## Abstract

This paper applies the notion of relative Cauchy–Riemann (CR) embeddings to study two related questions. First, it answers negatively the question posed by Penrose whether every shear-free null rotating congruence is analytic. Second, it proves that, given any shear-free null rotating congruence in Minkowski space, there exists a null electromagnetic field that is null with respect to the given congruence. In the course of answering these questions, we introduce some new techniques for studying null electromagnetic fields and shear-free congruences, in general, based on the notion of a relative CR embedding.

## 1. Introduction

A null geodesic congruence in (signature (1,3)) Minkowski space is a foliation of an open set in Minkowski space by oriented null geodesics. Such a foliation generates a tangent vector field *k*^{ a}. The congruence is said to be *shear free*, if there exists a complex null vector field *m*^{a} (called a *connecting vector* for the congruence) such that
The congruence is *rotating*, if in addition . Shear-free congruences were introduced in Robinson [1] and linked to the study of null solutions of the electromagnetic fields. They remain an important tool in the study of algebraically special solutions of Einstein's equations [2–5]. Trautman [6] gives a detailed history.

An electromagnetic field is a two-form *F* such that
where ⋆ is the duality operator. A non-zero field *F* is said to be *null* if
where *F*=*F*_{ab} d*x*^{a}∧d*x*^{b} and *F*_{ab}=−*F*_{ba}. One can show that a field *F* is null, if and only if the energy–momentum tensor splits as an outer product *T*_{ac}=*k*_{a}*k*_{c} for some non-zero null vector *k*^{ a}, the principal null direction of the field. If *F* is a null electromagnetic field, then *k*^{ a} is the tangent vector for some shear-free congruence (lemma 7.1). In that case, *F* is said to be *adapted to* *k*^{ a}.

This paper answers two distinct but related questions. First, it establishes a foundation for the theory of shear-free congruences of geodesics in Minkowski space. We prove that a given shear-free congruence corresponds to a certain CR submanifold *N* of the indefinite hyperquadric of (projective twistor space in this context; figure 1). Kerr, as cited in Penrose [7], has shown that the aforementioned result holds in the case of analytic congruences, where the shear-free congruence may be represented as a holomorphic surface in ; the CR manifold *N* is then the intersection of this holomorphic surface with the hyperquadric. In this paper, we drop the assumption of analyticity and work in the smooth setting and show that one still obtains this CR manifold *N* (theorem 5.2). Conversely, given a CR submanifold *N* of the hyperquadric (which obeys some genericity criteria) one obtains a shear-free congruence (theorem 5.3).

In the smooth category, there does not exist, in general, a holomorphic surface associated with *N* (theorem 6.1); however, we prove here that, if the Levi form of *N* is non-degenerate, then there is in twistor space a unique complex two-surface *Z* *with boundary* such that *N* is the boundary of *Z*. The key notion here is that of a *relative embedding* of CR manifolds, defined in §2. While, by a result of Jacobwitz & Trevès [8], not every three-dimensional strictly pseudo-convex CR manifold is embeddable, it is known that a solution of the vacuum Einstein equations equipped with a shear-free congruence defines a three-dimensional CR manifold that is (abstractly) two-sided embeddable [9]. However, §3 constructs a CR manifold that is a co-dimension two submanifold of the signature (2,2) real hyperquadric in that can only be *ambiently* realized as the boundary of a complex two-surface on one side. Such a relative embedding is termed an *essential one-sided embedding*.

It is significant that, in general, the CR manifold *N* need not be real analytic and yet corresponds to a shear-free *rotating* congruence. The construction of §3 exhibits such a non-analytic *N*, contrary to the observation in Penrose [7], p. 361: ‘These (non-analytic) exceptional cases appear to occur only when the rotation also vanishes’. Subsequent to Penrose [10], it seems unlikely that he would have retained this perspective. In fact, in Penrose & Rindler [11], the possibility of the existence of non-analytic rotating shear-free congruences is acknowledged (without providing a construction), and their hypothetical properties are discussed. In spite of this, the assumption of analyticity seems to have entered the lore of the subject, particularly in regards to the complex worldline approach of Newman and others [12]. Quite recently, non-analytic congruences have been studied in Baird & Eastwood [13], although the congruences there have zero rotation. Non-analytic congruences with zero rotation are readily constructed from the null rays emanating from a non-analytic worldline in real Minkowski space. It is substantially more difficult to construct non-analytic congruences with non-zero rotation.

Second, the paper applies these results to a class of solutions of the electromagnetic field equations, addressing the following question: given a shear-free rotating congruence *k*^{ a}, what is the space of solutions of the field equations which are adapted to *k*^{ a}? We prove that the space of solutions corresponds to the space of all holomorphic sections of the canonical bundle of the complex surface (with boundary) *Z* in a neighbourhood of the boundary, or alternatively to (2,0)-forms on the CR manifold *N* which are closed with respect to the tangential Cauchy–Riemann operator (lemma 8.2 and theorem 8.3). A special case of this result is the existence of local solutions to the electromagnetic field equations adapted to *k*^{ a}. This special case is discussed in Tafel [14] using rather different techniques. In Lewandowski and co-workers [15,16], special solutions of the Einstein equations are also algebraically found by similar methods.

