of 9
Single Kerr-Schild metric for Taub-NUT instanton
Joon-Hwi Kim
Walter Burke Institute for Theoretical Physics,
California Institute of Technology
,
Pasadena, California 91125
(Received 2 July 2024; accepted 17 December 2024; published 22 January 2025)
It is shown that a complex coordinate transformation maps the Taub-Newman-Unti-Tamburino instanton
metric to a Kerr-Schild metric. This metric involves a semi-infinite line defect as the gravitational analog of
the Dirac string, much like the original metric. Moreover, it facilitates three versions of classical double
copy correspondence with the self-dual dyon in electromagnetism, one of which involves a nonlocal
operator. The relevance to the Newman-Janis algorithm is briefly noted.
DOI:
10.1103/PhysRevD.111.L021703
Introduction.
Recently, an exciting program dubbed
classical double copy
[1
17]
has refined analogies between
gravity and electromagnetism into concrete mappings
between classical solutions while being rooted in the study
of scattering amplitudes
[17
21]
. The prime example is the
Kerr-Schild (KS) classical double copy, established in the
seminal work
[1]
. Namely, it is shown that a stationary KS
solution
g
¼
η
þ
Φ
l
l
in general relativity maps to a
stationary solution in Maxwell theory described by the
gauge potential
A
¼
Φ
l
, by taking a
single copy
of the
null one-form
l
. Conversely, one envisions a gravitational
counterpart
g
¼
η
þ
Φ
l
l
of a stationary solution
A
¼
Φ
l
in electromagnetism, by taking a
double copy
of
l
. Well-known instances of this correspondence are
Schwarzschild to point charge and Kerr to a rotating
solution dubbed
ffiffiffiffiffiffiffiffiffi
Kerr
p
[1,22
24]
.
From this angle, it is natural to expect that the Taub-
Newman-Unti-Tamburino (NUT) solution
[25,26]
will
correspond to a dyon, given the interpretation of the NUT
charge as the gravitomagnetic monopole
[16,27
33]
.
Unfortunately, only a
double
KS metric
g
¼
η
þ
Φ
1
l
1
l
1
þ
Φ
2
l
2
l
2
has been explicitly known for Taub-NUT
[34,35]
, so a mapping to electromagnetic dyons had to be
achieved by a
variant
of the KS double copy
[6,16]
.
Remarkably, in the present note we point out that the
straightforward application of the
original
KS 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 has gained a revived interest
[36
40]
because
of its intriguing physical properties
[41
45]
. The crucial
fact is that the SDTN solution remarkably admits a KS
metric in the
ordinary
sense:
g
¼
η
þ
Φ
l
l
.
First, a KS metric is obtained by KS double copying the
gauge potential of the self-dual dyon (SDD) in electro-
magnetism. We then show that it describes the SDTN
solution by explicitly constructing a coordinate transfor-
mation to the well-known Gibbons-Hawking
[41,42]
instanton metric.
Second, the physical interpretation of the metric as a
gravitational dyon is established within the KS description.
Most importantly, we confirm the characteristic
Misner
string
geometry: the gravitational analog
[27,28]
of the
Dirac string
[46,47]
. Notably, finding its distributional
source
[28,29]
becomes a simple problem in KS coordi-
nates, thanks to a linearization
[48
50]
.
Third, the classical double copy is established in two
more ways. We first show how our KS description
concretely validates the Weyl double copy proposed by
Refs.
[2
4]
, which has been stated without referencing an
explicit metric. We then propose an exotic form of classical
double copy employing a
nonlocal
operator, which arises
from propagators between dyonic matter
[51
55]
.
Lastly, we briefly remark on how we originally arrived at
this KS metric, which was, amusingly, from the Newman-
Janis shift property
[24]
of the Kerr solution.
Derivation of SDTN metric from double copy.
The gauge
potential of a dyon carrying electric and magnetic charges
ð
q;
iq
Þ
is given by
A
¼
q
4
π
r
ð
dt
ir
ð
1
cos
θ
Þ
d
φ
Þ
;
ð
1
Þ
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
¼
dA
is self-dual:

