#### Perturbative spacetimes from Yang-Mills theory

Received: January
Perturbative spacetimes from Yang-Mills theory
Andr´es Luna 0 2 6 7 8 9 10 11 12
Ricardo Monteiro 0 2 4 7 8 9 10 11 12
Isobel Nicholson 0 2 5 7 8 9 10 11 12
Alexander Ochirov 0 2 5 7 8 9 10 11 12
Donal O'Connell 0 2 5 7 8 9 10 11 12
Niclas Westerberg 0 1 2 5 7 8 9 10 11 12
Chris D. White 0 2 3 7 8 9 10 11 12
Edinburgh 0 2 7 8 9 10 11 12
U.K. 0 2 7 8 9 10 11 12
0 School of Physics and Astronomy, The University of Edinburgh
1 Institute of Photonics and Quantum Sciences
2 Glasgow G12 8QQ, Scotland, U.K
3 Centre for Research in String Theory
4 Theoretical Physics Department , CERN
5 Higgs Centre for Theoretical Physics
6 School of Physics and Astronomy, University of Glasgow
7 Open Access , c The Authors
8 327 Mile End Road , London E1 4NS, U.K
9 School of Physics and Astronomy, Queen Mary University of London
10 School of Engineering and Physical Sciences, Heriot-Watt University
11 Edinburgh EH9 3JZ, Scotland, U.K
12 [3] Z. Bern, T. Dennen, Y.-T. Huang and M. Kiermaier, Gravity as the square of gauge theory
The double copy relates scattering amplitudes in gauge and gravity theories. In this paper, we expand the scope of the double copy to construct spacetime metrics through a systematic perturbative expansion. The perturbative procedure is based on direct calculation in Yang-Mills theory, followed by squaring the numerator of certain perturbative diagrams as specified by the double-copy algorithm. The simplest spherically symmetric, stationary spacetime from the point of view of this procedure is a particular member of the Janis-Newman-Winicour family of naked singularities. Our work paves the way for applications of the double copy to physically interesting problems such as perturbative black-hole scattering.
1 Introduction
Review of the BCJ double copy
Linear gravitons from Yang-Mills fields
General linearised vacuum solutions
The linear fat graviton for Schwarzschild
Solutions with linearised dilatons
Perturbative corrections
Perturbative metrics from gauge theory
Relating fat and skinny fields: gauge transformations and field redefinitions
The perturbative corrections to the JNW fields
Non-abelian gauge and gravity theories describe very different physics. The former
govern much of high energy physics, including applications to particle colliders. The latter
underpin most of astrophysics and cosmology. In both types of theory, the ever
advancing experimental frontier demands theoretical precision, including the development of new
computational techniques. Recently, an intriguing new relationship between scattering
amplitudes in gauge and gravity theories has been discovered by Bern, Carrasco and Johansson
(BCJ) [1–3]. There are two elements in the BCJ story. The first is the colour-kinematics
duality, which is the statement that it is possible to organise the numerators of
perturbative Feynman-like diagrams so that the kinematic numerator of a given diagram obeys the
same algebraic relations as the colour factor of that diagram (for an arbitrary choice of
gauge group). These relations include Jacobi relations, which lead to three-term identities
connecting planar and non-planar diagrams in gauge theory. Furthermore, the presence
of Jacobi relations for kinematic objects hints at the existence of an algebraic structure
underlying the gauge theory [4].
The second major element of the BCJ story is the double copy [1–3]. This states that
gauge theory amplitudes can be straightforwardly modified to yield gravity amplitudes,
essentially by replacing the colour factor of the gauge amplitude with a second copy of the
kinematic numerator. At tree level, both the colour-kinematics duality and the double copy
are proven to be valid [3–11], and the latter is known to be equivalent to the celebrated
KLT relations [12], derived from string theory. However, the BCJ story is remarkable in
that it also appears to apply at loop level, and in different types of theory [2, 13–53].
The existence of the double copy hints at a profound relationship between gauge and
gravity theories, that should transcend perturbative amplitudes. To this end, refs. [54–57]
have generalised the notion of the double copy to exact classical solutions. That is, a large
family of gravitational solutions was found that could be meaningfully associated with a
gauge theory solution, such that the relationship between them was consistent with the BCJ
double copy. These solutions all had the special property that they linearised the Einstein
and Yang-Mills equations, so that the graviton and gauge field terminate at first order in
the coupling constant, with no higher-order corrections. A special choice of coordinates
(Kerr-Schild coordinates) must be chosen in the gravity theory, reminiscent of the fact that
the amplitude double copy is not manifest in all gauge choices. An alternative approach
exists, in a wide variety of linearised supersymmetric theories, of writing the graviton as
a direct convolution of gauge fields [58–63]. This in principle works for general gauge
choices, but it is not yet clear how to generalise this prescription to include nonlinear
effects. One may also consider whether the double copy can be generalised to intrinsically
non-perturbative solutions, and first steps have been taken in ref. [64].
