text:
\section{Introduction}
In general relativity, the gravitational fields of isolated systems are modeled by asymptotically flat spacetimes.  Such systems emit gravitational waves \cite{LIGO} which travel at the speed of light and eventually reach future null infinity of the spacetime.  In the study of gravitational waves, it is desirable to find a suitable notion of conserved quantities defined at null infinity. While we have the well-accepted Bondi-Sachs energy-momentum, there is no such consensus for angular momentum or center-of-mass. To shed light on this problem, we calculate the limits of the quasi-local angular momentum and center-of-mass defined by Chen-Wang-Yau \cite{CWY} for a family of spacelike two-spheres approaching future null infinity in an asymptotically flat spacetime $N$ admitting a Bondi-Sachs coordinates. We start this introduction by briefly reviewing the Bondi-Sachs coordinates and Chen-Wang-Yau quasi-local conserved quantities.

Concurrent work of Bondi, van der Burg, and Metzner \cite{BVM} and Sachs \cite{Sachs} introduced the Bondi-Sachs coordinates in order to clarify the nature of gravitational radiation. The coordinate system is built on the geometry of outgoing null hypersurfaces. In particular, the affine parameter of null geodesics $r$ is taken as a coordinate function. Let $N$ be a vacuum spacetime with metric $g$ given in Bondi-Sachs coordinates $(u,r,x^{A})$ by
\[g = -UVdu^2 - 2Ududr+\sigma_{AB}(dx^{A}+W^{A}du)(dx^{B}+W^{B}du),\]
where we demand
\[\det \left(\frac{\sigma_{AB}}{r^2}\right) = \det(\mathring\sigma_{AB}),\]
where $\mathring\sigma_{AB}$ is the round metric on the unit sphere. Denote the level sets of $u$ by $\mathcal{C}_u$. Assuming that the metric coefficients can be expanded into power series in $\frac{1}{r}$, the above determinant condition implies
\[ \sigma_{AB} = r^2 \mathring\sigma_{AB} + r C_{AB} + O(1),\]
where $C_{AB}(u, x^D)$ is a symmetric traceless two-tensor on $S^2$, referred to as the shear tensor.  Moreover, the null vacuum constraint equations enjoy a remarkable hierarchy that allows us to solve the metric coefficients on $\mathcal{C}_u$ order by order, with $C_{AB}$ being free data. In particular,
\begin{align*}
V & = 1 - \frac{2m}{r} + O(r^{-2})\\
\sigma_{AB} W^B &= \frac{1}{2}\mathring\nabla^{B}C_{AB}\\
 &+ \frac{1}{2r}\left(\frac{4}{3}N_{A} +\frac{4u}{3}\mathring\nabla_{A}m - \frac{1}{8}\mathring\nabla_{A}\left(C_{DE}C^{DE}\right)\right) + O(r^{-2})
\end{align*}
Here and hereafter, tensor contraction is performed with respect to $\mathring\sigma_{AB}$, unless otherwise noted.  The function $m(u,x^A)$ and the spherical co-vector $N_A(u,x^D)$ appear as free data in the aforementioned solution scheme, and are referred to as the mass aspect and the angular momentum aspect, respectively.  By comparison with static solutions, the Bondi-Sachs energy-momentum 4-vector $(e, p^1, p^2, p^3)$ is defined by
\begin{align}
e &= \frac{1}{4\pi}\int_{S^2} m, \label{BondiMass}\\
p^i &= \frac{1}{4\pi}\int_{S^2} m \tilde X^i, \label{LinearMomentum}
\end{align}
where $\tilde X^i$ are the first eigenfunctions on $(S^2, \mathring\sigma)$.  The positive mass theorem states that the Bondi-Sachs energy-momentum 4-vector is future-directed timelike if there is a complete spacelike hypersurface $\Sigma$ intersecting null infinity in the given cut such that the dominant energy condition is satisfied on $\Sigma$, and if the domain of dependence $D(\Sigma)$ of $\Sigma$ is nonflat. In particular, $e > 0$.  See Szabados \cite{Szabados} for further details.

Next, we describe the quasi-local quantities to be considered, per Chen-Wang-Yau \cite{CWY}.  Let $(\Sigma, \sigma)$ be a closed embedded spacelike two-sphere in a spacetime $(N, g)$, with spacelike mean curvature vector $H$.  The physical data used in the quasi-local definition consists of the triple $(\sigma, |H|, \alpha_{H})$, where $\sigma$ is the induced metric of $\Sigma$, $|H|$ is the norm of its mean curvature vector, and $\alpha_{H}$ is the connection one-form of the normal bundle with respect to the mean curvature vector; that is,
\[\alpha_{H}(\cdot) = g(\nabla^{N}_{(\cdot)}\frac{J}{|H|}, \frac{H}{|H|}),\]
where $J$ is the future-directed timelike vector obtain via reflection of $H$ through the incoming light cone in the normal bundle.

The unphysical data used in the quasi-local definition is specified with respect to an isometric embedding $X : (\Sigma,\sigma) \hookrightarrow (\mathbb{R}^{3,1}, \eta)$ of $\Sigma$ into the Minkowski spacetime.  In terms of the image surface $X(\Sigma)$ in $\mathbb{R}^{3,1}$, we have the unphysical data $(|H_0|, \alpha_{H_0})$, analogous to the physical data.

Let $(t,x^1, x^2, x^3)$ be a Cartesian coordinate system on the Minkowski spacetime $(\mathbb{R}^{3,1},\eta)$.  We take $T_0$ to be a future-directed, unit timelike vector field in the Minkowski spacetime, interpreted as an observer, and split $X = (X^0, X^1, X^2, X^3)$ into temporal and spatial components.  We define the height function
\begin{equation}\label{tauDef}
\tau := -\langle T_0, X \rangle
\end{equation}
in terms of $T_0$ and $X$.  Further, we define the density function
\begin{equation}\label{densityDef}
\rho := \frac{\sqrt{|H_0|^2 + \frac{\Delta \tau}{1+|\nabla \tau|^2}} - \sqrt{|H|^2 + \frac{\Delta \tau}{1+|\nabla \tau|^2}}}{\sqrt{1+|\nabla\tau|^2}}
\end{equation}
and the current one-form
\begin{equation}\label{currentDef}
j := \rho\nabla \tau -\nabla\left[\sinh^{-1}\left(\frac{\rho\Delta\tau}{|H_0||H|}\right)\right] -\alpha_{H_0} + \alpha_{H}.
\end{equation}

There are many such choices of isometric embedding $X$ and observer $T_0$ in the Minkowski spacetime.  Per Chen-Wang-Yau \cite{CWY}, we consider only those pairs $(X, T_0)$ with associated data satisfying the optimal isometric embedding equation
\begin{equation}\label{oiee}
div_{\sigma}j = 0.
\end{equation}

The quasi-local center-of-mass and angular momentum are defined with respect to boost and rotation Killing fields in Minkowski spacetime, respectively the images of the Lorentz Killing vector fields
\begin{equation}
K_i := x^i\partial_{t} + t\partial_{i},\ i = 1,2,3
\end{equation}
and
\begin{equation}
K_{i,j} := x^i\partial_{j} - x^j\partial_{i},\ i, j = 1, 2, 3;\  i < j
\end{equation}
under Lorentz transformations.

Per Chen-Wang-Yau \cite{CWY}, the quasi-local center-of-mass and angular momentum are defined as follows:
\begin{definition}\label{qlCharge}
Given a surface $(\Sigma, \sigma)$ in $(N,g)$, suppose that the pair $(X, T_0)$ provides an isometric embedding of $\Sigma$ into Minkowski spacetime such that the optimal isometric embedding equation \eqref{oiee} is satisfied.  Writing $T_0 = A(\partial_{t}) = A((1,0,0,0))$ for a Lorentz transformation $A$, we define the components of the quasi-local center-of-mass by
\begin{equation}\label{qlCOM}
C^i(\Sigma,X,T_0)\\ =-\frac{1}{8\pi e}\int_{\Sigma}[\langle A(K_{i}), T_0 \rangle \rho + (A(K_{i}))^{T}\cdot j]d\Sigma
\end{equation}
and the components of the quasi-local angular momentum by
\begin{equation}\label{qlAM}
J^k(\Sigma,X,T_0)\\ =-\frac{\epsilon^{ijk}}{8\pi}\int_{\Sigma}[\langle A(K_{i,j}), T_0 \rangle \rho + (A(K_{i,j}))^{T}\cdot j]d\Sigma,
\end{equation}
where $\langle \cdot, \cdot \rangle$ denotes the Minkowskian inner product and $K^{T}$ denotes the projection of a Lorentz Killing field $K$ onto the tangent space of the image $X(\Sigma)$, such that
\begin{equation}\label{projectionFormula}
K^{T} = \langle K, \partial_{A}X \rangle \sigma^{AC}\partial_{C}.
\end{equation}
In the angular momentum expression, we make use of the volume form $\epsilon_{ijk}$ of $\mathbb{R}^3$ written with respect to the coordinates $(x^1, x^2, x^3)$, such that $k = 1, 2,3; k \neq i, j$.
\end{definition}

In their simplest form, our main results, Theorems \ref{main_com} and \ref{main_angmom}, state the following:
\begin{theorem}
 Fix a null hypersurface $\{ u=u_0 \}$ and denote the level set of $r$ on $\{ u=u_0\}$ by $\Sigma_r$.  Under the assumption that $m$ is independent of angular variables and $C_{AB}$ is closed, the limits of $C^i(\Sigma_r, X, T_0)$ and $J^k(\Sigma_r, X, T_0)$ are
\[ \frac{1}{8\pi} \int_{S^2} Y^A N_A, \]
where $Y^A$ denotes rotation Killing vector fields on $S^2$ for the angular momentum and the conformal Killing vector fields on $S^2$ for the center-of-mass.
\end{theorem}
We remark that Theorems \ref{main_com} and \ref{main_angmom} provide more general expressions for the center-of-mass and angular momentum, without restrictions on the mass aspect $m$ or the shear tensor $C_{AB}$.

To close the introduction, we selectively review previous definitions of angular momentum for a given cut $\mathcal{S}$ of the null infinity. For a comprehensive treatment, see Szabados \cite{Szabados}, especially section 3.2. First we have the Komar-type integral of Winicour-Tamburino \cite{Winicour}
\[ L_{\mathcal{S}}[K^a] := \frac{1}{8\pi G} \int_{\mathcal{S}} \left( \nabla^{[c} K^{d]} + \nabla_e K^e \mbox{}^\perp \varepsilon^{cd} \right) \frac{1}{2} \varepsilon_{cdab}. \]
Here $\int_{\mathcal{S}}$ is understood as the integral over a family of surfaces approaching $\mathcal{S}$ and $K^a$ is a vector field approaching some element in BMS Lie algebra. Whenever the definition involves a BMS vector field, it suffers from the ``supertranslation ambiguity".  Namely, while we can identify a distinguished four-dimensional subgroup isometric to the translation Killing fields, there are infinitely many subgroups isometric to the Lorentz group.

Influenced by the classic paper of Bergmann-Thomson \cite{Bergmann}, physicists tend to analogize angular momentum and spin. A word on the denomination. In this paragraph, we follow their convention in using ``spin angular momentum" for all  six conserved quantities associated with $O(1,3)$ symmetry. For the spinor approach, we replace the Killing equation by the {\it twistor equation}.  However, since the twistor equation has only the trivial solution for general spacetimes, suitable modifications are needed.  The first such changes were proposed by Bramson \cite{Bramson}, who suggested the expression
\begin{align*}
J^{\underline{A}\underline{B}} = \frac{1}{8\pi G} \int_{\mathcal{S}} - i \lambda^{\underline{A}}_{(A } \lambda^{\underline{B}}_{B)} \epsilon_{A'B'},
\end{align*}
where $\underline{A}, \underline{B} = 0,1$ and an asymptotic twistor equation is imposed on the spinor dyad $\lambda^{\underline{A}}$.  Penrose \cite{Pen_quasilocal} later suggested a general construction $Q_{\mathcal{S}}[K^a]$ associated with any BMS vector field $K^a$ and introduced the two-surface twistor equation.  His definition suffers from the supertranslation ambiguity, although a resolution is discussed in Dain-Moreschi \cite{DaMo}. We remark that Penrose's construction is quasi-local, with a definition at null infinity achieved by taking a suitable limit; indeed, this approach is the same as that of Chen-Wang-Yau \cite{CWY2} and the present work.  Other notable quasi-local definitions appear in the work of Ludvigsen-Vickers \cite{LV} and of Dougan-Mason \cite{DouMa}.

Finally, under the framework for stability of Minkowski spacetime introduced by Christodoulou-Klainerman \cite{CK}, Rizzi proposed a definition of angular momentum at null infinity \cite{R_thesis, R_def}
\begin{align*}
\frac{1}{8\pi} \lim_{s\rightarrow\infty} \int_{S_{u,s}} \zeta_A \Omega^A_{(i)} d\mu_\gamma,
\end{align*}
where $S_{u,s},\ 0\leq s < \infty$ is a foliation of an outgoing null hypersurface $\mathcal{C}_u$,with induced metric $\gamma$. Here $\zeta_A = g(D_A l, \underline{l})$ is the torsion with respect to null normal vectors $l,\underline{l}$ of $S_{u,s}$ normalized by $g(l,\underline{l}) = -2$ and $\Omega_{(i)}$ is a ``rotation" vector field. At this moment, it is unclear to the authors how to compare the above definitions with our result.

The rest of the paper is organized as follows. In Section 2, we expand the metric coefficients to the order needed. In Sections 3 and 4, we compute the physical data $(\sigma_{r}, |H_{r}|, \alpha_{H,r})$ for $\Sigma_r$ and the unphysical data $(|H_{0,r}|, \alpha_{H_0,r})$ respectively. In Section 5, we discuss the center-of-mass-frame. The total center-of-mass and angular momentum are evaluated in Sections 6 and 7 respectively. In Section 8, it is shown that in Kerr spacetime, we recover the expected values.

\section{Bondi-Sachs Coordinates and Asymptotic Expansion}

Per Chen-Wang-Yau \cite{CWY2}, let $N$ be a spacetime with metric $g$ given in Bondi-Sachs coordinates $(u,r,x^{A})$ by
\begin{equation}\label{CWYform}
g = -UVdu^2 - 2Ududr+\sigma_{AB}(dx^{A}+W^{A}du)(dx^{B}+W^{B}du),
\end{equation}
where we demand
\begin{equation}\label{areaNorm}
\det \left(\frac{\sigma_{AB}}{r^2}\right) = \det(\mathring\sigma_{AB}),
\end{equation}
where $\mathring\sigma_{AB}$ is the round metric on the unit sphere.  In this way, the spacetime metric above has exactly six degrees of freedom.  For future reference, we denote the covariant derivatives associated with $\sigma_{AB}$ and $\mathring\sigma_{AB}$ by $\nabla_{A}$ and $\mathring\nabla_{A}$, respectively.  We also note that, for vector fields $X^{A}$ on the sphere, we have
\begin{equation}\label{divergenceRelation}
 div_{\sigma} X^{A} = div_{\mathring\sigma} X^{A}
 \end{equation}
 owing to the determinant condition \eqref{areaNorm}.

Assuming a Bondi-Sachs expansion from null infinity in powers of $\frac{1}{r}$, we have the metric form of Hawking-Perry-Strominger \cite{HPS},
\begin{align}\label{HPSexpansion}
\begin{split}
g = &-du^2 - 2dudr +r^2\mathring\sigma_{AB}dx^{A}dx^{B}\\
&+\frac{2m}{r}du^2 +rC_{AB}dx^{A}dx^{B} + \left(\mathring\nabla^{A}C_{AB}\right)dudx^{B}\\
&+\frac{1}{8r^2}\left(C_{DE}C^{DE}\right)dudr\\
&+\frac{1}{r}\left(\frac{4}{3}N_{A} +\frac{4u}{3}\mathring\nabla_{A}m - \frac{1}{8}\mathring\nabla_{A}\left(C_{DE}C^{DE}\right)\right)dudx^{A}\\
&+\frac{1}{4}\left(C_{DE}C^{DE}\right)\mathring\sigma_{AB}dx^{A}dx^{B} +\hdots.
\end{split}
\end{align}

The function $m(u,x^{A})$ and the spherical co-vector $N_{A}(u,x^{D})$ are referred to as the mass aspect and the angular momentum aspect, respectively.  The symmetric traceless spherical two-tensor $C_{AB}(u,x^{D})$ is referred to as the shear tensor.

The expansion above is derived from the Einstein equations expressed in the Bondi-Sachs variables.  In particular, there are a hierarchy of initial data equations which can be systematically integrated to yield higher terms in the metric expansion, assuming expressions in powers of $\frac{1}{r}$ as mentioned at the outset.  See M\"{a}dler-Winicour \cite{MW} for details.

