# Asymptotic Weyl double copy in Newman-Penrose formalism

Pujian Mao and Weicheng Zhao

Asymptotic Weyl double copy in Newman-Penrose formalism

Pujian Mao and Weicheng Zhao

Center for Joint Quantum Studies and Department of Physics,

School of Science, Tianjin University, 135 Yaguan Road, Tianjin 300350, China

Emails<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract: 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.

##  1 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

$\Psi_{ABCD}=\frac{3c}{S}\Phi_{(AB}\Phi_{CD)}.$ (1)

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.

##  2 Weyl double copy in NP formalism

In the NP formalism,111See Appendix A for our conventions. 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]:

$\begin{split}&\Psi_{4}=\frac{3c}{S}\phi_{2}\phi_{2},\phantom{10000}\Psi_{3}=\frac{3c}{S}\phi_{1}\phi_{2},\phantom{10000}\Psi_{2}=\frac{c}{S}(\phi_{0}\phi_{2}+2\phi_{1}\phi_{1}),\\\
&\phantom{100000000}\Psi_{1}=\frac{3c}{S}\phi_{0}\phi_{1},\phantom{10000}\Psi_{0}=\frac{3c}{S}\phi_{0}\phi_{0}.\end{split}$
(2)

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:

$\begin{split}&\Psi_{0}^{\prime}\rightarrow\Psi_{0}+4b\Psi_{1}+6b^{2}\Psi_{2}+4b^{3}\Psi_{3}+b^{4}\Psi_{4},\\\
&\Psi_{1}^{\prime}\rightarrow\bar{a}\Psi_{0}+(1+4\bar{a}b)\Psi_{1}+3b(1+2\bar{a}b)\Psi_{2}+b^{2}(3+4\bar{a}b)\Psi_{3}+b^{3}(1+\bar{a}b)\Psi_{4},\\\
&\Psi_{2}^{\prime}\rightarrow\bar{a}^{2}\Psi_{0}+2\bar{a}(1+2\bar{a}b)\Psi_{1}+(1+6\bar{a}b+6\bar{a}^{2}b^{2})\Psi_{2}\\\
&\hskip
142.26378pt+2b(1+3\bar{a}b+2\bar{a}^{2}b^{2})\Psi_{3}+b^{2}(1+{a}b)^{2}\Psi_{4},\\\
&\Psi_{3}^{\prime}\rightarrow\bar{a}^{3}\Psi_{0}+\bar{a}^{2}(3+4\bar{a}b)\Psi_{1}+3\bar{a}(1+3\bar{a}b+2\bar{a}^{2}b^{2})\Psi_{2}\\\
&\hskip
113.81102pt+(1+6\bar{a}b+9\bar{a}^{2}b^{2}+4\bar{a}^{3}b^{3})\Psi_{3}+b(1+\bar{a}b)^{3}\Psi_{4},\\\
&\Psi_{4}^{\prime}\rightarrow\bar{a}^{4}\Psi_{0}+4\bar{a}^{3}(1+\bar{a}b)\Psi_{1}+6\bar{a}^{2}(1+\bar{a}b)^{2}\Psi_{2}+4\bar{a}(1+\bar{a}b)^{3}\Psi_{3}+(1+\bar{a}b)^{4}\Psi_{4},\\\
&\phi_{2}^{\prime}\rightarrow\bar{a}^{2}\phi_{0}+2\bar{a}(1+\bar{a}b)\phi_{1}+(1+\bar{a}b)^{2}\phi_{2},\\\
&\phi_{1}^{\prime}\rightarrow\bar{a}\phi_{0}+(2\bar{a}b+1)\phi_{1}+b(1+\bar{a}b)\phi_{2},\\\
&\phi_{0}^{\prime}\rightarrow\phi_{0}+2b\phi_{1}+b^{2}\phi_{2},\end{split}$
(3)

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,

$\begin{split}&\Psi_{4}^{\prime}=\frac{3c}{S}\phi_{2}^{\prime}\phi_{2}^{\prime},\phantom{10000}\Psi_{3}^{\prime}=\frac{3c}{S}\phi_{1}^{\prime}\phi_{2}^{\prime},\phantom{10000}\Psi_{2}^{\prime}=\frac{c}{S}(\phi_{0}^{\prime}\phi_{2}^{\prime}+2\phi_{1}^{\prime}\phi_{1}^{\prime}),\\\
&\phantom{100000000}\Psi_{1}^{\prime}=\frac{3c}{S}\phi_{0}^{\prime}\phi_{1}^{\prime},\phantom{10000}\Psi_{0}^{\prime}=\frac{3c}{S}\phi_{0}^{\prime}\phi_{0}^{\prime}.\end{split}$
(4)

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

$\begin{split}&\phi_{2}^{\prime}=\phi_{2},\\\
&\phi_{1}^{\prime}=\phi_{1}+b\phi_{2},\\\
&\phi_{0}^{\prime}=\phi_{0}+2b\phi_{1}+b^{2}\phi_{2}.\end{split}$ (5)

For the condition $\phi_{0}+2b\phi_{1}+b^{2}\phi_{2}=0$, if $b$ has a double
root, we can use a second class null rotation to make
$\phi_{0}^{\prime}=\phi_{1}^{\prime}=0$. Accordingly, the gravitational field
constructed from Maxwell field from (2) in the new frame system is manifest of
Type N. If $\phi_{0}+2b\phi_{1}+b^{2}\phi_{2}=0$ has two distinct roots, then
we can always use another first class null rotation to make
$\phi_{0}^{\prime\prime}=\phi_{2}^{\prime\prime}=0$. Accordingly, the
gravitational solution is of type D.

##  3 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 $x^{\mu}=(u,r,z,\bar{z})$ [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

$\text{d}s^{2}=-2\text{d}u\text{d}r+2r^{2}\text{d}z\text{d}\bar{z}.$ (6)

The only non-vanishing NP variables are $\rho=-\frac{1}{r}$ and
$L^{\bar{z}}=\frac{1}{r}$. Correspondingly, the directional derivatives are
given by

$\displaystyle D=\frac{\partial}{\partial
r},\quad\Delta=\frac{\partial}{\partial
u},\quad\delta=\frac{1}{r}\frac{\partial}{\partial\bar{z}}.$ (7)

We define $\partial=\partial_{z}$ and $\bar{\partial}=\partial_{\bar{z}}$ for
notational brevity.

