while $M$ and $L_{A}$ are related to the Weyl tensor components (Eq. 2.24)
through

$\displaystyle\sigma_{AB}=-\tfrac{1}{2}C_{AB}$ $\displaystyle\,,\quad
N_{AB}=-\partial_{u}C_{AB}\,,$ (6.10)
$\displaystyle\mathcal{P}=-2M+\tfrac{1}{4}C_{AB}N^{AB}$
$\displaystyle\,,\quad\mathcal{J}_{A}=3L_{A}-\tfrac{3}{32}\mathscr{D}_{A}(C_{BC}C^{BC})-\tfrac{3}{4}C_{A}{}^{B}\mathscr{D}^{C}C_{BC}\,.$

Imposing the Einstein equation to higher order in $\Omega$ either relates the
higher order metric components to the lower order ones or gives evolution
equations along $u$ — for instance, one gets equations for $\partial_{u}M$ and
$\partial_{u}L_{A}$ which are equivalent to Eq. 2.25 using Eq. 6.10. We will
not need the explicit form of these higher order equations in our analysis.

The conformal Bondi-Sachs coordinates defined above can be used to define the
_physical Bondi-Sachs coordinates_ $(r,u,x^{A})$ which are often used in the
asymptotic analysis near $\mathscr{I}$ (see, for instance, Flanagan:2015pxa ).
Define the physical “radial coordinate” $r\mathrel{\mathop{:}}=\Omega^{-1}$,
and using $(r,u,x^{A})$ as coordinates, the physical metric
$\hat{g}_{ab}=\Omega^{-2}g_{ab}=r^{2}g_{ab}$ has the line element

$\displaystyle d\hat{s}^{2}\equiv-
Ue^{2B}du^{2}-2e^{2B}dudr+r^{2}h_{AB}(dx^{A}-U^{A}du)(dx^{B}-U^{B}du)\,,$
(6.11)

where, from Eqs. 6.2, 6.5, 6.7 and 6.9, we have the asymptotic expansions

$\displaystyle U=r^{2}W$ $\displaystyle=1-\tfrac{2}{r}M+O(1/r^{2})\,,$ (6.12)
$\displaystyle B$
$\displaystyle=-\tfrac{1}{32r^{2}}C^{AB}C_{AB}+O(1/r^{3})\,,$ $\displaystyle
U^{A}$
$\displaystyle=-\tfrac{1}{2r^{2}}\mathscr{D}_{B}C^{AB}+\tfrac{2}{r^{3}}L^{A}+O(1/r^{4})\,,$
$\displaystyle h_{AB}$
$\displaystyle=s_{AB}+\tfrac{1}{r}C_{AB}+\tfrac{1}{4r^{2}}s_{AB}C^{CD}C_{CD}+O(1/r^{3})\,.$

Note that since $\Omega$, and hence $r$, is chosen so that $\det h=\det s$,
the area (in the physical metric) of the $2$-spheres of constant $u$ and $r$
is precisely $4\pi r^{2}$. Thus $r$ is a “radial coordinate” along the
outgoing null surfaces $\mathcal{N}_{u}$ as constructed by Bondi and van der
Burg BBM . We emphasize that the Bondi-Sachs asymptotic form of the physical
metric (Eqs. 6.11 and 6.12) directly follows from the covariant definition of
asymptotic flatness in a particular choice of coordinate system and _does not_
involve any additional assumptions.

Having chosen the conformal Bondi-Sachs coordinate system, the asymptotic BMS
symmetries are coordinate transformations which preserve the Bondi-Sachs form
of the unphysical metric. We shall only consider the infinitesimal coordinate
transformations, i.e., we take $(\Omega,u,x^{A})$ and
$(\Omega^{\prime},u^{\prime},{x^{\prime}}^{A})$ to be two conformal Bondi-
Sachs coordinates, as constructed above, related by an infinitesimal
coordinate transformation parametrized by a vector field $\xi^{a}$. We next
obtain an expression for this vector field in the coordinate system
$(\Omega,u,x^{A})$.

Since in both coordinates $(\Omega,u,x^{A})$ and
$(\Omega^{\prime},u^{\prime},{x^{\prime}}^{A})$, null infinity $\mathscr{I}$
lies at $\Omega=\Omega^{\prime}=0$, the component $\xi^{\Omega}$ vanishes at
$\Omega=0$, i.e., $\xi^{a}$ must be tangent to $\mathscr{I}$. Thus, $\xi^{a}$
can be written in the coordinate system $(\Omega,u,x^{A})$ as

$\displaystyle\xi^{a}\equiv\beta\partial_{u}+X^{A}\partial_{A}+\Omega
Z^{a}\partial_{a}+\Omega^{2}W^{a}\partial_{a}+O(\Omega^{3})\,,$ (6.13)

where each of $\beta,X^{A},Z^{a}$ and $W^{a}$ are functions of $(u,x^{A})$.

Next we note that, since we are using the conformal factor itself as a
coordinate, an infinitesimal change in the coordinate system is accompanied by
an infinitesimal change of the conformal factor parametrized by the component
$\xi^{\Omega}$. Thus, when changing the coordinate system, the unphysical
metric changes infinitesimally by
$\pounds_{\xi}g_{ab}-2\Omega^{-1}\xi^{\Omega}g_{ab}$; note that this is finite
at $\mathscr{I}$ since $\xi^{\Omega}$ vanishes there. We now require that this
change preserve the Bondi-Sachs form of the metric obtained in Eq. 6.2 along
with Eq. 6.5 and the equations below.

As discussed above, the metric on $\mathscr{I}$, given by Eq. 6.1, is
universal and so we require that at $O(\Omega^{0})$,
$\pounds_{\xi}g_{ab}-2\Omega^{-1}\xi^{\Omega}g_{ab}\mathrel{\mathop{\widehat{=}}}0$.
This gives us the following conditions: $X^{A}$ is constant along $u$ and
satisfies the conformal Killing equation on the cross-sections, that is,

$\displaystyle\mathscr{D}_{(A}X_{B)}=\tfrac{1}{2}q_{AB}\mathscr{D}_{C}X^{C}\,,\quad\partial_{u}X^{A}=0\,,$
(6.14)

while the components of $Z^{a}$ satisfy

$\displaystyle
Z^{\Omega}=\partial_{u}\beta=\tfrac{1}{2}\mathscr{D}_{A}X^{A}\,,\quad
Z^{u}=0\,,\quad Z_{A}=-\mathscr{D}_{A}\beta\,.$ (6.15)

The first condition in Eq. 6.15 allows us to solve for the $u$-dependence of
$\beta$ to get

$\displaystyle\beta=f+\tfrac{1}{2}(u-u_{0})\mathscr{D}_{A}X^{A}\,,$ (6.16)

where $f$ is an arbitrary function of $x^{A}$ which denotes the value of
$\beta$ at some choice of cross-section with $u=u_{0}$.

At $O(\Omega)$, requiring that the form of the unphysical metric be preserved,
we further obtain

$\displaystyle W^{u}=0\,,\quad
W_{A}=\tfrac{1}{2}C_{AB}\mathscr{D}^{B}\,\beta\,,$ (6.17)

and the requirement that the metric component $C_{AB}$ remain trace-free with
respect to the unit round sphere metric gives us

$\displaystyle
W^{\Omega}=\tfrac{1}{2}\mathscr{D}_{A}Z^{A}=-\tfrac{1}{2}\mathscr{D}^{2}\beta\,.$
(6.18)

At this order one also obtains the transformation of $C_{AB}$ under a BMS
symmetry which, upon using Eq. 6.10, coincides with Eq. 4.33 (see also Eq.
2.18b of Flanagan:2015pxa ).

Putting all of this together, we obtain

$\displaystyle\xi^{a}\equiv\beta\partial_{u}+X^{A}\partial_{A}+\Omega(-\mathscr{D}^{A}\beta\,\partial_{A}+\tfrac{1}{2}\mathscr{D}_{A}X^{A}\partial_{\Omega})+\tfrac{1}{2}\Omega^{2}(-\mathscr{D}^{2}\beta\,\partial_{\Omega}+C^{AB}\mathscr{D}_{B}\beta\,\partial_{A})+O(\Omega^{3})\,.$
(6.19)

Up to $O(\Omega)$, this expression agrees with the covariant expression
derived in Sec. 4. The form of the $O(\Omega^{2})$ terms here is fixed by the
choice of the conformal Bondi-Sachs coordinates and that also matches Eq. 2.16
of Flanagan:2015pxa . One could continue this computation to higher orders in
$\Omega$, which gives expressions for higher order terms of the components of
$\xi^{a}$ and also the transformation laws for the various metric components,
but we will not require these expressions.

Using the relations Eq. 6.10 in Eq. 5.44 the WZ charge on _any_ cross-section
$S$ of $\mathscr{I}$ can be written in terms of the metric components in the
Bondi-Sachs form of the metric to get

$\displaystyle\mathcal{Q}[\xi,S]$
$\displaystyle\mathrel{\mathop{\widehat{=}}}-\frac{1}{8\pi}\int\limits_{S}\boldsymbol{\varepsilon}_{2}\bigg{[}-2M\beta+X^{A}\big{(}3L_{A}-\tfrac{1}{32}\mathscr{D}_{A}(C_{BC}C^{BC})-\tfrac{1}{2}C_{A}{}^{B}\mathscr{D}^{C}C_{BC}\big{)}\bigg{]}\,.$
(6.20)

The above charge expression matches the charge expression given by Flanagan
and Nichols in Eq. 3.5 of Flanagan:2015pxa , even though their expression was
calculated on a cross section of $\mathscr{I}$ where the News vanishes. A
similar expression was also obtained by Barnich and Troessaert BT  with non-
vanishing News but they do not find the integrable charge.

From the above expression we also see that the function $M$ determines the
Bondi mass at any cross-section of $\mathscr{I}$ and can be called the _mass
aspect_. In the Bondi-Sachs coordinates it coincides with the (constant) mass
parameter of Kerr spacetimes. Similarly, in the usual choice of Bondi-Sachs
coordinates in Kerr spacetime the angular momentum parameter $a$ appears in
the metric component $L_{A}$ Venter:2005cs . Note however, on cross-sections
with shear, there are other terms containing the $C_{AB}$ in the charge
formulae. In any case, since there is no preferred Lorentz subalgebra in the
BMS symmetries (see Remark 4.2), there is no preferred notion of “angular
momentum aspect” — so identifying any particular metric component with “the
angular momentum” is a moot point.

###  2 Conformal Gaußian null coordinates

Instead of extending the coordinates $(\Omega,u,x^{A})$ away from
$\mathscr{I}$ along null hypersurfaces, we can extend them into the
(unphysical) spacetime along affine null geodesics, transverse to
$\mathscr{I}$. We recall this construction below which leads to the _conformal
Gaußian null coordinates_ in a neighborhood of $\mathscr{I}$.

