Deriving Weyl double copies with sources

Kymani<EMAIL_ADDRESS>Nathan
<EMAIL_ADDRESS>and Chris D.
<EMAIL_ADDRESS>
a Centre for Theoretical Physics, Department of Physics and Astronomy,

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

The Weyl double copy is a relationship between classical solutions in gauge
and gravity theories, and has previously been applied to vacuum solutions in
both General Relativity and its generalisations. There have also been
suggestions that the Weyl double copy should extend to solutions with non-
trivial sources. In this paper, we provide a systematic derivation of sourced
Weyl double copy formulae, using spinorial methods previously established for
${\cal N}=0$ supergravity. As a cross-check, we rederive the same formulae
using a tensorial approach, which then allows us to extend our arguments to
sources containing arbitrary powers of the inverse radial coordinate. We also
generalise our results to include the Kerr-Newman black hole, clarifying
previous alternative double copy formulae presented in the literature. Our
results extend the validity of the Weyl double copy, and may be useful for
further astrophysical applications of this correspondence.

##  1 Introduction

Recent years have seen intense interest in the relationships between different
field theories, such as the gauge and gravity theories underlying fundamental
interactions in our universe. One such relationship is the double copy, which
first arose in the study of scattering amplitudes [1, 2], itself inspired by
previous results in string theory [3]. This states that certain quantities in
gauge theories can be straightforwardly recycled to produce gravity results,
and it is by now well-known that the idea can be applied to classical
solutions of each theory, where certain examples can be mapped exactly [4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27]. Even if this is not the case, perturbative classical double copies
are possible [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54], which have direct applications to
gravitational wave physics. It is not currently known how general the double
copy is, and in particular whether it extends beyond (weak coupling)
perturbation theory (see e.g. refs. [55, 56, 57, 58, 59, 60, 61, 62, 63, 64,
65, 47, 66, 67, 68] for non-perturbative aspects). Recent reviews of the
double copy may be found in refs. [69, 70, 71, 72, 73, 74].

Given the above remarks, there is clear scope for studies which aim to broaden
the remit of the double copy, by studying cases or examples which go beyond
what has been studied before. One such frontier consists of non-vacuum
classical solutions of gauge theories or gravity. A number of exact classical
double copies are known, such as the Kerr-Schild double copy of ref. [4], the
Weyl double copy of ref. [13], and the twistor double copy of refs. [17, 19]
(see ref. [75] for a discussion of how these various schemes are related). In
each of the original incarnations, only certain vacuum solutions were
considered, and the question then naturally arises as to whether such
conditions can be relaxed. Early work on the inclusion of sources in the Kerr-
Schild double copy can be found in ref. [6]. More recently, refs. [76, 23]
have looked at non-vacuum solutions in the context of the Weyl double copy of
ref. [13], which we will review below. The authors of refs. [76, 23] found
various relationships between particular non-vacuum gauge and gravity
solutions. Explicit examples included those in Einstein-Maxwell theory, in
which General Relativity is coupled to a photon, and whose solutions include
the well-known Kerr-Newman black hole. The double copy formulae presented in
refs. [76, 23] take the form of product relationships between spinor fields in
electromagnetic theories, and their counterparts in gravity. Given the
particular nature of the solutions considered, however, it is not yet clear
whether such formulae are generally applicable to all solutions of these
theories. This motivates trying to derive such formulae from first principles,
which is the subject of this paper.

A general prescription for deriving Weyl double copy formulae has been
developed in refs. [17, 19, 21, 20, 75, 77, 78], also taking inspiration from
ref. [79]. By starting with solutions of classical equations in momentum
space, one may perform an inverse Fourier transform to position space, but
where one splits the inverse transform into two stages. The first takes one
into an intermediate twistor space, such that classical solutions are
represented by certain functions there. The second part of the transform then
corresponds to the well-known Penrose transform [80, 81] that relates twistor
space functions (strictly speaking, representatives of cohomology classes
[82]) to spinorial fields in spacetime. The latter can then be seen to obey
the Weyl double copy where relevant, and thus this recipe can be used to
“derive” Weyl double copy formulae whenever they exist. Applications have so
far included the original Weyl double copy of ref. [13], a novel variant known
as the Cotton double copy [83, 84] in three spacetime dimensions [77] (see
also refs.[85, 86, 87, 88]), and an extension of the double copy that applies
to ${\cal N}=0$ supergravity [78]. Here, we will adapt this scheme to probe
Weyl double copies for sourced fields, finding a number of complications with
respect to previous cases. However, we will succeed in deriving Weyl double
copy formulae for a particular non-zero source. We will then cross-check our
results by showing how they can be obtained starting from the conventional
tensorial formulation of field theory, which will also allow us to generalise
our result to arbitrary spherically symmetric sources. Next, we show that
certain axially symmetric examples from ref. [23] can also be obtained, such
as the Kerr-Newman black hole [89]. We furthermore clarify the relationship
between apparently different doubly copy formulae presented there. We hope
that our results prove useful in extending the double copy, and also provide
useful data for future studies in this area.

The structure of our paper is as follows. In section 2, we review the Weyl
double copy, and also the conjectures of refs. [76, 23] regarding how it may
extend to sourced fields. In section 3, we take a particular type of source,
corresponding to the Reissner-Nordstrom black hole. We show how to extend our
previous formalism for deriving Weyl double copies, yielding to explicit
spacetime formulae linking gauge and gravity results. In section 4, we derive
similar results using tensor methods, and also argue how they generalise to
more complicated sources. We examine the Kerr-Newman black hole in section 5,
before discussing our results and concluding in section 6.

##  2 Review of the Weyl double copy

In this section, we review salient details regarding the Weyl double copy,
which are needed for what follows. Our starting point is the spinorial
formalism of field theory, which states that all tensorial quantities such as
field strengths and curvature tensors can be converted into multi-index spinor
objects via contraction with Infeld-van-der-Waerden symbols. If we are working
in Minkowski spacetime with line element

$ds^{2}=dt^{2}-dx^{2}-dy^{2}-dz^{2},$ (1)

we will follow the conventions of ref. [79] in choosing

$\sigma^{\mu}_{A\dot{A}}=\left(I,\sigma_{z},\sigma_{x},\sigma_{y}\right)_{A\dot{A}},$
(2)

where $I$ is the identity matrix in spinor space, and $\\{\sigma_{i}\\}$ are
the Pauli matrices. We will also have occasion to work in (2,2) signature,
with line element

$ds^{2}=dt^{2}-dx^{2}-dy^{2}+dz^{2},$ (3)

such that eq. (2) is modifed to

$\sigma^{\mu}_{A\dot{A}}=\left(I,\sigma_{z},\sigma_{x},i\sigma_{y}\right)_{A\dot{A}}.$
(4)

Whereas tensor indices are raised and lowered with the metric $\eta^{\mu\nu}$,
spinorial indices can be raised / lowered with the Levi-Civita symbols

$\displaystyle\epsilon_{AB}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix},\quad\epsilon^{AB}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},$
(5)

according to e.g.

$\displaystyle q_{A}=q^{B}\epsilon_{BA},\quad q^{A}=\epsilon^{AB}q_{B}.$ (6)

Upon translating to the spinor language, an $n$-index tensor
$V_{\alpha\beta\ldots\gamma}$ is associated with a $2n$-index spinor:

$V_{\alpha\beta\ldots\gamma}\sigma^{\alpha}_{A\dot{A}}\sigma^{\beta}_{B\dot{B}}\ldots\sigma^{\gamma}_{C\dot{C}}=V_{A\dot{A}B\dot{B}\ldots
C\dot{C}},$ (7)

where we keep the symbol $V$ for the tensor or spinor quantity, given that the
nature of indices makes this unambiguous. For spacetime vectors, we note the
useful identity

$V\cdot W=\frac{1}{2}V^{A\dot{A}}W_{A\dot{A}}.$ (8)

Also, the spinorial translation of a spacetime vector satisfies

${\rm det}\left[V_{A\dot{A}}\right]=V^{2},$ (9)

which vanishes for a null vector, such that its spinorial translation may be
written as an outer product of two spinors:

$V_{A\dot{A}}=\eta_{A}\tilde{\eta}_{\dot{A}}.$ (10)

Armed with these definitions, the familiar tensor quantities of field theory
can all be translated into multi-index spinors. This turns out to be
advantageous in many circumstances due to two particularly useful properties.
First, a general multi-index spinor can always be decomposed into sums of
products of spinors which are fully symmetric in their indices, and Levi-
Civita tensors. For example, the field strength tensor $F_{\mu\nu}$ yields a
spinor field $F_{A\dot{A}B\dot{B}}$, which can be decomposed as [90, 91]

$F_{A\dot{A}B\dot{B}}(x)=\Phi_{AB}(x)\epsilon_{\dot{A}\dot{B}}+\bar{\Phi}_{\dot{A}\dot{B}}(x)\epsilon_{AB}.$
(11)

