B}_{a_{1}\cdots a_{d-3}b}\neq 0$, we focus only on asymptotically
globally Minkowski space-time in this paper. In order to impose the condition
(26) consistently with Eq. (11), we must require a further additional
condition

$\Omega^{(5-d)/2}\,{}^{(d-1)}\hat{C}^{a}{}_{bcd}\;\hat{=}\;0$ (27)

as one of the conditions in the definition of asymptotic flatness. Note that
${}^{(d-1)}C_{abcd}$ vanishes automatically in four dimensions. By the way,
the condition (26) is not preserved under the supertranslation. To preserve
the condition (26), we realise that one must impose

$\hat{D}_{a}\hat{D}_{b}\alpha+\alpha\hat{h}_{ab}\;\hat{=}\;0\;.$ (28)

As in four dimensions, we can write down the solution to Eq. (28) as
$\alpha=\hat{\omega}_{a}\hat{\eta}^{a}$, where $\hat{\omega}_{a}$ is a fixed
vector at $i^{0}$. The number of independent solutions is the number of
dimensions. Thus, we can regard the transformation generated by $\alpha$
satisfying Eq. (28) as translation. Then, the asymptotic symmetry reduces to
the Poincare group which is constituted of the Lorentz group and the
translation group, and we can define conserved quantities associated with this
Poincare symmetry.

Now, it is ready to define conserved quantities. First, we define $d$-momentum
$P_{a}$ for translation $\hat{\omega}^{a}$ as

$P_{a}\omega^{a}\equiv\frac{-1}{8\pi
G_{d}(d-3)}\int_{S^{d-2}}\\!\\!\\!\\!\\!\\!\\!\hat{E}_{ab}\hat{\omega}^{a}\hat{\epsilon}^{b}{}_{e_{1}\cdots
e_{d-2}m}\hat{\eta}^{m}dS^{e_{1}\cdots e_{d-2}},$ (29)

where $dS^{e_{1}\cdots e_{d-2}}$ is the volume element on $(d-2)$-dimensional
unit sphere $S^{d-2}$ on $i^{0}$. From Eq. (10), we get
$\hat{D}_{a}\hat{E}^{ab}\hat{=}0$ since $\hat{E}_{ab}$ is traceless. Then, the
integral of Eq. (29) is independent of the choice of time slice at $i^{0}$,
and thus $P_{a}\omega^{a}$ is conserved. After tedious calculations, we can
show that Eq. (29) agrees with the ADM formula (see Appendix D.1 and D.2).

Next, in order to define angular momentum using the magnetic part of the Weyl
tensor, we consider the next-to-leading order part of $\hat{B}_{a_{1}\cdots
a_{d-3}b}\,$:

$\hat{\beta}_{a_{1}\cdots
a_{d-3}b}\;\hat{=}\;\Omega^{4-d}\hat{\epsilon}_{a_{1}\cdots
a_{d-3}mpq}\hat{C}^{pq}{}_{bn}\hat{\eta}^{m}\hat{\eta}^{n}.$ (30)

Since $\hat{\beta}_{a_{1}\cdots a_{d-3}b}$ satisfies
$\hat{D}_{b}\hat{\beta}^{ba_{2}\cdots a_{d-3}c}\hat{=}0$ due to Eq. (11) and
the traceless property of $\hat{B}_{a_{1}\cdots a_{d-3}b}$, we can define
conserved quantity $M_{ab}$ which is regarded as angular momentum:

$\displaystyle M_{ab}F^{ab}\equiv\frac{-1}{8\pi
G_{d}(d-2)!}\int_{S^{d-2}}\hat{\beta}_{a_{1}\cdots a_{d-3}b}$
$\displaystyle\xi^{a_{1}\cdots a_{d-3}}\;$ (31)
$\displaystyle\times\hat{\epsilon}^{b}_{~{}e_{1}\cdots e_{d-2}m}$
$\displaystyle\hat{\eta}^{m}dS^{e_{1}\cdots e_{d-2}},$

where

$\xi^{a_{1}\cdots a_{d-3}}\equiv\hat{\epsilon}^{a_{1}\cdots
a_{d-3}mpq}\hat{\eta}_{m}F_{pq}$ (32)

and $F_{ab}$ is any skew tensor in $\cal S$. The coefficients in (31) so that
angular momentum $M_{ab}$ transforms properly under translation
$\hat{\omega}_{a}$, including the coefficient:

$M_{ab}\rightarrow M_{ab}^{\prime}=M_{ab}+2P_{[a}\hat{\omega}_{b]}\;.$ (33)

See Appendix D.3 for details of the coefficient determination.

##  IV Summary and discussion

In this paper, we gave a definition of asymptotic flatness, and constructed
conserved quantities, $d$-momentum and angular momentum in $d$ dimensions. As
in four dimensions, by imposing an additional constraints on the behavior of
the “magnetic” part of the Weyl tensor, we can remove the supertranslational
ambiguity. Then, the asymptotic symmetry of the space-time reduces to the
Poincare symmetry, which is a symmetry of “background” flat metric, and we can
construct conserved quantities associated with this Poincare symmetry. It can
be shown that the expressions of these conserved quantities agree with the ADM
formulae.