###  3.1 Maxwell theory

Maxwell’s equations in the flat null coordinates are organized as

$\partial_{r}(r^{2}\phi_{1})=r\partial\phi_{0},\quad\partial_{r}(r\phi_{2})=\partial\phi_{1},\quad\partial_{u}\phi_{0}=\frac{1}{r}\bar{\partial}\phi_{1},\quad\partial_{u}\phi_{1}=\frac{1}{r}\bar{\partial}\phi_{2}.$
(8)

Suppose that $\phi_{0}$ is given as initial data in series expansion,

$\phi_{0}=\frac{\phi_{0}^{0}}{r^{3}}+\sum_{i=1}^{\infty}\frac{\phi_{0}^{i}}{r^{i+3}}.$
(9)

Then $\phi_{1}$ and $\phi_{2}$ can be solved as

$\displaystyle\phi_{1}=\frac{\phi_{1}^{0}}{r^{2}}-$
$\displaystyle\frac{\partial\phi_{0}^{0}}{r^{3}}-\sum_{i=1}^{\infty}\frac{1}{(i+1)}\frac{\partial\phi_{0}^{i}}{r^{i+3}},$
(10)
$\displaystyle\phi_{2}=\frac{\phi_{2}^{0}}{r}-\frac{\partial\phi_{1}^{0}}{r^{2}}+$
$\displaystyle\frac{\partial^{2}\phi_{0}^{0}}{2r^{3}}+\sum_{i=1}^{\infty}\frac{1}{(i+1)(i+2)}\frac{\partial^{2}\phi_{0}^{i}}{r^{i+3}}.$
(11)

The evolution of the initial data is controlled by

$\begin{split}&\partial_{u}\phi_{1}^{0}=\bar{\partial}\phi_{2}^{0},\\\
&\partial_{u}\phi_{0}^{0}=\bar{\partial}\phi_{1}^{0},\\\
&\partial_{u}\phi_{0}^{i}=-\frac{1}{i}\bar{\partial}\partial\phi_{0}^{i-1},\quad
i\geq 1.\end{split}$ (12)

###  3.2 Linearized gravity theory

Bianchi identities after the linearization are given by222We use $\psi_{i}$ to
denote the Weyl scalars in the linearized theory.

$\begin{split}&\partial_{r}(r^{4}\psi_{1})=r^{3}\partial\psi_{0},\quad\partial_{r}(r^{3}\psi_{2})=r^{2}\partial\psi_{1},\quad\partial_{r}(r^{2}\psi_{3})=r\partial\psi_{2},\quad\partial_{r}(r\psi_{4})=\partial\psi_{3},\\\
&\partial_{u}\psi_{0}=\frac{1}{r}\bar{\partial}\psi_{1},\quad\partial_{u}\psi_{1}=\frac{1}{r}\bar{\partial}\psi_{2},\quad\partial_{u}\psi_{2}=\frac{1}{r}\bar{\partial}\psi_{3},\quad\partial_{u}\psi_{3}=\frac{1}{r}\bar{\partial}\psi_{4}.\end{split}$
(13)

Suppose that $\psi_{0}$ is given as initial data in series expansion,

$\psi_{0}=\frac{\psi_{0}^{0}}{r^{5}}+\sum_{i=1}^{\infty}\frac{\psi_{0}^{i}}{r^{i+5}}.$
(14)

Then $\psi_{1}$-$\psi_{4}$ can be solved as

$\begin{split}\psi_{1}=\frac{\psi_{1}^{0}}{r^{4}}-&\frac{\partial\psi_{0}^{0}}{r^{5}}-\sum_{i=1}^{\infty}\frac{1}{(i+1)}\frac{\partial\psi_{0}^{i}}{r^{i+5}},\\\
\psi_{2}=\frac{\psi_{2}^{0}}{r^{3}}-\frac{\partial\psi_{1}^{0}}{r^{4}}+&\frac{\partial^{2}\psi_{0}^{0}}{2r^{5}}+\sum_{i=1}^{\infty}\frac{1}{(i+1)(i+2)}\frac{\partial^{2}\psi_{0}^{i}}{r^{i+5}},\\\
\psi_{3}=\frac{\psi_{3}^{0}}{r^{2}}-\frac{\partial\psi_{2}^{0}}{r^{3}}+\frac{\partial^{2}\psi_{1}^{0}}{2r^{4}}-&\frac{\partial^{3}\psi_{0}^{0}}{6r^{5}}-\sum_{i=1}^{\infty}\frac{1}{(i+1)(i+2)(i+3)}\frac{\partial^{3}\psi_{0}^{i}}{r^{i+5}},\\\
\psi_{4}=\frac{\psi_{4}^{0}}{r}-\frac{\partial\psi_{3}^{0}}{r^{2}}+\frac{\partial^{2}\psi_{2}^{0}}{2r^{3}}-\frac{\partial^{3}\psi_{1}^{0}}{6r^{4}}+&\frac{\partial^{4}\psi_{0}^{0}}{24r^{5}}+\sum_{i=0}^{\infty}\frac{1}{(i+1)(i+2)(i+3)(i+4)}\frac{\partial^{4}\psi_{0}^{i}}{r^{i+5}}.\end{split}$
(15)

The evolution of the initial data is controlled by

$\begin{split}&\partial_{u}\psi_{3}^{0}=\bar{\partial}\psi_{4}^{0},\\\
&\partial_{u}\psi_{2}^{0}=\bar{\partial}\psi_{3}^{0},\\\
&\partial_{u}\psi_{1}^{0}=\bar{\partial}\psi_{2}^{0},\\\
&\partial_{u}\psi_{0}^{0}=\bar{\partial}\psi_{1}^{0},\\\
&\partial_{u}\psi_{0}^{i}=-\frac{1}{i}\bar{\partial}\partial\psi_{0}^{i-1},\quad
i\geq 1.\end{split}$ (16)

