# Asymptotic symmetry algebra of Einstein gravity and Lorentz generators

Oscar Fuentealba1<EMAIL_ADDRESS>Marc Henneaux1,2
<EMAIL_ADDRESS>Cédric Troessaert1,3<EMAIL_ADDRESS>1Université Libre de Bruxelles and International Solvay Institutes, ULB-Campus
Plaine CP231, B-1050 Brussels, Belgium  2Collège de France, 11 place Marcelin
Berthelot, 75005 Paris, France  3Haute-Ecole Robert Schuman, Rue Fontaine aux
Mûres, 13b, B-6800, Belgium

###### Abstract

The asymptotic symmetry algebra of four-dimensional Einstein gravity in the
asymptotically flat context has been shown recently to be the direct sum of
the Poincaré algebra and of an infinite-dimensional abelian algebra (with
central charge) that includes the Bondi-Metzner-Sachs supertranslations. This
result, obtained within the Hamiltonian formalism, yields a supertranslation
invariant definition of the Lorentz generators (angular momentum and boosts).
Definitions of Lorentz generators free from the “supertranslation ambiguities”
have also been proposed recently at null infinity. We prove the equivalence of
the two approaches for redefining the charges.

In a recent paper Fuentealba:2022xsz , we showed that the Bondi-Metzner-Sachs
symmetry Bondi:1962px ; Sachs:1962wk ; Sachs:1962zza  of asymptotically flat
spacetimes in four spacetime dimensions could be extended by including a
certain class of logarithmic supertranslations. This was achieved by
considering more general asymptotic conditions for the metric
$g_{ij}(\mathbf{x})$ and its conjugate momentum $\pi^{ij}(\mathbf{x})$ than
those of Henneaux:2018hdj , which were shown to provide a non trivial and
faithful Hamiltonian realization of the BMS symmetry. These more general
boundary conditions schematically take the form

$g_{ij}(\mathbf{x})\longrightarrow_{r\rightarrow\infty}\Delta_{ij}^{\textrm{log}}+\Delta_{ij}^{\textrm{Diff}}+g_{ij}^{\textrm{RT}}$
(1)
$\pi^{ij}(\mathbf{x})\longrightarrow_{r\rightarrow\infty}\Gamma^{ij}_{\textrm{log}}+\Gamma^{ij}_{\textrm{Diff}}+\pi^{ij}_{\textrm{RT}}$
(2)

where (i) $\Delta_{ij}^{\textrm{log}}$ and $\Gamma^{ij}_{\textrm{log}}$ are
the changes in the metric and its conjugate momentum due to finite
diffeomorphisms of order $\mathcal{O}(\ln r)$ of the form considered in
Fuentealba:2022xsz , (ii) $\Delta_{ij}^{\textrm{Diff}}$ and
$\Gamma^{ij}_{\textrm{Diff}}$ are the changes in the metric and its conjugate
momentum due to finite diffeomorphisms of order $\mathcal{O}(1)$ of the form
considered in Henneaux:2018hdj , and (iii) $g_{ij}^{\textrm{RT}}$ and
$\pi^{ij}_{\textrm{RT}}$ obey the boundary conditions of Regge:1974zd ,
namely,

$g_{ij}^{\textrm{RT}}=\delta_{ij}+\frac{h_{ij}(\mathbf{n})}{r}+o(r^{-1}),\quad\pi^{ij}_{\textrm{RT}}=\frac{p^{ij}(\mathbf{n})}{r^{2}}+o(r^{-2}),$
(3)

with $h_{ij}(\mathbf{n})$ even and $p^{ij}(\mathbf{n})$ odd under the
antipodal map of the sphere, $h_{ij}(\mathbf{-n})=h_{ij}(\mathbf{n})$,
$p^{ij}(\mathbf{-n})=-p^{ij}(\mathbf{n})$. Here, $\mathbf{n}$ is the unit
normal to the sphere at infinity,
$(n^{i})=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$. The
boundary conditions of Henneaux:2018hdj  have
$\Delta_{ij}^{\textrm{log}}=\Gamma^{ij}_{\textrm{log}}=0$ but
$\Delta_{ij}^{\textrm{Diff}}\not=0$, $\Gamma^{ij}_{\textrm{Diff}}\not=0$.
Those of Regge:1974zd  have
$\Delta_{ij}^{\textrm{log}}=\Gamma^{ij}_{\textrm{log}}=\Delta_{ij}^{\textrm{Diff}}=\Gamma^{ij}_{\textrm{Diff}}=0$.
As it is customary, $\mathcal{O}(r^{-n})$ stands for terms with a decay rate
equal or faster than $r^{-n}$ in the limit $r\rightarrow\infty$, while by
$o(r^{-n})$ we refer to a fall-off strictly faster than $r^{-n}$.

