# On the angular momentum of compact binary coalescence

Xiaokai He<EMAIL_ADDRESS>School of Mathematics and Statistics,
Hunan First Normal University, Changsha 410205, China  Xiaoning Wu
<EMAIL_ADDRESS>Institute of Mathematics, Academy of Mathematics and Systems
Science and Hua Loo-Keng Key Laboratory, Chinese Academy of Sciences, Beijing
100190, China  Naqing Xie<EMAIL_ADDRESS>School of Mathematical Sciences,
Fudan University, Shanghai 200433, China

###### Abstract

The supertranslation ambiguity issue of angular momentum is a long-standing
problem in general relativity. Recently, there appeared the first definition
of angular momentum at null infinity that is supertranslation invariant.
However, in the compact binary coalescence community, supertranslation
ambiguity is often ignored. This paper demonstrates that we have the happy
circumstance that the newly defined angular momentum coincides with the
classical definition at the quadrupole level.

###### pacs:

04.30.-w, 04.20.Cv

##  I Introduction

Gravitational radiation is an important prediction of general relativity (GR).
Due to the complexity of diffeomorphisms, the existence of gravitational waves
(GW) became theoretically controversial since the genesis of GR Ken07 . Bondi,
van der Burg, Metzner and Sachs established the mathematical foundations of
the theory of GW in the 1960s Bondi1962 ; Sachs1962 . The Bondi-Sachs
framework describes the asymptotic structure and asymptotic symmetries near
the future null infinity of asymptotic flat (AF) spacetimes. In their seminal
works, Bondi et al. successfully defined the mass of an AF spacetime and
obtained the well-known Bondi mass loss formula. However, the asymptotic
symmetric group is not the Poincaré group but an infinite dimensional analogue
thereof, the Bondi-Metzner-Sachs (BMS) group. While the BMS group is
structurally similar to the Poincaré group, the subgroup of translations is
extended by a subgroup of supertranslations. The BMS group has a 4-dimensional
Abelian canonical subgroup which is called the translation subgroup. This
allows the well-defined notions of the gravitational energy and the linear
momentum and it ensures that the mass loss formula is invariant under
supertranslations. Absence of the canonical $SO(3)$ subgroup causes the
ambiguity in the definition of angular momentum and consequently the Bondi
angular momentum flux is not supertranslation invariant. The subtle
supertranslation ambiguity of the angular momentum has been emphasized in the
mathematical general relativity community over the past couple of decades
Penrose1982 . There exist many candidates for the notions of the angular
momentum in the literature Wini68 ; Bram75 ; Pri77 ; Str78 ; Ge81 ; Asht81 ;
Asht82 ; Dr84 ; Dr85 ; Wa00  and unfortunately they are not supertranslation
invariant. Recently, Po-Ning Chen, Mu-Tao Wang, Ye-Kai Wang and Shing-Tung Yau
investigated the first definition of angular momentum at null infinity that is
supertranslation invariant CWY2021-2 . Different from the existing notions of
the angular momentum, there is an important correction term in their newly
defined expression. This additional term, which comes from solving the optimal
isometric embedding equation for the Wang-Yau quasilocal mass WY09 , is
closely related to the Bondi mass aspect and the closed potential of the Bondi
shear tensor.

More and more gravitational events have been confirmed and a completely new
window on the universe has been opened in the astronomical physics
Abbott2016-1 ; Abbott2016-2 ; Abbott2017-1 ; Abbott2019 ; CCG2017 ; HJC2019 .
In particular, on September 14, 2015 at 09:50:45 UTC, the two detectors of the
Laser Interferometer Gravitational-wave Observatory (LIGO) observed the binary
black hole merge Abbott2016-1 . Binary systems are among the most important
sources of GW which are expected to be detected by the current or foreseeable
gravitational wave detectors. The supertranslation ambiguity is often ignored
in the compact binary coalescence (CBC) community in practice. The
supermomentum contribution is small for the normally considered kick velocity.
The detailed analysis was addressed in Asht20-1 ; Asht20-2 .

In this paper, we prove that, in the restricted context of compact binary
coalescence, the newly defined angular momentum coincides with the classical
definition at the quadrupole level. This result explicitly shows that we have
the happy circumstance that the supertranslation ambiguity can be safely
ignored in practice albeit it is conceptually important and mathematically
difficult.

We always assume that the spacetime admits a conformal completion at null
infinity à la Penrose Pen65 . Throughout this paper, we make use of the
convention that the speed of light and the universal gravitational constant
are taken as fundamental units in a system. The signature of the spacetime
metric is assumed to be $(-,+,+,+)$. Unless otherwise stated, Greek indices
are used to label spacetime dimensions, lower case Latin indices are reserved
for three-dimensional objects, and capital ones are for two-dimensional
objects.

The paper is organized as follows. The notions of the total angular momentum,
both the classical and the newly defined ones, are recalled in Section II. In
Section III, we derive the coordinate transformation between harmonic
coordinates and the Bondi coordinates, since the known results of the linear
gravitational wave are formulated in the harmonic gauge while the notion of
the angular momentum relies on the Bondi coordinates. In Section IV, we prove
that, in the linearised theory of GW for the compact binary coalescence, the
correction term in the new expression indeed gives zero contribution in the
flux integral at the quadrupole level. Conclusions and discussions are
presented in the last section.

