N ov 2 01 9 QMUL-PH-1932 S-duality and the Double Copy

The double copy formalism provides an intriguing connection between gauge theories and gravity. It was first demonstrated in the perturbative context of scattering amplitudes but recently the formalism has been applied to exact classical solutions in gauge theories such as the monopole and instanton. In this paper we will investigate how duality symmetries in the gauge theory double copy to gravity and relate these to solution generating transformations and the action of Sl(2, R) in general relativity.


Introduction
The double copy was first investigated in a series of works [1][2][3], as a relationship between perturbative scattering amplitudes in gauge theory and gravity. This has been proven at tree level [3][4][5][6][7][8][9][10][11], where it has a stringy origin [12]. However, there is still no non-perturbative proof of the double copy though the evidence is mounting with a series of works showing double copy behaviour for amplitudes for higher loops [2,.
In a recent set of works the double copy/single copy was applied to a class of exact classical solutions. Double copy refers to moving from gauge theory to gravity while single copy is the inverse map from gravity to gauge theory. The Schwarzschild solution was shown to single copy to an electric charge [44] and the Taub NUT solution single copy to a magnetic monopole [45]. Subsequent to that the single copy of the Eguchi Hanson solution has been mapped to a self-dual gauge field [46]. More general topologically non-trivial solutions have been double copied in the work of [47]. (Non-exact classical solutions have also appeared in a series of works [48][49][50][51][52][53][54][55][56].) To futher investigate the double copy formalism beyond the perturbative regime, we examine how non-perturbative symmetries in the gauge theory are double copied to gravity. In particular, gauge theories exhibit electromagnetic duality which exchanges electric and magnetic charges. This symmetry, often also called S-duality, was first discovered for classical Maxwell theory but then became a crucial ingredient in the study of properties of Yang-Mills theory [57]. General relativity is not known to exhibit S-duality which leads to the question of what S-duality could double copy to. We will address this at the non-perturbative classical level by identifying the solution generating symmetry in general relativity that single copies to electromagnetic duality.
We will show that the double copy of electromagnetic duality is identified as the Ehlers transformation in general relativity. It is worth commenting further that we are working with exact and thus classically non-perturbative solutions in the double copy and the metrics corresponding to the electric and magnetic solutions are mapped between each other using an exact non-local transformation. We will then demonstrate how another solution generating symmetry discovered by Buchdhal [58], is single copied to the charge conjugation.
We adopt two complementary calculational approaches. First we will use the Kerr-Schild form of the metric where the metric in this form is related to gauge fields in the single copy. Second we will use the correspondence developed in [59] where the Weyl curvature in gravity is related to a combination of field strengths in the single copy. In each case we will examine how the electromagnetic transformation in the single copy is related to a transform in relativity.
To make the paper as self contained as possible and make clear our conventions for the double/single copy we begin with a description of the Kerr-Schild form in GR and then the single copy prescription. 5 We give the detailed examples of the Schwarzschild black hole and the Taub-NUT solution and their single copies as described first in [44,45]. Then we describe the Ehlers transform in general before moving on to its application to Schwarzchild and its single copy. After discussing the role of the the Buchdahl transform and its single copy again using Kerr-Schild form we move to the double copy in terms of curvatures and field strengths, known as the Weyl double copy [59] where we can again examine the role of electromagnetic duality from a complementary approach.
Note added: While this manuscript was in preparation [60] appeared which has a substantial overlap with this paper.

