Twistor initial data characterisation of pp-waves

The existence of symmetries encoded through Killing objects (spinors, vectors, tensors, Killing–Yano tensors) in a spacetime is a strong constraint that can be exploited for obtaining geometric characterisations of spacetimes of physical interest. In the context of the Cauchy problem in General Relativity, whether an initial data for the Einstein field equations will develop into a spacetime admitting one of these Killing objects can be determined through Killing initial data equations. The prototypical examples are the Killing vector and Killing spinor initial data equations of [1, 2], respectively. These initial data conditions can, in turn, be exploited to obtain initial data characterisations of particular spacetimes. Examples of this construction include the characterisation of the Kerr spacetime in [3], exploiting the (valence-2) Killing spinor initial data equations, and the simpler characterisation of the Minkowski spacetime through twistors (valence-1 Killing spinor) in [4]. In this note, we show that, by augmenting the conditions imposed for the standard twistor initial data of [2], one can derive an initial data characterisation of pp-wave spacetimes, an approach which can be seen as the spinorial analogue of the pp-wave initial data characterisation via conformal Killing vectors of [5]. As pointed out in [5], an initial data characterisation of pp-waves can be obtained by employing Killing spinors, and has been carried out in length in [6] —albeit in a different language and with a different scope. In this note it is shown how a similar characterisation can be obtained concisely through a simple modification of twistor initial data equations of [2, 4].

Let $(\mathcal{M},{\boldsymbol{g}})$ be a 4-dimensional manifold equipped with a Lorentzian metric g of signature $(+,-,-,-)$ and a spinor structure. Any non-trivial spinor κ_A satisfying

$$\begin{align} \nabla_{A^{^{\prime}}}\left(A\kappa_{B}\right) = 0, \end{align} \tag{ 1 }$$

will be referred to as a valence-1 Killing spinor or simply as a twistor. The integrability condition for the last equation is $\Psi_{ABCD} \, \kappa^D = 0$ which restricts the spacetime to be of Petrov type N or O.

From this point onward, unless otherwise stated, it will be assumed that the vacuum Einstein field equations (without cosmological constant) are satisfied. In other words, it will be assumed that $(\mathcal{M},{\boldsymbol{g}})$ is Ricci-flat $R_{ab} = 0$, which, in spinorial Newman–Penrose notation, is encoded through the vanishing of trace-free Ricci spinor and the Ricci scalar ($\Phi_{AA^{^{\prime}}BB^{^{\prime}}} = \Lambda = 0$).One can give a spacetime characterisation of the Minkowski spacetime through the existence of a twistor as follows:

Proposition 1 (Bäckdahl & Valiente-Kroon). If κ_A is a twistor in an asymptotically flat spacetime $(\mathcal{M},{\boldsymbol{g}})$ and $\eta_A : = \nabla_A{}^{A^{^{\prime}}}\bar{\kappa}_{A^{^{\prime}}} \neq 0$ at some point $p\in \mathcal{M}$, then the spacetime is the Minkowski spacetime.

Proof. In short, the proof of this proposition is based on the observation that, if κ_A is a twistor for which $\eta_{A}\neq 0$, then one has $\Psi_{ABCD}\kappa^A = \Psi_{ABCD}\eta^A = 0$. These conditions imply that one can construct an adapted spin dyad $\{\kappa, \eta\}$ such that $\Psi_{ABCD} = 0$ —see [4] for the detailed proof. □

A plane-fronted wave with parallel rays, or pp-wave for short, is a solution to the Einstein field equations in vacuum characterised by the existence of a null covariantly constant vector k^a —see [5]. This in turn implies that there exist a local coordinate system $(u,r,x^i)$ with $i = 1,2$ for which the metric reads

$$\begin{align*} g_{H} = 2\mathcal{H}\left(u,x^i\right)\mbox{d}u^2+2\mbox{d}u\mbox{d}r- \delta_{ij}\mbox{d}x^i\mbox{d}x^j \end{align*}$$

where δ is the 2-dimensional Euclidean metric, and

$$\begin{equation} \delta^{ij}\partial_i\partial_j \mathcal{H} = 0. \end{equation} \tag{ 2 }$$

It is well-known that not every pp-wave spacetime is globally hyperbolic—see [7]. However, global hyperbolicity of this class of spacetimes strongly depends on the behaviour of $\mathcal{H}$ at spatial infinity. In [8] the conditions for a pp-wave to be strongly hyperbolic have been established.