##  4 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,

$\begin{split}\psi_{4}=F\phi_{2}\phi_{2},\quad&\psi_{3}=F\phi_{1}\phi_{2},\quad\psi_{2}=\frac{1}{3}F(\phi_{0}\phi_{2}+2\phi_{1}\phi_{1}),\\\
\quad&\psi_{1}=F\phi_{0}\phi_{1},\quad\psi_{0}=F\phi_{0}\phi_{0},\end{split}$
(17)

where $F$ can be expanded in inverse powers of $r$ as

$F=F_{0}r+\sum_{i=1}^{\infty}\frac{F_{i}}{r^{i-1}}.$ (18)

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,

$\begin{split}&\phi_{1}^{0}\partial_{u}(F_{0}\phi_{2}^{0})=\phi_{2}^{0}\bar{\partial}(F_{0}\phi_{2}^{0}),\\\
&2(\phi_{1}^{0})^{2}\partial_{u}F_{0}+\phi_{0}^{0}\partial_{u}(F_{0}\phi_{2}^{0})=-F_{0}\bar{\partial}(\phi_{1}^{0}\phi_{2}^{0})+3\phi_{2}^{0}\bar{\partial}(F_{0}\phi_{1}^{0}),\\\
&3\partial_{u}F_{0}\phi_{1}^{0}\phi_{0}^{0}=-2F_{0}\phi_{0}^{0}\bar{\partial}\phi_{2}^{0}+F_{0}\phi_{2}^{0}\bar{\partial}\phi_{0}^{0}+\phi_{0}^{0}\phi_{2}^{0}\bar{\partial}F_{0}+F_{0}\phi_{1}^{0}\bar{\partial}\phi_{1}^{0}+2(\phi_{1}^{0})^{2}\bar{\partial}F_{0},\\\
&\phi_{0}^{0}\partial_{u}(\phi_{0}^{0}F_{0})=\phi_{1}^{0}\bar{\partial}(F_{0}\phi_{0}^{0}).\end{split}$
(19)

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, $\phi_{1}^{0}=\phi_{0}^{0}=\bar{\partial}F_{0}=0,\,\phi_{2}^{0}\neq
0$ and
$\phi_{2}^{0}=\phi_{0}^{0}=\partial_{u}F_{0}=\bar{\partial}F_{0}=0,\,\phi_{1}^{0}\neq
0$. 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

$\begin{split}&\phi_{0}=\sum_{i=1}^{\infty}\frac{\phi_{0}^{i}}{r^{i+3}},\\\
&\phi_{1}=-\sum_{i=1}^{\infty}\frac{1}{(i+1)}\frac{\partial\phi_{0}^{i}}{r^{i+3}},\\\
&\phi_{2}=\frac{\phi_{2}^{0}(u,z)}{r}+\sum_{i=1}^{\infty}\frac{1}{(i+1)(i+2)}\frac{\partial^{2}\phi_{0}^{i}}{r^{i+3}}.\end{split}$
(20)

Then, WDC relations yield that

$\begin{split}&\psi_{4}=\frac{F_{0}\phi_{2}^{0}\phi_{2}^{0}}{r}+O(r^{-4}),\quad\psi_{3}=-\frac{F_{0}\phi_{2}^{0}\partial\phi_{0}^{1}}{2r^{4}}+O(r^{-5}),\quad\psi_{2}=\frac{F_{0}\phi_{2}^{0}\phi_{0}^{1}}{r^{4}}+O(r^{-5}),\\\
&\phantom{100000000000}\psi_{1}=-\frac{F_{0}\phi_{0}^{1}\partial\phi_{0}^{1}}{r^{7}}+O(r^{-8}),\quad\psi_{0}=\frac{F_{0}\phi_{0}^{1}\phi_{0}^{1}}{r^{7}}+O(r^{-8}).\end{split}$
(21)

Clearly, the WDC construction set that
$\psi_{3}^{0}=\psi_{2}^{0}=\psi_{1}^{0}=\psi_{0}^{0}=\psi_{0}^{1}=0$. Hence,
$\psi_{0}$ begins at order $O(r^{-7})$. As evident from the solution space of
the linearized gravity (15), those conditions lead to that all the Weyl
scalars should begin at order $O(r^{-7})$ except for $\psi_{4}$. Consequently,
the expression of $\psi_{2}$ in (21) indicates that $\phi_{0}^{1}=0$. In fact,
since $\phi_{2}^{0}\neq 0$, the leading term of $\psi_{2}$ is
$\frac{F_{0}\phi_{2}^{0}\phi_{0}^{i}}{r^{i+3}}$, and the leading term of
$\psi_{0}$ is $\frac{F_{0}\phi_{0}^{i}\phi_{0}^{i}}{r^{2i+5}}$ from the
asymptotic WDC formula (21), where $\phi_{0}^{i}$ is the leading non-vanishing
order of $\phi_{0}$. However, in the solution space (15), it is clear that
$\psi_{2}$ and $\psi_{0}$ should begin at the same order when
$\psi_{2}^{0}=0$. This yields that $\phi_{0}^{i}=0$ for any $i$, namely
$\phi_{0}=0$. Hence, $\phi_{1}=0$. 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

$\begin{split}&\phi_{0}=\phi_{1}=0,\quad\phi_{2}=\frac{\phi_{2}^{0}(u,z)}{r},\\\
&\psi_{0}=\psi_{1}=\psi_{2}=\psi_{3}=0,\quad\psi_{4}=\frac{\psi_{4}^{0}(u,z)}{r},\\\
&F=F_{0}r,\quad
F_{0}=\frac{\psi_{4}^{0}}{\phi_{2}^{0}\phi_{2}^{0}}.\end{split}$ (22)

Going back to the original WDC relations in (2), we obtain
$S=\frac{3c\phi_{2}^{0}\phi_{2}^{0}}{\psi_{4}^{0}r}$, 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

