# Parallel spinors, pp-waves, and gravitational perturbations

Bernardo Araneda111Email<EMAIL_ADDRESS>
Max-Planck-Institut für Gravitationsphysik
(Albert Einstein Institut), Am Mühlenberg 1,
D-14476 Potsdam, Germany

###### Abstract

We prove that any real, vacuum gravitational perturbation of a 4-dimensional
vacuum pp-wave space-time can be locally expressed, modulo gauge
transformations, as the real part of a Hertz/Debye potential, where the scalar
potential satisfies the wave equation. We discuss relations with complex
perturbations, complex space-times, non-linear structures, and real spaces
with split (ultra-hyperbolic/Kleinian) signature. Motivated by generalized
notions of parallel spinors, we also discuss generalizations of the result to
other space-times.

##  1 Introduction

Pp-wave space-times are exact solutions to the Einstein equations modelling
gravitational radiation. These space-times are interesting both physically and
mathematically for many reasons: they are relevant for gravitational wave
physics; they satisfy, in appropriate cases, a linear superposition principle;
they represent a universal limit for general relativity in that, as shown by
Penrose [1], every Lorentzian space-time looks like a pp- (plane222Plane waves
are a special case of pp-waves: the former have an isometry group that is at
least 5-dimensional, while the latter possess in general only one Killing
vector.) wave near a null geodesic; all their curvature invariants vanish
(which is relevant e.g. for string theory); etc. In addition, closer to our
motivation, they represent the simplest case of a 4-dimensional Lorentzian
geometry that admits a parallel spinor field [2]. In this work we study vacuum
gravitational perturbations of pp-waves in four dimensions, and the problem of
representations of solutions to linearized gravity in terms of the so-called
Hertz/Debye potentials.

The general question we want to address is: can any real vacuum gravitational
perturbation be represented, modulo gauge, in terms of a Hertz/Debye
potential? This is conjectured to be true, locally, for perturbations of all
algebraically special vacuum spaces, cf. the introduction in [3]; but, as far
as we know, the problem has only been completely solved for the case of
Minkowski space-time [4], [5], [6, Section 5.7]. Our main result is given in
sections 3.1 and 4.

Although in this work we study the special case of pp-waves, the techniques we
use also apply to perturbations of the above more general class of solutions.
This is because our procedure is based on exploiting the existence of special
geometric structures called $\alpha$\- and $\beta$-surfaces, or simply twistor
surfaces, that are present, in particular, for any algebraically special
Einstein space-time. Pp-waves have the advantage that, while having a very
simple curvature structure that facilitates computations, the conceptual
difficulties one has to deal with in the other more complicated cases are
already present in this class. We illustrate this point by studying the more
general case of a “half-Kähler” vacuum space-time (see sections 2.2 and 4).

Furthermore, the fact that our method is based on twistor surfaces allows us
to give, in the pp-wave case, a precise description of the close connection
that exists between the Hertz/Debye representation of real linear
gravitational perturbations and the fully non-linear geometry of a complex
analogue of a pp-wave: a complex 4-manifold admitting a parallel spinor field.
Notably, the situation can also be understood in terms of real geometry, but
for a metric with split (also called ultra-hyperbolic, Kleinian, or neutral)
signature.

Finally, parallel spinors constitute the major motivation in this work, since
as detailed in section 2.2 below, a pp-wave is the simplest case of a general
scheme in which special geometries (including e.g. black hole space-times) are
characterized in terms of “generalized parallel spinors”. Our approach
exploits a simple link between generalized parallel spinors and complex
geometry, and it has direct connections to the twistor programme and the
heavenly formalisms of Penrose, Newman and Plebański; see section 2.

Summary. In section 2 we give an elementary review of spinors in 4d; present
our motivation relating parallel spinors and complex geometry; and deduce the
structure of a 4-geometry that admits a parallel spinor in Lorentz signature
and also for complex metrics. Our main result is presented in section 3 where
we study gravitational perturbations. In section 4 we study a generalization
of this result, to the case of a “half-Kähler” vacuum space-time. Some final
remarks are given in section 5. We include appendix A with additional details
of calculations. We follow the notation and conventions of Penrose and Rindler
[6, 7]; in particular, we use abstract indices.

##  2 Parallel spinors, real and complex space-times

###  2.1 Preliminaries

Given a 4d complex vector space with a metric $g_{ab}$ and an orientation, the
orthogonal group is

$\displaystyle{\rm SO}(4,\mathbb{C})=({\rm SL}(2,\mathbb{C})\times{\rm
SL}(2,\mathbb{C}))/\mathbb{Z}_{2}.$ (2.1)

The relation between the two sides of (2.1) is understood by fixing an
isomorphism $\sigma$ between $\mathbb{C}^{4}$ and
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$, i.e. one writes a column vector $v$ in
$\mathbb{C}^{4}$ as a matrix $\sigma(v)$ in
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. Then (2.1) means that, for any
orthogonal transformation $\Lambda\in{\rm SO}(4,\mathbb{C})$, there are
elements $L$ and $R$ in ${\rm SL}(2,\mathbb{C})$ such that $\sigma(\Lambda
v)=L\sigma(v)R^{\rm t}$.

Elements in each copy of $\mathbb{C}^{2}$ are called spinors. Since each
$\mathbb{C}^{2}$ has an independent action of ${\rm SL}(2,\mathbb{C})$, there
are two different kinds of spinors. We say that the two kinds have opposite
‘chirality’. In abstract indices, these are distinguished by primed and
unprimed indices, e.g. $\psi^{A^{\prime}}$ and $\varphi^{A}$, and the
isomorphism $\sigma$ is $v^{a}\to\sigma(v)^{AA^{\prime}}\equiv
v^{AA^{\prime}}$. We usually omit $\sigma$, so that we identify $v^{a}\equiv
v^{AA^{\prime}}$. This way we have the usual identification of indices
$a=AA^{\prime}$, $b=BB^{\prime}$, etc., which we follow in this work. From the
relation $g(v,v)=2\det\sigma(v)$ one deduces that the metric is
$g_{ab}=\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}$, where $\epsilon_{AB}$
is the natural volume element of $\mathbb{C}^{2}$.

Without any reality conditions, spinors of opposite chirality are independent.
Real forms of (2.1) corresponding to different metric signatures are recovered
by using different reality structures. These structures can in turn be
understood as operations on spinors, that we call ‘spinor conjugations’, and
they may or may not lead to relations between chiralities.

For Lorentzian reality conditions, spinor conjugation interchanges chirality,
so the action of the two factors in the RHS of (2.1) is not independent, and
one recovers the Lorentz group $\rm{SO}(1,3)={\rm
SL}(2,\mathbb{C})/\mathbb{Z}_{2}$. We denote Lorentzian spinor conjugation
with an overbar, e.g. $\varphi^{A}\to\bar{\varphi}^{A^{\prime}}$. A spinor
$\varphi^{A}$ and its complex conjugate $\bar{\varphi}^{A^{\prime}}$ produce a
real null vector $N^{a}=\varphi^{A}\bar{\varphi}^{A^{\prime}}$. Given a basis
of $\mathbb{C}^{2}$, $\\{o^{A},\iota^{A}\\}$, one can consider the complex
conjugate basis $\\{\bar{o}^{A^{\prime}},\bar{\iota}^{A^{\prime}}\\}$ and
construct four linearly independent null vectors as

$\displaystyle\ell^{a}=o^{A}\bar{o}^{A^{\prime}},\qquad
n^{a}=\iota^{A}\bar{\iota}^{A^{\prime}},\qquad
m^{a}=o^{A}\bar{\iota}^{A^{\prime}},\qquad\bar{m}^{a}=\iota^{A}\bar{o}^{A^{\prime}}.$
(2.2)

If the basis $\\{o^{A},\iota^{A}\\}$ is normalized by
$\epsilon_{AB}o^{A}\iota^{B}=1$, then the vectors (2.2) satisfy the usual
conditions for a null tetrad: $g_{ab}\ell^{a}n^{b}=1=-g_{ab}m^{a}\bar{m}^{b}$,
and the rest vanishes.

For Euclidean (/Riemannian) reality conditions, spinor conjugation $\dagger$
preserves chirality, but a spinor $\varphi^{A}$ and its complex conjugate
$\varphi^{\dagger A}$ are linearly independent: if $\varphi^{A}$ has
components $(a,b)$ relative to some basis, then $\varphi^{\dagger A}$ has
components $(-\bar{b},\bar{a})$. Since $\dagger$ is anti-linear and it holds
$\dagger^{2}=-1$, this is really a quaternionic structure. The Euclidean form
of (2.1) is ${\rm SO}(4,\mathbb{R})=({\rm SU}(2)\times{\rm
SU}(2))/\mathbb{Z}_{2}$, and chiralities are independent. Given a spinor
$o^{A}$, one has a spin basis $\\{o^{A},o^{\dagger A}\\}$, but unlike Lorentz
signature, this does not give a basis for the opposite chirality.

Finally, the restriction to real elements in ${\rm SL}(2,\mathbb{C})$
corresponds to a metric with split signature. The isomorphism (2.1) becomes
${\rm SO}(2,2)=({\rm SL}(2,\mathbb{R})\times{\rm
SL}(2,\mathbb{R}))/\mathbb{Z}_{2}$, spinors are real and chiralities are
independent.

Over an open neighbourhood on a smooth manifold $M$ equipped with a metric
$g_{ab}$, one constructs the primed and unprimed spinor bundles
$\mathbb{S}^{\prime}$, $\mathbb{S}$, and the considerations above apply
pointwise on each fiber. A spinor field is a (local) section of $\mathbb{S}$
or $\mathbb{S}^{\prime}$ (or tensor products of them). If $x^{a}$ are local
coordinates on $M$, we use the identification of indices $a=AA^{\prime}$, etc.
to write e.g. ${\rm d}x^{a}\equiv{\rm d}x^{AA^{\prime}}$, so the metric is
$g=\epsilon_{AB}\epsilon_{A^{\prime}B^{\prime}}{\rm
d}x^{AA^{\prime}}\otimes{\rm d}x^{BB^{\prime}}$. Similarly, the Levi-Civita
connection is $\nabla_{a}=\nabla_{AA^{\prime}}$. If $(M,g_{ab})$ is real, the
operator $\nabla_{AA^{\prime}}$ is also real.

###  2.2 Motivation: parallel spinors

A parallel (or covariantly constant) spinor is a spinor field $o^{A}$ that
satisfies

$\nabla_{AA^{\prime}}o^{B}=0.$ (2.3)

The existence of a non-trivial solution to (2.3) imposes strong restrictions
on the geometry. Specific restrictions depend on the metric signature, see
[2].

In Lorentz signature, complex conjugation of (2.3) gives a parallel spinor
with opposite chirality, $\nabla_{AA^{\prime}}\bar{o}^{B^{\prime}}=0$. The
real null vector $\ell^{b}=o^{B}\bar{o}^{B^{\prime}}$ is therefore covariantly
constant, so the geometry is a pp-wave. In this work we are interested in this
case, see section 2.3. In Riemann signature, the complex conjugate of (2.3) is
$\nabla_{AA^{\prime}}o^{\dagger B}=0$. One then has a parallel spin frame, so
the manifold must be hyper-Kähler. We will not focus on this case. In split
signature, a (real) solution to (2.3) is equivalent to a null Kähler
structure, see [8, 9].

Our interest in parallel spinors actually arises from “generalized” versions
of them, where one considers connections more general than the Levi-Civita
connection. Such generalizations are important both in physics and in
mathematics. For example, these objects appear in supergravity in relation to
the existence of supersymmetries; and they are also relevant in certain areas
of pure geometry, for instance concerning different definitions of ‘mass’. See
e.g. [10] 333We are interested in parallel Weyl spinors, while in supergravity
and related areas one considers Dirac spinors..

But our major motivation is the connection that generalized parallel spinors
turn out to have with complex geometry. For example, a Kähler manifold can be
characterized by the existence of a parallel (pure) projective spinor, cf.
[11]. In four dimensions (where all spinors are pure), this can be expressed
in terms of the Riemannian version of a connection that is well-known in
general relativity, the so-called ‘GHP’ connection $\Theta_{AA^{\prime}}$.
Interestingly enough, a Hermitian manifold can be similarly defined via
parallel spinors, using a generalization of $\Theta_{AA^{\prime}}$, that we
may call ‘complex-conformal connection’ or ‘conformally invariant GHP
connection’, and we denote by $\mathcal{C}_{AA^{\prime}}$, cf. [12, 13]. We
summarize the situation in table 1.

