#  Angular momentum at null infinity in higher dimensions

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

###### Abstract

We define the angular momentum at null infinity in higher dimensions. The
asymptotic symmetry at null infinity becomes the Poincaré group in higher
dimensions. This fact implies that the angular momentum can be defined without
any ambiguities such as supertranslation in four dimensions. Indeed we can
show that the angular momentum in our definition is transformed covariantly
with respect to the Poincaré group.

###### pacs:

04.20.-q, 04.20.Ha

††preprint: YITP-12-13

##  I introduction

Motivated by the string theory and the scenario with large extra dimensions
such as the TeV scale gravity ArkaniHamed:1998rs ; Antoniadis:1998ig , the
gravitational theory in higher dimensions has been investigated PTP . Then it
has been realized that the higher dimensional gravity has much different
features from that in four dimensions. As one of such differences, there is an
issue of the asymptotic structure of the spacetime Hollands:2003ie ;
Hollands:2003xp ; Ishibashi:2007kb ; Tanabe:2011es ; Tanabe:2009va ;
Tanabe:2010rm ; Tanabe:2009xb . The asymptotically flat spacetime has two
asymptotic infinities: spatial infinity and null infinity. At spatial infinity
we can define the global conserved quantities of spacetime such as the mass
and angular momentum. In addition the multipole moments of spacetime is also
defined at spatial infinity Hansen ; Tanabe:2010ax . These multipole moments
can be used to classify the black hole solutions. At null infinity, the
asymptotic structure describes dynamical properties of the spacetime because
gravitational waves can reach at null infinity. Then the study of the
asymptotic structure at null infinity is indispensable when one considers the
dynamical phenomena such as the perturbation for black holes and the formation
of higher dimensional black holes in particle accelerators. As a fundamental
aspect of the general relativity, the notion of the asymptotic flatness at
null infinity is also necessary for the rigorous definition of black hole
Hawking:1973uf .

The asymptotic structure at null infinity in higher dimensions has been
investigated using the conformal method in Refs Hollands:2003ie ;
Hollands:2003xp ; Ishibashi:2007kb . In the conformal method, spacetime is
conformally embedded into the compact region of the another spacetime and the
null infinity is defined as the boundary of the spacetime. The asymptotic
structure at null infinity can be investigated using the introduced conformal
factor $\Omega\sim 1/r$ as a coordinate. Therein the null infinity is defined
on $\Omega=0$. However, there is one problem in this treatment. The
gravitational waves behave near null infinity with a half integer power of
$\Omega$ in odd dimensions. At first glance, this shows the non-smoothness of
the gravitational fields at null infinity in odd dimensions. Because of this
non-smooth behavior of the gravitational fields, using the conformal method,
we cannot define the asymptotic flatness at null infinity in odd dimensions.

In the Bondi coordinate method Refs. Tanabe:2011es ; Bondi:1962px ;
Sachs:1962wk ; Tanabe:2009va ; Tanabe:2010rm , on the other hand, we can
define the asymptotic flatness at null infinity in arbitrary higher dimensions
and safely investigate the asymptotic structure at null infinity. In the
analysis of the asymptotic structure, it was found that the Bondi mass always
decreases due to the gravitational waves and the asymptotic symmetry at null
infinity is the Poincaré group.

It is reminded that the asymptotic symmetry is not the Poincaré group in four
dimensions. The asymptotic symmetry in four dimensions is the semi-direct
group of the Lorentz group and supertranslation. The supertranslation has the
functional degree of freedom and then it is the infinite dimensional group.
This means that there are infinite directions of the translation which causes
the ambiguities in the definition of the angular momentum. Although there were
many efforts to define the angular momentum Prior:1977 ; Streubel:1978 ;
Winicour:1980 ; Geroch:1981ut ; Dray:1984 , there is no sharp definition
without any ambiguities in four dimensions.

In higher dimensions it was shown that the asymptotic symmetry at null
infinity is the Poincaré group Tanabe:2011es ; Tanabe:2009va . The
$n$-dimensional Poincaré group has the $n$ directions of the translation. In
this paper, we define the angular momentum at null infinity in higher
dimensions and shows that it has no ambiguities. In fact the angular momentum
is transformed covariantly with respect to the Poincaré group. Note that the
study of the angular momentum at null infinity was performed in five
dimensions Tanabe:2010rm . In this paper, we generalize this analysis to
arbitrary higher dimensions following our previous work of Ref. Tanabe:2011es
.

The organization of this paper is as follows. In the next section, we review
our previous work on null infinity Tanabe:2011es . Therein, using the Bondi
coordinates, we introduce the definition of null infinity in arbitrary
dimensions. We also discussed the asymptotic symmetry at the null infinities
briefly. In Sec. III, we define the Bondi angular momentum together with the
Bondi mass/momentum and show its radiation formulae using the Einstein
equations. In Sec. IV, it will be shown that the angular momentum defined here
is transformed covariantly under the transformation generated by the
asymptotic symmetry. Finally we give the summary and outlook in Sec. V.

##  II Review of our previous work

In this section we review our previous work Tanabe:2011es . First we introduce
the Bondi coordinates adopted here and write down some components of the
Einstein equation explicitly. Solving them we specify the boundary condition
which gives us the definition of the null infinity. Then we discuss the
asymptotic symmetries at the null infinity.

###  II.1 Bondi coordinates and Einstein equations

We introduce the Bondi coordinates in $n$ dimensions. First we assume the
function $u$ which satisfies $\hat{\nabla}_{a}u\hat{\nabla}^{a}u=0$ where
$\hat{\nabla}_{a}$ denotes the covariant derivative with respect to
$n$-dimensional metric $g_{ab}$. $u$ is used as the time coordinate. Next the
angular coordinate $x^{I}$ is defined as
$\hat{\nabla}^{a}u\hat{\nabla}_{a}x^{I}=g^{uI}=0$. We define the radial
coordinate $r$ as $\sqrt{\det{g}_{IJ}}=r^{n-2}\omega_{n-2}$ where
$\omega_{n-2}$ is the volume element of the unit $(n-2)$-dimensional sphere
$S^{n-2}$. Then the metric in the Bondi coordinates $x^{a}=(u,r,x^{I})$ can be
written as

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle g_{ab}dx^{a}dx^{b}$ (1)
$\displaystyle=$ $\displaystyle-
Ae^{B}du^{2}-2e^{B}dudr+\gamma_{IJ}(dx^{I}+C^{I}du)(dx^{J}+C^{J}du).$

In this coordinate system, the null infinity is defined at $r=\infty$ and its
topology is ${\mathbf{R}}\times S^{n-2}$. For the convenience of our
discussion, we define $h_{IJ}$ as $\gamma_{IJ}=r^{2}h_{IJ}$ with the following
gauge condition

$\sqrt{\det{h_{IJ}}}\,=\,\omega_{n-2}.$ (2)

We provide the Einstein equations in the Bondi coordinates. The vacuum
Einstein equations can be decomposed into the constraint equation without the
$u$ derivative terms and evolution equations with the $u$ derivative terms.

The constraint equations are $\hat{R}_{rr}=0,\hat{R}_{aI}\gamma^{IJ}=0$ and
$\hat{R}_{IJ}\gamma^{IJ}=0$. $\hat{R}_{ab}$ is the Ricci tensor with respect
to $g_{ab}$. Using the formulae in Appendix A and Sec. II in Ref.
Tanabe:2011es , we can write the equation $\hat{R}_{rr}=0$ as

$B^{\prime}=\frac{r}{4(n-2)}h_{IJ}^{\prime}h_{KL}^{\prime}h^{IK}h^{JL},$ (3)

where the prime denotes the $r$ derivative. The equation
$\hat{R}_{rJ}\gamma^{IJ}=0$ yields

$\frac{1}{r^{n-2}}(r^{n}e^{-B}h_{IJ}{C^{J}}^{\prime})^{\prime}=-{}^{(h)}\nabla_{I}B^{\prime}+\frac{n-2}{r}{}^{(h)}\nabla_{I}B+{}^{(h)}\nabla^{J}h_{IJ}^{\prime},$
(4)

where ${}^{(h)}\nabla_{I}$ is the covariant derivative with respect to
$h_{IJ}$.