We fix the conformal factor to leading order away from $\mathscr{I}$ as
follows. Consider the _expansion_ of $l^{a}$ at $\mathscr{I}$ defined by

$\displaystyle\vartheta\mathrel{\mathop{:}}=Q^{ab}\nabla_{a}l_{b}\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}Q^{ab}\pounds_{l}g_{ab}\,.$
(6.21)

We can set this expansion to vanish by suitably choosing the conformal factor
away from $\mathscr{I}$ as follows. Note that the conformal factor on
$\mathscr{I}$ has already been fixed so that the induced metric on cross-
sections of $\mathscr{I}$ is the unit-sphere metric. So consider a new
conformal factor $\tilde{\Omega}=\omega\Omega$ (with
$\omega\mathrel{\mathop{\widehat{=}}}1$) so that
$\tilde{g}_{ab}=\omega^{2}g_{ab}$. Then, the expansion of the new auxiliary
normal $\tilde{l}^{a}\mathrel{\mathop{\widehat{=}}}l^{a}$ can be computed to
be (the behaviour of the auxiliary normal away from $\mathscr{I}$ is not
relevant here)

$\displaystyle\tilde{\vartheta}\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}\tilde{Q}^{ab}\pounds_{\tilde{l}}\tilde{g}_{ab}\mathrel{\mathop{\widehat{=}}}\vartheta+2\pounds_{l}\omega\,.$
(6.22)

Then, choosing $\omega$ to be any solution of $\vartheta+2\pounds_{l}\omega=0$
with $\omega\mathrel{\mathop{\widehat{=}}}1$ we can set
$\tilde{\vartheta}\mathrel{\mathop{\widehat{=}}}0$. In the rest of this
section we work the choice of conformal factor where the auxiliary normal is
expansion-free at $\mathscr{I}$ — we drop the “tilde” from the notation.

Having made this choice, we then extend the vector field
$l^{a}\mathrel{\mathop{\widehat{=}}}-\partial/\partial\Omega$ away from
$\mathscr{I}$ to be the generator of affine null geodesics so that
$l^{b}\nabla_{b}l^{a}=0$. We can further use the remaining freedom in the
conformal factor $\Omega$ so that $\Omega$ is the affine parameter along these
null geodesics generated by $l^{a}$. To summarize, we can always choose the
conformal factor $\Omega$ and extend the auxiliary normal away from
$\mathscr{I}$ so that in a neighbourhood of $\mathscr{I}$ we have

$\displaystyle
l^{a}=-\frac{\partial}{\partial\Omega}\,,\quad\vartheta\mathrel{\mathop{\widehat{=}}}0\,,\quad
l^{a}l_{a}=0\,,\quad l^{b}\nabla_{b}l^{a}=0.$ (6.23)

Finally, we extend $(u,x^{A})$ into the spacetime by parallel-transport along
$l^{a}$, that is, we require

$\displaystyle l^{a}\nabla_{a}u=l^{a}\nabla_{a}x^{A}=0\,.$ (6.24)

This construction gives us the conformal Gaußian null coordinates in a
neighbourhood of $\mathscr{I}$.

The most general form of the unphysical metric in these coordinates is given
by

$\displaystyle ds^{2}$ $\displaystyle=2du(d\Omega-\alpha
du-\beta_{A}dx^{A})+h_{AB}dx^{A}dx^{B}\,,$ (6.25)

To see why this is the most general form, note that $g_{\Omega\Omega}=0$ by
$l^{a}l_{a}=0$ and $g_{\Omega A}=0$ by $l^{a}\nabla_{a}x^{A}=0$. Then,
$l^{b}\nabla_{b}l^{a}=0$ gives $\frac{\partial g_{u\Omega}}{\partial\Omega}=0$
which implies $g_{u\Omega}=1$ by the condition
$l^{a}n_{a}\mathrel{\mathop{\widehat{=}}}-1$. Further,

$\displaystyle\alpha=O(\Omega)\,,\quad\beta_{A}=O(\Omega)\,,\quad
h_{AB}=s_{AB}+O(\Omega)\,,$ (6.26)

since the metric at $\mathscr{I}$ is given by Eq. 6.1. Imposing the Bondi
condition $\nabla_{a}n_{b}\mathrel{\mathop{\widehat{=}}}0$ leads to the
following conditions

$\displaystyle\alpha=O(\Omega^{2})\,,\quad\beta_{A}=O(\Omega^{2})\,.$ (6.27)

We therefore consider the asymptotic expansions

$\displaystyle\alpha=\Omega^{2}\alpha^{(2)}+\Omega^{3}\alpha^{(3)}+O(\Omega^{4})\,,\quad\beta_{A}=\Omega^{2}\beta^{(2)}_{A}+\Omega^{3}\beta_{A}^{(3)}+O(\Omega^{4})$
(6.28) $\displaystyle h_{AB}=s_{AB}+\Omega
C_{AB}+\Omega^{2}h^{(2)}_{AB}+O(\Omega^{3})\,.$

The condition that the expansion of $l^{a}$ vanishes on $\mathscr{I}$ gives us

$\displaystyle s^{AB}C_{AB}=0\,.$ (6.29)

We then impose the Einstein equation Eq. 2.1. At $O(\Omega^{0})$, this gives
the conditions

$\displaystyle\alpha^{(2)}=\tfrac{1}{2}\,,\quad\beta^{(2)}_{A}=-\tfrac{1}{2}\mathscr{D}^{B}C_{AB}\,,\quad
s^{AB}h^{(2)}_{AB}=\tfrac{1}{4}C^{AB}C_{AB}\,,$ (6.30)

while at $O(\Omega)$, it implies

$\displaystyle\mathscr{D}^{B}h^{(2)}_{AB}=\tfrac{1}{8}\mathscr{D}_{A}(C_{BC}C^{BC})\,,\quad\partial_{u}h^{(2)}_{AB}=\tfrac{1}{8}s_{AB}\partial_{u}(C_{CD}C^{CD})\,.$
(6.31)

As in the Bondi-Sachs case, this implies that $h^{(2)}_{AB}$ is pure trace.

The metric coefficients $C_{AB}$, $\alpha^{(3)}$ and $\beta_{A}^{(3)}$ are
directly related to the shear and the Weyl tensor components as follows

$\displaystyle\sigma_{AB}=-\tfrac{1}{2}C_{AB}\,,\quad\mathcal{P}=2\alpha^{(3)}\,,\quad\mathcal{J}_{A}=\tfrac{3}{2}\beta_{A}^{(3)}\,.$
(6.32)

One could continue computing the Einstein equations to higher orders but we
will not need to do so.

To write an expression for the physical metric define
$\lambda\mathrel{\mathop{:}}=\Omega^{-1}$ so that in the coordinates
$(\lambda,u,x^{A})$ the physical metric
$\hat{g}_{ab}=\Omega^{-2}g_{ab}=\lambda^{2}g_{ab}$ has the components

$\displaystyle\hat{g}_{\lambda\lambda}$
$\displaystyle=0\,,\quad\hat{g}_{A\lambda}=0\,,\quad\hat{g}_{u\lambda}=-1\,,\quad$
(6.33) $\displaystyle\hat{g}_{uu}$
$\displaystyle=-1-\frac{1}{\lambda}\mathcal{P}+O(1/\lambda^{2})\,,\quad$
$\displaystyle\hat{g}_{uA}$
$\displaystyle=\tfrac{1}{2}\mathscr{D}^{B}C_{AB}-\tfrac{2}{3\lambda}\mathcal{J}_{A}+O(1/\lambda^{2})\,,\quad$
$\displaystyle\hat{g}_{AB}$ $\displaystyle=\lambda^{2}s_{AB}+\lambda
C_{AB}+\tfrac{1}{8}s_{AB}C_{CD}C^{CD}+O(1/\lambda)\,.$

Note that the vector field

$\displaystyle\hat{l}^{a}\equiv\frac{\partial}{\partial\lambda}=-\Omega^{2}\frac{\partial}{\partial\Omega}=\Omega^{2}l^{a}\,,$
(6.34)

generates outgoing null geodesics which are affinely parametrized with respect
to the physical metric $\hat{g}_{ab}$ with the affine parameter being
$\lambda$. Eq. 6.33 is consistent with Eqs. 5 and 58 of HIW  put together
(with the additional condition that our $C_{AB}$ is tracefree since we picked
$l^{a}$ to be expansion-free; see Remark 6.1). This also gives the physical
metric in the coordinates used in the _affine-null_ form used in an-Sachs ;
an-W , as well as the Newman-Unti coordinates NU .

We can also derive the form of the BMS vector fields by considering
infinitesimal coordinate transformations between two conformal Gaußian null
coordinate systems and demanding that the conditions on the unphysical metric
derived above be preserved. Since the computation proceeds exactly as in the
case of Bondi-Sachs coordinates detailed above, we skip the details. The end
result is that the BMS vector field in conformal Gaussian null coordinates
takes the same form as Eq. 6.19 above — the difference in the form of the BMS
vector fields written in conformal Gaußian null coordinates and conformal
Bondi-Sachs coordinates only appears at $O(\Omega^{3})$ and higher. Note that
this form is different than the one obtained by HIW  since out coordinates
differ slightly from theirs as explained in Remark 6.1 below. The WZ charge
(Eq. 5.44) can be straightforwardly written in these coordinates using Eq.
6.32.

######  Remark 6.1 (Comparison of different conformal Gaussian null
coordinates).

The conformal Gaußian null coordinates constructed in this section are closely
related to, but not the same as, the ones used in Thorne ; Hol-Th ; HIW . Note
that these references use the freedom in the conformal factor to set the
metric coefficient $\alpha^{(2)}={{}^{1}\\!\\!/\\!_{2}}$ (as in Eq. 6.30) and
then the Einstein equations imply that the expansion $\vartheta$ of the
auxiliary normal $l^{a}$ is constant along $n^{a}$ on $\mathscr{I}$, i.e.,
$\pounds_{n}\vartheta\mathrel{\mathop{\widehat{=}}}0$. In contrast, we used
the conformal freedom to set $\vartheta\mathrel{\mathop{\widehat{=}}}0$ and
then $\alpha^{(2)}={{}^{1}\\!\\!/\\!_{2}}$ follows from the Einstein equation.

##  7 Discussion

The aim of this paper was to obtain, using the Wald-Zoupas prescription,
manifestly covariant expressions for the charges and fluxes corresponding to
the Bondi-Metzner-Sachs (BMS) symmetries at null infinity in asymptotically
flat spacetimes in vacuum general relativity. While (special cases of) these
expressions have appeared in various places in the literature, they are
usually written in a way that obscures their covariant and conformally-
invariant nature, for example, by restricting to specific coordinate frames
near null infinity. The expressions we obtained are manifestly covariant,
conformally-invariant and do not rely on a preferred choice of foliation of
null infinity or on any arbitrary extension of the BMS symmetries away from
null infinity. We also recast our charge expression in two specific choices of
coordinate frames near null infinity and showed, for example, that in Bondi-
Sachs coordinates, it reduces to the expression obtained by Flanagan and
Nichols, even on cross-sections of $\mathscr{I}$ where the News tensor is non-
vanishing. In Secs. C.1, C.3 and C.2, we compare the Wald-Zoupas charge and
flux formulae to some other expressions that have appeared in the literature,
including the Ashtekar-Streubel flux formula, the Komar and linkage formulae
and Penrose’s twistor charge formula. In particular, we explicitly show that
the flux of the Wald-Zoupas charge matches the flux expression of Ashtekar and
Streubel.

