††institutetext: Center for Cosmology and Particle Physics, Department of
Physics, New York University, 726 Broadway, New York, NY 10003, USA

# Subleading BMS Charges and The Lorentz Group

Uri Kol<EMAIL_ADDRESS>
###### Abstract

The extended BMS group includes supertranslation, dual supertranslation and
Lorentz transformations. The generators of these symmetries can be classified
according to their parity into ”electric” and ”magnetic” types. Using a
multipole expansion of gravitational sources in the Newman-Penrose formalism,
we associate each one of these charges with a particular moment. At leading
order, the ”electric” and ”magnetic” monopole moments correspond respectively
to supertranslations and dual supertranslations. At the first subleading
order, the ”electric” and ”magnetic” dipole moments correspond respectively to
boosts and rotations, therefore comprising the entire Lorentz group. Electric-
magnetic type of duality then rotates the boost and rotation generators into
each other.

##  1 Introduction

The isometry group of asymptotically flat spacetimes, known as the Bondi-
Metzner-Sachs (BMS) group, includes Lorentz transformations and
supertranslations Bondi:1960jsa ; Bondi:1962px ; Sachs:1962wk . Recently, a
new _dual supertranslation_ symmetry of asymptotically flat spacetimes was
discovered in Kol:2019nkc . The corresponding dual supertranslation charges
Kol:2019nkc ; Godazgar:2018qpq  are akin to the large magnetic charges in QED.
The _global_ dual supertranslation charge is a topological invariant on the
space of asymptotic metrics, in analogy with the magnetic charge which
partitions the space of gauge fields into distinct topological sectors. The
new dual supertranslation symmetry does not correspond to spacetime
diffeomorphisms. Instead, evidence was presented Kol:2020ucd  suggesting that
it is a redundant gauge symmetry of the metric. In particular, all S-matrix
elements are invariant under dual supertranslations and there is no Ward
identity associated with them. As a result, dual supertranslations factorize
the Hilbert space of asymptotic states into distinct superselection sectors.
These properties distinguish the new dual supertranslation symmetry from
standard supertranslations.

The charges in the _extended BMS group_ therefore include supertranslations,
dual supertranslations and Lorentz transformations. These charges can be
classified according to their transformation law under spatial reflections
(parity) into ”electric”-types and ”magnetic”-types. The standard and dual
supertranslation charge densities are respectively given by the
”electric”-type and ”magnetic”-type Coulomb components of the gravitational
field. The Coulomb components of the gravitational field are described, in
turn, by the leading order terms in the asymptotic expansion of the complex
Weyl scalar $\psi_{2}$. These leading order monopole charges are analogous to
the large electric and magnetic charges in electrodynamics. In gravity, the
dipole moments describe subleading BMS charges and are related to the first
subleading terms in the asymptotic expansion of $\psi_{2}$. As with the
leading order charges, the subleading ones can be classified into ”electric”
and ”magnetic” types according to their parity. The ”magnetic”-type dipole
moment describes the Arnowitt, Deser, and Misner (ADM) angular momentum and it
generates rotations ADM ; Newman:1968uj ; Prior:1977 . In this paper, we will
show that the ”electric”-type gravitational dipole moment describes the Beig-Ó
Murchadha-Regge-Teitelboim (BORT) center of mass Beig:1987zz ; Regge:1974zd ;
Chen:2014uma  that generates boosts. Altogether, the gravitational monopole
and dipole moments therefore span the entire extended BMS group. We describe
this structure in table 1.

| Multipole Moment |  |  |  | ”Electric”-type |  |  | ”Magnetic”-type |
---|---|---|---|---|---|---|---|---|---
| Monopole |  |  |  |  Supertranslations (Mass) |  |  |  Dual Supertranslations (Dual Mass) |
| Dipole |  |  |  |  Boosts (Center of Mass) |  |  |  Rotations (Angular Momentum) |
Table 1: The extended BMS algebra includes supertranslations, dual
supertranslations and Lorentz transformations. This group of symmetries can be
classified using two properties: parity and multipole moments. The charge
generators of the symmetry operations are mentioned in parentheses.

The dual supertranslation charge gives rise to a duality operation that mixes
it with the standard supertranslations generator Huang:2019cja . The structure
described in table 1 therefore suggests that the same gravitational duality
operation rotates the dipole generators into each other. In other words,
angular momentum and the center of mass are mapped into each other under this
operation. These results suggest that boosts and rotations are related to each
other by an ”electric-magnetic” type of duality.

The paper is organized as follows. In section 2 we review the multipole
expansion of gravitational sources in the Newman-Penrose formalism. We show
how the supertranslation, dual supertranslation and angular momentum charges
are related to the multipole moments and discuss concrete examples. In section
3 we describe the asymptotic expansion of the metric and of the Einstein
equations. In section 4 we compute the first few leading order coefficients in
the asymptotic expansions of the Weyl scalars in terms of the asymptotic data.
Using these results, we then show in section 5 that the ”electric”-type
gravitational dipole describes the center of mass, which generates boost
transformations on the phase space. We end in section 6 with a discussion.

##  2 Multipole Expansion

To set up the stage, we start with a review of the multipole expansion in
electrodynamics using the Newman-Penrose formalism. The electromagnetic case
will serve as a familiar example that will be useful to keep in mind when we
later study the multipole expansion in gravity.

In the Newman-Penrose formalism Newman:1961qr ; Janis:1965tx ; Newman:1966ub ;
Newman:1968uj , the electromagnetic field strength $F_{\mu\nu}$ is organized
in terms of three complex scalars

$\displaystyle\phi_{0}$ $\displaystyle=F_{\mu\nu}\ell^{\mu}m^{\nu},$ (1)
$\displaystyle\phi_{1}$
$\displaystyle=\frac{1}{2}F_{\mu\nu}\left(\ell^{\mu}n^{\nu}-\bar{m}^{\mu}m^{\nu}\right),$
$\displaystyle\phi_{2}$ $\displaystyle=F_{\mu\nu}\bar{m}^{\mu}n^{\nu},$

where $e^{\mu}_{a}=\\{\ell^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu}\\}$ is a basis
of complex null tetrads in Minkowski background. The real and imaginary parts
of the three scalars encode their transformation law under spatial reflections
(parity). The real parts correspond to field components that are even under
parity and therefore of an electric type, while the imaginary parts are odd
under parity and therefore are of a magnetic type.

The asymptotic behaviors of the complex Newman-Penrose scalars at large
distances are given by

$\displaystyle\phi_{0}$
$\displaystyle=\frac{\phi_{0}^{(0)}}{r^{3}}+\frac{\phi_{0}^{(0)}}{r^{4}}+\dots,$
(2) $\displaystyle\phi_{1}$
$\displaystyle=\frac{\phi_{1}^{(0)}}{r^{2}}+\frac{\phi_{1}^{(0)}}{r^{3}}+\dots,$
$\displaystyle\phi_{2}$
$\displaystyle=\frac{\phi_{2}^{(0)}}{r}+\frac{\phi_{2}^{(0)}}{r^{2}}+\dots.$

Spacetime can then be classified into three zones - near, intermediate and far
- according to the different decay rates of the Newman-Penrose scalars.
$\phi_{0}$ is dominant in the near zone. $\phi_{1}$ describes the Coulomb
components of the field and is dominant in the intermediate zone. $\phi_{2}$
describes the radiative components of the field and is dominant in the far
zone.

The physical meaning of the coefficients $\phi_{n}^{(m)}$ in the expansion of
the Weyl scalars was studied in Janis:1965tx . The first few leading
coefficients correspond to the following solutions:

  1. 1.

Monopole

$\displaystyle\phi_{0}$ $\displaystyle=0,$ (3) $\displaystyle\phi_{1}$
$\displaystyle=\frac{a_{0}}{r^{2}},$ $\displaystyle\phi_{2}$
$\displaystyle=0.$

  2. 2.

Dipole

$\displaystyle\phi_{0}$ $\displaystyle=\frac{a_{1}(u)\sin\theta}{r^{3}},$ (4)
$\displaystyle\phi_{1}$
$\displaystyle=-\frac{\sqrt{2}\dot{a}_{1}\cos\theta}{r^{2}}-\frac{\sqrt{2}a_{1}\cos\theta}{r^{3}},$
$\displaystyle\phi_{2}$
$\displaystyle=-\frac{\ddot{a}_{1}\sin\theta}{r}-\frac{\dot{a}_{1}\sin\theta}{r^{2}}-\frac{a_{1}\sin\theta}{2r^{3}}.$

  3. 3.

