QMUL-PH-21-35

The double copy of the multipole expansion

Erick<EMAIL_ADDRESS>Andrés
<EMAIL_ADDRESS>and Chris D.
<EMAIL_ADDRESS>
Centre for Research in String Theory, School of Physics and Astronomy,

Queen Mary University of London, 327 Mile End Road, London E1 4NS, UK

Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and
Astronomy, UCLA, Los Angeles, CA 90095

We consider the classical double copy, that relates solutions of biadjoint
scalar, gauge and gravity theories. Using a recently developed twistor
expression of this idea, we use well-established techniques to show that the
multipole moments of arbitrary vacuum type D gravity fields are
straightforwardly mapped to their counterparts in gauge and biadjoint scalar
theory by the single and zeroth copies. We cross-check our results using
previously obtained results for the Kerr metric. Our results provide further
physical intuition of how the double copy operates.

##  1 Introduction

There is mounting evidence that our various theories of fundamental physics
are more closely connected than previously thought. In this paper, we will
focus on a particular correspondence – the classical double copy – that
relates solutions of the field equations in (non-)abelian gauge theories and
gravity, as well as in a novel scalar theory with two different types of
colour charge (biadjoint scalar field theory). Inspired by the original double
copy for scattering amplitudes in the corresponding quantum field theories [1,
2] (which itself has a string theoretic origin [3]), the first classical
double copy to appear was the Kerr-Schild double copy of ref. [4] (see refs.
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] for further developments).
An alternative exact double copy procedure is the Weyl double copy of ref.
[19] (see also refs. [20, 14, 21, 22, 23]). This uses the spinorial rather
than tensorial formalism of General Relativity, and includes the Kerr-Schild
double copy as a special case. To date, it constitutes the most general exact
statement of the classical double copy, although other formalisms also offer
useful alternative insights [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35].

As well as practical applications of this correspondence (see e.g. ref. [36]),
there are also important conceptual issues to address, including understanding
the ultimate origin of the classical double copy itself. To this end, refs.
[37, 38] recently showed how one can derive both the form and scope of the
Weyl double copy using well-established ideas from twistor theory [39, 40, 41]
(see refs. [42, 43, 44] for pedagogical reviews of this subject), as well as
showing that the Weyl copy is more general than previously thought. As well as
applying new mathematical techniques, another useful method for extending our
understanding of the double copy is to take known physical or mathematical
properties in biadjoint scalar, gauge and gravity theories, and to see how
they match up (or otherwise). Recent examples include properties of solutions
at strong coupling [45, 46, 47, 48], symmetries [49, 33, 50, 51, 52], and
geometric / topological information [8, 53, 54]. Even more simply, the
original Kerr-Schild double copy of ref. [4] told us that mass and energy map
to charge in the gauge theory, which nicely mirrors the replacement of
kinematic by colour information in scattering amplitudes [1, 2].

In this paper, we extend these ideas by considering the well-known multipole
expansion of classical solutions. For any given solution, one may define a
series of higher-rank multipole tensors, which completely characterise the
spatial and temporal distribution of charge (energy / momentum) in the gauge
(gravity) theory respectively. The lowest-order results in this expansion
constitute the total charge and mass mentioned above, and one may then ask
whether higher-multipole moments are strict double copies of each other. It
turns out that this can be easily addressed in the twistor picture of refs.
[37, 38]. As has been argued by Curtis [55], the multipole tensors can be
replaced by multi-index spinor fields, each of which satisfies the twistor
equation. As a consequence, one may instead describe the multipole moments of
a given field purely in terms of higher-rank twistors, which are
straightforwardly defined from the twistor-space functions describing the
fields. We will combine this with the twistor-space double copy of refs. [37,
38], and thus obtain an explicit statement that the multipole expansion
double-copies, for arbitrary type $D$ vacuum solutions. Our results provide a
useful physical insight into how the double copy operates, and may well be
relevant for thinking about further applications.

