Kerr-Schild double copy for Kundt spacetimes of any dimension

We show that vacuum type N Kundt spacetimes in an arbitrary dimension admit a Kerr-Schild (KS) double copy. This is mostly done in a coordinate-independent way using the higher-dimensional Newman-Penrose formalism. We also discuss two kinds of non-uniqueness of an electromagnetic ﬁeld corresponding to a given KS metric (i

Recently, there has been a great interest in the notion of double copy, an approach in which gravity can be understood as two copies of a gauge theory.This approach originated from relating perturbative scattering amplitudes in a non-abelian gauge theory and gravity [1,2].Subsequently, it has been extended to relate exact vacuum solutions of Einstein gravity with solutions to the Maxwell equations in a flat spacetime [3].The essential assumption for the original formulation [3] of the double copy for classical GR is that an n-dimensional spacetime metric can be expressed in the so-called Kerr-Schild (KS) form where the "background" metric ds 2 flat = η ab dx a dx b is flat (although not necessarily expressed in the Minkowskian coordinates), the KS covector field ℓ is null (with respect to both metrics η and g), and a, b . . .= 0, . . ., n − 1.
In vacuum (i.e., in the Ricci-flat case, considering the Einstein equations), several results for KS metrics in arbitrary dimension were obtained in [4].In particular, the null congruence defined by ℓ must be geodesic, which will thus be understood from now on.The KS ansatz (1) then restricts possible algebraic types of KS spacetimes.In fact in arbitrary dimension, Ricci-flat KS metrics are Weyl-type 1 D or II when the KS vector ℓ is expanding.When ℓ is non-expanding, the KS congruence becomes Kundt, and the Weyl type is N (or O for a flat spacetime).
Thus clearly, the KS ansatz is compatible only with a subset of solutions to the Einstein equations.However, it includes several spacetimes of great physical importance, such as Schwarzschild and Kerr black holes, their higher-dimensional counterparts [7,8] and type N pp -waves.Note that while in four dimensions all Ricci-flat KS spacetimes are known [9][10][11][12][13], 2 it is not so in higher dimensions.
It has been shown [3] that for stationary KS spacetimes, using an appropriate normalization of the KS covector field ℓ, the vacuum Einstein equations for the metric (1) imply 3 that the Maxwell equations for an electromagnetic field hold in the flat background spacetime ds 2 flat .Thus the electromagnetic field is constructed by multiplying the scalar φ by a single copy of the covector ℓ, while the gravitational field (i.e., the "perturbation" −2φℓ ⊗ ℓ of the flat metric ds 2 flat in (1)) is obtained by multiplying the scalar φ by two copies of ℓ.Recently, it has been pointed out [14] (see also [15]) that in fact the KS double copy of stationary, expanding spacetimes can be traced back to the well-known relation between test Maxwell fields and Killing vector fields [16,17] (at least in four dimensions this applies, more generally, to all expanding KS metrics [14], since they necessarily possess at least one Killing vector field [10] -not necessarily timelike).It is thus interesting to study the KS double copy also in time-dependent spacetimes, 4 for which the method of [16,17] cannot in general be applied.Although the KS double copy has been extended also to certain time-dependent solutions such as pp -waves [3] (see also [18]), works devoted to the KS double copy have focused mostly on stationary Weyl type D KS spacetimes [3,19], since these are of particular relevance in the context of black holes (see also [14] for the discussion of an example of a type II KS spacetime in the context of the double copy).On the other hand, from the viewpoint of scattering amplitudes [1,2], one would be more interested in the role of gravitons and thus in gravitational waves.
In this paper, we will focus on the detailed analysis of the only remaining algebraic type compatible with the KS ansatz (1) -namely, we will study Weyl type N spacetimes of arbitrary dimension in the context of KS double copy.