F
¼þ
iF
.
Published by the American Physical Society under the terms of
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Further distribution of this work must maintain attribution to
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3
.
PHYSICAL REVIEW D
111,
L021703 (2025)
Letter
2470-0010
=
2025
=
111(2)
=
L021703(9)
L021703-1
Published by the American Physical Society
Notably, adding a total derivative
ð
q=
4
π
Þ
d
log
ð
r
þ
z
Þ
gauge transforms this to a KS form:
A
¼
q
4
π
r

dt
þ
x
þ
iy
r
þ
z
dx
þ
y
ix
r
þ
z
dy
þ
dz

;
ð
2
Þ
where
x
¼
r
sin
θ
cos
φ
,
y
¼
r
sin
θ
sin
φ
,
z
¼
r
cos
θ
are
the Cartesian coordinates. Importantly, the terms inside the
bracket comprise a null one-form.
With this understanding, the KS classical double copy
[1]
maps Eq.
(2)
to the following line element, by literally
taking
two copies
of the null one-form:
ds
2
¼
dt
2
þ
dx
2
þ
dy
2
þ
dz
2
þ
m
4
π
r

dt
þ
x
þ
iy
r
þ
z
dx
þ
y
ix
r
þ
z
dy
þ
dz

2
:
ð
3
Þ
Here we have adopted
8
π
G
¼
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
.
The diffeomorphism.
To prove this, we simply construct an
explicit coordinate transformation to the previously known
metric. It is helpful to employ complex coordinates
ζ
¼ð
x
þ
iy
Þ
=
ð
r
þ
z
Þ
,
̃
ζ
¼ð
x
iy
Þ
=
ð
r
þ
z
Þ
that arise from
stereographically projecting the two-sphere with respect to
the south pole, so Eq.
(3)
translates to
ds
2
¼
dt
2
þ
dr
2
þ
r
2
4
d
̃
ζ
d
ζ
ð
1
þ
̃
ζζ
Þ
2
þ
m
4
π
r

dt
þ
dr
r
2
̃
ζ
d
ζ
1
þ
̃
ζζ

2
:
ð
4
Þ
Amusingly, applying a diffeomorphism given by
t
t; r
r
þ
m
4
π
;
ζ
ζ
;
̃
ζ
1
1
þð
m=
4
π
r
Þð
1
þ
̃
ζζ
Þ
̃
ζ
;
ð
5
Þ
we obtain
ds
2
¼