The terms $\Delta_{ij}^{\textrm{log}}$, $\Delta_{ij}^{\textrm{Diff}}$,
$\Gamma^{ij}_{\textrm{log}}$ and $\Gamma^{ij}_{\textrm{Diff}}$ are
respectively of orders $\mathcal{O}(\frac{\ln r}{r})$,
$\mathcal{O}(\frac{1}{r})$, $\mathcal{O}(\frac{\ln r}{r^{2}})$ and
$\mathcal{O}(\frac{1}{r^{2}})$. While having the same orders as
$g_{ij}^{\textrm{RT}}-\delta_{ij}$ and $\pi^{ij}_{\textrm{RT}}$, the
diffeomorphism terms $\Delta_{ij}^{\textrm{Diff}}$ and
$\Gamma^{ij}_{\textrm{Diff}}$ have opposite parities, which permits to
distinguish them. The explicit expressions for $\Delta_{ij}^{\textrm{log}}$,
$\Delta_{ij}^{\textrm{Diff}}$, $\Gamma^{ij}_{\textrm{log}}$ and
$\Gamma^{ij}_{\textrm{Diff}}$, which can be found in Fuentealba:2022xsz , will
not be needed here Footnote0 .

With the enlarged boundary conditions (1)-(2), the asymptotic symmetry is
enhanced and contains, besides supertranslations, “logarithmic
supertranslations” parametrized by one function on the sphere (with no
$\ell=0$ or $\ell=1$ modes, which act trivially and can be assumed to be
absent). Furthermore, the charges $L^{\alpha}$ generating the logarithmic
supertranslations are canonically conjugate to those of the pure
supertranslations $S_{\beta}$, i.e.,
$\\{L^{\alpha},S_{\beta}\\}=\delta^{\alpha}_{\beta}$, with $\alpha$, $\beta$
continuous variables parametrizing the sphere, $(\alpha)\equiv(x^{A})$,
$L^{\alpha}\equiv L(x^{A})$ where $x^{A}$ are the angles Fuentealba:2022xsz .
This key feature enables one to disentangle the logarithmic and pure
supertranslations from the Poincaré algebra by a redefinition “à la Darboux”
of the Lorentz generators, in such a way that the redefined Lorentz generators
have vanishing brackets with both the logarithmic and pure supertranslations.
These new generators involve quadratic corrections bilinear in the logarithmic
and supertranslations charges and are thus not linear combinations of the
original generators, evading Lie algebra obstructions pointed out in
Sachs:1962zza . That the proper framework for discussing charges and symmetry
generators is that of (nonlinear) Poisson manifolds is of course a well-
studied chapter of Hamiltonian dynamics. The charges themselves might in fact
provide sometimes “nonlinear algebras”. Recent nonlinear examples in the BMS
context have been given in Fuentealba:2021xhn ; Fuentealba:2021yvo ;
Fuentealba:2022yqt , but there exist many earlier ones, see e.g.
Henneaux:1999ib  where a nonlinear redefinition of the charges was also found
to be useful.

Now, similar redefinitions of the Lorentz generators were performed recently
at null infinity with the same purpose of eliminating the “supertranslation
ambiguity” Mirbabayi:2016axw ; Bousso:2017dny ; Javadinezhad:2018urv ;
Javadinezhad:2022hhl ; Compere:2019gft ; Compere:2021inq ; Compere:2023qoa ;
Chen:2021szm ; Chen:2021kug ; Chen:2021zmu ; Chen:2022fbu ; Footnote1 . We
prove in this letter that our charges coincide with those given in these works
in the (non-radiative) past limit of future null infinity
(“$u\rightarrow-\infty$”), where the comparison can be made. Since the
equivalence of the various definitions of the null-infinity supertranslation-
invariant Lorentz charges at the past boundary of future null infinity has
been established in the corresponding literature, it is sufficient to compare
our approach with any one of them. For definiteness, we will compare here our
approach with the definition given in the appendix E of Compere:2023qoa .