## 2. One-sided embeddings of Cauchy–Riemann manifolds

We follow here standard definitions of CR manifolds and the tangential CR complex (cf. [17]). A CR manifold is a smooth manifold *M* of dimension 2*n*+1 together with a distribution of complex *n*-planes with complex conjugate bundle *T*^{0,1}*M* such that *T*^{1,0}*M*∩*T*^{0,1}*M*=0 and which is integrable in the sense of Frobenius (i.e. the Lie bracket of two local sections of *T*^{1,0}*M* is again a section of *T*^{1,0}*M*). If *M* is a CR manifold, then the annihilator of *T*^{1,0}*M* in is denoted by *Ω*^{0,1}*M* and its complex conjugate by *Ω*^{1,0}*M*. Because *T*^{1,0}*M* is integrable in the sense of Frobenius, *Ω*^{1,0}*M* and *Ω*^{0,1}*M* each generate differential ideals and , respectively, in .

The *tangential Cauchy–Riemann complex* is defined as follows in Hill & Nacinovich [17]. Let
Then, *Q*^{0,1}(*M*) is the dual space of *T*^{0,1}*M*. For , define to be the co-set of *df* modulo *Ω*^{1,0}(*M*). More generally, because , we have for all integers *k*≥0 (where , and is the ideal generated by the *k*-fold wedge products of sections of ). Upon passing to the quotient, *d* induces a mapping
Let *Q*^{p,j}(*M*) be the graded part of of degree (*p*+*j*). Thus
by definition. Note that
because *d* is a graded differential of degree +1. We write and call the tangential Cauchy–Riemann operator. The tangential Cauchy–Riemann (or just CR) complex is then the complex
where by convention Let *Q* be the bigraded module . This is indeed a complex, for is given as the composition *dd*=0 modulo a suitable ideal. When a distinction is needed in the action of on the *Q*^{p,j}, denote by the (graded) part of the co-boundary that maps .

The *Levi form* is the Hermitian form
defined on sections *v*,*w* by
Note that behaves as a tensor under change in basis sections, and therefore represents a well-defined bundle map. If only for the zero vector, then *M* is called strictly pseudo-convex.

If *M* and *N* are two CR manifolds, then a morphism of CR structures is a smooth function such that
A morphism is an *embedding* if it is a smooth embedding in the sense of differential topology.

The following definition is due to Hill [18]. (Manifolds with boundary are all smooth, and smooth up to the boundary, which is regarded as part of the manifold.)

### Definition 2.1

A complex manifold with abstract boundary is a real 2*n*-dimensional manifold *M* with boundary, together with an involutive distribution of complex *n*-planes on *M*, denoted *T*^{1,0}*M*, such that .

In definition 2.1, *T*^{1,0}*M* will be called the holomorphic structure of *M*. Let *Ω*^{0,1}*M* be the sheaf of annihilators of *T*^{1,0}*M*. A mapping between two complex manifolds with boundary is *holomorphic* if *f*_{*}*T*^{1,0}*M*⊂*T*^{1,0}*N*. If *M* is a manifold with boundary, let ∂*M* be the boundary of *M* and *M*^{o}=*M*−∂*M* be the interior of *M*. Note well that *M* is a complex manifold by the Newlander–Nirenberg theorem, and that the tangential part of *T*^{1,0}*M* along ∂*M* determines a CR structure *T*^{1,0}(∂*M*), so that the boundary of a complex manifold with boundary is a CR manifold.

### Definition 2.2

A complex manifold with concrete boundary is a complex manifold with abstract boundary which is locally isomorphic to some domain with boundary in .

Hill [18] proves the inequivalence of these two definitions. For the rest of this paper, ‘manifold with boundary’ shall mean ‘manifold with concrete boundary’.

### Definition 2.3

Let *N* be a CR manifold and *Y* a complex manifold with boundary. Let be an embedding of real manifolds with boundary such that *f*(*N*)⊂∂*Y* and *f**(*Ω*^{1,0}(∂*Y*))⊂*Ω*^{1,0}(*N*). Then, *f* is said to be a *one-sided embedding* of *N* in *Y* .

If *X* is a complex manifold, and *N* is a (real) hypersurface in *X* with the inclusion, then *f* is also a local one-sided embedding of *N* in the following sense. For each *p*∈*N*, there exists an open neighbourhood *U* of *p* in *X* such that *U*−*N* has two components, call them *U*^{+o} and *U*^{−o}. Let *U*^{+} and *U*^{−} denote the closures of *U*^{+o} and *U*^{−o} in *U*. Then, *U*^{+} and *U*^{−} are complex manifolds with boundary *N*∩*U*, and are CR embeddings.

### Definition 2.4

Let *Y* be an *n*+1-dimensional complex manifold, *Q* an embedded CR submanifold of *Y* of real dimension 2*n*+1, and *N* a CR manifold embedded in *Q*.

The triple (

*Y*,*Q*,*N*) is called a two-sided (resp. one-sided) relative embedding if there exists a complex submanifold (resp. complex submanifold with boundary)*Z*of*Y*such that*N*=*Z*∩*Q*. We call*Z*the two-sided (resp. one-sided) extension of*N*in*Y*.The triple (

*Y*,*Q*,*N*) is called an essential one-sided relative embedding if it is a one-sided relative embedding that cannot be extended to a two-sided relative embedding.