$\begin{split}&\phi_{0}=\frac{\phi_{0}^{1}(z,\bar{z})}{r^{4}}+O(r^{-5}),\\\
&\phi_{1}=\frac{\phi_{1}^{0}(z)}{r^{2}}-\frac{1}{2}\frac{\partial\phi_{0}^{1}}{r^{4}}+O(r^{-5}),\\\
&\phi_{2}=-\frac{\partial\phi_{1}^{0}(z)}{r^{2}}+\frac{1}{6}\frac{\partial^{2}\phi_{0}^{1}}{r^{4}}+O(r^{-5}).\end{split}$
(23)

The WDC relations imply that

$\begin{split}&\psi_{0}=\frac{F_{0}\phi_{0}^{1}\phi_{0}^{1}}{r^{7}}+O(r^{-8}),\quad\psi_{1}=\frac{F_{0}\phi_{0}^{1}\phi_{1}^{0}}{r^{5}}+O(r^{-6}),\quad\psi_{2}=\frac{2F_{0}\phi_{1}^{0}\phi_{1}^{0}}{3r^{3}}+O(r^{-4}),\\\
&\phantom{10000000000}\psi_{3}=-\frac{F_{0}\phi_{1}^{0}\partial\phi_{1}^{0}}{r^{3}}O(r^{-4}),\quad\psi_{4}=\frac{F_{0}(\partial\phi_{1}^{0})^{2}}{r^{3}}+O(r^{-4}).\end{split}$
(24)

To be a solution of linearized gravity, one must set $F_{0}=F_{0}(z)$, and
$\psi_{4}^{0}=\psi_{3}^{0}=\psi_{1}^{0}=\psi_{0}^{0}=\psi_{0}^{1}=0$. Because
of $\phi_{1}^{0}\neq 0$ and $\phi_{2}^{0}=\phi_{0}^{0}=0$, the leading order
of $\psi_{1}$ is $\frac{F_{0}\phi_{1}^{0}\phi_{0}^{i}}{r^{i+4}}$, while that
of $\psi_{0}$ is $\frac{F_{0}\phi_{0}^{i}\phi_{0}^{i}}{r^{2i+5}}$. Similar to
the case of the previous option, one must have that $\phi_{0}=0$. Then the
solutions of the Maxwell theory are

$\phi_{0}=0,\quad\phi_{1}=\frac{\phi_{1}^{0}(z)}{r^{2}},\quad\phi_{2}=-\frac{\partial\phi_{1}^{0}}{r^{2}}.$
(25)

The gravitational fields constructed from the WDC relations are

$\psi_{0}=\psi_{1}=0,\quad\psi_{2}=\frac{2}{3}\frac{F_{0}(\phi_{1}^{0})^{2}}{r^{3}},\quad\psi_{3}=-\frac{F_{0}\phi_{1}^{0}\partial\phi_{1}^{0}}{r^{3}},\quad\psi_{4}=\frac{F_{0}(\partial\phi_{1}^{0})^{2}}{r^{3}}.$
(26)

The solution of the linearized gravitational theory (15) finally determines
that

$\phi_{1}^{0}=\frac{k_{2}}{(z+k_{3})^{2}},\quad F_{0}=\frac{k_{1}}{z+k_{3}},$
(27)

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 $\phi_{2}=0$. Then, the gravitational solution is in the
manifest form of type D.

##  5 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.

###  5.1 Type N solution

In this subsection, we will demonstrate that, when choosing the boundary
conditions $\phi_{1}^{0}=\phi_{0}^{0}=0$ and $\phi_{2}^{0}\neq 0$ 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 $\phi_{0}^{0}=\phi_{1}^{0}=0$, $\phi_{2}^{0}\neq 0$, then
$\phi_{0}^{1}=0$.

Proof: In this case, the leading terms of the Maxwell scalars on an
asymptotically flat background spacetime are

$\phi_{2}=\frac{\phi_{2}^{0}}{r}+O(r^{-3}),\quad\phi_{1}=-\frac{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\eth$\kern
0.0pt}}}\phi_{0}^{1}}{2r^{4}}+O(r^{-5}),\quad\phi_{0}=\frac{\phi_{0}^{1}}{r^{4}}+O(r^{-5}).$
(28)

where the definitions of the $\eth$ and  $\eth$ operators are presented below
in (35). The WDC relations yield the Weyl scalars as

$\Psi_{2}=\frac{F_{0}\phi_{2}^{0}\phi_{0}^{1}}{r^{4}}+O(r^{-5}),\quad\Psi_{1}=-\frac{F_{0}\phi_{0}^{1}\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\eth$\kern
0.0pt}}}\phi_{0}^{1}}{2r^{7}}+O(r^{-8}),\quad\Psi_{0}=\frac{F_{0}(\phi_{0}^{1})^{2}}{r^{7}}+O(r^{-8}).$
(29)

However, the two radial equations from the Bianchi identity (68) and (69),
with asymptotically flat fall-off conditions [19], determine that
$\Psi_{2}={\cal O}(r^{-7})$. This requires that $\phi_{0}^{1}=0$.

Lemma 2: If $\phi_{1}^{0}=0$ and $\phi_{0}=0$, 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
$\Psi_{0}=\Psi_{1}=0$. Therefore, the Goldberg-Sachs theorem yields that
$\sigma=0$, hence $\rho=-\frac{1}{r}$. The Maxwell’s equation (89) yields that
$\phi_{1}=0$. Consequently, both the Maxwell and the corresponding
gravitational field are type N and $l$ is the principle null direction.

Lemma 3: If $\phi_{1}^{0}=\phi_{0}^{0}=0$, and
$\phi_{0}^{1}=...=\phi_{0}^{i}=0$, then $\phi_{1}^{1}=...=\phi_{1}^{i}=0$ for
$i\geq 1$.

