In this paper we generalize the Weyl double copy to fivedimensional spacetime. We show that a special class of fivedimensional type N vacuum solutions admits a special class of degenerate Maxwell field that squares to give the Weyl tensor. The fivedimensional Weyl double copy relation defines a scalar field that satisfies the sourcefree KleinGordon equation on the curved background.
['High Energy Physics - Theory', 'General Relativity and Quantum Cosmology']
Five dimensional Weyl double copy
{'Weicheng Zhao a 1 , Pu-Jian Mao a , and Jun-Bao Wu a,b': 'a Center for Joint Quantum Studies and Department of Physics,  \nSchool of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China  \nb Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026, China  \nEmails: zhaoweichengok@tju.edu.cn, pjmao@tju.edu.cn, junbao.wu@tju.edu.cn  \nABSTRACT: In this paper, we generalize the Weyl double copy to fivedimensional spacetime. We show that a special class of five-dimensional type N vacuum solutions admits a special class of degenerate Maxwell field that squares to give the Weyl tensor. The five-dimensional Weyl double copy relation defines a scalar field that satisfies the source-free Klein-Gordon equation on the curved background.', '1 Introduction': "Duality is a very fascinating phenomenon in fundamental physics. In particular, the connection between gauge and gravity theories is at the center of attention of many researchers. The double copy relation is the most remarkable realization of the gauge/gravity duality, which is inspired by the Kawai-Lewellen-Tye relation [1] and formulated by the color/kinematics duality [2, 3] of the scattering amplitude at the quantum level. The double copy perspective offers a completely different way to interpret gravity from the gauge theory side and vice versa. The success of the double copy relation has also motivated investigations at the classical level connecting classical solutions of gauge and gravity theories which pioneers a classical double copy relation from the Kerr-Schild construction [4].  \nIn this paper, we will concentrate on the second exact classical double copy, namely the Weyl double copy (WDC) [5], which connects the Weyl tensor of the gravity theory  \nand the field strength tensor of the Maxwell theory. The gauge invariant nature of the WDC makes it very efficient to reveal the mathematical connections of the two theories and it has drawn intensive attentions from various perspectives [6-24]. However, the investigations of the WDC relation are now restricted in four-dimensional spacetime. 2 It is somehow mysterious to construct a WDC relation in higher dimensions considering its twistorial foundation [8, 10]. In contrast, the amplitude double copy relation does not rely on the dimension of the spacetime [2, 3] and the Kerr-Schild construction also holds for generic dimensions, see, e.g., concrete examples of the Kerr-Schild double copy in dimensions other than four [27-33]. Hence, there is definitely an urgent need for new perspectives on how to generalize the Weyl double copy to higher dimensions, which is the primary goal of this work.  \nIn this paper, we initialize the study of a five-dimensional Weyl double copy relation. The construction of the four-dimensional WDC formula constricts that it is only valid for two classes of algebraically special spacetime, namely the type D and type N spacetime. Westart from the simplest type in five dimensions, the type N vacuum solution of Einstein gravity in the Coley-Milson-Pravda-Pravdova (CMPP) classification [34-37] in the null frame. The spinor formalism in five dimensions [38] naturally provides an algebraic construction of a possible five-dimensional WDC formula for the type N solution. However, it is very difficult to prove that the Weyl tensor of any type N solution admits a decomposition to the square of a degenerate Maxwell field (solution of the source-free Maxwell's equation) and some scalar fields that satisfy the wave equation either on the curved background or on Minkowski spacetime. Nevertheless, we find a self-contained reduction of the type N Weyl tensor and the degenerate Maxwell field, where one can confirm that any solution of this special class admits a special class of degenerate Maxwell field that squares to give the Weyl tensor. Moreover, a complex scalar field is defined from this fivedimensional WDC relation and the scalar field satisfies the wave equation on the curved background. Hence, a concrete WDC relation is uncovered in five dimensions. We then present the WDC relation for a special class of the five-dimensional pp-wave and Kundt solutions.", '2 Null frame and CMPP classification': "In four-dimensional spacetime, the Newman-Penrose formalism [39] is very effective for addressing problems in General Relativity. In higher-dimensional spacetime, it is of convenience to construct a similar system, namely the null frame. The basis vectors are  \ntypically chosen as  \nl ' e 0 ' e 1 , n ' e 1 ' e 0 , m i ' e i ' m i ' e i , i ' 2 , ..., n ' 1 , (1)  \nwith two null vectors l and n , and n ' 2 spacelike vectors m i . They are required to satisfy the orthogonal conditions  \nl l ' n n ' l m i ' n m i ' 0 , (2)  \nand the normalization conditions  \nl n ' 1 , m i m j ' δ ij . (3)  \nThe spin coefficients are determined from the following defined quantities  \nL µν ' ∇ ν l µ , N µν ' ∇ ν n µ , M k µν ' ∇ ν m k µ . (4)  \nBased on the above null frame, CMPP [34-37] proposed an algebraic classification method for tensors. Similar to the four-dimensional case, the CMPP classification is based on the maximum boost order of tensors. For any tensor T µν...ρ , it can be projected onto the chosen null frame, and the boost weight of the null frame component T ab...c ' e µ a e ν b ...e ρ c T µν...ρ is defined as the number of the index 0 minus the number of the index 1 in the null frame component indices. The maximum boost weight of all non-zero null frame components is called as the boost order of the tensor T . It could be proved that the boost order of a tensor depends only on the choice of null direction l . The essence of the CMPP classification is to choose a null vector l to minimizes the maximum boost weight and another null vector n that maximizes the minimum boost weight.  \nIn general, the Weyl tensor could be decomposed as  \nC µνρσ ' 4 C 0 i 0 j n t µ m i ν n ρ m j σ u boost weight ' 2 ' 8 C 010 i n t µ l ν n ρ m i σ u ' 4 C 0 ijk n t µ m i ν m j ρ m k σ ' 1  \n' 8 C 101 i l t µ n ν l ρ m i σ u ' 4 C 1 ijk l t µ m i ν m j ρ m k σ u l jh n ' 1 ' 4 C 1 i 1 j l t µ m i ν l ρ m j σ u l jh n ' 2 . (5)  \nl jh n u l jh n ' 4 C 0101 n t µ l ν n ρ l σ u ' 4 C 01 ij n t µ l ν m i ρ m j σ u ' 8 C 0 i 1 j n t µ m i ν l ρ m j σ u ' C ijkl m i t µ m j ν m k ρ m l σ u l jh n 0  \nwhere the notation C µνρσ ' C t µνρσ u ' 1 2 p C r µν sr ρσ s ' C r ρσ sr µν s q should be understood. Specifically, a type N spacetime in the CMPP classification is defined when its Weyl tensor have the following form  \nC µνρσ ' 4 C 1 i 1 j l t µ m i ν l ρ m j σ u ' 8 ψ ij l t µ m i ν l ρ m j σ u . (6)  \nIn the null frame, the Maxwell tensor can be decomposed as  \nF µν ' F 0 i p n µ m i ν ' n ν m i µ q ' F ij m i µ m j ν ' F 01 p n µ l ν ' l µ n µ q ' F 1 i p l µ m i ν ' l ν m i µ q , (7)  \nwhere the boost weights of the right hand side terms are 1, 0, 0, -1, respectively. A