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#  Note on the Taub-NUT Instanton Metric

Joon-Hwi Kim  Walter Burke Institute for Theoretical Physics, California
Institute of Technology, Pasadena, CA 91125

###### Abstract

It is shown that a complex coordinate transformation maps the Taub-NUT
instanton metric to a Kerr-Schild metric. This metric involves a semi-infinite
line defect as a gravitational analog of the Dirac string, much like the
original metric. Also, it corresponds to self-dual dyon in electromagnetism
through three versions of classical double copy, one of them being a novel
nonlocal operator form. The relevance to Newman-Janis algorithm is briefly
noted.

††preprint: CALT-TH 2024-020

paracolorIntroduction.—Recently, an exciting program dubbed classical double
copy [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] has refined
analogies between gravity and electromagnetism into concrete mappings between
classical solutions while being rooted in the study of scattering amplitudes
[18, 19, 20, 17, 21]. The prime example is the Kerr-Schild classical double
copy, established in the seminal work [1]. Namely, it is shown that a
stationary Kerr-Schild solution $g=\eta+\Phi\hskip 1.00006pt\ell\otimes\ell$
in general relativity maps to a stationary solution in Maxwell theory
described by the gauge potential $A=\Phi\hskip 1.00006pt\ell$, by taking a
“single copy” of the null one-form $\ell$. Conversely, one envisions a
gravitational counterpart $g=\eta+\Phi\hskip 1.00006pt\ell\otimes\ell$ of a
stationary solution $A=\Phi\hskip 1.00006pt\ell$ in electromagnetism, by
taking a “double copy” of $\ell$. Well-known instances of this correspondence
are Schwarzschild to point charge and Kerr to a rotating solution dubbed
$\kern-0.74997pt\sqrt{\text{Kerr\hskip 0.50003pt}}$ [1, 22].

From this angle, it is natural to expect that the Taub-NUT solution [23, 24]
will correspond to a dyon, given the interpretation of the NUT charge as the
gravitomagnetic monopole [25, 26, 27, 28, 29, 30, 31, 16]. However, only a
“double Kerr-Schild” metric $g=\eta+\Phi_{1}\hskip
0.50003pt\ell_{1}\otimes\ell_{1}+\Phi_{2}\hskip
1.00006pt\ell_{2}\otimes\ell_{2}$ has been explicitly known for Taub-NUT [32,
33]. Thus, a mapping to electromagnetic dyons had to be achieved by a variant
of the Kerr-Schild double copy [6, 16], which yet offered an intriguing new
perspective.

Remarkably, in the present note we point out that the straightforward
application of the original Kerr-Schild double copy is in fact possible for
the self-dual Taub-NUT (SDTN) solution, which is an extremal case of the Taub-
NUT solution that is gaining a revived interest [34, 35, 36, 37, 38] due to
its intriguing physical properties [39, 40]. The crucial fact is that the SDTN
solution remarkably admits a Kerr-Schild metric in the ordinary sense:
$g=\eta+\Phi\hskip 1.00006pt\ell\otimes\ell$.

Firstly, a Kerr-Schild metric is obtained by Kerr-Schild double copying the
gauge potential of a self-dual dyon (SDD) in electromagnetism. Then we show
that it describes the SDTN solution by explicitly constructing a coordinate
transformation to the instanton metric of Gibbons and Hawking [41, 42, 43, 44,
45]. Curiously, this change of coordinates turns out to be inherently complex-
geometrical in Lorentzian signature, which is consistent with the
identification of the SDTN solution as a complex saddle.

Secondly, the physical interpretation of the solution as a gravitational dyon
is established within the Kerr-Schild description. Most importantly, we
confirm the characteristic Misner string geometry—the gravitational analog
[25, 26] of the Dirac string [46, 47]. Notably, finding its distributional
source [26, 27] becomes a simple problem in Kerr-Schild coordinates, thanks to
a linearization [48].

Thirdly, the classical double copy is established in two more ways. We first
show that the Weyl double copy relations given by Refs. [2, 4, 3], established
directly in terms of curvature tensors without referencing an explicit gauge
potential nor metric, are concretely reproduced in our explicit Kerr-Schild
description. This check clarifies a subtlety in the previous constructions [3,
4] and also illustrates that our Kerr-Schild coordinates are secretly the
“Cartesian” coordinates used in the twistor theory context [4, 3, 49, 50, 50,
51, 52, 53, 34, 54, 55]. Next, we propose an exotic new form of classical
double copy employing a nonlocal operator, which implements the Dirac strings
and traces back to propagators between dyonic sources [56, 57, 58, 59, 60].

Lastly, we briefly remark on how we originally arrived at this metric, which
was by deciphering the Newman-Janis shift—the magical property of spinning
black holes noticed in the curious “Note on the Kerr Spinning Particle Metric”
[61].

## Derivation of SDTN metric from double copy

The gauge potential of a dyon carrying electric and magnetic charges $(q,-iq)$
is given by

$\displaystyle\begin{split}A=\frac{q}{4\pi r}\hskip 1.00006pt\Big{(}\hskip
0.50003pt{-dt-i\hskip 0.50003ptr\hskip 0.50003pt(1{\hskip 1.00006pt-\hskip
1.00006pt}\cos\theta)\hskip 1.00006ptd\phi}\hskip
1.00006pt\Big{)}\,,\end{split}$ (1)

where we have set up spherical coordinates and oriented the Dirac string [46,
47] along the negative $z$-axis (south pole). This describes a complexified
solution where the field strength $F=dA$ is self-dual: $*F=+i\hskip
0.50003ptF$ 111 Technically, this holds off the Dirac string or when the Dirac
string is hidden. .

Now adding a total derivative $(q/4\pi)\hskip 1.00006ptd\log(r{\,+\,}z)$ gauge
transforms this to a Kerr-Schild form:

$\displaystyle A=\frac{q}{4\pi r}\hskip
1.00006pt\bigg{(}{-dt+\frac{x{\,+\,}iy}{r{\,+\,}z}\hskip
1.00006ptdx+\frac{y{\,-\,}ix}{r{\,+\,}z}\hskip 1.00006ptdy+dz}\bigg{)}\,,$ (2)

where $x=r\sin\theta\cos\phi$, $y=r\sin\theta\sin\phi$, $z=r\cos\theta$ are
the Cartesian coordinates. Importantly, the terms inside the bracket comprise
a null one-form.

With this understanding, the Kerr-Schild classical double copy [1] maps Eq.
(2) to the following line element, by literally taking “two copies” of the
null one-form:

$\displaystyle\begin{split}\kern-3.00003ptds^{2}={}&{}{-dt^{2}+dx^{2}+dy^{2}+dz^{2}}\\\
{}&{}+\frac{m}{4\pi r}\hskip
1.00006pt\bigg{(}{-dt+\frac{x{\,+\,}iy}{r{\,+\,}z}\hskip
1.00006ptdx+\frac{y{\,-\,}ix}{r{\,+\,}z}\hskip 1.00006ptdy+dz}\hskip
1.00006pt\bigg{)}^{2}\,.\kern-3.50006pt\end{split}$ (3)

Here we have adopted $8\pi G{\,=\,}1$ units and replaced the charge $q$ with
the mass $m$. Remarkably, Eq. (3) is a metric of the SDTN solution whose mass
and NUT charge are $m$ and $-im$, where complexified vacuum Einstein’s
equations are satisfied off the negative $z$-axis.

