aainstitutetext: Department of Physics, Arizona State University, Tempe, AZ
85281, USA

# Aligned Fields Double Copy to Kerr-NUT-(A)dS

Samarth Chawla a and Cynthia Keeler<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We find Abelian gauge fields that double copy to a large class of black hole
spacetimes with spherical horizon topology known as the Kerr-NUT-(A)dS family.
Using a multi-Kerr-Schild prescription, we extend the previously-known double
copy structure for arbitrarily rotating general dimension black holes, to
include NUT charges and an arbitrary cosmological constant. In all cases,
these single copy gauge fields are ‘aligned fields’, because their nonzero
components align with the principal tensor which generates the Killing
structure of the spacetime. In five dimensions, we additionally derive the
same single-copy field strengths via the Weyl double copy procedure.

##  1 Introduction

The classical double copy, first presented in Monteiro:2014cda , maps
classical Yang-Mills solutions to classical gravity solutions via a ‘squaring’
procedure. This procedure is based on the color-kinematics duality for
amplitudes (see Travaglini:2022uwo ; Bern:2019prr ; Adamo:2022dcm ). As in
color-kinematics duality, the classical gravitational solution is built from
two copies of the classical Yang-Mills solution, so the Yang-Mills solution is
called the ‘single copy’. Since the original formulation eight years ago, many
other examples of the classical double copy have been built Luna:2015paa ;
Luna:2016hge ; Monteiro:2018xev ; Ridgway:2015fdl ; Adamo:2017nia ; Bahjat-
Abbas:2017htu ; Carrillo-Gonzalez:2017iyj ; Ilderton:2018lsf ; Gurses:2018ckx
; CarrilloGonzalez:2019gof ; Lee:2018gxc ; Bah:2019sda ; Goldberger:2019xef ;
Kim:2019jwm ; Luna:2020adi ; Keeler:2020rcv ; Easson:2020esh ; Berman:2018hwd
; Alkac:2021seh ; Mkrtchyan:2022ulc ; Adamo:2022rob ; Alfonsi:2020lub ;
Adamo:2020qru ; Gonzo:2021drq . Recent reviews include Kosower:2022yvp  and
chapter 7 of Bern:2019prr .

The simplest example of the classical double copy starts with the Yang-Mills
field arrangement due to a single color charge at the origin. When the dilaton
and two-form field are tuned to zero Kim:2019jwm ; Luna:2020adi , this field
double-copies to the Schwarzschild black hole metric. Indeed, the first
classical double copy paper Monteiro:2014cda  proposes a single copy field
strength which double copies to the broader class of Kerr black holes in
higher dimensions: they include general-dimension black holes with arbitrary
spins but flat asymptotics.

In this paper, we extend this result to include the full Kerr-NUT-(A)dS class
of spacetimes Chen:2006xh . Our extension is possible because these spacetimes
all possess a series of Killing objects (vectors and higher tensors), all
related to the ‘principal tensor’ of the spacetime (Kubiznak:2006kt ; for a
recent review, see Frolov:2017kze ). The principal tensor, a non-degenerate
closed conformal Killing-Yano 2-form, not only provides for separability due
to the conserved quantities from the Killing objects (see Frolov:2017kze  and
citations therein, as well as Krtous:2018bvk ), it also ensures the spacetime
is algebraically special Mason:2010zzc . Indeed, the existence of the
principal tensor, together with the assumption that the metric is a vacuum
solution to Einstein’s equation, uniquely identifies the Kerr-NUT-(A)dS class
Krtous:2008tb .

As we show, the single copy for the full class of these spacetimes is an
‘aligned field’, so called because its components align with those of the
principal tensor. These aligned fields, found fully in Krtous:2007xg , have
charges which we set proportional to the mass and NUT charges of the
spacetime. We argue for our single copy proposal using a multi-Kerr-Schild
ansatz.

Specifically in five dimensions, we additionally show that our proposed
single-copy field matches the Weyl double copy. In the Weyl double copy, a
single copy field strength and zeroth copy scalar are proposed which combine
to form the Weyl tensor directly. Following the first proposal for Petrov type
D spacetimes in Luna:2018dpt , the Weyl double copy was extended to other
Petrov types Godazgar:2020zbv  and has found justification from amplitudes
directly Elor:2020nqe ; Farnsworth:2021wvs ; White:2020sfn ; Chacon:2021wbr ;
Chacon:2021lox ; Han:2022ubu ; Easson:2021asd ; Luna:2022dxo ;
Monteiro:2020plf . Aside from the spinor formalism (extended to five
dimensions in Monteiro:2018xev ), we additionally verify our five-dimensional
copy via the Weyl doubling formula of Alawadhi:2020jrv . For $D\geq 6$, we
leave a thorough exploration of a Weyl double copy to future work.

In the next section, we review the classical double copy and the spinor
formalism needed for its Weyl formulation, as well as the Kerr-NUT-(A)dS
spacetimes and their properties. In Section 3, we construct the five-
dimensional single copy using the Weyl spinor, and show its equivalence to the
multi-Kerr-Schild ansatz. We additionally note that this field is aligned to
the principal tensor of the spacetime. In Section 4, we use a multi-Kerr-
Schild approach, showing that the single copy of any Kerr-NUT-(A)dS spacetime
is an aligned field. Finally, in Section 5, we discuss our results and
directions for future work.

##  2 Review

###  2.1 Classical Double Copy

As we now review, the double copy is a map between the dynamics of gauge
theory and gravity, where gravity is realized in terms of two copies of a
gauge field. The earliest formulation Monteiro:2014cda  of this map takes as
its starting point exact solutions which can be written in Kerr-Schild form:

$g_{\mu\nu}=\eta_{\mu\nu}+\phi k_{\mu}k_{\nu},$ (1)

where $k_{\mu}$ is a null geodesic congruence with respect to the flat
background $\eta_{\mu\nu}$. Such solutions are of interest as they linearize
Einstein’s equations in terms of the field $h_{\mu\nu}=\phi k_{\mu}k_{\nu}$.

The single copy field is the Abelian gauge field $A_{\mu}=\phi k_{\mu}$, and
$\phi$ is the zeroth copy scalar field. They satisfy the Maxwell equations and
the Klein-Gordon wave equation on the flat background $\eta_{\mu\nu}$.111 Much
of the double copy literature uses a flat background owing to its origins in
the study of amplitudes in flat spacetime. However, the classical double copy
can still be performed in curved spacetimes, and the curved equations of
motion can be used; for relevant literature, see Alawadhi:2020jrv ;
Gurses:2018ckx ; Bahjat-Abbas:2017htu ; Carrillo-Gonzalez:2017iyj ;
Prabhu:2020avf ; Adamo:2017nia ; Alkac:2021bav ; Han:2022mze ; Alkac:2021bav ;
Han:2022ubu .

The Weyl formulation Luna:2018dpt  of the classical double copy is expressed
instead in terms of the Weyl curvature and the field strength tensor. The
index structure of these objects is simpler when written as spinors, in which
case the double copy relation is

$\Psi_{ABCD}=\frac{1}{S}f_{(AB}f_{CD)},$ (2)

where $\Psi_{ABCD}$ is the Weyl spinor, $f_{AB}$ is the single copy Maxwell
spinor and $S$ is the zeroth copy scalar field. The tensor translation of the
preceding equation is Alawadhi:2019urr ; Alawadhi:2020jrv

$W_{\mu\nu\rho\sigma}=\frac{1}{S}\Big{(}F_{\mu\nu}F_{\rho\sigma}+F_{\mu\rho}F_{\nu\sigma}-\frac{6}{D-2}g_{\mu\rho}F_{\nu\alpha}F_{\sigma}^{\phantom{\sigma}\alpha}\\\
+\frac{3}{(D-1)(D-2)}g_{\mu\rho}g_{\nu\sigma}F_{\alpha\beta}F^{\alpha\beta}\Big{)}\bigg{|}_{s},$
(3)

where “$\big{|}_{s}$” indicates that the expression must be anti-symmetrized
pairwise in $\mu,\nu$ and in $\rho,\sigma$ with unit weight.

While the Kerr-Schild single copy of the higher dimensional Kerr solution was
described already in Monteiro:2014cda , we aim to investigate whether
instances of a classical Weyl double copy continue to exist in dimensions
greater than four. We propose, as a starting point, a class of well-understood
black hole spacetimes known as the Kerr-NUT-(A)dS family, described in the
next section. We will find also a satisfying multi-Kerr-Schild description of
the single copy of Kerr-NUT-(A)dS in arbitrary dimension, which includes as a
special case the Kerr-Schild single copy of higher dimensional Kerr.

###  2.2 Higher dimensional black holes

The Kerr-NUT-(A)dS family of vacuum spacetimes includes all stationary black
holes with a spherical horizon topology and arbitrary mass, angular momentum
and NUT parameters. Following Frolov:2017kze , we set $D=2n+\varepsilon$,
where $\varepsilon=0$ for even $D$ and $\varepsilon=1$ for odd $D$. The Kerr-
NUT-(A)dS metric attains its most symmetric form in _canonical coordinates_
$(x_{k},\psi_{j})$,

$ds^{2}=\sum_{k=1}^{n}\left[\frac{U_{k}}{X_{k}}dx_{k}^{2}+\frac{X_{k}}{U_{k}}\left(\sum_{j=0}^{n-1}A_{k}^{(j)}d\psi_{j}\right)^{2}\right]-\varepsilon\frac{\mathcal{A}^{(n-1+\varepsilon)}}{A^{(n)}}\left(\sum_{j=0}^{n-1+\varepsilon}d\psi_{j}\right)^{2}.$
(4)

The $\psi_{j}$ coordinates for $i=0$ to $n-1+\epsilon$ are the Killing
directions, and are linear combinations of the azimuthal angles $\phi_{j}$ and
time $t$. The $x_{k}$ coordinates correspond to the remaining angles
$\theta_{k}$, except $x_{n}=ir$, which is the Wick-rotated radial coordinate.

Here, the functions $U_{k}$, $X_{k}$, $A_{k}^{(j)}$ and $A^{(j)}$ are
polynomials in $x_{k}$, given by

$\displaystyle X_{k}$ $\displaystyle=\frac{-(1+\lambda
x_{k}^{2})}{(-x_{k}^{2})^{\varepsilon}}\prod_{m=1}^{n-1+\varepsilon}(a_{m}^{2}-x_{k}^{2})+2M_{k}(-x_{k})^{1-\varepsilon},$
$\displaystyle U_{k}=\prod_{\begin{subarray}{c}j=1\\\ j\neq
k\end{subarray}}^{n}(x_{j}^{2}-x_{k}^{2}),$ $\displaystyle A_{k}^{(j)}$
$\displaystyle=\sum_{\begin{subarray}{c}1\leq i_{1}<i_{2}<\dots<i_{j}\leq n\\\
i_{1},i_{2},\dots i_{j}\neq k\end{subarray}}x_{i_{1}}^{2}x_{i_{2}}^{2}\dots
x_{i_{j}}^{2},$ $\displaystyle A^{(k)}=\sum_{1\leq i_{1}<i_{2}<\dots<i_{k}\leq
n}x_{i_{1}}^{2}x_{i_{2}}^{2}\dots x^{2}_{i_{k}},$ (5)

and $\mathcal{A}_{m}^{(l)}$, $\mathcal{A}^{(l)}$ are polynomials in $a_{m}$

$\mathcal{A}_{m}^{(l)}=\sum_{\begin{subarray}{c}1\leq
i_{1}<i_{2}<\dots<i_{l}\leq n-1+\varepsilon\\\ i_{1},i_{2},\dots i_{l}\neq
m\end{subarray}}a_{i_{1}}^{2}a_{i_{2}}^{2}\dots
a_{i_{l}}^{2},\qquad\mathcal{A}^{(l)}=\sum_{1\leq i_{1}<i_{2}<\dots<i_{l}\leq
n-1+\varepsilon}a_{i_{1}}^{2}\dots a_{i_{l}}^{2}.$ (6)

The constants $a_{j}$ are the independent spins of the black hole, and
$\lambda$ is related to the cosmological constant via
$\Lambda=\frac{1}{2}(D-2)(D-1)\lambda$. The parameters $M_{k}$ are NUT
parameters, except for $M_{n}$, which is the mass in odd dimensions and $i$
times the mass in even dimensions.