The Kerr-Schild double copy
First let use introduce the Kerr-Schild form of the metric and examine the behaviour of Einstein's field equations. Writing the the metric in this form is a crucial part in making the double/single copy procedure manifest. In what follows we shall use the procedure outlined in [44]. We take the metric η µν = diag(−1, +1, +1, +1) throughout.
A solution is called "Kerr-Schild" if a set of coordinates may be found such that the spacetime metric g µν may be put in the form where φ is a scalar field and k µ is a covector that satisfies the following property: i.e. null with respect to both the full and background metric. The inverse metric then takes the form In terms of the scalar field φ and covector k µ the Ricci tensor and Ricci scalar are where ∂ µ = η µν ∂ ν . In the stationary spacetime case (∂ 0 φ = ∂ 0 k µ = 0) one may take the time component of the Kerr-Schild vector as k 0 = 1, with the dynamics in the time component contained in φ. As a consequence, the components of the Ricci tensor simplify to where latin indicies indicate the spatial components.
Now if one defines a gauge field A µ = φk µ that satisfies the Abelian field strength F µν = ∂ µ A ν − ∂ µ A ν . Taking the stationary case of the vacuum Einstein's equations R µν = 0 one finds that the gauge field satisfies the Abelian Maxwell equations The remarkable thing about the double copy is that if now one considers a non-Abelian gauge field A a µ where the index a is related to the gauge group of the theory, there is still a single copy/double copy relationship. The recipe is to take the quantity φk µ k ν of a given gravity solution and strip off one of the Kerr-Schild vectors and dress with a gauge group index to get the corresponding gauge field A a µ = c a φk µ . There is no derivation as such for this procedure but by now there is a compelling amount of evidence as listed in the introduction. Thus, the basic statement of the double/single copy we will be applying is: If g µν= η µν + φk µ k ν is a stationary solution of Einstein's equations, then A a µ = c a φk µ is a solution of the Yang-Mills equations which is linearised by the Kerr-Schild coordinates allowing an arbitrary choice for the constant c a .

Schwarzschild and NUT spacetimes
We shall demonstrate the double/single procedure on two important gravity solutions, the Schwarzschild blackhole and the Taub NUT spacetime.

Schwarzschild spacetime
The Schwarzschild solution is where dΩ 2 = dθ 2 + sin θdφ 2 is the line element of the unit sphere, G is Newton's constant, and r 2 = x 2 + y 2 + z 2 is the radial distance from the origin.
In order to apply the single copy procedure, one must to write this metric in Kerr-Schild form. To do this we apply the coordinate transformation l = t +r with dr = dr 1− 2GM r so that the metric takes the form a further coordinate transformation of the form l =t + r is applied so the metric becomes Notice that the first three terms are just the flat Minkowski metric in spherical coordinates. Using the definition of r given above to transform into 'Cartesian' coordinates the metric becomes where the null vector k µ is defined by The metric is now in Kerr-Schild form: Then comparing with a metric: The single copy then yields where we have followed the same single copy prescription used for scattering amplitudes with the replacements: The first one is just choosing the right coupling constant, the second one being the corresponding charge between the two theories.

Taub-NUT spacetime
The solution known as Taub-NUT was first derived by Taub [61] and generalised by Newman, Tamubrino, and Unti [62] in 1963. The NUT solution is a non-asymptotically flat, axially symmetric stationary solution.
Following Ortin [63], the Taub-NUT metric can be written in the form where This can be thought of as a generalisation of the Schwarzschild solution with an additional topological charge. This solution exhibits the following interesting properties: • The solution is not trivial in the limit M → 0.
• Taking the Newtonian limit shows that M is indeed the mass of the source. The NUT charge has no Newtonian analogue.
• The solution defines its own class of asymptotic behaviour labeled by N and is associated with the non-vanishing at infinity g tφ component of the metric.
• The solution admits Dirac-like singularities at θ = 0, π which forces us to introduce two coordinate patches to get rid of them.
We have seen in the previous sections that the mass in gravity single copies to an electric or colour charge. What about the NUT charge in N which the Taub-NUT metric has as an additional parameter? Thanks to this new charge the Taub-NUT exhibits magnetic monopole-like behaviour.
We will see that the NUT charge N does indeed single copy to a magnetic monopole on the gauge theory side.
The Taub-NUT metric given by Plebanski [64] has been shown to exhibit a double Kerr-Schild form in [65]. The metric takes the form The vectors k µ and l µ satisfy the following conditions where all the contractions and covariant derivatives can be taken either with respect the background or the full metric. This metric linearises the Einstein equations as with the Kerr-Schild form. The null vectors in these coordinates are where the scalar functions φ and ψ are written as The generalisation of the single copy prescription for the gauge field is then Following [45], we have made the following substitutions The field strength is then, The first term is exactly the pure electric charge corresponding to a Coulomb solution that was derived in the Schwarzschild case. The second is the dual of a pure electric charge of strength is a magnetic monopole charge which single copied from the NUT contribution on the gravity side.
In summary, the Schwarzschild single copies to a Coulomb solution and the NUT charge to a magnetic monopole.