As is hopefully clear from the above discussion, it is not yet known how to formulate
the double copy for arbitrary field solutions, and in particular for those which are nonlinear.
However, such a procedure would have highly useful applications. Firstly, the calculation
of metric perturbations in classical general relativity is crucial for a plethora of
astrophysical applications, but is often cumbersome. A nonlinear double copy would allow one to
calculate gauge fields relatively simply, before porting the results to gravity. Secondly,
ref. [55] provided hints that the double copy may work in a non-Minkowski spacetime.
This opens up the possibility to obtain new insights (and possible calculational techniques)
The aim of this paper is to demonstrate explicitly how the BCJ double copy can be
used to generate nonlinear gravitational solutions order-by-order in perturbation theory,1
from simpler gauge theory counterparts. This is similar in spirit to refs. [65–67], which
extracted both classical and quantum gravitational corrections from amplitudes obtained
from gauge theory ingredients; and to refs. [68, 69], which used tree-level amplitudes to
construct perturbatively the Schwarszchild spacetime. Very recently, ref. [70] has studied
the double copy procedure for classical radiation emitted by multiple point charges. Here
we take a more direct approach, namely to calculate the graviton field generated by a
given source, rather than extracting this from a scattering amplitude. Another recent
work, ref. [71], proposes applications to cosmological gravitational waves, pointing out a
double copy of radiation memory.
As will be explained in detail in what follows, our scheme involves solving the
YangMills equations for a given source order-by-order in the coupling constant. We then copy
1This is the post-Minkowskian expansion, as opposed to the post-Newtonian expansion where the
nonrelativistic limit is also taken.
this solution by duplicating kinematic numerators, before identifying a certain product of
certain gauge transformation and field redefinition in general.
The structure of our paper is as follows. In section 2, we briefly review the BCJ
double copy. In section 3, we work at leading order in perturbation theory, and outline
our procedure for obtaining gravity solutions from Yang-Mills fields. In section 4, we work
to first and second subleading order in perturbation theory, thus explicitly demonstrating
how nonlinear solutions can be generated in our approach. Finally, we discuss our results
and conclude in section 5.
Review of the BCJ double copy
Our aim in this section is to recall salient details about the BCJ double copy [1–3], that
will be needed in what follows. Since we will be dealing with solutions to the classical
theories, we are only concerned with the tree-level story, which is well established, whereas
at loop level the BCJ proposal is a conjecture. First, we recall that an m-point tree-level
amplitude in non-abelian gauge theory may be written in the general form
Am = gm−2 X
nator arises from propagators associated with each internal line, and ci is a colour factor
obtained by dressing each vertex with structure constants. Finally, ni is a kinematic
numerator, composed of momenta and polarisation vectors. Note that the sum over graphs involves
cubic topologies only, despite the fact that non-abelian gauge theories include quartic
interaction terms for the gluon. These can always be broken up into cubic-type graph
contributions, so that eq. (2.1) is indeed fully general. The form is not unique, however, owing to the
fact that the numerators {ni} are modified by gauge transformations and / or field
redefinitions, neither of which affect the amplitude. A compact way to summarise this is that one is
free to modify each individual numerator according to the generalised gauge transformation
= 0,
where the latter condition expresses the invariance of the amplitude.
The set of cubic graphs in eq. (2.1) may be divided into overlapping sets of three,
where the colour factors ci are related by Jacobi identities, associated to the Lie algebra of
the colour group. Remarkably, it is possible to choose the numerators ni so that they obey
similar Jacobi identities, which take the form of coupled functional equations. This property
is known as colour-kinematics duality, and hints at an intriguing correspondence between
colour and kinematic degrees of freedom that is still not fully understood, although progress
has been made in the self-dual sector of the theory [4]. More generally, the field-theory limit
of superstring theory has been very fruitful for understanding colour-kinematics duality [10,
51, 72] and there has been recent progress on more formal aspects of the duality [73–75].
Given a gauge theory amplitude in BCJ-dual form, the double copy prescription states
is an m-point gravity amplitude, where
Mm = i
constant.2 This result is obtained from eq. (2.1) by replacing the gauge theory coupling
constant with its gravitational counterpart, and colour factors with a second set of kinematic
numerators n˜i. Therefore, the procedure modifies the numerators of amplitudes term by
term, but leaves the denominators in eqs. (2.1), (2.3) intact. A similar phenomenon occurs
in the double copy for exact classical solutions of refs. [54–56], in which scalar propagators
play a crucial role.