From $\hat{R}_{IJ}\gamma^{IJ}=0$, we obtain

$\displaystyle(n-2)\frac{(r^{n-3}A)^{\prime}}{r^{n-2}}$ $\displaystyle=$
$\displaystyle-{}^{(h)}\nabla_{I}{C^{I}}^{\prime}-\frac{2(n-2)}{r}{}^{(h)}\nabla_{I}C^{I}-\frac{r^{2}e^{-B}}{2}h_{IJ}{C^{I}}^{\prime}{C^{J}}^{\prime}$
(5)
$\displaystyle-\frac{e^{B}}{2r^{2}}h^{IJ}{}^{(h)}\nabla_{I}B{}^{(h)}\nabla_{J}B-\frac{e^{B}}{r^{2}}{}^{(h)}\nabla_{I}(h^{IJ}{}^{(h)}\nabla_{J}B)+\frac{e^{B}}{r^{2}}{}^{(h)}R,$

where ${}^{(h)}R$ is the Ricci scalar of $h_{IJ}$. Once $h_{IJ}$ is given on a
surface $u=u_{0}$, we can obtain the metric functions $A,B$ and $C^{I}$ by
solving the constraint equations (3), (4) and (5) on the surface.

The evolution equation is contained in
$\hat{R}_{ab}\gamma_{I}{}^{a}\gamma_{J}{}^{b}=0$ as

$\displaystyle
e^{-B}\Bigg{[}r^{2}\dot{h}_{IJ}^{{}^{\prime}}+\frac{n-2}{2}r\dot{h}_{IJ}-\frac{r^{2}}{2}\dot{h}_{IK}h^{{}^{\prime}}_{JL}h^{KL}-\frac{r^{2}}{2}\dot{h}_{JK}h^{{}^{\prime}}_{IL}h^{KL}\Bigg{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}-\frac{Ae^{-B}}{2}\Big{[}r^{2}h^{{}^{\prime\prime}}_{IJ}+(n-2)rh^{{}^{\prime}}_{IJ}+2(n-3)h_{IJ}{-}r^{2}h^{KL}h^{{}^{\prime}}_{IK}h^{{}^{\prime}}_{JL}\Big{]}-\frac{A^{{}^{\prime}}e^{-B}}{2}\Big{[}r^{2}h^{{}^{\prime}}_{IJ}+2rh_{IJ}\Big{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}-\frac{e^{-B}}{2}\Big{[}2r^{2}\mathcal{L}_{C}h^{{}^{\prime}}_{IJ}+(n-2)r\mathcal{L}_{C}h_{IJ}+r^{2}\mathcal{L}_{C^{{}^{\prime}}}h_{IJ}-{r^{2}}h^{{}^{\prime}}_{JL}h^{KL}\mathcal{L}_{C}h_{IK}$
$\displaystyle~{}~{}~{}~{}~{}~{}-{r^{2}}h^{{}^{\prime}}_{IL}h^{KL}\mathcal{L}_{C}h_{JK}+{}^{(h)}\nabla_{K}C^{K}(r^{2}h^{{}^{\prime}}_{IJ}+2rh_{IJ})\Big{]}$
$\displaystyle~{}~{}~{}~{}~{}~{}-\frac{e^{-2B}}{2}r^{4}h_{IK}h_{JL}{C^{K}}^{\prime}{C^{L}}^{\prime}-{}^{(h)}\nabla_{I}{}^{(h)}\nabla_{J}B-\frac{1}{2}{}^{(h)}\nabla_{I}B{}^{(h)}\nabla_{J}B+{}^{(h)}R_{IJ}\,=\,0,$
(6)

where ${}^{(h)}R_{IJ}$ is the Ricci tensor with respect to $h_{IJ}$ and the
dot denotes the $u$ derivative. The evolution of $h_{IJ}$ in the Bondi
coordinates is determined by Eq. (6).

###  II.2 Asymptotic flatness at null infinity

The asymptotic flatness at null infinity is defined by the boundary condition
at null infinity in the Bondi coordinates (1). In $n$ dimensions, the boundary
conditions for the asymptotic flatness at null infinity are

$h_{IJ}\,=\,\omega_{IJ}+O\left(\frac{1}{r^{n/2-1}}\right),$ (7)

where $\omega_{IJ}$ is the unit round metric on $S^{n-2}$. The boundary
conditions for the other metric functions are determined by the constraint
equations of Eqs. (3), (4) and (5) as 111The condition for $B$ will be relaxed
to $B=O(r^{-n/2})$ for non-vacuum cases Godazgar:2012zq . In our present
paper, we will focus on the vacuum cases. It is easy to extend our result to
non-vacuum cases.

$\displaystyle A\,=\,1+O\left(\frac{1}{r^{n/2-1}}\right),\quad
B\,=\,O\left(\frac{1}{r^{n-2}}\right),\quad
C^{I}\,=\,O\left(\frac{1}{r^{n/2}}\right).$ (8)

Let us see the above solving the constraint equations explicitly. We will use
some equations for later discussions. First, we expand $h_{IJ}$ near null
infinity as

$h_{IJ}\,=\,\omega_{IJ}+\sum_{k=0}\frac{h^{(k+1)}_{IJ}}{r^{n/2+k-1}},$ (9)

where the summation is taken over $k\in\bf{Z}$ in even dimensions and
$2k\in\bf{Z}$ in odd dimensions. The indices $I,J,\dots$ are raised and
lowered by $\omega_{IJ}$. From the gauge condition of Eq. (2), we find that
$h^{(k+1)}_{IJ}$ is traceless $\omega^{IJ}h^{(k+1)}_{IJ}=0$ for $k<n/2-1$ and,
for $k=n/2-1$,

$\omega^{IJ}h^{(n/2)}_{IJ}\,=\,\frac{1}{2}h^{(1)IJ}h^{(1)}_{IJ}.$ (10)

Solving the constraint equation (3), we have

$B\,=\,\frac{B^{(1)}}{r^{n-2}}+O\left(\frac{1}{r^{n-3/2}}\right),$ (11)

where

$B^{(1)}\,=\,-\frac{1}{16}\omega^{IK}\omega^{JL}h^{(1)}_{IJ}h^{(1)}_{KL}.$
(12)

From Eq. (4), $C^{I}$ is obtained as

$C^{I}\,=\,\sum_{k=0}^{k<n/2-1}\frac{C^{(k+1)I}}{r^{n/2+k}}+\frac{j^{I}}{r^{n-1}}+O\left(\frac{1}{r^{n-1/2}}\right),$
(13)

where, for $k<n/2-1$,

$C^{(k+1)I}\,=\,\frac{2(n+2k-2)}{(n+2k)(n-2k-2)}\nabla_{J}h^{(k+1)IJ}$ (14)

and $\nabla_{I}$ is the covariant derivative with respect to $\omega_{IJ}$.
$j^{I}$ is the integration function in the $r$ integration of Eq. (4). As seen
later, we can see that $j^{I}$ represents the angular momentum of the
spacetime at null infinity.

