Asymptotic Weyl double copy in Newman-Penrose formalism

In this paper, we provide a self-contained investigation of the Weyl double copy in the Newman-Penrose formalism. We examine the Weyl double copy constraints for the general asymptotically flat solution in the Newman-Unti gauge. We find that two transparent solutions of the asymptotic Weyl double copy constraints lead to truncated solutions for both linearized and Einstein gravity theory where the solutions are in the manifest form of Petrov type N or type D in the Newman-Unti gauge.


Introduction
Weyl double copy (WDC) [1] is a fascinating realization of the classical double copy relation [2] which in general connects solutions in gravitational theories with gauge theories.The WDC formula involves only curvatures, namely the Weyl tensor and the Maxwell field strength tensor, rather than the gauge fields.Hence, it is adorable from its gauge invariant nature and has been an active research topic [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17].The WDC relations are best expressed in spinor language as The formula of the WDC is very elegant, the applications are somewhat restrictive.On the one hand, by construction, the WDC formula is especially for Petrov type D [1] and Petrov type N [4] solutions.On the other hand, the spinorial description restricts its extensions to theories in dimensions other than 4 or to other types of gravitational theory.
In this paper, we provide an alternative probe of the WDC by exploring its constraints on the solutions of gravity and Maxwell theory in the Newman-Penrose (NP) formalism [18].
The investigation in this work is mainly inspired by the asymptotic WDC [8].In the NP formalism, there is a well known solution space in the form of a series expansion in the inverse radial coordinate in the Newman-Unti (NU) gauge [19].Solutions of the Maxwell theory in the NP formalism are also known in the same type of series expansion [20,21].The main concern of this work is to examine the consequence of the WDC relations on those two solution spaces asymptotically.We first present a direct verification in the NP formalism that the WDC formula is invariant under the local Lorentz transformations and the solution satisfying the WDC formula must be of either Petrov type D or type N.Those properties are transparent in the spinor language.Nevertheless, they are still meaningful for a self-contained investigation in the NP formalism.We then list the asymptotic WDC constraints on the solution space of Maxwell theory and linearized gravity.There are several reasons for linearization for the gravitational theory.The solution space of the linearized theory is much simpler and tractable which will manifest the WDC constraints.We linearize the theory on Minkowski background in the present work.Nonetheless, it is straightforward to perform a similar investigation on a curved background which extends the original scope of the WDC.The classical double copy is originated from amplitude double copye relation [22][23][24] where gravity amplitude is defined perturbatively on Minkowski background.So the linearization at the classical level may better approach the quantum feature.There are four equations from the asymptotic WDC relations constraining the time evolution of the scalar field in the WDC formula.A generic solution to those constraints is a very challenging problem.Nevertheless, there are some interesting special solutions.We check two special cases.Interestingly, those two solutions lead to significant simplification of the solution space and result in truncated solutions in a manifest form of type N or type D in the NU gauge.Remarkably, those two simplifications with truncation sustain at the non-linear level.When we choose the asymptotic WDC conditions as initial data, the solutions of the NP system that satisfy the WDC relations in the NU gauge truncate and are in the exact form of type N or type D.
The organization of this paper is as follows.In the next section, we examine the properties of the WDC in the NP formalism.In Section 3, we list the solution space of Maxwell theory and linearized gravity on Minkowski background.In Section 4, we present the asymptotic WDC constraints on the solution space and find that two solutions of the WDC constraints lead to truncation of the solution space.In Section 5, we show that the same type of truncation also happens for full Einstein gravity.We conclude in the last section.There is one Appendix that details our conventions for the NP formalism.