degenerate (type N) Maxwell tensor in the CMPP classification should have the following form  \nF µν ' F 1 i p l µ m i ν ' l ν m i µ q ' φ i p l µ m i ν ' l ν m i µ q . (8)", '3 5d WDC relation for type N solutions': "The WDC relation is better appriciated in spinorial formalism [5,7]. In four-dimensional spacetime, the WDC relation is interpreted by the decomposition of the Weyl spinor  \nψ ABCD ' 3 c S φ p AB φ CD q , (9)  \nwhere φ AB is a totally symmetric 2-spinor defined from the Maxwell tensor F µν and S is a scalar field that solves the wave equation. The WDC relation (9) can be naturally extended to higher dimensional theory with a spinorial description. Though, the spinorial formalism and the classification of spinors have been defined in 5d spacetime [38], the Bianchi identity of the Weyl tensor and the Maxwell equation are not known in spinorial formalism in 5d. So we need to convert the WDC relation in null frame to examine the 5d WDC relation.  \nA key relation in 4d to convert relations in spinorial formalism to tetrad formalism is the decomposition of the 2d Levi-Civita tensor  \n/epsilon1 AB ' o A ι B ' o B ι A , /epsilon1 A 1 B 1 ' ¯ o A 1 ¯ ι B 1 ' ¯ o B 1 ¯ ι A 1 . (10)  \nIn 5d spinorial formalism, a similar 2d tensor Ω AB is defined to raise or lower the spinor index [38]. One can show that Ω AB can be decomposed as  \nΩ AB ' p k Aa n Bb ' k Ba n Bb q /epsilon1 ab ' k Aa n a B ' k Ba n a A , Ω AB ' p k A a n B b ' k B a n A a q /epsilon1 ab ' k A a n Ba ' k B a n Aa , (11)  \nwhere we would refer to [38] for the notations and definitions. One can verify that the decomposition in (11) satisfies all the properties that Ω AB should obey. Then the Kronecker symbol can be written as  \nδ B A ' Ω AC Ω BC ' k Aa n Ba ' n a A k B a . (12)  \nFor the 5d type N spacetime, the only non-zero Weyl scalar is ψ ij ' C µνρσ n µ m iν n ρ m jσ in null frame [34-37]. Equivalently, the only non-zero little group bi-spinors in the spinorial  \nformalism for Weyl tensor of a type N solution is ψ p 4 q abcd ' C µνρσ σ µν AB σ ρσ CD n A a n B b n C c n D d [38], where  \nσ µν AB ' 1 2 p γ µ AC γ νC B ' γ ν AC γ µC B q , (13)  \nand the γ µ AC matrices are defined in [38].  \nA natural generalization of the WDC formula to 5d for the type N spacetime can be written as  \nψ ABCD n A a n B b n C c n D d 9 φ AB φ CD n p A a n B b n C c n D q d . (14)  \nHere we do not specify the scalar field as the case in 4d. Because there are more non-zero components of the Weyl spinor in 5d even for the simplest case, the type N spacetime. In principle, there could be more scalar fields involved. Note that the 4d case of the WDC relation is only valid for type D and type N spacetime. There is only one (complex) nonzero component of the Weyl spinor for both cases. The 5d WDC formula for the Weyl scalar of the type N case is  \nψ p 4 q abcd 9 φ p 2 q p ab φ p 2 q cd q , (15)  \nwhere φ p 2 q ab is the only non-zero little group bi-spinors for Maxwell tensor which is defined by φ p 2 q ab ' F µν σ µν AB n A a n B b [38]. More explicitly we have  \nψ p 4 q 1111 ' 3 c S 1 φ p 2 q 11 φ p 2 q 11 , ψ p 4 q 2222 ' 3 c S 1 1 φ p 2 q 22 φ p 2 q 22 , ψ p 4 q 1112 ' 3 c S 2 φ p 2 q 11 φ p 2 q 12 , ψ p 4 q 2221 ' 3 c S 1 2 φ p 2 q 22 φ p 2 q 21 , ψ p 4 q 1212 ' c S 3 r φ p 2 q 11 φ p 2 q 22 ' 2 p φ p 2 q 12 q 2 s . (16)  \nHere we have introduced two independent complex scalar fields S 1 , S 2 ( S 1 1 , S 1 2 are their complex conjugate) and one real scalar field S 3 . To convert to the null frame, we need to apply the following relation  \nγ µ AC γ νC B n A a n B b ' ' γ µ AD n A a n Dd γ ν EB k E d n B b ' γ µ AD n A a k D d γ ν EB n Ed n B b ' ' ? 