Integrating Eq. (5) we find

$A\,=\,1+\sum_{k=0}^{k<n/2-2}\frac{A^{(k+1)}}{r^{n/2+k-1}}-\frac{m}{r^{n-3}}+O(r^{-(n-5/2)}),$
(15)

where, for $k<n/2-2$,

$\displaystyle A^{(k+1)}\,=$
$\displaystyle\,-\frac{2(n+2k-4)}{(n-2k-4)(n+2k-2)}\nabla^{I}C_{I}^{(k+1)}$
(16) $\displaystyle\,=$
$\displaystyle\,-\frac{4(n+2k-4)}{(n+2k)(n-2k-2)(n-2k-4)}\nabla^{I}\nabla^{J}h^{(k+1)}_{IJ}.$

$m$ is the integration function and reflects the energy-momentum of the
spacetime at null infinity. For $k=n/2-2$, the left-hand sides of Eqs. (4) and
(5) vanish. Then the right-hand sides of Eqs. (4) and (5) provide the
following constraint equations

$\displaystyle\nabla_{I}C^{(n/2-1)I}\,=\,0,$ (17)
$\displaystyle\nabla^{I}\nabla^{J}h^{(n/2-1)}_{IJ}\,=\,0.$ (18)

In addition, for $k=n/2-1$, we have 222We found a minor error in Eq. (32) of
Ref. Tanabe:2011es , corresponding to Eq. (19) in the current paper. But it
does not affect the results/all equations presented there except for Eq. (32).

$\nabla^{J}h^{(n/2)}_{IJ}\,=\,2\nabla_{I}B^{(1)}{+\nabla_{J}\left(\frac{1}{2}h^{(1)}_{IK}h^{(1)JK}+\frac{1}{8}\omega_{I}{}^{J}h^{(1)}_{KL}h^{(1)KL}\right)}.$
(19)

We can solve the evolution equation of Eq. (6) as

$\displaystyle(k+1)\dot{h}^{(k+2)}_{IJ}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(n-2k-4\right)A^{(k+1)}\omega_{IJ}+\frac{1}{8}\Big{[}n^{2}-6n-(4k^{2}+4k-16)\Big{]}h^{(k+1)}_{IJ}$
(20)
$\displaystyle+\frac{1}{2}\left(-\nabla^{2}h^{(k+1)}_{IJ}+2\nabla_{(I}\nabla^{K}h^{(k+1)}_{J)K}\right)-\frac{1}{2}(n-2k-4)\nabla_{(I}C^{(k+1)}_{J)}-\nabla^{K}C^{(k+1)}_{K}\omega_{IJ}$

for $k<n/2-1$. The solutions for the higher order of $k\geq n/2-1$ are not
important in the following analysis. For the convenience of later discussions,
we derive the evolution equations for $A^{(k+1)}$ and $C^{(k+1)I}$.
Contracting $\nabla^{J}$ with Eq. (20), we obtain the evolution equation for
$C^{(k+1)I}$ as

$\displaystyle\frac{(k+1)(n+2k+2)}{2(n+2k)}\dot{C}^{(k+2)}_{I}$
$\displaystyle=$
$\displaystyle-\frac{n-4}{2(n+2k-4)}\nabla_{I}A^{(k+1)}-\frac{1}{4}\nabla^{2}C^{(k+1)}_{I}$
(21)
$\displaystyle+\frac{1}{16}\Big{[}n^{2}-6n-(4k^{2}+4k-12)\Big{]}C^{(k+1)}_{I}.$

Contracting $\nabla^{I}$ with Eq. (21) and using the solutions of the
constraint equations (14) and (16), we have the evolution equation for
$A^{(k+1)}$ as

$\dot{A}^{(k+2)}\,=\,-\frac{n+2k-2}{2(k+1)(n+2k+2)}\nabla^{2}A^{(k+1)}+\frac{(n+2k-2)^{2}(n-2k-4)}{8(k+1)(n+2k+2)}A^{(k+1)}.$
(22)

###  II.3 Asymptotic symmetry

The asymptotic symmetry is the global symmetry at null infinity generated by
the coordinate transformations preserving the gauge and boundary conditions in
the Bondi coordinates (1). The variation of the metric $\delta g_{ab}$ due to
the coordinate transformation generated by $\xi^{a}$ is given by

$\delta g_{ab}\,=\,\hat{\nabla}_{a}\xi_{b}+\hat{\nabla}_{b}\xi_{a},$ (23)

where $\hat{\nabla}_{a}$ is the covariant derivative with respect to $g_{ab}$.
From Eqs. (1) and (2), the gauge conditions to be satisfied are

$\delta g_{rr}\,=\,0\,,\quad\delta g_{rI}\,=\,0\,,\quad g^{IJ}\delta
g_{IJ}\,=\,0.$ (24)

From Eqs. (7) and (8), the boundary conditions to be preserved by the
coordinate transformations are

$\delta g_{uu}\,=\,O(r^{-(n/2-1)})\,,\quad\delta
g_{uI}\,=\,O(r^{-(n/2-2)})\,,\quad\delta g_{IJ}\,=\,O(r^{n/2-3}).$ (25)

To satisfy Eq. (24), the generator of the asymptotic symmetry $\xi$ becomes

$\displaystyle\xi^{u}\,=\,f(u,x^{I}),$ (26)
$\displaystyle\xi^{I}\,=\,f^{I}(u,x^{I})+\int
dr\frac{e^{B}}{r^{2}}h^{IJ}\nabla_{J}f,$ (27)
$\displaystyle\xi^{r}\,=\,-\frac{r}{n-2}\left(C^{I}\nabla_{I}f+\nabla_{I}\xi^{I}\right).$
(28)

$f(u,x^{I})$ and $f^{I}(u,x^{I})$ are the integration functions in the $r$
integration of the equation $\delta g_{rr}=0$ and $\delta g_{rI}=0$. The
asymptotic symmetry is the group generated by $f$ and $f^{I}$.

The boundary conditions (25) give the equations which $f$ and $f^{I}$ should
satisfy as

$\displaystyle\partial_{u}f^{I}\,=\,0,$ (29)
$\displaystyle\nabla_{I}f_{J}+\nabla_{J}f_{I}\,=\,\frac{2\nabla_{K}f^{K}}{n-2}\omega_{IJ},\quad\nabla_{I}f^{I}=(n-2)\frac{\partial
f}{\partial u},$ (30)
$\displaystyle\nabla_{I}\nabla_{J}f\,=\,\frac{\nabla^{2}f}{n-2}\omega_{IJ}.$
(31)

Note that Eq. (31) is required only in $n>4$ dimensions. In the following, for
the moment, we discuss the asymptotic symmetry in $n>4$ dimensions. We will
comment on the four dimensional case later.

From Eq. (29), we find $f^{I}=f^{I}(x^{I})$. $f^{I}$ is the vector on
$S^{n-2}$ and Eq. (30) implies that $f^{I}$ generates the conformal isometry
on $S^{n-2}$. The conformal group of $S^{n-2}$ is $\mathrm{SO}(1,n-1)$, which
is the Lorentz group. Thus $f^{I}$ is the generator of the Lorentz group.

Integrating the trace part of Eq. (30), we obtain

$\displaystyle f\,=\,\frac{F(x^{I})}{n-2}u+\alpha(x^{I}),$ (32)

where $F\,\equiv\,\nabla_{I}f^{I}$ and $\alpha(x^{I})$ is the integration
function on $S^{n-2}$. Note that the transverse part $f^{\text{(tra)}I}$ of
$f^{I}$ which satisfies $\nabla_{I}f^{\text{(tra)}I}=0$ is nothing but the
Killing vector on $S^{n-2}$, that is, the generator of $\mathrm{SO}(n-1)$.
This Killing vector plays an important role in defining the angular-momentum
later.

Now Eq. (31) gives the equation which $\alpha$ should satisfy as

$\displaystyle\nabla_{I}\nabla_{J}\alpha\,=\,\frac{1}{n-2}\omega_{IJ}\nabla^{2}\alpha.$
(33)

The general solutions of this equation are the $l=0$ and $l=1$ modes of the
scalar harmonics on $S^{n-2}$. Note that the $l=1$ modes satisfy
$\nabla_{I}\nabla_{J}\alpha=-\alpha\omega_{IJ}$ too. From Eq. (31), we find
that $F(x^{I})$ should satisfy $\nabla^{2}F+(n-2)F=0$. The solutions of this
equation are the $l=1$ modes of the scalar harmonics on $S^{n-2}$. These
results mean that the functions $\alpha(x^{I})$ and $F(x^{I})$ are the
generators of the translation and Lorentz boost respectively, and $f$
represents the semi-direct property of the Lorentz group and translation. Then
it turns out that the asymptotic symmetry is the semi-direct group of the
Lorentz group and translation, which is the Poincaré group, in $n>4$
dimensions.

