aainstitutetext: Department of Physics, Arizona State University, Tempe, AZ
85287-1504, USAbbinstitutetext: Brown Theoretical Physics Center, Department
of Physics, Brown University, Providence, RI 02912, USA

# Isometries and the double copy

Damien A. Easson b Gabriel Herczeg b Tucker Manton a,b and Max Pezzelle

###### Abstract

In the standard derivation of the Kerr-Schild double copy, the geodicity of
the Kerr-Schild vector and the stationarity of the spacetime are presented as
assumptions that are necessary for the single copy to satisfy Maxwell’s
equations. However, it is well known that the vacuum Einstein equations
_imply_ that the Kerr-Schild vector is geodesic and shear-free, and that the
spacetime possesses a distinguished vector field that is simultaneously a
Killing vector of the full spacetime and the flat background, but need not be
timelike with respect to the background metric. We show that the gauge field
obtained by contracting this distinguished Killing vector with the Kerr-Schild
graviton solves the vacuum Maxwell equations, and that this definition of the
Kerr-Schild double copy implies the Weyl double copy when the spacetime is
Petrov type D. When the Killing vector is taken to be timelike with respect to
the background metric, we recover the familiar Kerr-Schild double copy, but
the prescription is well defined for any vacuum Kerr-Schild spacetime and we
present new examples where the Killing vector is null or spacelike. While most
examples of physical interest are type D, vacuum Kerr-Schild spacetimes are
generically of Petrov type II. We present a straightforward example of such a
spacetime and study its double copy structure. Our results apply to real
Lorentzian spacetimes as well as complex spacetimes and real spacetimes with
Kleinian signature, and provide a simple correspondence between real and self-
dual vacuum Kerr-Schild spacetimes. This correspondence allows us to study the
double copy structure of a self-dual analog of the Kerr spacetime. We provide
evidence that this spacetime may be diffeomorphic to the self-dual Taub-NUT
solution.

##  1 Introduction

Our understanding of the interplay between gauge and gravity theories has
progressed considerably in the past few decades. Notably, a duality called the
double copy has attracted significant interest. The double copy has its
historical roots in the Kawai-Lewellen-Tye (KLT) relations Kawai:1985xq ,
which provide a procedure for obtaining tree-level scattering amplitudes of
closed strings from two open string amplitudes. In the field theory limit,
excitations of closed and open strings correspond to gravitons and non-Abelian
gauge fields respectively, suggesting that amplitudes describing spin-1 and
spin-2 theories are intimately related.

Decades later, Bern, Carrasco and Johansson introduced the notion of color-
kinematics duality Bern:2008qj , which utilizes a kinematic analog of the
Jacobi identity for color factors to recast gravity tree amplitudes in terms
of QCD amplitudes, shedding new light on the KLT relations, and initiating a
program that has become the state of the art in perturbative gravity, with
applications ranging from higher loop graviton scattering Bern:2012uf  to
efficiently computing conservative potentials for compact binary systems
bern2019black .

To briefly summarize, the Yang-Mills amplitude can be written schematically as
Bern:2019prr ; Monteiro:2014cda
$\mathcal{A}\sim\sum_{i}\frac{n_{i}c_{i}}{\text{props}}$. The functions
$n_{i}$ are so-called kinematic numerators (functions of polarizations and
momenta), $c_{i}$ are color factors (contracted products of gauge theory
structure constants), and the ‘props’ denominator contains all relevant
propagator momenta. From gauge-invariance, it follows that the color factors
satisfy the Jacobi identity $c_{i}+c_{j}+c_{k}=0.$ The key observation is that
when the Feynman diagrams are organized such that a set of kinematic
numerators, say $\tilde{n}_{i}$, satisfy a Jacobi identity analogously to the
color factors, $\tilde{n}_{i}+\tilde{n}_{j}+\tilde{n}_{k}=0,$ then the $c_{i}$
can be replaced with the $\tilde{n}_{i}$ to obtain
$\mathcal{M}\sim\sum_{i}\frac{n_{i}\tilde{n}_{i}}{\text{props}}$. This object
is the graviton amplitude corresponding to the graviton scattering process at
the same loop order as the Yang-Mills amplitude. Since we replaced the color
factor with a copy of the kinematic numerator, $\mathcal{M}$ is referred to as
the double copy of $\mathcal{A}$, and $\mathcal{A}$ is referred to as the
single copy of $\mathcal{M}$. It can be shown that the Jacobi identity for the
kinematic numerators arises due to linear diffeomorphism invariance, as
expected for any genuine graviton amplitude Bern:2019prr . We can instead
strip the kinematic numerator from $\mathcal{A}$ and replace it with a second
color factor $\tilde{c}_{i}$ (again satisfying the Jacobi identity) to obtain
the gauge-invariant object
$\mathcal{T}\sim\sum_{i}\frac{c_{i}\tilde{c}_{i}}{\text{props}}$. This
describes the scattering of scalar fields $\phi^{aa^{\prime}}$, which
transform in the adjoint representation of two Lie algebras. The scalar fields
are referred to as the zeroth copy and are described by the cubic Lagrangian
$\mathcal{L}_{\text{scalar}}=\frac{1}{2}\partial_{\mu}\phi^{aa^{\prime}}\partial^{\mu}\phi_{aa^{\prime}}+\frac{y}{3}f_{abc}f_{a^{\prime}b^{\prime}c^{\prime}}\phi^{aa^{\prime}}\phi^{bb^{\prime}}\phi^{cc^{\prime}}.$

The utility of the double copy has become increasingly clear as the program
has continued to advance. It has played a central role in computing four-loop
amplitudes in $\mathcal{N}=4$ super Yang-Mills and $\mathcal{N}=8$
supergravity Bern:2012uf , and it has taken on a surprisingly prominent role
in computing gravitational observables Goldberger:2017frp ; Luna:2017dtq ;
Goldberger:2017vcg ; Shen:2018ebu ; Cheung:2018wkq ; Kosower:2018adc ;
Bern:2019nnu ; Antonelli:2019ytb ; Bern:2019crd ; Kalin:2019rwq , which has
become more important than ever due to the discovery of gravitational waves
LIGOScientific:2016aoc ; LIGOScientific:2017vwq . Despite these significant
advances, a fundamental theoretical understanding of the double copy remains
elusive. While some efforts have been made to develop a Lagrangian formulation
of the double copy Bern:2010yg ; Tolotti:2013caa ; Cheung:2016say , there is
much that remains obscure regarding why the double copy works, and its scope
of applicability.

One approach to grappling with these shortcomings is to try to understand the
double copy at the classical level. The first attempt to understand the
classical aspects of the double copy was initiated by Monteiro, O’Connell and
White in Monteiro:2014cda , where it was observed that a particular subset of
solutions of Einsteins equations—those of the Kerr-Schild form—could be mapped
to solutions of the flat-space Maxwell equations using a procedure that
closely mimics color-kinematics duality. Specifically, associated to a
stationary solution of the Einstein equations that admits the decomposition

$g_{\mu\nu}=g^{(0)}_{\mu\nu}+\phi k_{\mu}k_{\nu},$ (1)

where $g^{(0)}_{\mu\nu}$ is a background metric usually taken to be flat space
and the vector $k_{\mu}$ is null and geodesic with respect to $g^{\mu\nu}$,
and $g_{(0)}^{\mu\nu}$ is a gauge potential

$\displaystyle A_{\mu}^{a}=T^{a}\phi k_{\mu}$ (2)

that is a solution to the linearized Yang-Mills equations (Maxwell theory)
over $g_{\mu\nu}^{(0)}$. The color factor $T^{a}$ is trivial in Maxwell
theory. Identifying the linear graviton as $\kappa h_{\mu\nu}=\phi
k_{\mu}k_{\nu},$ we observe that the gauge potential is obtained from the
graviton by replacing a null vector $k_{\mu}$ with a color factor $T^{a}$,
analogous to replacing a kinematic numerator in the graviton amplitude
$\mathcal{M}$ with a color factor to obtain the Yang-Mills amplitude
$\mathcal{A}$. We can further replace the second null vector with another
color vector $T^{a^{\prime}}$ to obtain a scalar
$\phi^{aa^{\prime}}=T^{a}T^{a^{\prime}}\phi,$ which is a solution to the flat
space wave equation $\Box^{(0)}\phi^{aa^{\prime}}=0,$ i.e. the linearized bi-
adjoint scalar theory.111In the main text, we slightly alter the notation for
the zeroth copy in order to stay consistent with our more general approach to
the copying program. We will also neglect the passive color indices, as is
common in the classical double copy literature. We note that an alternative
but equivalent prescription for identifying the single and zeroth copy fields
from the graviton for stationary spacetimes is by simply contracting the
timelike Killing vector $X=\partial_{t}$ as $A_{\mu}=h_{\mu\nu}X^{\nu}$,
$\phi=h_{\mu\nu}X^{\mu}X^{\nu}$. Such an approach was considered for timelike
and null Killing vectors for stationary and wave solutions respectively in,
e.g., Carrillo-Gonzalez:2017iyj ; Bah:2019sda , and a special ‘screw’ Killing
vector played a significant role in the analysis in Ilderton:2018lsf . This
perspective is particularly useful for the more general prescription that we
present in section 4, which we will elaborate on momentarily.

A complementary realization of the classical double copy was presented in
Luna:2018dpt , and is referred to as the Weyl double copy. The Weyl double
copy is formulated using spinorial language where the key gravitational
variable is the completely symmetric Weyl spinor $\Psi_{ABCD}$. The original
construction of the Weyl double copy applies to Petrov type D and type N
vacuum spacetimes, and the statement is that for a Weyl spinor associated to
such a vacuum solution, the relationship

$\Psi_{ABCD}=\frac{1}{S}\varphi_{(AB}\varphi_{CD)}$ (3)

identifies a Maxwell spinor $\varphi_{AB}$ and (complex) scalar field $S$ that
respectfully satisfy the vacuum Maxwell and wave equations about an
appropriate flat limit of the full spacetime. The Weyl double copy has since
been shown to emerge from the so-called Penrose transform using the twistor
formalism White:2020sfn ; Chacon:2021wbr ; Chacon:2021lox ;
CarrilloGonzalez:2022ggn  and can be applied to higher Petrov types, and more
recently, it has been applied to certain non-vacuum solutions using both the
twistor Armstrong-Williams:2023ssz  and spinor Easson:2021asd ; Easson:2022zoh
constructions.