The two metric expressions \eqref{CWYform} and \eqref{HPSexpansion} are related by
\begin{align}\label{CWYexpansion}
\begin{split}
U &= 1-\frac{1}{16r^2}\left(C_{DE}C^{DE}\right) + O(r^{-3}),\\
V &=1-\frac{2m}{r} + O(r^{-2}),\\
W_{A} &=\frac{1}{2}\mathring\nabla^{B}C_{AB}\\
 &+ \frac{1}{2r}\left(\frac{4}{3}N_{A} +\frac{4u}{3}\mathring\nabla_{A}m - \frac{1}{8}\mathring\nabla_{A}\left(C_{DE}C^{DE}\right)\right) + O(r^{-2}),\\
 \sigma_{AB} &= r^2\mathring\sigma_{AB} + rC_{AB} + \frac{1}{4}\left(C_{DE}C^{DE}\right)\mathring\sigma_{AB} +O(r^{-1}).
\end{split}
\end{align}

Indeed, we need a further term in the expansion of $V$ in calculating the quasi-local quantities.
\begin{proposition}
The metric function $V$ expands as
\begin{align}\label{Vexpansion}
\begin{split}
V &= 1 - \frac{2m}{r} - \frac{1}{2r^2}R^{(2)} + \frac{1}{16r^2}\left(C_{DE}C^{DE}\right) - \frac{1}{32r^2}\mathring\Delta\left(C_{DE}C^{DE}\right)\\
&+\frac{1}{4r^2}\left(\frac{4}{3}\mathring\nabla^{A}N_{A} + \frac{4u}{3}\mathring\Delta m - \frac{1}{8}\mathring\Delta\left(C_{DE}C^{DE}\right)\right)\\
&-\frac{1}{4r^2}C_{AB}\mathring\nabla^{A}\mathring\nabla^{D}C_{D}^{B} + O(r^{-3}),
\end{split}
\end{align}
where
\begin{align}\label{scalarCurv}
\begin{split}
R^{(2)} &:= \frac{1}{2}\left(C_{DE}C^{DE}\right) +\frac{1}{2}\mathring\nabla_{A}C_{BD}\mathring\nabla^{B}C^{AD} + \frac{1}{4}\mathring\Delta\left(C_{DE}C^{DE}\right)\\
&- \mathring\nabla_{A}\left(C^{AB}\mathring\nabla^{D}C_{BD} + C^{BD}\mathring\nabla_{B}C_{D}^{A}\right) - \frac{1}{4}\left(\mathring\nabla_{A}C_{BD}\mathring\nabla^{A}C^{BD}\right).
\end{split}
\end{align}
\end{proposition}
\begin{proof}
Per M\"{a}dler-Winicour \cite{MW}, we have the initial data equation
\begin{align}
\begin{split}
\partial_{r}(rV) &= \frac{r^2}{2}UR(\sigma) - \frac{r^2}{2}\Delta U + \frac{r^2}{4U}\nabla_A U\nabla^A U\\
& -\frac{1}{2r^2}\nabla_{A}\left(\partial_{r}(r^4W^{A})\right) - \frac{r^2}{4U}\sigma_{AB}\left(\partial_{r}W^{A}\right)\left(\partial_{r}W^{B}\right),
\end{split}
\end{align}
with $R(\sigma)$ the scalar curvature of $\sigma$.

The scalar curvature expands as
\[R(\sigma) = \frac{2}{r^2} + \frac{1}{r^3}\mathring\nabla^{D}\mathring\nabla^{E}C_{DE} + \frac{1}{r^4}R^{(2)} + O(r^{-5}),\]
where we use the shorthand $R^{(2)}$ \eqref{scalarCurv}.

Raising indices via
\begin{equation}\label{invMetricExpansion}
\sigma^{AB} = \frac{1}{r^2}\mathring\sigma^{AB} - \frac{1}{r^3}C^{AB} + O(r^{-4}),
\end{equation}
we calculate
\begin{align}\label{Wexpansion}
\begin{split}
W^{A} &= \frac{1}{2r^2}\mathring\nabla^{D}C_{D}^{A} + \frac{1}{2r^3}\left(\frac{4}{3}N^{A} + \frac{4u}{3}\mathring\nabla^{A}m - \frac{1}{8}\mathring\nabla^{A}\left(C_{DE}C^{DE}\right)\right)\\
& - \frac{1}{2r^3}C^{A}_{B}\mathring\nabla^{D}C_{D}^{B} + O(r^{-4}).
\end{split}
\end{align}
Using the divergence relation \eqref{divergenceRelation} and the calculations above, we compute the expansion of $V$ \eqref{Vexpansion}.
\end{proof}

\section{The Physical Data}
We study the surfaces $(\Sigma_{r}, \sigma_{r})$ in $(N,g)$ with $u$ and $r$ fixed.  Per Chen-Wang-Yau \cite{CWY2} and the Bondi-Sachs expansion \eqref{HPSexpansion}, the physical data is specified by
\begin{equation}\label{metricExpansion}
\sigma_{AB,r} = r^2\mathring\sigma_{AB} + rC_{AB} + \frac{1}{4}(C_{DE}C^{DE})\mathring\sigma_{AB} + O(r^{-1}),
\end{equation}
\begin{equation}\label{meanCurv}
|H_{r}|^2 = \frac{4}{Ur}\left(\frac{V}{r}+div_{\sigma}W\right),
\end{equation}
\begin{align}\label{connectionForm}
\begin{split}
g(\nabla^{N}_{\partial_{A}}J_{r},H_{r}) &= -\frac{2}{rU}\partial_{A}\left(\frac{V}{r}+div_{\sigma} W\right)\\
&+ \frac{2}{rU^2}\left(\frac{V}{r}+div_{\sigma} W\right)\sigma_{AC}\partial_{r}W^{C}.
\end{split}
\end{align}

We rewrite \eqref{meanCurv} and \eqref{connectionForm} in terms of the Bondi-Sachs expansion \eqref{HPSexpansion} in the following proposition:

\begin{proposition}
With respect to the Bondi-Sachs expansion \eqref{HPSexpansion}, the norm of the mean curvature \eqref{meanCurv} expands as
\begin{align}\label{H}
\begin{split}
&|H_{r}| = \frac{1}{r}|H|^{(-1)} + \frac{1}{r^2}|H|^{(-2)} + \frac{1}{r^3}|H|^{(-3)} + O(r^{-4})\\
&= \frac{2}{r} - \frac{2m}{r^2} + \frac{1}{2r^2}\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right) + \frac{1}{8r^3}\left(C_{DE}C^{DE}\right)\\
& - \frac{1}{32r^3}\mathring\Delta\left(C_{DE}C^{DE}\right) +\frac{3}{4r^3}\left(\frac{4}{3}\mathring\nabla^{E}N_{E} + \frac{4u}{3}\mathring\Delta m - \frac{1}{8}\mathring\Delta\left(C_{DE}C^{DE}\right)\right)\\
&-\frac{1}{4r^3}\left(C^{EA}\mathring\nabla_{E}\mathring\nabla^{D}C_{AD}\right) - \frac{1}{2r^3}\mathring\nabla_{E}\left(C^{EA}\mathring\nabla^{D}C_{AD}\right)-\frac{1}{2r^3}R^{(2)}\\
&- \frac{m^2}{r^3} - \frac{1}{16r^3}\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right)^2 + \frac{1}{2r^3}m\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right) + O(r^{-4}),
\end{split}
\end{align}
and the connection one-form \eqref{connectionForm} expands as
\begin{align}\label{alphaH}
\begin{split}
&\alpha_{H,r}(\partial_{B}) =  \frac{1}{r}\alpha_{H}^{(-1)}(\partial_{B}) + \frac{1}{r^2}\alpha_{H}^{(-2)}(\partial_{B}) + O(r^{-3})\\
&=\frac{1}{r}\mathring\nabla_{B}m - \frac{1}{4r}\mathring\nabla_{B}\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right) - \frac{1}{2r}\mathring\nabla^{A}C_{AB}\\
&-\frac{1}{32r^2}\mathring\nabla_{B}\left(C^{DE}C_{DE}\right) + \frac{1}{64r^2}\mathring\nabla_{B}\mathring\Delta\left(C^{DE}C_{DE}\right)  + \frac{1}{4r^2}\mathring\nabla_{B}R^{(2)}\\
&-\frac{3}{8r^2}\mathring\nabla_{B}\left(\frac{4}{3}\mathring\nabla^{C}N_{C} + \frac{4u}{3}\mathring\Delta m - \frac{1}{8}\mathring\Delta\left(C^{DE}C_{DE}\right)\right)\\
&+\frac{3}{8r^2}\mathring\nabla_{B}\left(C^{DA}\mathring\nabla_{D}\mathring\nabla^{E}C_{AE}\right)  +  \frac{1}{4r^2}\mathring\nabla_{B}\left(\mathring\nabla_{D}C^{DA}\mathring\nabla^{E}C_{AE}\right)\\
&-\frac{3}{4r^2}\left(\frac{4}{3}N_{B}+\frac{4u}{3}\mathring\nabla_{B}m-\frac{1}{8}\mathring\nabla_{B}\left(C^{DE}C_{DE}\right)\right)\\
&+\frac{1}{4r^2}C_{B}^{A}\mathring\nabla^{D}C_{AD}+\frac{1}{r^2}\mathring\nabla_{B}\left(m^2\right)-\frac{1}{2r^2}\mathring\nabla_{B}\left(m\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right)\\
&+\frac{1}{16r^2}\mathring\nabla_{B}\left(\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right)^2\right) + O(r^{-3}).
\end{split}
\end{align}
\end{proposition}

\begin{proof}
For the norm of the mean curvature, we substitute the expansions (\ref{CWYexpansion},\ \ref{Wexpansion},\ \ref{Vexpansion}) into \eqref{meanCurv} and find \eqref{H}.

Regarding the connection one-form, we rewrite \eqref{connectionForm} as
\begin{align*}
U^2g(\nabla^{N}_{\partial_{A}}J_{r},H_{r}) &= -\frac{2}{r}U\partial_{A}\left(\frac{V}{r}+div_{\sigma} W\right) \\
&+ \frac{2}{r}\left(\frac{V}{r}+div_{\sigma} W\right)\sigma_{AC}\partial_{r}W^{C},
\end{align*}
and expand the two terms using (\ref{CWYexpansion},\ \ref{Wexpansion},\ \ref{Vexpansion}):
\begin{align*}
&\frac{2}{r}U\partial_{B}\left(\frac{V}{r}+div_{\sigma} W\right) = -\frac{4}{r^3}\mathring\nabla_{B}m +\frac{1}{r^3}\mathring\nabla_{B}\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right)\\
&+\frac{1}{8r^4}\mathring\nabla_{B}\left(C_{DE}C^{DE}\right) -\frac{1}{16r^4}\mathring\nabla_{B}\mathring\Delta\left(C_{DE}C^{DE}\right)\\
&+\frac{3}{2r^4}\mathring\nabla_{B}\left(\frac{4}{3}\mathring\nabla^{C}N_{C} + \frac{4u}{3}\mathring\Delta m - \frac{1}{8}\mathring\Delta\left(C_{DE}C^{DE}\right)\right)\\
&-\frac{3}{2r^4}\mathring\nabla_{B}\left(C^{EA}\mathring\nabla_{E}\mathring\nabla^{D}C_{AD}\right)\\
& - \frac{1}{r^4}\mathring\nabla_{B}\left(\mathring\nabla_{E}C^{EA}\mathring\nabla^{D}C_{AD}\right) -\frac{1}{r^4}\mathring\nabla_{B}R^{(2)} + O(r^{-5}),
\end{align*}
\begin{align*}
&- \frac{2}{r}\left(\frac{V}{r}+div_{\sigma} W\right)\sigma_{BC}\partial_{r}W^{C} \\
&=\frac{2}{r^3}\mathring\nabla^{A}C_{AB} + \frac{3}{r^4}\left(\frac{4}{3}N_{B}+\frac{4u}{3}\mathring\nabla_{B}m-\frac{1}{8}\mathring\nabla_{B}\left(C_{DE}C^{DE}\right)\right)\\
&-\frac{1}{r^4}C_{B}^{A}\mathring\nabla^{D}C_{DA} - \frac{4m}{r^4}\mathring\nabla^{A}C_{AB} + \frac{1}{r^4}\mathring\nabla^{A}C_{AB}\mathring\nabla^{D}\mathring\nabla^{E}C_{DE} + O(r^{-5}).
\end{align*}

From these expressions, we calculate the connection one-form expansion \eqref{alphaH}.

In both calculations we make use of the divergence relation \eqref{divergenceRelation}.
\end{proof}

\section{The Unphysical Data}

We consider isometric embeddings $X_r : (\Sigma_r,\sigma_r) \hookrightarrow (\mathbb{R}^{3,1}, \eta)$, with components $X_{r} = (X^0_{r}, X^1_{r}, X^2_{r}, X^3_r)$ expanding according to the ansatz
\begin{align}\label{embeddingExpansion}
\begin{split}
X^0_{r} &= X^{0(0)} + \frac{1}{r}X^{0(-1)} + O(r^{-2}),\\
X^i_{r} &= r\tilde{X}^i + X^{i(0)} + \frac{1}{r}X^{i(-1)} + O(r^{-2}),
\end{split}
\end{align}
where $\tilde{X} = (\tilde{X}^1,\tilde{X}^2,\tilde{X}^3)$ are the standard coordinate functions on the unit sphere.  The isometric embedding equation
\begin{equation}\label{IE}
dX^i_{r,A} \cdot dX^i_{r,B} = \sigma_{AB,r} + dX^0_{r,A}\cdot dX^0_{r,B},
\end{equation}
and the metric expansion \eqref{metricExpansion} imply the linearized equation
\begin{equation}\label{linIE}
d\tilde{X}^i_{A} \cdot dX^{i(0)}_{B} + dX^{i(0)}_{A} \cdot d\tilde{X}^i_{B} = C_{AB}.
\end{equation}

Recall from the introduction that the shear tensor $C_{AB}(u,x^{D})$ is symmetric and traceless; generally, $C_{AB}(u,x^{D})$ has the form
\begin{align}\label{CPotentials}
\begin{split}
C_{AB}(u,x^{A}) &= \left(\mathring\nabla_{A}\mathring\nabla_{B} - \frac{1}{2}\mathring\sigma_{AB}\mathring\Delta\right)c(u,x^{D})\\
&+ \frac{1}{2}\left(\mathring{\epsilon}_{AD}\mathring\nabla^{D}\mathring\nabla_{B} + \mathring{\epsilon}_{BD}\mathring\nabla^{D}\mathring\nabla_{A}\right)\underline{c}(u,x^{D}),
\end{split}
\end{align}
with scalar potentials $c(u,x^{D})$ and $\underline{c}(u,x^{D})$ and $\mathring\epsilon_{AB}$ the area form of the standard unit sphere.  Without loss of generality, we can assume that the potentials $c$ and $\underline{c}$ have spherical harmonic expansions with support in $\ell \geq 2$.