Alternatively, the spacetime metric can be described in canonical coordinates
by exhibiting an orthonormal frame of vectors

$\begin{split}\mathbf{e}_{k}&=\sqrt{\frac{X_{k}}{U_{k}}}\partial_{x_{k}},\\\
\mathbf{\hat{e}}_{k}&=\sqrt{\frac{U_{k}}{X_{k}}}\sum_{j=0}^{n-1+\varepsilon}(-x_{j}^{2})^{n-1-j}\partial_{\psi_{j}},\\\
\mathbf{\hat{e}}_{0}&=\sqrt{-\mathcal{A}^{(n-1+\varepsilon)}A^{(n)}}\partial_{\psi_{n}},\end{split}$
(7)

which is dual to the orthonormal coframe

$\begin{split}\mathbf{e}^{k}&=\sqrt{\frac{U_{k}}{X_{k}}}dx_{k},\\\
\mathbf{\hat{e}}^{k}&=\sqrt{\frac{X_{k}}{U_{k}}}\sum_{j=0}^{n-1}A_{k}^{(j)}d\psi_{j},\\\
\mathbf{\hat{e}}^{0}&=\sqrt{\frac{-\mathcal{A}^{(n-1+\varepsilon)}}{A^{(n)}}}\sum_{j=0}^{n}A^{(j)}d\psi_{j}.\end{split}$
(8)

The negative sign in the square roots will be canceled by the one from
squaring $x_{n}=ir$. With the appropriate coordinate ranges,
$\mathbf{\hat{e}}^{n}$ will turn out to be purely imaginary, thus playing the
role of the timelike element of the basis.

Explicitly, the metric in this frame is

$ds^{2}=\sum_{k=1}^{n}\left(\mathbf{e}^{k}\mathbf{e}^{k}+\mathbf{\hat{e}}^{k}\mathbf{\hat{e}}^{k}\right)+\varepsilon\mathbf{\hat{e}}^{0}\mathbf{\hat{e}}^{0}.$
(9)

These spacetimes are distinguished in possessing a full _Killing tower_ : a
non-degenerate set of $n+\varepsilon$ Killing vectors and $n$ Killing tensors.
These geometric objects are generators of explicit and hidden symmetries,
which make geodesic motion in these spacetimes completely integrable.

The Killing tower of Kerr-NUT-(A)dS spacetimes is generated using a single
$2$-form, known as the principal tensor. The existence of a principal tensor
is a nontrivial constraint on spacetime geometry, and the Kerr-NUT-(A)dS
metric is (locally) the most general vacuum spacetime which admits a principal
tensor Houri:2007xz ; Frolov:2017kze . In terms of the orthornormal coframe,
the principal tensor for the geometry (4) is

$\mathbf{h}=\sum_{k=1}^{n}x_{k}\mathbf{e}^{k}\wedge\mathbf{\hat{e}}^{k}.$ (10)

####  2.2.1 Transformation to Boyer-Lindquist and Myers-Perry

While canonical coordinates are well-suited to examining symmetries of the
spacetime for generic spins, mass, cosmological constant and NUT parameters,
taking any special limit of the spin parameters or the cosmological constant
in this coordinate system requires a great deal of care, since the
transformation to familiar coordinates is parameter-dependent.

Specifically, the coordinate transformations from $\psi_{j}$ to azimuthal
angles and the Killing time, and from $x_{k}$ to $\theta_{k}$, depend non-
trivially on the spin parameters and $\lambda$. The coordinate transformation
from the $n+\varepsilon$ coordinates $\psi_{j}$ to $t$ and the
$n-1+\varepsilon$ angles $\phi_{k}$ defines _generalized Boyer-Lindquist
coordinates_ Frolov:2017kze

$\begin{split}t&=\sum_{k=0}^{n-1+\varepsilon}\mathcal{A}^{(k)}\psi_{k},\\\
\phi_{j}&=a_{j}\sum_{k=0}^{n-1+\varepsilon}\left(\lambda\mathcal{A}_{j}^{(k)}-\mathcal{A}_{j}^{(k-1)}\right)\psi_{k}.\end{split}$
(11)

We can make a further transformation to Myers-Perry coordinates
$(t,r,\mu_{i},\phi_{j})$, defining $r$ by $x_{n}=ir$ and the latitude
coordinates $\mu_{i}$ by222When indexing the latitude coordinates $\mu_{i}$ we
follow the conventions of Chen:2006xh  instead of Frolov:2017kze .

$\mu_{i}^{2}\prod_{\begin{subarray}{c}j=1\\\ j\neq
i\end{subarray}}^{n}(a_{i}^{2}-a_{j}^{2})=\prod_{k=1}^{n-1}(a_{i}^{2}-x_{k}^{2}).$
(12)

In even dimensions, the expression above includes the extraneous parameter
$a_{n}$. To obtain the correct transformation, $a_{n}$ is set to $0$ in even
dimensions. The $n$ coordinates $\mu_{i}$ involve only the $n-1$ coordinates
$x_{k}$, and are thus not all independent. They satisfy the constraint

$\sum_{i=1}^{n}\mu_{i}^{2}=1,$ (13)

and are best thought of as direction cosines for the independent rotational
$2-$planes.

We will now discuss the $D=4$ and $D=5$ cases in detail. They are instructive
as to the general form of the coordinate transformations, and we will later
perform new explicit computations in five dimensions, as well as comparisons
with the extensive prior classical double copy literature in four dimensions.
In $D=4$ the transformation from canonical to Myers-Perry coordinates is
(setting $a_{1}=a$)

$\begin{gathered}t=\psi_{0}+a^{2}\psi_{1},\qquad r=-ix_{2},\qquad\phi=\lambda
a\psi_{0}-a\psi_{1},\\\
\mu_{1}^{2}=\frac{(a^{2}-x_{1}^{2})}{a^{2}},\qquad\mu_{2}^{2}=\frac{x_{1}^{2}}{a^{2}}.\end{gathered}$
(14)

The redundant latitude coordinates $\mu_{1}$ and $\mu_{2}$, in terms of the
familiar co-latitude angle $\theta$, are simply $\sin\theta$ and $\cos\theta$.
Solving for $x_{1}$ in terms of $\theta$,

$x_{1}=a\cos\theta.$ (15)

In the Schwarzschild limit $a\to 0$, $x_{1}$ ceases to be a well-defined
coordinate. The coordinate transformation to $\phi$ is also singular in this
limit, and the canonical angles $\psi_{0},\psi_{1}$ are ill-defined.

In $D=5$, the relation between the Myers-Perry and canonical coordinate
systems is

$\displaystyle t$
$\displaystyle=\psi_{0}+(a_{1}^{2}+a_{2}^{2})\psi_{1}+a_{1}^{2}a_{2}^{2}\psi_{2},$
$\displaystyle r$ $\displaystyle=-ix_{2},$
$\displaystyle\frac{\phi_{1}}{a_{1}}$ $\displaystyle=\lambda\psi_{0}+(\lambda
a_{2}^{2}-1)\psi_{1}-a_{2}^{2}\psi_{2},$ $\displaystyle\frac{\phi_{2}}{a_{2}}$
$\displaystyle=\lambda\psi_{0}+(\lambda
a_{1}^{2}-1)\psi_{1}-a_{1}^{2}\psi_{2},$ (16) $\displaystyle\mu_{1}^{2}$
$\displaystyle=\frac{a_{1}^{2}-x_{1}^{2}}{a_{1}^{2}-a_{2}^{2}},$
$\displaystyle\mu_{2}^{2}$
$\displaystyle=\frac{a_{2}^{2}-x_{1}^{2}}{a_{2}^{2}-a_{1}^{2}}.$

Again, setting $\mu_{1}$ and $\mu_{2}$ to $\sin\theta$ and $\cos\theta$, the
inverse transformation is

$x_{1}^{2}=a_{1}^{2}\cos^{2}\theta+a_{2}^{2}\sin^{2}\theta.$ (17)

Here, both the spinless limit and the equal-spin limit $a_{2}\to a_{1}$ set
$x_{1}$ to a constant (either $0$ or $a_{1}$) for any value of $\theta$. In
either case, $x_{1}$ is an ill-defined coordinate for the spacetime.

In the following two sections, we describe two properties of Kerr-NUT-(A)dS
that make it amenable to a double copy description: the existence of a multi-
Kerr-Schild form of the metric and the algebraic speciality of the Weyl
tensor.

####  2.2.2 Multi-Kerr-Schild form

A spacetime whose metric is in the Kerr-Schild form (1) has a simple proposal
for its single copy. The vacuum Einstein equations further guarantee that the
single copy field is on-shell. The Kerr-NUT-(A)dS metric (4) can be written in
a similar way: a multi-Kerr-Schild form about a flat (or dS/AdS for nonzero
$\lambda$) background metric $\mathring{\mathbf{g}}$ as333The background
$\mathbf{\mathring{g}}$ matches the Kerr-NUT-(A)dS metric when all $M_{k}$
vanish, although it does not take the same form in canonical coordinates.
There is a coordinate system in which it does have the same metric components
as the corresponding Kerr-NUT-(A)dS metric; see Frolov:2017kze  for details.
Such “Kerr-Schild” coordinates are convenient when computing the flat limit of
derivatives.

$\mathbf{g}=\mathring{\mathbf{g}}+2\sum_{k=1}^{n}\frac{M_{k}x_{k}^{1-\varepsilon}}{U_{k}}\boldsymbol{\mu}^{k}\boldsymbol{\mu}^{k},$
(18)

where the 1-forms $\boldsymbol{\mu}^{k}$ are null and geodesic. Note that
these 1-forms are unrelated to the Myers-Perry latitude coordinates $\mu_{i}$
despite their usage of the same symbol. Their explicit form in terms of the
orthonormal coframe (8) is

$\boldsymbol{\mu}^{k}=\sum_{j=0}^{n-1}A_{k}^{(j)}d\psi_{j}+i\frac{U_{k}}{X_{k}}dx_{k}=\sqrt{\frac{U_{k}}{X_{k}}}\left(\mathbf{\hat{e}}^{k}+i\mathbf{e}^{k}\right).$
(19)

In canonical coordinates, the explicit form of the flat metric
$\mathring{\mathbf{g}}$ is444This is essentially eq (4.75) from Frolov:2017kze
, with a couple of typos corrected (in the most up-to-date version as of
writing) and $\mathbf{\hat{e}}^{0}$ used instead of its scaled version
$\mathbf{\hat{\epsilon}}^{0}$.