##  II Review of the notions of the angular momentum

In this section, we give a quick review of the notions of the angular
momentum. Within a Bondi-Sachs (BS) coordinate system $(u,r_{\rm
BS},x^{2},x^{3})$, the physical spacetime metric takes the form

$\begin{split}\mathrm{d}s^{2}=&-UV\mathrm{d}u^{2}-2U\mathrm{d}u\mathrm{d}r_{\rm
BS}\\\
&+r^{2}h_{AB}(\mathrm{d}x^{A}+W^{A}\mathrm{d}u)(\mathrm{d}x^{B}+W^{B}\mathrm{d}u)\end{split}$
(II.1)

where $A,B=2,3$. The future null infinity, denoted by $\mathscr{I}^{+}$,
corresponds to the null hypersurface $r_{\rm BS}=\infty$. The outgoing
radiation condition implies the following expansions in inverse powers of
$r_{\rm BS}$ CJK2002 ; CWY2021-2 :

$\begin{split}h_{AB}=&q_{AB}+\frac{C_{AB}}{r_{\rm BS}}+\frac{1}{4r_{\rm
BS}^{2}}|C|^{2}q_{AB}+O(r_{\rm BS}^{-3}),\\\ W^{A}=&\frac{1}{2r_{\rm
BS}^{2}}\nabla_{B}C^{AB}+\frac{1}{r_{\rm
BS}^{3}}\bigg{(}\frac{2}{3}N^{A}-\frac{1}{16}\nabla^{A}|C|^{2}\\\
&-\frac{1}{2}C^{AB}\nabla^{D}C_{BD}\bigg{)}+O(r_{\rm BS}^{-4}),\\\
U=&1-\frac{1}{16r_{\rm BS}^{2}}|C|^{2}+O(r_{\rm BS}^{-3}),\\\
V=&1-\frac{2m}{r_{\rm BS}}+\frac{1}{r_{\rm
BS}^{2}}\bigg{(}\frac{1}{3}\nabla^{A}N_{A}+\frac{1}{4}\nabla^{A}C_{AB}\nabla_{D}C^{BD}\\\
&+\frac{1}{16}|C|^{2}\bigg{)}+O(r_{\rm BS}^{-3}).\end{split}$ (II.2)

Here $q_{AB}$ is the round metric on the unit sphere $S^{2}$, i.e.,

$q_{AB}dx^{A}dx^{B}=\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}$
(II.3)

and $\nabla_{A}$ denotes the covariant derivative associated with $q_{AB}$.
The indices are contracted, raised and lowered with respect to $q_{AB}$.
Moreover, $m=m(u,x^{A})$ is the mass aspect, $N_{A}=N_{A}(u,x^{A})$ is the
angular aspect and $C_{AB}=C_{AB}(u,x^{A})$ is the shear tensor of this BS
coordinates. The news tensor $N_{AB}$ is defined as

$N_{AB}=\partial_{u}C_{AB}.$ (II.4)

It is well known that the evolution of the mass aspect function is

$\partial_{u}m=-\frac{1}{8}N_{AB}N^{AB}+\frac{1}{4}\nabla^{A}\nabla^{B}N_{AB}.$
(II.5)

The evolution of $N_{A}$ reads CJK2002 ; CWY2021-1

$\begin{split}\partial_{u}N_{A}=&\nabla_{A}m-\frac{1}{4}\nabla^{D}(\nabla_{D}\nabla^{E}C_{EA}-\nabla_{A}\nabla^{E}C_{ED})\\\
&+\frac{1}{4}\nabla_{A}(C_{BE}N^{BE})-\frac{1}{4}\nabla_{B}(C^{BD}N_{DA})\\\
&+\frac{1}{2}C_{AB}\nabla_{D}N^{DB}.\end{split}$ (II.6)

The standard formula for the BS energy-momentum at a $u$ cut along
$\mathscr{I}^{+}$ are

$\begin{split}E(u)&=\frac{1}{8\pi}\int_{S^{2}}2m(u,\cdot),\\\
P^{k}(u)&=\frac{1}{8\pi}\int_{S^{2}}2m(u,\cdot)\tilde{X}^{k},\ \
(k=1,2,3),\end{split}$ (II.7)

where

$\tilde{X}^{k}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$
(II.8)

is the position vector of the standard unit sphere $S^{2}$ in
$\mathbb{R}^{3}$.

A BMS field $Y$ is called a rotation BMS field if in a BS coordinate system,

$Y=\hat{Y}^{A}\frac{\partial}{\partial x^{A}}$ (II.9)

where $\hat{Y}^{A}$ is a rotation Killing vector field on $S^{2}$. Precisely,
the three rotation Killing vector fields on $S^{2}$ are

$\begin{split}\hat{Y}_{1}&=\sin\varphi\frac{\partial}{\partial\theta}+\cot\theta\cos\varphi\frac{\partial}{\partial\varphi},\\\
\hat{Y}_{2}&=\cos\varphi\frac{\partial}{\partial\theta}-\cot\theta\sin\varphi\frac{\partial}{\partial\varphi},\\\
\hat{Y}_{3}&=\frac{\partial}{\partial\varphi}.\end{split}$ (II.10)

For a rotation BMS field $Y$ that is tangent to $u$ cuts on $\mathscr{I}^{+}$,
the classical Bondi angular momentum of a $u$ cut is defined as

$\tilde{J}(u,Y)=\frac{1}{8\pi}\int_{S^{2}}Y^{A}\bigg{(}N_{A}-\frac{1}{4}C_{AB}\nabla_{D}C^{DB}\bigg{)}.$
(II.11)

This definition is that of Dray-Streubel in Dr84  and it does not satisfy the
supertranslation invariance property. Very recently, Po-Ning Chen, Mu-Tao
Wang, Ye-Kai Wang and Shing-Tung Yau investigated a new notion of the angular
momentum that is supertranslation invariant CWY2021-2 . Aiming at eliminating
the supertranslation ambiguity, they noticed that the shear tensor (as a
symmetric traceless tensor) can be decomposed into

$C_{AB}=\nabla_{A}\nabla_{B}c-\frac{1}{2}q_{AB}\Delta
c+\frac{1}{2}(\epsilon_{A}^{\
E}\nabla_{E}\nabla_{B}\underline{c}+\epsilon_{B}^{\
E}\nabla_{E}\nabla_{A}\underline{c})$ (II.12)

where $\epsilon_{AB}$ is the volume form of $q_{AB}$. Here $c$ and
$\underline{c}$ are the so-called closed and co-closed potentials of $C_{AB}$.
The potentials $c$ and $\underline{c}$ are uniquely determined by $C_{AB}$ if
they are chosen to be of $l\geq 2$ harmonic modes.