1
þ
m
4
π
r

1

dt
þ
m
4
π
r

dr
r
2
̃
ζ
d
ζ
1
þ
̃
ζζ

2
þ

1
þ
m
4
π
r

dr
2
þ
r
2
4
d
̃
ζ
d
ζ
ð
1
þ
̃
ζζ
Þ
2

;
ð
6
Þ
which is the Gibbons-Hawking metric
[41,42]
for the
SDTN solution with mass
m
.
This diffeomorphism is surely invertible. In fact, as is
explicated in the Appendix, it turns out that Eq.
(5)
traces
back to the flow along the null geodesic congruence
associated with the KS metric. Thus, the inverse map is
simply Eq.
(5)
with
m
replaced with
m
.
This completes our
constructive
proof that the KS and
Gibbons-Hawking metrics in Eqs.
(3)
and
(6)
are diffeo-
morphic. Consequently, we establish that the SDTN sol-
ution admits a construction from the KS classical double
copy. Previously, these facts were only recently
conjectured
by Ref.
[56]
in split signature.
Meanwhile, the astute reader may have noticed an un-
usual feature of the diffeomorphism in Eq.
(5)
: it describes
a
complexified diffeomorphism
since
ζ
is frozen while
̃
ζ
gets scrambled, hence the notation
̃
ζ
instead of
̄
ζ
.Yet
crucially, this is a legitimate operation since the Lorentzian-
signature SDTN solution is inherently a complex saddle as
a self-dual solution
[36]
. Just as the SDD gauge potential
described in Eq.
(2)
is a solution to complexified Maxwell
s
equations due to the imaginary magnetic charge
iq
, the
SDTN metric in Eq.
(3)
solves complexified Einstein
s
equations governing holomorphic metrics
[36,57
60]
.Asa
consequence of this interpretation,
ζ
and
̃
ζ
can be trans-
formed independently as
holomorphic
coordinates.
Physical interpretation.
We shall now examine the physical
validity of the KS metric in Eq.
(3)
. To begin with, we point
out that Eq.
(3)
itself
is quite unusual as a KS metric, in fact.
While the KS potential is simply
Φ
¼
m=
4
π
r
, the null
one-form
l
involves a
semi-infinite line defect
along the
negative
z
axis, which is exactly how the gravitomagnetic
charge is encoded in this metric:
l
¼
dt
þ
x
þ
iy
r
þ
z
dx
þ
y
ix
r
þ
z
dy
þ
dz:
ð
7
Þ
This is intriguing, as the null one-form is usually regular
wherever the KS potential is regular
which is especially
true for the double KS construction of Taub-NUT
[6]
.
Crucially, as in the original metric in Eq.
(6)
, the
interpretation of this line defect should be the
Misner
string
[27,28]
: 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
[27,28]
describes a gravita-
tional analog of a dyon: (a) the Misner string is invisible as
a coordinate artifact if the gravitational analog of the Dirac
quantization condition holds
[27]
, (b) the Misner string
implements a flux tube of time monodromy
[16,30,31]
, and
(c) the distributional stress-energy of the string describes an
energyless thin solenoid of mass current
[28,29]
. Our goal
now is to show that all of these properties hold in the KS
description as well.
First, we derive the Dirac quantization condition.
For the SDD gauge potential in Eq.
(2)
, adding a term
ð
q=
2
π
Þ
d
log
ζ
repositions its Dirac string along the posi-
tive
z
axis (north pole). This encodes the complexified
group-valued transition function
ζ
ieq=
2
π
, where
e
is the
JOON-HWI KIM
PHYS. REV. D
111,
L021703 (2025)
L021703-2
electromagnetic coupling. Demanding its single-valuedness
on the
equator
[61]
implies
ieq
2
π
Z
.
The gravitational Dirac quantization can be derived in a
similar fashion, i.e., by hiding the strings with overlapping
coordinate patches
[16,27,30,31]
. In the KS description, we
find that the diffeomorphism
t
t
þ
m
2
π
log
ζ
;
̃
ζ
1
ð
m=
4
π
r
Þð
1
þ
̃
ζζ
Þ
=
̃
ζζ
1
þð
m=
4
π
r
Þð
1
þ
̃
ζζ
Þ
̃
ζ
ð
8
Þ
with
r
and
ζ
fixed transforms the KS metric in Eq.
(4)
to
another KS metric:
ds
2
¼
dt
2
þ
dr
2
þ
r
2
4
d
̃
ζ
d
ζ
ð
1
þ
̃
ζζ
Þ
2
þ
m
4
π
r