## 3. An essential one-sided embedding

Let denote the complex projective three-space, equipped with projective coordinates (*z*,*v*,*u*,*w*). The aim of this section is to prove the following.

### Theorem 3.1

*Let Q′ be the hyperquadric |z|*^{2}*+|v|*^{2}*−|u|*^{2}*−|w|*^{2}*=0 in* *. There exists an essential one-sided relative embedding of a strictly pseudo-convex CR manifold into Q′.*

For convenience, let us work on the affine subset of given by *v*≠0, and normalize coordinates so that *v*=1. Let *Q* be the locus of points (*u*,*w*,*z*) such that

The CR submanifold *N* of *Q* shall be given as a graph
where *g* is a function defined on the whole complex plane. We impose the following conditions on *g*:

The domain of analyticity of

*g*is the complex plane slit from 1 to along the positive real axis: .*g*(*u*) is continuous at*u*=1.*g*(*u*) vanishes to infinite order at*u*=1 in the sense that for every positive integer*k*. Furthermore, the restriction of*g*to is at*u*=1.*g*(*u*) does not vanish away from 1.on its domain.

An example of *g*(*u*) satisfying these requirements is the following:
along the branch of the logarithm for which . Then, *g*(*u*) is analytic at every point of the slitted plane and continuous at *u*=1. Furthermore, writing 1−*u*=*r* *e*^{iθ} with *r*≥0 and *θ*∈[−*π*,*π*), it follows that
so that evidently and moreover, because on −*π*≤*θ*<*π*,
whence .

Let *E* be the closed region in defined by 2|*u*|^{2}+|*w*|^{2}≤2. Let
and let *T*={(*u*,*w*)∈*E*:*t*(*u*,*w*)=0}. Let *N*′ be the graph of *z*=*wg*(*u*) over *T*; thus

### Lemma 3.2

*There is a neighbourhood U of* (*u*,*w*,*z*)=(1,0,0) *such that N′∩U is a smooth submanifold of Q containing the point* (*u*,*w*,*z*)=(1,0,0).

### Proof.

The fact that *N*′∩*U* contains the point (1,0,0) is immediately verified from the definition. Because *N*′ is a smooth graph over *T*, it suffices to show that *T* is a submanifold with boundary of *E* in a neighbourhood the point (*u*,*w*)=(1,0). On the boundary of *E*, with equality if and only if |*u*|=1 and *w*=0, by the conditions on *g* and by the definition of the region *E*. At the point *q*=(1,0), we have (by the conditions on *g* again): Hence, for any vector *v* at *q* pointing towards the interior of *E*, we have *i*_{v} d*t*(*q*)>0. The restriction of *t* to *E* is . By the Whitney extension theorem, choose a extension *h* of *t*|_{E} to all of . It follows by the implicit function theorem that the zero locus of *h* in a sufficiently small neighbourhood of *q* is a submanifold of that neighbourhood. Now d*h*=d*t* on *E*, and because *i*_{v} d*h*(*q*)>0 for interior-pointing vectors at *q*, and *t*≤0 on the boundary of *E*, it follows that *t*<0 for points outside *E* sufficiently near *q*. Hence, a portion of the zero locus of *h* in a sufficiently small neighbourhood of *q* lies entirely within *E*. Moreover, within that neighbourhood, it touches the boundary only at *q*. ■

Now let *N*=*N*′∩*U* as in lemma 3.2.

### Lemma 3.3

*The triple* *is an essential one-sided relative embedding*.

### Proof.

*T* *is strictly pseudo-convex*. The Levi form of *T* is represented by As , , so this is a negative definite form in a sufficiently small neighbourhood of (*u*,*w*)=(1,0), and so *T* is strictly pseudo-convex.

*The smooth embedding of T in Q given by j:(u,w)↦(u,w,wg(u)) is a CR embedding.* The CR structure

*T*

^{1,0}(

*T*) is generated by the vector field Note in particular that

*L*annihilates

*t*. Now, because the function

*j*is holomorphic, it suffices to check that

*j*

_{*}

*L*annihilates the defining relation for

*Q*, namely we need

*j*

_{*}

*Lρ*=0. However, this is precisely the statement that

*L*annihilates

*t*, because

*t*=

*ρ*°

*j*.

*The triple is a one-sided relative embedding.* Indeed, the embedding *j* of *E* into is holomorphic, and up to the boundary. Also, *N*=*j*(*E*)∩*Q*∩*U* (for some open set *U*). So *j*(*E*) is a one-sided holomorphic extension of *N* into the interior of *Q*.

*The one-sided relative embedding is essential.* Assume that *N* had a two-sided holomorphic extension *Z*, say. Because d*u* and d*w* are linearly independent on restriction to *N*, they remain linearly independent on *Z* locally in a neighbourhood of *p*. By the implicit function theorem, *Z* may be given locally as a graph *z*=*q*(*u*,*w*) with *q* holomorphic in a polydisc *Δ*_{r}((1,0)) for polyradius *r*=(*r*,*r*) sufficiently small. Consider the function *wg*(*u*)−*q*(*u*,*w*) in *Δ*_{r}((1,0)). This function vanishes identically on the hypersurface *T*∩*Δ*_{r}((1,0)). But *T* was proven to be pseudo-convex. Therefore, by the Lewy extension theorem [19], *wg*(*u*)−*q*(*u*,*w*) vanishes identically in *E*∩*Δ*_{r}((1,0)) (after possibly shrinking *r* if necessary). But, by the choice of *g*, for all integers *k* we have It follows that *q*(*u*,*w*) and all of its derivatives must vanish at (1,0), which implies that *q*(*u*,*w*) is identically zero in *Δ*_{r}((0,1)). Thus, *g* vanishes identically in a neighbourhood of *u*=1, which is contrary to the choice of *g*. ■

The proof of theorem 3.1 is now complete.