In four dimensions, Eqs. (29) and (30) are required while Eq. (31) is not.
Therefore $f^{I}$ generates the Lorentz group and $f$ can be written as Eq.
(32) in $n=4$ dimensions. However, there are no constraints on $\alpha$ in
four dimensions because of the absence of Eq. (31). Thus, $\alpha(x^{I})$ is
the arbitrary function on $S^{2}$ and generates so called supertranslation,
not translation. The asymptotic symmetry in four dimensions is the semi-direct
group of the Lorentz group and the supertranslation. This supertranslation
leads the ambiguity to the definition of the angular momentum at null infinity
in four dimensions.

##  III Bondi angular momentum and radiation formula

In this section, we will define the Bondi angular momentum. In the pedagogical
aspect, we describe the definition of the Bondi mass too given in our previous
work Tanabe:2011es .

###  III.1 Bondi mass and angular momentum

We define the Bondi mass and angular momentum. In the Bondi coordinates,
$g_{uu}$ and $g_{uI}$ can be expanded near null infinity as

$\displaystyle
g_{uu}\,=\,-1-\sum_{k=0}^{k<n/2-2}\frac{A^{(k+1)}}{r^{n/2+k-1}}+\frac{m(u,x^{I})}{r^{n-3}}+O(r^{-(n-5/2)})$
(34)

and

$\displaystyle
g_{uI}\,=\,\sum_{k=0}^{k<n/2-1}\frac{C_{I}^{(k+1)}}{r^{n/2+k-2}}+\frac{j_{I}(u,x^{I})+h^{(1)}_{IJ}C^{(1)J}}{r^{n-3}}+O(r^{-(n-5/2)}).$
(35)

The functions $m$ and $j_{I}$ are the integration functions and they are free
functions on the initial surface $u=u_{0}$.

The Bondi mass $M_{\text{Bondi}}$ and momentum $P^{i}_{\text{Bondi}}$ are
defined by Tanabe:2011es

$\displaystyle M_{\text{Bondi}}(u)$
$\displaystyle\equiv\frac{n-2}{16\pi}\int_{S^{n-2}}md\Omega,$ (36)
$\displaystyle P^{i}_{\text{Bondi}}(u)$
$\displaystyle\equiv\frac{n-2}{16\pi}\int_{S^{n-2}}m\hat{x}^{(i)}d\Omega,$

where $\hat{x}^{(i)}$ is the scalar function on $S^{n-2}$ satisfying
$\nabla_{I}\nabla_{J}\hat{x}^{(i)}+\omega_{IJ}\hat{x}^{(i)}\,=\,0$. These
functions are the $l=1$ modes of the scalar harmonic on $S^{n-2}$, which are
defined by $\hat{x}^{(i)}=x^{(i)}/\rho$ in the Cartesian coordinates
$\\{x^{(i)}\\}$ of the $(n-1)$-dimensional Euclidean flat space. Here
$S^{n-2}$ is embedded into the $(n-1)$-dimensional Euclidean flat space as
$\rho^{2}=\sum_{i=1}^{n-1}(x^{(i)})^{2}$. The indices $i$ represent the
directions of the translation. Note that $A^{(k+1)}$ for $k<n/2-2$ does not
contribute to the global quantities at null infinities because $A^{(k+1)}$ and
$\hat{x}^{(i)}A^{(k+1)}$ are written as the form of total derivative [see Eq.
(16)]. For the details, see Eq. (80) and Appendix B in Ref. Tanabe:2011es .

The Bondi energy-momentum vector
$P^{\mu}_{\text{Bondi}}=(M_{\text{Bondi}},P^{i}_{\text{Bondi}})$ is defined as
the $n$-dimensional vector at null infinity. In the definition of
$P^{\mu}_{\text{Bondi}}$, we introduce the $n$-dimensional vector by
$\hat{x}^{\mu}=(1,\hat{x}^{(i)})$. This vector can be naturally identified to
the bases in the $n$-dimensional Minkowski spacetime as mentioned in the next
section. Thus the Bondi energy-momentum vector $P^{\mu}_{\text{Bondi}}$ can be
also regarded as a vector in the Minkowski spacetime. In the following, the
Greek indices represent the index in Minkowski spacetime.

The Bondi angular momentum $J^{\text{Bondi}}_{(p)}$ is defined by

$J^{\text{Bondi}}_{(p)}\,=\,-\frac{n-1}{16\pi
G}\int_{S^{n-2}}\varphi^{I}_{(p)}j_{I}d\Omega,$ (37)

where $\varphi^{I}_{(p)}$ is the Killing vector of the round metric
$\omega_{IJ}$ on $S^{n-2}$. $p$ labels the Killing vectors and $1\leq
p\leq(n-1)(n-2)/2$. Note that the Bondi angular momentum in five dimensions
were defined for the Killing vectors which commute mutually in Ref.
Tanabe:2010rm . In this paper, we generalized this to define the Bondi angular
momentum in arbitrary dimensions for all Killing vectors. The
$\lfloor\frac{n-1}{2}\rfloor$ independent angular momenta are, of course,
given by the mutually commuting Killing vectors.

Here we show that the first term in Eq. (35) does not contribute to the Bondi
angular momentum. This is because $\varphi^{I}_{(p)}C^{(k+1)}_{I}$ for
$k<n/2-1$ can be written as the total derivative as

$\displaystyle\varphi^{I}_{(p)}C^{(k+1)}_{I}$ $\displaystyle=$
$\displaystyle\frac{2(n+2k-2)}{(n+2k)(n-2k-2)}\varphi^{I}_{(p)}\nabla^{J}h^{(k+1)}_{IJ}$
(38) $\displaystyle=$
$\displaystyle\frac{2(n+2k-2)}{(n+2k)(n-2k-2)}\nabla^{J}\left(\varphi^{I}_{(p)}h^{(k+1)}_{IJ}\right),$

where we used Eq. (14) and the Killing equation
$\nabla_{I}\varphi_{(p)J}+\nabla_{J}\varphi_{(p)I}=0$. The term of
$h_{IJ}^{(1)}C^{(1)J}$ in Eq. (35) is nothing, but it just comes from the
lowering of the index of the metric. Therefore, we will not think that it
contributes to the angular momentum. This will be also confirmed later when
one considers the transformation property generated by asymptotic symmetry at
null infinity (Sec. IV).

###  III.2 Radiation formula

The functions $m$ and $j_{I}$ are free functions on the initial surface
$u=u_{0}$. The evolutions of these quantities are determined from the Einstein
equations. The Einstein equation $\hat{R}^{rr}=0$ (see Eq. (16) in Ref.
Tanabe:2011es ) can be expanded near null infinity as

$\hat{R}^{rr}\,=\,\sum_{k=0}^{k<n/2-2}\frac{(\hat{R}^{rr})^{(k+1)}}{r^{n/2+k-1}}+\frac{(\hat{R}^{rr})^{(n/2-1)}}{r^{n-3}}+O\left(\frac{1}{r^{n-5/2}}\right).$
(39)

The equations $(\hat{R}^{rr})^{(k+1)}=0$ for $k<n/2-2$ provide us Eq. (22)
again and has no new informations. This feature is guaranteed by the Bianchi
identity. The equation $(\hat{R}^{rr})^{(n/2-1)}=0$ describes the evolution of
the function $m$ as

$\displaystyle\dot{m}\,=\,-\frac{1}{2(n-2)}\dot{h}_{IJ}^{(1)}\dot{h}^{(1)IJ}+\frac{n-5}{n-2}\nabla^{I}C^{(n/2-2)}_{I}+\frac{1}{n-2}\nabla^{2}A^{(n/2-2)}.$
(40)

Integrating this equation on the unit $(n-2)$-dimensional sphere, we obtain
the Bondi mass-loss law

$\displaystyle\frac{d}{du}M_{\text{Bondi}}\,=\,-\frac{1}{32\pi}\int_{S^{n-2}}\dot{h}_{IJ}^{(1)}\dot{h}^{(1)IJ}d\Omega\leq
0.$ (41)

The above implies that the Bondi mass always decreases by radiating the
gravitational waves in any dimensions. In other words, the gravitational waves
carry the positive energy flux to null infinity in any dimensions.