Solution generating transformations in general relativity
A solution generating transformation is simply a recipe for obtaining a new solution to Einstein's equations from a known one. This is particularly impressive since Einstein's equations are a set of ten second order coupled nonlinear differential equations that are notoriously difficult to solve. Of course the symmerties in Einstein's equations make some transformations trivial. There are however "hidden symmetries" that allow us to generate very non-trivial solutions to Einstein's equations through this transformation technique. This approach to solving Einstein's equations was originally developed by Buchdahl, Ehlers, Geroch and Ernst [66]. The transformations require a Killing symmetry in the spacetime. When such a symmetry is present then there are a set of solution generating techniques which we describe below.
The Ehlers transformation is a transformation that acts on the parameters of a static solution of Einstein gravitational field equations and generates other solutions of the field equations which need not be static but are stationary. In what follows we describe how the Ehlers transformation works [67].

Ehlers transformation
In this section we will closely follow [68]. It is assumed the spacetime possesses a time like Killing vector ξ that generates an isometry. First perform a (1+3) decomposition of the metric and choose coordinates x µ = {x 0 , x i } to put the line element in the following form where we define and The Ehlers transformation states that if with dl 2 = e 2U dℓ 2 , is the metric of a static spacetime, then whereγ ab is the conformal spatial metric. So using this method, generating a stationary solution from a static one is done by finding the potential A a which is related to a given static potential U . Let's see how that comes about by applying the procedure described above to the Schwarzschild metric.
Start with the Schwarzschild metric in the form (32) with the potential taking the form and where C is a constant and U (x a ) only depends on the spatial coordinates. Then the Ehlers equation (34) reads One may choose a gauge such that A θ,φ = 0. Then substituting (35) in and solving for the field A φ we find The resulting metric is then given by (33) as where Recovering the Schwarzschild metric as α → 0 requires us to set Now apply the following changes of variables and redefinitions of constants; to get which is the metric for the NUT spacetime. Remarkably, the constant C in the potential U , while having no effect on the Schwarzschild spacetime, plays an important role in generating solutions when Ehlers transformation is applied. Had we started with C = 0 we would have ended up with the pure NUT spacetime (M = 0). This means that the seed metric for both the NUT with mass and pure NUT metric is the Schwarzschild metric.