$\mathring{\mathbf{g}}=\sum_{k=1}^{n}\frac{X_{k}}{U_{k}}\bigg{|}_{M_{k}=0}\boldsymbol{\mu}^{k}\boldsymbol{\mu}^{k}-i\sum_{k=1}^{n}\left(\boldsymbol{\mu}^{k}dx_{k}+dx_{k}\
\boldsymbol{\mu}^{k}\right)+\varepsilon\
\mathbf{\hat{e}}^{0}\mathbf{\hat{e}}^{0}.$ (20)

Since the single copy and zeroth copy fields (usually, see footnote 1) solve
equations of motion on a flat background, an unambiguous definition of the
flat metric is needed. The multi-Kerr-Schild form of the metric provides an
unambiguous flat (or dS/AdS) limit of geometric quantities. In particular, the
flat limit of covariant derivatives ($\nabla^{(0)}_{\mu}$) is defined as the
covariant derivative with respect to $\mathbf{\mathring{g}}$.

In section 4, we will use the multi-Kerr-Schild form to propose a single copy
gauge field for the generic Kerr-NUT-(A)dS black hole in arbitrary dimensions.

####  2.2.3 Type D

The generic Kerr-NUT-(A)dS spacetime in general dimension is algebraically
special, with a type D Weyl curvature. We will explain the characteristics of
type D spacetimes shortly. For more on algebraic speciality in higher
dimensions see Coley:2004jv ; Coley:2007tp ; Pravda:2007ty ; Ortaggio:2012jd ;
Milson:2004jx .

To express the type D condition, we must first define the boost weight of a
tensor in a null-orthonormal basis
$(\mathbf{k},\mathbf{n},\mathbf{e}^{i})_{i=1}^{D-2}$ where $\mathbf{k}$ and
$\mathbf{n}$ are null vectors such that $k_{\mu}n^{\mu}=-1$ and the spacelike
vectors $\mathbf{e}_{i}$ are mutually orthogonal as well as orthogonal to
$\mathbf{k}$ and $\mathbf{n}$. Any symmetric tensor $T_{\mu\nu}$, for
instance, can be expanded in this basis as

$T_{\mu\nu}=\underbrace{T_{\mathbf{kk}}k_{\mu}k_{\nu}}_{2}+\underbrace{T_{\mathbf{k}i}k_{(\mu}e_{\nu)}^{i}}_{1}+\underbrace{T_{\mathbf{kn}}k_{(\mu}n_{\nu)}+T_{ij}e^{i}_{(\mu}e^{j}_{\nu)}}_{0}+\underbrace{T_{\mathbf{n}i}n_{(\mu}e_{\nu)}^{i}}_{-1}+\underbrace{T_{\mathbf{nn}}n_{\mu}n_{\nu}}_{-2},$
(21)

where the different terms have been labeled by their boost weight. The boost
weight of each term is the power of $\Lambda$ it scales by under a Lorentz
boost in the $(\mathbf{k},\mathbf{n})$ subspace, which scales $\mathbf{k}$ and
$\mathbf{n}$ as

$\mathbf{k}\to\Lambda\mathbf{k},\qquad\mathbf{n}\to\Lambda^{-1}\mathbf{n}.$
(22)

So, for example, term labeled by the boost weight $2$ transforms as

$T_{\mathbf{kk}}k_{\mu}k_{\nu}\to T_{\mathbf{kk}}(\Lambda k_{\mu})(\Lambda
k_{\nu})=\Lambda^{2}\ T_{\mathbf{kk}}k_{\mu}k_{\nu}.$ (23)

The Weyl tensor of a spacetime $W_{\mu\nu\rho\sigma}$, too, can be expanded in
this basis and its components classified by boost weight. For a type D
spacetime, there always exists a null-orthonormal basis in which its only
nonzero components have boost weight $0$. This can be confirmed by checking
that the following conditions hold: Hamamoto:2006zf

$\begin{split}W^{\mu\nu\rho\sigma}k_{\mu}e_{\nu}^{i}e_{\rho}^{j}e_{\sigma}^{k}&=W^{\mu\nu\rho\sigma}n_{\mu}e_{\nu}^{i}e_{\rho}^{j}e_{\sigma}^{k}=0,\\\
W^{\mu\nu\rho\sigma}k_{\mu}e_{\nu}^{i}k_{\rho}e_{\sigma}^{j}&=W_{\mu\nu\rho\sigma}n_{\mu}e_{\nu}^{i}n_{\rho}e_{\sigma}^{j}=0,\\\
W^{\mu\nu\rho\sigma}k_{\mu}n_{\nu}k_{\rho}e_{\sigma}^{i}&=W^{\mu\nu\rho\sigma}n_{\mu}k_{\nu}n_{\rho}e_{\sigma}^{i}=0,\end{split}$
(24)

for the basis choice

$\mathbf{k}=\frac{\mathbf{e}_{n}+i\mathbf{\hat{e}}_{n}}{\sqrt{2}},\qquad\mathbf{n}=\frac{\mathbf{e}_{n}-i\mathbf{\hat{e}}_{n}}{\sqrt{2}}.$
(25)

The vectors $\mathbf{k}$ and $\mathbf{n}$ are real null vectors, since
$\mathbf{\hat{e}}_{n}$ is imaginary once the Wick rotation to Lorentzian
signature is carried out. We will refer to any basis that satisfies (24) as a
Weyl-aligned frame.

###  2.3 Spinors in five dimensions

In four dimensions, the Weyl formulation of the double copy is most compactly
expressed using spinor representations of the Weyl tensor and the Maxwell
field strength. This continues to hold in five dimensions, as we demonstrate
in section 3. The following is a summary of relevant spinor definitions from
Monteiro:2018xev .

In five dimensions the objects which translate between tensor and spinor
representations are gamma matrices, which are a representation of the Dirac
algebra

$\gamma^{(\mu}_{AB}\gamma^{\nu)}_{CD}\Omega^{BC}=g^{\mu\nu}\Omega_{AD},$ (26)

where $A=1,2,3,4$ is a spinor index and $\mu=0,1,2,3,4$ is a tensor index. We
work with the metric $g^{\mu\nu}$ in $(-++++)$ signature. The representation
of gamma matrices used is exactly the one used in Monteiro:2018xev . We use
also the antisymmetric bispinor $\Omega_{AB}$ defined in Monteiro:2018xev  to
provide an antisymmetric product of two arbitrary spinors $\xi^{A}$ and
$\chi^{A}$ via

$\xi_{A}\chi^{A}=\xi^{A}\Omega_{AB}\chi^{B}=\xi_{A}\Omega^{AB}\chi_{B}.$ (27)

We use the following convention for raising and lowering spinor indices

$\chi_{A}=\chi^{B}\Omega_{BA},\qquad\chi^{A}=\Omega^{AB}\chi_{B},$ (28)

which in turn implies

$\Omega^{AC}\Omega_{BC}=\mathbbm{1}^{A}_{B}.$ (29)

Since our aim will be to work with spinor representations of tensor fields, we
will find it of use to work in a spinor basis that is associated to a (null)
orthonormal vector basis.

####  2.3.1 Spinor basis and little group

The spinor basis we work in is defined in terms of a null frame of vectors
comprising of real null vectors $k^{\mu}$ and $n^{\mu}$ satisfying
$k^{\mu}n_{\mu}=-1$, a complex null vector $m^{\mu}$ and its complex conjugate
$\overline{m}^{\mu}$ which obey $m^{\mu}\overline{m}_{\mu}=1$ and a spacelike
unit vector $e_{0}^{\mu}$.

In other words, the inverse metric in terms of the null frame is

$g^{\mu\nu}=-k^{\mu}n^{\nu}-n^{\mu}k^{\nu}+m^{\mu}\overline{m}^{\nu}+\overline{m}^{\mu}m^{\nu}+e_{0}^{\mu}e_{0}^{\nu}.$
(30)

The spinor basis $(k^{A}_{a},n^{A}_{a})$ associated to the null frame is
defined by

$k_{\mu}\gamma^{\mu}_{AB}k^{A}_{a}=0,\qquad
n_{\mu}\gamma^{\mu}_{AB}n^{A}_{a}=0,$ (31)

where $a=1,2$, and

$k_{a}^{A}\Omega_{AB}n^{B}_{b}=\epsilon_{ab}.$ (32)

The convention used for the two-dimensional Levi-Civita symbol $\epsilon_{ab}$
is $\epsilon^{12}=-\epsilon_{12}=1$. It is used to lower and raise lowercase
latin indices in the same way $\Omega_{AB}$ is used to lower and raise spinor
indices.

This definition does not uniquely fix a basis relative to a particular null
frame $(k^{\mu},n^{\mu},m^{\mu},\overline{m}^{\mu},e_{0}^{\mu})$. Any two
bases which satisfy (31) and (32) are related by an $SU(2)$ action on
lowercase latin indices. This $SU(2)$ is referred to as the little group.

Contractions of any spinor with the spinor basis will carry $SU(2)$ indices
and thus transform under the little group. We briefly look at contractions of
the Weyl spinor and the resulting little group objects in the next subsection.

####  2.3.2 Weyl little group objects

As hinted at in the beginning of this section, we will find it convenient to
work with the spinor representation of the Weyl tensor. Owing to the
symmetries of the Weyl tensor $W_{\mu\nu\rho\lambda}$, all its independent
components can be packaged into a four-index spinor $\Psi_{ABCD}$ which is
symmetric under any permutation of its indices. This Weyl spinor is
constructed with the aid of the spinor Lorentz generators

$\sigma^{\mu\nu}_{AB}=\gamma^{[\mu}_{AC}\gamma^{\nu]}_{DB}\Omega^{CD},$ (33)

in the following way

$\Psi_{ABCD}=W_{\mu\nu\rho\lambda}\sigma^{\mu\nu}_{AB}\sigma^{\rho\lambda}_{CD}.$
(34)

Following Monteiro:2018xev  the Weyl spinor can be expanded in our chosen
frame-associated spinor basis

$\Psi_{ABCD}=\Psi^{(0)}_{abcd}\
k^{a}_{(A}k^{b}_{B}k^{c}_{C}k^{d}_{D)}+4\Psi^{(1)}_{abcd}\
k^{a}_{(A}k^{b}_{B}k^{c}_{C}n^{d}_{D)}+6\Psi^{(2)}_{abcd}\
k^{a}_{(A}k^{b}_{B}n^{c}_{C}n^{d}_{D)}\\\ +4\Psi^{(3)}_{abcd}\
k^{a}_{(A}n^{b}_{B}n^{c}_{C}k^{d}_{D)}+\Psi^{(4)}_{abcd}\
n^{a}_{(A}n^{b}_{B}n^{c}_{C}n^{d}_{D)}.$ (35)

The components $\Psi^{(i)}_{abcd}$ carry only $SU(2)$ indices ${a,b,c,d}$ and
are dubbed “little group objects”. Based on their index symmetry, each
component can be broken into their respective irreducible representations
(irreps) under $SU(2)$.