As a shorthand, we write
\begin{align}
F_{AB}(u,x^{A}) &:= \left(\mathring\nabla_{A}\mathring\nabla_{B} - \frac{1}{2}\mathring\sigma_{AB}\mathring\Delta\right)c(u,x^{A}),\\
\underline{F}_{AB}(u,x^{A}) &:= \frac{1}{2}\left(\mathring{\epsilon}_{AD}\mathring\nabla^{D}\mathring\nabla_{B} + \mathring{\epsilon}_{BD}\mathring\nabla^{D}\mathring\nabla_{A}\right)\underline{c}(u,x^{A}).
\end{align}

With this notation in place, the equation \eqref{linIE} implies
\begin{align}\label{linearizedEmbedding}
\begin{split}
X^{i(0)} &= \frac{1}{2}\left(\mathring\nabla^{A}c + \mathring\epsilon^{AB}\mathring\nabla_{B}\underline{c}\right)\mathring\nabla_{A}\tilde{X}^{i} - \frac{1}{4}\mathring\Delta c \tilde{X}^{i},\\
\mathring\nabla_{D}X^{i(0)} &= \frac{1}{2}F_{D}^{B}\mathring\nabla_{B}\tilde{X}^{i} -\frac{1}{2}\left(\mathring\nabla^{A}F_{AD}\right)\tilde{X}^{i}\\
&+\frac{1}{2}\mathring\epsilon^{AB}\mathring\nabla_{D}\mathring\nabla_{B}\underline{c}\mathring\nabla_{A}\tilde{X}^{i} - \frac{1}{2}\mathring{\epsilon}_{DB}\mathring\nabla^{B}\underline{c}\tilde{X}^{i},\\
\mathring\Delta X^{i(0)} &= -\frac{1}{2}\left(\mathring\nabla^{D}\mathring\nabla^{E}F_{DE}\right)\tilde{X}^{i} + \frac{1}{2}\mathring\epsilon^{AB}\mathring\nabla_{B}\mathring\Delta\underline{c}\mathring\nabla_{A}\tilde{X}^{i}.
\end{split}
\end{align}

Using these calculations, we proceed to expand the norm of the unphysical mean curvature and the connection one-form, as in the previous section.  We present the norm of the mean curvature in the following proposition:
\begin{proposition}
\begin{align}\label{H0}
\begin{split}
|H_{0,r}| &= \frac{1}{r}|H_0|^{(-1)} + \frac{1}{r^2}|H_0|^{(-2)} + \frac{1}{r^3}|H_0|^{(-3)} + O(r^{-4})\\
&=\frac{2}{r} + \frac{1}{2r^2} \mathring{\nabla}_A \mathring{\nabla}_B C^{AB} - \frac{1}{4r^3} \left( \mathring\Delta X^{0(0)}\right)^2 - \frac{1}{r^3} \tilde X^i \mathring\Delta X^{i(-1)} \\
&\quad - \frac{1}{2r^3} \mathring{\nabla}_A \left( C^{AB} \mathring{\nabla}^D F_{DB}\right) - \frac{1}{2r^3} \mathring{\nabla}_A C^{AB} \mathring{\nabla}_D \underline{F}^D_B \\
&\quad - \frac{1}{2r^3} C^{AB} \underline{F}_{AB}  + \frac{1}{16r^3} \mathring\nabla_{A}\mathring\Delta \underline{c}\mathring\nabla^{A}\mathring\Delta \underline{c} + \frac{1}{4r^3} \mathring{\nabla}_A C^{AB} \mathring{\nabla}_D C^D_B.
\end{split}
\end{align}
\end{proposition}
\begin{proof}
The unphysical mean curvature is given by $H_{0,r} = (\Delta X^0_{r}, \Delta X^{i}_{r})$, with associated expansions
\begin{align*}
&\Delta X^0_{r} = \frac{1}{r^2}\mathring\Delta X^{0(0)} + \frac{1}{r^3}\mathring\Delta X^{0(-1)} - \frac{1}{r^3}\mathring\nabla_{B}\left(C^{AB}\mathring\nabla_{A}X^{0(0)}\right) + O(r^{-4}),\\
&\Delta X^{i}_{r} = -\frac{2}{r}\tilde{X}^{i} + \frac{1}{r^2}\mathring\Delta X^{i(0)} -\frac{1}{r^2}\left(\mathring\nabla_{B}C^{AB}\right)\mathring\nabla_{A}\tilde{X}^{i} \\
&\quad+  \frac{1}{r^3} \mathring{\Delta} X^{i(-1)} - \frac{1}{r^3}\mathring{\nabla}_A \left( C^{AB} \mathring{\nabla}_B X^{i(0)} \right) + \frac{1}{4r^3} \mathring{\nabla}_A \left( C_{DE}C^{DE} \mathring{\nabla}^A \tilde X^i \right),
\end{align*}
per the inverse metric expansion \eqref{invMetricExpansion} and divergence relation \eqref{divergenceRelation}.
The norm of the mean curvature has expansion
\begin{align*}
|H_{0,r}|
&= \frac{2}{r} +\frac{1}{2r^2}\left(\mathring\nabla^{D}\mathring\nabla^{E}F_{DE}\right) - \frac{1}{r^3}  \tilde X^j \mathring\Delta X^{j(-1)} - \frac{1}{4r^3} \left( \mathring\Delta X^{0(0)}\right)^2 \\
&\quad + \frac{1}{r^3} \tilde X^i \mathring{\nabla}_A \left( C^{AB} \mathring{\nabla}_B X^{i(0)} \right) + \frac{1}{2r^3} C_{AB}C^{AB}\\
&\quad + \frac{1}{4r^3} \left( \mathring\Delta X^{i(0)} - \mathring{\nabla}_A C^{AB} \mathring{\nabla}_B \tilde X^i \right)^2 - \frac{1}{16r^3} \left( \mathring{\nabla}_A \mathring{\nabla}_B C^{AB} \right)^2. \\
\end{align*}

Substituting for $X^{i(0)}$ per \eqref{linearizedEmbedding},
\begin{align*}
&\tilde X^i \mathring{\nabla}_A \left( C^{AB} \mathring{\nabla}_B X^{i(0)} \right) = \\
& -\frac{1}{2} C^{AB} C_{AB} - \frac{1}{2} \mathring{\nabla}_A \left( C^{AB} \mathring{\nabla}^D F_{DB} \right) - \frac{1}{2} \mathring{\nabla}_A \left( C^{AB} \mathring\epsilon_{BD} \mathring{\nabla}^D \underline{c} \right)
\end{align*}
and
\begin{align*}
&\frac{1}{4} \left( \mathring\Delta X^{i(0)} - \mathring{\nabla}_A C^{AB} \mathring{\nabla}_B \tilde X^i \right)^2 = \\
&\frac{1}{16} \left( \mathring{\nabla}_A \mathring{\nabla}_B C^{AB} \right)^2 + \frac{1}{16} \mathring\nabla_{A} \mathring\Delta \underline{c} \mathring\nabla^{A}\mathring\Delta \underline{c} + \frac{1}{4} \mathring{\nabla}_A C^{AB} \mathring{\nabla}_D C^D_B - \frac{1}{4} \mathring{\nabla}_A C^{AB} \mathring\epsilon_{BD} \mathring{\nabla}^D \mathring\Delta \underline{c}.
\end{align*}
Adding the last term in each of these relations, we further rewrite
\begin{align*}
&- \frac{1}{4} \mathring{\nabla}_A C^{AB} \mathring\epsilon_{BD} \mathring{\nabla}^D \mathring\Delta \underline{c} - \frac{1}{2} \mathring{\nabla}_A \left( C^{AB} \mathring\epsilon_{BD} \mathring{\nabla}^D \underline{c} \right) \\
&=-\frac{1}{2} C^{AB} \underline{F}_{AB} - \frac{1}{2} \mathring{\nabla}_A C^{AB} \mathring{\nabla}_D \underline{F}_B^D.
\end{align*}
Substituting these reductions, we obtain \eqref{H0}.
\end{proof}

Turning to the unphysical connection one-form, we have the expansion:
\begin{proposition}
\begin{align}\label{alphaH0}
\begin{split}
&\alpha_{H_0,r}(\partial_{B}) = \frac{1}{r}\alpha_{H_0}^{(-1)}(\partial_{B}) + \frac{1}{r^2}\alpha_{H_0}^{(-2)}(\partial_{B}) + O(r^{-3})\\
&=\frac{1}{2r}\mathring\nabla_{B}\left(\mathring\Delta + 2\right)X^{0(0)} + \frac{1}{2r^2}\mathring\nabla_{B}\left(\mathring\Delta + 2\right)X^{0(-1)} \\
&-\frac{1}{2r^2}\mathring\nabla_{B}\left(\mathring\nabla_{A}\underline{F}^{AD}\mathring\nabla_{D}X^{0(0)}\right) -\frac{1}{2r^2}\left(\mathring\nabla_{B}\mathring\nabla_{E}X^{0(0)}\right)\left(\mathring\nabla_{D}F^{DE}\right)\\
& -\frac{1}{2r^2}\left(\mathring\nabla_{B}\mathring\nabla_{E}X^{0(0)}\right)\left(\mathring\epsilon^{ED}\mathring\nabla_{D}\underline{c}\right)-\frac{1}{8r^2}\mathring\nabla_{B}\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\mathring\Delta X^{0(0)}\right)\\
& - \frac{1}{2r^2}\mathring\nabla_{B}\left(C^{AD}\mathring\nabla_{A}\mathring\nabla_{D}X^{0(0)}\right)- \frac{1}{2r^2}C^{D}_{B}\mathring\nabla_{D}X^{0(0)} \\
&+ \frac{1}{2r^2}\mathring\nabla_{B}\left(\mathring\nabla^{D}X^{0(0)}\mathring{\epsilon}_{DE}\mathring\nabla^{E}\underline{c}\right)+ O(r^{-3}).
\end{split}
\end{align}
\end{proposition}
\begin{proof}
First, we expand
\[ |H_{0,r}|^{-1} = \frac{r}{2} - \frac{1}{8}\mathring\nabla^{D}\mathring\nabla^{E}C_{DE} + O(r^{-1}),\]
and
\begin{align*}
&\frac{H_{0,r}}{|H_{0,r}|}\\
&= \Bigg(\frac{1}{2r}\mathring\Delta X^{0(0)} + \frac{1}{r^2}\left(\frac{1}{2}\mathring\Delta X^{0(-1)} - \frac{1}{2}\mathring\nabla_{D}\left(C^{DE}\mathring\nabla_{E}X^{0(0)}\right) - \frac{1}{8}\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\mathring\Delta X^{0(0)}\right),\\
&-\tilde{X}^{i}+\frac{1}{r}\left(\frac{1}{2}\mathring\Delta X^{i(0)} - \frac{1}{2}\mathring\nabla_{D}C^{DE}\mathring\nabla_{E}\tilde{X}^{i} + \frac{1}{4}\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\tilde{X}^{i}\right)\Bigg)\\
&+ \Bigg(O(r^{-3}), O(r^{-2})\Bigg).
\end{align*}

Following the ideas of Chen-Wang-Yau \cite{CWY}, we note that the vector
\begin{equation}
w_r := (1+\sigma^{AB}\partial_{A}X_{r}^0\partial_{B}X_{r}^0,\sigma^{AB}\partial_{A}X_{r}^0\partial_{B}X_{r}^{i}),
\end{equation}
is normal to the embedded surface, owing to the isometric embedding equation.  Shifting by an appropriate factor, we find that
\begin{equation}
v_r := w_r - \Bigg\langle w_r , \frac{H_{0,r}}{|H_{0,r}|}\Bigg\rangle \frac{H_{0,r}}{|H_{0,r}|}
\end{equation}
is parallel to $J_{0,r}$, with length
\[ \sqrt{-\langle v_r, v_r \rangle} = 1 + O(r^{-2}).\]
Hence we have
\[\sqrt{-\langle v_r, v_r \rangle}\alpha_{H_0,r}(\partial_{B}) =\Bigg\langle\nabla^{\mathbb{R}}_{B} v_r, \frac{H_{0,r}}{|H_{0,r}|} \Bigg\rangle = \Bigg\langle \nabla^{\mathbb{R}}_{B}w_r, \frac{H_{0,r}}{|H_{0,r}|} \Bigg\rangle -\nabla^{\mathbb{R}}_{B}\Bigg\langle w_r, \frac{H_{0,r}}{|H_{0,r}|}\Bigg\rangle.\]

Expanding
\begin{align*}
w_r &= \Bigg(1 + O(r^{-2}),\ \frac{1}{r}\mathring\nabla^{A}\tilde{X}^{i}\mathring\nabla_{A}X^{0(0)}\\
&+ \frac{1}{r^2}\left(\mathring\nabla^{A}X^{0(0)}\mathring\nabla_{A}X^{i(0)} + \mathring\nabla^{A}X^{0(-1)}\mathring\nabla_{A}\tilde{X}^{i} - C^{AB}\mathring\nabla_{A}X^{0(0)}\mathring\nabla_{B}\tilde{X}^{i}\right) + O(r^{-3})\Bigg),
\end{align*}
we have
\begin{align*}
&\Bigg\langle w_r, \frac{H_{0,r}}{|H_{0,r}|} \Bigg\rangle =\\
&-\frac{1}{2r}\mathring\Delta X^{0(0)} + \frac{1}{r^2}\Big(-\frac{1}{2}\mathring\Delta X^{0(-1)} + \frac{1}{2}\mathring\nabla_{A}\left(C^{AB}\mathring\nabla_{B}X^{0(0)}\right) + \frac{1}{8}\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\mathring\Delta X^{0(0)}\\
&-\mathring\nabla^{A}X^{0(0)}\mathring\nabla_{A}X^{i(0)}\tilde{X}^{i} +\frac{1}{2}\mathring\Delta X^{i(0)}\mathring\nabla^{A}\tilde{X}^{i}\mathring\nabla_{A}X^{0(0)} - \frac{1}{2}\mathring\nabla_{A}X^{0(0)}\mathring\nabla_{D}C^{DA}\Big),
\end{align*}
\begin{align*}
&\Bigg\langle \nabla^{\mathbb{R}}_{B}w_r, \frac{H_{0,r}}{|H_{0,r}|} \Bigg\rangle = \\
& \frac{1}{r}\mathring\nabla_{B}X^{0(0)} + \frac{1}{r^2}\Big(\mathring\nabla_{B}\left(\mathring\nabla^{A}\tilde{X}^{i}\mathring\nabla_{A}X^{0(0)}\right)\left(\frac{1}{2}\mathring\Delta X^{i(0)} - \frac{1}{2}\mathring\nabla_{D}C^{DE}\mathring\nabla_{E}\tilde{X}^{i} + \frac{1}{4}\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\tilde{X}^{i}\right)\\
&-C_{B}^{D}\mathring\nabla_{D}X^{0(0)} + \mathring\nabla_{B}X^{0(-1)} - \mathring\nabla_{B}\left(\mathring\nabla_{D}X^{0(0)}\mathring\nabla^{D}X^{i(0)}\right)\tilde{X}^{i}\Big) + O(r^{-3}).
\end{align*}

Substituting for $X^{i(0)}$ via \eqref{linearizedEmbedding}, we have
\[-\mathring\nabla^{A}X^{0(0)}\mathring\nabla_{A}X^{i(0)}\tilde{X}^{i} +\frac{1}{2}\mathring\Delta X^{i(0)}\mathring\nabla^{A}\tilde{X}^{i}\mathring\nabla_{A}X^{0(0)} = \frac{1}{2}\mathring\nabla_{A}X^{0(0)}\mathring\nabla_{D}C^{DA},\]
such that
\begin{align*}
&\Bigg\langle w_r, \frac{H_{0,r}}{|H_{0,r}|} \Bigg\rangle =\\
&-\frac{1}{2r}\mathring\Delta X^{0(0)} + \frac{1}{r^2}\Big(-\frac{1}{2}\mathring\Delta X^{0(-1)} + \frac{1}{2}\mathring\nabla_{A}\left(C^{AB}\mathring\nabla_{B}X^{0(0)}\right) + \frac{1}{8}\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\mathring\Delta X^{0(0)}\Big).
\end{align*}

Per \eqref{linearizedEmbedding}, we have moreover
\begin{align*}
&\frac{1}{2}\mathring\Delta X^{i(0)} - \frac{1}{2}\mathring\nabla_{D}C^{DE}\mathring\nabla_{E}\tilde{X}^{i} + \frac{1}{4}\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\tilde{X}^{i}\\
&= -\frac{1}{2}\mathring\nabla_{D}F^{DE}\mathring\nabla_{E}\tilde{X}^{i} - \frac{1}{2}\mathring\epsilon^{ED}\mathring\nabla_{D}\underline{c}\mathring\nabla_{E}\tilde{X}^{i},
\end{align*}
\begin{align*}
&- \mathring\nabla_{B}\left(\mathring\nabla_{D}X^{0(0)}\mathring\nabla^{D}X^{i(0)}\right)\tilde{X}^{i}\\
&= - \mathring\nabla_{B}\left(\mathring\nabla_{D}X^{0(0)}\mathring\nabla^{D}X^{i(0)}\tilde{X}^{i}\right)+\mathring\nabla_{D}X^{0(0)}\mathring\nabla^{D}X^{i(0)}\mathring\nabla_{B}\tilde{X}^{i}\\
&= \frac{1}{2}\mathring\nabla_{B}\left(\mathring\nabla^{D}X^{0(0)}\mathring\nabla^{A}F_{AD}\right) + \frac{1}{2}\mathring\nabla_{B}\left(\mathring\nabla^{D}X^{0(0)}\mathring{\epsilon}_{DE}\mathring\nabla^{E}\underline{c}\right)+\frac{1}{2}C_{B}^{D}\mathring\nabla_{D}X^{0(0)},
\end{align*}
such that
\begin{align*}
&\Bigg\langle \nabla^{\mathbb{R}}_{B}w_r, \frac{H_{0,r}}{|H_{0,r}|} \Bigg\rangle = \\
& \frac{1}{r}\mathring\nabla_{B}X^{0(0)} + \frac{1}{r^2}\Big(-\frac{1}{2}\mathring\nabla_{B}\mathring\nabla_{A}X^{0(0)}\left(\mathring\nabla_{D}F^{DA} + \mathring\epsilon^{AD}\mathring\nabla_{D}\underline{c}\right)\\
&-\frac{1}{2}C_{B}^{D}\mathring\nabla_{D}X^{0(0)} + \mathring\nabla_{B}X^{0(-1)} +\frac{1}{2}\mathring\nabla_{B}\left(\mathring\nabla^{D}X^{0(0)}\mathring\nabla^{A}F_{AD}\right)\\
&+ \frac{1}{2}\mathring\nabla_{B}\left(\mathring\nabla^{D}X^{0(0)}\mathring{\epsilon}_{DE}\mathring\nabla^{E}\underline{c}\right)\Big) + O(r^{-3}).
\end{align*}

Collecting terms, we have the connection one-form expansion \eqref{alphaH0}.
\end{proof}

\section{The Center-of-Mass Frame}

We assume that the isometric embeddings $X_{r} = (X^0_{r}, X^i_{r})$ expanding as \eqref{embeddingExpansion} satisfy the optimal isometric embedding equation \eqref{oiee} to second order with respect to the observers
\begin{equation}\label{T0expansion}
T_{0,r} =  (1,0,0,0) + \frac{1}{r}(0,b_1,b_2,b_3) + O(r^{-2}).
\end{equation}
That is, given
\begin{align}\label{tauExpansion}
\begin{split}
\tau_{r} := -\langle T_{0,r}, X_{r} \rangle &= \tau^{(0)} + \frac{1}{r}\tau^{(-1)} + O(r^{-2}),\\
&= (X^{0(0)} - b_{i}\tilde{X}^{i}) + \frac{1}{r}(X^{0(-1)} - b_{i}X^{i(0)}) + O(r^{-2}),
\end{split}
\end{align}
together with the data (\ref{H},\ \ref{alphaH},\ \ref{H0},\ \ref{alphaH0}), we assume that $\tau_{r}$ satisfies the equation
\[div_{\sigma}j = div_{\sigma}\left(\rho_{r}\nabla \tau_{r} -\nabla\left[\sinh^{-1}\left(\frac{\rho_{r}\Delta\tau_{r}}{|H_{0,r}||H_{r}|}\right)\right] -\alpha_{H_0,r} + \alpha_{H,r}\right) = 0\]
up to second order.  As we shall see, the embedding and observer ansatze (\ref{embeddingExpansion},\ \ref{T0expansion}) are justified assuming two simple conditions on the mass aspect function $m$ in the metric expansion \eqref{HPSexpansion}.