In four dimensions, the additional constraint is only $\hat{B}_{ab}=0$ to
realize the Poincare symmetry as the asymptotic symmetry, and it is satisfied
if there is a Killing vector in $M$, such as timelike Killing vector
$(\partial/\partial t)$ or rotational Killing vector
$(\partial/\partial\varphi)$ AM2 . On the other hand, in higher dimensions,
due to the evolution equation (11) of $\hat{B}_{a_{1}\cdots a_{d-3}b}$, we
need to impose a further condition
$\Omega^{(5-d)/2}\,{}^{(d-1)}\hat{C}^{a}{}_{bcd}\hat{=}0$ to remove the
supertranslational ambiguity and realize the Poincare symmetry. As in four
dimensions, $\hat{B}_{a_{1}\cdots a_{d-3}b}=0$ would be satisfied in
stationary or axisymmetric space-time in higher dimensions. However, it might
be interesting to investigate asymptotic symmetry under more general
conditions which $\Omega^{(5-d)/2}\,{}^{(d-1)}\hat{C}^{a}{}_{bcd}\hat{=}0$
does not hold.

In this paper, we focused only on spatial infinity. However, it is interesting
to explore the full asymptotic structure including null infinity. As our
future work, we would investigate the relationship between the Bondi energy
formula at null infinity and Weyl tensor formula in this paper at spatial
infinity. We also would like to consider asymptotic structure and its symmetry
at null infinity, and investigate its connection to the supertranslation at
spatial infinity.

Another future issue is the preparation for the uniqueness theorem in
stationary black hole space-times. As mentioned in the introduction, at first
glance, the uniqueness theorem does not hold in higher dimensions, although
there are some partial achievements Hollands ; Morisawa ; Hollands2 ;
Morisawa2 . However, we would guess that the reason why we fail to prove it is
due to lack of asymptotic boundary conditions. If we can specify the boundary
condition appropriately, we will be able to prove the uniqueness theorem. The
mass, charge and angular momentum are not enough to specify the black hole
space-time uniquely. The additional information for the uniqueness may be
higher multipole moments. Therefore, the study on higher multipole moments in
stationary space-time will be useful.

###### Acknowledgements.

The work of TS was supported by Grant-in-Aid for Scientific Research from
Ministry of Education, Science, Sports and Culture of Japan (Nos. 19GS0219 and
20540258). NT was supported by JSPS Grant-in-Aid for Scientific Research No.
20$\cdot$56381\. This work was supported by the Grant-in-Aid for the Global
COE Program ”The Next Generation of Physics, Spun from Universality and
Emergence” from the Ministry of Education, Culture, Sports, Science and
Technology (MEXT) of Japan.

##  Appendix A directional dependence

In the conformal completion method, spatial infinity which has a non-zero size
in the physical space-time $M$ contracts to a point $i^{0}$ in the unphysical
space-time $\hat{M}$. Hence, the definition of differentiability and
continuity of physical fields (e.g. electromagnetic fields or gravitational
fields) on $i^{0}$ is more subtle. In this appendix, we give the notion of
directional dependent limit and $C^{>n}$ class.

First, the tensor $\hat{T}^{a\cdots b}_{c\cdots d}$ is said to have
directional dependent limit at $i^{0}$ if $\hat{T}^{a\cdots b}_{c\cdots d}$
satisfies the following conditions:

  1. 1.

$\displaystyle\lim_{\rightarrow i^{0}}\hat{T}^{a\cdots b}_{c\cdots
d}=\hat{T}^{a\cdots b}_{c\cdots d}(\hat{\eta})\;,$
where $\hat{\eta}$ is a vector on tangential space at $i^{0}$, which is
tangent to the curve arriving at $i^{0}$.

  2. 2.

The derivative coefficients at $i^{0}$ defined by

$\left(\Omega^{1/2}\hat{\nabla}_{e_{1}}\right)\cdots\left(\Omega^{1/2}\hat{\nabla}_{e_{n}}\right)\hat{T}^{a\cdots
b}_{c\cdots d}$

are regular.

The first condition says that, since $i^{0}$ has a non-zero size ($S^{d-2}$)
in $M$, $\hat{T}^{a\cdots b}_{c\cdots d}$ may have an angular dependence even
in the limit $r\rightarrow\infty$. The operator $\Omega^{1/2}\hat{\nabla}_{a}$
in the second condition gives regular derivative coefficients, since an
application of a derivative operator $\hat{\nabla}_{a}$ in $\hat{M}$
corresponds to a multiplication of $r$ near $i^{0}$ (see Appendix B). The
second condition says that these regular derivative coefficients should be
finite and regular.

Next, we define $C^{>n}$ class. A tensor $\hat{T}^{a\cdots b}_{c\cdots d}$ is
$C^{>n}$ at $i^{0}$ if the $n+1$ derivatives of $\hat{T}^{a\cdots b}_{c\cdots
d}$ have directional dependent limit at $i^{0}$. For example, when we set
$\hat{g}_{ab}$ to be $C^{>n}$ at $i^{0}$, the behavior of $\hat{g}_{ab}$ near
$i^{0}$ is

$\hat{g}_{ab}\sim\text{const.}+\frac{f(\theta,\varphi,\cdots)}{r^{n+1}}\;,$
(34)

where the dots stand for other angular coordinates.

##  Appendix B conformal completion for Minkowski space-time

In this appendix, we discuss conformal completion for Minkowski space-time.
This analysis tells us how we can define asymptotically flat space-time in
general. First, we introduce coordinates $(U,V)$ such that

$\displaystyle ds^{2}=$ $\displaystyle-dt^{2}+dr^{2}+r^{2}d\Omega_{d-2}^{2}$
$\displaystyle=$ $\displaystyle-dudv+\frac{(u-v)^{2}}{4}d\Omega_{d-2}^{2}$
$\displaystyle=$
$\displaystyle-\frac{dUdV}{\cos^{2}U\cos^{2}V}+\frac{\sin^{2}(U-V)}{4\cos^{2}U\cos^{2}V}d\Omega_{d-2}^{2}\;,$
(35)

where

$u=t-r=\tan U~{},~{}v=t+r=\tan V\;,$ (36)

and $d\Omega_{d-2}^{2}$ is a metric on unit $S^{d-2}$. Let us take
$\Omega\equiv\cos U\cos V$ as a conformal factor. In this case, we can see
that

$\hat{\nabla}_{a}\hat{\nabla}_{b}\Omega\;\hat{=}\;2\hat{g}_{ab}$ (37)

holds at $i^{0}$. The unit normal vector $\hat{\eta}_{a}$ to
$\Omega=\text{constant}$ surface becomes

$\hat{\eta}_{a}\hat{=}\hat{\nabla}_{a}\Omega^{1/2}$ (38)

on $i^{0}$.