As mentioned above, results of [4] imply that Ricci-flat KS type N spacetimes are necessarily Kundt.By definition, Kundt spacetimes admit a null geodesic vector field with vanishing optical scalars shear, twist, and expansion. 5In fact, [4] also shows that all type N Ricci-flat Kundt spacetimes are compatible with the KS ansatz (1).Thus in arbitrary dimension for Ricci-flat type N spacetimes, being Kundt and admitting KS metric (1) are equivalent conditions.In contrast with the expanding stationary case, for Kundt spacetimes, the KS double copy cannot be in general traced to electromagnetic test fields constructed from Killing vectors and thus in this case the double copy is a genuinely distinct procedure. 6 Kundt metrics [20,21] have played an important role as exact solutions describing, in particular, plane [22] and pp -waves [23], as well as the gravitational field produced by light-like sources [24][25][26].Higher-dimensional extensions have been studied, e.g., in [27][28][29] (cf.also [6] and references therein). 1The Weyl type is a generalization of the Petrov type to higher dimensions in the algebraic classification of [5] (see also [6] for a review).
2 Some errors in [12] are amended in [13]. 3Note, however, that the Maxwell equations themselves do not guarantee that metric (1) obeys the vacuum Einstein equations, see [3]. 4 More precisely, in settings such that the potential (17) is not gauge-equivalent to a Killing covector. 5For Ricci-flat KS spacetimes, vanishing expansion implies that shear and twist vanish as well and it is thus a sufficient condition for a spacetime being Kundt [4]. 6For example, generic pp -waves admit only one Killing vector ℓ, which is covariantly constant and thus cannot be used to produce a non-trivial test Maxwell field with the method of [16,17] (i.e., F = dA vanishes if A = ℓ).
Type N Ricci-flat Kundt spacetimes consist of two disjoint classes of gravitational waves -Kundt waves and pp -waves.Denoting by ℓ the Kundt vector field, pp -waves can be characterized by the property ℓ a;b = ℓ a p ,b , where p is a scalar function (ℓ is determined up to a rescaling, which can be used to set p = 0), while for Kundt waves one has ℓ a;b = ℓ a p b + q a ℓ b , where p a and q a are covector fields with q a ℓ a = 0 = q a (in this case, the scaling freedom can be used to set, e.g., p a = q a or p a = 0, if desired) -cf.also section 3.2 and [6].
In this paper, we establish that the double copy holds for all type N Ricci-flat Kundt spacetimes in an arbitrary dimension.In order not to be limited by a specific choice of coordinates for Kundt spacetimes, first we approach the problem and express, e.g., the double-copy compatibility conditions using the higher-dimensional Newman-Penrose (NP) formalism [30][31][32], resorting to coordinates only at a later stage.
We also study the non-uniqueness of the single-copy electromagnetic field corresponding to a given KS metric.Two distinct reasons for this non-uniqueness are: i) the possibility of rescaling the KS vector ℓ (with a simultaneous appropriate rescaling of the KS function φ); ii) the non-uniqueness of the splitting of the KS metric in the flat part and the KS part of the metric.The first nonuniqueness i) leads to the infinite non-uniqueness of an electromagnetic field arising as a single copy from a given KS Kundt metric.As a consequence of ii), a single copy of a pp -wave can be either a null or a non-null electromagnetic field.
The KS double copy has been considered also in the context of nonlinear electrodynamics, especially as a tool for constructing regular black holes [33][34][35].It is therefore interesting to point out that the KS-Kundt double copy demonstrated in the present paper also applies to higher-order theories (except for the non-null fields of section 5.2).Namely, Ricci-flat Kundt metrics of Weyl type N also solve Lagrangian theories L = L(g ab , R abcd , ∇ a1 R bcde , . . ., ∇ a1...ap R bcde ) constructed from arbitrary powers of the Riemann tensor and its covariant derivatives and are thus "universal" [36] (earlier results in special cases include [37][38][39][40][41]).Similarly, their single-copied null electromagnetic fields are immune to corrections expressed in terms of arbitrary powers and derivatives of the field strength [42][43][44] (see also the earlier works [37,38,45,46]). 7However, note that higher-order theories may admit also additional solutions on top of the universal ones (e.g., non-Ricci-flat metrics or non-Maxwellian electromagnetic fields), for which a possible double copy would need to be studied on a case-by-case basis.