The gravity theory associated with the scattering amplitudes (2.3) depends on the two
gauge theories from which the numerators {ni}, {n˜i} are taken. In this paper, both will
by an axion in four spacetime dimensions). The action for these fields is
S =
dDx√
−g
R − 2(D − 2)
solutions of this theory around Minkowski space. The starting point is to consider linearised
fields, for which the equations of motion are
∂2hμν − ∂μ∂ρhρν − ∂ν ∂ρhρμ + ∂μ∂ν h + ημν ∂ρ∂σhρσ − ∂2h = 0,
as it is common in perturbation theory [76]. In terms of this gothic graviton field, the de
2We work in the mostly plus metric convention.
metric perturbations are simply related:
and the linear gauge transformation generated by xμ → xμ − κ ξμ is
hμν → hμν = hμν + ∂μξν + ∂ν ξμ − ημν ∂ · ξ.
p · ǫi = 0,
q · ǫi = 0,
This transformation is more convenient in what follows than the standard gauge
transfor∂2hμν − ∂μ∂ρhρν − ∂ν ∂ρhρμ + ημν ∂ρ∂σhρσ = 0.
Linear gravitons from Yang-Mills fields
the fat graviton is the field whose interactions are directly dictated by the double copy
from gauge theory. In this section, we will discuss in some detail the mapping between
the skinny fields and the fat graviton at the linearised level. Indeed, we will see that
there is an invertible map, so that the fat graviton may be constructed from skinny fields
the fields beginning with the simplest case: linearised waves.
As a prelude to obtaining non-linear gravitational solutions from Yang-Mills theory, we first
discuss linear solutions of both theories. The simplest possible solutions are linear waves.
These are well-known to double copy between gauge and gravity theories (see e.g. [77]). This
property is crucial for the double copy description of scattering amplitudes, whose incoming
and outgoing states are plane waves. Here, we use linear waves to motivate a prescribed
relationship between fat and skinny fields, which will be generalised in later sections.
Let us start by considering a gravitational plane wave in the de Donder gauge. The free
p2 = 0,
satisfying the orthogonality conditions
of freedom for an on-shell massless vector boson. These polarisation vectors are a complete
set, so they satisfy a completeness relation
where fi/tj is a traceless symmetric matrix. Thus, the linearised gravitational waves have
polarisation states which can be constructed from outer products of vector waves, times
traceless symmetric matrices.
where f˜ij is a constant antisymmetric matrix. Meanwhile the free equation of motion for
D − 2
which explicitly constructs the fat graviton from skinny fields. Working in position space
for constant q, this becomes
where we have defined the projection operator
D − 2
q · ∂
which will be important throughout this article.3
Our goal in this work is not to construct fat gravitons from skinny fields, but on the
contrary to determine skinny fields using a perturbative expansion based on the double
copy and the fat graviton. Therefore it is important that we can determine the skinny
fields given knowledge of the fat graviton. To that end, recall that we have been able to
choose a gauge so that the trace, h, of the metric perturbation vanishes. Therefore the
trace of the fat graviton determines the dilaton:
We may now use symmetry to determine the skinny graviton and antisymmetric tensor
from the fat graviton:
into its antisymmetric, traceless symmetric, and trace parts.
It is worth dwelling on the decomposition of the fat graviton into skinny fields a little
transformation of the skinny graviton:
q · ∂
q · ∂
Thus, up to a gauge transformation, the skinny graviton is the symmetric part of the
gauge transformation, which is, of course, a particular diffeomorphism.
be gauge invariant for diffeomorphisms at this order.
We will see below that the perturbative expansion for fat gravitons is much simpler
than the perturbative expansion for the individual skinny fields. But before we embark on
that story, it is important to expand our understanding of the relationship between the fat
graviton and the skinny fields beyond the sole case of plane waves.
For plane waves, the fat graviton is given in terms of skinny fields in eq. (3.10), and at first
glance this equation is not surprising: one may always choose to decompose an arbitrary
rank two tensor into its symmetric traceless, antisymmetric and trace parts. However,
eq. (3.10) contains non-trivial physical content, namely that the various terms on the r.h.s.
are the genuine propagating degrees of freedom associated with each of the skinny fields.
with the definition of physical polarisation vectors, and thus can be used to project out
physical degrees of freedom in the gravity theory. One may then ask whether eq. (3.10)
generalises for arbitrary solutions of the linearised equations of motion. There is potentially
a problem in that the relationship becomes ambiguous: the trace of the skinny graviton
may be nonzero (as is indeed the case in general gauges), and one must then resolve how
will work when non-zero sources are present in the field equations. In order to use the
double copy in physically relevant applications, we must consider this possibility.