Using the mean curvature formulae \eqref{H} and \eqref{H0} together with the fact that $\tau_{r}$ is $O(1)$, the density \eqref{densityDef} expands as
\begin{align}\label{densityFirst}
\begin{split}
\rho_{r} &= \frac{1}{r^2}\rho^{(-2)} + \frac{1}{r^3}\rho^{(-3)} + O(r^{-4}),\\
&= \frac{2m}{r^2} + \frac{1}{r^3}\left(|H_0|^{(-3)} - |H|^{(-3)}\right) + O(r^{-4}).
 \end{split}
\end{align}

Substituting the expansions (\ref{H},\ \ref{alphaH},\ \ref{H0},\ \ref{alphaH0},\ \ref{tauExpansion},\ \ref{densityFirst}), the current \eqref{currentDef} takes the form
\begin{align}\label{jDef}
\begin{split}
&j_{A,r} = \frac{1}{r}j_{A}^{(-1)} + \frac{1}{r^2}j_{A}^{(-2)} + O(r^{-3})\\
&= \frac{1}{r}\left(\alpha_{H}^{(-1)}(\partial_{A}) - \alpha_{H_0}^{(-1)}(\partial_{A})\right) + \frac{1}{r^2}\Big(2m\mathring\nabla_{A}\tau^{(0)} - \frac{1}{2}\mathring\nabla_{A}(m\mathring\Delta\tau^{(0)})\\
& - \alpha_{H_0}^{(-2)}(\partial_{A}) + \alpha_{H}^{(-2)}(\partial_{A})\Big) + O(r^{-3})\\
&= \frac{1}{r}\left(\mathring\nabla_{A}\left(m - \frac{1}{4}\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right) - \frac{1}{2}(\mathring\Delta +2)X^{0(0)}\right)- \frac{1}{2}\mathring\nabla^{B}C_{AB}\right)\\
&+\frac{1}{r^2}\Big(2m\mathring\nabla_{A}\tau^{(0)} - \frac{1}{2}\mathring\nabla_{A}(m\mathring\Delta\tau^{(0)}) - \alpha_{H_0}^{(-2)}(\partial_{A}) + \alpha_{H}^{(-2)}(\partial_{A})\Big)\\
& + O(r^{-3}).
\end{split}
\end{align}

Finally, we note that the observers $T_{0,r}$ have the form $T_{0,r} = A_{r}((1,0,0,0))$, where $A_{r}$ are Lorentz transformations expanding as
\begin{equation}\label{boostExpansion}
A_{r} = Id + \frac{1}{r}A^{(-1)} + O(r^{-2}),
\end{equation}
with
\[A^{(-1)} =
\begin{bmatrix}
 0 & b_1 & b_2 & b_3\\
 b_1 & 0 & -a_{12} & -a_{13}\\
 b_2 & a_{12} & 0 & -a_{23}\\
 b_3 & a_{13} & a_{23} & 0
 \end{bmatrix}.\]

According to \eqref{jDef}, we use only the connection one-form expansions \eqref{alphaH} and \eqref{alphaH0}, along with the divergence relation \eqref{divergenceRelation}, to calculate the linear term of the optimal isometric embedding equation:
\begin{align}\label{firstOrder}
\begin{split}
&\mathring\Delta(\mathring\Delta + 2)X^{0(0)} - \mathring\Delta (2m) + \frac{1}{2}(\mathring\Delta + 2)(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}) = 0,\\
&\mathring\Delta\left[(\mathring\Delta +2)X^{0(0)} - 2m + \frac{1}{4}(\mathring\Delta +2)(\mathring\Delta +2)c\right] = 0.
\end{split}
\end{align}

We solve \eqref{firstOrder} for $X^{0(0)}$, complementing our earlier calculation of the $X^{i(0)}$ \eqref{linearizedEmbedding} per the linearized isometric embedding equations.  Integrating against the eigenfunctions $\tilde{X}^{i}$, we deduce that the equation is solvable for $X^{0(0)}$ if and only if the mass aspect satisfies
\begin{equation}\label{zeroLinMom}
\int_{S^2} m(u, x^{A})\tilde{X}^{i} = 0.
\end{equation}
That is, solvability is guaranteed by vanishing of linear momentum \eqref{LinearMomentum} at future null infinity, placing the spacetime into what is often referred to as the center-of-mass frame.  The center-of-mass frame will turn out to be essential in the calculation of center-of-mass.  On the other hand, the calculation for angular momentum can be modified to accommodate other linear momenta profiles at future null infinity by means of different choices of observer $T_{0,r}$.

Owing to \eqref{zeroLinMom}, we can solve for $M(u,x^{A})$ satisfying
\begin{equation}\label{MDef}
(\mathring{\Delta} + 2) M = 2m,
\end{equation}
specified up to the addition of a linear combination of first eigenfunctions $f_{i}\tilde{X}^{i}$.

Substituting into \eqref{firstOrder}, we find
\begin{equation}\label{firstOrder2}
\mathring\Delta(\mathring\Delta +2)\left[X^{0(0)} - M + \frac{1}{4}(\mathring\Delta +2)c\right] = 0,
\end{equation}
such that
\begin{equation}\label{X00}
X^{0(0)} = M -\frac{1}{4}\left(\mathring\Delta  + 2\right)c + h,
\end{equation}
with $h$ an arbitrary constant.  We remark that, substituting $X^{0(0)}$ above, it is possible to solve the isometric embedding equation \eqref{IE} at the next order, obtaining the embedding term $X^{i(-1)}$.

Raising indices of the current \eqref{jDef} via \eqref{invMetricExpansion} and applying the divergence relation \eqref{divergenceRelation}, we calculate the next order term of the optimal isometric embedding equation \eqref{oiee}:
\begin{equation}\label{secondOrder}
-\frac{1}{2}\mathring\Delta\left(\mathring\Delta + 2\right)X^{0(-1)}-2b_{i}\mathring\nabla^{A}(m\mathring\nabla_{A}\tilde{X}^{i}) - b_{i}\mathring\Delta(m\tilde{X}^{i}) + S = 0,
\end{equation}
where we have used the connection one-form expansions \eqref{alphaH} and \eqref{alphaH0} in addition to that for $\tau$ \eqref{tauExpansion}.  Here $S$ is a shorthand for terms expressible in terms of the physical observables in the metric expansion \eqref{HPSexpansion} and $X^{0(0)}$ \eqref{X00}.

We solve \eqref{secondOrder} for $X^{0(-1)}$, with integration against the eigenfunctions $\tilde{X}^{j}$ yielding necessary and sufficient conditions for solvability:
\begin{align*}
&\int_{S^2}\left[S\tilde{X}^{j} - 2b_{i}\mathring\nabla^{A}(m\mathring\nabla_{A}\tilde{X}^{i})\tilde{X}^{j} - b_{i}\mathring\Delta(m\tilde{X}^{i})\tilde{X}^{j}\right] = 0,\\
&\int_{S^2}\left[S\tilde{X}^{j} + 2b_{i}m\mathring\nabla_{A}\tilde{X}^{i}\mathring\nabla^{A}\tilde{X}^{j} + 2b_{i}m\tilde{X}^{i}\tilde{X}^{j}\right] = 0,\\
&\int_{S^2}S\tilde{X}^{j} +2b_{j}\int_{S^2}m = 0,
\end{align*}
where we have used the pointwise relation
\[ \mathring\nabla^{A}\tilde{X}^{i}\mathring\nabla_{A}\tilde{X}^{j} = \delta^{ij} - \tilde{X}^{i}\tilde{X}^{j}.\]

Assuming positivity of the Bondi mass \eqref{BondiMass}, such that $\int_{S^2} m > 0$, and any $S$, solvability follows from an appropriate choice of $b_{j}$, which we leave implicit.  In generating $X^{0(0)}$, $X^{0(-1)}$, and the $b_{j}$, the analysis of \eqref{firstOrder} and \eqref{secondOrder} allows us to solve for $\tau^{(0)}$ and $\tau^{(-1)}$ in \eqref{tauExpansion}.  In principle, we could also solve the isometric embedding equation \eqref{IE} to the next order, obtaining the embedding term $X^{i(-2)}$.

Determination of the $b_{i}$ is necessary in proper calculation of the terms appearing in the angular momentum calculation, though it turns out that such terms vanish after integration by parts.  On the other hand, the calculation for quasi-local center-of-mass relies only upon the condition \eqref{zeroLinMom} allowing solvability and application of the linearized equation \eqref{firstOrder}.

Solvability of higher-order terms appearing in the optimal isometric embedding equation \eqref{oiee} and the isometric embedding equation \eqref{IE} is accomplished in the work of Chen-Wang-Yau \cite{CWY2}; see Theorem 3.  The authors show that, given the same assumption on the Bondi mass \eqref{BondiMass}, such that $\int_{S^2} m > 0$, it is possible to solve \eqref{oiee} and \eqref{IE} inductively with respect to embeddings $X_{r}$ and observers $T_{0,r} = A_{r}((1,0,0,0))$ expanding according to \eqref{embeddingExpansion}, \eqref{T0expansion}, and \eqref{boostExpansion}, adding lower order terms to the Lorentz transformations $A_{r}$ as necessary.

To summarize the section above, given our family of surfaces $\Sigma_{r}$ in a spacetime $(N, g)$ with mass aspect function $m$ satisfying
\begin{align*}
\int_{S^2} m &> 0,\\
\int_{S^2} m\tilde{X}^{i} &= 0,
\end{align*}
the pairs $(X_r, T_{0,r})$, with embeddings $X_{r}$ and observers $T_{0,r}$ expanding as \eqref{embeddingExpansion} and \eqref{T0expansion}, respectively, satisfy the optimal isometric embedding equation \eqref{oiee} and isometric embedding equation \eqref{IE} at all orders.

\section{Limit of Quasi-local Center-of-Mass}

We evaluate the quasi-local center-of-mass formula \eqref{qlCOM} with respect to the surfaces $(\Sigma_{r}, \sigma_{r})$ in a spacetime $(N,g)$ with vanishing linear momentum at null infinity \eqref{zeroLinMom} and with positive Bondi mass, such that the pairs $(X_{r}, T_{0,r})$ described above satisfy both the isometric embedding equation \eqref{IE} and optimal isometric embedding equation \eqref{oiee} to all orders.  Taking limits as $r$ approaches infinity, we recover the components of the center-of-mass at future null infinity.  In doing so, we make use of the data (\ref{H},\ \ref{alphaH},\ \ref{H0},\ \ref{alphaH0}), the derived expansions (\ref{tauExpansion},\ \ref{densityFirst},\ \ref{jDef},\ \ref{boostExpansion}), the linearized optimal isometric embedding equation \eqref{firstOrder}, and the expression for $X^{0(0)}$ \eqref{X00} arising from its analysis.

For simplicity, we begin by considering the case where the observers $T_{0,r} = (1,0,0,0)$, such that $b_i = 0$ in its expansion \eqref{T0expansion}.  According to Definition \ref{qlCharge}, we consider the boosts
\begin{equation}
K_{i,r} := x^{i}\partial_{t} + t\partial_{i}
\end{equation}
associated with $T_0 = (1,0,0,0).$  Here $K_{i,r}$ denotes the restriction of the Lorentz boost to the embedded surface $X_{r}(\Sigma_{r})$.

On the embedded surfaces $X_{r}(\Sigma_{r})$, the $K_{i,r}$ satisfy
\begin{equation}\label{innerProdExpansion}
\langle K_{i,r}, T_0 \rangle = -r\tilde{X}^{i} - X^{i(0)} + O(r^{-1}),
\end{equation}
\begin{align}\label{projectionExpansion}
\begin{split}
(K^{T}_{i,r})^{B} &= \left(\frac{1}{r^2}\mathring\sigma^{AB} - \frac{1}{r^3}C^{AB} + O(r^{-4})\right)\\
&\Bigg\langle \left(r\tilde{X}^{i} +X^{i(0)}\right)\partial_{t} + X^{0(0)}\partial_{i} + O(r^{-1}),\\
&\mathring\nabla_{A}X^{0(0)}\partial_{t} + \left(r\mathring\nabla_{A}\tilde{X}^{j} + \mathring\nabla_{A}X^{j(0)}\right)\partial_{j} + O(r^{-1})\Bigg\rangle\\
&=\frac{1}{r}\left(-\mathring\nabla^{B}\left(\tilde{X}^{i}X^{0(0)}\right) + 2\mathring\nabla^{B}\tilde{X}^{i}X^{0(0)}\right) + O(r^{-2}),
 \end{split}
 \end{align}
 where in the second expression we apply the projection formula \eqref{projectionFormula}.

Expanding the center-of-mass formula \eqref{qlCOM}, we find
 \begin{equation}
 C^i(\Sigma_{r},X_{r},T_0)  = \frac{r}{4\pi e}\int_{S^2}m\tilde{X}^{i} + O(1),
 \end{equation}
 with the seemingly divergent top-order term annihilated by our assumption of vanishing linear momentum at null infinity \eqref{zeroLinMom}.

Expanding \eqref{qlCOM} to the next order and taking the limit as $r$ approaches infinity, the center-of-mass at future null infinity is given by components
\begin{align}\label{comFirst}
\begin{split}
C^i = \frac{1}{8\pi e} \int_{S^2} &\Big[\tilde X^i \left( |H_0|^{(-3)} - |H|^{(-3)} \right)\\
 &+ 2mX^{i(0)}- 2 \mathring{\nabla}^A \tilde X^i X^{0(0)} j_A^{(-1)}\Big],
 \end{split}
\end{align}
where we have applied the linearized optimal isometric embedding equation \eqref{firstOrder}, amounting to a divergence-free condition on $j_{A}^{(-1)}$, to integrate away its contraction with the first term in the expansion \eqref{projectionExpansion}.

As mentioned in the previous section, owing to the form of the components of \eqref{comFirst}, calculation of the center-of-mass does not rely upon application of the second order term \eqref{secondOrder} in the optimal isometric embedding equation.