Since the appearance of Monteiro:2014cda , numerous other approaches to the
classical double copy have proliferated Luna:2015paa ; White:2016jzc ;
Goldberger:2016iau ; DeSmet:2017rve ; Adamo:2017nia ; Luna:2018dpt ;
Berman:2018hwd ; Lee:2018gxc ; Gurses:2018ckx ; Bahjat-Abbas:2018vgo ;
Bah:2019sda ; Cho:2019ype ; Alawadhi:2019urr ; Banerjee:2019saj ; Kim:2019jwm
; Huang:2019cja ; Arkani-Hamed:2019ymq ; CarrilloGonzalez:2019gof ;
Luna:2020adi ; Keeler:2020rcv ; Easson:2020esh ; Elor:2020nqe ;
prabhu2020classical ; alawadhi2020s ; cheung2020scattering ; delaCruz:2020bbn
; Emond:2020lwi ; White:2020sfn ; 1969JMP….10.1842D ; Godazgar:2020zbv ;
Chacon:2020fmr ; Berman:2020xvs ; Ben-Shahar:2021zww ; Monteiro:2021ztt ;
guevara2021reconstructing ; Gonzalez:2021bes ; Chacon:2021hfe ; Cho:2021nim ;
Chacon:2021wbr ; Easson:2021asd ; Campiglia:2021srh ; Farnsworth:2021wvs ;
guevara2021worldsheet ; Adamo:2021dfg ; Godazgar:2021iae ; Kosower:2022yvp ;
CarrilloGonzalez:2022mxx ; Alkac:2021seh ; Alkac:2021bav ; Chacon:2021lox ;
Chen:2021chy ; Gonzalez:2021ztm ; Angus:2021zhy ; Moynihan:2021rwh ;
Alkac:2022tvc ; Han:2022ubu ; Mao:2021kxq ; Nagy:2022xxs ; Armstrong-
Williams:2022apo ; CarrilloGonzalez:2022ggn ; Ben-Shahar:2022ixa ;
Didenko:2022qxq ; Easson:2022zoh ; Chawla:2022ogv ; Armstrong-Williams:2023ssz
; chawla2023black ; lipstein2023self ; gonzalez2023mini ; Bonezzi:2023pox ;
Brown:2023zxm ; Borsten:2023paw ; Ball:2023xnr  —however, the main purpose of
this work is to revisit the original Kerr-Schild double copy of
Monteiro:2014cda  with the goal of clarifying some technical points that have
been overlooked up until now. In particular, the geodicity of the Kerr-Schild
vector and the existence of a timelike Killing vector were treated as
assumptions in the original presentation of the Kerr-Schild double copy.
However, the geodicity of the Kerr-Schild vector and the existence of a
Killing vector are _consequences_ of the vacuum conditions Stephani:2003tm ;
Bini:2010hrs . This Killing vector can be spacelike, timelike or null with
respect to the background flat metric, and the timelike character of the
Killing vector is not necessary to define the double copy. This leads us to
propose a modified definition of the Kerr-Schild double copy for four-
dimensional vacuum spacetimes that applies to all real vacuum spacetimes with
Lorentzian signature222Technically, our definition only applies to spacetimes
with an expanding Kerr-Schild vector, a caveat that excludes PP wave
spacetimes. as well as a subset of complex vacuum spacetimes and real
spacetimes with Kleinian signature within the Kerr-Schild class. Moreover, the
modified definition of the Kerr-Schild double copy that we present implies the
Weyl double copy Luna:2018dpt  when either the self-dual or anti-self-dual
part of the Weyl tensor is type D. The modified definition we propose is very
simple: denoting the distinguished Killing vector associated with any vacuum
Kerr-Schild spacetime by $X^{\mu}$ and the graviton by
$h_{\mu\nu}:=g_{\mu\nu}-\eta_{\mu\nu}$, the proposed single copy gauge field
and zeroth copy scalar field are $A_{\mu}=h_{\mu\nu}X^{\nu}$ and
$\phi=h_{\mu\nu}X^{\mu}X^{\nu}$.

Vacuum Kerr-Schild spacetimes can be characterized very conveniently in terms
of a suitable null tetrad using the spin coefficient formalism of Newman and
Penrose. Since this formalism may not be entirely familiar to our audience, we
devote section §2 to a brief review of this material with a particular
emphasis on the role played by differential forms. In section §3 we apply this
formalism to vacuum Kerr-Schild spacetimes, and in section §4 we present our
proposal for defining the Kerr-Schild double copy for vacuum spacetimes in
terms of the distinguished null vector admitted by any such spacetime, and
show that this proposal leads to single and zeroth copies that satisfy the
vacuum Maxwell equations and wave equation respectively. In section §5 we show
that our proposal for the Kerr-Schild double copy for vacuum spacetimes
implies the Weyl double copy when the left or right Weyl spinor is Petrov type
D. Finally, in section §6 we provide a number of examples of spacetimes that
can be studied with our approach, including spacetimes with Killing vectors
that are spacelike or null with respect to the background flat metric, self-
dual spacetimes, and a simple, explicit example with Petrov type II.

##  2 Spin coefficient formalism

Our approach to expanding on the Kerr-Schild double copy will make use of a
version of the spin coefficient formalism that emphasizes the role of
differential forms mcintosh1985complex . It is somewhat regrettable that this
formalism seems to be the one that is best suited for the problems we wish to
address, since we expect that it is not well known to many of the readers who
may be most interested in our results. Nevertheless, we feel obliged to employ
it for three compelling reasons:

  1. 1.

It is the only formalism that admits an explicit procedure for constructing a
broad class of vacuum Kerr-Schild spacetimes.

  2. 2.

It provides a very natural language for discussing the connection between the
Kerr-Schild and Weyl double copies.

  3. 3.

It provides a new framework for studying self-dual solutions in the context of
the double copy.

Throughout the article, we will restrict our attention to four-dimensional
spacetimes. However, we will allow spacetime to be complex in general, and we
make use of the convention that quantities decorated with a tilde should be
understood as the complex conjugate of the corresponding quantity when the
spacetime is real and Lorentzian, and otherwise should be regarded as
independent. We will refer to quantities without tildes as having _left_
chirality, while quantities with tildes will be said to have _right_
chirality. The primary object of study in this formalism is a null co-tetrad
$\theta^{a}$, where $a$ runs from 1 to 4. The metric can be recovered from the
co-tetrad as

$ds^{2}=2(\theta^{1}\theta^{2}-\theta^{3}\theta^{4}).$ (4)

The co-tetrad is related to the standard tetrad
$\lambda_{a}=(l,n,m,\tilde{m})$ by
$\theta^{a}_{\mu}\lambda_{b}^{\mu}=\delta^{a}_{b}$, and obeys

$\theta^{1}_{\mu}=n_{\mu},\qquad\theta^{2}_{\mu}=l_{\mu},\qquad\theta^{3}_{\mu}=-\tilde{m}_{\mu},\qquad\theta^{4}_{\mu}=\tilde{\theta}^{3}_{\mu}=-m_{\mu}.$
(5)

We also define the directional derivatives

$D=l^{\mu}\partial_{\mu},\qquad\Delta=n^{\mu}\partial_{\mu},\qquad\delta=m^{\mu}\partial_{\mu},\qquad\tilde{\delta}=\tilde{m}^{\mu}\partial_{\mu}.$
(6)

The exterior derivative can be expressed in terms of the co-tetrad and
directional derivatives as

$d=\theta^{1}D+\theta^{2}\Delta+\theta^{3}\delta+\theta^{4}\tilde{\delta}.$
(7)

It is often convenient to expand differential forms in terms of wedge products
of the co-tetrad one-forms, for which we use the compact notation

$\theta^{ab...d}\equiv\theta^{a}\wedge\theta^{b}\wedge\ldots\wedge\theta^{d}.$
(8)

It is also often useful to decompose two-forms into self-dual and anti-self-
dual parts. For this, we define the following basis of two-forms:

$Z^{0}=\theta^{13},\qquad Z^{1}=\theta^{12}-\theta^{34},\qquad
Z^{2}=\theta^{42},$ (9)
$\tilde{Z}^{0}=\theta^{14},\qquad\tilde{Z}^{1}=\theta^{12}+\theta^{34},\qquad\tilde{Z}^{2}=\theta^{32},$
(10)

where $Z^{j}$ are self-dual and $\tilde{Z}^{j}$ are anti-self-dual—i.e.
$*Z^{j}=iZ^{j}$ and $*\tilde{Z}^{j}=-i\tilde{Z}^{j}$.333Note that this self-
duality condition is appropriate for real spacetimes with Lorentzian
signature. For other cases, this condition must be modified accordingly.

Now the spin coefficients can be computed either from the first Cartan
structure equations

$\displaystyle d\theta^{1}$ $\displaystyle=$
$\displaystyle(\gamma+\tilde{\gamma})\theta^{12}+(\tilde{\alpha}+\beta-\tilde{\pi})\theta^{13}+(\alpha+\tilde{\beta}-\pi)\theta^{14}-\tilde{\nu}\theta^{23}-\nu\theta^{24}-(\mu-\tilde{\mu})\theta^{34}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle d\theta^{2}$ $\displaystyle=$
$\displaystyle(\epsilon+\tilde{\epsilon})\theta^{12}+(\tau-\tilde{\alpha}-\beta)\theta^{23}+(\tilde{\tau}-\alpha-\tilde{\beta})\theta^{24}+\kappa\theta^{13}+\tilde{\kappa}\theta^{14}-(\rho-\tilde{\rho})\theta^{34}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle d\theta^{3}$ $\displaystyle=$
$\displaystyle-(\tilde{\tau}+\pi)\theta^{12}-(\tilde{\rho}+\epsilon-\tilde{\epsilon})\theta^{13}-\tilde{\sigma}\theta^{14}+(\mu+\tilde{\gamma}-\gamma)\theta^{23}+\lambda\theta^{24}+(\alpha-\tilde{\beta})\theta^{34}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle d\theta^{4}$ $\displaystyle=$
$\displaystyle-(\tau+\tilde{\pi})\theta^{12}-(\rho+\tilde{\epsilon}-\epsilon)\theta^{14}-\sigma\theta^{13}+(\tilde{\mu}+\gamma-\tilde{\gamma})\theta^{24}+\tilde{\lambda}\theta^{23}-(\tilde{\alpha}-\beta)\theta^{34}~{}~{}~{}~{}~{}~{}~{}~{}$
(11)

or from the the bivector equations444An earlier version of this article
contained a number of typographical errors in equations (11)-(12). We are
grateful to Michael Graesser for bringing this to our attention.

$\displaystyle dZ^{0}$ $\displaystyle=$
$\displaystyle(2\gamma-\mu)\theta^{123}-\lambda\theta^{124}+(\pi-2\alpha)\theta^{134}+\nu\theta^{234}$
$\displaystyle dZ^{1}$ $\displaystyle=$
$\displaystyle-2\tau\theta^{123}+2\pi\theta^{124}+2\rho\theta^{134}-2\mu\theta^{234}$
$\displaystyle dZ^{2}$ $\displaystyle=$
$\displaystyle\sigma\theta^{123}+(\rho-2\epsilon)\theta^{124}-\kappa\theta^{134}+(2\beta-\tau)\theta^{234},$
(12)

along with the corresponding expressions for $d\tilde{Z}^{j}$. For real
spacetimes, the bivector equations for the self-dual two-forms $Z^{j}$ are
sufficient to determine all of the spin coefficients, while for complex
spacetimes the bivector equations conveniently decouple the left and right
spin coefficients.

The Newman-Penrose equations, which encode the dynamics of general relativity
in the spin coefficient formalism, and the Bianchi identities can also be cast
in terms of equations involving bivectors and spin coefficients, but since we
will not use them, we refer the reader to mcintosh1985complex  for details.
The Newman-Penrose equations and Bianchi identities can be found in
newman1962approach ; Stephani:2003tm  and many other places, so we will not
record them here. We will now examine the application of this formalism to a
broad class of vacuum Kerr-Schild spacetimes.