Here we will restrict ourselves to skinny gravitons that are in de Donder gauge.
However, we will relax the traceless condition on the skinny graviton which was natural in the
previous section. To account for the trace, we postulate that eq. (3.10) should be replaced
Hμν (x) = hμν (x) + Bμν (x) + Pμqν (φ − h).
To be useful, this definition of the fat graviton must be invertible. First, note that the
Finally, the traceless symmetric part of the fat graviton is
(Hμν + Hνμ) − Pμqν H = hμν (x) − Pμqν h = h′μν (x),
in de Donder gauge, since
D − 2
q · ∂
h = − D − 2 q · ∂
Our relationship between skinny and fat fields still holds only for linearised fields;
we will explicitly find corrections to eq. (3.19) at higher orders in perturbation theory
in section 4. Before doing so, however, it is instructive to illustrate the above general
discussion with some specific solutions of the linear field equations, showing how the fat
and skinny fields are mutually related.
The linear fat graviton for Schwarzschild
One aim of our programme is to be able to describe scattering processes involving black
holes. To this end, let us see how to extend the above results in the presence of point-like
masses. It is easy to construct a fat graviton for the linearised Schwarzschild metric: we
the projector (3.11) in position space in full. A computation gives
Going in the other direction, it is easy to compute the skinny fields given this fat
graviton. While this result seems to be at odds with (3.22), recall that they differ only
skinny graviton we recover is traceless, as we would expect from eq. (3.20).
It may not seem that we have gained much by passing to eq. (3.24) from eq. (3.22).
However, it is our contention that it is simpler to compute perturbative corrections to
metrics using the formalism of the fat graviton than with the traditional approach. We
will illustrate this in a specific example later in this paper.
Solutions with linearised dilatons
The linearised Schwarzschild metric corresponds to a somewhat complicated fat graviton.
which corresponds to inserting a singularity at the origin. We will see that this solution
has the physical interpretation of a point mass which is also a source for the scalar dilaton.
Indeed, the dilaton contained in the fat graviton is given by its trace:
Again, a linearised diffeomorphism can give the skinny graviton the same form as the fat
coupling Y as
− Y 2)uμuν + (M 2 + Y 2)rˆμrˆν + O(κ5), (3.31)
It is natural to ask what is the non-perturbative static spherically-symmetric solution
for which we are finding the linearised fields. Exact solutions of the Einstein equations
minimally coupled to a scalar field of this form were discussed by Janis, Newman and
Winicour (JNW) [78] and have been extensively studied in the literature [78–84]. The
complete solution is, in fact, a naked singularity, consistent with the no-hair theorem. The
general JNW metric and dilaton can be expressed as
ds2 = −
particularly natural object from the point of view of the perturbative double copy. At
large distances from the singularity, both the metric perturbation and the scalar field fall
corrections to the JNW metric using fat gravitons, and, in the case of the first correction,
the exact solution associated to the linearised fat graviton (3.25).
We can also ask what fat graviton would be associated to the general JNW family
of solutions, with M and Y generic. Since we are dealing with linearised fields, we can
superpose contributions, and so we arrive at
M uμuν + (M − Y ) (ημν − qμlν − qν lμ) .
The gauge theory “single copy” associated to this field is simply the Coulomb solution,
which presents an apparent puzzle: ref. [54] argued that the double copy of the Coulomb
solution is a pure Schwarzschild black hole, with no dilaton field. Above, however, the
double copy produces a JNW solution. The latter was also found in ref. [70], which thus
concluded that the Schwarzschild solution is not obtained by the double copy, but can
only be true in certain limits (such as the limit of an infinite number of dimensions). The
resolution of this apparent contradiction is that one can choose whether or not the dilaton
is sourced upon taking the double copy. It is well-known in amplitude calculations, for
example, that gluon amplitudes can double copy to arbitrary combinations of amplitudes
for gravitons, dilatons and/or B-fields. A simple example are amplitudes for linearly
polarised gauge bosons: the double copied “amplitude” involves mixed waves of gravitons
and dilatons. Thus, the result in the gravity theory depends on the linear combinations
of the pairs of gluon polarisations involved in the double copy. Here, we may say that the
Schwarzschild solution is a double copy of the Coulomb potential, as given by the
KerrSchild double copy [54], just as one may say that appropriate combinations of amplitudes of
gluons lead to amplitudes of pure gravitons. The analogue of more general gravity
amplitudes with both gravitons and dilatons, obtained via the double copy, is the JNW solution.