The condition $\eta_{A} = 0$ (explicitly excluded in proposition 1) is key for the characterisation of pp-wave spacetimes. To see this and to set up the notation, let $H_{A^{^{\prime}}AB}: = 2\nabla_{A^{^{\prime}}(A}\kappa_{B)}$ and $\bar{\eta}_{A^{^{\prime}}}: = \nabla_{A^{^{\prime}}}^Q\kappa_Q$. Then, the irreducible decomposition of $\nabla_{AA^{^{\prime}}}\kappa_{B}$ reads

$$\begin{align} 2\nabla_{A^{^{\prime}}A}\kappa_B = H_{A^{^{\prime}}AB}+ \epsilon_{AB}\bar{\eta}_{A^{^{\prime}}}. \end{align} \tag{ 3 }$$

If κ_A is a twistor then $H_{A^{^{\prime}}AB} = 0$ and, if in addition, $\bar{\eta}_{A^{^{\prime}}} = 0$, then $\nabla_{AA^{^{\prime}}}\kappa_B = 0$. Consequently, $\kappa^A \bar{\kappa}^{A^{^{\prime}}}$ is a covariantly constant vector. Hence, the condition $\nabla_{AQ^{^{\prime}}} \kappa^{Q^{^{\prime}}} = 0$ ensures that proposition 1 does not apply and that $\kappa^{A} \bar{\kappa}^{A^{^{\prime}}}$ is covariantly constant. One then concludes that thespacetime is a pp-wave. This discussion is summarised in the following:

Proposition 2. If κ_A is a twistor for which $\bar{\eta}_{A^{^{\prime}}} : = \nabla_{A^{^{\prime}}}{}^{A}\kappa_{A} = 0$, then $(\mathcal{M},{\boldsymbol{g}})$ is a pp-wave spacetime for some function $\mathcal{H}$ satisfying (2).

Proposition (2) amounts to a spacetime characterisation; one can, however, obtain a characterisation at the level of initial data by slightly modifying the twistor initial data equations of [2]. Before doing so, we first give a brief discussion of the derivation of the twistor initial data equations. The twistor initial data result of [2] is based on the following identities which hold in Ricci-flat spacetimes:

$$\begin{align} \square H_{A^{^{\prime}}AB} & = 2\nabla_{A^{^{\prime}}}\left(A\square \kappa_{B}\right) + 2\Psi_{AB}{}^{PQ}H_{A^{^{\prime}}PQ} \end{align} \tag{ 4 }$$

$$\begin{align} \square \kappa_A & = \frac{2}{3}\nabla^{PP^{^{\prime}}}H_{P^{^{\prime}}PA} \end{align} \tag{ 5 }$$

where $\square: = g^{ab}\nabla_a\nabla_b$. Assume that the twistor-candidate equation

$$\begin{equation} \square \kappa_A = 0. \end{equation} \tag{ 6 }$$

holds. Then equation (4) reduces to

$$\begin{equation} \square {H}{_{A^{^{\prime}}AB}} = 2{\Psi}{_{AB}{}^{PQ}} {H}{_{A^{^{\prime}}PQ}} \end{equation} \tag{ 7 }$$

so that if one provides trivial initial data for $H_{A^{^{\prime}}AB}$ on a Cauchy hypersurface Σ₀:

$$\begin{equation} H_{A^{^{\prime}}AB}|_{\Sigma_0} = 0, \qquad \nabla_{EE^{^{\prime}}}H_{A^{^{\prime}}AB}|_{\Sigma_0} = 0, \end{equation} \tag{ 8 }$$

