# On the geometry of Petrov type II spacetimes

Steffen Aksteiner<EMAIL_ADDRESS>Albert Einstein Institute, Am
Mühlenberg 1, D-14476 Potsdam, Germany  ,  Lars Andersson
<EMAIL_ADDRESS>Albert Einstein Institute, Am Mühlenberg 1, D-14476
Potsdam, Germany  ,  Bernardo Araneda<EMAIL_ADDRESS>Facultad de
Matemática, Astronomía, Física y Computación, Universidad Nacional de Córdoba,
Instituto de Física Enrique Gaviola, CONICET, Ciudad Universitaria, (5000)
Córdoba, Argentina  and  Bernard Whiting<EMAIL_ADDRESS>Department of
Physics, University of Florida, 2001 Museum Road, Gainesville, FL 32611-8440,
USA

###### Abstract.

In general, geometries of Petrov type II do not admit symmetries in terms of
Killing vectors or spinors. We introduce a weaker form of Killing equations
which do admit solutions. In particular, there is an analog of the Penrose-
Walker Killing spinor. Some of its properties, including associated
conservation laws, are discussed. Perturbations of Petrov type II Einstein
geometries in terms of a complex scalar Debye potential yield complex
solutions to the linearized Einstein equations. The complex linearized Weyl
tensor is shown to be half Petrov type N. The remaining curvature component on
the algebraically special side is reduced to a first order differential
operator acting on the potential.

##  1\. Introduction

A remarkable property of vacuum spacetimes of Petrov type D is the existence
of `hidden symmetries', namely, appropriate generalizations of Killing vectors
such as Killing tensors and conformal Killing-Yano tensors. Penrose and Walker
have shown [30] the existence of a valence 2 Killing spinor, from which one
can obtain the above mentioned symmetries. Because of the Goldberg-Sachs
theorem, the vacuum type D condition is equivalent to the existence of two
independent null geodesic congruences that are shear-free. On the other hand,
in general, vacuum spacetimes of Petrov type II do not possess any symmetries
or hidden symmetries. However they do admit a shear-free null geodesic
congruence. In this paper we show that this can be used to define a weaker
version of the Killing equations, and we show that they are solved by a
generalization of the Penrose-Walker Killing spinor.

The construction of solutions to the field equations for linear spinning
fields in terms of scalar, tensorial or spinorial potentials has a long
history and has been widely studied. In the case of the Maxwell field, the
main names associated to this procedure are Debye and Hertz. Although the
usage in the literature is not consistent, see Stewart [26], we shall here
refer to scalar potentials of the above mentioned type as Debye potentials. In
this paper we focus on the spin-2 case, and consider the construction of
solutions to the linearized Einstein equations on backgrounds of Petrov type
II, in terms of Debye potentials. The analogous construction on backgrounds of
Petrov type D, including the Schwarzschild and Kerr spacetimes has been widely
studied, see e.g. [15, 29, 9], and plays an important role in the study of
black hole perturbations [16], [11] and the self-force problem [17], [8]. The
construction of solutions to the linearized Einstein equation by the Debye
potential method on backgrounds of Petrov type II is possible and is closely
analogous to the type D case. Kegeles and Cohen [15] and Stewart [26] have
carried out a systematic study of the Debye potential construction in this
case, and in particular, Stewart calculated the tetrad components of the
linearized Weyl tensor. In this work we show that the linearized Weyl tensor
is half type N111The linearized metric generated from a Debye potential is
naturally complex and therefore there are two Petrov classifications and half
types possible, see below for details..

It is worth recalling briefly the situation for vacuum type D spacetimes. We
shall use the conventions and notation of [18, 19]. A spacetime of Petrov type
D admits two repeated principal null directions $l^{a},n^{a}$, which have
corresponding principal spin dyad $o^{A},\iota^{A}$. In terms of this
principal frame, the only non-vanishing Weyl scalar is $\Psi_{2}$. As was
shown in [30], vacuum spacetimes (for the more general situation see [4]) of
Petrov type D admit a Killing spinor of the form

$\mathring{K}_{AB}=\Psi^{-1/3}_{2}o_{(A}\iota_{B)}$ (1.1)

(the ring `` $\mathring{}$ '' here is notation only intended to separate the
type D case from the type II case that we discuss below), which satisfies the
equation

$\nabla_{A^{\prime}(A}\mathring{K}_{BC)}=0.$ (1.2)

Different kinds of symmetries are associated to the object (1.1), see [12],
[19, Section 6.7]. In particular in vacuum, the vector field defined by

$\mathring{\xi}_{AA^{\prime}}=\nabla^{B}{}_{A^{\prime}}\mathring{K}_{AB}$
(1.3)

is a Killing vector, $\nabla_{(a}\mathring{\xi}_{b)}=0$. In the Kerr-NUT
class, from (1.1) one can also construct a Killing tensor $H_{ab}$ and a
second Killing vector $\eta_{a}=H_{ab}\mathring{\xi}^{b}$, see [12] for
details.

Consider an Einstein spacetime, i.e. the Einstein tensor being proportional to
the metric or equivalently vacuum with cosmological constant, of Petrov type
II. By [13], a spacetime with a Killing spinor of valence 2, cf. eq. (1.2),
has Weyl tensor of type D, N or O, so a type II geometry does not admit a
Killing spinor of valence 2. Instead, we prove in section 3 below:

######  Theorem 1.

Let $(\mathcal{M},g_{ab})$ be a real Einstein spacetime of Petrov type II, and
let $o_{A},\iota_{A}$ be a spin dyad, such that $o_{A}$ is a repeated
principal spinor. Let $\Psi_{i}$ be the corresponding Weyl scalars. Define

$\displaystyle K_{AB}={}$
$\displaystyle\Psi^{-1/3}_{2}o_{(A}\iota_{B)}-\tfrac{1}{3}\Psi^{-4/3}_{2}\Psi_{3}o_{A}o_{B},$
(1.4) $\displaystyle\xi_{AA^{\prime}}={}$
$\displaystyle\nabla^{B}{}_{A^{\prime}}K_{AB}.$ (1.5)

Then

  1. (1)

$K_{AB}$ solves the ``projected'' Killing spinor equation

$o^{A}\nabla_{A^{\prime}(A}K_{BC)}=0.$ (1.6)

  2. (2)

$\xi_{AA^{\prime}}$ solves the ``projected'' Killing equation

$\displaystyle
o^{A}(\nabla_{AA^{\prime}}\xi_{BB^{\prime}}+\nabla_{BB^{\prime}}\xi_{AA^{\prime}})=0.$
(1.7)

Now, the Debye potential construction produces complex solutions to the
linearized Einstein equations, which leads to the possibility of having ``half
types'' for the Weyl tensor. Recall first that in a real, four-dimensional
orientable manifold with a metric of Lorentzian signature, the Hodge star
operator $*$ acting on 2-forms satisfies $*^{2}=-1$, thus it has eigenvalues
$\pm i$. As a consequence, the eigenspaces of $*$ are complex, i.e. a self-
dual (SD) or anti-self-dual (ASD) 2-form is necessarily complex. Any real
2-form can be written as the sum of a SD part and an ASD part, and these
pieces are complex conjugates of each other. Since the Weyl tensor $C_{abcd}$
can be regarded as a tensor-valued 2-form, a similaRózgar discussion applies
to it. Namely, $C_{abcd}$ can be written as the sum of a SD piece
$C^{+}_{abcd}$ and an ASD piece $C^{-}_{abcd}$, where $C^{+}_{abcd}$ and
$C^{-}_{abcd}$ are complex conjugates of each other. In spinor terms (see
section 2 for notation and details), these are
$C^{-}_{abcd}=\Psi_{ABCD}\bar{\epsilon}_{A^{\prime}B^{\prime}}\bar{\epsilon}_{C^{\prime}D^{\prime}}$
and
$C^{+}_{abcd}=\bar{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}\epsilon_{AB}\epsilon_{CD}$,
where $\Psi_{ABCD}$ is the Weyl curvature spinor, see [18, Section 4.6].

On the other hand, if the spacetime metric is complex, the above decomposition
still holds, but the pieces $C^{+}_{abcd}$ and $C^{-}_{abcd}$ are now
independent entities, so one has two independent Weyl spinors $\Psi_{ABCD}$
and $\tilde{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}$, see [19,
Section 6.9]. In particular, the Petrov types of $\Psi_{ABCD}$ and
$\tilde{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}$ are independent; for
example, one part may be algebraically special while the other one is
algebraically general222Note however that due to results by Rózga [23], in
order for a complex 4-dimensional spacetime to admit a real, Lorentzian slice,
the algebraic type of $\Psi_{ABCD}$ and $\tilde{\Psi}_{ABCD}$ must be the
same. See also [31].. This also applies to linearized gravity, where even if
the background metric is real, a complex perturbation will in general have
independent SD and ASD linearized curvatures. In what follows we denote the SD
and ASD linearized curvature spinors for a complex perturbation by
$\dot{\tilde{\Psi}}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}$ and
$\dot{\Psi}_{ABCD}$ respectively, see section 2 for details.

We shall make use of the Geroch-Held-Penrose (GHP) formalism [10]. For
algebraically special spacetimes, the method of adjoint operators introduced
by Wald in [29] (see section 4 for a brief review) can be used to show that if
a scalar field $\chi$ of GHP weight $\\{-4,0\\}$ solves the Debye equation

$\displaystyle\left(\left(\operatorname{\text{}}^{\prime}-\bar{\rho}^{\prime}\right)\left(\operatorname{\text{}}+3\rho\right)-\left(\operatorname{\eth}^{\prime}-\bar{\tau}\right)\left(\operatorname{\eth}+3\tau\right)-3\Psi_{2}\right)\chi=0,$
(1.8)

then the complex tensor field $h_{ab}=\mathcal{S}^{{\dagger}}(\chi)_{ab}$,
where $\mathcal{S}^{{\dagger}}$ is the adjoint of the operator $\mathcal{S}$
defined in eq. (4.2a) below, is a solution to the linearized Einstein vacuum
equations (possibly with cosmological constant). For perturbations of vacuum
type D spacetimes, it was shown in [15] that the ASD linearized Weyl spinor
$\dot{\Psi}_{ABCD}$ has special algebraic structure, viz. it is of Petrov type
N,

$\dot{\Psi}_{ABCD}=o_{A}o_{B}o_{C}o_{D}\dot{\Psi}_{4},$ (1.9)

whereas the SD linearized Weyl spinor may be algebraically general.

Furthermore, while the linearized Weyl scalars associated to a Debye potential
$\mathring{\chi}$ are given in general by fourth order differential operators
applied to $\mathring{\chi}$, it was shown in [15], see also [3], that the
scalar field $\dot{\Psi}_{4}$ in (1.9) is given by the simple expression,

$\dot{\Psi}_{4}=c\mathcal{L}_{\mathring{\xi}}\mathring{\chi},$ (1.10)

with (possibly complex) constant $c$. Here, $\mathcal{L}_{\mathring{\xi}}$ is
the Lie derivative along $\mathring{\xi}^{a}$, which itself is given by (1.5)
for the type D Killing spinor. In the Kerr spacetime, (1.10) is essentially
the time derivative of $\mathring{\chi}$, see also [16]. For Petrov type D,
this reduction was done by Kegeles and Cohen [15] for the vacuum case and by
Torres del Castillo [27] including a cosmological constant333One of the
Authors, B.W., also did this tedious computation including a cosmological
constant in 1983, but did not publish it..

One may think that the remarkably simple structure (1.9)-(1.10) is associated
to the very special symmetry properties of vacuum type D spacetimes, i.e. to
the existence of the ``hidden'' symmetry (1.1) and the two associated
isometries mentioned before. In this note we generalize these results to
Petrov type II spacetimes, which in general do not possess any isometries:

######  Theorem 2.

Consider an Einstein spacetime of Petrov type II with repeated principal
spinor $o^{A}$. Let $h_{ab}$ be a complex solution to the linearized Einstein
vacuum equations generated by a Debye potential $\chi$. Then

  1. (1)

the ASD Weyl spinor of $h_{ab}$ is of Petrov type N,

$\dot{\Psi}_{ABCD}=o_{A}o_{B}o_{C}o_{D}\dot{\Psi}_{4}.$ (1.11)

  2. (2)

the non-vanishing component of (1.11) is given by

$\displaystyle\dot{\Psi}_{4}={}$
$\displaystyle-(\Psi_{2}^{4/3}\xi^{a}\Theta_{a}+3\Psi_{2}^{2}+6\Psi_{2}\Lambda)\chi.$
(1.12)

Here $\xi^{a}$ is given by (1.5), where $K_{AB}$ is the projected Killing
spinor (1.4), $\Theta_{a}$ is the GHP connection and $6\Lambda$ corresponds to
the cosmological constant.

######  Remark 3.

Upon finishing this work, we found a virtually unknown preprint by Jeffryes,
[14], about half-algebraically special geometries and potentials for field
equations. Although Theorem 2 can alternatively be obtained from §8 of that
work, the interpretation of the derivative in terms of a projected Killing
vector has not been given there. In fact the focus of that work was on the
non-linear case coupled to Yang-Mills and the relation to the situation here
is quite intricate, see remark 39 for further details.

In the case of linearized gravity in Minkowski spacetime, it was shown in [28]
that real solutions of the linearized Einstein vacuum equations are in one-to-
one correspondence with complex solutions with half-flat curvature, which in
turn are in one-to-one correspondence with solutions of the scalar wave
equation. The result of Theorem 2 tells us that, while the complex metric
perturbation generated by a Debye potential in a type II space is not half-
flat, the linearized curvature has a simple structure since it is half type N.

For vacuum type D spacetimes, it is sometimes assumed that, up to gauge, all
real solutions of the linearized Einstein vacuum equations can be obtained,
locally, as the real part of a metric generated by a Debye potential; for
recent advances in the Schwarzschild and Kerr cases see respectively [20] and
[11]. For the more general vacuum type II case, from these considerations we
expect the result of Theorem 2 to be of relevance for addressing the following
conjecture:

######  Conjecture.

All real solutions of the linearized Einstein vacuum equations on a vacuum
type II background can be locally obtained, up to gauge, as the real part of
the metric generated by a Debye potential.

We also point out, that Jeffryes in [14] made remarks supporting the validity
of this conjecture.

######  Remark 4.

Metrics generated from a Debye potential are always in radiation gauge. It is
known that any perturbation of Petrov type II geometries can be transformed
into radiation gauge, see [21].

Theorem 2 generalizes the result (1.9)-(1.10) in the type D case to type II.
We stress that a generic type II spacetime does not possess any ordinary or
hidden symmetries, but only the more general objects (1.4) and (1.5)
introduced in this work. While it can be shown that the ordinary, valence-2
Killing spinor equation is equivalent to the real, conformal Killing-Yano
equation, which is itself a generalization to differential forms of the
conformal Killing equation, for the `projected' Killing spinor equation (1.6)
no such equivalence exists. We will give a geometric interpretation to the
origin of (1.6) in terms of spinors that are parallel under a suitable
connection especially adapted to the geometry, which is the conformal-GHP
connection. This also allows us to generalize the result (1.4)-(1.6) to non-
vacuum spacetimes in the real-analytic case, and to derive conservation laws
associated to projected Killing spinors.

In appendix A, we also review the Robinson-Trautman reduction of the Einstein
equations which admits solutions of various Petrov types. In particular there
is a Petrov type II solution which we use in example 18 to compute the
projected Killing vector $\xi^{a}$.

Most computations were performed with Spinframes [2], based on the symbolic
computer algebra package xAct for Mathematica.