##  3 Vacuum Kerr-Schild spacetimes

We now direct our attention to Kerr-Schild spacetimes, which are characterized
by a metric that can be expressed in the following form

$g_{\mu\nu}=g^{(0)}_{\mu\nu}+2Vl_{\mu}l_{\nu},$ (13)

where $g^{(0)}_{\mu\nu}$ is a flat metric, $V$ is a scalar function, and
$l_{\mu}=\theta^{2}_{\mu}$ is null, and expanding.555In terms of spin
coefficients, the condition for $l_{\mu}$ to be geodesic is
$\kappa=\tilde{\kappa}=0$, the shear-free condition is
$\sigma=\tilde{\sigma}=0$, and the expanding condition is
$\rho+\tilde{\rho}\neq 0$. We also note that we have employed a slightly
different notation for the Kerr-Schild form in (13)compared with (1), because
here we use a different scaling for the Kerr-Schild vector than the one
typically used in the Kerr-Schild double copy. The vacuum Einstein equations
then imply that $l_{\mu}$ is also geodesic and shear-free with respect to both
the full metric and the background flat metric (see theorem (32.1) of
Stephani:2003tm  and Chapter 6 section 57 of Chandrasekhar:1985kt ). Note that
$l_{\mu}$ is null with respect to $g_{\mu\nu}$ if and only if it is null with
respect to $g^{(0)}_{\mu\nu}$. Much of what appears in this section can be
found in mcintosh1988single ; mcintosh1989kerr  as well as chapter 32 of
Stephani:2003tm . A co-tetrad corresponding to the metric (13) is given by

$\displaystyle\theta^{1}$ $\displaystyle=$ $\displaystyle dv+V\theta^{2}$
$\displaystyle\theta^{2}$ $\displaystyle=$ $\displaystyle
du+\tilde{Y}d\zeta+Yd\tilde{\zeta}+Y\tilde{Y}dv$ $\displaystyle\theta^{3}$
$\displaystyle=$ $\displaystyle d\tilde{\zeta}+\tilde{Y}dv$
$\displaystyle\theta^{4}$ $\displaystyle=$ $\displaystyle d\zeta+Ydv.$ (14)

where $Y$, $\tilde{Y}$ and $V$ are generically complex functions, and all of
the coordinates $u$, $v$, $\zeta$, $\tilde{\zeta}$ are generically complex and
independent. When the spacetime is real, $Y$ and $\tilde{Y}$ are complex
conjugates and V is real; $u,v$ are real light-cone coordinates, and
$\zeta,\tilde{\zeta}$ are complex conjugate coordinates related to the usual
Cartesian coordinates $(t,x,y,z)$ by

$u=\tfrac{1}{\sqrt{2}}(t-z),\quad
v=\tfrac{1}{\sqrt{2}}(t+z),\quad\zeta=\tfrac{1}{\sqrt{2}}(x+iy).$ (15)

In these coordinates, the flat metric appearing in (13) is

$ds_{0}^{2}=g^{(0)}_{\mu\nu}dx^{\mu}dx^{\nu}=2(dudv-d\zeta d\tilde{\zeta}).$
(16)

The directional derivatives associated with (14) are

$\displaystyle D$ $\displaystyle=$
$\displaystyle\partial_{v}-Y\partial_{\zeta}-\tilde{Y}\partial_{\tilde{\zeta}}+Y\tilde{Y}\partial_{u}\,$
$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle\partial_{u}-VD\,$
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\partial_{\tilde{\zeta}}-Y\partial_{u}\,$
$\displaystyle\tilde{\delta}$ $\displaystyle=$
$\displaystyle\partial_{\zeta}-\tilde{Y}\partial_{u}.$ (17)

The conditions that $l_{\mu}=\theta^{2}_{\mu}$ be geodesic and shear-free are
encoded in the vanishing of the spin coefficients $\kappa=DY$, $\sigma=\delta
Y$, $\tilde{\kappa}=D\tilde{Y}$, and $\tilde{\sigma}=\tilde{\delta}\tilde{Y}$,
with the latter two conditions following from the first two for real
spacetimes. Note that because $\kappa$ and $\sigma$ do not depend on $V$,
$l_{\mu}$ is shear-free and geodesic with respect to the full spacetime if and
only if it is shear-free and geodesic with respect to the background flat
spacetime. Setting $\kappa=\sigma=0$ is then equivalent to the following pair
of partial differential equations for $Y$

$Y_{,v}=YY_{,\zeta}\,,\qquad Y_{,\tilde{\zeta}}=YY_{,u}\,,$ (18)

with analogous expressions for $\tilde{Y}$ following from
$\tilde{\kappa}=\tilde{\sigma}=0$. Note that the pair of equations above also
implies that $Y$ is a solution of the flat-space wave equation
$\square^{(0)}Y=0$, where
$\square^{(0)}=2(\partial_{u}\partial_{v}-\partial_{\zeta}\partial_{\tilde{\zeta}})$.
The general solution to these equations is found using Kerr’s theorem
Penrose:1967wn ; cox1976conventional  which characterizes the most general
shear-free, null geodesic congruence on Minkowksi space by specifying $Y$
implicitly as the solution of the zero locus of a homogeneous function of
twistor variables. Alternatively, equations (18) can be solved via the method
of characteristics john1982partial . In either case, one finds that $Y$ solves
(18) if and only if it is a solution of an _algebraic_ equation

$f(Y,u+Y\tilde{\zeta},\zeta+Yv)=0$ (19)

for an arbitrary analytic function $f$. Having imposed $\kappa=\sigma=0$, the
remaining non-vanishing left spin coefficients are

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\tilde{\delta}Y,\,\,\,\,\,\qquad\tau=\Delta
Y,\,\,\,\,\,\qquad\mu=V\rho,$ $\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[DV+V(\rho-\tilde{\rho})\right],\qquad\nu=\tilde{\delta}V-V\tilde{\tau},$
(20)

with analogous expressions holding for the right spin coefficients, as always.

Now the vacuum Newman-Penrose equations can be organized into three groups
herlt1980kerr ; Stephani:2003tm . The first group

$D\rho=\rho^{2},\quad
D\tau=\tau\rho,\quad\delta\tau=\tau^{2},\quad\delta\rho=(\rho-\tilde{\rho})\tau,\quad\tilde{\delta}\tau-\Delta\rho=\tau\tilde{\tau}+V\rho^{2},$
(21)

is satisfied automatically after taking into account commutator identities for
the directional derivatives (6). The second group

$\displaystyle\delta\gamma=\tau(\mu+\gamma)$ $\displaystyle
D\gamma=\mu\rho+\gamma(\rho-\tilde{\rho})$ $\displaystyle
D\mu=\mu(\rho+\tilde{\rho})+\gamma(\rho-\tilde{\rho})$
$\displaystyle\delta\nu-\Delta\mu=\mu^{2}+(\gamma+\tilde{\gamma})\mu+\tau\nu$
(22)

remain to be solved, and the third group determines the non-vanishing Weyl
scalars in terms of the spin coefficients:

$\displaystyle\Psi_{2}$ $\displaystyle=$
$\displaystyle\mu\rho+\gamma(\rho-\tilde{\rho})$ $\displaystyle\Psi_{3}$
$\displaystyle=$
$\displaystyle\tilde{\delta}\gamma+\rho\nu-\tilde{\tau}\gamma$
$\displaystyle=$ $\displaystyle D\nu-\tilde{\tau}\mu$ $\displaystyle=$
$\displaystyle-\tilde{\delta}\mu+(\rho-\tilde{\rho})\nu$
$\displaystyle\Psi_{4}$ $\displaystyle=$
$\displaystyle\tilde{\delta}\nu-\tilde{\tau}\nu.$ (23)

The vacuum conditions solving (22) were provided by Debney, Kerr and Schild in
1969JMP….10.1842D . First, a particular form for the analytic function in (19)
is required:

$f(Y,u+Y\tilde{\zeta},\zeta+Yv)=\Phi(Y)+(\tilde{c}Y+a)(\zeta+Yv)-(bY+c)(u+Y\tilde{\zeta})$
(24)

where $\Phi(Y)$ is an arbitrary function, and $a,b,c,\tilde{c}$ are constants.
Second, the function $V$ is fixed as

$V=\frac{m}{2P^{3}}(\rho+\tilde{\rho}),$ (25)

where

$P=a+\tilde{c}Y+c\tilde{Y}+bY\tilde{Y}.$ (26)

Note that $DY=D\tilde{Y}=0$ implies that $DP=0$. Now the vacuum condition
(25), together with the Newman-Penrose equation $D\rho=\rho^{2}$ from (21)
imply that

$\gamma=\frac{m\rho^{2}}{2P^{3}}.$ (27)

Acting with $\Delta$ and $\tilde{\delta}$ on (19) with $f$ given by (24) lead
to, respectively

$\tau=\frac{bY+c}{f_{Y}},\qquad\rho=-\frac{P}{f_{Y}},$ (28)

and eliminating $f_{Y}$ leads to

$\tau=-\frac{\rho}{P}(bY+c).$ (29)

Now consider the vector field

$X=a\partial_{u}+b\partial_{v}+c\partial_{\zeta}+\tilde{c}\partial_{\tilde{\zeta}}.$
(30)

Acting with $X$ on (19) with $f$ given by (24) leads to $X(Y)=0$ which
together with the right-handed analog $X(\tilde{Y})=0$ implies that $X(P)=0$.
Also, note that $X(Y)=X(\tilde{Y})=0$ implies
$[X,\delta]=[X,\tilde{\delta}]=0$, which in turn implies that
$X(\rho)=X(\tilde{\rho})=0$. Now, acting with $X$ on (25) results in $X(V)=0$.
Finally, since $X$ is clearly a Killing vector for the background metric
$g^{(0)}_{\mu\nu}$, and $X(Y)=X(\tilde{Y})=0$ implies
$\mathcal{L}_{X}\theta^{2}=0$, we conclude that $X$ is a Killing vector for
the full spacetime.

##  4 Kerr-Schild double copy from the distinguished Killing vector $X$

Here we show that the Kerr-Schild double copy can be defined for any vacuum
Kerr-Schild spacetime with an expanding and shear-free Kerr-Schild vector.
Denoting the Kerr-Schild graviton by

$h_{\mu\nu}=2Vl_{\mu}l_{\nu},$

the familiar recipe for the Kerr-Schild double copy can be expressed as

$A_{\mu}=h_{\mu\nu}X^{\nu}$

where $X=\frac{\partial}{\partial t}$ is a timelike Killing vector. Here we
make the same ansatz for the distinguished Killing vector $X$ defined by (30),
which leads to

$A_{\mu}=2PVl_{\mu}.$ (31)

We now show that this is indeed a solution of the flat-space vacuum Maxwell
equations when the vacuum Newman-Penrose equations (21), (22), (23) are
satisfied. The proof is somewhat technical, and the reader who is only
interested in making use of our results may wish to proceed directly to
section 3. However, since it is a core result, we present the details of most
calculations explicitly in this section, relegating only the most tedious
steps to appendix A. To begin, we calculate the field strength $F\equiv dA$:

$\displaystyle F=\varphi_{i}Z^{i}+\tilde{\varphi}_{i}\tilde{Z}^{i}$ (32)

where

$\varphi_{0}=0,\qquad\varphi_{1}=2P\gamma,\qquad\varphi_{2}=2P\nu+2V\tilde{\delta}P,$
(33)

and where we used equation (7) and the first Cartan structure equations (11)
to evaluate the exterior derivatives, and equations (3) and (9) to simplify
the result. Taking the self-dual projection
$\mathcal{F}\equiv\frac{1}{2}(F-i*F)$ leads to

$\mathcal{F}=\varphi_{1}Z^{1}+\varphi_{2}Z^{2}.$ (34)

It’s worth noting that we get the same result for the self-dual projection
whether we use the Hodge star associated with the full spacetime or the one
associated with the background flat spacetime. Hence, $F$ is a solution of the
vacuum Maxwell equations on the full spacetime, if and only if it is a
solution on the background spacetime if and only if $d\mathcal{F}=0$. Strictly
speaking, at this stage we are done. We know that $X_{\mu}=g_{\mu\nu}X^{\nu}$
is a solution of Maxwell’s equations on the spacetime with metric $g_{\mu\nu}$
whenever $X^{\nu}$ is a Killing vector for $g_{\mu\nu}$. On the other hand,
$A_{\mu}=h_{\mu\nu}X^{\nu}$ is gauge equivalent to $X_{\mu}$, so it must also
be a solution of Maxwell’s equations on the full spacetime with metric
$g_{\mu\nu}$ and hence, it must also be a solution on the background flat
spacetime. Nevertheless, it’s a worthwhile exercise to show how
$d\mathcal{F}=0$ follows from the Newman-Penrose equations newman1962approach
by carrying out an explicit calculation.