The Hamiltonian charges of the BMS algebra and of the logarithmic
supertranslations involve explicitly the asymptotic fields switched on by
performing a logarithmic diffeomorphism, which appear in
$\Delta_{ij}^{\textrm{log}}$ and $\Gamma^{ij}_{\textrm{log}}$
Fuentealba:2022xsz . Such logarithmic diffeomorphisms are not considered in
the null infinity analysis, however. For that reason, we shall compare the
redefinitions of the charges when
$\Delta_{ij}^{\textrm{log}}=\Gamma^{ij}_{\textrm{log}}=0$ (which does not
imply that the logarithmic charges themselves vanish, see below). This turns
out to be rather immediate, because the asymptotic form of the metric and its
conjugate momentum reduce then exactly to those studied in Henneaux:2018hdj
(see also Henneaux:2019yax ).

With $\Delta_{ij}^{\textrm{log}}=\Gamma^{ij}_{\textrm{log}}=0$, the asymptotic
form of the metric and its conjugate momentum reads explicitly, in polar
coordinates ($r,x^{A}$),

$\displaystyle g_{rr}=$ $\displaystyle 1+\frac{1}{r}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{rr}+\mathcal{O}(r^{-2})\,,\quad g_{rA}=\mathcal{O}(r^{-1})\,,$ (4)
$\displaystyle g_{AB}=r^{2}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}_{AB}+r\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{AB}+\mathcal{O}(1)\,,$ (5)

and

$\displaystyle\pi^{rr}$ $\displaystyle=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rr}+\mathcal{O}(r^{-1})\,,\quad\pi^{rA}=\frac{1}{r}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rA}+\mathcal{O}(r^{-2})\,,$ (6)
$\displaystyle\pi^{AB}=\frac{1}{r^{2}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{AB}+\mathcal{O}(r^{-3})\,,$ (7)

where the leading orders of the metric and the momenta are subject to the
conditions:

$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$h$\kern 0.0pt}}}_{rr}$ $\displaystyle=(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{rr})^{\text{even}},$ (8) $\displaystyle\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern 0.0pt}}}_{AB}$
$\displaystyle=(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$h$\kern 0.0pt}}}_{AB})^{\text{even}}+2(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}U+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}_{AB}U)\,,$ (9)

and

$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rr}$ $\displaystyle=(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rr})^{\text{odd}}-\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern 0.0pt}}}V\,,$
(10) $\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rA}$ $\displaystyle=(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{rA})^{\text{even}}-\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}V\,,$ (11)
$\displaystyle\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{AB}$ $\displaystyle=(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}^{AB})^{\text{odd}}+\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}^{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}^{B}V-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}^{AB}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern 0.0pt}}}V)\,.$
(12)

Here, $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}$ is the covariant derivative on the unit sphere with round metric
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}_{AB}$ and  $\triangle$ its Laplacian. All coefficients in the $1/r$
expansions are functions on the sphere (i.e., functions of the angles $x^{A}$
only). They are arbitrary except for the parity assignments on
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{ij}$ and $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{ij}$ just written, completed by the conditions that
the function $U$ is odd while the function $V$ is even Henneaux:2018hdj ;
Henneaux:2019yax . Without loss of generality, we can assume that $U$ contains
no $\ell=1$ mode and $V$ contains no $\ell=0$ mode, since these drop from the
above expressions. The functions $U$ and $V$ parametrize the terms
$\Delta_{ij}^{\textrm{Diff}}$ and $\Gamma^{ij}_{\textrm{Diff}}$.

It was proved in Henneaux:2018hdj  that shifts of $U$ by an odd function
$W^{\text{odd}}$ and of $V$ by an even function $T^{\text{even}}$, which leave
these boundary conditions invariant by construction, are precisely the pure
BMS supertranslations. Under the conditons
$\Delta_{ij}^{\textrm{log}}=\Gamma^{ij}_{\textrm{log}}=0$, the corresponding
charges reduce to (modulo weakly vanishing bulk terms that we do not write
explicitly)

$Q_{T,W}=\oint d^{2}x\Big{[}2\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,T^{\text{even}}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{rr}+2W^{\text{odd}}\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rr}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}\Big{)}\Big{]}\,,$ (13)

where we have also included the energy (zero mode $\ell=0$ of
$T^{\text{even}}$) and the linear momentum ($\ell=1$ mode of
$W^{\text{odd}}$), which, even though not acting effectively on $U$ and $V$,
have a non trivial action on the other canonical fields. We use units such
that $16\pi G=1$.