then, by local existence and uniqueness of symmetric hyperbolic systems, one has that $H_{A^{^{\prime}}AB} = 0$ on a spacetime neighbourhood $\mathcal{U} \subset \mathcal{D}^{+}(\Sigma_0)$, where $D^{+}(\Sigma_0)$ denotes the future domain of dependence of Σ₀. The initial conditions (8) can be translated in terms of κ_A as:

$$\begin{align} \nabla_{A^{^{\prime}}}\left(A\kappa_{B}\right)|_{\Sigma_0} = 0, \qquad \nabla_{EE^{^{\prime}}}\nabla_{A^{^{\prime}}}\left(A\kappa_{B}\right)|_{\Sigma_0} = 0, \end{align} \tag{ 9 }$$

and are regarded as initial data constraints for equation (6). To obtain conditions intrinsic to Σ₀ one needs to perform a 1+3 spinor split. Although this is analogous to the standard 3+1 split in tensors, in general, the spacespinor split is not adapted to a foliation but rather to a congruence of timelike curves with tangent $\tau^{AA^{^{\prime}}}$ normalised so that $\tau^{AA^{^{\prime}}}\tau_{AA^{^{\prime}}} = 2$. The Levi-Civita covariant derivative of any spinor µ_C splits as:

$$\begin{equation} \nabla_{AA^{^{\prime}}}\mu_{C} = \tfrac{1}{2} \tau _{AA^{^{\prime}}} \nabla_{\tau }\mu_{C} - \tau ^{B}{}_{A^{^{\prime}}} \mathcal{D} _{BA}\mu _{C} \end{equation} \tag{ 10 }$$

where $\nabla_{\tau}: = \tau^{AA^{^{\prime}}}\nabla_{AA^{^{\prime}}}$ and $\mathcal{D}_{AB}: = \tau_{(A}{}^{A^{^{\prime}}}\nabla_{B)A^{^{\prime}}}$ is the Sen connection relative to τ^a —see [9]. The spacetime covariant derivative of τ^a is determined in terms of the acceleration χ_AB and Weingarten spinors χ_ABCD through:

$$\begin{equation} \nabla_{AA^{^{\prime}}}\tau _{CC^{^{\prime}}} = - \tfrac{1}{\sqrt{2}} \chi _{CD} \tau _{AA^{^{\prime}}} \tau ^{D}{}_{C^{^{\prime}}} + \sqrt{2} \chi _{ABCD} \tau ^{B}{}_{A^{^{\prime}}} \tau ^{D}{}_{C^{^{\prime}}}. \end{equation} \tag{ 11 }$$

If $\tau^{AA^{^{\prime}}}$ is hypersurface orthogonal then $\chi_{ABCD} = \chi_{AB(CD)}$ and corresponds to the second fundamental form of a foliation $\Sigma_{\tau}$. In addition, the 3-dimensional Levi-Civita connection on $\Sigma_{\tau}$ is given by,

$$\begin{equation} D_{AB} \mu_{C} = \mathcal{D}_{AB}\mu_{C} + \frac{1}{\sqrt{2}}\chi_{\left(AB\right)C}{}^{Q}\mu_{Q}. \end{equation} \tag{ 12 }$$

Using the spacespinor formalism, in [4], it was shown that the equations (9) are reduced to the following conditions:

$$\begin{align} \mathcal{D} _{\left(AB\kappa _{C}\right)}& = 0, \end{align} \tag{ 13a }$$

$$\begin{align} \Psi_{ABCD}\kappa^A& = 0 \end{align} \tag{ 13b }$$

$$\begin{align} \nabla_\tau \kappa_A & = -\frac{2}{3}\mathcal{D}_{A}{}^{B}\kappa_B. \end{align} \tag{ 13c }$$