### Remark

The example constructed in this section fails to be two-sided embeddable *at one point only*. Ideally, one should be able to construct an example of a CR submanifold *N* of the hyperquadric which fails to be two-sided embeddable at *all* of its points, perhaps by using a method analogous to that of Lewy [20]. However, it is not known whether such a construction is possible.

## 4. The relative extension theorem

### Lemma 4.1

*Suppose that N is an abstract CR manifold of dimension* 2*k*−1. *Suppose also that there exists a CR embedding* . *Then, there exists locally a CR embedding* *and a holomorphic submersion* *such that Ψ*=*π*°*Φ*.

### Proof.

We work locally at a point of *Φ*(*N*), which we shall choose to be the origin of , by a suitable translation. By the rank theorem, it is possible to choose coordinates
such that *Φ*(*N*) may be written as a graph
(where *z*=*x*+*iy*) with *G* and *g* smooth functions defined on which vanish at the origin, together with their first partials. (These last two statements are furnished by taking a suitable affine change of the coordinates.)

Thus, we may define a hypersurface in , *N*_{0} say, by
Now, *N* may be written as a graph over *N*_{0} in the obvious way,
Define the mapping *π* by *π*(*ζ*,*z*,*w*)=(*z*,*w*). Then, *π*|_{Φ(N)} is the inverse of *Ψ*. But *π* itself is holomorphic, so that *Ψ* and *π*|_{Φ(N)} are CR isomorphisms onto their respective images. ■

Now suppose a given CR manifold *Q* is embedded as a hypersurface in with non-degenerate Levi form of signature (*k*,*l*), with *k*≥2, *l*≥1, in a neighbourhood of a point *x*_{0}∈*Q*. Suppose also that we have a (2*k*−1)-dimensional strictly pseudo-convex CR manifold *N* which is CR embedded in *Q*, with *x*_{0} lying in *N*. Then, lemma 4.2 holds:

### Lemma 4.2

*Near x*_{0}, *N has a one-sided extension into* . *The direction of the extension is the direction of the concavity of N*.

### Proof.

*N* is given as a CR submanifold of . As in lemma 4.1, we may choose a mapping *π* and an embedding *Ψ* such that *Ψ*=*π*°*Φ*. In particular, there exist holomorphic coordinate functions (*z*^{1},…,*z*^{n}) on and (*w*^{1},…,*w*^{k}) on with
Also, for *i*=*k*+1,…,*n*, the functions *z*^{i} can be pulled through to define functions on *Ψ*(*N*). These functions will be CR on *Ψ*(*N*), because the *z*^{i} are holomorphic. But, *Ψ*(*N*) is a strictly pseudo-convex CR submanifold of near *Ψ*(*x*_{0}). Hence, by the Lewy extension theorem, each *z*^{i}, for *i*=*k*+1,…,*n*, extends to a holomorphic function on the side of the concavity of *Ψ*(*N*), say . Now defines a holomorphic graph in over the concave side of *Ψ*(*N*), which gives the one-sided extension. ■

It should be remarked that the given local extension is *unique* (or at least its germ at a point of *N* is unique), for if two extensions were given, then the difference of their graphing functions must be zero on *Φ*(*N*), and thus identically zero everywhere by the Lewy extension theorem.

### Theorem 4.3

*N has a global one-sided extension. Furthermore, any two such extensions agree on another possibly smaller extension.*

### Proof.

*N* may be covered by a locally finite system {*U*_{α}} of neighbourhoods, such that each *U*_{α} admits a one-sided extension (by the lemma). Because of the uniqueness of local one-sided extensions, these local one-sided extensions must agree on the overlaps *U*_{α}∩*U*_{β}, and so patch together to give a global extension. ■

The same set of arguments, only using the two-sided Lewy extension theorem, prove the following.

### Theorem 4.4

*If N is Levi non-degenerate and indefinite (instead of pseudo-convex), then, near x*_{0}*, N has a two-sided extension into* *.*

## 5. Twistor space

The purpose of this section is to review some standard facts about twistor space (in the flat case), and to fix notation for later sections. More detailed accounts can be found in Penrose & Rindler [11] or Huggett & Tod [21].

Let be a four-dimensional complex vector space (called twistor space) equipped with the standard representation of . One of the basic constructions of twistor theory is to exploit the isomorphism of with , the simply connected cover of the conformal group in four dimensions in order to express data on Minkowski space in terms of data on twistor space and vice versa. Concretely, the structure on is realized by fixing a non-zero element . This, in turn, defines a complex bilinear form on via *q*(*X*,*Y*)=*ϵ*(*X*∧*Y*), thus realizing the isomorphism . In the projectivization, , the zero locus of *q* is none other than the Klein quadric, and is identified with the Grassmannian of two-planes in , . The metric associated to *q* is degenerate on *q*=0, but only in the complex scaling direction (along which *q* is preserved up to scale). So *q* defines a complex conformal structure on . The quadratic form *q* is conformally flat in any affine slice of the Klein quadric, so is locally conformally flat. In the setting of interest, is called conformal compactified complexified Minkowski space, and is denoted .