For a rotation BMS field $Y$ that is tangent to $u$ cuts on $\mathscr{I}^{+}$,
the new angular momentum of a $u$ cut is defined as CWY2021-2

$J(u,Y)=\frac{1}{8\pi}\int_{S^{2}}Y^{A}\bigg{(}N_{A}-\frac{1}{4}C_{AB}\nabla_{D}C^{DB}-c\nabla_{A}m\bigg{)}.$
(II.13)

Comparing with the classical angular momentum (II.11), there appears an
additional correction term in the new expression of the angular momentum

$\delta
J(u,Y):=J(u,Y)-\tilde{J}(u,Y)=-\frac{1}{8\pi}\int_{S^{2}}cY^{A}\nabla_{A}m.$
(II.14)

##  III Bondi coordinates for the linear gravitational wave

Consider a perturbation of the Minkowski spacetime in which the metric can be
decomposed as

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},\ \ \ |h_{\mu\nu}|\ll 1.$ (III.1)

It is convenient to introduce the trace-reversed perturbation
$\bar{h}_{\mu\nu}$ by

$\bar{h}_{\mu\nu}:=h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h.$ (III.2)

Up to the linear order in $h_{\mu\nu}$, the Einstein equation

$Ric(g)_{\mu\nu}-\frac{1}{2}Scal(g)g_{\mu\nu}=8\pi T_{\mu\nu}$ (III.3)

becomes

$\Box\bar{h}_{\mu\nu}+\eta_{\mu\nu}\partial^{\rho}\partial^{\sigma}\bar{h}_{\rho\sigma}-\partial^{\rho}\partial_{\nu}\bar{h}_{\mu\rho}-\partial^{\rho}\partial_{\mu}\bar{h}_{\nu\rho}=-{16\pi}T_{\mu\nu}.$
(III.4)

where $\Box=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$ and the indices are
raised and lowered by $\eta_{\mu\nu}.$

Under the harmonic gauge condition

$\partial^{\nu}\bar{h}_{\mu\nu}=0,$ (III.5)

the linearised Einstein equation reads

$\Box\bar{h}_{\mu\nu}=-{16\pi}T_{\mu\nu}.$ (III.6)

By using the Green function method, the linearised equation (III.6) can be
solved and the result is

$\bar{h}_{\mu\nu}(t,\vec{x})={4}\int\frac{T_{\mu\nu}(t-{|\vec{x}-\vec{x}^{\prime}|},\vec{x}^{\prime})}{|\vec{x}-\vec{x}^{\prime}|}\mathrm{d}^{3}\vec{x}^{\prime}.$
(III.7)

At large distance from the compact source, we have the expansion

$\bar{h}_{\mu\nu}(t,\vec{x})=\frac{4}{r}\int
T_{\mu\nu}(t-r+{\vec{x}^{\prime}\cdot\hat{n}},\vec{x}^{\prime})+O(\frac{1}{r^{2}}),$
(III.8)

where $r=|\vec{x}|$ and $\hat{n}=\frac{\vec{x}}{r}.$

In terms of the momenta of the energy-momentum tensor $T_{\mu\nu}$,

$\begin{split}S^{\mu\nu}(t)&:=\int T^{\mu\nu}(t,\vec{x})\mathrm{d}^{3}x,\\\
S^{\mu\nu,k}(t)&:=\int x^{k}T^{\mu\nu}(t,\vec{x})\mathrm{d}^{3}x,\\\
S^{\mu\nu,kl}(t)&:=\int x^{k}x^{l}T^{\mu\nu}(t,\vec{x})\mathrm{d}^{3}x,\\\
\end{split}$ (III.9)

it follows from (III.8) that

$\bar{h}^{\mu\nu}(t,\vec{x})=\frac{4}{r}\bigg{[}S^{\mu\nu}+n_{m}\dot{S}^{\mu\nu,m}+\frac{1}{2}n_{m}n_{p}\ddot{S}^{\mu\nu,mp}+\cdots\bigg{]}_{\rm
ret}$ (III.10)

where the subscript ‘ret’ means the quantities are evaluated at the retarded
time $t_{R}=t-{r}$ Maggiore2008 . Furthermore, via the linearised conservation
law

$\partial_{\mu}T^{\mu\nu}=0,$ (III.11)

one shows that the leading term of $\bar{h}_{ij}$ is

$\bar{h}_{ij}(t,\vec{x})=\frac{2}{r}\ddot{M}^{ij}(t_{R})$ (III.12)

and

$M^{ij}=\int x^{i}x^{j}{T^{00}}\mathrm{d}^{3}x$ (III.13)

is the mass quadrupole of the source.

Now we consider a binary system with rest masses $M_{1}$ and $M_{2}$ and
further assume that the relative coordinate is performing a circular motion.
By appropriately choosing the $(x,y,z)$ coordinates so that the orbit lies in
the $(x,y)$ plane, the relative motion $\vec{x}_{0}:=\vec{x}_{1}-\vec{x}_{2}$
is given by

$\begin{split}x_{0}(t)&=R_{0}\cos(\omega_{s}t+\frac{\pi}{2}),\\\
y_{0}(t)&=R_{0}\sin(\omega_{s}t+\frac{\pi}{2}),\\\ z_{0}(t)&=0,\end{split}$
(III.14)

where $R_{0}$ is the distance between the two bodies and $\omega_{s}$ is the
frequency of the motion. It is well known that, in the center-of-mass (CM)
frame, the mass quadrupole of a binary system is given by Maggiore2008

$M^{ij}(t)=\mu x_{0}^{i}x_{0}^{j}$ (III.15)

where $\mu=\frac{M_{1}M_{2}}{M_{1}+M_{2}}$ is the reduced mass of the binary
system. Combining (III.12), (III.14) and (III.15), one finds that the nonzero
components of $\bar{h}_{ij}$ are