Since the field strength and its self-dual projection are written naturally in
terms of spin coefficients and self-dual forms on the full spacetime, we will
use the tetrad for the full spacetime rather than the background spacetime for
our calculations. One can compute the exterior derivative in this basis using
(7) and (12) to find

$\displaystyle\frac{1}{2}d\mathcal{F}$ $\displaystyle\equiv$ $\displaystyle J$
(35) $\displaystyle=$ $\displaystyle
J_{123}\theta^{123}+J_{124}\theta^{124}+J_{134}\theta^{134}+J_{234}\theta^{234},$

where the three-form current components are

$\displaystyle J_{123}$ $\displaystyle=$ $\displaystyle\delta(P\gamma)-2\tau
P\gamma$ (36) $\displaystyle J_{124}$ $\displaystyle=$
$\displaystyle\tilde{\delta}(P\gamma)-D(P\nu+V\tilde{\delta}P)+\rho(P\nu+V\tilde{\delta}P)$
(37) $\displaystyle J_{134}$ $\displaystyle=$ $\displaystyle-D(P\gamma)+2\rho
P\gamma$ (38) $\displaystyle J_{234}$ $\displaystyle=$
$\displaystyle-\Delta(P\gamma)-2V\rho
P\gamma+\delta(P\nu+V\tilde{\delta}P)-\tau(P\nu+V\tilde{\delta}P).$ (39)

Let’s look at this component by component, starting with $J_{123}$. We have

$J_{123}=\gamma\delta P+P\delta\gamma-2\tau P\gamma.$ (40)

Using the shear-free and geodesic conditions $\kappa=\sigma=0\iff DY=\delta
Y=0$, we have $\delta P=\tilde{\rho}(bY+c)$, and from the Newman-Penrose
equations (22), we have $\delta\gamma=\tau(\gamma+\mu)=\tau(\gamma+\rho V)$.
Using these equations along with (29)

$J_{123}=(bY+c)\left[\gamma(\rho+\tilde{\rho})-\rho^{2}V\right],$ (41)

which vanishes in light of (27). The next simplest component of $J$ is

$J_{134}=-PD\gamma+2\rho P\gamma,$ (42)

where we used that $DY=D\tilde{Y}=0\implies DP=0$. Now, from the Newman-
Penrose equations (22) we have

$\displaystyle D\gamma$ $\displaystyle=$
$\displaystyle\mu\rho+\gamma(\rho-\tilde{\rho})$ (43) $\displaystyle=$
$\displaystyle\rho^{2}V+\gamma(\rho-\tilde{\rho}),$ (44)

so that

$J_{134}=P\left[\gamma(\rho+\tilde{\rho})-\rho^{2}V\right]$ (45)

which once again vanishes taking into account (27).

To compute $J_{124}$, we note that from (21) we have the Newman-Penrose
equation $D\rho=\rho^{2}$, which implies that
$D\tilde{\delta}P=\rho\tilde{\delta}P$. We also use two of the Newman-Penrose
equations (23)

$\displaystyle\tilde{\delta}\gamma$ $\displaystyle=$
$\displaystyle-\rho\nu+\tilde{\tau}\gamma+\Psi_{3}$ (46) $\displaystyle D\nu$
$\displaystyle=$ $\displaystyle\tilde{\tau}\mu+\Psi_{3},$

and the fact that the expression for $\gamma$ in (3) implies $\gamma-
DV=-\tilde{\gamma}$.

Now we compute

$\displaystyle J_{124}$ $\displaystyle=$
$\displaystyle\tilde{\delta}(P\gamma)-PD\nu-\tilde{\delta}PDV-
VD\tilde{\delta}P+\rho(P\nu+V\tilde{\delta}P)$ (47) $\displaystyle=$
$\displaystyle
P\tilde{\delta}\gamma+\gamma\tilde{\delta}P-PD\nu-\tilde{\delta}PDV-V\rho\tilde{\delta}P+\rho
P\nu+\rho V\tilde{\delta}P$ $\displaystyle=$ $\displaystyle
P\tilde{\tau}(\gamma-\rho V)-\tilde{\delta}P\tilde{\gamma}$ $\displaystyle=$
$\displaystyle(b\tilde{Y}+\tilde{c})(\rho\tilde{\rho}V-\tilde{\rho}\gamma-\rho\tilde{\gamma})$

which once again vanishes by virtue of (27).

The calculation for the last component is a bit more involved. We have

$\displaystyle J_{234}$ $\displaystyle=$ $\displaystyle-\Delta(P\gamma)-2V\rho
P\gamma+\delta(P\nu+V\tilde{\delta}P)-\tau(P\nu+V\tilde{\delta}P)$ (48)
$\displaystyle=$ $\displaystyle P\Delta(\mu-\gamma)-\gamma\Delta P-2V\rho
P\gamma+\delta P\nu+P(\mu^{2}+(\gamma+\tilde{\gamma})\mu)+\delta
V\tilde{\delta}P+V\delta\tilde{\delta}P-\tau V\tilde{\delta}P$
$\displaystyle=$ $\displaystyle
P\Delta\left(\frac{m}{2P^{3}}\rho\tilde{\rho}\right)-\gamma\Delta P+\delta
P\nu+P\rho\tilde{\rho}V^{2}+\tilde{\nu}\tilde{\delta}P+V\delta\tilde{\delta}P$
$\displaystyle=$ $\displaystyle
P\Delta\left(\frac{m}{2P^{3}}\rho\tilde{\rho}\right)+|bY+c|^{2}\frac{\rho\tilde{\rho}}{P}V+(bY+c)\tilde{\rho}\nu+P\rho\tilde{\rho}V^{2}+\tilde{\nu}(b\tilde{Y}+\tilde{c})\rho+Vb\rho\tilde{\rho},$

where we have defined $|bY+c|^{2}\equiv(bY+c)(b\tilde{Y}+\tilde{c})$. In going
from the first line to the second line, we have used the vacuum Newman-Penrose
equation

$\delta\nu-\Delta\mu=\mu^{2}+(\gamma+\tilde{\gamma})\mu+\tau\nu.$ (49)

In going from the second line to the third, we have used $\mu=\rho V$ and the
right-handed analog of $\nu=\tilde{\delta}V-V\tilde{\tau}$ from (3), as well
as (25) and (27). In going from the fourth line to the fifth, we evaluated all
of the derivatives of $P$ except those in the first term, which involved
making use of the Newman-Penrose equation $\delta\rho=(\rho-\tilde{\rho})\tau$
from (21), and we have simplified the result using (25), (27) and (29) . We
now turn our attention to the very first term:

$\displaystyle P\Delta\left(\frac{m}{2P^{3}}\rho\tilde{\rho}\right)$
$\displaystyle=$ $\displaystyle\frac{m}{2P^{3}}\left[-3\rho\tilde{\rho}\Delta
P+P(\tilde{\rho}\Delta\rho+\rho\Delta\tilde{\rho})\right]$ (50)
$\displaystyle=$ $\displaystyle-3\frac{V}{P}|bY+c|^{2}\rho\tilde{\rho}$
$\displaystyle-\tilde{\rho}(bY+c)\nu+\frac{m}{P^{4}}|bY+c|^{2}\rho\tilde{\rho}^{2}-b\gamma\tilde{\rho}-\frac{m}{2P^{2}}V\tilde{\rho}\rho^{2}$
$\displaystyle-\rho(b\tilde{Y}+\tilde{c})\tilde{\nu}+\frac{m}{P^{4}}|bY+c|^{2}\tilde{\rho}\rho^{2}-b\tilde{\gamma}\rho-\frac{m}{2P^{2}}V\rho\tilde{\rho}^{2}$
$\displaystyle=$
$\displaystyle-\frac{V}{P}|bY+c|^{2}\rho\tilde{\rho}-\tilde{\rho}(bY+c)\nu-\rho(b\tilde{Y}+\tilde{c})\tilde{\nu}-b\rho\tilde{\rho}V-\rho\tilde{\rho}PV^{2}.$

In going from the first line to the second we have made repeated use of (25),
(27) and (29), as well as the equation (LABEL:DeltaRhoCalc) for $\Delta\rho$
which is derived in appendix A from a Bianchi identity and some of the Newman-
Penrose equations. In going from the second line to the third, we have again
made use of (25) and (27). Substituting the last line of (50) into the last
line of (48), we see that $J_{234}=0$. This concludes the proof that the
vacuum Newman-Penrose equations imply that the gauge field (31) is a solution
of the flat space vacuum Maxwell equations.

We may further show that the zeroth copy defined by
$\phi=h_{\mu\nu}X^{\mu}X^{\nu}$ satisfies the flat-space vacuum wave equation
$\square^{(0)}\phi=0$. First, we note that the Killing vector $X$ is parallel
with respect to the background flat metric, i.e $\nabla^{(0)}_{\mu}X^{\nu}=0$.
Now we calculate:

$\displaystyle\square^{(0)}\phi$ $\displaystyle=$
$\displaystyle\square^{(0)}(h_{\mu\nu}X^{\mu}X^{\nu})$ (51) $\displaystyle=$
$\displaystyle X^{\mu}\nabla^{(0)\nu}\nabla^{(0)}_{\nu}A_{\mu}$
$\displaystyle=$ $\displaystyle
X^{\mu}\nabla^{(0)\nu}\nabla^{(0)}_{\mu}A_{\nu}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{X}\big{(}\nabla^{(0)\nu}A_{\nu}\big{)}$
$\displaystyle=$ $\displaystyle
g_{(0)}^{\mu\nu}\mathcal{L}_{X}\big{(}\nabla^{(0)}_{(\mu}A_{\nu)}\big{)}$
$\displaystyle=$ $\displaystyle
g_{(0)}^{\mu\nu}\mathcal{L}_{X}\big{(}\tfrac{1}{2}\mathcal{L}_{A}\hskip
0.85358ptg^{(0)}_{\mu\nu}\big{)}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}g_{(0)}^{\mu\nu}\big{(}\mathcal{L}_{[X,A]}\hskip
0.85358ptg^{(0)}_{\mu\nu}+\mathcal{L}_{A}\mathcal{L}_{X}\hskip
0.85358ptg^{(0)}_{\mu\nu}\big{)}=0.$

In the above, we used the background flat metric to raise and lower indices.
In going from the second line to the third line, we used the fact that
$A_{\mu}$ satisfies the vacuum Maxwell equations, and in the last line we used
$[X,A]^{\mu}=\mathcal{L}_{X}(g_{(0)}^{\mu\nu}A_{\nu})=0$.