Quadrupole

$\displaystyle\phi_{0}$
$\displaystyle=\frac{\dot{a}_{2}\sin\theta\cos\theta}{2r^{3}}+\frac{a_{2}(u)\sin\theta\cos\theta}{r^{4}},$
(5) $\displaystyle\phi_{1}$
$\displaystyle=-\frac{\ddot{a}_{2}(3\cos^{2}\theta-1)}{6\sqrt{2}r^{2}}-\frac{\dot{a}_{2}(3\cos^{2}\theta-1)}{2\sqrt{2}r^{3}}-\frac{a_{2}(3\cos^{2}\theta-1)}{2\sqrt{2}r^{4}},$
$\displaystyle\phi_{2}$
$\displaystyle=-\frac{a^{(3)}_{2}\sin\theta\cos\theta}{6r}-\frac{\ddot{a}_{2}\sin\theta\cos\theta}{2r^{2}}-\frac{\dot{a}_{2}\sin\theta\cos\theta}{4r^{3}}-\frac{a_{2}\sin\theta\cos\theta}{2r^{4}}.$

Here $a_{0}$ is constant while $a_{1}(u),a_{2}(u)$ are functions of the
retarded null time $u=t-r$. The dot above a letter represents a derivative
with respect to retarded null time $\dot{a}_{n}\equiv\partial_{u}a_{n}$. The
real and imaginary parts of $a_{0}$ are proportional to the electric and
magnetic charges, respectively. In a covariant form these charges are given by
the real and imaginary parts of

$q+ig=\int_{S^{2}}d\Omega\,\phi_{1}^{(0)},$ (6)

where $q$ and $g$ are the electric and magnetic charges. Large Gauge
Transformations (LGT) are then generated by

$q(f)+ig(f)=\int_{S^{2}}d\Omega\,f(\Omega)\,\phi_{1}^{(0)},$ (7)

where $f(\Omega)$ is a real function of the angles on the sphere
Strominger:2013lka ; He:2014cra ; Campiglia:2015qka ; Strominger:2017zoo .
Similarly, the real and imaginary parts of $a_{1}$ and $a_{2}$ correspond to
the electric and magnetic components of the dipole and the quadrupole moments,
respectively.

Here we would like to make a comment about the interplay between LGT and the
multipole expansion of $\phi_{1}$. The LGT transformation parameter
$f(\Omega)$ can be expanded in multipoles as well. However, it is important to
notice that these two multipole expansions are different. For example,
evaluating the charges in (7) on the dipole solution (4) we have

$q(f_{\ell,m})+ig(f_{\ell,m})\sim\dot{a}_{1}\delta_{\ell 1}\delta_{m0},$ (8)

where $f_{\ell,m}$ are the multipole moments of $f(\Omega)$. We therefore see
that by choosing $f(\Omega)$ with the spherical modes $\ell=1$ and $m=0$, the
resulting large gauge charges are proportional to $\dot{a}_{1}$, which is the
time derivative of $\phi_{1}$’s dipole moment and not the dipole moment
$a_{1}$ itself. Therefore we should distinguish the two multipole expansions,
though there is a clear interplay between the two.

After reviewing the more familiar case of electrodynamics we will now describe
the multipole expansion in gravity using the Newman-Penrose formalism. The
lessons learned from the electromagnetic case will serve as a guidance when
analogous concepts are introduced in the gravitational case. In the Newman-
Penrose formalism, the Weyl tensor $C_{\mu\nu\rho\sigma}$ is organized in
terms of five complex scalars

$\displaystyle\psi_{0}$
$\displaystyle=C_{\mu\nu\rho\sigma}\ell^{\mu}m^{\nu}\ell^{\rho}m^{\sigma},$
(9) $\displaystyle\psi_{1}$
$\displaystyle=C_{\mu\nu\rho\sigma}\ell^{\mu}n^{\nu}\ell^{\rho}m^{\sigma},$
$\displaystyle\psi_{2}$
$\displaystyle=C_{\mu\nu\rho\sigma}\bar{m}^{\mu}n^{\nu}\ell^{\rho}m^{\sigma},$
$\displaystyle\psi_{3}$
$\displaystyle=C_{\mu\nu\rho\sigma}\bar{m}^{\mu}n^{\nu}\ell^{\rho}n^{\sigma},$
$\displaystyle\psi_{4}$
$\displaystyle=C_{\mu\nu\rho\sigma}\bar{m}^{\mu}n^{\nu}\bar{m}^{\rho}n^{\sigma}.$

Here $\\{\ell^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu}\\}$ is a basis of complex
null tetrads in the curved background. As in the electromagnetic case, the
real and imaginary parts of the five Weyl scalars encode their transformation
law under spatial reflections. The real parts correspond to field components
that are even under parity and therefore of an ”electric”-type, while the
imaginary parts are odd under parity and therefore are of a ”magnetic”-type.

The five complex Weyl scalars describe the ten independent components of the
Weyl tensor. They can be expanded at large distances as

$\displaystyle\psi_{0}$
$\displaystyle=\frac{\psi_{0}^{(0)}}{r^{5}}+\frac{\psi_{0}^{(1)}}{r^{6}}+\dots,$
(10) $\displaystyle\psi_{1}$
$\displaystyle=\frac{\psi_{1}^{(0)}}{r^{4}}+\frac{\psi_{1}^{(1)}}{r^{5}}+\dots,$
$\displaystyle\psi_{2}$
$\displaystyle=\frac{\psi_{2}^{(0)}}{r^{3}}+\frac{\psi_{2}^{(1)}}{r^{4}}+\dots,$
$\displaystyle\psi_{3}$
$\displaystyle=\frac{\psi_{3}^{(0)}}{r^{2}}+\frac{\psi_{3}^{(1)}}{r^{3}}+\dots,$
$\displaystyle\psi_{4}$
$\displaystyle=\frac{\psi_{4}^{(0)}}{r}+\frac{\psi_{4}^{(1)}}{r^{2}}+\dots.$

One could therefore classify spacetime into five regions according to the
decay rates of the Weyl scalars. $\psi_{0}$ and $\psi_{1}$ describe near-zone
fields and are dominant at small distances. $\psi_{2}$ describes the Coulomb
components of the gravitational field and is dominant in the intermediate
zone. $\psi_{3}$ and $\psi_{4}$ describe the radiative components of the field
and are dominant at large distances.

In addition to the five complex Weyl scalars $\psi_{n}$, the Newman-Penrose
formalism is described by twelve complex spin coefficients which describe the
change in the tetrad from point to point. Of these twelve spin coefficients,
the shear

$\sigma\equiv-m^{\mu}m^{\nu}\nabla_{\nu}\ell_{\mu}$ (11)

plays a significant role in the analysis of asymptotically flat spacetimes.
The shear can be expanded at large distances

$\sigma=\frac{\sigma^{0}}{r^{2}}+\mathcal{O}(r^{-3}),$ (12)

where $\sigma^{0}$ is called the _asymptotic shear_. The time derivative of
the shear is called the _news_ and it characterizes gravitational radiation.

The physical meaning of the coefficients $\psi_{n}^{(m)}$ in the expansion of
the Weyl scalars was studied in Janis:1965tx . The few leading coefficients
correspond to the following solutions:

  1. 1.

Monopole

$\displaystyle\psi_{0}$ $\displaystyle=0,$ (13) $\displaystyle\psi_{1}$
$\displaystyle=0,$ $\displaystyle\psi_{2}$
$\displaystyle=\frac{a_{0}}{r^{3}},$ $\displaystyle\psi_{3}$
$\displaystyle=0,$ $\displaystyle\psi_{4}$ $\displaystyle=0,$

where $a_{0}$ is a complex constant.

  2. 2.

Dipole

$\displaystyle\psi_{0}$ $\displaystyle=0,$ (14) $\displaystyle\psi_{1}$
$\displaystyle=-\frac{a_{1}\sin\theta}{r^{4}},$ $\displaystyle\psi_{2}$
$\displaystyle=\frac{\sqrt{2}\dot{a}_{1}\cos\theta}{r^{3}}+\frac{\sqrt{2}a_{1}\cos\theta}{r^{4}}$
$\displaystyle\psi_{3}$
$\displaystyle=\frac{\dot{a}_{1}\sin\theta}{r^{3}}+\frac{a_{1}\sin\theta}{2r^{4}},$
$\displaystyle\psi_{4}$ $\displaystyle=0,$

where $\dot{a}_{1}$ is a complex constant.

  3. 3.

Quadrupole

$\displaystyle\psi_{0}$ $\displaystyle=\frac{3a_{2}\sin^{2}\theta}{r^{5}},$
(15) $\displaystyle\psi_{1}$ $\displaystyle=-\frac{3\dot{a}_{2}\sin
2\theta}{\sqrt{2}r^{4}}-\frac{6a_{2}\sin 2\theta}{\sqrt{2}r^{5}},$
$\displaystyle\psi_{2}$ $\displaystyle=\frac{\ddot{a}_{2}(1+3\cos
2\theta)}{2r^{3}}+\frac{3\dot{a}_{2}(1+3\cos
2\theta)}{2r^{4}}+\frac{3a_{2}(1+3\cos 2\theta)}{2r^{5}}$
$\displaystyle\psi_{3}$ $\displaystyle=\frac{a_{2}^{(3)}\sin
2\theta}{\sqrt{2}r^{2}}+\frac{3\ddot{a}_{2}\sin
2\theta}{\sqrt{2}r^{3}}+\frac{9\dot{a}_{2}\sin
2\theta}{\sqrt{2}2r^{4}}+\frac{3a_{2}\sin 2\theta}{\sqrt{2}r^{5}},$
$\displaystyle\psi_{4}$
$\displaystyle=\frac{a_{2}^{(4)}\sin^{2}\theta}{2r}+\frac{a_{2}^{(3)}\sin^{2}\theta}{r^{2}}+\frac{3\ddot{a}_{2}\sin^{2}\theta}{2r^{3}}+\frac{3\dot{a}_{2}\sin^{2}\theta}{2r^{4}}+\frac{a_{2}\sin^{2}\theta}{4r^{5}},$

where $a_{2}^{(n)}$ is the $n$-th derivative of $a_{2}(u)$ with respect to
$u$.