Physically, $\phi_{AB}$ and $\phi_{\dot{A}\dot{B}}$ represent the self-dual
and anti-self-dual parts of the electromagnetic field respectively. In
Lorentzian signature, these quantities are related by complex conjugation,
whereas they constitute independent real degrees of freedom in split
signature. Gravity is described in terms of the Riemann curvature tensor,
whose spinorial translation has the decomposition [92, 90, 91]

$\begin{gathered}R_{ABCD\dot{A}\dot{B}\dot{C}\dot{D}}=\Psi_{ABCD}\epsilon_{\dot{A}\dot{B}}\epsilon_{\dot{C}\dot{D}}+\bar{\Psi}_{\dot{A}\dot{B}\dot{C}\dot{D}}\epsilon_{AB}\epsilon_{CD}+\Phi_{AB\dot{C}\dot{D}}\epsilon_{\dot{A}\dot{B}}\epsilon_{CD}+\bar{\Phi}_{\dot{A}\dot{B}CD}\epsilon_{AB}\epsilon_{\dot{C}\dot{D}}\\\
+2\Lambda\left(\epsilon_{AC}\epsilon_{BD}\epsilon_{\dot{A}\dot{C}}\epsilon_{\dot{B}\dot{D}}-\epsilon_{AD}\epsilon_{BC}\epsilon_{\dot{A}\dot{D}}\epsilon_{\dot{B}\dot{C}}\right).\end{gathered}$
(12)

For vacuum solutions, all but the first two terms on the right-hand side are
absent, and the Riemann tensor itself reduces to the Weyl tensor
$C_{\alpha\beta\gamma\delta}$, with spinorial translation

$\begin{gathered}C_{ABCD\dot{A}\dot{B}\dot{C}\dot{D}}=\Psi_{ABCD}\epsilon_{\dot{A}\dot{B}}\epsilon_{\dot{C}\dot{D}}+\bar{\Psi}_{\dot{A}\dot{B}\dot{C}\dot{D}}\epsilon_{AB}\epsilon_{CD}.\end{gathered}$
(13)

Analogous to the case of electromagnetism, the quantity $\Psi_{ABCD}$
($\bar{\Psi}_{\dot{A}\dot{B}\dot{C}\dot{D}}$) describes the (anti-)self-dual
part of the field respectively, and is known as the (conjugate) Weyl spinor.

The second useful property in the spinorial formalism is that fully symmetric
spinors can themselves be decomposed in terms of single-index principal
spinors. For example, a given Weyl spinor may be decomposed as

$\Psi_{ABCD}=\alpha_{(A}\beta_{B}\gamma_{C}\delta_{D)}.$ (14)

Some or all of the principal spinors may be mutually proportional, and the
different patterns of degeneracy are known as Petrov types. The Weyl double
copy can now be phrased as follows. For certain vacuum solutions of
electromagnetism and gravity, the Weyl spinor of the latter can be expressed
as

$\Psi_{ABCD}=\frac{\Phi_{(AB}\Phi_{CD)}}{\phi},$ (15)

where the brackets denote symmetrisation over indices, $\phi_{AB}$ is the
relevant electromagnetic spinor, and $\phi$ is a solution of a linearised
scalar theory satisfying

$\partial^{2}\phi=0.$ (16)

The corresponding relationship for the conjugate Weyl spinor can be obtained
by replacing all quantities in eq. (15) with their conjugates. Equation (15)
was argued to hold for general vacuum Petrov type D solutions in ref. [13],
and extended to type N solutions in ref. [16]. Already in ref. [13], the
possibility of mixed Weyl double copy formulae was given, namely that one may
take a pair of different electromagnetic spinors and combine them according to

$\Psi_{ABCD}=\frac{\Phi^{(1)}_{(AB}\Phi^{(2)}_{CD)}}{\phi},$ (17)

also yielding a valid Weyl spinor. In refs. [17, 19], a derivation of the Weyl
double copy was given using ideas from twistor theory, which also showed that
the formalism could be generalised to other Petrov types, albeit at linearised
level in the vacuum field equations. Reference [75], inspired by ref. [79],
clarified matters further by showing that the momentum-space amplitudes double
copy amounts to the same thing as the twistor and Weyl approaches.
Furthermore, these three double copies can be systematically related to each
other by well-defined integral transforms. We will see these arguments in
detail in section 3, and indeed generalise them. For now, we note that a
similar scheme has been used to derive analogues of the Weyl double copy for
vacuum solutions of topologically massive gauge and gravity theories in three
dimensions [77, 83, 84], and for ${\cal N}=0$ supergravity. In the latter, the
generalised Riemann tensor has a decomposition similar to eq. (12), which
contains mixed-index spinors in addition to the Weyl spinors. We will refer to
$\Phi_{AB\dot{C}\dot{D}}$ as the Ricci spinor in what follows.

The presence of sources in the Weyl double copy was first considered in refs.
[76, 23], where the basic idea is to replace eq. (15) with a series of terms:

$\Psi_{ABCD}=\sum_{n=1}^{m}\frac{1}{\phi^{(n)}}\Phi^{(n)}_{(AB}\Phi^{(n)}_{CD)},$
(18)

and similarly for the conjugate Weyl spinor. The right-hand side contains a
tower of electromagnetic spinors $\Phi^{(n)}_{AB}$ and scalar fields
$\phi^{(n)}$, such that the $n=1$ term corresponds to the traditional vacuum
Weyl double copy. Each $n>1$ electromagnetic spinor corresponds to a field
strength $F_{\mu\nu}^{(n)}$ satisfying a non-vacuum Maxwell equation

$\partial^{\mu}F_{\mu\nu}^{(n)}=j^{(n)}_{\mu}.$ (19)

Likewise, each scalar field $\phi^{(n)}$ satisfies a non-vacuum equation

$\partial^{2}\phi^{(n)}=\rho_{S}^{(n)}$ (20)

for some charge density $\rho_{S}^{(n)}$. The physical interpretation of eq.
(18) is then that one has decomposed a given gravity solution into $n$
distinct terms, each of which has a well-defined single and zeroth copy. One
particular example in refs. [76, 23] consists of expanding a given gravity
solution in inverse powers of the radial coordinate $r$. Each power then
corresponds to a distinct term in eq. (18). Another example concerns Einstein-
Maxwell theory, namely General Relativity coupled to electromagnetism. For
certain solutions, one may truncate the sum in eq. (18) at $n=2$:

$\Psi_{ABCD}=\frac{1}{\phi^{(1)}}\Phi^{(1)}_{(AB}\Phi^{(1)}_{CD)}+\frac{1}{\phi^{(2)}}\Phi^{(2)}_{(AB}\Phi^{(2)}_{CD)}.$
(21)

Consistent with the above remarks, the gauge field $A_{\mu}^{(1)}$
corresponding to $\phi^{(1)}_{AB}$ satisfies a vacuum equation, as does the
scalar field $\phi^{(1)}$. However, the fields $A_{\mu}^{(2)}$ and
$\phi^{(2)}$ will satisfy non-vacuum equations, where the corresponding charge
density and currents must be physically analogous to the source in the gravity
theory. An explicit case is that of the Kerr-Newman black hole, for which one
finds (up to an overall constant factor in each case) [23]

$\displaystyle\phi^{(1)}=\frac{1}{r+ia\cos\theta},\quad\Phi^{(1)}_{AB}=\frac{1}{(r+ia\cos\theta)^{2}}o_{(A}\imath_{B)}$
(22)

and

$\displaystyle\phi^{(2)}=\frac{1}{(r+ia\cos\theta)(r-ia\cos\theta)},\quad\Phi^{(2)}_{AB}=\frac{1}{(r+ia\cos\theta)^{2}(r-ia\cos\theta)}o_{(A}\imath_{B)}.$
(23)

Here $r$ and $\theta$ are radial and polar angular coordinates, and $a$ a
constant parameter related to the angular momentum of the black hole in
gravity.

The corresponding scalar, electromagnetic and gravitational charge / energy
densities are then found to be

$\rho^{(2)}_{S}\propto\rho_{e}^{(2)}\propto\rho_{\rm
grav.}^{(2)}\propto\frac{(r^{2}+a^{2})+a^{2}\sin^{2}\theta}{\rho^{6}},\quad\rho^{2}=r^{2}+a^{2}\cos^{2}\theta.$
(24)

For a non-vacuum solution, the Riemann tensor will have additional components
in addition to the Weyl tensor. In terms of the spinorial decomposition of eq.
(12), this means that the mixed-index spinors will be nonzero. Reference [23]
then conjectured the following double copy formula for the Kerr-Newman
solution:

$\Phi_{AB\dot{C}\dot{D}}=\frac{1}{\phi^{(2)}}\Phi^{(2)}_{AB}\bar{\Phi}^{(2)}_{\dot{C}\dot{D}}.$
(25)

An alternative formula was also given in ref. [23], that expresses the Ricci
spinor as a product of the vacuum-like electromagnetic spinors:

$\Phi_{AB\dot{C}\dot{D}}\propto\Phi^{(1)}_{AB}\bar{\Phi}^{(1)}_{\dot{C}\dot{D}}.$
(26)

This form was argued to be more generally applicable, including to spacetimes
that are non-asymptotically flat. Given the previous studies that have derived
(generalised) Weyl double copies from first principles [75, 77, 78], we may
now ask whether similar methods may be used in deriving sourced Weyl double
copy formulae directly. We commence this investigation in the following
section.