2 n µ δ d a ε ν db ' ? 2 ε µ ad n ν δ D b ' ? 2 ε r µ ab n ν s , (17)  \nwhere ε µ ab ' k A a γ µ AC n C b [38]. In particular,  \nε µ 11 ' 1 ? 2 p m 2 µ ' im 3 µ q , ε µ 22 ' 1 ? 2 p m 2 µ ' im 3 µ q , ε µ 12 ' ε µ 21 ' im 4 µ , (18)  \nwhere m 2 µ , m 3 µ , m 4 µ are three spacelike vectors in the null frame. Finally, we uncover the connections bewteen the little group bi-spinors and the spacetime tensors as  \nψ p 4 q abcd ' C µνρσ σ µν AB σ ρσ CD n A a n B b n C c n D d ' 2 C r µν sr ρσ s ε µ ab n ν ε ρ cd n σ , (19)  \nand  \nφ p 2 q ab ' F µν ε µ ab n ν . (20)  \nThen the WDC formulas can be established in the null frame as  \nψ 22 ' ψ 33 ' 2 iψ 23 ' 3 c S 1 rp φ 2 q 2 'p φ 3 q 2 ' 2 iφ 2 φ 3 s , ψ 22 ' ψ 33 ' 2 iψ 23 ' 3 c S 1 1 rp φ 2 q 2 'p φ 3 q 2 ' 2 iφ 2 φ 3 s , ψ 34 ' iψ 24 ' 3 c S 2 p φ 3 φ 4 ' iφ 2 φ 4 q , ψ 34 ' iψ 24 ' 3 c S 1 2 p φ 3 φ 4 ' iφ 2 φ 4 q , ψ 44 ' c S 3 rp φ 2 q 2 'p φ 3 q 2 ' 4 p φ 4 q 2 s . (21)  \nThe above equations present a natural extension of the WDC formulas to 5d for type N spacetime. However the formulas in (21) are simply algebraic constructions between a Weyl-like rank four tensor and an antisymmetric rank two tensor. The central dogma of the WDC relation for the present case is that if a type N vacuum solution always admits a degenerate Maxwell field that can square to give the Weyl tensor and the scalar field satisfies the wave equation. An alternative statement of the WDC relation is that if any degenerate Maxwell field on the curved vacuum type N background can always give the Weyl tensor and the scalar field satisfies the wave equation. We are not able to prove the central dogma for a generic type N spacetime from the Bianchi identity for the Weyl tensor and the Maxwell's equation for the field strength. In the next section, we will adapt the WDC formulas (21) to realize a concrete Weyl double copy relation for a special class of type N solutions.", '4 Reduced type N solution and the WDC relation': "We now consider a 4d-like reduction of the 5d type N spacetime where we impose the following restrictions for the Weyl scalars  \nψ 4 i ' 0 . (22)  \nThe traceless property of the Weyl tensor then yields ψ 22 ' ' ψ 33 . Correspondingly, the single and zeroth copies are required to be  \nφ 4 ' 0 , S 3 Ñ8 , (23)  \nwhere the divergent scalar field S 3 is chosen to turn off the extra component from the doubling in 5d. We postpone commenting more on this curious point by the end of this  \nsection. A higher dimensional Goldberg-Sachs-like theorem [40, 41] guarantees that for the CMPP type N spacetime the Weyl aligened null direction is geodesic (but no longer shear-free in general). Thus we can affine parameterize the geodesic and then take an appropriate spin transformation and a null rotation to make other basis vectors parallely propagated along l . Consequently, the following spin coefficients will vanish in such arrangement  \nL i 0 ' 0 , L 10 ' 0 , M i j 0 ' 0 , N i 0 ' 0 . (24)  \nThe Bianchi identities in the null frame [36] are now reduced to  \nDψ ij ' ' ψ ik L kj , δ r k ψ j s i ' ' ψ il M l r jk s ' M l i r j ψ k s l ' ψ i r j L k s 1 ' 2 L 1 r j ψ k s i , L k r i ψ j s k ' 0 , L k r j ψ m s i ' L i r m ψ j s k ' 0 , ψ i r k ψ jm s ' 0 , (25)  \nwhere D , ∆ , and δ i are the directional derivatives associated to the basis vectors l , n , and m i , respectively. The Bianchi identities in the reduced case yield that  \nL 22 ' L 33 , L 32 ' ' L 23 , L i 4 ' L 4 i ' 0 , M 2 44 ' M 3 44 ' 0 , M 2 42 ' M 3 43 , M 2 43 ' ' M 3 42 . (26)  \nWith the above reduction, the Maxwell's equation (the full equations in the null frame are listed in the Appendix) is now reduced to  \nDφ i ' ' φ j L ji , δ j φ i ' δ i φ j ' ' φ i L 1 j ' φ j L 1 i ' φ j L i 1 ' φ i L j 1 ' φ k M k ji ' φ k M k ij , δ i φ i ' ' φ i L 1 i ' φ i L i 1 φ i M i jj . (27)  \nWe introduce the following definitions  \nΨ 4 ' 'p ψ 22 ' iψ 23 q , ¯ Ψ 4 ' 'p ψ 22 ' iψ 23 q , ϕ 2 ' ' 1 ? 