The conditions (13a ) and (13b ) are called the twistor initial data equations. Notice that Ψ_ABCD on Σ₀ can be written in terms of its electric and magnetic part respect to τ^a which in turn can be expressed in terms of the initial data set $(\Sigma_0, {\boldsymbol{h}}, {\boldsymbol{\chi}})$ where h and χ are the first and second fundamental forms of Σ₀ —see [4] for further details and [2] for an alternative way of expressing these conditions. If the twistor initial data equations are solved for κ_A on Σ₀ and its time derivative on Σ₀ is prescribed according to equation (13c ) one obtains the initial data for the twistor-candidate equation (6) that ensures that κ^A is an actual twistor in $\mathcal{U}\subset \mathcal{D}^{+}(\Sigma_0)$.

This discussion is summarised in the following propositions.

Proposition 3 (García-Parrado & Valiente-Kroon). If a spinor κ^A satisfies the conditions (9) and solves the vacuum twistor candidate wave equation (6) then κ^A is twistor in a Ricci-flat open set $\;\mathcal{U}\subset \mathcal{D}^{+}(\Sigma_0)$.

Proposition 4 (Bäckdahl & Valiente-Kroon). Let $(\Sigma_0,{\boldsymbol{h}},{\boldsymbol{\chi}})$ be an initial data set for the vacuum Einstein field equations (without cosmological constant) where h is the 3-metric on a spacelike Cauchy hypersurface Σ₀ and χ is the second fundamental form. If there exist a non-trivial spinor $\kappa^A_{*}$ satisfying the twistor initial data equations (13a ) and (13b ) on Σ₀, then the spacetime development of $(\Sigma_0,{\boldsymbol{h}},{\boldsymbol{\chi}})$ will posses a twistor κ_A in an open set $\mathcal{U}\subset \mathcal{D}^{+}(\Sigma_0)$. The twistor κ^A is obtained by solving the twistor candidate equation (6) with initial data $(\kappa^A_{*}, \nabla_\tau \kappa^A_{*})$ on Σ₀ prescribed according to equations (13a )–(13c ).

Generally, initial data for the Einstein field equations satisfying conditions (13a )–(13c ) is not sufficient to ensure that the development will be a pp-wave spacetime, as proposition 4 does not guarantee that $\nabla_{A^{^{\prime}}Q}\kappa^Q = 0$. To see whether imposing further conditions conditions on the initial data is enough so that $\nabla_{A^{^{\prime}}Q}\kappa^Q = 0$ on $\mathcal{U}\subset \mathcal{D}^{+}(\Sigma_0)$, it suffices to construct a propagation equation for the quantity $\bar{\eta}_{A^{^{\prime}}}: = \nabla_{A^{^{\prime}}}^Q\kappa_Q$. Commuting covariant derivatives and using the spinorial-Ricci identities, a calculation similar to that leading to equation (4) gives

$$\begin{align*} \square \bar{\eta}_{A^{^{\prime}}} = -2 \Lambda \bar{\eta}_{A^{^{\prime}}} + 8\kappa^A\nabla_{AA^{^{\prime}}}\Lambda. \end{align*}$$

Thus, if $\Lambda = 0$,

$$\begin{equation} \square \bar{\eta}_{A^{^{\prime}}} = 0. \end{equation} \tag{ 14 }$$

Hence, by providing trivial initial data for equation (14):

$$\begin{align} \bar{\eta}_{A^{^{\prime}}}|_{\Sigma_0} = 0, \qquad \nabla_{EE^{^{\prime}}}\bar{\eta}_{A^{^{\prime}}}|_{\Sigma_0} = 0, \end{align} \tag{ 15 }$$