Condition | Riemann signature | Lorentz signature | Split signature
---|---|---|---
$\nabla_{AA^{\prime}}o^{B}=0$ | hyper-Kähler | pp-wave | null-Kähler
$\Theta_{AA^{\prime}}o^{B}=0$ | Kähler | “half-Kähler” | no name
$\mathcal{C}_{AA^{\prime}}o^{B}=0$ | Hermitian | “half-Hermitian” | no name
Table 1: Different notions of parallel spinors give different special
4-geometries. $\Theta_{AA^{\prime}}$ is the ‘GHP connection’, and
$\mathcal{C}_{AA^{\prime}}$ is a conformally invariant version of it. Apart
from $\nabla_{AA^{\prime}}o^{B}=0$, the other equations are non-linear, since
the connections depend on $o^{A}$. The terminology in the Lorentzian case is
perhaps not standard, although similar names have been used by Flaherty [14].
In the split case, the conditions can be related to para-complex geometry.

The operators $\Theta_{AA^{\prime}}$ and $\mathcal{C}_{AA^{\prime}}$ are well-
defined in any signature444One needs a pair of spinors in the construction of
$\Theta_{AA^{\prime}}$, $\mathcal{C}_{AA^{\prime}}$. In the Riemannian case a
single spinor is enough since its complex conjugate gives the other. In the
other cases the extra spinor can be chosen at will.. Our interest in the
(generalized) parallel spinor equations presented in table 1 is that they
imply the existence of twistor surfaces, which are the basic object that give
integration procedures. The kind of algebraic and differential manipulations
that one has to follow in these procedures is essentially the same in all
cases, which is why we find the parallel spinors viewpoint attractive: it is
both conceptually (geometrically) meaningful and computationally practical. In
this paper we are interested in the simplest case, eq. (2.3), and its
applications to the linearized gravity problem in general relativity. We will
also discuss the “half-Kähler” case, see section 4. For the treatment of
$\mathcal{C}_{AA^{\prime}}o^{B}=0$ in conformal geometry, see [13]
(perturbations are not treated in this reference).

###  2.3 Lorentz signature: pp-wave space-times

We define a pp-wave space-time as a 4-dimensional Lorentzian manifold
$(M,g_{ab})$ that admits a non-trivial parallel real null vector $N^{a}$,
$N_{a}N^{a}=0$, $\nabla_{a}N^{b}=0$, and such that the Ricci tensor is
$R_{ab}\propto N_{a}N_{b}$. As shown in [15], any such geometry admits a
parallel spinor, that in the rest of this work we denote by $o^{A}$. The
associated parallel null vector is denoted by
$\ell^{a}=o^{A}\bar{o}^{A^{\prime}}$.

The following result is just the standard characterization of pp-waves in
terms of Brinkmann coordinates, and it is well-known:

######  Proposition 2.1.

Let $(M,g_{ab})$ be a Lorentzian space-time admitting a non-trivial parallel
spinor field $o^{A}$, eq. (2.3). Then there exist (locally) a coordinate
system $(u,v,\zeta,\bar{\zeta})$ and a real scalar field
$H=H(v,\zeta,\bar{\zeta})$ such that the metric is

$g=2({\rm d}u{\rm d}v-{\rm d}\zeta{\rm d}\bar{\zeta})+H{\rm d}v^{2}.$ (2.4)

The Ricci scalar vanishes, and the rest of the curvature is given by

$\displaystyle\Phi_{ABA^{\prime}B^{\prime}}={}$
$\displaystyle\tfrac{1}{2}H_{\zeta\bar{\zeta}}o_{A}o_{B}\bar{o}_{A^{\prime}}\bar{o}_{B^{\prime}},$
(2.5) $\displaystyle\Psi_{ABCD}={}$
$\displaystyle\tfrac{1}{2}H_{\bar{\zeta}\bar{\zeta}}o_{A}o_{B}o_{C}o_{D}.$
(2.6)

It is instructive to look at the derivation of this result from the
perspective of twistor surfaces; we will do this in the rest of this
subsection.

Consider the 2-dimensional complex distribution in $TM\otimes\mathbb{C}$ given
by
$D=\\{o^{A}\beta^{A^{\prime}}\;|\;\beta^{A^{\prime}}\in\mathbb{S}^{\prime}\\}$.
The condition for this to be involutive (i.e. $[D,D]\subset D$) is the shear-
free equation $o^{A}o^{B}\nabla_{AA^{\prime}}o_{B}=0$ (cf. [7, Section 7.3]),
which is certainly satisfied if (2.3) holds. This implies that there exist
complex 2-surfaces in the complexified space-time $\mathbb{C}M$, called
$\beta$-surfaces, such that their tangent bundle is $D$. Analogously, the
distribution
$\bar{D}=\\{\bar{o}^{A^{\prime}}\alpha^{A}\;|\;\alpha^{A}\in\mathbb{S}\\}$ is
involutive, and is the tangent bundle to a different kind of complex
2-surfaces in $\mathbb{C}M$, called $\alpha$-surfaces. Let us focus on the
former. The $\beta$-surfaces are labelled by two complex coordinates
$(v,\zeta)$ that are constant on them, namely $o^{A}\nabla_{AA^{\prime}}v=0$,
$o^{A}\nabla_{AA^{\prime}}\zeta=0$ (see [7, Lemma (7.3.15)]). From these two
equations we deduce that there are two spinor fields, say
$v_{A^{\prime}},\bar{\iota}_{A^{\prime}}$, such that
$\nabla_{AA^{\prime}}v=o_{A}v_{A^{\prime}}$,
$\nabla_{AA^{\prime}}\zeta=o_{A}\bar{\iota}_{A^{\prime}}$. Since
$\ell_{a}=o_{A}\bar{o}_{A^{\prime}}$ is covariantly constant, it is in
particular closed, so we can take $v_{A^{\prime}}=\bar{o}_{A^{\prime}}$. So
$(v,\zeta)$ are defined by

${\rm d}v=o_{A}\bar{o}_{A^{\prime}}{\rm d}x^{AA^{\prime}},\qquad{\rm
d}\zeta=o_{A}\bar{\iota}_{A^{\prime}}{\rm d}x^{AA^{\prime}}.$ (2.7)

We see that $v$ is real, whereas $\zeta$ is complex. They are functionally
independent, which means that $\bar{o}_{A^{\prime}}\bar{\iota}^{A^{\prime}}=N$
for some scalar field $N\neq 0$.

From the condition ${\rm d}^{2}\zeta=0$ we deduce that
$o^{A}\nabla_{AA^{\prime}}\bar{\iota}_{B^{\prime}}=0$. Therefore,
$o^{A}\nabla_{AA^{\prime}}N=0$, which implies that $N$ is a holomorphic
function of $v,\zeta$ (i.e. ${\rm d}N$ is a linear combination of ${\rm d}v$
and ${\rm d}\zeta$), so it can be set to 1 by a coordinate transformation
$\zeta\to\zeta^{\prime}(v,\zeta)$. We drop the prime and denote again by
$\zeta$ the new coordinate, with
$\bar{o}_{A^{\prime}}\bar{\iota}^{A^{\prime}}=1$.

Notice that $v,\bar{\zeta}$ satisfy
$\bar{o}^{A^{\prime}}\nabla_{AA^{\prime}}v=0$,
$\bar{o}^{A^{\prime}}\nabla_{AA^{\prime}}\bar{\zeta}=0$, so these scalars are
constant on $\alpha$-surfaces. This is a generic feature of Lorentz signature:
spinor complex conjugation interchanges $\alpha$\- and $\beta$-surfaces.

Using some of the previous identities, one can show that the vector fields
$o^{A}\bar{o}^{A^{\prime}}$ and $o^{A}\bar{\iota}^{A^{\prime}}$ commute, so
the two scalar fields $u,w$ defined by
$\partial_{u}=o^{A}\bar{o}^{A^{\prime}}\partial_{AA^{\prime}}$,
$\partial_{w}=o^{A}\bar{\iota}^{A^{\prime}}\partial_{AA^{\prime}}$ are
functionally independent. We see that $u$ is real and $w$ is complex. These
are coordinates along the $\beta$-surfaces. The coordinate $w$ is however not
functionally independent of $(v,\bar{\zeta})$, since a short calculation gives
$\partial_{\bar{\zeta}}=-o^{A}\bar{\iota}^{A^{\prime}}\partial_{AA^{\prime}}$.
Summarizing, we have

$\partial_{u}=o^{A}\bar{o}^{A^{\prime}}\partial_{AA^{\prime}},\qquad\partial_{\bar{\zeta}}=-o^{A}\bar{\iota}^{A^{\prime}}\partial_{AA^{\prime}}.$
(2.8)

So $\alpha$\- and $\beta$-surfaces give a coordinate system
$(u,v,\zeta,\bar{\zeta})$ for $M$, that we illustrate in Fig. 1. For pp-waves
these are simply Brinkmann coordinates, so the interpretation is known: the
integral curves of $\ell^{a}=o^{A}\bar{o}^{A^{\prime}}$ are the rays of the
wave and $u$ is an affine parameter along them, the hypersurfaces $v={\rm
constant}$ are ‘wave surfaces’, and the real and imaginary parts of $\zeta$
are coordinates transverse to the direction of propagation of the wave.

real light ray $\gamma\subset M$$\beta$-surface
$\subset\mathbb{C}M$$(v,\zeta)=$constant$\alpha$-surface
$\subset\mathbb{C}M$$(v,\bar{\zeta})=$constant$\partial_{u}$$\partial_{\zeta}$$\partial_{\bar{\zeta}}$
Figure 1: An $\alpha$-surface, a $\beta$-surface, and a real (Lorentzian)
space-time intersect in a real light ray $\gamma$, that has tangent vector
$\ell^{a}=o^{A}\bar{o}^{A^{\prime}}=\partial^{a}_{u}$. The coordinate system
defined by these twistor surfaces coincides, in the pp-wave case, with
Brinkmann coordinates.

With the above information, the structure of the metric can be deduced from
the expression $g=\epsilon_{AB}\bar{\epsilon}_{A^{\prime}B^{\prime}}{\rm
d}x^{AA^{\prime}}\otimes{\rm d}x^{BB^{\prime}}$, by replacing
$\epsilon_{AB}=o_{A}\iota_{B}-\iota_{A}o_{B}$, its complex conjugate, and the
definition of the coordinates (2.7). When doing this computation, one finds
that the only piece of information missing at this point is an expression for
the 1-form $\iota_{A}\bar{\iota}_{A^{\prime}}{\rm d}x^{AA^{\prime}}$. This can
be obtained as follows. For any function $f$, we have ${\rm
d}f=(\nabla_{AA^{\prime}}f){\rm d}x^{AA^{\prime}}$. Using the identities
$\nabla_{AA^{\prime}}=\delta_{A}^{B}\bar{\delta}_{A^{\prime}}^{B^{\prime}}\nabla_{BB^{\prime}}$
and $\delta^{B}_{A}=o_{A}\iota^{B}-\iota_{A}o^{B}$, together with definitions
(2.7), (2.8), and putting $f=u$, we get

$\displaystyle\iota_{A}\bar{\iota}_{A^{\prime}}{\rm d}x^{AA^{\prime}}={\rm
d}u-(\iota^{A}\bar{\iota}^{A^{\prime}}\nabla_{AA^{\prime}}u){\rm d}v.$

The expression (2.4) for the metric then follows straightforwardly, by
defining the real scalar field

$H:=-2\iota^{A}\bar{\iota}^{A^{\prime}}\nabla_{AA^{\prime}}u.$ (2.9)

This function $H$ represents the wave profile, and the case $H=0$ reduces to
Minkowski space-time.

We also notice that in the coordinate system $(u,v,\zeta,\bar{\zeta})$, the
wave operator acting on an arbitrary scalar field $\varphi$ is

$\Box\varphi=2(\varphi_{uv}-\varphi_{\zeta\bar{\zeta}})-H\varphi_{uu}.$ (2.10)

There is a natural spin frame: the parallel spinor $o_{A}$, and the spinor
$\iota_{A}$ used in (2.7). The associated connection 1-form has only one non-
trivial component:

$\nabla_{AA^{\prime}}\iota_{B}=-\kappa^{\prime}o_{A}o_{B}\bar{o}_{A^{\prime}},$
(2.11)

where
$\kappa^{\prime}:=-\iota^{A}\iota^{B}\bar{\iota}^{B^{\prime}}\nabla_{BB^{\prime}}\iota_{A}$.
In order to show this, notice first that, from (2.7) and ${\rm
d}^{2}\bar{\zeta}=0$, it follows that
$\bar{o}^{A^{\prime}}\nabla_{AA^{\prime}}\iota_{B}=0$. In addition, from (2.7)
one deduces that $o_{A}=-\bar{o}^{A^{\prime}}\nabla_{AA^{\prime}}\zeta$ and
$\iota_{A}=\bar{\iota}^{A^{\prime}}\nabla_{AA^{\prime}}\bar{\zeta}$. These
identities can then be used to show that
$o^{A}\nabla_{AA^{\prime}}\iota_{B}=0$, so (2.11) follows. In terms of $H$,
the expression for $\kappa^{\prime}$ is

$\kappa^{\prime}=\tfrac{1}{2}H_{\zeta}.$ (2.12)

This can be shown by using (2.9), which gives
$o^{A}\nabla_{AA^{\prime}}H=2\iota^{B}\bar{\iota}^{B^{\prime}}\nabla_{BB^{\prime}}\bar{\iota}_{A^{\prime}}$.