##  3 A Weyl double copy for the Reissner-Nordstrom black hole

According to eq. (18), a sourced Weyl double copy states that a gravitational
Weyl spinor can be written as a sum of products of electromagnetic spinors,
where each term has a certain scalar field divided out. In order to probe this
relationship, we may take a single well-defined term in such a sum, and then
ascertain whether or not we can reproduce it using similar methods to those
used for other Weyl(-like) double copies [75, 77]. That is, we may pick a set
of scalar, electromagnetic and graviton fields whose sources are physically
related as follows:

$\rho_{S}=\rho(x),\quad j^{\mu}(x)=\rho(x)u^{\mu},\quad
T^{\mu\nu}(x)=\rho(x)u^{\mu}u^{\nu},$ (27)

where we will assume static and spherically symmetric sources, such that
$u^{\mu}=\delta^{\mu}_{0}$ is a vector pointing in the time direction. In the
scalar and gauge theories, $\rho(x)$ represents the appropriate charge
density, whereas it will be an energy density in gravity. We have furthermore
ignored dimensionful coupling constants in our definitions, for brevity.
Reference [75] provided a systematic derivation of the Weyl double copy using
the following three-step procedure:

  1. (i)

One may express spinorial solutions of (linearised) classical equations for
scalar, gauge and gravity theory as inverse Fourier transforms of momentum-
space solutions.

  2. (ii)

One may transform the momentum integral to certain spinor variables, and carry
out some of the integrals to yield an intermediate twistor-space
representation of each classical solution. This step reproduces the twistor
double copy first explored in refs. [17, 19].

  3. (iii)

The remaining integral can be carried out to yield position-space
representations of each classical solution, with their principal spinors
clearly identified. These solutions are then found to obey the Weyl double
copy.

Here, we will apply a similar scheme to sourced solutions, for which we must
begin by expressing classical solutions in terms of momentum space integrals.
Due to the static nature of each source in eq. (27), their Fourier transforms
may be written as

$\tilde{\rho}_{S}(k)=\delta(u\cdot k){\cal
J}(\mbox{\boldmath$k$}),\quad\tilde{j}^{\mu}(k)=\delta(u\cdot k){\cal
J}(\mbox{\boldmath$k$})u^{\mu},\quad\tilde{T}^{\mu\nu}(k)=\delta(u\cdot
k){\cal J}(\mbox{\boldmath$k$})u^{\mu}u^{\nu},$ (28)

where we have introduced the 4-momentum $k^{\mu}=(k^{0},\mbox{\boldmath$k$})$,
and noted that $k^{0}=u\cdot k$. The classical solution for the scalar field
given the source in eq. (27) can then be written as

$\phi(x)=\int\frac{d^{4}k}{(2\pi)^{4}}\delta(u\cdot k)\frac{{\cal
J}(\mbox{\boldmath$k$})}{k^{2}}e^{-ik\cdot x}.$ (29)

Likewise, the gauge field (in Lorenz gauge $\partial\cdot A=0$) has the form

$A^{\mu}(x)=\int\frac{d^{4}k}{(2\pi)^{4}}u^{\mu}\delta(u\cdot k)\frac{{\cal
J}(\mbox{\boldmath$k$})}{k^{2}}e^{-ik\cdot x},$ (30)

leading to the gauge-invariant field-strength tensor

$F^{\mu\nu}(x)=\int\frac{d^{4}k}{(2\pi)^{4}}k^{[\mu}u^{\nu]}\delta(u\cdot
k)\frac{{\cal J}(\mbox{\boldmath$k$})}{k^{2}}e^{-ik\cdot x}.$ (31)

Finally, we must consider the corresponding gravity solution, which we may
assume to be a solution of the linearised Einstein equations. The linearised
Riemann tensor can be expressed in terms of the graviton as

$\displaystyle
R^{\mu\nu\rho\sigma}(x)=\frac{\kappa}{2}\left(\partial^{\sigma}\partial^{[\mu}h^{\nu]\rho}+\partial^{\rho}\partial^{[\nu}h^{\mu]\sigma}\right),$
(32)

where the graviton itself is given by

$\displaystyle
h^{\mu\nu}(x)-\frac{1}{2}\eta^{\mu\nu}h(x)=\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\tilde{T}^{\mu\nu}(k)}{k^{2}}e^{-ik\cdot
x}.$ (33)

Taking the trace and inverting, one finds

$\displaystyle
h^{\mu\nu}(x)=\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\tilde{T}^{\mu\nu}(k)-\frac{1}{2}\eta^{\mu\nu}\tilde{T}(k)}{k^{2}}e^{-ik\cdot
x},$ (34)

and using this result together with the source of eq. (28) yields

$R^{\mu\nu\rho\sigma}(x)=\kappa\int\frac{d^{4}k}{(2\pi)^{4}}\left[U^{\rho[\mu}k^{\nu]}k^{\sigma}-U^{\sigma[\mu}k^{\nu]}k^{\rho}\right]\frac{{\cal
J}(\mbox{\boldmath$k$})}{k^{2}}e^{-ik\cdot x},$ (35)

where

$\displaystyle U^{\nu\rho}=u^{\nu}u^{\rho}-\frac{1}{2}\eta^{\nu\rho}.$ (36)

In the above results, we can recognise that the electromagnetic and
gravitational expressions can be rewritten in terms of derivative operators
acting on the scalar field $\phi$. In particular, comparison of eqs. (31) and
eq. (29) shows that the field strength tensor associated with a given scalar
solution is given by

$\displaystyle F_{\mu\nu}^{(n)}=2u_{[\mu}\partial_{\nu]}\phi^{(n)}.$ (37)

Furthermore, the graviton and Riemann curvature tensor can be written as

$\displaystyle
h^{(n)}_{\mu\nu}=\left(u_{\mu}u_{\nu}-\frac{1}{2}\eta_{\mu\nu}\right)\phi^{(n)}(r).$
(38)

and

$\displaystyle R^{(n)}_{\mu\nu\rho\sigma}(x)$
$\displaystyle=-2\left[U_{\rho[\mu}\partial_{\nu]}\partial_{\sigma}-U_{\sigma[\mu}\partial_{\nu]}\partial_{\rho}\right]\phi^{(n)}(r).$
(39)

Having seen how to obtain the tensor forms of various quantities, we next need
to translate them into the spinor formalism, which is the subject of the
following section.

###  3.1 Scalar field

Considering the scalar momentum integral of eq. (29), we can introduce a
lightcone decomposition

$k^{\mu}=\omega\ell^{\mu}+\xi q^{\mu},$ (40)

where $\ell^{2}=q^{2}=0$; the parameters $\omega$ and $\xi$ have dimensions of
energy; and

$q^{\mu}=\frac{1}{2}(0,0,-1,1)$ (41)

is a fixed 4-vector. From eq. (10), the spinorial translations of $l^{\mu}$
and $q^{\mu}$ will each decompose into an outer product of two spinors, so
that one may write

$k_{A\dot{A}}=\omega\lambda_{A}\tilde{\lambda}_{\dot{A}}+\xi
q_{A}\tilde{q}_{\dot{A}}.$ (42)

The delta function in eq. (29) imposes the condition

$u\cdot k=0\quad\Rightarrow\quad u\cdot l=0,$ (43)

where the second condition follows from eq. (40), and the fact that our choice
of $q^{\mu}$ implies $u\cdot q=0$. In the spinor language, this amounts to

$u^{A\dot{A}}\lambda_{A}\tilde{\lambda}_{\dot{A}}=0,$ (44)

which in turn implies

$\lambda_{A}\propto{u_{A}}^{\dot{A}}\tilde{\lambda}_{\dot{A}}.$ (45)

We can turn this into an equality by parametrising the spinors appearing in
eq. (40) as follows:

$\lambda_{A}=\left(\begin{array}[]{c}\sqrt{1/z}\\\
\sqrt{z}\end{array}\right),\quad\tilde{\lambda}_{\dot{A}}=\left(\begin{array}[]{c}\sqrt{-1/\tilde{z}}\\\
-\sqrt{-\tilde{z}}\end{array}\right).$ (46)