Therefore the double copy of the Coulomb solution is somewhat ambiguous: in fact, it is
any member of the JNW family of singularities, including the Schwarzschild metric. Note
that the Kerr-Schild double copy is applicable only in the Schwarzschild special case since
the other members of the JNW family of spacetimes do not admit Kerr-Schild coordinates.
For the vacuum Kerr-Schild solutions studied in [54], in particular for the Schwarzschild
black hole, it was possible to give an exact map between the gauge theory solution and the
exact graviton field, making use of Kerr-Schild coordinates (as opposed to the de Donder
gauge used here). For the general JNW solution, the double copy correspondence was
inferred above from the symmetries of the problem and from the perturbative results. A
more general double copy map would also be able to deal with the exact JNW solution.
This remains an important goal, but one which is not addressed in this paper.
Perturbative corrections
Now that we have understood how to construct fat gravitons in several cases, let us finally
put them to use. In this section, we will construct nonlinear perturbative corrections to
spacetime metrics and/or dilatons using the double copy. Thus, we will map the problem
of finding perturbative corrections to a simple calculation in gauge theory.
Perturbative metrics from gauge theory
Since the basis of our calculations is the perturbative expansion of gauge theory, we begin
with the vacuum Yang-Mills equation
where g is the coupling constant, while the field strength tensor is
can be written as a power series in the coupling:
on the coupling g. We use a similar notation for the perturbation series for the skinny and
We can construct solutions in perturbation theory in a straightforward manner. To
For our present purposes, two basic solutions of this equation will be of interest: wave
solutions, and Coulomb-like solutions with isolated singularities.
first order in g:
∂2A(ν1)a = −2f abcA(0)bμ∂μA(ν0)c + f abcA(0)bμ∂ν A(μ0)c.
The double copy is most easily understood in Fourier (momentum) space. To simplify our
notation, we define
Z d−DpF (p) ≡
Using this notation, we may write the solution for the first perturbative correction in
Fourier space in the familiar form
× h(p1 − p2)γ ημβ + (p2 − p3)μηβγ + (p3 − p1)βηγμi A(0)b(p2)A(γ0)c(p3).
β
Notice that the factor in square brackets in this equation obeys the same algebraic
symmetries as the colour factor, f abc, appearing in the equation. This is a requirement of
colour-kinematics duality. Before using the double copy, it is necessary to ensure that this
The power of the double copy is that it is now completely trivial to compute the
ing [1–3], is to square the numerator in eq. (4.9), ignore the colour structure, and assemble
1 Z d−Dp2d−Dp3−δD(p1 + p2 + p3)
× h(p1 − p2)γ ημβ + (p2 − p3)μηβγ + (p3 − p1)βηγμi
× h(p1 − p2)γ′ ημ′β′ + (p2 − p3)μ′ ηβ′γ′ + (p3 − p1)β′ ηγ′μ′ i Hβ(0β)′ (p2)Hγ(0γ)′ (p3).
Notice that the basic structure of the perturbative calculation is that of gauge theory. The
double copy upgrades the gauge-theoretic perturbation into a calculation appropriate for
gravity, coupled to a dilaton and an antisymmetric tensor.
As a simple example of this formalism at work, let us compute the first order correction
to the simple fat graviton eq. (3.25) corresponding to a metric and scalar field. To begin,
It is now straightforward to integrate this expression using spherical symmetry and the
known boundary conditions to find
It is interesting to pause for a moment to contrast this calculation with its analogue in
Yang-Mills theory. The simplest gauge counterpart of the JNW linearised fat graviton is
Inserting this into our expression for H(1), eq. (4.10), we quickly find
κ 2 M 2 Z d−3p2 d−3p3 −δ4(p1 + p2 + p3)
02 = 0 = p03, and consequently p
we need only calculate the spatial components H(1)ij . To do so, it is convenient to Fourier
For future use, we note that
= −
(p2 − p3)i(p2 − p3)j
To what extent is the first non-linear correction to the Yang-Mills equation similar to
the equivalent in our double-copy theory? The answer to this question is clear: they are
information from one theory to the other is unclear, but as a mathematical statement there
is no issue with using the double copy to simplify gravitational calculations.