We begin our simplification of \eqref{comFirst} by rewriting terms in the integral of the unphysical mean curvature norm \eqref{H0}:
\begin{lemma}\label{X(-1)}
\begin{align*}
&\int_{S^2} \tilde X^i \left( - \tilde X^j \mathring{\Delta} X^{j(-1)} - \frac{1}{4} \left( \mathring{\Delta} X^{0(0)}\right)^2 \right) \\
&= \int_{S^2} \tilde X^i \Bigg( \frac{1}{4}(C_{DE}C^{DE}) + \frac{1}{4}(\underline{F}_{DE}\underline{F}^{DE}) -\frac{1}{4}\mathring\nabla_{A}\mathring\nabla_{B}\underline{c}\mathring\nabla^{A}\mathring\nabla^{B}\underline{c} - \frac{1}{2} \mathring{\nabla}^B F_{BD} \mathring\epsilon^{DA} \mathring{\nabla}_A \underline{c}\Bigg)\\
&\quad +\int_{S^@}\tilde{X}^{i}\Bigg(- \frac{1}{4} \mathring\nabla_{A}\underline{c}\mathring\nabla^{A}\underline{c} -\frac{1}{16} \left( \mathring{\nabla}^A \mathring{\nabla}^B C_{AB}\right)^2 -m^2 + \frac{1}{2} m  (\mathring{\Delta} + 2) c + \frac{1}{2} m \mathring{\nabla}^A \mathring{\nabla}^B C_{AB} \Bigg).
\end{align*}
\end{lemma}
\begin{proof}
Suppose $X^{j(-1)} = \alpha^A \mathring{\nabla}_A \tilde X^j + \beta \tilde X^j$.  With this notation and the surface metric expansion \eqref{metricExpansion}, the second order term of the isometric embedding equation \eqref{IE} takes the form
\begin{align*}
&\mathring{\nabla}_A \alpha_B + \mathring{\nabla}_B \alpha_A + 2 \beta \mathring{\sigma}_{AB} \\
&= \frac{1}{4}(C_{DE}C^{DE}) \mathring{\sigma}_{AB} - \mathring{\nabla}_A X^{j(0)} \mathring{\nabla}_B X^{j(0)} + \mathring{\nabla}_A X^{0(0)} \mathring{\nabla}_B X^{0(0)} =: \delta\sigma_{AB}.
\end{align*}
We compute
\[ - \tilde X^j \mathring{\Delta} X^{j(-1)} = \mathring\sigma^{AB} \delta\sigma_{AB} - (\mathring{\Delta}+2)\beta. \]
Substituting for $X^{i(0)}$ \eqref{linearizedEmbedding} and $X^{0(0)}$ \eqref{X00}, we have
\begin{align*}
\mathring\sigma^{AB} \delta\sigma_{AB} &= \frac{1}{2}(C_{AB}C^{AB}) -\frac{1}{4}(F_{AB}F^{AB}) - \frac{1}{2} (F_{AB} \underline{F}^{AB}) - \frac{1}{4} \mathring\nabla_{A}\mathring\nabla_{B}\underline{c}\mathring\nabla^{A}\mathring\nabla^{B}\underline{c}\\
&- \frac{1}{2} \mathring{\nabla}^B F_{BD} \mathring\epsilon^{DA} \mathring{\nabla}_A \underline{c} - \frac{1}{4}\mathring\nabla_{A}\underline{c}\mathring\nabla^{A}\underline{c}  + \mathring\nabla_{A}M\mathring\nabla^{A}M - \mathring{\nabla}^A M \mathring{\nabla}^D F_{AD}.
\end{align*}
By \eqref{X00}, we also have
\[ -\frac{1}{4} \left( \mathring\Delta X^{0(0)}\right)^2 = - \frac{1}{4} \left( \mathring\Delta M \right)^2 + \frac{1}{4} \mathring\Delta M \mathring{\nabla}^A \mathring{\nabla}^B C_{AB} - \frac{1}{16}\left( \mathring{\nabla}^A \mathring{\nabla}^B C_{AB}\right)^2.\]
We simplify the terms involving $M$ via \eqref{MDef}:
\begin{align*}
&\int_{S^2} \tilde X^i \left( \mathring\nabla^{A}M\mathring\nabla_{A}M - \frac{1}{4} (\mathring{\Delta} M)^2 - \mathring{\nabla}_A M \mathring{\nabla}^D F^A_D + \frac{1}{4} \mathring{\Delta} M \mathring{\nabla}^A \mathring{\nabla}^B C_{AB} \right)\\
&= \int_{S^2} \Big[- \mathring{\nabla}^A \tilde X^i M \mathring{\nabla}_A M - \tilde X^i M \mathring{\Delta} M - \frac{1}{4} \tilde X^i (\mathring{\Delta} M)^2 \\
&\quad +\frac{1}{2} \mathring{\nabla}_A \tilde X^i M \mathring{\nabla}^A (\mathring{\Delta} + 2) c + \frac{1}{4} \tilde X^i M \mathring{\Delta} (\mathring{\Delta} + 2) c + \frac{1}{4} \tilde X^i (\mathring{\Delta}+2) M \mathring{\nabla}^A \mathring{\nabla}^B C_{AB}\Big]\\
&= \int_{S^2}\Big[ - \tilde X^i M^2 - \frac{1}{2} \tilde X^i M \mathring{\Delta} M - \frac{1}{4} \tilde X^i
 \mathring{\Delta }M (\mathring{\Delta}+2) M\\
 &\quad+ \tilde X^i M (\mathring{\Delta}+2)c + \frac{1}{4} (-2 \tilde X^i M + \tilde X^i \mathring{\Delta}M) (\mathring{\Delta}+ 2 )c + \frac{1}{2} \tilde X^i m \mathring{\nabla}^A \mathring{\nabla}^B C_{AB}\Big]\\
&= \int_{S^2} \Big[- \tilde X^i m^2 + \frac{1}{2} \tilde X^i m (\mathring{\Delta} + 2) c + \frac{1}{2} \tilde X^i m \mathring{\nabla}^A \mathring{\nabla}^B C_{AB}\Big].
\end{align*}
\end{proof}

Applying the mean curvature expansions \eqref{H} and \eqref{H0} together with Lemma \ref{X(-1)}, we find
\begin{align*}
&\int_{S^2} \tilde X^i \left( |H_0|^{(-3)} - |H|^{(-3)}\right) \\
&= \int_{S^2} \left[\mathring{\nabla}^A \tilde X^i N_A + \frac{1}{2} \tilde X^i m (\mathring\Delta + 2) c -u\tilde{X}^{i}\mathring\Delta m + \frac{1}{8}\tilde{X}^{i}(\mathring\Delta +2)(C_{DE}C^{DE})\right]\\
&+\int_{S^2} \tilde{X}^{i}\left(\frac{1}{2}R^{(2)} + \frac{1}{4}\mathring\nabla_{E}\left(C^{EA}\mathring\nabla^{D}C_{AD}\right) - \frac{1}{8}(C_{DE}C^{DE})\right)\\
&+ \int_{S^2} \tilde X^i \left( \frac{1}{4}(\underline{F}_{DE}\underline{F}^{DE}) -\frac{1}{4} (\mathring\nabla_{A}\mathring\nabla_{B}\underline{c}\mathring\nabla^{A}\mathring\nabla^{B}\underline{c}) - \frac{1}{2} \mathring{\nabla}^B F_{BD} \mathring\epsilon^{DA} \mathring{\nabla}_A \underline{c} - \frac{1}{4}\mathring\nabla^{A}\underline{c}\mathring\nabla^{A}\underline{c} \right)\\
&+ \int_{S^2} \tilde X^i \left( -\frac{1}{2} \mathring{\nabla}_A C^{AB} \mathring{\nabla}_D \underline{F}_B^D + \frac{1}{2} \mathring{\nabla}_A \left( C^{AB} \mathring{\nabla}^D \underline{F}_{DB} \right) - \frac{1}{2} C^{AB} \underline{F}_{AB} + \frac{1}{16} \mathring\nabla_{A}\mathring\Delta \underline{c} \mathring\nabla^{A}\mathring\Delta \underline{c} \right)\\
&= \int_{S^2} \left[\mathring{\nabla}^A \tilde X^i N_A + \frac{1}{2} \tilde X^i m (\mathring\Delta + 2) c\right] \\
&+\int_{S^2} \tilde{X}^{i}\left(\frac{1}{2}R^{(2)} + \frac{1}{4}\mathring\nabla_{E}\left(C^{EA}\mathring\nabla^{D}C_{AD}\right) - \frac{1}{8}(C_{DE}C^{DE})\right)\\
&+ \int_{S^2} \tilde X^i \Bigg( - \frac{1}{4}(\underline{F}_{DE}\underline{F}^{DE}) -\frac{1}{4} (\mathring\nabla_{A}\mathring\nabla_{B}\underline{c}\mathring\nabla^{A}\mathring\nabla^{B}\underline{c}) - \frac{1}{4} \mathring\nabla^{A}\underline{c}\mathring\nabla^{A}\underline{c}\\
&\quad\quad - \frac{1}{2} \mathring{\nabla}_A \underline{F}^{AB} \mathring{\nabla}_D \underline{F}_B^D + \frac{1}{2} \mathring{\nabla}_A \left( \underline{F}^{AB} \mathring{\nabla}^D \underline{F}_{DB} \right)+ \frac{1}{16} \mathring\nabla_{A}\mathring\Delta \underline{c} \mathring\nabla^{A}\mathring\Delta \underline{c} \Bigg)\\
&+ \int_{S^2} \tilde X^i \left( - \frac{1}{2} \mathring{\nabla}^B F_{BD} \mathring\epsilon^{DA} \mathring{\nabla}_A \underline{c} -\frac{1}{2} \mathring{\nabla}_A F^{AB} \mathring{\nabla}_D \underline{F}_B^D + \frac{1}{2} \mathring{\nabla}_A \left( F^{AB} \mathring{\nabla}^D \underline{F}_{DB} \right) - \frac{1}{2} (F^{AB} \underline{F}_{AB}) \right).
\end{align*}

In the last identity, we separate $\underline{F} \cdot \underline{F}$ and $F \cdot \underline{F}$ terms. Note that the last two terms in the first line vanish via integration by parts, with vanishing of the first of these terms relying upon our assumption on vanishing of linear momentum \eqref{zeroLinMom}.

We reduce each of the last three terms above in a sequence of lemmae; in doing so, we make use of the tensor identities contained in the following proposition:
\begin{proposition}\label{interchange}
\begin{align}
&\mathring{\nabla}_A F_{BD} - \mathring{\nabla}_B F_{AD} = \mathring{\nabla}^E F_{BE} \mathring\sigma_{AD} - \mathring{\nabla}^E F_{AE} \mathring\sigma_{BD}, \label{interchangeOne}\\
&\mathring{\nabla}_A \underline{F}_{BD} - \mathring{\nabla}_B \underline{F}_{AD} = -\frac{1}{2} \mathring\epsilon_{AB} \mathring\Delta \mathring{\nabla}_D \underline{c} + \frac{1}{2} \mathring\epsilon_{DA} \mathring{\nabla}_B \underline{c} - \frac{1}{2} \mathring\epsilon_{DB} \mathring{\nabla}_A \underline{c}, \label{interchangeTwo}\\
&\mathring{\nabla}^D \mathring{\nabla}_A C_{BD} + \mathring{\nabla}^D \mathring{\nabla}_B C_{AD} - \mathring\Delta C_{AB} = \mathring\sigma_{AB} \mathring{\nabla}^D \mathring{\nabla}^E C_{DE} + 2 C_{AB},\label{interchangeThree}\\
&\mathring\epsilon^{AB} \mathring{\nabla}_A F_{BD} = \frac{1}{2}\mathring\epsilon_{DB} \mathring{\nabla}^B (\mathring\Delta + 2) c,\label{interchangeFour}\\
&\mathring\epsilon^{AB} \mathring{\nabla}_A \underline{F}_{BD} = - \frac{1}{2}\mathring{\nabla}_D (\mathring\Delta + 2) \underline{c}.\label{interchangeFive}
\end{align}
\end{proposition}
\begin{proof}
To verify the second identity at a given point $p$, let $\{ e_1, e_2\}$ be a local oriented orthonormal frame around $p$ with $\mathring{\nabla}_{e_A} e_B(p)=0$. It suffices to check the case $e_A = e_1, e_B = e_2$. At $p$, we have
\begin{align*}
&\mathring{\nabla}_{e_1} \underline{F}(e_2, e_D) - \mathring{\nabla}_{e_2} \underline{F} (e_1, e_D) \\
&= -\frac{1}{2} \left( \mathring{\nabla}_{e_1} \mathring{\nabla}_{e_1} \mathring{\nabla}_{e_D} \underline{c} + \mathring{\nabla}_{e_2} \mathring{\nabla}_{e_2} \mathring{\nabla}_{e_D} \underline{c} \right) + \frac{1}{2}\mathring\epsilon_{DE} (\mathring{\nabla}_{e_1}\mathring{\nabla}_{e_E} \mathring{\nabla}_{e_2} \underline{c} - \mathring{\nabla}_{e_2}\mathring{\nabla}_{e_E}\mathring{\nabla}_{e_1} \underline{c})\\
&= -\frac{1}{2} \left( \mathring{\nabla}_{e_1} \mathring{\nabla}_{e_1} \mathring{\nabla}_{e_D} \underline{c} + \mathring{\nabla}_{e_2} \mathring{\nabla}_{e_2} \mathring{\nabla}_{e_D} \underline{c} \right) - \frac{1}{2} \mathring\epsilon_{D2} \mathring{\nabla}_{e_1}\underline{c} + \frac{1}{2} \mathring\epsilon_{D1} \mathring{\nabla}_{e_2} \underline{c}.
\end{align*}
\end{proof}
\begin{remark}
The last two identities can be combined into $\mathring\epsilon^{AB} \mathring{\nabla}_A C_{BD} = \mathring\epsilon_{DA} \mathring{\nabla}_B C^{AB}$.
\end{remark}

\begin{lemma}\label{magical identity}
Using Proposition \ref{interchange} and \eqref{scalarCurv}, we have the following identity:
\begin{equation}
\frac{1}{2} R^{(2)} + \frac{1}{4} \mathring{\nabla}_A \left( C^{AB} \mathring{\nabla}^D C_{BD} \right) + \frac{1}{16} \mathring{\Delta}(C_{DE}C^{DE})=0.
\end{equation}
\end{lemma}
\begin{proof}
Using (\ref{interchangeOne},\ \ref{interchangeTwo},\ \ref{interchangeFour},\ \ref{interchangeFive}) in Proposition \ref{interchange}, we have
\begin{align}
\mathring{\nabla}_A C_{BD} \mathring{\nabla}^A C^{BD} = \mathring{\nabla}_A C_{BD} \mathring{\nabla}^B C^{AD} + \mathring{\nabla}^B C_{AB} \mathring{\nabla}_D C^{AD}.
\end{align}
Indeed, \begin{align*}
&\mathring{\nabla}_A C_{BD} \mathring{\nabla}^A C^{BD} \\
&= \mathring{\nabla}_A C_{BD} \left( \mathring{\nabla}^B C^{AD} - \frac{1}{2} \mathring\epsilon^{AB} \mathring\Delta \mathring{\nabla}^D \underline{c} + \frac{1}{2} \mathring\epsilon^{DA} \mathring{\nabla}^B \underline{c} + \mathring{\nabla}^E F_{BE} \mathring\sigma_{AD} \right)\\
&= \mathring{\nabla}_A C_{BD} \mathring{\nabla}^B C^{AD} + \frac{1}{4} \left( \mathring{\nabla}_D (\mathring\Delta + 2) \underline{c} - \mathring\epsilon_{DB} \mathring{\nabla}^B(\mathring\Delta+2) c \right) \mathring\Delta \mathring{\nabla}^D \underline{c}\\
&\quad + \frac{1}{4} \left( \mathring{\nabla}_B (\mathring\Delta + 2) \underline{c}\ - \mathring\epsilon_{BD} \mathring{\nabla}^D(\mathring\Delta + 2)c \right) \mathring{\nabla}^B \underline{c}\\
&\quad + \frac{1}{2} \mathring{\nabla}^E F_{BE} \left( \mathring{\nabla}^B(\mathring\Delta + 2) c + \mathring\epsilon^{BD} \mathring{\nabla}_D (\mathring\Delta + 2) \underline{c} \right) \\
&= \mathring{\nabla}_A C_{BD} \mathring{\nabla}^B C^{AD} + \frac{1}{4} \mathring{\nabla}_{A}(\mathring\Delta + 2) \underline{c}\mathring{\nabla}^{A}(\mathring\Delta +2)\underline{c}\\
& + \frac{1}{2} \mathring\epsilon^{BD} \mathring{\nabla}_B (\mathring\Delta + 2) c \mathring{\nabla}_D (\mathring\Delta + 2) \underline{c} + \frac{1}{4} \mathring{\nabla}_{A}(\mathring\Delta +2) c\mathring{\nabla}^{A}(\mathring\Delta+2)c\\\
&= \mathring{\nabla}_A C_{BD} \mathring{\nabla}^BC^{AD} + \mathring{\nabla}^B C_{AB} \mathring{\nabla}_D C^{AD}.
\end{align*}
Moreover, contracting \eqref{interchangeThree} in Proposition \ref{interchange} with $C^{AB}$, we obtain
\[C^{AB} \mathring\Delta C_{AB} = 2 C^{AB} \mathring{\nabla}^D\mathring{\nabla}_A C_{BD} - 2(C_{DE}C^{DE}).\]
Applying \eqref{scalarCurv}, we deduce
\begin{align*}
&R^{(2)} + \frac{1}{2} \mathring{\nabla}_A \left( C^{AB} \mathring{\nabla}^D C_{BD}\right) + \frac{1}{8} \mathring\Delta(C_{DE}C^{DE})\\
&= \frac{1}{2}(C_{DE}C^{DE}) + \frac{3}{4} \left( \mathring\Delta C_{AB} C^{AB} + \mathring{\nabla}_A C_{BD} \mathring{\nabla}^A C^{BD} \right) + \frac{1}{4} \mathring{\nabla}_A C_{BD} \mathring{\nabla}^B C^{AD}\\
&\quad- \frac{1}{4} \mathring{\nabla}^B C_{AB} \mathring{\nabla}_D C^{AD} - \frac{1}{2} \mathring{\nabla}_A \left( C^{AB} \mathring{\nabla}^D C_{BD} \right) - \mathring{\nabla}_A \left( C^{BD} \mathring{\nabla}_B C^A_D \right)\\
&= \frac{1}{2}(C_{DE}C^{DE})+ \frac{3}{2} C^{AB} \mathring{\nabla}^D \mathring{\nabla}_A C_{BD} - \frac{3}{2}(C_{DE}C^{DE}) + \mathring{\nabla}_A C_{BD} \mathring{\nabla}^B C^{AD}\\
& + \frac{1}{2} \mathring{\nabla}^B C_{AB} \mathring{\nabla}_D C^{AD}  - \frac{1}{2} \mathring{\nabla}_A \left( C^{AB} \mathring{\nabla}^D C_{BD} \right) - \mathring{\nabla}_A \left( C^{BD} \mathring{\nabla}_B C^A_D \right)\\
&= - (C_{DE}C^{DE}) + \frac{1}{2} C^{AB} \mathring{\nabla}^D \mathring{\nabla}_A C_{BD} -\frac{1}{2} C^{AB} \mathring{\nabla}_A \mathring{\nabla}^D C_{BD} \\
&= 0.
\end{align*}
\end{proof}