The Einstein equations $\hat{R}^{rI}=0$ contain the evolution equations of
$j_{I}$. $\hat{R}^{rI}$ can be expanded near null infinity as (see Eq. (19) in
Ref. Tanabe:2011es )

$\varphi_{(p)I}\hat{R}^{rI}\,=\,\sum_{k=0}^{k<n/2-1}\frac{\varphi_{(p)I}(\hat{R}^{rI})^{(k+1)}}{r^{n/2+k-1}}+\frac{\varphi_{(p)I}(\hat{R}^{rI})^{(n/2)}}{r^{n-2}}+O\left(\frac{1}{r^{n-3/2}}\right).$
(42)

The equations $\varphi_{(p)I}(\hat{R}^{rI})^{(k+1)}=0$ for $k<n/2-1$ provide
us Eq. (21) again. It is also guaranteed by the Bianchi identity. The equation
$\varphi_{(p)I}(\hat{R}^{rI})^{(n/2)}=0$ presents the evolution equation of
$j_{I}$ as

$\displaystyle-(n-1)\varphi^{I}_{(p)}\dot{j}_{I}$ $\displaystyle=$
$\displaystyle\varphi^{I}_{(p)}\Bigg{[}\partial_{u}(h^{(1)}_{IJ}\nabla_{K}h^{(1)JK})+h^{(1)JK}\nabla_{J}\dot{h}^{(1)}_{IK}+\nabla_{K}h^{(1)JK}\dot{h}^{(1)}_{IJ}+\frac{1}{2}\dot{h}^{(1)JK}\nabla_{I}h^{(1)}_{JK}\Bigg{]}$
(43)
$\displaystyle-\varphi^{I}_{(p)}\Big{[}\nabla_{I}m{-}\nabla_{I}\dot{B}^{(1)}{+}\nabla^{J}\dot{h}^{(n/2)}_{IJ}{+}(n-3)\nabla^{J}h^{(n/2-1)}_{IJ}\Big{]}+2\varphi^{I}_{(p)}\nabla^{J}\nabla_{(I}C^{(n/2-1)}_{J)}$
$\displaystyle=$
$\displaystyle\varphi^{I}_{(p)}\Bigg{[}2\dot{h}^{(1)}_{IJ}\nabla_{K}h^{(1)JK}-\nabla^{K}h^{(1)IJ}\dot{h}^{(1)}_{JK}+\frac{1}{2}\dot{h}^{(1)JK}\nabla_{I}h^{(1)}_{JK}\Bigg{]}$
$\displaystyle+\nabla_{I}\Bigg{[}\varphi^{J}_{(p)}\partial_{u}(h^{(1)}_{JK}h^{(1)IK})+\varphi^{I}_{(p)}(-m+\dot{B}^{(1)})-\varphi_{(p)J}\dot{h}^{(n/2)IJ}$
$\displaystyle-(n-3)\varphi_{(p)J}h^{(n/2-1)IJ}+\varphi^{J}_{(p)}\nabla^{I}C^{(n/2-1)}_{J}-C^{(n/2-1)}_{J}\nabla^{I}\varphi^{J}_{(p)}\Bigg{]},$

where we used the Killing equation,
$\nabla_{I}\varphi_{(p)J}+\nabla_{J}\varphi_{(p)I}=0$.

Then, we can obtain the radiation formula of the Bondi angular momentum
$J^{\text{Bondi}}_{(p)}$ as

$\displaystyle\frac{d}{du}J^{\text{Bondi}}_{(p)}\,=\,\frac{1}{16\pi
G}\int_{S^{n-2}}\varphi^{I}_{(p)}\Bigg{[}2\dot{h}^{(1)}_{IJ}\nabla_{K}h^{(1)JK}-\nabla_{K}h_{IJ}^{(1)}\dot{h}^{(1)JK}+\frac{1}{2}\dot{h}^{(1)JK}\nabla_{I}h^{(1)}_{JK}\Bigg{]}d\Omega.$
(44)

This equation shows that the Bondi angular momentum is changed when the
spacetime has time and angular dependences. The radiation formula (44) is
natural in this sense.

##  IV Poincaré covariance

In this section we consider the transformation of our Bondi mass
$M_{\text{Bondi}}$ and angular momentum $J^{\text{Bondi}}_{(p)}$ generated by
the asymptotic symmetry. The validity of our definitions of the Bondi mass and
angular momentum will be supported by the fact that the Bondi mass and angular
momentum are transformed covariantly with respect to the asymptotic symmetry.

###  IV.1 Poincaré covariance

Let us investigate the transformation rule of the Bondi energy-momentum
$P^{\mu}_{\text{Bondi}}$ and angular momentum $J^{\text{Bondi}}_{(p)}$ by the
asymptotic symmetry. In particular, we focus the cases with $f=\alpha$ and
$f^{I}=0$, which is the translation of the Poincaré group in $n>4$ dimensions.

As we mentioned in Sec. II.3, $\alpha(x^{I})$ can be decomposed into the $l=0$
and $l=1$ modes of the scalar harmonics on $S^{n-2}$. Using these harmonics as
bases $\hat{x}^{\mu}=(1,\hat{x}^{(i)})$, we can naturally introduce the
translational vector $\alpha_{\mu}$ in the $n$-dimensional Minkowski spacetime
defined by $\alpha(x^{I})=\alpha_{\mu}\hat{x}^{\mu}$. Moreover, because the
asymptotic symmetry at null infinity is the Poincaré group, we can identify
asymptotic structure at null infinity with the $n$-dimensional Minkowski
spacetime and then obtain a natural map between quantities at null infinity
and those of vector spaces in the Minkowski spacetime. Then we can discuss the
transformations of $P^{\mu}_{\text{Bondi}}$ and $J^{\text{Bondi}}_{(p)}$ by
the translational vector $\alpha_{\mu}$ in the Minkowski spacetime.

In general, the energy-momentum vector $P_{\mu}$ and angular momentum
$M_{\mu\nu}$ in the Minkowski spacetime are expected to be transformed by
translation of the Poincaré group as

$\displaystyle P_{\mu}$ $\displaystyle\rightarrow P_{\mu},$ (45)
$\displaystyle M_{\mu\nu}$ $\displaystyle\rightarrow
M_{\mu\nu}-2P_{[\mu}\alpha_{\nu]},$

where $\alpha_{\mu}$ is a translational vector. However, since the
gravitational waves carry the energy and angular momentum to null infinity,
the Bondi energy-momentum $P^{\mu}_{\text{Bondi}}$ and angular momentum
$J^{\text{Bondi}}_{(p)}$ are changed under the translation. Then, taking these
effects into account, $P^{\mu}_{\text{Bondi}}$ and $J^{\text{Bondi}}_{(p)}$
should be transformed as

$\displaystyle P_{\text{Bondi}}^{\mu}$ $\displaystyle\rightarrow
P_{\text{Bondi}}^{\mu}+\alpha_{\nu}\frac{d}{du}P^{\mu\nu}_{\text{Bondi}},$
(46) $\displaystyle M^{\text{Bondi}}_{\mu\nu}$ $\displaystyle\rightarrow
M^{\text{Bondi}}_{\mu\nu}-2P^{\text{Bondi}}_{[\mu}\alpha_{\nu]}+\alpha^{\rho}\frac{d}{du}M^{\text{Bondi}}_{\mu\nu\rho},$

instead of Eq. (45). Note that the each space-space component of
$M^{\text{Bondi}}_{\mu\nu}$ corresponds to $J^{\text{Bondi}}_{(p)}$. From now
on, we will confirm these equations. The last terms in each transformations
come from the effect of radiations and the concrete expressions will be given
later.