dt
þ
dr
þ
r
2
d
ζ
ζ
ð
1
þ
̃
ζζ
Þ

2
;
ð
9
Þ
which develops a string defect along the positive
z
axis
(north pole). For this diffeomorphism to be single-valued
within the overlapping region, the time coordinate should
be periodic as
t
t
þ
im
Z
, given the
2
π
i
periodicity of
log
ζ
. Hence the energy
e
of a test particle is quantized as
iem
2
π
Z
. This is precisely the gravitational Dirac quan-
tization condition due to the NUT charge
im
.
Second, we study the near-string geometries in an
explicit manner. Let
ξ
x
þ
iy
,
̃
ξ
x
iy
be complex
coordinates for the (complexified)
x
y
plane and suppose
a positive infinitesimal
ε
. Applying the replacement
z
1
=
ε
þ
z
to the gauge potential in Eq.
(2)
and then
taking the limit
ε
0
, we obtain
A
¼ð
q=
2
π
Þ
d
log
̃
ξ
. This
precisely describes an infinite Dirac string generated by a
large gauge transformation
[62,63]
.
For the line element in Eq.
(3)
, the same limiting
procedure leaves out a divergent part that can be compen-
sated by a shift
ξ
ξ
þðð
m=
π
Þð
1
=
ε
z
Þ
3
m
2
=
4
π
2
Þ
=
̃
ξ
.
Amusingly, the resulting line element is simply the
Minkowski line element, but with a multivalued time:
t
0
¼
t
þ
m
2
π
log
̃
ξ
:
ð
10
Þ
Namely, in the vicinity of the string, the geometry
approaches to the image of empty spacetime under a large
diffeomorphism. Here, a Sagnac interferometer measures a
monodromy in time as the gravitational Aharanov-Bohm
phase
[31,32,64,65]
, localized along the
z
axis (as
̃
ξ
¼
0
),
see Fig.
1
. This precisely describes an infinite Misner string
as a thin tube of gravitomagnetic flux
im
[66]
.
Third, we derive the distributional stress-energy tensor of
this string. Among various approaches
[70
72]
, we par-
ticularly consider the
harmonic-gauge linearization
of
Ref.
[50]
, which stationary KS metrics enjoy as a luxury.
Provided the congruence of
l
μ
μ
is geodesic and shear-free
(GSF), this approach identifies
T
μν
¼
u
ð
μ
J
ν
Þ
as the source
for the double copy solution
[50]
. Here
J
μ
is the single copy
source, while
u
μ
¼
δ
μ
0
is the unit timelike Killing vector.
For our case, the GSF property of the congruence asso-
ciated with the null one-form in Eq.
(7)
can be established
from an identity
μ
l
ν
¼
1
2
r

η
μν
þ
l
ð
μ
ð
2
u
l
Þ
ν
Þ
i
ε
μνρσ
u
ρ
l
σ

;
ð
11
Þ
while the single copy source
J
μ
easily follows by acting on
the Laplacian
≔−
η
μν
μ
ν
to Eq.
(1)
. Applying the for-
mula
T
μν
¼
u
ð
μ
J
ν
Þ
then immediately reproduces Bonnor
s
stress-energy tensor
[28,29]
, describing a point mass
m
attached to a massless semi-infinite rotating rod.
To sum up, we have convinced ourselves from multiple
angles that the KS 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 KS 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 sol-
ution. First, we explicate how our KS metric solidly verifies
the
Weyl double copy
correspondence
[3,4]
.
The Weyl double copy directly constructs the Weyl cur-
vatures of algebraically special spacetimes by
squaring
electromagnetic field strengths
[2
5]
while not referencing
a metric nor a gauge potential. For example, take the
Coulomb and Schwarzschild solutions. In the spinor
language, their field strength and Weyl tensor are
F
̇
α
̇
β
¼
2
q
4
π
r
2
̃
o
ð
̇
α
̃
ι
̇
β
Þ
;C
̇
α
̇
β
̇
γ
̇
δ
¼
6
m
4
π
r
3
̃
o
ð
̇
α
̃
o
̇
β
̃
ι
̇
γ
̃
ι
̇
δ
Þ
;
ð
12
Þ
where we have omitted the anti-self-dual counterparts.
The Weyl double copy is the squaring relation
C
̇
α
̇
β
̇
γ
̇
δ
¼
F
ð
̇
α
̇
β
F
̇
γ
̇
δ
Þ
=S
[2]
, which holds with
S
1
=r
.
FIG. 1. The Misner string is a
screw dislocation
in spacetime,
inducing time monodromy as an Aharanov-Bohm phase.
SINGLE KERR-SCHILD METRIC FOR TAUB-NUT INSTANTON
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Crucially, this relation is
chiral
: there is no mixing
between the self-dual and anti-self-dual parts. Thus the
Weyl double copy for SDTN seems to simply follow from
that of Schwarzschild, if one posits that the SDTN solution
is the
self-dual part
of the Schwarzschild solution in the
sense that its self-dual Weyl curvature and the principal
spinors exactly equal those of the Schwarzschild solu-
tion
[3,4]
. This physical anticipation, however, may appear
a bit subtle due to the nonlinearity of gravity. However
luckily, straightforward calculations with the explicit KS
metric in Eq.
(3)
concretely verifies this claim, so the Weyl
double copy construction of the SDTN solution is com-
pletely validated at the fully nonlinear order.
Specifically, the principal spinors for the KS metric in
Eq.
(3)
are found as
̃
o
̇
α
¼ð
1
;
ζ
Þ
and
̃
ι
̇
α
¼ð
̃
ζ
;
1
Þ
=
ð
1
þ
̃
ζζ
Þ
,
one of which being encoded in the null one-form:
l
μ
¼ð
σ
μ
Þ
α
̇
α
̃
o
̇
α
o
α
;
̃
o
̇
α
¼