##  2 The twistor space double copy

In this section, we review salient details of the Weyl double copy, together
with its twistor space incarnation, referring the reader to refs. [37, 38] for
full details. First, we recall that massless free spacetime fields can be
represented by multi-index spinors $\phi_{AB\ldots C}$
($\bar{\phi}_{A^{\prime}B^{\prime}\ldots C^{\prime}}$), representing the anti-
self-dual (self-dual) parts of the field respectively. Index values run from 0
to 1, and may be raised, lowered and / or contracted using the Levi-Civita
symbols $\epsilon^{AB}$ etc. There are $2n$ indices for a spin-$n$ field, and
the resulting quantities then satisfy a special case of the general massless
free field equation

$\nabla^{AA^{\prime}}\bar{\phi}_{A^{\prime}\ldots
C^{\prime}}=0,\quad\nabla^{AA^{\prime}}\phi_{AB\ldots C}=0,$ (1)

where $\nabla^{AA^{\prime}}$ is the appropriate translation of the spacetime
covariant derivative. These fields can be reinterpreted in twistor space
$\mathbb{T}$, corresponding to solutions of the twistor equation

$\nabla_{A^{\prime}}^{(A}\Omega^{B)}=0\quad\Rightarrow\quad\Omega^{A}=\omega^{A}-ix^{AA^{\prime}}\pi_{A^{\prime}}.$
(2)

In the second equality we have written the general solution in Minkowski
space, in terms of constant spinors which may be grouped together to make a
4-component twistor

$Z^{\alpha}=(\omega^{A},\pi_{A^{\prime}}).$ (3)

A non-local map between spacetime and twistor space is established by
requiring that the field in eq. (2) vanish, such that the twistor components
satisfy the incidence relation

$\omega^{A}=ix^{AA^{\prime}}\pi_{A^{\prime}}.$ (4)

This is invariant under rescalings $Z^{\alpha}\rightarrow\lambda Z^{\alpha}$,
$\lambda\in{\mathbb{C}}$, such that we need only consider projective twistor
space $\mathbb{PT}$. A point in spacetime corresponds to a Riemann sphere in
$\mathbb{PT}$, also referred to as a (complex) line. An important result known
as the Penrose transform expresses massless free spacetime fields satisfying
eq. (1) via the contour integrals

$\bar{\phi}_{A^{\prime}B^{\prime}\ldots C^{\prime}}(x)=\frac{1}{2\pi
i}\oint_{\Gamma}d\pi_{E^{\prime}}d\pi^{E^{\prime}}\pi_{A^{\prime}}\pi_{B^{\prime}}\ldots\pi_{C^{\prime}}[\rho_{x}f(Z^{\alpha})],$
(5)

where $\rho_{x}$ restricts all twistors to obey the incidence relation
corresponding to spacetime point $x$, and the contour $\Gamma$ lies on the
appropriate Riemann sphere. The combined integrand and measure must be
invariant under rescalings $Z^{\alpha}\rightarrow\lambda Z^{\alpha}$, which
fixes the (holomorphic) function $f(Z^{\alpha})$ to have homogeneity $-(n+2)$
for a spin-$n$ field. The above remarks imply that twistor functions of
homogeneity $-2$, $-4$ and $-6$ correspond to spacetime gravity fields in
scalar, gauge and gravity theory respectively. Denoting the respective twistor
functions by the subscripts $\\{{\rm scal.},{\rm EM},{\rm grav.}\\}$
respectively, refs. [37, 38] argued that one may define a gravity twistor
function via444We have here skimmed over the fact that the twistor functions
used throughout are not unique, and are instead representatives of cohomology
classes. The product of eq. (6) is then interpreted to apply only to
particular chosen representatives, as discussed in detail in ref. [38].