Based on the work of Troessaert:2017jcm , the Hamiltonian description of the
supertranslations in terms of even and odd functions on the sphere has been
shown in Henneaux:2018hdj  to be equivalent to the familiar description at
null infinity. Not only do the supertranslation vector fields match, but also
their corresponding charges.

It is easy to see that the vector fields of the homogeneous Lorentz generators
are identical. Since the generators of (true) symmetries are determined in the
canonical formalism by the corresponding transformations up to a constant -
easily adjusted to zero in our case - , the matching of the Lorentz
transformations implies the matching of the Lorentz charges in the Hamiltonian
description at spatial infinity with those of the null infinity description
provided these are constructed following moment map methods (in the non
radiative past limit $u\rightarrow-\infty$ where the charges are integrable
and unambiguously defined).

We will explicitly use the matching procedure that identifies the
supertranslations vector fields of the Hamiltonian description on Cauchy
hypersurfaces with the supertranslation vector fields at null infinity. For
that reason, we briefly recall it here. The idea is to go to hyperbolic
coordinates following Ashtekar:1978zz ; BeigSchmidt ; Compere:2011ve  and to
integrate the asymptotic equations of motion with Cauchy data given on the
slice “hyperbolic time $s=0$” Troessaert:2017jcm ,Henneaux:2018hdj . In
hyperbolic coordinates,

$\eta=\sqrt{-t^{2}+r^{2}},\qquad s=\frac{t}{r}.$ (14)

the metric asymptotically goes to

$ds^{2}\rightarrow
d\eta^{2}+\eta^{2}h^{0}_{ab}dx^{a}dx^{b}\,,\qquad(x^{a})\equiv(s,x^{A})$ (15)

where the metric on the unit hyperboloid is

$h^{0}_{ab}dx^{a}dx^{b}=-\frac{1}{(1-s^{2})^{2}}ds^{2}+\frac{1}{(1-s^{2})}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}_{AB}dx^{A}dx^{B}\,.$ (16)

The slice $s=0$ is indeed a Cauchy surface on which the above asymptotic
conditions therefore hold. To leading order, the relevant asymptotic analysis
becomes linear and the linearized equations are sufficient for our purposes.
The matching with null infinity of the solutions in hyperbolic coordinates is
then performed using the method of Fried1 ; Friedrich:1999wk ;
Friedrich:1999ax . The Hamiltonian parity conditions on $s=0$ are equivalent
to parity conditions relating the values of the fields at the point
$(s,x^{A})$ and at its hyperboloid antipodal image $(-s,-x^{A})$
Compere:2011ve , in particular, the past of future null infinity ($s=1$) is
related with the future of past null infinity ($s=-1$).

The functions $W^{\text{odd}}$ and $T^{\text{even}}$ that parametrize the
supertranslations in the Hamiltonian formalism combine to form an odd function
$\omega(s,x^{A})=-\omega(-s,-x^{A})$ which is a solution of the equation
$(\mathcal{D}^{2}+3)\omega=0$ on the hyperboloid and matches the null infinity
supertranslation parameter $\alpha(x^{A})=\lim_{s\rightarrow
1}\sqrt{1-s^{2}}\omega$ Troessaert:2017jcm . Here, $\mathcal{D}^{2}$ is the
d’Alembertian on the hyperboloid. One has $W^{\text{odd}}=\omega(0,x^{A})$ and
$T^{\text{even}}=\partial_{s}\omega(0,x^{A})$. The detailed transformation
between $\alpha$ and $W^{\text{odd}}$ and $T^{\text{even}}$ is given in
Henneaux:2018cst .

For the matching of the supertranslation charges, one notes that the conjugate
momenta $\pi^{ij}$ are related to the time derivatives $\partial_{s}g_{ij}$ in
the usual way. One then finds that the Hamiltonian variables
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{rr}$ and $\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\pi$\kern 0.0pt}}}^{rr}-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}$ combine to form a function
$\sigma(s,x^{A})=\sigma(-s,-x^{A})$ which obeys the same equation as $\omega$
on the hyperboloid, namely, $(\mathcal{D}^{2}+3)\sigma=0$, but with the
opposite parity. The function $\sigma$ matches the Bondi mass aspect
$m(x^{A})$ at null infinity Troessaert:2017jcm . The Hamiltonian
supertranslation generator (13) is thus equal to

$Q_{\alpha(T,W)}=4\oint d^{2}x\,\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}}\,\alpha\,m\,,$ (17)

where $\alpha$ is the supertranslation parameter defined by $T^{\text{even}}$
and $W^{\text{odd}}$.