The generator of the translation $f=\alpha$ and $f^{I}=0$ can be expanded near
null infinity as

$\displaystyle\xi^{u}$ $\displaystyle=$ $\displaystyle\alpha(x^{I}),$ (47)
$\displaystyle\xi^{I}$ $\displaystyle=$
$\displaystyle-\frac{1}{r}\nabla^{I}\alpha+\sum_{k=0}^{k<n/2-1}\frac{2h^{(k+1)IJ}\nabla_{J}\alpha}{n+2k}\frac{1}{r^{n/2+k}}$
(48)
$\displaystyle-\frac{1}{n-1}\frac{1}{r^{n-1}}\left(B^{(1)}\nabla^{I}\alpha-h^{(n/2)IJ}\nabla_{J}\alpha{+}h^{(1)IL}h^{(1)J}_{L}\nabla_{J}\alpha\right)+O(r^{-(n-1/2)}),$
$\displaystyle\xi^{r}$ $\displaystyle=$
$\displaystyle\frac{\nabla^{2}\alpha}{n-2}-\sum_{k=0}^{k<n/2-1}\frac{2}{n+2k-2}\frac{{C^{(k+1)I}\nabla_{I}\alpha}}{r^{n/2+k-1}}+O(r^{-(n-2)}).$
(49)

###  IV.2 Covariance of Bondi energy-momentum

Following Ref. Tanabe:2011es , we briefly sketch the argument to show the
covariance of the Bondi energy-momentum. The Bondi energy-momentum
$P^{\mu}_{\text{Bondi}}$ is defined from $g_{uu}$ as in Eq. (36). To find the
variation of $m$, we look at the variation $\delta g_{uu}$. $\delta g_{uu}$
can be expanded near null infinity as

$\displaystyle\delta g_{uu}$ $\displaystyle=$ $\displaystyle
2\hat{\nabla}_{u}\xi_{u}$ (50) $\displaystyle=$
$\displaystyle\sum_{k=0}^{k<n/2-2}\delta
g^{(k+1)}_{uu}r^{-(n/2+k-1)}+\frac{\delta m}{r^{n-3}}+O(r^{-(n-5/2)}),$

where

$\displaystyle\delta g^{(k+1)}_{uu}$ $\displaystyle=$
$\displaystyle\frac{2}{n+2k}[\nabla^{2}(\alpha A^{(k)})+(n-2)\alpha
A^{(k)}]+\frac{4}{(n+2k)(n-2k-2)}\nabla^{I}\nabla^{J}(\alpha\dot{h}^{(k+1)}_{IJ})$
(51)
$\displaystyle-\frac{2(n+2k-6)}{(n+2k)(n-2k-2)}[\nabla^{I}\nabla^{J}(\nabla_{I}\alpha
C^{(k)}_{J})+C^{(k)}_{I}\nabla^{I}\alpha]$

for $0\leq k<n/2-2$. $\delta m$ is given by

$\displaystyle\delta m$ $\displaystyle=$
$\displaystyle\,\alpha\dot{m}+\frac{2}{n-3}\nabla^{I}\alpha\dot{C}_{I}^{(n/2-1)}-(n-4)\alpha
A^{(n/2-2)}+\nabla^{I}\alpha\nabla_{I}A^{(n/2-2)}$ (52) $\displaystyle=$
$\displaystyle-\frac{\alpha}{2(n-2)}\dot{h}^{(1)}_{IJ}\dot{h}^{(1)IJ}+\frac{1}{n-2}\Big{[}\nabla^{2}(\alpha
A^{(n/2-2)})+(n-2)\alpha A^{(n/2-2)}\Big{]}$
$\displaystyle-\frac{n-5}{n-2}\Big{[}\nabla^{I}\nabla^{J}(\nabla_{I}\alpha
C_{J}^{(n/2-2)})+C^{(n/2-2)}_{I}\nabla^{I}\alpha\Big{]},$

where we used Eqs. (16) and (22). From Eq. (51) and the fact that
$\hat{x}^{(i)}$ satisfies
$\nabla_{I}\nabla_{J}\hat{x}^{(i)}+\hat{x}^{(i)}\omega_{IJ}=0$, we see

$\displaystyle\int_{S^{n-2}}\delta g_{uu}^{(k+1)}\,=\,0$ (53)

and

$\displaystyle\int_{S^{n-2}}\hat{x}^{(i)}\delta g_{uu}^{(k+1)}\,=\,0$ (54)

for $k<n/2-2$. This means that $g_{uu}^{(k+1)}$ for $k<n/2-2$ does not
contribute to the global quantities in the transformed Bondi coordinates.

The variation of the Bondi energy-momentum $\delta P^{\mu}_{\text{Bondi}}$ can
be obtained by integrating Eq. (52) as

$\displaystyle\delta P^{\mu}_{\text{Bondi}}$ $\displaystyle=$
$\displaystyle\frac{n-2}{16\pi G}\int_{S^{n-2}}\hat{x}^{\mu}\delta md\Omega$
(55) $\displaystyle=$ $\displaystyle-\frac{1}{16\pi
G}\int_{S^{n-2}}\alpha\hat{x}^{\mu}\dot{h}^{(1)}_{IJ}\dot{h}^{(1)IJ}d\Omega$
$\displaystyle=$ $\displaystyle-\alpha_{\nu}\frac{1}{16\pi
G}\int_{S^{n-2}}\hat{x}^{\mu}\hat{x}^{\nu}\dot{h}^{(1)}_{IJ}\dot{h}^{(1)IJ}d\Omega.$

In the last line of the above, we used $\alpha=\alpha_{\mu}\hat{x}^{\mu}$.
Then we regard the right-hand side of this equation as
$\alpha_{\nu}dP^{\mu\nu}_{\text{Bondi}}/du$ in Eq. (46)

$\displaystyle\frac{d}{du}P^{\mu\nu}_{\text{Bondi}}=-\frac{1}{16\pi
G}\int_{S^{n-2}}\hat{x}^{\mu}\hat{x}^{\nu}\dot{h}^{(1)}_{IJ}\dot{h}^{(1)IJ}d\Omega.$
(56)

This means that the Bondi energy-momentum defined is transformed covariantly
with respect to the Poincaré group. In particular, since the time-component
becomes $dP^{\mu 0}_{\text{Bondi}}/du=dP^{\mu}_{\text{Bondi}}/du$, we have

$P^{\mu}_{\text{Bondi}}\rightarrow
P_{\text{Bondi}}^{\mu}+\alpha\frac{d}{du}P_{\text{Bondi}}^{\mu},$ (57)

for the time-translation.

###  IV.3 Poincaré covariance of Bondi angular momentum

Next we investigate the variation of the Bondi angular momentum
$J^{\text{Bondi}}_{(p)}$ by the translation. The Bondi angular momentum
$J^{\text{Bondi}}_{(p)}$ is identified with a space-space component of
$M^{\text{Bondi}}_{\mu\nu}$ in Eq. (46). Thus, in the following, we consider
the space-space components. Note that the time-space component of
$M^{\text{Bondi}}_{\mu\nu}$ represents the Lorentz boost. The $l=0$ mode of
$\alpha$ generates the time-translation and the $l=1$ modes generate the
translations in the spatial directions.