$f_{\rm grav.}(Z^{\alpha})=\frac{f^{(1)}_{\rm EM}(Z^{\alpha})f^{{(2)}}_{\rm
EM}(Z^{\alpha})}{f_{\rm scal.}(Z^{\alpha})},$ (6)

leading to the spacetime Weyl double copy formula

$\phi_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}(x)=\frac{\phi^{(1)}_{(A^{\prime}B^{\prime}}(x)\phi^{(2)}_{C^{\prime}D^{\prime})}(x)}{\phi(x)}$
(7)

first presented in ref. [19]. Here $\phi$ is a biadjoint scalar field,
$\phi^{(i)}_{A^{\prime}B^{\prime}}$ an electromagnetic spinor, and
$\phi_{A^{\prime}B^{\prime}C^{\prime}D}$ a Weyl spinor. The above discussion
applies to the case of primed spinor fields in spacetime. For unprimed fields,
one may consider the conjugate of eq. (2), whose solutions are associated with
dual twistors $W_{\alpha}$. The notion of the Penrose transform can be
straightforwardly adapted from eq. (8):

$\phi_{AB\ldots C}(x)=\frac{1}{2\pi
i}\oint_{\Gamma}d\lambda_{E}d\lambda^{E}\lambda_{A}\lambda_{B}\ldots\lambda_{C}[\rho_{x}f(W_{\alpha})],$
(8)

and the twistor double copy of eq. (6) similarly generalises. We will work
with dual twistors by default in what follows, in order to match conventions
with ref. [55].

##  3 Multipoles and the double copy

The idea of multipoles is familiar from Newtonian physics in three-dimensional
Euclidean space. A stationary electrostatic or Newtonian potential $\phi$ in a
sourceless region satisfies Laplace’s equation $\nabla^{2}\phi=0$, and may be
expanded as555Throughout the paper we use lower-case Latin, upper-case Latin
and Greek indices for spacetime tensors, spacetime spinors and twistors
respectively. Note, however, that the indices in eq. (9) run only over spatial
components i.e. from 1 to 3.

$\phi=\frac{M}{r}+\frac{M_{i}x^{i}}{r^{3}}+\frac{M_{ij}x^{i}x^{j}}{r^{5}}+\ldots,$
(9)

where $r=(x^{i}x^{i})^{\frac{1}{2}}$, and the multipole tensors
$\\{M_{ij\ldots k}\\}$ are constant tensors defined in terms of derivatives of
the potential, evaluated at the origin ${\cal O}$. Upon shifting to a
different point, the multipole moments change in a way that involves only
lower-order multipoles. The extension of these ideas to General Relativity has
been discussed in refs. [56, 57, 58], for general asymptotically flat
spacetimes. We will not need the full complication of the latter, given that
we will be concerned with solutions of the massless free field equation of eq.
(1) in Minkowski space. Given a constant unit timelike vector $t^{a}$, one may
then consider the 3-space orthogonal to this, with induced metric

$h_{ab}=\eta_{ab}-t_{a}t_{b}.$ (10)

Reference [56] then showed that an appropriate generalisation of the multipole
tensors appearing in eq. (9) is provided by symmetric, trace-free tensor
fields $Q^{a_{1}\ldots a_{n}}(x)$ satisfying

$t_{a_{1}}Q^{a_{1}\ldots a_{n}}=0,\quad\nabla^{m}Q^{a_{1}\ldots
a_{n}}=\frac{n(2n-1)}{3}h^{m(a_{1}}Q^{a_{2}\ldots
a_{n})}-\frac{n(n-1)}{3}Q^{m(a_{3}\ldots a_{n}}h^{a_{1}a_{2})},$ (11)

where the $n^{\rm th}$ such quantity is referred to as the $2^{n}$-multipole
tensor, and the second condition requires that the derivatives of multipole
tensors depend only upon lower multipoles. This is the analogue of the
“shifting the origin” property mentioned for Newtonian multipoles above, and
ensures that the set of tensors $\\{Q^{a_{1}\ldots a_{n}}\\}$ corresponds to
the same solution of the field equation.

In the spinorial formalism, each spacetime index in eq. (11) will become a
pair of spinor indices. Contracting with the timelike vector appearing there,
one may define the symmetric spinor field

$\omega^{A^{\prime}_{1}\ldots A^{\prime}_{2n}}=(6i)^{n}Q^{A^{\prime}_{1}\ldots
A^{\prime}_{n}B_{1}\ldots B_{n}}t_{B_{1}}^{A^{\prime}_{n+1}}\ldots
t_{B_{n}}^{A^{\prime}_{2n}},$ (12)

which turns out to solve a higher-rank generalisation of the twistor equation
of eq. (2):