It is worth mentioning that, after the KS approach of [3], a different formulation of the classical double copy has been put forward in [48] (see also the early results [49,50]), namely the Weyl double copy.The latter has been established for all twistfree type N vacua in four dimensions in [51], thus in particular for Kundt metrics of type N (see also [48] for the special case of pp -waves).A natural question to ask is whether this result extends to higher dimensions.However, since the set-up of [48] relies heavily on using 2-spinors in four-dimensional spacetime, this issue will deserve further investigation (see [52] for some results in this direction).
This paper is organized as follows.Section 2 contains some preliminaries.In section 3, we express the Einstein and Maxwell equations in a coordinate-free manner using the higher-dimensional NP formalism, derive the double-copy compatibility conditions, and discuss a special gauge simplifying these equations.Section 4 establishes the double copy for Kundt waves.The main part of the proof is presented using the NP formalism, however, a result concerning the existence of a certain preferred null frame needed for the proof in the NP approach is obtained using the Kundt coordinates.Section 5 is devoted to pp -waves.In particular, the existence of both null and non-null single copies is shown using the NP formalism.For completeness, a discussion in the Kundt coordinates is also included.Finally, section 6 returns to the discussion of the non-uniqueness of the single copy encountered already, e.g., in section 5.
Various auxiliary results are presented in the appendices.Appendix A contains a concise summary of the higher-dimensional Newman-Penrose formalism used throughout the paper.Appendix B presents the Ricci rotation coefficients and the curvature for KS-Kundt spacetimes.Appendix C extends the useful coordinate system and frame defined for n = 4 KS-Kundt spacetimes in [12,13] to arbitrary dimension, and further discusses their role in connection to the general analysis of sections 4 and 5. Finally, in Appendix D we illustrate the above mentioned "universal" character of the KS-Kundt double copy using specific examples of modified theories of gravity and electromagnetism.
Since ℓ is geodesic (cf.above), without loss of generality we may take it to be affinely parametrized (which is equivalent to setting L 10 to zero by a boost, see (11) and (A1)) with an affine parameter r such that Then employing a frame parallelly transported along ℓ, one has This leaves the freedom of spins (A3), null rotation about ℓ (A2) and boosts (11) (cf.[5]), provided all the transformation parameters are constant along ℓ.
The optical matrix is defined as see, e.g., [6,30].In particular, the null congruence defined by ℓ is expansionfree, twistfree, and shearfree precisely when L ij = 0, which characterizes the Kundt class of spacetimes.Covariant derivatives along the frame vectors are denoted as so that and their commutators are given in (A4)-(A7).
The box operator (acting on scalar functions) can be written as where we used L 10 = 0 = N i0 (cf.( 6)) and, in the second equality, the commutator (A4).
For the KS metric (1), the null vector ℓ is not unique since it can be rescaled together with the metric function φ without changing the metric (φℓ a ℓ b = φl a lb , where la = λℓ a , φ = λ −2 φ).The rescaling of ℓ corresponds to a boost with λ being a real function.Ricci rotation coefficients transform under the boost (11) according to (A1), and we need to require Dλ = 0 if we want to preserve the condition L 10 = 0.
Throughout the paper, we will consider only KS spacetimes for which the KS vector field ℓ is Kundt, i.e., from now on we assume along with (6).Under (12), the operator (10) can thus be rewritten compactly as for an arbitrary function f , where ˜ is the box operator associated to the flat metric η. (Eq.( 13) follows readily from the comments in appendix B, cf.(B1)-(B3).)In particular, on a function f which satisfies Df = 0, one clearly has f = ˜ f , which will be useful for later purposes.Cf. also [12] in four dimensions.