2 p φ 2 ' iφ 3 q , ¯ ϕ 2 ' ' 1 ? 2 p φ 2 ' iφ 3 q , δ ' 1 ? 2 p δ 2 ' iδ 3 q , ¯ δ ' 1 ? 2 p δ 2 ' iδ 3 q , δ 1 ' iδ 4 , S ' S 1 c . (28)  \nThen, the Bianchi identity and Maxwell's equation could be rewritten as  \nD Ψ 4 ' 'p L 22 ' iL 23 q Ψ 4 , δ Ψ 4 ' ' 1 ? 2 rp 2 L 12 ' 2 M 2 33 ' L 21 q ' i p L 31 ' 2 L 13 ' 2 M 2 32 qs Ψ 4 , δ 1 Ψ 4 ' rp M 3 42 ' 2 M 3 24 q ' i p M 2 42 ' L 41 ' 2 L 14 qs Ψ 4 , (29)  \nand  \nDϕ 2 ' 'p L 22 ' iL 23 q ϕ 2 , δϕ 2 ' ' 1 ? 2 rp L 12 ' M 2 33 ' L 21 q ' i p L 31 ' M 2 32 ' L 13 qs ϕ 2 , δ 1 ϕ 2 ' rp M 3 42 ' M 3 24 q ' i p L 41 ' L 14 ' M 2 42 qs ϕ 2 . (30)  \nThe WDC formula in the reduced case is given in a more compact form by  \nΨ 4 ' 1 S p ϕ 2 q 2 . (31)  \nNowit is clear that the previously chosen divergent scalar field S 3 is to prevent the mixing term constructed from ϕ 2 ¯ ϕ 2 in the Weyl doubling. Such term is not involved at all in the 4d WDC relation. So it is reasonable to have a special treatment in the 5d WDC formula. Here we initialize the 5d investigation by simply adapting the WDC relation and imposing that there is no Weyl scalar associated to ϕ 2 ¯ ϕ 2 term.  \nSubstituting the WDC relation (31) into the Bianchi's identities and simplifying them with the Maxwell's equation, we obtain the equations for the scalar field  \nD log S ' 'p L 22 ' iL 23 q , δ log S ' 1 ? 2 p L 21 ' iL 31 q , δ 1 log S ' M 3 42 ' i p M 2 42 ' L 41 q . (32)  \nFollowing closely the treatment in [9], one can prove the following integrability conditions for the above differential equations,  \np Dδ ' δD q log S ' 1 ? 2 D p L 21 ' iL 31 q ' δ p L 22 ' iL 23 q , p Dδ 1 ' δ 1 D q log S ' 1 ? 2 D p L 21 ' iL 31 q ' δ p L 22 ' iL 23 q , p δδ 1 ' δ 1 δ q log S ' δ p M 3 42 ' iM 2 42 ' iL 41 q ' 1 ? 2 δ 1 p L 21 ' iL 31 q , (33)  \nwhere the commutators of the operators D , δ , δ 1 in [42] and the Ricci identities in [43] are applied. Finally, it is straightforward to verify that the scalar field S solves the KleinGordon equation in curved spacetime  \nl S ' p ∇ µ l µ ∆ ' ∇ µ n µ D ' ∇ µ m iµ δ i ' D ∆ ' ∆ D ' δ i δ i q S ' 0 , (34)  \nwhere we have applied the equations in (32), the commutators of the operators and the Ricci identities. This completes the zeroth copy of the 5d WDC relation for the reduced type N solutions. In the next section, we will present two concrete examples of the 5d WDCrelation.", '5.1 5d pp-wave solution': "The line-element of the 5d PP-wave solution in the convention of the CMPP null frame is given by [44]  \nd s 2 ' 2 d u r d v ' H p u, x 2 , x 3 , x 4 q d u s ' d x 2 2 ' d x 2 3 ' d x 2 4 . (35)  \nAdapted to the reduction of the previous section, one obtains  \nH p u, x 2 , x 3 , x 4 q ' f p u q x 4 ' G p u, x 2 , x 3 q , (36)  \nwhere G p u, x 2 , x 3 q ' g p u, z q ' ¯ g p u, ¯ z q and z ' x 2 ' ix 3 . We choose the same null frame system as [33]. Then, it is easy to show  \nL 22 ' L 23 ' L 21 ' L 31 ' L 41 ' M 2 42 ' M 3 42 ' 0 . (37)  \nA generic solution to (32) for the 5d pp-wave solution is  \nS ' S p u, ¯ z q . (38)  \nThe Weyl scalar is given by  \nΨ 4 ' B x 2 B ¯ z ¯ g p u, ¯ z q ' B 2 ¯ z ¯ g p u, ¯ z q , (39)  \nwhich allows us to determine the Maxwell scalar as  \nϕ 2 ' b S p u, ¯ z qB 2 ¯ z ¯ g p u, ¯ z q . (40)  \nWehave checked directly that the Maxwell scalar satisfies the Maxwell's equation. Hence we uncover a concrete WDC relation for the 5d pp-wave solution.", '5.2 5d Kundt solution': "The type