Since the spacetimes we will consider are type D, we note that the only
nonzero little group object is $\Psi^{(2)}_{abcd}$ if $k^{\mu}$ and $n^{\mu}$
are chosen such that the type D conditions (24) hold; see Appendix A. The
irreps contained within $\Psi^{(2)}_{abcd}$ are the fully symmetric objects
$\psi^{(2)_{abcd}}$ and $\chi^{(2)}_{ab}$, and the scalar
$\Psi^{(2)}_{\mathrm{tr}}$. In terms of these irreps, $\Psi^{(2)}_{abcd}$ is

$\Psi^{(2)}_{abcd}=\psi^{(2)}_{(abcd)}-\frac{1}{4}\left(\epsilon_{ac}\chi^{(2)}_{(bd)}+\epsilon_{ad}\chi^{(2)}_{(bc)}+\epsilon_{bc}\chi^{(2)}_{(ad)}+\epsilon_{bd}\chi^{(2)}_{(ac)}\right)+\\\
\frac{1}{6}\left(\epsilon_{ac}\epsilon_{bd}+\epsilon_{ad}\epsilon_{bc}\right)\Psi^{(2)}_{\mathrm{tr}}.$
(36)

Similarly to the expansion of $\Psi^{(2)}_{ABCD}$, in general the little group
objects $\psi^{(2)}_{abcd}$ and $\chi^{(2)}_{ab}$ can be expanded in a little
group basis $(\alpha_{a},\beta_{b})$ as

$\begin{split}\psi^{(2)}_{abcd}=&\
\psi_{0}^{(2)}\alpha_{a}\alpha_{b}\alpha_{c}\alpha_{d}+4\psi_{1}^{(2)}\alpha_{(a}\alpha_{d}\alpha_{c}\beta_{d)}+6\psi^{(2)}_{2}\alpha_{(a}\alpha_{b}\beta_{c}\beta_{d)}+4\psi^{(2)}_{3}\alpha_{(a}\beta_{b}\beta_{c}\beta_{d)}\\\
&+\psi^{(2)}_{4}\beta_{a}\beta_{b}\beta_{c}\beta_{d},\\\ \chi^{(2)}_{ab}=&\
\chi_{0}^{(2)}\alpha_{a}\alpha_{b}+2\chi_{1}^{(2)}\alpha_{(a}\beta_{b)}+\chi_{2}^{(2)}\beta_{a}\beta_{b}.\end{split}$
(37)

Analogous to how algebraic speciality indicates the existence of a spinor
basis in terms of which various little group objects $\Psi^{(i)}_{abcd}$ may
vanish, within each algebraically special class we can make a finer
classification based on the existence of little group bases
$(\alpha_{a},\beta_{a})$ for which little group irreps take special forms. For
instance, for certain spacetimes within the type D class it may be possible to
choose a basis $(\alpha_{a},\beta_{a})$ such that the only nonzero components
of $\psi^{(2)}_{abcd}$ and $\chi^{(2)}_{ab}$ are $\psi_{2}^{(2)}$ and
$\chi^{(2)}_{1}$. Precisely such a fine-grained notion of algebraic
speciality, within the coarse type D class, will be of use in proposing a
single copy of Kerr-NUT-(A)dS in five dimensions. That these finer
classifications could be important was anticipated already in Monteiro:2018xev
.

##  3 Single copy of five dimensional black holes

The general Kerr-NUT-(A)dS spacetime in five dimensions includes all
stationary black holes with spherical horizon topology and arbitrary values of
mass, angular momentum, NUT charge and cosmological constant. In this section
we compute the Weyl spinor for these spacetimes and show that it can be
written as the double copy of a Maxwell field. If we interpret the field as
living on the curved spacetime instead of on the flat background (as is
otherwise usual for single copies, see Alawadhi:2020jrv ; Bahjat-Abbas:2017htu
; Carrillo-Gonzalez:2017iyj ; Prabhu:2020avf ; Adamo:2017nia ; Alkac:2021bav ;
Han:2022mze  for examples where the field lives on a curved background), it
turns out to belong to a previously known class of Maxwell fields on Kerr-
NUT-(A)dS which are aligned along the principal tensor of the spacetime
Krtous:2007xg .

In fact, many previously known examples of the double copy map — in particular
the Schwarzschild-Tangherlini and Myers-Perry black holes in general dimension
and the $D=4$ Taub-NUT solution — can be described as aligned fields double-
copying to Kerr-NUT-(A)dS solutions.

###  3.1 Curvature tensors and Weyl spinor

We now exhibit the Weyl curvature of the five-dimensional Kerr-NUT-(A)dS black
hole, both in terms of the Weyl tensor and the Weyl spinor. We show that for
this class of spacetimes, the Weyl spinor has a well-defined square root,
which is algebraically necessary for a double-copy description that involves a
map to exactly one single copy field.

Beginning from the components of the Riemann and Ricci curvature tensors for
the general Kerr-NUT-(A)dS spacetime computed in Hamamoto:2006zf , we can
compute the Weyl tensor $W_{\mu\nu\rho\sigma}$ using

$W_{\mu\nu\rho\sigma}=R_{\mu\nu\rho\sigma}-\frac{1}{D-2}(R_{\mu\rho}g_{\nu\sigma}-R_{\nu\rho}g_{\mu\sigma}+R_{\nu\sigma}g_{\mu\rho}-R_{\mu\sigma}g_{\nu\rho})\\\
+\frac{R}{(D-1)(D-2)}(g_{\mu\rho}g_{\nu\sigma}-g_{\nu\rho}g_{\mu\sigma}).$
(38)

In the Darboux basis (8) the Weyl tensor is

$\begin{split}\mathbf{W}=&\sum_{i=1}^{n}\left(-\frac{1}{2}\partial_{i}^{2}\hat{Q}-\frac{2r_{i}}{3}+\frac{\sum_{k}r_{k}}{6}\right)\left(\mathbf{e}^{i}\wedge\mathbf{\hat{e}}^{i}\right)^{2}\\\
&+\sum_{i=1}^{n}\left(-\frac{1}{2x_{i}}\partial_{i}\hat{Q}-\frac{r_{0}r_{i}}{3}+\frac{r_{0}+2\sum_{k}r_{k}}{12}\right)\left[\left(\mathbf{e}^{i}\wedge\mathbf{\hat{e}}^{0}\right)^{2}+\left(\mathbf{\hat{e}}^{i}\wedge\mathbf{\hat{e}}^{0}\right)^{2}\right]\\\
&+\sum_{\begin{subarray}{c}1\leq i<j\leq
n\end{subarray}}\left(\frac{x_{j}\partial_{j}\hat{Q}-x_{i}\partial_{i}\hat{Q}}{2(x_{i}^{2}-x_{j}^{2})}-\frac{r_{i}+r_{j}}{3}+\frac{\sum_{k}r_{k}}{6}\right)\begin{bmatrix}\left(\mathbf{e}^{i}\wedge\mathbf{e}^{j}\right)^{2}+\left(\mathbf{e}^{i}\wedge\hat{\mathbf{e}}^{j}\right)^{2}\\\
+\left(\hat{\mathbf{e}}^{i}\wedge\hat{\mathbf{e}}^{j}\right)^{2}+\left(\hat{\mathbf{e}}^{i}\wedge\mathbf{e}^{j}\right)^{2}\end{bmatrix}\\\
&+\sum_{\begin{subarray}{c}1\leq i<j\leq
2\end{subarray}}\frac{x_{i}\partial_{j}\hat{Q}-x_{j}\partial_{i}\hat{Q}}{x_{i}^{2}-x_{j}^{2}}\begin{bmatrix}\left(\mathbf{e}^{i}\wedge\mathbf{e}^{j}\right)\odot\left(\hat{\mathbf{e}}^{i}\wedge\hat{\mathbf{e}}^{j}\right)\\\
+\left(\mathbf{e}^{i}\wedge\hat{\mathbf{e}}^{j}\right)\odot\left(\mathbf{e}^{j}\wedge\hat{\mathbf{e}}^{i}\right)\\\
+2\left(\mathbf{e}^{i}\wedge\hat{\mathbf{e}}^{i}\right)\odot\left(\mathbf{e}^{j}\wedge\hat{\mathbf{e}}^{j}\right)\end{bmatrix}.\end{split}$
(39)

Here, the $\odot$ represents the symmetrized tensor product, in this case of
$2$-forms. The square of a $2$-form, such as that of
$\mathbf{e}^{i}\wedge\mathbf{\hat{e}}^{i}$ in the first line, is shorthand for
a tensor product of two copies of the $2$-form. The symbol $\wedge$ denotes
the standard exterior product of differential forms. The $r_{i}$ are
components of the Ricci tensor

$\mathbf{Ric}=\sum_{i=1}^{n}r_{i}\left[\mathbf{e}^{i}\mathbf{e}^{i}+\mathbf{\hat{e}}^{i}\mathbf{\hat{e}}^{i}\right]+r_{0}\mathbf{\hat{e}}^{0}\mathbf{\hat{e}}^{0},$
(40)

given by

$r_{i}=-\frac{1}{2}\partial_{i}^{2}\hat{Q}+\sum_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{n}\frac{x_{j}\partial_{j}\hat{Q}-x_{i}\partial_{i}\hat{Q}}{x_{i}^{2}-x_{j}^{2}},\qquad
r_{0}=\sum_{i=1}^{n}-\frac{1}{x_{i}}\partial_{i}\hat{Q}.$ (41)

$\hat{Q}$ is defined in terms of metric functions (5) as555Our definition of
$\hat{Q}$ is consistent with the definitions of the similarly named functions
$Q_{T}$ in even dimensions and $\hat{Q}_{T}$ in odd dimensions in
Hamamoto:2006zf .

$\hat{Q}=\sum_{k=1}^{n}\frac{X_{k}}{U_{k}}-\varepsilon\frac{\prod_{j=1}^{n-1+\varepsilon}a_{j}^{2}}{\prod_{i=1}^{n}x_{i}^{2}}.$
(42)

The Weyl tensor (39) for Kerr-NUT-(A)dS exhibits extra algebraic symmetries
beyond those required of all Weyl curvature tensors. For instance, swapping
$\mathbf{e}^{i}$ and $\mathbf{\hat{e}}^{i}$ leaves the Weyl tensor unchanged.

We show in the next section that the $D=5$ Weyl spinor exhibits these extra
symmetries in a particularly simple form. A straightforward argument then
shows that a well-defined square root $f_{AB}$ in the sense of (2) exists,
which allows for its interpretation as a single copy field.

###  3.2 Constructing the single copy

Since the spinor definitions used are dimension-dependent, we now specialize
to the five dimensional case throughout this section. We follow the
conventions of Monteiro:2018xev , and provide a brief review in section 2.3
with a few further details in Appendix A.