The real part of $a_{0}$ corresponds to an ”electric”-type monopole. An
explicit realization of the ”electric”-type monopole is the Schwarzschild
solution, for which

$\displaystyle a_{0}$ $\displaystyle=m,$ (16) $\displaystyle\sigma^{0}$
$\displaystyle=0,$

where $m$ is the standard Schwarzschild mass parameter. The imaginary part of
$a_{0}$ is a ”magnetic”-type monopole. An example for a solution with a
”magnetic”-type monopole is the Taub-NUT metric, for which

$\displaystyle a_{0}$ $\displaystyle=m+i\ell,$ (17) $\displaystyle\sigma^{0}$
$\displaystyle=\ell\frac{1+\tan^{4}\frac{\theta}{2}}{2\tan^{2}\frac{\theta}{2}},$

where $\ell$ is known as the NUT parameter and $\theta$ is the polar angle on
the two sphere. The singularity of the shear at $\theta=0$ and $\theta=\pi$ is
due to the Misner string Kol:2019nkc .

In covariant form, the monopole charges are given by

$M+i\tilde{M}=\frac{1}{4\pi
G}\int_{S^{2}}d\Omega\,\left(\psi_{2}^{(0)}+\sigma^{0}\partial_{u}\bar{\sigma}^{0}\right),$
(18)

corresponding to the mass and the dual mass, respectively. Here $G$ is
Newton’s constant. $M$ is the ADM mass, which coincides under some assumptions
with the Komar mass Komar:1958wp  and $\tilde{M}$ is the dual Komar mass. For
an extensive discussion about the mass and the dual mass we refer the reader
to Kol:2020zth . The formula (18) is akin to the Coulomb charges in
electrodynamics (6). On the Taub-NUT solution (17), the charges (18) are then
evaluated to be

$\displaystyle M$ $\displaystyle=\frac{m}{G},$ (19) $\displaystyle\tilde{M}$
$\displaystyle=\frac{\ell}{G}.$

The Taub-NUT solution is therefore characterized by both ”electric”-type and
”magnetic”-type monopole moments. The Schwarzschild solution is recovered when
the NUT parameter is set to zero. A pure ”magnetic”-type monopole is recovered
similarly by setting the Schwarzschild mass $m$ to zero.

Next, $a_{1}$ represents a gravitational dipole moment, whose real and
imaginary parts correspond respectively to ”electric”-type dipole and
”magnetic”-type dipole. An example for a solution that contains a
”magnetic”-type dipole is the Kerr solution, for which

$\displaystyle a_{0}$ $\displaystyle=m,$ (20) $\displaystyle a_{1}$
$\displaystyle=i\frac{3G}{\sqrt{2}}J,$ $\displaystyle\sigma^{0}$
$\displaystyle=0,$

where $J$ is Kerr’s angular momentum

$J\equiv\frac{am}{G}.$ (21)

Here $a$ is the spin parameter. It is therefore evident that the
”magnetic”-type gravitational dipole moment describes angular momentum, which
is a conserved charge in gravity. Note that while in electrodynamics only the
monopole moments are conserved, in gravity the dipole moments are conserved as
well. From a field theory perspective this could seem strange, but we shall
not forget that gravity is a theory of Lorentz transformations.

The generalization of the Kerr angular momentum (21) to a covariant expression
for the charge is known as the ADM angular momentum. In terms of the Newman-
Penrose coefficients it is given by the following integral over
$\psi_{2}^{(1)}$ and the shear

$J(\Phi)=-\frac{1}{16\pi G}\int_{S^{2}}d\Omega\,\Phi(\Omega)\,{\rm
Im}\,\left(\psi_{2}^{(1)}+\eth\left(\bar{\sigma}^{0}\bar{\eth}\sigma^{0}\right)\right).$
(22)

The differential operators $\eth,\bar{\eth}$ encode the covariant derivative
on the sphere and will be discussed soon in detail. $\Phi$ is an $\ell=1$
spherical mode and the integral in (22) is over a large two sphere. The first
term in (22) was suggested to describe angular momentum in the linearized
theory already by Newman and Penrose Newman:1968uj . However, to the best of
our knowledge Prior Prior:1977  was the first to derive the exact expression
for angular momentum using the Weyl scalars in the non-linear theory. The
angular momentum (21) for the Kerr solution (20) is reproduced with

$\displaystyle\Phi(\Omega)=-8\sqrt{\frac{\pi}{3}}Y_{\ell=1,m=0}(\theta,\varphi)=-4\cos\theta,$
(23)

but more generally it can point in any other direction. In addition to the
”magnetic”-type dipole moment, the Kerr solution also contains an
”electric”-type monopole moment. In principle, one could isolate the dipole
moment by taking the limit $m\rightarrow 0$ while keeping the angular momentum
$J=am$ fixed. From the asymptotic point of view it is natural to consider such
a limit, but since it violates the bound $a\leq m$, a naked singularity will
develop in the bulk.

Finally, an example for a solution that contains an ”electric”-type dipole
moment is given by the Kerr-Taub-NUT metric Emond:2020lwi , for which

$\displaystyle\psi_{2}$ $\displaystyle=\frac{m+i\ell}{(r-i\ell-
ia\cos\theta)^{3}},$ (24) $\displaystyle\sigma^{0}$
$\displaystyle=\ell\frac{1+\tan^{4}\frac{\theta}{2}}{2\tan^{2}\frac{\theta}{2}}.$

The first few leading orders’ coefficients in the multipole expansion are
therefore given by

$\begin{aligned} \psi_{2}^{(0)}&=m+i\ell,\\\
\psi_{2}^{(1)}&=3ia(m+i\ell)\cos\theta,\end{aligned}\qquad\Longrightarrow\qquad\begin{aligned}
a_{0}&=m+i\ell,\\\ a_{1}&=-\frac{3G}{\sqrt{2}}\left(K-iJ\right),\end{aligned}$
(25)

where we have defined the _dual angular momentum_ by Henneaux:2004jw ;
Bunster:2006rt ; Argurio:2009xr

$K\equiv\frac{a\ell}{G}.$ (26)

Evidently, the Kerr-Taub-NUT solution contains all four moments: ”electric”
monopole, ”magnetic” monopole, ”electric” dipole and ”magnetic” dipole. When
$\ell=0$, the solution reduces to the Kerr metric and correspondingly only the
”electric”-type monopole and ”magnetic”-type dipole survive. When $a=0$, the
solution reduces to the Taub-NUT metric and correspondingly only the monopole
moments survive. When both $\ell=0$ and $a=0$ the solution reduces to the
Schwarzschild metric and only the ”electric”-type monopole survives. When both
$m=0$ and $a=0$ the solution reduces to the pure-NUT metric and only the
”magnetic”-type monopole survives. Note that the asymptotic shear is
proportional to the NUT parameter and is real. It therefore vanishes for the
Schwarzschild and Kerr metrics, but is different from zero in the presence of
a dual, ”magnetic”, mass.

The discussion in this section now naturally leads to the following question:
what is the physics of the ”electric”-type dipole moment? More precisely -
does it give rise to a conserved charge and if yes - what is its action on the
phase space? As we will see, the ”electric”-type dipole indeed gives rise to a
conserved charge known in the literature as the BORT center of mass
Beig:1987zz ; Regge:1974zd ; Chen:2014uma , which generates boosts on the
phase space.

###  2.1 Symmetry Operations

We have seen that the monopole and dipole moments of the Weyl scalar
$\psi_{2}$ correspond to four conserved charges: mass, dual mass, angular
momentum and the center of mass. Each one of these charge acts differently on
the phase space. First, the mass $M$ in (18) generates time-translations. When
mass density is smeared over the two sphere using a weight function
$f(\Omega)$

$T(f)=-\frac{1}{4\pi G}\int_{S^{2}}d\Omega\,f(\Omega)\,{\rm
Re}\,\left(\psi_{2}^{(0)}+\sigma^{0}\partial_{u}\bar{\sigma}^{0}\right),$ (27)

it generates spatial translations if $f(\Omega)$ is an $\ell=1$ spherical
mode. When $f(\Omega)$ is a spherical mode with $\ell>1$, the charge (27)
generates supertranslations, which are analogues to the large gauge
transformation generated by (7) Bondi:1962px ; Sachs:1962wk ;
Strominger:2013jfa ; He:2014laa .

The dual mass $\tilde{M}$ in (18), on the other hand, acts trivially on the
phase space Kol:2019nkc . However, when smeared over the two sphere

$\tilde{T}(f)=-\frac{1}{4\pi G}\int_{S^{2}}d\Omega\,f(\Omega)\,{\rm
Im}\,\left(\psi_{2}^{(0)}+\sigma^{0}\partial_{u}\bar{\sigma}^{0}\right),$ (28)

it generates non-trivial dual supertranslation transformations, provided that
$f(\Omega)$ contains spherical modes with $\ell>1$ Godazgar:2018qpq ;
Kol:2019nkc . The $\ell=0,1$ modes of dual supertranslations are trivial,
which means that there are no ”magnetic”-type dual transformations of the
standard four translations.