N Kundt line-element is given by [45]  \nd s 2 ' 2 d u r d v ' H p u, v, x k q d u ' W i p u, v, x k q d x i s ' δ ij d x i d x j , (41)  \nwhere i, j, k ' 2 , 3 , 4 , and  \nH p u, v, x k q ' v 2 2 y 2 ' H p 0 q p u, x i q , y ' δ ij x i x j , W 2 ' ' 2 v y , W m p u, v, x k q ' x n B nm p u q ' C m p u q , m,n ' 3 , 4 . (42)  \nThe reductions from ψ 24 ' ψ 34 ' ψ 44 ' 0 yield that  \nH p 0 q ' f p u, x 2 , x 3 q ' g p u, x 2 , x 4 q , g p u, x 2 , x 4 q ' 1 2 B 43 p u q 2 x 2 4 ' g 1 p u q x 2 x 4 ' C 3 p u q x 4 ' g 2 p u, x 2 q . (43)  \nProjecting the solution into the null frame system defined in [33], one can obtain  \nL 22 ' L 23 ' L 31 ' L 41 ' M 2 42 ' M 3 42 ' 0 , L 21 ' ' 1 y . (44)  \nA generic solution to (32) for the 5d Kundt solution is  \nS ' P p u, x 2 ' ix 3 q x 2 . (45)  \nWeyl scalar of the Kundt solution is given by  \nΨ 4 ' 1 p x 2 q 3 ' p x 2 ' ix 3 q B 43 p u q 2 ' iB 43 p u q C 4 p u q ' i B x 3 f p u, x 2 , x 3 q ' x 2 B 2 x 3 f p u, x 2 , x 3 q ' ix 2 B x 2 B x 3 f p u, x 2 , x 3 q  . (46)  \nThen, one can recover the Maxwell scalar from taking the square-root ϕ 2 ' ? S Ψ 4 . It can be verified directly that the Maxwell scalar satisfies the Maxwell's equation. Thus, a concrete WDC relation has been uncovered for the 5d Kundt solution.", '6 Concluding remarks': 'In this paper, we establish a 5d WDC relating the curvature of a special class of the type N spacetime to an electromagnetic field strength. Our proposal, for the first time, extends the 4d WDC relation to higher dimensions. A higher-dimensional spinorial formalism [38] allows one to construct the possible WDC formula in a universal way in any dimension. Hence, our investigation can be in principle extended, with certain adaptions, to any dimension for a special class of type N or type D spacetime, which would significantly enlarge the scope of the WDC relation.', 'Acknowledgments': 'The authors thank Hong Lu for useful discussions and Ricardo Monteiro for useful correspondence on the integrability conditions. This work is supported in part by the National Natural Science Foundation of China (NSFC) under Grants No. 11935009, No. 12375066, and No. 12247103.', "A Maxwell's equation in the null frame": "DF 01 ' δ i F 0 i ' F 0 i p M i 10 ' M i 01 ' M i jj q ' F 01 p L 10 ' L jj q ' F 0 i p M i 01 ' N 0 i q ' F ij M j 0 i ' F 01 N 00 ' F 1 i M i 00 ' 0 , (47)  \n∆ F 01 ' δ i F 1 i ' F 01 p N 01 ' N jj q ' F 1 i p M i 10 ' M i 01 ' M i jj q ' F 0 i M i 11 ' F ij M j 1 i ' F 01 L 11 ' F 1 i p M i 10 ' L 1 i q ' 0 , (48)  \n∆ F 0 k ' δ i F ik ' DF 1 k ' F ok p N 01 ' N jj q ' F 1 k p L 10 ' L jj q ' F ik p M i 10 ' M i 01 ' M i jj q ' F 0 i p M i k 1 ' N ki q ' F ij M j ki ' F 01 p L k 1 ' N k 0 q ' F 1 i p M i k 0 ' L ki q ' 0 , (49)  \nDF 1 i ' ∆ F i 0 ' δ i F 01 ' F 01 L 1 i ' F j 1 L ji ' F 01 N 0 i ' F 0 j N ji ' F 1 i N 00 ' F ji N j 0 ' F 10 M i 10 ' F 1 j M i j 0 ' F i 0 L 11 ' F ij L j 1 ' F 10 M i 01 ' F j 0 M i j 1 , (50)  \nDF ij ' δ j F 0 i ' δ i F j 0 ' F 0 i L 1 j ' F ki L kj ' F 01 M i 0 j ' F 0 k M i kj ' F 1 j M i 00 ' F 0 j M i 10 ' F kj M i k 0 ' F i 1 M j 00 ' F i 0 M j 10 ' F ik M j k 0 ' F 10 M j 0 i ' F k 0 M j ki ' F j 0 L 1 i ' F jk L ki , (51)  \n∆ F ij ' δ j F 1 i ' δ i F j 1 ' F 1 i N 0 j ' F ki N kj ' F 10 M i 1 j ' F 1 k M i kj ' F 0 j M i 11 ' F 1 j M i 01 ' F kj M i k 1 ' F i 0 M j 11 ' F i 1 M j 01 ' F ik M j k 1 ' F 01 M j 1 i ' F k 1 M j ki ' F j 1 N 0 i ' F jk N ki , (52)  \nδ k F ij ' F i 1 M j 0 k ' F i 0 M j 1 k ' F il M j kl ' F 1 j M i 0 k ' F 0 j M i 1 k ' F lj M i lk , (53)  \nwhere the permutation of the indices i, j, k for the last equation should be understood.", 'References': "- [1] H. Kawai, D. C. Lewellen, and S. H. H. Tye, 'A Relation Between Tree Amplitudes of Closed and Open Strings,' Nucl. Phys. B 269 (1986) 1-23.