##  5 Weyl Double copy

The Weyl double copy is typically written in spinorial form as

$\Psi_{ABCD}=\frac{1}{S}\varphi_{(AB}\varphi_{CD)}$ (52)

where $\Psi_{ABCD}$ is the Weyl spinor, $\varphi_{AB}$ is the field-strength
spinor, and $S$ is a scalar satisfying the flat-space, vacuum wave equation
$\square^{(0)}S=0$. In terms of the Newman-Penrose Weyl scalars $\Psi_{I}$ and
Maxwell scalars $\varphi_{i}$, (52) becomes

$\Psi_{0}=\tfrac{1}{S}\varphi_{0}^{2},\quad\Psi_{1}=\tfrac{1}{S}\varphi_{0}\varphi_{1},\quad\Psi_{2}=\tfrac{1}{3S}\left(2\varphi_{1}^{2}+\varphi_{0}\varphi_{2}\right),\quad\Psi_{3}=\tfrac{1}{S}\varphi_{1}\varphi_{2},\quad\Psi_{4}=\tfrac{1}{S}\varphi_{2}^{2}.$
(53)

Here we show that our proposal (31) for the Kerr-Schild double copy obeys the
Weyl double copy (53) when the spacetime is type D. Since
$\varphi_{0}=\Psi_{0}=\Psi_{1}=0$, the first two equations in (53) are
satisfied automatically. The first non-trivial equation is
$\Psi_{2}=\frac{2}{3S}\varphi_{1}^{2}$. Using
$\Psi_{2}=\mu\rho+\gamma(\rho-\tilde{\rho})$ from (23), $\mu=\rho V$ from (3),
along with (25) and (27) leads to $\Psi_{2}=m\frac{\rho^{3}}{P^{3}}$. On the
other hand, from (33) we have $\varphi_{1}=2P\gamma$ and using (27) leads to
$\varphi_{1}=\frac{m\rho^{2}}{P^{2}}.$ Using these expressions for $\Psi_{2}$
and $\varphi_{1}$ leads to

$S=\frac{2m\rho}{3P}.$ (54)

We now check that $S$ satisfies the flat-space wave equation, which is
entirely plausible, since we have already shown that the zeroth copy $\phi$ is
harmonic, and we now see that $\phi=\frac{3}{4}(S+\tilde{S})$. In order to
show that $S$ is harmonic, we note from (29) that
$\frac{\rho}{P}=-\frac{\tau}{bY+c}$. Since $DY=0$, we write $\tau=\Delta
Y=(\partial_{u}-VD)Y=Y_{,u}$. Now we have
$\frac{\rho}{P}=-\frac{1}{b}\partial_{u}\log(bY+c)$, and we calculate

$\displaystyle\square^{(0)}S$ $\displaystyle=$
$\displaystyle-\frac{4m}{3b}\partial_{u}(\partial_{u}\partial_{v}-\partial_{\zeta}\partial_{\tilde{\zeta}})\log(bY+c)$
(55) $\displaystyle=$
$\displaystyle-\frac{4m}{3b}\partial_{u}\left[\frac{b(Y_{,uv}-Y_{,\zeta\tilde{\zeta}})}{bY+c}-\frac{b^{2}(Y_{,u}Y_{,v}-Y_{,\zeta}Y_{,\tilde{\zeta}})}{(bY+c)^{2}}\right]=0,$
(56)

where in the second line we used equations (18) to eliminate the second term
and their corollary $\square^{(0)}Y=0$ to eliminate the first.

We now turn our attention to $\Psi_{3}$, which can be expressed most simply as
$\Psi_{3}=\tilde{\delta}\gamma+\rho\nu-\tilde{\tau}\gamma$ from the Newman-
Penrose equation (23). Using the vacuum condition
$\gamma=\frac{m\rho^{2}}{2P^{3}}$ from (27), this becomes

$\Psi_{3}=-3\gamma\frac{\tilde{\delta}P}{P}+2\gamma\frac{\tilde{\delta}\rho}{\rho}+\rho\nu-\tilde{\tau}\gamma.$
(57)

From (3), we have $\nu=\tilde{\delta}V-V\tilde{\tau}$, and applying the vacuum
condition $V=\frac{m}{2P^{3}}(\rho+\tilde{\rho})$ from (25) leads to

$\nu=-3V\frac{\tilde{\delta}P}{P}+\frac{\gamma}{\rho^{2}}\tilde{\delta}\rho-\frac{2\tilde{\tau}\gamma}{\rho}.$
(58)

Substituting (58) into (57), and using (25), (27), and the right-handed analog
of (29) leads to

$\Psi_{3}=3\gamma\Big{(}\,\frac{\tilde{\delta}\rho}{\rho}-2\frac{\tilde{\delta}P}{P}\,\Big{)}.$
(59)

On the other hand, from the expressions (33) for the Maxwell scalars and
equation (54) for $S$ and again applying the vacuum condition
$\gamma=\frac{m\rho^{2}}{2P^{3}}$ from (27), we have

$\frac{1}{S}\varphi_{1}\varphi_{2}=3\rho\Big{(}\nu+V\frac{\tilde{\delta}P}{P}\Big{)}.$
(60)

Inserting (58) into (60), and once again using (25), (27), and the right-
handed analog of (29) establishes that
$\Psi_{3}=\frac{1}{S}\varphi_{1}\varphi_{2}$, as in (53).

Since vacuum Kerr-Schild spacetimes are not generally of Petrov type D, one
cannot expect the Weyl double copy to hold in general. As a consequence, we
cannot assume that the last equation in (53) for $\Psi_{4}$ will be
automatically satisfied, unlike the equations for the other Weyl scalars.
However, for any type D spacetime with a tetrad satisfying
$\Psi_{0}=\Psi_{1}=0$, the Weyl scalars must satisfy

$3\Psi_{2}\Psi_{4}=2\Psi_{3}^{2}.$ (61)

Putting $\Psi_{2}=\frac{2}{3S}\varphi_{1}^{2}$ and
$\Psi_{3}=\frac{1}{S}\varphi_{1}\varphi_{2}$ into (61) immediately gives
$\Psi_{4}=\frac{1}{S}\varphi_{2}^{2}$. This completes the proof that our
proposal (31) for the Kerr-Schild double copy implies the Weyl double copy
when the spacetime is type D. In fact, what we have shown is somewhat
stronger; since we have made no assumptions about the reality of the metric or
about its signature, the left and right Weyl scalars are in general
independent. The full Petrov type of such a spacetime is thus characterized by
a pair of symbols $(\cdot,\tilde{\cdot})$, indicating the Petrov types of the
left and right Weyl scalars. The argument we have made for the validity of the
Weyl double copy applies separately to the left and right Weyl scalars, and so
applies to complex spacetimes or real spacetimes in Kleinian signature with
Petrov types $(D,\tilde{X})$ or $(X,\tilde{D})$, where $X\in\\{II,D,O\\}$.

##  6 Applications

###  6.1 Kerr

Here we calculate the Kerr-Schild metric with Killing vector
$X=(\partial_{u}+\epsilon\partial_{v})/\sqrt{2}$ and $\Phi(Y)=-iaY$ with $a$ a
constant. We see that $\epsilon=1$ corresponds to $X=\partial_{t}$ and
$\epsilon=-1$ to $X=-\partial_{z}$. The roots of (24) are thus

$Y=\frac{v-\epsilon u-ia\pm\sqrt{(v-\epsilon
u-ia)^{2}+4\epsilon\zeta\tilde{\zeta}}}{2\epsilon\tilde{\zeta}},$ (62)

and from here on out we choose to work with the $(-)$ solution. We find that

$\rho=-2\left(v-\epsilon u-ia+\sqrt{(v-\epsilon
u-ia)^{2}+4\epsilon\zeta\tilde{\zeta}}\right)^{-1}$ (63)

and that $P=-\rho\sqrt{(v-\epsilon u-ia)^{2}+4\epsilon\zeta\tilde{\zeta}}$.
With these one can now find the form of the metric. The Kerr metric can be
obtained by letting $X$ be timelike, in which case $\epsilon=+1$. Its
construction using the Newman-Penrose formalism for Kerr-Schild metrics can be
read in mcintosh1988single . Here we use the final result, namely that in
coordinates $\\{t,x,y,z\\}$,

$l_{\mu}dx^{\mu}=\frac{\sqrt{2}r}{r+z}\Big{[}dt-\frac{rx-
ay}{r^{2}+a^{2}}dx-\frac{ax+ry}{r^{2}+a^{2}}dy-\frac{z}{r}dz\Big{]}$ (64)

where $r$ is given implicitly by $(x^{2}+y^{2})/(r^{2}+a^{2})+z^{2}/r^{2}=1$,
and that

$2V=-\frac{mr(r+z)^{2}}{r^{4}+a^{2}z^{2}}.$ (65)

Therefore the single copy is

$A=h_{\mu\nu}X^{\nu}X^{\mu}=-\frac{2mr^{3}}{r^{4}+a^{2}z^{2}}\Big{[}dt-\frac{rx-
ay}{r^{2}+a^{2}}dx-\frac{ax+ry}{r^{2}+a^{2}}dy-\frac{z}{r}dz\Big{]},$ (66)

which is, as expected, the result found via the standard stationary Kerr-
Schild approach Monteiro:2014cda .

###  6.2 Schwarzschild and family

With $a=0$, the Kerr-Schild graviton with a general signature Killing vector
is

$h_{\mu\nu}dx^{\mu}dx^{\nu}=-\frac{2\sqrt{2}m}{\epsilon^{2}\sqrt{(v-\epsilon
u)^{2}+4\epsilon\zeta\tilde{\zeta}}}\Bigg{[}\frac{1}{\sqrt{2}}d(\sqrt{(v-\epsilon
u)^{2}+4\epsilon\zeta\tilde{\zeta}})-\frac{1}{\sqrt{2}}d(v+\epsilon
u)\Bigg{]}^{2}.$ (67)

This metric is diffeomorphic to equation (28.21) of Stephani:2003tm . For a
timelike Killing vector, this solution becomes Schwarzschild. However, we find
another solution that can be thought of as a cousin of Schwarzschild, one with
a spacelike instead of a timelike Killing vector. Let $X=-\partial_{z}$ be the
spacelike Killing vector. Therefore

$Y=\frac{-v-u-\sqrt{(v+u)^{2}-4\zeta\tilde{\zeta}}}{2\tilde{\zeta}}$ (68)

is the scalar function in null coordinates. The final metric can be written in
a simpler way using cylindrical coordinates $\\{t,\rho,\theta,z\\}$. The new
coordinates are $\rho=\sqrt{x^{2}+y^{2}}$ and $\theta=\arctan({y/x})$. The
resulting metric is

$ds^{2}=dt^{2}-d\rho^{2}-\rho^{2}d\theta^{2}-dz^{2}-\frac{2m}{\sqrt{t^{2}-\rho^{2}}}\Big{[}\frac{t}{\sqrt{t^{2}-\rho^{2}}}dt-\frac{\rho}{\sqrt{t^{2}-\rho^{2}}}d\rho+dz\Big{]}^{2}.$
(69)

Notice that this metric is cylindrically symmetric about the $z$-axis since
$\partial_{z}$ and $\partial_{\theta}=x\partial_{y}-y\partial_{x}$ are Killing
vectors. There are also two boost Killing vectors that are most easily
expressed in Cartesian coordinates as $\xi_{x}=x\partial_{t}+t\partial_{x}$
and $\xi_{y}=y\partial_{t}+t\partial_{y}$. Now the symmetry algebra satisfies
the commutation relations $[\xi_{x},\xi_{y}]=\partial_{\theta}$,
$[\xi_{x},\partial_{\theta}]=\xi_{y}$, and
$[\xi_{y},\partial_{\theta}]=-\xi_{x}$, with $X=-\partial_{z}$ being a central
element that commutes with all others. Note that $X$ is also central in the
Schwarzschild and Kerr cases.

The single copy is

$A=\frac{q}{\sqrt{t^{2}-\rho^{2}}}\Big{[}\frac{t}{\sqrt{t^{2}-\rho^{2}}}dt-\frac{\rho}{\sqrt{t^{2}-\rho^{2}}}d\rho+dz\Big{]},$
(70)

which is gauge-equivalent to

$A^{\prime}=\frac{q}{\sqrt{t^{2}-\rho^{2}}}dz.$ (71)

The zeroth copy is given by
$\phi=h_{\mu\nu}X^{\mu}X^{\nu}=A_{\mu}X^{\mu}=q/\sqrt{t^{2}-\rho^{2}}$. This
solution can be thought of as an infinitely long cylindrically symmetric
singular shell that expands outward at the speed of light. The coordinates
used above do not cover the region outside the shell (where the metric becomes
complex). Inside the shell and around the $z$-axis, as $t$ increases, the
components of the metric asymptotically approach their flat Minkowskian form,
$\mathrm{diag}(1,-1,-\rho^{2},-1)$.