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
$\phi_{1}^{0}=\phi_{0}^{0}=0$ require that the solutions are of type N and $l$
is the principle null direction. We know from the WDC relations that
$\Psi_{2}=\frac{1}{3}F\phi_{2}\phi_{0}+\frac{2}{3}F(\phi_{1})^{2}$, therefore
the leading term of $\Psi_{2}$ could be either
$\frac{F_{0}\phi_{2}^{0}\phi_{0}^{i}}{3r^{i+3}}$ or
$\frac{2F_{0}(\phi_{1}^{j})^{2}}{3r^{2j+3}}$, where $\phi_{0}^{i}$ and
$\phi_{1}^{j}$ are the leading terms of $\phi_{0}$ and $\phi_{1}$,
respectively. The Lemma 3 requires that the second term is always order-
suppressed. Then the leading term of $\Psi_{2}$ is
$\frac{F_{0}\phi_{2}^{0}\phi_{0}^{i}}{3r^{i+3}}$. While the WDC relation
yields that the leading term of $\Psi_{0}$ is
$\frac{F_{0}(\phi_{0}^{i})^{2}}{r^{2i+5}}$. Thus the peeling-off property
encoded in the radial equations (68) and (69) yields that $\phi_{0}^{i}=0$ for
any $i>0$, 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

$\Psi_{0}=0,\quad\Psi_{1}=0,\quad\Psi_{2}=0,\quad\Psi_{3}=0.$ (30)

The solution space is given by

$\begin{split}&\sigma=0,\quad\rho=-\frac{1}{r},\quad
L^{\bar{z}}=\frac{P}{r},\quad\alpha=\frac{\alpha^{0}}{r},\quad\alpha^{0}=\frac{1}{2}\bar{P}\partial\ln
P,\quad\beta=-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\alpha$\kern 0.0pt}}},\\\ &\tau=0,\quad\omega=0,\quad
X^{A}=0,\quad\lambda=0,\quad\mu=\frac{\mu^{0}}{r},\quad\mu^{0}=-\eth\alpha^{0}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\eth$\kern
0.0pt}}}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\alpha$\kern 0.0pt}}}^{0}=\eth\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\eth$\kern 0.0pt}}}\varphi,\\\
&\gamma=\gamma^{0},\quad\gamma^{0}=-\frac{1}{2}\partial_{u}\ln\bar{P},\quad\nu=\nu^{0},\quad\nu^{0}=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\eth$\kern
0.0pt}}}(\gamma^{0}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\gamma$\kern 0.0pt}}}^{0})=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\eth$\kern 0.0pt}}}\partial_{u}\varphi,\\\
&U=-(\gamma^{0}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\gamma$\kern
0.0pt}}}^{0})r+\mu^{0},\quad\Psi_{4}=\frac{\Psi_{4}^{0}}{r},\quad\Psi_{4}^{0}=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\eth$\kern
0.0pt}}}^{2}\partial_{u}\varphi,\end{split}$ (31)

where we define $P\bar{P}=2e^{-2\varphi}$. The constraints on $\varphi$ from
the NP equations are

$\eth^{2}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\eth$\kern 0.0pt}}}\varphi=0=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\eth$\kern 0.0pt}}}^{2}\eth\varphi,$ (32)

which yields the following equation

$\eth\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\eth$\kern
0.0pt}}}\varphi=2e^{-2\varphi}\partial\bar{\partial}\varphi=f_{0}(u).$ (33)

Note that

$\eth\Psi_{4}^{0}=(\partial_{u}+4\gamma^{0}+2\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\gamma$\kern
0.0pt}}}^{0})\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\eth$\kern 0.0pt}}}^{2}\eth\varphi=0,$ (34)

which does not lead to a new constraint on $\varphi$. The “$\eth$” operator is
defined by

$\begin{split}&\eth\eta^{(s)}=P\bar{P}^{-s}\bar{\partial}(\bar{P}^{s}\eta^{(s)})=P\bar{\partial}\eta^{(s)}+2s\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\alpha$\kern
0.0pt}}}^{0}\eta^{(s)},\\\ &\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\eth$\kern
0.0pt}}}\eta^{(s)}=\bar{P}P^{s}\partial(P^{-s}\eta^{(s)})=\bar{P}\partial\eta^{(s)}-2s\alpha^{0}\eta^{(s)},\end{split}$
(35)

where $s$ is the spin weight of the field $\eta^{(s)}$. The spin weights of
relevant fields are listed in Table 1.

Table 1: Spin weights | $\eth$ | $\partial_{u}$ | $\alpha^{0}$ | $\gamma^{0}$ | $\nu^{0}$ | $\mu^{0}$ | $\sigma^{0}$ | $\lambda^{0}$ | $\Psi^{0}_{4}$ | $\Psi^{0}_{3}$ | $\Psi^{0}_{2}$ | $\Psi^{0}_{1}$ | $\Psi_{0}^{0}$ | $\phi_{2}$ | $\phi_{1}$ | $\phi_{0}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
s | $1$ | $0$ | $-1$ | $0$ | $-1$ | $0$ | $2$ | $-2$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $-1$ | $0$ | $1$

The operators $\eth$, $\overline{\eth}$ raise and lower the spin weight by one
unit. Note that $P$ is of spin weight $1$ and “holomorphic”,
$\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\eth$\kern
0.0pt}}}P=0$ and the commutator of the operators is

$[\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\eth$\kern
0.0pt}}},\eth]\eta^{(s)}=\frac{s}{2}R\eta^{(s)},$ (36)

with $R=-4\mu^{0}$. We also have their commutation relation with the time
derivative [33],

$[\partial_{u},\eth]\eta^{(s)}=-2(\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\gamma$\kern
0.0pt}}}^{0}\eth+s\eth\gamma^{0})\eta^{(s)},\quad[\partial_{u},\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\eth$\kern
0.0pt}}}]\eta^{(s)}=-2(\gamma^{0}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\eth$\kern 0.0pt}}}-s\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\eth$\kern
0.0pt}}}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\gamma$\kern 0.0pt}}}^{0})\eta^{(s)}.$ (37)

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 $\mu_{0}=0,\pm 1$.