To see this, note that an explicit calculation gives

$\delta(u\cdot
k)=\frac{2}{\omega}\delta\left(\frac{1+z\tilde{z}}{\sqrt{-z\tilde{z}}}\right),$
(47)

and hence

$z=-\frac{1}{\tilde{z}}.$ (48)

From eq. (46), one may also calculate

${u_{A}}^{\dot{A}}\tilde{\lambda}_{\dot{A}}=\left(\begin{array}[]{c}\sqrt{-\tilde{z}}\\\
1/\sqrt{-\tilde{z}}\end{array}\right),$ (49)

which indeed yields $\lambda_{A}$ upon using eq. (48). Returning to eq. (29),
we can now transform the momentum integration to the variables
$(\omega,\xi,z,\tilde{z})$. The Jacobian is found to be

$J=\frac{\omega^{2}\sqrt{-\tilde{z}}}{4\tilde{z}^{2}\sqrt{z}},$ (50)

and one also finds

$k^{2}=-|\mbox{\boldmath$k$}|^{2}=\frac{\xi\omega}{\tilde{z}},$ (51)

such that eq. (29) becomes

$\phi(x)=\int\frac{d\omega d\xi
dzd\tilde{z}}{2(2\pi)^{4}}\frac{\omega\sqrt{-\tilde{z}}}{\tilde{z}\sqrt{z}}\delta\left(\frac{1+z\tilde{z}}{\sqrt{-z\tilde{z}}}\right)e^{-i\xi
q\cdot
x}\exp\left[-\frac{i\omega}{2}\tilde{\lambda}_{\dot{A}}\tilde{\lambda}_{\dot{B}}x^{A\dot{A}}{u_{A}}^{\dot{B}}\right]{\cal
J}(\mbox{\boldmath$k$}).$ (52)

A similar integral occured in deriving the original Weyl double copy in ref.
[75], whose starting point was to note that certain vacuum classical solutions
could be expressed as inverse Fourier transforms of on-shell momentum-space
scattering amplitudes. In that case, the integral over $\xi$ becomes trivial,
in that $\xi$ is fixed to be zero, corresponding to the on-shellness of the
emitted radiation. This can also be seen from eq. (40): if $\xi=0$, then the
second term vanishes, and the remaining term is null. For general non-zero
sources, as represented by ${\cal J}(\mbox{\boldmath$k$})$, the $\xi$ integral
instead becomes non-trivial, corresponding physically to the fact that one
must explicitly account for off-shell modes of the emitted radiation. This is
not a problem in practice, and to make progress we will study a particular
choice for ${\cal J}(\mbox{\boldmath$k$})$, corresponding to arguably the
simplest solution of Einstein-Maxwell theory, namely the Reissner-Nordstrom
black hole. The scalar field associated with the non-vacuum term is given in
the $a\rightarrow 0$ limit of eq. (23), as

$\phi^{(2)}\propto\frac{1}{r^{2}},$ (53)

One then finds

$\partial^{2}\phi^{(2)}\propto\frac{1}{r^{4}}\quad\Rightarrow\quad\rho_{S}(x)\propto\frac{1}{r^{4}}.$
(54)

In momentum space, this implies that we consider a source term

${\cal J}(\mbox{\boldmath$k$})=2\pi^{2}|\mbox{\boldmath$k$}|.$ (55)

Substituting this into eq. (52) and using eq. (51), one may straightforwardly
evaluate the $\xi$, $\omega$ and $z$ integrals to obtain

$\phi^{(2)}(x)=\frac{1}{\sqrt{32}\pi}\frac{1}{\sqrt{q\cdot x}}\int
d\tilde{z}\frac{1}{\tilde{z}^{3/2}}\frac{1}{\left(\tilde{\lambda}_{\dot{A}}\tilde{\lambda}_{\dot{B}}x^{A\dot{A}}{u_{A}}^{\dot{B}}\right)^{3/2}}.$
(56)

It is now convenient to define the rescaled spinor

$\tilde{\chi}_{\dot{A}}=\sqrt{-\tilde{z}}\tilde{\lambda}_{\dot{A}}=\left(\begin{array}[]{c}1\\\
\tilde{z}\end{array}\right),$ (57)

such that eq. (56) can be written more compactly as

$\phi^{(2)}(x)=\frac{1}{\sqrt{32}\pi}\frac{1}{\sqrt{q\cdot x}}\int
d\tilde{z}\frac{1}{\left(\tilde{\chi}_{\dot{A}}\tilde{\chi}_{\dot{B}}x^{A\dot{A}}{u_{A}}^{\dot{B}}\right)^{3/2}}.$
(58)

Similar integrals were obtained in the derivation of the Weyl double copy of
ref. [75], where they could be interpreted in projective twistor space
$\mathbb{PT}$. Points in $\mathbb{PT}$ consist of a pair of spinors collected
into a single 4-component twistor

$Z^{\alpha}=\left(\omega^{A},\tilde{\chi}_{\dot{A}}\right),$ (59)

where the two spinors satisfy the so-called incidence relation

$\omega^{A}=x^{A\dot{A}}\tilde{\chi}_{\dot{A}}.$ (60)

This relation is invariant under a simultaneous (complex) rescaling of both
sides, which amounts to an overall scaling $Z^{\alpha}\rightarrow\alpha
Z^{\alpha}$ in eq. (59). It is for this reason that twistors obeying the
incidence relation are associated with projective twistor space. With the
parametrisation of eq. (57), the measure in eq. (58) can be written in the
invariant form

$d\tilde{z}=\tilde{\chi}_{\dot{B}}d\tilde{\chi}^{\dot{B}}.$ (61)

Then eq. (58) can be written in terms of twistor variables as

$\phi^{(2)}(x)=\frac{\lambda}{\sqrt{q\cdot
x}}\int\tilde{\chi}_{\dot{B}}d\tilde{\chi}^{\dot{B}}\frac{1}{\left(Q_{\alpha\beta}Z^{\alpha}Z^{\beta}\right)^{3/2}},\quad
Q_{\alpha\beta}=\left(\begin{array}[]{cc}0&{u_{A}}^{\dot{B}}\\\
{u_{A}}^{\dot{B}}&0\end{array}\right).$ (62)

However, unlike the previous cases of refs. [75, 78], the integral of eq. (62)
cannot be consistently interpreted as a contour integral in projective twistor
space, as the combined measure and integrand is not invariant under rescalings
of $Z^{\alpha}$. This is not unexpected, given that the conventional Penrose
transform is for vacuum solutions only. Furthermore, we will not need to
formally pursue a twistorial interpretation of eq. (58) in what follows.

Another complication of eq. (58) compared to the related integrals in refs.
[75, 78] is that its singularity structure involves branch cuts rather than
simple poles, and it is not then immediately clear how to choose an
appropriate contour. However, we can instead carry out the inverse Fourier
transform in eq. (29) exactly for our chosen source to obtain

$\displaystyle\phi^{(2)}(x)$
$\displaystyle=\int\frac{d^{4}k}{(2\pi)^{4}}\frac{2\pi^{2}\delta(u\cdot
k)|\mbox{\boldmath$k$}|}{k^{2}}e^{-ik\cdot x}$ (63)
$\displaystyle=\frac{1}{(x^{2}-(u\cdot x)^{2})}=\frac{1}{r^{2}}.$ (64)

Comparison with eq. (58) then yields

$\int
d\tilde{z}\frac{1}{\left(\tilde{\chi}_{\dot{A}}\tilde{\chi}_{\dot{B}}x^{A\dot{A}}{u_{A}}^{\dot{B}}\right)^{3/2}}=\frac{\sqrt{32}\pi\sqrt{q\cdot
x}}{r^{2}}.$ (65)

It will turn out that all spinor integrals needed for the electromagnetic and
gravitational spinors in subsequent sections can be expressed in terms of eq.
(65). In anticipation of this, let us note that the spinor combination
appearing in the denominator of eq. (65) is given by

$\tilde{\chi}_{\dot{A}}\tilde{\chi}_{\dot{B}}x^{A\dot{A}}{u_{A}}^{\dot{B}}=\tilde{z}^{2}(y+z)+2x\tilde{z}+z-y.$
(66)

The integrals we will need to evaluate can then all be written in terms of the
family

$I_{m,n}=(-1)^{m+1}\int
d\tilde{z}\frac{\tilde{z}^{n}}{[\tilde{z}^{2}(y+z)+2x\tilde{z}+z-y]^{3/2+m}},$
(67)

where e.g. the scalar field above is given by

$\phi^{(2)}(x)=\frac{1}{\sqrt{32}\pi}\frac{1}{\sqrt{q\cdot x}}I_{0,0}.$ (68)

with

$I_{0,0}=\frac{\sqrt{32}\pi\sqrt{q\cdot
x}}{r^{2}}=\frac{4\pi\sqrt{y+z}}{x^{2}+y^{2}-z^{2}}.$ (69)

By differentiating eq. (67) with respect to the spatial coordinates $(x,y,z)$,
one can derive the recurrence relations

$\displaystyle\frac{\partial}{\partial x}I_{m,n}=-(3+2m)I_{m+1,n+1},$
$\displaystyle\left(\frac{\partial}{\partial y}+\frac{\partial}{\partial
z}\right)I_{m,n}=-(3+2m)I_{m+1,n+2},$
$\displaystyle\left(\frac{\partial}{\partial y}-\frac{\partial}{\partial
z}\right)I_{m,n}=(3+2m)I_{m+1,n},$ (70)

which we will use repeatedly in what follows.

