K\"ahler metrics via Lorentzian Geometry in dimension four

Given a semi-Riemannian $4$-manifold $(M,g)$ with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of K\"ahler metrics $g_K$ is constructed, defined on an open set in $M$, which coincides with $M$ in many typical examples. Under certain conditions $g$ and $g_K$ share various properties, such as a Killing vector field or a vector field with a geodesic flow. In some cases the K\"ahler metrics are complete. The Ricci and scalar curvatures of $g_K$ are computed under certain assumptions in terms of data associated to $g$. Many examples are described, including classical spacetimes in warped products, for instance de Sitter spacetime, as well as gravitational plane waves, metrics of Petrov type $D$ such as Kerr and NUT metrics, and metrics for which $g_K$ is an SKR metric. For the latter an inverse ansatz is described, constructing $g$ from the SKR metric.


Introduction
In one of his open problem collections, S. T. Yau concludes a problem with the question "Can one go from complete Kähler manifolds back to physically interesting spacetimes?" ( [33,Problem 89]). This study makes a contribution mostly in the opposite direction, by constructing Kähler metrics from certain classes of Lorentzian 4-manifolds, equipped with associated data.
To be sure, Yau's question is written in the context of the effort to make sense of the transformation between metrics known as Wick rotation. In contrast, the construction method presented in this study, which applies in arbitrary signature, is different, and more involved, than Wick rotation. But it is invariantly defined, and its various variants can be carried out on a variety of classical spacetimes, such as de Sitter, Kerr and gravitational plane waves. For one class of Kähler metrics, which includes the extremal metric conformal to the Page metric, we give a kind of inverse construction: the Kähler metric is the input data for an ansatz producing Lorentzian 4-manifolds, for which the construction method recovers the Kähler metric on an open dense set.
It is well-known that a Lorentzian metric is never compatible with any given almost complex structure. Thus relating notions of Lorentzian and complex geometry requires some sophistication. The long history of attempts to achieve this dates back at least to the 1960's, and one of its most well-known outcomes is the invention of twistor theory. Connections relating Lorentzian geometry specifically to Kähler geometry have also been made, some focusing on analogous structures in the two geometries, partly based on considerations from spin geometry [17,4,24]. Our construction is given via more direct differential geometric data, but some relations do hold with those earlier works. This is particularly true of Flaherty's manuscript [15], and so we devote, aside from further remarks here, several introductory comments in Section 11, to a description of the relations between our work and his.
In more detail, given an oriented four-manifold with a semi-Riemannian metric g and two distinguished vector fields k, t satisfying certain properties, we construct an exact symplectic form and an integrable almost complex structure. On some open set these two are compatible and so yield a Kähler metric g K (see The symplectic form in question belongs, in fact, to a family, parameterized by real valued functions on a domain in R. We will consider this parameter function fixed in the following discussion. Such families form a generalization of the symplectic forms defined by the first author in [1], which constituted the original motivation for this study. As noted there, the main condition guaranteeing the nondegeneracy of such a symplectic form is that one of the two vector fields has a nowhere vanishing twist operator. The twist operator is a so-called optical invariant, first introduced in relativity theory [28]. Note that our notion of an optical invariant is somewhat more general than the standard versions (see subsection 2.1).
The almost complex structure is defined directly by specifying how it intertwines the two vector fields, and requiring it to be compatible with the restriction of g to the orthogonal complement H of the pointwise span V of k and t, and to respect the orientation. Integrability is guaranteed when a "horizontal" and a "vertical" condition hold, where these terms refer essentially to the projection of the Nijenhaus tensor on H and V, respectively (see Theorem 1).
The horizontal condition is given in terms of the shear operators of both k and t, shear being another optical invariant. A special case occurs when both shear operators vanish, and this is called the shear-free case. In that case there is a second "dual" almost complex structure which is also integrable, and the Kähler metric is in fact ambihermitian, in the sense of [3].
In geometric settings pertaining to a single vector field, the shear-free condition has been much studied in the literature, including the determination of its relation to integrability of almost complex structures [7,6]. It is thus of special interest that the horizontal condition is in fact more general, and in Section 12 we give an example of a Lie group with a Lorentzian metric that admits a Kähler metric in accordance with our construction, with neither shear operator vanishing. Note that subsection 3.5 contains a few pointers on how the relevant theorem in [7] can be viewed from the perspective of this paper.