With the type N spacetime (31) as the background and the conditions
$\phi_{0}=\phi_{1}=0$, it is straightforward to solve the Maxwell’s equations.
The solution is

$\phi_{2}=\frac{\phi_{2}^{0}}{r},\quad\eth\phi_{2}^{0}=0,\quad\phi_{2}^{0}=\bar{P}f_{\phi}(u,z),$
(38)

where $f_{\phi}(u,z)$ is an arbitrary function. The WDC relation
$\Psi_{4}=F\phi_{2}\phi_{2}$ yields

$F=\frac{\Psi_{4}^{0}}{\phi_{2}^{0}\phi_{2}^{0}}r.$ (39)

From the original WDC relations in (2),
$S=\frac{3c\phi_{2}^{0}\phi_{2}^{0}}{\Psi_{4}^{0}r}$. Note that
$\eth\Psi_{4}^{0}=0$. This yields that
$\Psi_{4}^{0}=\bar{P}^{2}f_{\Psi}(u,z)$. Hence, $S$ must be of the form
$\frac{\zeta(u,z)}{r}$, which is a solution of the equation of motion for a
scalar field on both Minkowski spacetime and the curved spacetime (31).

###  5.2 Type D solution

In this subsection, we will demonstrate for the case
$\phi_{2}^{0}=\phi_{0}^{0}=0$, $\phi_{1}^{0}\neq 0$. The asymptotic form of
the Maxwell fields are now

$\phi_{0}=\frac{\phi_{0}^{1}}{r^{4}}+O(r^{-5}),\quad\phi_{1}=\frac{\phi_{1}^{0}}{r^{2}}+O(r^{-4}),\quad\phi_{2}=-\frac{\bar{\eth}\phi_{1}^{0}}{r^{2}}+O(r^{-3}).$
(40)

The WDC relations lead to

$\Psi_{0}=\frac{F_{0}(\phi_{0}^{1})^{2}}{r^{7}}+O(r^{-8}),\quad\Psi_{1}=\frac{F_{0}\phi_{1}^{0}\phi_{0}^{1}}{r^{5}}+O(r^{-6}).$
(41)

The peeling-off property encoed in (68) requires that $\phi_{0}^{1}=0$ when
$\phi_{1}^{0}\neq 0$. Following the same argument, any $\phi_{0}^{i}$ are thus
equal to zero, therefore $\phi_{0}=0$. Then, the WDC relations determine that
$\Psi_{0}=\Psi_{1}=0$, and the Goldberg-Sachs theorem yields that $\sigma=0$,
hence $\rho=-\frac{1}{r}$. Substituting these results back into the Maxwell’s
equations, we obtain

$\phi_{1}=\frac{\phi_{1}^{0}}{r^{2}}.$ (42)

We can use a first class of null rotation to turn off $\phi_{2}$. Then $l$ and
the rotated basis $n^{\prime}$ 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 $\pi$, which are out of the NU
gauge.

Nevertheless, we will still adopt the NU gauge to derive the gravitational
solution space. Correspondingly, we will have

$\sigma=0,\quad\Psi_{0}=0,\quad\Psi_{1}=0,\quad\Psi_{3}={\cal
O}(r^{-3}),\quad\Psi_{4}={\cal O}(r^{-3}).$ (43)

The solution space of gravitational theory is given by

$\begin{split}&\sigma=0,\quad\rho=-\frac{1}{r},\quad
L^{\bar{z}}=\frac{P}{r},\quad\alpha=\frac{\alpha^{0}}{r},\quad\alpha^{0}=\frac{1}{2}\bar{P}\partial\ln
P,\quad\beta=-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\alpha$\kern 0.0pt}}},\\\ &\tau=0,\quad\omega=0,\quad
X^{A}=0,\quad\lambda=0,\quad\Psi_{3}=0,\quad\Psi_{4}=0,\\\
&\mu=\frac{\mu^{0}}{r}-\frac{\Psi_{2}^{0}}{r^{2}},\quad\mu^{0}=-\eth\alpha^{0}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\eth$\kern
0.0pt}}}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\alpha$\kern 0.0pt}}}^{0}=\eth\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\eth$\kern
0.0pt}}}\varphi,\quad\varphi=\varphi_{0}(u)+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\varphi$\kern
0.0pt}}}(z,\bar{z}),\\\
&\Psi_{2}=\frac{\Psi_{2}^{0}(u)}{r^{3}},\quad\Psi_{2}^{0}=C_{\Psi}e^{-3\varphi_{0}},\quad\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\Psi$\kern
0.0pt}}}_{2}^{0}=\Psi_{2}^{0},\\\
&\gamma=\gamma^{0}-\frac{\Psi_{2}^{0}}{2r^{2}},\quad\gamma^{0}=-\frac{1}{2}\partial_{u}\ln\bar{P},\quad\nu=0,\\\
&U=-(\gamma^{0}+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\gamma$\kern 0.0pt}}}^{0})r+\mu^{0}-\frac{\Psi_{2}^{0}}{r},\end{split}$
(44)

where  $\varphi$ is constrained by $e^{-2\hbox{\vbox{\hrule height=0.5pt\kern
0.90417pt\hbox{\kern 0.0pt$\varphi$\kern
0.0pt}}}}\partial\bar{\partial}\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\varphi$\kern 0.0pt}}}=C_{\varphi}$, and
$C_{\varphi}$, $C_{\Psi}$ are two arbitrary constants.

With the spacetime (44) as the background and the conditions
$\phi_{0}=0=\phi_{2}^{0}$, it is straightforward to solve the Maxwell’s
equations as

$\phi_{1}=\frac{\phi_{1}^{0}}{r^{2}},\quad\phi_{2}=-\frac{\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\eth$\kern
0.0pt}}}\phi_{1}^{0}}{r^{2}},$ (45)

where $\phi_{1}^{0}$ is constrained by

$\eth\phi_{1}^{0}=0,\quad\partial_{u}\phi_{1}^{0}+2(\gamma^{0}+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\gamma$\kern
0.0pt}}}^{0})\phi_{1}^{0}=0.$ (46)

The WDC relations incorporated with the gravity solution (44), yield that
$\phi_{2}=0$. Hence, $\phi_{1}^{0}$ can be fixed as

$\phi_{1}^{0}=C_{\phi}e^{-2\varphi_{0}}.$ (47)

Finally, the scalar field in the WDC relations is determined as

$F=\frac{3e^{\varphi_{0}}C_{\Psi}}{2C_{\phi}^{2}}r.$ (48)

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

$\phi_{2}=0=\phi_{0},\quad\phi_{1}=\frac{C_{\phi}}{r^{2}},$ (49)

which leads to $\varphi_{0}=0$ for the gravitational solution by the WDC
relations. Hence,

$F=\frac{3C_{\Psi}}{2C_{\phi}^{2}}r,$ (50)

and the corresponding scalar function $S$ satisfies the equation of motion on
the flat background.