By definition of the quadratic form *q*, two points are null-related if and only if *X*∧*Y* =0. In particular, two points of are null-related if and only if the corresponding two-planes in share a line in common. This incidence relation allows to be recovered from as one of the two (topologically distinct) sets of completely null two-planes in : the α-planes are those whose annihilator at each point of is an anti-selfdual two-form on , and the β-planes are those annihilated by a selfdual two-form.

The tautological bundle of is the sub-bundle of whose fibre over a point is the two-plane in defined by *x*. This bundle is denoted by , and is called the primed co-spin bundle. Let *S* be the quotient bundle of by . This is the spin bundle. An essential fact is that the tangent bundle of is naturally identified with . Indeed, more generally if *Taut* denotes the tautological bundle for the Grassmannian and *Taut*^{op} is the quotient , then there is a natural isomorphism

Breaking the conformal invariance can be achieved by selecting a point . Then the metric, depending on away from the null cone of *I*, given by *g*_{X}=*q*(*dX*,*dX*)/*q*(*I*,*X*)^{2} is flat. The stabilizer of *I* is the Poincaré group plus dilations (the semidirect product ).

Real Minkowski space can also be described from the twistor point of view. Here, we introduce a (2,2) signature Hermitian form *h* on that is compatible with the chosen structure, giving a reduction of the structure group to *SU*(2,2). The idea is then to exploit the isomorphism of *SU*(2,2) with the spin group *spin*(2,4) associated to the conformal group of a real Lorentzian metric in four dimensions (via a 4 : 1 cover). To this end, let be the null cone of *h* in . The space of complex two-planes lying in , denoted , is compactified real Minkowski space. Then, is a four-dimensional real submanifold of . The induced metric is real and Lorentzian, if the point *I* is required to be on and hence real.

The process of associating the real Minkowski space to the real quadric is also reversible. In the real case, the real hypersurface is a CR manifold with Levi signature (1,1). It can be identified with the space of null geodesics in . The projective spheres associated to the null cones at each point in each carry a canonical complex structure. The CR structure on is the unique CR structure such that the image of each of these spheres under the null geodesic spray is a holomorphic curve. The rest of the section is devoted to establishing this fact.

### (a) Complex structure on the projective sphere

Let be an oriented four-dimensional real vector space. Suppose that is equipped with a time-oriented Lorentzian structure whose future null cone is the set . The sphere is the quotient of (minus the vertex) by the multiplicative group of positive dilations.

Let *H* be the tangent vector field to that generates the positive dilations, and let *H*^{⊥} be the orthogonal complement of the vector field *H* with respect to the Lorentzian structure on . The tangent bundle of the sphere is identified with sections of the quotient bundle that are homogeneous of degree zero. The exterior product with *H* defines an isomorphism of with the two-dimensional space *H*∧*H*^{⊥}. The duality operator ⋆ preserves this subspace, and satisfies ⋆^{2}=−*Id*. This gives the sphere its complex structure. In particular, splits into a direct sum of a pair of one-dimensional eigenspaces of eigenvalues *i* and −*i*, respectively.

It is convenient to have a section of in coordinates. Suppose that carries linear coordinates (*v*^{0},*v*^{1},*v*^{2},*v*^{3}) in terms of which the Lorentzian quadratic form is
5.1and the metric is therefore . Then, modulo *H* and up to an overall scaling, a section of has the form
5.2

### (b) The tangent bundle and null geodesic spray

Let be a four-dimensional manifold and its tangent bundle with the zero section deleted. Let be the projection map. Local coordinates *x*^{i} defined on induce fibre coordinates *v*^{i} on defined by writing a vector field *X* in terms of the partials ∂/∂*x*^{i}:*X*=*v*^{i}(*X*)(∂/∂*x*^{i}). The double tangent bundle is a bundle over with a local basis of 2*n* vector fields ∂/∂*x*^{i},∂/∂*v*^{i}. The vertical subspace of , denoted by , is the kernel of . This is locally spanned by the vector fields ∂/∂*v*^{i}.

There is a canonical diffeomorphism-invariant endomorphism of the double tangent bundle that is the generator of translations in the fibres of . It is given in these local coordinates by *λ*=d*x*^{i}⊗∂/∂*v*^{i}. Note that the image and kernel of *λ* are both . In particular *λ*^{2}=0. The endomorphism *λ* allows us to define differentiation up the vertical direction. To wit, if and *f* is a function on , then define . The other relevant diffeomorphism-invariant structure on is the Euler vector field *H*, which is the generator of scaling in the fibres. In local coordinates *H*=*v*^{i}∂/∂*v*^{i}.

Suppose now that is equipped with a Lorentzian metric *g*. This defines a quadratic form *G* on . The zero locus of *G*, denoted by , is the fibration of null cones. The one-form *α* defined by *α*(*X*)=*D*_{X}*G* is a symplectic potential on . The geodesic spray is the Hamiltonian vector field *V* defined by . Note that *λV* =*H*. The integral curves of *V* project to affinely parametrized geodesics on . Because *V* is everywhere tangent to , geodesics that start on must stay on . These are the null geodesics, and the restriction of *V* to is the null geodesic spray.