When we set $\epsilon=0$, then $X=\partial_{u}$ and $P=1$. Since the
$\epsilon\to 0$ limit is undefined in $h_{\mu\nu}$ above, we must start from
scratch with the vacuum Kerr-Schild conditions. Doing so, we find that
$Y=-v/\zeta$ and so $\rho=v/\zeta^{2}$. Thus
$2V=mv(\zeta^{2}+\tilde{\zeta}^{2})/(\zeta\tilde{\zeta})^{2}$, and the Kerr-
Schild graviton is

$h_{\mu\nu}dx^{\mu}dx^{\nu}=\frac{mv(\zeta^{2}+\tilde{\zeta}^{2})}{(\zeta\tilde{\zeta})^{2}}\Big{[}du-\frac{v}{\zeta\tilde{\zeta}}(\zeta
d\zeta+\tilde{\zeta}d\tilde{\zeta})+\frac{v^{2}}{\zeta\tilde{\zeta}}dv\Big{]}^{2}.$
(72)

The single copy is then

$A=\frac{mv(\zeta^{2}+\tilde{\zeta}^{2})}{(\zeta\tilde{\zeta})^{2}}\Big{[}du-\frac{v}{\zeta\tilde{\zeta}}(\zeta
d\zeta+\tilde{\zeta}d\tilde{\zeta})+\frac{v^{2}}{\zeta\tilde{\zeta}}dv\Big{]},$
(73)

and the zeroth copy is

$\phi=\frac{mv(\zeta^{2}+\tilde{\zeta}^{2})}{(\zeta\tilde{\zeta})^{2}}.$ (74)

This solution is a special case of a Kasner metric and, like the two examples
above, diffeomorphic to a limit of equation (28.21) in Section 28.1 of
Stephani:2003tm ; here it is their $K=0$ limit.

###  6.3 Self-dual Solutions

A particularly interesting class of complex spacetimes are those that are
_self-dual_. In the Newman-Penrose formalism, self-duality is characterized by
the vanishing of the right-handed Weyl scalars $\tilde{\Psi}_{I}=0$. Of
course, for real Lorentzian spacetimes, this implies that the left-handed Weyl
scalars also vanish, since they are simply the complex conjugates of the
right-handed Weyl scalars. In this case, the spacetime is conformally flat and
the only possible self-dual solutions are maximally symmetric. However, as we
have emphasized repeatedly, for complex spacetimes or real spacetimes with
Kleinian signature, $\Psi_{I}$ and $\tilde{\Psi}_{I}$ are independent, and
non-trivial self-dual solutions are possible. Similarly, in electromagnetism,
self-duality can be characterized by the vanishing of the right-handed Maxwell
scalars $\tilde{\varphi}_{i}=0$, and self-dual solutions are generically
complex. Because the equations of motion are linear, real solutions of
Maxwell’s equations are in one-to-one correspondence with self-dual solutions.
As one might expect, this is not the case for solutions of Einstein’s
equation, which are non-linear. However, it is well-known that for Kerr-Schild
spacetimes with a geodesic Kerr-Schild vector, the Einstein equations
linearize, so perhaps it is not surprising that there is a very simple
correspondence between real vacuum Kerr-Schild spacetimes and self-dual
spacetimes. Remarkably, the correspondence between real and self-dual vacuum
Kerr-Schild spacetimes, as presented here, appears to have gone unnoticed in
the existing literature until now.

In section §3, we found that vacuum Kerr-Schild spacetimes are characterized
by a function $Y$ that is a solution to $f(Y,u+Y\tilde{\zeta},\zeta+Yv)=0$,
where the function $f$ is given by (24), and a second function $\tilde{Y}$
that satisfies the right-handed analog of the same equation. For real
Lorentzian spacetimes, $\tilde{Y}$ is simply the complex conjugate of $Y$, so
the solution is completely characterized by $Y$. However, for complex
spacetimes, $Y$ and $\tilde{Y}$ are independent, and it turns out that the
choice $\tilde{Y}=0$ always leads to a self-dual solution of the vacuum
Einstein equations.

To see that $\tilde{Y}=0$ always leads to a self-dual solution, we focus on
the right-handed spin coefficients. From the right-handed version of (3), we
can see immediately that $\tilde{\rho}$ and $\tilde{\tau}$ are zero, and since
$\tilde{\rho}$ is zero, so is $\tilde{\mu}$. From the right-handed version of
(27), we also see that $\tilde{\gamma}=0$. Finally, from the right-handed
version of (3) we have $\tilde{\nu}=\delta V-V\tau$ and from (21)
$\delta\rho=\rho\tau$, which leads to $\tilde{\nu}=0$ after using the vacuum
condition (25). We conclude that for solutions with $\tilde{Y}=0$, all of the
right-handed spin coefficients vanish. Putting this result into the right-
handed version of (23) leads to $\tilde{\Psi}_{I}=0$, so the spacetime is
self-dual. It is simple to check that when $\tilde{Y}=0$, the right handed
version of the Maxwell scalars (33) also vanish, so the corresponding single
copy is also self-dual.

####  6.3.1 Self-dual analog of Kerr

We illustrate the method described above to obtain the self-dual analog of the
Kerr metric. One obtains the Kerr metric by setting the Killing vector
$X=(\partial_{u}+\partial_{v})/\sqrt{2}$ and setting $\Phi(Y)=-iaY$, where $a$
is the rotation parameter. $Y$ is thus a solution to a quadratic equation, and
choosing the negative root gives

$Y=\frac{v-u-ia-\sqrt{(v-u-ia)^{2}+4\zeta\tilde{\zeta}}}{2\tilde{\zeta}}.$
(75)

Since $\tilde{Y}=0$, we can write
$\theta^{2}=\mathrm{d}u+Y\mathrm{d}\tilde{\zeta}$. Furthermore the spin
coefficient
$\rho=\tilde{\delta}Y=\partial_{\zeta}Y-\tilde{Y}\partial_{u}Y=\partial_{\zeta}Y$,
and $P=l_{\mu}X^{\mu}=2^{-1/2}$. From these we compute the metric:

$ds^{2}=2dudv-2d\zeta d\tilde{\zeta}-\frac{2\sqrt{2}m}{\sqrt{(v-u-
ia)^{2}+4\zeta\tilde{\zeta}}}\Big{[}du+\frac{v-u-ia-\sqrt{(v-u-
ia)^{2}+4\zeta\tilde{\zeta}}}{2\tilde{\zeta}}d\tilde{\zeta}\Big{]}^{2}.$ (76)

This is the self-dual analog of the Kerr solution obtained via setting
$\tilde{Y}=0$. A few comments are in order. First, we note that analytically
continuing $a\to-ia$, and restricting to real values of all of the
coordinates, results in a real metric with Kleinian signature. Furthermore,
unlike in the case of the Kerr black hole, the rotation parameter does not
correspond to a physical angular momentum in the self-dual analog, as it can
be removed from the metric by performing the coordinate transformation
$u^{\prime}=u+ia$ or $v^{\prime}=v-ia$ (before analytically continuing to
imaginary $a$). The resulting metric can be expressed in Cartesian coordinates
as

$ds^{2}=dt^{2}-dx^{2}+dy^{2}-dz^{2}-\frac{m}{r}\Big{[}dt-
dz-\frac{r-z}{x-y}d(x-y)\Big{]}^{2}$ (77)

where $r^{2}\equiv x^{2}-y^{2}+z^{2}$.

This situation is reminiscent of the surprising discovery by Crawley, Guevara,
Miller and Strominger that self-dual Kerr-Taub-NUT is diffeomorphic to Taub-
NUT in Kleinian signature crawley2022black  (see also crawley2023self ). While
we have not been able to find an explicit diffeomorphism taking one metric to
the other, we suspect that the self-dual analog of Kerr we have presented here
is in fact diffeomorphic to self-dual Taub-NUT. An analysis of a few
quadratic, cubic and quartic curvature invariants (which we have included in
appendix B) seems to support this conjecture. Establishing this conjecture
would be an important result, since to our knowledge, it is not known that
self-dual Taub-NUT is a single Kerr-Schild metric (although its double Kerr-
Schild form is well known plebanski1976rotating ; Chong:2004hw ). Conversely,
if this indeed represents a newly discovered self-dual black hole, it would
undoubtedly be of significant interest in its own regard. Additionally, it
would indicate that the property of removing the rotation parameter through a
diffeomorphism may persist more broadly in self-dual black hole spacetimes.

Returning to the double copy context, we obtain the single copy gauge field by
contracting $h_{\mu\nu}$ with $X^{\nu}$:

$A=-\frac{2m}{\sqrt{(v-u-
ia)^{2}+4\zeta\tilde{\zeta}}}\Big{[}du+\frac{v-u-a-\sqrt{(v-u-a)^{2}+4\zeta\tilde{\zeta}}}{2\tilde{\zeta}}d\tilde{\zeta}\Big{]}.$
(78)

Similarly, the zeroth copy is

$\phi=-\frac{\sqrt{2}m}{\sqrt{(v-u-ia)^{2}+4\zeta\tilde{\zeta}}}.$ (79)

####  6.3.2 The Eguchi-Hanson metric

The Eguchi-Hanson metric is a self-dual, vacuum solution of Einstein’s
equation which is of the Kerr-Schild form and which has an expanding repeated
principle null direction. This solution has been studied in the context of the
double copy in Berman:2018hwd  and Luna:2018dpt . Nevertheless, the Eguchi-
Hanson metric does _not_ fall into the family of solutions we have studied in
this article, which inlcudes all real Lorentzian vacuum Kerr-Schild spacetimes
with an expanding Kerr-Schild vector, but only a subset of the complex and
Kleinian signature spacetimes with the same properties. Although the Eguchi-
Hanson metric lies outside the scope of the general results we have presented,
we will find that examining the Killing vectors of this metric can still offer
valuable insights and yield intriguing results.