$\nabla_{L}^{(L^{\prime}}\omega^{A^{\prime}_{1}\ldots A^{\prime}_{2n})}=0.$
(13)

Thus, we can associate the spinors of eq. (12) with multi-index multipole
twistors $\\{Q_{\alpha_{1}\ldots\alpha_{2n}}\\}$. To see how this works in
practice, consider a given physical spin-$n$ field $\Psi_{A_{1}A_{2}\ldots
A_{2n}}$. Then one may define higher-spin fields iteratively by taking
derivatives and contracting with the timelike vector appearing in eq. (12):

$\Psi^{(n)}_{A_{1}\ldots
A_{2n}}=t_{A^{\prime}A_{1}}\nabla^{A^{\prime}}_{A_{2}}[\Psi^{(n-1)}_{A_{3}\ldots
A_{2n}}].$ (14)

These constitute a spinorial analogue of the multiple derivatives appearing in
the Newtonian formalism of eq. (14), whereby higher multipole moments contain
more derivatives of the original potential. For a spin-1 field, one may write
an explicit twistor space integral for the total conserved charge producing
the field [59]:

$Q=-\frac{i}{4\pi^{2}}\oint f(W_{\alpha})d^{4}W,\quad
d^{4}W=\frac{1}{4!}\epsilon^{\alpha\beta\gamma\delta}dW_{\alpha}\,dW_{\beta}\,dW_{\gamma}\,dW_{\delta},$
(15)

where an appropriate contour must be chosen, and where $f(W_{\alpha})$ is the
twistor function corresponding to the spacetime field. Given a higher-spin
field as in eq. (14), we can form multiple spin-1 fields by contracting with
solutions of the twistor equation666That the fields of eq. (16) indeed satisfy
the massless free field equation of eq. (1) follows from eq. (13).
$\\{\alpha^{A_{1}\ldots A_{2n}}\\}$:

$\Phi^{(n)}_{AB}=-i^{n}\alpha^{A_{1}\ldots A_{2n}}[\Psi^{(n+1)}_{ABA_{1}\ldots
A_{2n}}].$ (16)

Each of these fields will have a conserved charge according to eq. (15), and
we may collect together all such charges in the twistor-covariant form

$q(A^{\alpha_{1}\ldots\alpha_{2n}})=\frac{i^{n+1}}{4\pi^{2}}\oint
W_{\alpha_{1}}\ldots
W_{\alpha_{2n}}A^{\alpha_{1}\ldots\alpha_{2n}}f_{n+1}(W_{\alpha})d^{4}W,$ (17)

for symmetric twistors $\\{A^{\alpha_{1}\ldots\alpha_{2n}}\\}$, where
$f_{n+1}(W_{\alpha})$ is the twistor function corresponding to the spacetime
higher-spin field ${{}_{(n+1)}}\Psi_{ABA_{1}\ldots A_{2n}}$, and multipole
index $n$. Equation (17) defines a set of quantities dual to the
$\\{A^{\alpha_{1}\ldots\alpha_{2n}}\\}$:

$Q_{\alpha_{1}\ldots\alpha_{2n}}=\frac{i^{n+1}}{4\pi^{2}}\oint
W_{\alpha_{1}}\ldots W_{\alpha_{2n}}f_{n+1}(W_{\alpha})d^{4}W,$ (18)

which are the multipole twistors we have been seeking. Note that the iterative
structure of the higher-spin fields in eq. (14) implies that the twistor
functions $\\{f_{n+1}\\}$ in eq. (18) can also be constructed iteratively, and
there are various ways that this can be written. A fully invariant condition
is [55]

$f_{n}=i(R_{\alpha}W_{\beta}I^{\alpha\beta})^{-1}R_{\gamma}P^{\gamma}_{\delta}\frac{\partial
f_{n-1}}{\partial W_{\delta}},$ (19)

where we have introduced the so-called infinity twistors for Minkowski
spacetime:

$I_{\alpha\beta}=\left(\begin{array}[]{cc}0&0\\\
0&\epsilon^{A^{\prime}B^{\prime}}\end{array}\right),\quad
I_{\alpha\beta}=\left(\begin{array}[]{cc}\epsilon^{AB}&0\\\
0&0\end{array}\right),$ (20)

$R_{\alpha}$ is an arbitrary twistor, and we have introduced the projector
[55]

$\lambda P^{\alpha}_{\beta}=I^{\alpha\gamma}Q_{\gamma\beta},$ (21)

where $\lambda$ is the relevant mass or charge parameter for a given theory.
One thus has $\lambda=m$ in gravity, where $m$ is the total mass of the
system. In gauge or biadjoint theory, it will be the total charge of the
system that is creating the field, which we will denote by $q$ and $y$
respectively.