##  2\. Preliminaries

###  2.1. The 2-spinor formalism

In this paper we shall make extensive use of the 2-spinor formalism, following
the notation and conventions in [18, 19]. The spinor bundles
$\mathbb{S}\to\mathcal{M}$ and $\mathbb{S}^{\prime}\to\mathcal{M}$ are rank-2
vector bundles with symplectic forms $\epsilon_{AB}$ and
$\bar{\epsilon}_{A^{\prime}B^{\prime}}$, such that
$T\mathcal{M}\otimes\mathbb{C}\cong\mathbb{S}\otimes\mathbb{S}^{\prime}$ and
$g_{ab}=\epsilon_{AB}\bar{\epsilon}_{A^{\prime}B^{\prime}}$. The spaces of SD
and ASD 2-forms are written in spinor terms as
$\Lambda^{2}_{+}\cong\mathbb{S}^{\prime*}\odot\mathbb{S}^{\prime*}$ and
$\Lambda^{2}_{-}\cong\mathbb{S}^{*}\odot\mathbb{S}^{*}$ (with $\mathbb{S}^{*}$
the dual of $\mathbb{S}$, etc.); in other words, a real 2-form
$F_{ab}=F_{[ab]}$ has the spinor decomposition

$F_{ab}=\phi_{AB}\bar{\epsilon}_{A^{\prime}B^{\prime}}+\bar{\phi}_{A^{\prime}B^{\prime}}\epsilon_{AB},$
(2.1)

where $F^{-}_{ab}=\phi_{AB}\bar{\epsilon}_{A^{\prime}B^{\prime}}$ and
$F^{+}_{ab}=\bar{\phi}_{A^{\prime}B^{\prime}}\epsilon_{AB}$ are the SD and ASD
parts of $F_{ab}$ respectively, and $\phi_{AB}=\phi_{(AB)}$. The Riemann
tensor admits a similar decomposition [18, Eq. (4.6.38)]:

$\displaystyle R_{abcd}={}$
$\displaystyle\Psi_{ABCD}\bar{\epsilon}_{A^{\prime}B^{\prime}}\bar{\epsilon}_{C^{\prime}D^{\prime}}+\bar{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}\epsilon_{AB}\epsilon_{CD}$
$\displaystyle+\Phi_{ABC^{\prime}D^{\prime}}\bar{\epsilon}_{A^{\prime}B^{\prime}}\epsilon_{CD}+\Phi_{A^{\prime}B^{\prime}CD}\epsilon_{AB}\bar{\epsilon}_{C^{\prime}D^{\prime}}$
$\displaystyle+2\Lambda(\epsilon_{AC}\epsilon_{BD}\bar{\epsilon}_{A^{\prime}C^{\prime}}\bar{\epsilon}_{B^{\prime}D^{\prime}}-\epsilon_{AD}\epsilon_{BC}\bar{\epsilon}_{A^{\prime}D^{\prime}}\bar{\epsilon}_{B^{\prime}C^{\prime}}),$
(2.2)

where $\Psi_{ABCD}=\Psi_{(ABCD)}$ is the Weyl conformal spinor,
$\Phi_{ABC^{\prime}D^{\prime}}=\Phi_{(AB)(C^{\prime}D^{\prime})}$ is the
trace-free Ricci spinor (which is real, i.e. $\bar{\Phi}_{ab}=\Phi_{ab}$), and
$\Lambda=R/24$ represents the scalar curvature (which is also real,
$\bar{\Lambda}=\Lambda$)444In the Einstein case, we have
$\Phi_{ABC^{\prime}D^{\prime}}=0$ and cosmological constant $6\Lambda$..

###  2.2. Self-duality

Let $(\mathcal{M},g_{ab})$ be a real, four-dimensional Lorentzian manifold.
Throughout we make use of the abstract index notation. We assume the spacetime
to be orientable so that there is a volume form $\varepsilon_{abcd}$ and the
associated Hodge star $*:\Lambda^{k}\to\Lambda^{4-k}$, where $\Lambda^{k}$ is
the space of $k$-forms. For 2-forms, $*$ satisfies $*^{2}=-1$; this induces a
decomposition $\Lambda^{2}=\Lambda^{2}_{+}\oplus\Lambda^{2}_{-}$, where
$\Lambda^{2}_{+}$ (resp. $\Lambda^{2}_{-}$) is the rank 3 eigenbundle of $*$
associated to the eigenvalue $+i$ (resp. $-i$). Elements of $\Lambda^{2}_{+}$
are called self-dual (SD) 2-forms, and those of $\Lambda^{2}_{-}$ are anti-
self-dual (ASD) 2-forms. Since the Riemann curvature tensor has the symmetries
$R_{abcd}=R_{[ab][cd]}$, one can also apply the Hodge duality operation to it,
in particular this defines the left- and right-dual Riemann tensors by
${}^{*}R_{abcd}=\frac{1}{2}\varepsilon_{ab}{}^{ef}R_{efcd}$ and
$R^{*}_{abcd}=\frac{1}{2}\varepsilon_{cd}{}^{ef}R_{abef}$, respectively. For
the Weyl tensor, the left- and right-duals coincide:
${}^{*}C_{abcd}=C^{*}_{abcd}$. One then defines the SD and ASD Weyl tensors by

$C^{\pm}_{abcd}:=\tfrac{1}{2}(C_{abcd}\mp i{}^{*}C_{abcd}).$ (2.3)

These tensors satisfy $*C^{\pm}_{abcd}=\pm iC^{\pm}_{abcd}$.

In spinor terms, the SD and ASD Weyl tensors (2.3) are

$C^{-}_{abcd}=\Psi_{ABCD}\bar{\epsilon}_{A^{\prime}B^{\prime}}\bar{\epsilon}_{C^{\prime}D^{\prime}},\qquad
C^{+}_{abcd}=\bar{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}\epsilon_{AB}\epsilon_{CD}.$
(2.4)

In particular, the SD and ASD Weyl tensors are complex conjugate of each
other, which follows from (2.3).

###  2.3. Complex geometries

For complex geometries, the above decompositions still apply, but now a pair
of complex conjugate quantities that appear together in a real expression such
as (2.1), is replaced by two independent entities; see [19, Section 6.9]. For
example, for a complex 2-form, the spinor $\bar{\phi}_{A^{\prime}B^{\prime}}$
in (2.1) is replaced by a spinor $\psi_{A^{\prime}B^{\prime}}$ which is no
longer the complex conjugate of $\phi_{AB}$. Similarly, the Riemann tensor of
a complex metric has a spinor decomposition analogous to (2.2), but where
$\bar{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}$ is now replaced by
another spinor $\widetilde{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}$
which is no longer the complex conjugate of $\Psi_{ABCD}$. The Ricci spinor
$\Phi_{ABC^{\prime}D^{\prime}}$ and the scalar curvature $\Lambda$ do not
acquire ``tilded'' versions because of the original reality conditions
$\bar{\Phi}_{ab}=\Phi_{ab}$ and $\bar{\Lambda}=\Lambda$, which are a
consequence of the symmetries of the Riemann tensor; they simply become
complex objects.

The fact that $\Psi_{ABCD}$ and
$\widetilde{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}$ are now two
independent entities implies that one correspondingly has two independent
algebraic classification schemes for the Weyl curvature spinors, so we can
have for example conformally `half-flat' manifolds if, say, $\Psi_{ABCD}=0$
and $\widetilde{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}\neq 0$, or
`half-algebraically special' solutions if $\Psi_{ABCD}$ is algebraically
special while $\widetilde{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}$ is
general. These remarks about complex geometries apply of course also to
complex, linear perturbations of real geometries.

###  2.4. Conformal-GHP connections

In section 3 we shall discuss some geometric aspects of projected Killing
spinors. In order to do this we need an extension of the GHP derivative that
includes conformal transformations. We call such an extension the `conformal-
GHP' connection, and give a brief review of some aspects of the construction
that are relevant for this work.

Let $(\mathcal{M},g_{ab})$ be a Lorentzian spacetime, with Levi-Civita
connection $\nabla_{a}$. Let $(o^{A},\iota^{A})$ be a general spin dyad. We
allow two kinds of transformations: $(i)$ `GHP transformations'
$o^{A}\to\gamma o^{A}$, $\iota^{A}\to\gamma^{-1}\iota^{A}$, with $\gamma$ a
non-vanishing complex scalar field, and $(ii)$ conformal transformations
$o^{A}\to o^{A}$, $\iota^{A}\to\Omega^{-1}\iota^{A}$, where $\Omega$ is a
positive scalar field. The transformation $(ii)$ is induced by a
transformation of the spin metric $\epsilon^{AB}\to\Omega^{-1}\epsilon^{AB}$,
which is in turn induced by a conformal transformation of the metric,
$g^{ab}\to\Omega^{-2}g^{ab}$.

Let $\mathbb{S}_{\\{p,q\\}}[w]$ be the vector bundle of conformally weighted
spinors with GHP weight $\\{p,q\\}$ and conformal weight $w$, and with an
arbitrary index structure. This means that a section
$\varphi^{A...A^{\prime}...}_{B...B^{\prime}...}\in\Gamma(\mathbb{S}_{\\{p,q\\}}[w])$
transforms as
$\varphi^{A...A^{\prime}...}_{B...B^{\prime}...}\to\gamma^{p}\bar{\gamma}^{q}\varphi^{A...A^{\prime}...}_{B...B^{\prime}...}$
under GHP transformations, and as
$\varphi^{A...A^{\prime}...}_{B...B^{\prime}...}\to\Omega^{w}\varphi^{A...A^{\prime}...}_{B...B^{\prime}...}$
under conformal transformations. For example, we have
$o^{A}\in\Gamma(\mathbb{S}_{\\{1,0\\}}[0])$ and
$\iota^{A}\in\Gamma(\mathbb{S}_{\\{-1,0\\}}[-1])$. The conformal-GHP covariant
derivative is a linear connection
$\mathcal{C}_{AA^{\prime}}:\Gamma(\mathbb{S}_{\\{p,q\\}}[w])\to\Gamma(T^{*}\mathcal{M}\otimes\mathbb{S}_{\\{p,q\\}}[w])$.
In this work all objects considered have vanishing `$q$-weight', so we shall
restrict to quantities of GHP weight $\\{p,0\\}$. For the general case and
more details see [5] and references therein555When acting on scalar
quantities, the projection of $\mathcal{C}_{AA^{\prime}}$ on a null tetrad
reduces to the conformally invariant GHP operators of Penrose and Rindler [18,
Eq. (5.6.36)]. We note however that, for general spinors and tensors, the
operators in [18, Eq. (5.6.36)] do not map conformal densities to conformal
densities.. The action of $\mathcal{C}_{AA^{\prime}}$ on, say, a spinor field
$\varphi^{BB^{\prime}}_{CC^{\prime}}$ with GHP weight $\\{p,0\\}$ and
conformal weight $w$, is given by

$\displaystyle\mathcal{C}_{AA^{\prime}}\varphi^{BB^{\prime}}_{CC^{\prime}}={}$
$\displaystyle\nabla_{AA^{\prime}}\varphi^{BB^{\prime}}_{CC^{\prime}}+(wf_{AA^{\prime}}+p(\omega_{AA^{\prime}}+B_{AA^{\prime}}))\varphi^{BB^{\prime}}_{CC^{\prime}}$
$\displaystyle+\epsilon_{A}{}^{B}f_{QA^{\prime}}\varphi^{QB^{\prime}}_{CC^{\prime}}+\epsilon_{A^{\prime}}{}^{B^{\prime}}f_{AQ^{\prime}}\varphi^{BQ^{\prime}}_{CC^{\prime}}-f_{CA^{\prime}}\varphi^{BB^{\prime}}_{AC^{\prime}}-f_{AC^{\prime}}\varphi^{BB^{\prime}}_{CA^{\prime}},$
(2.5)

with

$\displaystyle\omega_{a}:=-\epsilon
n_{a}+\epsilon^{\prime}\ell_{a}-\beta^{\prime}m_{a}+\beta\bar{m}_{a},$ (2.6a)
$\displaystyle B_{a}:=-\rho n_{a}+\tau\bar{m}_{a},$ (2.6b) $\displaystyle
f_{a}:=\rho n_{a}+\rho^{\prime}\ell_{a}-\tau^{\prime}m_{a}-\tau\bar{m}_{a},$
(2.6c)

where we are using standard GHP notation for spin coefficients. For spinors
with a different index structure, the corresponding action of
$\mathcal{C}_{AA^{\prime}}$ can be deduced from (2.5) by linearity and the
Leibniz rule. By construction, the connection (2.5) is covariant under
combined conformal and GHP transformations,

$\mathcal{C}_{a}\varphi^{B...B^{\prime}...}_{C...C^{\prime}...}\to\gamma^{p}\Omega^{w}\mathcal{C}_{a}\varphi^{B...B^{\prime}...}_{C...C^{\prime}...}.$

In particular, applying (2.5) to the spin frame $(o_{A},\iota_{A})$ and taking
into account that $o_{A}\in\Gamma(\mathbb{S}_{\\{1,0\\}}[1])$ and
$\iota_{A}\in\Gamma(\mathbb{S}_{\\{-1,0\\}}[0])$, one finds

$\displaystyle\mathcal{C}_{AA^{\prime}}o_{B}=(o^{C}o^{D}\nabla_{CA^{\prime}}o_{D})\iota_{A}\iota_{B},$
(2.7a)
$\displaystyle\mathcal{C}_{AA^{\prime}}\iota_{B}=(\iota^{C}\iota^{D}\nabla_{CA^{\prime}}\iota_{D})o_{A}o_{B}.$
(2.7b)

######  Remark 5.

Analogously to the usual GHP connection, the conformal-GHP connection
$\mathcal{C}_{AA^{\prime}}$ depends on the choice of a spin dyad
$(o^{A},\iota^{A})$. In what follows, the dyad $(o^{A},\iota^{A})$ will always
be understood to be the one associated to $\mathcal{C}_{AA^{\prime}}$.

##  3\. Special geometry and parallel spinors

In this section we provide a geometric interpretation for projected Killing
spinors and discuss general properties such as integrability conditions, the
solution space and a relation to conservation laws. Except for some specific
examples, we do not assume the geometry to be Ricci-flat. Several results
below need a complexification of spacetime and in those situations we impose
real-analyticity; this will be explicitly indicated in each case.

We will first show that the projected Killing spinor equation can be
understood as a consequence of the existence of spinors that are parallel
under the conformal-GHP connection.

######  Remark 6.

The formulation in this section is conformally invariant (except for the
specific examples in which we assume the Einstein condition, see (2) in Lemma
12 and examples 17 and 22). This means that all results below are valid not
only for a specific metric but for the equivalence class of metrics
conformally related to each other, i.e. for conformal structures. For
simplicity, however, we shall state the results in terms of spacetimes and not
conformal structures.

Let $(\mathcal{M},g_{ab})$ be a Lorentzian spacetime, and let $o^{A}$ be a
spinor field satisfying

$\displaystyle o^{A}o^{B}\nabla_{AA^{\prime}}o_{B}=0.$ (3.1)

Such an $o^{A}$ will be called a shear-free ray (SFR). If $o_{A}$ is SFR, it
follows from (2.7a) that it is parallel under $\mathcal{C}_{AA^{\prime}}$. In
GHP notation, (3.1) is equivalent to $\kappa=\sigma=0$. The condition
$\kappa=0$ encodes that the null congruence associated to the vector field
$o^{A}\bar{o}^{A^{\prime}}$ is geodesic, whereas $\sigma=0$ states that this
congruence is shear-free. Thus, a non-trivial solution to (3.1) imposes the
existence of a shear-free null geodesic congruence.

######  Remark 7.

Several results below require a complexified spacetime, for which we need to
impose real-analyticity. The complex extension of $\mathcal{M}$ will be
denoted $\mathbb{C}\mathcal{M}$. Our interest here is in the case where
$\mathbb{C}\mathcal{M}$ arises as the complexification of a real spacetime
with Lorentzian signature, see [19, Section 6.9].