The vertical condition is given in terms of Lie bracket relations involving the two vector fields. Some of these relations hold in special cases: when the vector fields have a geodesic or a pre-geodesic flow and are of constant length, or when they are Killing. In order to keep track of these three possibilities, we define three similar notions of an admissible Lorentzian metric, each giving rise to an integrable almost complex structure (Definition 3.6). Interestingly, under certain conditions the Kähler metrics we construct share many of the properties of the admissible Lorentzian metrics they are constructed from (subsection 5.1). The underlying reason for this is, roughly, that the metric-independent condition of the vanishing of the Nijenhaus tensor singles out a class of semi-Riemannian metrics to which both the Lorentzian and the Kähler metrics belong.
We also show that an enhanced version of our vertical condition, together with the horizontal one, are necessary and sufficient for integrability in the case where the almost complex structure is self-adjoint with respect to the restriction of the metric to V (Theorem 2), so that together with its hermitian character on H it may be called split-adjoint, a condition that we regard as an alternative to the standard hermitian requirement appearing in complex Riemannian geometry, suitable especially for Lorentzian metrics. Formula (VIII.5) in [15] contains a Newman-Penrose version of the conditions of Theorem 2, suitable for the special type of frame called a null tetrad. Our version applies to cases where the vector fields k, t are not part of a null tetrad. One of these cases is described in Remark 3.8 and pertains to the Lorentzian metrics associated to SKR metrics.
The relation between the curvatures of g and g K is not simple. For instance, even with the simplifying assumptions that k and t are g-orthogonal and have opposite g-magnitudes, the two metrics are related by a biconformal change composed on a Wick rotation. Biconformal changes have been considered in studies related to harmonic morphisms. For example, [10] includes formulas for the relation between the Levi-Civita connections of the two metrics, though not general formulas relating their curvatures. The latter are, however, cumbersome to work with, and we chose not to employ them in this work. Instead, under certain assumptions, most crucially that the manifold is contained in the total space of a holomorphic line bundle, we give formulas for the Ricci and scalar curvatures of g K in terms of data associated with g, k and t (Section 6). For the type of admissible manifold where our vector fields have geodesic flow, we also relate the assumption of completeness of g K with the g-geodesic completeness of the integral curves of k and t. Conversely, results from [8] also imply that unboundedness of the magnitude of one of these vector fields implies incompleteness of g K .
After discussing various mechanisms where one varies an admissible Lorentzian metric and still produces the same Kähler metric g K (Section 7), we begin our discussion of examples with the above-mentioned case where we produce, via an ansatz, an admissible Lorentzian metric from a given Kähler one. The Kähler metrics for which this construction holds are called SKR, and were first introduced in [12], in the context of the classification of conformally-Einstein Kähler metrics (Section 8). Examples of SKR metrics on compact manifolds exist, and, as mentioned above, for those the ansatz produces an admissible Lorentzian metric on an appropriate open and dense set.
In our next class of examples we give a general construction of two types of admissible Lorentzian metrics, given as a warped product with a one-dimensional base, where the fiber is any 3-manifold possessing a geodesic vector field with certain prescribed optical invariants. A number of examples of this construction are then given, one of which is de Sitter spacetime, and another derived from a fairly general pp-wave metric in dimension four (Section 9). We then describe a non-warped example where the admissible Lorentzian metric is a gravitational plane wave (Section 10).
For metrics of Petrov type D, we give in Section 11 three examples in which the theory is implemented and produces Kähler metrics: the Kerr metric, a metric conformal to the Kerr metric and a class of NUT metrics. Only the second of these fits one of the above-mentioned admissible categories. Thus for the first and third cases we need a different variant of our construction of an associated Kähler metric. The NUT metric we study is the only one which is both globally hyperbolic and has an associated Kähler metric g K defined on the whole manifold. The complex structure for the NUT metric was first described in [15]. Dixon [14] has recently studied another Kähler metric Wickrotated from the Kerr metric on a domain in Kerr spacetime, and showed it is ambitoric (see also [2]).
In Table 1 Table 1. Examples and their main properties. The "warped from pp-wave" refers to a warped product R × N , where the 3-manifold N has a metric constructed via a pp-wave metric in dimension four (see Section 9.4). The Kerr and conformally Kerr examples refer to the rapidly rotating version of this metric. The complex structure of the NUT example already appears in [15].