Weyl double copy in NP formalism
In the NP formalism, 1 after choosing an appropriate system of null tetrads, the spinor form of the WDC can be translated into the connections of the Weyl scalars and the Maxwell scalars as follows [8]: Since the null vector l can be transformed to any null direction through a single second class null rotation and a single first class null rotation, we can verify that the WDC relations are invariant under such gauge transformations.Under a combined first and second classes of null rotations, the Weyl scalars and Maxwell scalars transform into: where a and b are complex functions that characterize first and second class null rotations respectively.It is straightforward to verify that, once the WDC relations are satisfied in the original frame, they sustain in the new frame, Assuming that S is invariant under the null rotations, this is a direct confirmation in the NP formalism that the WDC formula is invariant under local Lorentz transformations.Then, it is easy to verify that any gravitational solution that satisfies the WDC formula must necessarily be of either Petrov type D or Petrov type N.This is because the WDC formula requires that the Weyl scalars can be expressed as products of Maxwell scalars.Under the second class null rotation, the Maxwell scalars transform as ( For the condition φ 0 `2bφ 1 `b2 φ 2 " 0, if b has a double root, we can use a second class null rotation to make φ 1 0 " φ 1 1 " 0. Accordingly, the gravitational field constructed from Maxwell field from (2) in the new frame system is manifest of Type N. If φ 0 `2bφ 1 b2 φ 2 " 0 has two distinct roots, then we can always use another first class null rotation to make φ 2 0 " φ 2 2 " 0. Accordingly, the gravitational solution is of type D.

Solution space
In this section, we will review the solution space for Maxwell theory and linearized gravity with adaption to our conventions.Relevant results were presented in [20,21], also in [25,26].For computational simplicity, we use flat null coordinates x µ " pu, r, z, zq [27] where the celestial sphere at null infinity is mapped to a 2d plane [28][29][30][31].The line element of the Minkowski spacetime in this coordinates system is The only non-vanishing NP variables are ρ " ´1 r and L z " 1 r .Correspondingly, the directional derivatives are given by We define B " B z and B " B z for notational brevity.

Maxwell theory
Maxwell's equations in the flat null coordinates are organized as Suppose that φ 0 is given as initial data in series expansion, Then φ 1 and φ 2 can be solved as The evolution of the initial data is controlled by

Linearized gravity theory
Bianchi identities after the linearization are given by 2 B r pr 4 ψ 1 q " r 3 Bψ 0 , B r pr 3 ψ 2 q " r 2 Bψ 1 , B r pr 2 ψ 3 q " rBψ 2 , B r prψ 4 q " Bψ 3 , Suppose that ψ 0 is given as initial data in series expansion, Then ψ 1 -ψ 4 can be solved as The evolution of the initial data is controlled by

WDC constraints and linearized gravity
In this section, we will examine the asymptotic WDC constraints on the solution of Maxwell and linearized gravitational theory asymptotically.For our purpose, it is convenient to have the scalar field in the WDC formula on the numerator, 2 We use ψ i to denote the Weyl scalars in the linearized theory.
where F can be expanded in inverse powers of r as The asymptotic WDC was proposed in [8] by requiring that the leading orders of all Weyl scalars and Maxwell scalars satisfy the WDC formula.If we assume that the Maxwell scalars already satisfy the Maxwell's equations and construct the Weyl scalars via the asymptotic WDC, then the Bianchi identities in ( 16) yield the following constraints at the leading orders, The above four equations can all be seen as evolution equations for F 0 .Since the system is way more overdetermined, direct solutions are challenging.Nevertheless, we will consider some special cases.There are two transparent solutions, φ 0 1 " φ 0 0 " BF 0 " 0, φ 0 2 ‰ 0 and φ 0 2 " φ 0 0 " B u F 0 " BF 0 " 0, φ 0 1 ‰ 0. Surprisingly, as we will demonstrate below, those two simple solutions lead to truncation of the solution space when combined with the WDC relations.For the first option, the Maxwell fields are reduced to Then, WDC relations yield that Clearly, the WDC construction set that ψ 0 3 " ψ 0 2 " ψ 0 1 " ψ 0 0 " ψ 1 0 " 0. Hence, ψ 0 begins at order Opr ´7q.As evident from the solution space of the linearized gravity (15), those conditions lead to that all the Weyl scalars should begin at order Opr ´7q except for ψ 4 .Consequently, the expression of ψ 2 in (21) indicates that φ 1 0 " 0. In fact, since φ 0 2 ‰ 0, the leading term of ψ 2 is , and the leading term of ψ 0 is from the asymptotic WDC formula (21), where φ i 0 is the leading non-vanishing order of φ 0 .However, in the solution space (15), it is clear that ψ 2 and ψ 0 should begin at the same order when ψ 0 2 " 0. This yields that φ i 0 " 0 for any i, namely φ 0 " 0. Hence, φ 1 " 0. Thus, the electromagnetic solution is type N and l is the principle null direction.The WDC formula implies that any asymptotically type N electromagnetic solution is manifestly type N. Finally, the full solutions fulfilling the WDC relations are Going back to the original WDC relations in (2), we obtain S " 3cφ 0 2 φ 0 2 ψ 0 4 r , which is just the news function of a scalar field [32].It is obvious that S satisfies the equation of motion of a scalar field on Minkowski spacetime.
For the second option, the Maxwell fields are reduced to The WDC relations imply that To be a solution of linearized gravity, one must set F 0 " F 0 pzq, and ψ 0 4 " ψ 0 3 " ψ 0 1 " ψ 0 0 " ψ 1 0 " 0. Because of φ 0 1 ‰ 0 and φ 0 2 " φ 0 0 " 0, the leading order of ψ 1 is Similar to the case of the previous option, one must have that φ 0 " 0. Then the solutions of the Maxwell theory are The gravitational fields constructed from the WDC relations are The solution of the linearized gravitational theory (15) finally determines that with k 1 , k 2 , k 3 being constants.The corresponding scalar field S will satisfy the equation of motion.One can apply a first class of null rotation to have φ 2 " 0.Then, the gravitational solution is in the manifest form of type D.