Because the integral curves of *V* are geodesics, in local coordinates we have
5.3where are the Christoffel symbols of the metric on .

### (c) Twistor space

Assume that is also oriented. Twistor space is the space of (unparametrized) null geodesics in , which is the quotient of by the two-parameter group of diffeomorphisms generated by the null geodesic spray *V* and Euler vector field *H*. Let be the complex line bundle of vertical vector fields on that gives the complex structure on each of its fibres. Let *M* be given by (5.2) in local coordinates. If is conformally flat, then the complex vector fields *M*,[*V*,*M*] are in involution with *V* and *H*, and therefore descend to a CR structure on the quotient. Involutivity follows at once by using a local coordinate chart in which the metric has the form (5.1), and applying (5.3) and (5.2).

### (d) The connection and horizontal vectors

The Levi–Civita connection associated to the metric *g* on allows us to lift any *C*^{1} curve in through a point to a *C*^{1} curve *γ*(*t*) through any point in the fibre over *x* for small time, by parallel translation of *X* along *γ*. This is the horizontal lift of the curve to the tangent bundle. The tangent vector to the lifted curve is a vector in , and is called a horizontal vector. A vector is horizontal if and only if it is in the kernel of the operator given by
Invariantly, . The space of horizontal vectors is a complementary subspace to in . The operator *P* has image and satisfies *P*^{2}=*P*, and so is the projection operator associated to this splitting. Note in particular that the dynamical vector field *V* is horizontal.

### (e) Umbral bundle

We describe here a vector bundle, known as the *umbral bundle* [22], on that is needed to develop the theory of shear-free congruences. The pullback bundle is isomorphic via *Id*−*P* to the bundle of horizontal vectors. The metric *g* on lifts to a metric in the pullback bundle, and so defines a metric on horizontal vectors denoted by . Along , *V* is a null vector with respect to , and so *V* lies in its own orthogonal complement . Define the umbral bundle to be the vector bundle *E*=*V* ^{⊥}/*span*(*V*). The bundle inherits an orientation from . The metric and orientation define a duality operator on which preserves the subspace *V* ∧*E*. Select a horizontal complex vector field *M*′ in such that ⋆(*V* ∧*M*′)=−*i*(*V* ∧*M*′). This vector field is defined uniquely up to an overall factor, and modulo multiples of *V* .

### Lemma 5.1

*M*′≡0 (mod *V*,[*V*,*M*]).

### Proof.

Modulo vertical directions, the identity *M*′=−[*V*,*λM*′] holds. However, *λM*′ is a vertical vector such that *H*∧*λM*′ is anti-selfdual, and therefore *λM*′ is a representative of a section of . Thus, *λM*′=*aH*+*bM* for some scalars *a* and *b*. Taking Lie brackets with *V* , using the fact that [*H*,*V* ]=*V* , and projecting out the vertical directions establishes the lemma. ■

### (f) Null geodesic congruences

A null geodesic congruence is a non-vanishing null geodesic vector field *k* throughout a region of space–time. Equivalently, this is a smooth (local) section of the null cone bundle over whose image is foliated by the null geodesic spray. The pullback of the umbral bundle associated along *s*_{k} is the bundle of two-planes *k*^{⊥}/*span*(*k*) consisting of vectors orthogonal to *k*, modulo multiples of *k* itself [22]. Note that the metric descends to a negative definite form in *k*^{⊥}/*span*(*k*).

The umbral bundle associated to *s*_{k} carries a natural complex structure. Wedging with *k* gives a natural linear isomorphism *k*^{⊥}/*k*≅*k*∧*k*^{⊥}. The duality operator preserves *k*∧*k*^{⊥}, because the orthogonal complement of *k*∧*k*^{⊥} is and this also annihilates *k*∧*k*^{⊥} under the wedge product. Because ⋆^{2}=−1, the eigenspaces of ⋆ define a complex structure on *k*∧*k*^{⊥} and therefore also on *k*^{⊥}/*k*. Let *m* be a null section of such that the co-set of *m* modulo *k* is an element of the +*i* eigenspace of the star operator, scaled so that . Now, there exists a unique null vector field *n* such that *g*(*m*,*n*)=0, *g*(*k*,*n*)=1. The quadruple of null vectors is a *null tetrad*.

The distortion tensor ∇_{a}*k*_{b} descends to a bilinear form on *k*^{⊥}/*k*. The trace-free symmetric part is called the *shear* of the congruence. The skew part is called the *rotation*.

A shear-free congruence *k* is called *regular* in an open set if the image *N* of *U* in under the natural projection along *k* is a three-dimensional smooth submanifold, and the natural projection is a submersion of *U* onto *N*. The following theorem is well known in the analytic category, going back at least to the work of Sommers [5]. See, for instance, references [9,11,23].

### Theorem 5.2

*A regular shear-free null geodesic congruence in an open set* *defines an embedded CR submanifold N of* *. The Levi form of this CR manifold is non-degenerate if and only if the rotation of the congruence is non-zero.*

### Proof.

Complete the vector *k* to a null tetrad basis such that the only non-vanishing inner products are and with *k*∧*m*, and all anti-selfdual and , *n*∧*m* and all selfdual. In terms of the basis, the shear and rotation are the scalars
(Here, the notation *k*^{♭}:=*g*(*k*,−) means that the metric has been used to lower the index of the vector field *k*.)