we get $\bar{\eta}_{A^{^{\prime}}} = 0$ on $\mathcal{U}\subset \mathcal{D}^{+}(\Sigma_0)$. Thus, by augmenting the requirements on the initial data implied by equation (8) to include those encoded in equation (15), one ensures that $\nabla_{A^{^{\prime}}Q}\kappa^Q = 0$ on $\mathcal{U} \subset \mathcal{D}^{+}(\Sigma_0)$. To translate the latter to intrinsic conditions on Σ₀ one needs to perform a 1+3 spacespinor split as follows. Let $\zeta _{A} = \bar{\eta} _{A^{^{\prime}}} \tau _{A}{}^{A^{^{\prime}}}$. Equivalently, $\bar{\eta}_{B^{^{\prime}}} = - \zeta _{A} \tau ^{A}{}_{B^{^{\prime}}}$. Then the conditions (15), are equivalent to impose $\zeta_A = 0$ and $\nabla_\tau \zeta_A = 0$. A direct calculation shows that

$$\begin{align} \zeta _{A} = - \tfrac{1}{2} \nabla_{\tau }\kappa _{A} + \mathcal{D} _{A}{}^{B}\kappa _{B}. \end{align} \tag{ 16 }$$

Hence using the condition (13c ) one gets

$$\begin{align} \zeta_A = \frac{4}{3}\mathcal{D} _{A}{}^{B}\kappa_{B}. \end{align} \tag{ 17 }$$

Similarly, a direct calculation shows that

$$\begin{equation} \nabla_{\tau }\bar{\eta} _{A^{^{\prime}}} = \tfrac{4}{3} \tau ^{A}{}_{A^{^{\prime}}} \nabla_{\tau }\mathcal{D} _{AB}\kappa ^{B} - \tfrac{4 \sqrt{2}}{3} \chi ^{A}{}_{C} \tau ^{C}{}_{A^{^{\prime}}} \mathcal{D} _{AB}\kappa ^{B}. \end{equation} \tag{ 18 }$$

Now, recall that the commutator $[\nabla_{\tau },\mathcal{D} _{AB}]$ acting on any spinor µ_C reads

$$\begin{align} \left[\nabla_{\tau },\mathcal{D} _{AB}\right]\mu_{C} & = \Psi _{ABCD} \mu^{D} - 2\Lambda\mu_{\left(A \epsilon _{B}\right)}C- \Phi _{CDA^{^{\prime}}B^{^{\prime}}} \mu^{D} \tau _{A}{}^{A^{^{\prime}}} \tau _{B}{}^{B^{^{\prime}}} - \tfrac{1}{\sqrt{2}}\chi _{AB} \nabla_{\tau}\mu_{C} \nonumber \\ & \quad + \tfrac{2}{\sqrt{2}}\chi _{\left(A{}^{D} \mathcal{D} _{B}\right)}D\mu_{C} - \sqrt{2} \chi _{\left(AB\right)DF} \mathcal{D} ^{DF}\mu_{C}. \end{align} \tag{ 19 }$$

Taking $\mu_{C} = \kappa_{C}$ and using equations (13c ) and (17), a long but straightforward calculation renders

$$\begin{align} \nabla_{\tau}\zeta _{A} & = - \sqrt{2} \chi _{AB} \zeta ^{B} + 4 \Lambda\kappa _{A} - \tfrac{2}{3} \sqrt{2} \zeta ^{B} \chi _{A}{}^{C}{}_{BC} - \tfrac{4}{3} \Phi _{BCA^{^{\prime}}B^{^{\prime}}} \kappa^{B} \tau _{A}{}^{A^{^{\prime}}} \tau ^{CB^{^{\prime}}} \nonumber \\ & \quad + \tfrac{2}{3} \mathcal{D}_{AB}\zeta^{B} + \tfrac{2 \sqrt{2}}{3} \chi ^{BC} \mathcal{D}_{\left(AB\kappa _C\right)} - \tfrac{4}{3} \sqrt{2} \chi _{A}{}^{BCD} \mathcal{D} _{\left(BC\kappa _D\right)}. \end{align} \tag{ 20 }$$

Assuming vacuum and using condition (13a ), simplifies the latter expression to

$$\begin{align} \nabla_{\tau }\zeta _{A} & = - \sqrt{2}\chi _{AB} \zeta ^{B} - \tfrac{2}{3} \sqrt{2} \zeta ^{B} \chi _{A}{}^{C}{}_{BC} + \tfrac{2}{3} \mathcal{D} _{AB}\zeta ^{B} . \end{align} \tag{ 21 }$$