###  3.2 Electromagnetic field

Given the scalar field of the previous section, we can consider an
electromagnetic field strength tensor given by eq. (31), where we again take
the source appearing on the right-hand side to be given by eq. (55). The
corresponding electromagnetic spinor can be obtained by projecting out the
relevant components, following contraction of the tensorial field strength
with the Infeld-van-der-Waerden symbols:

$\Phi^{(2)}_{AB}(x)=\epsilon^{\dot{B}\dot{A}}\sigma^{\mu}_{A\dot{A}}\sigma^{\nu}_{B\dot{B}}F_{\mu\nu}(x).$
(71)

Carrying out the contractions and performing the change of variables outlined
in the previous section yields the following result:

$\Phi^{(2)}_{\dot{A}\dot{B}}=-\frac{3}{4\pi}\frac{1}{(x^{0\dot{1}})^{1/2}}\int
d\tilde{z}\frac{\tilde{\chi}_{\dot{A}}\tilde{\chi}_{\dot{B}}}{(\tilde{\chi}_{\dot{A}}\tilde{\chi}_{\dot{B}}x^{A\dot{A}}{u_{A}}^{\dot{B}})^{5/2}}+\frac{1}{4\pi}\frac{1}{(x^{0\dot{1}})^{3/2}}\int
d\tilde{z}\frac{{u_{\dot{A}}}^{A}q_{A}\tilde{q}_{\dot{B}}+{u_{\dot{B}}}^{B}q_{B}\tilde{q}_{\dot{A}}}{(\tilde{\chi}_{\dot{A}}\tilde{\chi}_{\dot{B}}x^{A\dot{A}}{u_{A}}^{\dot{B}})^{3/2}}.$
(72)

In contrast to previously studied cases, this has two distinct terms,
involving different powers of the spinor combination appearing in the
denominator. Indeed, the second term can be directly traced to the fact that
off-shell modes of the gauge field are included, given that the vanishing of
the $q^{\mu}$ term in eq. (40) is associated with an on-shell gauge field, as
remarked above. In order to evaluate the integrals in eq. (72), one may
substitute in the explicit forms of each spinor, after which one finds that
the matrix form of $\Phi^{(2)}_{AB}$ is

$\Phi^{(2)}_{\dot{A}\dot{B}}=\frac{1}{4\pi}\left[\frac{1}{(q\cdot
x)^{3/2}}\left(\begin{array}[]{cc}0&0\\\
0&-I_{0,0}\end{array}\right)-\frac{6}{\sqrt{q\cdot
x}}\left(\begin{array}[]{cc}I_{1,0}&I_{1,1}\\\
I_{1,1}&I_{1,2}\end{array}\right)\right],$ (73)

in terms of the family of integrals introduced in eq. (67). The recurrence
relations of eq. (70) yield

$\displaystyle I_{1,1}$ $\displaystyle=-\frac{1}{3}\frac{\partial}{\partial
x}I_{0,0},$ $\displaystyle I_{1,0}$
$\displaystyle=\frac{1}{3}\left(\frac{\partial}{\partial
y}-\frac{\partial}{\partial z}\right)I_{0,0},$ $\displaystyle I_{1,2}$
$\displaystyle=-\frac{1}{3}\left(\frac{\partial}{\partial
y}+\frac{\partial}{\partial z}\right)I_{0,0},$ (74)

such that substituting eq. (69) gives

$\displaystyle I_{1,0}$ $\displaystyle=-\frac{32\pi(q\cdot x)^{3/2}}{3r^{4}},$
$\displaystyle I_{1,1}$ $\displaystyle=\frac{16\pi x(q\cdot x)}{3r^{4}},$
$\displaystyle I_{1,2}$ $\displaystyle=\frac{16\pi}{3}(y-z)\frac{\sqrt{q\cdot
x}}{r^{4}}-\frac{1}{6}\frac{I_{00}}{q\cdot x}.$ (75)

Putting things together, the electromagnetic spinor becomes

$\Phi^{(2)}_{AB}=-\frac{2}{r^{4}}\left(\begin{array}[]{cc}-y-z&x\\\
x&-y+z\end{array}\right),$ (76)

which may also be written in the convenient form

$\Phi^{(2)}_{AB}=2\frac{{u_{(\dot{A}}}^{B}r_{\dot{B})B}}{r^{4}},$ (77)

where

$r_{B\dot{B}}=(x-(u\cdot
x)u)\cdot\sigma_{B\dot{B}}=\left(\begin{array}[]{cc}-x&z-y\\\
-z-y&x\end{array}\right).$ (78)

###  3.3 Gravitational field

In the previous two sections, we have assumed a scalar and electromagnetic
source that mimics the source for the Reissner-Nordstrom black hole. We can
now finally calculate the various gravitational spinors for this solution, and
hence ascertain whether or not there is a non-vacuum generalisation of the
Weyl double copy for this particular solution. First, we recall that the
Riemann curvature for a given source is given by eq. (35). Upon translating to
the spinorial language via

$R_{ABCD\dot{A}\dot{B}\dot{C}\dot{D}}=\sigma^{\mu}_{A\dot{A}}\sigma^{\nu}_{B\dot{B}}\sigma^{\rho}_{C\dot{C}}\sigma^{\lambda}_{D\dot{D}}R_{\mu\nu\rho\lambda},$
(79)

one may find the multi-index spinors appearing in the decomposition of eq.
(12) via the contractions

$\displaystyle\Psi_{ABCD}$
$\displaystyle=\frac{1}{4}R_{(A\dot{X}B}{}^{\dot{X}}{}_{C\dot{Y}D)}{}^{\dot{Y}}$
$\displaystyle\Phi_{AB\dot{C}\dot{D}}$
$\displaystyle=\frac{1}{4}R_{(A\dot{X}B)}{}^{\dot{X}}{}_{Y(\dot{C}\dot{D})}.$
(80)

For the scalar field $\phi^{(2)}$, carrying out all necessary contractions and
changing to spinor variables as before yields the result

$\displaystyle\Psi^{(2)}_{\dot{A}\dot{B}\dot{C}\dot{D}}$
$\displaystyle=\frac{3\kappa}{2}\int
d\tilde{z}\left[\frac{5\tilde{\chi}_{\dot{A}}\tilde{\chi}_{\dot{B}}\tilde{\chi}_{\dot{C}}\tilde{\chi}_{\dot{D}}}{(\tilde{\chi}_{\dot{A}}\tilde{\chi}_{\dot{B}}x^{A\dot{A}}{u_{A}}^{\dot{B}})^{7/2}(q\cdot
x)^{1/2}}+\frac{2\tilde{\chi}_{(\dot{A}}\tilde{\chi}_{\dot{B}}(u\cdot
q)_{\dot{C}}\tilde{q}_{\dot{D})}}{(\tilde{\chi}_{\dot{A}}\tilde{\chi}_{\dot{B}}x^{A\dot{A}}{u_{A}}^{\dot{B}})^{5/2}(q\cdot
x)^{3/2}}\right.$ (81) $\displaystyle\left.\qquad\qquad\qquad+\frac{(u\cdot
q)_{(\dot{A}}\tilde{q}_{\dot{B}}(u\cdot
q)_{\dot{C}}\tilde{q}_{\dot{D})}}{(\tilde{\chi}_{\dot{A}}\tilde{\chi}_{\dot{B}}x^{A\dot{A}}{u_{A}}^{\dot{B}})^{3/2}(q\cdot
x)^{5/2}}\right].$

Comparison with the integral form of the electromagnetic spinor in eq. (72)
shows that it is not at all obvious whether the Weyl spinor of eq. (81) is
related by any form of double copy. However, the latter (if it applies) must
be true for the integrated Weyl spinor, and we may carry out the integrals by
expressing them in terms of the master integrals of eq. (67), and using the
recurrence relations of eq. (70). Using these, one finds that the Weyl spinor
requires integrals of the form $I_{2,n}$, which we can related to the lower-
spin integrals via

$\displaystyle I_{2,n}$
$\displaystyle=\frac{1}{5}\left(\frac{\partial}{\partial
Y}-\frac{\partial}{\partial Z}\right)I_{1,n}$ (82)
$\displaystyle=\frac{1}{15}\left(\frac{\partial}{\partial
Y}-\frac{\partial}{\partial Z}\right)^{2}I_{0,n}.$