While our analysis was limited to null infinity in asymptotically flat
spacetimes in vacuum general relativity, the Wald-Zoupas prescription is in
fact much more general and can be used to obtain local and covariant charges
for arbitrary diffeomorphism covariant Lagrangian theories of gravity
including gravity coupled to electromagnetism Bonga2019  and Brans-Dicke
theory Hou2020 . The Wald-Zoupas prescription has also been applied to the
context of symmetries and charges associated with finite null surfaces and
horizons CFP ; Chandrasekaran:2019ewn  and spatial infinity PS  in
asymptotically flat spacetimes in vacuum general relativity. We hope that the
explicit computations presented here will be useful for similar analyses in
other contexts.

## Acknowledgements

We thank Éanna É. Flanagan for useful discussions. A.M.G. thanks David Nichols
for helpful conversations, as well as for sharing preliminary results from
work with Arwa Elhashash. This work is supported in part by the NSF grant
PHY-1707800 to Cornell University. K.P. is supported in part by the NSF grant
PHY-1801805.

##  Appendix A General choices of conformal factor

In the body of the paper, we worked in a conformal frame where

$\displaystyle\Phi\mathrel{\mathop{:}}=\tfrac{1}{4}\nabla_{a}n^{a}\mathrel{\mathop{\widehat{=}}}0\,,$
(A.1)

which, as a result of the Einstein equation (Eq. 2.1), implies the conditions
Eqs. 2.6 and 2.7. This choice, however, was made purely for convenience and is
not essential to the results of this paper. In this appendix, we state some of
our main results in general conformal frames where $\Phi\neq 0$ and therefore
the Bondi condition does not hold. In this context, one is allowed more
general conformal transformations of the form

$\displaystyle\Omega\mapsto\omega\Omega\,,\quad\mbox{where}\quad\pounds_{n}\omega\not\mathrel{\mathop{\widehat{=}}}0\,.$
(A.2)

Using the fact that $S_{ab}$ is smooth at $\mathscr{I}$, Eq. 2.1 implies that
in general

$\displaystyle\nabla_{a}n_{b}\mathrel{\mathop{\widehat{=}}}\Phi
g_{ab}\,,\quad\lim_{\to\mathscr{I}}\Omega^{-1}n_{a}n^{a}=2\Phi\,.$ (A.3)

which generalize Eqs. 2.6 and 2.7. In these conformal frames, the pullback of
the unphysical metric to $\mathscr{I}$, $q_{ab}$, satisfies

$\displaystyle\pounds_{n}q_{ab}\mathrel{\mathop{\widehat{=}}}2\Phi q_{ab}\,.$
(A.4)

It is important to note that $\Phi$ is universal in the sense of Sec. 3, that
is, it is independent of the physical spacetime under consideration and, can,
without any loss of generality, be picked to be the same for the conformal
completion of any asymptotically-flat physical spacetime. Hence, $\delta\Phi$
=0 on phase space.

In these general conformal frames, evolution equations for components of the
Weyl tensor on $\mathscr{I}$ given in Eq. 2.25 also get generalized and may be
found in Eq. 3.3 of KP-GR-match  (see also Eq. B.13 for their expressions in
the GHP formalism). Moreover, the definition of the News tensor is generalized
to

$N_{ab}\mathrel{\mathop{:}}=2Q_{a}{}^{c}Q_{b}{}^{d}(\pounds_{n}-\Phi)\sigma_{cd}\,,$
(A.5)

(given by Eq. B.15 in the GHP formalism) where $\sigma_{ab}$ is (still)
defined by Eq. 2.16 and has conformal weight 1. In addition, Eq. 2.36 is
generalized to

$\displaystyle
N_{ab}\mathrel{\mathop{\widehat{=}}}\operatorname{STF}\left[S_{ab}-2\Phi\sigma_{ab}+2\left(\mathscr{D}_{a}+\tau_{a}\right)\tau_{b}\right]\,,$
(A.6)

(which corresponds to Eq. B.17 in the GHP formalism). Note that these
expressions for the News tensor are defined in any conformal frame and are
conformally invariant under all conformal transformations, including those of
the form Eq. A.2, whereas, for example, the expression due to Geroch,
discussed in Eq. 2.39, is only defined in frames where the Bondi condition
holds.

In general conformal frames where $\Phi\neq 0$, a BMS symmetry is given by
$\xi^{a}\mathrel{\mathop{\widehat{=}}}\beta n^{a}+X^{a}$ where
$X^{a}l_{a}\mathrel{\mathop{\widehat{=}}}0$. Here, $\beta\,,X^{a}$ satisfy the
relations (which are derived using the same method as for Eq. 4.21)

$\displaystyle(\pounds_{n}-\Phi)\beta\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}(\mathscr{D}_{a}-2\tau_{a})X^{a}\,,\quad\pounds_{n}X^{a}\mathrel{\mathop{\widehat{=}}}n^{a}X_{b}\tau^{b}\,,\quad\operatorname{STF}\mathscr{D}_{a}X_{b}\mathrel{\mathop{\widehat{=}}}0\,,\quad\alpha_{(\xi)}\mathrel{\mathop{\widehat{=}}}\Phi\beta+\tfrac{1}{2}\mathscr{D}_{a}X^{a}\,,$
(A.7)

which correspond to Eq. B.22 in the GHP formalism. It is worth emphasizing
that this is _not_ a different symmetry algebra than the one considered in the
body of the paper. The observation that, for example, in these conformal
frames a pure supertranslation, $\xi^{a}\mathrel{\mathop{\widehat{=}}}fn^{a}$,
is no longer constrained to satisfy
$\pounds_{n}f\mathrel{\mathop{\widehat{=}}}0$ and that $f$ can have a non-
trivial functional dependence on the normal direction along $\mathscr{I}$,
should not be taken to mean that symmetry algebra has become bigger. The point
is simply that supertranslations are associated with conformally-weighted
functions which have different-looking functional forms in different conformal
frames.

With these considerations all of our computations can be carried out in a
similar manner; one only has to keep track of the chosen $\Phi$ through the
calculations. For any BMS symmetry with $\beta$ and $X^{a}$ subject to Eq. A.7
and $N_{ab}$ given by Eq. A.5, the expressions for the WZ flux (Eq. 5.56) and
charge (Eq. 5.44), remain unchanged in these conformal frames and are given by

$\displaystyle\mathcal{F}[\xi;\Delta\mathscr{I}]\mathrel{\mathop{\widehat{=}}}-\frac{1}{16\pi}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}N^{ab}[\tfrac{1}{2}\beta
N_{ab}+(\mathscr{D}_{a}+\tau_{a})(\mathscr{D}_{b}-\tau_{b})\beta+\pounds_{X}\sigma_{ab}-\tfrac{1}{2}(\mathscr{D}_{c}X^{c})\sigma_{ab}]\,,$
(A.8)

and

$\displaystyle\mathcal{Q}[\xi;S]\mathrel{\mathop{\widehat{=}}}-\frac{1}{8\pi}\int\limits_{S}\boldsymbol{\varepsilon}_{2}\left[\beta(\mathcal{P}+\tfrac{1}{2}\sigma^{ab}N_{ab})+X^{a}\mathcal{J}_{a}+X^{a}\sigma_{ab}(\mathscr{D}_{c}-\tau_{c})\sigma^{bc}-\tfrac{1}{4}\sigma_{ab}\sigma^{ab}(\mathscr{D}_{c}-2\tau_{c})X^{c}\right]\,,$
(A.9)

respectively. These expressions are invariant under all conformal
transformations including those of the form Eq. A.2.

##  Appendix B The GHP formalism at $\mathscr{I}$

In this appendix, we include a brief review of the Geroch-Held-Penrose (GHP)
formalism (see NP ; GHP ; KP-GR-match  for details) applied at null infinity.
We also present some of the expressions written in the body of the paper,
including those for the WZ charge and flux, in this formalism.

In order to be consistent with our conventions for the signature and Riemann
curvature of the metric, we will use the sign conventions of SK .111111Note
however that $l^{a}$ in the notation of SK  corresponds to our $n^{a}$ while
their $k^{a}$ corresponds to our $l^{a}$. We pick a null tetrad
$(n^{a},l^{a},m^{a},\overline{m}^{a})$ which is normalised such that
$n_{a}l^{a}=-m_{a}\overline{m}^{a}=-1$, with all other inner products
vanishing. The metric then takes the form
$g_{ab}=-2n_{(a}l_{b)}+2m_{(a}\overline{m}_{b)}$. It is easy to see that the
form of the metric is invariant under the following transformations,

$\displaystyle l^{a}\mapsto\lambda\overline{\lambda}l^{a}\,,\quad
n^{a}\mapsto(\lambda\overline{\lambda})^{-1}n^{a}\,,\quad
m^{a}\mapsto\lambda(\overline{\lambda})^{-1}m^{a}\,,$ (B.1)

for any complex scalar $\lambda$. These are called _GHP transformations_. A
scalar field $\xi$, associated with a choice of null tetrad, is said to have
_GHP weight_ $(p,q)$ if, under the GHP transformations in Eq. B.1, $\xi$
transforms as

$\displaystyle\xi\mapsto\lambda^{p}\overline{\lambda}{}^{q}\xi\,.$ (B.2)

We will indicate the GHP weight of such fields by writing $\xi\circeq(p,q)$.