$\begin{split}\bar{h}_{11}(t,\vec{x})&=-\bar{h}_{22}(t,\vec{x})=\frac{8\mu
R_{0}^{2}\omega_{s}^{2}}{r}\cos 2\omega_{s}t_{R},\\\
\bar{h}_{12}(t,\vec{x})&=\bar{h}_{21}(t,\vec{x})=\frac{8\mu
R_{0}^{2}\omega_{s}^{2}}{r}\sin 2\omega_{s}t_{R}.\end{split}$ (III.16)

Moreover, the leading order term of $\bar{h}_{00}$ and $\bar{h}_{0i}$ can be
computed from (III.10) and the results are

$\begin{split}\bar{h}_{00}&=\frac{4}{r}(M+\frac{1}{2}\ddot{M}_{ij}n^{i}n^{j}),\\\
\bar{h}_{0i}&=-\frac{2}{r}\ddot{M}_{ij}n^{j}.\end{split}$

More explicitly,

$\begin{split}\bar{h}_{00}&=\frac{4}{r}(M+F),\\\ \bar{h}_{01}&=-\frac{8\mu
R_{0}^{2}\omega_{s}^{2}}{r}\cos(2\omega_{s}t_{R}-\varphi)\sin\theta,\\\
\bar{h}_{02}&=-\frac{8\mu
R_{0}^{2}\omega_{s}^{2}}{r}\sin(2\omega_{s}t_{R}-\varphi)\sin\theta,\\\
\bar{h}_{03}&=0,\end{split}$ (III.17)

with

$M=M_{1}+M_{2}$

and

$\begin{split}F&=\frac{1}{2}\ddot{M}_{ij}n^{i}n^{j}\\\ &=\mu
R_{0}^{2}\omega^{2}_{s}\bigg{[}\cos(2\omega_{s}t_{R}-2\varphi)-\frac{1}{2}\cos(2\omega_{s}t_{R}-2\theta-2\varphi)\\\
&-\frac{1}{2}\cos(2\omega_{s}t_{R}+2\theta-2\varphi)\bigg{]}.\end{split}$

Eqs. (III.16) and (III.17) give the form of the metric as

$g_{\mu\nu}^{(H)}=\left(\begin{array}[]{cccc}-1+\frac{2M}{r}+\frac{2F}{r}&-\frac{8\mu
R_{0}^{2}\omega_{s}^{2}\cos(2\omega_{s}t_{R}-\varphi)\sin\theta}{r}&-\frac{8\mu
R_{0}^{2}\omega_{s}^{2}\sin(2\omega_{s}t_{R}-\varphi)\sin\theta}{r}&0\\\
-\frac{8\mu
R_{0}^{2}\omega_{s}^{2}\cos(2\omega_{s}t_{R}-\varphi)\sin\theta}{r}&1+\frac{8\mu
R_{0}^{2}\omega_{s}^{2}\cos
2\omega_{s}t_{R}}{r}+\frac{2M}{r}+\frac{2F}{r}&\frac{8\mu
R_{0}^{2}\omega_{s}^{2}\sin 2\omega_{s}t_{R}}{r}&0\\\ -\frac{8\mu
R_{0}^{2}\omega_{s}^{2}\sin(2\omega_{s}t_{R}-\varphi)\sin\theta}{r}&\frac{8\mu
R_{0}^{2}\omega_{s}^{2}\sin 2\omega_{s}t_{R}}{r}&1-\frac{8\mu
R_{0}^{2}\omega_{s}^{2}\cos
2\omega_{s}t_{R}}{r}+\frac{2M}{r}+\frac{2F}{r}&0\\\
0&0&0&1+\frac{2M}{r}+\frac{2F}{r}\\\ \end{array}\right).$ (III.18)

Here the script ‘(H)’ means that the metric is expressed in the harmonic
coordinates. To calculate the correction term $\delta J$ in the new angular
momentum expression, the above metric form should be transformed into certain
Bondi-Sachs (BS) form. In a BS form,

$\begin{split}g_{(\rm BS)}^{uu}&=0,\ \ g_{(\rm BS)}^{uA}=0,\\\ g^{ur}_{(\rm
BS)}&=-1+O(\frac{1}{r_{\rm BS}^{2}}),\\\ g^{rr}_{(\rm BS)}&=1-\frac{2m}{r_{\rm
BS}}+O(\frac{1}{r_{\rm BS}^{2}}),\\\ g^{rA}_{(\rm BS)}&=\frac{1}{2r_{\rm
BS}^{2}}\nabla_{B}C^{AB}+O(\frac{1}{r_{\rm BS}^{3}}),\\\ g^{AB}_{(\rm
BS)}&=\frac{1}{r_{\rm BS}^{2}}q^{AB}-\frac{1}{r_{\rm
BS}^{3}}C^{AB}+O(\frac{1}{r_{\rm BS}^{4}}).\end{split}$ (III.19)

The coordinate transformation can be done in the following way. Within the
harmonic coordinate system, in general, we have

$\begin{split}g^{(H)}_{00}=&-1+\frac{2M}{r}+\frac{1}{r}H_{00}(t-r,\theta,\varphi),\\\
g^{(H)}_{0i}=&\frac{1}{r}H_{0i}(t-r,\theta,\varphi),\\\
g_{ij}^{(H)}=&\delta_{ij}+\frac{2M}{r}\delta_{ij}+\frac{1}{r}H_{ij}(t-r,\theta,\varphi).\end{split}$
(III.20)

For the discussion of gravitational radiation, we only need to work up to the
order of $\frac{1}{r}$ in the harmonic coordinates and determine the
corresponding BS coordinates such that the Bondi gauge is imposed up to the
order of $\frac{1}{r_{B}}$ beyond the leading order metric. To perform the
coordinate transformation, it is more convenient to work with the components
of the inverse metric. In the harmonic gauge, direct calculation shows that
the components of the inverse metric are given by

$\begin{split}g^{00}_{(H)}=&-1-\frac{2M}{r}-\frac{1}{r}H^{00}+O(\frac{1}{r^{2}}),\\\
g^{0i}_{(H)}=&-\frac{1}{r}H^{0i}+O(\frac{1}{r^{2}}),\\\
g^{ij}_{(H)}=&(1-\frac{2M}{r})\delta_{ij}-\frac{H^{ij}}{r}+O(\frac{1}{r^{2}}).\end{split}$
(III.21)

where

$H^{\mu\nu}=\eta^{\mu\alpha}\eta^{\nu\beta}H_{\alpha\beta}.$ (III.22)

The coordinate transformations from the harmonic gauge to the BS gauge are
split into two steps. Firstly, we imposes the gauge conditions on the inverse
Bondi metric so that $g^{uu}_{({\rm BS})}$ and $g^{uA}_{({\rm BS})}$ vanish
with the accuracy of order $\frac{1}{r_{\rm BS}}$. It also makes $r_{\rm BS}$
an areal radius and relates the BS coordinate $\tilde{u}$ to the harmonic
gauge retarded time $t-r$.