## The diffeomorphism

To show this, we simply construct an explicit coordinate transformation to the
known metric of the SDTN solution. It is convenient to employ complex
coordinates $\zeta=(x{\,+\,}iy)/(r{\,+\,}z)$,
$\smash{{\tilde{\zeta}}}=(x{\,-\,}iy)/(r{\,+\,}z)$ that arise from
stereographically projecting the two-sphere with respect to the south pole so
that Eq. (3) translates to

$\displaystyle\begin{split}ds^{2}={}&{}{-dt^{2}+dr^{2}+r^{2}\hskip
1.00006pt\frac{4\hskip 0.50003ptd\smash{{\tilde{\zeta}}}\hskip
0.50003ptd\zeta}{(1{\,+\,}{\tilde{\zeta}}\zeta)^{2}}}\\\ {}&{}+\frac{m}{4\pi
r}\hskip 1.00006pt\bigg{(}{-dt+dr-r\hskip
1.00006pt\frac{2{\tilde{\zeta}}\hskip
0.50003ptd\smash{\zeta}}{1{\,+\,}{\tilde{\zeta}}\zeta}}\hskip
1.00006pt\bigg{)}^{2}\,.\end{split}$ (4)

Amusingly, applying a diffeomorphism given by

$\displaystyle\begin{split}t\,\mapsto\hskip 1.00006ptt&\,,\quad
r\,\mapsto\hskip 1.00006ptr+\frac{m}{4\pi}\,,\\\ \zeta\,\mapsto\hskip
1.00006pt\zeta&\,,\quad{\tilde{\zeta}}\,\mapsto\hskip
1.00006pt\frac{1}{1+(m/4\pi r)(1{\,+\,}{\tilde{\zeta}}\zeta)}\hskip
1.00006pt{\tilde{\zeta}}\,,\end{split}$ (5)

we obtain

$\displaystyle ds^{2}={}$ $\displaystyle{}-\bigg{(}\hskip
0.50003pt{1+\frac{m}{4\pi r}}\hskip 0.50003pt\bigg{)}^{-1}\bigg{(}\hskip
1.00006pt{dt+\frac{m}{4\pi r}\hskip 1.00006pt\bigg{(}\hskip
1.00006pt{dr-r\hskip 1.00006pt\frac{2{\tilde{\zeta}}\hskip
0.50003ptd\smash{\zeta}}{1{\,+\,}{\tilde{\zeta}}\zeta}}\hskip
1.00006pt\bigg{)}}\hskip 1.00006pt\bigg{)}^{2}$ $\displaystyle+\bigg{(}\hskip
0.50003pt{1+\frac{m}{4\pi r}}\hskip 0.50003pt\bigg{)}\bigg{(}\hskip
1.00006pt{dr^{2}+r^{2}\hskip 0.50003pt\frac{4\hskip
1.00006ptd{\tilde{\zeta}}d\zeta}{(1{\,+\,}{\tilde{\zeta}}\zeta)^{2}}}\hskip
1.00006pt\bigg{)}\,,$ (6)

which is the Gibbons-Hawking metric [41, 42] 222 Note that this metric has a
residual freedom of gauge transforming $t$.  of the SDTN solution with mass
and NUT charge $m$ and $-im$!

This diffeomorphism is surely invertible. In fact, as is elaborated on in the
appendix, it turns out that Eq. (5) traces back to the flow along the null
geodesic congruence associated with the Kerr-Schild metric. As a result, the
inverse map is simply Eq. (5) with $m$ replaced with $-m$.

This completes our proof that the Kerr-Schild and Gibbons-Hawking metrics in
Eqs. (3) and (The diffeomorphism) are diffeomorphic. In turn, we establish
that the SDTN solution admits a construction from the Kerr-Schild classical
double copy. Previously, these facts had been only conjectured by Ref. [64] in
split signature: see the note added.

Meanwhile, the astute reader may have noticed an unusual feature of the
diffeomorphism in Eq. (5). Namely, it describes a genuine complexified
diffeomorphism, since $\zeta$ is frozen while $\smash{{\tilde{\zeta}}}$ gets
scrambled; hence the notation $\smash{{\tilde{\zeta}}}$ instead of
$\smash{\bar{\zeta}}$. However crucially, this is a legitimate operation
because the Lorentzian signature SDTN solution is inherently a complexified
solution as a self-dual solution. In fact, the SDD gauge potential described
in Eq. (2) is a solution to complexified Maxwell’s equations, due to the
imaginary magnetic charge $-iq$. In the precise same sense, the SDTN spacetime
is a solution to “complex general relativity” as a holomorphic field theory
governing holomorphic metrics of complex four-manifolds [54, 65, 66, 67]
(i.e., belongs to the “holomorphic, complexified category” [34]). Especially,
the imaginary NUT charge $-im$ essentially makes the metric complex-valued. As
a consequence of this interpretation, $\zeta$ and $\smash{{\tilde{\zeta}}}$
can be transformed independently as holomorphic coordinates.

## Physical interpretation

We shall now examine the physical validity of the Kerr-Schild metric in Eq.
(3). To begin with, we point out that Eq. (3) itself is unusual as a Kerr-
Schild metric, in fact. While the Kerr-Schild potential is simply $\Phi=m/4\pi
r$, the null one-form $\ell$ involves a semi-infinite line defect along the
negative $z$-axis, which is exactly how the gravitomagnetic charge is encoded:

$\displaystyle\ell=-dt+\frac{x{\,+\,}iy}{r{\,+\,}z}\hskip
1.00006ptdx+\frac{y{\,-\,}ix}{r{\,+\,}z}\hskip 1.00006ptdy+dz\,.$ (7)

This is intriguing because the null one-form is usually regular wherever the
Kerr-Schild potential is regular, which is especially true for the double
Kerr-Schild construction of Taub-NUT in Ref. [6].

Crucially, as in the original metric in Eq. (The diffeomorphism), the
interpretation of this line defect should be the Misner string [25, 26]: the
gravitational analog of the Dirac string, inputting a gravitomagnetic flux of
$-im$ at the origin. To show this, we recall that there are at least three
senses in which the Taub-NUT metric [25, 26] is a gravitational analog of a
dyon: (a) the Misner string is invisible as a coordinate artifact if the
gravitational analog of the Dirac quantization condition holds [25], (b) the
Misner string implements a flux tube of time monodromy [28, 29, 16], and (c)
the distributional stress-energy of the string describes an energy-less thin
solenoid of mass current [26, 27]. Our goal now is to show that all of these
properties hold in the Kerr-Schild description as well.

First, we derive the Dirac quantization condition. As a primer, consider the
SDD gauge potential in Eq. (2). Adding a term $(q/2\pi)\hskip
1.00006ptd\log\zeta$ repositions its Dirac string along the positive $z$-axis
(north pole). This encodes the complexified group-valued transition function
$\zeta{}^{\hskip 0.35002ptieq/2\pi}$, where $e$ is the electromagnetic
coupling [16]. Demanding its single-valuedness within the overlap of the two
patches covering either north or south poles [68] implies $ieq\in
2\pi\mathbb{Z}$, which is precisely the Dirac quantization condition from the
imaginary magnetic charge $-iq$.

The gravitational Dirac quantization has been derived in a similar fashion,
namely by hiding the string with overlapping coordinate patches [25]. In the
Kerr-Schild description, we find that the diffeomorphism

$\displaystyle\begin{split}t\,\mapsto\hskip
1.00006ptt+\frac{m}{2\pi}\log\zeta\,,\quad{\tilde{\zeta}}\,\mapsto\hskip
1.00006pt\frac{1-(m/4\pi
r)(1{\,+\,}{\tilde{\zeta}}\zeta)/{\tilde{\zeta}}\zeta}{1+(m/4\pi
r)(1{\,+\,}{\tilde{\zeta}}\zeta)}\end{split}$ (8)

with $r$ and $\zeta$ fixed transforms Eq. (4) to

$\displaystyle\begin{split}ds^{2}={}&{}{-dt^{2}+dr^{2}+r^{2}\hskip
1.00006pt\frac{4\hskip 0.50003ptd\smash{{\tilde{\zeta}}}\hskip
0.50003ptd\zeta}{(1{\,+\,}{\tilde{\zeta}}\zeta)^{2}}}\\\ {}&{}+\frac{m}{4\pi
r}\hskip 1.00006pt\bigg{(}{-dt+dr+r\hskip 1.00006pt\frac{2\hskip
0.50003ptd\smash{\zeta}}{\zeta(1{\,+\,}{\tilde{\zeta}}\zeta)}}\hskip
1.00006pt\bigg{)}^{2}\,,\end{split}$ (9)

which develops a string defect along the positive $z$-axis (north pole) 333
Note that repositioning the string commutes with the double copy: the single
copy of (9) exactly describes the SDD with north-pointing Dirac string. . For
this diffeomorphism to be single-valued within the overlapping region, the
time coordinate should be periodic as $t\sim t+im\mathbb{Z}$, given the $2\pi
i$ periodicity of $\log\zeta$. As a result, the energy $e$ 444 defined as a
constant since the spacetime is stationary  of a test particle is quantized as
$iem\in 2\pi\mathbb{Z}$. This is precisely the gravitational analog of the
Dirac quantization condition in the presence of the imaginary NUT charge $-im$
[29, 28, 16].