###  3.1 The double copy of the multipole expansion

The multipole twistors introduced above allow us to address the double copy of
the multipole expansion in a particularly compact and elegant way. Consider
twistor functions corresponding to a biadjoint scalar, electromagnetic and
gravity solution respectively, which we label by $f_{X}(W_{\alpha})$,
$X\in\\{{\rm scal.},{\rm EM},{\rm grav.}\\}$. From each of these, one may
define a set of higher-spin twistor functions according to the iterative
procedure of eq. (19), denoted here by $f_{X}^{(n)}(W_{\alpha})$. By eq. (18),
this immediately leads to a set of multipole twistors for each original
spacetime field. This construction is shown in table 1, where each column
contains twistor functions of the same homogeneity, leading to the same spin
fields in position space. We show the multipole twistors that arise from these
in table 2. Note that a gravitational monopole contribution is not directly
obtained from the corresponding twistor functions. As the total mass, however,
it is obtainable from the angular momentum twistor $Q^{\rm
grav.}_{\alpha_{1}\alpha_{2}}$. For the $n=1$ case, one finds integrals
expressing the total charge generated by the biadjoint or EM field. From $n=2$
upwards, there are multipole twistors in all three theories. For a given set
of functions $\\{f_{X}\\}$ related by the twistor-space double copy, we can
then associate each column of table 2 with a classical double copy triple, as
shown in figure 1.

Theory | Multipole index $n\rightarrow$
---|---
Biadjoint scalar | $f_{\rm scal.}$ | $f^{(1)}_{\rm scal.}$ | $f^{(2)}_{\rm scal.}$ | $f^{(3)}_{\rm scal.}$ | $f^{(4)}_{\rm scal.}$
Gauge |  | $f_{\rm EM}$ | $f^{(2)}_{\rm EM}$ | $f^{(3)}_{\rm EM}$ | $f^{(4)}_{\rm EM}$
Gravity |  |  | $f_{\rm grav.}$ | $f^{(3)}_{\rm grav.}$ | $f^{(4)}_{\rm grav.}$
Table 1: Twistor functions occuring in the multipole expansion for three different theories. Here $f_{X}$ is a twistor function corresponding to physical spacetime field in theory $X$, and $f_{X}^{(n)}$ is a higher-spin field generated from this by the iterative procedure of eq. (19). Theory | Multipole index $n$
---|---
| 0 | 1 | 2 | 3
Biadjoint scalar | $Q^{\rm scal.}$ | $Q^{\rm scal.}_{\alpha_{1}\alpha_{2}}$ | $Q^{\rm scal.}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}$ | $Q^{\rm scal.}_{\alpha_{1}\ldots\alpha_{6}}$
Gauge | $Q^{\rm EM}$ | $Q^{\rm EM}_{\alpha_{1}\alpha_{2}}$ | $Q^{\rm EM}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}$ | $Q^{\rm EM}_{\alpha_{1}\ldots\alpha_{6}}$
Gravity |  | $Q^{\rm grav.}_{\alpha_{1}\alpha_{2}}$ | $Q^{\rm grav.}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}$ | $Q^{\rm grav.}_{\alpha_{1}\ldots\alpha_{6}}$
Table 2: Multipole twistors arising from the twistor functions of table 1.

Figure 1: Double copy structure of the multipole twistors appearing in a
single column in table 2.

The physical interpretation of the identifications in figure 1 is
straightforward. Consider, for example, the 2-multipole tensors
$Q^{X}_{\alpha\beta}$. This represents the angular momentum in gravity [40],
whereas in electromagnetism it is the charge dipole tensor, as expected given
that the single copy turns mass into charge. Likewise, for the higher
multipoles, the single copy replaces the relevant spatiotemporal distribution
of mass / momentum with that of charge, with a further replacement to
“biadjoint charge” in the zeroth copy.