It will be useful for discussions in the main text to look how the
differential operators behave:

$\hat{\nabla}_{U}\sim\frac{\partial}{\partial U}\sim
r^{2}\frac{\partial}{\partial r}\;,$ (39)

i.e. an application of $\hat{\nabla}_{a}$ corresponds to a multiplication of
$r$. When we say that $\hat{g}_{ab}$ is $C^{>n}$ at $i^{0}$, by the way, we
should take differentiation in the coordinates $(U,V)$, and so this condition
implies that the metric in the unphysical space-time is given by

$\hat{g}_{ab}=\hat{\eta}_{ab}\left(1+\frac{f(\theta,\varphi,\cdots)}{r^{n+1}}\right),$
(40)

where

$\hat{\eta}_{ab}dx^{a}dx^{b}\equiv-
dUdV+\frac{\sin^{2}\left(U-V\right)}{4}d\Omega^{2}_{d-2}$ (41)

is the unphysical space-time metric corresponding to the flat metric (35) in
the physical space-time.

##  Appendix C Derivations of Eqs. (10), (11), (18) and (19)

In this appendix, we give detailed derivations of Eqs. (10), (11), (18) and
(19). Since we compute quantities only at spatial infinity, we omit “hat” and
$\lim_{\rightarrow i^{0}}$ throughout this appendix for convenience.

###  C.1 Derivation of Eqs. (10) and (11)

First, from Eqs. (8) and (9), we obtain

$\displaystyle\Omega^{1/2}\nabla_{[m}\mathcal{X}_{ab]cd}$
$\displaystyle\;\;=\Omega^{-1/2}\Bigl{(}g_{c[m}\mathcal{X}_{ab]pd}\nabla^{p}\Omega+g_{d[m}\mathcal{X}_{ab]cp}\nabla^{p}\Omega$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+\frac{5-d}{2}(\nabla_{[m}\Omega)\mathcal{X}_{ab]cd}\Bigr{)},$
(42)

where $\mathcal{X}_{abcd}\equiv\Omega^{(5-d)/2}C_{abcd}$ .

Multiplying $\eta^{b}\eta^{d}h_{e}^{m}h_{f}^{a}h_{g}^{c}$ to the above, the
left-hand side becomes

$\displaystyle\Omega^{1/2}\eta^{b}\eta^{d}h_{e}^{m}h_{f}^{a}h_{g}^{c}\nabla_{[m}\mathcal{X}_{ab]cd}$
$\displaystyle\;\;\;\;=\frac{1}{3}(D_{e}E_{fg}-D_{f}E_{eg})-\frac{1}{3}(2W_{feg}+W_{gef}-W_{gfe}),$
(43)

where we used the fact that

$\Omega^{1/2}\nabla_{a}\eta_{b}=g_{ab}-\eta_{a}\eta_{b}=h_{ab}\;,$ (44)

and the definition

$W_{abc}\equiv h_{a}^{e}h_{b}^{f}h_{c}^{g}\mathcal{X}_{efgd}\eta^{d}.$ (45)

In addition, we used the fact that $\mathcal{X}_{abcd}$ has directional
dependent limit and thus $\eta^{e}\nabla_{e}\mathcal{X}_{abcd}$ vanishes. In
the right-hand side of Eq. (42), the second and third terms become

$\Omega^{-1/2}\eta^{b}\eta^{d}h_{e}^{m}h_{f}^{a}h_{g}^{c}g_{d[m}\mathcal{X}_{ab]cp}\nabla^{p}\Omega=\frac{2}{3}W_{efg}$
(46)

and

$\frac{5-d}{2}\Omega^{-1/2}\eta^{b}\eta^{d}h_{e}^{m}h_{f}^{a}h_{g}^{c}\nabla_{[m}\Omega\mathcal{X}_{ab]cd}=\frac{5-d}{3}W_{efg}.$
(47)

The first term vanishes since $\nabla^{p}\Omega=2\Omega^{1/2}\eta^{p}$.
Finally, we obtain Eq. (10) from Eq. (42), that is

$\displaystyle D_{e}E_{fg}-D_{f}E_{eg}$
$\displaystyle=(d-5)W_{feg}+W_{gef}+W_{fge}$ $\displaystyle=(d-4)W_{feg}\;,$
(48)

where we used $W_{[abc]}=0$ in the second line.