As in the electromagnetic case, we would like to emphasize that two different
multipole expansion are being discussed here. The first is the multipole
expansion of the Weyl scalar $\psi_{2}$, while the second is of the
transformation parameter $f(\Omega)$. For example, evaluating the charges in
(27)-(28) on the dipole solution (14) we have

$T(f_{\ell,m})+i\tilde{T}(f_{\ell,m})\sim\dot{a}_{1}\delta_{\ell
1}\delta_{m0},$ (29)

where $f_{\ell,m}$ are the multipole moments of $f(\Omega)$. We therefore see
that by choosing $f(\Omega)$ with the spherical modes $\ell=1$ and $m=0$, the
resulting charges are proportional to $\dot{a}_{1}$, which is the time
derivative of $\psi_{2}$’s dipole moment and not the dipole moment $a_{1}$
itself. For example, when angular momentum is conserved ${\rm
Im}\,\dot{a}_{1}=0$ Janis:1965tx . Therefore we should distinguish the two
multipole expansions, though there is a clear interplay between the two.

Angular momentum (22) generates rotations over the phase space and similarly
the BORT center of mass generates boosts. We will therefore see that the
gravitational dipole charges generate the entire group of Lorentz
transformations. Those can be further generalized to super-Lorentz
transformations Barnich:2009se ; Barnich:2010eb ; Barnich:2011ct ;
Barnich:2011mi ; Troessaert:2017jcm ; Campiglia:2014yka ; Campiglia:2015yka ;
Campiglia:2015kxa  by promoting their transformation parameters to include
$\ell>1$ spherical modes, but we will not discuss this generalization here.
The four leading gravitational moments - monopoles and dipoles - therefore
span the entire extended BMS group. Our goal now is to study the physics of
the ”electric”-type gravitational dipole.

##  3 Asymptotic Analysis

A general metric can be written in retarded null coordinates using the Bondi
gauge Bondi:1960jsa ; Bondi:1962px ; Sachs:1962wk

$ds^{2}=-Ve^{2\beta}du^{2}-2e^{2\beta}dudr+r^{2}g_{AB}\Big{(}d\Theta^{A}+\frac{1}{2}U^{A}du\Big{)}\Big{(}d\Theta^{B}+\frac{1}{2}U^{B}du\Big{)},$
(30)

where $\Theta^{A}$ with $A=1,2$ are two coordinates that parametrize a two
dimensional compact space and

$\partial_{r}\det g_{AB}=0.$ (31)

The gauge condition (31), together with $g_{rr}=g_{rA}=0$, completely fix the
local diffeomorphism invariance. The metric components $g_{AB}$ can be
parameterized _à la_ Sachs Sachs:1962wk  as

$2g_{AB}d\Theta^{A}d\Theta^{B}=(e^{2\alpha}+e^{2\lambda})d\theta^{2}+4\sin\theta\sinh(\alpha-\lambda)d\theta
d\varphi+\sin^{2}\theta(e^{-2\alpha}+e^{-2\lambda})d\varphi^{2}$ (32)

using the standard angle variables. In this parameterization the gauge
condition (31) is satisfied automatically since the determinant of $g_{AB}$ is
equal to $\sin^{2}\theta$. Here $V,\beta,\alpha,\lambda,U^{A}$ are any six
functions of all coordinates and they parametrize the most general solution.

The Bondi metric can be expanded systematically around flat space. In this
expansion, the leading order is the Minkowski metric, which is given in
retarded null coordinates by

$ds^{2}_{\text{Minkowski}}=-du^{2}-2dudr+2r^{2}\gamma_{z\bar{z}}dzd\bar{z}.$
(33)

Here we choose to work with the following complex variables

$\Theta^{A}=(z,\bar{z}),\qquad z=\tan\frac{\theta}{2}\,\,e^{i\varphi}$ (34)

to parametrize the coordinates on the two sphere. In this system of
coordinates, the metric on the unit two sphere is given by

$\gamma_{AB}=\left(\begin{matrix}0&\gamma_{z\bar{z}}\\\
\gamma_{z\bar{z}}&0\end{matrix}\right),$ (35)

where

$\gamma_{z\bar{z}}\equiv\frac{2}{(1+z\bar{z})^{2}}.$ (36)

Note that in complex coordinates

$\gamma\equiv\det\gamma_{AB}=\gamma_{z\bar{z}}^{2}.$ (37)

Using complex coordinates, the metric (32) then takes the form

$\displaystyle g_{AB}d\Theta^{A}d\Theta^{B}$
$\displaystyle=\left[\Big{(}\sinh(2\alpha)+\sinh(2\lambda)-2i\sinh(\alpha-\lambda)\Big{)}\frac{\gamma_{z\bar{z}}\bar{z}dz^{2}}{2z}+\text{c.c.}\right]$
(38)
$\displaystyle+\Big{(}\cosh(2\alpha)+\cosh(2\lambda)\Big{)}\gamma_{z\bar{z}}dzd\bar{z},$

The asymptotic behavior determines the fall-off conditions of the metric
components, which can be expanded around $r=\infty$ as follows

$\displaystyle V(u,r,x^{A})$
$\displaystyle=1+\frac{V_{0}(u,x^{A})}{r}+\frac{V_{1}(u,x^{A})}{r^{2}}+\dots,$
(39) $\displaystyle\beta(u,r,x^{A})$
$\displaystyle=\frac{\beta_{0}(u,x^{A})}{r^{2}}+\frac{\beta_{1}(u,x^{A})}{r^{3}}+\dots,$
$\displaystyle U_{A}(u,r,x^{A})$
$\displaystyle=\frac{U_{A}^{(0)}(u,x^{A})}{r^{2}}+\frac{U_{A}^{(1)}(u,x^{A})}{r^{3}}+\dots,$
$\displaystyle g_{AB}$
$\displaystyle=\gamma_{AB}+\frac{C_{AB}}{r}+\gamma_{AB}\frac{C^{2}}{4r^{2}}+\dots.$

Here we have defined

$C^{2}\equiv C_{AB}C^{AB},$ (40)

where

$C^{AB}\equiv\gamma^{AC}\gamma^{BD}C_{CD}.$ (41)

We are using a notation where the indices of $C_{AB}$ are raised and lowered
using the metric $\gamma_{AB}$, while otherwise the full metric is used to
raise and lower indices. This mixed notation could be confusing but we adopt
it since it is common in the literature. The metric in Sachs’ parametrization
(38) is then expanded as Godazgar:2018vmm ; Godazgar:2018dvh

$\displaystyle\alpha(u,r,\Theta^{A})$
$\displaystyle=\frac{\alpha_{0}\left(u,\Theta^{A}\right)}{r}+\frac{\alpha_{2}\left(u,\Theta^{A}\right)}{r^{3}}+\dots,$
(42) $\displaystyle\lambda(u,r,\Theta^{A})$
$\displaystyle=\frac{\lambda_{0}\left(u,\Theta^{A}\right)}{r}+\frac{\lambda_{2}\left(u,\Theta^{A}\right)}{r^{3}}+\dots,$

where $\alpha_{0}$ and $\lambda_{0}$ are related to $C_{AB}$ by

$C_{AB}=\gamma_{z\bar{z}}\left(\begin{matrix}(1-i)\frac{\bar{z}}{z}(\alpha_{0}+i\lambda_{0})&0\\\
0&(1+i)\frac{z}{\bar{z}}(\alpha_{0}-i\lambda_{0})\end{matrix}\right).$ (43)

In particular, $C_{z\bar{z}}=0$ and therefore

$C^{2}=2C_{zz}C^{zz}.$ (44)

###  3.1 The Newman-Penrose formalism

In the Newman-Penrose formalism the metric is expressed using a basis of
complex null tetrads

$e^{\mu}_{a}=\\{\ell^{\mu},n^{\mu},m^{\mu},\bar{m}^{\mu}\\},$ (45)

where $\bar{m}^{\mu}$ is the complex conjugate of $m^{\mu}$ and

$\ell_{\mu}n^{\mu}=-1,\qquad m_{\mu}\bar{m}^{\mu}=1,\qquad
m_{\mu}\ell^{\mu}=m_{\mu}n^{\mu}=0.$ (46)

The inverse metric is given by

$g^{\mu\nu}=\eta^{ab}e^{\mu}_{a}e^{\nu}_{b}$ (47)

with

$\eta^{ab}=\left(\begin{matrix}0&-1&0&0\\\ -1&0&0&0\\\ 0&0&0&1\\\
0&0&1&0\end{matrix}\right),$ (48)

such that

$g^{\mu\nu}=-\ell^{\mu}n^{\nu}-\ell^{\nu}n^{\mu}+m^{\mu}\bar{m}^{\nu}+m^{\nu}\bar{m}^{\mu}.$
(49)

The Bondi metric (30) can then be described in the Newman-Penrose formalism
using the following basis

$\displaystyle\ell^{\mu}$ $\displaystyle=\delta^{\mu}_{r},\qquad
n^{\mu}=e^{-2\beta}\Big{(}\delta^{\mu}_{u}-\frac{1}{2}V\,\delta^{\mu}_{r}+U^{A}\delta_{A}^{\mu}\Big{)},$
(50) $\displaystyle m^{\mu}$
$\displaystyle=\frac{1}{r}(\cosh\alpha+i\cosh\lambda)\sqrt{\frac{z}{\bar{z}}}\gamma^{z\bar{z}}dz-\frac{1}{r}(\sinh\alpha+i\sinh\lambda)\sqrt{\frac{\bar{z}}{z}}\gamma^{z\bar{z}}d\bar{z}.$