It is one thing to formally identify the multipole twistors in different
theories, as we have done in figure 1. It is quite another thing to say that
the multipole twistors in the different theories are the same, up to simple
mass and charge replacements. Remarkably, this strong statement indeed turns
out to be true for fields related by the original type D Weyl double copy of
ref. [19], as we discuss in the following section.

###  3.2 Multipole moments of type D solutions

As discussed in ref. [60] and reviewed in refs. [37, 38], all vacuum type D
solutions arise from twistor functions of the form

$f_{\rm grav.}=(A^{\alpha\beta}W_{\alpha}W_{\beta})^{-3},$ (22)

where $A^{\alpha\beta}$ is a constant twistor that can be taken to be
symmetric. We see that eq. (22) has homogeneity $-6$ under rescalings of
$W_{\alpha}$, as required for a gravity solution. Furthermore, it has two
poles in twistor space, which give rise, after performing the Penrose
transform of eq. (8) to position space, to the two two-fold degenerate
principal spinors of the Weyl spinor that characterise it as being of type D.
It turns out that the twistor $A^{\alpha\beta}$ can be straightforwardly
related to the 2-multipole twistor for this field. Substituting eq. (22) into
eq. (18) for $n=1$, one may carry out the integral using a special case of

$\oint W_{\alpha_{1}}\ldots
W_{\alpha_{2n}}(W_{\alpha}W_{\beta}A^{\alpha\beta}))^{-(n+2)}d^{4}W=\frac{\pi^{3}i}{\Delta}\frac{(2n)!}{2^{2n-1}(n+1)!n!}B_{(\alpha_{1}\alpha_{2}}\ldots
B_{\alpha_{2n-1}\alpha_{2n})},$ (23)

where $B_{\alpha\beta}$ is the inverse of $A^{\alpha\beta}$, and $\Delta$ the
determinant of the latter. One finds

$Q_{\alpha\beta}=\frac{\pi}{8i\Delta}B_{\alpha\beta},\quad
Q^{\alpha\beta}=\frac{8i\Delta}{\pi}A^{\alpha\beta}.$ (24)

Given the general type D gravity twistor function of eq. (22), one may also
identify the single and zeroth copies, giving rise to a gauge and biadjoint
scalar field in spacetime respectively. As explained in refs. [37, 38], these
are

$f_{\rm scal.}={\cal N}_{0}(A^{\alpha\beta}W_{\alpha}W_{\beta})^{-1},\quad
f_{\rm EM}={\cal N}_{1}(A^{\alpha\beta}W_{\alpha}W_{\beta})^{-2}.\quad$ (25)

We have here included arbitrary constant normalisation factors in the scalar
and electromagnetic functions, which are in any case not fixed in the Weyl
double copy of ref. [19]. Physically, one may absorb such constants by
redefining the total amount of charge in a particular solution, but we will
fix them shortly. Let us now construct and compare the multipole twistors from
these solutions. For each field, we may construct higher-spin twistor
functions using the procedure of eq. (19). Starting with the gravity function
from eq. (22), one finds

$\displaystyle f^{(3)}_{\rm grav.}$
$\displaystyle=-\frac{3i}{\alpha}(R_{\alpha}W_{\beta}I^{\alpha\beta})^{-1}(A^{\rho\lambda}W_{\rho}W_{\lambda})^{-4}R_{\gamma}I^{\gamma\tau}Q_{\tau\delta}A^{\delta\sigma}W_{\sigma},$
(26)

where we have used eq. (21). We may now use eq. (24), which yields

$\displaystyle f^{(3)}_{\rm grav.}$ $\displaystyle=3\left(-\frac{\pi}{4\Delta
m}\right)(A^{\rho\lambda}W_{\rho}W_{\lambda})^{-4},$ (27)

such that iterating this procedure leads to the formula

$f^{(n)}_{\rm grav.}=\left(-\frac{\pi}{4\Delta
m}\right)^{n-2}\frac{n!}{2}(W_{\alpha}W_{\beta}A^{\alpha\beta})^{-(n+1)},$
(28)

as quoted in ref. [55]. Note that placing $n=2$ in this formula reproduces the
original gravity twistor function $f_{\rm grav.}(W_{\alpha})$ itself. We may
find the multipole twistors of eq. (18) using eq. (23, 24), yielding