We find the following required integrals for the Weyl spinor:

$\displaystyle I_{2,0}$ $\displaystyle=\frac{256\pi(q\cdot x)^{5/2}}{15r^{6}}$
(83) $\displaystyle I_{2,1}$ $\displaystyle=-\frac{128\pi x(q\cdot
x)^{3/2}}{15r^{6}}$ $\displaystyle I_{2,2}$
$\displaystyle=\frac{16\pi\sqrt{q\cdot x}}{5r^{4}}-\frac{128\pi(y-z)(q\cdot
x)^{3/2}}{15r^{6}}$ $\displaystyle I_{2,3}$ $\displaystyle=\frac{64\pi
x(y-z)\sqrt{q\cdot x}}{15r^{6}}-\frac{8\pi x}{15r^{4}\sqrt{q\cdot x}}$
$\displaystyle I_{2,4}$ $\displaystyle=\frac{64\pi(y-z)^{2}\sqrt{q\cdot
x}}{15r^{6}}-\frac{16\pi(y-z)}{15r^{4}\sqrt{q\cdot
x}}-\frac{2\pi}{15r^{2}(q\cdot x)^{3/2}}.$

We can use these to solve the integrals that come up in each component, and
find the following result for the Weyl spinor:

$\displaystyle\Psi^{\dot{A}\dot{B}\dot{C}\dot{D}}_{(2)}$
$\displaystyle=\frac{1}{6r^{6}}\Bigg{[}u_{C}^{(\dot{A}}u_{B}^{\dot{D})}r^{\dot{B}B}r^{\dot{C}C}-u_{B}^{(\dot{C}}u_{A}^{\dot{D})}r^{\dot{A}A}r^{\dot{B}B}-u_{C}^{(\dot{B}}u_{A}^{\dot{D})}r^{\dot{A}A}r^{\dot{C}C}$
$\displaystyle-\left(u_{D}^{\dot{C}}\left(u_{A}^{\dot{B}}r^{\dot{A}A}+u_{B}^{\dot{A}}r^{\dot{B}B}\right)+u_{D}^{\dot{B}}\left(u_{A}^{\dot{C}}r^{\dot{A}A}+u_{C}^{\dot{A}}r^{\dot{C}C}\right)+u_{D}^{\dot{A}}\left(u_{B}^{\dot{C}}r^{\dot{B}B}+u_{C}^{\dot{B}}r^{\dot{C}C}\right)\right)r^{\dot{D}D}\Bigg{]},$
(84)

where $r_{\dot{B}B}$ is defined as in eq. (78). Remarkably, the various terms
combine in such a way as to yield the combination

$\Psi^{(n)}_{\dot{A}\dot{B}\dot{C}\dot{D}}=-2\frac{\Phi^{(2)}_{(\dot{A}\dot{B}}\Phi^{(2)}_{\dot{C}\dot{D})}}{\phi^{(2)}},$
(85)

in terms of the electromagnetic spinor of eq. (77). Thus, a Weyl double copy
does indeed hold for the integrated Weyl spinor, even in the presence of a
non-trivial source. We can also consider the mixed index spinor, for which the
integrated result turns out to be

$\displaystyle\Phi^{(2)}_{AB\dot{C}\dot{D}}$
$\displaystyle=-\frac{2}{r^{6}}\left(u_{\dot{A}}^{B}r^{A\dot{A}}+u_{\dot{B}}^{A}r^{B\dot{B}}\right)\left(u_{C}^{\dot{D}}r^{C\dot{C}}+u_{C}^{\dot{C}}r^{C\dot{D}}\right),$
(86)

which reduces to the form

$\Phi^{(2)}_{AB\dot{C}\dot{D}}=-2\frac{\Phi^{(2)}_{AB}\bar{\Phi}^{(2)}_{\dot{C}\dot{D}}}{\phi^{(2)}},$
(87)

in terms of the electromagnetic (conjugate) spinors and scalar field of
sections 3.1 and 3.2. Thus, we also find a double copy formula for the mixed
index spinor, and we may immediately compare this with the conjectured form of
refs. [76, 23], given here in eqs. (25). We indeed find agreement, which
confirms the conjecture for the special case of the momentum-space source
distribution of eq. (55). In order to go beyond this, we can in principle
repeat the arguments of this section for different sources. In the following
section, however, we see that an approach exists, based directly on the
tensorial double copy ideas of refs. [93].

##  4 Sourced Weyl double copy from tensor methods

In the previous section, we have obtained a Weyl double copy formula for the
Reissner-Nordstrom black hole, which has a particular source profile for the
associated scalar field, as shown in eq. (54). Our results (including for the
mixed index Ricci spinor) agree with the more generally applicable formula
conjectured in refs. [76, 23], and thus we now look at generalising our
conclusions to include sources which have an arbitrary inverse power of the
radial coordinate $r$. To do so, we will introduce an alternative argument
that complements the spinorial approach of the previous sections, and recovers
our former results as a special case. Our starting point is a tensorial
formula for the conventional Weyl double copy, obtained in ref. [93]:

$W^{\mu\nu\rho\sigma}=\frac{1}{\phi}\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}F^{\tau\lambda}F^{\eta\omega}.$
(88)

The Weyl tensor appears on the left-hand side, and this is then expressed as a
certain product of electromagnetic field-strength tensors, contracted with a
projector $\mathcal{P}$. The latter appears in the definition of the Weyl
tensor in terms of the Riemann curvature tensor:

$\displaystyle
W^{\mu\nu\rho\sigma}=\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}R^{\tau\lambda\eta\omega},$
(89)

and we thus see that the job of the projector is to remove traces from the
latter. It is given by

$\displaystyle\mathcal{P}_{\tau\lambda\eta\omega}^{\mu\nu\rho\sigma}=\delta_{\tau}^{\mu}\delta_{\lambda}^{\nu}\delta_{\eta}^{\rho}\delta_{\omega}^{\sigma}+\frac{1}{2}g_{\tau\eta}\delta_{\lambda}^{[\mu}g^{\nu][\rho}\delta_{\omega}^{\sigma]}+\frac{1}{6}g_{\tau\eta}g_{\lambda\omega}g^{\mu[\rho}g^{\sigma]\nu},$
(90)

where $g^{\mu\nu}$ is the full metric which, given that we are considering
solutions of the linearised Einstein equation, can be replaced with the
Minkowski metric $\eta^{\mu\nu}$. For the sourced double copy, and based on
eq. (18), it is natural to generalise eq. (88) to

$\displaystyle
W^{\mu\nu\rho\sigma}=\sum_{n=1}^{m}\frac{f(n)}{\phi^{(n)}}\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}F^{(n),\tau\lambda}F^{(n),\eta\omega}.$
(91)

That is, we conjecture that the Weyl tensor can be written as a sum of terms
similar to eq. (88), and where we also allow for some function $f(n)$ that
introduces a potential normalisation constant that can be different for each
term. For each term, we will consider a scalar field of the form

$\phi^{(n)}=\frac{1}{r^{n}},$ (92)

where $r^{2}=x^{2}-(u\cdot x)^{2}$ the timelike vector associated with the
source. For static sources with $u^{\mu}=(1,\mbox{\boldmath$0$})$, $r$ then
reduces to the spatial radial coordinate. Each field in eq. (92) satisfies an
equation of motion

$\displaystyle\partial^{2}\phi^{(n)}=\frac{n(n-1)}{r^{2+n}}=\rho^{(n)}(r),$
(93)

with $n=1$ being the vacuum case consistent with the standard Weyl double
copy. For the case of general $n$, we can use eqs. (37, 38, 39) to verify that
the conjectured tensorial double copy of eq. (91) indeed holds. First, from
eq. (37), the electromagnetic field strength tensor associated with the scalar
field of eq. (92) is

$\displaystyle
F_{\mu\nu}^{(n)}=n\frac{u_{\mu}r_{\nu}-u_{\nu}r_{\mu}}{r^{2+n}},$ (94)

with associated equation of motion

$\displaystyle\partial^{\nu}F_{\mu\nu}^{(n)}=\frac{n(n-1)}{r^{2+n}}u_{\mu}=J_{\mu}^{(n)}=\rho^{(n)}(r)u_{\mu}.$
(95)

Again we see that the vacuum solution is given by $n=1$, which acts as a
cross-check of this expression. Furthermore, the source term has the form
expected from eq. (27). Finally, one may associate a gravity solution with the
scalar field of eq. (92), where the Riemann tensor will have the form of eq.
(39). One may then write this in terms of the projector in eq. (90), which
gives