As we indicated, the terms $\Delta_{ij}^{\textrm{Diff}}$ and
$\Gamma^{ij}_{\textrm{Diff}}$ in (1) and (2) are the changes in the metric and
its conjugate due to finite diffeomorphisms of order $\mathcal{O}(1)$
respecting the boundary conditions, i.e., the supertranslations. The odd
function $U\equiv U^{\text{odd}}$ and the even function $V\equiv
V^{\text{even}}$ provide therefore a parametrization of the orbits of the
supertranslation subgroup through given ($g_{ij}^{\textrm{RT}}$,
$\pi^{ij}_{\textrm{RT}}$). This parametrization is abelian because the
nonlinear terms in the diffeomorphism variations of the fields are of highest
order in $1/r$.

It follows that one can repeat verbatim for $U^{\text{odd}}$ and
$V^{\text{even}}$ the above derivation of the matching of $W^{\text{odd}}$ and
$T^{\text{even}}$ with null infinity supertranslations. The functions
$U^{\text{odd}}$ and $V^{\text{even}}$ combine to form an odd function
$\psi(s,x^{A})=-\psi(-s,-x^{A})$ which is a solution of the equation
$(\mathcal{D}^{2}+3)\psi=0$ on the hyperboloid and matches the null infinity
supertranslation parameter $C(x^{A})=\lim_{s\rightarrow 1}\sqrt{1-s^{2}}\psi$
which parametrizes the supertranslation part (“electric parity part”)
$\Delta^{\textrm{Diff}}C_{AB}=(-2\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{B}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}_{AB}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\triangle$\kern 0.0pt}}})C$ of the “Bondi shear”
$C_{AB}$ at null infinity as $u\rightarrow-\infty$. We recall that the Bondi
shear is the leading perturbation term of the angular part of the metric and
is explicitly given in terms of $C(x^{A})$ by
$\lim_{u\rightarrow-\infty}C_{AB}(u,x^{C})=(-2\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern
0.0pt}}}_{A}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$D$\kern 0.0pt}}}_{B}+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern 0.0pt}}}_{AB}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern 0.0pt}}})C\,+$
“magnetic parity part”. (The magnetic part is often assumed to be zero in this
limit but this is not necessary.) One has $U^{\text{odd}}=\psi(0,x^{A})$ and
$V^{\text{even}}=\partial_{s}\psi(0,x^{A})$. The detailed transformation
between the parameters $C$ and ($U^{\text{odd}}$, $V^{\text{even}}$) of the
supertranslation part of the metric is the same as the one given in
Henneaux:2018cst  for $\alpha$, $W^{\text{odd}}$ and $T^{\text{even}}$. The
$\ell=0$ and $\ell=1$ modes of $C$ do not contribute to $C_{AB}$ and can be
assumed to be zero, in agreement with what we found for $U^{\text{odd}}$ and
$V^{\text{even}}$.

The key observation that enables one to connect the redefinitions of the
Lorentz charges at spatial infinity and at null infinity is that the functions
$U^{\text{odd}}$ and $V^{\text{even}}$ are also the logarithmic charges
Fuentealba:2022xsz . One has indeed, for the generators of logarithmic
supertranslations (again modulo weakly vanishing bulk terms that we do not
write)

$Q_{T_{\log},W_{\log}}=\oint d^{2}x\Big{(}2\sqrt{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,T_{\log}U^{\text{odd}}-2\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$g$\kern
0.0pt}}}}\,W_{\log}V^{\text{even}}\Big{)}\,,$ (18)

where $T_{\log}$ and $W_{\log}$ are the parameters of the logarithmic
supertranslations of Fuentealba:2022xsz . It is crucial to note that even
though the terms $\Delta_{ij}^{\textrm{log}}$ and $\Gamma^{ij}_{\textrm{log}}$
are set to zero in the asymptotic form of the fields, the values of the
generators of the logarithmic supertranslations do not vanish. There is no
contradiction. This is similar to the fact that the value of the generators of
shifts in $q$ in classical mechanics, namely $p$, does not vanish even if one
freezes the coordinate $q$ to $q=0$.