Let

$\begin{split}\tilde{u}=&t-{r}-{2M}\ln\frac{r}{r_{0}},\\\ \tilde{r}=&r+{M},\\\
\tilde{\theta}=&\theta=\arccos\frac{z}{r},\\\
\tilde{\varphi}=&\varphi=\arctan\frac{y}{x}.\end{split}$ (III.23)

The coordinates $\\{\tilde{u},\tilde{r},\tilde{\theta},\tilde{\varphi}\\}$ are
not precisely the BS coordinates up to the order of $1/\tilde{r}$ since the
metric does not satisfy all the requirements of the Bondi gauge Tahura2021 .
We will thus denote by $g^{\mu\nu}_{(I)}$ this ‘intermediate’ metric. More
precisely, $g^{\mu\nu}_{(I)}$ can be computed from the harmonic gauge metric
(III.21) using the transformation law

$g^{\mu\nu}_{(I)}=g^{\alpha\beta}_{(H)}\frac{\partial\tilde{x}^{\mu}}{\partial
x^{\alpha}}\frac{\partial\tilde{x}^{\nu}}{\partial x^{\beta}}.$ (III.24)

Direct calculation gives

$g^{00}_{(I)}=-\frac{1}{r}[H_{00}+2H_{0i}n^{i}+H_{ij}n^{i}n^{j}]+O(\frac{1}{r^{2}}).$
(III.25)

For the metric (III.18), one can read off $H_{00}$, $H_{0i}$ and $H_{ij}$ from
(III.20) and this immediately yields $H_{00}+2H_{0i}n^{i}+H_{ij}n^{i}n^{j}=0$.

In general, recall that the harmonic gauge condition gives Tahura2021

$H_{00}+H=n^{i}n^{j}H_{ij},$ (III.26)

and

$H_{0i}=-n^{j}H_{ij}+\frac{H}{2}n^{i}.$ (III.27)

Then $g^{00}_{(I)}$ can be simplified as

$\begin{split}g^{00}_{(I)}&=-\frac{1}{r}[H_{00}+2(-n^{i}n^{j}H_{ij}+\frac{H}{2})+H_{ij}n^{i}n^{j}]+O(\frac{1}{r^{2}})\\\
&=-\frac{1}{r}[H_{00}-n^{i}n^{j}H_{ij}+H]+O(\frac{1}{r^{2}})\\\
&=O(\frac{1}{r^{2}})=O(\frac{1}{\tilde{r}^{2}}),\end{split}$ (III.28)

which implies that the Bondi gauge condition ${g^{uu}=0}$ is satisfied at the
required order. Similarly, we have

$\begin{split}g^{0A}_{(I)}&=O(\frac{1}{\tilde{r}^{3}}),\\\
g^{01}_{(I)}&=-1+{\frac{1}{\tilde{r}}\frac{H}{2}}+O(\frac{1}{\tilde{r}^{2}}),\\\
g^{11}_{(I)}&=1-\frac{1}{\tilde{r}}(2M+{H_{ij}n^{i}n^{j}})+O(\frac{1}{\tilde{r}^{2}}),\\\
g^{1A}_{(I)}&=\frac{1}{\tilde{r}^{2}}{H_{0i}\partial^{A}n^{i}}+O(\frac{1}{\tilde{r}^{3}}),\\\
g^{AB}_{(I)}&=\frac{1}{\tilde{r}^{2}}q^{AB}-{\frac{1}{\tilde{r}^{3}}H_{ij}(\partial^{A}n^{i})(\partial^{B}n^{j})}+O(\frac{1}{\tilde{r}^{4}}).\end{split}$
(III.29)

Here $\partial^{A}=q^{AB}\partial_{B}$ denotes the derivative on the sphere.

So far, the metric in the
$\\{\tilde{u},\tilde{r},\tilde{\theta},\tilde{\varphi}\\}$ coordinate system
looks similar to a BS form up to order $\frac{1}{r}$. The metric can be
further put into the Bondi gauge in a perturbative manner. For this purpose,
we parameterise the perturbative coordinate transformation in terms of a
vector $\xi^{\mu}$,

$x^{\mu}_{({\rm BS})}=\tilde{x}^{\mu}+\xi^{\mu}(\tilde{x}).$ (III.30)

This coordinate transformation will bring the metric $g^{\mu\nu}_{(I)}$ to the
Bondi-Sachs form $g^{\mu\nu}_{({\rm BS})}$ via the transformation

$g^{\mu\nu}_{({\rm BS})}=g^{\alpha\beta}_{(I)}\frac{\partial
x^{\mu}_{B}}{\partial\tilde{x}^{\alpha}}\frac{\partial
x^{\nu}_{B}}{\partial\tilde{x}^{\beta}}=g^{\alpha\beta}_{(I)}\bigg{(}\delta^{\mu}_{\alpha}+\frac{\partial\xi^{\mu}}{\partial\tilde{x}^{\alpha}}\bigg{)}\bigg{(}\delta^{\nu}_{\beta}+\frac{\partial\xi^{\nu}}{\partial\tilde{x}^{\beta}}\bigg{)}.$
(III.31)