$\displaystyle
R_{(n)}^{\mu\nu\rho\sigma}=\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\partial^{[\tau}u^{\lambda]}\partial^{[\eta}u^{\omega]}\phi^{(n)}(x)-\left[\left(u^{\mu}u^{[\sigma}\eta^{\rho]\nu}-u^{\nu}u^{[\sigma}\eta^{\rho]\mu}\right)+\frac{1}{3}\left(\eta^{\mu\sigma}\eta^{\nu\rho}-\eta^{\mu\rho}\eta^{\nu\sigma}\right)\right]\partial^{2}\phi^{(n)}(r).$
(96)

The use of this is that in operating with the projector to obtain the Weyl
tensor, as given in eq. (89), the second term on the right-hand side of eq.
(96) vanishes. Thus, the Weyl tensor is simply given by

$\displaystyle W_{(n)}^{\mu\nu\rho\sigma}$
$\displaystyle=\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\partial^{[\tau}u^{\lambda]}\partial^{[\eta}u^{\omega]}\phi^{(n)}(r)$
(97)
$\displaystyle=\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}u^{[\lambda}u^{[\omega}\partial^{\tau]}\partial^{\eta]}\phi^{(n)}(r),$

where the derivatives are understood as acting on all factors to the right. We
can relate the double derivative of the scalar to a combination of single
derivatives:

$\displaystyle\partial_{\mu}\partial_{\nu}\phi^{(n)}=\frac{(2+n)}{n}\frac{\partial_{\mu}\phi^{(n)}\partial_{\nu}\phi^{(n)}}{\phi^{(n)}}-nr^{2n-2}(\phi^{(n)})^{3}(\eta_{\mu\nu}-u_{\mu}u_{\nu}).$
(98)

The second term is projected away when forming the Weyl tensor, and thus we
finally arrive at the result:

$\displaystyle W^{\mu\nu\rho\sigma}$
$\displaystyle=\sum_{n}\frac{4(2+n)}{n}\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\frac{\partial^{[\tau}u^{\lambda]}\phi^{(n)}\partial^{[\eta}u^{\omega]}\phi^{(n)}}{\phi^{(n)}}$
(99)
$\displaystyle=\sum_{n}\frac{(2+n)}{n}\mathcal{P}^{\mu\nu\rho\sigma}_{\tau\lambda\eta\omega}\frac{F^{(n),\tau\lambda}F^{(n),\eta\omega}}{\phi^{(n)}}.$

In other words, we confirm the sourced tensorial Weyl double copy of eq. (91),
where the function entering each term is found to be

$f(n)=\frac{2+n}{n}.$ (100)

Let us now connect this result with the spinorial analysis of the previous
section. Computing the Maxwell spinor for a given $n$ from eq. (94) we find

$\displaystyle\Phi_{AB}^{(n)}=-n\frac{u_{(A}^{~{}~{}\dot{B}}r_{B)\dot{B}}}{r^{2+n}}.$
(101)

Similarly, the Weyl spinor is found to be

$\displaystyle\Psi^{ABCD}_{(n)}$
$\displaystyle=\frac{1}{24}\frac{n(2+n)}{r^{4+n}}\Bigg{[}u_{\dot{C}}^{(A}u_{\dot{B}}^{D)}r^{B\dot{B}}r^{C\dot{C}}-u_{\dot{B}}^{(C}u_{\dot{A}}^{D)}r^{A\dot{A}}r^{B\dot{B}}-u_{\dot{C}}^{(B}u_{\dot{A}}^{D)}r^{A\dot{A}}r^{C\dot{C}}$
(102)
$\displaystyle-\left(u_{\dot{D}}^{C}\left(u_{\dot{A}}^{B}r^{A\dot{A}}+u_{\dot{B}}^{A}r^{B\dot{B}}\right)+u_{\dot{D}}^{B}\left(u_{\dot{A}}^{C}r^{A\dot{A}}+u_{\dot{C}}^{A}r^{C\dot{C}}\right)+u_{\dot{D}}^{A}\left(u_{\dot{B}}^{C}r^{B\dot{B}}+u_{\dot{C}}^{B}r^{C\dot{C}}\right)\right)r^{D\dot{D}}\Bigg{]}$
$\displaystyle=-\frac{2+n}{n}\frac{\varphi_{n}^{(AB}\varphi_{n}^{CD)}}{\phi^{(n)}},$

and for the mixed index case

$\displaystyle\Phi^{AB\dot{C}\dot{D}}_{(n)}$
$\displaystyle=-\frac{1}{4}n(2+n)r^{-4-n}\left(u_{\dot{A}}^{B}r^{A\dot{A}}+u_{\dot{B}}^{A}r^{B\dot{B}}\right)\left(u_{C}^{\dot{D}}r^{C\dot{C}}+u_{C}^{\dot{C}}r^{C\dot{D}}\right)$
(103)
$\displaystyle=-\frac{2+n}{n}\frac{\varphi_{n}^{AB}\bar{\varphi}_{n}^{\dot{C}\dot{D}}}{\phi^{(n)}}.$

We see then that these spinors agree with the ones derived in the previous
section for the case of $n=2$, up to overall constants. However, the results
are much broader than this, in that eqs. (102, 103) provide a generalisation
of the spinorial version of the sourced Weyl double copy for scalar sources
involving arbitrary powers of the inverse radial coordinate. We thus fully
confirm the conjectures presented in ref. [76] for such solutions, but also go
beyond the results in showing that the Ricci spinor has the general double-
copy form

$\Phi^{(n)}_{AB\dot{C}\dot{D}}=\sum_{n=1}^{m}\frac{1}{\phi^{(n)}}\Phi^{(n)}_{AB}\Phi^{(n)}_{\dot{C}\dot{D}},$
(104)

where normalisation constants associated with each term have been absorbed in
the scalar field. This is the mixed-index equivalent of the generalised Weyl
double copy formula of ref. (18).

##  5 Spinning Sources and the Weyl Double Copy

So far, we have considered the Weyl double copy for scalar sources, which are
spherically symmetric. However, our results can be generalised beyond this to
include other cases considered in refs. [76, 23], including the Kerr-Newman
black hole. The latter is a non-spherically symmetric charged black hole
solution, with non-zero intrinsic angular momentum. Its Weyl double copy is
straightforwardly inherited from the results of the previous section, due to
the existence of the well-known _Newman-Janis algorithm_ [94], which allows
spinning gravity solutions to be obtained from their non-spinning counterparts
via a complex translation. A similar procedure can be applied to the scalar
field and electromagnetic solutions, acting directly in the latter case on the
Maxwell spinor $\Phi_{AB}$ and its conjugate. Starting with the
electromagnetic and scalar fields corresponding to the Reissner-Nordstrom
black hole, we may then construct Weyl and Ricci spinors using eqs. (18, 25),
and verify that they reproduce the known results for the Kerr-Newman black
hole.

In more detail, the Newman-Janis prescription tells us to complexify the
radial coordinate in some particular way, and then to take $x^{\mu}\rightarrow
x^{\mu}+ia^{\mu}$, where $a^{\mu}$ is a vector describing the angular momentum
per unit mass. The Newman-Janis algorithm is usually applied at the level of
the metric, in the case of gravity, or the gauge field in the case of
electromagnetism. One prescription in these cases is to complexify the radial
coordinate as [95]

$\displaystyle r^{n}\rightarrow\frac{[\operatorname{Re}(r)]^{n+2}}{|r|^{2}},$
(105)

before performing the shift $x^{\mu}\rightarrow x^{\mu}+ia^{\mu}$. We are
interested in scalar, gauge and gravitational potentials of the form
$1/r^{n}$, complexified as above. In the scalar case, this is simply

$\displaystyle\phi^{(n)}=\frac{[\operatorname{Re}(r)]^{2-n}}{|r|^{2}},$ (106)

while for gauge fields, we have

$\displaystyle
A_{\mu}^{(n)}=u_{\mu}\frac{[\operatorname{Re}(r)]^{2-n}}{|r|^{2}}.$ (107)

We can use this to compute the relevant field-strengths. For $n=1$, we have

$\displaystyle F^{(1)}_{\mu\nu}=4\frac{u_{[\mu}\hat{r}_{\nu]}}{r^{2}},$ (108)

while for $n=2$ it is given by

$\displaystyle F^{(2)}_{\mu\nu}=4\frac{u_{[\mu}\hat{r}_{\nu]}}{r|r|^{2}},$
(109)

where we have expressed the position vector in spherical polar coordinates as

$\displaystyle
r_{\nu}=r\hat{r}_{\nu}=r(0,\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta).$
(110)

From these field-strengths, we can compute the Maxwell spinors, and then apply
the complex translation to find the spinning solutions. For the $n=1$ case, we
find the Maxwell spinor is given by

$\displaystyle\Phi^{(1)}_{AB}=\frac{1}{(\tilde{r}+ia\cos\theta)^{2}}o_{(A}\iota_{B)}.$
(111)

while for $n=2$, we find

$\displaystyle\Phi^{(2)}_{AB}=\frac{1}{(\tilde{r}^{2}+a^{2}\cos^{2}\theta)(\tilde{r}+ia\cos\theta)}o_{(A}\iota_{B)}.$
(112)

where we have chosen $a^{\mu}=(0,0,0,a)$ and $\tilde{r}$ is the radial
coordinate in oblate-spheroidal coordinates, defined by