For the curvature, eq. (2.3) implies $[\nabla_{a},\nabla_{b}]o^{D}=0$, so it
follows that $\Lambda=0$,
$\Phi_{ABA^{\prime}B^{\prime}}=\Phi_{22}o_{A}o_{B}\bar{o}_{A^{\prime}}\bar{o}_{B^{\prime}}$,
and $\Psi_{ABCD}=\Psi_{4}o_{A}o_{B}o_{C}o_{D}$, where $\Phi_{22}$ and
$\Psi_{4}$ are defined by contractions with
$\iota^{A},\bar{\iota}^{A^{\prime}}$ on the left-hand sides. Expressions for
$\Phi_{22}$ and $\Psi_{4}$ in terms of $H$ can be deduced, for example, by
using the Newman-Penrose equations and the fact that all spin coefficients
except $\kappa^{\prime}$ vanish: equations $(4.11.12)(a^{\prime})$ and
$(4.11.12)(b^{\prime})$ in [6] give $\Phi_{22}=-\delta\kappa^{\prime}$ and
$\Psi_{4}=-\delta^{\prime}\kappa^{\prime}$ (in Newman-Penrose notation). Using
then $\delta=-\partial_{\bar{\zeta}}$, $\delta^{\prime}=-\partial_{\zeta}$
(which follow from (2.8)), and (2.12), one obtains the expressions (2.5),
(2.6).

###  2.4 Complex space-times

We now consider a complex space-time with a parallel spinor. Here, we are
referring to a genuinely complex 4-manifold, not to a complexified pp-wave
space-time. See [7, Section 6.9] for the distinction between ‘complex’ and
‘complexified’ space-time. In the complexified case we still have two parallel
spinors of opposite chirality ($o^{A}$ and $\bar{o}^{A^{\prime}}$ become
independent but they are both parallel), whereas in the genuinely complex case
we only retain one parallel spinor.

The situation is very similar to the case of split signature. This is because
in that case, spinors are real and the two chiralities are independent. The
following result is for complex space-times, but if one replaces “complex” by
“real” everywhere then exactly the same holds true for split signature
metrics, as was already shown in [8, 9]:

######  Proposition 2.2 (See [8, 9]).

Let $(\mathbb{C}M,g^{\mathbb{C}}_{ab})$ be a complex space-time with a non-
trivial parallel spinor field $o^{A}$. Then there exist, locally, a complex
coordinate system $(u,v,\zeta,w)$ and a complex scalar field $\Theta$ such
that the metric is

$g^{\mathbb{C}}=2({\rm d}u{\rm d}v+{\rm d}\zeta{\rm d}w)-2\Theta_{ww}{\rm
d}v^{2}+4\Theta_{wu}{\rm d}v{\rm d}\zeta-2\Theta_{uu}{\rm d}\zeta^{2}.$ (2.13)

The Ricci scalar vanishes, and the rest of the curvature is given by

$\displaystyle\Phi_{ABA^{\prime}B^{\prime}}={}$ $\displaystyle
o_{A}o_{B}\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}f,$ (2.14)
$\displaystyle\Psi_{ABCD}={}$
$\displaystyle-\tfrac{1}{2}o_{A}o_{B}o_{C}o_{D}\Box f,$ (2.15)
$\displaystyle\tilde{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}={}$
$\displaystyle-\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}\tilde{\nabla}_{C^{\prime}}\tilde{\nabla}_{D^{\prime}}\Theta,$
(2.16)

where $\tilde{\nabla}_{A^{\prime}}:=o^{A}\nabla_{AA^{\prime}}$, $\Box$ is the
wave operator associated to (2.13), and

$f=\Theta_{uv}+\Theta_{\zeta w}+\Theta_{uu}\Theta_{ww}-\Theta_{uw}^{2}.$
(2.17)

In coordinates, the wave operator $\Box$ acting on an arbitrary scalar field
$\varphi$ is

$\Box\varphi=2(\varphi_{uv}+\varphi_{w\zeta}+\Theta_{uu}\varphi_{ww}+\Theta_{ww}\varphi_{uu}-2\Theta_{uw}\varphi_{uw}).$
(2.18)

From Prop. 2.2 we see that the vacuum Einstein equations are now more
complicated than in the pp-wave case: the Ricci-flat condition is equivalent
to $\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}f=0$, which in the
coordinate system of the proposition reads $f_{uu}=f_{uw}=f_{ww}=0$. The
solution to this is $f=p(v,\zeta)u+q(v,\zeta)w+r(v,\zeta)$, where $p,q,r$ are
arbitrary functions of $(v,\zeta)$, so in terms of the “potential” $\Theta$,
the Einstein equations are

$\Theta_{uv}+\Theta_{\zeta
w}+\Theta_{uu}\Theta_{ww}-\Theta_{uw}^{2}=p(v,\zeta)u+q(v,\zeta)w+r(v,\zeta),$
(2.19)

see [8]. This is a very special case of the hyper-heavenly equation of
Plebański and Robinson [16]. The non-trivial right hand side in (2.19) (i.e.
$f\neq 0$) complicates the analysis of the integrability properties of this
equation. The special case $f\equiv 0$ is Plebański’s second heavenly
equation, and notice from (2.15) that this case gives a self-dual (half-flat)
space, which is an integrable system by virtue of the non-linear graviton
twistor construction of Penrose.

It is useful to briefly discuss the structures involved in the derivation of
the result of Prop. 2.2. As in section 2.3, the condition
$\nabla_{AA^{\prime}}o^{B}=0$ implies that the distribution
$D=\\{o^{A}\beta^{A^{\prime}}\;|\;\beta^{A^{\prime}}\in\mathbb{S}^{\prime}\\}$
is involutive, and this gives origin to $\beta$-surfaces in $\mathbb{C}M$,
which are labelled by two complex coordinates $v,\zeta$ defined by
$o^{A}\nabla_{AA^{\prime}}v=0=o^{A}\nabla_{AA^{\prime}}\zeta$. Unlike the
Lorentzian case, there are no $\alpha$-surfaces now. In addition, both
coordinates $v,\zeta$ are now complex. There are two independent spinor fields
$v_{A^{\prime}},\zeta_{A^{\prime}}$, with
$v_{A^{\prime}}\zeta^{A^{\prime}}=N\neq 0$, such that

$\displaystyle{\rm d}v=o_{A}v_{A^{\prime}}{\rm d}x^{AA^{\prime}},\qquad{\rm
d}\zeta=o_{A}\zeta_{A^{\prime}}{\rm d}x^{AA^{\prime}}.$ (2.20)

From the conditions ${\rm d}^{2}v=0={\rm d}^{2}\zeta$, it follows that
$\tilde{\nabla}_{A^{\prime}}v_{B^{\prime}}=\tilde{\nabla}_{A^{\prime}}\zeta_{B^{\prime}}=0$,
which give $\tilde{\nabla}_{A^{\prime}}N=0$, so we can set $N\equiv 1$ by a
coordinate transformation.

Using the above information, a short calculation shows that the vector fields
$o^{A}v^{A^{\prime}}$ and $o^{A}\zeta^{A^{\prime}}$ commute, so the two
complex scalar fields $u$, $w$ defined by

$o^{A}v^{A^{\prime}}\partial_{AA^{\prime}}=\partial_{u},\qquad
o^{A}\zeta^{A^{\prime}}\partial_{AA^{\prime}}=\partial_{w}$ (2.21)

are functionally independent, and correspond to complex coordinates along the
$\beta$-surfaces. Thus, we see again that twistor surfaces produce a natural
coordinate system $(v,\zeta,u,w)$ for $\mathbb{C}M$: these are the coordinates
used in Prop. 2.2, and they generalize the Brinkmann coordinates of the pp-
wave case of Prop. 2.1.

The structure (2.13) of the metric can be deduced in a similar way to what we
did in section 2.3: there is a flat (complex) metric $\eta_{ab}$ and a
symmetric spinor field $H_{A^{\prime}B^{\prime}}$ such that

$g_{ab}=\eta_{ab}+o_{A}o_{B}H_{A^{\prime}B^{\prime}}.$ (2.22)

The components of $H_{A^{\prime}B^{\prime}}$ generalize the pp-wave profile
function $H$ (2.9). In addition, a short calculation shows that
$H_{A^{\prime}B^{\prime}}$ satisfies
$\tilde{\nabla}^{A^{\prime}}H_{A^{\prime}B^{\prime}}=0$, so there exists a
scalar field $\Theta$ (see Remark 3.2 below) such that

$H_{A^{\prime}B^{\prime}}=-2\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}\Theta.$
(2.23)

The equations (2.22)-(2.23) give a coordinate-free expression for (2.13). The
Einstein equation (2.19) is, in coordinate-free terms:

$\Box\Theta-2(\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}\Theta)(\tilde{\nabla}^{A^{\prime}}\tilde{\nabla}^{B^{\prime}}\Theta)=f,\qquad\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}f=0.$
(2.24)

##  3 Perturbations

We will now study real gravitational perturbations of a real, Lorentzian,
vacuum pp-wave space-time, and connections with complex space-times. We recall
that the structure of the background pp-wave space-time is described in Prop.
2.1: one has Brinkmann coordinates $(u,v,\zeta,\bar{\zeta})$ defined by
$\alpha$\- and $\beta$-surfaces, the spinor field $o^{A}$ is parallel, the
spinor $\iota^{A}$ is defined in eq. (2.7), and all the information of the
geometry is encoded in the real scalar field $H$. In addition, the vacuum
condition for the background implies that $H_{\zeta\bar{\zeta}}=0$.

###  3.1 Main result

######  Theorem 3.1.

Let $(M,g_{ab})$ be a vacuum pp-wave space-time, eq. (2.4) with
$H_{\zeta\bar{\zeta}}=0$. For any real metric perturbation $h_{ab}$ satisfying
the linearized Einstein vacuum equations, there exist, locally, a real vector
field $V_{a}$ and a complex scalar field $\Phi$, such that $h_{ab}$ can be
written as

$h_{ab}=2{\rm Re}(h^{\rm H}_{ab})+\nabla_{a}V_{b}+\nabla_{b}V_{a}$ (3.1)

where $h^{\rm H}_{ab}$ is given by

$h^{\rm
H}_{ab}=o_{A}o_{B}\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}\Phi=\Phi_{\bar{\zeta}\bar{\zeta}}\ell_{a}\ell_{b}+2\Phi_{\bar{\zeta}u}\ell_{(a}m_{b)}+\Phi_{uu}m_{a}m_{b}$
(3.2)

with $\tilde{\nabla}_{A^{\prime}}=o^{A}\nabla_{AA^{\prime}}$, and $\Phi$
satisfies the wave equation

$\Box\Phi=0,$ (3.3)

where $\Box$ is the wave operator associated to $g_{ab}$, eq. (2.10).

We prove this result in section 3.2 below. Note that in tensor terms, the
tensor field (3.2) can also be written as

$h^{\rm
H}_{ab}=\nabla_{c}\nabla_{d}[\mathcal{H}_{(a}{}^{cd}{}_{b)}\Phi],\qquad\mathcal{H}_{abcd}=4\ell_{[a}m_{b]}\ell_{[c}m_{d]}.$
(3.4)

This is the usual expression for a Hertz/Debye potential in perturbation
theory, particularized to the special background of a pp-wave. (The
superscript “${\rm H}$” is from “Hertz”.)

Combining the result of theorem 3.1 with the discussion of section 2.4, we see
some sort of correspondence between a real linear problem and a complex non-
linear one: modulo gauge, the linearized gravity problem for real pp-wave
space-times would seem to be a “linear version” (see below) of the structure
of a complex space with a parallel spinor. Furthermore, as observed in section
2.4, the non-linear structures can actually be understood in a real context,
by going to a real space with a split signature metric.