We now adapt the choice of the null tetrad to $\mathscr{I}$. Note that the
conformal factor $\Omega$, and therefore $\nabla^{a}\Omega$, are independent
of the choice of null tetrad. To use the GHP formalism consistently with the
transformations Eq. B.1, we therefore need to define (on $\mathscr{I}$) a
scalar field $A$ such that

$\displaystyle
An^{a}\mathrel{\mathop{\widehat{=}}}\nabla^{a}\Omega\quad\text{where
}A\circeq(1,1)\,.$ (B.3)

When converting to tensor expressions used in the body of the paper, we set
$A=1$ at the end of the calculation. As in Sec. 2.1, we choose $l^{a}$ to be
the auxiliary null normal to a foliation of $\mathscr{I}$. The tetrads
$(m^{a},\overline{m}^{a})$ are then a _complex_ orthonormal basis for the
cross-sections of $\mathscr{I}$ in the chosen foliation, and are related to
$Q_{ab}$ and $\varepsilon_{ab}$ by

$\displaystyle
Q_{ab}\mathrel{\mathop{\widehat{=}}}2m_{(a}\overline{m}_{b)}\,,\quad\varepsilon_{ab}\mathrel{\mathop{\widehat{=}}}2im_{[a}\overline{m}_{b]}\,.$
(B.4)

The definition of the GHP spin coefficients in our conventions can be found in
Eq.7.2 of SK ; see also footnote 11 above. This definition differs by a sign
from the ones used in NP ; GHP . At $\mathscr{I}$, using
$\nabla_{a}\nabla_{b}\Omega|_{\mathscr{I}}=\Phi g_{ab}$ (from Eq. A.3) and Eq.
B.3 we obtain

$\displaystyle\begin{aligned}
\kappa^{\prime}&\mathrel{\mathop{\widehat{=}}}\sigma^{\prime}\mathrel{\mathop{\widehat{=}}}\tau^{\prime}\mathrel{\mathop{\widehat{=}}}0\,,\quad&\rho^{\prime}&\mathrel{\mathop{\widehat{=}}}-A^{-1}\Phi\,,\\\\[6.45831pt]
\epsilon&\mathrel{\mathop{\widehat{=}}}-\tfrac{1}{2}(n^{a}l^{b}\nabla_{b}l_{a}-\overline{m}^{a}l^{b}\nabla_{b}m_{a})\,,\quad&\beta&\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}(m^{a}\nabla_{a}\ln
A+\overline{m}^{a}m^{b}\nabla_{b}m_{a})\,,\\\
\epsilon^{\prime}&\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}(-\pounds_{n}\ln
A+A^{-1}\Phi+m^{a}n^{b}\nabla_{b}\overline{m}_{a})\,,\quad&\beta^{\prime}&\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}(-\overline{m}^{a}\nabla_{a}\ln
A+m^{a}\overline{m}^{b}\nabla_{b}\overline{m}_{a})\,,\end{aligned}$ (B.5)

while the spin coefficients $\kappa,\rho,\sigma,\tau$ are arbitrary. Note that
when the conformal factor is chosen to satisfy the Bondi condition we have
$\Phi\mathrel{\mathop{\widehat{=}}}0$ and thus
$\rho^{\prime}\mathrel{\mathop{\widehat{=}}}0$. Also, Eq. B.5 implies that

$\displaystyle(\operatorname{\text{\rm\th}}^{\prime}+\rho^{\prime})A\mathrel{\mathop{\widehat{=}}}\operatorname{\text{\rm\dh}}A\mathrel{\mathop{\widehat{=}}}0\,,$
(B.6)

which is Eq. 9.8.26 of PR2 .

Further, since $l_{a}$ is chosen to be normal to the cross-sections of
$\mathscr{I}$, we have $\rho=\overline{\rho}$ (i.e., the twist of $l_{a}$
vanishes), and the expansion of $l_{a}$ is given by

$\displaystyle\vartheta\mathrel{\mathop{\widehat{=}}}-2{\rm
Re}\rho\mathrel{\mathop{\widehat{=}}}-2\rho\,.$ (B.7)

The spin coefficient $\kappa$ depends on the choice of extension of $l_{a}$
off of $\mathscr{I}$, while $\sigma$ and $\tau$ are related to the tensors
$\sigma_{ab}$ and $\tau_{a}$ defined in Eqs. 2.17 and 2.16 by

$\displaystyle\sigma_{ab}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}-\sigma\overline{m}_{a}\overline{m}_{b}-\overline{\sigma}m_{a}m_{b}\,,\quad$
$\displaystyle\sigma$
$\displaystyle\mathrel{\mathop{\widehat{=}}}-\sigma_{ab}m^{a}m^{b}\,,$ (B.8)
$\displaystyle\tau_{a}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}-\tau\overline{m}_{a}-\overline{\tau}m_{a}\,,\quad$
$\displaystyle\tau$
$\displaystyle\mathrel{\mathop{\widehat{=}}}-\tau_{a}m^{a}\,.$

Note that the spin coefficients
$\epsilon,\beta,\epsilon^{\prime},\beta^{\prime}$ are not GHP-weighted
quantities, and in fact their GHP transformations contain derivatives of
$\lambda$. These are used to define the _GHP derivative operators_
$(\operatorname{\text{\rm\th}},\operatorname{\text{\rm\th}}^{\prime},\operatorname{\text{\rm\dh}},\operatorname{\text{\rm\dh}}^{\prime})$
(see Eq. 7.41 of SK ) which map GHP-weighted scalars to (other) GHP-weighted
scalars.

Recall that, under a change of conformal factor $\Omega\mapsto\omega\Omega$,
the transformations of the metric and null tetrad on $\mathscr{I}$ are given
by

$\displaystyle
g_{ab}\mapsto\omega^{2}g_{ab}\,,\quad(n^{a},l^{a},m^{a},\overline{m}^{a})\mapsto\omega^{-1}(n^{a},l^{a},m^{a},\overline{m}^{a})\,,$
(B.9)

where $\omega$ has GHP-weight $(0,0)$. Under this transformation, a scalar
field $\xi$ associated with the choice of null tetrad will have an additional
conformal weight $w$ if it transforms as $\xi\mapsto\omega^{w}\xi$. We will
indicate the combined _conformal-GHP weight_ of such scalars by writing
$\xi\circeq(p,q;w)$.

Under the conformal transformation in Eq. B.9, the GHP derivatives of a
conformally-weighted GHP scalar $\xi$ will, in general, pick up derivatives of
$\omega$. However, using the fact that the spin coefficients $\rho^{\prime}$
and $\tau$ are not conformally-weighted and transform as (see also Eqs. 2.5
and 2.30)

$\displaystyle\rho^{\prime}\mapsto\omega^{-1}(\rho^{\prime}-\operatorname{\text{\rm\th}}^{\prime}\ln\omega)\,,\quad\tau\mapsto\omega^{-1}(\tau-\operatorname{\text{\rm\dh}}\ln\omega)\,,$
(B.10)

we can “correct” the conformal behaviour of the GHP derivatives by adding
suitable combinations of $\rho^{\prime}$ and $\tau$ to define the _conformal-
GHP derivatives_. For our purposes we only need the conformal-GHP derivatives
tangential to $\mathscr{I}$, which are given by (see Eq. 5.6.36 of PR1 , where
we set $w_{0}=w_{1}=-{{}^{1}\\!\\!/\\!_{2}}$ as in Eq. 5.6.26 (iii) of PR1
for compatibility with the conformal transformations of the null tetrad given
in Eq. B.9)

$\displaystyle\begin{split}\left[\operatorname{\text{\rm\th}}^{\prime}+(w+\tfrac{1}{2}(p+q))\rho^{\prime}\right]\xi&\circeq(p-1,q-1;w-1)\,,\\\
\left[\operatorname{\text{\rm\dh}}+(w-\tfrac{1}{2}(p-q))\tau\right]\xi&\circeq(p+1,q-1;w-1)\,,\\\
\left[\operatorname{\text{\rm\dh}}^{\prime}+(w+\tfrac{1}{2}(p-q))\overline{\tau}\right]\xi&\circeq(p-1,q+1;w-1)\,,\end{split}$
(B.11)

where $\xi\circeq(p,q;w)$ is any conformally-weighted GHP scalar. We note that
the spin coefficient $\tau^{\prime}$ is, in our formalism, conformally-
invariant on $\mathscr{I}$, contrary to its transformation given in Eq. 5.6.27
(iii) of PR1 . This difference arises because, under
$\Omega\mapsto\omega\Omega$, the normal $n_{a}$, in our definition, transforms
away from $\mathscr{I}$ as $n_{a}\mapsto\omega n_{a}+\Omega
A^{-1}\nabla_{a}\omega$. For this reason, we have defined the conformal-GHP
derivative corresponding to $\operatorname{\text{\rm\dh}}^{\prime}$ with
$\overline{\tau}$ instead of $\tau^{\prime}$. This definition corresponds to
the conformal operators
$\tfrac{1}{2}(\operatorname{\text{\rm\dh}}_{\mathscr{C}}+\overline{\operatorname{\text{\rm\dh}}}^{\prime}_{\mathscr{C}})$
and
$\tfrac{1}{2}(\overline{\operatorname{\text{\rm\dh}}}_{\mathscr{C}}+\operatorname{\text{\rm\dh}}^{\prime}_{\mathscr{C}})$
that were introduced in Eq. 5.6.36 of PR1 . The conformal-GHP derivatives are
useful for translating expressions from a Bondi frame to arbitrary choices of
conformal factor and foliation, as follows: in the Bondi frame, where
$\rho^{\prime}=\tau=0$, the conformal-GHP derivatives Eq. B.11 are equivalent
to the usual GHP derivatives. Thus, to generalise any expression in the Bondi
frame to arbitrary conformal frames, we replace the usual GHP derivatives in
that expression with the conformal-GHP derivatives, taking into account the
appropriate weights. We will summarize the essential relations below; see
Appendix A of KP-GR-match  for details of this procedure.

By the peeling theorem (see Theorem 9.6.41 of PR2 ) the usual GHP components
$\Psi_{i}$ of the Weyl tensor vanish at $\mathscr{I}$ and the components of
$\Omega^{-1}C_{abcd}$ are smooth at $\mathscr{I}$. We denote the components of
$\Omega^{-1}C_{abcd}$ in the chosen tetrad at $\mathscr{I}$, which we call the
(unphysical) _Weyl scalars_ , by

$\displaystyle\begin{aligned}
\psi_{4}&\mathrel{\mathop{:}}=\mathcal{R}_{ab}\overline{m}^{a}\overline{m}^{b}&&\circeq(-4,0;-3)\,,\\\
\psi_{3}&\mathrel{\mathop{:}}=\mathcal{S}_{a}\overline{m}^{a}&&\circeq(-2,0;-3)\,,\\\
\psi_{2}&\mathrel{\mathop{:}}=\tfrac{1}{2}(\mathcal{P}+i\mathcal{P}^{*})&&\circeq(0,0;-3)\,,\\\
\psi_{1}&\mathrel{\mathop{:}}=\mathcal{J}_{a}m^{a}&&\circeq(2,0;-3)\,,\\\
\psi_{0}&\mathrel{\mathop{:}}=\mathcal{I}_{ab}m^{a}m^{b}&&\circeq(4,0;-3)\,.\end{aligned}$
(B.12)

The relations between the spin coefficients, $\Psi_{i}=\Omega\psi_{i}$, and
components of the Ricci tensor in the GHP formalism can be found in GHP .