Let us remark that the leading order terms in the asymptotic expansion of the
tetrad $m^{\mu}$ are given by

$m^{\mu}=\frac{1+i}{2r}\sqrt{2\gamma^{z\bar{z}}\frac{z}{\bar{z}}}\left(dz-\frac{1}{r}C^{\bar{z}\bar{z}}d\bar{z}+\mathcal{O}\left(r^{-2}\right)\right).$
(51)

###  3.2 The Shear

Using the Sachs’ parametrization (38) one can evaluate the shear (11)

$\sigma=\frac{1+i\cosh(\alpha-\lambda)}{2}\partial_{r}\alpha-\frac{1-i\cosh(\alpha-\lambda)}{2}\partial_{r}\lambda.$
(52)

The asymptotic shear (12) is then given in complex coordinates by

$\sigma^{0}=\frac{i}{2}\frac{z}{\bar{z}}\gamma^{z\bar{z}}C_{zz}.$ (53)

We see that the tensor $C_{AB}$ is directly related to the shear and therefore
it is called the _shear tensor_. We also have

$\sigma^{0}\bar{\sigma}^{0}=\frac{1}{8}C^{2}.$ (54)

For example, in the case of the Kerr-Taub-NUT (24) the asymptotic shear in
complex coordinates is given by

$\sigma^{0}=\ell\frac{1+|z|^{4}}{2|z|^{2}}.$ (55)

Notice that the asymptotic shear is real and proportional to the NUT parameter
only. It therefore vanishes when the solution reduces the the Schwarzschild or
Kerr metrics.

###  3.3 Spin-weighted derivatives

Newman and Penrose Newman:1961qr  introduced the concept of _spin-weight_.
Under rotations in the plane span by $\\{m^{\mu},\bar{m}^{\mu}\\}$

$m^{\mu}\rightarrow e^{i\omega}m^{\mu},\qquad\omega\in\mathbb{R},$ (56)

a quantity $\eta^{(s)}(z,\bar{z})$ is said to have spin-weight $s$ if it
transforms as

$\eta^{(s)}\rightarrow e^{is\omega}\eta^{(s)}.$ (57)

Complex conjugation flips the sign of the spin weight. The covariant
derivative on the two surface is then encoded in the operators

$\displaystyle\eth\eta^{(s)}$
$\displaystyle=-\frac{1+i}{2}z\,P^{1-s}\partial_{z}\left(P^{s}\eta^{(s)}\right),$
(58) $\displaystyle\bar{\eth}\eta^{(s)}$
$\displaystyle=-\frac{1-i}{2}\bar{z}\,P^{1+s}\partial_{\bar{z}}\left(P^{-s}\eta^{(s)}\right),$

where

$P=-\frac{1+z\bar{z}}{|z|}.$ (59)

$\eth$ and $\bar{\eth}$ raise and lower, respectively, the spin weight by one
unit. The commutator of the two operators obeys

$\left[\bar{\eth},\eth\right]\eta^{(s)}=\frac{s}{2}R_{0}\eta^{(s)},$ (60)

where $R_{0}$ is the curvature of the celestial space ($R_{0}=2$ for the two
sphere). We have summarized the spin-weights of different objects in table 2.

|  |  |  | $\eth$ | $\partial_{u}$ | $\sigma^{0}$ | $\psi_{4}^{(0)}$ | $\psi_{3}^{(0)}$ | $\psi_{2}^{(0)}$ | $\psi_{1}^{(0)}$ | $\psi_{0}^{(0)}$
---|---|---|---|---|---|---|---|---|---|---|---
| $s$ |  |  | $1$ | $0$ | $2$ | $-2$ | $-1$ | $0$ | $1$ | $2$
Table 2: Spin-weights.

###  3.4 Einstein Equations

With the asymptotic expansion described above we can now analyze the Einstein
equations perturbatively. We will assume that the following components of the
stress-energy tensor fall off at large distances as

$T_{rA}^{M}=\mathcal{O}(r^{-3}),\qquad T_{rr}^{M}=\mathcal{O}(r^{-6}),\qquad
T_{ur}^{M}=\mathcal{O}(r^{-5}).$ (61)

With these assumptions on the fall-off conditions for the stress-energy
tensor, the $G_{rA}$ components of the Einstein equations at order $r^{-2}$
imply that

$U_{A}^{(0)}=-\frac{1}{2}D^{B}C_{AB}.$ (62)

The $G_{rr}$ component of the Einstein equations at orders $r^{-4}$ and
$r^{-5}$ imposes

$\displaystyle\beta_{0}$ $\displaystyle=$ $\displaystyle-\frac{1}{32}C^{2},$
(63) $\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle 0.$ (64)

The $G_{ur}$ component of the Einstein equations at order $r^{-4}$ gives

$V_{1}=-\frac{1}{2}D^{A}U^{(1)}_{A}+\frac{3}{32}\left(\Delta-2\right)C^{2}-\frac{1}{2}(D_{A}C^{AB})(D^{C}C_{CB})-\frac{1}{8}D^{A}C^{BC}D_{A}C_{BC},$
(65)

where the Laplacian on the two sphere is $\Delta\equiv D_{A}D^{A}$.

###  3.5 The Mass Aspect

The Bondi mass aspect is defined by

$m_{B}(u,\Theta^{A})\equiv-\frac{1}{2}V_{0}(u,\Theta^{A})$ (66)

Using this definition, the constraint equation $G_{uu}$ at order $r^{-2}$
encodes the time development of the mass

$\partial_{u}m_{B}=+\frac{1}{4}D_{A}D_{B}N^{AB}-T_{uu},$ (67)

where

$T_{uu}\equiv\frac{1}{8}N_{AB}N^{AB}+4\pi\lim_{r\rightarrow\infty}r^{2}T_{uu}^{M}$
(68)

and

$N_{AB}\equiv\partial_{u}C_{AB}$ (69)

is the Bondi news.

###  3.6 The Angular Momentum Aspect

Following the notations of Barnich and Troessaert Barnich:2009se ;
Barnich:2010eb ; Barnich:2011ct ; Barnich:2011mi ; Troessaert:2017jcm , the
angular momentum aspect $N_{A}(u,\Theta^{A})$ is defined by

$U^{(1)}_{A}=-\frac{2}{3}N_{A}-\frac{1}{6}C_{AB}D_{C}C^{BC}.$ (70)

The constraint equations $G_{uA}$ at order $r^{-2}$ then encodes the time
development of the angular momentum aspect

$\partial_{u}N_{A}=-\frac{1}{4}D^{B}\left(D_{B}D^{C}C_{AC}-D_{A}D^{C}C_{BC}\right)+\partial_{A}m-T_{uA},$
(71)

where

$\displaystyle T_{uA}$
$\displaystyle\equiv\frac{1}{4}N_{BC}D_{A}C^{BC}+\frac{1}{4}D_{B}\left(N_{AC}C^{BC}-C_{AC}N^{BC}\right)-\frac{1}{16}\partial_{A}\left(C_{BD}N^{BD}\right)$
(72) $\displaystyle+8\pi\lim_{r\rightarrow\infty}r^{2}T_{uA}^{M}.$

##  4 Weyl Scalars

In section 2 we have emphasized the role of the Weyl scalar $\psi_{2}$ in the
discussion about conserved charges. In particular, we have demonstrated that
the first two leading orders’ coefficients in its asymptotic expansion,
$\psi_{2}^{(0)}$ and $\psi_{2}^{(1)}$, correspond to the monopole and dipole
moments. Our ultimate goal is therefore to evaluate these two coefficients in
terms of the asymptotic data described in the previous section 3. We would
then be able to associate the multipole moments with conserved charges.

###  4.1 Leading Order

We start by evaluating the leading order coefficients in the expansion of the
Weyl scalars (10). First of all, our main interest is in the Coulomb field,
which is given at leading order by

$\psi_{2}^{(0)}+\sigma^{0}\partial_{u}\bar{\sigma}^{0}=-m_{B}-i\tilde{m}_{B}.$
(73)

Here we have defined the dual mass aspect by

$\tilde{m}_{B}\equiv\frac{1}{4}D^{A}D^{B}\tilde{C}_{AB}$ (74)

and the dual shear tensor as

$\tilde{C}_{AB}\equiv\tensor{\epsilon}{{}_{A}^{C}}C_{CB}.$ (75)

The covariant, two-dimensional, Levi-Civita tensor is

$\epsilon_{AB}=\left(\begin{matrix}0&+1\\\
-1&0\end{matrix}\right)\sqrt{\gamma},\qquad\epsilon^{AB}=\left(\begin{matrix}0&+1\\\
-1&0\end{matrix}\right)\frac{1}{\sqrt{\gamma}}$ (76)

and in complex coordinates we then have

$\tensor{\epsilon}{{}_{A}^{B}}=\left(\begin{matrix}i&0\\\
0&-i\end{matrix}\right),$ (77)

The dual mass aspect can then be written explicitly as

$\tilde{m}_{B}=-\frac{i}{4}\left(D_{z}^{2}C^{zz}-D_{\bar{z}}^{2}C^{\bar{z}\bar{z}}\right)=\frac{1}{2}{\rm
Im}\,\left(D_{z}^{2}C^{zz}\right).$ (78)