Note that this correspondence can be established only after one proves theorem
3.1: we want to show that there exists a scalar potential for the
gravitational perturbation, while in a linear version of (2.22)–(2.24) one is
already assuming that a potential exists. Actually, a closer look at the
linear version of (2.22)–(2.24) reveals that the situation is subtle:

  * •

By “linear version” we mean that, in the complex metric (2.22)-(2.23) and in
the complex Einstein equations (2.19), one formally replaces $\Theta$ by
$\Theta+\varepsilon\Phi$ (where $\varepsilon$ is a parameter), and one keeps
only linear terms in $\varepsilon$. Then the perturbation to the background
complex metric is exactly (3.2), and the scalar field $\Phi$ would seem to
satisfy the wave equation $\Box\Phi=0$. In addition, using the general
expression [6, Eq. (5.7.15)] for the perturbed Weyl spinor, it is not
difficult to show that for the complex perturbation (3.2) one has

$\dot{\Psi}_{ABCD}[h^{\rm
H}]=\tfrac{1}{8}o_{A}o_{B}o_{C}o_{D}\Box\Box\Phi,\qquad\dot{\tilde{\Psi}}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}[h^{\rm
H}]=\tfrac{1}{2}\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}\tilde{\nabla}_{C^{\prime}}\tilde{\nabla}_{D^{\prime}}\Phi,$
(3.5)

which resemble (2.15), (2.16).

  * •

However, the linear version of the complex Einstein equations is the fourth
order equation
$\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}\Box\Phi=0$, see
(2.14). Analogously to the discussion around eq. (2.19), this implies that
$\Box\Phi=F$, where $F$ is a function such that $F_{uu}=F_{uw}=F_{ww}=0$, so
one does not really get the homogeneous wave equation for $\Phi$. We will
encounter a similar issue in our proof of theorem 3.1 below, where we will
show that one can get rid of inhomogeneous terms by considering gauge
transformations.

  * •

Even if one manages to get the homogeneous wave equation, the background real
and complex geometries, eqs. (2.4) and (2.13) respectively, are different,
which means that the wave equations, while formally equal, are different in
practice. Explicitly, the wave operators of the real and complex geometries
are given by equations (2.10) and (2.18).

###  3.2 Proof of theorem 3.1

####  3.2.1 Preliminaries

We consider a smooth mono-parametric family of real space-times
$(M,g_{ab}(\varepsilon))$, where $g_{ab}\equiv g_{ab}(0)$ is the background
space-time and is assumed to satisfy the vacuum Einstein equations $R_{ab}=0$.
The background Levi-Civita connection is denoted by $\nabla_{a}$, and the
linearization of the metric is $h_{ab}=\frac{\rm d}{{\rm
d}\varepsilon}|_{\varepsilon=0}[g_{ab}(\varepsilon)]$. The linearizations of
the Ricci tensor and of the curvature scalar are linear operators acting on
$h_{ab}$. They will be denoted by $\dot{R}_{ab}[h]$ and $\dot{R}[h]$
respectively, and explicit expressions for them are (see e.g. [17])

$\displaystyle\dot{R}_{ab}[h]={}$ $\displaystyle-\tfrac{1}{2}\Box
h_{ab}-\tfrac{1}{2}\nabla_{a}\nabla_{b}(g^{cd}h_{cd})+\tfrac{1}{2}\nabla^{c}\nabla_{a}h_{bc}+\tfrac{1}{2}\nabla^{c}\nabla_{b}h_{ac},$
(3.6) $\displaystyle\dot{R}[h]={}$
$\displaystyle\nabla^{a}\nabla^{b}h_{ab}-\Box(g^{ab}h_{ab}).$ (3.7)

Calculations are greatly simplified by using spinors. We emphasize that we do
not perturb spinors themselves, we just use the spinor structure of the
background space-time. Since all perturbations are tensor fields, they can be
written in spinor language, using the usual dictionary between tensor indices
and pairs of spinor indices (see section 2.1). For example, using the
background Levi-Civita connection $\nabla_{a}=\nabla_{AA^{\prime}}$, for the
perturbed Ricci tensor we can write
$\dot{R}_{ab}[h]\equiv\dot{R}_{ABA^{\prime}B^{\prime}}[h]$, with

$\dot{R}_{ABA^{\prime}B^{\prime}}[h]=-\frac{1}{2}\left[\Box
h_{ABA^{\prime}B^{\prime}}+\nabla_{AA^{\prime}}\nabla_{BB^{\prime}}(g^{cd}h_{cd})-\nabla^{CC^{\prime}}\nabla_{AA^{\prime}}h_{BB^{\prime}CC^{\prime}}-\nabla^{CC^{\prime}}\nabla_{BB^{\prime}}h_{AA^{\prime}CC^{\prime}}\right].$
(3.8)

Notice that this does not mean that we are linearizing a spinor field. In this
work, the meaning of “perturbed field” is the ordinary one in perturbation
theory in general relativity (see e.g. [17]).

Let $\\{o^{A},\iota^{A}\\}$ be a spin frame for the background space-time,
$\epsilon_{AB}o^{A}\iota^{B}=1$. For the calculations in this section, it is
useful to define the operators

$\tilde{\nabla}_{A^{\prime}}:=o^{A}\nabla_{AA^{\prime}},\quad\nabla_{A^{\prime}}:=\iota^{A}\nabla_{AA^{\prime}},\quad\bar{\nabla}_{A}:=\bar{o}^{A^{\prime}}\nabla_{AA^{\prime}},\quad\nabla_{A}:=\bar{\iota}^{A^{\prime}}\nabla_{AA^{\prime}}.$
(3.9)

For the particular case of a pp-wave background, from the discussion of
section 2.3 we have
$\tilde{\nabla}_{A^{\prime}}o^{B}=\nabla_{A^{\prime}}o^{B}=0$,
$\tilde{\nabla}_{A^{\prime}}\iota^{B}=0$ and
$\nabla_{A^{\prime}}\iota^{B}=-\kappa^{\prime}\bar{o}_{A^{\prime}}o^{B}$.

####  3.2.2 The radiation gauge

As is well-known, diffeomorphism invariance in general relativity implies that
in linearized gravity, any metric perturbation $h_{ab}$ is physically
equivalent to $h_{ab}+K[\xi]_{ab}$, where
$K[\xi]_{ab}=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}$ and $\xi_{a}$ is arbitrary.
For a vacuum background, it identically holds $\dot{R}_{ab}[K[\xi]]\equiv 0$
for any $\xi_{a}$. For a pp-wave, in appendix A.1 we give explicit expressions
for the components of $K[\xi]_{ab}$.

For a background space-time possessing a null vector $\ell^{a}$ associated to
a geodesic shear-free congruence (which is certainly the case for the pp-waves
studied in this work), one can impose (see [18]) the so-called radiation
gauge:

$\ell^{a}h_{ab}=0,\qquad g^{ab}h_{ab}=0.$ (3.10)

A short calculation then shows that in terms of a null tetrad
$\\{\ell_{a},n_{a},m_{a},\bar{m}_{a}\\}$, it holds

$h_{ab}=h_{nn}\ell_{a}\ell_{b}-2h_{n\bar{m}}\ell_{(a}m_{b)}-2h_{nm}\ell_{(a}\bar{m}_{b)}+h_{\bar{m}\bar{m}}m_{a}m_{b}+h_{mm}\bar{m}_{a}\bar{m}_{b}$
(3.11)

where $h_{nn}=n^{a}n^{b}h_{ab}$, etc. Replacing the expression (2.2) for the
null vectors, in spinor language we get

$h_{ab}=o_{A}o_{B}\mathring{X}_{A^{\prime}B^{\prime}}+\bar{o}_{A^{\prime}}\bar{o}_{B^{\prime}}\bar{\mathring{X}}_{AB},$

where

$\mathring{X}_{A^{\prime}B^{\prime}}=\tfrac{1}{2}h_{nn}\bar{o}_{A^{\prime}}\bar{o}_{B^{\prime}}-2h_{n\bar{m}}\bar{o}_{(A^{\prime}}\bar{\iota}_{B^{\prime})}+h_{\bar{m}\bar{m}}\bar{\iota}_{A^{\prime}}\bar{\iota}_{B^{\prime}}.$

Now, let $\psi$ be an arbitrary real scalar field. Then we have, trivially,

$\displaystyle h_{ab}={}$ $\displaystyle
o_{A}o_{B}\mathring{X}_{A^{\prime}B^{\prime}}+\bar{o}_{A^{\prime}}\bar{o}_{B^{\prime}}\bar{\mathring{X}}_{AB}+i\psi
o_{A}o_{B}\bar{o}_{A^{\prime}}\bar{o}_{B^{\prime}}-i\psi
o_{A}o_{B}\bar{o}_{A^{\prime}}\bar{o}_{B^{\prime}}$ $\displaystyle={}$
$\displaystyle
o_{A}o_{B}\underbrace{(\mathring{X}_{A^{\prime}B^{\prime}}+i\psi\bar{o}_{A^{\prime}}\bar{o}_{B^{\prime}})}_{=:X_{A^{\prime}B^{\prime}}}+\bar{o}_{A^{\prime}}\bar{o}_{B^{\prime}}\underbrace{(\bar{\mathring{X}}_{AB}-i\psi
o_{A}o_{B})}_{=\bar{X}_{AB}},$ (3.12)

so the tensor field (3.11) is

$h_{ab}=\gamma_{ab}+\bar{\gamma}_{ab},\qquad\gamma_{ab}=o_{A}o_{B}X_{A^{\prime}B^{\prime}}.$
(3.13)

The reason for including the arbitrary scalar field $\psi$ will become clear
in the next section. The relation between the components of
$X_{A^{\prime}B^{\prime}}$ and the components in (3.11) is

$X_{0^{\prime}0^{\prime}}=h_{\bar{m}\bar{m}},\qquad
X_{0^{\prime}1^{\prime}}=h_{n\bar{m}},\qquad
X_{1^{\prime}1^{\prime}}=\tfrac{1}{2}h_{nn}+i\psi.$ (3.14)

It is important to note that the conditions (3.10) do not exhaust the gauge
freedom. In Appendix A.1.1 we analyse the residual gauge transformations under
which (3.10) is preserved. This plays an important role below.

####  3.2.3 Potentials

We now assume that we are given a real metric perturbation $h_{ab}$ in
radiation gauge, eq. (3.13), that satisfies the linearized Einstein vacuum
equations:

$\dot{R}_{ABA^{\prime}B^{\prime}}[h]=\dot{R}_{ABA^{\prime}B^{\prime}}[\gamma]+\dot{R}_{ABA^{\prime}B^{\prime}}[\bar{\gamma}]=0.$
(3.15)

Notice that this equation does not imply that
$\dot{R}_{ABA^{\prime}B^{\prime}}[\gamma]$ vanishes. Using (3.8), (3.9), and
(3.13), after some calculations we find the following expressions for the
linearized Ricci tensor and Ricci scalar of the tensor field
$\gamma_{ab}=o_{A}o_{B}X_{A^{\prime}B^{\prime}}$:

$\displaystyle\dot{R}[\gamma]={}$
$\displaystyle\tilde{\nabla}^{A^{\prime}}\tilde{\nabla}^{B^{\prime}}X_{A^{\prime}B^{\prime}},$
(3.16a) $\displaystyle\dot{R}_{ABA^{\prime}B^{\prime}}[\gamma]={}$
$\displaystyle-\frac{1}{2}\left[o_{A}o_{B}\Box
X_{A^{\prime}B^{\prime}}+o_{B}\tilde{\nabla}^{A^{\prime}}\nabla_{AA^{\prime}}X_{B^{\prime}C^{\prime}}+o_{A}\tilde{\nabla}^{A^{\prime}}\nabla_{BB^{\prime}}X_{A^{\prime}C^{\prime}}\right].$
(3.16b)

For the complex conjugate
$\bar{\gamma}_{ab}=\bar{o}_{A^{\prime}}\bar{o}_{B^{\prime}}\bar{X}_{AB}$, the
corresponding formulas are obtained by simply taking the complex conjugate of
the above. Note that, regardless of (3.15), it follows immediately that

$o^{A}o^{B}\dot{R}_{ABA^{\prime}B^{\prime}}[\gamma]\equiv
0,\qquad\bar{o}^{A^{\prime}}\bar{o}^{B^{\prime}}\dot{R}_{ABA^{\prime}B^{\prime}}[\bar{\gamma}]\equiv
0.$ (3.17)