1.1. Acknowledgements. We thank Akito Futaki for the suggestion to phrase our findings in the general semi-Riemannian context. We also acknoweldge and thank Vestislav Apostolov for reading and commenting on an earlier version of this work, especially with regard to matters related to ambihermitian metrics.

Shear and twist
In this preliminary section we introduce variants of the notions of shear and twist, the optical invariants that will have a significant role in what follows. After describing their expressions in appropriate frames, we compare our version to the more standard one for null vector fields with a geodesic or pre-geodesic flow. We then describe a few known application valid especially for 3-manifolds, which will be needed in Section 9.
2.1. Relative versions of shear and twist. Our notions of shear and twist will differ somewhat from their common usage in the Physics literature, and also from mathematical references such as [7]. The need for these atypical definitions arises, in small part, from their application to vector fields on semi-Riemannian manifolds which may not be null, or even of constant length. But more importantly, the difference is attributed to the fact that we consider two distinguished vector fields, rather than just one. Thus our shear and twist will be defined not with respect to the orthogonal complement of a single vector field, but relative to a decomposition of the tangent bundle into an orthogonal direct sum of distributions, one of which contains these two vector fields. We therefore sometimes employ the terms relative shear and twist, when emphasizing this decomposition. Specifically, for a semi-Riemannian manifold M , let be a orthogonal decomposition of the tangent bundle into two (necessarily nondegenerate) distributions V, H. Let π : T M → H be the projection relative to this decomposition. Then for any nowhere-vanishing vector field X with values in V, the (relative) shear operator and (relative) twist operator of X are defined, respectively, as the H → H operators given by where ∇X refers to the linear operator v → ∇ v X on the tangent bundle, with ∇ the Levi-Civita covariant derivative on M . If the shear operator vanishes, we will say X is shear-free, and similarly for the twist operator. In all our applications the rank of H will be two. In this case the following related entity is real-valued, and will play an important role. The twist function is |ι| = |ι X | := 2 det(∇ s X). The reason for the notation |ι| will be made clear in the next subsection. In our most common application, the manifold will be Lorentzian of dimension four, and admit an almost complex structure. Then V will be the complex span of some vector field X. As mentioned above, to ensure that (1) holds, one requires that V be non-degenerate at each point. In the Lorenztian case, we will in fact always choose it to be timelike.

Frame representation.
A fact that will be of great value in our applications is that, as we will see below, the projection π does not appear explicitly in the matrix representation of the shear and twist operators with respect to a local orthonormal frame for the (rank two) distribution H. In fact, in the setting of the previous subsection, assume H has rank two, and let x , y be an ordered orthonormal frame for H. Then at each point p, the matrix of ∇X H with respect to {x p , y p } is given by Thus the shear operator is given, dropping p from the notation, as where the entries are the shear coefficients Note of course, that these coefficients are frame-dependent, while σ 2 1 + σ 2 2 is just the frame-independent quantity − det ∇ o X. Another way to give the shear coefficients is in terms of Lie bracket relations. Let d X denote differentiation in the direction of X. As g(∇ X x , x ) = d X (g(x , x ))/2 = 0, and similarly for y replacing x , and g(∇ X x , y ) + g(x , ∇ X y) = d X (g(x , y )) = 0, we can also write the shear coefficients as The twist operator, on the other hand, is given as where ι is given below, and its absolute value is the twist function: Similarly to the shear coefficients, ι 2 is invariant, being the smooth function 4 det ∇ s X, whereas |ι| may be only continuous at the zeros of ι. Note that relations (5) have interesting consequences. For example, if X is a vertical field with respect to a submersion, while H is its horizontal distribution, then X is shear-free, because the bracket of a vertical vector field with a horizontal one is vertical.
2.3. Null pre/geodesic vector fields. To compare our versions of twist and shear with more standard notions (see, e.g., [31] and [26,Chapter 5]), recall that a vector field k on a semi-Riemannian manifold (M, g) is null if k is nowhere vanishing and g(k, k) = 0. The fact that k is null, or more generally its having a constant length 1 , implies that the ∇k is well-defined as an operator k ⊥ −→ k ⊥ , where k ⊥ is the orthogonal complement of k. The analog of the operators on H from previous sections is here an operator on the quotient bundle: where the "division by k" refers to taking a quotient by the span of k at each point, and [v] denotes the equivalence class containing v. But there is a caveat: in order for this quotient bundle endomorphism to be well-defined, k must have pre/geodesic flow, or equivalently ∇ k k = αk for some smooth function α on M .