Next, we multiply $\Omega^{(4-d)/2}\mathcal{E}_{a_{1}\cdots
a_{d-3}}{}^{fcd}\eta_{f}\eta^{b}h_{g}^{m}h_{h}^{a}$ to Eq. (42), where
$\mathcal{E}_{a_{1}\cdots a_{d-3}}{}^{fcd}\equiv h_{a_{1}}^{b_{1}}\cdots
h_{a_{d-3}}^{b_{d-3}}\epsilon_{b_{1}\cdots b_{d-3}}{}^{fcd}$, and then obtain

$\displaystyle\frac{1}{3}\left(D_{g}B_{a_{1}\cdots
a_{d-3}h}-D_{h}B_{a_{1}\cdots a_{d-3}g}\right)$
$\displaystyle-\frac{1}{3}{\cal E}_{a_{1}\cdots
a_{d-3}}{}^{fcd}\Omega^{(4-d)/2}$
$\displaystyle\;\;\;\;\times\left(h_{fg}h^{a}_{h}\eta^{b}+h_{g}^{b}h_{h}^{a}\eta_{f}-h_{hf}h_{g}^{a}\eta^{b}-h_{h}^{b}h_{g}^{a}\eta_{f}\right)\mathcal{X}_{abcd}$
$\displaystyle=\frac{1}{3}\Bigl{[}4\left(h_{dg}E_{hc}-h_{dh}E_{gc}\right)+(5-d)h_{g}^{a}h_{h}^{b}\mathcal{X}_{abcd}\Bigr{]}$
$\displaystyle~{}~{}~{}\times\Omega^{(4-d)/2}{\cal E}_{a_{1}\cdots
a_{d-3}}{}^{fcd}\eta_{f}\;.$ (49)

From this equation, we obtain Eq. (11):

$\displaystyle\\!\\!D_{g}B_{a_{1}\cdots a_{d-3}h}-D_{h}B_{a_{1}\cdots
a_{d-3}g}$
$\displaystyle=-(d-3)\Omega^{(9-2d)/2}h_{g}^{p}h_{h}^{q}\eta_{f}\mathcal{E}_{a_{1}\cdots
a_{d-3}}^{~{}~{}~{}~{}~{}~{}~{}~{}~{}fcd}\,{}^{(d-1)}C_{pqcd}\;,$ (50)

where ${}^{(d-1)}C_{abcd}$ is the $(d-1)$-dimensional Weyl tensor on
$\Omega=\text{constant}$ surface at $i^{0}$. To transform Eq. (49) to Eq.
(50), we used the following relations

$\displaystyle{\cal E}_{a_{1}\cdots
a_{d-3}}{}^{fcd}h_{hf}h_{g}^{a}\eta^{b}\mathcal{X}_{abcd}$
$\displaystyle~{}={\cal E}_{a_{1}\cdots
a_{d-3}}{}^{fcd}h_{hf}h_{g}^{a}\eta^{b}(h_{c}^{i}+\eta_{c}\eta^{i})(h_{d}^{j}+\eta_{d}\eta^{j})\mathcal{X}_{abij}$
$\displaystyle~{}=2{\cal E}_{a_{1}\cdots a_{d-3}}{}^{fcd}h_{hf}E_{gc}\eta_{d}$
(51)

and

$\displaystyle\mathcal{X}_{abcd}h_{w}^{~{}a}h_{x}^{~{}b}h_{y}^{~{}c}h_{z}^{~{}d}=$
$\displaystyle\;\Omega^{(5-d)/2}{}^{(d-1)}C_{wxyz}$
$\displaystyle-\frac{2}{d-3}\left(E_{w[y}h_{z]x}-E_{x[y}h_{z]w}\right).$ (52)

In Eqs. (51) and (52), we used the fact that the extrinsic curvature of
$\Omega={\rm constant}$ surface at $i^{0}$ is

$\displaystyle\pi_{ab}$ $\displaystyle\equiv(1/2)\mbox{\pounds}_{\eta}h_{ab}$
$\displaystyle=\frac{1}{2}(\eta^{c}\nabla_{c}h_{ab}+h_{ac}\nabla_{b}\eta^{c}+h_{bc}\nabla_{a}\eta^{c})$
$\displaystyle=\Omega^{-1/2}h_{ab}\;.$ (53)

For the derivation of Eq. (52), see Eq. (A6) in SMS . (Note that the magnetic
part defined there is different from ours.)

###  C.2 Derivation of Eqs. (18) and (19)

Hereafter in this appendix, we derive Eqs. (18) and (19). Firstly, to
facilitate the derivation, we derive the following relation:

$\mathcal{X}_{abcm}\eta^{m}=\frac{1}{d-2}\Bigl{[}\Omega^{1/2}\nabla_{[b}T_{a]c}+(d-5)\eta_{[b}T_{a]c}\Bigr{]},$
(54)

where $T_{ab}\equiv\Omega^{(5-d)/2}S_{ab}$ is a tensor which have directional
dependent limit at $i^{0}$, and $S_{ab}$ is defined in Eq. (14). The
manipulation of $g^{md}\times$ Eq. (9) implies

$\nabla_{m}C_{abc}{}^{m}=(d-3)\Omega^{-1}C_{abcp}\nabla^{p}\Omega\;.$ (55)

Note that Eq. (8) was derived from the Bianchi identity in the physical vacuum
spacetime ($\nabla_{[m}C_{ab]cd}=0$). On the other hand, from the Bianchi
identity in the unphysical spacetime ($\hat{\nabla}_{[m}\hat{R}_{ab]cd}=0$),
we can derive

$\nabla_{m}C_{abc}{}^{m}+\frac{2(d-3)}{d-2}\nabla_{[a}S_{b]c}=0~{}.$ (56)

From these two equations, we can see that

$C_{abcm}\eta^{m}=-\frac{1}{d-2}\Omega^{1/2}\nabla_{[a}S_{b]c}\;.$ (57)

It is easy to see that Eq. (54) holds from this equation.