Eq. 2.3a implies that the Weyl scalars $\psi_{i}$ in the unphysical spacetime
satisfy the same Bianchi identity at $\mathscr{I}$ as the Weyl scalars
$\hat{\Psi}_{i}$ defined by the physical Weyl tensor, with all the physical
Ricci tensor components set to zero by the vacuum Einstein equations (see Eqs.
9.10.1 and 9.10.2 of PR2 ). Taking the prime transform (defined by Eq. 2.7 of
GHP ) of the Bianchi identities Eqs. 2.33–2.36 of GHP , with their $\Psi_{i}$
replaced by $\psi_{i}$ and the Ricci tensor terms set to zero, we have

$\displaystyle(\operatorname{\text{\rm\th}}^{\prime}-4\rho^{\prime})\psi_{3}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}(\operatorname{\text{\rm\dh}}-\tau)\psi_{4}\,,$
(B.13a)
$\displaystyle(\operatorname{\text{\rm\th}}^{\prime}-3\rho^{\prime})\psi_{2}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}(\operatorname{\text{\rm\dh}}-2\tau)\psi_{3}+\sigma\psi_{4}\,,$
(B.13b)
$\displaystyle(\operatorname{\text{\rm\th}}^{\prime}-2\rho^{\prime})\psi_{1}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}(\operatorname{\text{\rm\dh}}-3\tau)\psi_{2}+2\sigma\psi_{3}\,,$
(B.13c)
$\displaystyle(\operatorname{\text{\rm\th}}^{\prime}-\rho^{\prime})\psi_{0}$
$\displaystyle\mathrel{\mathop{\widehat{=}}}(\operatorname{\text{\rm\dh}}-4\tau)\psi_{1}+3\sigma\psi_{2}\,.$
(B.13d)

If the conformal factor satisfies the Bondi condition (i.e.,
$\rho^{\prime}=0$), these reduce to Eqs. 9.10.4–9.10.7 of PR2 . Using Eqs.
B.12 and B.5 and setting $A=1$, we see that Eq. B.13 is equivalent to Eq.
2.25.

The spin coefficient $\sigma$ has conformal-GHP weight $(3,-1;-1)$. The
_complex News function_ $N$ is defined by

$\displaystyle\overline{N}\mathrel{\mathop{:}}=-\operatorname{\text{\rm\th}}^{\prime}\sigma\circeq(2,-2;-2)\,.$
(B.14)

The _conformally-invariant_ News tensor $N_{ab}$ is given by

$\displaystyle
N_{ab}\mathrel{\mathop{\widehat{=}}}2Nm_{a}m_{b}+2\overline{N}\overline{m}_{a}\overline{m}_{b}\,,\quad
2\overline{N}\mathrel{\mathop{\widehat{=}}}N_{ab}m^{a}m^{b}\,.$ (B.15)

Using Eqs. B.8, B.17, B.14 and B.15, we reproduce the definition of the News
tensor in Eq. 2.32. Note that Eq. B.14 differs from the definition of the News
function in Eq. 9.8.73 of PR2  by terms involving the spin coefficient $\tau$,
which vanishes in the Bondi frame. In arbitrary conformal frames we choose Eq.
B.14 as the definition of the News function, since the corresponding News
tensor in Eq. B.15 is conformally-invariant.

The News function is related to the Weyl scalars by

$\displaystyle
A\psi_{4}\mathrel{\mathop{\widehat{=}}}(\operatorname{\text{\rm\th}}^{\prime}-2\rho^{\prime})N\,,\quad
A\psi_{3}\mathrel{\mathop{\widehat{=}}}\operatorname{\text{\rm\dh}}N\,.$
(B.16)

Rewriting the above in terms of tensors using Eqs. B.12 and B.15 and setting
$\rho^{\prime}\mathrel{\mathop{\widehat{=}}}0$ and $A=1$, we get Eq. 2.37.
Moreover, using Eq. 2.25 of GHP , along with Eqs. B.5 and B.14, we have

$\displaystyle\overline{N}\mathrel{\mathop{\widehat{=}}}\Phi_{02}-\rho^{\prime}\sigma-(\operatorname{\text{\rm\dh}}-\tau)\tau\,,$
(B.17)

where, in our conventions,
$\Phi_{02}\mathrel{\mathop{:}}=\tfrac{1}{2}S_{ab}m^{a}m^{b}$ SK ; this
corresponds to Eq. 2.36 when $\rho^{\prime}\mathrel{\mathop{\widehat{=}}}0$,
or Eq. A.6 when the Bondi condition is not imposed.

Using $\Psi_{i}=\Omega\psi_{i}$ with Eqs. B.17 and B.5 in Eq. 2.34 of GHP , we
also get

$\displaystyle A{\rm Im}~{}\psi_{2}\mathrel{\mathop{\widehat{=}}}{\rm
Im}\left[(\operatorname{\text{\rm\dh}}^{\prime}-\overline{\tau})(\operatorname{\text{\rm\dh}}^{\prime}+\overline{\tau})\sigma+\sigma
N\right]\,,$ (B.18)

which corresponds to Eq. 2.38. Similarly, from Eq. 2.21 of GHP  we have

$\displaystyle\Phi_{01}\mathrel{\mathop{\widehat{=}}}\operatorname{\text{\rm\dh}}\rho-\operatorname{\text{\rm\dh}}^{\prime}\sigma\,,$
(B.19)

where $\Phi_{01}=\tfrac{1}{2}S_{ab}l^{a}m^{b}$; using Eq. B.7, this
corresponds to Eq. C.5.

For any vector $\xi^{a}$ tangent to $\mathscr{I}$, we have the tetrad
components

$\displaystyle\xi^{a}\mathrel{\mathop{\widehat{=}}}\beta\nabla^{a}\Omega+X^{a}\mathrel{\mathop{\widehat{=}}}(A\beta)n^{a}+Xm^{a}+\overline{X}\overline{m}^{a}\,,$
(B.20)

where the conformal-GHP weights of $\beta$ and $X$ are:

$\displaystyle A\beta\circeq(1,1;1)\,,\quad X\circeq(-1,1;1)\,.$ (B.21)

The conditions for $\xi^{a}$ to be a BMS symmetry are given by

$\displaystyle(\operatorname{\text{\rm\th}}^{\prime}+2\rho^{\prime})(A\beta)\mathrel{\mathop{\widehat{=}}}{\rm
Re}[(\operatorname{\text{\rm\dh}}+2\tau)X]\,,$ (B.22)
$\displaystyle(\operatorname{\text{\rm\th}}^{\prime}+2\rho^{\prime})X\mathrel{\mathop{\widehat{=}}}0\,,\quad\operatorname{\text{\rm\dh}}^{\prime}X\mathrel{\mathop{\widehat{=}}}0\,.$

For a supertranslation $(\beta=f,X=0)$ to be a translation we have the
additional condition

$\displaystyle(\operatorname{\text{\rm\dh}}-\tau)(\operatorname{\text{\rm\dh}}+\tau)(Af)\mathrel{\mathop{\widehat{=}}}0\,.$
(B.23)

Using Eq. B.5 and setting $A=1$, we reproduce Eqs. A.7 and 4.24; when
$\rho^{\prime}=0$, the former reduces to Eq. 4.21.

Finally, the WZ charge in Eq. 5.44, in GHP notation, takes the form

$\displaystyle\mathcal{Q}[\xi;S]\mathrel{\mathop{\widehat{=}}}-\tfrac{1}{4\pi}{\rm
Re}\int\limits_{S}\boldsymbol{\varepsilon}_{2}~{}\bigg{[}\beta(\psi_{2}-A^{-1}\sigma
N)+A^{-1}X\psi_{1}+A^{-2}X\sigma(\operatorname{\text{\rm\dh}}+\tau)\overline{\sigma}-\tfrac{1}{2}A^{-2}\sigma\overline{\sigma}(\operatorname{\text{\rm\dh}}+2\tau)X\bigg{]},$
(B.24)

while the WZ flux is given by

$\begin{split}\mathcal{F}[\xi;\Delta\mathscr{I}]\mathrel{\mathop{\widehat{=}}}-\tfrac{1}{4\pi}{\rm
Re}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}~{}N\Big{[}&\tfrac{1}{2}\beta\overline{N}+(\operatorname{\text{\rm\dh}}-\tau)(\operatorname{\text{\rm\dh}}+\tau)\beta\\\
&-2A^{-1}({\rm
Re}[X\operatorname{\text{\rm\dh}}]\sigma+\sigma\operatorname{\text{\rm\dh}}X)+A^{-1}\sigma{\rm
Re}[\operatorname{\text{\rm\dh}}X]\Big{]}.\end{split}$ (B.25)

##  Appendix C Comparison to other charge formulae

In this appendix we compare the Wald-Zoupas prescription to other formulations
of the charge and fluxes associated with the BMS symmetries. In particular, we
consider the Hamiltonian formulation of Ashtekar and Streubel, the
Noether/Komar charge formula and their linkage versions and also the charges
defined by considering twistors at null infinity.

###  1 Ashtekar-Streubel flux and charge

In this appendix, we compare the expression for the WZ flux given in Eq. 5.55
to the flux given by Ashtekar and Streubel AS . Rewriting the expression for
the flux given in Eq. 4.14 of AS  in our conventions, we obtain

$\displaystyle\mathcal{F}^{\rm(AS)}[\xi;\Delta\mathscr{I}]$
$\displaystyle\mathrel{\mathop{\widehat{=}}}-\frac{1}{16\pi}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}N^{ab}\left(\pounds_{\xi}D_{a}-D_{a}\pounds_{\xi}\right)l_{b}\,,$
(C.1)

where $D_{a}$ is the derivative operator induced on $\mathscr{I}$ by the
unphysical derivative operator $\nabla_{a}$ as defined on pp. 46 of Geroch-
asymp . The derivative operator $D_{a}$ satisfies

$\displaystyle D_{a}n^{b}\mathrel{\mathop{\widehat{=}}}0\,,\quad
D_{a}q_{bc}\mathrel{\mathop{\widehat{=}}}0\,,\quad
D_{a}\underleftarrow{v_{b}}\mathrel{\mathop{\widehat{=}}}(\delta_{a}{}^{c}+n_{a}l^{c})(\delta_{b}{}^{d}+n_{d}l^{b})\nabla_{c}v_{d}\,,$
(C.2)

where $v_{a}$ is any covector field in in the unphysical spacetime and
$\underleftarrow{v_{a}}$is its pullback to $\mathscr{I}$.