The Bondi angular momentum is defined using a part of $g_{uI}$ as Eq. (37).
The variation of the Bondi angular momentum is given by

$\displaystyle\delta J_{(p)}^{\text{Bondi}}=-\frac{n-1}{16\pi
G}\int_{S^{n-2}}\varphi^{I}_{(p)}\delta j_{I}d\Omega.$ (58)

The variation $\delta g_{uI}$ can be expanded as

$\displaystyle\delta g_{uI}$ $\displaystyle=$
$\displaystyle\hat{\nabla}_{u}\xi_{I}+\hat{\nabla}_{I}\xi_{u}$ (59)
$\displaystyle=:$ $\displaystyle\sum_{k=0}^{k<n/2-1}\frac{\delta
g^{(k+1)}_{uI}}{r^{n/2+k-2}}+\frac{\delta
g^{(n/2)}_{uI}}{r^{n-3}}+O(r^{-(n-5/2)}),$

where

$\displaystyle\delta g_{uI}^{(k+1)}$ $\displaystyle=$
$\displaystyle-A^{(k)}\nabla_{I}\alpha+\frac{2}{n+2k-4}\nabla_{I}(C^{(k)J}\nabla_{J}\alpha)-C^{(k)J}\nabla_{I}\nabla_{J}\alpha-\nabla^{J}\alpha\nabla_{J}C^{(k)}_{I}$
(60)
$\displaystyle-\frac{n+2k-6}{2(n-2)}C^{(k)}_{I}\nabla^{2}\alpha+\alpha\dot{C}_{I}^{(k+1)}+\frac{2}{n+2k}\dot{h}^{(k+1)}_{IJ}\nabla^{J}\alpha$

for $k<n/2-1$. Then we find

$\displaystyle\varphi^{I}_{(p)}\delta g_{uI}^{(k+1)}$ $\displaystyle=$
$\displaystyle\nabla_{I}\Bigg{[}\frac{2}{n+2k-4}\varphi^{I}_{(p)}C^{(k)J}\nabla_{J}\alpha-\frac{4(n-4)}{(n+2k)(n+2k-6)}\varphi^{I}_{(p)}\alpha
A^{(k)}$ (61)
$\displaystyle~{}~{}+\frac{2\Big{[}n^{2}+(4k-10)n+(4k^{2}-12k+16)\Big{]}}{(n+2k)(n-2k-2)(n+2k-4)}C^{(k)I}\varphi^{J}_{(p)}\nabla_{J}\alpha$
$\displaystyle~{}~{}-\frac{2}{n+2k}\Big{[}\alpha\varphi_{(p)J}\nabla^{I}C^{(k)J}-\varphi^{J}_{(p)}C^{(k)}_{J}\nabla^{I}\alpha-\alpha
C^{(k)}_{J}\nabla^{I}\varphi^{J}_{(p)}\Big{]}-\varphi_{(p)}^{J}C_{J}^{(k)}\nabla^{I}\alpha$
$\displaystyle~{}~{}+\frac{n+2k-4}{n+2k}\alpha\varphi_{(p)J}h^{(k)IJ}+\frac{2(n+2k-6)}{(n+2k)(n-2k-2)}h^{(k)IJ}\nabla^{K}\alpha\nabla_{K}\varphi_{(p)J}$
$\displaystyle~{}~{}+\frac{n^{2}-8n-4k^{2}+20}{(n-2)(n+2k)(n-2k-2)}\varphi_{(p)J}h^{(k)IJ}\nabla^{2}\alpha\Bigg{]},$

where we used Eqs. (14), (16) and (21). Also we used the Killing equation
$\nabla_{I}\varphi_{(p)J}+\nabla_{J}\varphi_{(p)I}=0$ and
$\nabla_{I}\nabla_{J}\alpha=\omega_{IJ}\nabla^{2}\alpha/(n-2)$. Thus we could
confirm again that $g_{uI}^{(k+1)}$ for $k<n/2-1$ does not contribute to the
global quantities because it can be written by the total derivative as Eq.
(61). For $k=n/2-1$, on the other hand, the variation becomes

$\displaystyle\delta g_{uI}^{(n/2)}$ $\displaystyle=$
$\displaystyle\delta(j_{I}+h^{(1)}_{IJ}C^{(1)J})$ (62) $\displaystyle=$
$\displaystyle\alpha\partial_{u}(j_{I}+h^{(1)}_{IJ}C^{(1)J})+\frac{2}{n}h^{(1)}_{IJ}\partial_{u}h^{(1)JL}\nabla_{L}\alpha+{\frac{1}{n-3}}\nabla_{I}(C^{(n/2-1)}_{J}\nabla^{J}\alpha)$
$\displaystyle+m\nabla_{I}\alpha-C^{(n/2-1)J}\nabla_{I}\nabla_{J}\alpha-\frac{1}{n-1}\left(\partial_{u}B^{(1)}\nabla_{I}\alpha-\dot{h}^{(n/2)}_{IJ}\nabla^{J}\alpha{+}\partial_{u}(h^{(1)}_{IJ}h^{(1)JK})\nabla_{K}\alpha\right)$
$\displaystyle-\frac{n-4}{n-2}C^{(n/2-1)}_{I}\nabla^{2}\alpha-\nabla^{J}\alpha\nabla_{J}C^{(n/2-1)}_{I}.$

We must evaluate $\delta j_{I}$ to see the variation of the Bondi angular
momentum. Therefore, we should subtract the variation
$\delta(h^{(1)}_{IJ}C^{(1)J})$ from Eq. (62). The variation $\delta C^{(1)I}$
is given by

$\displaystyle\delta
C^{(1)I}\,=\,\frac{2}{n}\nabla_{J}\alpha\dot{h}^{(1)IJ}+\alpha\dot{C}^{(1)I},$
(63)

from Eq. (60) for $k=0$. Since $\delta g_{IJ}$ is

$\displaystyle\delta g_{IJ}$ $\displaystyle=$
$\displaystyle\hat{\nabla}_{I}\xi_{J}+\hat{\nabla}_{J}\xi_{I}$ (64)
$\displaystyle=$ $\displaystyle
r^{2}\left(\frac{\alpha\dot{h}^{(1)}_{IJ}}{r^{n/2-1}}+O(r^{-(n/2-1/2)})\right),$

we find $\delta h^{(1)}_{IJ}=\alpha\dot{h}^{(1)}_{IJ}$. Then we have

$\delta(h^{(1)}_{IJ}C^{(1)J})\,=\,{\alpha\left(\dot{h}^{(1)}_{IJ}C^{(1)J}+h^{(1)}_{IJ}\dot{C}^{(1)J}\right)+\frac{2}{n}h^{(1)}_{IJ}\dot{h}^{(1)JK}\nabla_{K}\alpha.}$
(65)

