Twistor initial data characterisation of pp-waves

This note gives a concise derivation of a twistor-initial-data characterisation of pp-wave spacetimes in vacuum. The construction is based on a similar calculation for the Minkowski spacetime in Bäckdahl and Kroon [2011 Class. Quantum Grav. 28 075010]. The key difference is that for the Minkowski spacetime a necessary condition is that ∇AA′κ¯A′≠0 . In this note it is shown that if ∇AA′κ¯A′=0 then the development is a pp-wave spacetime. Furthermore, it is shown that such condition propagates off the initial hypersurface, which, in turn, gives a 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 [2] and [3], 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 [4], exploiting the (valence-2) Killing spinor initial data equations, and the simpler characterisation of the Minkowski spacetime through twistors (valence-1 Killing spinor) in [1].In this note, we show that, by augmenting the conditions imposed for the standard twistor initial data of [3], 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 [7] -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 [3,1].
Let (M, 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 will be referred to as a valence-1 Killing spinor or simply as a twistor.The integrability condition for the last equation is Ψ ABCD κ 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 (M, 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 (Φ AA ′ BB ′ = Λ = 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 (M, g) and η A := ∇ A A ′ κA ′ = 0 at some point p ∈ 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 η A = 0, then one has Ψ ABCD κ A = Ψ ABCD η A = 0.These conditions imply that one can construct an adapted spin dyad {κ, η} such that Ψ ABCD = 0 -see [1] 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 where δ is the 2-dimensional Euclidean metric, and It is well-known that not every pp-wave spacetime is globally hyperbolic -see [8].However, global hyperbolicity of this class of spacetimes strongly depends on the behaviour of H at spatial infinity.In [9] the conditions for a pp-wave to be strongly hyperbolic have been established.
The condition η 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 If κ A is a twistor then H A ′ AB = 0 and, if in addition, ηA ′ = 0, then ∇ AA ′ κ B = 0. Consequently, κ A κA ′ is a covariantly constant vector.Hence, the condition ∇ AQ ′ κ Q ′ = 0 ensures that Proposition 1 does not apply and that κ A κA ′ is covariantly constant.One then concludes that the spacetime is a pp-wave.This discussion is summarised in the following: 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 [3].Before doing so, we first give a brief discussion of the derivation of the twistor initial data equations.The twistor initial data result of [3] is based on the following identities which hold in Ricci-flat spacetimes: where := g ab ∇ a ∇ b .Assume that the twistor-candidate equation holds.Then equation ( 4) reduces to so that if one provides trivial initial data for H A ′ AB on a Cauchy hypersurface Σ 0 : then, by local existence and uniqueness of symmetric hyperbolic systems, one has that H A ′ AB = 0 on a spacetime neighbourhood U ⊂ D + (Σ 0 ), where D + (Σ 0 ) denotes the future domain of dependence of Σ 0 .The initial conditions (8) can be translated in terms of κ A as: and are regarded as initial data constraints for equation (6).To obtain conditions intrinsic to Σ 0 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 τ AA ′ normalised so that τ AA ′ τ AA ′ = 2.The Levi-Civita covariant derivative of any spinor µ C splits as: where ∇ τ := τ AA ′ ∇ AA ′ and D AB := τ (A A ′ ∇ B)A ′ is the Sen connection relative to τ a -see [10].The spacetime covariant derivative of τ a is determined in terms of the acceleration χ AB and Weingarten spinors χ ABCD through: If τ AA ′ is hypersurface orthogonal then χ ABCD = χ AB(CD) and corresponds to the second fundamental form of a foliation Σ τ .In addition, the 3-dimensional Levi-Civita connection on Σ τ is given by, Using the spacespinor formalism, in [1], it was shown that the equations ( 9) are reduced to the following conditions: The conditions (13a) and (13b) are called the twistor initial data equations.Notice that Ψ ABCD on Σ 0 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 (Σ 0 , h, χ) where h and χ are the first and second fundamental forms of Σ 0 -see [1] for further details and [3] for an alternative way of expressing these conditions.If the twistor initial data equations are solved for κ A on Σ 0 and its time derivative on Σ 0 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 U ⊂ D + (Σ 0 ).This discussion is summarised in the following propositions.Proposition 4 (Bäckdahl & Valiente-Kroon).Let (Σ 0 , h, χ) 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 Σ 0 and χ is the second fundamental form.If there