##  6 Concluding remarks

In this paper, we study the WDC constraints on the solutions of Maxwell theory
and gravity. This is an extension of the original scope of the WDC by
selecting from a larger solution for the part that satisfies the WDC formula.
We find that two straightforward solutions to the asymptotic WDC constraints
lead to interesting truncation of the solution space. In particular, our
computation shows that, when turning off the leading order of the Weyl scalar
in the NP formalism in the NU gauge, WDC constraints will turn off the Weyl
scalar at any order. We also present a linearized version of the WDC by
checking the relations of the linearized Weyl tensor and the Maxwell field
strength tensor. In this way, one may better appreciate the connection of the
WDC and the Kerr-Schild double copy formula and may seek for a unified
treatment for the classical double copy relation, which could be a future
direction explored elsewhere.

## Acknowledgments

This work is supported in part by the National Natural Science Foundation of
China (NSFC) under Grants No. 11905156 and No. 11935009.

##  Appendix A NP formalism and NU gauge

The NP formalism [18] is a special tetrad formalism in four dimensions with
all basis vectors null, denoted as
$e_{1}=l=e^{2},\;e_{2}=n=e^{1},\;e_{3}=m=-e^{4},\;e_{4}=\bar{m}=-e^{3}$. The
null basis vectors satisfy the following orthogonality and normalization
conditions,

$l\cdot m=l\cdot\bar{m}=n\cdot m=n\cdot\bar{m}=0,\quad l\cdot n=1,\quad
m\cdot\bar{m}=-1,$ (51)

where $l$ and $n$ are real, while $m$ and $\bar{m}$ are complex conjugates of
each other. The 24 components of spin connection are interpreted as 12 complex
scalars labeled by Greek symbols. The components of the Weyl tensor are
represented by five complex scalars, denoted as $\Psi_{0}$, $\Psi_{1}$,
$\Psi_{2}$, $\Psi_{3}$, $\Psi_{4}$. We would refer to [35] for the full
notations. We will deal with vacuum solution in this work where the Ricci
tensor vanishes.

In the NP formalism, it is of great convenience to impose the UN gauge [19],
where one chooses

$\displaystyle\pi=\kappa=\epsilon=0,\,\,\;\;\rho=\bar{\rho},\;\;\,\,\tau=\bar{\alpha}+\beta.$
(52)

The geometric meaning of the NU gauge is that $l$ is tangent to a null
geodesic with affine parameter and is the gradient of a scalar field.
Moreover, other null basis vectors are parallel transported along $l$. Then,
the coordinate system is selected as follows. The scalar field defining $l$ is
chosen as coordinate $u=x^{1}$ and the affine parameter of the geodesic of $l$
is chosen as coordinate $r=x^{2}$. The rest two angular coordinates are the
stereographic coordinates $A=(z,\bar{z})$ and they are related to the usual
angular variables $(\theta,\phi)$ by $z=\cot\frac{\theta}{2}e^{i\phi}$. In
this coordinate system, the tetrad and the co-tetrad satisfying conditions in
(51) must have the forms

$\begin{split}&n=\frac{\partial}{\partial u}+U\frac{\partial}{\partial
r}+X^{A}\frac{\partial}{\partial x^{A}},\\\ &l=\frac{\partial}{\partial r},\\\
&m=\omega\frac{\partial}{\partial r}+L^{A}\frac{\partial}{\partial
x^{A}}.\end{split}$ (53)

The basis vectors are assigned with special symbols as diretional derivatives,

$\displaystyle
D=l^{\mu}\partial_{\mu},\;\;\;\;\Delta=n^{\mu}\partial_{\mu},\;\;\;\;\delta=m^{\mu}\partial_{\mu}.$
(54)

The NP equations are organized as follows:

Radial equations

$\displaystyle D\rho=\rho^{2}+\sigma\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\sigma$\kern 0.0pt}}},$ (55) $\displaystyle
D\sigma=2\rho\sigma+\Psi_{0},$ (56) $\displaystyle
D\tau=\tau\rho+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\tau$\kern 0.0pt}}}\sigma+\Psi_{1},$ (57) $\displaystyle
D\alpha=\rho\alpha+\beta\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\sigma$\kern 0.0pt}}},$ (58) $\displaystyle
D\beta=\alpha\sigma+\rho\beta+\Psi_{1},$ (59) $\displaystyle
D\gamma=\tau\alpha+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\tau$\kern 0.0pt}}}\beta+\Psi_{2},$ (60) $\displaystyle
D\lambda=\rho\lambda+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\sigma$\kern 0.0pt}}}\mu,$ (61) $\displaystyle
D\mu=\rho\mu+\sigma\lambda+\Psi_{2},$ (62) $\displaystyle
D\nu=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\tau$\kern 0.0pt}}}\mu+\tau\lambda+\Psi_{3},$ (63) $\displaystyle
DU=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\tau$\kern
0.0pt}}}\omega+\tau\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\omega$\kern 0.0pt}}}-(\gamma+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\gamma$\kern 0.0pt}}}),$ (64) $\displaystyle
DX^{A}=\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\tau$\kern 0.0pt}}}L^{A}+\tau\bar{L}^{A},$ (65) $\displaystyle
D\omega=\rho\omega+\sigma\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\omega$\kern 0.0pt}}}-\tau,$ (66) $\displaystyle
DL^{A}=\rho L^{A}+\sigma\bar{L}^{A},$ (67) $\displaystyle
D\Psi_{1}-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\delta$\kern 0.0pt}}}\Psi_{0}=4\rho\Psi_{1}-4\alpha\Psi_{0},$ (68)
$\displaystyle D\Psi_{2}-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\delta$\kern
0.0pt}}}\Psi_{1}=3\rho\Psi_{2}-2\alpha\Psi_{1}-\lambda\Psi_{0},$ (69)
$\displaystyle D\Psi_{3}-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\delta$\kern
0.0pt}}}\Psi_{2}=2\rho\Psi_{3}-2\lambda\Psi_{1},$ (70) $\displaystyle
D\Psi_{4}-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\delta$\kern
0.0pt}}}\Psi_{3}=\rho\Psi_{4}+2\alpha\Psi_{3}-3\lambda\Psi_{2},$ (71)