Then, with these assumptions, the conditions $\zeta_A = \nabla_\tau \zeta_A = 0$ reduce to the requirement that $\mathcal{D} _{A}{}^{B}\kappa_{B} = 0$. Altogether the condition $\mathcal{D} _{(AB}\kappa _{C)} = 0$ and $\mathcal{D} _{A}{}^{B}\kappa_{B} = 0$ can be encoded simply as $D_{AB}\kappa_{C} = 0$.

Remark 1. Observe that $\kappa_A = 0$ trivially solves the condition $D_{AB}\kappa_{C} = 0$. Hence, in the sequel we will be concerned only with non-trivial initial data, namely a spinor $\kappa_{*A} \neq 0$ everywhere on Σ₀ that satisfies $D_{AB}\kappa_{*C} = 0$.

This discussion is summarised in the following:

Proposition 5. Let $(\Sigma_0,{\boldsymbol{h}},{\boldsymbol{\chi}})$ be an initial data set for the vacuum Einstein field equations (without cosmological constant) where h is the 3-metric on a Cauchy hypersurface Σ₀ and χ is the second fundamental form. If the conditions

$$\begin{align} \mathcal{D} _{AB}\kappa _{C}& = 0 \end{align} \tag{ 22 }$$

$$\begin{align} \Psi_{ABCD}\kappa^A& = 0 \end{align} \tag{ 23 }$$

are satisfied by a non-trivial spinor $\kappa_{*}^A$ on Σ₀, then a covariantly constant spinor κ^A in some open set $\mathcal{U}\subset \mathcal{D}^{+}(\Sigma_0)$ is obtained by solving the twistor candidate equation (6) with initial data $(\kappa^A_{*}, \nabla_\tau \kappa^A_{*})$ on Σ₀ prescribed according to equation (13c ).

Remark 2 (Continuity argument). Notice that proposition 5 does not exclude the possibility that the covariantly constant spinor κ^A becomes trivial ($\kappa^A = 0$) at some point in the evolution. In other words, although by assumption the initial data for the wave equation (6) is non-trivial, $\kappa_{*}^{A} \neq 0$, this condition alone does not guarantee that $\kappa^A \neq 0$ in the whole domain of dependence $\mathcal{D}^{+}(\Sigma_0)$. However, if the solution κ_A is a classical solution (C²) of the wave equation (6), then by continuity, it follows that in a small spacetime neighbourhood of the initial hypersurface Σ₀ the solution is non-trivial: $\kappa_A \neq 0$ in $\mathcal{U}\subset \mathcal{D}^{+}(\Sigma_0)$. Therefore, although we cannot control the size of the spacetime neighbourhood $\mathcal{U}$ where the solution is non-trivial, it is clear by continuity that, by shrinking the size of the set $\mathcal{U}\subset \mathcal{D}^{+}(\Sigma_0)$, proposition (5) implies, in conjuction with proposition 2, that the development of $(\Sigma_0,{\boldsymbol{h}},{\boldsymbol{\chi}})$ will be a pp-wave spacetime on a small spacetime neighbourhood $\mathcal{U}$ close to the initial hypersurface.

The requirement on the regularity of κ_A (needed for the continuity argument of remark 2) can be transformed to a condition at the level of initial data by employing a standard existence and uniqueness theorem for wave equations. Although there could be other options, in the following we will make use of the local existence and uniqueness result for wave equations of [10] with initial data in some suitable Sobolev space H ^m . We use this result in the form presented in theorem 2 in appendix E of [11].