To seek the required perturbative gauge $\xi^{\mu}$, we set

$\xi^{\mu}=\bigg{(}\frac{\xi^{0}_{(1)}(\tilde{u},\tilde{\theta},\tilde{\varphi})}{\tilde{r}},\frac{\xi^{1}_{(1)}(\tilde{u},\tilde{\theta},\tilde{\varphi})}{\tilde{r}},\frac{\xi^{2}_{(1)}(\tilde{u},\tilde{\theta},\tilde{\varphi})}{\tilde{r}^{2}},\frac{\xi^{3}_{(1)}(\tilde{u},\tilde{\theta},\tilde{\varphi})}{\tilde{r}^{2}}\bigg{)}.$
(III.32)

Equivalently,

$\begin{split}u_{B}&=\tilde{u}+\frac{\xi^{0}_{(1)}(\tilde{u},\tilde{\theta},\tilde{\varphi})}{\tilde{r}},\\\
r_{B}&=\tilde{r}+\frac{\xi^{1}_{(1)}(\tilde{u},\tilde{\theta},\tilde{\varphi})}{\tilde{r}},\\\
\theta_{B}&=\tilde{\theta}+\frac{\xi^{2}_{(1)}(\tilde{u},\tilde{\theta},\tilde{\varphi})}{\tilde{r}^{2}},\\\
\varphi_{B}&=\tilde{\varphi}+\frac{\xi^{2}_{(1)}(\tilde{u},\tilde{\theta},\tilde{\varphi})}{\tilde{r}^{2}}.\end{split}$
(III.33)

Straightforward calculation shows

$\begin{split}g^{00}_{({\rm BS})}&=O(\frac{1}{r_{\rm BS}^{2}}),\ \
g^{0A}_{({\rm BS})}=O(\frac{1}{r_{\rm BS}^{3}}),\\\ g^{01}_{({\rm
BS})}&=-1+\frac{1}{r_{\rm
BS}}\bigg{[}\frac{H}{2}-\frac{\partial\xi^{0}_{(1)}}{\partial\tilde{u}}\bigg{]}+O(\frac{1}{r_{\rm
BS}^{2}}),\\\ g^{11}_{({\rm BS})}&=1-\frac{1}{r_{\rm
BS}}\bigg{[}2M+H_{ij}n^{i}n^{j}+2\frac{\partial\xi^{1}_{(1)}}{\partial\tilde{u}}\bigg{]}+O(\frac{1}{r_{\rm
BS}^{2}}),\\\ g^{12}_{({\rm BS})}&=\frac{1}{r_{\rm
BS}^{2}}\bigg{[}-H_{ij}n^{i}\frac{\partial
n^{j}}{\partial\theta}-\frac{\partial\xi^{2}_{(1)}}{\partial\tilde{u}}\bigg{]}+O(\frac{1}{r_{\rm
BS}^{3}}),\\\ g^{13}_{({\rm BS})}&=\frac{1}{r_{\rm
BS}^{2}}\bigg{[}-\frac{1}{\sin^{2}\theta}H_{ij}n^{i}\frac{\partial
n^{j}}{\partial\varphi}-\frac{\partial\xi^{3}_{(1)}}{\partial\tilde{u}}\bigg{]}+O(\frac{1}{r_{\rm
BS}^{3}}),\\\ g^{AB}_{({\rm BS})}&=\frac{1}{r_{\rm
BS}^{2}}(\partial^{A}n^{i})(\partial^{B}n^{i})-\frac{1}{r_{\rm
BS}^{3}}H_{ij}(\partial^{A}n^{i})(\partial^{B}n^{j})+O(\frac{1}{r_{\rm
BS}^{4}})\\\ &=\frac{q_{AB}}{r_{\rm BS}^{2}}-\frac{1}{r_{\rm
BS}^{3}}H_{ij}(\partial^{A}n^{i})(\partial^{B}n^{j})+O(\frac{1}{r_{\rm
BS}^{4}}).\end{split}$ (III.34)

From the above results, we find that the Bondi shear is

${C^{AB}=H_{ij}(\partial^{A}n^{i})(\partial^{B}n^{j})}.$ (III.35)

To achieve the Bondi gauge, we should choose $\xi_{(1)}^{0},\xi_{(1)}^{A}$
such that

$\frac{H}{2}-\frac{\partial\xi^{0}_{(1)}}{\partial\tilde{u}}=0,$ (III.36)

and

$-H_{ij}n^{i}\partial^{A}n^{j}-\frac{\partial\xi^{A}_{(1)}}{\partial\tilde{u}}=\frac{1}{2}\nabla_{B}C^{AB}.$
(III.37)

Moreover, one requires that
$\frac{\partial\xi^{1}_{(1)}}{\partial\tilde{u}}=\frac{1}{2}H_{ij}n^{i}n^{j}$
to preserve (II.5) and the Bondi mass aspect now becomes

$m={M+H_{ij}n^{i}n^{j}}.$ (III.38)

By using the harmonic gauge condition again, it follows that the Bondi shear
tensor $C^{AB}$ is only related to the transverse-traceless part of $H_{ij}$
as Tahura2021

${C^{AB}=H^{\rm TT}_{ij}(\partial^{A}n^{i})(\partial^{B}n^{j})},$ (III.39)

where

$H^{\rm TT}_{ij}=\Lambda_{ijkl}H^{kl}$ (III.40)

with

$\begin{split}\Lambda_{ijkl}&=P_{ik}P_{jl}-\frac{1}{2}P_{ij}P_{kl},\\\
P_{ij}&=\delta_{ij}-n_{i}n_{j}.\end{split}$ (III.41)

Recently, Blanchet et al. transformed the metric of an isolated matter source
in the multipolar post-Minkowskian (MPM) approximation from the harmonic
coordinates to the Newman-Unti coordinates (which are equivalent to the Bondi-
Sachs coordinates). They also obtained the mass and angular momentum aspects,
as well as the Bondi shear in terms of the canonical multipole moments
Blanchet2021 . Up to the linear order of $G$, the Bondi shear tensor $C^{AB}$
reads

${C^{AB}=H^{\rm TT}_{ij}(\partial^{A}n^{i})(\partial^{B}n^{j})},$ (III.42)

and the Bondi mass aspect is

$m=\sum_{l=0}^{+\infty}\frac{(l+1)(l+2)}{2l!}n_{L}M_{L}^{(l)},$ (III.43)

which reduces to

$m={M+3\ddot{M}_{ij}n^{i}n^{j}}$ (III.44)

at the quadrupole level.