, so that w = v + ak for some constant a, then which equals D ([v]) if and only if ∇ k k is proportional to k. Let us henceforth assume this is the case. Observe that the Lorentzian metric g descends to a Riemannian metricḡ on The shear and twist operators of k are now defined as before, as the trace-free symmetric or antisymmetric parts, respectively, of the operator D.
The relation between these definitions and the relative ones, is that H is the image of a chosen embedding of k ⊥ /k into k ⊥ , which yields an isometry (k ⊥ /k,ḡ) → (H, g H ). In this non-relative case, the optical invariants are independent of the choice of H; whereas they do, in general, vary with H in the relative case of subsection 2.1.
Remark 2.1. Throughout the paper, if equation (7) holds, we will call k itself pre/geodesic; if α is not identically zero we will call it pre-geodesic, and geodesic if ∇ k k = 0.
2.4. The twist and shear on a Riemannian 3-manifold. Let (N, g) be a Riemannian 3-manifold, k a vector field on N of unit length. We then have an orthognonal decomposition of T N into the pointwise span of k and its orthogonal complement k ⊥ . In a similar manner to the case of a Lorentzian null geodesic vector field, the endomorphism is well defined because k has constant length. Geometrically, it may be seen as a second order approximation, via the "screen" k ⊥ p at a point p, of the flow of k. The names shear and twist represent the distorsion effects of these respective operators on a typical disk in the screen centered at p. In terms of an orthonormal frame {x p , y p } of k ⊥ p , the shear and twist of k are still given by formulas (4), (5) and (6). The matrix of D is given, by means of the divergence δk of k, the twist function (up to sign) ι, and the shear coefficients σ 1 , and σ 2 , as follows Note that, because k has constant length, its divergence δk is the trace of D: 2.5. Preliminary applications in dimensions 3 and 4. For a Riemannian 3-manifold (N, g) with k, x , y as in the previous section, Frobenius' theorem implies that ι = ι k , given as in (6) vanishes identically if and only if the orthogonal complement k ⊥ of k is integrable. We will be interested in the diametrically opposed situation where the frame independent twist function |ι| is nowhere vanishing, so that k ⊥ is nowhere integrable: at any p ∈ M , there is no embedded submanifold S containing p such that T q S = k ⊥ q for all q ∈ S. In this circumstance we sometimes say that the flow of k is everywhere twisting. We record this analysis together with a related result, proven in [19].
Lemma 2.2. If a unit length vector field k on a Riemannian 3-manifold is complete, has geodesic flow, and Ric(k, k) > 0, then its orthogonal complement k ⊥ is nowhere integrable. The latter occurs if and only if k is everywhere twisting.
Next, recall that the vanishing of the shear operator or the divergence of a vector field k are often described by the statement that k is shear-free or divergencefree, respectively. We record the following well-known lemma, omitting its standard proof. Lemma 2.3. A unit length vector field on a Riemannian 3-manifold is a Killing vector field if and only if it is geodesic, divergence-free, and shear-free.
The last two propositions will be applied only in Section 9. Returning to the setting of a null vector field k with pre/geodesic flow on a Lorentzian 4-manifold (M, g), analogous results hold. First, integrability of k ⊥ is still equivalent to the vanishing of the twist function of k. Indeed, letting {k, x , y } denote, as before, a local frame of the orthogonal complement k ⊥ , with x , y being two orthonormal vector fields orthogonal to k, it follows by Frobenius' theorem that k ⊥ is integrable if and only if However, as k is null and pre/geodesic, g(k, [k, x ]) = g(k, [k, y ]) = 0, as can be seen by writing the Lie brackets using the (torsion-free) Levi-Civita connection, and applying its compatibility with the metric. Therefore, just as in the threedimensional Riemannian setting, on a Lorentzian 4-manifold the integrability of k ⊥ of a null vector field k with pre-geodesic flow is completely determined by the vanishing of the twist function |ι| = |g(k, [x , y ])|. Moreover, an identical result to Lemma 2.2 holds.