$Q^{\rm
grav.}_{\alpha_{1}\ldots\alpha_{2n}}=\frac{1}{2}\frac{1}{(2m)^{n-1}}\frac{(2n)!}{n!}Q^{\rm
grav.}_{(\alpha_{1}\alpha_{2}}\ldots Q^{\rm
grav.}_{\alpha_{2n-1}\alpha_{2n})},$ (29)

In principle, one may convert these multipole twistors back into multipole
tensors. For the Kerr solution, a set of scalar multipole moments has been
defined in the literature [58]. Let $\tilde{z}^{a}$ be a vector aligned with
the axis of rotation of the black hole, and $\Lambda$ be the point at infinity
after conformal compactification of the spacetime. Then the multipole moments
are given by

$Q_{n}=\frac{1}{n!}Q_{a_{1}\ldots a_{n}}\tilde{z}^{a_{1}}\ldots
z^{a_{n}}\Big{|}_{\Lambda},$ (30)

where the notation on the right-hand side denotes that this be evaluated at
$\Lambda$ itself. As stated in ref. [55], the multipole twistors of eq. (29)
do indeed reproduce the known multipole moments of the Kerr solution, first
found in ref. [58].

We may carry out the above procedure for the biadjoint scalar and gauge theory
twistor functions of eq. (25), and the resulting higher spin twistor functions
are given by

$\displaystyle f^{(n)}_{\rm scal.}$ $\displaystyle={\cal
N}_{0}\left(-\frac{\pi}{4\Delta
y}\right)^{n}\frac{n!}{2}(W_{\alpha}W_{\beta}A^{\alpha\beta})^{-(n+1)}$
$\displaystyle f^{(n)}_{\rm EM}$ $\displaystyle={\cal
N}_{1}\left(-\frac{\pi}{4\Delta
q}\right)^{n-1}\frac{n!}{2}(W_{\alpha}W_{\beta}A^{\alpha\beta})^{-(n+1)},$
(31)

where we have replaced the mass $m$ in the gravity solution with the charge
$q$ in gauge theory, and biadjoint charge $y$ in the scalar theory. These
functions reproduce the original fields for $n=0$ and $n=1$ respectively. We
may choose to fix the arbitrary normalisation constants ${\cal N}_{i}$ by
requiring that the 2-multipole (dipole) tensor in each theory is simply
related by replacing

$m\rightarrow q\rightarrow y.$ (32)

in going from gravity to gauge theory, to biadjoint theory. This determines

${\cal N}_{0}=\left(-\frac{\pi}{4\Delta y}\right)^{-2},\quad{\cal
N}_{1}=\left(-\frac{\pi}{4\Delta q}\right)^{-1},$ (33)

after which comparison of eq. (31) with eq. (28) shows that all higher-spin
twistor functions agree across all three theories, so that one may simply
replace the multipole twistors of eq. (29) with the gauge and biadjoint scalar
counterparts

$\displaystyle Q^{\rm scal.}_{\alpha_{1}\ldots\alpha_{2n}}$
$\displaystyle=\frac{1}{2}\frac{1}{(2y)^{n-1}}\frac{(2n)!}{n!}Q^{\rm
scal.}_{(\alpha_{1}\alpha_{2}}\ldots Q^{\rm
scal.}_{\alpha_{2n-1}\alpha_{2n})}$ $\displaystyle Q^{\rm
EM}_{\alpha_{1}\ldots\alpha_{2n}}$
$\displaystyle=\frac{1}{2}\frac{1}{(2q)^{n-1}}\frac{(2n)!}{n!}Q^{\rm
EM}_{(\alpha_{1}\alpha_{2}}\ldots Q^{\rm EM}_{\alpha_{2n-1}\alpha_{2n})},$
(34)