1
ζ

;o
α
¼

1
0

:
ð
13
Þ
Crucially, the
real
combinations
½
̃
o
̇
α


̃
o
̇
β
,
½
̃
ι
̇
α


̃
ι
̇
β
of these
spinors exactly reproduce the null one-forms
dt

dr
associated with the KS etrics of the Schwarzschild solution
as stated in Refs.
[3,4,73]
.
Weyl double copy in twistor space.
Meanwhile, the Weyl
double copy has been also discussed in
twistor space
[3,4]
.
This is essentially a statement about the
linearized
Weyl
tensor as a massless spin-2 field on flat background con-
structed from Penrose transform
[74,76,77]
. The Penrose
transform for a massless spin-
h
field is given by
[78]
φ
̇
α
1

̇
α
2
h
ð
x
Þ¼
I
̃
π
̇
γ
d
̃
π
̇
γ
2
π
i
̃
π
̇
α
1

̃
π
̇
α
2
h
ð
f
ð
̃
π
;
̃
π
x
ÞÞ
h
þ
1
;
ð
14
Þ
and the self-dual linearized Weyl tensor of the
Schwarzschild and SDTN solutions arises from Eq.
(14)
for
h
¼
2
by taking the twistor function
[4,80
83]
f
ð
̃
Z
Þ¼
̃
ω
α
u
α
̇
α
̃
π
̇
α
;u
α
̇
α
u
μ
ð
σ
μ
Þ
α
̇
α
;
ð
15
Þ
whose zeros encode the principal spinors
̃
o
̇
α
;
̃
ι
̇
α
as per the
Kerr theorem
[74,84,85]
. The twistor-space Weyl double
copy implies that taking
h
¼
1
with the same function
derives the single copy field strength in Eq.
(12) [3,4]
.
Here, the crucial clarification, which the explicit KS
metric provides, is that this twistor-space construction from
Penrose transform has, in fact, described the
exact
Weyl
tensor of the SDTN solution at the fully nonlinear level.
Namely, the Cartesian coordinates of the flat background
employed in the Penrose transform secretly described KS
coordinates for the curved SDTN geometry.
Note that the
position vector
x
, which one identifies
from the Killing spinor as
̃
χ
̇
α
̇
β
¼
1
2
ð
x
·
σ
Þ
̇
α
̇
β
[36]
, also
describes the (spatial part of) our KS coordinates.
Self-dual double copy.
Moreover, our KS coordinates are
also the Pleba
ń
ski coordinates
[57,86]
for the second
heavenly equation, on which the
self-dual double copy
[1,7,17]
is based. Notably, the self-dual double copy would
imply fascinating conclusions such as
SDTN
is
a large-
N
dyon
in the context of Moyal deformation of the heavenly
equation
[87
92]
. Strangely, however, the SDTN metric
is not straightforwardly the self-dual double copy of
SDD in terms of the Pleba
ń
ski scalar field
[93]
:
A
α
̇
α
¼
o
α
0
̇
α
ð
q
ζ
=
4
π
Þ
,
h
α
̇
αβ
̇
β
¼
o
α
o
β
0
̇
α
0
̇
β
ð
m
ð
2
r
þ
z
Þ
ζ
2
=
24
π
Þ
,if
h
μν
¼
g
μν
η
μν
denotes the KS metric perturbation. This
puzzle deserves further investigation, in comparison with
the Eguchi-Hanson case
[7]
.
Operator KS double copy.
Interestingly, we still find that an
operator KS double copy is viable, but in a more exotic
form. The operator KS double copy promotes the null one-
form
l
μ
in the KS ansatz to a differential operator
ˆ
k
μ
, which
may factorize as
1
2
ð
σ
μ
Þ
α
̇
α
ˆ
k
μ
¼
o
α
ˆ
k
̇
α
so that the fields are
manifestly put in the light cone gauge
[1,7,8]
. Intriguingly,
we find that Eqs.
(2)
and
(3)
precisely arise in this way if the
operator
ˆ
k
̇
α
is given by
ˆ
k
̇
α
¼