Lemma 2.4. If a null vector field k with pre/geodesic flow on a Lorentzian 4manifold is complete, has geodesic flow, and Ric(k, k) > 0, then k ⊥ is nowhere integrable. The latter occurs if and only if k is everywhere twisting.
One proof of this, which can be found in [1], employs the Bochner-type formulas obtained via the Newman-Penrose formalism (cf. [26])

Almost complex structures and integrability
In this section we introduce a key component of this work, namely an almost complex structure attached to a semi-Riemannian 4-manifold equipped with certain data involving two vector fields. We investigate its integrability and related properties. We also introduce three closely related notions of an admissible manifold. For all of them, the almost complex structure will be integrable. They differ from each other in the characteristics of one of the distinguished vector fields on the manifold.
3.1. Definition of J. Let (M, g) be an oriented semi-Riemannian 4-manifold, with two vector fields k + , k − . Let V := span(k + , k − ) denote the distribution spanned at each point p by k + | p , k − | p . We assume that and H := span(k + , k − ) ⊥ is spacelike.
Remark 3.1. Condition (10) is equivalent to having both nondegeneracy of the restriction g V of g to V at each point, and linear independence of k ± everywhere. In particular k ± then have no zeros. Condition (11) means that g H is positive definite at each point, in particular it is pointwise nondegenerate, which also implies that g V is pointwise non-degenerate. Thus (10) and (11) are equivalent to linear independence of k ± and (11). Finally, if g is Lorentzian, (11) is equivalent to i.e. g V has Lorentzian signature at each point. In that case (12) implies We consider an almost complex structure 2 J on M defined as follows. First, we set Jk + = k − , Jk − = −k + and extend these relations linearly on V. Second, as the restriction of g to H := span(k + , k − ) ⊥ is positive definite, we choose J H to be the unique almost complex structure making g| H hermitian 3 , such that J respects the orientation on M . Finally, we extend J linearly on T M = V ⊕ H.

3.2.
Integrability. Integrability of an almost complex structure implies the manifold admits complex coordinates. It is defined by the vanishing of the Nijenhuis tensor We now give sufficient conditions for integrability of the almost complex structure defined in the previous subsection.
Theorem 1. An almost complex structure J defined as in subsection 3.1 is integrable if the following three conditions hold: The notation of i) means that the Lie bracket operation with k + or k − sends any vector field in H to another such vector field. The shear notation in ii) is as in Section 2.
Remark 3.2. Note that if J instead takes k + to a linear combination of k + and k − (with coefficients which are constant, at least in directions tangent to H), then conditions i) imply the same conditions on the pair k + , Jk + , so that integrability of this J will still follow from the theorem.
Proof. We examine the Nijenhuis tensor of J for a frame {k ± , x ± }, where {x + , x − =Jx + } is an ordered orthonormal frame for g H . Clearly N vanishes on any pair a, Ja. The relation N (a, b) = JN (a, Jb) along with the antisymmetry of N (a, b) imply that it is enough to check the vanishing for the pair k + , x + . We thus analyze Conditions (14)i) along with the J-invariance of H imply via (15) that N (k + , x + ) is a section of H. Next, taking the inner product of the right-hand side of (15) with x + , while employing the fact that g H is hermitian, we arrive at the following expression: Referring now to the shear coefficient expressions (5), we thus arrive at the following two equalities, the second obtained in analogy with the first: where the shear-related notations are as in Section 2.3. Since the action of the shear matrix (3) on each of the standard basis vectors in R 2 yields (−σ 1 , σ 2 ) and (σ 2 , σ 1 ), respectively, the last two equations yield the invariant formula In fact, they yield equality of both sides on x + , and we obtain it on x − because of both N (k + , x − ) = −JN (k + , x + ) and the fact that J H anticommutes with any trace-free symmetric operator P acting on H. This last fact holds since J makes g H hermitian, so that the adjoint of P J is −JP , while the trace-free condition implies g(P Jx + , x − ) = g(P x − , x − ) = −g(P x + , x + ) = g(P Jx − , x + ), so that P J is also self-adjoint. By applying the last mentioned fact to (16), the theorem follows.