As mentioned in section 2.3.2 and shown explicitly in Appendix A, for a type D
spacetime in a Weyl-aligned frame (i.e. a frame that satisfies the type D
conditions (24)) the only nonzero little group object is $\Psi^{(2)}$ (35).
Thus, in the $(k^{a}_{A},n^{a}_{A})$ basis the Weyl spinor is

$\Psi_{ABCD}=6\Psi^{(2)}_{abcd}k^{a}_{(A}k^{b}_{B}n^{c}_{C}n^{d}_{D)}.$ (43)

The irreps (36) of $\Psi^{(2)}_{ABCD}$ are

$\begin{split}\psi^{(2)}_{abcd}&=\frac{128(M-N)x_{1}^{2}}{(r^{2}+x_{1}^{2})^{3}}\alpha_{(a}\alpha_{b}\beta_{c}\beta_{d)},\\\
\chi^{(2)}_{ab}&=-i\frac{128(M-N)rx_{1}}{(r^{2}+x_{1}^{2})^{3}}\alpha_{(a}\beta_{b)},\\\
\Psi^{(2)}_{\mathrm{tr}}&=\frac{32(M-N)(x_{1}^{2}-3r^{2})}{(r^{2}+x_{1}^{2})^{3}},\end{split}$
(44)

where

$\alpha=\begin{bmatrix}1&0\end{bmatrix},\qquad\beta=\begin{bmatrix}0&1\end{bmatrix}.$
(45)

In the $(\alpha_{a},\beta_{a})$ basis, the symmetric four-index object
$\psi^{(2)}_{(abcd)}$ has a component only along
$\alpha_{(a}\alpha_{b}\beta_{c}\beta_{d)}$ and $\chi^{(2)}_{(ab)}$ is only
along $\alpha_{(a}\beta_{b)}$. In terms of the general expansion of
$\psi^{(2)}$ and $\chi^{(2)}$ (37), the only nonzero components are
$\psi^{(2)}_{2}$ and $\chi^{(2)}_{1}$. Additionally, $\psi^{(2)}$ is the
“square” of $\chi^{(2)}$:
$\psi^{(2)}_{abcd}\sim\chi^{(2)}_{(ab}\chi^{(2)}_{cd)}$.

The algebraic structure of the these irreps suggests the single copy ansatz
$f_{AB}=f^{(1)}_{ab}k^{a}_{(A}n^{b}_{B)}$ where
$f^{(1)}_{ab}=f^{(1)}_{1}\alpha_{a}\beta_{b}+f^{(1)}_{2}\beta_{a}\alpha_{b}$.
The Weyl polynomial $\mathcal{W}$ provides a straightforward method for
computing the ‘square root’ and thus finding $f_{AB}$. It is defined using a
formal parameter $\xi^{A}$:

$\mathcal{W}=\Psi_{ABCD}\xi^{A}\xi^{B}\xi^{C}\xi^{D}.$ (46)

Since the Weyl spinor $\Psi_{ABCD}$ is symmetric in all indices, its
contraction with four copies of $\xi^{A}$ is an equivalent object. In terms of
the variables

$K^{a}=k^{a}_{A}\xi^{A},\qquad N^{a}=n^{a}_{A}\xi^{A},$ (47)

the Weyl polynomial is

$\mathcal{W}=\frac{192(N-M)}{(r^{2}+x_{1}^{2})^{3}}\left[(r+ix_{1})K^{1}N^{2}-(r-ix_{1})K^{2}N^{1}\right]^{2}.$
(48)

Its form suggests the single copy

$f_{AB}=\sqrt{\frac{192(N-M)S}{(r^{2}+x_{1}^{2})^{3}}}\left[(r+ix_{1})k^{1}_{(A}n^{2}_{B)}-(r-ix_{1})k^{2}_{(A}n^{1}_{B)}\right].$
(49)

Here, $S$ is the scalar factor that appears in the double copy relation (2),
which we fix666Our procedure to fix $S$ closely follows the treatment for type
N spacetimes in Godazgar:2020zbv , which also finds a set of first order
differential equations for $S$ that are integrable. This approach differs
slightly from the one used for type D spacetimes in Luna:2018dpt , which
relies on prior work showing how the double, single and zeroth copies can all
be written in terms of a single Killing spinor that always exists in type D.
by requiring that $F_{\mu\nu}=\frac{1}{8}f_{AB}\sigma^{AB}_{\mu\nu}$ satisfy
Maxwell’s equations on the Kerr-NUT-(A)dS background.

Away from sources, either the Maxwell equations $\nabla^{\mu}F_{\mu\nu}=0$ or
their dual $\nabla_{[\mu}F_{\nu\rho]}=0$ are equivalent to the two independent
first order equations

$\displaystyle r\sqrt{S}+\left(r^{2}+x_{1}^{2}\right)\frac{\partial}{\partial
r}\sqrt{S}$ $\displaystyle=0,$ (50) $\displaystyle
x_{1}\sqrt{S}+\left(r^{2}+x_{1}^{2}\right)\frac{\partial}{\partial
x_{1}}\sqrt{S}$ $\displaystyle=0.$ (51)

These equations have the unique solution (up to an overall constant)

$S=\frac{q^{2}}{192(N-M)(r^{2}+x_{1}^{2})}.$ (52)

With this solution for $S$, the field strength $F_{\mu\nu}$ becomes

$F_{\mu\nu}=\frac{q}{2(r^{2}+x_{1}^{2})^{2}}\left(x_{1}\mathbf{e}^{1}_{[\mu}\mathbf{\hat{e}}^{1}_{\nu]}-ir\mathbf{e}^{2}_{[\mu}\mathbf{\hat{e}}^{2}_{\nu]}\right).$
(53)

This field strength belongs to a class of field strengths known in the Kerr-
NUT-(A)dS literature as _aligned fields_. These are on-shell electromagnetic
fields that, in the orthonormal coframe, have the same structure as the
principal tensor (10). That is, they have the expansion Krtous:2007xg

$\mathbf{F}=\sum_{k=1}^{n}f_{k}\mathbf{e}^{k}\wedge\mathbf{\hat{e}}^{k},$ (54)

where

$\begin{split}f_{k}&=\frac{e_{k}}{U_{k}}+2x_{k}\sum_{\begin{subarray}{c}j=1\\\
j\neq
k\end{subarray}}^{n}\frac{1}{U_{j}}\frac{e_{j}x_{j}-e_{k}x_{k}}{x_{j}^{2}-x_{k}^{2}};\qquad\text{when
}D=2n\\\
f_{k}&=2x_{k}\sum_{j=1}^{n}\frac{1}{U_{j}}\frac{e_{j}-e_{k}}{x_{j}^{2}-x_{k}^{2}};\qquad\text{when
}D=2n+1.\end{split}$ (55)

Aligned fields in general depend on $n$ parameters $e_{k}$; the constant $q$
here corresponds to $2(e_{1}-e_{2})$.

Also, we find that the scalar $S$ satisfies the source-free wave equation only
in the flat limit (defined in terms of the multi-Kerr-Schild expansion (18))

$\square^{(0)}S=0,$ (56)

where $\square^{(0)}$ signifies the Laplacian on the flat background.

It is curious that the field strength satisfies Maxwell’s equations on the
full (curved) background as well as on the flat background, while the scalar
$S$ satisfies the source-free wave equation only when the flat limit is taken.
Rather than investigate this issue further, we now instead compare our
proposal of the single copy with known examples.

###  3.3 Single copy from Weyl prescription consistent with Kerr-Schild and
with Weyl Doubling

The first work Monteiro:2014cda 777See especially section 4.2. on the
classical double copy described the single copy map for an arbitrary spin,
asymptotically flat black hole with no NUT parameters in general dimension.
Translating their proposal for the single copy gauge field to Myers-Perry
coordinates, we find

$\mathbf{A}=\frac{2M}{U}\left(dt+\frac{U}{V-2M}dr+a_{1}\mu_{1}^{2}d\phi_{1}+a_{2}\mu_{2}^{2}d\phi_{2}\right),$
(57)

where

$V=\frac{1}{r^{1+\varepsilon}}\prod_{j=1}^{n-1+\varepsilon}(r^{2}+a_{j}^{2}),\qquad
U=V\left(1-\sum_{k=1}^{n-1+\varepsilon}\frac{a_{k}^{2}\mu_{k}^{2}}{r^{2}+a_{k}^{2}}\right).$
(58)

Transforming further to canonical coordinates, we have

$\mathbf{A}=\frac{2M}{U_{2}}\left(d\psi_{0}+x_{1}^{2}d\psi_{1}\right)+i\frac{2M}{X_{2}}dx_{2}.$
(59)

The second term here is pure gauge, since $X_{2}$ is a function of $x_{2}$
only.888The Kerr-Schild prescription for the double copy is sensitive to gauge
choice on the single copy side. The original single copy for Schwarzschild,
for instance, was in an unusual gauge Monteiro:2014cda . Godazgar:2021iae
discusses this gauge issue in the Weyl double copy context, while Bahjat-
Abbas:2020cyb  relates monopoles in the double copy to singular gauge
transformations. The first term matches the aligned field gauge potential
found in Krtous:2007xg  when $e_{1}=0,\ e_{2}=2M$. Thus, the single copy for
$D=5$ Myers-Perry proposed in Monteiro:2014cda  is an aligned field.

In addition to matching with the previously known Kerr-Schild single copy, the
tensor-language formalism proposed in Alawadhi:2020jrv  also agrees with our
single copy prescription found using spinors. We take the generic aligned
field strength in $D=5$,

$\mathbf{F}=\frac{2x_{1}}{(x_{1}^{2}+r^{2})^{2}}(e_{1}-e_{2})\mathbf{e}^{1}\wedge\mathbf{\hat{e}}^{1}+\frac{2ir}{(x_{1}^{2}+r^{2})^{2}}(e_{2}-e_{1})\mathbf{e}^{2}\wedge\mathbf{\hat{e}}^{2},$
(60)

and plug it into the Weyl doubling relation (3). We find that the relation is
satisfied for the scalar choice

$S=\frac{(e_{1}-e_{2})^{2}}{2(M_{1}-M_{2})(x_{1}^{2}+r^{2})}.$ (61)

Our observation that the single copy for Kerr-NUT-(A)dS black holes is an
aligned field is not restricted to $D=5$. Previously known single copies for
$D=4$ Schwarzschild, Kerr and Taub-NUT spacetimesMonteiro:2014cda ;
Luna:2018dpt ; Luna:2015paa  (all of which are in the Kerr-NUT-(A)dS class)
are also aligned fields, as we now briefly demonstrate.

The single copy of the Kerr solution is the gauge field Luna:2018dpt
999Luna:2018dpt  claims a slightly different gauge field since it sets $M=0$
in order to take the flat background limit. Even though the $dr$ term is pure
gauge anyway, it is useful to transform to coordinates
$(\mathring{t},r,\theta,\mathring{\phi})$, whose relation to the Kerr-Schild
coordinates $(x_{k},\mathring{\psi}_{j})$ defined in equation (4.77) of
Frolov:2017kze  parallels the transformation to Boyer-Lindquist coordinates
from canonical coordinates (11). Then, we can view $\mathbf{A}$ as living on
the flat background with metric $\mathbf{\mathring{g}}$. Written in these
coordinates, the gauge potential takes precisely the form in Luna:2018dpt
with no parameter limits post hoc.