Given our expression, eq. (4.14), for the fat graviton, it is now straightforward to
extract the trace and the symmetric fields:
≡ H(1) = −
first order correction to the metric in some well-known gauge. The double copy is only
guaranteed to compute quantities which are field redefinitions or gauge transformations of
the graviton and dilaton. This suggests structuring calculations to compute only quantities
which are invariant under field redefinitions and gauge transformations [65–67, 70, 85, 86].
However, if desired, it is nevertheless possible to determine explicitly the relevant field
redefinitions and gauge transformations. This is the topic of the next section.
Relating fat and skinny fields: gauge transformations and field
redefiniIn section 3, we argued that the relationship between the fat and skinny fields in linear
Beyond linear theory, we can expect perturbative corrections to this formula, so that
Hμ(0ν)(x) = h(μ0ν)(x) + Bμ(0ν)(x) + Pμqν (φ(0)(x) − h(0)(x)).
Hμν (x) = hμν (x) + Bμν (x) + Pμqν (φ(x) − h(x)) + O(κ).
Hμ(1ν)(x) = h(μ1ν)(x) + Bμ(1ν)(x) + Pμqν (φ(1)(x) − h(1)(x)) + Tμ(ν1).
Hμ(nν)(x) = h(μnν)(x) + Bμ(nν)(x) + Pμqν (φ(n)(x) − h(n)(x)) + Tμ(νn)(h(αmβ), Bα(mβ ), φ(m)),
(n) iteratively in perturbation theory.
(n) rests on two facts. Firstly, the double copy is
(1) =
known to work to all orders in perturbation theory for tree amplitudes. Secondly, the
classical background field which we have been discussing is a generating function for tree
scattering amplitudes. Therefore it must be the case that scattering amplitudes computed
from the classical fat graviton background fields equal their known expressions. So consider
Bμ(nν)(x) − Pμqν (φ(n)(x) − h(n)(x)) ≡ Tμν
(n) must vanish upon use of the LSZ procedure. We
computing scattering amplitudes: gauge transformations and field redefinitions. Indeed,
gauge for the skinny graviton.
transformation function is
choice of gauge, we do not expect a particularly simple form for it. Nevertheless, to compare
explicit skinny gravitons computed via the double copy with standard metrics, it may be
through its definition, at the expense of perturbatively solving the coupled Einstein, scalar
and antisymmetric tensor equations of motion. For example, consider the fat graviton
We find that when ∂μh(0)μν = ∂μH(0)μν = 0, then the
H2(0α)βH3(0)αβp1μpν1 + 8p2αH3(0α)βH2(0)β(μpν)
1
+8p2 · p3 H2(0)μαH3(0)αν − 2ημν p2 · p3 H2(0α)βH3(0)αβ + 4ημν p2αH3(0α)βH2(0)βγ p3γ
− 4(D − 2)p2αH3(0α)βH2(0)βγ p3γ i , (4.22)
where we have used a convenient short-hand notation
≡ 2
While the information in the transformation function contains little content of physical
interest, it may be of some interest from the point of view of the mathematics of
colourkinematics duality. Indeed, in the special case of the self-dual theory, it is known how to
choose an explicit parameterisation of the metric perturbation so that the double copy is
spacetimes. Once the relevant variables have been chosen, then the kinematic algebra in
the self-dual case was manifest at the level of the equation of motion of self-dual gravity:
the algebra is one of area-preserving diffeomorphisms. Perhaps it is the case that an
understanding of the transformation function in the general case will open the way towards
a simple understanding of the full kinematic algebra.
since p · u = 0 for a stationary source. Thus T
M 2 Z d−4p2d−4p3−δ4(p1 + p2 + p3) 4p21
× 8p2 · p3uμuν − p1μpν1 + 2ημν p2 · p3 + Pqμν [4p2 · p3] ,
in D = 4. Performing the Fourier transform, we find
Let us now extract the skinny fields in de Donder gauge from our fat graviton, eq. (4.14).
The relation between the fat and skinny fields is now given by
h(μ1ν)(x) + Pμqν hφ(1)(x) − h(1)(x)i = Hμ(1ν)(x) − Tμ(ν1)(x)
= −
Thus, the dilaton vanishes as anticipated in section 3.4, since
(1)(x) = 0.
The metric is easily seen to be
h(1)(x) = −
consistent with the anticipated trace, and in agreement with the known result for the JNW
metric, eq. (3.31), when M = Y .
In section 4.1, we saw how fat graviton fields can be obtained straightforwardly from
perturbative solutions of the Yang-Mills equations. These can then be translated to skinny
fields, if necessary, after obtaining the relevant transformation functions T
briefly describe how this procedure generalises to higher orders.