The vector field *m* descends up to scale to the three-dimensional space of geodesics that form the congruence if and only if [*k*,*m*]∧*k*∧*m*=0. Write . Because *k* is a null geodesic and *m* is orthogonal to *k*, it follows that *g*([*k*,*m*],*k*)=0. Hence, . Thus, *k* is shear free if and only if *m* descends to a complex direction field on the quotient space. This generates a CR structure. The Levi form of the CR structure is given by . Writing in the tetrad, the Levi form is , the rotation of the congruence.

It remains to show that the CR structure of a shear-free congruence is compatible with the CR structure on twistor space. The null vector field *k* defines a section of the null cone bundle. The null geodesic spray is tangent to this section, because *k* is a geodesic, and *s*_{*}*k*=*V* . The vector *s*_{*}*m* is in *V* ^{⊥}, and satisfies ⋆(*V* ∧*s*_{*}*m*)=−*i*(*V* ∧*s*_{*}*m*). Because this property also characterizes *M*′, . It is therefore sufficient to show that *Ps*_{*}*m*≡0 (mod *H*,*M*). It is shown in Holland & Sparling [22] that *Ps*_{*}*m*=*λs*_{*}(*m*^{a}∇_{a}*k*). This is a linear combination of *H*=*λs*_{*}*k* and *M*=*λs*_{*}*m* if and only if *m*^{a}∇_{a}*k* is a linear combination of *k* and *m*, if and only if *k* is shear free. ■

The converse is also true in the rotating case. Let be the quotient by the null geodesic spray, and be the natural projection onto Minkowksi space. The null geodesic in associated to each point *L*∈*N* is the subset .

### Theorem 5.3

*Let N be a pseudo-convex CR submanifold of* *. For each L∈N and each point* *there is an open neighbourhood* *of π*_{2}*(p) and* *of L such that the null geodesics in V ∩N define a regular rotating shear-free congruence in U.*

### Proof.

It is sufficient to prove that the restriction of *π*_{2} to is a submersion, as the statement of the theorem then follows by an application of the inverse function theorem. Let *k* be the restriction of the null geodesic spray *V* to the fibres of *π*_{1} over *N* and let *X* be a complex vector field in a neighbourhood of *p* that commutes with *k* such that (*π*_{1})_{*}*X* is a non-vanishing local section of *T*^{1,0}*N*. We claim that are linearly independent.

Choose a complex vector field *M* on that is a representative of a non-vanishing section of the line bundle , and let *M*′ be a section of the umbral bundle *E*. These can each be chosen so that . Because the Levi form of is non-degenerate, the six vector fields form a basis of the tangent space of , and, in particular, the last four of these must project down to a basis of the tangent space of because the first two are in the kernel of this projection. Because *N* is CR embedded into , and *M* and *M*′ descend to a basis of , *X* has the form *X*=*αM*+*βM*′. Because *X* commutes with *k*, *β* cannot vanish, and so (*π*_{2})_{*}*X* is a non-zero multiple of (*π*_{2})_{*}*M*′. Likewise, is a non-zero multiple of . Finally, because the Levi form of *N* is non-degenerate, is independent of , and so has a non-trivial component in the direction of . ■

## 6. Non-analytic congruences

In this section, we prove the main result of this work:

### Theorem 6.1

*There exist non-analytic congruences of null geodesics whose shear vanishes, and whose rotation is non-zero.*

We need the following fact (attributed to Kerr in Penrose [7]) about the relationship between the CR submanifold and the structure of the congruence:

### Lemma 6.2

*An embedded CR submanifold* *is the intersection of* *with a complex-analytic submanifold of* *if and only if the associated congruence is a real-analytic shear-free congruence of null geodesics*.

### Proof.

A real-analytic congruence is by definition a null geodesic whose tangent vector field *k* is a real-analytic function of the coordinates of Minkowski space in some open set *U* in the real Minkowski space. We may thus extend *k* to a complex-analytic null vector field on a neighbourhood of *U* in the complexified Minkowski space. When *k* is descended to the leaves of the twistor foliation, we obtain a complex-analytic submanifold of , as required. The converse follows by reversing the line of argument. ■

### Proof of theorem 6.1.

By theorem 3.1, there exists an essentially one-sided embeddable CR manifold with non-degenerate Levi form. Such a manifold is not the intersection of with a complex analytic submanifold. Hence, by lemma 6.2, the shear-free congruence associated to *N* is non-analytic, and by theorem 5.2 the rotation of the associated congruence is non-vanishing. ■

## 7. Null electromagnetic fields

A two-form *F* on an open set in is a source-free (or homogeneous) electromagnetic field if the following field equations hold:
where ⋆ is the duality operation associated to the metric. If we let *G*=*F*+*i*⋆*F*, then the equations assume the simple form *dG*=0, and because *F* is real, *G* determines *F* completely. In this language, solutions of the homogeneous electromagnetic field equations correspond to complex two-forms *G* that obey *dG*=0 and which are anti-selfdual (⋆*G*=−*iG*). We concentrate on the case where the field tensor is null, i.e.
or in terms of *G*, A field *G* satisfying this equation is also called *null*. Thus, *G* is null if and only if *F* is null.