From (3.16a) and its complex conjugate, we find

$\displaystyle\dot{R}[h]={}$
$\displaystyle\tilde{\nabla}^{A^{\prime}}\tilde{\nabla}^{B^{\prime}}X_{A^{\prime}B^{\prime}}+\bar{\nabla}^{A}\bar{\nabla}^{B}\bar{X}_{AB}$
$\displaystyle={}$
$\displaystyle(\tilde{\nabla}^{A^{\prime}}\tilde{\nabla}^{B^{\prime}}\mathring{X}_{A^{\prime}B^{\prime}}+i\partial^{2}_{u}\psi)+(\bar{\nabla}^{A}\bar{\nabla}^{B}\overline{\mathring{X}}_{AB}-i\partial^{2}_{u}\psi)=0,$
(3.18)

where in the second line we used the definition of $X_{A^{\prime}B^{\prime}}$
given in eq. (3.12). We now see the reason for including the arbitrary scalar
field $\psi$: since it is free, we can choose it so as to satisfy

$\partial^{2}_{u}\psi=i\tilde{\nabla}^{A^{\prime}}\tilde{\nabla}^{B^{\prime}}\mathring{X}_{A^{\prime}B^{\prime}},$
(3.19)

or more explicitly:

$\partial^{2}_{u}\psi=i\left[\partial^{2}_{\bar{\zeta}}h_{\bar{m}\bar{m}}+2\partial_{u}\partial_{\bar{\zeta}}h_{n\bar{m}}+\tfrac{1}{2}\partial^{2}_{u}h_{nn}\right].$
(3.20)

(One still has the freedom $\psi\to\psi+\chi$ with $\partial^{2}_{u}\chi=0$,
but we will not need this.) The choice (3.20) of $\psi$ has the consequence
that

$\tilde{\nabla}^{A^{\prime}}\tilde{\nabla}^{B^{\prime}}X_{A^{\prime}B^{\prime}}=0,$
(3.21)

which implies that there exists, locally, a spinor field $Y_{A^{\prime}}$ such
that

$X_{A^{\prime}B^{\prime}}=\tilde{\nabla}_{(A^{\prime}}Y_{B^{\prime})}.$ (3.22)

######  Remark 3.2.

The argument for deducing (3.22) from (3.21) is essentially a variant of the
argument given by Penrose in [5, Section 4]. It is always true locally, and it
can be extended globally to a region that has vanishing first and second
homotopy groups. As explained by Penrose, this topological restriction
accounts for the impossibility of finding global potentials in certain cases,
such as for Coulomb fields.

In coordinate terms, the above means that, by virtue of the vanishing of the
perturbed Ricci scalar of $h_{ab}$, and by choosing $\psi$ in the form (3.20),
one can locally find two fields $Y_{0^{\prime}}$, $Y_{1^{\prime}}$ (which are
the components of a spinor field $Y_{A^{\prime}}$ in the spin frame
$\\{\bar{o}_{A^{\prime}},\bar{\iota}_{A^{\prime}}\\}$) such that

$\displaystyle\partial_{u}Y_{0^{\prime}}={}$ $\displaystyle
h_{\bar{m}\bar{m}},$ (3.23a)
$\displaystyle\partial_{u}Y_{1^{\prime}}-\partial_{\bar{\zeta}}Y_{0^{\prime}}={}$
$\displaystyle 2h_{n\bar{m}},$ (3.23b)
$\displaystyle\partial_{\bar{\zeta}}Y_{1^{\prime}}={}$
$\displaystyle-(\tfrac{1}{2}h_{nn}+i\psi).$ (3.23c)

From these equations we see that there is some freedom in $Y_{0^{\prime}}$,
$Y_{1^{\prime}}$: one can check that the equations are invariant under
$Y_{0^{\prime}}\to Y_{0^{\prime}}+\tau_{0^{\prime}}$, $Y_{1^{\prime}}\to
Y_{1^{\prime}}+\tau_{1^{\prime}}$, where
$\tau_{0^{\prime}}=p(v,\zeta)\bar{\zeta}+q_{0^{\prime}}(v,\zeta)$ and
$\tau_{1^{\prime}}=p(v,\zeta)u+q_{1^{\prime}}(v,\zeta)$, with
$p,q_{0^{\prime}},q_{1^{\prime}}$ arbitrary functions of $v,\zeta$.
Alternatively, this is seen from integrating equations (3.23), which gives

$\displaystyle Y_{0^{\prime}}={}$ $\displaystyle\int{\rm
d}uh_{\bar{m}\bar{m}}+p(v,\zeta)\bar{\zeta}+q_{0^{\prime}}(v,\zeta),$ (3.24a)
$\displaystyle Y_{1^{\prime}}={}$ $\displaystyle-\int{\rm
d}\bar{\zeta}(\tfrac{1}{2}h_{nn}+i\psi)+p(v,\zeta)u+q_{1^{\prime}}(v,\zeta).$
(3.24b)

From a coordinate-free perspective, the freedom in
$p,q_{0^{\prime}},q_{1^{\prime}}$ corresponds to the fact that eq. (3.22) is
invariant under $Y_{A^{\prime}}\to Y_{A^{\prime}}+\tau_{A^{\prime}}$, where
$\tau_{A^{\prime}}$ is any solution to
$\tilde{\nabla}_{(A^{\prime}}\tau_{B^{\prime})}=0$. We will not need to use
this freedom.

In view of (3.22), the original real metric perturbation is

$h_{ab}=\gamma_{ab}+\bar{\gamma}_{ab},\qquad\gamma_{ab}=o_{A}o_{B}\tilde{\nabla}_{(A^{\prime}}Y_{B^{\prime})}.$
(3.25)

The linearized Ricci operator for tensor fields of the form
$\gamma_{ab}=o_{A}o_{B}\tilde{\nabla}_{(A^{\prime}}Y_{B^{\prime})}$ is, of
course, a special case of (3.16b). After some calculations, we find

$-2\dot{R}_{ABA^{\prime}B^{\prime}}[\gamma]=o_{A}o_{B}\left[2\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}\nabla^{C^{\prime}}Y_{C^{\prime}}-\nabla_{(A^{\prime}}\tilde{\nabla}_{B^{\prime})}\tilde{\nabla}^{C^{\prime}}Y_{C^{\prime}}\right]-o_{(A}\iota_{B)}\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}\tilde{\nabla}^{C^{\prime}}Y_{C^{\prime}}.$
(3.26)

Summarizing, so far we have only imposed the vanishing of the perturbed Ricci
scalar, and we used this to deduce the structure (3.25) of the metric
perturbation. Using (3.26) and its complex conjugate, the rest of the Einstein
equations (3.15) is

$\displaystyle
o_{A}o_{B}\left[2\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}\nabla^{C^{\prime}}Y_{C^{\prime}}-\nabla_{(A^{\prime}}\tilde{\nabla}_{B^{\prime})}\tilde{\nabla}^{C^{\prime}}Y_{C^{\prime}}\right]-o_{(A}\iota_{B)}\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}\tilde{\nabla}^{C^{\prime}}Y_{C^{\prime}}+\rm{c.c}=0.$
(3.27)

This equation is automatically satisfied if
$\nabla^{C^{\prime}}Y_{C^{\prime}}$ and
$\tilde{\nabla}^{C^{\prime}}Y_{C^{\prime}}$ vanish. In this case the result of
theorem 3.1 would follow immediately: the equation
$\tilde{\nabla}^{C^{\prime}}Y_{C^{\prime}}=0$ would imply that
$Y_{C^{\prime}}=\tilde{\nabla}_{C^{\prime}}\Phi$ for some (locally defined)
complex scalar field $\Phi$, and $\nabla^{C^{\prime}}Y_{C^{\prime}}=0$ would
give the wave equation $\Box\Phi=0$. However, the converse of the above
statement is not necessarily true: the equation
$\dot{R}_{ABA^{\prime}B^{\prime}}[h]=0$ does not imply that
$\nabla^{C^{\prime}}Y_{C^{\prime}}$ and
$\tilde{\nabla}^{C^{\prime}}Y_{C^{\prime}}$ vanish.

The non-vanishing of $\nabla^{C^{\prime}}Y_{C^{\prime}}$ and
$\tilde{\nabla}^{C^{\prime}}Y_{C^{\prime}}$ makes the completion of the proof
of theorem 3.1 more difficult. What we will show is that these fields can be
set to zero by a gauge transformation. To this end, recall that we mentioned
in section 3.2.2 that we still have the freedom to perform residual gauge
transformations, i.e. transformations of the form

$h_{ab}\to h^{\prime}_{ab}=h_{ab}-K[\xi]_{ab}$ (3.28)

where $K[\xi]_{ab}=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}$ satisfies
$\ell^{a}K[\xi]_{ab}=0=g^{ab}K[\xi]_{ab}$. We analyse this residual freedom in
appendix A.1.1, where we show that there exists a spinor field
$g_{A^{\prime}}$ such that $K[\xi]_{ab}$ can be written as in eq. (A.11). The
gauge-transformed metric is then

$h^{\prime}_{ab}=\gamma^{\prime}_{ab}+\bar{\gamma}^{\prime}_{ab},$ (3.29)

where

$\gamma^{\prime}_{ab}=o_{A}o_{B}\tilde{\nabla}_{(A^{\prime}}Z_{B^{\prime})},\qquad
Z_{B^{\prime}}=Y_{B^{\prime}}-g_{B^{\prime}}.$ (3.30)

Since (3.28)-(3.29) is a gauge transformation, we have
$\dot{R}_{ABA^{\prime}B^{\prime}}[h]=\dot{R}_{ABA^{\prime}B^{\prime}}[h^{\prime}]$,
thus, the Einstein equations (3.27) are equivalently

$\displaystyle
o_{A}o_{B}\left[2\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}\nabla^{C^{\prime}}Z_{C^{\prime}}-\nabla_{(A^{\prime}}\tilde{\nabla}_{B^{\prime})}\tilde{\nabla}^{C^{\prime}}Z_{C^{\prime}}\right]-o_{(A}\iota_{B)}\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}\tilde{\nabla}^{C^{\prime}}Z_{C^{\prime}}+\rm{c.c}=0.$
(3.31)

######  Proposition 3.3.

The gauge transformation (3.28)-(3.29) can be chosen such that the spinor
field $Z_{A^{\prime}}$ satisfies the neutrino equation

$\nabla^{AA^{\prime}}Z_{A^{\prime}}=0.$ (3.32)

We defer the proof of this proposition to appendix A.2. Now, any solution to
(3.32) can be written (locally) as
$Z_{A^{\prime}}=\tilde{\nabla}_{A^{\prime}}\Phi$ for some complex scalar field
that satisfies the wave equation. To see this, first contract (3.32) with
$o_{A}$, which gives $\tilde{\nabla}^{A^{\prime}}Z_{A^{\prime}}=0$. This
implies that there is, locally, a complex scalar field $\Phi$ such that
$Z_{A^{\prime}}=\tilde{\nabla}_{A^{\prime}}\Phi$ (see Remark 3.2). Contracting
now (3.32) with $\iota_{A}$, we get
$\nabla^{A^{\prime}}\tilde{\nabla}_{A^{\prime}}\Phi=0$, which is the same as
$\Box\Phi=0$.

Summarizing, the original real metric perturbation is
$h_{ab}=h^{\prime}_{ab}+K[\xi]_{ab}$, where
$h^{\prime}_{ab}=\gamma^{\prime}_{ab}+\bar{\gamma}^{\prime}_{ab}$,
$\gamma^{\prime}_{ab}$ is given by

$\gamma^{\prime}_{ab}=o_{A}o_{B}\tilde{\nabla}_{A^{\prime}}\tilde{\nabla}_{B^{\prime}}\Phi,$
(3.33)

and $\Phi$ satisfies the wave equation $\Box\Phi=0$ on the background pp-wave
space-time. This concludes the proof of theorem 3.1.

######  Remark 3.4.

  1. 1.

The perturbation
$h^{\prime}_{ab}=\gamma^{\prime}_{ab}+\bar{\gamma}^{\prime}_{ab}$ is both in
radiation gauge and in Lorenz gauge: one can check that
$\ell^{a}h^{\prime}_{ab}=0=g^{ab}h^{\prime}_{ab}$ as well as
$\nabla^{a}h^{\prime}_{ab}=0$.

  2. 2.

The residual radiation gauge freedom is essential for the proof of theorem
3.1. Note that this must also be used if one wants to apply the same method to
even the simplest case of Minkowski space-time, which can be obtained by
simply setting $H=0$ in our formulas555 Note that the formulations of the
Minkowski problem in [4], [5], [6, Section 5.7] are very different to our
method and only apply when the background is flat..

  3. 3.