Now we are ready to derive Eqs. (18) and (19). Let us first take the
manipulation of $h_{p}^{a}h_{q}^{c}\eta^{b}\times$ Eq. (54), which results in

$\displaystyle E_{pq}$
$\displaystyle=\frac{1}{d-2}h_{p}^{a}h_{q}^{c}\eta^{b}\Bigl{[}\Omega^{1/2}\nabla_{[b}T_{a]c}+(d-5)\eta_{[b}T_{a]c}\Bigr{]}$
$\displaystyle=-\frac{1}{2(d-2)}\Bigl{[}D_{p}Q_{q}+h_{pq}E+(4-d)U_{pq}\Bigr{]}$
$\displaystyle=-\frac{1}{2(d-2)}\Bigl{[}\frac{1}{d-3}D_{p}D_{q}E+h_{pq}E$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\;+(4-d)U_{pq}\Bigr{]}.$
(58)

This is Eq. (18). In the last line, we used the relation derived by the
manipulation of $h^{a}_{e}\eta^{b}\eta^{c}\times$ Eq. (54):

$D_{e}E=(d-3)Q_{e}\;.$ (59)

Next, let us apply $\Omega^{(4-d)/2}\mathcal{E}_{a_{1}\cdots
a_{d-3}}{}^{fab}\eta_{f}$ to Eq. (54). Then we obtain Eq. (19):

$\displaystyle B_{a_{1}\cdots a_{d-3}c}$
$\displaystyle~{}~{}=\frac{1}{d-2}\mathcal{E}_{a_{1}\cdots
a_{d-3}}{}^{fab}\eta_{f}\Omega^{(4-d)/2}(D_{b}U_{ac}+h_{bc}Q_{a})$
$\displaystyle~{}~{}=-\frac{1}{d-2}\mathcal{E}_{a_{1}\cdots
a_{d-3}}{}^{fab}\eta_{f}\Omega^{(4-d)/2}D_{a}\Bigl{(}U_{bc}-\frac{h_{bc}}{d-3}E\Bigr{)}.$
(60)

##  Appendix D $(d-1)+1$ decomposition

In this appendix, we show that $d$-momentum defined in Eq. (29) agrees with
the ADM formulae for energy and momentum:

$\displaystyle E$ $\displaystyle=\frac{1}{16\pi
G_{d}}\lim_{r_{0}\rightarrow\infty}\int_{S^{d-2}}\left(\partial^{a}h_{ab}-\partial_{b}h^{a}_{~{}a}\right)dS^{b}\Big{|}_{r=r_{0}},$
(61) $\displaystyle Q_{N^{a}}$ $\displaystyle=\frac{-1}{8\pi
G_{d}}\lim_{r_{0}\rightarrow\infty}\int_{S^{d-2}}\left(K_{ab}-K^{m}_{\,\,m}h_{ab}\right)N^{a}dS^{b}\Big{|}_{r=r_{0}},$
(62)

where $h_{ab}\equiv g_{ab}+t^{a}t^{b}$ and $K_{ab}\equiv
h_{a}^{~{}c}h_{bd}\nabla_{c}t^{d}$ are the induced metric and the extrinsic
curvature of a $t=\text{constant}$ surface whose unit normal is $t^{a}$, and
$\partial_{a}$ is a coordinate derivative with respect to asymptotic Cartesian
coordinates. $N^{a}$ is an asymptotic spacelike translational Killing vector
such that $D_{a}N_{b}\rightarrow 0$ as $r\rightarrow\infty$, where $D_{a}$ is
the connection for $h_{ab}$.

We also show in this appendix that the angular momentum defined in Eq. (31)
transforms in translational transformation as Eq. (33). This appendix may be
regarded as an extension of the work by Ashtekar and Magnon in four dimensions
AM . We will describe in much detail because it is very hard to check their
result.

###  D.1 Energy

First, let us consider the energy. Let $\hat{\Sigma}$ be a spacelike
hypersurface in $\hat{M}$ on $i^{0}$ which has unit timelike vector
$\hat{t}^{a}$ as its normal. Then, the energy defined by Eq. (29) becomes

$-P_{a}\hat{t}^{a}=-\frac{1}{8\pi
G_{d}(d-3)}\int_{S^{d-2}}\hat{E}_{ab}\hat{t}^{a}\hat{t}^{b}dS\;,$ (63)

where $dS$ is the volume element of a $(d-2)$-dimensional unit sphere
$S^{d-2}$. In order to compare the above with the ADM formula, we must write
it down in terms of quantities in physical space-time $M$. To do so, we
introduce a spacelike hypersurface $\Sigma$ in $M$, unit timelike vector
$t^{a}$ normal to $\Sigma$, and a unit radial vector $\eta^{a}=\partial^{a}r$.
$t^{a}$ and $\eta^{a}$ are related to $\hat{t}^{a}$ and the unit radial vector
in the unphysical space-time $\hat{\eta}^{a}$ as
$\lim_{r_{0}\rightarrow\infty}\Omega^{-1}t^{a}=\hat{t}^{a}$ and
$\lim_{r_{0}\rightarrow\infty}\Omega^{-1}\eta^{a}=\hat{\eta}^{a}$,
respectively. Then, the above expression of energy (63) becomes

$\displaystyle-P_{a}\hat{t}^{a}=$ $\displaystyle-\frac{1}{8\pi
G_{d}(d-3)}\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}\\!\\!\\!\\!\\!\\!\\!\\!r^{d-1}C_{abcd}\eta^{b}\eta^{d}t^{a}t^{c}dS\,,$
(64)

where we used the fact that $\Omega\simeq 1/r^{2}$ near $i^{0}$.