\begin{lemma}
Applying Proposition \ref{interchange}, we have
\begin{align*}
&\int_{S^2} \tilde X^i \Bigg( - \frac{1}{4}(\underline{F}_{DE}\underline{F}^{DE}) -\frac{1}{4} (\mathring\nabla_{A}\mathring\nabla_{B}\underline{c}\mathring\nabla^{A}\mathring\nabla^{B}\underline{c}) - \frac{1}{4} \mathring\nabla^{A}\underline{c}\mathring\nabla^{A}\underline{c}\\
&\quad\quad - \frac{1}{2} \mathring{\nabla}_A \underline{F}^{AB} \mathring{\nabla}_D \underline{F}_B^D + \frac{1}{2} \mathring{\nabla}_A \left( \underline{F}^{AB} \mathring{\nabla}^D \underline{F}_{DB} \right)+ \frac{1}{16} \mathring\nabla_{A}\mathring\Delta \underline{c} \mathring\nabla^{A}\mathring\Delta \underline{c} \Bigg)\\
& = \int_{S^2} -\frac{1}{16} \tilde X^i \mathring{\nabla}_{A}(\mathring\Delta + 2) \underline{c}\mathring{\nabla}^{A}(\mathring\Delta+2)\underline{c}.
\end{align*}
\end{lemma}
\begin{proof} We note the cancellation
\begin{align*}
&\int_{S^2} \frac{1}{2} \tilde X^i \mathring{\nabla}_A \left( \underline{F}^{AB} \mathring{\nabla}^D \underline{F}_{DB} \right) \\&= \int_{S^2} -\frac{1}{2} \mathring{\nabla}_A \tilde X^i \underline{F}^{AB} \mathring{\nabla}^D \underline{F}_{DB}\\
&= \int_{S^2} \left[- \frac{1}{2} \tilde X^i(\underline{F}_{DE}\underline{F}^{DE}) + \frac{1}{2} \mathring{\nabla}_A \tilde X^i \mathring{\nabla}^D \underline{F}^{AB} \underline{F}_{DB}\right]\\
&= \int_{S^2} \left[- \frac{1}{2} \tilde X^i(\underline{F}_{DE}\underline{F}^{DE}) + \frac{1}{2} \mathring{\nabla}_A \tilde X^i \left( \mathring{\nabla}^A \underline{F}^{BD} - \frac{1}{2} \mathring\epsilon^{DA} \mathring\Delta \mathring{\nabla}^B \underline{c} + \frac{1}{2} \mathring\epsilon^{BD} \mathring{\nabla}^A \underline{c} - \frac{1}{2} \mathring\epsilon^{BA} \mathring{\nabla}^D \underline{c} \right) \underline{F}_{DB}\right]\\
&= \int_{S^2} -\frac{1}{4} \mathring{\nabla}_A \tilde X^i \mathring\underline{F}_{DB} \mathring{\epsilon}^{DA} \mathring{\nabla}^B (\mathring\Delta + 2) \underline{c} \\
&= \int_{S^2} \frac{1}{8} \tilde X^i \left( (\mathring\Delta + 2)\underline{c} \right)^2,
\end{align*}
where we have used \eqref{interchangeTwo}.  Moreover, we rewrite
\begin{align*}
&\int_{S^2} - \frac{1}{4} \tilde X^i(\underline{F}_{DE}\underline{F}^{DE}) \\
&= \int_{S^2} -\frac{1}{4} \tilde X^i \underline{F}^{AB} \mathring\epsilon_{AD} \mathring{\nabla}_B \mathring{\nabla}^D \underline{c}\\
&= \int_{S^2} \left[\frac{1}{4} \mathring{\nabla}^D \tilde X^i \mathring\epsilon_{AD} \underline{F}^{AB} \mathring{\nabla}_B \underline{c} + \frac{1}{8} \tilde X^i \mathring{\nabla}^B (\mathring\Delta + 2) \underline{c} \mathring{\nabla}_B \underline{c}\right]\\
&= \int_{S^2} \left[-\frac{1}{8}\underline{c} \mathring{\nabla}^D \tilde X^i \mathring{\nabla}_D (\mathring\Delta + 2) \underline{c}  - \frac{1}{8} \mathring{\nabla}^B \tilde X^i (\mathring\Delta + 2) \underline{c} \mathring{\nabla}_B \underline{c} - \frac{1}{8} \tilde X^i (\mathring\Delta + 2) \underline{c} \mathring\Delta \underline{c}\right]\\
&= \int_{S^2} -\frac{1}{8} \tilde X^i \left( (\mathring\Delta + 2) \underline{c} \right)^2,
\end{align*}
using \eqref{interchangeFive}.

Next, we simplify
\begin{align*}
&\int_{S^2} -\frac{1}{4} \tilde X^i (\mathring\nabla_{A}\mathring\nabla_{B}\underline{c}\mathring\nabla^{A}\mathring\nabla^{B}\underline{c})\\
&= \int_{S^2} \left[\frac{1}{4} \mathring{\nabla}_A \tilde X^i \mathring{\nabla}_B \underline{c} \mathring{\nabla}^A \mathring{\nabla}^B \underline{c} + \frac{1}{4} \tilde X^i \mathring{\nabla}_B \underline{c} \mathring\Delta \mathring{\nabla}^B \underline{c}\right]\\
& = \int_{S^2} \left[\frac{1}{2} \tilde X^i \mathring{\nabla}_{A}\underline{c}\mathring{\nabla}^{A}\underline{c} + \frac{1}{4} \tilde X^i \mathring{\nabla}_B \underline{c} \mathring{\nabla}^B \mathring\Delta \underline{c}\right]
\end{align*}
and
\begin{align*}
\int_{S^2} -\frac{1}{2} \tilde X^i \mathring{\nabla}_A \underline{F}^{AB} \mathring{\nabla}_D \underline{F}^D_B &= \int_{S^2} - \frac{1}{8} \tilde X^i \mathring{\nabla}_{A}(\mathring\Delta + 2) \underline{c}\mathring{\nabla}^{A}(\mathring\Delta+2)\underline{c}.
\end{align*}
\end{proof}

\begin{lemma}
Per Proposition \ref{interchange},
\begin{align*}
&\int_{S^2} \tilde X^i \left( - \frac{1}{2} \mathring{\nabla}^B F_{BD} \mathring\epsilon^{DA} \mathring{\nabla}_A \underline{c} -\frac{1}{2} \mathring{\nabla}_A F^{AB} \mathring{\nabla}_D \underline{F}_B^D + \frac{1}{2} \mathring{\nabla}_A \left( F^{AB} \mathring{\nabla}^D \underline{F}_{DB} \right) - \frac{1}{2} F^{AB} \underline{F}_{AB} \right)\\ &= \int_{S^2} \tilde X^i \left( -\frac{1}{4} F^{AB} \underline{F}_{AB} - \mathring{\nabla}_A F^{AB} \mathring{\nabla}_D \underline{F}^D_B \right).
\end{align*}
\end{lemma}
\begin{proof}
We simplify
\begin{align*}
\int_{S^2} \tilde X^i \left( \mathring{\nabla}^B F_{BD} \mathring\epsilon^{DA} \mathring{\nabla}_A \underline{c} + F^{AB} \underline{F}_{AB} \right) &= \int_{S^2} - \mathring{\nabla}^B \tilde X^i F_{BD} \mathring\epsilon^{DA} \mathring{\nabla}_A \underline{c}\\
&= \int_{S^2} -\frac{1}{2} \underline{c}\mathring{\nabla}^B \tilde X^i \mathring\epsilon_{BA} \mathring{\nabla}^A (\mathring\Delta + 2) c \\
&= \int_{S^2} -\frac{1}{2} \tilde X^i \mathring{\nabla}^B (\mathring\Delta + 2) c\ \mathring\epsilon_{BA} \mathring{\nabla}^A \underline{c}\\
&= \int_{S^2} - \tilde X^i \mathring{\nabla}^B F_{BD} \mathring\epsilon^{DA} \mathring{\nabla}_A \underline{c}
\end{align*}
and
\begin{align*}
\int_{S^2} \tilde X^i \mathring{\nabla}_A \left( F^{AB} \mathring{\nabla}^D \underline{F}_{DB}\right) &= \int_{S^2} -\frac{1}{2} \mathring{\nabla}_A \tilde X^i F^{AB} \mathring\epsilon_{BD} \mathring{\nabla}^D (\mathring\Delta + 2) \underline{c} \\
&= \int_{S^2} -\frac{1}{4} \mathring{\nabla}_A \tilde X^i \mathring\epsilon^{AB} \mathring{\nabla}_B (\mathring\Delta + 2) c (\mathring\Delta + 2)\underline{c}\\
&= \int_{S^2} -\tilde X^i \mathring{\nabla}^D F_{BD} \mathring{\nabla}_A \underline{F}^{AB},
\end{align*}
using \eqref{interchangeFour} in each reduction.
\end{proof}

Finally, we evaluate the second part of the center-of-mass integral \eqref{comFirst}.
\begin{lemma}\label{last_two_terms}
\begin{align*}
\int_{S^2} &\left[2mX^{i(0)}- 2 \mathring{\nabla}^A \tilde X^i X^{0(0)} j_A^{(-1)}\right] \\
= \int_{S^2} &\Big[\frac{1}{2} \tilde X^i \mathring{\nabla}_D F^{AD} \mathring{\nabla}^B \underline{F}_{AB} - \mathring{\nabla}_A \tilde X^i c \mathring{\nabla}^A m + 2 \tilde X^i c m \\
&- \frac{1}{2} \tilde X^i (\mathring\Delta c) m + 2 \mathring\epsilon^{AB} \mathring{\nabla}_A \tilde X^i (\mathring{\nabla}_B \underline{c}) m\Big].
\end{align*}
\end{lemma}
\begin{proof}
Per the connection-one form expansions \eqref{alphaH} and \eqref{alphaH0} applied to $j_{A}$ \eqref{jDef}, we have
\begin{align}\label{jFirst}
\begin{split}
j_{A}^{(-1)} &= \mathring\nabla_{A}m - \frac{1}{4}\mathring\nabla_{A}\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right)\\
& - \frac{1}{2}\mathring\nabla^{D}C_{DA} - \frac{1}{2}\mathring\nabla_{A}(\mathring\Delta +2)X^{0(0)},\\
&= -\frac{1}{2}\mathring\nabla^{D}\underline{F}_{DA},
\end{split}
\end{align}
where we have substituted for $X^{0(0)}$ via \eqref{X00}.

The result follows from \eqref{jFirst} along with the expressions for $X^{i(0)}$ \eqref{linearizedEmbedding} and $X^{0(0)}$ \eqref{X00}.
\end{proof}

\begin{theorem}\label{main_com}
Suppose $(N,g)$ is a spacetime with Bondi-Sachs expansion \eqref{HPSexpansion}.  Further assuming that $(N,g)$ has vanishing linear momentum at null infinity \eqref{zeroLinMom} and positive Bondi mass, the center-of-mass at future null infinity has components
\begin{align}\label{comFinal}
\begin{split}
C^i = \frac{1}{8\pi e} \int_{S^2} &\Big[\mathring\nabla^{A}\tilde{X}^{i} N_A - \mathring\nabla^{A}\tilde{X}^{i} c \mathring{\nabla}_A m  + 3 \tilde X^i cm\\
 &+ 2 \mathring\epsilon^{AB} \mathring{\nabla}_A \tilde X^i (\mathring{\nabla}_B \underline{c}) m -\frac{1}{16} \tilde X^i  \mathring{\nabla}_{A}(\mathring\Delta + 2) \underline{c}\mathring{\nabla}^{A}(\mathring\Delta+2)\underline{c}\\
&- \frac{1}{2} \tilde{X}^i\mathring{\nabla}_D F^{AD} \mathring{\nabla}^B \underline{F}_{AB} -\frac{1}{4} \tilde{X}^i F^{AB}\underline{F}_{AB} \Big].
\end{split}
\end{align}
In particular, if $C_{AB}$ is closed, then
\[C^i =\frac{1}{8\pi e} \int_{S^2} \left[\mathring\nabla^{A}\tilde{X}^{i} N_A  - \mathring\nabla^{A}\tilde{X}^{i} c \mathring{\nabla}_A m  + 3 \tilde X^i cm\right].\]
In addition, if $C_{AB}$ is closed and $m$ is angular independent, then
\[C^{i} = \frac{1}{8\pi e} \int_{S^2} \mathring\nabla^{A}\tilde{X}^{i}N_{A}.\]
\end{theorem}
\begin{proof}
We have calculated \eqref{comFinal} assuming an observer $T_0 = (1,0,0,0)$.  More generally, the condition \eqref{zeroLinMom} allows for observers $T_{0,r}$ expanding according to \eqref{T0expansion}.  Following Definition \ref{qlCharge}, we write $T_{0,r} = A_{r}((1,0,0,0))$ for Lorentz transformations $A_{r}$ and measure the quasi-local center-of-mass \eqref{qlCOM} with respect to the Lorentz boost $A_{r}(K_{i,r})$, taking limits as $r$ goes to infinity to recover the components of the center-of-mass at future null infinity.