The Eguchi Hanson metric is Eguchi:1978xp

$ds^{2}=ds_{0}^{2}+\frac{\lambda
v^{2}}{(uv-\zeta\tilde{\zeta})^{3}}\left(du-\frac{\zeta}{v}d\tilde{\zeta}\right)^{2},$
(80)

where $ds_{0}^{2}=2(dudv-d\zeta d\tilde{\zeta})$, and $\lambda$ is a real
parameter, which is neither a mass nor a cosmological constant. Taking all
coordinates to be real results in the metric (80) in Kleinian signature. We
may read off the metric functions

$Y=-\zeta/v,\qquad\tilde{Y}=0,\qquad V=\frac{\lambda
v^{2}}{2(uv-\zeta\tilde{\zeta})^{3}}.$ (81)

Now we can see why these metric functions do not define a metric of the type
we have been considering: they do not obey the vacuum condition (25), which,
given the above expressions for $Y$ and $\tilde{Y}$, should give
$V=-\frac{m}{2v}$. To see this, note that $Y=-\zeta/v$ corresponds to putting
$\Phi(Y)=0$ and $X=\partial_{u}$ in (24), thus $P=1$, $\rho=-1/v$, and
$\tilde{\rho}=0$. Evidently, this construction does not lead to the Eguchi-
Hanson metric, for which $\partial_{u}$ is not even a Killing vector. Rather,
the Killing vectors for (80) may be written as

$\displaystyle K_{0}$ $\displaystyle=$ $\displaystyle
u\partial_{u}-v\partial_{v}-\zeta\partial_{\zeta}+\tilde{\zeta}\partial_{\tilde{\zeta}}$
$\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle\zeta\partial_{v}+u\partial_{\tilde{\zeta}}$ $\displaystyle
K_{2}$ $\displaystyle=$
$\displaystyle\tilde{\zeta}\partial_{u}+v\partial_{\zeta}$ $\displaystyle
K_{3}$ $\displaystyle=$ $\displaystyle
u\partial_{u}-v\partial_{v}+\zeta\partial_{\zeta}-\tilde{\zeta}\partial_{\tilde{\zeta}}=\tilde{K}_{0}.$
(82)

Note that all of the $K_{i}$ above are also Killing vectors of the flat metric
$ds_{0}^{2}$ and that $K_{0}$ is a central element of the symmetry algebra for
the full metric (80), i.e. $[K_{0},K_{i}]=0$. Similarly, the distinguished
Killing vector $X$ was central in all of the examples we have studied.
Furthermore, for the Eguchi-Hanson metric (80), we can write

$V=-\frac{\lambda}{2Q^{3}}(\rho+\tilde{\rho})$ (83)

where we have defined $Q\equiv l_{\mu}K_{0}^{\mu}=u-\zeta\tilde{\zeta}/v$ in
analogy with $P=l_{\mu}X^{\mu}$. In view of this structure, it is tempting to
consider both the ‘standard’ gauge field $A_{\mu}=h_{\mu\nu}X^{\nu}$ and the
modified gauge field $A^{\textrm{mod}}_{\mu}=h_{\mu\nu}K_{0}^{\nu}$. For the
standard gauge field, we have

$A_{\mu}=\frac{\lambda v^{2}}{(uv-\zeta\tilde{\zeta})^{3}}l_{\mu},$ (84)

while for the modified gauge field we have

$A^{\textrm{mod}}_{\mu}=\frac{\lambda v}{(uv-\zeta\tilde{\zeta})^{2}}l_{\mu},$
(85)

both of which are self-dual solutions of the flat-space vacuum Maxwell
equations!666And neither of which agrees with the self-dual single copy in
Berman:2018hwd .

We may also consider a real Lorentzian analog of the Eguchi-Hanson metric;
however, since the Eguchi-Hanson metric is not contained in the family of
solutions we have considered in our general discussion, there is no guarantee
that the real analog will be a solution of the vacuum Einstein equations.
Indeed, since the family we have considered in this article contains all real
Lorentzian vacuum Kerr-Schild solutions with an expanding repeated principle
null direction, one would expect that the real Lorentzian analog of the
Eguchi-Hanson metric must not be a vacuum solution. Let us consider the line
element

$ds^{2}=ds_{0}^{2}+\tfrac{\lambda
v^{2}}{(uv-\zeta\tilde{\zeta})^{3}}\left(du-\tfrac{\tilde{\zeta}}{v}d\zeta-\tfrac{\zeta}{v}d\tilde{\zeta}+\tfrac{\zeta\tilde{\zeta}}{v^{2}}dv\right)^{2},$
(86)

which corresponds to a pure-radiation stress tensor

$T_{\mu\nu}=\frac{24\pi G\lambda}{(uv-\zeta\tilde{\zeta})^{4}}l_{\mu}l_{\nu}.$
(87)

The line element (86) has metric functions

$Y=-\zeta/v,\qquad\tilde{Y}=-\tilde{\zeta}/v,\qquad V=\frac{\lambda
v^{2}}{(uv-\zeta\tilde{\zeta})^{3}},$ (88)

where once again $V$ satisfies (83) because the new terms that have been
introduced in $l_{\mu}$ to make the line element real are orthogonal to
$K_{0}^{\mu}$. The line element (86) has four Killing vectors, three of which
are also Killing vectors of the self-dual metric (80), namely
$K^{\prime}_{0}=K_{0}$,
$K^{\prime}_{1}=\zeta\partial_{u}+v\partial_{\tilde{\zeta}}=\tilde{K}_{2}$,
$K^{\prime}_{2}=K_{2}$, $K^{\prime}_{3}=K_{3}=\tilde{K}_{0}$. A few points are
worth noting here. First, $K_{0}=K^{\prime}_{0}$ remains a Killing vector of
the new metric (86), but it is no longer central, as it does not commute with
$K^{\prime}_{1}$. Second, even though $K_{0}$ is complex, its imaginary part
is orthogonal to $l_{\mu}$, so it can still be used to construct a _real_
gauge field $A^{\textrm{mod}}_{\mu}=h_{\mu\nu}K_{0}^{\nu}$ in addition to the
standard gauge field $A_{\mu}=h_{\mu\nu}X^{\nu}$ with $X=\partial_{u}$. We
have

$A_{\mu}=\frac{2\lambda v^{2}}{(uv-\zeta\tilde{\zeta})^{3}}l_{\mu},$ (89)

and

$A^{\textrm{mod}}_{\mu}=\frac{2\lambda
v}{(uv-\zeta\tilde{\zeta})^{2}}l_{\mu}.$ (90)

As one might expect, these are not solutions of the vacuum Maxwell equations,
but rather correspond to sources $J_{\mu}=\nabla^{(0)\nu}F_{\mu\nu}$ and
$J^{\textrm{mod}}_{\mu}=\nabla^{(0)\nu}F^{\textrm{mod}}_{\mu\nu}$ given by

$J_{\mu}=\frac{6\lambda v^{2}}{(uv-\zeta\tilde{\zeta})^{4}}l_{\mu},$ (91)

and

$J^{\textrm{mod}}_{\mu}=\frac{4\lambda
v}{(uv-\zeta\tilde{\zeta})^{3}}l_{\mu}.$ (92)

In either case, we see that the double copy for fields leads to a double copy
for sources of the form $T_{\mu\nu}\propto J_{\mu}J_{\nu}\propto
J^{\textrm{mod}}_{\mu}J^{\textrm{mod}}_{\nu}$, which has been observed in the
double copy analysis of Kinnersley’s photon rocket spacetime Kinnersley:1969zz
which was carried out in Luna:2016due .

###  6.4 A simple solution with Petrov type II

The Petrov type of the solution is determined by the arbitrary function
$\Phi(Y)$, with the solution having Petrov type II if and only if
$\Phi^{\prime\prime\prime}(Y)\neq 0$ mcintosh1988single . One can see why
simple examples of closed form Kerr-Schild solutions with Petrov type II are
in short supply: one needs to solve equation (24) for some choice of $\Phi(Y)$
with $\Phi^{\prime\prime\prime}(Y)\neq 0$, but the latter requirement
typically makes the equation of cubic order or higher; or worse, requires us
to solve a non-polynomial equation for $Y$. However, there is a simple case
where the equation can be solved explicitly and exactly. First notice that
when $X=\partial_{u}$, the equation for $Y$ becomes

$\Phi(Y)+\zeta+vY=0.$ (93)

Next we make the choice

$\Phi(Y)=a\sqrt{Y}$ (94)

where $a$ is a real constant with dimensions of length. With this choice,
equation (93) becomes a quadratic equation for $\sqrt{Y}$, which leads to the
solution

$Y=\left(\frac{a}{2v}\right)^{2}\left(1\pm\sqrt{1-4\frac{v\zeta}{a^{2}}}\right)^{2}.$
(95)

Now using equation (25), we find

$V=\frac{m}{2v}+\frac{ma}{4v}\left(\frac{1}{\sqrt{a^{2}-4v\zeta}}+\frac{1}{\sqrt{a^{2}-4v\tilde{\zeta}}}\right).$
(96)

The spin coefficients are:

$\displaystyle\rho=-\frac{1}{v}\left(1+\frac{a}{\sqrt{{a}^{2}-4\,v\zeta}}\right)$
(97) $\displaystyle\gamma=\frac{m\rho^{2}}{2}$ (98) $\displaystyle\mu=\rho V$
(99)
$\displaystyle\nu=-{\frac{ma}{2}\left({a}^{2}-4\,v\zeta\right)^{-{\frac{3}{2}}}}$
(100) $\displaystyle\kappa=\sigma=\alpha=\beta=\tau=\lambda=\epsilon=0.$ (101)

The Weyl scalars are:

$\displaystyle\Psi_{0}=\Psi_{1}=0$ (102)
$\displaystyle\Psi_{2}=-\frac{m}{{v}^{3}}\,{\frac{\left(a+\sqrt{{a}^{2}-4\,v\zeta}\right)^{3}}{\left({a}^{2}-4\,v\zeta\right)^{3/2}}}$
(103)
$\displaystyle\Psi_{3}=\frac{3ma}{v}{\frac{\left(a+\sqrt{{a}^{2}-4\,v\zeta}\right)}{\left({a}^{2}-4\,v\zeta\right)^{2}}}$
(104)
$\displaystyle\Psi_{4}=-{\frac{6mav}{\left({a}^{2}-4\,v\zeta\right)^{5/2}}}.$
(105)

It can be checked that $2\Psi_{3}^{2}-3\Psi_{2}\Psi_{4}\neq 0$, so the
solution is indeed of Petrov type II. Since we have taken $X=\partial_{u}$, we
have $P=1$, and the single copy is simply $A_{\mu}=2Vl_{\mu}$, which can be
checked to solve the flat space vacuum Maxwell equations. The zeroth copy is
simply $\phi=V$, which clearly solves the vacuum wave equation as
$\partial_{u}\partial_{v}V$ = $\partial_{\zeta}\partial_{\tilde{\zeta}}V$ = 0.

##  7 Discussion

We have shown that infinitesimal isometries play a crucial role in the Kerr-
Schild double copy construction. In the context of four-dimensional vacuum
spacetimes, the geodesic and shear-free character of the Kerr-Schild vector
and the existence of a distinguished Killing vector allow us to define a
classical double copy without making any additional assumptions. In
particular, it can be defined even when the Killing vector is null or
spacelike with respect to the background metric, allowing us to dispense with
the restriction that the spacetime be stationary. In hindsight, there are
numerous reasons to be suspicious of this requirement. Even in the simplest
example of the Schwarzschild black hole, the spacetime is only stationary
outside the event horizon. Meanwhile, the Kerr spacetime contains a stationary
region with closed timelike curves where the timelike Killing vector tangent
to these curves does _not_ lead to a solution of Maxwell’s equations via
contraction with the graviton. This state of affairs might lead one to
speculate that the Killing vector must be timelike with respect to the
background metric, however our results clearly indicate otherwise.