x
þ
i
y
z
;
1

;
ð
16
Þ
which is
nonlocal
. Specifically, it follows from
ˆ
k
̇
α
ð
1
=r
Þ¼
ð
1
=r
Þ
̃
o
̇
α
and
ˆ
k
̇
α
ˆ
k
̇
β
ð
1
=r
Þ¼ð
1
=r
Þ
̃
o
̇
α
̃
o
̇
β
that a double copy
correspondence between SDD and SDTN is realized as
A
α
̇
α
¼
o
α
ˆ
k
̇
α
q
4
π
r
;h
α
̇
αβ
̇
β
¼
o
α
o
β
ˆ
k
̇
α
ˆ
k
̇
β
m
4
π
r
:
ð
17
Þ
Notably, this new form of classical double copy admits a
nice physical explanation from an electric-magnetic dual
perturbation theory. The key fact is that the SDD can be
viewed as an object in the
extended
Maxwell theory
where the violation of Bianchi identity by
magnetic matter
J
is allowed by the introduction of the dual gauge poten-
tial
A
[47]
:
F
¼
dA
þ
dA
. In this picture, the distri-
butional source for the SDD is
J
μ
¼
ν
F
μν
¼
qu
μ
δ
ð
3
Þ
ð
x
Þ
;
J
μ
¼
ν

F
μν
¼
iqu
μ
δ
ð
3
Þ
ð
x
Þ
:
ð
18
Þ
We may redescribe these in the
helicity basis
as
J
μ
þ
¼
J
μ
iJ
μ
¼
0
and
J
μ
¼
J
μ
þ
iJ
μ
¼
2
qu
μ
δ
ð
3
Þ
ð
x
Þ
, while
having
A