Second, we study the near-string geometries in an explicit manner. Let
$\xi:=x+iy$, $\smash{{\tilde{\xi}}}:=x-iy$ be complex coordinates for the
(complexified) $x$-$y$ plane, and suppose a positive infinitesimal $\epsilon$.
Applying the replacement $z\mapsto-1/\epsilon+z$ to the gauge potential in Eq.
(2) and then taking the limit $\epsilon\to 0$, we obtain $A=(q/2\pi)\hskip
1.00006ptd\log\smash{{\tilde{\xi}}}$. This precisely describes an infinite
Dirac string as a thin tube of magnetic flux $iq$ flowing in the $-z$
direction, which should be clear from its field strength $F=dA$:

$\displaystyle F=\frac{q}{2\pi}\hskip 1.00006ptdd\log{\tilde{\xi}}=iq\hskip
1.00006pt\delta^{(2)}({\tilde{\xi}},\bar{{\tilde{\xi}}})\hskip
1.00006ptd\smash{\bar{{\tilde{\xi}}}}{\hskip
1.00006pt{\wedge}\,}d{\tilde{\xi}}\,.$ (10)

Here $\delta^{(2)}({\tilde{\xi}},\bar{{\tilde{\xi}}})=(1/2\pi i)\hskip
1.00006pt\partial(1/{\tilde{\xi}})/\partial\bar{{\tilde{\xi}}}$ describes the
Dirac delta distribution peaked at ${\tilde{\xi}}=0$ 555 Note that
$\bar{{\tilde{\xi}}}\neq\xi$. .

For the line element in Eq. (3), the same limiting procedure leaves out a
divergent part, which interestingly can be compensated by a shift
$\smash{\xi}\mapsto\smash{\xi}+((m/\pi)\hskip
1.00006pt(1/\epsilon{\,-\,}z)-3m^{2}/4\pi^{2})/\smash{{\tilde{\xi}}}$.
Amusingly, the metric then becomes simply Minkowski, $ds^{2}=-dt^{\prime
2}+dz^{\prime 2}+d\smash{{\tilde{\xi}}}\hskip 1.00006ptd\xi$, but with
multivalued time:

$\displaystyle t^{\prime}=t+\frac{m}{2\pi}\hskip
1.00006pt\log{\tilde{\xi}}\,,\quad z^{\prime}=z-\frac{m}{\pi}\hskip
1.00006pt\log{\tilde{\xi}}\,.$ (11)

In other words, in the vicinity of the string the geometry approaches to the
image of empty spacetime upon a periodically multivalued diffeomorphism (large
gauge transformation [72, 73]). In this geometry, a Sagnac interferometer
measures the “time translation monodromy” $\oint dt^{\prime}=im$ as the
gravitational Aharanov-Bohm phase [74, 29, 30, 75] 666 To understand this
quickly, note that the Coriolis force $2m\vec{v}{\,\times\hskip
1.00006pt}\vec{\omega}$ is analogous to the Lorentz force
$q\vec{v}{\,\times}\smash{\vec{B}}$ [74]. . Clearly, the flux of this
monodromy is localized along $\smash{{\tilde{\xi}}}=0$:

$\displaystyle ddt^{\prime}=\frac{m}{2\pi}\hskip
1.00006ptdd\log{\tilde{\xi}}=im\hskip
1.00006pt\delta^{(2)}({\tilde{\xi}},\bar{{\tilde{\xi}}})\hskip
1.00006ptd\bar{{\tilde{\xi}}}{\hskip 1.00006pt{\wedge}\,}d{\tilde{\xi}}\,,$
(12)

which is curiously the “double copy” of Eq. (10) 777 In this sense the “field
strength” of gravity is torsion: note that (12) can be interpreted as the
torsion two-form generated by the multivalued diffeomorphism
($ddx^{\mu}=\Gamma^{\mu}{}_{\nu\rho}\hskip 1.00006ptdx^{\rho}{\hskip
1.00006pt{\wedge}\,}dx^{\nu}=\frac{1}{2}\hskip
1.00006ptT^{\mu}{}_{\rho\sigma}\hskip 1.00006ptdx^{\rho}{\hskip
1.00006pt{\wedge}\,}dx^{\sigma}$) [138, 139, 140]. This might be suggesting a
realization of color-kinematics duality in terms of Einstein-Cartan theory or
teleparallel gravity. . Therefore, we conclude that the string defect of Eq.
(3) indeed describes a thin flux tube of time monodromy.

Figure 1:  The Misner string is a “screw dislocation” in spacetime, inducing
time monodromy as an Aharanov-Bohm phase.

Finally, we explicitly derive the distributional stress-energy tensor of this
string. In general one can take various approaches [78, 79, 80], but here we
particularly consider the distributional stress-energy that follows from the
“harmonic-gauge linearization” of Ref. [48] which the Kerr-Schild description
enables as a luxury.

The harmonic-gauge linearization employs a crucial property that the
congruence of $\ell^{\mu}\partial_{\mu}$ can enjoy: geodesic and shear-free
(GSF). Given such assumptions, it follows that
$T^{\mu\nu}{\,=\,}u^{(\mu}J^{\nu)}$ can be identified as the source for the
gravitational solution [48]. Here $J^{\mu}$ denotes the source for the
electromagnetic solution, and $u^{\mu}{\,=\,}\delta^{\mu}{}_{0}$ is the unit
timelike Killing vector. Applying this formula, we immediately reproduce
Bonnor’s stress-energy tensor [26, 27], describing a point mass $m$ attached
to an energy-less 888 in the sense that $T^{00}$ vanishes  half-infinite
rotating rod! The GSF property of our null one-form in Eq. (7) can be shown
from an identity $\partial_{\mu}\ell_{\nu}{\,=\,}(1/2r)\hskip
1.00006pt(\eta_{\mu\nu}+u_{\mu}u_{\nu}-(\ell{\hskip 1.00006pt-\hskip
1.00006pt}u)_{\mu}(\ell{\hskip 1.00006pt-\hskip 1.00006pt}u)_{\nu}+i\hskip
1.00006pt\varepsilon_{\mu\nu\rho\sigma}u^{\rho}\ell^{\sigma})$, while the
single copy source $J^{\mu}$ easily follows by acting on the Laplacian
$\square:=-\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$ to Eq. (1) This shows
that the string defect of Eq. (3) indeed describes a thin solenoid of mass
current.

To sum up, we have convinced ourselves from multiple angles that the Kerr-
Schild metric in Eq. (3) indeed exhibits the geometrical and physical
characteristics of a Taub-NUT spacetime arising from the Misner string. This
analysis demonstrates that the Kerr-Schild description of the solution is not
merely a formal construct but physical. Moreover, the quick derivation of the
distributional source demonstrates its usefulness as well.

## Weyl double copy in position space

Finally, let us unlock more classical double copy relations for the SDTN
solution. First of all, we consider the Weyl double copy and explicate how the
Kerr-Schild metric in Eq. (3) resolves a subtlety in the previous
constructions [3, 4].

We begin with a brief review. The Weyl double copy directly constructs the
Weyl curvatures of algebraically special spacetimes by “squaring”
electromagnetic field strengths [2, 3, 4, 5], while not necessarily
referencing a metric or a gauge potential. For example, take the Coulomb and
Schwarzschild solutions. In the spinor language, their field strength and Weyl
tensor are given as the following:

$\displaystyle
F_{\smash{{\dot{\alpha}}{\dot{\beta}}}\vphantom{\beta}}=\frac{2q}{4\pi
r^{2}}\hskip
1.00006pt\tilde{o}_{\smash{({\dot{\alpha}}}\vphantom{\beta}}\tilde{\iota}_{\smash{{\dot{\beta}})}\vphantom{\beta}}\,,\,\,\,\,C_{\smash{{\dot{\alpha}}{\dot{\beta}}{\dot{{\gamma}}}{\dot{{\delta}}}}\vphantom{\beta}}=-\frac{6m}{4\pi
r^{3}}\hskip
1.00006pt\tilde{o}_{\smash{({\dot{\alpha}}}\vphantom{\beta}}\tilde{o}_{\smash{{\dot{\beta}}}\vphantom{\beta}}\hskip
0.50003pt\tilde{\iota}_{\smash{{\dot{{\gamma}}}}\vphantom{\beta}}\tilde{\iota}_{\smash{{\dot{{\delta}}})}\vphantom{\beta}}\,,$
(13)

where we have omitted the anti-self-dual counterparts. The Weyl double copy is
the squaring relation
$C_{\smash{{\dot{\alpha}}{\dot{\beta}}{\dot{{\gamma}}}{\dot{{\delta}}}}\vphantom{\beta}}=F_{\smash{({\dot{\alpha}}{\dot{\beta}}}\vphantom{\beta}}F_{\smash{{\dot{{\gamma}}}{\dot{{\delta}}})}\vphantom{\beta}}/S$
[2], which holds with $S{\,\propto\,}1/r$.