The above considerations enable one to match immediately the logarithmic
supertranslation charges to null infinity quantities: as we have just
explained, these charges, being identical with the supertranslation parameters
($U^{\text{odd}}$, $V^{\text{even}}$) combine through the hyperboloid function
$\psi$ to form the null infinity potential $C(x^{A})$ for the electric part of
the Bondi shear $C_{AB}$ along the above lines. This completes incidentally
the derivation, from the initial data, of the matching conditions between the
future of past null infinity and the past of future null infinity. Besides the
matching of the supertranslation parameters and charges recalled above, one
has also the correct matching of the $C$ field Strominger:2013jfa ;
Strominger:2017zoo  since $C$ coincides with the odd function
$\sqrt{1-s^{2}}\psi$ on the hyperboloid Goldstone  (for a meaningful
comparison with the literature, one must take into account the conventions for
the orientation of the null generators at past and future null infinities,
which depend on the authors).

We can then compare the redefinitions of the Lorentz generators made
respectively at null infinity and spatial infinity to achieve supertranslation
invariance. The formulas (9.18) and (9.19) of Fuentealba:2022xsz  for the new
Lorentz charges $Q_{b,Y}^{\textrm{inv}}$ can be written,

$Q_{b,Y}^{\textrm{inv}}=Q_{b,Y}^{\textrm{old}}-Q_{T_{b,Y},W_{b,Y}}^{\textrm{corr}}\,,$
(19)

where the correcting term $Q_{T_{b,Y},W_{b,Y}}^{\textrm{corr}}$ is the
generator of the following field-dependent supertranslation (pure
supertranslation and ordinary translation)

$Q_{T_{b,Y},W_{b,Y}}^{\text{corr}}=2\oint
d^{2}x\Big{[}\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern 0.0pt}}}}\,T_{b,Y}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$h$\kern
0.0pt}}}_{rr}+W_{b,Y}\Big{(}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\pi$\kern 0.0pt}}}^{rr}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\pi$\kern
0.0pt}}}\Big{)}\Big{]}\,.$ (20)

(The fields turned on by logarithmic supertranslations (appearing in
$\Delta_{ij}^{\textrm{log}}$ and $\Gamma^{ij}_{\textrm{log}}$) have been set
equal to zero in order to perform the comparison.) Here, the parameters

$(T_{b,Y},W_{b,Y})=\delta_{M}(U,V)$ (21)

are the variations of $U$ and $V$ under Lorentz transformations

$W_{b,Y}=\mathcal{L}_{Y}U-bV,\quad
T_{b,Y}=\mathcal{L}_{Y}V-(3b+\partial_{A}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}+b\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\triangle$\kern 0.0pt}}})U$ (22)

(see formulas (9.16) and (9.17) of Fuentealba:2022xsz ). Even though $V$ has
no mode $\ell<2$, $W_{b,Y}$ has a $\ell=1$ component through $bV$. This term
is necessary to remove the ordinary translations from the brackets of the
Lorentz generators with the pure supertranslations, as explained in the
general algebraic discussion in section 9.1 of Fuentealba:2022xsz .

Applying the translation rules from spatial infinity to null infinity, one
first finds that $\delta_{M}(U,V)$ becomes the Lorentz variation of the
corresponding null infinity field $C$, known to be
$\delta_{M}C=Y^{A}\partial_{A}C-\frac{1}{2}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}Y^{A}C$ where
the $Y^{A}$’s stand here for all conformal Killing vectors of the $2$-sphere
(related to spatial rotations $Y^{A}_{\text{rot}}$ and boosts $b$ through
$Y^{A}=Y^{A}_{\text{rot}}-\frac{1}{2}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}^{A}b$) and where the variation
$Y^{A}\partial_{A}C-\frac{1}{2}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}Y^{A}C$ includes the needed
$\ell=1$ term, i.e. $(T_{b,Y},W_{b,Y})\rightarrow
Y^{A}\partial_{A}C-\frac{1}{2}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$D$\kern 0.0pt}}}_{A}Y^{A}C.$ Applying now the
formula (17), one then gets,

$Q_{Y}^{\textrm{inv}}=Q_{Y}^{\textrm{old}}-4\oint
d^{2}x\,\sqrt{\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$g$\kern
0.0pt}}}}\big{(}Y^{A}\partial_{A}C-\frac{1}{2}D_{A}Y^{A}C\big{)}\,m$ (23)

which coincides with the formula (E.6) of Compere:2023qoa  (see also formula
(12) of Chen:2022fbu  for spatial rotations).