Field equations and double-copy compatibility conditions
In this section, we work out the Einstein and Maxwell equations for KS-Kundt solutions in the Newman-Penrose formalism, as well as their compatibility conditions in the set-up of the double copy (as described above in section 1).We also briefly discuss a non-uniqueness of the KS vector field related to boost freedom (further comments will be given in section 6), and use it to identify a particular gauge that will be convenient for later purposes.These results will be employed in the subsequent sections 4 and 5.

Field equations
In the parallelly transported frame defined above obeying ( 6) and ( 12), the components of the vacuum Einstein equations R ab = 0 read (cf.(B7)-(B9)) For the Maxwell field F = dA we take the ansatz where α is a spacetime function.The non-zero components of F thus are The field F is null (cf., e.g., [31,42,53]) iff Dα = 0.
Recalling that the quantities L i1 are invariant under boosts (while the L 1i are not, cf.(A1)), it will be useful to study separately the two invariantly defined cases of "Kundt waves", defined by L i1 = 0, and of "pp -waves", corresponding to L i1 = 0 (cf.also [6] and references therein) in sections 4 and 5, respectively.

Boost freedom
Before proceeding, it is useful to comment on the gauge freedom mentioned in sections 1 and 2. Namely, one can perform a boost (11) of the vector ℓ with Dλ = 0 and a simultaneous rescaling of φ in such a way that the KS metric remains unchanged, along with the conditions (6), see (A1).Naturally, the Einstein equations ( 14)-( 16) are invariant under this boost-rescaling transformation.In contrast, the Maxwell equations ( 20), (21) are not invariant under this transformation (whilst (19) is).The electromagnetic potential changes as A a = φℓ a → Âa = φl a = λ −1 A a .Demanding that both A and Â obey the Maxwell equations leads to a differential condition containing λ and its (up to 2nd) derivatives (cf.section 6.1).Therefore, in principle, distinct electromagnetic fields could be related to a given KS metric via the KS double copy, see the discussion in section 6. See also [57] for related comments.

Gauge L 1i = 0
Here we will use the boost freedom described above to identify a particularly convenient gauge.Namely, given an affinely parametrized Kundt vector field ℓ, one can always use a boost (11) with Dλ = 0 to set L1i = 0, and thus also δ i r = 0, cf.(23) (the Ricci identity for δ [j| L 1|i] (A8) and the commutator (A7) ensure that the necessary integrability conditions are satisfied; one should also redefine r = λ −1 r in order to have l = ∂ r ).This can be done at once both for the background spacetime (i.e., for φ = 0) and for the full metric [4] (cf. (B2)).There still remains residual boost freedom with Dλ = 0 = δ i λ.In the rest of this section, this boosted frame will be understood and hats above boosted quantities will thus be dropped.
With the gauge choice L 1i = 0, the compatibility conditions (30)-( 32) reduce to while the Einstein equations ( 24) and ( 25) become This gauge will be chosen in sections 4 and 5.1 below.
4 Case L i1 = 0: higher-dimensional Kundt waves In this section, we prove the KS double copy for Kundt waves in arbitrary dimension.First, we present a general, coordinate-independent analysis in the Newman-Penrose formalism (section 4.1), building on the set-up of section 3. We next give a more explicit demonstration of how the double copy is realized in the canonical Kundt coordinates of [20,21,58] (section 4.2).

Proof of the KS double copy for all Ricci-flat Kundt waves
It follows from the compatibility conditions (see (30)) and the comments given at the end of section 3.2, that here The compatibility conditions ( 30)-( 32) thus reduce to while the Einstein equations ( 24), ( 25) become simply Using (38), the latter can be rewritten as From now on we choose the gauge (cf.section 3.3) (This can be done without affecting (37).)Hence the compatibility eq. ( 38) becomes an identity and the Einstein eq. ( 39) reduces to Since the compatibility conditions are now satisfied identically, this demonstrates the double copy for Kundt waves with φ (1) = 0 (eq.( 37)).Recall that this condition followed from the Einstein equations together with the compatibility conditions and the Kundt waves property L i1 = 0.
It thus remains to clarify whether the KS double copy holds for general type N vacuum Kundt waves (that could have, in principle, φ (1) = 0).Indeed, one can argue that φ (1) does not contribute to the spacetime curvature and can thus always be reabsorbed into the η part of the metric or transformed away by a coordinate transformation.This is illustrated in section 4.2.3 (cf.eq. ( 68)) and appendix C (eqs.(C32), (C33)).Therefore, we conclude that the KS double copy holds for all Ricci flat type N Kundt waves.An independent, coordinate-based proof will be given in section 4.2.1.
Let us further note that the residual boost freedom with Dλ = 0 = δ i λ mentioned in section 3.3.2gives rise to a special instance of the case 1. of the non-uniqueness discussed in section 6.1.

Kundt waves in Kundt coordinates
This section focuses on analyzing the KS double copy for Kundt waves using the canonical Kundt coordinates.For some applications, this may be a useful addition to the discussion in Newman-Penrose formalism given above.