$\mathbf{A}=\frac{Qr}{r^{2}+x_{1}^{2}}\left(dt+\frac{r^{2}+x_{1}^{2}}{r^{2}+a^{2}-2Mr}\
dr+a\left(1-\frac{x_{1}^{2}}{a^{2}}\right)d\phi\right).$ (62)

Transforming to canonical coordinates,

$\mathbf{A}=\frac{Qr}{r^{2}+x_{1}^{2}}\left(d\psi_{0}+x_{1}^{2}d\psi_{1}\right)+\frac{Qx_{2}}{X_{2}}dx_{2}.$
(63)

As in $D=5$, this is the gauge potential for an aligned field with
$e_{1}=0,e_{2}=-iQ$ up to the second (pure gauge) term.

The single copy field strength of the (non-rotating) Taub-NUT solution is (up
to an overall constant) Luna:2015paa

$\mathbf{F}=\frac{M}{r^{2}}dt\wedge dr+N\sin\theta d\theta\wedge d\phi,$ (64)

where $M$ is the mass and $N$ is the NUT charge. The easiest way to see that
this is an aligned field is to convert the generic aligned field to Myers-
Perry coordinates and set the spin $a$ and the cosmological constant $\lambda$
to $0$. A term-by-term comparison then shows that the aligned field charges
$e_{1}$ and $e_{2}$ must be set to $N$ and $-iM$.

In the next section we show that aligned fields continue to serve as single
copies for Kerr-NUT-(A)dS spacetimes in arbitrary dimensions, with aligned
field charges set by the NUT charges and the mass.

##  4 Towards a single copy of general dimension Kerr-NUT-(A)dS

We have shown that the single copy of Kerr-NUT-(A)dS black holes in $D=5$ is
an aligned electromagnetic field. We also demonstrated that the various $D=4$
single copy results for black holes in the Kerr-NUT-(A)dS class are aligned
fields too. These observations lead us to ask: is the single copy of a Kerr-
NUT-(A)dS black hole always an aligned field?

One way to propose a single copy is to use the Kerr-Schild prescription. We
start with the multi-Kerr-Schild form of the Kerr-NUT-(A)dS metric (18) and
propose the single copy gauge field

$\mathbf{A}=\sum_{k=1}^{n}\frac{2Q_{k}x_{k}^{1-\varepsilon}}{U_{k}}\boldsymbol{\mu}^{k},$
(65)

where $\boldsymbol{\mu}^{k}$ are the null geodesic $1$-forms (19) and $Q_{k}$
are constant parameters which satisfy $Q_{k}=QM_{k},$ where $Q$ is an
arbitrary proportionality constant.

Expanding each $\boldsymbol{\mu}^{k}$ in canonical coordinates,

$\begin{split}\mathbf{A}&=\sum_{k=1}^{n}\frac{2Q_{k}x_{k}^{1-\varepsilon}}{U_{k}}\left(\sum_{j=0}^{n-1}A_{k}^{(j)}d\psi_{j}+i\frac{U_{k}}{X_{k}}dx_{k}\right)\\\
&=\sum_{k=1}^{n}2Q_{k}x_{k}^{1-\varepsilon}\left(\sum_{j=0}^{n-1}\frac{A_{k}^{(j)}}{U_{k}}d\psi_{j}+i\frac{dx_{k}}{X_{k}}\right).\end{split}$
(66)

The second term inside the parentheses is gauge trivial since $X_{k}$ is a
function of $x_{k}$ only. Thus, this field is gauge equivalent to

$\mathbf{A}=\sum_{k=1}^{n}\frac{2Q_{k}x_{k}^{1-\varepsilon}}{U_{k}}\sum_{j=0}^{n-1}A_{k}^{(j)}d\psi_{j},$
(67)

which is the gauge potential of an aligned field Krtous:2007xg . Analogous to
how the Kerr-Schild double copy was extended to allow for the single copy of
the $D=4$ Taub-NUT spacetime in Luna:2015paa , this gauge field is the multi-
Kerr-Schild prescription for the single copy of Kerr-NUT-(A)dS in any
dimension.

In the case of vanishing NUT charges, the corresponding prescription would be
$Q_{k}=0$ for $k=1$ to $n-1$; only $Q_{n}\propto M$ remains. The single copy
gauge field is then exactly the same as the one proposed as the single copy
for higher dimensional Kerr spacetimes in Monteiro:2014cda .

Instead of constructing higher dimensional spinor versions of the Weyl tensor
and the aligned field, we now use our proposal to test a straightforward
extension of the Weyl doubling prescription (3). However, plugging in the Weyl
tensor components (39) and the components of the aligned field from (54) and
(55), there is no scalar that satisfies (3) for any value of aligned field
parameters $Q_{k}$ given arbitrary mass and NUT charges.

Even setting all NUT charges to zero (where our proposal matches
Monteiro:2014cda ), there is no zeroth copy scalar which satisfies the Weyl
doubling prescription (3). We expect that the Weyl doubling formula may need
additional terms in $D>5$ in order to describe the Weyl double copy structure
of Kerr-NUT-(A)dS. Alawadhi:2020jrv .

##  5 Discussion

We have shown that the single copies of the Kerr-NUT-(A)dS spacetimes map onto
the ‘aligned fields’ in the same spacetimes, with charges proportional to the
mass and NUT charges. In five dimensions, we further showed that this aligned
field single copy matches with the Weyl double copy in spinor language and in
tensor language via the Weyl doubling procedure.

Importantly, this five-dimensional single copy which follows the Weyl doubling
formula has arbitrary spins (as well as a possible nonzero NUT charge and
cosmological constant), in contrast to the comment in Alawadhi:2020jrv  that
only the singly-rotating solution was possible. However, we do note that our
zeroth-copy scalar only satisfies the flat-space Klein-Gordon equation (the
aligned field itself satisfies both the curved and flat-space equations).
Similarly, the obstruction we found to a higher-dimensional application of the
Weyl doubling formula to our multi-Kerr-Schild single copy was in the
inability to find the necessary scalar.

We also want to highlight the importance of the principal tensor as reviewed
in Frolov:2017kze . Spacetimes possessing a principal tensor are always
algebraically special (type D). If they are vacuum solutions, then they also
have aligned fields: electromagnetic fields with the same components as the
principal tensor. These fields have $(D-\epsilon)/2$ free parameters; setting
these charges proportional to the mass and $(D-2-\epsilon)/2$ NUT charges
results in our single copy field. In some sense the principal tensor controls
the form of the metric, so it is not too surprising that it is related so
closely to the metric’s single copy. A natural extension of our work would be
to include non-vacuum metrics which still possess a principal tensor.

Since the class of vacuum spacetimes which possess a principal tensor is the
same Kerr-NUT-(A)dS class we consider here, any attempt to consider a broader
set of vacuum solutions (e.g. bumpy black holes, black rings, etc) may require
significant extension. Indeed, the argument for Weyl locality in position
space given in Luna:2022dxo  indicates that non-algebraically special
spacetimes cannot have a local Weyl single copy, and spacetimes possessing a
principal tensor are always algebraically special. Nevertheless, we hope that
our work here may provide an approach to make further spacetimes at least
perturbatively accessible.

We additionally hope our work may provide a framework to further study
gravitational transformations and dualities in the double copy Bahjat-
Abbas:2020cyb ; Arkani-Hamed:2019ymq ; Alawadhi:2019urr ; Banerjee:2019saj ;
Huang:2019cja ; Emond:2020lwi , since it includes the full class of Kerr-
NUT-(A)dS spacetimes. Similarly, perhaps the principal tensor can be studied
in the single-copy picture, just as NUT charges, geodesics, and Ricci flow
have been Alfonsi:2020lub ; Alawadhi:2022gwy ; Gonzo:2021drq ;
Alawadhi:2021uie . Finally, for, the Kerr-NUT solutions in our family, we
would be interested to see if the Weyl double copy in general dimensions can
be understood asymptotically, as at null infinity in four dimensions
Godazgar:2021iae ; Luna:2016due ; Gonzo:2022tjm ; Chacon:2021hfe ;
Adamo:2021dfg ; Campiglia:2021srh .

###### Acknowledgements.

The authors thank Damien Easson, Andres Luna, Tucker Manton, and Silvia Nagy
for helpful discussions. The authors are supported by the U.S. Department of
Energy under grant number DE-SC0019470 and by the Heising-Simons Foundation
“Observational Signatures of Quantum Gravity” collaboration grant 2021-2818.

##  Appendix A Little group structure of type D Weyl spinor

This brief appendix is a companion to section 2.3 and gives details of
calculations that use the spinor set-up used in this paper. As an illustrative
exercise in the spinor conventions used, we will lay out an argument for why
only $\Psi^{(2)}\neq 0$ for a type D spacetime.

First, we describe how the spinor basis $(k^{A}_{a},n^{A}_{a})$ can be used to
give back members of the null frame (30). Effectively, this inverts the
definition of the basis (31) in terms of the null frame. The vectors $k^{\mu}$
and $n^{\mu}$ can be obtained by contracting two copies of $k^{A}_{a}$ and
$n^{A}_{a}$ respectively with gamma matrices

$\begin{split}k_{a}^{A}\gamma^{\mu}_{AB}k_{b}^{B}&=-\sqrt{2}k^{\mu}\epsilon_{ab},\\\
n_{a}^{A}\gamma^{\mu}_{AB}n_{b}^{B}&=-\sqrt{2}n^{\mu}\epsilon_{ab}.\end{split}$
(68)

The remaining contractions of $\gamma^{\mu}$ with the spinor basis produces
the remaining members of the null frame

$\begin{split}k_{a}^{A}\gamma^{\mu}_{AB}n^{B}_{b}&=-n_{a}^{A}\gamma^{\mu}_{AB}k^{B}_{b}\\\
&=\begin{pmatrix}\sqrt{2}m^{\mu}&ie_{0}^{\mu}\\\
ie_{0}^{\mu}&\sqrt{2}\overline{m}^{\mu}\end{pmatrix},\end{split}$ (69)

which, following Monteiro:2018xev , we dub the polarization tensor
$\varepsilon^{\mu}_{ab}$.