Remark 3.3. If J satisfies the conditions of this theorem, the almost complex structure defined just as J, but with respect to the opposite orientation, will also be integrable if k ± are shear-free. This follows since condition i) of Theorem 1 is independent of orientation, so still holds, while condition ii) follows because the new almost complex structure still sends k + to k − and the shears of these vector fields are independent of orientation (though not their representation in appropriate ordered frames). Thus in the equation J∇ o k + = ∇ o k − , applied to vectors in H, only the left hand side acquires a minus sign when switching the almost complex structure, hence both sides remain zero.
While Theorem 1 gives sufficient conditions for integrability, one can also make some statement regarding necessary conditions. A further generalization of this theorem will appear in subsection 3.4. For now, first note that (10) implies that i) of (14) is equivalent to the four conditions We now have and express the following vector fields in our standard frame: The coefficients a, b may be obtained by applying the inverse of A to the vector (g([k − , On the other hand similar coefficients for [k + , x ± ] all vanish by our assumptions. Subtituting the above expressions for the Lie bracket terms in N , we see that As N = 0, these coefficients of k ± vanish, and together with the above method of obtaining a, b, c, d this gives the two equations qB = rC, −rB = qC, where . These equations in turn imply B 2 + C 2 = 0, or B = C = 0. We thus see that g([k − , ·], k + ) vanishes on H. Then (ii) of (14) follows as in Theorem 1.

3.3.
Admissibility. We now consider special circumstances in which many of conditions (17) hold automatically.
Remark 3.5. For k = k ± , relation g([k, ·], k) = 0 holds on H in two significant situations: a) k is a pre/geodesic vector field of constant length, or b) k is Killing.
The first of these follows as g( and so is orthogonal to k. The second holds similarly since g( Note here that a pre-geodesic vector field of constant length is necessarily null, essentially since 0 = d k (g(k, k))/2 = g(∇ k k, k) = αg(k, k).
We do not consider special circumstances where the fourth condition, namely g([k − , ·], k + ) = 0, holds on H. However, if (19)i) holds, it can be translated into the form . When both k + , k − are pre/geodesic, and H is integrable, condition (20) guarantees that V is the horizontal distribution for a semi-Riemannian submersion. But integrability of H will never, in fact, occur in the circumstances we will be considering later. As condition (20), unlike the geodesic condition, mixes k + and k − , we will describe it by the phrase the distribution V satisfies the mixed condition.
In light of the considerations in Remark 3.5, we make the following definition, which singles out manifolds that obey the special conditions noted in this remark. Note that there are in fact three variants of this definition encapsulated below.
Definition 3.6. An oriented semi-Riemannian 4-manifold (or its metric) with two everywhere linearly independent vector fields k = k + , t = k − and associated distribution V = span(k, t), will be called geodesic-admissible (or pregeodesic-admissible or Killing-admissible) if k is a geodesic (or pre-geodesic) of constant length (or Killing); t is a pre/geodesic vector field of constant length; V satisfies the mixed condition; Note that the type of admissibility depends entirely on the type of k. We will see in later sections many examples of admissible manifolds. The point of this definition is Corollary 3.7. The almost complex structure associated, as in subsection 3.1, to any geodesic-, pre-geodesic-or Killing-admissible manifold, is integrable.

3.4.
Integrability with J V self-adjoint. We present here a generalization of Theorem 1. In that theorem we only gave a necessary condition for integrability of J. A condition that is also sufficient is too unwieldy to write down in full generality. Instead we consider another special case, which involves an assumption on the relation between g V and J V . Recall that in the Lorentzian case, the usual requirement that the metric is hermitian, is not available, except on H. Instead we require that J V is self-adjoint. This gives a new and very workable set-up for examining integrability.
Theorem 2. Let (M, g) be semi-Riemmanian with vector fields k ± satisfying (10) and (11), and J the associated almost complex structure of Section 3. If J V is self-adjoint with respect to g V , then it is integrable if and only if the following conditions hold: where x in i) is any vector field lying in H.
Proof. Recall from Theorem 1 that it is enough to analyze the vanishing of Taking the inner product of this expression with x ± has been carried out in Theorem 1, and led to the shearcondition. This remains unchanged. Condition i) in the theorem is obtained by first taking the inner product of N (k + , x + ) with k ± , and then employing the self-adjointness of J V , to shift J in the middle two terms from the Lie bracket to the other vector field in the metric expression. Note finally that condition i) is tensorial on H.