Line-element and double copy
All Weyl type N Ricci-flat Kundt metrics belong to the so-called VSI (vanishing scalar invariants) class of spacetimes [31] admitting a metric of the form [58] where i, j = 2, . . ., n − 1 and The canonical Kundt covector field is given by ℓ a dx a = du, for which ℓ a;b = ǫv y 2 + H (1) du 2 − ǫ y (dudy + dydu) [42,58].Consider a null electromagnetic field F = dA with Then the Maxwell equations in any spacetime (42) reduce to [42] −detg where △ stands for the Euclidean Laplace operator in the (n − 2)-dimensional flat space spanned by the coordinates x i (not to be confused with the Newman-Penrose symbol ∆ defined in ( 8)). 10he VSI metrics ( 42)-( 44) are in general of Weyl and Ricci type III and thus contain also some non-KS spacetimes.However, here we are interested in Kundt waves, i.e., in the Weyl type N, Ricciflat subclass of metrics ( 42)- (44) with L i1 = 0.This is given by the family ǫ = 1 for which, after using some of the Einstein equations and coordinate freedom, eqs.(43) and (44) simplify to [58] (see also [27]) where C m and B nm = −B mn are arbitrary functions of u (in the special case n = 4, one thus has B nm = 0, corresponding to the canonical form of [20]).
The only non-trivial component of the Einstein vacuum equations reads The metric ( 42) with ( 47)-( 49) is flat for H given in (49) with Now defining φ (0) the metric takes the KS form where flat solves (50), the Einstein equation ( 50) reduces to △ψ = 0.
Now rewriting the metric (53) in the form where φ = ψ/y and l = ydu and taking potential A = φl = ψdu (45) (A u = ψ = A v ), the Maxwell equations ( 46) both in the curved ( 47)-( 49) and flat backgrounds, are equivalent to the Einstein equations ( 54), which establishes the KS double copy for higherdimensional Kundt waves.An equivalent observation in four dimensions was made some time ago [20,59].
Since the standard natural null frame for Kundt metrics (42) [6,58], is not, for Kundt waves, parallelly transported along ℓ (since it gives N i0 = 0), let us conclude this section by presenting two frames that are parallelly propagated along ℓ.This will enable us to make contact with the analysis of section 4.1.
To conclude this section, let us observe that, without loss of generality, one can in fact set W m = 0 in (42) and (48): C m can be set to zero by redefining x m → x m +f m (u) with f m , u = −C m , while B nm can be set to zero with a rotation x m → R m n (u)x n , δ pq R p m R q n = δ mn with R p m,u = B pn R n m .This observation seems to have been overlooked in the literature so far.Under such a coordinate fixing, formulae (50), (51), and (58) simplify accordingly.

Parallelly transported frame with
By a null rotation (A2) (with z 2 = v/y, z m = 0) of the frame (57)-( 59), one obtains a parallelly transported frame (i.e., for which Since L 12 = 0 this frame clearly does not correspond to the gauge L 1i = 0 employed in section 3.3.2,which we implement in the next step.

Parallelly transported frame with
By boosting (eq.( 11) with λ = y) the above frame ( 60), (61), one obtains a parallelly transported frame for which The existence of the frame ( 63)-( 64) has been employed in section 4.1 (for four-dimensional Kundt waves, a parallelly transported frame with L 1i = 0 was also constructed in [13] using different coordinates, see appendix C).
A further null rotation with z 2 = 0, z m = −W m /y accompanied by the coordinate transformation brings the derivative operators to the form (dropping hats on the operators in the new frame) in agreement with (55).This gives an explicit illustration of the parallelly transported frame introduced in a coordinate-independent way in section 4.1 and proves that for all Kundt waves, one can set φ (1) = 0 while using such a frame.
5 Case L i1 = 0: higher-dimensional pp -waves )), i.e., ℓ is a recurrent vector field [6,60].It is not difficult to see that ℓ is, in fact, always proportional to a covariantly constant null vector field 12 and therefore the corresponding spacetime is a pp -wave.In contrast to the case of Kundt waves of section 4, now two cases need to be studied separately, depending on whether φ (1) vanishes or not.

Case φ (1) = 0: null electromagnetic field
Here the compatibility conditions ( 30)-( 32) become a single equation and the Einstein equations ( 24), ( 25) reduce to 5.1.1Gauge L 1i = 0 Using the gauge L 1i = 0 (see section 3.3), eq. ( 69) is satisfied identically and eq.( 70) gives simply which demonstrates the null field double copy for all pp -waves (see also section 5.3 for a coordinate approach and appendix C; in particular, the arguments given at the end of section 4.1 about the generality of the condition φ (1) = 0 apply also here).
5.2 Case φ (1) = 0: non-null electromagnetic field In this section, we consider the case φ (1) = 0 in (22).Using ( 9), we see that for the electromagnetic tensor F ab = A b , a −A b , a , where A a = φℓ a , the boost-weight zero component F 01 = Dφ = φ (1) = 0 and thus F ab is non-null, cf. ( 18), (22), and ( 27).The compatibility conditions ( 30)-( 32) reduce to while the Einstein equations ( 24), ( 25) become Provided ( 72) is fulfilled, then the Einstein and Maxwell equations are equivalent.Since we already have Dφ (1) = 0, it follows from the second of ( 72) and ( 73) that φ (1) must be a (nonzero) constant.It is proven in sections 5.3, 6.2, and appendix C that, for all vacuum pp -waves of type N, one can indeed find a frame such that (72) and φ (1) = const = 0 are both satisfied (while maintaining our earlier "gauge fixings" ( 6) and (23); note that here the first of ( 72) is not a gauge choice but a consequence of the compatibility condition (30)).This proves the non-null-field KS double copy for pp -waves (to our knowledge, previous literature has considered only the case of null fields).An explicit example of a pp -wave metric and the corresponding non-null electromagnetic field in arbitrary dimension, using the Kundt coordinates, is given in section 6.2.