\n- [2] Z. Bern, J. J. M. Carrasco, and H. Johansson, 'New Relations for Gauge-Theory Amplitudes,' Phys. Rev. D 78 (2008) 085011, arXiv:0805.3993 [hep-ph] .\n- [3] Z. Bern, J. J. M. Carrasco, and H. Johansson, 'Perturbative Quantum Gravity as a Double Copy of Gauge Theory,' Phys. Rev. Lett. 105 (2010) 061602, arXiv:1004.0476 [hep-th] .\n- [4] R. Monteiro, D. O'Connell, and C. D. White, 'Black holes and the double copy,' JHEP 12 (2014) 056, arXiv:1410.0239 [hep-th] .\n- [5] A. Luna, R. Monteiro, I. Nicholson, and D. O'Connell, 'Type D Spacetimes and the Weyl Double Copy,' Class. Quant. Grav. 36 (2019) 065003, arXiv:1810.08183 [hep-th] .  \n- [6] R. Alawadhi, D. S. Berman, and B. Spence, 'Weyl doubling,' JHEP 09 (2020) 127, arXiv:2007.03264 [hep-th] . [7] H. Godazgar, M. Godazgar, R. Monteiro, D. Peinador Veiga, and C. N. Pope, 'Weyl Double Copy for Gravitational Waves,' Phys. Rev. Lett. 126 no. 10, (2021) 101103, arXiv:2010.02925 [hep-th] . [8] C. D. White, 'Twistorial Foundation for the Classical Double Copy,' Phys. Rev. Lett. 126 no. 6, (2021) 061602, arXiv:2012.02479 [hep-th] . [9] R. Monteiro, D. O'Connell, D. Peinador Veiga, and M. Sergola, 'Classical solutions and their double copy in split signature,' JHEP 05 (2021) 268, arXiv:2012.11190 [hep-th] . [10] E. Chac'on, S. Nagy, and C. D. White, 'The Weyl double copy from twistor space,' JHEP 05 (2021) 2239, arXiv:2103.16441 [hep-th] . [11] H. Godazgar, M. Godazgar, R. Monteiro, D. Peinador Veiga, and C. N. Pope, 'Asymptotic Weyl double copy,' JHEP 11 (2021) 126, arXiv:2109.07866 [hep-th] . [12] T. Adamo and U. Kol, 'Classical double copy at null infinity,' Class. Quant. Grav. 39 no. 10, (2022) 105007, arXiv:2109.07832 [hep-th] . [13] D. A. Easson, T. Manton, and A. Svesko, 'Sources in the Weyl Double Copy,' Phys. Rev. Lett. 127 no. 27, (2021) 271101, arXiv:2110.02293 [gr-qc] . [14] E. Chac'on, S. Nagy, and C. D. White, 'Alternative formulations of the twistor double copy,' JHEP 03 (2022) 180, arXiv:2112.06764 [hep-th] . [15] S. Han, 'Weyl double copy and massless free-fields in curved spacetimes,' Class. Quant. Grav. 39 no. 22, (2022) 225009, arXiv:2204.01907 [gr-qc] . [16] S. Han, 'The Weyl double copy in vacuum spacetimes with a cosmological constant,' JHEP 09 (2022) 238, arXiv:2205.08654 [gr-qc] . [17] A. Luna, N. Moynihan, and C. D. White, 'Why is the Weyl double copy local in position space?,' JHEP 12 (2022) 046, arXiv:2208.08548 [hep-th] . [18] D. A. Easson, T. Manton, and A. Svesko, 'Einstein-Maxwell theory and the Weyl\n- double copy,' Phys. Rev. D 107 no. 4, (2023) 044063, arXiv:2210.16339 [gr-qc] .  \n- [19] D. A. Easson, G. Herczeg, T. Manton, and M. Pezzelle, 'Isometries and the double copy,' JHEP 09 (2023) 162, arXiv:2306.13687 [gr-qc] .\n- [20] G. Alkac, M. K. Gumus, O. Kasikci, M. A. Olpak, and M. Tek, 'Regularized Weyl double copy,' Phys. Rev. D 109 no. 8, (2024) 084047, arXiv:2310.06048 [hep-th] .\n- [21] P. Mao and W. Zhao, 'Asymptotic Weyl double copy in Newman-Penrose formalism,' JHEP 02 (2024) 171, arXiv:2312.17160 [hep-th] .\n- [22] Y.-R. Liu, J.-R. Zhang, and Y.-L. Zhang, 'Slowly rotating charges from Weyl double copy for Kerr black hole with Chern-Simons correction,' Commun. Theor. Phys. 76 no. 8, (2024) 085405, arXiv:2404.07888 [gr-qc] .\n- [23] S. Chawla, K. Fransen, and C. Keeler, 'The Penrose limit of the Weyl double copy,' arXiv:2406.14601 [hep-th] .\n- [24] K. Armstrong-Williams, N. Moynihan, and C. D. White, 'Deriving Weyl double copies with sources,' arXiv:2407.18107 [hep-th] .