μ
1
2
ð
A
μ

iA
μ
Þ
for the potentials.
As a dynamical field theory, a formulation of this
extended Maxwell theory has been given by Zwanziger
[51
55]
. We note that the propagators of this theory can be
summarized into a single formula in the spinor notation:
JOON-HWI KIM
PHYS. REV. D
111,
L021703 (2025)
L021703-4
ˆ
Δ
þ
α
̇
αβ
̇
β
¼
n
α
̇
β
β
̇
α
n
·
1
;
ð
19
Þ
where
n
is an auxiliary four-vector that traces back to the
Dirac string. Equation
(19)
describes the position-space
propagator from
A
β
̇
β
to
A
þ
α
̇
α
, while the
h
A
A
i
and
h
A
þ
A
þ
i
propagators vanish. A simple calculation then shows that
the SDD gauge potential in Eq.
(2)
is precisely reproduced
by this propagator, if one takes
n
α
̇
α
¼
o
α
̄
o
̇
α
for the
auxiliary vector so that
ˆ
k
̇
α
¼
1
̇
α
=
0
̇
0
.
Amusingly, the graviton field arises in the same fashion
by using the linearized gravity equivalent of Zwanziger
propagator, which is given by doubling Eq.
(19)
as is
advocated in Ref.
[55]
. In this case, the sources in the two-
potential theory are
T
μν
þ
¼
0
and
T
μν
¼
2
mu
μ
u
ν
δ
ð
3
Þ
ð
x
Þ
.
The resulting linearized metric perturbation happens
to be also the full metric perturbation by taking the KS
form
[48
50]
.
In summary, the double copy structure proposed in
Eq.
(17)
can be traced back to propagators between dyonic
matter in electrodynamics and linearized gravity:
A
þ
α
̇
α
¼
ˆ
Δ
þ
α
̇
αβ
̇
β
J
̇
ββ
;h
þ
α
̇
αβ
̇
β
¼
ˆ
Δ
þ
α
̇
αγ
̇
γ
ˆ
Δ
þ
β
̇
βδ
̇
δ
T
̇
γγ
̇
δδ
:
ð
20
Þ
A few comments are in order. First, it is instructive
to translate the propagator in Eq.
(19)
into the vector
notation, where one can identify a combination of operators
ð
n
×
!
=
n
·
!
Þ
connecting between the Gilbertian and
Amp`
erian implementations
[95]
of multipoles. This implies
that the nonlocality of the operator in Eq.
(16)
has encoded
the Dirac string structures. Second, our construction could
be justified from the on-shell perspective. The
numerator
of the off-shell propagator in Eq.
(19)
correctly factorizes
into the null polarization vectors so that the double copy of
sources from
J
μ
¼
2
qu
μ
δ
ð
3
Þ
ð
x
Þ
to
T
μν
¼
2
mu
μ
u
ν
δ
ð
3
Þ
ð
x
Þ
encodes the squaring relation of the massive on-shell
amplitudes
[55,96
98]
. In this perspective, the nonlocality
traces back to that of the
x
factor. Lastly, it is interesting to
apply this approach to construct generic Taub-NUT space-
times, in which case one naturally obtains a double KS
metric perturbation.
Conclusions.
In this work, we showed that a complexified
diffeomorphism reveals a KS metric for the SDTN solution.
The validity of this diffeomorphism and the metric was
then verified while highlighting their complex nature.
Notably, the KS metric which also exhibits the Misner
string defect, supporting its physical interpretation as a
gravitational dyon. Further, we established three concrete
senses in which the SDTN solution is precisely the gravi-
tational counterpart of the SDD in electromagnetism: KS,
Weyl, and nonlocal operator KS classical double copies.
Notably, the last one adds to the inventory of classical
double copies as a new entry, whose physical origin is the
propagators between dyonic mater.
The most distinctive charm of this work lies in the
exciting development along the very first three equations.
The usual punchline of classical double copy has been
starting with a gravitational solution and then deducing the
electromagnetic counterpart. This work, in contrast, takes a
reverse path: we constructed an electromagnetic solution
and then applied the double copy to yield a new statement
in general relativity.
It is also insightful to frame this work in the angle of
classical double copy of gravitational instantons.
The
only instance has been the self-dual double copy for the
Eguchi-Hanson instanton
[99
101]
provided by the work
[7]
, which traces back to Refs.
[102,103]
. This work adds
another instance to the discourse, namely the Taub-NUT
instanton. It is amusing that the correspondence between
self-dual objects in electromagnetism and gravity, which
the Gibbons-Hawking ansatz
[41,42]
had suggested a long
time ago, has been strengthened from new perspectives by
the advent of the classical double copy program.
The KS metric of the SDTN solution conveys an
aesthetic elegance in terms of its minimalist and spinorial