$\displaystyle
x+iy=\tilde{r}e^{i\theta}\sin\theta,~{}~{}~{}~{}~{}z=\tilde{r}\cos\theta.$
(113)

The radial coordinate $\tilde{r}$ is now implicitly a solution to the equation

$\displaystyle\frac{x^{2}+y^{2}}{\tilde{r}^{2}+a^{2}}+\frac{z^{2}}{\tilde{r}^{2}}=1.$
(114)

A particular solution to the equation (114) is given by

$\displaystyle\tilde{r}=\frac{1}{\sqrt{2}}\sqrt{r^{2}-a^{2}+\sqrt{4a^{2}\cos^{2}\theta+(r^{2}-a^{2})^{2}}},$
(115)

which has been chosen such that $\lim_{a\rightarrow 0}\tilde{r}=r$. We note
that in these coordinates, the basis spinors are given by

$\displaystyle
o^{A}=\binom{\cos(\theta/2)}{e^{i\phi}\sin(\theta/2)},\quad\imath^{A}=\binom{-\sin\left(\frac{\theta}{2}\right)}{e^{i\phi}\cos(\theta/2)},$
(116)

which satisfy $o_{A}\iota^{A}=1$ and the relations

$\displaystyle u^{A\dot{A}}$
$\displaystyle=o^{A}\bar{o}^{\dot{A}}+\iota^{A}\bar{\iota}^{\dot{A}}$ (117)
$\displaystyle\hat{r}^{A\dot{A}}$
$\displaystyle=o^{A}\bar{o}^{\dot{A}}-\iota^{A}\bar{\iota}^{\dot{A}}.$

We now have all of the ingredients we need to compute the gravitational
spinors via the double copy, defined as before. We find that the Weyl spinor
for the $n=1$ case is given by

$\displaystyle\Psi^{ABCD}_{(1)}$
$\displaystyle=-3\frac{\Phi^{(AB}_{(1)}\Phi^{CD)}_{(1)}}{\phi_{(1)}}$ (118)
$\displaystyle=-3\frac{GM}{(\tilde{r}+ia\cos\theta)^{3}}o^{(A}\iota^{B}o^{C}\iota^{D)},$

while for $n=2$ it is

$\displaystyle\Psi^{ABCD}_{(2)}$
$\displaystyle=-2\frac{\Phi^{(AB}_{(2)}\Phi^{CD)}_{(2)}}{\phi_{(2)}}$ (119)
$\displaystyle=-2\frac{1}{(\tilde{r}^{2}+a^{2}\cos^{2}\theta)(\tilde{r}+ia\cos\theta)^{2}}o^{(A}\iota^{B}o^{C}\iota^{D)}.$

The sum of eqs. (118, 119) indeed reproduces the known Weyl spinor for the
Kerr-Newman solution, whilst also confirming the form of eq. (25), as
conjectured in ref. [23]. We can also construct the Ricci spinor via eq. (25),
finding

$\Phi_{AB\dot{C}\dot{D}}=\frac{1}{(r^{2}+a^{2}\cos^{2}\theta)}o_{(A}\iota_{B)}\bar{o}_{(\dot{C}}\bar{\iota}_{\dot{D})},$
(120)

which again agrees with the known result.

As discussed in section 2, an alternative double-copy form for the Ricci
spinor is presented in ref. [23], namely that of eq. (26), and which is argued
to be more generally applicable than eq. (25), in that it applies to solutions
that are not asymptotically flat. In fact, it is possible to give a general
argument for this alternative formula, as follows. First, one may consider the
tensorial equations of motion for Einstein-Maxwell theory:

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$
$\displaystyle=F_{\mu\alpha}F_{\nu}^{~{}\alpha}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}$
(121) $\displaystyle=\frac{1}{2}F_{\mu\alpha}^{+}F_{\nu}^{~{}\alpha-},$

where in the second line we have written the right-hand side explicitly in
terms of (anti-)self-dual field strengths. Taking the trace of eq. (121)
yields $R=0$, due to the tracelessness of the electromagnetic field strength
tensor. This in turn simplifies the calculation of the Ricci spinor, as

$\displaystyle
R_{\mu\nu}\sigma^{\mu}_{A\dot{A}}\sigma^{\nu}_{B\dot{B}}=\frac{1}{2}F_{\mu\alpha}^{+}F_{\nu}^{~{}\alpha-}\sigma^{\mu}_{A\dot{A}}\sigma^{\nu}_{B\dot{B}}.$
(122)

The spinorial translations of the (anti-)self-dual field strength tensors are
simply given by

$\frac{1}{2}F^{+}_{A\dot{A}B\dot{B}}=\Phi_{AB}\epsilon_{\dot{A}\dot{B}},\quad\frac{1}{2}F^{-}_{A\dot{A}B\dot{B}}=\bar{\Phi}_{\dot{A}\dot{B}}\epsilon_{AB}.$
(123)

One then finds that the Ricci spinor is given by

$\displaystyle\Phi_{AB\dot{A}\dot{B}}=2\Phi_{AB}\bar{\Phi}_{\dot{A}\dot{B}}.$
(124)

Note that this is not yet a double-copy formula: eq. (124) relates the Ricci
spinor of an Einstein-Maxwell solution to the electromagnetic spinors of the
same solution in the same theory. This is different to the double copy, which
relates quantities in a gravitational theory (possibly with extra fields) to
quantities in a different “single-copy” theory. However, it is often the case
that the purely electromagnetic part of an Einstein-Maxwell solution, obtained
by setting the gravitational constant $G_{N}\rightarrow 0$, corresponds with
the single-copy gauge field. This is certainly true for the Kerr-Newman black
hole [4]. For such solutions, we may therefore also interpret eq. (124) as a
genuine double-copy formula, relating a gravitational quantity on the left, to
single-copy quantities on the right.

It may seem that eq. (124) contradicts the double copy formula of eq. (104).
However, the fact that $\Phi_{AB}$ above is associated with the vacuum part of
the single-copy (electromagnetic) solution means that it corresponds to the
$n=1$ term in eq. (18). Equation (104) instead involves the $n=2$ spinors, and
it is indeed the case that

$\displaystyle
R_{AB\dot{A}\dot{B}}=2\Phi^{(1)}_{AB}\bar{\Phi}^{(1)}_{\dot{A}\dot{B}}=2\frac{\Phi^{(2)}_{AB}\bar{\Phi}^{(2)}_{\dot{A}\dot{B}}}{\phi^{(2)}}.$
(125)

We thus obtain the two double copy prescriptions for the Ricci spinor (where
applicable) of ref. [23].

##  6 Conclusion

In this paper, we have addressed the systematic derivation of Weyl double copy
formulae for sourced (non-vacuum) classical solutions. In such cases, the
spinorial translation of the Riemann curvature tensor gives rise to a number
of different spinor quantities, including a mixed-index Ricci spinor in
addition to the Weyl spinor. References [76, 23] conjectured a generalised
Weyl double copy formula for the Weyl tensor, involving a sum of double-copy
terms, as shown here in eqs. (18). In each term, the gravity, gauge and scalar
theories are linked by having physically similar source terms, and we here
confirm the conjecture by focusing on sources that can be expanded in
arbitrary powers of the inverse radial coordinate. We demonstrate the double
copy using two methods: (i) a spinorial approach inspired by the twistor
double copy approach of refs. [17, 19, 21, 20, 75, 77, 78]; (ii) a tensorial
approach inspired by ref. [93], and which reproduces the same results as the
spinorial approach where our analyses overlap. Furthermore, we present a
generalised double copy formula for the Ricci spinor, involving a similar sum
of terms with physically similar sources. This generalises the results
presented in refs. [76, 23], which were limited to Einstein-Maxwell theory,
for which only two terms are required in the sum. Going beyond spherically
symmetric sources, we obtained the Kerr-Newman black hole by taking a Newman-
Janis shift of our previous results. This in turn clarified the fact that the
two alternative double copy formulae for the Ricci spinor given in ref. [23]
are indeed equivalent.

There are a number of directions for further work. One interesting avenue
would be to explore whether non-vacuum double copies may have applications to
astrophysics / cosmology, or to the generation of new gravitational solutions.
Also, our results are for linearised gauge / gravity theories only, and one
might think of trying to generalise the Weyl double copy – or alternative
classical double copy approaches – to non-linear level. We hope that our
results prove useful in thinking about these, and other, issues.

## Acknowledgments

We are very grateful to Tucker Manton for numerous useful 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”. KAW is supported by a studentship from the UK Engineering and
Physical Sciences Research Council (EPSRC).

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