Non-radial equations

$\displaystyle\Delta\lambda=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\delta$\kern 0.0pt}}}\nu-(\mu+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}})\lambda-(3\gamma-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\gamma$\kern 0.0pt}}})\lambda+2\alpha\nu-\Psi_{4},$
(72) $\displaystyle\Delta\rho=\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\delta$\kern 0.0pt}}}\tau-\rho\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}-\sigma\lambda-2\alpha\tau+(\gamma+\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\gamma$\kern
0.0pt}}})\rho-\Psi_{2},$ (73) $\displaystyle\Delta\alpha=\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\delta$\kern
0.0pt}}}\gamma+\rho\nu-(\tau+\beta)\lambda+(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\gamma$\kern
0.0pt}}}-\gamma-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\mu$\kern 0.0pt}}})\alpha-\Psi_{3},$ (74)
$\displaystyle\Delta\mu=\delta\nu-\mu^{2}-\lambda\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}-(\gamma+\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\gamma$\kern 0.0pt}}})\mu+2\beta\nu,$ (75)
$\displaystyle\Delta\beta=\delta\gamma-\mu\tau+\sigma\nu+\beta(\gamma-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\gamma$\kern
0.0pt}}}-\mu)-\alpha\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\lambda$\kern 0.0pt}}},$ (76)
$\displaystyle\Delta\sigma=\delta\tau-\sigma\mu-\rho\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}-2\beta\tau+(3\gamma-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\gamma$\kern 0.0pt}}})\sigma,$ (77)
$\displaystyle\Delta\omega=\delta U+\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\nu$\kern 0.0pt}}}-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\omega$\kern 0.0pt}}}+(\gamma-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\gamma$\kern 0.0pt}}}-\mu)\omega,$ (78)
$\displaystyle\Delta L^{A}=\delta X^{A}-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\lambda$\kern
0.0pt}}}\bar{L}^{A}+(\gamma-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\gamma$\kern 0.0pt}}}-\mu)L^{A},$ (79)
$\displaystyle\delta\rho-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\delta$\kern
0.0pt}}}\sigma=\rho\tau-\sigma(3\alpha-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\beta$\kern 0.0pt}}})-\Psi_{1},$ (80)
$\displaystyle\delta\alpha-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\delta$\kern
0.0pt}}}\beta=\mu\rho-\lambda\sigma+\alpha\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\alpha$\kern 0.0pt}}}+\beta\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\beta$\kern
0.0pt}}}-2\alpha\beta-\Psi_{2},$ (81)
$\displaystyle\delta\lambda-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\delta$\kern 0.0pt}}}\mu=\mu\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\tau$\kern
0.0pt}}}+\lambda(\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\alpha$\kern 0.0pt}}}-3\beta)-\Psi_{3},$ (82)
$\displaystyle\delta\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\omega$\kern 0.0pt}}}-\bar{\delta}\omega=\mu-\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\mu$\kern
0.0pt}}}-(\alpha-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\beta$\kern 0.0pt}}})\omega+(\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\alpha$\kern 0.0pt}}}-\beta)\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\omega$\kern 0.0pt}}},$ (83)
$\displaystyle\delta\bar{L}^{A}-\bar{\delta}L^{A}=(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\alpha$\kern
0.0pt}}}-\beta)\bar{L}^{A}-(\alpha-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\beta$\kern 0.0pt}}})L^{A},$ (84)
$\displaystyle\Delta\Psi_{0}-\delta\Psi_{1}=(4\gamma-\mu)\Psi_{0}-(4\tau+2\beta)\Psi_{1}+3\sigma\Psi_{2},$
(85)
$\displaystyle\Delta\Psi_{1}-\delta\Psi_{2}=\nu\Psi_{0}+(2\gamma-2\mu)\Psi_{1}-3\tau\Psi_{2}+2\sigma\Psi_{3},$
(86)
$\displaystyle\Delta\Psi_{2}-\delta\Psi_{3}=2\nu\Psi_{1}-3\mu\Psi_{2}+(2\beta-2\tau)\Psi_{3}+\sigma\Psi_{4},$
(87)
$\displaystyle\Delta\Psi_{3}-\delta\Psi_{4}=3\nu\Psi_{2}-(2\gamma+4\mu)\Psi_{3}+(4\beta-\tau)\Psi_{4},$
(88)

Maxwell’s equations

$\displaystyle D\phi_{1}-\hbox{\vbox{\hrule height=0.5pt\kern
1.29167pt\hbox{\kern 0.0pt$\delta$\kern
0.0pt}}}\phi_{0}=2\rho\phi_{1}-2\alpha\phi_{0},$ (89) $\displaystyle
D\phi_{2}-\hbox{\vbox{\hrule height=0.5pt\kern 1.29167pt\hbox{\kern
0.0pt$\delta$\kern 0.0pt}}}\phi_{1}=\rho\phi_{2}-\lambda\phi_{0},$ (90)
$\displaystyle\Delta\phi_{0}-\delta\phi_{1}=(2\gamma-\mu)\phi_{0}-2\tau\phi_{1}+\sigma\phi_{2},$
(91)
$\displaystyle\Delta\phi_{1}-\delta\phi_{2}=\nu\phi_{0}-2\mu\phi_{1}-(\hbox{\vbox{\hrule
height=0.5pt\kern 1.29167pt\hbox{\kern 0.0pt$\alpha$\kern
0.0pt}}}-\beta)\phi_{2}.$ (92)

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