Note that $\tilde{m}_{B}$ is real. In terms of the shear we have

$\displaystyle
2\,i\,\tilde{m}_{B}=\eth^{2}\bar{\sigma}^{0}-\bar{\eth}^{2}\sigma^{0}.$ (79)

In addition to the Coulomb field, we also have

$\displaystyle\psi_{3}^{(0)}$
$\displaystyle=\eth\partial_{u}\bar{\sigma}^{0},$ (80)
$\displaystyle\psi_{4}^{(0)}$
$\displaystyle=-\partial_{u}^{2}\bar{\sigma}^{0}.$

The shear is therefore effectively a double integral of the leading radiation
field at large distances. Finally, there is a set of Bianchi identities
involving the rest of the leading order coefficients

$\displaystyle\partial_{u}\psi_{3}^{(0)}$ $\displaystyle=-\eth\psi_{4}^{(0)},$
(81) $\displaystyle\partial_{u}\psi_{2}^{(0)}$
$\displaystyle=-\eth\psi_{3}^{(0)}+\sigma^{0}\psi_{4}^{(0)},$
$\displaystyle\partial_{u}\psi_{1}^{(0)}$
$\displaystyle=-\eth\psi_{2}^{(0)}+2\sigma^{0}\psi_{3}^{(0)},$
$\displaystyle\partial_{u}\psi_{0}^{(0)}$
$\displaystyle=-\eth\psi_{1}^{(0)}+3\sigma^{0}\psi_{2}^{(0)}.$

###  4.2 Subleading Order

At the first sub-leading order we will focus on the coefficient
$\psi_{2}^{(1)}$, which we evaluate to be

$\psi_{2}^{(1)}=D^{z}N_{z}-\frac{3}{2}\gamma^{z\bar{z}}(D^{z}C_{zz})(D^{\bar{z}}C_{\bar{z}\bar{z}})+\frac{1}{4}D^{z}\left(C_{zz}D_{z}C^{zz}\right)+\frac{3}{64}\Delta
C^{2}.$ (82)

In terms of the shear the last equation can be written as

$\psi_{2}^{(1)}+\eth\left(\bar{\sigma}^{0}\bar{\eth}\sigma^{0}\right)+\eth\sigma^{0}\bar{\eth}\bar{\sigma}^{0}+5\bar{\eth}\sigma^{0}\eth\bar{\sigma}^{0}-\frac{3}{2}\eth\bar{\eth}\left(\sigma^{0}\bar{\sigma}^{0}\right)+2\sigma^{0}\bar{\sigma}^{0}=D^{z}N_{z}+\frac{1}{64}\Delta
C^{2}.$ (83)

We have used that

$\eth\left(\bar{\sigma}^{0}\bar{\eth}\sigma^{0}\right)=\frac{1}{4}D^{\bar{z}}\left(C_{\bar{z}\bar{z}}D_{\bar{z}}C^{\bar{z}\bar{z}}\right)$
(84)

and

$\displaystyle\eth\sigma^{0}\bar{\eth}\bar{\sigma}^{0}$
$\displaystyle=\frac{1}{8}D^{B}C_{AB}D_{C}C^{AC}-\frac{1}{4}D^{A}\left(C_{AB}D_{C}C^{BC}\right)+\frac{1}{16}(\Delta-4)C^{2},$
(85) $\displaystyle\bar{\eth}\sigma^{0}\eth\bar{\sigma}^{0}$
$\displaystyle=\frac{1}{8}D^{B}C_{AB}D_{C}C^{AC},$
$\displaystyle\eth\bar{\eth}\left(\sigma^{0}\bar{\sigma}^{0}\right)$
$\displaystyle=\bar{\eth}\eth\left(\sigma^{0}\bar{\sigma}^{0}\right)=\frac{1}{2}\Delta\left(\sigma^{0}\bar{\sigma}^{0}\right)=\frac{1}{16}\Delta
C^{2},$ $\displaystyle\sigma^{0}\bar{\sigma}^{0}$
$\displaystyle=\frac{1}{8}C^{2}.$

Note that all the expressions in (85) are real, while (84) is a complex
function. We see that the dipole moment $\psi_{2}^{(1)}$ depends on the
angular momentum aspect $N_{A}$ and on the asymptotic shear $\sigma^{0}$.

##  5 Lorentz Generators

In the previous section we evaluated the first two leading orders’
coefficients of the Weyl scalar $\psi_{2}$ in terms of the asymptotic data. We
are therefore in a good position to study the relation between these
coefficients and the asymptotic charges.

Let us start by reviewing the connection between the leading order coefficient
$\psi_{2}^{(0)}$ and the corresponding leading order charges. Using (73) we
see that the supertranslation charge (27), corresponding to the
”electric”-type monopole moment ${\rm Re}\,\psi_{2}^{(0)}$, can be evaluated
in terms of the Bondi mass

$T(f)=\frac{1}{4\pi G}\int_{S^{2}}d^{2}\Theta\sqrt{\gamma}\,f\,m_{B}.$ (86)

This is, of course, a standard result in the literature. Similarly, the dual
supertranslation charge (28), corresponding to the ”magnetic”-type monopole
moment ${\rm Im}\,\psi_{2}^{(0)}$, can be evaluated in terms of the dual mass
aspect

$\tilde{T}(f)=\frac{1}{4\pi
G}\int_{S^{2}}d^{2}\Theta\sqrt{\gamma}\,f\,\tilde{m}_{B}.$ (87)

The result is the dual supertranslation charge discussed in Kol:2019nkc ;
Godazgar:2018qpq .

With the results (83) at hand, we can now address the question: what are the
charges associated with the dipole moment $\psi_{2}^{(1)}$? As discussed in
section 2, we already know that the charge associated with the ”magnetic”-type
dipole moment ${\rm Im}\,\psi_{2}^{(1)}$ is the angular momentum (22). Angular
momentum, in turn, generates rotations on the phase space, which are part of
the Lorentz group. In the notations of Barnich and Troessaert Barnich:2009se ;
Barnich:2010eb ; Barnich:2011ct ; Barnich:2011mi  the Lorentz generators are
given by

$Q(Y)=\frac{1}{32\pi
G}\int_{S^{2}}d^{2}\Theta\sqrt{\gamma}\left[Y^{A}\left(N_{A}+\frac{1}{32}\partial_{A}C^{2}\right)\right].$
(88)

Here the six generators of Lorentz transformations $Y^{A}$ obey the conformal
Killing equation

$D_{A}Y_{B}+D_{B}Y_{A}=\gamma_{AB}\left(D\cdot Y\right),$ (89)

where $D\cdot Y\equiv D_{A}Y^{A}$. This can be further generalized to super-
Lorentz transformations if $Y^{A}$ obeys the conformal Killing equation only
_locally_ , but we will not discuss this generalization here. Within the
solutions to (89), the three rotation generators are uniquely defined as the
divergence-free subgroup

$\text{Rotations:}\qquad D\cdot Y=0.$ (90)

The three boost generators, on the other hand, are uniquely defined as the
curl-free subgroup

$\text{Boosts:}\qquad\epsilon^{AB}\partial_{A}Y_{B}=0.$ (91)

A general Lorentz generator can therefore be decomposed as Compere:2019gft

$Y^{A}=\gamma^{AB}\partial_{B}\Psi+\epsilon^{AB}\partial_{B}\Phi,$ (92)

where $\Psi$ and $\Phi$ are both $\ell=1$ spherical modes that respectively
parametrize boosts and rotations.

Using the decomposition (92) of the conformal Killing vectors into boost and
rotation generators, we can now rewrite the Lorentz charge as

$\displaystyle Q(Y)$ $\displaystyle=-\frac{1}{32\pi
G}\int_{S^{2}}d^{2}\Theta\sqrt{\gamma}\,\left[+\Psi\,D^{A}\left(N_{A}+\frac{1}{32}\partial_{A}C^{2}\right)\right.$
(93)
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad\,\,\,+i\,\Phi\,\epsilon^{AB}\partial_{B}\left(N_{A}+\frac{1}{32}\partial_{A}C^{2}\right)\right].$

To derive the last expression we have used integration by parts. In complex
coordinates the Lorentz charge then takes the form

$\displaystyle Q(Y)$ $\displaystyle=-\frac{1}{16\pi
G}\int_{S^{2}}d^{2}\Theta\sqrt{\gamma}\,\left[+\Psi\,{\rm
Re}\,\left(D^{z}N_{z}+\frac{1}{64}\Delta C^{2}\right)\right.$ (94)
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad\,\,\,+i\,\Phi\,{\rm
Im}\,\left(D^{z}N_{z}+\frac{1}{64}\Delta C^{2}\right)\right].$

For brevity of the presentation, let us now define

$\tilde{\psi}_{2}^{(1)}\equiv\psi_{2}^{(1)}+\eth\left(\bar{\sigma}^{0}\bar{\eth}\sigma^{0}\right)+\eth\sigma^{0}\bar{\eth}\bar{\sigma}^{0}+5\bar{\eth}\sigma^{0}\eth\bar{\sigma}^{0}-\frac{3}{2}\eth\bar{\eth}\left(\sigma^{0}\bar{\sigma}^{0}\right)+2\sigma^{0}\bar{\sigma}^{0},$
(95)

such that equation (83) takes the compact form

$\tilde{\psi}_{2}^{(1)}=D^{z}N_{z}+\frac{1}{64}\Delta C^{2}.$ (96)

It is now evident, based on (96), that in complex coordinates the Lorentz
generators are given by

$\displaystyle Q(Y)=-\frac{1}{16\pi
G}\int_{S^{2}}d^{2}\Theta\sqrt{\gamma}\,\left[\Psi\,{\rm
Re}\,\tilde{\psi}_{2}^{(1)}+i\,\Phi\,{\rm Im}\,\tilde{\psi}_{2}^{(1)}\right].$
(97)

We conclude that the real and imaginary parts of $\tilde{\psi}_{2}^{(1)}$
respectively generate boosts and rotations. Altogether,
$\tilde{\psi}_{2}^{(1)}$ generates the entire Lorentz group.