Next, we explain how to compute the components of the spinor Lorentz
generators in the $(k^{A}_{a},n^{A}_{a})$ basis. Take, as an example, the
contraction $\sigma^{\mu\nu}_{AB}k^{A}_{a}k^{B}_{b}$. Noting the definition of
the Lorentz generators in terms of gamma matrices (33), we have

$\sigma_{AB}^{\mu\nu}k^{A}_{a}k^{a}_{b}=\gamma^{[\mu}_{AE}\gamma^{\nu]}_{FB}\Omega^{EF}k^{A}_{a}k^{B}_{b}.$
(70)

The normalization of the spinor basis (32) is equivalent to the following
expansion of $\Omega^{AB}$

$\Omega^{AB}=\epsilon^{ab}\left(k^{A}_{a}n^{B}_{b}+n^{A}_{a}k^{B}_{b}\right).$
(71)

Substituting this back in (70), and keeping in mind (68) and (69) we obtain

$\begin{split}\sigma_{AB}^{\mu\nu}k^{A}_{a}k^{B}_{b}&=\epsilon^{cd}\gamma^{[\mu}_{AE}\gamma^{\nu]}_{FB}\left(k^{E}_{c}n^{F}_{d}+n^{E}_{c}k^{F}_{d}\right)k^{A}_{a}k^{B}_{b}\\\
&=\epsilon^{cd}\left(k^{A}_{a}\gamma^{[\mu|}_{AE}k^{E}_{c}n^{F}_{d}\gamma^{|\nu]}_{FB}k^{B}_{b}+k^{A}_{a}\gamma^{[\mu|}_{AE}n^{E}_{c}k^{F}_{d}\gamma^{|\nu]}_{FB}k^{B}_{b}\right)\\\
&=\epsilon^{cd}\left(\epsilon_{ac}(-\sqrt{2}k^{[\mu})(-\varepsilon^{\nu]}_{bd})+(\varepsilon^{[\mu}_{ac})(-\sqrt{2}k^{\nu]}\epsilon_{db})\right)\\\
&=2\sqrt{2}k^{[\mu}\varepsilon^{\nu]}_{ab}.\end{split}$ (72)

Similarly the other components are

$\begin{split}\sigma_{AB}^{\mu\nu}k^{A}_{a}n^{B}_{b}&=2k^{[\mu}n^{\nu]}\epsilon_{ab}+\varepsilon^{[\mu}_{ac}\varepsilon^{\nu]}_{bd}\epsilon^{cd},\\\
\sigma_{AB}^{\mu\nu}n^{A}_{a}k^{A}_{a}&=2n^{[\mu}k^{\nu]}\epsilon_{ab}+\varepsilon^{[\mu}_{ac}\varepsilon^{\nu]}_{bd}\epsilon^{cd},\\\
\sigma_{AB}^{\mu\nu}n^{A}_{a}n^{B}_{b}&=-2\sqrt{2}n^{[\mu}\varepsilon^{\nu]}_{ab}.\end{split}$
(73)

We are now prepared to tackle the little group structure of a type D Weyl
spinor. Inverting the expansion of $\Psi_{ABCD}$ (35), we obtain

$\begin{split}\Psi^{(0)}_{abcd}&=\Psi_{ABCD}n^{A}_{a}n^{B}_{b}n^{C}_{c}n^{D}_{d},\\\
\Psi^{(1)}_{abcd}&=\Psi_{ABCD}n^{a}_{a}n^{b}_{b}n^{c}_{c}k^{d}_{d},\\\
\Psi^{(2)}_{abcd}&=\Psi_{ABCD}n^{a}_{a}n^{b}_{b}k^{c}_{c}k^{d}_{d},\\\
\Psi^{(3)}_{abcd}&=\Psi_{ABCD}n^{a}_{a}k^{b}_{b}k^{c}_{c}k^{d}_{d},\\\
\Psi^{(4)}_{abcd}&=\Psi_{ABCD}k^{a}_{a}k^{b}_{b}k^{c}_{c}k^{d}_{d}.\end{split}$
(74)

Next, we rewrite $\Psi_{ABCD}$ in terms of the Weyl tensor
$W_{\mu\nu\rho\lambda}$ using (34)

$\begin{split}\Psi^{(0)}_{abcd}&=W_{\mu\nu\rho\lambda}\sigma^{\mu\nu}_{AB}\sigma^{\rho\lambda}_{CD}n^{A}_{a}n^{B}_{b}n^{C}_{c}n^{D}_{d},\\\
\Psi^{(1)}_{abcd}&=W_{\mu\nu\rho\lambda}\sigma^{\mu\nu}_{AB}\sigma^{\rho\lambda}_{CD}n^{a}_{a}n^{b}_{b}n^{c}_{c}k^{d}_{d},\\\
\Psi^{(2)}_{abcd}&=W_{\mu\nu\rho\lambda}\sigma^{\mu\nu}_{AB}\sigma^{\rho\lambda}_{CD}n^{a}_{a}n^{b}_{b}k^{c}_{c}k^{d}_{d},\\\
\Psi^{(3)}_{abcd}&=W_{\mu\nu\rho\lambda}\sigma^{\mu\nu}_{AB}\sigma^{\rho\lambda}_{CD}n^{a}_{a}k^{b}_{b}k^{c}_{c}k^{d}_{d},\\\
\Psi^{(4)}_{abcd}&=W_{\mu\nu\rho\lambda}\sigma^{\mu\nu}_{AB}\sigma^{\rho\lambda}_{CD}k^{a}_{a}k^{b}_{b}k^{c}_{c}k^{d}_{d}.\end{split}$
(75)

Finally, substituting the components of $\sigma^{\mu\nu}_{AB}$ we just found
in (70), (73),

$\begin{split}\Psi^{(0)}_{abcd}&=8W_{\mu\nu\rho\lambda}k^{\mu}\varepsilon^{\nu}_{ab}k^{\rho}\varepsilon^{\lambda}_{cd},\\\
\Psi^{(1)}_{abcd}&=2\sqrt{2}W_{\mu\nu\rho\lambda}k^{[\mu}\varepsilon^{\nu]}_{ab}\left(2k^{[\rho}n^{\lambda]}\epsilon_{cd}+\epsilon^{gh}\varepsilon^{[\rho}_{cg}\varepsilon^{\lambda]}_{hd}\right),\\\
\Psi^{(2)}_{abcd}&=-8W_{\mu\nu\rho\lambda}k^{[\mu}\varepsilon^{\nu]}_{ab}n^{[\rho}\varepsilon^{\lambda]}_{cd},\\\
\Psi^{(3)}_{abcd}&=-2\sqrt{2}W_{\mu\nu\rho\lambda}\left(2k^{[\mu}n^{\nu]}\epsilon_{ab}+\epsilon^{gh}\varepsilon^{[\mu}_{ag}\varepsilon^{\nu]}_{hb}\right)n^{[\rho}\varepsilon^{\lambda]}_{cd},\\\
\Psi^{(4)}_{abcd}&=8W_{\mu\nu\rho\lambda}n^{\mu}\varepsilon^{\nu}_{ab}n^{\rho}\varepsilon^{\lambda}_{cd}.\end{split}$
(76)

Now, remembering that $\varepsilon^{\mu}_{ab}$ is orthogonal to both $k^{\mu}$
and $n^{\mu}$, a comparison with (24) shows that with a properly chosen null
frame $\Psi^{(0)},\Psi^{(1)},\Psi^{(3)}$ and $\Psi^{(4)}$ all vanish for a
type D spacetime.