The Weyl double copy correspondences between the SDD and SDTN solutions are
established in Refs. [3, 4] as the “self-dual parts” of the Coulomb and
Schwarzschild solutions. That is, the self-dual Weyl curvature (and the
principal spinors) of the SDD/SDTN solution is simply postulated to be that of
the Coulomb/Schwarzschild solution. This physical anticipation, however, may
appear a little nontrivial if one recalls the nonlinearity of gravity 999 In
particular, the Newman-Janis shift is essentially the same argument applied to
the Kerr solution [34, 61, 129, 131, 1, 130], but concretely verifying that
SDTN is the “self-dual part” of Kerr requires a nontrivial discussion [132,
133, 36, 134]. . Especially, the spin-2 field
$\phi_{\smash{{\dot{\alpha}}{\dot{\beta}}{\dot{{\gamma}}}{\dot{{\delta}}}}\vphantom{\beta}}$
obtained from the Penrose transform describes only the linearized Weyl tensor
in the flat background [50]. To promote it to the fully nonlinear solution one
necessarily has to appeal to a linearization, which is famously possible in
the Kerr-Schild gauge [83, 84, 48].

Certainly, the explicit Kerr-Schild metric we have established clearly
resolves this subtlety and concretely validates the Weyl double copy
constructions in Refs. [3, 4]. Through a straightforward computation of spin
connection coefficients in a tetrad frame, we explicitly checked that the
self-dual Weyl tensor of the nonlinear SDTN solution as defined in Eq. (3)
indeed is equal to that of the Schwarzschild solution with the same principal
spinors. Specifically, the null one-form in Eq. (7) boils down to

$\displaystyle-\frac{1}{2}\hskip
1.00006pt\ell_{\mu}(\sigma^{\mu})_{\alpha{\dot{\alpha}}}=o_{\alpha}\hskip
0.50003pt\tilde{o}_{\dot{\alpha}}\,,\quad\tilde{o}^{\dot{\alpha}}=\begin{pmatrix}1\\\
\zeta\end{pmatrix}\,,\quad o^{\alpha}=\begin{pmatrix}1\\\ 0\end{pmatrix}\,,$
(14)

where $\tilde{o}^{\dot{\alpha}}$ is one of the principal spinors in Eq. (13).
Computing the Weyl tensor then reveals the other principal spinor,
$\tilde{\iota}^{\dot{\alpha}}{\,=\,}(-\smash{{\tilde{\zeta}}},1)/(1{\,+\,}\smash{{\tilde{\zeta}}}\zeta)$.
Crucially, the real combinations $[\tilde{o}_{\dot{\alpha}}]^{*}\hskip
0.50003pt\tilde{o}_{\dot{\beta}}$, $[\tilde{\iota}_{\dot{\alpha}}]^{*}\hskip
0.50003pt\tilde{\iota}_{\dot{\beta}}$ of these principal spinors exactly
reproduce the null one-forms $-dt\pm dr$ associated with the Kerr-Schild
metrics of the Schwarzschild solution [4, 3], hence confirming the statement
that SDTN is the “self-dual part” of Schwarzschild.

Note that this relationship between SDTN and Schwarzschild could be traced
back to the fact that the condition for a null vector field
$\smash{\tilde{\pi}}^{\dot{\alpha}}(x)\hskip 1.00006pt\pi^{\alpha}(x)$ to be
GSF is given by a chiral equation for each copy of the spinors: in flat
background, the equation for the right-handed reads [50, 85]

$\displaystyle\smash{\tilde{\pi}}^{\dot{\alpha}}(x)\hskip
1.00006pt\smash{\smash{\tilde{\pi}}^{\dot{\beta}}(x)}\hskip 1.00006pt(\hskip
0.12497pt\smash{\partial_{\alpha{\dot{\alpha}}}\smash{\tilde{\pi}}_{\dot{\beta}}(x)})=0\,.$
(15)

If $\smash{\tilde{\pi}}^{\dot{\alpha}}(x)$ is a solution to Eq. (15), then
both the complexified, self-dual combination $o_{\alpha}\hskip
1.00006pt\smash{\tilde{\pi}}_{\dot{\beta}}(x)$ (with constant $o_{\alpha}$)
and the real, non-self-dual combination
$[\smash{\tilde{\pi}}_{\dot{\alpha}}(x)]^{*}\hskip
0.50003pt\smash{\tilde{\pi}}_{\dot{\beta}}(x)$ define null GSF one-forms, from
which the idea of taking the “self-dual part” is validated 101010 It would be
interesting to experiment a further mix-and-match of null GSF congruences,
such as one copy of SDTN congruence for the right-handed tensored with one
copy of Robinson congruence for the left-handed. .

## Weyl double copy in twistor space

Next, let us revisit the Weyl couble copy from twistor space [4, 3], which
eventually illuminates how the Kerr-Schild coordinates have been secretly used
in the twistor theory context.

To smoothly motivate the twistor space, we consider a curious theorem due to
Kerr [87, 50, 88]: any solution to Eq. (15) arises as a zero locus of a
holomorphic function $f(\smash{\tilde{Z}})$ homogeneous in its four complex
arguments $\smash{\tilde{Z}}^{\mathrm{A}}{\,\hskip 1.00006pt=\hskip
1.00006pt\,}(\smash{\tilde{\pi}}_{\dot{\alpha}},\tilde{\omega}^{\alpha})$ upon
plugging in the “incidence relation”
$\tilde{\omega}^{\alpha}{\,=\,}\smash{\tilde{\pi}}_{\dot{\alpha}}\hskip
0.50003ptx^{{\dot{\alpha}}\alpha}$ [87]. This curious complex variable
$\smash{\tilde{Z}}^{\mathrm{A}}$ is called a twistor [87] and is known to
represent a null ray in complexified Minkowski space [87].

Taking inspirations from Refs. [89, 90, 91], Ref. [4] proposed the following
function for constructing the GSF congruences associated with the SDTN
solution 111111 This function can be written as $i\hskip
1.00006pt\smash{\tilde{Z}}^{\mathrm{A}}\hskip
1.00006ptu_{\mu}(\gamma^{\mu})_{\mathrm{A}}{}^{\mathrm{C}}\hskip
1.00006pt\bar{I}_{{\mathrm{C}}{\mathrm{B}}}\hskip
1.00006pt\smash{\tilde{Z}}^{\mathrm{B}}$, where
$\bar{I}_{{\mathrm{A}}{\mathrm{B}}}$ denotes infinity twistor. :

$\displaystyle f(\smash{\tilde{Z}})=-\tilde{\omega}^{\alpha}\hskip
1.00006ptu_{\alpha{\dot{\alpha}}}\hskip
1.00006pt\smash{\tilde{\pi}}^{\dot{\alpha}}\,,\quad
u_{\alpha{\dot{\alpha}}}:=u_{\mu}(\sigma^{\mu})_{\alpha{\dot{\alpha}}}\,.$
(16)

Indeed, $f(\smash{\tilde{\pi}},\smash{\tilde{\pi}}x)=0$ with the above
function exactly yields our principal spinors
$\tilde{o}_{\dot{\alpha}},\tilde{\iota}_{\dot{\alpha}}$ as
$\smash{\tilde{\pi}}_{\dot{\alpha}}(x)\propto\tilde{o}_{\dot{\alpha}},\tilde{\iota}_{\dot{\alpha}}$.