Buchdahl's reciprocal transformation
In [58] Buchdahl showed that if a solution of Einstein's equations admits a Killing field one can obtain a new solution by applying the so called "reciprocal transformation". For coordinates adapted to the Killing direction such that the Killing vector is given by ∂ ∂x α , then the line element maybe written: The reciprocal transformation then generates the following line element The transformation may be written acting on metric components in this adapted coordinate system as (The reader familiar with string theory will note that Buchdahl's reciprocal transformation is essentially T-duality but actually predates T-duality some thirty years as a solution generating transform.) We shall apply it to the Schwarzschild metric. As noted above, the d = 4 Schwarzschild metric admits a timelike killing vector ∂/∂t which will be our 'Killing coordinate t = a'. Applying the transformation yields which upon substitution into the metric we get The Schwarzschild metric admits another killing field ∂/∂φ which is spatial. Applying the same transformation but now with α = φ we get the following metric This new solution is completely unrelated to the original seed metric unlike (48) which was obtained using the timelike Killing symmetry. However, the metric (48) is related to the Schwarzschild metric by a simple coordinate transformation R = r−2M whereby once used one obtains the Schwarzschild metric but with negative mass parameter (M → −M ) If we write the negative mass Schwarzschild metric in in Kerr-Schild form and then do the usual single copy procedure where −M → −c a T a we arrive at which has the opposite sign of the gauge field A µ compared to the single copy the positive mass Schwarzschild solution [44] ie.
This indicates that the Buchdahl reciprocal transformation associated with the timelike Killing vector in Schwarzschild is the gravitational analogue of charge conjugation on the gauge theory side.
We now examine Buchdahl tranformation acting on the Taub-NUT solution by first using Schwarzschild as our seed metric on which we act with the Buchdahl transfrom followed by the Ehler's transformation. First note that Buchdahl transformation corresponds to U →Û = −U when one writes the Schwarzschild metric in the form So a Buchdahl transformed Schwarzschild metric reads which as before is just the Schwarzschild metric with negative mass as shown before. Now lets construct the Ehlers transformed metric as before. Using the Ehlers equation for the reciprocal which upon comparing it to the standard one (34) we find Then following exactly the same steps as before, i.e. solving for A µ while imposing axial symmetry we find: and as before the NUT charge is given byN which as one can see has acquired a negative sign upon comparing the field A µ to Eq. (38). The reciprocal Taub-NUT metric then reads Upon applying the single copy procedure [45] as in the previous sections using the multi-Kerr-Schild form of the metric one finds again that the resulting gauge field has negative electric and magnetic monopole charges.
Here From the action of the Ehlers transformation on the charges in the double copy one obtain that the single copy of the Ehlers transformation is electromagnetic duality in the gauge theory or alternatively the double copy of electromagnetic duality is generated by the Ehlers transformation in gravity.
More formally, the Ehlers transformation is an element of SL(2, R) as introduced by Geroch [69]. However when this acts on the Taub NUT solution the quantisation of NUT charge means that the group must be broken to SL(2, Z) so as to preserve the quantisation condition. This exactly follows what happens with the electromagnetic duality group acting on the dyon spectrum. Classically the group is SL(2, R) but this reduces to SL(2, Z) in order to maintain Dirac quantisation for the magnetic charges.

Definitions
The double copy as described above is defined in terms of the gauge connection and the metric. It is a natural question to ask whether one might study a double copy directly in terms of field strengths and curvatures. This was investigated in [59], where a particularly nice form of the double copy was obtained using spinors -writing the spinor corresponding to the Weyl tensor in terms of the spinor for the Maxwell tensor as where S is a suitable function related to the "zeroth copy". This "Weyl double copy" was shown to be consistent with the previously known Kerr-Schild double copy, and resolved some of ambiguities in that formulation. It also presented new double copy interpretations of the Eguchi-Hanson instanton, and the C-metric, relating the latter to the Liénard-Wiechert potential for a pair of uniformly accelerated charges. Extending this four-dimensional result to higher dimensions requires an appropriate study of spinors and curvature invariants in higher dimensions, and the latter has been explored recently in [70].
A higher-dimensional Weyl double copy might also be investigated in terms of a tensor version of the spinor Weyl double copy. One can obtain this by translation from the spinor equations of course; more directly one can note that this must involve writing the Weyl tensor as a sum of terms quadratic in the Maxwell tensor. Keeping in mind the algebraic symmetries of the Weyl tensor, in four dimensions there are two independent expressions that one can write down: where F 2 = F λδ F λδ , F.F = F λδF λδ , andF µν = 1 2 √ g ǫ µνρσ F ρσ with ǫ µνρσ the numerical alternating symbol. In equations like those above, the symbol "| s " applies to the expression on the right-hand side of the equation, and it means to anti-symmetrise in the indices µ, ν and in ρ, σ, with unit weight. In D dimensions there is no equivalent of D µνρσ [F ] and one just has We make some comments on the higher-dimensional double copy at the end of the paper and hereon consider four dimensions. A four-dimensional Weyl tensor double copy must involve a linear sum of the two expressions C µνρσ and D µνρσ , with suitable coefficients which will in general be functions of the relevant variables and constants. We note some useful properties of these expressions below: for any coefficients a, b. Define the self-dual and anti-self-dual parts of a two form F via F ± = 1 2 (F ±F ) with a similar formula for C ± , with the action either on the left or right pair of indices. Then