Extension to Einstein gravity
In the previous section, we have shown that two types of boundary conditions which are transparent solutions to the asymptotic WDC constraints lead to interesting truncated solution space for Maxwell and linear gravitational theory when combined with the WDC relations.In this section, we will prove that the same types of truncation also hold for the Einstein theory and Maxwell theory on the curved background spacetime as a probe.

Type N solution
In this subsection, we will demonstrate that, when choosing the boundary conditions φ 0 1 " φ 0 0 " 0 and φ 0 2 ‰ 0 in asymptotically flat spacetime, the WDC relations (2) yield a truncated solution space which is manifestly of type N in the NU gauge.We first prove the following three lemmas from the WDC relations.
Lemma 1: If φ 0 0 " φ 0 1 " 0, φ 0 2 ‰ 0, then φ 1 0 " 0. Proof: In this case, the leading terms of the Maxwell scalars on an asymptotically flat background spacetime are where the definitions of the ð and ð operators are presented below in (35).The WDC relations yield the Weyl scalars as However, the two radial equations from the Bianchi identity (68) and (69), with asymptotically flat fall-off conditions [19], determine that Ψ 2 " Opr ´7q.This requires that φ 1 0 " 0. Lemma 2: If φ 0 1 " 0 and φ 0 " 0, then both the Maxwell fields and the gravitational fields are in the manifest form of Type N in the NU gauge.
Proof: According to the WDC relations, it is obvious from the assumptions that Ψ 0 " Ψ 1 " 0. Therefore, the Goldberg-Sachs theorem yields that σ " 0, hence ρ " ´1 r .The Maxwell's equation (89) yields that φ 1 " 0. Consequently, both the Maxwell and the corresponding gravitational field are type N and l is the principle null direction.
Lemma 3: If φ 0 1 " φ 0 0 " 0, and φ 1 0 " ... " φ i 0 " 0, then φ 1 1 " ... " φ i 1 " 0 for i ě 1. Proof: We just need to apply the radial equation ( 89) and the fact that the background spacetime is asymptotically flat.Lemma 3 can be proved easily.Now we can prove that the WDC relations combined with the boundary condition φ 0 1 " φ 0 0 " 0 require that the solutions are of type N and l is the principle null direction.We know from the WDC relations that Ψ 2 " 1 3 F φ 2 φ 0 `2 3 F pφ 1 q 2 , therefore the leading term of Ψ 2 could be either , where φ i 0 and φ j 1 are the leading terms of φ 0 and φ 1 , respectively.The Lemma 3 requires that the second term is always ordersuppressed.Then the leading term of Ψ 2 is While the WDC relation yields that the leading term of Ψ 0 is F 0 pφ i 0 q 2 r 2i`5 .Thus the peeling-off property encoded in the radial equations ( 68) and (69) yields that φ i 0 " 0 for any i ą 0, the proof is similar to the proof of Lemma 1 or the linearized case.Finally, according to Lemma 2, the gravitational field and the Maxwell field are of type N and l is the principle null direction.Now we consider that l is the principle null direction as initial data and derive the general solution space of the NP system in the NU gauge.Correspondingly, we should set Ψ 0 " 0, Ψ 1 " 0, Ψ 2 " 0, Ψ 3 " 0.