Remarkably, this twistor function turns out to encode not only the congruences
but the entire geometry of the (linearized) SDTN spacetime through a contour
integral formula known as Penrose transform [93, 94], where the “localization”
of $\smash{\tilde{\pi}}_{\dot{\alpha}}$ to zeroes of
$f(\smash{\tilde{\pi}},\smash{\tilde{\pi}}x)$ arises from a residue
computation [4, 3, 50]:

$\displaystyle\phi_{{\dot{\alpha}}_{1}\cdots{\dot{\alpha}}_{2h}}(x)\hskip
1.00006pt=\oint\frac{\smash{\tilde{\pi}}_{\dot{{\gamma}}}\hskip
0.50003ptd\smash{\tilde{\pi}}^{\dot{{\gamma}}}}{2\pi
i}\,\frac{\smash{\tilde{\pi}}_{{\dot{\alpha}}_{1}}\cdots\smash{\tilde{\pi}}_{{\dot{\alpha}}_{2h}}}{({f(\smash{\tilde{\pi}},\smash{\tilde{\pi}}x)})^{h+1}\vphantom{\tilde{Z}}}\,.$
(17)

Evaluating these contour integrals for $h{\,=\,}1,2$ readily reproduces the
spin-1 and spin-2 fields in Eq. (13). The calculations are almost immediate if
one notes that
$f(\smash{\tilde{\pi}},\smash{\tilde{\pi}}x)=-\smash{\smash{\tilde{\pi}}_{\dot{\alpha}}\hskip
0.50003pt\tilde{\chi}^{\dot{\alpha}}{}_{\dot{\beta}}\hskip
0.50003pt\smash{\tilde{\pi}}^{\dot{\beta}}}=-r\sigma$ with
$\tilde{\chi}_{\smash{{\dot{\alpha}}{\dot{\beta}}}\vphantom{\beta}}=r\hskip
1.00006pt\tilde{o}_{\smash{({\dot{\alpha}}}\vphantom{\beta}}\tilde{\iota}_{\smash{{\dot{\beta}})}\vphantom{\beta}}$
when the contour is parameterized as
$\smash{\tilde{\pi}}_{\dot{\alpha}}=\tilde{o}_{\dot{\alpha}}{\,+\,}\sigma\hskip
1.00006pt\tilde{\iota}_{\dot{\alpha}}$, where

$\displaystyle\tilde{\chi}^{\dot{\alpha}}{}_{\dot{\beta}}=-x^{\mu}u^{\nu}(\tilde{\sigma}_{\mu\nu})^{\dot{\alpha}}{}_{\dot{\beta}}=-{\textstyle\frac{1}{2}}\hskip
1.00006pt(\vec{x}{\hskip 1.00006pt\cdot\hskip
1.00006pt}\vec{\sigma})^{\dot{\alpha}}{}_{\dot{\beta}}$ (18)

describes the self-dual part of the bivector $2x^{[\mu}u^{\nu]}$.

In this manner, the curvature tensors arise as the Penrose transform of
functions $\mathcal{M}_{h}(\smash{\tilde{Z}})=(f(\smash{\tilde{Z}}))^{-h-1}$
intwistor space, which are in fact scattering amplitudes of a massive object
absorbing a negative-helicity force carrier [49] 121212 Ref. [49] has half-
Fourier transformed only the massless leg to the twistor space. For the
possibility of half-Fourier transforming the massive legs as well, see Ref.
[141]. . The twistor-space Weyl double copy is the squaring relation
$\mathcal{M}_{2}(\smash{\tilde{Z}})=({\mathcal{M}_{1}(\smash{\tilde{Z}})})^{2}/\mathcal{M}_{0}(\smash{\tilde{Z}})$
between these “twistor-space amplitudes.”

Crucially, this rederivation of the twistor-space construction of the SDTN
solution is not merely a review. Throughout, the coordinates
$x^{\mu}=(\sigma^{\mu})_{\alpha{\dot{\alpha}}}\hskip
1.00006ptx^{{\dot{\alpha}}\alpha}$ have always referred to the Kerr-Schild
coordinates defined by the metric in Eq. (3). This explicitly shows that our
Kerr-Schild coordinates are exactly the Cartesian coordinates which the
twistor-space construction of the SDTN solution employs in terms of Penrose
transform. In fact, this may not be surprising because our Kerr-Schild
coordinates turn out to be just Plebański’s coordinates [54, 55] leading to
the second heavenly equation. To conclude, twistor-theoretic perspectives on
the SDTN solution have presumed the Kerr-Schild coordinates, even if the
explicit metric may have not been written down as in Eq. (3) or shown to be
equivalent to the previously documented form (Gibbons-Hawking [41] or Taub-NUT
[23, 24]).

As a side note, we also remark that
$\tilde{\chi}^{\dot{\alpha}}{}_{\dot{\beta}}$ in Eq. (18) is in fact the
Killing spinor [50, 51, 52, 53] of the SDTN solution, from which a “position
vector” $\vec{x}$ is identified [34]. The identity of this $\vec{x}$ should be
evident now.

## Operator Kerr-Schild double copy

Lastly, we propose a nonlocal operator Kerr-Schild double copy as a new
result, which interestingly differs from the self-dual double copy previously
established in Refs. [17, 1].

In operator Kerr-Schild double copy, the null one-form $\ell_{\mu}$ in the
Kerr-Schild ansatz is promoted to a differential operator
$\smash{\hat{k}_{\mu}}$. Especially, it can factorize as
$(\sigma^{\mu})_{\alpha{\dot{\alpha}}}\hskip
1.00006pt\smash{\hat{k}_{\mu}}=-2\hskip
1.00006pto_{\alpha}\smash{\hat{k}_{\dot{\alpha}}}$ so that the fields are
manifestly described in the lightcone gauge [1, 7, 8]. Intriguingly, we find
that the gauge potential in Eq. (2) and the metric in Eq. (3) precisely arise
in this way if the operator $\hat{k}_{\dot{\alpha}}$ is given by

$\displaystyle\hat{k}_{\dot{\alpha}}\hskip 1.00006pt=\hskip
0.50003pt\bigg{(}\hskip 1.00006pt{-\frac{\partial_{x}{\hskip
1.00006pt+\,}i\partial_{y}}{\partial_{z}}\hskip 1.00006pt,\hskip
1.00006pt1}\hskip 1.00006pt\bigg{)}\,,$ (19)

which is nonlocal. Specifically, direct computation shows that
$\smash{\hat{k}_{\dot{\alpha}}}(1/r)=(1/r)\hskip
1.00006pt\tilde{o}_{\dot{\alpha}}$ and
$\smash{\hat{k}_{\dot{\alpha}}}\smash{\hat{k}_{\dot{\beta}}}(1/r)=\smash{(1/r)\hskip
1.00006pt\tilde{o}_{\dot{\alpha}}\tilde{o}_{\dot{\beta}}}$, so a classical
double copy correspondence between SDD and SDTN is realized in the following
form:

$\displaystyle A_{\alpha{\dot{\alpha}}}=-2\hskip
1.00006pto_{\alpha}\smash{\hat{k}_{\dot{\alpha}}}\hskip 1.00006pt\frac{q}{4\pi
r}\,,\quad
h_{\smash{\alpha{\dot{\alpha}}\beta{\dot{\beta}}}\vphantom{\beta}}=4\hskip
1.00006pto_{\alpha}o_{\beta}\hskip
1.00006pt\smash{\hat{k}_{\dot{\alpha}}}\smash{\hat{k}_{\dot{\beta}}}\hskip
1.00006pt\frac{m}{4\pi r}\,,$ (20)

where $h_{\mu\nu}:=g_{\mu\nu}{\,-\,}\eta_{\mu\nu}$ denotes the “graviton
field.”

Interestingly, we find that this exotic new form of classical double copy is
demystified from an electric-magnetic dual perturbation theory of dyons [56,
57, 58, 59, 60]. Note first that the distributional source of the SDD is
explicitly given by

$\displaystyle J^{\mu}=\partial_{\nu}F^{\mu\nu}=\bigg{(}\hskip
1.00006pt{1\hskip 1.00006pt,\hskip 1.00006pt-i\hskip
1.00006pt\frac{\vec{n}{\hskip 1.00006pt\times\hskip
0.50003pt}{\vec{\partial}}}{\vec{n}{\hskip 1.00006pt\cdot\hskip
1.00006pt}{\vec{\partial}}}}\hskip 1.00006pt\bigg{)}\,q\hskip
1.00006pt\delta^{(3)}(\vec{x})\,,$ (21)

where $\vec{n}$ is the unit vector along the $z$-axis 131313 In fact, the
orientation of the Dirac string is determined by not only the three-vector
$\vec{n}$ here but also the prescription for the anti-derivative
$\smash{(\vec{n}{\hskip 1.00006pt\cdot\hskip
1.00006pt}{\vec{\partial}})^{-1}}$. In particular, it is placed along the
negative $z$-axis as in (1) if $\smash{\vec{n}{\hskip 1.00006pt\cdot\hskip
1.00006pt}{\vec{\partial}}}=\partial_{z}$ and
$(\partial_{z})^{-1}(1/r)=\log(r{\,+\,}z)$. . This describes an extended
object: the anti-derivative $\smash{(-\vec{n}{\hskip 1.00006pt\cdot\hskip
1.00006pt}{\vec{\partial}})^{-1}}$ ${=\,}(-\partial_{z})^{-1}$ stacks magnetic
dipoles $(-\vec{n}{\hskip 1.00006pt\times}\smash{{\vec{\partial}}})(-iq\hskip
1.00006pt\delta^{(3)}(\vec{x}))$ along the negative $z$-axis, implementing a
thin solenoid. On the other hand, physically we know that this extended object
mimics an isolated point dyon $(q,-iq)$ localized at the origin, which is
described by the following source:

$\displaystyle\begin{split}J^{\mu}=\partial_{\nu}F^{\mu\nu}&=qu^{\mu}\hskip
1.00006pt\delta^{(3)}(\vec{x})\,,\\\
J^{\star\mu}=-\partial_{\nu}{*}F^{\mu\nu}&=-iqu^{\mu}\hskip
1.00006pt\delta^{(3)}(\vec{x})\,.\end{split}$ (22)

Specifically, $F_{\mu\nu}$ sourced by Eq. (21) is identical to $F_{\mu\nu}$
sourced by Eq. (22) at all points outside the solenoid 141414 This traces back
to the well-known fact in electromagnetism that the magnetic field of
Gilbertian and Ampèrian dipoles is identical up to the contact term [142]. In
fact, the operator $(\vec{n}{\hskip 1.00006pt\times\hskip
0.50003pt}{\vec{\partial}})/(\vec{n}{\hskip 1.00006pt\cdot\hskip
1.00006pt}{\vec{\partial}})$ in $\smash{\vec{J}}$ given by (21) connects
between the Gilbertian (separated opposite magnetic monopoles) and Ampèrian
pictures (electric current loop) of magnetic dipole, so dropping it gives the
magnetic charge density $J^{\star 0}$ given by (22). . Yet crucially, Eq. (22)
involves a bonafide violation of the Bianchi identity:
$\partial_{\smash{[\rho}\vphantom{\beta}}F_{\smash{\mu\nu]}\vphantom{\beta}}{\,\neq\,}0$.
Hence it does not define a source in the Maxwell theory but instead belongs to
its extension admitting both electric and magnetic sources considered by Dirac
[98].

As a dynamical field theory, a Lagrangian formulation of this extended Maxwell
theory is given by Zwanziger’s “two-potential electrodynamics” where the dual
photon field $A^{\star}_{\mu}$ is employed to incorporate the Bianchi
violation [99, 56]. This means the gauge potential of the SDD could have been
derived from the dyonic source in Eq. (22) by using the propagators of
Zwanziger theory!

The Zwanziger propagators can be found in Refs. [56, 57, 58, 59]. Here, we
note that a single spinor equation succinctly summarizes them 151515 As will
be clear from the exposition in Ref. [60], this propagator essentially traces
back to the XZC presentation [143, 144] of massless propagator employing an
auxiliary massless vector. :

$\displaystyle\begin{split}\hat{\Delta}^{+-}_{{\smash{\alpha{\dot{\alpha}}\beta{\dot{\beta}}}\vphantom{\beta}}}=\frac{n_{{\smash{\alpha{\dot{\beta}}}\vphantom{\beta}}}\hskip
0.50003pt\partial_{{\smash{\beta{\dot{\alpha}}}\vphantom{\beta}}}}{-n{\hskip
1.00006pt\cdot\hskip 1.00006pt}\partial}\frac{1}{\square}\,,\end{split}$ (23)

where $n_{\smash{\alpha{\dot{\alpha}}}\vphantom{\beta}}$ is an auxiliary four-
vector that traces back to the Dirac string. Eq. (23) describes the position-
space propagator from $A^{-}_{\mu}$ to $A^{+}_{\mu}$ where
$A^{\pm}_{\mu}:={\textstyle\frac{1}{2}}\hskip
1.00006pt(A^{\vphantom{+}}_{\mu}{\,\pm\,}iA^{\star}_{\mu})$, while other
propagators between $A^{\pm}_{\mu}$ vanish. Meanwhile, the sources coupling to
$A^{\pm}_{\mu}$ are given by $J_{\pm}^{\mu}=J^{\mu}\mp i\hskip
1.00006ptJ^{\star\mu}$, which are $J_{+}^{\mu}=0$,
$J_{-}^{\mu}=2qu^{\mu}\hskip 1.00006pt\delta^{(3)}(\vec{x})$ for Eq. (22). As
a result, the photon field
$A_{\mu}=\smash{A^{+}_{\mu}}{\,+\,}\smash{A^{-}_{\mu}}$ sourced by Eq. (22) is
found as

$\displaystyle
A^{\vphantom{\smash{+}}}_{\smash{\alpha{\dot{\alpha}}}\vphantom{\beta}}=A^{\smash{+}}_{\smash{\alpha{\dot{\alpha}}}\vphantom{\beta}}=\frac{n_{{\smash{\alpha{\dot{\beta}}}\vphantom{\beta}}}\hskip
0.50003pt\smash{u^{{\dot{\beta}}\beta}}\partial_{{\smash{\beta{\dot{\alpha}}}\vphantom{\beta}}}}{n{\hskip
1.00006pt\cdot\hskip 1.00006pt}\partial}\hskip 1.00006pt\frac{q}{4\pi r}\,.$
(24)

Then a simple calculation shows that Eqs. (19) and (20) are exactly reproduced
when the auxiliary vector is taken to be null as
$n_{\smash{\alpha{\dot{\alpha}}}\vphantom{\beta}}={-o_{\alpha}{\bar{o}}_{\dot{\alpha}}}$,
yielding
$\smash{\hat{k}}_{\dot{\alpha}}={-\partial_{\smash{1{\dot{\alpha}}}\vphantom{\beta}}/\partial_{\smash{0\dot{0}}\vphantom{\beta}}}$.

Amusingly, the graviton field arises in the same fashion by using the
linearized gravity equivalent of Zwanziger propagator, which is simply given
by double copying Eq. (23) as is advocated in Ref. [60]! In the gravitational
case the source dual to $T^{\mu\nu}=u^{(\mu}J^{\nu)}$ is given by
$T_{+}^{\mu\nu}=0$ and $T_{-}^{\mu\nu}=2mu^{\mu}u^{\nu}\hskip
1.00006pt\delta^{(3)}(\vec{x})$ 161616 This is the gravitational analog of
“Gilbert-Ampère duality” described in footnote [96]. Note that a static
gravitational dyon $(m,-im)$ is precisely described by
$T^{\mu\nu}=mu^{\mu}u^{\nu}\hskip 1.00006pt\delta^{(3)}(\vec{x})$ and
$T^{\star\mu\nu}=-imu^{\mu}u^{\nu}\hskip 1.00006pt\delta^{(3)}(\vec{x})$. ,
describing a static self-dual gravitational dyon. As a result, the linearized
graviton field $h^{\vphantom{+}}_{\mu\nu}=h^{+}_{\mu\nu}{\,+\,}h^{-}_{\mu\nu}$
is found as

$\displaystyle
h^{\vphantom{\smash{+}}}_{\smash{\alpha{\dot{\alpha}}\beta{\dot{\beta}}}\vphantom{\beta}}=h^{\smash{+}}_{\smash{\alpha{\dot{\alpha}}\beta{\dot{\beta}}}\vphantom{\beta}}=\frac{n_{{\smash{\alpha{\dot{{\gamma}}}}\vphantom{\beta}}}\hskip
0.50003pt\smash{u^{{\dot{{\gamma}}}{\gamma}}}\partial_{{\smash{{\gamma}{\dot{\alpha}}}\vphantom{\beta}}}}{n{\hskip
1.00006pt\cdot\hskip 1.00006pt}\partial}\hskip
1.00006pt\frac{n_{{\smash{\beta{\dot{{\delta}}}}\vphantom{\beta}}}\hskip
0.50003pt\smash{u^{{\dot{{\delta}}}{\delta}}}\partial_{{\smash{{\delta}{\dot{\beta}}}\vphantom{\beta}}}}{n{\hskip
1.00006pt\cdot\hskip 1.00006pt}\partial}\hskip 1.00006pt\frac{m}{4\pi r}\,,$
(25)

which is in turn identified as the full metric perturbation
$g_{\mu\nu}{\,-\,}\eta_{\mu\nu}$ by virtue of the linearization in the Kerr-
Schild gauge [83, 84, 48]. This precisely explains the double copy structure
proposed in Eq. (20) as we claimed earlier.