To compare the flux Eq. C.1 to the WZ flux we need to compute the quantity
$\operatorname{STF}[\left(\pounds_{\xi}D_{a}-D_{a}\pounds_{\xi}\right)l_{b}]$
for a BMS symmetry $\xi^{a}$. First, note that
$\left(\pounds_{\xi}D_{a}-D_{a}\pounds_{\xi}\right)l_{b}$ can be expressed in
terms of the Riemann tensor of $D_{a}$ which can, in turn, be written in terms
of $S_{ab}$ using Eq. 3.5 of AS . This gives

$\displaystyle\left(\pounds_{\xi}D_{a}-D_{a}\pounds_{\xi}\right)l_{b}\mathrel{\mathop{\widehat{=}}}\xi^{c}\left(q_{b[c}S_{a]}{}^{d}+S_{b[c}\delta_{a]}{}^{d}\right)l_{d}-l_{c}D_{a}D_{b}\xi^{c}\,.$
(C.3)

For a BMS symmetry $\xi^{a}\mathrel{\mathop{\widehat{=}}}\beta n^{a}+X^{a}$,
we get

$\displaystyle\operatorname{STF}[\left(\pounds_{\xi}D_{a}-D_{a}\pounds_{\xi}\right)l_{b}]\mathrel{\mathop{\widehat{=}}}\operatorname{STF}\left[\tfrac{1}{2}X_{b}S_{a}{}^{c}l_{c}+\tfrac{1}{2}S_{ab}\beta+D_{a}D_{b}\beta-
l_{c}D_{a}D_{b}X^{c}\right]\,.$ (C.4)

Next, we use the following relation (which follows from Eq. B.19 in the GHP
notation)

$\displaystyle\tfrac{1}{2}Q_{b}{}^{c}S_{ac}l^{a}\mathrel{\mathop{\widehat{=}}}\mathscr{D}^{a}\sigma_{ab}-\tfrac{1}{2}\mathscr{D}_{b}\vartheta\,,$
(C.5)

where $\vartheta\mathrel{\mathop{\widehat{=}}}Q^{ab}\nabla_{a}l_{b}$ is the
expansion of $l^{a}$. Therefore, the first term in Eq. C.4 gives

$\displaystyle\operatorname{STF}[\tfrac{1}{2}X_{b}S_{a}{}^{c}l_{c}]\mathrel{\mathop{\widehat{=}}}\operatorname{STF}\left[X_{b}\mathscr{D}^{c}\sigma_{ac}-\tfrac{1}{2}X_{b}\mathscr{D}_{a}\vartheta\right]\,.$
(C.6)

Using Eq. 2.36, the second term in Eq. C.4 is

$\displaystyle\operatorname{STF}[\tfrac{1}{2}S_{ab}\beta]\mathrel{\mathop{\widehat{=}}}\beta\operatorname{STF}[\tfrac{1}{2}N_{ab}-(\mathscr{D}_{a}\tau_{b}+\tau_{a}\tau_{b})]\,.$
(C.7)

The third term in Eq. C.4 can be expressed as

$\displaystyle\operatorname{STF}D_{a}D_{b}\beta$
$\displaystyle\mathrel{\mathop{\widehat{=}}}\operatorname{STF}(\mathscr{D}_{a}\mathscr{D}_{b}\beta-\pounds_{n}\beta\nabla_{a}l_{b})=\operatorname{STF}\left[\mathscr{D}_{a}\mathscr{D}_{b}\beta-\tfrac{1}{2}(\mathscr{D}_{c}-2\tau_{c})X^{c}\sigma_{ab}\right]\,,$
(C.8)

where we have used
$\pounds_{n}\beta\mathrel{\mathop{\widehat{=}}}\frac{1}{2}(\mathscr{D}_{a}-2\tau_{a})X^{a}$
(Eq. 4.21a). For the fourth term in Eq. C.4, using
$l_{a}X^{a}\mathrel{\mathop{\widehat{=}}}0$ we have
$-l_{c}D_{a}D_{b}X^{c}=D_{a}(X^{c}D_{b}l_{c})+D_{b}X^{c}D_{a}l_{c}$. We
substitute
$D_{a}l_{b}\mathrel{\mathop{\widehat{=}}}\sigma_{ab}+\frac{1}{2}Q_{ab}\vartheta-
l_{a}\tau_{b}$ and take the $\operatorname{STF}$ of the resulting expression.
Using $\operatorname{STF}\mathscr{D}_{a}X_{b}\mathrel{\mathop{\widehat{=}}}0$
(Eq. 4.21c), we get

$\displaystyle\operatorname{STF}[-l_{c}D_{a}D_{b}X^{c}]\mathrel{\mathop{\widehat{=}}}\operatorname{STF}[\mathscr{D}_{a}(X^{c}\sigma_{bc})+\tfrac{1}{2}X_{b}\mathscr{D}_{a}\vartheta-X^{c}\tau_{c}\sigma_{ab}+\sigma_{a}{}^{c}\mathscr{D}_{b}X_{c}]\,.$
(C.9)

Putting together Eqs. C.6, C.7, C.8 and C.9 we get

$\displaystyle\operatorname{STF}[\left(\pounds_{\xi}D_{a}-D_{a}\pounds_{\xi}\right)l_{b}]$
$\displaystyle\mathrel{\mathop{\widehat{=}}}\operatorname{STF}\bigg{[}\tfrac{1}{2}\beta
N_{ab}+(\mathscr{D}_{a}+\tau_{a})(\mathscr{D}_{b}-\tau_{b})\beta$ (C.10)
$\displaystyle\qquad+X_{b}\mathscr{D}^{c}\sigma_{ac}-\tfrac{1}{2}\mathscr{D}_{c}X^{c}\sigma_{ab}+\mathscr{D}_{a}(X^{c}\sigma_{bc})+\sigma_{a}{}^{c}\mathscr{D}_{b}X_{c}\bigg{]}\,.$

Finally, we simplify the second line above using Prop. D.1. As a result, we
find

$\displaystyle\mathcal{F}^{\rm(AS)}[\xi;\Delta\mathscr{I}]\mathrel{\mathop{\widehat{=}}}-\frac{1}{16\pi}\int\limits_{\Delta\mathscr{I}}\boldsymbol{\varepsilon}_{3}~{}N^{ab}\left[\tfrac{1}{2}\beta
N_{ab}+(\mathscr{D}_{a}+\tau_{a})(\mathscr{D}_{b}-\tau_{b})\beta+\pounds_{X}\sigma_{ab}-\tfrac{1}{2}\sigma_{ab}\mathscr{D}_{c}X^{c}\right]\,.$
(C.11)

This is the same as the WZ flux formula given in Eq. 5.56. A charge expression
whose flux is given by the Ashtekar-Streubel formula, for the case where
$\tau_{a}=\vartheta=0$, was also considered in AK . Our charge expression,
when adapted to $\tau_{a}=\vartheta=0$, differs from the one in AK  which we
believe to be a result of sign errors in some of the terms given in AK .

###  2 Komar formulae and linkage charges

In this section we review the Komar formulae for conserved charges associated
with Killing vector fields and the linkage charges with generalize these to
asymptotic BMS symmetries. We show that for translations and Lorentz
symmetries these reproduce the WZ charge formula (up to proportionality
constants), however for supertranslations the linkage charge does not coincide
with the WZ charge.

For physical spacetimes with a Killing vector field a prescription for a
corresponding conserved quantity was given by Komar Komar . For stationary
spacetimes (with a timelike Killing field $\hat{t}^{a}$) and axisymmetric
spacetimes (with an axial Killing field $\hat{\phi}^{a}$) this gives a
prescription for the Komar mass $\mathcal{M}^{\rm(K)}$ and Komar angular
momentum $\mathcal{J}^{\rm(K)}$, respectively, defined as

$\displaystyle\mathcal{M}^{\rm(K)}\mathrel{\mathop{:}}=-\frac{1}{8\pi}\int\limits_{S}\hat{\varepsilon}_{abcd}\hat{\nabla}^{c}\hat{t}^{d}\,,\quad\mathcal{J}^{\rm(K)}\mathrel{\mathop{:}}=\frac{1}{16\pi}\int\limits_{S}\hat{\varepsilon}_{abcd}\hat{\nabla}^{c}\hat{\phi}^{d}\,.$
(C.12)

Using the vacuum Einstein equation and the Killing equation it can be checked
that these expressions are identically conserved and so can be evaluated on
any $2$-sphere $S$ in the spacetime. The relative difference in the sign
arises because $\hat{t}^{a}$ is timelike while $\hat{\phi}^{a}$ is spacelike
with the choice dictated by standard conventions for fields in Minkowski
spacetime. The relative “factor of $2$” between the two formulae is
essentially put in “by hand” to match the standard notions of mass and angular
momentum in the case of the Kerr spacetime.

Note that the integrands in Eq. C.12 are proportional to the Noether charge
$2$-form (Eq. 5.9) evaluated for the respective Killing fields. Thus, for
asymptotic BMS symmetries we could define the asymptotic charges using the
Noether charge to get

$\displaystyle\mathcal{Q}^{\rm(N)}[\xi;S]=-\frac{1}{16\pi}\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\hat{\varepsilon}_{abcd}\hat{\nabla}^{c}\xi^{d}=-\frac{1}{16\pi}\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\varepsilon_{abcd}\nabla^{c}(\Omega^{-2}\xi^{d})\,.$
(C.13)

It can be shown that the limit of the integrals defined above is finite and is
independent of the manner in which $S^{\prime}$ limits to $S$ (this is similar
to the proof of Lemma 5.1; see GW ). However, a major drawback of using the
Noether charge as the charge associated with a BMS symmetry is that Eq. C.13
depends on how the BMS symmetry is extended away from $\mathscr{I}$: if
${\xi^{\prime}}^{a}$ and $\xi^{a}$ are equivalent representatives of a BMS
symmetry, so that ${\xi^{\prime}}^{a}=\xi^{a}+\Omega^{2}W^{a}$ (from Prop.
4.1) then we have

$\displaystyle\mathcal{Q}^{\rm(N)}[\xi^{\prime};S]-\mathcal{Q}^{\rm(N)}[\xi;S]=-\frac{1}{16\pi}\int\limits_{S}\varepsilon_{abcd}\nabla^{c}W^{d}\,.$
(C.14)

It is clear that one can obtain _any_ value for the Noether charge on $S$
associated with the BMS symmetry simply by using different extensions of the
symmetry into the spacetime. Note, if $\xi^{a}$ is an exact Killing vector
field of the physical spacetime then $\xi^{a}$ is determined everywhere by its
value on $\mathscr{I}$ by the Killing equation in the physical spacetime, or
equivalently the conformal Killing equation in the unphysical spacetime and
this ambiguity does not arise. Thus, for a Killing vector field the Noether
charge (i.e., the Komar formula) at $S$ is unambiguously defined. But for
asymptotic BMS symmetries which need not be exact Killing fields the Noether
charge cannot be unambiguously defined.

A way of avoiding this dependence on the extension of the BMS vector fields
would be to, suitably modify the Noether charge formula (so that it reduces to
the Noether charge for a Killing field), and/or, prescribe a choice of
extension for the BMS vector field. Such a prescription is given by the
_linkage charge_ which we summarize next.