As we explained in section 2, the validity of the double copy relies on writing Yang-Mills
diagrams such that colour-kinematics duality is satisfied. But, in general, a perturbative
solution of the conventional Yang-Mills equations will not satisfy this property. So before
using the double copy, one must reorganise the perturbative solution of the theory so that,
firstly, only three-point interaction vertices between fields occur, and secondly, the
numerators of these three-point diagrams satisfy the same algebraic identities (Jacobi relations and
antisymmetry properties) as the colour factors. The Jacobi identities can be enforced by
using an explicit Yang-Mills Lagrangian designed for this purpose [3, 87]. It is known how
to construct this Lagrangian to arbitrary order in perturbation theory. This Lagrangian is
non-local and contains Feynman vertices with an infinite number of fields. If desired, it is
possible to obtain a local Lagrangian containing only three point vertices at the expense
of introducing auxiliary fields. For now, we will restrict ourselves to four-point order. At
this order Bern, Dennen, Huang and Kiermaier (BDHK) introduced [3] an auxiliary field
LBDHK =
1 Aaμ∂2Aaμ + Baμνρ∂2Bμaνρ − gf abc ∂μAν − ∂ρBρaμν AbμAcν .
a
To illustrate the procedure in a non-trivial example, let us compute the second order
calculation remarkably straightforward. Firstly, the momentum space equation of motion
for the auxiliary field appearing in the BDHK Lagrangrian, eq. (4.30), is
i f abc Z d−4p2d−4p3−δ4(p1 +p2 +p3)p1μ [ηνβηργ − ηνγ ηρβ] A(0)bβ(p2)A(0)cγ (p3).
associated colour structure is antisymmetric under interchange of b and c. A consequence
of this simple fact is that, in the double copy, the auxiliary field vanishes in the JNW case
(to this order of perturbation theory). In fact, two auxiliary fields appear in the double
copy: one can take two copies of the field B, or one copy of B times one copy of the
gauge boson A. In either case, the expression for an auxiliary field in the double copy in
momentum space will contain a factor
p1μ [ηνβηργ − ηνγ ηρβ] H(0)ββ′ (p2)H(0)γγ′ (p3)
because of the antisymmetry of the vertex in square brackets, and the factorisability of
the tensor structure of the zeroth order JNW expression.
Consequently, the Yang-Mills four-point vertex plays no role in the the double copy
for JNW at second order. Thus the Yang-Mills equation to be solved is simply
p21A(2)aμ(−p1) = if abc Z d−4p2d−p3−δ4(p1 + p2 + p3)
× h(p1 − p2)γ ημβ + (p2 − p3)μηβγ + (p3 − p1)βηγμi A(0)b(p2)A(γ1)c(p3), (4.33)
β
Z d−4p2d−4p3−δ4(p1 + p2 + p3)Hμ(0μ)′ (p2) p2αHα(1β)(p3)p2β.
We find it convenient to Fourier transform back to position space, where we must solve the
simple differential equation
Inserting explicit expressions for H(0), eq. (3.25) and H(1), eq. (4.14), and bearing in mind
that the situation is static, the differential equation simplifies to
using the symmetry of the expression under interchange of p2 and p3. Thus, H(2) is the
1 Z d−4p2d−4p3−δ4(p1 + p2 + p3)
× h(p1 − p2)γ ημβ + (p2 − p3)μηβγ + (p3 − p1)βηγμi
×h(p1 −p2)γ′ ημ′β′ +(p2 −p3)μ′ ηβ′γ′ +(p3 −p1)β′ ηγ′μ′ i Hβ(0β)′ (p2)Hγ(1γ)′ (p3).
We could now, if we wished, extract the metric perturbation and scalar field corresponding
to this expression. Indeed, it is always possible to convert fat gravitons into ordinary metric
perturbations in a specified gauge.
It is possible to continue to continue this calculation to higher orders. In that case,
more work is required in order to satisfy the requirement of colour-kinematics duality.
It is possible to supplement the BDHK Lagrangian by higher-order effective operators
involving the gluon field, constructed order-by-order in perturbation theory, which act to
enforce colour-kinematics duality. Furthermore, one may introduce further auxiliary fields
so that only cubic interaction terms appear in the Lagrangian. This procedure is explained
in detail in refs. [3, 87], and can be carried out to arbitrary perturbative order. The fat
graviton equation of motion is constructed as a term-by-term double copy of the fields in the
colour-kinematics satisfying Yang-Mills Lagrangian. In this way, it is possible to calculate
perturbative fat gravitons to any order using Yang-Mills theory and the double copy.