Remark 3 (Regularity of initial data). Introduce some local coordinates $x = (x^\mu) = (\tau,x^i)$ in $\mathcal{U}$, with $\mu = 0,1,2,3$ and $i = 1,2,3$ so that Σ₀ is described by τ = 0. Denote the components of κ_A as κ . Then, the wave equation (6) written in local coordinates reads:

$$\begin{equation*} g^{\mu\nu}\partial_\mu\partial_\nu {\boldsymbol{\kappa}} = F\left(x,{\boldsymbol{\kappa}}, \partial{\boldsymbol{\kappa}}\right), \end{equation*}$$

where F is linear in κ and $\partial {\boldsymbol{\kappa}}$ and g_µν is a Lorentzian metric. We will consider non-trivial solutions ${\boldsymbol{\kappa}}_{*}$ to the initial data equations (22) and (23) which satisfy the following regularity conditions. For $m \unicode{x2A7E} 4$ :

$$\begin{align} & {\boldsymbol{\kappa}}_{*}\in H^m\left(\Sigma_0, \mathbb{C}^2\right) \quad \text{and} \quad \partial_\tau{\boldsymbol{\kappa}}_{*}\in H^m\left(\Sigma_0, \mathbb{C}^2\right),\\ & \left({\boldsymbol{\kappa}}_{*},\; \partial_\tau{\boldsymbol{\kappa}}_{*}\right) \in D_{\delta} \qquad \text {for some}\, \delta>0\, \mathrm{where} \nonumber \end{align} \tag{ 24a }$$

$$\begin{align} & D_\delta \equiv \left\{\left(w_1,w_2\right)\in H^m\left(\Sigma_0,\mathbb{C}^2\right)\times H^m\left(\Sigma_0,\mathbb{C}^2\right) \; | \; \delta < | \det g_{\mu\nu} | \right\}. \end{align} \tag{ 24b }$$

With these assumptions, then using point (i) of theorem 2 of [11] one concludes that there exists T > 0 and a unique solution to the Cauchy problem defined on $[0,T)\times \Sigma_0$ such that

$$\begin{equation*} {\boldsymbol{\kappa}} \in C^{m-2}([0,T)\times \Sigma_0, \mathbb{C}^2). \end{equation*}$$

Combining propositions 5 and 2, and aided with the discussion of remarks 2 and 3 one obtains the following:

Theorem 1. Let $(\Sigma_0,{\boldsymbol{h}},{\boldsymbol{\chi}})$ be an initial data set for the vacuum Einstein field equations (without cosmological constant) where h is the 3-metric on a Cauchy hypersurface Σ₀ and χ is the second fundamental form. If the conditions

$$\begin{equation} \mathcal{D} _{AB}\kappa _{C} = 0 \end{equation} \tag{ 25 }$$

$$\begin{equation} \Psi_{ABCD}\kappa^A = 0, \end{equation} \tag{ 26 }$$

are satisfied by a non-trivial spinor $\kappa_{*}^A$ on Σ₀ satisfying the regularity conditions of remark 3. Then there exist a open set $\mathcal{U}\subset \mathcal{D}^{+}(\Sigma_0)$ —a possibly very small spacetime neighbourhood of the initial hypersurface Σ₀— for which the development of $(\Sigma_0,{\boldsymbol{h}},{\boldsymbol{\chi}})$ on $\mathcal{U}$ is a pp-wave spacetime for some function $\mathcal{H}$ satisfying equation (2) in $\mathcal{U}$.

E Gasperín holds a FCT (Portugal) investigator Grant 2020.03845.CEECIND. We have profited from scientific discussions with Filipe C Mena. This Project was supported by the H2020-MSCA-2022-SE Project EinsteinWaves, GA No. 101131233. E Gasperín has also benefited from the RISE Project at CENTRA through the European Union's H2020 ERC Advanced Grant 'Black holes: gravitational engines of discovery' Grant Agreement No. Gravitas-101052587, also the Villum Foundation (Grant No. VIL37766) and the DNRF Chair program (Grant No. DNRF162) by the Danish National Research Foundation and the Marie Sklodowska-Curie Grant Agreement Nos. 101007855 and 101131233. We are grateful for the comments of the anonymous referees on this note.

No new data were created or analysed in this study.