When $\Psi=0$, we recover the result (22) of Prior for angular momentum that
is the generator of rotations

$\displaystyle Q(Y)\Big{|}_{\Psi=0}=iJ(\Phi)=-\frac{i}{16\pi
G}\int_{S^{2}}d^{2}\Theta\sqrt{\gamma}\,\,\Phi\,{\rm
Im}\,\tilde{\psi}_{2}^{(1)},$ (98)

since

${\rm Im}\,\tilde{\psi}_{2}^{(1)}={\rm
Im}\,\left(\psi_{2}^{(1)}+\eth\left(\bar{\sigma}^{0}\bar{\eth}\sigma^{0}\right)\right).$
(99)

When $\Phi=0$, on the other hand, the Lorentz charge (97) reduces to

$\displaystyle Q(Y)\Big{|}_{\Phi=0}=K(\Psi)\equiv-\frac{1}{16\pi
G}\int_{S^{2}}d^{2}\Theta\sqrt{\gamma}\,\Psi\,{\rm
Re}\,\tilde{\psi}_{2}^{(1)}.$ (100)

Since $K(\Psi)$ generates boosts it therefore coincides with the BORT center
of mass. We therefore see that the ”electric and ”magnetic” type dipole
moments of $\psi_{2}$ are associated with the the real and imaginary parts of
the Lorentz charge

$\displaystyle Q(Y)=K(\Psi)+iJ(\Phi),$ (101)

which in turn coincide with the BORT center of mass and the ADM angular
momentum, respectively.

As we did for the angular momentum, we would now like to evaluate the center
of mass explicitly for a given solution. As an example, we will use the Kerr-
Taub-NUT metric (24)-(25) for which we compute

$\displaystyle\tilde{\psi}_{2}^{(1)}=\psi_{2}^{(1)}+\ell^{2}\Big{(}4\sin^{-2}\theta\left(3+3\sin^{-2}\theta-4\sin^{-4}\theta\right)-11\Big{)}.$
(102)

For brevity, we are now using the angle variables on the two sphere since the
expression above takes a complicated form in complex coordinates. As can be
seen, the contribution to $\tilde{\psi}_{2}^{(1)}$ due the shear is singular
at $\theta=0,\pi$. This is nothing but a manifestation of the Misner string.
However, when plugged into the expression for the center of mass (100), the
divergent contributions cancel each other. One way to see this is by
regulating the integral around the location of the string at $\theta=0$

$\displaystyle\int^{\theta=\epsilon}d^{2}\Theta\sqrt{\gamma}\,\Psi\,\Big{(}\tilde{\psi}_{2}^{(1)}-\psi_{2}^{(1)}\Big{)}=\delta_{m,0}\times
6\ell^{2}\sqrt{\frac{3}{\pi}}\Big{(}\log\epsilon+\frac{1}{3\epsilon^{4}}-\frac{5}{18\epsilon^{2}}-\frac{23}{270}+\dots\Big{)}$
(103)

where the index $m$ refers to $\Psi$’s azimuthal moment and the ellipsis stand
for terms that vanish as $\epsilon\rightarrow 0$. It turns out that by
regulating the integral around the location of the string singularity at
$\theta=\pi$, exactly the same singular terms arise and therefore

$\displaystyle\lim_{\epsilon\rightarrow
0}\quad\int^{\theta=\pi-\epsilon}_{\theta=\epsilon}d^{2}\Theta\sqrt{\gamma}\,\Psi\,\Big{[}\tilde{\psi}_{2}^{(1)}-\psi_{2}^{(1)}\Big{]}_{\text{Kerr-
Taub-NUT}}=0.$ (104)

Hence we find that for the Kerr-Taub-NUT solution, the center of mass is given
by

$\displaystyle K(\Psi)=-\frac{1}{16\pi
G}\int_{S^{2}}d^{2}\Theta\sqrt{\gamma}\,\Psi\,{\rm Re}\,\psi_{2}^{(1)},$ (105)

effectively casting out the shear from the expression. Now, for the particular
choice

$\displaystyle\Psi(\Omega)=8\sqrt{\frac{\pi}{3}}Y_{\ell=1,m=0}(\theta,\varphi)=4\cos\theta,$
(106)

the BORT center of mass will coincide with the dual angular momentum defined
in (26). More generally, the center of mass can describe boosts in any other
direction.

##  6 Discussion

In this paper we have studied the extended BMS group, which includes
supertranslations, dual supertranslations and Lorentz transformations, using
the Newman-Penrose formalism. The charges in the extended BMS group are
classified according to their parity into ”electric” and ”magnetic” types.
Supertranslations and dual supertranslations are closely related to the
monopole moment $\psi_{2}^{(0)}$ and therefore we refer to them as the
_leading BMS charges_. The first corresponds to the ”electric”-type monopole
${\rm Re}\,\psi_{2}^{(0)}$ while the later corresponds to the ”magnetic”-type
monopole ${\rm Im}\,\psi_{2}^{(0)}$. The dipole moment $\psi_{2}^{(1)}$, in
turn, describes Lorentz transformations. The ”magnetic”-type dipole ${\rm
Im}\,\psi_{2}^{(1)}$ corresponds to the angular momentum, which generates
rotations over the phase space. Our main result is that the ”electric”-type
dipole ${\rm Re}\,\psi_{2}^{(1)}$ corresponds to the BORT center of mass,
which generates boosts. Altogether, the gravitational monopole and dipole
moments span the entire extended BMS group.

We have studied the Kerr-Taub-NUT metric as an example for a solution that
contains all the first two leading multipole moments. In this case, the
”electric”-type monopole corresponds to the Schwarzschild ADM mass, the
”magnetic”-type monopole corresponds to the NUT parameter, the ”magnetic”-type
dipole corresponds to the Kerr angular momentum and finally, the
”electric”-type dipole correspond to the BORT center of mass (which is
sometimes referred to as the dual angular momentum). The charges of the Kerr-
Taub-NUT solution are summarized in table 3.

| Multipole Moment |  |  |  | ”Electric”-type |  |  | ”Magnetic”-type |
---|---|---|---|---|---|---|---|---|---
| Monopole |  |  |  |  $M=\frac{m}{G}$ |  |  |  $\tilde{M}=\frac{\ell}{G},$ |
| Dipole |  |  |  |  $K=\frac{a\ell}{G}$ |  |  |  $J=\frac{am}{G}$ |
Table 3: Asymptotic charges of the Kerr-Taub-NUT solution. The leading BMS
charges are described by the monopole moments of the Weyl scalar $\psi_{2}$
while the subleading BMS charges are described by its dipole moment. At each
order the charges can be classify into ”electric” and ”magnetic” types
according to their parity.

Let us comment on the relation of our results to the works Godazgar:2018vmm ;
Godazgar:2018dvh  (see also Oliveri:2020xls ), where subleading BMS charges in
the Newman-Penrose formalism were studied using the Barnich-Brandt
prescription Barnich:2001jy . The first subleading BMS charges, which are
associated with the dipole moments $\psi_{2}^{(1)}$ were dismissed in
Godazgar:2018vmm ; Godazgar:2018dvh  as total derivatives. Indeed, the charge
densities of $K(\Psi)$ and $J(\Phi)$ in (98) and (100) are given by total
derivative terms and would integrate to zero if the transformation parameters
$\Psi$ and $\Phi$ were taken to be constants. However, when $\Psi$ and $\Phi$
are given by the $\ell=1$ spherical modes, the corresponding charges do not
integrate to zero and, as we have shown in this paper, their action on the
phase space correspond to Lorentz transformations. We believe that taking
careful consideration of total derivative terms in the Barnich-Brandt
prescription will yield the correct Lorentz charges at the first subleading
order.

We conclude with some proposals for future research directions. In General
Relativity, the Lorentz generators do not commute with supertranslations,
which parametrize in turn a moduli space of vacua. For this reason, the
angular momentum of the vacuum is not well defined in General Relativity
Chen:2014uma ; Javadinazhed:2018mle . Here we would like to suggest that the
discovery of dual supertranslations, the resulting ”electric-magnetic”-type of
duality and the multipole structure of charges could potentially shed new
light on this old problem. This direction is particularly appealing given that
angular momentum is described by a _”magnetic”_ -type dipole moment. Perhaps,
addressing this question in the extended Einstein-Cartan theory could lead to
a better understanding of the issue, in the spirit of the resolution suggested
in Kol:2020zth  for the ”magnetic”-type monopole. Finally, it would be
interesting to understand the subleading ”electric-magnetic”-type duality in
relation to soft theorems (along the lines of Conde:2016csj ; Conde:2016rom )
and to dressing of asymptotic states Choi:2017bna ; Choi:2017ylo ;
Choi:2018oel ; Choi:2019rlz ; Choi:2019fuq ; Choi:2019sjs .