As a by-product of the above construction, we obtained a method to generate
solutions to the linearized Einstein vacuum equations from solutions to the
neutrino equation.

  4. 4.

If the potential $\Phi$ is independent of $u$, i.e. $\Phi_{u}=0$, then the
perturbation actually gives a solution to the full (non-linear) Einstein
equations. This can be seen from eqs. (3.2), (3.3), (2.5), (2.10): $\Phi$ is
just a perturbation to the background wave profile $H$.

##  4 Generalization to a “half-Kähler” vacuum space-time

In this section we briefly show how the ideas of the previous sections can be
carried over to a more general (real, Lorentzian) space-time: the “half-
Kähler” case of table 1. This is defined by the condition that there is a
parallel projective spinor, that is, a ‘spinor field up to scale’ that is
parallel. We are not aware of a description of this space-time (or its complex
generalization) analogous to the one given in propositions 2.1, 2.2. In the
Euclidean case, the manifold must be Kähler, so the Lorentzian version might
be of interest on its own right666Perhaps a closer Lorentzian analogue to a
Kähler manifold would be a space-time where we have both
$\Theta_{AA^{\prime}}o^{B}=0$ and $\Theta_{AA^{\prime}}\iota^{B}=0$. This may
be seen as a complexified version of Kähler geometry. Black holes (in
particular) are conformal to this space-time.. In addition, a real version of
the complex result of Prop. 2.2 for this case would correspond again to a
split signature metric. Here we restrict ourselves merely to the description
of gravitational perturbations.

A convenient way of expressing the existence of a spinor up to scale that is
parallel is to use GHP language (cf. [6, Section 4.12]). As is known, the use
of spinors/vectors up to scale in relativity brings about the notion of “GHP
weight”. Let $\\{o^{A},\iota^{A}\\}$ be two spinor fields in a Lorentzian
space-time $(M,g_{ab})$, with $o_{A}\iota^{A}=1$. A (scalar/tensor/spinor)
field $\eta$ is said to have GHP weight $\\{p,q\\}$ if, under the rescaling
$o^{A}\to\lambda o^{A}$, $\iota^{A}\to\lambda^{-1}\iota^{A}$ (with $\lambda$ a
complex scalar different from zero), it transforms as
$\eta\to\lambda^{p}\bar{\lambda}^{q}\eta$. A derivative operator that is
covariant under this transformation is the GHP connection
$\Theta_{a}=\nabla_{a}+p\omega_{a}+q\bar{\omega}_{a}$, where
$\omega_{a}:=\iota_{B}\nabla_{a}o^{B}$. The existence of a parallel projective
spinor can then be expressed as the condition $\Theta_{AA^{\prime}}o^{B}=0$.

If $o^{A}$ satisfies $\Theta_{AA^{\prime}}o^{B}=0$, then a few calculations
using $[\Theta_{a},\Theta_{b}]o^{C}=0$ show that the space-time must be of
Petrov type II. If in addition, we impose the vacuum condition
$\Phi_{ABA^{\prime}B^{\prime}}=0=\Lambda$, then $\Psi_{2}=0$,
$\nabla_{A(A^{\prime}}\omega^{A}_{B^{\prime})}=0$ and
$\nabla_{A^{\prime}(A}\omega^{A^{\prime}}_{B)}=-\Psi_{3}o_{A}o_{B}$. The GHP
connection is then self-dual and algebraically special. Notice that
$\Psi_{2}=0$ excludes the type D case, so in particular, black holes are not
included in this vacuum class.

The vector field $\ell^{b}=o^{B}\bar{o}^{B^{\prime}}$ satisfies
$\Theta_{a}\ell^{b}=0$, so it is tangent to a null congruence that is both
geodesic and shear-free. The radiation gauge for gravitational perturbations
can then be imposed [18]. The discussion from now on is analogous to what we
did in sections 3.2.2 and 3.2.3, the only extra point to keep in mind is that
all fields now carry GHP weights. If $h_{ab}$ is a perturbation in radiation
gauge, we can write it as in (3.12), that is
$h_{ab}=\gamma_{ab}+\bar{\gamma}_{ab}$, with
$\gamma_{ab}=o_{A}o_{B}X_{A^{\prime}B^{\prime}}$. The spinor field
$X_{A^{\prime}B^{\prime}}$ has weights $\\{-2,0\\}$. The linearized Einstein
equations are then
$\dot{R}_{ab}[h]=\dot{R}_{ab}[\gamma]+\dot{R}_{ab}[\bar{\gamma}]=0$, and a
calculation shows that

$-2\dot{R}_{ab}[\gamma]=o_{A}o_{B}[\Box^{\Theta}X_{A^{\prime}B^{\prime}}+2\tilde{\Theta}^{C^{\prime}}\Theta_{(A^{\prime}}X_{B^{\prime})C^{\prime}}]-2o_{(A}\iota_{B)}\tilde{\Theta}^{C^{\prime}}\tilde{\Theta}_{(A^{\prime}}X_{B^{\prime})C^{\prime}}-\tfrac{1}{2}g_{ab}\dot{R}[\gamma],$
(4.1)

where

$\dot{R}[\gamma]=\tilde{\Theta}^{A^{\prime}}\tilde{\Theta}^{B^{\prime}}X_{A^{\prime}B^{\prime}},$
(4.2)

and we defined

$\Box^{\Theta}:=g^{ab}\Theta_{a}\Theta_{b},\qquad\tilde{\Theta}_{A^{\prime}}:=o^{A}\Theta_{AA^{\prime}},\qquad\Theta_{A^{\prime}}:=\iota^{A}\Theta_{AA^{\prime}}.$
(4.3)

Choosing the free scalar field $\psi$ in (3.12) so that
$\tilde{\Theta}^{A^{\prime}}\tilde{\Theta}^{B^{\prime}}X_{A^{\prime}B^{\prime}}=0$,
we deduce that there is a spinor field $Y_{A^{\prime}}$, with weights
$\\{-3,0\\}$, such that
$X_{A^{\prime}B^{\prime}}=\tilde{\Theta}_{(A^{\prime}}Y_{B^{\prime})}$. Thus,
after some calculations, the linearized Einstein equations become (compare to
(3.27))

$\displaystyle
o_{A}o_{B}\left[2\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\Theta^{C^{\prime}}Y_{C^{\prime}}-\Theta_{(A^{\prime}}\tilde{\Theta}_{B^{\prime})}\tilde{\Theta}^{C^{\prime}}Y_{C^{\prime}}\right]-o_{(A}\iota_{B)}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\tilde{\Theta}^{C^{\prime}}Y_{C^{\prime}}+\rm{c.c}=0.$
(4.4)

For a residual radiation gauge transformation $h_{ab}\to
h^{\prime}_{ab}=h_{ab}-K[\xi]_{ab}$, (where
$K[\xi]_{ab}=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}$ satisfies (A.5)) there is a
spinor $g_{A^{\prime}}$ such that
$K[\xi]_{ab}=o_{A}o_{B}\tilde{\Theta}_{(A^{\prime}}g_{B^{\prime})}+{\rm c.c}$.
So the Einstein equations for $h^{\prime}_{ab}$ are the same as (4.4) with
$Y_{A^{\prime}}$ replaced by $Z_{A^{\prime}}=Y_{A^{\prime}}-g_{A^{\prime}}$.
The analysis of the residual gauge is analogous to the pp-wave case discussed
in appendix A.1.1, where instead of coordinate derivatives we use GHP
operators. For example, instead of (A.7), we find
$\operatorname{\text{\rm\th}}\xi_{\ell}=0$,
$\operatorname{\text{\rm\th}}^{2}\xi_{m}=\operatorname{\text{\rm\th}}^{2}\xi_{\bar{m}}=\operatorname{\text{\rm\th}}^{2}\xi_{n}=0$,
$\operatorname{\text{\rm\dh}}\operatorname{\text{\rm\dh}}^{\prime}\xi_{\ell}=0$.
Choosing then the gauge such that
$\tilde{\Theta}^{A^{\prime}}g_{A^{\prime}}=\tilde{\Theta}^{A^{\prime}}Y_{A^{\prime}}$,
$\Theta^{A^{\prime}}g_{A^{\prime}}=\Theta^{A^{\prime}}Y_{A^{\prime}}$, we
obtain $\tilde{\Theta}^{A^{\prime}}Z_{A^{\prime}}=0$ and
$\Theta^{A^{\prime}}Z_{A^{\prime}}=0$, or equivalently a weighted neutrino
equation

$\Theta^{AA^{\prime}}Z_{A^{\prime}}=0.$ (4.5)

From $\tilde{\Theta}^{A^{\prime}}Z_{A^{\prime}}=0$ we deduce that there is,
locally, a complex scalar field $\Phi$, with weights $\\{-4,0\\}$, such that
$Z_{A^{\prime}}=\tilde{\Theta}_{A^{\prime}}\Phi$, and from
$\Theta^{A^{\prime}}Z_{A^{\prime}}=0$ we deduce that $\Box^{\Theta}\Phi=0$.

In summary, we see that any real gravitational perturbation to a “half-Kähler”
vacuum space-time, once written in radiation gauge and assuming that it
satisfies the linearized Einstein vacuum equations, can be locally expressed
as

$h_{ab}=o_{A}o_{B}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\Phi+{\rm
c.c}+2\nabla_{(a}\xi_{b)},$ (4.6)

where $\Phi$ satisfies the GHP weighted wave equation

$\Box^{\Theta}\Phi=0.$ (4.7)

This generalizes the pp-wave result, theorem 3.1. Notice that, analogously to
the pp-wave case, the Hertz potential
$h^{\prime}_{ab}=o_{A}o_{B}\tilde{\Theta}_{A^{\prime}}\tilde{\Theta}_{B^{\prime}}\Phi+{\rm
c.c}$ is both in radiation gauge and in Lorenz gauge (i.e.
$\nabla^{a}h^{\prime}_{ab}=0$).

##  5 Conclusions

In this work we have shown that any real, linear gravitational perturbation of
a (real, Lorentzian) vacuum pp-wave space-time can be locally expressed,
modulo gauge transformations, as the real part of a Hertz/Debye potential,
where the scalar Debye potential satisfies the wave equation of the background
pp-wave solution. This is believed to hold for more general backgrounds as
well (replacing the wave equation by, e.g., the Teukolsky equation), but to
our knowledge, the result has been completely proven only for perturbations of
Minkowski [4], [5], [6, Section 5.7]. We stress that our result is local, cf.
Remark 3.2 and also [19].

We also showed the connections between the Hertz/Debye representation for
perturbations of pp-waves and the non-linear structure of a complex space-time
with a parallel spinor. This illustrates the formal relation between this
representation and a particular case of the hyper-heavenly construction of
Plebański and Robinson [16]. In addition, we argued that a linear problem in a
real space with a Lorentzian metric is related to a non-linear problem also in
a real space but with a split signature metric. This is interesting in view of
modern developments where physics in split signature is relevant, especially
in the context of scattering amplitudes and connections to gravitation, see
[20], [21].

Our approach relied on using special complex 2-surfaces in the complexified
space-time, called $\alpha$\- and $\beta$-surfaces, which are the basic object
of twistor theory, cf. [7]. These surfaces are present, in particular, for
(complexifications of) any algebraically special, vacuum, real, Lorentzian
space-time. Thus, the method employed in this work can also be applied to the
analysis of linearized gravity on more general backgrounds. We illustrated
this by generalizing our result to perturbations of a “half-Kähler” vacuum
space-time. Explicit computations in more general backgrounds are more
involved due to the complicated structure of the curvature. The interpretation
of the coordinates defined by twistor surfaces is also more difficult than in
the pp-wave case (where these coordinates are simply Brinkmann coordinates).

From a physical point of view, our motivation came from perturbation theory in
general relativity and its applications to gravitational wave physics,
concerning the Hertz/Debye potential representation of perturbations, and
gauge issues. While the currently most interesting space-times for
gravitational wave physics are more general than pp-waves, representing e.g.
single or binary black holes, the case of pp-waves already presents conceptual
difficulties similar to those that appear in the other more general cases.
This can be seen from our study of perturbations to “half-Kähler” space-times.
The application of these ideas to the general class of Petrov type II vacuum
solutions (including type D and the Kerr solution) is left for future work
[22].