A few comments are in order. Firstly, it is instructive to revisit this
construction from the on-shell perspective. It is easy to check that the
“numerator factor” of the off-shell propagator in Eq. (23) correctly
factorizes into the null polarization vectors with
$\smash{n_{\smash{\alpha{\dot{\alpha}}}\vphantom{\beta}}}{\,=\,}\smash{-o_{\alpha}{\bar{o}}_{\dot{\alpha}}}$,
from which its double copy can be argued [60]. The double copy of sources from
$\smash{J_{-}^{\mu}}=\smash{2qu^{\mu}\hskip 1.00006pt\delta^{(3)}(\vec{x})}$
to $\smash{T_{-}^{\mu\nu}}=\smash{2mu^{\mu}u^{\nu}\hskip
1.00006pt\delta^{(3)}(\vec{x})}$ realizes the squaring relation of the massive
on-shell amplitudes in terms of the $x$-factor [102, 103, 104, 60]. Further,
one may also explicitly connect the off-shell constructed fields in Eqs. (24)
and (25) to the Penrose transform in Eq. (17) through a parameterization of
off-shell momentum employed in Ref. [49].

Secondly, it will be interesting to examine whether this “Zwanziger double
copy” persists for generic Taub-NUT 171717 Duff construction [145] from the
Gilbertian linearized sources might be not straightforward by the failure of
electric-magnetic duality due to the inherent nonlinearity of gravity [146].
Meanwhile, switching between the Ampèrian and Gilbertian pictures at the
nonlinear level might be realized by the duality observed in Ref. [147]. . Its
failure would signal the significance of not only linearization by Kerr-Schild
gauge but also self-duality.

Lastly, we report a strange fact that the SDTN solution fails to be the self-
dual double copy [17, 1, 7] of SDD. Such a dissonance is disturbing,
especially since the self-dual double copy is arguably the most natural option
for self-dual solutions and will imply fascinating conclusions such as “SDTN
black hole is a large-$N$ dyon” by being rooted on Moyal deformation [106,
107, 108, 109] of Plebański [54]’s second heavenly equation [110, 111].
Specifically, the local operator
$\smash{\hat{k}_{\dot{\alpha}}}=\partial_{0{\dot{\alpha}}}$ of the self-dual
double copy makes the Plebański scalar field different for each helicity:
$A_{\alpha{\dot{\alpha}}}=-2\hskip
1.00006pto_{\alpha}\partial_{0{\dot{\alpha}}}(\hskip
0.50003pt{q\zeta/4\pi}\hskip 0.50003pt)$,
$h_{\smash{\alpha{\dot{\alpha}}\beta{\dot{\beta}}}\vphantom{\beta}}=4\hskip
1.00006pto_{\alpha}o_{\beta}\hskip
1.00006pt\partial_{\smash{0{\dot{\alpha}}}\vphantom{\beta}}\partial_{\smash{0{\dot{\beta}}}\vphantom{\beta}}$
$(\hskip 0.50003pt{m(2r{\,+\,}z)\hskip 1.00006pt\zeta^{2}/24\pi}\hskip
0.50003pt)$ 181818 Naively, incorporating the nonabelian color structure seems
to not solve the puzzle: as a diagnosis one can check whether the color
Killing vector equation [148] in the single copy configuration is mapped to
the Killing equation for gravity and find failure. Fundamentally, there seems
to be a difference in how the mass dimension increases by one when the gauge
potential is mapped to the graviton field, between the self-dual double copy
construction of the Eguchi-Hanson instanton in Ref. [7] and the double copy
constructions of Taub-NUT instanton presented here. The former keeps the
“matter coupling” (denoted $\lambda$ in Ref. [7]) the same while
$\smash{\hat{k}}_{\dot{\alpha}}=\partial_{0{\dot{\alpha}}}$ imparts a mass
dimension. However, the latter imparts the mass dimension by changing the
matter coupling from $q$ to $m$. . This puzzle seems to deserve a further
investigation, possibly by comparing with the case of Eguchi-Hanson instanton
[7].

## Conclusions

In this note, we have shown that a complexified diffeomorphism reveals a Kerr-
Schild metric for the SDTN solution. Crucially, the validity of this
diffeomorphism and the metric is verified by highlighting their intrinsically
complex nature. Especially, we have verified that the Kerr-Schild metric also
exhibits the Misner string defect, supporting its physical interpretation as a
gravitational dyon. Further explicitly, we have established three concrete
senses in which the SDTN solution is precisely the gravitational counterpart
of a SDD in electromagnetism: Kerr-Schild, Weyl, and operator Kerr-Schild
classical double copy. Notably, the last one adds to the inventory of
classical double copies as a new entry. We have identified its physical origin
as the double copy of propagators between dyonic sources.

It is interesting to realize that the Gibbons-Hawking ansatz had already
suggested a sense in which self-dual electromagnetic objects correspond to
self-dual gravitational objects, far before the advent of the classical double
copy program. It is amusing to see how this correspondence has become
strengthened for SDTN and SDD from three new perspectives. Especially, the
idea that the Misner string is a gravitational analog of Dirac string is
concretely confirmed in the first and third double copy constructions. On a
related note, we can also frame this work from the angle of “classical double
copy constructions of gravitational instantons.” A noteworthy example has been
given by Ref. [7], where the Eguchi-Hanson instanton [113, 114, 115] is
constructed from the self-dual double copy (cf. Ref. [116]). This work adds
another instance to the discourse, i.e., the “Taub-NUT instanton.”

The Kerr-Schild metric of the SDTN solution conveys an aesthetic elegance in
terms of its minimalist and spinorial nature as manifested in Eq. (14). To
investigate what facilitates this remarkable gem, we could examine its
relation to the self-dual limit of the double Kerr-Schild metric employed in
Ref. [6]. Finding a diffeomorphism bridging between the two would shed an
insight 191919 We thank Andres Luna for suggesting this direction. .

Meanwhile, as we have clarified the Kerr-Schild coordinates themselves have
been already employed in the spinorial/twistorial formalisms. In this light,
the metric in Eq. (3) can be thought of as providing an explicit construction
of a “potential” for the Weyl curvature which the Penrose transform or Killing
spinor has described.

Yet, we believe there is a value in having a more explicit description. In
particular, it will be interesting to explore further consequences and
applications of our Kerr-Schild metric: asymptotic symmetries [118, 119]
202020 The Kerr-Schild metric in (3) nicely falls into the BMS category of
asymptotically flat metrics [149, 150, 151], with mass aspect $m$, angular
momentum aspects zero, and $\smash{C_{\smash{\zeta\zeta}\vphantom{\beta}}}=$
$\smash{4M{\tilde{\zeta}}{}^{2}/(1+{\tilde{\zeta}}\zeta)^{2}}$,
$\smash{C_{{\tilde{\zeta}}{\tilde{\zeta}}}}=0$. The Gibbons-Hawking metric in
(The diffeomorphism), in contrast, fails to satisfy the falloff conditions as
$\smash{g_{{\tilde{\zeta}}\zeta}}{\,=\,}4r^{2}\smash{(1+{\tilde{\zeta}}\zeta)^{-2}}+\mathcal{O}(r^{1})$.
, revisiting the integrability of test-particle motion [39, 40], black hole
perturbation theory [35, 34, 121, 122], or finding exact graviton propagators
[123, 124, 125] to carry out self-force calculations [126, 127, 128] in an
idealized set-up, to list a few directions. Especially, explorations in the
scattering context are likely to shed insights on Kerr black hole scattering
as a certain integrable sector on account of the Newman-Janis shift: the
“self-dual part” of the Kerr solution is a SDTN solution displaced along an
imaginary direction [61, 129, 130, 131, 1, 22, 35, 132, 133, 36, 34].

Figure 2:  The very physical origin of the Newman-Janis shift is the fact that
the Kerr black hole is secretly a “dumbbell” of self-dual and anti-self-dual
Taub-NUT instantons [134].

Amusingly, about a year ago we found a vivid physical picture that explains
this fact explicitly by means of complex diffeomorphisms. Namely, as depicted
in Fig. 2, we realized that a simple complex coordinate transformation maps
the Kerr metric in Kerr-Schild coordinates to another Kerr-Schild metric
describing two point masses joined by a Misner string. By zooming into the
upper tip of the string, we obtained—by construction—a Kerr-Schild metric,
which was precisely the metric in Eq. (3). And crucially, this had to be
describing a Taub-NUT instanton according to the Newman-Janis shift. The
complex diffeomorphism in Eq. (5) was precisely a result of explicitly
verifying this strong conviction. This is how we originally derived the Kerr-
Schild metric reported in this note. We hope to release more details soon
[134].