###### Acknowledgements.

I would like to thank William Emond, Yu-tin Huang, Reza Javadinezhad, Nathan
Moynihan, Donal O’Connell, Massimo Porrati, Jacob Sonnenschein and Shimon
Yankielowicz for useful discussions on related subjects.

## References

  * (1) H. Bondi, _Gravitational Waves in General Relativity_ , _Nature_ 186 (1960) 535.
  * (2) H. Bondi, M. van der Burg and A. Metzner, _Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems_ , _Proc. Roy. Soc. Lond. A_ A269 (1962) 21.
  * (3) R. Sachs, _Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times_ , _Proc. Roy. Soc. Lond. A_ 270 (1962) 103.
  * (4) U. Kol and M. Porrati, _Properties of Dual Supertranslation Charges in Asymptotically Flat Spacetimes_ , _Phys. Rev._ D100 (2019) 046019 [1907.00990].
  * (5) H. Godazgar, M. Godazgar and C.N. Pope, _New Dual Gravitational Charges_ , _Phys. Rev._ D99 (2019) 024013 [1812.01641].
  * (6) U. Kol and M. Porrati, _Gravitational Wu-Yang Monopoles_ , _Phys. Rev. D_ 101 (2020) 126009 [2003.09054].
  * (7) R. Arnowitt, S. Deser and C.W. Misner, _Republication of: The dynamics of general relativity_ , _General Relativity and Gravitation_ 40 (2008) 1997.
  * (8) E.T. Newman and R. Penrose, _New conservation laws for zero rest-mass fields in asymptotically flat space-time_ , _Proc. Roy. Soc. Lond._ A305 (1968) 175.
  * (9) C. Prior, _Angular momentum in general relativity I. Definition and asymptotic behaviour_ , _Proc. Roy. Soc. Lond. A_ 354 (1977) 379.
  * (10) R. Beig and N. Ó Murchadha, _The Poincare Group as the Symmetry Group of Canonical General Relativity_ , _Annals Phys._ 174 (1987) 463.
  * (11) T. Regge and C. Teitelboim, _Role of Surface Integrals in the Hamiltonian Formulation of General Relativity_ , _Annals Phys._ 88 (1974) 286.
  * (12) P.-N. Chen, L.-H. Huang, M.-T. Wang and S.-T. Yau, _On the Validity of the Definition of Angular Momentum in General Relativity_ , _Annales Henri Poincare_ 17 (2016) 253 [1401.0597].
  * (13) Y.-T. Huang, U. Kol and D. O’Connell, _Double copy of electric-magnetic duality_ , _Phys. Rev. D_ 102 (2020) 046005 [1911.06318].
  * (14) E. Newman and R. Penrose, _An Approach to gravitational radiation by a method of spin coefficients_ , _J. Math. Phys._ 3 (1962) 566.
  * (15) A. Janis and E. Newman, _Structure of Gravitational Sources_ , _J. Math. Phys._ 6 (1965) 902.
  * (16) E.T. Newman and R. Penrose, _Note on the Bondi-Metzner-Sachs Group_ , _J. Math. Phys._ 7 (1966) 863.
  * (17) A. Strominger, _Asymptotic Symmetries of Yang-Mills Theory_ , _JHEP_ 07 (2014) 151 [1308.0589].
  * (18) T. He, P. Mitra, A.P. Porfyriadis and A. Strominger, _New Symmetries of Massless Qed_ , _JHEP_ 10 (2014) 112 [1407.3789].
  * (19) M. Campiglia and A. Laddha, _Asymptotic symmetries of QED and Weinberg’s soft photon theorem_ , _JHEP_ 07 (2015) 115 [1505.05346].
  * (20) A. Strominger, _Lectures on the Infrared Structure of Gravity and Gauge Theory_ , 1703.05448.
  * (21) A. Komar, _Covariant conservation laws in general relativity_ , _Phys. Rev._ 113 (1959) 934.
  * (22) U. Kol, _Dual Komar Mass, Torsion and Riemann-Cartan Manifolds_ , 2010.07887.
  * (23) W.T. Emond, Y.-t. Huang, U. Kol, N. Moynihan and D. O’Connell, _Amplitudes from Coulomb to Kerr-Taub-Nut_ , 2010.07861.
  * (24) M. Henneaux and C. Teitelboim, _Duality in linearized gravity_ , _Phys. Rev._ D71 (2005) 024018 [gr-qc/0408101].
  * (25) C.W. Bunster, S. Cnockaert, M. Henneaux and R. Portugues, _Monopoles for gravitation and for higher spin fields_ , _Phys. Rev. D_ 73 (2006) 105014 [hep-th/0601222].
  * (26) R. Argurio and F. Dehouck, _Gravitational Duality and Rotating Solutions_ , _Phys. Rev._ D81 (2010) 064010 [0909.0542].
  * (27) A. Strominger, _On BMS Invariance of Gravitational Scattering_ , _JHEP_ 07 (2014) 152 [1312.2229].
  * (28) T. He, V. Lysov, P. Mitra and A. Strominger, _BMS supertranslations and Weinberg’s soft graviton theorem_ , _JHEP_ 05 (2015) 151 [1401.7026].
  * (29) G. Barnich and C. Troessaert, _Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited_ , _Phys. Rev. Lett._ 105 (2010) 111103 [0909.2617].
  * (30) G. Barnich and C. Troessaert, _Aspects of the BMS/CFT correspondence_ , _JHEP_ 05 (2010) 062 [1001.1541].
  * (31) G. Barnich and C. Troessaert, _Supertranslations call for superrotations_ , _PoS_ CNCFG2010 (2010) 010 [1102.4632].
  * (32) G. Barnich and C. Troessaert, _BMS charge algebra_ , _JHEP_ 12 (2011) 105 [1106.0213].
  * (33) C. Troessaert, _The BMS4 algebra at spatial infinity_ , _Class. Quant. Grav._ 35 (2018) 074003 [1704.06223].
  * (34) M. Campiglia and A. Laddha, _Asymptotic symmetries and subleading soft graviton theorem_ , _Phys. Rev. D_ 90 (2014) 124028 [1408.2228].
  * (35) M. Campiglia and A. Laddha, _New symmetries for the Gravitational S-matrix_ , _JHEP_ 04 (2015) 076 [1502.02318].
  * (36) M. Campiglia and A. Laddha, _Asymptotic symmetries of gravity and soft theorems for massive particles_ , _JHEP_ 12 (2015) 094 [1509.01406].
  * (37) H. Godazgar, M. Godazgar and C. Pope, _Subleading BMS charges and fake news near null infinity_ , _JHEP_ 01 (2019) 143 [1809.09076].
  * (38) H. Godazgar, M. Godazgar and C.N. Pope, _Tower of Subleading Dual Bms Charges_ , _JHEP_ 03 (2019) 057 [1812.06935].
  * (39) G. Compére, R. Oliveri and A. Seraj, _The Poincaré and Bms Flux-Balance Laws with Application to Binary Systems_ , 1912.03164.
  * (40) R. Oliveri and S. Speziale, _A note on dual gravitational charges_ , 2010.01111.
  * (41) G. Barnich and F. Brandt, _Covariant Theory of Asymptotic Symmetries, Conservation Laws and Central Charges_ , _Nucl. Phys. B_ 633 (2002) 3 [hep-th/0111246].
  * (42) R. Javadinezhad, U. Kol and M. Porrati, _Comments on Lorentz Transformations, Dressed Asymptotic States and Hawking Radiation_ , _JHEP_ 01 (2019) 089 [1808.02987].
  * (43) E. Conde and P. Mao, _Remarks on asymptotic symmetries and the subleading soft photon theorem_ , _Phys. Rev. D_ 95 (2017) 021701 [1605.09731].
  * (44) E. Conde and P. Mao, _BMS Supertranslations and Not So Soft Gravitons_ , _JHEP_ 05 (2017) 060 [1612.08294].
  * (45) S. Choi, U. Kol and R. Akhoury, _Asymptotic Dynamics in Perturbative Quantum Gravity and BMS Supertranslations_ , _JHEP_ 01 (2018) 142 [1708.05717].
  * (46) S. Choi and R. Akhoury, _BMS Supertranslation Symmetry Implies Faddeev-Kulish Amplitudes_ , _JHEP_ 02 (2018) 171 [1712.04551].
  * (47) S. Choi and R. Akhoury, _Soft Photon Hair on Schwarzschild Horizon from a Wilson Line Perspective_ , _JHEP_ 12 (2018) 074 [1809.03467].
  * (48) S. Choi and R. Akhoury, _Subleading soft dressings of asymptotic states in QED and perturbative quantum gravity_ , _JHEP_ 09 (2019) 031 [1907.05438].
  * (49) S. Choi, S. Sandeep Pradhan and R. Akhoury, _Supertranslation Hair of Schwarzschild Black Hole: A Wilson Line Perspective_ , _JHEP_ 01 (2020) 013 [1910.05882].
  * (50) S. Choi and R. Akhoury, _Magnetic soft charges, dual supertranslations, and ’t Hooft line dressings_ , _Phys. Rev. D_ 102 (2020) 025001 [1912.02224].