The reason why we need to complexify $\mathcal{M}$ is the following important
result about SFRs:

######  Lemma 8 (Proposition (7.3.18) in [19]).

Let $o^{A}$ be a spinor, and let $\mu^{A^{\prime}}$, $\nu^{A^{\prime}}$ be a
primed spin dyad. Then the complex vector fields
$X^{a}=o^{A}\mu^{A^{\prime}}$, $Y^{a}=o^{A}\nu^{A^{\prime}}$ on
$\mathbb{C}\mathcal{M}$ are in involution if and only if $o^{A}$ is SFR.

Hence by Frobenius' theorem, the distribution defined by $\\{X^{a},Y^{a}\\}$
is surface-forming in $\mathbb{C}\mathcal{M}$. The surfaces associated to this
distribution are complex, and they are called $\beta$-surfaces, see e.g. [19,
pp. 309-310]:

######  Definition 9 ($\beta$-surface).

A $\beta$-surface is a complex 2-dimensional surface in
$\mathbb{C}\mathcal{M}$ whose tangent vectors at any one point are all of the
form $o^{A}\mu^{A^{\prime}}$ for some $\mu^{A^{\prime}}$, where $o^{A}$ is
fixed and satisfies equation (3.1).

For the following it is convenient to introduce some additional notation.

######  Definition 10.

Let $\mathcal{C}_{AA^{\prime}}$ be the conformal-GHP connection associated to
a spin dyad $(o^{A},\iota^{A})$. We define the operators

$\tilde{\mathcal{C}}_{A^{\prime}}:=o^{A}\mathcal{C}_{AA^{\prime}},\qquad\mathcal{C}_{A^{\prime}}:=\iota^{A}\mathcal{C}_{AA^{\prime}}.$
(3.2)

It is worth discussing some properties of $\tilde{\mathcal{C}}_{A^{\prime}}$.
It is a linear operator that satisfies the Leibniz rule, and which maps a
section of $\mathbb{S}_{\\{p,0\\}}[w]$ into a section of
$\mathbb{S}^{\prime}\otimes\mathbb{S}_{\\{p+1,0\\}}[w]$, where
$\mathbb{S}^{\prime}$ is the primed spin bundle introduced in section 2.
Consider now a $\beta$-surface $\Sigma$ in $\mathbb{C}\mathcal{M}$. Any
element of the tangent bundle $T\Sigma$ is of the form
$(x,o^{A}\mu^{A^{\prime}})$, where $x\in\Sigma$, $o^{A}$ is fixed, and
$\mu^{A^{\prime}}$ is some primed spinor at $x$. Therefore we can identify
$T\Sigma$ with the restriction of the primed spin bundle to $\Sigma$,
$\mathbb{S}^{\prime}|_{\Sigma}$. Similarly, the cotangent bundle $T^{*}\Sigma$
can be identified with the dual $\mathbb{S}^{\prime*}|_{\Sigma}$. This means
that a primed spinor field $\psi_{A^{\prime}}$ can be thought of as a
``1-form'' in $T^{*}\Sigma$. Therefore, restricting to $\beta$-surfaces, the
operator $\tilde{\mathcal{C}}_{A^{\prime}}$ is
$\tilde{\mathcal{C}}_{A^{\prime}}:\Gamma(\mathbb{S}_{\\{p,0\\}}[w])\to\Gamma(T^{*}\Sigma\otimes\mathbb{S}_{\\{p+1,0\\}}[w])$,
from which we see that it can be regarded as a connection, in the usual sense,
on conformally and GHP-weighted vector bundles over $\beta$-surfaces on
$\mathbb{C}\mathcal{M}$, see [6].

We have the following result, which does not require analyticity:

######  Lemma 11.

Let $(\mathcal{M},g_{ab})$ be a Lorentzian spacetime with
$\tilde{\mathcal{C}}_{A^{\prime}}$ given by (3.2), where $o^{A}$ satisfies
(3.1). Then

$[\tilde{\mathcal{C}}_{A^{\prime}},\tilde{\mathcal{C}}_{B^{\prime}}]=0$ (3.3)

if and only if $o^{A}$ is a repeated principal spinor of $\Psi_{ABCD}$, that
is

$\Psi_{ABCD}o^{B}o^{C}o^{D}=0.$ (3.4)

This result follows from Lemmas 3.4 and 3.5 in [6]. As mentioned, the proof of
Lemma 11 does not require analyticity. However, from the discussion above we
know that on $\beta$-surfaces, which live in $\mathbb{C}\mathcal{M}$, we can
interpret $\tilde{\mathcal{C}}_{A^{\prime}}$ as a connection. In that case the
commutator
$[\tilde{\mathcal{C}}_{A^{\prime}},\tilde{\mathcal{C}}_{B^{\prime}}]$ is the
curvature of $\tilde{\mathcal{C}}_{A^{\prime}}$. Thus, the result of Lemma 11
tells us that, as long as conditions (3.1) and (3.4) hold,
$\tilde{\mathcal{C}}_{A^{\prime}}$ is a flat connection on $\beta$-surfaces.
From [6, eq. (3.27)] we see that one can associate a (twisted) de Rham complex
to $\tilde{\mathcal{C}}_{A^{\prime}}$. Local exactness of a de Rham complex
allows to find (local) potentials in specific situations; for example, we find
the following:

######  Lemma 12.

Suppose that $o^{A}$ is SFR (3.1) and a repeated principal spinor (3.4).

  1. (1)

Assume $(\mathcal{M},g_{ab})$ is real-analytic with complexification
$\mathbb{C}\mathcal{M}$. Then there exist scalar fields $\phi$ and $\eta$ on
$\mathbb{C}\mathcal{M}$, whose GHP and conformal weights are $\\{0,0\\}$,
$w=-1$ and $\\{-2,0\\}$, $w=-1$ respectively, such that $f_{a}$, given in
(2.6c), takes the form

$f_{AA^{\prime}}=\nabla_{AA^{\prime}}\log\phi-
o_{A}\tilde{\mathcal{C}}_{A^{\prime}}\eta.$ (3.5)

  2. (2)

Assume that $(\mathcal{M},g_{ab})$ is real and that $g_{ab}$ is Einstein.
Then, with Weyl scalars $\Psi_{i}$, (3.5) holds for $\phi$ and $\eta$ given by

$\displaystyle\phi=\Psi^{1/3}_{2},\qquad\eta=\tfrac{1}{3}\Psi^{-1}_{2}\Psi_{3}.$
(3.6)

The proof of eq. (3.5), along with other results and applications, will be
given in a forthcoming publication. This proof involves the existence of
$\beta$-surfaces and that is the reason why it is formulated in the real-
analytic setting. On the other hand, (3.6) can be easily demonstrated in any
Einstein spacetime, not necessarily analytic, by using the GHP form of the
Bianchi identities666Since Bianchi identities are not conformally invariant,
the expression (3.6) breaks conformal invariance..

######  Remark 13.

Consider the complexified spacetime $\mathbb{C}\mathcal{M}$.

  * •

Since $\tilde{\mathcal{C}}_{A^{\prime}}$ is a flat connection, the equation
$\tilde{\mathcal{C}}_{A^{\prime}}\lambda=0$ has non-trivial solutions for any
weights $\\{p,0\\}$ and $w$. In particular, from (2.5) we see that, choosing
$w=p=-1$, we have

$0=\tilde{\mathcal{C}}_{A^{\prime}}\lambda=o^{A}\partial_{AA^{\prime}}\lambda-o^{A}\omega_{AA^{\prime}}\lambda,$
(3.7)

where we used that $o^{A}f_{AA^{\prime}}=-o^{A}B_{AA^{\prime}}$. It then
follows that such $\lambda$ satisfies

$o^{A}\partial_{AA^{\prime}}\log\lambda=o^{A}\omega_{AA^{\prime}}.$ (3.8)

  * •

Let $\psi$ be a scalar field on $\mathbb{C}\mathcal{M}$ with GHP weight
$\\{p,0\\}$ and conformal weight $w$. Using (2.5), (3.5) and (3.8), we see
that the action of $\tilde{\mathcal{C}}_{A^{\prime}}$ reduces to

$\tilde{\mathcal{C}}_{A^{\prime}}\psi=\phi^{-(w-p)}\lambda^{-p}o^{A}\partial_{AA^{\prime}}[\phi^{w-p}\lambda^{p}\psi].$
(3.9)

Using formula (3.9), it follows that if the weighted scalar field $\psi$
satisfies $\tilde{\mathcal{C}}_{A^{\prime}}\psi=0$, then defining
$F=\phi^{w-p}\lambda^{p}\psi$, where $\phi$ and $\lambda$ are defined in (3.5)
and (3.8) respectively, this is equivalent to

$o^{A}\partial_{AA^{\prime}}F=0.$ (3.10)

On $\mathbb{C}\mathcal{M}$, these functions are said to be `constant on
$\beta$-surfaces', since the operator $o^{A}\partial_{AA^{\prime}}$ represents
translations along the $\beta$-surfaces associated to $o^{A}$. That is to say,
there exist linearly independent spinor fields $\mu^{A^{\prime}}$,
$\nu^{A^{\prime}}$ such that
$o^{A}\mu^{A^{\prime}}\partial_{AA^{\prime}}\equiv\partial_{\tilde{z}}$ and
$o^{A}\nu^{A^{\prime}}\partial_{AA^{\prime}}\equiv\partial_{\tilde{w}}$, where
$\tilde{z},\tilde{w}$ are coordinates along the $\beta$-surfaces. Then
equation (3.10) can be interpreted as

$F=F(z,w),$ (3.11)

where $z,w$ are coordinates constant on each $\beta$-surface. Notice that
(3.11) is a function on $\mathbb{C}\mathcal{M}$.

######  Remark 14.

Let $\iota^{A}$ be any spinor field independent of $o^{A}$. Then the equation

$\iota^{A}\partial_{AA^{\prime}}f=0$ (3.12)

has only constant functions as solutions. To see this, note that if (3.12)
holds we must also have
$[\iota^{A}\nabla_{AA^{\prime}},\iota^{B}\nabla_{BB^{\prime}}]f=0$, but

$0=[\iota^{A}\nabla_{AA^{\prime}},\iota^{B}\nabla_{BB^{\prime}}]f=-\epsilon_{A^{\prime}B^{\prime}}(\iota^{A}\iota^{B}\nabla_{A}{}^{C^{\prime}}\iota_{B})o^{C}\nabla_{CC^{\prime}}f.$
(3.13)

By definition we know that
$\iota^{A}\iota^{B}\nabla_{A}{}^{C^{\prime}}\iota_{B}=0$ if and only if
$\iota^{A}$ is SFR, but this is not the case since $\iota^{A}$ is arbitrary,
so we get $o^{C}\nabla_{CC^{\prime}}f=0$, which, when combined with (3.12),
implies $\nabla_{AA^{\prime}}f=0$, i.e. $f$ is constant.

In terms of this geometric setup, the projected Killing spinor equation can be
deduced as follows:

######  Proposition 15.

Let $(\mathcal{M},g_{ab})$ be a Lorentzian spacetime. Let $(o_{A},\iota_{A})$
be the dyad associated to $\mathcal{C}_{AA^{\prime}}$ (cf. Remark 5), and
suppose that $o_{A}$ is SFR.

  1. (1)

$o_{A}$ and $\iota_{A}$ are parallel spinors for (3.2):

$\tilde{\mathcal{C}}_{A^{\prime}}o_{B}=0,\qquad\tilde{\mathcal{C}}_{A^{\prime}}\iota_{B}=0.$
(3.14)

  2. (2)

Suppose in addition that $(\mathcal{M},g_{ab})$ is real-analytic and that
$o^{A}$ is also a repeated principal spinor, so that eq. (3.5) holds. Then the
spinor field

$K_{AB}=\phi^{-1}[o_{(A}\iota_{B)}-\eta o_{A}o_{B}]$ (3.15)

satisfies the projected Killing spinor equation,

$o^{A}\nabla_{A^{\prime}(A}K_{BC)}=0.$ (3.16)

###### Proof.

Equation (3.14) follows simply from (2.7a)–(2.7b), so we see that analyticity
is not required for this item. For the second item, applying the definition
(2.5) to the field $o_{B}\iota_{C}$, whose weights are $\\{0,0\\}$ and $w=1$,
we have

$\mathcal{C}_{A^{\prime}A}(o_{B}\iota_{C})=\nabla_{A^{\prime}A}(o_{B}\iota_{C})+f_{A^{\prime}A}o_{B}\iota_{C}-f_{A^{\prime}B}o_{A}\iota_{C}-f_{A^{\prime}C}o_{B}\iota_{A}.$
(3.17)

But from (2.7a)–(2.7b) we see that

$\mathcal{C}_{A^{\prime}A}(o_{B}\iota_{C})=\sigma^{\prime}_{A^{\prime}}o_{A}o_{B}o_{C},$
(3.18)

thus
$\mathcal{C}_{A^{\prime}A}(o_{B}\iota_{C})=\mathcal{C}_{A^{\prime}(A}(o_{B}\iota_{C)})$.
Combining with (3.17), we get

$\mathcal{C}_{A^{\prime}A}(o_{B}\iota_{C})=\mathcal{C}_{A^{\prime}(A}(o_{B}\iota_{C)})=(\nabla_{A^{\prime}(A}-f_{A^{\prime}(A})o_{B}\iota_{C)}.$
(3.19)

Using (3.5), this is

$\displaystyle\mathcal{C}_{A^{\prime}A}(o_{B}\iota_{C})={}$
$\displaystyle\phi\nabla_{A^{\prime}(A}[\phi^{-1}o_{B}\iota_{C)}]+o_{(A}o_{B}\iota_{C)}\tilde{\mathcal{C}}_{A^{\prime}}\eta$
$\displaystyle={}$
$\displaystyle\phi\left[\nabla_{A^{\prime}(A}[\phi^{-1}o_{B}\iota_{C)}]+\iota_{(A}\tilde{\mathcal{C}}_{|A^{\prime}|}(\phi^{-1}\eta
o_{B}o_{C)})\right],$

where in the second line we used that
$\tilde{\mathcal{C}}_{A^{\prime}}o_{B}=0$,
$\tilde{\mathcal{C}}_{A^{\prime}}\phi=0$. Contracting this equation with
$o^{A}$ and using (3.18) gives

$0=o^{A}\nabla_{A^{\prime}(A}[\phi^{-1}o_{B}\iota_{C)}]+o^{A}\iota_{(A}\tilde{\mathcal{C}}_{|A^{\prime}|}(\phi^{-1}\eta
o_{B}o_{C)}).$ (3.20)

Let us compute the second term in the right hand side. Using
$o^{A}\tilde{\mathcal{C}}_{A^{\prime}}(\cdot)=\tilde{\mathcal{C}}_{A^{\prime}}(o^{A}\cdot)$,
we get

$o^{A}\iota_{(A}\tilde{\mathcal{C}}_{|A^{\prime}|}(\phi^{-1}\eta
o_{B}o_{C)})=-\tfrac{1}{3}\tilde{\mathcal{\mathcal{C}}}_{A^{\prime}}(\phi^{-1}\eta
o_{B}o_{C})=-o^{A}\mathcal{C}_{A^{\prime}(A}[\phi^{-1}\eta o_{B}o_{C)}].$

Noticing that the weights of the field $\phi^{-1}\eta o_{B}o_{C}$ are
$\\{0,0\\}$ and $w=2$, and using the definition (2.5), we have

$\mathcal{C}_{A^{\prime}(A}[\phi^{-1}\eta
o_{B}o_{C)}]=\nabla_{A^{\prime}(A}[\phi^{-1}\eta o_{B}o_{C)}].$ (3.21)

Therefore

$o^{A}\iota_{(A}\tilde{\mathcal{C}}_{|A^{\prime}|}(\phi^{-1}\eta
o_{B}o_{C)})=-o^{A}\nabla_{A^{\prime}(A}(\phi^{-1}\eta o_{B}o_{C)})$

and thus

$0=o^{A}\nabla_{A^{\prime}(A}\left[\phi^{-1}o_{B}\iota_{C)}-\phi^{-1}\eta
o_{B}o_{C)}\right],$ (3.22)

hence the result follows. ∎

######  Remark 16.

Any spinor field of the form $\omega_{A}=fo_{A}$, where $f$ is of type
$\\{-1,0\\}$, $w=0$, and satisfies $\tilde{\mathcal{C}}_{A^{\prime}}f=0$, is a
solution to the ``projected twistor equation''

$o^{A}\nabla_{A^{\prime}(A}\omega_{B)}=0.$ (3.23)

This follows from the identity
$o^{A}\mathcal{\mathcal{C}}_{A^{\prime}(A}\omega_{B)}=o^{A}\nabla_{A^{\prime}(A}\omega_{B)}$
replacing $\omega_{A}=fo_{A}$.