\n- [25] M. Carrillo Gonz'alez, A. Momeni, and J. Rumbutis, 'Cotton double copy for gravitational waves,' Phys. Rev. D 106 no. 2, (2022) 025006, arXiv:2202.10476 [hep-th] .\n- [26] M. Carrillo Gonz'alez, W. T. Emond, N. Moynihan, J. Rumbutis, and C. D. White, 'Mini-twistors and the Cotton double copy,' JHEP 03 (2023) 177,  \narXiv:2212.04783 [hep-th]  \n.  \n- [27] M. Carrillo-Gonz'alez, R. Penco, and M. Trodden, 'The classical double copy in maximally symmetric spacetimes,' JHEP 04 (2018) 028, arXiv:1711.01296 [hep-th] .\n- [28] M. Carrillo Gonz'alez, B. Melcher, K. Ratliff, S. Watson, and C. D. White, 'The classical double copy in three spacetime dimensions,' JHEP 07 (2019) 167, arXiv:1904.11001 [hep-th] .\n- [29] M. K. Gumus and G. Alkac, 'More on the classical double copy in three spacetime dimensions,' Phys. Rev. D 102 no. 2, (2020) 024074, arXiv:2006.00552 [hep-th] .\n- [30] G. Alkac, M. K. Gumus, and M. Tek, 'The Kerr-Schild Double Copy in Lifshitz Spacetime,' JHEP 05 (2021) 214, arXiv:2103.06986 [hep-th] .  \n- [31] G. Alkac, M. K. Gumus, and M. A. Olpak, 'Kerr-Schild double copy of the Coulomb solution in three dimensions,' Phys. Rev. D 104 no. 4, (2021) 044034, arXiv:2105.11550 [hep-th] .\n- [32] M. Ortaggio and A. Srinivasan, 'Charging Kerr-Schild spacetimes in higher dimensions,' Phys. Rev. D 110 no. 4, (2024) 044035, arXiv:2309.02900 [gr-qc] .\n- [33] M. Ortaggio, V. Pravda, and A. Pravdova, 'Kerr-Schild double copy for Kundt spacetimes of any dimension,' JHEP 2024 no. 02, (2024) 069, arXiv:2312.00706 [gr-qc] .\n- [34] A. Coley, R. Milson, V. Pravda, and A. Pravdova, 'Classification of the Weyl tensor in higher dimensions,' Class. Quant. Grav. 21 (2004) L35-L42, arXiv:gr-qc/0401008 .\n- [35] R. Milson, A. Coley, V. Pravda, and A. Pravdova, 'Alignment and algebraically special tensors in Lorentzian geometry,'\n- Int. J. Geom. Meth. Mod. Phys. 2 (2005) 41-61, arXiv:gr-qc/0401010 .\n- [36] V. Pravda, A. Pravdova, A. Coley, and R. Milson, 'Bianchi identities in higher dimensions,' Class. Quant. Grav. 21 (2004) 2873-2898, arXiv:gr-qc/0401013 . [Erratum: Class.Quant.Grav. 24, 1691 (2007)].\n- [37] M. Ortaggio, V. Pravda, and A. Pravdova, 'Algebraic classification of higher dimensional spacetimes based on null alignment,' Class. Quant. Grav. 30 (2013) 013001, arXiv:1211.7289 [gr-qc] .\n- [38] R. Monteiro, I. Nicholson, and D. O'Connell, 'Spinor-helicity and the algebraic classification of higher-dimensional spacetimes,' Class. Quant. Grav. 36 (2019) 065006, arXiv:1809.03906 [gr-qc] .\n- [39] E. Newman and R. Penrose, 'An Approach to gravitational radiation by a method of spin coefficients,' J. Math. Phys. 3 (1962) 566-578.\n- [40] M. Durkee and H. S. Reall, 'A Higher-dimensional generalization of the geodesic part of the Goldberg-Sachs theorem,' Class. Quant. Grav. 26 (2009) 245005, arXiv:0908.2771 [gr-qc] .\n- [41] M. Ortaggio, V. Pravda, A. Pravdova, and H. S. Reall, 'On a five-dimensional version of the Goldberg-Sachs theorem,' Class. Quant. Grav. 29 (2012) 205002, arXiv:1205.1119 [gr-qc] .  \n- [42] A. Coley, R. Milson, V. Pravda, and A. Pravdova, 'Vanishing scalar invariant spacetimes in higher dimensions,' Class. Quant. Grav. 21 (2004) 5519-5542, arXiv:gr-qc/0410070 .\n- [43] M. Ortaggio, V. Pravda, and A. Pravdova, 'Ricci identities in higher dimensions,' Class. Quant. Grav. 24 (2007) 1657-1664, arXiv:gr-qc/0701150 .\n- [44] A. Coley, R. Milson, N. Pelavas, V. Pravda, A. Pravdova, and R. Zalaletdinov, 'Generalized pp wave space-times in higher dimensions,' Phys. Rev. D 67 (2003) 104020, arXiv:gr-qc/0212063 .\n- [45] A. Coley, A. Fuster, S. Hervik, and N. Pelavas, 'Higher dimensional VSI spacetimes,' Class. Quant. Grav. 23 (2006) 7431-7444, arXiv:gr-qc/0611019 ."}