Note added. Shortly after the initial release of this work, the author was
informed that a split-signature version of the Kerr-Schild metric in Eq. (3)
was previously conjectured in Ref. [64]. Then, about a week after this notice,
Ref. [135] appeared, providing an existence proof for a real diffeomorphism in
split signature but “without determining an explicit diffeomorphism carrying
one line element to the other” [135]. In contrast, this work directly provides
a concrete constructive proof in the form of an explicit diffeomorphism, for
all signatures by taking the complexified point of view 212121 Note that this
approach of first working with complexified spacetime and fields and then
specializing to particular real sections to derive results in any signature is
in accordance with the common practice in twistor theory (see [152], for
example). . Especially, it is interesting to observe that our constructive
proof in fact resolves the question left by Ref. [135]: taking the real slice
$t,x,iy,z\in\mathbb{R}$ makes the two metrics in Eqs. (3) and (The
diffeomorphism) as well as the diffeomorphism in (5) all real for real mass
$m{\,\in\,}\mathbb{R}$, while also exactly reproducing Ref. [64]’s conjectured
metric, Eq. (6.16) 222222 To be explicit, the line element
$-dt^{2}+dx^{2}-d{\tilde{y}}^{2}+dz^{2}+(K/r)\big{(}\hskip
1.00006pt{-dt+(x{\,+\,}\j{\tilde{y}})(dx{\,-\,}\j\hskip
1.00006ptd{\tilde{y}})/(r{\,+\,}z)+dz}\hskip 1.00006pt\big{)}^{2}$ maps to
$-(1{\,+\,}K/r)^{-1}\smash{\big{[}}\hskip 1.00006pt\hskip
0.50003ptdt+(K/r)\big{(}\hskip
1.00006pt{(x{\,+\,}\j{\tilde{y}})(dx{\,-\,}\j\hskip
1.00006ptd{\tilde{y}})/(r{\,+\,}z)+dz}\hskip 1.00006pt\big{)}\hskip
1.00006pt\big{]}^{2}$ $+(1{\,+\,}K/r)(dx^{2}-d{\tilde{y}}^{2}+dz^{2})$ through
$t\mapsto t$, $z\mapsto z+K$, $x\mapsto x+K\hskip
1.00006pt({x{\,+\,}\j{\tilde{y}}})/({r{\,+\,}z})$,
${\tilde{y}}\mapsto{\tilde{y}}+\j^{-1}\hskip 1.00006ptK\hskip
1.00006pt({x{\,+\,}\j{\tilde{y}}})/({r{\,+\,}z})$, with
$r:=(x^{2}{\,-\,}{\tilde{y}}^{2}{\,+\,}z^{2})^{1/2}$ and $\j^{2}=1$. The sign
choice for $\j=\pm 1$ depends on the sign of the NUT charge or the orientation
of the volume form, while $\j=+1$ reproduces Eq. (6.16) of Ref. [64]. It could
be interesting if a connection to split-complex numbers can be made. .

Acknowledgements. We would like to thank Clifford Cheung, Jung-Wook Kim,
Andres Luna, Donal O’Connell, Nabha Shah, Justin Vines, and Chirs D. White for
discussions and insightful comments. We are also grateful to Gabriel Herczeg,
Max Pezzelle, and Jash Desai for bringing the author’s attention to the works
[64, 135] and the follow-up discussions. This material is based upon work
supported by the U.S. Department of Energy, Office of Science, Office of High
Energy Physics, under Award Number DE-SC0011632. J.-H. K. is also supported by
Ilju Academy and Culture Foundation.

paracolor Appendix: Kerr-Schild to Gibbons-Hawking
paracolor by the null geodesic flow

Suppose a vector field $\ell^{\mu}$ in complexified Minkowski space that
defines a geodesic congruence: $\ell^{\mu}{}_{,\nu}\hskip
1.00006pt\ell^{\nu}=0$. The time-$\epsilon$ flow generated by this vector
field is simply $x^{\mu}\mapsto x^{\mu}+\epsilon\hskip
1.00006pt\ell^{\mu}(x)$, as geodesics are straight lines in flat space. The
pull-back of the flat metric $\eta=\eta_{\mu\nu}\hskip
1.00006ptdx^{\mu}{\hskip 1.00006pt\otimes\hskip 1.00006pt}dx^{\nu}$ is
$\eta+2\epsilon\hskip 1.00006pt\ell_{\smash{(\mu,\nu)}\vphantom{\beta}}\hskip
1.00006ptdx^{\mu}{\hskip 1.00006pt\otimes\hskip
1.00006pt}dx^{\nu}+\epsilon^{2}\hskip 1.00006pt\eta_{\rho\sigma}\hskip
1.00006pt\ell^{\rho}{}_{,\mu}\hskip
1.00006pt\ell^{\sigma}{}_{,\nu}dx^{\mu}{\hskip 1.00006pt\otimes\hskip
1.00006pt}dx^{\nu}$.

The null vector field associated with the Kerr-Schild metric of the SDTN
solution, $\ell^{\mu}=(1,\zeta,-i\zeta,1)$, is stationary, geodesic, and
shear-free. With the self-duality condition, they together implies that
$2\hskip 0.50003pt\partial_{\nu}\ell^{\mu}=\rho\hskip
1.00006pt\tilde{m}^{\mu}m_{\nu}$, when described with a null tetrad
$(\ell^{\mu},n^{\mu}=2u^{\mu}{\hskip
1.00006pt-\,}\ell^{\mu},m^{\mu},{\tilde{m}}^{\mu})$ where
$u^{\mu}:=\delta^{\mu}{}_{0}$. The complex expansion computes to $\rho=1/r$.

As a result, it follows that the flat metric transforms to
$\eta+\epsilon\hskip 1.00006pt\rho\hskip 1.00006pt{\tilde{m}}{\hskip
1.00006pt\odot\hskip 1.00006pt}m$ if the geodesic flow $x^{\mu}\mapsto
x^{\mu}+\epsilon\hskip 1.00006pt\ell^{\mu}(x)$ is used as a diffeomorphism,
where $\odot$ denotes the symmetrized tensor product. In turn, the Kerr-Schild
metric $\eta+\phi\hskip 1.00006pt\ell^{2}$ transforms to

$\displaystyle\eta\,+\,\epsilon\hskip 1.00006pt\rho\hskip
1.00006pt{\tilde{m}}{\hskip 1.00006pt\odot\hskip
1.00006pt}m\,+\,\phi^{\prime}\hskip 1.00006pt\ell^{2}\,,$ (26)

where $\phi^{\prime}(x):=\phi(\hskip 1.00006ptx{\,+\,}\epsilon\hskip
1.00006pt\ell(x))$.

Meanwhile, using $\eta=-2u{\hskip 1.00006pt\odot\hskip
1.00006pt}\ell+\ell^{2}+{\tilde{m}}{\hskip 1.00006pt\odot\hskip 1.00006pt}m$,
the Gibbons-Hawking metric $-(1{\,+\,}\phi)^{-1}\hskip 1.00006pt(\hskip
0.50003pt{u-\phi\hskip 1.00006pt(\ell{\,-\,}u)}\hskip
0.50003pt)^{2}+(1{\,+\,}\phi)\hskip 1.00006pt(u^{2}{\,+\,}\eta)$ boils down to

$\displaystyle\begin{split}\eta\,+\,\phi\hskip 1.00006pt{\tilde{m}}{\hskip
1.00006pt\odot\hskip 1.00006pt}m\,+\,(\phi^{-1}{\,+\,}1)^{-1}\hskip
1.00006pt\ell^{2}\,.\end{split}$ (27)

Therefore, if we can flow along the geodesic congruence such that
$\phi^{-1}{\,+\,}1=\phi^{\prime-1}$ and $\epsilon\hskip 1.00006pt\rho=\phi$,
then the Kerr-Schild metric is mapped to the Gibbons-Hawking metric.
Evidently, $\epsilon=m/4\pi$ satisfies both of these conditions. Thus the
diffeomorphism from the Kerr-Schild to Gibbons-Hawking metrics is given by
$t\mapsto t+m/4\pi$, $x\mapsto x+m\zeta/4\pi$, $y\mapsto y-im\zeta/4\pi$,
$z\mapsto z+m/4\pi$, which is equivalent to Eq. (5) up to dropping the
constant shift in the time coordinate. It could be interesting if this “trick”
also proves to be useful for more general cases such as non-self-dual
solutions or multi-centered Gibbons-Hawking metrics.

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