Ehlers and the double copy: the Schwarzschild case
We would like to explore how the Ehlers transformation described earlier may be understood in terms of the double copy. Consider as starting point the Schwarzschild metric. We will find it useful to work in coordinates (u, v, p, q) with the metric The single copy Maxwell tensor is then [59] It is then straightforward to show that the Weyl tensor C S for the Schwarzschild metric is given by where | e→m means to replace e by m on the right-hand side of the equation. This result may be written simply as where α S = −i(p+iq) e . Now let us consider the Taub-NUT metric in the corresponding coordinate system: where n is the Taub-NUT charge.
The single copy Maxwell tensor in this case is This can be expressed simply in terms of the Schwarzschild Maxwell single copy tensor as Now, if we make the replacements on the right-hand side of (70), and then make the replacements e → m, g → n then we find a tensor that we will call C T which is where α T = −i(p+iq) (e−ig) . It can be checked that C T is the Weyl tensor for the Taub-NUT metric (71). (Note that we implicitly assumed that the charge e is complex prior to the shift, and that in going from (70) to (75) the metric dependence in C µνρσ [F ] also needs to shift from (67) to (71).) Thus we see that the Ehlers transformation which takes one from the Schwarzschild to the Taub-NUT spacetime can be seen via the Weyl double copy as a simple duality transformation (74) (combined with identifying (e, g) with (m, n)) which maps between the two Weyl double copy curvatures. It is instructive to return to the spinor form of the Weyl double copy (61) in the light of this (see Section 4 of [59]). The transformation eF → (e−ig)F induces the shifts eF ± → (e∓ig)F ± . The Maxwell field strength spinor f AB depends only on the self-dual part of the Maxwell tensor and thus transforms according to this formula. The scale function eS in (61) transforms to (e − ig)S and hence the double copy formula yields correctly mapping the Schwarzschild Weyl spinor to the Taub-NUT one.

Type D metrics
The Taub-NUT example considered above is a special case of the general vacuum type D solution with vanishing cosmological constant [71], as given in [59]: with where the parameters m, n, γ, ǫ are related to the angular momentum and acceleration (see [72] for a discussion of the various limits and definitions which enable the identifications in different cases).
The single copy Maxwell tensor in this case is the same as the one for the Taub-NUT metric (71). It is then natural to investigate the Weyl double copy in this case and, indeed, one finds that the Weyl tensor C D for the type D metric (77) is given by the same formula as that for the TN case: . Note that the metric (77) enters the right-hand side of (79) so that this doesn't simply reproduce C T µνρσ . One can then ask if an Ehlers transformation will take one from the spacetime with Type D metric (77) with n = 0, to that with nonzero n. To see this, consider the type D metric g D 0 with vanishing NUT charge. This satisfies Then if we make the replacements F S → F S − ig eF S and e → e − ig in the right-hand side of the above, and shift the metric from g D 0 to (77), we find that we reproduce (79). 5 The sl(2, IR) transformations