As a direct consequence, the multipole moments of the gauge and biadjoint
scalar fields corresponding to a given gravity field from eq. (22) precisely
match, after making the necessary mass-to-charge replacements. Our twistor
analysis has applied for an arbitrary quadratic form in eq. (22) which, as
explained in refs. [37, 38], is a general statement for any (vacuum type D)
spacetime entering the original Weyl double copy of ref. [19]. In particular,
this must apply to the Kerr solution, and there is a novel cross-check one may
perform. The gauge theory counterpart of this solution is the $\sqrt{\rm
Kerr}$ solution discussed above, and its electromagnetic monopole moments have
not previously been calculated directly. However, one may instead consider a
charged Kerr black hole, otherwise known as a Kerr-Newman black hole [61, 62].
This is a solution of the Einstein-Maxwell equations, and as such consists of
a metric plus a gauge field. The $\sqrt{\rm Kerr}$ solution can be obtained by
setting the mass of the solution to zero, leaving a gauge field living in
Minkowski space, and which is known to correspond to the single copy of the
gravity solution. The combined gravitational and electromagnetic multipole
moments of the Kerr-Newman solution have been calculated in ref. [63]. The
gravity moments agree with the pure Kerr solution, and the electromagnetic
ones are simply obtained by the mass-to-charge replacement of eq. (32). This
indeed verifies our results. Further support comes from recent studies showing
that the multipole expansion for $\sqrt{\rm Kerr}$ and Kerr can be obtained
from scattering amplitudes that manifestly double-copy [64].

##  4 Discussion

In this paper, we have considered whether the multipole expansions of fields
in biadjoint scalar, gauge and gravity theory can be related by the classical
double copy. By combining a twistor formulation of the multipole expansion
[55] with a recently developed twistor language for the classical double copy
[37, 38], we have shown that the multipole moments for arbitrary type D vacuum
solutions indeed match up in different theories, subject to appropriate mass /
charge replacements.

Our results provide a nice illustration of the efficiency of the twistor
double copy, but are of interest in their own right. It is often the case that
a single copy of a given gravity solution can be found, but not easily
interpreted. A canonical case of this is the single copy of the Kerr black
hole, first formally identified in ref. [4], and denoted as $\sqrt{\rm Kerr}$
in subsequent literature (see e.g. refs. [65, 66, 64]). It is known that this
solution occurs by replacing the source for the Kerr black hole (a rotating
disk of mass) with a similar gauge theory source (a rotating disk of charge).
However, the nature of the sources is subtly different in the two theories
[4], such that it is not clear what impact this has on the fields themselves.
Multipole moments, however, allow us to fully characterise the structure of
fields in a gauge-invariant way. Thus, the fact that the multipole moments for
the Kerr and $\sqrt{\rm Kerr}$ solutions are essentially identical tells us a
great deal of information about how to physically interpret the single copy,
by recycling our intuition gathered from the Kerr black hole. Furthermore, the
fact that our results apply for any type D vacuum solution makes this a rather
powerful statement, that may well help in interpreting and extending the
double copy in future.

Another nice aspect of our results is that the multipole expansion in
biadjoint theory also matches that in the gauge and gravity theories, for the
wide class of solutions we have considered. This adds a powerful weight to the
observations made in refs. [37, 38], namely that the twistor double copy
allows us to understand the inverse zeroth copy from biadjoint scalar theory
to gauge theory. That is, we have seen directly that the multipoles of vacuum
type D gravity solutions and their single copies are essentially inherited
directly from a much simpler scalar theory! It is interesting to ponder what
other physical quantities can be phrased in such an appealing manner.

## Acknowledgments

We are very grateful to Donal O’Connell and Justin Vines for discussions. This
work has been supported by the UK Science and Technology Facilities Council
(STFC) Consolidated Grant ST/P000754/1 “String theory, gauge theory and
duality”, and by the European Union Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No. 764850 “SAGEX”.
EC is supported by the National Council of Science and Technology (Conacyt).
AL is supported by the U.S. Department of Energy (DOE) under award number DE-
SC0009937, and by the Mani L. Bhaumik Institute for Theoretical Physics.

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