exist a non-trivial spinor κ A * satisfying the twistor initial data equations (13a)-(13b) on Σ 0 , then the spacetime development of (Σ 0 , h, χ) will posses a twistor κ A in an open set U ⊂ D + (Σ 0 ).The twistor κ A is obtained by solving the twistor candidate equation (6) with initial data (κ A * , ∇ τ κ A * ) on Σ 0 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 ∇ A ′ Q κ Q = 0. To see whether imposing further conditions conditions on the initial data is enough so that ∇ A ′ Q κ Q = 0 on U ⊂ D + (Σ 0 ), it suffices to construct a propagation equation for the quantity ηA Commuting covariant derivatives and using the spinorial-Ricci identities, a calculation similar to that leading to equation ( 4) gives Hence, by providing trivial initial data for equation ( 14): we get ηA ′ = 0 on U ⊂ D + (Σ 0 ).Thus, by augmenting the requirements on the initial data implied by equation ( 8) to include those encoded in equation ( 15), one ensures that To translate the latter to intrinsic conditions on Σ 0 one needs to perform a 1+3 spacespinor split as follows.Let Then the conditions (15), are equivalent to impose ζ A = 0 and ∇ τ ζ A = 0.A direct calculation shows that Hence using the condition (13c) one gets Similarly, a direct calculation shows that Now, recall that the commutator [∇ τ , D AB ] acting on any spinor µ C reads Taking µ C = κ C and using equations (13c) and (17), a long but straightforward calculation renders Assuming vacuum and using condition (13a), simplifies the latter expression to Then, with these assumptions, the conditions ζ A = ∇ τ ζ A = 0 reduce to the requirement that D A B κ B = 0. Altogether the condition D (AB κ C) = 0 and D A B κ B = 0 can be encoded simply as Remark 1. Observe that κ A = 0 trivially solves the condition D AB κ C = 0. Hence, in the sequel we will be concerned only with non-trivial initial data, namely a spinor κ * A = 0 everywhere on Σ 0 that satisfies D AB κ * C = 0.This discussion is summarised in the following: Proposition 5. Let (Σ 0 , h, χ) be an initial data set for the vacuum Einstein field equations (without cosmological constant) where h is the 3-metric on a Cauchy hypersurface Σ 0 and χ is the second fundamental form.If the conditions are satisfied by a non-trivial spinor κ A * on Σ 0 , then a covariantly constant spinor κ A in some open set U ⊂ D + (Σ 0 ) is obtained by solving the twistor candidate equation (6) with initial data (κ A * , ∇ τ κ A * ) on Σ 0 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 (κ 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, κ A * = 0, this condition alone does not guarantee that κ A = 0 in the whole domain of dependence D + (Σ 0 ).However, if the solution κ A is a classical solution (C 2 ) of the wave equation ( 6), then by continuity, it follows that in a small spacetime neighbourhood of the initial hypersurface Σ 0 the solution is non-trivial: κ A = 0 in U ⊂ D + (Σ 0 ).Therefore, although we cannot control the size of the spacetime neighbourhood U where the solution is non-trivial, it is clear by continuity that, by shrinking the size of the set U ⊂ D + (Σ 0 ), proposition (5) implies, in conjuction with proposition 2, that the development of (Σ 0 , h, χ) will be a pp-wave spacetime on a small spacetime neighbourhood U close to the initial hypersurface.
The requirement on the regularity of κ A 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 [11] 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 [6].
Remark 3 (Regularity of initial data).Introduce some local coordinates x = (x µ ) = (τ, x i ) in U, with µ = 0, 1, 2, 3 and i = 1, 2, 3 so that Σ 0 is described by τ = 0. Denote the components of κ A as κ.Then, the wave equation ( 6) written in local coordinates reads: where F is linear in κ and ∂κ and g µν is a Lorentzian metric.We will consider non-trivial solutions κ * to the initial data equations (25) and (26) which satisfy the following regularity conditions.For m ≥ 4 : for some δ > 0 where With these assumptions, then using point (i) of Theorem 2 of [6] one concludes that there exists T > 0 and a unique solution to the Cauchy problem defined on [0, T ) × Σ 0 such that κ ∈ C m−2 ([0, T ) × Σ 0 , C 2 ).
Combining propositions 5 and 2, and aided with the discussion of Remarks 2 and 3 one obtains the following: Theorem 1.Let (Σ 0 , h, χ) be an initial data set for the vacuum Einstein field equations (without cosmological constant) where h is the 3-metric on a Cauchy hypersurface Σ 0 and χ is the second fundamental form.If the conditions are satisfied by a non-trivial spinor κ A * on Σ 0 satisfying the regularity conditions of Remark 3. Then there exist a open set U ⊂ D + (Σ 0 ) -a possibly very small spacetime neighbourhood of the initial hypersurface Σ 0 -for which the development of (Σ 0 , h, χ) on U is a pp-wave spacetime for some function H satisfying equation (2) in U.