Note that ðΨ 0 4 " pB u `4γ 0 `2γ 0 qð 2 ðϕ " 0, which does not lead to a new constraint on ϕ.The "ð" operator is defined by ðη psq " P P ´sB p P s η psq q " P Bη psq `2sα 0 η psq , ðη psq " P P s BpP ´sη psq q " P Bη psq ´2sα 0 η psq , Table 1: Spin weights where s is the spin weight of the field η psq .The spin weights of relevant fields are listed in Table 1.The operators ð, ð raise and lower the spin weight by one unit.Note that P is of spin weight 1 and "holomorphic", ðP " 0 and the commutator of the operators is with R " ´4µ 0 .We also have their commutation relation with the time derivative [33], rB u , ðsη psq " ´2pγ 0 ð `sðγ 0 qη psq , rB u , ðsη psq " ´2pγ 0 ð ´sðγ 0 qη psq .
The solution (31) is the type N Robinson-Trautman solution [34].The Weyl double copy of the Robinson-Trautman solution is revealed in [4] for a special case by µ 0 " 0, ˘1.
With the type N spacetime (31) as the background and the conditions φ 0 " φ 1 " 0, it is straightforward to solve the Maxwell's equations.The solution is where f φ pu, zq is an arbitrary function.The WDC relation Ψ 4 " F φ 2 φ 2 yields From the original WDC relations in (2), S " 3cφ 0 2 φ 0 2 Ψ 0 4 r .Note that ðΨ 0 4 " 0. This yields that Ψ 0 4 " P 2 f Ψ pu, zq.Hence, S must be of the form ζpu,zq r , which is a solution of the equation of motion for a scalar field on both Minkowski spacetime and the curved spacetime (31).

Type D solution
In this subsection, we will demonstrate for the case φ 0 2 " φ 0 0 " 0, φ 0 1 ‰ 0. The asymptotic form of the Maxwell fields are now The WDC relations lead to The peeling-off property encoed in (68) requires that φ 1 0 " 0 when φ 0 1 ‰ 0. Following the same argument, any φ i 0 are thus equal to zero, therefore φ 0 " 0.Then, the WDC relations determine that Ψ 0 " Ψ 1 " 0, and the Goldberg-Sachs theorem yields that σ " 0, hence ρ " ´1 r .Substituting these results back into the Maxwell's equations, we obtain We can use a first class of null rotation to turn off φ 2 .Then l and the rotated basis n 1 are the principle null directions and the gravitational solution is of type D in the manifest form.However, the null rotation will turn on the spin coefficient π, which are out of the NU gauge.
The WDC relations incorporated with the gravity solution (44), yield that φ 2 " 0. Hence, φ 0 1 can be fixed as φ 0 1 " C φ e ´2ϕ 0 . (47) Finally, the scalar field in the WDC relations is determined as Clearly, the corresponding scalar function S satisfies the equation of motion on Minkowski spacetime.However, it is not a solution on the curved spacetime (44).One can consider an alternative option to realize the WDC relations which involves the Maxwell theory on Minkowski spacetime [1].The solution space of the Maxwell theory for such case has been presented in Section 3. The WDC relations yield that which leads to ϕ 0 " 0 for the gravitational solution by the WDC relations.Hence, and the corresponding scalar function S satisfies the equation of motion on the flat background.