  1. (1)

One formulation of the linkage charge is given as follows TW ; Winicour :
Consider a _fixed_ cross-section $S$ of $\mathscr{I}$, and let $\mathcal{N}$
be an outgoing null hypersurface which intersects $\mathscr{I}$ in $S$. Let
$l^{a}$ be a null geodesic vector field (defined on $\mathcal{N}$) which is
tangent to $\mathcal{N}$ and coincides with the auxiliary normal at $S$. Given
a BMS vector field $\xi^{a}$ on $S$ we extend it along the null surface
$\mathcal{N}$ by the condition

$\displaystyle\left(\gamma_{ab}^{(\xi)}-\tfrac{1}{2}\gamma^{(\xi)}g_{ab}\right)l^{b}\big{|}_{\mathcal{N}}=0\,,$
(C.15)

where $\gamma_{ab}^{(\xi)}$ is the unphysical metric perturbation generated by
$\xi^{a}$ (see Eqs. 4.2 and 4.5) and $\gamma^{(\xi)}$ is its trace. Then the
linkage charge at $S$ is defined by

$\displaystyle\mathcal{Q}^{\rm(L)}[\xi;S]\mathrel{\mathop{:}}=-\frac{1}{16\pi}\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\left[\varepsilon_{abcd}\nabla^{c}(\Omega^{-2}\xi^{d})+\Omega^{-1}\gamma^{(\xi)}\varepsilon_{ab}\right]\,,$
(C.16)

where the integral is evaluated on $2$-sphere cross-sections $S^{\prime}$ of
the null surface $\mathcal{N}$, with area-element
$\varepsilon_{ab}\mathrel{\mathop{:}}=\varepsilon_{abcd}n^{c}l^{d}(l^{e}n_{e})^{-1}$,
and then the limit is taken to $S$ along $\mathcal{N}$.

  2. (2)

Another formulation of the linkage charge, which does not require fixing a
cross-section and a null surface was given by Geroch and Winicour GW . In this
formulation the BMS vector field $\xi^{a}$ on $\mathscr{I}$ is extended away
from $\mathscr{I}$ by the condition

$\displaystyle\gamma^{(\xi)}=0\,,$ (C.17)

and the linkage charge is defined by

$\displaystyle\mathcal{Q}^{\rm(L)}[\xi;S]\mathrel{\mathop{:}}=-\frac{1}{16\pi}\lim_{S^{\prime}\to
S}\int\limits_{S^{\prime}}\varepsilon_{abcd}\nabla^{c}(\Omega^{-2}\xi^{d})\,,$
(C.18)

where now the integral is evaluated on _any_ family of $2$-spheres
$S^{\prime}$ in $M$ and then the limit is taken as $S^{\prime}$ limits to $S$.

It was shown in GW  that both formulations of the linkage charge give the same
charge on _any_ choice of cross-section $S$ (when the respective conditions in
each formulation are satisfied). The advantages and disadvantages of both
formulations are also discussed in GW . An explicit expression for the linkage
charge in terms of fields on $\mathscr{I}$ was computed by Winicour Winicour .
After integration-by-parts on the cross-section $S$, the expression for the
linkage charge given in Eq. 5.6 of Winicour  (in the GHP notation) can be
written as121212Note that the function $\overline{N}$ defined in Eq. 3.30 of
Winicour  is in fact $\overline{\Phi_{02}}$ in our notation and we have used
Eq. B.17 to convert to our notation. Similarly, the quantity $P$ defined in
Eq. 3.31 of Winicour  vanishes at $\mathscr{I}$ due to the Bondi condition. We
have also restored factors of $A$ (defined in Eq. B.3) to make the expression
invariant under GHP transformations (Eq. B.1).

$\displaystyle\mathcal{Q}^{\rm(L)}[\xi;S]\mathrel{\mathop{\widehat{=}}}\tfrac{1}{4\pi}{\rm
Re}$
$\displaystyle\int\limits_{S}\boldsymbol{\varepsilon}_{2}~{}\bigg{[}\tfrac{1}{2}\beta\big{(}\psi_{2}-A^{-1}\sigma
N\big{)}-\tfrac{1}{2}\beta
A^{-1}(\operatorname{\text{\rm\dh}}-\tau)(\operatorname{\text{\rm\dh}}+\tau)\overline{\sigma}$
(C.19)
$\displaystyle\qquad\qquad+A^{-1}X\psi_{1}+A^{-2}X\sigma(\operatorname{\text{\rm\dh}}+\tau)\overline{\sigma}-\tfrac{1}{2}A^{-2}\sigma\overline{\sigma}(\operatorname{\text{\rm\dh}}+2\tau)X\bigg{]}\,.$

The linkage charge expression is conformally-invariant and thus only depends
on the physical spacetime and _not_ on the choice of the conformal-completion.
If we pick $\beta|_{S}=0$, i.e. $\xi^{a}$ is a Lorentz vector field at $S$,
the linkage charge coincides with the WZ charge Eq. B.24; this was argued by
Wald and Zoupas WZ . For a BMS translation (with $\beta=f$ satisfying Eq. B.23
and $X=0$) note that the third term in the first line of Eq. C.19 vanishes
upon integration-by-parts (also using Eq. B.6) and we get _half_ of the WZ
charge. Thus, the linkage charges also have a “factor of $2$” discrepancy
between the mass and angular momentum similar to the Komar formulae Eq. C.12.
Note that the WZ charge gives the correct relative factor between mass and
angular momentum without any need to fix these factors in an ad hoc manner.
Since exact Killing vector fields in the physical spacetime can be shown to be
part of some Poincaré algebra at $\mathscr{I}$ (see Remark 4.2), we conclude
that the WZ charge of an exact Killing vector field agrees with the Komar
formula.

However for supertranslations which are not translations the third term in Eq.
C.19 does not vanish when the shear $\sigma$ on the cross-section $S$ is non-
zero, and the linkage charge _does not_ coincide with the WZ charge or the
Geroch supermomentum Geroch-asymp . Thus, the linkage charge for
supertranslations is non-zero even in Minkowski spacetime when the cross-
section is not shear-free. This also implies that the flux of the linkage
charge between arbitrary cross-sections does not vanish in Minkowski
spacetime; though the flux does vanish when computed between two shear-free
cross-sections. A detailed comparison of the linkage flux and the Ashtekar-
Streubel flux (which equals the WZ flux, see Sec. C.1), with similar
conclusions, was given in AW-linkage .

###  3 Twistor charge

Another expression for a charge associated with the BMS symmetries was given
by Penrose Penrose-charges  motivated by twistor theory (see also DS ; Shaw ;
PR2 ). To compare to these references we will work with the GHP formalism, set
$\rho^{\prime}\mathrel{\mathop{\widehat{=}}}0$ (i.e. the Bondi condition is
satisfied) and $\tau\mathrel{\mathop{\widehat{=}}}0$ (equivalent to a choice
of foliation where $\tau_{a}\mathrel{\mathop{\widehat{=}}}0$).

With these choices being made, we choose a cross-section $S$ of $\mathscr{I}$
and define $2$-surface twistors $(\omega^{0},\omega^{1})$ on $S$
satisfying131313The pair $(\omega^{0},\omega^{1})$ are the components of a
$2$-spinor defined at $S$ in a spin frame compatible with the GHP tetrad frame
at $S$; see Shaw .

$\displaystyle\operatorname{\text{\rm\dh}}^{\prime}\omega^{0}\mathrel{\mathop{\widehat{=}}}0\,,\quad\operatorname{\text{\rm\dh}}\omega^{1}\mathrel{\mathop{\widehat{=}}}\sigma\omega^{0}\,,$
(C.20)
$\displaystyle\omega^{0}\circeq(-1,0;{{}^{1}\\!\\!/\\!_{2}})\,,\quad\omega^{1}\circeq(1,0;{{}^{1}\\!\\!/\\!_{2}})\,,$

Given two solutions $(\omega^{0},\omega^{1})$ and
$(\tilde{\omega}^{0},\tilde{\omega}^{1})$ of Eq. C.20 we can form the
(complex) vector field

$\displaystyle\xi^{a}_{\mathbb{C}}=(A\beta)n^{a}+Xm^{a}\,,\quad\beta=-i(\omega^{0}\tilde{\omega}^{1}+\omega^{1}\tilde{\omega}^{0})\,,\quad
A^{-1}X=-2i\omega^{0}\tilde{\omega}^{0}\,,$ (C.21)

where $A$ is as defined in Eq. B.3. Then Eq. C.20 implies

$\displaystyle\operatorname{\text{\rm\dh}}^{\prime}X$
$\displaystyle\mathrel{\mathop{\widehat{=}}}0\,,$ (C.22a)
$\displaystyle\operatorname{\text{\rm\dh}}^{2}(A\beta)$
$\displaystyle\mathrel{\mathop{\widehat{=}}}\tfrac{1}{2}\sigma\operatorname{\text{\rm\dh}}X+\operatorname{\text{\rm\dh}}(\sigma
X)\,.$ (C.22b)

The first equation says that $Xm^{a}$ generates a (complex) Lorentz
transformation on $S$, while the second is a restriction to a (complex)
Poincaré subalgebra at $S$. The “complexification” is necessary since Eq.
C.22b in general does not have any real solutions for $\beta$. A fixed choice
of cross-section $S$ is necessary since Eq. C.22b is _not_ preserved along the
null generators of $\mathscr{I}$ unless $\beta=0$ or $\overline{N}=0$, i.e.
the News vanishes. For this choice of subalgebra the (complex) twistor charge
at $S$ proposed by Penrose Penrose-charges  is

$\displaystyle\mathcal{Q}^{\rm(T)}[\xi_{\mathbb{C}};S]\mathrel{\mathop{\widehat{=}}}-\tfrac{i}{4\pi}\int\limits_{S}\boldsymbol{\varepsilon}_{2}\left[(\omega^{0}\tilde{\omega}^{1}+\omega^{1}\tilde{\omega}^{0})\psi_{2}+\omega^{0}\tilde{\omega}^{0}\psi_{1}+\omega^{1}\tilde{\omega}^{1}\psi_{3}\right]\,.$
(C.23)

Using $A\psi_{3}=\operatorname{\text{\rm\dh}}N$ (Eq. B.16), and integrating by
parts on $S$ we can write this charge in terms of the vector field
$\xi^{a}_{\mathbb{C}}$, defined in Eq. C.21, as

$\displaystyle\mathcal{Q}^{\rm(T)}[\xi_{\mathbb{C}};S]\mathrel{\mathop{\widehat{=}}}\tfrac{1}{8\pi}\int\limits_{S}\boldsymbol{\varepsilon}_{2}\left[2\beta(\psi_{2}-A^{-1}\sigma
N)+A^{-1}X\psi_{1}\right]\,.$ (C.24)

Using Eq. C.22 and integrating-by-parts on $S$, Eq. C.24 can be written as
(see also Eq. 11 of DS )