Owing to preservation under Lorentz transformations, the inner product expansion \eqref{innerProdExpansion} remains the same.  Likewise, the density terms in the expansion \eqref{densityFirst} and the current term $j_{A}^{(-1)}$ \eqref{jFirst} are unchanged.  Using the expansion of the Lorentz transformations $A_{r}$ \eqref{boostExpansion}, we have
\[ A_r(K_{i,r}) = x^i\partial_{t} + t\partial_{i} + \frac{1}{r}\left(x^{i}b_{j}\partial_j + tb_i\partial_t \pm t a_{ij}\partial_{j}\right) + O(r^{-2}),\]
from which application of the projection formula \eqref{projectionFormula} gives the expansion
\[(A_{r}(K_{i,r}))^{T,B} =\frac{1}{r}\left(-\mathring\nabla^{B}\left(\tilde{X}^{i}X^{0(0)}\right) + 2\mathring\nabla^{B}\tilde{X}^{i}X^{0(0)} + \tilde{X}^{i}b_j\mathring\nabla^{B}\tilde{X}^{j}\right) + O(r^{-2}),\]
differing from the earlier \eqref{projectionExpansion} in its final term.  The new terms in the center-of-mass component amount to
\begin{align*}
&\frac{1}{16\pi e}\int_{S^2} \tilde{X}^{i}b_{j}\mathring\nabla^{B}\tilde{X}^{j}\mathring\nabla^{D}\underline{F}_{DB}\\
&=\frac{-b_{j}}{32\pi e}\int_{S^2} \mathring\epsilon_{BE}\mathring\nabla^{E}\tilde{X}^{i}\mathring\nabla^{B}\tilde{X}^{j}(\mathring\Delta +2)\underline{c}\\
&=\frac{-b_{j}}{32\pi e}\int_{S^2} \underline{c}(\mathring\Delta +2)\left(\mathring\epsilon_{BE}\mathring\nabla^{E}\tilde{X}^{i}\mathring\nabla^{B}\tilde{X}^{j}\right) = 0.
\end{align*}
In this way, the earlier calculation of \eqref{comFinal} is preserved.
\end{proof}

\begin{remark}
The third term in \eqref{comFinal} can be rewritten as follows:
\begin{align*}
&\int \mathring\epsilon^{AB} \mathring{\nabla}_A \tilde X^i \mathring{\nabla}_B \underline{c} (\mathring\Delta + 2)M \\
&= \int_{S^2} \left[-\mathring\epsilon^{AB} \mathring{\nabla}_A \tilde X^i \mathring{\nabla}^D \mathring{\nabla}_B \underline{c} \mathring{\nabla}_D M + \mathring\epsilon^{AB} \tilde X^i \mathring{\nabla}_B \underline{c} \mathring{\nabla}_A M + 2 \mathring\epsilon^{AB} \mathring{\nabla}_A \tilde X^i \mathring{\nabla}_B \underline{c} M\right]\\
&= \int_{S^2} \left[\mathring\epsilon^{AB} \mathring{\nabla}_A \tilde X^i \mathring\Delta \mathring{\nabla}_B \underline{c} M + \mathring\epsilon^{AB} \mathring{\nabla}_A \tilde X^i \mathring{\nabla}_B \underline{c} M\right]\\
&= \int_{S^2} \mathring\epsilon^{AB} \mathring{\nabla}_A \tilde X^i \mathring{\nabla}_B (\mathring\Delta + 2) \underline{c} M.
\end{align*}
\end{remark}

\section{Limit of Quasi-local Angular Momentum}

We evaluate the quasi-local angular momentum formula \eqref{qlAM} with respect to the surfaces $(\Sigma_{r}, \sigma_{r})$ in a spacetime $(N,g)$ with vanishing linear momentum at null infinity \eqref{zeroLinMom} and with positive Bondi mass, such that the pairs $(X_{r}, T_{0,r})$ described above satisfy both the isometric embedding equation \eqref{IE} and optimal isometric embedding equation \eqref{oiee} to all orders.  In particular, we can make appropriate choices of $b_{i}$ in the observer expansion of $T_{0,r}$ \eqref{T0expansion} guaranteeing solvability of the optimal isometric embedding equation at first order \eqref{firstOrder} and second order \eqref{secondOrder}.  Taking limits as $r$ approaches infinity, we recover the components of the angular momentum at future null infinity.  In doing so, we make use of the data (\ref{H},\ \ref{alphaH},\ \ref{H0},\ \ref{alphaH0}), the derived expansions (\ref{tauExpansion},\ \ref{densityFirst},\ \ref{jDef},\ \ref{boostExpansion}), the linearized optimal isometric embedding equation \eqref{firstOrder}, and the expression for $X^{0(0)}$ \eqref{X00} arising from its analysis.

As with the calculation of center-of-mass, for simplicity we begin by considering the case where the observer $T_0 = (1,0,0,0)$; in particular, $b_i = 0$ in its expansion \eqref{T0expansion}.  According to Definition \ref{qlCharge}, we consider the angular momentum vector fields
\begin{equation}
K_{i,j,r} = x^{i}\partial_{j} - x^{j}\partial_{i},
\end{equation}
associated with $T_0 = (1,0,0,0).$  Here $K_{i,j,r}$ denotes the restriction of the Lorentz boost to the embedded surface $X_{r}(\Sigma_{r})$.

Evaluating on $X_{r}(\Sigma_{r})$, we have
\begin{equation}\label{innerProdVanishing}
\langle K_{i,j,r}, T_0 \rangle = 0.
\end{equation}

Considering the projection on the embedded surface $X_{r}(\Sigma_{r})$, we calculate
\begin{align*}
&(K^{T}_{i,j,r})^{B} = \left(\frac{1}{r^2}\mathring\sigma^{AB} - \frac{1}{r^3}C^{AB} + O(r^{-4})\right)\cdot\\
&\Bigg\langle \left(\Big(\mathring\nabla_{A}X^{0(0)} + \frac{1}{r}\mathring\nabla_{A}X^{0(-1)} + O(r^{-2})\Big)\partial_{t} + \left(r\mathring\nabla_{A}\tilde{X}^{k} + \mathring\nabla_{A}X^{k(0)} + O(r^{-1})\right)\partial_{k}\right),\\
&\left(r\tilde{X}^{i}\partial_{j}+X^{i(0)}\partial_{j} - r\tilde{X}^{j}\partial_{i}-X^{j(0)}\partial_{i} + O(r^{-1})\right)\Bigg\rangle\\
&=  \tilde{X}^{i}\mathring\nabla^{B}\tilde{X}^{j} - \tilde{X}^{j}\mathring\nabla^{B}\tilde{X}^{i} - \frac{1}{r}C^{B}_{A}(\tilde{X}^{i}\mathring\nabla^{A}\tilde{X}^{j} - \tilde{X}^{j}\mathring\nabla^{A}\tilde{X}^{i} )\\
&+ \frac{1}{r}\mathring\nabla^{B}\left(\tilde{X}^{j}X^{i(0)}-\tilde{X}^{i}X^{j(0)}\right)+\frac{2}{r}\left(\tilde{X}^{i}\mathring\nabla^{B}X^{j(0)} - \tilde{X}^{j}\mathring\nabla^{B}X^{i(0)}\right) + O(r^{-2}),
 \end{align*}
 which simplifies to
 \begin{align}\label{projectionExpansion2}
 \begin{split}
 (K^{T}_{i,j,r})^{B} &= Y_{i,j}^{B} - \frac{1}{r}\underline{F}^{B}_{D}Y_{i,j}^{D} + \frac{1}{r}\mathring\epsilon_{AD}\mathring\nabla^{B}\mathring\nabla^{D}\underline{c}Y_{i,j}^{A}\\
 &+  \frac{1}{r}\mathring\nabla^{B}\left(\tilde{X}^{j}X^{i(0)}-\tilde{X}^{i}X^{j(0)}\right) + O(r^{-2}),
\end{split}
\end{align}
where
\begin{equation}\label{YDef}
Y_{i,j}^{A} := \tilde{X}^{i}\mathring\nabla^{A}\tilde{X}^{j} - \tilde{X}^{j}\mathring\nabla^{A}\tilde{X}^{i} = \epsilon_{ijq}\mathring\epsilon^{AB}\mathring\nabla_{B}\tilde{X}^{q},
\end{equation}
and we have substituted for the $X^{i(0)}$ in the last term via \eqref{linearizedEmbedding}.

Note that $Y_{i,j}^{A}$ satisfies the Killing equation
\begin{equation}\label{YKilling}
\mathring\nabla^{A}Y_{i,j}^{B} = -\mathring\nabla^{B}Y_{i,j}^{A},
\end{equation}
which implies the further identities
\begin{align}
\mathring\nabla_{A}Y_{i,j}^{A} &= 0, \label{angMomOne}\\
\mathring\Delta Y_{i,j}^{A} &= -Y_{i,j}^{A}. \label{angMomTwo}
\end{align}

With the above calculations, the quasi-local angular momentum \eqref{qlAM} expands as
\begin{align}
\begin{split}
J^{k}(\Sigma_{r},X_{r},T_0) = \frac{r\epsilon^{ijk}}{16\pi}&\int_{S^2} Y_{i,j}^{A}\mathring\nabla^{D}\underline{F}_{DA} + O(1),
\end{split}
\end{align}
with worrisome, possibly divergent behavior arising from the top-order term of $j_{A}$ \eqref{jFirst}.  This top-order term vanishes via integration by parts and application of the Killing equation \eqref{YKilling}.

Expanding \eqref{qlAM} to the next order, we have vanishing of integrals of exact terms in the current $j_{A}^{(-2)}$ \eqref{jDef} contracted with $Y^{A}_{i,j}$, owing to integration by parts and application of \eqref{angMomOne}.  In addition, we have vanishing of the term
\[ \int_{S^2} \mathring\nabla^{B}\left(\tilde{X}^{j}X^{i(0)}-\tilde{X}^{i}X^{j(0)}\right)j_{B}^{(-1)} = -\frac{1}{2}\int_{S^2}\mathring\nabla^{B}\left(\tilde{X}^{j}X^{i(0)}-\tilde{X}^{i}X^{j(0)}\right)\mathring\nabla^{D}\underline{F}_{DB},\]
owing to integration by parts and
\[ \mathring\nabla^{B}\mathring\nabla^{D}\underline{F}_{BD} = 0,\]
following from co-closedness.  We could equally well have applied the linearized optimal isometric embedding equation \eqref{firstOrder}, amounting to a divergence-free condition on $j_{A}^{(-1)}.$

Applying these reductions and taking the limit as $r$ tends to infinity, we calculate the components of the angular momentum at future null infinity:\begin{align}\label{angmomFirst}
\begin{split}
J^k = -\frac{\epsilon^{ijk}}{8\pi}&\int_{S^2}\Bigg[- Y_{i,j}^{A}N_{A} + 2mY_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}+\frac{1}{4}Y_{i,j}^{A}C_{A}^{B}\mathring\nabla^{D}C_{DB}\\
& + \frac{1}{2}Y_{i,j}^{A}C_{A}^{D}\mathring\nabla_{D}X^{0(0)} +\frac{1}{2}Y_{i,j}^{A}\left(\mathring\nabla_{A}\mathring\nabla_{E}X^{0(0)}\right)\left(\mathring\nabla_{D}F^{DE}\right)\\
&+\frac{1}{2}Y_{i,j}^{A}\mathring\nabla_{A}\mathring\nabla_{D}X^{0(0)}\mathring\epsilon^{DE}\mathring\nabla_{E}\underline{c}+\frac{1}{2}Y_{i,j}^{A}\underline{F}_{A}^{B}\mathring\nabla^{D}\underline{F}_{DB}\\
&- \frac{1}{2}Y_{i,j}^{A}\mathring\epsilon_{AD}\mathring\nabla^{B}\mathring\nabla^{D}\underline{c}\mathring\nabla^{E}\underline{F}_{EB}\Bigg].
\end{split}
\end{align}

We simplify the terms involving $X^{0(0)}$ in the following lemma:
\begin{lemma}\label{ang_X0}
\begin{align*}
&\int_{S^2} \Bigg[ - 2mY_{i,j}^{A}\mathring\nabla_{A}X^{0(0)} - \frac{1}{2}Y_{i,j}^{A}C_{A}^{D}\mathring\nabla_{D}X^{0(0)} \\ &\qquad -\frac{1}{2}Y_{i,j}^{A}\left(\mathring\nabla_{A}\mathring\nabla_{E}X^{0(0)}\right)\left(\mathring\nabla_{D}F^{DE}\right)
-\frac{1}{2}Y_{i,j}^{A}\mathring\nabla_{A}\mathring\nabla_{D}X^{0(0)}\mathring\epsilon^{DE}\mathring\nabla_{E}\underline{c} \Bigg]\\
&= \int_{S^2} \left[- cY_{i,j}^{A}\mathring\nabla_{A}m +\frac{1}{4}Y_{i,j}^{A}\mathring\Delta X^{0(0)}\mathring\epsilon_{AD}\mathring\nabla^{D}\underline{c} \right]
\end{align*}
\end{lemma}
\begin{proof}

With repeated integration by parts and application of the properties of $Y^{A}_{i,j}$ (\ref{YKilling},\ \ref{angMomOne},\ \ref{angMomTwo}), we calculate
\begin{align*}
&\int_{S^2}-\frac{1}{2}Y_{i,j}^{A}\left(\mathring\nabla_{A}\mathring\nabla_{E}X^{0(0)}\right)\left(\mathring\nabla_{D}F^{DE}\right)\\
=&\int_{S^2}\left[\frac{1}{4}\left(\mathring\nabla^{E}Y_{i,j}^{A}\right)\mathring\nabla_{A}X^{0(0)}\mathring\nabla_{E}\left(\mathring\Delta + 2\right)c + \frac{1}{2}Y_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}\left(\mathring\nabla^{D}\mathring\nabla^{E}F_{DE}\right)\right]\\
=&\int_{S^2}\left[-\frac{1}{4}\mathring\Delta Y_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}\left(\mathring\Delta + 2\right)c + \frac{1}{2}Y_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right)\right]\\
=&\int_{S^2}\left[-\frac{1}{2}Y_{i,j}^{A}X^{0(0)}\mathring\nabla^{D}F_{DA} + \frac{1}{2}Y_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right)\right]\\
=&\int_{S^2}\left[\frac{1}{2}Y_{i,j}^{A}\mathring\nabla_{D}X^{0(0)}F^{D}_{A} + \frac{1}{2}Y_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right)\right].\\
\end{align*}
Hence, the original integral reduces to
\begin{align*}
\int_{S^2} \Bigg[&-2mY_{i,j}^{A}\mathring\nabla_{A}X^{0(0)} + \frac{1}{2}Y_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right) \\
& -\frac{1}{2} Y^A_{i,j} \underline{F}_A^D
 \mathring{\nabla}_D X^{0(0)} -\frac{1}{2}Y_{i,j}^{A}\mathring\nabla_{A}\mathring\nabla_{D}X^{0(0)}\mathring\epsilon^{DE}\mathring\nabla_{E}\underline{c} \Bigg].
\end{align*}
Substituting the linearized equation \eqref{firstOrder} into the first line, we calculate
\begin{align*}
&\int_{S^2}\left[-2mY_{i,j}^{A}\mathring\nabla_{A}X^{0(0)} + \frac{1}{2}Y_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}\left(\mathring\nabla^{D}\mathring\nabla^{E}C_{DE}\right)\right]\\
&= \int_{S^2} \left[-(\mathring\Delta +2)X^{0(0)}Y_{i,j}^{A}\mathring\nabla_{A}X^{0(0)} - \frac{1}{2}(\mathring\Delta +2)cY_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}\right],
\end{align*}
the terms of which reduce  as
\begin{align*}
&\int_{S^2} \left(\mathring\Delta + 2\right)X^{0(0)}Y_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}\\
= &\int_{S^2} \left[Y_{i,j}^{A}\mathring\nabla_{A}\left(X^{0(0)}\right)^2 -\mathring\nabla^{B}Y_{i,j}^{A}\mathring\nabla_{B}X^{0(0)}\mathring\nabla_{A}X^{0(0)} - \frac{1}{2}Y_{i,j}^{A}\mathring\nabla_{A}\left(\mathring\nabla^{B}X^{0(0)}\mathring\nabla_{B}X^{0(0)}\right)\right] = 0,
\end{align*}
and
\begin{align*}
&\int_{S^2}\left(\mathring\Delta + 2\right)cY_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}\\
&= \int_{S^2} \left[\mathring\Delta\left(Y_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}\right)c + 2Y_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}c\right] \\
&=\int_{S^2}\left[\mathring\Delta Y_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}c + Y_{i,j}^{A}\mathring\Delta\mathring\nabla_{A}X^{0(0)}c + 2Y_{i,j}^{A}\mathring\nabla_{A}X^{0(0)}c\right]\\
&=\int_{S^2}cY_{i,j}^{A}\mathring\nabla_{A}\left(\mathring\Delta + 2\right)X^{0(0)}\\
&=\int_{S^2} 2cY_{i,j}^{A}\mathring\nabla_{A}m ,
\end{align*}
where in the last line we have applied \eqref{firstOrder} and integrated away the quadratic terms in $c$, as in the previous reduction.  Throughout, we have made use of the propreties of $Y_{i,j}^{A}$ (\ref{YKilling},\ \ref{angMomOne},\ \ref{angMomTwo}).