Since we have fixed a null tetrad suitable for vacuum Kerr-Schild spacetimes
from the outset and made use of the spin coefficient formalism of Newman and
Penrose, our approach is particularly well-suited to make contact with the
Weyl double copy. Although the Weyl double copy is usually formulated in
spinorial language, it admits a very simple rewriting in terms of the Newman-
Penrose Weyl scalars and Maxwell scalars, allowing us to prove that the
reformulation of the Kerr-Schild double copy we have proposed implies the Weyl
double copy when the spacetime is Petrov type D. Interestingly, four out of
the five complex scalar equations encoding the Weyl double copy in Newman-
Penrose language are satisfied without making any assumption about the Petrov
type of the spacetime, and the type D condition is only required in order to
satisfy the component of the Weyl double copy corresponding to $\Psi_{4}$,
which is associated with radiation. This suggests that it may be possible to
formulate an exact Weyl double copy for Kerr-Schild spacetimes of any Petrov
type by including an additional radiation term along the lines of
$\Psi_{ABCD}=\frac{1}{S_{\tiny{\textrm{Coul}}}}\varphi^{\tiny{\textrm{Coul}}}_{(AB}\varphi^{\tiny{\textrm{Coul}}}_{CD)}+\frac{1}{S_{\tiny{\textrm{Rad}}}}\varphi^{\tiny{\textrm{Rad}}}_{(AB}\varphi^{\tiny{\textrm{Rad}}}_{CD)}$,
where the Coulombic part is the standard Weyl double copy and the radiation
part vanishes for type D spacetimes. Such a scheme is also reminiscent of
recent explorations into the Weyl double copy with sources Easson:2021asd ;
Easson:2022zoh . In terms of the Newman-Penrose Maxwell scalars, this would
correspond to radiation field with
$\varphi^{\tiny{\textrm{Rad}}}_{0}=\varphi^{\tiny{\textrm{Rad}}}_{1}=0$.
Indeed, the twistorial approach to the classical double copy White:2020sfn ;
Chacon:2021wbr  can be applied to solutions of _linearized_ gravity with any
Petrov type, while vacuum Kerr-Schild spacetimes are simultaneously solutions
of linearized gravity and the full non-linear Einstein equations. It stands to
reason that the twistorial double copy should imply an exact Weyl double copy
for Kerr-Schild spacetimes, and our results hint at what this exact relation
may look like.

We have presented a number of examples of vacuum Kerr-Schild spacetimes and
their single copy gauge fields that have not previously been studied in this
context. Perhaps the most interesting example we have studied is the self-dual
analog of the Kerr spacetime in Kleinian signature. While we have speculated
that this spacetime is diffeomorphic to the Kleinian self-dual Taub-NUT
metric, we have not been successful in finding an explicit diffeomorphism that
confirms this conjecture. Either confirming or refuting this conjecture would
be an important result, as the former would establish that Kleinian self-dual
Taub-NUT is a single Kerr-Schild spacetime, while the latter would indicate
that the self-dual analog of Kerr we have presented here is in fact a
genuinely new self-dual black hole spacetime.

Our work suggests a number of straightforward extensions. Developing a deeper
understanding of the map between real Lorentzian and self-dual vacuum Kerr-
Schild spacetimes in general and the self-dual analog of Kerr in particular
would be worthwhile endeavors. The self-dual setting also offers the
possibility of developing a non-abelian classical double copy via the close
analogy between Penrose’s non-linear graviton proposal penrose1976nonlinear
and the Penrose-Ward correspondence for Yang-Mills instantons
atiyah1977instantons . Pursuing this approach offers the possibility of making
closer contact between the classical and quantum versions of the double copy,
and deepening our understanding of the role of twistor theory in the double
copy paradigm. Additionally, there exist some simple generalizations of the
null tetrad we have used that accommodate solutions of Einstein-Maxwell theory
as well as spacetimes with a pure radiation stress tensor. In the case of
Kerr-Schild spacetimes with pure radiation stress tensors, examples of
interest, including the real analog of the Eguchi-Hanson metric presented here
and Kinnersley’s photon rocket, exhibit a double copy for sources that mimics
the double copy structure of the fields—it would be interesting to see whether
this feature persists in a more systematic analysis of such non-vacuum
spacetimes. Since we have made use of Newman-Penrose formalism, our results
are necessarily restricted to four-dimensional spacetimes. It would be a major
improvement on our work if at least some of the results could be extended to
spacetimes of arbitrary dimension. Finally, the tetrad we have used here is
the same one used in the study of the so-called Newman-Penrose map
Elor:2020nqe ; Farnsworth:2021wvs . It would be interesting to use the results
of this work to test the degree to which the Newman-Penrose map is compatible
with the Kerr-Schild double copy in a general setting.

##  8 Acknowledgements

We are grateful to Kara Farnsworth, Michael Graesser, and Cindy Keeler for
helpful discussions. DAE is supported in part by the U.S. Department of
Energy, Office of High Energy Physics, under Award Number DE-SC0019470. TM is
supported by the Simons Foundation, Award 896696.

##  Appendix A Calculation of $\Delta\rho$

We have the Bianchi identity

$\delta\Psi_{2}=3\tau\Psi_{2}$ (106)

along with the following expression for $\Psi_{2}$ which is valid in vacuum:

$\Psi_{2}=\mu\rho+\gamma(\rho-\tilde{\rho}).$ (107)

Acting with $\delta$ on (107) gives

$\displaystyle\delta\Psi_{2}$ $\displaystyle=$
$\displaystyle\rho\delta\mu+\mu\delta\rho+\delta\gamma(\rho-\tilde{\rho})+\gamma\delta(\rho-\tilde{\rho})$
$\displaystyle=$
$\displaystyle\rho\delta(V\rho)+(2\mu+\gamma)(\rho-\tilde{\rho})\tau+\gamma\delta\rho-\gamma\delta\tilde{\rho}$
$\displaystyle=$ $\displaystyle\rho^{2}\delta V+\rho
V\delta\rho+(2\mu+\gamma)(\rho-\tilde{\rho})\tau+\gamma\delta\rho-\gamma\delta\tilde{\rho}$
$\displaystyle=$ $\displaystyle\rho^{2}(\tilde{\nu}+V\tau)+\rho
V(\rho-\tilde{\rho})\tau+(2\mu+\gamma)(\rho-\tilde{\rho})\tau+\gamma(\rho-\tilde{\rho})\tau-\gamma\delta\tilde{\rho}$
$\displaystyle=$
$\displaystyle\rho^{2}(\tilde{\nu}+V\tau)+(3\mu+2\gamma)(\rho-\tilde{\rho})\tau-\gamma\delta\tilde{\rho}$

where we used $\mu=\rho V$, $\tilde{\nu}=\delta V-V\tau$ from (3), and the
(vacuum) Newman-Penrose equations $\delta\rho=(\rho-\tilde{\rho})\tau$ from
(21) and $\delta\gamma=(\mu+\gamma)\tau$ from (22). Meanwhile, putting (107)
into the right hand side of (106) gives

$\displaystyle\delta\Psi_{2}$ $\displaystyle=$ $\displaystyle
3\tau\mu\rho+3\tau\gamma(\rho-\tilde{\rho})$ (109) $\displaystyle=$
$\displaystyle 3\tau(\mu+\gamma)(\rho-\tilde{\rho})+3\tau\mu\tilde{\rho}.$

Combining (LABEL:deltaPhi2-1) and (109) leaves us with an expression for
$\delta\tilde{\rho}$ which is not found in the Newman-Penrose equations:

$\delta\tilde{\rho}=\frac{1}{\gamma}\Big{(}\rho^{2}(\
\tilde{\nu}+V\tau)-3\tau\mu\tilde{\rho}-\gamma(\rho-\tilde{\rho})\tau\Big{)}.$
(110)

Now from (21), we have the (vacuum) Newman-Penrose equation

$\Delta\rho=\tilde{\delta}\tau-\tau\tilde{\tau}-V\rho^{2}$ (111)

from which we can isolate an expression for $\Delta\rho$. First, we recall
that

$\tau=-\frac{\rho}{P}(bY+c).$ (112)

Acting on the equation above with $\tilde{\delta}$ leads to

$\displaystyle\tilde{\delta}\tau$ $\displaystyle=$
$\displaystyle\tilde{\delta}\left(-\frac{\rho}{P}(bY+c)\right)$ (113)
$\displaystyle=$
$\displaystyle-\frac{\tilde{\delta}\rho}{P}(bY+c)+\frac{\rho\delta
P}{P^{2}}(bY+c)-b\frac{\rho^{2}}{P}$ $\displaystyle=$
$\displaystyle-\frac{1}{P}(bY+c)\frac{1}{\tilde{\gamma}}\left[\tilde{\rho}^{2}(\nu+V\tilde{\tau})-3\tilde{\tau}\tilde{\mu}\rho+\tilde{\gamma}(\rho-\tilde{\rho})\tilde{\tau}\right]+\frac{\rho^{2}}{P^{2}}|bY+c|^{2}-b\frac{\rho^{2}}{P}.$

Substituting this last equation into (111) then gives us

$\displaystyle\Delta\rho$ $\displaystyle=$
$\displaystyle-\frac{1}{P}(bY+c)\left[\frac{2P^{3}}{m}\nu+2\rho\tilde{\tau}+\frac{3}{P}(b\tilde{Y}+\tilde{c})\rho(\rho+\tilde{\rho})\right]+\frac{\rho^{2}}{P^{2}}|bY+c|^{2}-b\frac{\rho^{2}}{P}-\tau\tilde{\tau}-V\rho^{2}.$

##  Appendix B Curvature invariants for self-dual black holes

Quadratic, cubic and quartic curvature invariants for the self-dual analog of
Kerr (76) can be computed straightforwardly with the help of an appropriate
symbolic algebra system such as Maple’s differential geometry package, or
Mathematica’s xAct suite. The results are summarized below:

$\displaystyle R_{\mu\nu\sigma\rho}R^{\mu\nu\sigma\rho}$ $\displaystyle=$
$\displaystyle\frac{192\,m^{2}}{R^{6}}$ (115) $\displaystyle
R_{\mu\nu}{}^{\alpha\beta}R_{\alpha\beta}{}^{\rho\sigma}R_{\rho\sigma}{}^{\mu\nu}$
$\displaystyle=$ $\displaystyle\frac{768\sqrt{2}m^{3}}{R^{9}}$ (116)
$\displaystyle
R_{\alpha\beta\gamma\rho}R^{\alpha\beta\gamma}{}_{\sigma}R_{\mu\nu\kappa}{}^{\rho}R^{\mu\nu\kappa\sigma}$
$\displaystyle=$ $\displaystyle\frac{9216m^{4}}{R^{12}}\,,$ (117)

where we have defined $R^{2}=u^{2}+v^{2}-2uv+4\zeta\tilde{\zeta}$.
The metric for the self-dual Taub-NUT spacetime can be written in the form
crawley2022black :

$ds^{2}=\frac{r-m}{r+m}(dt-2M\cosh\theta\,d\phi)^{2}+\frac{r+m}{r-m}\,dr^{2}-(r^{2}-m^{2})(d\theta^{2}+\sinh^{2}\theta\,d\phi^{2}).$
(118)

The corresponding curvature invariants for the self-dual Taub-NUT metric are:

$\displaystyle R_{\mu\nu\sigma\rho}R^{\mu\nu\sigma\rho}$ $\displaystyle=$
$\displaystyle\frac{96\,M^{2}}{(r+M)^{6}}$ (119) $\displaystyle
R_{\mu\nu}{}^{\alpha\beta}R_{\alpha\beta}{}^{\rho\sigma}R_{\rho\sigma}{}^{\mu\nu}$
$\displaystyle=$ $\displaystyle\frac{384M^{3}}{(r+M)^{9}}$ (120)
$\displaystyle
R_{\alpha\beta\gamma\rho}R^{\alpha\beta\gamma}{}_{\sigma}R_{\mu\nu\kappa}{}^{\rho}R^{\mu\nu\kappa\sigma}$
$\displaystyle=$ $\displaystyle\frac{2304M^{4}}{(r+M)^{12}}.$ (121)

The above invariants match for the two metrics under the identification

$R\rightarrow r+M\,,\qquad m\rightarrow\frac{M}{\sqrt{2}}$ (122)

Furthermore, the self-dual black hole (76) only has two obvious Killing
vectors, yet the above curvature invariants depend only on a radial
coordinate, suggesting that the solution may indeed be fully spherically
symmetric as is the self-dual Taub-NUT metric. Even still, finding an explicit
diffeomorphism between (76) and (118) has proven to be surprisingly subtle,
and we leave a definitive resolution of this problem for future work.

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