$\displaystyle\mathcal{Q}^{\rm(T)}[\xi_{\mathbb{C}};S]$
$\displaystyle\mathrel{\mathop{\widehat{=}}}\tfrac{1}{8\pi}\int\limits_{S}\boldsymbol{\varepsilon}_{2}~{}\big{[}2\beta(\psi_{2}-A^{-1}\sigma
N-iA^{-1}{\rm
Im}\operatorname{\text{\rm\dh}}^{\prime}\operatorname{\text{\rm\dh}}^{\prime}\sigma)$
(C.25)
$\displaystyle\qquad\qquad+A^{-1}X\psi_{1}+A^{-2}X\sigma\operatorname{\text{\rm\dh}}\overline{\sigma}-\tfrac{1}{2}A^{-2}\sigma\overline{\sigma}\operatorname{\text{\rm\dh}}X\big{]}\,.$

To compare this expression to the WZ charge we can use Eq. B.18 (with
$\tau\mathrel{\mathop{\widehat{=}}}0$) to get

$\displaystyle\mathcal{Q}^{\rm(T)}[\xi_{\mathbb{C}};S]\mathrel{\mathop{\widehat{=}}}\tfrac{1}{8\pi}\int\limits_{S}\boldsymbol{\varepsilon}_{2}\left[2\beta{\rm
Re}(\psi_{2}-A^{-1}\sigma
N)++A^{-1}X\psi_{1}+A^{-2}X\sigma\operatorname{\text{\rm\dh}}\overline{\sigma}-\tfrac{1}{2}A^{-2}\sigma\overline{\sigma}\operatorname{\text{\rm\dh}}X\right]\,.$
(C.26)

Comparing to Eq. B.24 we see that the twistor charge is the same as the WZ
charge for the complex vector field $\xi^{a}_{\mathbb{C}}=\beta n^{a}+Xm^{a}$
in the Poincaré subalgebra at $S$ chosen by Eq. C.22. The flux of the twistor
charge was shown to be equivalent to the (complex) Ashtekar-Streubel flux in
Shaw .

##  Appendix D Symmetric and tracefree tensors in two dimensions

This appendix collects some useful identities for symmetric, tracefree tensors
in two dimensions. In the main body of the paper these are applied to tensors
on a cross-section $S$ of $\mathscr{I}$, but they hold for any $2$-dimensional
orientable manifold $S$ with a Riemannian metric $Q_{ab}$; as such, for
simplicity, we drop the hats on all equalities.

The results of this appendix primarily rely on the following relation between
the $2$-dimensional Riemannian metric $Q_{ab}$ and the corresponding area-
element $\varepsilon_{ab}$, (see Eq. B.2.12 of Wald Wald-book ):

$\varepsilon^{ab}\varepsilon_{cd}=2Q^{[a}{}_{c}Q^{b]}{}_{d}.$ (D.1)

For a tensor field $A_{ab}$, let $\operatorname{STF}A_{ab}$ be the symmetric
tracefree part as defined in Eq. 2.13. Then, we can decompose $A_{ab}$ into an
antisymmetric part, a pure trace part, and a symmetric, tracefree part:

$A_{ab}=A_{[ab]}+\tfrac{1}{2}A^{c}{}_{c}Q_{ab}+\operatorname{STF}A_{ab}.$
(D.2)

Most of the tensors on $S$ which arise in our main analysis are symmetric and
tracefree, so we collect the identities satisfied by such tensors in the
following proposition:

######  Proposition D.1.

Let $A_{ab}$ and $B_{ab}$ be symmetric and tracefree tensors on $S$, then the
following identities hold:

$\displaystyle\operatorname{STF}\left(A_{ac}B^{c}{}_{b}\right)=0\,.$ (D.3)
$\displaystyle\mathscr{D}_{[a}A_{b]c}=Q_{c[a}\mathscr{D}^{d}A_{b]d}\,.$ (D.4)

where $\mathscr{D}$ is the covariant derivative compatible with $Q_{ab}$.

$\displaystyle\operatorname{STF}\left(\pounds_{X}A_{ab}-\tfrac{1}{2}A_{ab}\mathscr{D}_{c}X^{c}\right)=\operatorname{STF}\left[X_{a}\mathscr{D}^{c}A_{bc}+\mathscr{D}_{a}(X^{c}A_{bc})+A_{a}{}^{c}\mathscr{D}_{[b}X_{c]}\right]\,.$
(D.5)

for any vector field $X^{a}$ on $S$.

###### Proof.

For any tensor $A_{ab}$ on $S$, define a notion of left and right dual,
respectively, by

$(^{*}A)_{ab}\mathrel{\mathop{:}}=\varepsilon_{ca}A^{c}{}_{b}\,,\quad(A^{*})_{ab}\mathrel{\mathop{:}}=A_{a}{}^{c}\varepsilon_{cb}.$
(D.6)

It follows from Eq. D.1 that

$(^{**}A)_{ab}=-A_{ab}\,,\quad(A^{**})_{ab}=-A_{ab}\,,\quad(^{*}A^{*})_{ab}=A_{c}{}^{c}Q_{ab}-A_{ba}.$
(D.7)

Using Eqs. D.7 and D.7, we therefore can characterize $A_{ab}$ by how its left
and right duals are related:

$(A^{*})_{ab}=\begin{cases}({}^{*}A)_{ab}&\iff
A_{ab}=\operatorname{STF}A_{ab}\\\
-({}^{*}A)_{ab}&\iff\operatorname{STF}A_{ab}=0\end{cases}$ (D.8)

Now, let $A_{ab}$ and $B_{ab}$ be two symmetric, tracefree tensors, and
consider the left and right duals of $U_{ab}=A_{ac}B^{c}{}_{b}$. Using Eqs.
D.6 and D.8 we obtain

$(U^{*})_{ab}=A_{ac}(B^{*})^{c}{}_{b}=A_{ac}(^{*}B)^{c}{}_{b}=A_{ac}\varepsilon^{dc}B_{db}=-(A^{*})_{ac}B^{c}{}_{b}=-(^{*}A)_{ac}B^{c}{}_{b}=-(^{*}U)_{ab}\,,$
(D.9)

Then, using Eq. D.8 we get that $\operatorname{STF}(A_{ac}B^{c}{}_{b})=0$.

To show Eq. D.4 we start with the fact that

$\varepsilon^{cd}\varepsilon^{ab}\mathscr{D}_{a}A_{bc}=2\mathscr{D}_{a}(Q^{a[c}Q^{d]b}A_{bc})=\mathscr{D}_{c}A^{dc},$
(D.10)

where the first equality uses Eq. D.1 and the compatibility of
$\mathscr{D}_{a}$ and $\varepsilon^{ab}$, and the second follows from the fact
that $Q^{ab}A_{ab}=0$. As such, the trace of Eq. D.1 implies that

$\varepsilon^{ab}\mathscr{D}_{a}A_{bc}=\varepsilon_{cd}\mathscr{D}_{e}A^{de}.$
(D.11)

The result Eq. D.4 then follows by Eq. D.1.

Finally, consider the Lie derivative of $A_{ab}$ with respect to some vector
field $X^{a}$ on $S$. We have that

$\displaystyle\operatorname{STF}\left(\pounds_{X}A_{ab}-\tfrac{1}{2}A_{ab}\mathscr{D}_{c}X^{c}\right)$
$\displaystyle=\operatorname{STF}\left(X^{c}\mathscr{D}_{c}A_{ab}+2A_{a}{}^{c}\mathscr{D}_{b}X_{c}-\tfrac{1}{2}A_{ab}\mathscr{D}_{c}X^{c}\right)$
(D.12)
$\displaystyle=\operatorname{STF}\big{[}2X^{c}\mathscr{D}_{[c}A_{a]b}+\mathscr{D}_{a}(X^{c}A_{bc})+A_{a}{}^{c}\mathscr{D}_{b}X_{c}-\tfrac{1}{2}A_{ab}\mathscr{D}_{c}X^{c}\big{]},$

where the first equation follows by the definition of the Lie derivative and
the second from the Leibniz rule. Using Eq. D.4 for the first term in this
expression and rewriting the last two terms as

$\displaystyle\operatorname{STF}\left[A_{a}{}^{c}\mathscr{D}_{[b}X_{c]}+A_{a}{}^{c}\operatorname{STF}\mathscr{D}_{b}X_{c}\right]=\operatorname{STF}\left[A_{a}{}^{c}\mathscr{D}_{[b}X_{c]}\right]\,,$
(D.13)

using Eq. D.3, we find Eq. D.5. ∎

Finally, we show that divergence-free, symmetric, and trace-free tensors
$A_{ab}$ on $S$ must vanish. Unlike the above proposition, which holds for any
two-dimensional manifold $S$, this is a consequence of the fact that any
cross-section $S$ of $\mathscr{I}$ has the topology of $\mathbb{S}^{2}$. Our
proof is based on one given in Lemma 5 of Geroch-asymp  (for similar
discussion, see Sec. A.4 of AK  or Appendix C of Ashtekar-Magnon1984 ):

######  Proposition D.2.

Let $A_{ab}$ be a symmetric, trace-free tensor on $S$ that satisfies
$\mathscr{D}_{a}A^{ab}=0$. Then $A_{ab}$ vanishes.

###### Proof.

Consider any conformal Killing vector $\xi^{a}$ on $S$, and define

$v_{a}\mathrel{\mathop{:}}=A_{ab}\xi^{b}.$ (D.14)

First note that, since $A_{ab}$ is symmetric and trace-free and $\xi^{a}$ is a
conformal Killing field on $S$, $v_{a}$ is divergence-free:

$\mathscr{D}_{a}v^{a}=A^{ab}\mathscr{D}_{a}\xi_{b}=0.$ (D.15)

Next, we show that $v_{a}$ is curl-free. To do so, we use Eq. D.1 to show that

$\varepsilon_{ab}\varepsilon^{ac}A_{c}{}^{b}=Q_{ab}A^{ab}=0,\quad
Q_{ab}\varepsilon^{ac}A_{c}{}^{b}=\varepsilon_{ab}A^{ab}=0,$ (D.16)

so that $\varepsilon^{ac}A_{c}{}^{b}$ is also a symmetric, trace-free tensor.
Then, Prop. D.1 implies that

$\mathscr{D}_{a}(\varepsilon^{ac}A_{c}{}^{b})=\varepsilon^{ac}Q^{b}{}_{a}\mathscr{D}^{d}A_{cd}=0,$
(D.17)

and so

$\varepsilon^{ab}\mathscr{D}_{a}v_{b}=\mathscr{D}_{a}(\varepsilon^{ac}A_{c}{}^{b}\xi_{b})=\varepsilon^{ac}A_{c}{}^{b}\mathscr{D}_{a}\xi_{b}=0.$
(D.18)

As such, $v_{a}$ is both divergence- and curl-free, and so it must vanish (as
$S$ has the topology of $\mathbb{S}^{2}$): that is,

$A_{ab}\xi^{b}=0,$ (D.19)

for _any_ conformal Killing vector $\xi^{a}$ on $S$. However, at each point,
the set of conformal Killing vectors on $S$ spans the tangent space, and so we
find that $A_{ab}=0$. ∎

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