In this paper, we have addressed how classical solutions of gravitational theories can be
obtained by double-copying Yang-Mills solutions. These results go beyond the classical
double copies of refs. [54–63] in that the solutions are non-linear. However, the price
one pays is that they are no longer exact, but must be constructed order-by-order in
perturbation theory. We have concentrated on solutions obtained from two copies of pure
(non-supersymmetric) Yang-Mills theory, for which the corresponding gravity theory is
that we call the fat graviton, and which in principle can be decomposed into its constituent
Our procedure for calculating gravity solutions is as follows:
1. For a given distribution of charges, one may perturbatively solve the Yang-Mills
2. The solution for the fat graviton is given by double copying the gauge theory
solution expression according to the rules of refs. [1–3] once colour-kinematics duality is
satisfied. That is, one strips off all colour information, and duplicates the interaction
vertices, leaving propagators intact.
3. The fat graviton can in principle be translated into skinny fields using the
transformation law of eq. (4.21), which iteratively defines the transformation function T
function can be obtained from matching the fat graviton solution to a perturbative
can be used for arbitrary source distributions.
The presence of the transformation function T
always decompose the fat graviton in terms of its symmetric traceless, anti-symmetric
and trace degrees of freedom. Then one could simply define that these correspond to the
physical graviton, two-form and dilaton. However, one has the freedom to perform further
field redefinitions and gauge transformations of the skinny fields, in order to put these into
a more conventional gauge choice (e.g. de Donder). The role of T
redefinition. It follows that it carries no physical degrees of freedom itself, and indeed is
irrelevant for any physical observable.
We have given explicit examples of fat gravitons, and their relation to de Donder gauge
skinny fields, up to the first subleading order in perturbation theory. We took a stationary
point charge as our source, finding that one can construct either the Schwarzschild metric
(as in the Kerr-Schild double copy of ref. [54]), or the JNW solution [78] for a black hole
the choice of whether or not to source the dilaton upon performing the double copy. This
mirrors the well-known situation for amplitudes, namely that the choice of polarisation
states in gauge theory amplitudes determines whether or not a dilaton or two-form is
obtained in the corresponding gravity amplitudes at tree level. This clarifies the apparent
puzzle presented in ref. [70], regarding whether it is possible for the same gauge theory
solution to produce different gravity solutions.
Underlying the simplicity of the double copy is the mystery of the kinematic algebra.
While it is known that one can always find kinematic numerators for gauge theory
diagrams so that colour-kinematics duality is satisfied, it is not known whether an off-shell
algebraic structure exists in the general case which can compute these numerators. If this
algebra exists, it may further simplify the calculations we have described in this paper.
The kinematic algebra should allow for a more algebraic computation of the numerators
of appropriate gauge-theoretic diagrams, perhaps without the need for auxiliary fields.
Similarly, it seems possible that a detailed understanding of the kinematic algebra will go
the choice of gauge and field redefinition picked out by the double copy.
Our ultimate aim is to use the procedure outlined in this paper in astrophysical
applications, namely to calculate gravitational observables for relevant physical sources (a
motivation shared by ref. [70]). To this end, our fat graviton calculations must be
extended to include different sources, and also higher orders in perturbation theory. In order
to translate the fat graviton to more conventional skinny fields, one would then need to
calto calculate physical observables, which must be manifestly invariant under gauge
transformations and field redefinitions, directly from fat graviton fields, without referring to skinny
fields at all. Work on these issues is ongoing.
We are very grateful to Alex Anastasiou and Radu Roiban for useful discussions. We
especially thank John Joseph Carrasco for emphasising the relevance of the fat graviton (and
providing its name). DOC is supported in part by the STFC consolidated grant Particle
Physics at the Higgs Centre, while DOC and AO are supported in part by the Marie Curie
FP7 grant 631370. IN is supported by an STFC studentship. NW acknowledges support
from the EPSRC CM-CDT Grant No. EP/L015110/1. CDW is supported by the U.K.
Science and Technology Facilities Council (STFC), and AL by a Conacyt studentship and
a Lord Kelvin Fund travel scholarship. Both CDW and AL thank the Higgs Centre for
Theoretical Physics, University of Edinburgh, for warm hospitality. RM, DOC, AO and
CDW are grateful to Nordita for hospitality during the programme “Aspects of
Amplitudes.” RM, DOC, AL, AO and CDW would like to thank the Isaac Newton Institute for
Mathematical Sciences for its hospitality during the programme “Gravity, Twistors and
Amplitudes” which was supported by EPSRC Grant No. EP/K032208/1.
This article is distributed under the terms of the Creative Commons
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any medium, provided the original author(s) and source are credited.
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