From a geometric perspective, our motivation originated in the relations that
(generalized) parallel spinors have with complex geometry, as discussed in
section 2.2. We focused on perturbations to the simplest case of a parallel
spinor in Lorentz signature, a pp-wave metric. In Euclidean signature this
corresponds to hyper-Kähler manifolds, and linear perturbations in this
context have been studied e.g. in [23], [24]. Natural generalizations are the
other cases described in Table 1. We also studied the Lorentzian “half-Kähler”
case, which in a Riemannian setting would correspond to perturbations of
Kähler manifolds. A natural next step would be the study of perturbations to
the “half-Hermitian” case $\mathcal{C}_{AA^{\prime}}o^{B}=0$.

#### Acknowledgements

I am very grateful to Steffen Aksteiner, Lars Andersson and Bernard Whiting
for many discussions on the topics of this work, and for ongoing collaboration
on generalizations of the results presented in this manuscript. The author is
supported by the Alexander von Humboldt Foundation.

##  Appendix A Gauge issues

Throughout this appendix we assume a (real, Lorentzian) vacuum pp-wave
background, with the special spin frame $\\{o^{A},\iota^{A}\\}$, and its
complex conjugate $\\{\bar{o}^{A^{\prime}},\bar{\iota}^{A^{\prime}}\\}$,
introduced in section 2.3. The associated null tetrad
$\\{\ell^{a},n^{a},m^{a},\bar{m}^{a}\\}$ is defined as in eq. (2.2). In terms
of Brinkmann coordinates (see eq. (2.4)), we have

$\ell^{a}\partial_{a}=\partial_{u},\qquad
m^{a}\partial_{a}=-\partial_{\bar{\zeta}},\qquad\bar{m}^{a}\partial_{a}=-\partial_{\zeta},\qquad
n^{a}\partial_{a}=\partial_{v}-\tfrac{1}{2}H\partial_{u}.$ (A.1)

The connection coefficients are given by

$\nabla_{a}\ell^{b}=0,\qquad\nabla_{a}m^{b}=-\bar{\kappa}^{\prime}\ell_{a}\ell^{b},\qquad\nabla_{a}\bar{m}^{b}=-\kappa^{\prime}\ell_{a}\ell^{b},\qquad\nabla_{a}n^{b}=-\ell_{a}(\kappa^{\prime}m^{b}+\bar{\kappa}^{\prime}\bar{m}^{b}).$
(A.2)

###  A.1 The gauge operator

For an arbitrary covector field $\xi_{a}$, we define the “gauge operator” (or
Killing operator) $K$ by

$K[\xi]_{ab}=\nabla_{a}\xi_{b}+\nabla_{b}\xi_{a}.$ (A.3)

In terms of a null tetrad, this can be written as follows:

$\displaystyle K[\xi]_{ab}={}$ $\displaystyle
K_{nn}\ell_{a}\ell_{b}-2K_{n\bar{m}}\ell_{(a}m_{b)}-2K_{nm}\ell_{(a}\bar{m}_{b)}+K_{\bar{m}\bar{m}}m_{a}m_{b}+K_{mm}\bar{m}_{a}\bar{m}_{b}$
$\displaystyle+K_{\ell\ell}n_{a}n_{b}-2K_{\ell\bar{m}}n_{(a}m_{b)}-2K_{\ell
m}n_{(a}\bar{m}_{b)}+2K_{\ell
n}n_{(a}\ell_{b)}-2K_{m\bar{m}}m_{(a}\bar{m}_{b)}.$

For a pp-wave, using (A.1)-(A.2) we find:

$\displaystyle K_{\ell\ell}=2\partial_{u}\xi_{\ell},$ $\displaystyle
K_{m\bar{m}}=-(\partial_{\bar{\zeta}}\xi_{\bar{m}}+\partial_{\zeta}\xi_{m})$
(A.4a) $\displaystyle
K_{\ell\bar{m}}=\partial_{u}\xi_{\bar{m}}-\partial_{\zeta}\xi_{\ell},$
$\displaystyle
K_{nm}=(\partial_{v}-\tfrac{1}{2}H\partial_{u})\xi_{m}+\bar{\kappa}^{\prime}\xi_{\ell}-\partial_{\bar{\zeta}}\xi_{n},$
(A.4b) $\displaystyle K_{\ell
m}=\partial_{u}\xi_{m}-\partial_{\bar{\zeta}}\xi_{\ell},$ $\displaystyle
K_{n\bar{m}}=(\partial_{v}-\tfrac{1}{2}H\partial_{u})\xi_{\bar{m}}+\kappa^{\prime}\xi_{\ell}-\partial_{\zeta}\xi_{n},$
(A.4c) $\displaystyle K_{\bar{m}\bar{m}}=-2\partial_{\zeta}\xi_{\bar{m}},$
$\displaystyle K_{\ell
n}=\partial_{u}\xi_{n}+(\partial_{v}-\tfrac{1}{2}H\partial_{u})\xi_{\ell},$
(A.4d) $\displaystyle K_{mm}=-2\partial_{\bar{\zeta}}\xi_{m},$ $\displaystyle
K_{nn}=2(\partial_{v}-\tfrac{1}{2}H\partial_{u})\xi_{n}+2\kappa^{\prime}\xi_{m}+2\bar{\kappa}^{\prime}\xi_{\bar{m}}.$
(A.4e)

####  A.1.1 Residual radiation gauge freedom

The radiation gauge (3.10) is preserved by transformations in which the new
gauge vector $\xi_{a}$ satisfies

$\ell^{a}K[\xi]_{ab}=0,\qquad g^{ab}K[\xi]_{ab}=0.$ (A.5)

Equivalently: $K_{\ell\ell}=K_{\ell m}=K_{\ell\bar{m}}=K_{\ell
n}=K_{m\bar{m}}=0$. Using identities (A.4), this is

$\displaystyle\partial_{u}\xi_{\ell}={}$ $\displaystyle 0,$ (A.6a)
$\displaystyle\partial_{u}\xi_{\bar{m}}-\partial_{\zeta}\xi_{\ell}={}$
$\displaystyle 0,$ (A.6b)
$\displaystyle\partial_{u}\xi_{m}-\partial_{\bar{\zeta}}\xi_{\ell}={}$
$\displaystyle 0,$ (A.6c)
$\displaystyle\partial_{u}\xi_{n}+\partial_{v}\xi_{\ell}={}$ $\displaystyle
0,$ (A.6d)
$\displaystyle\partial_{\bar{\zeta}}\xi_{\bar{m}}+\partial_{\zeta}\xi_{m}={}$
$\displaystyle 0.$ (A.6e)

From here we deduce

$\partial^{2}_{u}\xi_{\bar{m}}=\partial^{2}_{u}\xi_{m}=\partial^{2}_{u}\xi_{n}=0,\qquad\partial_{\zeta}\partial_{\bar{\zeta}}\xi_{\ell}=0.$
(A.7)

Notice that in view of (A.4), and given that $\partial_{u}$ is a Killing
vector of the background space-time, from (A.7) it follows that
$\partial^{2}_{u}K_{\bf ab}=0$ for all ${\bf a,b}=u,v,\zeta,\bar{\zeta}$.

So we get the following general form for the components of the (real) gauge
vector $\xi_{a}$:

$\displaystyle\xi_{\ell}={}$ $\displaystyle
f_{1}(v,\zeta)+f_{2}(v,\bar{\zeta}),$ (A.8a) $\displaystyle\xi_{\bar{m}}={}$
$\displaystyle[\partial_{\zeta}f_{1}(v,\zeta)]u+f_{3}(v,\zeta,\bar{\zeta}),$
(A.8b) $\displaystyle\xi_{n}={}$
$\displaystyle-[\partial_{v}f_{1}(v,\zeta)+\partial_{v}f_{2}(v,\bar{\zeta})]u+f_{4}(v,\zeta,\bar{\zeta}),$
(A.8c)

for some functions $f_{1}(v,\zeta)$, $f_{2}(v,\bar{\zeta})$,
$f_{3}(v,\zeta,\bar{\zeta})$ and $f_{4}(v,\zeta,\bar{\zeta})$. Apart from
reality conditions for $\xi_{a}$, any restrictions on these functions will be
differential.

Given that we here impose the gauge operator (A.3) to satisfy (A.5), the same
reasoning that we used in section 3.2.2 to deduce (3.12) now gives

$K[\xi]_{ab}=o_{A}o_{B}G_{A^{\prime}B^{\prime}}+\bar{o}_{A^{\prime}}\bar{o}_{B^{\prime}}\bar{G}_{AB},$
(A.9)

where

$G_{A^{\prime}B^{\prime}}=(\tfrac{1}{2}K_{nn}+i\eta)\bar{o}_{A^{\prime}}\bar{o}_{B^{\prime}}-2K_{n\bar{m}}\bar{o}_{(A^{\prime}}\bar{\iota}_{B^{\prime})}+K_{\bar{m}\bar{m}}\bar{\iota}_{A^{\prime}}\bar{\iota}_{B^{\prime}}$
(A.10)

and we have included an arbitrary scalar field $\eta$. Explicit expressions
for $K_{nn}$, $K_{n\bar{m}}$, $K_{\bar{m}\bar{m}}$ in terms of $\xi_{a}$ are
given in (A.4). We can now express $G_{A^{\prime}B^{\prime}}$ in terms of a
1-index spinor by doing the same trick that we did in section 3.2.3. The
linearized Ricci scalar of (A.9) is
$\dot{R}[K[\xi]]=\tilde{\nabla}^{A^{\prime}}\tilde{\nabla}^{B^{\prime}}G_{A^{\prime}B^{\prime}}+\bar{\nabla}^{A}\bar{\nabla}^{B}\bar{G}_{AB}$
(which vanishes identically since the background is vacuum), where

$\tilde{\nabla}^{A^{\prime}}\tilde{\nabla}^{B^{\prime}}G_{A^{\prime}B^{\prime}}=\partial^{2}_{\bar{\zeta}}K_{\bar{m}\bar{m}}+2\partial_{u}\partial_{\bar{\zeta}}K_{n\bar{m}}+\tfrac{1}{2}\partial^{2}_{u}K_{nn}+i\partial^{2}_{u}\eta.$

A short calculation shows that
$\partial^{2}_{u}K_{nn}=0=\partial_{u}\partial_{\bar{\zeta}}K_{n\bar{m}}$,
while $\partial^{2}_{\bar{\zeta}}K_{\bar{m}\bar{m}}$ is independent of $u$.
Therefore, if we choose the arbitrary scalar $\eta$ in (A.10) in the form
$\eta=\tfrac{i}{2}[\partial^{2}_{\bar{\zeta}}K_{\bar{m}\bar{m}}]u^{2}$, then
$\tilde{\nabla}^{A^{\prime}}\tilde{\nabla}^{B^{\prime}}G_{A^{\prime}B^{\prime}}=0$,
so there is a spinor field $g_{A^{\prime}}$ such that
$G_{A^{\prime}B^{\prime}}=\tilde{\nabla}_{(A^{\prime}}g_{B^{\prime})}$, and

$K[\xi]_{ab}=o_{A}o_{B}\tilde{\nabla}_{(A^{\prime}}g_{B^{\prime})}+\bar{o}_{A^{\prime}}\bar{o}_{B^{\prime}}\bar{\nabla}_{(A}\bar{g}_{B)}.$
(A.11)

The relation between $g_{A^{\prime}}$ and $\xi_{a}$ is given by

$\displaystyle\partial_{u}g_{0^{\prime}}={}$ $\displaystyle
K_{\bar{m}\bar{m}},$ (A.12a)
$\displaystyle\partial_{u}g_{1^{\prime}}-\partial_{\bar{\zeta}}g_{0^{\prime}}={}$
$\displaystyle 2K_{n\bar{m}},$ (A.12b)
$\displaystyle\partial_{\bar{\zeta}}g_{1^{\prime}}={}$
$\displaystyle-(\tfrac{1}{2}K_{nn}+i\eta),$ (A.12c)

where in the right-hand sides one replaces the expressions (A.4). We deduce
from here that

$\partial^{3}_{u}g_{0^{\prime}}=0,\qquad\partial_{\bar{\zeta}}\partial^{2}_{u}g_{0^{\prime}}=0,\qquad\partial^{3}_{u}g_{1^{\prime}}=0.$
(A.13)

Thus, the general structure of $g_{0^{\prime}}$, $g_{1^{\prime}}$ is

$\displaystyle g_{0^{\prime}}={}$ $\displaystyle
A_{0^{\prime}}(v,\zeta)u^{2}+B_{0^{\prime}}(v,\zeta,\bar{\zeta})u+C_{0^{\prime}}(v,\zeta,\bar{\zeta}),$
(A.14a) $\displaystyle g_{1^{\prime}}={}$ $\displaystyle
A_{1^{\prime}}(v,\zeta,\bar{\zeta})u^{2}+B_{1^{\prime}}(v,\zeta,\bar{\zeta})u+C_{1^{\prime}}(v,\zeta,\bar{\zeta}),$
(A.14b)

for some functions $A_{0^{\prime}},...,C_{1^{\prime}}$ where the arguments are
as specified in the previous equations. Using (A.12) and (A.4), one can relate
these functions to the ones appearing in (A.8); this way we see, for example,
that they do not identically vanish. For instance, we get
$A_{0^{\prime}}(v,\zeta)=-\partial^{2}_{\zeta}f_{1}(v,\zeta)$. However, in
general the explicit expressions do not seem to be particularly enlightening.