The spacetime Ehlers group
We would now like to discuss how sl(2, IR) transformations act more generally in the context of the double copy. For the study of the spacetime Ehlers group we will use the analysis of [73], which we now summarise briefly. 6 Given a Killing vector field ξ = ξ µ ∂ µ and one-form W = W µ dx µ on a Lorentzian manifold with metric g µν , satisfying the vacuum Einstein equations, the spacetime Ehlers group is defined in [73] by the transformation where Ω 2 ≡ ξ µ W µ + 1 ≥ 1, and the inequality holds over the whole geometry. Define the Killing tensor two form F µν = 2∂ [µ ξ ν] (note that we have a factor of 2 here, and a factor of 1/2 in the definitions of the (anti-)self-dual parts of F , in comparison with [73]). Define also the twist potential ω µ = − det(g) ǫ µνσρ ξ ν ∇ σ ξ ρ and the Killing vector norm λ = −ξ µ ξ µ . Then the Ernst one-form is closed, following from the vanishing of the Ricci tensor, and so locally σ µ = ∇ µ σ for some complex function σ.
The spacetime Ehlers group is then defined for W satisfying where a bar denotes complex conjugation, and γ and δ are non-simultaneously vanishing real constants, which as a pair fix the gauge of W . The transformation defines an sl(2, IR) group action on the Ernst scalar by the Möbius map The self-dual part of the Killing tensor transforms as where W, σ are the one-forms defined above. The self-dual part of the Weyl tensor transforms as where (in our conventions) I µνρσ = 1 4 (g µρ g νσ − g νρ g µσ + ǫ µνρσ ) is the canonical metric in the space of self-dual two-forms and P αβ µν = Ω 2 δ α µ δ β ν − δ α µ ξ β W ν − ξ α W µ δ β ν . Notice that in terms of the definition in (62). The Mars-Simons tensor is then defined as for a suitable function Q. Finally, as discussed in Section 6 of [73], we note that the vanishing of the Mars-Simons tensor is maintained under the Ehlers transformation, with Q transforming appropriately.

The Taub-NUT case
Let us now consider applying these arguments in the context of the Weyl double copy described earlier. Consider first the Taub-NUT metric, in the real form where M and N are the mass and the NUT parameter. Consider the Killing vector ξ = ∂ t . Its associated two-form is This solves the Maxwell equations on the Taub-NUT background. The single copy of Taub-NUT was found in [45,59] and solves the flat-background Maxwell equations. The Ernst one-form is obtained from its definition In [73], it was proven that the Ernst one-form is exact, σ µ = ∂ µ σ, and the integration constant can be chosen such that Re(σ) = −ξ µ ξ µ Additionally, the fact that (89) has a Weyl double copy structure implies that with c = 1 and A = −(4(M − iN )) −1 . Next, we find W by solving (83). After this, we have everything we need to transform the original metric into (81) In order to interpret this new metric it is convenient to define polar coordinates in the parameter space Performing a change of coordinates and a charge redefinition the metric simplifies to Hence, it is still a member of the Taub-NUT family. The self dual part of F µν transforms as (85). The integrated Ernst 1-from transforms as After the transformation, (93) also holds with in agreement with (54) in [73].
Let us now study the implications for the single copy. The single copy of (89) can be written in flat spherical coordinates (t,r, θ, φ) [45] Hence, the single copy of the transformed space-time, on the same background reads Using (96) it can be checked that This corresponds to an electromagnetic duality rotation and a rescaling by ρ. The zeroth copy is affected similarly, transforming using M → M ′ , N → N ′ , leaving the double copy structure intact. The Weyl double copy where F + is the self-dual part of the single-copy Killing tensor and σ + = −2(m + in)/(r + in), transforms directly to the double copy in the transformed spacetime where F + is the self-dual part of the single-copy Killing tensor, now defined using the shifted metric (and the same Killing vector, although note of course that the co-vector differs in the new spacetime), and the transformed Ernst scalar We see that in terms of the action on the fields, a restricted set of the SL(2, R) transformations act in this case, and the orbit is within the Taub-NUT class of metrics. The two degrees of freedom are realised by the rotation parameter ζ and scaling ρ.