## References

  * (1) R. Monteiro, D. O’Connell, and C. D. White, Black holes and the double copy, Journal of High Energy Physics 2014 (Dec., 2014) 56, [arXiv:1410.0239].
  * (2) G. Travaglini, A. Brandhuber, P. Dorey, T. McLoughlin, S. Abreu, Z. Bern, N. E. J. Bjerrum-Bohr, J. Blümlein, R. Britto, J. J. M. Carrasco, D. Chicherin, M. Chiodaroli, P. H. Damgaard, V. Del Duca, L. J. Dixon, D. Dorigoni, C. Duhr, Y. Geyer, M. B. Green, E. Herrmann, P. Heslop, H. Johansson, G. P. Korchemsky, D. A. Kosower, L. Mason, R. Monteiro, D. O’Connell, G. Papathanasiou, L. Plante, J. Plefka, A. Puhm, A.-M. Raclariu, R. Roiban, C. Schneider, J. Trnka, P. Vanhove, C. Wen, and C. D. White, The SAGEX Review on Scattering Amplitudes, Mar., 2022.
  * (3) Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson, and R. Roiban, The Duality Between Color and Kinematics and its Applications, arXiv:1909.01358 [gr-qc, physics:hep-ph, physics:hep-th] (Sept., 2019) [arXiv:1909.01358].
  * (4) T. Adamo, J. J. M. Carrasco, M. Carrillo-González, M. Chiodaroli, H. Elvang, H. Johansson, D. O’Connell, R. Roiban, and O. Schlotterer, Snowmass White Paper: The Double Copy and its Applications, in 2022 Snowmass Summer Study, Apr., 2022.  arXiv:2204.06547.
  * (5) A. Luna, R. Monteiro, D. O’Connell, and C. D. White, The classical double copy for Taub-NUT spacetime, Physics Letters B 750 (Nov., 2015) 272–277, [arXiv:1507.01869].
  * (6) A. Luna, R. Monteiro, I. Nicholson, A. Ochirov, D. O’Connell, N. Westerberg, and C. D. White, Perturbative spacetimes from Yang-Mills theory, Journal of High Energy Physics 2017 (Apr., 2017) 69, [arXiv:1611.07508].
  * (7) R. Monteiro, I. Nicholson, and D. O’Connell, Spinor-helicity and the algebraic classification of higher-dimensional spacetimes, Classical and Quantum Gravity 36 (Mar., 2019) 065006, [arXiv:1809.03906].
  * (8) A. K. Ridgway and M. B. Wise, Static Spherically Symmetric Kerr-Schild Metrics and Implications for the Classical Double Copy, Physical Review D 94 (Aug., 2016) 044023, [arXiv:1512.02243].
  * (9) T. Adamo, E. Casali, L. Mason, and S. Nekovar, Scattering on plane waves and the double copy, Classical and Quantum Gravity 35 (Jan., 2018) 015004, [arXiv:1706.08925].
  * (10) N. Bahjat-Abbas, A. Luna, and C. D. White, The Kerr-Schild double copy in curved spacetime, Journal of High Energy Physics 2017 (Dec., 2017) 4, [arXiv:1710.01953].
  * (11) M. Carrillo-Gonzalez, R. Penco, and M. Trodden, The classical double copy in maximally symmetric spacetimes, arXiv:1711.01296 [gr-qc, physics:hep-th] (Apr., 2018) [arXiv:1711.01296].
  * (12) A. Ilderton, Screw-symmetric gravitational waves: A double copy of the vortex, May, 2018.
  * (13) M. Gurses and B. Tekin, Classical Double Copy: Kerr-Schild-Kundt metrics from Yang-Mills Theory, Physical Review D 98 (Dec., 2018) 126017, [arXiv:1810.03411].
  * (14) M. C. González, B. Melcher, K. Ratliff, S. Watson, and C. D. White, The classical double copy in three spacetime dimensions, Apr., 2019.
  * (15) K. Lee, Kerr-Schild Double Field Theory and Classical Double Copy, Journal of High Energy Physics 2018 (Oct., 2018) 27, [arXiv:1807.08443].
  * (16) I. Bah, R. Dempsey, and P. Weck, Kerr-Schild Double Copy and Complex Worldlines, Journal of High Energy Physics 2020 (Feb., 2020) 180, [arXiv:1910.04197].
  * (17) W. D. Goldberger and J. Li, Strings, extended objects, and the classical double copy, Journal of High Energy Physics 2020 (Feb., 2020) 92, [arXiv:1912.01650].
  * (18) K. Kim, K. Lee, R. Monteiro, I. Nicholson, and D. P. Veiga, The Classical Double Copy of a Point Charge, Journal of High Energy Physics 2020 (Feb., 2020) 46, [arXiv:1912.02177].
  * (19) A. Luna, S. Nagy, and C. White, The convolutional double copy: A case study with a point, Journal of High Energy Physics 2020 (Sept., 2020) 62, [arXiv:2004.11254].
  * (20) C. Keeler, T. Manton, and N. Monga, From Navier-Stokes to Maxwell via Einstein, arXiv:2005.04242 [gr-qc, physics:hep-th] (May, 2020) [arXiv:2005.04242].
  * (21) D. A. Easson, C. Keeler, and T. Manton, The classical double copy of non-singular black holes, Physical Review D 102 (Oct., 2020) 086015, [arXiv:2007.16186].
  * (22) D. S. Berman, E. Chacón, A. Luna, and C. D. White, The self-dual classical double copy, and the Eguchi-Hanson instanton, Journal of High Energy Physics 2019 (Jan., 2019) 107, [arXiv:1809.04063].
  * (23) G. Alkac, M. K. Gumus, and M. A. Olpak, Kerr-Schild Double Copy of the Coulomb Solution in Three Dimensions, Physical Review D 104 (May, 2021) 044034, [arXiv:2105.11550].
  * (24) K. Mkrtchyan and M. Svazas, Solutions in Nonlinear Electrodynamics and their double copy regular black holes, JHEP 09 (May, 2022) 012, [arXiv:2205.14187].
  * (25) T. Adamo, A. Cristofoli, and P. Tourkine, The ultrarelativistic limit of Kerr, Sept., 2022.
  * (26) L. Alfonsi, C. D. White, and S. Wikeley, Topology and Wilson lines: Global aspects of the double copy, Journal of High Energy Physics 2020 (July, 2020) 91, [arXiv:2004.07181].
  * (27) T. Adamo and A. Ilderton, Classical and quantum double copy of back-reaction, Journal of High Energy Physics 2020 (Sept., 2020) 200, [arXiv:2005.05807].
  * (28) R. Gonzo and C. Shi, Geodesics From Classical Double Copy, arXiv:2109.01072 [gr-qc, physics:hep-th] (Nov., 2021) [arXiv:2109.01072].
  * (29) D. A. Kosower, R. Monteiro, and D. O’Connell, The SAGEX Review on Scattering Amplitudes, Chapter 14: Classical Gravity from Scattering Amplitudes, arXiv:2203.13025 [hep-th] (Mar., 2022) [arXiv:2203.13025].
  * (30) W. Chen, H. Lu, and C. N. Pope, General Kerr-NUT-AdS Metrics in All Dimensions, hep-th/0604125.
  * (31) D. Kubiznak and V. P. Frolov, Hidden Symmetry of Higher Dimensional Kerr-NUT-AdS Spacetimes, Dec., 2006.
  * (32) V. P. Frolov, P. Krtous, and D. Kubiznak, Black holes, hidden symmetries, and complete integrability, Living Reviews in Relativity 20 (Dec., 2017) 6, [arXiv:1705.05482].
  * (33) P. Krtouš, V. P. Frolov, and D. Kubizňák, Separation of Maxwell equations in Kerr-NUT-(A)dS spacetimes, Nuclear Physics B 934 (Sept., 2018) 7–38, [arXiv:1803.02485].
  * (34) L. Mason and A. Taghavi-Chabert, Killing-Yano tensors and multi-hermitian structures, Journal of Geometry and Physics 60 (June, 2010) 907–923, [arXiv:0805.3756].
  * (35) P. Krtous, V. P. Frolov, and D. Kubiznak, Hidden Symmetries of Higher Dimensional Black Holes and Uniqueness of the Kerr-NUT-(A)dS spacetime, Physical Review D 78 (Sept., 2008) 064022, [arXiv:0804.4705].
  * (36) P. Krtous, Electromagnetic Field in Higher-Dimensional Black-Hole Spacetimes, Physical Review D 76 (Oct., 2007) 084035, [arXiv:0707.0002].
  * (37) A. Luna, R. Monteiro, I. Nicholson, and D. O’Connell, Type D Spacetimes and the Weyl Double Copy, Classical and Quantum Gravity 36 (Mar., 2019) 065003, [arXiv:1810.08183].
  * (38) H. Godazgar, M. Godazgar, R. Monteiro, C. N. Pope, and D. P. Veiga, The Weyl Double Copy for Gravitational Waves, arXiv:2010.02925 [gr-qc, physics:hep-th] (Oct., 2020) [arXiv:2010.02925].
  * (39) G. Elor, K. Farnsworth, M. L. Graesser, and G. Herczeg, The Newman-Penrose Map and the Classical Double Copy, July, 2020.
  * (40) K. Farnsworth, M. L. Graesser, and G. Herczeg, Twistor Space Origins of the Newman-Penrose Map, July, 2022.
  * (41) C. D. White, A Twistorial Foundation for the Classical Double Copy, arXiv:2012.02479 [gr-qc, physics:hep-ph, physics:hep-th, physics:math-ph] (Dec., 2020) [arXiv:2012.02479].
  * (42) E. Chacón, S. Nagy, and C. D. White, The Weyl double copy from twistor space, arXiv:2103.16441 [gr-qc, physics:hep-ph, physics:hep-th] (Mar., 2021) [arXiv:2103.16441].
  * (43) E. Chacón, S. Nagy, and C. D. White, Alternative formulations of the twistor double copy, Journal of High Energy Physics 2022 (Mar., 2022) 180, [arXiv:2112.06764].
  * (44) S. Han, Weyl double copy and massless free fields in curved spacetimes, Apr., 2022.
  * (45) D. A. Easson, T. Manton, and A. Svesko, Sources in the Weyl double copy, arXiv:2110.02293 [gr-qc, physics:hep-th] (Dec., 2021) [arXiv:2110.02293].
  * (46) A. Luna, N. Moynihan, and C. D. White, Why is the Weyl double copy local in position space?, Aug., 2022.
  * (47) R. Monteiro, D. O’Connell, D. P. Veiga, and M. Sergola, Classical Solutions and their Double Copy in Split Signature, Journal of High Energy Physics 2021 (May, 2021) 268, [arXiv:2012.11190].
  * (48) R. Alawadhi, D. S. Berman, and B. Spence, Weyl doubling, Journal of High Energy Physics 2020 (Sept., 2020) 127, [arXiv:2007.03264].
  * (49) S. G. Prabhu, The classical double copy in curved spacetimes: Perturbative Yang-Mills from the bi-adjoint scalar, arXiv:2011.06588 [gr-qc, physics:hep-th] (Nov., 2020) [arXiv:2011.06588].
  * (50) G. Alkac, M. K. Gumus, and M. Tek, The Kerr-Schild Double Copy in Lifshitz Spacetime, June, 2021.
  * (51) S. Han, The Weyl double copy in maximally symmetric spacetimes, arXiv:2205.08654.
  * (52) R. Alawadhi, D. S. Berman, B. Spence, and D. P. Veiga, S-duality and the Double Copy, Journal of High Energy Physics 2020 (Mar., 2020) 59, [arXiv:1911.06797].
  * (53) T. Houri, T. Oota, and Y. Yasui, Closed conformal Killing-Yano tensor and Kerr-NUT-de Sitter spacetime uniqueness, Physics Letters B 656 (Nov., 2007) 214–216, [arXiv:0708.1368].
  * (54) A. Coley, R. Milson, V. Pravda, and A. Pravdova, Classification of the Weyl Tensor in Higher Dimensions, Classical and Quantum Gravity 21 (Apr., 2004) L35–L41, [gr-qc/0401008].
  * (55) A. Coley, Classification of the Weyl Tensor in Higher Dimensions and Applications, arXiv:0710.1598.
  * (56) V. Pravda, A. Pravdova, and M. Ortaggio, Type D Einstein spacetimes in higher dimensions, arXiv:0704.0435.
  * (57) M. Ortaggio, V. Pravda, and A. Pravdova, Algebraic classification of higher dimensional spacetimes based on null alignment, Nov., 2012.
  * (58) R. Milson, A. Coley, V. Pravda, and A. Pravdova, Alignment and algebraically special tensors in Lorentzian geometry, International Journal of Geometric Methods in Modern Physics 02 (Feb., 2005) 41–61, [gr-qc/0401010].
  * (59) N. Hamamoto, T. Houri, T. Oota, and Y. Yasui, Kerr-NUT-de Sitter Curvature in All Dimensions, Journal of Physics A: Mathematical and Theoretical 40 (Feb., 2007) F177–F184, [hep-th/0611285].
  * (60) H. Godazgar, M. Godazgar, R. Monteiro, D. P. Veiga, and C. N. Pope, Asymptotic Weyl Double Copy, Journal of High Energy Physics 2021 (Nov., 2021) 126, [arXiv:2109.07866].
  * (61) N. Bahjat-Abbas, R. Stark-Muchão, and C. D. White, Monopoles, shockwaves and the classical double copy, Journal of High Energy Physics 2020 (Apr., 2020) 102, [arXiv:2001.09918].
  * (62) N. Arkani-Hamed, Y.-t. Huang, and D. O’Connell, Kerr Black Holes as Elementary Particles, Journal of High Energy Physics 2020 (Jan., 2020) 46, [arXiv:1906.10100].
  * (63) A. Banerjee, E. Ó. Colgáin, J. A. Rosabal, and H. Yavartanoo, Ehlers as EM duality in the double copy, Physical Review D 102 (Dec., 2020) 126017, [arXiv:1912.02597].
  * (64) Y.-t. Huang, U. Kol, and D. O’Connell, The Double Copy of Electric-Magnetic Duality, Physical Review D 102 (Aug., 2020) 046005, [arXiv:1911.06318].
  * (65) W. T. Emond, Y.-t. Huang, U. Kol, N. Moynihan, and D. O’Connell, Amplitudes from Coulomb to Kerr-Taub-NUT, arXiv:2010.07861 [gr-qc, physics:hep-th] (Oct., 2020) [arXiv:2010.07861].
  * (66) R. Alawadhi, Single copy of the Ricci flow, arXiv:2202.09874.
  * (67) R. Alawadhi, D. S. Berman, C. D. White, and S. Wikeley, The single copy of the gravitational holonomy, Journal of High Energy Physics 2021 (July, 2021) 229, [arXiv:2107.01114].
  * (68) A. Luna, R. Monteiro, I. Nicholson, D. O’Connell, and C. D. White, The double copy: Bremsstrahlung and accelerating black holes, Journal of High Energy Physics 2016 (June, 2016) 23, [arXiv:1603.05737].
  * (69) R. Gonzo, T. McLoughlin, and A. Puhm, Celestial holography on Kerr-Schild backgrounds, July, 2022.
  * (70) E. Chacón, A. Luna, and C. D. White, The double copy of the multipole expansion, arXiv:2108.07702 [gr-qc, physics:hep-th, physics:math-ph] (Aug., 2021) [arXiv:2108.07702].
  * (71) T. Adamo and U. Kol, Classical double copy at null infinity, Nov., 2021.
  * (72) M. Campiglia and S. Nagy, A double copy for asymptotic symmetries in the self-dual sector, Journal of High Energy Physics 2021 (Mar., 2021) 262, [arXiv:2102.01680].