Now, we define the usual electric part of the Weyl tensor $e_{ab}\equiv
C_{ambn}t^{m}t^{n}$ in the physical space-time $M$. This electric part can be
decomposed as

$\displaystyle e_{ab}=$ $\,{}^{(d-1)}R_{ab}-K_{a}^{~{}m}K_{bm}+KK_{ab}$
$\displaystyle-\frac{1}{d-2}\Bigl{(}(d-3)h_{a}^{~{}m}h_{b}^{~{}n}+h_{ab}h^{mn}\Bigr{)}S_{mn}\;.$
(65)

Taking into account of asymptotic behaviors $K_{ab}=\mathcal{O}(1/r^{d-2})$
and ${}^{(d-1)}R_{ab}=\mathcal{O}(1/r^{d-1})$ for $r\rightarrow\infty\,$, and
the vacuum Einstein equation $R_{ab}=0$, we obtain

$-P_{a}\hat{t}^{a}=-\frac{1}{8\pi
G_{d}(d-3)}\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}\\!\\!\\!\\!\\!\\!r^{d-1}\,{}^{(d-1)}R_{ab}\eta^{a}\eta^{b}dS.$
(66)

In order to integrate by parts in direction $r$, we rewrite the integral into
the following form:

$\displaystyle-P_{a}\hat{t}^{a}=$ $\displaystyle-\frac{1}{8\pi
G_{d}(d-3)}\lim_{r_{0}\rightarrow\infty}$ (67)
$\displaystyle\times\frac{1}{\Delta r}\int_{r=r_{0}}^{r=r_{0}+\Delta
r}\int_{S^{d-2}}r^{d-1}\,{}^{(d-1)}R_{ab}\eta^{a}\eta^{b}drdS,$

where we used the fact that the integrand in Eq. (66) is independent of $r$ at
large $r$. In this expression, the part which contribute to the integral is

${}^{(d-1)}R_{ab}\sim\frac{1}{2}\left(\partial^{c}\partial_{b}h_{ac}+\partial_{a}\partial^{c}h_{bc}-\partial^{c}\partial_{c}h_{ab}-\partial_{a}\partial_{b}h^{c}_{~{}c}\right).$
(68)

Substituting (68) into (67) and integrating by parts, we can get the desired
result. Since this calculation is a little difficult, we describe carefully.
First, we integrate the first part in (68) by parts:

$\displaystyle\frac{1}{\Delta r}\\!\\!\int_{S^{d-2}\times\Delta
r}r\left(\partial^{c}\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}dV$ (69)
$\displaystyle=$ $\displaystyle\frac{1}{\Delta
r}\\!\\!\int_{S^{d-2}\times\Delta
r}\\!\Bigl{[}\partial^{c}\bigl{(}r\left(\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}\bigr{)}\\!-\\!\left(\partial_{b}h_{ac}\right)\partial^{c}(r\eta^{a}\eta^{b})\Bigr{]}dV,$

where $dV\equiv r^{d-2}drdS$. The first term in the right-hand side becomes

$\displaystyle\frac{1}{\Delta r}$ $\displaystyle\int_{S^{d-2}\times\Delta
r}\partial^{c}\bigl{(}r\left(\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}\bigr{)}dV$
$\displaystyle=\;\;\;\,\frac{1}{\Delta
r}\int_{S^{d-2}}r\left(\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}dS^{c}\Big{|}_{r=r_{0}+\Delta
r}$ $\displaystyle\;\;\;\;-\frac{1}{\Delta
r}\int_{S^{d-2}}r\left(\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}dS^{c}\Big{|}_{r=r_{0}}$
$\displaystyle=\int_{S^{d-2}}\left(\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}dS^{c}\Big{|}_{r=r_{0}}\quad,$
(70)

where $dS^{c}\equiv\eta^{c}r^{d-2}dS$. In the first and the second equalities,
we used the Gauss theorem, and the fact that
$\left(\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}r^{d-2}$ is independent of $r$
in the limit of $r_{0}\to\infty$. The second term in the right-hand side of
Eq. (69) becomes

$\displaystyle\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta
r}\left(\partial_{b}h_{ac}\right)\partial^{c}(r\eta^{a}\eta^{b})dV$
$\displaystyle=\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta
r}\\!\\!\\!\\!\\!\left(\partial_{b}h_{ac}\right)(\eta^{a}\eta^{b}\eta^{c}+q^{ac}\eta^{b}+q^{bc}\eta^{a})r^{d-2}drdS$
$\displaystyle=\int_{S^{d-2}}\\!\\!\left(\partial_{b}h_{ac}\right)(\eta^{a}\eta^{b}\eta^{c}+q^{ac}\eta^{b}+q^{bc}\eta^{a})r^{d-2}dS.$
(71)

To transform the second into the third line, we used the fact that the
integrand in the second line does not depend on $r$. Then, we obtain

$\displaystyle\int_{S^{d-2}}r^{d-1}\left(\partial^{c}\partial_{b}h_{ac}\right)\eta^{a}\eta^{b}dS$
$\displaystyle\;\;=-\int_{S^{d-2}}\left(\partial_{b}h_{ac}\right)(\eta^{a}q^{bc}+\eta^{b}q^{ac})r^{d-2}dS.$
(72)

Here, we defined a metric $q_{ab}$ on $r=\text{constant}$ surface such that
$\partial_{a}\eta_{b}=(h_{ab}-\eta_{a}\eta_{b})/r\equiv q_{ab}/r$. In the same
way, the other terms in (68) are transformed as