From Proposition 15 we have that the spinor field (3.15) is a projected
Killing spinor in the general class of spacetimes where $o_{A}$ is SFR and a
repeated PND but not necessarily Einstein, although in the non-Einstein case
it must be real-analytic. This includes, in particular, the Kerr-(A)dS and
(analytic) Kerr-Newman-(A)dS spacetimes, where $K_{AB}$ satisfies the usual,
i.e. non-projected, Killing spinor equation.

######  Example 17.

Suppose that $(\mathcal{M},g_{ab})$ is Einstein, i.e. $g_{ab}$ satisfies the
vacuum Einstein equations with cosmological constant. Then, using (3.6), we
find that (3.15) reduces to the expression (1.4) given in the introduction.
Furthermore, the vector field

$\displaystyle\xi_{AA^{\prime}}=\nabla^{B}{}_{A^{\prime}}K_{AB}$ (3.24)

solve the projected Killing equation

$\displaystyle
o^{A}(\nabla_{AA^{\prime}}\xi_{BB^{\prime}}+\nabla_{BB^{\prime}}\xi_{AA^{\prime}})=0.$
(3.25)

In GHP notation, (3.24) takes the form

$\displaystyle\xi_{a}={}$
$\displaystyle\frac{3}{2\Psi_{2}^{1/3}}\left(\rho^{\prime}l_{a}-\rho
n_{a}-\tau^{\prime}m_{a}+\tau\bar{m}_{a}\right)+\frac{(\operatorname{\eth}-4\tau)\Psi_{3}}{2\Psi_{2}^{4/3}}l_{a}-\frac{(\operatorname{\text{}}-4\rho)\Psi_{3}}{2\Psi_{2}^{4/3}}m_{a},$
(3.26)

and from the Bianchi identities it follows that

$\displaystyle\xi^{a}\nabla_{a}\Psi_{2}=0.$ (3.27)

To give an explicit example, we review the Robinson-Trautman geometries in
appendix A. There are known solutions of various Petrov types in this class
and in particular there is one of Petrov type II.

######  Example 18.

For the type II Robinson-Trautman solution given in (A.10), the vector (3.26)
is of the form

$\displaystyle\xi^{a}=\frac{-2(-1)^{2/3}}{3m^{4/3}}\left((\zeta+\bar{\zeta})^{3}(\partial_{\zeta})^{a}+m(\partial_{u})^{a}\right).$
(3.28)

We note that it does not reduce to $\partial_{u}$, which is a Killing vector
of that solution.

Let us now discuss the space of projected Killing spinors. In type D
spacetimes one can show that the space of valence–2 Killing spinors (1.1) is
1-dimensional, see e.g. [13]. For type II spacetimes, we first show the
following:

######  Proposition 19.

Let $(\mathcal{M},g_{ab})$ be a Lorentzian spacetime, where $o^{A}$ is SFR and
a repeated principal spinor. Let

$\displaystyle K_{AB}={}$ $\displaystyle
K_{2}o_{A}o_{B}-2K_{1}o_{(A}\iota_{B)}+K_{0}\iota_{A}\iota_{B}$ (3.29)

be a projected Killing spinor, i.e. solving (1.6). Then the components satisfy

$\displaystyle K_{0}={}$ $\displaystyle 0,$ (3.30a)
$\displaystyle\tilde{\mathcal{C}}_{A^{\prime}}K_{1}={}$ $\displaystyle 0,$
(3.30b)
$\displaystyle\tilde{\mathcal{C}}_{A^{\prime}}K_{2}+2\mathcal{C}_{A^{\prime}}K_{1}={}$
$\displaystyle 0.$ (3.30c)

###### Proof.

We first derive integrability conditions for (1.6) by introducing

$\displaystyle Q_{BCA^{\prime}}=o^{A}\nabla_{A^{\prime}(A}K_{BC)}.$ (3.31)

For any geometry satisfying (3.1) we find

$\displaystyle\Theta_{(A}{}^{A^{\prime}}Q_{BC)A^{\prime}}=-4(\Psi_{3}K_{0}-\Psi_{1}K_{2})o_{(A}o_{B}\iota_{C)}+4(\Psi_{2}K_{0}-\tfrac{2}{3}\Psi_{1}K_{1})o_{(A}\iota_{B}\iota_{C)}-2\Psi_{1}K_{0}\iota_{A}\iota_{B}\iota_{C}.$
(3.32)

The integrability conditions are then obtained by setting
$Q_{BCA^{\prime}}=0$, so the left hand side of (3.32) vanishes. If
$\Psi_{1}\neq 0$, it follows that $K_{AB}=0$ is the only solution to (1.6), so
we need to have at least $\Psi_{1}=0$ to have non-trivial solutions. Since we
are interested in type II, we can impose $\Psi_{1}=0$. In that case (3.32)
leads to the restrictions

$\displaystyle\Psi_{3}K_{0}=0,$ $\displaystyle\Psi_{2}K_{0}=0.$ (3.33)

If $K_{0}\neq 0$, the Weyl spinor would be of Petrov type N (i.e. only
$\Psi_{4}\neq 0$). In that case (3.32) yields no restrictions on $K_{AB}$. For
Petrov type D or II, we have $\Psi_{2}\neq 0$ which forces $K_{0}=0$. For
$K_{1}$ and $K_{2}$, a compact form of the equations to be satisfied can be
obtained in terms of the covariant derivative $\mathcal{C}_{AA^{\prime}}$ and
its projections $\tilde{\mathcal{C}}_{A^{\prime}}$, $\mathcal{C}_{A^{\prime}}$
given in (3.2). Using $K_{0}=0$, and requiring $K_{AB}$ to have conformal
weight $2$ so that the projected Killing spinor equation is conformally
invariant, eq. (1.6) is

$0=\tilde{\mathcal{C}}_{A^{\prime}}K_{BC}+o_{B}\mathcal{C}_{A^{\prime}C}K_{1}+o_{C}\mathcal{C}_{A^{\prime}B}K_{1}.$
(3.34)

Contraction with $o^{B}$ and with $\iota^{B}\iota^{C}$ leads to,

$\displaystyle\tilde{\mathcal{C}}_{A^{\prime}}K_{1}={}$ $\displaystyle 0,$
(3.35a)
$\displaystyle\tilde{\mathcal{C}}_{A^{\prime}}K_{2}+2\mathcal{C}_{A^{\prime}}K_{1}={}$
$\displaystyle 0,$ (3.35b)

or in GHP notation

$\displaystyle(\operatorname{\eth}+\tau)K_{1}={}$ $\displaystyle 0,$
$\displaystyle(\operatorname{\text{}}+\rho)K_{1}={}$ $\displaystyle 0,$
(3.36a) $\displaystyle
2(\operatorname{\text{}}^{\prime}+\rho^{\prime})K_{1}+(\operatorname{\eth}+2\tau)K_{2}={}$
$\displaystyle 0,$ $\displaystyle
2(\operatorname{\eth}^{\prime}+\tau^{\prime})K_{1}+(\operatorname{\text{}}+2\rho)K_{2}={}$
$\displaystyle 0.$ (3.36b)

∎

######  Remark 20 (Solution space of the projected Killing spinor equation).

In the real-analytic case, we see from eqs. (3.30b)-(3.30c) that the space of
solutions to the projected Killing spinor equation is infinite dimensional.
For example, one solution is given in (3.15), but we also see that, taking
$K_{1}\equiv 0$, equations (3.30b)-(3.30c) reduce to
$\tilde{\mathcal{C}}_{A^{\prime}}K_{2}=0$. It follows from equations (3.9),
(3.10) and (3.11) that there are infinitely many solutions to this equation.
Any function of the form $F=\lambda^{-2}\phi^{2}K_{2}$, see Remark 13 and
Lemma 11 for the definition of $\lambda$ and $\phi$, which is constant on
$\beta$-surfaces leads to a projected Killing spinor
$K_{AB}=\lambda^{2}\phi^{-2}Fo_{A}o_{B}$. We also mention that one could in
principle consider $K_{1}\neq 0$ and $K_{2}\equiv 0$ (or
$\tilde{\mathcal{C}}_{A^{\prime}}K_{2}=0$), but then (3.30b)-(3.30c) leads to
$\mathcal{C}_{AA^{\prime}}K_{1}=0$, which, taking into account that the
weights of $K_{1}$ are $p=0$ and $w=1$, is explicitly
$\nabla_{a}K_{1}+f_{a}K_{1}=0$. This implies $f_{a}=-\nabla_{a}\log K_{1}$ and
so $\nabla_{[a}f_{b]}=0$, which is a restriction on the geometry: this is
satisfied for vacuum type D but not for arbitrary type II spacetimes.

Finally, an interesting property of projected Killing spinors is that they
give rise to solutions of the vacuum Maxwell equations, and therefore to
conservation laws:

######  Lemma 21.

Let $(\mathcal{M},g_{ab})$ be real-analytic, and let $o^{A}$ be SFR and a
repeated principal spinor. Let $K_{AB}$ be a projected Killing spinor as in
(3.29). Assume $K_{1}=c\phi^{-1}$, where $\phi$ was introduced in (3.5) and
$c$ is an arbitrary constant, possibly zero. Then the spinor field

$\varphi_{AB}=-2\phi^{3}K_{AB}$ (3.37)

is a solution to the vacuum Maxwell equations.

The result of this Lemma is valid regardless of the Einstein condition, but it
imposes the real-analyticity restriction, because we need the existence of the
scalar field $\phi$ introduced in Lemma 12. If the Einstein condition is
assumed, then real-analyticity is not needed, see Example 22 below.

###### Proof of Lemma 21.

The proof is straightforward in terms of the conformal connection (2.5). Since
the field (3.37) has GHP weight $\\{0,0\\}$ and conformal weight $w=-1$, it
follows from (2.5) that
$\nabla_{A^{\prime}}{}^{A}\varphi_{AB}=\mathcal{\mathcal{C}}_{A^{\prime}}{}^{A}\varphi_{AB}$.
Replacing expression (3.29) for $K_{AB}$ with $K_{0}=0$, and using
$\mathcal{\mathcal{C}}_{AA^{\prime}}o_{B}=0$, we get

$\displaystyle-\tfrac{1}{2}\nabla_{A^{\prime}}{}^{A}\varphi_{AB}={}$
$\displaystyle-
o_{B}\tilde{\mathcal{C}}_{A^{\prime}}(\phi^{3}K_{2})+\tilde{\mathcal{C}}_{A^{\prime}}(\phi^{3}K_{1}\iota_{B})-o_{B}\mathcal{C}_{A^{\prime}}{}^{A}(\phi^{3}K_{1}\iota_{A})$
$\displaystyle={}$ $\displaystyle-
o_{B}\phi^{3}\tilde{\mathcal{C}}_{A^{\prime}}K_{2}+o_{B}\mathcal{C}_{A^{\prime}}(\phi^{3}K_{1})$
$\displaystyle={}$ $\displaystyle
2o_{B}\phi^{3}\mathcal{C}_{A^{\prime}}K_{1}+o_{B}\mathcal{C}_{A^{\prime}}(\phi^{3}K_{1})$
$\displaystyle={}$ $\displaystyle
3o_{B}\phi^{3}\mathcal{C}_{A^{\prime}}K_{1}+o_{B}K_{1}\mathcal{C}_{A^{\prime}}\phi^{3}$
$\displaystyle={}$ $\displaystyle 3o_{B}\phi^{2}\mathcal{C}_{A^{\prime}}(\phi
K_{1}),$ (3.38)

where in the second line we used
$0=\tilde{\mathcal{C}}_{A^{\prime}}\phi=\tilde{\mathcal{C}}_{A^{\prime}}K_{1}=\tilde{\mathcal{C}}_{A^{\prime}}\iota_{B}$
and also $\mathcal{\mathcal{C}}_{A^{\prime}}{}^{A}\iota_{A}=0$ (which follows
from (2.7b)), and in the third line we used eq. (3.30c). Now, the scalar $\phi
K_{1}$ has weights $\\{0,0\\}$ and $w=0$, so from the definition (2.5) we have
$\mathcal{\mathcal{C}}_{A^{\prime}}(\phi
K_{1})=\iota^{A}\nabla_{AA^{\prime}}(\phi K_{1})$. Therefore
$\nabla_{A^{\prime}}{}^{A}\varphi_{AB}=0$ if and only if
$\iota^{A}\nabla_{AA^{\prime}}(\phi K_{1})=0$. Since we are assuming a generic
type II spacetime with SFR $o^{A}$, this is true if and only if
$K_{1}=c\phi^{-1}$ for some constant $c$, which can be zero. This follows from
the discussion in Remark 14. ∎

The result of Lemma 21 tells us that, as long as $K_{1}=c\phi^{-1}$, we have
the conservation law

${\rm
d}\mathcal{F}=0,\qquad\mathcal{F}_{ab}=\varphi_{AB}\epsilon_{A^{\prime}B^{\prime}}.$
(3.39)

We note that, choosing $c=0$, the Maxwell field (3.37) is
$\varphi_{AB}=-2\phi^{3}K_{2}o_{A}o_{B}$, with
$\tilde{\mathcal{\mathcal{C}}}_{A^{\prime}}(\phi^{3}K_{2})=0$. These are the
Robinson null fields discussed for example in [19, Theorem 7.3.14].

######  Example 22 (Spin lowering).

If we assume that $(\mathcal{M},g_{ab})$ is Einstein, then from (3.6) we know
that $\phi=\Psi^{1/3}_{2}$ and we do not need to restrict to real-analytic
spacetimes. The Maxwell field (3.37) is $\varphi_{AB}=-2\Psi_{2}K_{AB}$.
Choosing $c=1$, it follows that

$\varphi_{AB}=\Psi_{ABCD}K^{CD}.$ (3.40)

Thus, in this case, the result of Lemma 21 can be interpreted as a form of
Penrose's spin lowering:

$\nabla^{AA^{\prime}}\varphi_{AB}=(\nabla^{AA^{\prime}}\Psi_{ABCD})K^{CD}+\Psi_{ABCD}(\nabla^{A^{\prime}(A}K^{CD)})=0$
(3.41)

where the first term on the right vanishes since
$\nabla^{AA^{\prime}}\Psi_{ABCD}=0$ are the Bianchi identities for an Einstein
space, and the vanishing of the second term follows from the fact that the
projected Killing spinor equation (1.6) implies
$\nabla^{A^{\prime}(A}K^{CD)}=A^{A^{\prime}}o^{A}o^{C}o^{D}$ for some
$A^{A^{\prime}}$, and then one uses the type II condition (3.4).