### Lemma 7.1

*Let G be a non-zero complex anti-selfdual null two-form. Then, G*=*k*^{♭}∧*m*^{♭}, *where k and m are mutually orthogonal null vectors with k real. If, in addition, G is closed, then k is tangent to a shear-free null geodesic congruence.*

A field *G* of the form *G*=*k*^{♭}∧*m*^{♭}, as in the lemma, is called *adapted to k*. Thus,

*G*is adapted to

*k*if and only if the associated Maxwell field

*F*is itself null and adapted to

*k*.

### Proof.

Because *G* is anti-selfdual, implies that *G*∧*G*=0, so *G* splits as an exterior product *G*=*u*^{♭}∧*v*^{♭} for some (complex) linearly independent vectors *u* and *v*. Now, *u*^{♭}∧*G*=0, so again by self-duality , and so *u* is null and orthogonal to *v*. Likewise, *v* must also be null. Next, we claim that at any point where *G* is not zero there is a real vector *k*, necessarily null, in the linear span of *u* and *v*. Because *G* is anti-selfdual, is anti-selfdual, so , whence the claim.

Now suppose that *dG*=0. It follows that . Using the decomposition *G*=*k*^{♭}∧*m*^{♭}, this implies , or . But, because is real, this implies that for some *α*. Now because *k* is null, . So *k*^{ a}∇_{a}*k*^{♭}=*αk*^{♭}. The constant *α* can be absorbed by rescaling *k* (and so also rescaling *m*), so that *k*^{ a}∇_{a}*k*^{♭}=0, and thus *k* is a null geodesic. Finally,
Applying duality gives 0=〈*k*,*m*^{a}∇_{a}*m*〉=*σ*/2. ■

## 8. Local existence

The classical theorem on local existence of solutions of the null field equations for flat space–time and *analytic* congruences is due to Robinson [1] and Sommers [5]:

### Theorem 8.1

*Let k be a vector field tangent to an analytic shear-free congruence of null geodesics. Then, there exists locally a solution of the electromagnetic field equations F that is adapted to k. The freedom in the solution is described by a holomorphic function of two complex variables.*

We may drop the assumption of analyticity of the congruence, provided that we assume that the congruence is rotating throughout its domain. Let *k* be a shear-free congruence. Assume that the domain of *k* is restricted to an open region such that the CR manifold and quotient mapping *π* from the domain of the congruence to *N* are smooth, as in §5*f*.

### Lemma 8.2

*Let k be a shear-free congruence. Let G be a anti-selfdual non-vanishing two-form. The following conditions are equivalent:*

*There is a*-*closed (2,0) form H on N such that G*=*π***H*.*G is a null solution of the electromagnetic field equation and is adapted to k*.

### Proof.

Locally, *N* is the quotient of an open set *U* in by the leaves of the foliation defined by *k* via a submersion *ϕ*:*U*→*N*.

Suppose that *H* is a given -closed (2,0) form on *N*, and let *G*=*ϕ***H*. If *H* is closed, then so is *G*. Recall the definition of where
The operator is the induced mapping of *d* on the indicated quotient spaces. However, in the above expression for *Q*^{2,1}, the space *Ω*^{1,0} is two dimensional, and so its triple exterior product vanishes. Thus
Hence, *dH*=0. So it follows that *G*=*ϕ***H* is closed. Now *k*^{♭} and *m*^{♭} descend to a basis of *Ω*^{1,0}, so *G* must have the form *μk*^{♭}∧*m*^{♭}. So condition 1 implies condition 2.

Conversely, if *G*=*μk*^{♭}∧*m*^{♭} is a closed form, then , and therefore *G* descends to a closed (2,0)-form *H* on *N*. ■

Assume now that *k* is rotating. Then, *N* is strictly pseudo-convex, and so there is a one-sided holomorphic extension *Z*. Now consider the sheaf of -closed (2,0)-forms on *Z* which are up to the boundary *N*. Any section of this sheaf in a neighbourhood of a point on *N* will induce a local solution of the electromagnetic field equations on *N*. Because *Z* can be locally embedded into , any local holomorphic section of the canonical line bundle of will induce a section of the aforementioned sheaf by restriction, and thus a local solution of the electromagnetic field equations which is null with respect to *k*^{ a}.

If now *G*_{1} and *G*_{2} are two non-zero null anti-selfdual electromagnetic fields adapted to *k*, then the associated (2,0)-forms *H*_{1} and *H*_{2} on *N* satisfy *H*_{2}=*fH*_{1} for some CR function *f* on *N*. By Lewy's extension theorem, *f* can be extended on one side to a holomorphic function on the complex surface *Z* whose boundary is *N*.

To summarize, we have:

### Theorem 8.3

*If k is a shear-free rotating congruence, then there exists locally a solution of the electromagnetic field equations which is null with respect to k. The freedom in the solution is described by the boundary value of a holomorphic function of two variables. A global solution exists if and only if there exists a global* *-closed (2,0)-form on the CR manifold N associated with the congruence.*

## Acknowledgements

The authors gratefully acknowledge the many helpful improvements suggested by the referees.

- Received October 4, 2012.
- Accepted December 19, 2012.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.