We can thus conclude that the constructions of supertranslation-invariant
Lorentz charges at spatial infinity and null infinity lead to the same
redefinitions. The interest of the Hamiltonian approach, which allows
logarithmic supertranslations by including them in the asymptotic form of the
fields through the orbit method, is that all terms appearing in the non-linear
redefinition of the Lorentz generators have a well-defined symplectic action.
The logarithmic charges $C$ are more than just definite asymptotic field
components but are true generators of symmetries, with well-defined Poisson
brackets (when supplemented by the appropriate constraint terms in the bulk).
These are the logarithmic supertranslations, i.e., the diffeomorphisms which
behave logarithmically at infinity and satisfy the prescribed parity
conditions of Fuentealba:2022xsz . Logarithmic supertranslations are on an
equal footing as the pure supertranslations and one can write equivalently the
correcting term in the redefined Lorentz generators as the generator of
logarithmic supertranslations with field-dependent coefficients given by the
Lorentz variation of the supertranslation charges (including ordinary
translations).

It is gratifying that all roads to the analysis of the asymptotic symmetries
lead to the same final redefinitions of the Lorentz generators. In that
context, the importance of including the non trivial diffeomorphism terms
$\Delta_{ij}^{\textrm{Diff}}$ and $\Gamma^{ij}_{\textrm{Diff}}$ in the
asymptotic expressions of the canonical variables in the Hamiltonian
formulation should not be underestimated. These diffeomorphism terms cannot be
set equal to zero because they are improper Benguria:1976in  and can be given
a direct physical interpretation.

Physical observers and detectors are not sitting at $r=\infty$ (where the
metric would not be flat in any case in a cosmological context) but at very
large distances such that the above developments in $1/r$ make physical sense.
Suppose that $\Delta_{ij}^{\textrm{Diff}}=\Gamma^{ij}_{\textrm{Diff}}=0$
initially for a detector that is “sufficiently far away”. At some later times,
waves hit the detector, after which the gravitational field close to the
detector returns to a situation that can again be described in asymptotically
flat terms. Not only will the ADM mass measured by the observer have changed
but also $\Delta_{ij}^{\textrm{Diff}}$ and $\Gamma^{ij}_{\textrm{Diff}}$ will
generically not return to zero, capturing the memory effect – well-known
indeed to be related to supertranslations as previous studies at null infinity
indicate (see e.g. Strominger:2017zoo ). For that reason, setting
$\Delta_{ij}^{\textrm{Diff}}=\Gamma^{ij}_{\textrm{Diff}}=0$ is not physically
acceptable if one wants to describe real observers at very large but finite
distances. An interesting question is whether one could measure a logarithmic
memory effect associated with non-vanishing $\Delta_{ij}^{\textrm{log}}$ and
$\Gamma^{ij}_{\textrm{log}}$, which define also improper gauge
transformations.

We close this letter with two comments. First, the fact that the asymptotic
symmetry has a direct sum structure “à la Coleman-Mandula” simplifies the
theory of its representations. The supertranslations simply provide additional
independent quantum numbers to the well-known representations of the Poincaré
algebra. Second, the discussion can be repeated in the case of
electromagnetism, where a similar construction of Lorentz generators free from
the angle-dependent gauge transformation ambiguities was performed in
Fuentealba:2023rvf , following Henneaux:2018gfi . What plays the role of the
memory field $(U,V)\leftrightarrow C$ is the even phase $\Phi$ that enters the
boundary conditions through the $\mathcal{O}(1)$ angular components of the
vector potential and the odd phase  $\Psi$ that enters its temporal component.
These match the pure phase part $\phi_{-}$ of $A_{A}$ at the past of future
null infinity He:2014cra  as can be seen by applying the rules for matching
the gauge transformations derived in Henneaux:2018gfi .

## Acknowledgements

We thank Geoffrey Compère and Ricardo Troncoso for important discussions. O.F.
is grateful to the Collège de France for kind hospitality while this article
was completed. This work was partially supported by a Marina Solvay Fellowship
(O.F.), by FNRS-Belgium (conventions FRFC PDRT.1025.14 and IISN 4.4503.15), as
well as by funds from the Solvay Family.

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