######  Example 23 (Sachs' conservation law).

In [24, eq.(5.23)], Sachs found a conservation law for vacuum type II
spacetimes, where he chose a tetrad rotation such that $\Psi_{3}=0$ to derive
the conservation law

$\displaystyle{\rm
d}\left(\Psi_{2}^{2/3}o_{(A}\iota_{B)}\bar{\epsilon}_{A^{\prime}B^{\prime}}\right)=0.$
(3.42)

From Example 22 we see that this conservation law can be interpreted as a form
of Penrose's spin lowering without the necessity of imposing the tetrad gauge
$\Psi_{3}=0$, since the components of the Maxwell field (3.40) are

$\displaystyle\varphi_{0}={}0,$ $\displaystyle\varphi_{1}={}\Psi_{2}^{2/3},$
$\displaystyle\varphi_{2}={}\frac{2\Psi_{3}}{3\Psi_{2}^{1/3}},$ (3.43)

which, using (3.39), shows the equivalence to the result of Sachs.

##  4\. Perturbation theory in terms of a Debye potential

In this section we review first order perturbation theory of Einstein
spacetimes of Petrov type II in terms of scalar potentials and prove Theorem
2. We make use of the adjoint operator method introduced by Wald in [29] to
generate solutions to the linearized Einstein equations from solutions of a
scalar wave-like equation.

First, recall the expression for the linearized ASD Weyl spinor
$\dot{\Psi}_{ABCD}$ in terms of the linearized metric for an Einstein
background with metric $g_{ab}$, as given by Penrose and Rindler in [18, Eq.
(5.7.15)],

$\dot{\Psi}_{ABCD}=\tfrac{1}{2}\nabla_{(A}{}^{A^{\prime}}\nabla_{B}{}^{B^{\prime}}h_{CD)A^{\prime}B^{\prime}}+\tfrac{1}{4}g^{ef}h_{ef}\Psi_{ABCD}.$
(4.1)

######  Remark 24.

We note that the spinor variational operator $\vartheta$ introduced in [7]
leads to a minus sign of the trace term in (4.1). Since the linearized metrics
generated from a scalar potential are trace-free, i.e. $g^{ef}h_{ef}=0$, see
(4.10) for details, the result would be the same for $\vartheta\Psi_{ABCD}$.
However, since we are only interested in the ASD Weyl curvature here, there is
no need to introduce the additional structures involving the $\vartheta$
variation.

To state the underlying operator identity of Wald's method, we begin with

######  Definition 25.

Let $x_{ab}$ be a (possibly complex) symmetric 2-tensor, $\Phi$ be a complex
scalar field of GHP weight $\\{4,0\\}$ and denote $l^{a}l^{b}x_{ab}$ by
$x_{ll}$ etc. for tetrad components. Define the differential operators
$\mathcal{S},\mathcal{E},\mathcal{O},\mathcal{T}$ by

$\displaystyle\mathcal{S}(x_{ab}):={}$
$\displaystyle(\operatorname{\eth}-4\tau-\bar{\tau}^{\prime})\left((\operatorname{\text{}}-2\bar{\rho})x_{lm}-(\operatorname{\eth}-\bar{\tau}^{\prime})x_{ll}\right)$
$\displaystyle+(\operatorname{\text{}}-4\rho-\bar{\rho})\left(\bar{\sigma}^{\prime}x_{ll}+(\operatorname{\eth}-2\bar{\tau}^{\prime})x_{lm}-(\operatorname{\text{}}-\bar{\rho})x_{mm}\right),$
(4.2a) $\displaystyle\mathcal{E}(x)_{ab}:={}$
$\displaystyle(-\tfrac{1}{2}\Box+6\Lambda)x_{ab}-\tfrac{1}{2}\nabla_{a}\nabla_{b}(g^{cd}x_{cd})+\nabla^{c}\nabla_{(a}x_{b)c}-\tfrac{1}{2}g_{ab}(\nabla^{c}\nabla^{d}x_{cd}-\Box(g^{cd}x_{cd})),$
(4.2b) $\displaystyle\mathcal{O}(\Phi):={}$ $\displaystyle
2[(\operatorname{\text{}}-4\rho-\bar{\rho})(\operatorname{\text{}}^{\prime}-\rho^{\prime})-(\operatorname{\eth}-4\tau-\bar{\tau}^{\prime})(\operatorname{\eth}^{\prime}-\tau^{\prime})-3\Psi_{2}]\Phi,$
(4.2c) $\displaystyle\mathcal{T}(x_{ab}):={}$
$\displaystyle\tfrac{1}{2}\left((\operatorname{\eth}-2\bar{\tau}^{\prime})\operatorname{\eth}-\bar{\sigma}^{\prime}(\operatorname{\text{}}-2\bar{\rho})-2(\operatorname{\text{}}\bar{\sigma}^{\prime})\right)x_{ll}+\tfrac{1}{2}(\operatorname{\text{}}-2\bar{\rho})\operatorname{\text{}}x_{mm}$
$\displaystyle-\left((\operatorname{\text{}}-2\bar{\rho})(\operatorname{\eth}-\bar{\tau}^{\prime})-(\operatorname{\eth}\bar{\rho})\right)x_{lm}.$
(4.2d)

######  Remark 26.

  1. (1)

Note that $\mathcal{E}$ defined in (4.2b) is the linearized Einstein operator
plus a cosmological term. This operator is formally self adjoint,

$\displaystyle\mathcal{E}^{\dagger}=\mathcal{E}.$ (4.3)

  2. (2)

Acting on a linearized metric $h_{ab}$, the operator $\mathcal{T}$ defined in
(4.2d) yields the 0-component of the linearized anti-self dual Weyl curvature
(4.1),

$\displaystyle\mathcal{T}(h_{ab})={}$ $\displaystyle
o^{A}o^{B}o^{C}o^{D}\dot{\Psi}_{ABCD}=\dot{\Psi}_{0}.$ (4.4)

  3. (3)

For any $h_{ab}$ such that $\mathcal{E}(h)_{ab}=0$, a decoupled wave-like
(Teukolsky) equation is given by

$\displaystyle\mathcal{O}\mathcal{T}(h_{ab})=\mathcal{O}(\dot{\Psi}_{0})=0.$
(4.5)

######  Theorem 27 (Wald [29]).

On Einstein spacetimes of Petrov type II with repeated principal spinor
$o^{A}$, the operators of definition 25 satisfy the identity

$\mathcal{S}\mathcal{E}=\mathcal{O}\mathcal{T}.$ (4.6)

Thus, in particular, if $\chi$ is a complex scalar field of GHP weight
$\\{-4,0\\}$ solving

$\displaystyle\mathcal{O}^{{\dagger}}\chi=0,$ (4.7)

then the complex metric $h_{ab}=\mathcal{S}^{{\dagger}}(\chi)_{ab}$ solves the
linearized Einstein equation

$\displaystyle\mathcal{E}(h)_{ab}=\mathcal{E}(\mathcal{S}^{{\dagger}}(\chi))_{ab}=0.$
(4.8)

In [29] the general idea was outlined and applied for Petrov type D, see
[20]777It seems that there is a factor 4 missing in the operator identity in
that reference. where the explicit form of $\mathcal{S}$ is given on Petrov
type II. The scalar field $\chi$ is called Debye potential and (4.7) is the
Debye equation.

######  Lemma 28.

Let $\chi$ be a complex scalar field of GHP weight $\\{-4,0\\}$.

  * •

The operators adjoint to $\mathcal{S}$ and $\mathcal{O}$ are given by

$\displaystyle\mathcal{S}^{\dagger}(\chi)_{ab}={}$
$\displaystyle\left(l_{(a}m_{b)}(\operatorname{\text{}}-\rho+\bar{\rho})-l_{a}l_{b}(\operatorname{\eth}-\tau)\right)(\operatorname{\eth}+3\tau)\chi$
$\displaystyle+\left(l_{(a}m_{b)}(\operatorname{\eth}-\tau+\bar{\tau}^{\prime})-l_{a}l_{b}\bar{\sigma}^{\prime}-m_{a}m_{b}(\operatorname{\text{}}-\rho)\right)(\operatorname{\text{}}+3\rho)\chi,$
(4.9a) $\displaystyle\mathcal{O}^{{\dagger}}\chi={}$ $\displaystyle
2\left(\left(\operatorname{\text{}}^{\prime}-\bar{\rho}^{\prime}\right)\left(\operatorname{\text{}}+3\rho\right)-\left(\operatorname{\eth}^{\prime}-\bar{\tau}\right)\left(\operatorname{\eth}+3\tau\right)-3\Psi_{2}\right)\chi.$
(4.9b)

  * •

If $\chi$ solves the Debye equation (4.7), then the complex metric
$h_{ab}=\mathcal{S}^{{\dagger}}(\chi)_{ab}$ solving the linearized Einstein
equation is given by

$\displaystyle h_{ab}={}$ $\displaystyle
h_{nn}l_{a}l_{b}-2h_{n\bar{m}}l_{(a}m_{b)}+h_{\bar{m}\bar{m}}m_{a}m_{b},$
(4.10)

with components

$\displaystyle h_{nn}={}$
$\displaystyle-(\operatorname{\eth}+2\tau)\operatorname{\eth}\chi-\bar{\sigma}^{\prime}\operatorname{\text{}}\chi,$
(4.11a) $\displaystyle h_{n\bar{m}}={}$
$\displaystyle-(\operatorname{\text{}}+\rho)\operatorname{\eth}\chi-(\tau+\bar{\tau}^{\prime})\operatorname{\text{}}\chi,$
(4.11b) $\displaystyle h_{\bar{m}\bar{m}}={}$
$\displaystyle-(\operatorname{\text{}}+2\rho)\operatorname{\text{}}\chi.$
(4.11c)

###### Proof.

The formal adjoints of the GHP operators are given by

$\displaystyle\operatorname{\text{}}^{\dagger}={}$
$\displaystyle-\operatorname{\text{}}+\rho+\bar{\rho},$
$\displaystyle\operatorname{\text{}}^{\prime\dagger}={}$
$\displaystyle-\operatorname{\text{}}^{\prime}+\rho^{\prime}+\bar{\rho}^{\prime},$
$\displaystyle\operatorname{\eth}^{\dagger}={}$
$\displaystyle-\operatorname{\eth}+\tau+\bar{\tau}^{\prime},$
$\displaystyle\operatorname{\eth}^{\prime\dagger}={}$
$\displaystyle-\operatorname{\eth}^{\prime}+\tau^{\prime}+\bar{\tau},$ (4.12)

see e.g. [1] for details. Using this, together with the rule
$(AB)^{\dagger}=B^{\dagger}A^{\dagger}$ for compositions, to compute the
adjoint of (4.2a) and (4.2c) leads to (4.9).

The equations (4.11) are the simplified components of (4.9a), using the
commutator

$\displaystyle[\operatorname{\eth},\operatorname{\text{}}]\chi={}$
$\displaystyle(\bar{\tau}^{\prime}\operatorname{\text{}}-\bar{\rho}\operatorname{\eth})\chi,$
(4.13)

and Ricci identities,

$\displaystyle\operatorname{\text{}}\rho={}$ $\displaystyle\rho^{2},$
$\displaystyle\operatorname{\eth}\tau={}$
$\displaystyle\tau^{2}-\bar{\sigma}^{\prime}\rho,$
$\displaystyle\operatorname{\text{}}\tau={}$
$\displaystyle\rho(\tau-\bar{\tau}^{\prime}),$
$\displaystyle\operatorname{\eth}\rho={}$
$\displaystyle\tau(\rho-\bar{\rho}),$ (4.14)

valid on type II Einstein spacetimes with repeated principal spinor $o^{A}$. ∎

Before proceeding to the proof of Theorem 2, we need to introduce some
additional identities and operators. First, similarly to Definition 10, we
introduce:

######  Definition 29.

Let $(o^{A},\iota^{A})$ be a spin dyad, $\Theta_{AA^{\prime}}$ the associated
GHP connection, and $f_{AA^{\prime}}$ the 1-form (2.6c). We define

$\displaystyle\tilde{\Theta}_{A^{\prime}}:=$ $\displaystyle
o^{A}\Theta_{AA^{\prime}},$ $\displaystyle\Theta_{A^{\prime}}:=$
$\displaystyle\iota^{A}\Theta_{AA^{\prime}},$ (4.15)
$\displaystyle\tilde{f}_{A^{\prime}}:=$ $\displaystyle o^{A}f_{AA^{\prime}},$
$\displaystyle f_{A^{\prime}}:=$ $\displaystyle\iota^{A}f_{AA^{\prime}},$
(4.16) $\displaystyle\sigma^{\prime}_{A^{\prime}}:=$
$\displaystyle\iota^{A}\iota^{B}\nabla_{AA^{\prime}}\iota_{B}.$ (4.17)

######  Proposition 30.

If $o^{A}$ is SFR, then

$\displaystyle\tilde{\Theta}_{A^{\prime}}o^{B}=0,\qquad\Theta_{A^{\prime}}o^{B}=-\tilde{f}_{A^{\prime}}\iota^{B},\qquad\tilde{\Theta}_{A^{\prime}}\iota^{B}=-f_{A^{\prime}}o^{B},\qquad\Theta_{A^{\prime}}\iota^{B}=\sigma^{\prime}_{A^{\prime}}o^{B}.$
(4.18)

######  Remark 31.

If $o^{A}$ is SFR and a repeated principal spinor, using (3.5) we note that
$\tilde{f}_{A^{\prime}}$ can be written as

$\tilde{f}_{A^{\prime}}=\tilde{\Theta}_{A^{\prime}}\log\phi.$ (4.19)

Since in this section we are interested in the Einstein case, we can use (3.6)
and replace $\phi=\Psi^{1/3}_{2}$.