##  IV Correction term $\delta J$ for linear gravitational wave

Now we are in the position to calculate the contribution of the additional
correction term $\delta J$ (II.14) in the new angular momentum expression.

Recall that the closed potential $c$ and co-closed potential $\underline{c}$
of $C_{AB}$ are related with $C_{AB}$ via

$C_{AB}=\nabla_{A}\nabla_{B}c-\frac{1}{2}q_{AB}\Delta
c+\frac{1}{2}(\epsilon_{A}^{\
E}\nabla_{E}\nabla_{B}\underline{c}+\epsilon_{B}^{\
E}\nabla_{E}\nabla_{A}\underline{c})$ (IV.1)

Then

$\begin{split}&\nabla^{A}\nabla^{B}C_{AB}\\\
=&\nabla^{A}\nabla^{B}\bigg{[}\nabla_{A}\nabla_{B}c-\frac{1}{2}q_{AB}\Delta
c+\frac{1}{2}(\epsilon_{A}^{\ E}\nabla_{E}\nabla_{B}\underline{c}\\\
&+\epsilon_{B}^{\ E}\nabla_{E}\nabla_{A}\underline{c})\bigg{]}\\\
=&\nabla^{A}\nabla^{B}\nabla_{A}\nabla_{B}c-\frac{1}{2}\Delta^{2}c+\frac{1}{2}\epsilon_{A}^{\
E}\nabla^{A}\nabla^{B}\nabla_{B}\nabla_{E}\underline{c}\\\
&+\frac{1}{2}\epsilon_{B}^{\
E}\nabla^{A}\nabla^{B}\nabla_{A}\nabla_{E}\underline{c}\\\
=&\nabla^{A}[\nabla_{A}\nabla^{B}\nabla_{B}c+R_{A}^{\
E}\nabla_{E}c]+\frac{1}{2}\epsilon_{A}^{\
E}\nabla^{A}\nabla^{B}\nabla_{E}\nabla_{B}\underline{c}\\\
&-\frac{1}{2}\Delta^{2}c+\frac{1}{2}\epsilon_{B}^{\
E}\nabla^{A}(\nabla_{A}\nabla^{B}\nabla_{E}\underline{c}+R^{B\ \ \ F}_{\
AE}\nabla_{F}\underline{c})\\\
=&\frac{1}{2}\Delta^{2}c+\nabla^{A}(\delta_{A}^{E}\nabla_{E}c)+\frac{1}{2}\epsilon_{A}^{\
E}\nabla^{A}(\nabla_{E}\nabla^{B}\nabla_{B}\underline{c}\\\ &+R^{\
F}_{E}\nabla_{F}\underline{c})+\frac{1}{2}\epsilon_{B}^{\
E}\nabla^{A}(\nabla_{A}\nabla^{B}\nabla_{E}\underline{c}R^{B\ \ \ F}_{\
EB}\nabla_{F}\underline{c})\\\ =&\frac{1}{2}\Delta^{2}c+\Delta
c+\frac{1}{2}\epsilon_{A}^{\ E}\nabla^{A}(\delta_{E}^{\
F}\nabla_{F}\underline{c})\\\ &+\frac{1}{2}\epsilon_{B}^{\ E}\nabla^{A}(R^{B\
\ \ F}_{\ EB}\nabla_{F}\underline{c})\\\ =&\frac{1}{2}\Delta^{2}c+\Delta
c+\frac{1}{2}\epsilon_{B}^{\ E}\nabla^{A}(\epsilon_{A}^{\ B}\epsilon_{E}^{\
F}\nabla_{F}\underline{c})\\\ =&\frac{1}{2}\Delta^{2}c+\Delta c,\end{split}$
(IV.2)

where we have used the fact that, for the round 2-sphere $(S^{2},q_{AB})$,

$R_{ABCD}=\epsilon_{AB}\epsilon_{CD},\ R_{A}^{B}=\delta^{B}_{A}.$ (IV.3)

To obtain the closed potential $c$ from $C_{AB}$, we expand
$\nabla^{A}\nabla^{B}C_{AB}$ and $c$ in terms of the spherical harmonics

$\begin{split}\nabla^{A}\nabla^{B}C_{AB}&=\sum_{l\geq
2}\sum_{m=-l}^{l}f_{lm}Y_{lm}(\theta,\varphi),\\\ c&=\sum_{l\geq
2}\sum_{m=-l}^{l}c_{lm}Y_{lm}(\theta,\varphi).\end{split}$ (IV.4)

Inserting (IV.4) into (IV.2), one has

$c_{lm}=\frac{2f_{lm}}{(l-1)l(l+1)(l+2)},\ \ (l\geq 2).$ (IV.5)

At the quadrupole level, $H_{ij}=2\ddot{M}_{ij}+2F\delta_{ij}$ and it follows
from (III.39) and (III.40) that

$C_{AB}=\left(\begin{array}[]{cc}H_{+}&H_{\times}\sin\theta\\\
H_{\times}\sin\theta&-H_{+}\sin^{2}\theta\\\ \end{array}\right)$ (IV.6)

where

$\begin{split}H_{+}=&\ddot{M}_{11}(\cos^{2}\theta\cos^{2}\varphi-\sin^{2}\varphi)+\ddot{M}_{12}(1+\cos^{2}\theta)\sin
2\varphi\\\ &-\ddot{M}_{13}\sin
2\theta\cos\varphi-\ddot{M}_{22}(\cos^{2}\varphi-\sin^{2}\varphi\cos^{2}\theta)\\\
&-\ddot{M}_{23}\sin 2\theta\sin\varphi+\ddot{M}_{33}\sin^{2}\theta,\\\
H_{\times}=&(\ddot{M}_{22}-\ddot{M}_{11})\cos\theta\sin
2\varphi+2\ddot{M}_{12}\cos\theta\cos 2\varphi\\\
&+2\ddot{M}_{13}\sin\theta\sin\varphi-2\ddot{M}_{23}\sin\theta\cos\varphi,\end{split}$
(IV.7)