$\displaystyle\int_{S^{d-2}}r^{d-1}\left(\partial_{a}\partial^{c}h_{bc}\right)\eta^{a}\eta^{b}dS=-(d-2)\int_{S^{d-2}}\partial^{c}h_{ac}dS^{a},$
$\displaystyle\int_{S^{d-2}}r^{d-1}\left(\partial^{c}\partial_{c}h_{ab}\right)\eta^{a}\eta^{b}dS$
$\displaystyle\qquad\qquad\qquad=-\int_{S^{d-2}}\left(\partial_{c}h_{ab}\right)(q^{ac}\eta^{b}+q^{bc}\eta^{a})r^{d-2}dS\;,$
$\displaystyle\int_{S^{d-2}}r^{d-1}\left(\partial_{a}\partial_{b}h^{c}_{~{}c}\right)\eta^{a}\eta^{b}dS=-(d-2)\int_{S^{d-2}}\\!\partial_{a}h^{c}_{~{}c}dS^{a}.$

Finally, we obtain the desired result:

$-P_{a}\hat{t}^{a}=\frac{1}{16\pi
G_{d}}\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}\left(\partial^{a}h_{ab}-\partial_{b}h^{a}_{~{}a}\right)dS^{b}\;.$
(73)

###  D.2 Momentum

Next, let us consider momentum. The components of $(d-1)$-momentum along a
spacelike vector $N^{a}$ at $i^{0}$ can be written as

$P_{a}\hat{N}^{a}=\frac{1}{8\pi
G_{d}(d-3)}\int_{S^{d-2}}\hat{E}_{ab}\hat{N}^{a}\hat{t}^{b}dS.$ (74)

In terms of quantities of physical space-time, this equation becomes

$P_{a}\hat{N}^{a}=\frac{1}{8\pi
G_{d}(d-3)}\lim_{r\rightarrow\infty}\int_{r=r_{0}}r^{d-1}C_{abcd}\eta^{b}\eta^{d}N^{a}t^{c}dS\,,$
(75)

where $N^{a}=\lim_{\rightarrow i^{0}}\Omega\hat{N}^{a}$. Using the Codacci
equation and the vacuum Einstein equation, this expression becomes

$\displaystyle P_{a}\hat{N}^{a}=\;$ $\displaystyle\frac{1}{8\pi
G_{d}(d-3)}\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}r^{d-1}$
$\displaystyle\times\left(D_{d}K_{ab}-D_{a}K_{db}\right)\eta^{b}\eta^{d}N^{a}dS.$
(76)

Using the fact that the leading part of $r^{d-1}D_{d}K_{ab}$ does not depend
on $r$, the first term in the right-hand side is reexpressed as volume
integral as

$\displaystyle\int_{r=r_{0}}r^{d-1}\left(D_{d}K_{ab}\right)\eta^{b}\eta^{d}N^{a}dS$
$\displaystyle\;\;=\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta
r}r\left(D_{d}K_{ab}\right)\eta^{b}\eta^{d}N^{a}dV$
$\displaystyle\;\;=\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta
r}\Bigl{[}\;D_{d}(rK_{ab}\eta^{b}\eta^{d}N^{a})$
$\displaystyle\;\;\qquad\qquad\qquad\quad\;\;-K_{ab}D_{d}(r\eta^{b}\eta^{d}N^{a})\;\Bigr{]}dV.$
(77)

Using the Gauss theorem to the first term in the last line, we see that

$\displaystyle\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta
r}D_{d}(rK_{ab}\eta^{b}\eta^{d}N^{a})dV$ $\displaystyle=\frac{1}{\Delta
r}\biggl{[}\int_{S^{d-2}}\\!\\!\\!\\!\\!rK_{ab}N^{a}dS^{b}\Big{|}_{r=r_{0}+\Delta
r}\\!-\\!\int_{S^{d-2}}\\!\\!\\!\\!\\!rK_{ab}N^{a}dS^{b}\Big{|}_{r=r_{0}}\biggr{]}$
$\displaystyle=\int_{S^{d-2}}K_{ab}N^{a}dS^{b}\Big{|}_{r=r_{0}}\;\;,$ (78)

where we used the fact that $K_{ab}N^{a}r^{d-2}$ does not depend on $r$. The
second term in Eq. (77) can be rearranged as

$\displaystyle-$ $\displaystyle\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta
r}K_{ab}D_{d}(r\eta^{b}\eta^{d}N^{a})dV$ $\displaystyle=-\frac{d-1}{\Delta
r}\int_{S^{d-2}\times\Delta r}K_{ab}\eta^{b}N^{a}dV$
$\displaystyle=-(d-1)\int_{S^{d-2}}K_{ab}N^{a}dS^{b}\Big{|}_{r=r_{0}}\;\;.$
(79)

In the same way, the second term of Eq. (76) is rearranged as

$\displaystyle\int_{r=r_{0}}r^{d-1}\left(D_{a}K_{db}\right)\eta^{b}\eta^{d}N^{a}dS$
$\displaystyle=2\int_{S^{d-2}}K_{ab}\eta^{a}\eta^{b}N^{c}dS_{c}-2\int_{S^{d-2}}K_{ab}N^{a}dS^{b}$
(80)

From this equation, we can obtain a relation

$\displaystyle 2$
$\displaystyle\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}r^{d-2}K_{ab}N_{c}\eta^{a}\eta^{b}\eta^{c}dS$
$\displaystyle=$
$\displaystyle\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}\Big{(}K_{ab}-(d-3)Kh_{ab}\Big{)}N^{a}dS^{b}.$
(81)

Derivation of this relation is a little non-trivial, so we describe it in
detail. Note that the Gauss theorem makes the surface integral into the volume
integral as