######  Proposition 32.

Let $(\mathcal{M},g_{ab})$ be a type II Einstein spacetime with repeated
principal spinor $o^{A}$. On spinors of GHP weight $\\{p,0\\}$ with an
arbitrary number of primed indices the operators in (4.15) satisfy the
following commutator relations,

$\displaystyle[\tilde{\Theta}_{A^{\prime}},\tilde{\Theta}_{B^{\prime}}]={}$
$\displaystyle 0,$ (4.20a)
$\displaystyle[\tilde{\Theta}_{A^{\prime}},\Theta_{B^{\prime}}]={}$
$\displaystyle\epsilon_{A^{\prime}B^{\prime}}\frac{p}{2}(2\tilde{\Theta}_{C^{\prime}}f^{C^{\prime}}-\tilde{f}_{C^{\prime}}f^{C^{\prime}}+6\Lambda)+\Box^{\Theta}_{A^{\prime}B^{\prime}}-f_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}+\tilde{f}_{B^{\prime}}\Theta_{A^{\prime}},$
(4.20b) $\displaystyle[\Theta_{A^{\prime}},\Theta_{B^{\prime}}]={}$
$\displaystyle\epsilon_{A^{\prime}B^{\prime}}\left(\frac{p}{2}(\Theta_{C^{\prime}}f^{C^{\prime}}-\tilde{\Theta}_{C^{\prime}}\sigma^{\prime
C^{\prime}}-\tilde{f}_{C^{\prime}}\sigma^{\prime C^{\prime}})-\sigma^{\prime
C^{\prime}}\tilde{\Theta}_{C^{\prime}}\right).$ (4.20c)

###### Proof.

We prove (4.20a) explicitly, the other equations follow analogously. In a
general spacetime, using definition (4.15) and acting on an arbitrary spinor
$\varphi^{E...E^{\prime}...}_{F...F^{\prime}...}$ with GHP weight $\\{p,q\\}$
we get

$[\tilde{\Theta}_{A^{\prime}},\tilde{\Theta}_{B^{\prime}}]\varphi^{E...E^{\prime}...}_{F...F^{\prime}...}=\bar{\epsilon}_{A^{\prime}B^{\prime}}(o^{A}o^{B}\nabla_{A}{}^{C^{\prime}}o_{B})\iota^{C}\Theta_{CC^{\prime}}\varphi^{E...E^{\prime}...}_{F...F^{\prime}...}+o^{A}o^{B}[\Theta_{AA^{\prime}},\Theta_{BB^{\prime}}]\varphi^{E...E^{\prime}...}_{F...F^{\prime}...}.$
(4.21)

For the second term in the right, using GHP notation we find

$\displaystyle
o^{A}o^{B}[\Theta_{AA^{\prime}},\Theta_{BB^{\prime}}]\varphi^{E...E^{\prime}...}_{F...F^{\prime}...}={}$
$\displaystyle\bar{\epsilon}_{A^{\prime}B^{\prime}}o^{A}o^{B}\Theta_{AC^{\prime}}\Theta_{B}{}^{C^{\prime}}\varphi^{E...E^{\prime}...}_{F...F^{\prime}...}$
$\displaystyle={}$
$\displaystyle\bar{\epsilon}_{A^{\prime}B^{\prime}}o^{A}o^{B}\Box_{AB}\varphi^{E...E^{\prime}...}_{F...F^{\prime}...}$
$\displaystyle+\left(p(-\Psi_{1}+\sigma\tau^{\prime}-\kappa\rho^{\prime})+q(-\Phi_{01}+\bar{\tau}^{\prime}\bar{\rho}-\bar{\kappa}\bar{\sigma}^{\prime})\right)\varphi^{E...E^{\prime}...}_{F...F^{\prime}...}$
(4.22)

where $\Box_{AB}=\nabla_{A^{\prime}(A}\nabla_{B)}{}^{A^{\prime}}$ is the usual
spinor curvature operator [18, Eq. (4.9.2)]. Suppose now that the spacetime is
Einstein and of Petrov type II. Restricting the identity (4.22) to primed
spinor fields $\varphi^{E^{\prime}...}_{F^{\prime}...}$ with GHP weight
$\\{p,0\\}$, we get that each of the terms in the right hand side vanishes:
$\Box_{AB}\varphi^{E^{\prime}...}_{F^{\prime}...}=0$ because this only
involves contractions with $\Phi_{ABA^{\prime}B^{\prime}}$, which vanishes
because of the Einstein condition; the term with $q$ vanishes because we are
restricting to weight $\\{p,0\\}$; and the term with $p$ vanishes because the
vacuum type II condition implies $\kappa=\sigma=\Psi_{1}=0$. Furthermore, from
this last condition we get $o^{A}o^{B}\nabla_{A}{}^{C^{\prime}}o_{B}=0$ (see
(3.1)), therefore (4.21) vanishes identically and we get the result (4.20a). ∎

######  Remark 33.

Since
$[\tilde{\Theta}_{A^{\prime}},\tilde{\Theta}_{B^{\prime}}]=2\tilde{\Theta}_{[A^{\prime}}\tilde{\Theta}_{B^{\prime}]}=-\bar{\epsilon}_{A^{\prime}B^{\prime}}\tilde{\Theta}^{C^{\prime}}\tilde{\Theta}_{C^{\prime}}$,
the result (4.20a) can be equivalently stated as

$\tilde{\Theta}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}\varphi^{E^{\prime}...}_{F^{\prime}...}=0$
(4.23)

for any primed spinor $\varphi^{E^{\prime}...}_{F^{\prime}...}$ (or scalar)
with weight $\\{p,0\\}$. This identity will be useful below.

######  Proposition 34.

Let $\tilde{\Theta}_{A^{\prime}}$ be as in (4.15), and let $\phi$ be given by
(3.6). The linearized metric (4.10) can be expressed as

$h_{AA^{\prime}BB^{\prime}}=o_{A}o_{B}X_{A^{\prime}B^{\prime}},$ (4.24)

where

$\displaystyle X_{A^{\prime}B^{\prime}}={}$
$\displaystyle-\phi^{-2}\tilde{\Theta}_{(A^{\prime}}\left(\phi^{2}\tilde{\Theta}_{B^{\prime})}\chi\right).$
(4.25)

###### Proof.

This can be shown by direct comparison with (4.10)-(4.11). First, from (4.24)
we see that the only non-trivial components are

$\displaystyle X_{1^{\prime}1^{\prime}}={}$
$\displaystyle-\iota^{A}\bar{\iota}^{A^{\prime}}\iota^{B}\bar{\iota}^{B^{\prime}}h_{AA^{\prime}BB^{\prime}}=-n^{a}n^{b}h_{ab}=-h_{nn},$
(4.26a) $\displaystyle X_{0^{\prime}1^{\prime}}={}$
$\displaystyle-\iota^{A}\bar{\iota}^{A^{\prime}}\iota^{B}\bar{o}^{B^{\prime}}h_{AA^{\prime}BB^{\prime}}=-n^{a}\bar{m}^{b}h_{ab}=-h_{n\bar{m}},$
(4.26b) $\displaystyle X_{0^{\prime}0^{\prime}}={}$
$\displaystyle-\iota^{A}\bar{o}^{A^{\prime}}\iota^{B}\bar{o}^{B^{\prime}}h_{AA^{\prime}BB^{\prime}}=-\bar{m}^{a}\bar{m}^{b}h_{ab}=-h_{\bar{m}\bar{m}},$
(4.26c)

so the general structure (4.10) is recovered. Now we check that these
components coincide with (4.11). To do this, we write
$X_{A^{\prime}B^{\prime}}$ as

$X_{A^{\prime}B^{\prime}}=-\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-2(\tilde{\Theta}_{(A^{\prime}}\log\phi)(\tilde{\Theta}_{B^{\prime})}\chi).$
(4.27)

Next, we project this expression over the primed spin dyad
$(\bar{o}^{A^{\prime}},\bar{\iota}^{A^{\prime}})$ to compute the components
(4.26). We will need the following identities:

$\operatorname{\text{}}\log\phi=\rho,\qquad\operatorname{\eth}\log\phi=\tau,$
(4.28)

which follow from the fact that $\phi=\Psi^{1/3}_{2}$ (eq. (3.6)) together
with the Bianchi identities for a Petrov type II spacetime. Then we have, for
example,

$\displaystyle X_{1^{\prime}1^{\prime}}={}$
$\displaystyle-\bar{\iota}^{A^{\prime}}\bar{\iota}^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-2\operatorname{\eth}\log\phi\operatorname{\eth}\chi$
$\displaystyle={}$
$\displaystyle-\operatorname{\eth}\operatorname{\eth}\chi+(\operatorname{\eth}\bar{\iota}^{B^{\prime}})\tilde{\Theta}_{B^{\prime}}\chi-2\tau\operatorname{\eth}\chi$
$\displaystyle={}$
$\displaystyle-(\operatorname{\eth}+2\tau)\operatorname{\eth}\chi-\bar{\sigma}^{\prime}\operatorname{\text{}}\chi$
(4.29)

where we used
$\operatorname{\eth}\bar{\iota}^{B^{\prime}}=-\bar{\sigma}^{\prime}\bar{o}^{B^{\prime}}$,
see [18, Eq. (4.12.28)]. The other components can be computed along the same
line. ∎

######  Proposition 35.

Let $(\mathcal{M},g_{ab})$ be a type II Einstein spacetime with repeated
principal spinor $o^{A}$. The Ricci, Bianchi and commutator identities yield

$\displaystyle\tilde{\Theta}_{A^{\prime}}\tilde{f}_{B^{\prime}}={}$
$\displaystyle\tilde{f}_{A^{\prime}}\tilde{f}_{B^{\prime}},$ (4.30a)
$\displaystyle\tilde{\Theta}^{A^{\prime}}f_{A^{\prime}}={}$
$\displaystyle\Psi_{2}+2\Lambda,$ (4.30b)
$\displaystyle\Theta^{A^{\prime}}\tilde{f}_{A^{\prime}}={}$
$\displaystyle-\Psi_{2}-2\Lambda,$ (4.30c)
$\displaystyle\tilde{\Theta}_{A^{\prime}}\Psi_{2}={}$ $\displaystyle
3\tilde{f}_{A^{\prime}}\Psi_{2},$ (4.30d)
$\displaystyle\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\phi^{-1}={}$
$\displaystyle 0,$ (4.30e)

with the scalar field $\phi$ defined in (3.6).

###### Proof.

These equations can be checked by GHP expansion using for example [18, Section
4.12]. ∎

The main result of this section is given in lemma 37 below, from which theorem
2 follows. The proof of lemma 37 involves long computations and we transfer
some intermediate steps into the following:

######  Proposition 36.

The Debye equation in this formulation is given by

$\displaystyle\mathcal{O}^{\dagger}\chi=2\Theta^{A^{\prime}}(\tilde{\Theta}_{A^{\prime}}+3\tilde{f}_{A^{\prime}})\chi-6\Psi_{2}\chi.$
(4.31)

Repeated application of commutators on $\chi$ of weight $\\{-4,0\\}$ leads to

$\displaystyle\tilde{\Theta}_{A^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi={}$
$\displaystyle-\tilde{f}^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+3\tilde{f}^{A^{\prime}}\Lambda\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{15}{2}\tilde{f}^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{3}{2}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi$
$\displaystyle-3f^{A^{\prime}}\tilde{f}_{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\Theta^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{3}{2}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi$
$\displaystyle+\tfrac{3}{2}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}f_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi-\tfrac{3}{2}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}f_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi.$
(4.32)
$\displaystyle\tilde{\Theta}_{B^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi={}$
$\displaystyle-\tilde{f}^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+3\tilde{f}^{A^{\prime}}\Lambda\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{15}{2}\tilde{f}^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{3}{2}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi$
$\displaystyle-3f^{A^{\prime}}\tilde{f}_{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\Theta^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{3}{2}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi$
$\displaystyle+\tfrac{3}{2}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}f_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi-\tfrac{3}{2}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}f_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi.$
(4.33)

$\displaystyle\Theta_{B^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi={}$
$\displaystyle-9\chi
f^{A^{\prime}}\tilde{f}_{A^{\prime}}\Lambda+18\chi\Lambda^{2}-\tfrac{3}{4}f^{A^{\prime}}\tilde{f}_{A^{\prime}}\mathcal{O}^{\dagger}{}\chi+\tfrac{9}{2}\Lambda\mathcal{O}^{\dagger}{}\chi+\tfrac{1}{4}\mathcal{O}^{\dagger}{}\mathcal{O}^{\dagger}{}\chi-9\chi
f^{A^{\prime}}\tilde{f}_{A^{\prime}}\Psi_{2}$
$\displaystyle-36\chi\Lambda\Psi_{2}+\tfrac{3}{2}\mathcal{O}^{\dagger}{}\chi\Psi_{2}-36\chi\Psi_{2}^{2}+3\tilde{f}^{A^{\prime}}\Lambda\Theta_{A^{\prime}}\chi-30\tilde{f}^{A^{\prime}}\Psi_{2}\Theta_{A^{\prime}}\chi$
$\displaystyle+\tfrac{3}{2}\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\mathcal{O}^{\dagger}{}\chi-18\chi\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\Psi_{2}-3\Lambda\Theta_{A^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi-6\Psi_{2}\Theta_{A^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi$
$\displaystyle-\tfrac{3}{2}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-\tfrac{9}{2}f^{A^{\prime}}\tilde{f}_{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{B^{\prime}}\chi-3\tilde{f}^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{B^{\prime}}\Theta_{A^{\prime}}\chi$
$\displaystyle+\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}\Theta^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{3}{2}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi-3\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\Theta_{A^{\prime}}\chi$
$\displaystyle+3\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\tilde{f}_{B^{\prime}}\Theta^{B^{\prime}}\chi-\Theta_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi\Theta^{B^{\prime}}\tilde{f}^{A^{\prime}}-3f^{A^{\prime}}\Lambda\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{3}{2}f^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi$
$\displaystyle+3\Theta^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{3}{2}\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}f^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{3}{2}\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}f^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi$
$\displaystyle-\tfrac{3}{2}f^{A^{\prime}}\Theta_{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{3}{2}\tilde{f}^{A^{\prime}}\Theta^{B^{\prime}}f_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-\tfrac{3}{2}f^{A^{\prime}}\Theta^{B^{\prime}}\tilde{f}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi$
$\displaystyle+3\Theta^{B^{\prime}}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\Theta_{A^{\prime}}\chi.$
(4.34)
$\displaystyle\Theta_{B^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi={}$
$\displaystyle-6\Lambda\Theta_{A^{\prime}}\chi-6\Psi_{2}\Theta_{A^{\prime}}\chi-\tfrac{1}{2}\Theta_{A^{\prime}}\mathcal{O}^{\dagger}{}\chi-6\chi\Theta_{A^{\prime}}\Psi_{2}+3\tilde{f}_{B^{\prime}}\Theta_{A^{\prime}}\Theta^{B^{\prime}}\chi$
$\displaystyle+3\Theta_{A^{\prime}}\tilde{f}_{B^{\prime}}\Theta^{B^{\prime}}\chi-\tfrac{3}{2}\tilde{f}_{B^{\prime}}\sigma^{\prime
B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{3}{2}\Theta_{B^{\prime}}f^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{3}{2}\tilde{\Theta}_{A^{\prime}}\chi\tilde{\Theta}_{B^{\prime}}\sigma^{\prime
B^{\prime}}$ $\displaystyle-\sigma^{\prime
B^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi.$ (4.35)
$\displaystyle\Theta_{A^{\prime}}\tilde{\Theta}^{A^{\prime}}\Theta^{B^{\prime}}\chi={}$
$\displaystyle-4\chi f^{B^{\prime}}\Lambda-2\chi
f^{B^{\prime}}\Psi_{2}+4\chi\tilde{f}^{A^{\prime}}\tilde{f}^{B^{\prime}}\sigma^{\prime}_{A^{\prime}}-2f^{B^{\prime}}\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\chi-2f^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{A^{\prime}}\chi$
$\displaystyle-6\chi\tilde{f}^{B^{\prime}}\Theta_{A^{\prime}}f^{A^{\prime}}-2\chi\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}f^{B^{\prime}}-2\chi
f^{A^{\prime}}\Theta_{A^{\prime}}\tilde{f}^{B^{\prime}}-\tilde{f}^{B^{\prime}}\Theta_{A^{\prime}}\Theta^{A^{\prime}}\chi-f^{A^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi$
$\displaystyle+\Theta_{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta^{A^{\prime}}\chi-2f^{A^{\prime}}\tilde{f}_{A^{\prime}}\Theta^{B^{\prime}}\chi-2\Lambda\Theta^{B^{\prime}}\chi-10\Psi_{2}\Theta^{B^{\prime}}\chi+2\chi\tilde{f}^{A^{\prime}}\Theta^{B^{\prime}}f_{A^{\prime}}$
$\displaystyle-2\chi
f^{A^{\prime}}\Theta^{B^{\prime}}\tilde{f}_{A^{\prime}}+3\Theta^{A^{\prime}}\chi\Theta^{B^{\prime}}\tilde{f}_{A^{\prime}}-\tfrac{1}{2}\Theta^{B^{\prime}}O^{\dagger}{}\chi-10\chi\Theta^{B^{\prime}}\Psi_{2}-3\tilde{f}^{A^{\prime}}\Theta^{B^{\prime}}\Theta_{A^{\prime}}\chi$
$\displaystyle-2\tilde{f}^{B^{\prime}}\sigma^{\prime
A^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+4\chi\tilde{f}^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\sigma^{\prime
A^{\prime}}-\sigma^{\prime
A^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi+\tfrac{3}{2}\tilde{f}^{A^{\prime}}\sigma^{\prime}_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi$
$\displaystyle-\tfrac{5}{2}\Theta_{A^{\prime}}f^{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi+\tfrac{3}{2}\tilde{\Theta}_{A^{\prime}}\sigma^{\prime
A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi.$ (4.36)
$\displaystyle\Theta_{B^{\prime}}\Theta^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi={}$
$\displaystyle\tfrac{1}{2}(-3\tilde{f}_{B^{\prime}}\sigma^{\prime
B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi-3\Theta_{B^{\prime}}f^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+3\tilde{\Theta}_{A^{\prime}}\chi\tilde{\Theta}_{B^{\prime}}\sigma^{\prime
B^{\prime}}-2\sigma^{\prime
B^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi).$
(4.37)