Further, we have

$\begin{split}&\nabla^{A}\nabla^{B}C_{AB}\\\
=&-\ddot{M}_{11}-\ddot{M}_{22}+2\ddot{M}_{33}-3(\ddot{M}_{11}+\ddot{M}_{22}-2\ddot{M}_{33})\cos
2\theta\\\
&-\frac{3}{2}(\ddot{M}_{11}-\ddot{M}_{22})\cos(2\theta-2\varphi)+6\ddot{M}_{23}\cos(2\theta-\varphi)\\\
&+3\ddot{M}_{11}\cos 2\varphi-3\ddot{M}_{22}\cos
2\varphi-\frac{3}{2}\ddot{M}_{11}\cos(2\theta+2\varphi)\\\
&+\frac{3}{2}\ddot{M}_{22}\cos(2\theta+2\varphi)-6\ddot{M}_{23}\cos(2\theta+\varphi)\\\
&+3\ddot{M}_{12}\sin(2\theta-2\varphi)+6\ddot{M}_{13}\sin(2\theta-\varphi)+6\ddot{M}_{12}\sin
2\varphi\\\
&-3\ddot{M}_{12}\sin(2\theta+2\varphi)+6\ddot{M}_{13}\sin(2\theta+\varphi),\end{split}$
(IV.8)

which yields

$\begin{split}c=&\frac{-1}{12}\bigg{[}\ddot{M}_{11}+\ddot{M}_{22}-2\ddot{M}_{33}+3(\ddot{M}_{11}+\ddot{M}_{22}-2\ddot{M}_{33})\cos
2\theta\\\
&+\frac{3}{2}(\ddot{M}_{11}-\ddot{M}_{22})\cos(2\theta-2\varphi)-6\ddot{M}_{23}\cos(2\theta-\varphi)\\\
&-3\ddot{M}_{11}\cos 2\varphi+3\ddot{M}_{22}\cos
2\varphi+\frac{3}{2}\ddot{M}_{11}\cos(2\theta+2\varphi)\\\
&-\frac{3}{2}\ddot{M}_{22}\cos(2\theta+2\varphi)+6\ddot{M}_{23}\cos(2\theta+\varphi)\\\
&-3\ddot{M}_{12}\sin(2\theta-2\varphi)-6\ddot{M}_{13}\sin(2\theta-\varphi)-6\ddot{M}_{12}\sin
2\varphi\\\
&+3\ddot{M}_{12}\sin(2\theta+2\varphi)-6\ddot{M}_{13}\sin(2\theta+\varphi)\bigg{]}.\end{split}$
(IV.9)

Moreover, at the quadrupole level, the Bondi mass aspect is

$\begin{split}m=&M+3n^{i}n^{j}\ddot{M}_{ij}\\\
=&M_{1}+M_{2}+3\big{(}\ddot{M}_{11}\sin^{2}\theta\cos^{2}\varphi+\ddot{M}_{12}\sin^{2}\theta\sin
2\varphi\\\ &+\ddot{M}_{13}\sin
2\theta\cos\varphi+\ddot{M}_{22}\sin^{2}\theta\sin^{2}\varphi\\\
&+\ddot{M}_{23}\sin
2\theta\sin\varphi+\ddot{M}_{33}\cos^{2}\theta\big{)}.\end{split}$ (IV.10)

By direct calculation, one finally finds that the integrals involving the
correction terms in the new angular momentum expression are

$\begin{split}&\int_{S^{2}}cY_{1}^{A}\nabla_{A}m=0,\\\
&\int_{S^{2}}cY_{2}^{A}\nabla_{A}m=0,\\\
&\int_{S^{2}}cY_{3}^{A}\nabla_{A}m=0,\end{split}$ (IV.11)

where $Y_{1}^{A},Y_{2}^{A}$ and $Y_{3}^{A}$ are the three rotation Killing
vector fields (II.10) on the round sphere $S^{2}$.

##  V Conclusions and Discussions

The notion of the angular momentum can be shifted by supertranslations. Such
ambiguity has been an obstacle to seek a satisfactory notion of the angular
momentum. Recently, Po-Ning Chen, Mu-Tao Wang, Ye-Kai Wang and Shing-Tung Yau
investigated the first definition of the angular momentum that is
supertranslation invariant in the full nonlinear theory of GR. On the other
side, in the compact binary coalescence community, supertranslation ambiguity
is often ignored in practice. Under certain assumptions normally made in the
CBC literature, the orders of the magnitude of the ambiguity contribution are
smaller than the statistical errors of GW detectors Asht20-1 . In this paper,
we investigate the supertranslation ambiguity issue in the restricted context
of compact binary coalescence. We have shown that, in the linearised theory of
GW, the new angular momentum coincides with the classical definition at the
quadrupole level.

However, there is a caveat in this current work. Within the frame of the
linearised theory of GW, we have not checked the contribution of the
additional correction term in the full nonlinear theory. The main difficulty
we have encountered comes from the lack of fully explicit expressions of the
radiative metrics (in terms of the BS coordinates). We believe that the
recently proposed angular momentum is a very nice definition and it should
deserve much more attention and investigation in the future. In particular,
one wishes to seek the physical origin and significance of the newly added
correction term. There is a question for the geometric analysis community: Can
one provide a concrete physical example of radiative spacetime in which the
correction term has nonzero contribution?

## Acknowledgements

X. He was partially supported by the Key Project of Education Department of
Hunan Province (21A0576). X. Wu was partially supported by the National
Natural Science Foundation of China (11731001). N. Xie was partially supported
by the National Natural Science Foundation of China (11671089).

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