###  A.2 Proof of proposition 3.3

We have the identity
$\nabla^{AA^{\prime}}Z_{A^{\prime}}=o^{A}\nabla^{A^{\prime}}Z_{A^{\prime}}-\iota^{A}\tilde{\nabla}^{A^{\prime}}Z_{A^{\prime}}$,
so $\nabla^{AA^{\prime}}Z_{A^{\prime}}=0$ iff
$\nabla^{A^{\prime}}Z_{A^{\prime}}=0$ and
$\tilde{\nabla}^{A^{\prime}}Z_{A^{\prime}}=0$. Since
$Z_{A^{\prime}}=Y_{A^{\prime}}-g_{A^{\prime}}$, we have

$\displaystyle\tilde{\nabla}^{A^{\prime}}Z_{A^{\prime}}={}$
$\displaystyle\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}}-\tilde{\nabla}^{A^{\prime}}g_{A^{\prime}},$
(A.15a) $\displaystyle\nabla^{A^{\prime}}Z_{A^{\prime}}={}$
$\displaystyle\nabla^{A^{\prime}}Y_{A^{\prime}}-\nabla^{A^{\prime}}g_{A^{\prime}},$
(A.15b)

so we want to show that, as long as the Einstein equations are satisfied, we
can choose the gauge transformation such that the associated spinor field
$g_{A^{\prime}}$ satisfies
$\tilde{\nabla}^{A^{\prime}}g_{A^{\prime}}=\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}}$
and $\nabla^{A^{\prime}}g_{A^{\prime}}=\nabla^{A^{\prime}}Y_{A^{\prime}}$.

The first observation is that any requirement for the function
$\tilde{\nabla}^{A^{\prime}}g_{A^{\prime}}$ restricts $g_{A^{\prime}}$ only up
to the addition of terms of the form $\tilde{\nabla}_{A^{\prime}}S$. In other
words, we can write
$g_{A^{\prime}}=V_{A^{\prime}}+2\tilde{\nabla}_{A^{\prime}}S$ where
$V_{A^{\prime}}$ and $S$ are independent, then (A.15) become

$\displaystyle\tilde{\nabla}^{A^{\prime}}Z_{A^{\prime}}={}$
$\displaystyle\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}}-\tilde{\nabla}^{A^{\prime}}V_{A^{\prime}},$
(A.16a) $\displaystyle\nabla^{A^{\prime}}Z_{A^{\prime}}={}$
$\displaystyle\nabla^{A^{\prime}}Y_{A^{\prime}}-\nabla^{A^{\prime}}V_{A^{\prime}}-\Box
S$ (A.16b)

(where we used the identity $\Box
S=2\nabla^{A^{\prime}}\tilde{\nabla}_{A^{\prime}}S$), and we want to show that
$V_{A^{\prime}}$ and $S$ can be chosen such that
$\tilde{\nabla}^{A^{\prime}}Z_{A^{\prime}}=0=\nabla^{A^{\prime}}Z_{A^{\prime}}$.
Restrictions on $V_{A^{\prime}}$ and $S$ arise from the fact that they come
from a gauge transformation: the general form of the components of
$g_{A^{\prime}}$ was obtained in (A.14). See (A.20) below.

In order to obtain expressions for the fields
$\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}}$,
$\nabla^{A^{\prime}}Y_{A^{\prime}}$, we use the linearized Einstein equations
(3.27). The non-trivial, independent components are:

$\displaystyle\partial^{2}_{u}(\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}})=0,$
(A.17a) $\displaystyle
2\partial^{2}_{u}(\nabla^{A^{\prime}}Y_{A^{\prime}})+\partial_{u}\partial_{\zeta}(\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}})=0,$
(A.17b)
$\displaystyle\partial_{u}\partial_{\bar{\zeta}}(\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}})+\partial_{u}\partial_{\zeta}(\bar{\nabla}^{A}\bar{Y}_{A})=0,$
(A.17c) $\displaystyle
2\partial_{u}\partial_{\bar{\zeta}}(\nabla^{A^{\prime}}Y_{A^{\prime}})+\tfrac{1}{2}(\partial_{u}\partial_{v}+\partial_{\zeta}\partial_{\bar{\zeta}})(\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}})-\tfrac{1}{2}\partial^{2}_{\zeta}(\bar{\nabla}^{A}\bar{Y}_{A})=0,$
(A.17d)
$\displaystyle\partial^{2}_{\bar{\zeta}}(\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}})-4\partial_{u}\partial_{\zeta}(\nabla^{A}\bar{Y}_{A})-2(\partial_{u}\partial_{v}+\partial_{\zeta}\partial_{\bar{\zeta}})(\bar{\nabla}^{A}\bar{Y}_{A})=0,$
(A.17e) $\displaystyle
2\partial^{2}_{\bar{\zeta}}(\nabla^{A^{\prime}}Y_{A^{\prime}})+\partial_{\bar{\zeta}}(\partial_{v}-\tfrac{1}{2}H\partial_{u})(\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}})+2\partial^{2}_{\zeta}(\nabla^{A}\bar{Y}_{A})+\partial_{\zeta}(\partial_{v}-\tfrac{1}{2}H\partial_{u})(\bar{\nabla}^{A}\bar{Y}_{A})=0.$
(A.17f)

Equations (A.17a)-(A.17b) correspond to
$\bar{o}^{A^{\prime}}\bar{o}^{B^{\prime}}\dot{R}_{ABA^{\prime}B^{\prime}}[h]=0$,
(A.17c)-(A.17d) correspond to
$\bar{o}^{A^{\prime}}\bar{\iota}^{B^{\prime}}\dot{R}_{ABA^{\prime}B^{\prime}}[h]=0$,
and (A.17e)-(A.17f) correspond to
$\bar{\iota}^{A^{\prime}}\bar{\iota}^{B^{\prime}}\dot{R}_{ABA^{\prime}B^{\prime}}[h]=0$.
Notice that (A.17a)-(A.17b) are the only equations that involve only
$\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}}$ and
$\nabla^{A^{\prime}}Y_{A^{\prime}}$ (not their complex conjugates); this is
because of the identities (3.17). We will then use (A.17a)-(A.17b) to deduce
the structure of $\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}}$,
$\nabla^{A^{\prime}}Y_{A^{\prime}}$.

From (A.17a) we deduce that

$\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}}=a(v,\zeta,\bar{\zeta})u+b(v,\zeta,\bar{\zeta})$
(A.18)

for some functions $a,b$. These functions can be written in terms of the
metric perturbation $h_{ab}$ (up to an arbitrary function of $v,\zeta$, that
we can set to zero), by noticing that
$\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}}=-(\partial_{u}Y_{1^{\prime}}+\partial_{\bar{\zeta}}Y_{0^{\prime}})$
and using eqs. (3.24a)-(3.24b). Taking a $u$-derivative in (A.17b) and using
(A.17a), we see that $\partial^{3}_{u}(\nabla^{A^{\prime}}Y_{A^{\prime}})=0$,
so $\nabla^{A^{\prime}}Y_{A^{\prime}}$ is quadratic in $u$. Using also (A.18),
it follows that

$\nabla^{A^{\prime}}Y_{A^{\prime}}=[-\tfrac{1}{4}a_{\zeta}(v,\zeta,\bar{\zeta})]u^{2}+c(v,\zeta,\bar{\zeta})u+d(v,\zeta,\bar{\zeta})$
(A.19)

for some functions $c,d$. The rest of the equations in (A.17) involve also the
complex conjugate fields, and they give additional restrictions on the
functions that appear in the right hand sides of (A.18)-(A.19).

Notice that, since the fields $\tilde{\nabla}^{A^{\prime}}g_{A^{\prime}}$,
$\nabla^{A^{\prime}}g_{A^{\prime}}$ come from a gauge transformation, the same
equations (A.17) hold for them. In other words, any restrictions on
$\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}}$,
$\nabla^{A^{\prime}}Y_{A^{\prime}}$ coming from (A.17) are also satisfied by
$\tilde{\nabla}^{A^{\prime}}g_{A^{\prime}}$,
$\nabla^{A^{\prime}}g_{A^{\prime}}$. But
$\tilde{\nabla}^{A^{\prime}}g_{A^{\prime}}$,
$\nabla^{A^{\prime}}g_{A^{\prime}}$ must also fulfil restrictions that come
from the gauge condition. These additional restrictions were analysed in
appendix A.1.1, where the general expressions (A.14) were found. In our
current context, we have
$g_{A^{\prime}}=V_{A^{\prime}}+2\tilde{\nabla}_{A^{\prime}}S$, or in
components $g_{0^{\prime}}=V_{0^{\prime}}+2\partial_{u}S$,
$g_{1^{\prime}}=V_{1^{\prime}}-2\partial_{\bar{\zeta}}S$. The restrictions
(A.13) together with the fact that $V_{A^{\prime}}$ and $S$ are independent
imply that
$\partial^{3}_{u}V_{0^{\prime}}=\partial_{\bar{\zeta}}\partial^{2}_{u}V_{0^{\prime}}=\partial^{3}_{u}V_{1^{\prime}}=0$
and $\partial^{4}_{u}S=\partial^{3}_{u}\partial_{\bar{\zeta}}S=0$. So we have
the following form:

$\displaystyle V_{0^{\prime}}={}$
$\displaystyle\alpha_{0^{\prime}}(v,\zeta)u^{2}+\beta_{0^{\prime}}(v,\zeta,\bar{\zeta})u+\gamma_{0^{\prime}}(v,\zeta,\bar{\zeta}),$
(A.20a) $\displaystyle V_{1^{\prime}}={}$
$\displaystyle\alpha_{1^{\prime}}(v,\zeta,\bar{\zeta})u^{2}+\beta_{1^{\prime}}(v,\zeta,\bar{\zeta})u+\gamma_{1^{\prime}}(v,\zeta,\bar{\zeta}),$
(A.20b) $\displaystyle S={}$ $\displaystyle
S_{3}(v,\zeta)u^{3}+S_{2}(v,\zeta,\bar{\zeta})u^{2}+S_{1}(v,\zeta,\bar{\zeta})u+S_{0}(v,\zeta,\bar{\zeta}).$
(A.20c)

After some tedious calculations, this gives (using in particular the
expression (2.10) for $\Box$):

$\displaystyle\tilde{\nabla}^{A^{\prime}}g_{A^{\prime}}={}$
$\displaystyle-(2\alpha_{1^{\prime}}+\partial_{\bar{\zeta}}\beta_{0^{\prime}})u-(\beta_{1^{\prime}}+\partial_{\bar{\zeta}}\gamma_{0^{\prime}}),$
(A.21) $\displaystyle\nabla^{A^{\prime}}g_{A^{\prime}}={}$
$\displaystyle[\partial_{v}\alpha_{0^{\prime}}+\partial_{\zeta}\alpha_{1^{\prime}}+6\partial_{v}S_{3}-2\partial_{\zeta}\partial_{\bar{\zeta}}S_{2}]u^{2}$
$\displaystyle+[\partial_{v}\beta_{0^{\prime}}-H\alpha_{0^{\prime}}+\partial_{\zeta}\beta_{1^{\prime}}+4\partial_{v}S_{2}-2\partial_{\zeta}\partial_{\bar{\zeta}}S_{1}-6HS_{3}]u$
$\displaystyle+[\partial_{v}\gamma_{0^{\prime}}-\tfrac{1}{2}H\beta_{0^{\prime}}+\partial_{\zeta}\gamma_{1^{\prime}}+2\partial_{v}S_{1}-2\partial_{\zeta}\partial_{\bar{\zeta}}S_{0}-2HS_{2}]$
(A.22)

Comparing these expressions to (A.18)-(A.19), we see that we can choose the
free functions in (A.20) so that
$\tilde{\nabla}^{A^{\prime}}g_{A^{\prime}}=\tilde{\nabla}^{A^{\prime}}Y_{A^{\prime}}$
and $\nabla^{A^{\prime}}g_{A^{\prime}}=\nabla^{A^{\prime}}Y_{A^{\prime}}$,
which is what we wanted to prove.

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