###### Proof.

The identities are verified by direct computation using the projected
operators $\Theta_{A^{\prime}},\tilde{\Theta}_{A^{\prime}}$ defined in (4.15)
and their commutator properties (4.20). The Debye equation (4.31) can be seen
to coincide with (4.9b) by GHP expansion. ∎

######  Lemma 37.

The ASD curvature components of a metric of the form (4.10),(4.11) are given
by

$\displaystyle\dot{\Psi}_{0}={}$ $\displaystyle 0$ (4.38a)
$\displaystyle\dot{\Psi}_{1}={}$ $\displaystyle 0$ (4.38b)
$\displaystyle\dot{\Psi}_{2}={}$ $\displaystyle 0$ (4.38c)
$\displaystyle\dot{\Psi}_{3}={}$
$\displaystyle-\tfrac{1}{4}(\tau\operatorname{\text{}}-\rho\operatorname{\eth})\mathcal{O}^{\dagger}{}\chi,$
(4.38d) $\displaystyle\dot{\Psi}_{4}={}$
$\displaystyle-(\Psi_{2}^{4/3}\xi^{a}\Theta_{a}+3\Psi_{2}^{2}+6\Psi_{2}\Lambda)\chi-(\tfrac{1}{8}\mathcal{O}^{\dagger}-\rho\operatorname{\text{}}^{\prime}+\tau\operatorname{\eth}^{\prime}+2\Psi_{2}+\tfrac{5}{2}\Lambda)\mathcal{O}^{\dagger}{}\chi.$
(4.38e)

###### Proof.

The result is verified by direct computation using the projected operators
$\Theta_{A^{\prime}},\tilde{\Theta}_{A^{\prime}}$ defined in (4.15) and their
commutator properties (4.20). The first step is to derive an appropriate form
for the components of the linearized ASD Weyl curvature spinor (4.1).
Replacing (4.24) in (4.1) and using (4.15), (4.16), (4.18), we find

$\displaystyle\dot{\Psi}_{0}={}$ $\displaystyle 0,$ (4.39a)
$\displaystyle\dot{\Psi}_{1}={}$ $\displaystyle 0,$ (4.39b)
$\displaystyle\dot{\Psi}_{2}={}$
$\displaystyle\frac{1}{12}(\tilde{\Theta}_{B^{\prime}}+2\tilde{f}_{B^{\prime}})(\tilde{\Theta}_{A^{\prime}}+2\tilde{f}_{A^{\prime}})X^{A^{\prime}B^{\prime}},$
(4.39c) $\displaystyle\dot{\Psi}_{3}={}$
$\displaystyle\frac{1}{8}\left((\Theta_{B^{\prime}}+f_{B^{\prime}})(\tilde{\Theta}_{A^{\prime}}+2\tilde{f}_{A^{\prime}})+(\tilde{\Theta}_{B^{\prime}}+3\tilde{f}_{B^{\prime}})\Theta_{A^{\prime}}\right)X^{A^{\prime}B^{\prime}},$
(4.39d) $\displaystyle\dot{\Psi}_{4}={}$
$\displaystyle\frac{1}{2}\left(\Theta_{B^{\prime}}\Theta_{A^{\prime}}-\sigma^{\prime}_{B^{\prime}}(\tilde{\Theta}_{A^{\prime}}+2\tilde{f}_{A^{\prime}})\right)X^{A^{\prime}B^{\prime}}.$
(4.39e)

So the first two equations, (4.39a) and (4.39b), follow from the algebraic
structure of (4.25). For $\dot{\Psi}_{2}$, rewrite the operator in (4.39c)
using (4.19) and insert (4.25),

$\displaystyle\dot{\Psi}_{2}={}\tfrac{1}{12}\phi^{-2}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}(\phi^{2}X^{A^{\prime}B^{\prime}})={}\tfrac{1}{12}\phi^{-2}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}^{A^{\prime}}(\phi^{2}\tilde{\Theta}^{B^{\prime}}\chi)={}0,$
(4.40)

where the last step follows from (4.23). Next we compute $\dot{\psi}_{3}$ by
first inserting (4.25) into (4.39d) and expanding out,

$\displaystyle\dot{\Psi}_{3}{}={}$
$\displaystyle-\tfrac{1}{8}\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi-\tfrac{1}{16}\Theta_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi+\tfrac{5}{8}\tilde{f}^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{3}{16}\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi$
$\displaystyle-\tfrac{1}{16}\Theta_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi+\tfrac{5}{16}\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{5}{8}\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{1}{4}\Theta^{B^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi\tilde{\Theta}_{A^{\prime}}\tilde{f}_{B^{\prime}}$
$\displaystyle-\tfrac{1}{8}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi-\tfrac{1}{16}\tilde{\Theta}_{A^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi-\tfrac{1}{8}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{1}{8}\Theta^{B^{\prime}}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi$
$\displaystyle-\tfrac{1}{8}\tilde{\Theta}_{A^{\prime}}\Theta_{B^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi-\tfrac{1}{4}f^{A^{\prime}}\tilde{f}_{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{1}{4}\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{1}{8}\Theta_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi$
$\displaystyle-\tfrac{3}{8}\tilde{f}^{A^{\prime}}\Theta^{B^{\prime}}\tilde{f}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{1}{8}\Theta^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-\tfrac{1}{4}\Theta_{A^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi\tilde{\Theta}_{B^{\prime}}\tilde{f}^{B^{\prime}}+\tfrac{1}{8}f^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi\tilde{\Theta}_{B^{\prime}}\tilde{f}^{B^{\prime}}$
$\displaystyle-\tfrac{1}{8}\tilde{\Theta}^{A^{\prime}}\chi\tilde{\Theta}_{B^{\prime}}\Theta_{A^{\prime}}\tilde{f}^{B^{\prime}}-\tfrac{1}{8}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi-\tfrac{1}{16}\tilde{\Theta}_{B^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi-\tfrac{1}{8}\Theta^{B^{\prime}}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi$
$\displaystyle+\tfrac{1}{16}f^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi+\tfrac{1}{8}f^{A^{\prime}}\tilde{f}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi-\tfrac{1}{4}\Theta_{A^{\prime}}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi+\tfrac{1}{16}f^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi$
$\displaystyle+\tfrac{1}{8}f^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{f}_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi.$
(4.41)

Using (4.30) and (4.20a) yields

$\displaystyle\dot{\Psi}_{3}{}={}$
$\displaystyle\tfrac{7}{8}\tilde{f}^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{1}{2}\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{3}{2}\tilde{f}^{A^{\prime}}\Lambda\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{9}{8}\tilde{f}^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{1}{8}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi$
$\displaystyle-\tfrac{1}{16}\tilde{\Theta}_{A^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi-\tfrac{3}{8}f^{A^{\prime}}\tilde{f}_{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{3}{8}\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-\tfrac{3}{8}\tilde{f}^{A^{\prime}}\Theta^{B^{\prime}}\tilde{f}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi$
$\displaystyle-\tfrac{1}{8}\tilde{\Theta}^{A^{\prime}}\chi\tilde{\Theta}_{B^{\prime}}\Theta_{A^{\prime}}\tilde{f}^{B^{\prime}}-\tfrac{1}{8}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi-\tfrac{1}{16}\tilde{\Theta}_{B^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{\Theta}^{A^{\prime}}\chi-\tfrac{1}{8}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi.$
(4.42)

To eliminate 4th order terms, we use (36) and (36), leading to

$\displaystyle\dot{\Psi}_{3}{}={}$
$\displaystyle\tilde{f}^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{1}{2}\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{9}{8}\tilde{f}^{A^{\prime}}\Lambda\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{33}{16}\tilde{f}^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{1}{8}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi$
$\displaystyle+\tfrac{3}{16}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{3}{8}\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-\tfrac{3}{8}\tilde{f}^{A^{\prime}}\Theta^{B^{\prime}}\tilde{f}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-\tfrac{1}{8}\tilde{\Theta}^{A^{\prime}}\chi\tilde{\Theta}_{B^{\prime}}\Theta_{A^{\prime}}\tilde{f}^{B^{\prime}}$
$\displaystyle-\tfrac{1}{8}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi-\tfrac{1}{8}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\Theta^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{1}{16}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{3}{16}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}f_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi$
$\displaystyle+\tfrac{3}{16}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}f_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi.$
(4.43)

Using the commutators (4.20a), (4.20b) and the Debye equation (4.31) together
with (4.30) takes care of third order terms,

$\displaystyle\dot{\Psi}_{3}{}={}$
$\displaystyle-\tfrac{13}{8}\tilde{f}^{A^{\prime}}\Lambda\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{13}{16}\tilde{f}^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{1}{4}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}\mathcal{O}^{\dagger}{}\chi+\tfrac{3}{8}\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-\tfrac{3}{8}\tilde{f}^{A^{\prime}}\Theta^{B^{\prime}}\tilde{f}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi$
$\displaystyle+\tfrac{7}{16}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}f_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi-\tfrac{7}{16}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}f_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi.$
(4.44) Finally, the irreducible decompositions
$\displaystyle\tilde{\Theta}_{A^{\prime}}f_{B^{\prime}}={}$
$\displaystyle\tilde{\Theta}_{(A^{\prime}}f_{B^{\prime})}+\tfrac{1}{2}\bar{\epsilon}_{A^{\prime}B^{\prime}}\tilde{\Theta}_{C^{\prime}}f^{C^{\prime}},$
(4.45a) $\displaystyle\Theta_{A^{\prime}}\tilde{f}_{B^{\prime}}={}$
$\displaystyle\Theta_{(A^{\prime}}\tilde{f}_{B^{\prime})}+\tfrac{1}{2}\bar{\epsilon}_{A^{\prime}B^{\prime}}\Theta_{C^{\prime}}\tilde{f}^{C^{\prime}},$
(4.45b)

lead to

$\displaystyle\dot{\Psi}_{3}{}={}$
$\displaystyle-\tfrac{1}{4}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}\mathcal{O}^{\dagger}{}\chi,$
(4.46)

which gives (4.38d) by GHP expanding
$\tilde{f}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}$.

To compute $\dot{\Psi}_{4}$, insert (4.25) into (4.39e) and use the identity
(36) together with the Debye equation (4.31) and (4.30) leading to

$\displaystyle\dot{\Psi}_{4}{}={}$ $\displaystyle\tfrac{9}{2}\chi
f^{A^{\prime}}\tilde{f}_{A^{\prime}}\Lambda-18\chi\Lambda^{2}+\tfrac{3}{8}f^{A^{\prime}}\tilde{f}_{A^{\prime}}\mathcal{O}^{\dagger}{}\chi-2\Lambda\mathcal{O}^{\dagger}{}\chi-\tfrac{1}{8}\mathcal{O}^{\dagger}{}\mathcal{O}^{\dagger}{}\chi+\tfrac{9}{2}\chi
f^{A^{\prime}}\tilde{f}_{A^{\prime}}\Psi_{2}+6\chi\Lambda\Psi_{2}$
$\displaystyle-\tfrac{7}{4}\mathcal{O}^{\dagger}{}\chi\Psi_{2}+\tfrac{15}{2}\chi\Psi_{2}^{2}+\tfrac{21}{2}\tilde{f}^{A^{\prime}}\Psi_{2}\Theta_{A^{\prime}}\chi+6\chi\Lambda\Theta_{A^{\prime}}\tilde{f}^{A^{\prime}}-\tfrac{3}{2}\chi\Psi_{2}\Theta_{A^{\prime}}\tilde{f}^{A^{\prime}}-\tfrac{1}{2}\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\mathcal{O}^{\dagger}{}\chi$
$\displaystyle+\tfrac{21}{2}\chi\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\Psi_{2}+\tfrac{3}{2}\chi\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\Theta_{B^{\prime}}\tilde{f}^{B^{\prime}}+\tfrac{3}{4}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{9}{4}f^{A^{\prime}}\tilde{f}_{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{B^{\prime}}\chi$
$\displaystyle+\tfrac{3}{2}\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\chi\Theta_{B^{\prime}}\tilde{f}^{B^{\prime}}+3\tilde{f}^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{B^{\prime}}\Theta_{A^{\prime}}\chi-\tfrac{1}{2}\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi-\tfrac{1}{2}\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}\Theta^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi$
$\displaystyle+\tfrac{3}{4}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{3}{2}\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}^{B^{\prime}}\Theta_{A^{\prime}}\chi-3\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\tilde{f}_{B^{\prime}}\Theta^{B^{\prime}}\chi+\Theta_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi\Theta^{B^{\prime}}\tilde{f}^{A^{\prime}}$
$\displaystyle+\tfrac{1}{2}\Theta_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi\Theta^{B^{\prime}}\tilde{f}^{A^{\prime}}+\tfrac{3}{2}f^{A^{\prime}}\Lambda\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{3}{4}f^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi-\Theta^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{3}{4}\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}f^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi$
$\displaystyle-\tfrac{3}{2}\tilde{f}^{A^{\prime}}\tilde{f}^{B^{\prime}}\sigma^{\prime}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{3}{4}\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}f^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{3}{4}f^{A^{\prime}}\Theta_{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{1}{2}\Theta_{A^{\prime}}\Theta^{B^{\prime}}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi$
$\displaystyle-\tfrac{3}{4}\tilde{f}^{A^{\prime}}\Theta^{B^{\prime}}f_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{3}{4}f^{A^{\prime}}\Theta^{B^{\prime}}\tilde{f}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-\tfrac{3}{2}\Theta^{B^{\prime}}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\Theta_{A^{\prime}}\chi+\tfrac{1}{2}\tilde{f}^{A^{\prime}}\sigma^{\prime
B^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi$
$\displaystyle-\tfrac{1}{4}\sigma^{\prime
A^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi.$
(4.47)

To convert third order terms we use identities (36), (36), (4.37). After a
commutator (4.20b) and (4.30) is used, we have

$\displaystyle\dot{\Psi}_{4}{}={}$ $\displaystyle\tfrac{9}{2}\chi
f^{A^{\prime}}\tilde{f}_{A^{\prime}}\Lambda+6\chi\Lambda^{2}+\tfrac{3}{8}f^{A^{\prime}}\tilde{f}_{A^{\prime}}\mathcal{O}^{\dagger}{}\chi-2\Lambda\mathcal{O}^{\dagger}{}\chi-\tfrac{1}{8}\mathcal{O}^{\dagger}{}\mathcal{O}^{\dagger}{}\chi+\tfrac{9}{2}\chi
f^{A^{\prime}}\tilde{f}_{A^{\prime}}\Psi_{2}+3\chi\Lambda\Psi_{2}$
$\displaystyle-\tfrac{7}{4}\mathcal{O}^{\dagger}{}\chi\Psi_{2}+3\tilde{f}^{A^{\prime}}\Lambda\Theta_{A^{\prime}}\chi-\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\mathcal{O}^{\dagger}{}\chi-\tfrac{3}{4}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{9}{4}f^{A^{\prime}}\tilde{f}_{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{B^{\prime}}\chi$
$\displaystyle+\tfrac{3}{4}f^{A^{\prime}}\tilde{f}^{B^{\prime}}\Theta_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{1}{2}\Theta_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi\Theta^{B^{\prime}}\tilde{f}^{A^{\prime}}+\tfrac{1}{2}\Theta_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi\Theta^{B^{\prime}}\tilde{f}^{A^{\prime}}+\tfrac{3}{2}f^{A^{\prime}}\Lambda\tilde{\Theta}_{A^{\prime}}\chi$
$\displaystyle+\tfrac{3}{4}f^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi-\Theta^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{3}{2}\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}f^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{3}{4}\tilde{f}^{A^{\prime}}\tilde{f}^{B^{\prime}}\sigma^{\prime}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi$
$\displaystyle+\tfrac{3}{4}\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}f^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-\tfrac{3}{4}f^{A^{\prime}}\Theta_{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{1}{2}\Theta_{A^{\prime}}\Theta^{B^{\prime}}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-\tfrac{3}{4}\tilde{f}^{A^{\prime}}\Theta^{B^{\prime}}f_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi$
$\displaystyle+\tfrac{3}{4}f^{A^{\prime}}\Theta^{B^{\prime}}\tilde{f}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{3}{4}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi\tilde{\Theta}_{B^{\prime}}\sigma^{\prime
B^{\prime}}-\tfrac{1}{4}\sigma^{\prime
A^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}^{B^{\prime}}\chi.$
(4.48)