Subtracting the above from Eq. (62), we obtain

$\displaystyle\varphi^{I}_{(p)}\delta j_{I}$ $\displaystyle=$
$\displaystyle\varphi^{I}_{(p)}\Bigg{[}\alpha\dot{j}_{I}+\frac{1}{n-3}\nabla_{I}(C^{(n/2-1)}_{J}\nabla^{J}\alpha)+m\nabla_{I}\alpha-C^{(n/2-1)J}\nabla_{I}\nabla_{J}\alpha$
(66)
$\displaystyle~{}~{}-\frac{1}{n-1}\left(\dot{B}^{(1)}\nabla_{I}\alpha-\dot{h}^{(n/2)}_{IJ}\nabla^{J}\alpha+\partial_{u}(h^{(1)}_{IJ}h^{(1)JK})\nabla_{K}\alpha\right)$
$\displaystyle~{}~{}-\frac{n-4}{n-2}C^{(n/2-1)}_{I}\nabla^{2}\alpha-\nabla^{J}\alpha\nabla_{J}C^{(n/2-1)}_{I}\Bigg{]}$
$\displaystyle=$
$\displaystyle-\frac{\alpha}{n-1}\varphi^{I}_{(p)}\Bigg{[}2\dot{h}^{(1)}_{IJ}\nabla_{K}h^{(1)JK}-\nabla_{K}h_{IJ}^{(1)}\dot{h}^{(1)JK}+\frac{1}{2}\dot{h}^{(1)JK}\nabla_{I}h^{(1)}_{JK}\Bigg{]}+\frac{n-2}{n-1}m\varphi^{I}_{(p)}\nabla_{I}\alpha$
$\displaystyle+\nabla_{I}\Bigg{[}-\frac{1}{n-1}\Big{[}\alpha\varphi^{I}_{(p)}(\dot{B}^{(1)}-m)-\alpha\varphi_{(p)J}\dot{h}^{(n/2)IJ}+\partial_{u}(h^{(1)}_{IJ}h^{(1)JK})\nabla_{K}\alpha$
$\displaystyle+\alpha\varphi^{J}_{(p)}\nabla^{I}C^{(n/2-1)}_{J}-\alpha
C^{(n/2-1)}_{J}\nabla^{I}\varphi^{J}_{(p)}\Big{]}+\frac{1}{n-3}\varphi^{I}_{(p)}C^{(n/2-1)J}\nabla_{J}\alpha-\frac{n-2}{n-1}\varphi^{J}C^{(n/2-1)}_{J}\nabla^{I}\alpha$
$\displaystyle+\frac{n-3}{n-1}\Big{[}\varphi_{(p)J}h^{(n/2-1)IJ}\alpha+\frac{1}{(n-2)^{2}}\varphi_{(p)J}h^{(n/2-1)IJ}\nabla^{2}\alpha+\frac{n-3}{n-2}h^{(n/2-1)IJ}\nabla_{K}\varphi_{(p)J}\nabla^{K}\alpha\Big{]}\Bigg{]},$

where we used Eq. (43), the Killing equation
$\nabla_{I}\varphi_{(p)J}+\nabla_{J}\varphi_{(p)I}=0$ and
$\nabla_{I}\nabla_{J}\alpha=\omega_{IJ}\nabla^{2}\alpha/(n-2)$.

Using Eq. (66), the variation of the Bondi angular momentum $\delta
J^{\text{Bondi}}_{(p)}$ becomes

$\displaystyle\delta J^{\text{Bondi}}_{(p)}$ $\displaystyle=$
$\displaystyle-\frac{n-1}{16\pi G}\int_{S^{n-2}}\varphi^{I}_{(p)}\delta
j_{I}d\Omega$ (67) $\displaystyle=$ $\displaystyle\frac{1}{16\pi
G}\int_{S^{n-2}}\alpha\varphi^{I}_{(p)}\Bigg{[}2\dot{h}^{(1)}_{IJ}\nabla_{K}h^{(1)JK}-\nabla_{K}h_{IJ}^{(1)}\dot{h}^{(1)JK}+\frac{1}{2}\dot{h}^{(1)JK}\nabla_{I}h^{(1)}_{JK}\Bigg{]}d\Omega$
$\displaystyle-\frac{n-2}{16\pi
G}\int_{S^{n-2}}m\varphi^{I}_{(p)}\nabla_{I}\alpha d\Omega.$

Note that the total derivative terms in Eq. (66) do not contribute to $\delta
J^{\text{Bondi}}_{(p)}$.

From the result of Eq. (67), we can show that Eq. (46) holds as follows. We
note that a rotational Killing vector $\varphi^{I}_{(p)}$ can be rewritten as
$\varphi^{I}_{(p)}=\varphi_{ij}\hat{x}^{(i)}\nabla^{I}\hat{x}^{(j)}$ where
$\varphi_{ij}$ is a constant anti-symmetric tensor with $(n-1)(n-2)/2$
independent components. Using the relation
$\nabla_{I}\hat{x}^{(i)}\nabla^{I}\hat{x}^{(j)}=\delta^{ij}-\hat{x}^{(i)}\hat{x}^{(j)}$,
we have
$\varphi^{I}_{(p)}\nabla_{I}\alpha=\varphi_{ij}\alpha^{j}\hat{x}^{(i)}$. Since
$J^{\text{Bondi}}_{(p)}$ is expressed by
$J^{\text{Bondi}}_{(p)}=\varphi^{ij}M^{\text{Bondi}}_{ij}$, Eq. (67) yields

$\varphi^{ij}M^{\text{Bondi}}_{ij}\rightarrow\varphi^{ij}\Big{[}M^{\text{Bondi}}_{ij}-2P^{\text{Bondi}}_{[i}\alpha_{j]}+\alpha^{\mu}\frac{d}{du}M^{\text{Bondi}}_{ij\mu}\Big{]},$
(68)

where we wrote

$\varphi^{ij}\frac{d}{du}M^{\text{Bondi}}_{ij\mu}=\frac{1}{16\pi
G}\int_{S^{n-2}}\hat{x}_{\mu}\varphi^{I}_{(p)}\Bigg{[}2\dot{h}^{(1)}_{IJ}\nabla_{K}h^{(1)JK}-\nabla_{K}h_{IJ}^{(1)}\dot{h}^{(1)JK}+\frac{1}{2}\dot{h}^{(1)JK}\nabla_{I}h^{(1)}_{JK}\Bigg{]}d\Omega.$
(69)

Note that the indices $\mu$, $\nu$, $\dots$ and $i$, $j$, $\dots$ are raised
and lowered by the $n$-dimensional Minkowski metric and the
$(n-1)$-dimensional Euclidean flat metric, respectively. Consequently, we
could show that the Bondi angular momentum $J^{\text{Bondi}}_{(p)}$ is
transformed covariantly as Eq. (46).

Here we have a comment on the four dimensional cases. Because there is no
condition on $\alpha$ in four dimensions, we cannot obtain the expressions
corresponding to Eqs. (66) and (67). Therefore the Bondi angular momentum is
not transformed as Eq. (68) in four dimensions. In fact, the variation of the
Bondi angular momentum has additional contributions from supertranslations,
which are given by $l>1$ modes of spherical harmonics in $\alpha$. This is
called the supertranslation ambiguity of the angular momentum at null
infinity. Hence, we cannot have well-defined notion of the angular momentum at
null infinity in four dimensions.

##  V summary and outlook

In this paper we defined the Bondi angular momentum at null infinity in
arbitrary higher dimensions and showed its covariant property with respect to
the asymptotic symmetry at null infinity. The asymptotic symmetry becomes the
Poincaré group in higher dimensions than four. This means that we can choose
the $n$ directions of the translation without any ambiguities at null
infinity. Then the angular momentum with the rotational axis can be defined.
In four dimensions, on the other hand, the asymptotic symmetry at null
infinities has the supertranslation, not the translation. The supertranslation
has the infinite directions of the translation. Hence there are ambiguities of
the choice of the rotational axis. This effects of the freedom in the
definition of the rotational axis due to the supertranslation cannot be
distinguished from the contributions of the variation of angular momentum by
gravitational waves. This is the reason why we cannot define the angular
momentum at null infinity in four dimensions.

As one of applications of our analysis, there is the investigation of the
peeling theorem in higher dimensions Bondi:1962px ; Sachs:1962wk ;
Godazgar:2012zq . The peeling property has played an important role in the
study of the gravity in four dimensions, such as the stability analysis of
black holes and construction of the exact solutions. We expect that the
peeling theorem is useful in higher dimensions too. Using our results, general
higher dimensional spacetimes with gravitational waves are classified by the
decaying rate of the Weyl tensor or some geometric quantities. The effort for
this direction has been reported Godazgar:2012zq .

## Acknowledgment

KT is supported by JSPS Grant-in-Aid for Scientific Research (No. 21-2105).
This work is supported in part by MEXT thorough Grant-in-Aid for Scientific
Research (A) No. 21244033 (TS) and Grant-in-Aid for Creative Scientific
Research No. 19GS0219 (TS and SK). This work is also supported in part by MEXT
through Grant-in-Aid for the Global COE Program “The Next Generation of
Physics, Spun from Universality and Emergence” at Kyoto University.

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