The Eguchi-Hansen metric
It is of interest to consider a Riemannian metric example and we turn to the Eguchi-Hanson metric with coordinates (u, v, X, Y ) and constant λ. The single-copy gauge potential and (self-dual) Maxwell tensor are [46,59] Consider the Killing vector and Killing two-form The single-copy Maxwell tensor is then given by We have the relations We now consider the equations (83) with σ → σ + ,σ → σ − and ξ the Killing vector (118). These are solved by The new metric is given by (81) with W given by the expression above. This is a complicated expression which we will not reproduce here. The single-copy Maxwell tensor K + µν transforms in the same way as F in (85), and the transformation of its dual is the conjugate of this. It can be checked that the transformed tensors are (anti-)self-dual with respect to the transformed metric (81), and agree with the new Killing two form obtained from (119) using the same Killing vector (118) but with the index lowered with the new metric. To gain some insight into the action of sl(2, IR) in this example, consider the transformations with δ = 0. The shifted Maxwell fields are given by and where ǫ = 1 − 8λγ 2 .
Considering the Weyl tensor, we have the double copy relation for the Eguchi-Hansen metric where K + is the (self-dual) single copy Maxwell tensor (120). The Weyl tensor for the Eguchi-Hansen metric is also self-dual. For the transformed metric in the case of δ = 0 we find that the new Weyl tensor is again self-dual and obeys an analogous double copy relationship where K − ′ µν is the shifted Maxwell tensor given in (125). It would be interesting to follow up with a full study of the action of the transformations when δ = 0.

Higher dimensions
Higher-dimensional double copies via a direct application of the formula (64) appear on first investigation to be limited to the Tangherlini metric in D dimensions. But in many cases of interest there is more structure in higher-dimensions that may play a key role in consideration of double copies. The review [76], for example, summarises work on symmetries of higher-dimensional Kerr-NUT-(A)dS black hole spacetimes. These symmetries are linked with the presence of Killing and Killing-Yano tensors, which feature strongly in the discussion of the special properties of these spacetimes, such as their algebraic type, the integrability of geodesic motion, and separability of the Hamilton-Jacobi, Klein-Gordon, and Dirac equations. The fundamental Killing object in these discussions is the "principal tensor" h, which generates a complete set of explicit and hidden symmetries and uniquely determines the geometry of the Kerr-NUT-(A)dS metric.
The principal tensor, and those other tensors generated by it, might be expected to also play a role in defining tensor double copies for such spacetimes in higher dimensions. If so, we would expect a relationship between any Maxwell tensor (or other two forms to be used in the double copy) and the principal tensor and its descendants. This might first be investigated in four dimensions. In the four-dimensional spacetime discussed in section 4.3 above, with the two null co-vectors the Maxwell single copy gauge field is given by A = 1 p 2 + q 2 g pK + e qL . (129) Its field strength is given by (72) above, i.e. F = 1 (p 2 + q 2 ) 2 α dpK + β dqL , with α = −2epq − e(p 2 − q 2 ), β = −2gpq + e(p 2 − q 2 ). The principal tensor h and its dual for this case are given by [76] h = 1 (1 − pq) 3 p dpK − q dqL , h = 1 (1 − pq) 3 q dpK + p dqL .
One can check that the Maxwell single copy (130) can be expressed in terms of the principal tensor as F = a h + bh , where a = Ω 3 (−3pq(gp + eq) + gq 3 + ep 3 ) and b = Ω 3 (−3pq(ep − gq) + eq 3 − gp 3 ), with Ω = 1−pq p 2 +q 2 . Curiously, the factors in the coefficients a, b (with e → m, g → n) also appear in some of the components of the Weyl tensor. The result (132) implies that this four-dimensional Weyl double copy can be written purely in terms of the principal tensor. It would be interesting to explore higher-dimensional Weyl double copies by seeing how the principal tensor and its descendants, along with the Maxwell tensor where this is independent, might be used in formulae like (64) to map to the Weyl tensor; spinor analogues might also be explored of course.

Discussion
This work examines the double copy formalism through the lens of solution generating symmetries in relativity. The essence of the paper is to see how the zoo of solution generating techniques in general relativity are related to hidden or duality symmetries in the single copy gauge theory. We have used two complementary techniques, the Kerr-Schild and the Weyl double copy formalisms. We hope that this provides more evidence that the double copy is far beyond just a perturbative symmetry for amplitudes but a fascinating relation between gravity and gauge theory.