pp -waves in Kundt coordinates
The canonical line-element in Kundt coordinates for Ricci-flat pp -waves of Weyl type N is given by ( 42) with ǫ = 0 and [23,27,58,64] 13 which gives The remaining Einstein vacuum equation reads This metric is flat for [58] where G and F i are arbitrary functions of u (they can be set to zero by a transformation [58,60], if desired).Now defining the metric takes the KS form The null vector ℓ = du is covariantly constant, ∇ a ℓ b = 0. (Here ℓ a ∂ a = ∂ v and L 1i = 0 = L i1 in any frame adapted to ℓ, so in this case the coordinate v can be identified with the affine parameter r (5) employed in sections 3, 5.1 and 5.2).H 0 flat is a solution of (78) and thus the Einstein equation ( 78) reduces to Thus identifying φ (0) with α in ( 45), the Maxwell equations (both in flat and curved backgrounds) (46) are equivalent to the Einstein equations ( 82).This establishes the null-field KS double copy in the coordinate form.The same conclusion also follows from the test-field limit of the results of [69] (cf.again [20,59] for related results in four dimensions).
The null field KS double copy in the context of higher-dimensional pp-waves has been already discussed in [3] in the coordinates (77) with a fixed gauge ℓ = du, in which ℓ is covariantly constant.
One can employ the coordinate transformation (90) to generate a non-vanishing H (1) = H (1) (u), see (91).Since H (1) does not enter the field equations one can either identify it as a part of the flat background H flat , see section 5.1, or as a part of the KS function as in section 5.2.The latter case is related to the non-null field KS double copy, see also section 6.2.

Non-uniqueness of the KS double copy
In section 5, we have found that one gravitational field can have multiple distinct electromagnetic single copies.This was limited to pp -waves.Now let us discuss the non-uniqueness of electromagnetic single copies in more detail and generality.
There are two distinct sources of non-uniqueness: 1.The first source of the non-uniqueness is tied to the non-uniqueness of the KS vector field ℓ under boosts (11) discussed in sections 2 and 3.While under the boost and rescaling transformation φℓ a ℓ b = φl a lb , where the KS metric (1) remains unchanged, the corresponding potential transforms as Distinct gauge fields can thus, in principle, correspond to the same metric via different choices of ℓ (but all defining the same null direction).If F is null then F is null too, as long as Dλ = 0. We shall show below (section 6.1) that this kind of non-uniqueness applies to all Kundt and pp -waves, giving rise, in both cases, to a continuous infinity of (null-field) single copies.
2. The second source of the non-uniqueness is tied to the non-uniqueness of the splitting of the KS metric in the flat part and KS part of the metric (1).In some cases, the KS metric (1) is flat not only for φ = 0 but also for some non-trivial φ = φ and then the term φℓ ⊗ ℓ can be either absorbed in the flat part of the metric ds 2 flat or left in the KS part φℓ ⊗ ℓ (see, e.g., [13,70] for related comments in four dimensions).In general, this can lead to distinct electromagnetic fields A = φℓ, even if the same ℓ is used in all cases.In particular, the resulting field strength can be both null and non-null, depending on the chosen splitting.Below in section 6.2, we will illustrate this type of non-uniqueness in the context of pp -waves and establish its connection to the non-null-field single copy discussed in section 5.2.