$\displaystyle\int_{r=r_{0}}r^{d-2}K_{ab}N_{c}\eta^{a}\eta^{b}\eta^{c}dS$
$\displaystyle\\!=\\!\frac{1}{\Delta r}\\!\int_{S^{d-2}\times\Delta
r}\partial^{a}\left(rK_{ab}N_{c}\eta^{b}\eta^{c}\right)dV$
$\displaystyle\\!=\\!\frac{1}{\Delta r}\\!\int_{S^{d-2}\times\Delta
r}\Bigl{[}r\left(D_{b}K\right)\eta^{b}N^{c}\eta_{c}\\!+\\!K_{ab}D^{a}\left(r\eta^{b}N^{c}\eta_{c}\right)\Bigr{]}dV,$
(82)

where we used the momentum constraint equation for the vacuum Einstein
equation, $D_{a}K^{a}_{b}-D_{b}K=0$, in the last line. The first and the
second terms are rearranged respectively as

$\displaystyle\frac{1}{\Delta r}$ $\displaystyle\int_{S^{d-2}\times\Delta
r}r\left(D_{b}K\right)\eta^{b}N^{c}\eta_{c}dV$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta
r}\Bigl{[}D_{b}\left(rK\eta^{b}N^{c}\eta_{c}\right)-KD_{b}\left(r\eta^{b}N^{c}\eta_{c}\right)\Bigr{]}dV$
$\displaystyle=$
$\displaystyle-(d-2)\int_{S^{d-2}}KN^{a}dS_{a}\Big{|}_{r=r_{0}}\;\;,$ (83)

and

$\displaystyle\frac{1}{\Delta r}\int_{S^{d-2}\times\Delta
r}K_{ab}D^{a}\left(r\eta^{b}N^{c}\eta_{c}\right)dV$
$\displaystyle\\!=\\!\int_{S^{d-2}}\\!\\!\left(KN^{a}dS_{a}+K_{ab}N^{b}dS^{a}-K_{ab}\eta^{a}\eta^{b}N^{c}dS_{c}\right)\Big{|}_{r=r_{0}}.$
(84)

Then, we proceed as

$\displaystyle\int_{r=r_{0}}r^{d-2}K_{ab}N_{c}\eta^{a}\eta^{b}\eta^{c}dS$
$\displaystyle~{}~{}=~{}\int_{S^{d-2}}\Bigl{(}K_{ab}N^{b}-(d-3)KN_{a}\Bigr{)}dS^{a}\Big{|}_{r=r_{0}}$
$\displaystyle~{}~{}~{}~{}-\int_{S^{d-2}}K_{ab}\eta^{a}\eta^{b}N^{c}dS_{c}\Big{|}_{r=r_{0}}\;\;.$
(85)

The last term in the right-hand side is the same with the left-hand side
except for the signature. Therefore, we have the relation of Eq. (81).

Substituting Eq. (81) into Eq. (80), we can show

$\displaystyle\int_{r=r_{0}}r^{d-1}(D_{d}K_{ab}-D_{a}K_{db})\eta^{b}\eta^{d}N^{a}dS$
$\displaystyle=-(d-3)\int_{S^{d-2}}(K_{ab}-Kh_{ab})N^{a}dS^{b}\Big{|}_{r=r_{0}}\;\;,$
(86)

and then, combining this equation with Eq. (76), we see that our formula (29)
for $(d-1)$-momentum becomes the ADM formula, that is

$P_{a}\hat{N}^{a}=-\frac{1}{8\pi
G_{d}}\lim_{r_{0}\rightarrow\infty}\int_{r=r_{0}}\left(K_{ab}-Kh_{ab}\right)N^{a}dS^{b}.$
(87)

###  D.3 Angular momentum

Finally, we consider translational transformation of angular momentum
$M_{ab}$. We consider translation $\omega_{a}$ which is a fixed vector at
$i^{0}$, and relate it with $\alpha$ as $\alpha=\omega_{a}\hat{\eta}^{a}$.
This translation transforms $\hat{\eta^{a}}$ as

$\hat{\eta}_{a}^{\prime}\;\hat{=}\;\hat{\eta}_{a}+\frac{1}{2}\Omega^{(d-3)/2}\left((d-2)\alpha\hat{\eta}_{a}+\Omega^{1/2}\hat{\nabla}_{a}\alpha\right)\;.$
(88)

Then, the magnetic part of the Weyl tensor $\hat{\beta}_{a_{1}\cdots
a_{d-3}b}$ transforms as

$\displaystyle\hat{\beta}_{a_{1}\cdots a_{d-3}b}^{\prime}\,\hat{=}$
$\displaystyle\;\hat{\beta}_{a_{1}\cdots
a_{d-3}b}+\frac{d-2}{d-3}\hat{\epsilon}_{a_{1}\cdots
a_{d-3}mpq}\hat{\eta}^{m}\hat{E}^{q}_{~{}b}\hat{D}^{p}\alpha$
$\displaystyle+\frac{1}{d-3}\hat{\epsilon}_{a_{1}\cdots
a_{d-3}mpb}\hat{\eta}^{m}\hat{E}^{pr}\hat{D}_{r}\alpha\;.$ (89)

We used the projection formulae of the Weyl tensor (51) and (52) to derive
this equation. Substituting (89) into (31), and noting that
$F_{ab}\hat{\epsilon}^{b}_{~{}e_{1}\cdots
e_{d-2}m}\hat{\eta}^{m}dS^{e_{1}\cdots e_{d-2}}$ vanishes since $d-1$ indices
of $\hat{\epsilon}$ are projected onto $(d-2)$-dimensional surface, we find
that usual translational transformation

$M_{ab}^{\prime}=M_{ab}+2P_{\,[a}\hat{\omega}_{b]}\;,$ (90)

where $\hat{D}_{a}\alpha=\hat{\omega}_{a}$, is correctly reproduced including
the coefficient, if we define the angular momentum as (31).

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