For the second line, using the properties \eqref{YKilling} and \eqref{angMomTwo}, we note
\begin{align*}
&\int_{S^2} Y_{i,j}^{A}\underline{F}_{A}^{D}\mathring\nabla_{D}X^{0(0)} \\
&=\int_{S^2} -\frac{1}{2}Y_{i,j}^{A}X^{0(0)}\mathring\epsilon_{AD}\mathring\nabla^{D}\left(\mathring\Delta + 2\right)\underline{c}\\
&=\int_{S^2}\left[-Y_{i,j}^{A}X^{0(0)}\mathring\epsilon_{AD}\mathring\nabla^{D}\underline{c} - \frac{1}{2}Y_{i,j}^{A}X^{0(0)}\mathring\epsilon_{AD}\mathring\nabla^{D}\mathring\Delta\underline{c}\right]\\
&=\int_{S^2}\left[-\frac{1}{2}Y_{i,j}^{A}X^{0(0)}\mathring\epsilon_{AD}\mathring\nabla^{D}\underline{c} - \frac{1}{2}\mathring\Delta\left(Y_{i,j}^{A}X^{0(0)}\right)\mathring\epsilon_{AD}\mathring\nabla^{D}\underline{c}\right]\\
&=\int_{S^2}\left[-\frac{1}{2}Y_{i,j}^{A}\mathring\Delta X^{0(0)}\mathring\epsilon_{AD}\mathring\nabla^{D}\underline{c} - \mathring\nabla^{B}Y_{i,j}^{A}\mathring\nabla_{B}X^{0(0)}\mathring\epsilon_{AD}\mathring\nabla^{D}\underline{c}\right]\\
&=\int_{S^2}\left[-\frac{1}{2}Y_{i,j}^{A}\mathring\Delta X^{0(0)}\mathring\epsilon_{AD}\mathring\nabla^{D}\underline{c} - Y_{i,j}^{B}\mathring\nabla^{A}\mathring\nabla_{B}X^{0(0)}\mathring\epsilon_{AD}\mathring\nabla^{D}\underline{c}\right].
\end{align*}

Putting the two reductions together, we obtain Lemma \ref{ang_X0}.
\end{proof}

Turning to the terms involving the shear tensor alone, we have the following lemma:
\begin{lemma}\label{ang_shear_alone}
\begin{align*}
&\int_{S^2} \Bigg[ -\frac{1}{4}Y_{i,j}^{A}C_{A}^{B}\mathring\nabla^{D}C_{DB} -\frac{1}{2}Y_{i,j}^{A}\underline{F}_{A}^{B}\mathring\nabla^{D}\underline{F}_{DB} + \frac{1}{2}Y_{i,j}^{A}\mathring\epsilon_{AD}\mathring\nabla^{B}\mathring\nabla^{D}\underline{c}\mathring\nabla^{E}\underline{F}_{EB}\Bigg]\\
&= \int_{S^2} \Bigg[ -\frac{1}{4}Y_{i,j}^{A}F_{A}^{B}\mathring\nabla^{D}\underline{F}_{DB}-\frac{1}{4}Y_{i,j}^{A}\underline{F}_{A}^{B}\mathring\nabla^{D}F_{DB}\Bigg].
\end{align*}
\end{lemma}
\begin{proof}
We have the vanishing
\begin{align*}
&\int_{S^2} Y_{i,j}^{A}F_{A}^{B}\mathring\nabla^{D}F_{DB} = \int_{S^2}\frac{1}{2}Y_{i,j}^{A}F_{A}^{B}\mathring\nabla_{B}\left(\mathring\Delta + 2\right)c\\
=& \int_{S^2} -\frac{1}{2}Y_{i,j}^{A}\mathring\nabla_{B}F_{A}^{B}\left(\mathring\Delta + 2\right)c = \int_{S^2}-\frac{1}{8}Y_{i,j}^{A}\mathring\nabla_{A}\left(\left(\mathring\Delta +2\right)c\right)^2 = 0,
\end{align*}
using the properties of $Y^{A}_{i,j}$ \eqref{YKilling} and \eqref{angMomOne}.

In addition, we compute
 \begin{align*}
\int_{S^2} \mathring\epsilon^{AE} \mathring{\nabla}_E \tilde X^k \underline{F}_A^B \mathring{\nabla}^D \underline{F}_{DB} &= \int_{S^2} \frac{1}{2} \mathring\epsilon^{AE} \mathring{\nabla}_E \tilde X^k \underline{F}_A^B \mathring\epsilon_{BD} \mathring{\nabla}^D (\mathring\Delta + 2)\underline{c}
\\ &= \int_{S^2} \frac{1}{2} \mathring\epsilon^{AE} \mathring{\nabla}_E \tilde X^k \mathring\epsilon_{BD} \mathring{\nabla}^B \underline{F}^D_A (\mathring\Delta + 2) \underline{c}\\
&= \int_{S^2} - \frac{1}{4} \mathring\epsilon^{AE} \mathring{\nabla}_E \tilde X^k \mathring{\nabla}_A (\mathring\Delta + 2) \underline{c}\ (\mathring\Delta + 2) \underline{c}\\
&=0,
\end{align*}
where we have applied \eqref{interchangeFive}, and
\begin{align*}
\int_{S^2} \mathring{\nabla}_D \tilde X^k \mathring{\nabla}^B \mathring{\nabla}^D \underline{c} \mathring{\nabla}^E \underline{F}_{EB} &= \int_{S^2} \frac{1}{2} \mathring{\nabla}_D \tilde X^k \mathring\epsilon_{BE} \mathring{\nabla}^B\mathring{\nabla}^D \underline{c}  \mathring{\nabla}^E (\mathring\Delta + 2) \underline{c}\\
&= \int_{S^2} \frac{1}{2} \tilde X^k \mathring\epsilon_{BE} \mathring{\nabla}^B \underline{c}  \mathring{\nabla}^E (\mathring\Delta + 2) \underline{c}\\
&= \int_{S^2} -\frac{1}{2} \mathring{\nabla}^E \tilde X^k \mathring\epsilon_{BE} \mathring{\nabla}^B \underline{c}  \mathring\Delta \underline{c}\\
&= \int_{S^2} \frac{1}{4} \mathring{\nabla}^E \tilde X^k \mathring\epsilon_{BE} \mathring{\nabla}^B(\mathring{\nabla}_{A}\underline{c}\mathring{\nabla}^{A}\underline{c})\\
&=0.
\end{align*}

Per \eqref{YDef}, $Y_{i,j}^A = \varepsilon_{ijq} \mathring\epsilon_{AB} \mathring{\nabla}^B \tilde X^q$, and the relations above imply
\begin{align*}
\int_{S^2} Y_{i,j}^A \underline{F}_A^B \mathring{\nabla}^D \underline{F}_{DB} =0,
\end{align*}
and
\begin{align*}
\int_{S^2} Y_{i,j}^A \mathring\epsilon_{AD} \mathring{\nabla}^B \mathring{\nabla}^D \underline{c} \mathring{\nabla}^E \underline{F}_{EB}=0.
\end{align*}
\end{proof}

Applying these reductions, we have a simplified formula for the angular momentum at future null infinity:
\begin{theorem}\label{main_angmom}
Suppose $(N,g)$ is a spacetime with Bondi-Sachs expansion \eqref{HPSexpansion}.  Further assuming that $(N,g)$ has vanishing linear momentum at null infinity \eqref{zeroLinMom} and positive Bondi mass, the angular momentum at null infinity has the components
\begin{align}\label{angmomFinal}
\begin{split}
J^{k} = \frac{\epsilon^{ijk}}{8\pi}\int_{S^2}&\Bigg[Y_{i,j}^{A}N_{A} - cY_{i,j}^{A}\mathring\nabla_{A}m +\frac{1}{4}Y_{i,j}^{A}\mathring\Delta X^{0(0)}\mathring\epsilon_{AD}\mathring\nabla^{D}\underline{c}\\
&-\frac{1}{4}Y_{i,j}^{A}F_{A}^{B}\mathring\nabla^{D}\underline{F}_{DB}-\frac{1}{4}Y_{i,j}^{A}\underline{F}_{A}^{B}\mathring\nabla^{D}F_{DB}\Bigg].
\end{split}
\end{align}
In particular, if $C_{AB}$ is closed, then
\[J^k =\frac{\epsilon^{ijk}}{8\pi} \int_{S^2} \left[Y_{i,j}^{A}N_{A} - cY_{i,j}^{A}\mathring\nabla_{A}m\right].\]
In addition, if $C_{AB}$ is closed and $m$ is angular independent, then
\[J^{k} = \frac{\epsilon^{ijk}}{8\pi} \int_{S^2} Y_{i,j}^{A}N_{A}.\]
\end{theorem}

\begin{proof}
We have calculated \eqref{angmomFinal} with respect to an observer $T_0 = (1,0,0,0)$, assuming that \eqref{firstOrder} and \eqref{secondOrder} are solvable.  More generally, we have solvability of the two for appropriate choice of $b_{i}$ in the observer expansion \eqref{T0expansion}, assuming vanishing of linear momentum at null infinity \eqref{zeroLinMom} and positivity of the Bondi mass.  Following Definition \ref{qlCharge}, we write $T_{0,r} = A_{r}((1,0,0,0))$, with $SO(3,1)$ transformations $A_{r}$, and measure the quasi-local angular momentum with respect to the Lorentz rotation $A_{r}(K_{i,j,r})$.

Owing to preservation under $SO(3,1)$ transformations, vanishing of the inner product \eqref{innerProdVanishing} still holds.  Using the expansion of the Lorentz transformations $A_{r}$ \eqref{boostExpansion}, we have
\[ A_r(K_{i,j,r}) = x^i\partial_{j} - x^{j}\partial_{i} + \frac{1}{r}\left((x^ib^j - x^jb^i)\partial_{t} -a_{ij}(x^i\partial_{i} + x^j\partial_{j})\right)+ O(r^{-2}).\]
from which application of the projection formula \eqref{projectionFormula} gives the projection expansion
 \begin{align*}
&(K^{T}_{i,j,r})^{B} = Y_{i,j}^{B} - \frac{1}{r}\underline{F}^{B}_{D}Y_{i,j}^{D} + \frac{1}{r}\mathring\epsilon_{AD}\mathring\nabla^{B}\mathring\nabla^{D}\underline{c}Y_{i,j}^{A}\\
 &+  \frac{1}{r}\mathring\nabla^{B}\left(\tilde{X}^{j}X^{i(0)}-\tilde{X}^{i}X^{j(0)} -\frac{a_{ij}}{2}((\tilde{X}^{i})^2+(\tilde{X}^{j})^2)\right) + O(r^{-2}),
\end{align*}
differing from the earlier expansion \eqref{projectionExpansion2} by an exact term.  The contribution to the quasi-local formula amounts to
\[ \frac{a_{ij}}{16\pi}\int_{S^2}\mathring\nabla^{B}\left((\tilde{X}^{i})^2+(\tilde{X}^{j})^2\right)j_{B}^{(-1)} = 0,\]
owing to integration by parts and the divergence-free property of $j_{B}^{(-1)}$, i.e. the linearized optimal isometric embedding equation.  One can also see this by substituting the formula for $j_{B}^{(-1)}$ \eqref{jFirst} and using the double-divergence-free property of $\underline{F}_{AB}$.

There are also changes in $j_{A}^{(-2)}$, via the definition of $j_{A}$ \eqref{jDef} and the expansion of $\tau$ \eqref{tauExpansion}.  The modifications to the quasi-local formula vanish, as
\begin{align*}
&\frac{1}{8\pi}\int_{S^2}\left[2mY_{i,j}^{A}\mathring\nabla_{A}b_{n}\tilde{X}^{n} + Y_{i,j}^{A}\mathring\nabla_{A}\left(m b_{n}\tilde{X}^{n}\right)\right]\\
&=\frac{b_{n}}{8\pi}\int_{S^2}(\mathring\Delta +2)M\epsilon_{ijq}\mathring\epsilon^{AB}\mathring\nabla_{B}\tilde{X}^{q}\mathring\nabla_{A}\tilde{X}^{n}\\
&=\frac{b_{n}}{8\pi}\int_{S^2}M\epsilon_{ijq}(\mathring\Delta +2)\left(\mathring\epsilon^{AB}\mathring\nabla_{A}\tilde{X}^{q}\mathring\nabla_{B}\tilde{X}^{n}\right) = 0,
\end{align*}
where we have used the definition of $Y^{A}_{i,j}$ \eqref{YDef} and its divergence-free property \eqref{angMomOne}.

With the vanishing of these terms and taking limits, we see that the earlier formula \eqref{angmomFinal} is preserved.
\end{proof}

\section{Example: Kerr Spacetime}

Barnich-Troessaert \cite{Barnich} provide a BMS coordinate system $(u,r,\theta,\phi)$ for the Kerr spacetime, such that
\[g_{rr} = g_{r\theta} = g_{r\theta} = 0,\]
and the spherical part of the metric satisfies the determinant condition \eqref{areaNorm}
\[\det g_{AB} = r^4 \sin^2\theta.\]
The remaining components expand as
\begin{align}
\begin{split}
g_{uu} &= -1 + 2Mr^{-1} + O(r^{-2}),\\
g_{ur} &= -1 + a^2(\frac{1}{2}-\cos^2\theta)r^{-2} + O(r^{-3}),\\
g_{u\theta} &= \frac{a\cos\theta}{2\sin^2\theta} + \frac{a\cos\theta}{4}\left(8M + \frac{a}{\sin^3\theta}\right)r^{-1} + O(r^{-2}),\\
g_{u\phi} &= -2aM\sin^2\theta r^{-1} + O(r^{-2}),\\
g_{\theta\theta} &= r^2 + \frac{a}{\sin\theta} r + \frac{a^2}{2\sin^2\theta} + O(r^{-1}),\\
g_{\theta\phi} &= O(r^{-1}),\\
g_{\phi\phi} &= r^2\sin^2\theta - a\sin\theta r + \frac{a^2}{2} + O(r^{-1}),
\end{split}
\end{align}
where $M$ is the mass and $a$ is the angular velocity.

Regarding the quantities appearing in the Hawking-Perry-Strominger expansion \eqref{HPSexpansion}, the mass aspect $m = M$ is constant, while the shear tensor $C_{AB}$ has closed form, with components
\begin{align}\label{shearKerr}
\begin{split}
C_{\theta\theta} &= \frac{a}{\sin\theta},\\
C_{\theta\phi} &= 0,\\
C_{\phi\phi} &= -a\sin\theta,
\end{split}
\end{align}
and potentials
\begin{align}
\begin{split}
c &= -2a\sin\theta,\\
\underline{c} &= 0.
\end{split}
\end{align}

Finally, the angular momentum aspect $N_{A}$ has components
\begin{align}\label{angAspectKerr}
\begin{split}
N_{\theta} &= 3Ma\cos\theta,\\
N_{\phi} &= -3Ma\sin^2\theta.
\end{split}
\end{align}
We also calculate the divergence and curl of $N_{A}$:
\begin{align}\label{angAspectDivCurl}
\begin{split}
\mathring\nabla^{A}N_{A} &= 3Ma\left(-2\sin\theta + \frac{1}{\sin\theta}\right) = 3Ma\mathring\Delta\sin\theta,\\
\mathring\epsilon^{AB}\mathring\nabla_{B}N_{A} &= 6Ma\cos\theta.
\end{split}
\end{align}

As the mass aspect is constant, the condition \eqref{zeroLinMom} holds, yielding vanishing of the linear momentum at future null infinity.  That is, the spacetime is in center-of-mass frame, and solvability of the linearized optimal isometric embedding equation \eqref{firstOrder} is ensured.  In particular, we find
\[ X^{0(0)} = -\frac{1}{4}(\mathring\Delta +2)c.\]
Considering the next order in the optimal isometric embedding equation, we can directly compute vanishing of the term $S$ in the discussion following \eqref{secondOrder}, using the mass aspect, the angular momentum aspect \eqref{angAspectKerr}, the shear tensor \eqref{shearKerr}, and the form of $X^{0(0)}$ obtained by solving the linearized equation \eqref{firstOrder}.  As a consequence, we consider the observer $T_0 = (1,0,0,0)$, such that the $b_{i} = 0$ in the observer expansion \eqref{T0expansion}.

In calculating the center-of-mass and angular momentum at future null infinity, it is helpful to express the first eigenfunctions in spherical coordinates:
\begin{align}
\begin{split}
\tilde{X}^1 &= \sin\theta\cos\phi,\\
\tilde{X}^2 &= \sin\theta\sin\phi,\\
\tilde{X}^3 &= \cos\theta.
\end{split}
\end{align}

Owing to the simplicity of the Kerr spacetime, with constant mass aspect and closed shear tensor, we apply the special case appearing below our general formula for the quasi-local center-of-mass at null infinity \eqref{comFinal} to deduce
\[ C^{i} = \frac{1}{8\pi} \int_{S^2} \mathring\nabla^{A}\tilde{X}^{i}N_{A} = 0,\]
via integration by parts and \eqref{angAspectDivCurl}.

On the other hand, the special case appearing below our general formula for the quasi-local angular momentum \eqref{angmomFinal} yields
\begin{align}
\begin{split}
J^1 &= -\frac{1}{8\pi}\int_{S^2} \tilde{X}^1 \mathring\epsilon^{AB}\mathring\nabla_{B}N_{A} = 0,\\
J^2 &= -\frac{1}{8\pi}\int_{S^2} \tilde{X}^2 \mathring\epsilon^{AB}\mathring\nabla_{B}N_{A} = 0,\\
J^3 &= \frac{1}{8\pi}\int_{S^2}\mathring\epsilon^{AB}\mathring\nabla_{B}\tilde{X}^{3} N_{A} = -Ma,
\end{split}
\end{align}
where
\[ \mathring\epsilon_{AB}\mathring\nabla^{B}\tilde{X}^3 = \partial_{\phi}\]
in spherical coordinates.  In this way, the calculations are precisely what we expect from the usual presentation of the Kerr spacetime.

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redpajama_set_name: RedPajamaArXiv


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