Case 1. -infinite non-uniqueness of the single copy
In sections 2, 3.3.1,and 4.1, we pointed out a boost freedom with Dλ = 0 in the definition of the KS vector field ℓ and its corresponding metric function φ in (1), which implies a non-uniqueness of the single-copy field in the double-copy procedure.Let us now show in detail how this non-uniqueness arises for both pp -waves and Kundt waves.
Using (85), eq. ( 87) can be rewritten as Provided λ satisfies (88), the boost produces a non-equivalent single copy F = d Â (w.r.t.l = λdu) of the same KS metric.In the special case λ = λ(u), 15 eq.( 86) reduces to Fiu = Fiu λ = φ,i λ , and F ab ;b = 0 implies F ab ;b = 0 (see [70] for related comments in four dimensions).Thus in this case, two null electromagnetic fields related by a rescaling by any function λ(u) can represent two different single copies of the same gravitational field.For example, when the u-dependence of φ is factorized, this freedom can be used to make F u-independent and thus clearly physically distinct from F .
From a complementary viewpoint, assume φ 1 and φ 2 are harmonic functions.Let A = φ 1 ℓ = φ 1 du be a single copy of the metric (81) (with ) is also a single copy corresponding to the same KS metric (see (87) with φ → φ 1 ).Since the previous observation applies to any solutions of the Laplace equation (except at spacetime points where φ 2 = 0), this also means that any type N pp -wave spacetime (81) corresponds via a double copy to any aligned null electromagnetic field and vice versa.

Acknowledgments
This work has been supported by research plan RVO: 67985840.
A Formulas of the higher-dimensional Newman-Penrose formalism employed in this paper Here, let us complement section 2 by giving a concise summary of the higher-dimensional Newman-Penrose formalism as needed throughout the paper.
with a corresponding relation for the frame vectors (in particular n a = ña + φℓ a and thus n a = ña − φℓ a ).In the special case of a scalar function f which satisfies Df = 0, one thus has ∆f = ∆f .The (possibly) non-zero Ricci rotation coefficients (3) for a KS-Kundt geometry (1), (12) where tilded quantities refer to the flat background geometry η (defined by φ = 0) evaluated in the tilded frame.Thanks to (12), the only non-zero components of the Riemann tensor are [4] R 0101 = D 2 φ, (B4) It follows that the Ricci tensor is given by [4] R 01 = −D 2 φ, (B7) (The Ricci identity [32] has also been used in (B9).)We observe that the Riemann and Ricci tensors are linear in φ (see [71,72] for a similar observation for the mixed coordinate components R a b of general KS spacetimes).

C Debney coordinates for all KS-Kundt metrics, and double copy
Here we show how to extend to arbitrary dimension the n = 4 coordinates and frame of [12,13], in connection to the general analysis of sections 4 and 5.

C.1 Cartesian coordinates and adapted frame
In Cartesian coordinates, the KS line-element (1) can we written as (cf.[10,12,13] for n = 4) where represents the most general (up to a boost (11)) null vector field, and φ, Y i are spacetime functions (for notational convenience we identify Y i = Y i , and similarly below for the functions X i = X i defined in (C26)).
Along with ℓ, a null frame (as defined in section 2) can be completed by taking The derivative operators then read For later use, let us observe that δ i r = 0.
In the above frame one finds (cf.[12,13] in four dimensions and [73] in any dimensions 17 ) Requiring ℓ to be Kundt means (cf.( 6), ( 12)) DY i = 0 = δ j Y i , such that The above Ricci rotation coefficients then simplify to We note that, by construction, the frame in use is parallelly transported and ℓ is affinely parametrized.The Riemann (B4)-(B6) and Ricci (B7)-(B9) tensors then reduce to where in R 11 we used the second of (A8).So far we have not imposed any field equations and the KS-Kundt metric (C11) (with (C18)) is, at this stage, of Weyl (and Riemann) type II.