nature. To investigate what facilitates this remarkable gem,
we could examine its relation to the self-dual limit of the
double
KS metric in Refs.
[6,104]
.
Lastly, we anticipate that the explicit KS metric we have
established here will contribute to future studies on the
SDTN solution, both conceptually and practically. In fact,
we have already seen that it largely clarifies the subtle
aspects of the Weyl double copy. For further consequences
or applications, we suggest the following directions:
asymptotic symmetries
[105
107]
, integrability of test-
particle motion
[43,44]
, black hole perturbation theory
[36,37,111,112]
, or exact graviton propagators
[113
115]
,
which will facilitate self-force calculations
[116
119]
in an
idealized setup. Especially, explorations in the scattering
context are likely to shed insights on the Kerr black hole
scattering for an integrable subsector, on account of the
Newman-Janis shift
[24]
: the
self-dual part
of Kerr is a
SDTN solution
[36,38,56,120
122]
.
Amusingly, about a year and a half ago, we deciphered a
complete physical picture behind the Newman-Janis algo-
rithm, generalizing the above statement. As is depicted in
Fig.
2
, we realized that a simple complex coordinate trans-
formation maps the Kerr metric in KS coordinates to
another KS metric describing two point masses joined by
a Misner string. By zooming to the upper tip of the string,
we
by construction
obtained a KS metric, which was
precisely Eq.
(3)
. Crucially, this had to be describing a
Taub-NUT instanton on account of the Newman-Janis shift.
The effort to verify this strong physical conviction gave
birth to this work. We will release more details in the
future
[123]
.
SINGLE KERR-SCHILD METRIC FOR TAUB-NUT INSTANTON
PHYS. REV. D
111,
L021703 (2025)
L021703-5
Note added.
The work
[124]
, which appeared recently,
provided an
existence
proof for a real diffeomorphism
connecting between the KS and Gibbons-Hawking metrics
in split signature. However, this paper has already provided
a
constructive
proof for all signatures by taking the
complexified point of view, e.g., the result and conjecture
of Refs.
[56,124]
are reproduced by taking the real
slice
t; x; iy; z
R
.
Acknowledgments.
We would like to thank Tim Adamo,
Clifford Cheung, Maciej Dunajski, Jung-Wook Kim, Andres
Luna, Donal O
Connell, Lionel Mason, 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
[56,124]
after the initial release of this paper and
also 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
No. DE-SC0011632. J.-H. K. is also supported by Ilju
Academy and Culture Foundation.
Appendix: Kerr-Schild to Gibbons-Hawking by null
geodesic flow.
Suppose a vector field
l
μ
in complexified
Minkowski space that defines a geodesic congruence:
l
μ
;
ν
l
ν
¼
0
. The time-
ε
flow generated by this vector
field is simply
x
μ
x
μ
þ
ε
l
μ
ð
x
Þ
, as geodesics are
straight lines in flat space. The pullback of the flat
metric
η
¼
η
μν
dx
μ
dx
ν
is
η
þ
2
ε
l
ð
μ
;
ν
Þ
dx
μ
dx
ν
þ
ε
2
η
ρσ
l
ρ
;
μ
l
σ
;
ν
dx
μ
dx
ν
.
The null vector field associated with the KS metric of the
SDTN solution,
l
μ
¼ð
1
;
ζ
;
i
ζ
;
1
Þ
, is stationary, geodesic,
and shear free. With the self-duality condition, they
together imply that
2
ν
l
μ
¼
ρ
̃
m
μ
m
ν
, when described by
a null tetrad
ð
l
μ
;n
μ
¼
2
u
μ
l
μ
;m
μ
;
̃
m
μ
Þ
where
u
μ
δ
μ
0
.
The complex expansion computes to
ρ
¼
1
=r
.
As a result, it follows that the flat metric transforms to
η
þ
ερ
̃
m
m
if the geodesic flow
x
μ
x
μ
þ
ε
l
μ
ð
x
Þ
is
used as a diffeomorphism, where
denotes the sym-
metrized tensor product. In turn, the KS metric
η
þ
φ
l
2
transforms to
η
þ
ερ
̃
m
m
þ
φ
0
l
2
;
ð
A1
Þ
where
φ
0
ð
x
Þ
φ
ð
x
þ
ε
l
ð
x
ÞÞ
.
Meanwhile, using
η
¼
2
u
l
þ
l
2
þ
̃
m
m
, the
Gibbons-Hawking metric
ð
1
þ
φ
Þ
1
ð
u
φ
ð
l
u
ÞÞ
2
þ
ð
1
þ
φ
Þð
u
2
þ
η
Þ
boils down to
η
þ
φ
̃
m
m
þð
φ
1
þ
1
Þ
1
l
2
:
ð
A2
Þ
Therefore, if we can flow along the geodesic congruence
such that
φ
1
þ
1
¼
φ
0
1
and
ερ
¼
φ
, then the KS metric
is mapped to the Gibbons-Hawking metric. Evidently,
ε
¼
m=
4
π
satisfies both of these conditions. Thus the
diffeomorphism from the KS to Gibbons-Hawking
metrics is given by
t
t
þ
m=
4
π
,
x
x
þ
m
ζ
=
4
π
,
y
y
im
ζ
=
4
π
,
z
z
þ
m=
4
π
, 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
multicentered Gibbons-Hawking metrics.
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