Now, the commutator (4.20a) together with the irreducible decomposition

$\displaystyle\Theta_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi={}$
$\displaystyle\Theta_{(A^{\prime}}\tilde{\Theta}_{B^{\prime})}\chi+\tfrac{1}{2}\bar{\epsilon}_{A^{\prime}B^{\prime}}\Theta_{C^{\prime}}\tilde{\Theta}^{C^{\prime}}\chi,$
(4.49)

and (4.31), (4.30) yields

$\displaystyle\dot{\Psi}_{4}{}={}$
$\displaystyle-\tfrac{5}{2}\Lambda\mathcal{O}^{\dagger}{}\chi-\tfrac{1}{8}\mathcal{O}^{\dagger}{}\mathcal{O}^{\dagger}{}\chi-6\chi\Lambda\Psi_{2}-2\mathcal{O}^{\dagger}{}\chi\Psi_{2}-3\chi\Psi_{2}^{2}-\tfrac{3}{2}\tilde{f}^{A^{\prime}}\Psi_{2}\Theta_{A^{\prime}}\chi-\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\mathcal{O}^{\dagger}{}\chi$
$\displaystyle+\tfrac{3}{2}f^{A^{\prime}}\Lambda\tilde{\Theta}_{A^{\prime}}\chi+\tfrac{3}{4}f^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{1}{2}\Theta^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{5}{4}\tilde{f}^{A^{\prime}}\Theta_{B^{\prime}}f^{B^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi$
$\displaystyle-\tfrac{1}{2}\tilde{f}^{A^{\prime}}\tilde{f}^{B^{\prime}}\sigma^{\prime}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{3}{4}\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}f^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-\tfrac{3}{4}f^{A^{\prime}}\Theta_{A^{\prime}}\tilde{f}^{B^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi-\tfrac{3}{4}\tilde{f}^{A^{\prime}}\Theta^{B^{\prime}}f_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi$
$\displaystyle+\tfrac{3}{4}f^{A^{\prime}}\Theta^{B^{\prime}}\tilde{f}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\chi+\tfrac{1}{2}\tilde{f}^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}\chi\tilde{\Theta}_{B^{\prime}}\sigma^{\prime
B^{\prime}}.$ (4.50)

The Ricci identity

$\displaystyle 2\Psi_{3}-\tilde{f}_{B^{\prime}}\sigma^{\prime
B^{\prime}}+\Theta_{A^{\prime}}f^{A^{\prime}}-\tilde{\Theta}_{B^{\prime}}\sigma^{\prime
B^{\prime}}={}$ $\displaystyle 0.$ (4.51)

and the irreducible decompositions (4.45b) and

$\displaystyle\Theta_{A^{\prime}}f_{B^{\prime}}={}$
$\displaystyle\Theta_{(A^{\prime}}f_{B^{\prime})}+\tfrac{1}{2}\bar{\epsilon}_{A^{\prime}B^{\prime}}\Theta_{C^{\prime}}f^{C^{\prime}},$
(4.52)

lead to

$\displaystyle\dot{\Psi}_{4}{}={}$
$\displaystyle-\tfrac{5}{2}\Lambda\mathcal{O}^{\dagger}{}\chi-\tfrac{1}{8}\mathcal{O}^{\dagger}{}\mathcal{O}^{\dagger}{}\chi-6\chi\Lambda\Psi_{2}-2\mathcal{O}^{\dagger}{}\chi\Psi_{2}-3\chi\Psi_{2}^{2}-\tfrac{3}{2}\tilde{f}^{A^{\prime}}\Psi_{2}\Theta_{A^{\prime}}\chi-\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}\mathcal{O}^{\dagger}{}\chi$
$\displaystyle+\tilde{f}^{A^{\prime}}\Psi_{3}\tilde{\Theta}_{A^{\prime}}\chi-\tfrac{1}{2}\Theta^{A^{\prime}}\Psi_{2}\tilde{\Theta}_{A^{\prime}}\chi.$
(4.53)

To bring it into the final form we use the Bianchi identity

$\displaystyle-3f_{A^{\prime}}\Psi_{2}+2\tilde{f}_{A^{\prime}}\Psi_{3}+\Theta_{A^{\prime}}\Psi_{2}-\tilde{\Theta}_{A^{\prime}}\Psi_{3}={}$
$\displaystyle 0,$ (4.54)

resulting in

$\displaystyle\dot{\Psi}_{4}{}={}$
$\displaystyle-\tfrac{3}{2}\Psi_{2}(f^{A^{\prime}}\tilde{\Theta}_{A^{\prime}}+\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}+2\Psi_{2}+4\Lambda)\chi+\tfrac{1}{2}(\tilde{\Theta}_{A^{\prime}}-4\tilde{f}_{A^{\prime}})\Psi_{3}\tilde{\Theta}^{A^{\prime}}\chi$
$\displaystyle-\left(\tfrac{1}{8}\mathcal{O}^{\dagger}{}+\tilde{f}^{A^{\prime}}\Theta_{A^{\prime}}+2\Psi_{2}+\tfrac{5}{2}\Lambda\right)\mathcal{O}^{\dagger}{}\chi.$
(4.55)

GHP expansion of
$\tilde{f}_{A^{\prime}},\Theta_{A^{\prime}},\tilde{\Theta}_{A^{\prime}}$ leads
to

$\displaystyle\dot{\Psi}_{4}={}$
$\displaystyle-\tfrac{3}{2}\Psi_{2}(\rho^{\prime}\operatorname{\text{}}-\rho\operatorname{\text{}}^{\prime}+\tau\operatorname{\eth}^{\prime}-\tau^{\prime}\operatorname{\eth}+2\Psi_{2}+4\Lambda)\chi+\tfrac{1}{2}((\operatorname{\text{}}-4\rho)\Psi_{3})\operatorname{\eth}\chi-\tfrac{1}{2}((\operatorname{\eth}-4\tau)\Psi_{3})\operatorname{\text{}}\chi$
$\displaystyle-(\tfrac{1}{8}\mathcal{O}^{\dagger}-\rho\operatorname{\text{}}^{\prime}+\tau\operatorname{\eth}^{\prime}+2\Psi_{2}+\tfrac{5}{2}\Lambda)\mathcal{O}^{\dagger}{}\chi.$
(4.56)

Comparison to the projected Killing vector defined in (3.26) shows (4.38e). ∎

###### Proof of Theorem 2.

The result follows from Lemma 37 by imposing the Debye equation (4.7). ∎

######  Remark 38.

In the special case of vacuum Petrov type D and for tetrads invariant under
$\xi^{a}$, (4.38e) reduces to $\dot{\Psi}_{4}=\xi^{a}\nabla_{a}\chi$, see [1].

######  Remark 39.

Let us finally compare to three references closely related to the results of
this section.

  1. (1)

In [15] Kegeles and Cohen discuss Debye potentials for algebraically special
geometries. They restricted to vacuum Petrov type D for the derivation of the
linearized Weyl spinor, see equation (5.28) in that reference. They reduced
the ASD Weyl curvature to type N and also the $\dot{\Psi}_{4}$ to first order.

  2. (2)

In [26], Stewart derived the linearized connection and curvature components
for vacuum type II perturbations in terms of a Debye potential. However, the
result was not fully simplified, see equation (4.27) of that reference, so
that the type N property could not be observed. The result was also presented
in terms of a real metric, which means that all terms involving $\bar{\chi}$
correspond to self dual Weyl curvature, while $\chi$ terms belong to anti-self
dual Weyl curvature.

It should also be noted that in general, linearized Dyad components differ
from dyad components of the linearized field. In this paper $\dot{\Psi}_{i}$
refers to the latter, while Stewart used the linearized Newman-Penrose
equations, i.e. the former. However, he made a special choice of linearized
tetrad for which the two sets of linearized Weyl components coincide.

  3. (3)

In [14], Jeffryes discusses a reduction to scalar potentials for algebraically
special solutions to the full non-linear Einstein-Yang-Mills equations.
Further it is shown that, to linear order, this construction reduces to the
Debye potential formulation. Remarkably, the ASD Weyl curvature can be
simplified already on the non-linear level, so that Theorem 2 we discuss here
follows from the linearized equations (8.75-77) of that reference.

### Acknowledgements

This work started while the authors were in residence at Institut Mittag-
Leffler in Djursholm, Sweden during the fall of 2019, supported by the Swedish
Research Council under grant no. 2016-06596. SA thanks Thomas Bäckdahl for
xAct support and Benjamin Jeffryes for comments on his preprint [14]. BA is
supported by a postdoctoral fellowship from Conicet (Argentina). BFW
acknowledges support from NSF grant PHY 1607323, sabbatical support from the
University of Florida and the Observaroire de Paris at Meudon, and the
Institut d'Astrophysique de Paris.

##  Appendix A Robinson-Trautman metrics

In 1968, Robinson and Trautman, [22], published a line element for which the
vacuum Einstein equations reduce essentially to a non-linear fourth order
equation for a real scalar function. It admits solutions of all Petrov types.
Here we briefly review the reduction and also the explicit example of Petrov
type II given in [22].

In coordinates $(u,r,\zeta,\bar{\zeta})$ and with real functions $H,P$, define
the tetrad

$\displaystyle l^{a}={}(\partial_{r})^{a},\qquad
n^{a}={}(\partial_{u})^{a}-H(\partial_{r})^{a},\qquad
m^{a}={}Pr^{-1}(\partial_{\zeta})^{a},\qquad\bar{m}^{a}={}Pr^{-1}(\partial_{\bar{\zeta}})^{a}.$
(A.1)

Due to the normalization $l^{a}n_{a}=1,m^{a}\bar{m}_{a}=-1$, the inverse is
given by

$\displaystyle l_{a}={}\mathrm{d}u_{a},\qquad
n_{a}={}\mathrm{d}r_{a}+H\mathrm{d}u_{a},\qquad
m_{a}={}-rP^{-1}\mathrm{d}\zeta_{a},\qquad\bar{m}_{a}={}-rP^{-1}\mathrm{d}\bar{\zeta}_{a},$
(A.2)

so that the metric $g_{ab}=2l_{(a}n_{b)}-2m_{(a}\bar{m}_{b)}$ is of the form

$\displaystyle g_{ab}={}$ $\displaystyle
2\mathrm{d}r_{(a}\mathrm{d}u_{b)}+2H\mathrm{d}u_{a}\mathrm{d}u_{b}-2r^{2}P^{-2}\mathrm{d}\zeta_{(a}\mathrm{d}\bar{\zeta}_{b)}.$
(A.3)

This metric is Ricci flat if $H$ is given by

$\displaystyle H={}$ $\displaystyle
P^{2}\partial_{\zeta}\partial_{\bar{\zeta}}\log(P)-r\partial_{u}\log(P)-mr^{-1},\qquad\text{with
}m=m(u),$ (A.4)

and $P$ being independent of $r$, solving

$\displaystyle
P^{2}\partial_{\zeta}\partial_{\bar{\zeta}}\left(P^{2}\partial_{\zeta}\partial_{\bar{\zeta}}\log(P)\right)-\partial_{u}m+3m\partial_{u}\log(P)={}$
$\displaystyle 0.$ (A.5)

For the connection and curvature we find

$\displaystyle\kappa=\sigma=\sigma^{\prime}=\tau=\tau^{\prime}=\epsilon=0,\qquad
R_{ab}=0,\qquad\Psi_{0}=\Psi_{1}=0,$ (A.6)

in particular the metric is algebraically special. The non-vanishing spin
coefficients are given by

$\displaystyle\kappa^{\prime}={}$ $\displaystyle-
Pr^{-1}\partial_{\bar{\zeta}}H,$ $\displaystyle\rho={}$
$\displaystyle-r^{-1},$ $\displaystyle\rho^{\prime}={}$ $\displaystyle
Hr^{-1}\partial_{u}\log(P),$ (A.7a) $\displaystyle\gamma={}$
$\displaystyle\tfrac{1}{2}\partial_{r}H,$ $\displaystyle\alpha={}$
$\displaystyle\tfrac{1}{2}r^{-1}\partial_{\bar{\zeta}}P,$
$\displaystyle\beta={}$ $\displaystyle-\tfrac{1}{2}r^{-1}\partial_{\zeta}P,$
(A.7b)

with $H$ given in (A.4) and the remaining Weyl components are of the form

$\displaystyle\Psi_{2}={}$ $\displaystyle-
mr^{-3},\qquad\Psi_{3}=-Pr^{-2}\partial_{\bar{\zeta}}\left(P^{2}\partial_{\zeta}\partial_{\bar{\zeta}}\log(P)\right),$
(A.8a) $\displaystyle\Psi_{4}={}$ $\displaystyle
r^{-2}\partial_{\bar{\zeta}}\left(P^{2}\partial_{\bar{\zeta}}\left(P^{2}\partial_{\zeta}\partial_{\bar{\zeta}}\log(P)-r\partial_{u}\log(P)\right)\right).$
(A.8b)

An explicit example of a type II geometry, found in [22], is given by

$\displaystyle P=(\zeta+\bar{\zeta})^{3/2},\qquad m=\text{const.},$ (A.9)

leading to $H=-m/r-3(\zeta+\bar{\zeta})/2$ and to the metric

$\displaystyle g_{ab}={}$ $\displaystyle
2\mathrm{d}r_{(a}\mathrm{d}u_{b)}-\left(3(\zeta+\bar{\zeta})+2m/r\right)\mathrm{d}u_{a}\mathrm{d}u_{b}-2r^{2}(\zeta+\bar{\zeta})^{-3}\mathrm{d}\zeta_{(a}\mathrm{d}\bar{\zeta}_{b)},$
(A.10)

see also [25, §28]. The spin coefficients reduce to

$\displaystyle\kappa^{\prime}=\frac{3(\zeta+\bar{\zeta})^{3/2}}{2r},$
$\displaystyle\rho=-\frac{1}{r},$
$\displaystyle\rho^{\prime}=-\frac{m}{r^{2}}-\frac{3(\zeta+\bar{\zeta})}{2r},$
$\displaystyle\gamma=\frac{m}{2r^{2}},$
$\displaystyle\alpha=-\beta=\frac{3(\zeta+\bar{\zeta})^{1/2}}{4r},$ (A.11)

and the curvature components are given by

$\displaystyle\Psi_{2}=-\frac{m}{r^{3}},$
$\displaystyle\Psi_{3}=\frac{3(\zeta+\bar{\zeta})^{3/2}}{2r^{2}},$
$\displaystyle\Psi_{4}=-\frac{9(\zeta+\bar{\zeta})^{2}}{2r^{2}}.$ (A.12)

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