C.2 Debney coordinates, vacuum solutions and double copy
Defining new coordinates (u, r = r, y i ) (cf. [12,13]) the basis of 1-forms (C12), (C13) takes the form and the derivative operators read simply In particular, r is the same coordinate as defined in (5).Note that in these coordinates Y i = Y i (u), dY i = X i ℓ a dx a and thus X i = 1 + X j y j Y i,u , which also gives 1 + X j y j = 1 − Y j,u y j −1 .We further have δ i r = 0, as in the set-up of section 3.3.2.There remains a freedom of boosts (11) with λ = λ(u) and spins (A3) with X i j = X i j (u), which both preserve (C19), (C20), except for L11 = λ −1 L 11 + λ −2 ∆λ (cf.(A1)).Since L i1 = X i is invariant under such transformations, ℓ can be (parallel to) a covariantly constant null vector field iff X i = 0 ⇔ Y i,u = 0.
By (C23), in vacuum one obtains φ = rφ (1) + φ (0) , (C28) with where f (u) is an integration function.Recall that the first of (C30) is equivalent to the wave equation for φ (0) in flat space (section 2).The only non-zero components of the curvature tensor become such that the Weyl type is N (as expected from the general results mentioned in section 1).Since the function φ (1) does not enter the curvature (C31), the most general vacuum KS-Kundt line element (C11) can be rewritten as ds 2 flat = 2 1 − Y j,u y j dudr + dy i + rY i ,u du dy i + rY i ,u du − 2rφ (1) ℓ ⊗ ℓ, where φ (1) and φ (0) obey (C29), (C30).The function φ (1) has been effectively absorbed into the background part of the metric and one can thus apply the (null field) double copy as described in section 4.1 for X i = 0 and in section 5.1 for X i = 0.As mentioned above, recall that ℓ can be boosted with an arbitrary λ(u), which corresponds to the case 1. of the non-uniqueness discussed in section 6.1 (see also section 4.1).The case X i = 0 corresponds to vacuum pp -waves (i.e., Y i,u = 0, cf.above), for which (C29), (C30) reduce to ,y i y i = 0. (C34) For certain applications, it is useful to recall that f (u) can be rescaled to an arbitrary constant (including zero) with a transformation of the type (90).Here ℓ = du and L 11 = −f (u), which can be set to zero by a suitable boost, thus obtaining a covariantly constant l.It is worth recalling that for pp -waves the double copy can additionally be applied also to the alternative form of the metric (with (C34) and f =const) ds 2 flat = 2dudr + dy i dy i , (C36) as shown in section 5.2.This gives rise to a non-null gauge field and corresponds to the case 2. of the non-uniqueness discussed in section 6.2.
To conclude, let us present the relation between the general (type N) Kundt coordinates constructed above and the particular canonical forms considered separately for Kundt and pp -waves in sections 4.2 and 5.3.

C.3.1 Kundt waves (Y i,u = 0)
The canonical form of ( 47)-( 49) of [58] can be obtained from (C40), (C41) (with (C29), (C30)) by performing a spatial rotation y i = R i j (u)ỹ j such that R i j Y i,u = A(u)δ 2,j (R i j is an orthogonal matrix and A 2 = Y i,u Y i,u ), followed by the redefinitions ỹ2 = x + A −1 (such that P = −Ax), v → v + g(u, x, ỹm ) (to get rid of a v-independent term in W 2 ; m = 3, . . ., n − 1) and finally a transformation of the form u → h(u), v → v/ ḣ (to get rid of a term proportional to v in H) [58,60].

D Double copy in modified theories: an example
Here we illustrate the "universal" character of the KS-Kundt double copy mentioned in section 1 using a specific modified theory as an example.On the gravity side, let us consider general relativity with an additional Gauss-Bonnet term, described by the action where κ and γ are a constant parameters.This gives rise to the gravity field equations (D44) Since the Kundt waves (42) (with ( 47)-( 50)) and the pp -waves ( 77), (78) are Ricci-flat spacetimes of Weyl type N, the Einstein and the Gauss-Bonnet terms of (D44) vanish separately, which implies that both such metrics are vacuum solutions of the theory (D43) [56,74,75] (see also [36] for more general results).
As for the electromagnetic theory, let us modify the standard Maxwellian action by an additional "F 4 " term (cf., e.g., [76,77] and references therein), namely where c 1 and c 2 are constants.This theory has non-linear equations of motion of the form The 2-form F = dA with (45) obeying ( 46) is a null field solution of Maxwell's equations, which implies that ∇ b F ab = 0, F cd F cd = 0 and F ac F cd F db = 0 separately (for the latter conclusion see [42] and Proposition 2.4 of [44]).Therefore F is also a solution of the modified theory (D45) (see also [42][43][44] for more general results).The fact that such F is a single copy of the metrics (42) (with ( 47)-( 50)) and ( 77), (78) follows as in sections 4.2.1 and 5.3.Note that, in the spirit of the KS double copy [3], the electromagnetic field has been treated as a test field above.However, the same arguments can be extended to include back-reaction, i.e., the coupled theory S G + S M (cf.Theorem 3.3 of [66]).