Remark 3.8. Theorem 2 is significant in two important cases in which J V is self-adjoint: if both k ± are null, or if they are orthogonal with lengths of opposite signs. As mentioned in the introduction, Flaherty [15] has given a Newman-Penrose version of Theorem 2 for the first case, while examples of the second case appear in Section 8.
Corollary 3.9. If k ± are null shear-free geodesic vector fields giving rise to J satisfying the assumptions in Theorem 1, then for nowhere vanishing smooth functions f 1 , f 2 , the vector fields f 1 k + , f 2 k − define, via the procedure described in subsection 3.1, an integrable almost complex structureJ if and only if the Proof. First, since the shear operator multiplies by a factor when the vector field does, the shear operators of so that as i) of Theorem 1 holds, if and only if d x (f 1 /f 2 ) = 0 for any vector field x lying in H, which is equivalent to the condition in the theorem. This in turn means that i) of Theorem 2 holds with k + , k − replaced by f 1 k + , J(f 1 k + )=f 2 k − . As ii) also holds as a result of the latter two vector fields being shear-free,J is integrable.
3.5. Basic characteristics. Some further properties of the vector fields k + , Jk + follow in the presence of integrablity.
Proposition 3.10. If J is the integrable complex structure in Theorem 1, then k + , Jk + are holomorphic vector fields with respect to it if and only if they commute and are shear-free.
a taking values in the frame used in the proof of Theorem 1. For a = k + or a = Jk + , vanishing of such an expression will occur if and only if k ± commute. For a takes values in H, note first that N (k + , Thus repeating the calculation after (15) for each summand separately, we have, for example g((L k + J)x − , x ± ) = 0 if and only if the shear coefficients of k + vanish, or equivalently the projection of (L k + J)x − to H vanishes. Since J(L k+ Jx − ) = (L k+ J)x + , this is also equivalent to the statement that the restriction of L k+ J to H vanishes. Similar consideration lead the the vanishing of the restriction of L k− J to H.
Note that in most, but not all our later examples, k + and Jk + are shear-free.
If integrability of J is not given, but k ± commute and are shear-free, the above argument shows that both are J-holomorphic and J is integrable. Similarly if one of them is J-holomorphic and both are shear-free, then both commute, the second one is thus also J-holomorphic and again J is integrable. This provides another approach to issues pertaining to a theorem in [7].
Since, for our usual frame, and similarly for k + replacing k − , we see that in the shear-free case, H, which is locally spanned by {x + , x − }, also satisfies (in a local sense) a mixed condition.
Also note that in the geodesic-admissible and pre-geodesic-admissible cases, if k + , k − commute, then V is totally geodesic. This follows since then

Kähler and Sasaki metrics induced by admissible metrics
This section gives the construction of Kähler metrics associated to admissible semi-Riemannian 4-manifolds. In a more limited context a relation of admissibility to Sasaki metrics is also described.
4.1. Symplectic forms. We first give certain families of symplectic forms which under certain conditions give rise to Kähler metrics on an oriented semi-Riemannian manifold associated to vector fields and complex structures as in Section 3. Some such forms were first considered in [1].
Let (M, g) be one of the three types of admissible semi-Riemannian 4-manifolds given in Definition 3.6, with vector fields k, t. We will employ its notations below, and additionally, in the pre-geodesic case, denote by α the function for which ∇ k k = αk holds, and by G the determinant in (13). Finally, we will say k is proper if ∇k restricts to an operator H → k ⊥ (equivalently, if the gradient of g(k, k) lies in V). We consider 2-forms on M of the form where k ♭ = g(k, ·), and f is a smooth function defined on the range of τ . Among the different choices of the "parameter function" f , our interest will lie mainly in the case where f is affine in τ , or else is e τ .
Recalling that ι = ι k is, up to sign, the twist function of k, we have To check this we note that ω can be written in the form Proof. We only need to determine whether ω is nondegenerate. First, in all three cases, if x ′ is a vector field lying in H, we have ω(k, x ′ ) = 0. In fact, this certainly holds for the first term in (23), while for the second we note that This is clearly zero for the first two admissible types, while for the third it equals −2g(∇ x ′ k, k), which is zero by the properness of k. Consider the ordered frame {v i } = {k, x , y , t}, where x , y are a (possibly local) ordered orthonormal frame for H. With respect to this ordering, the matrix with entries ω(v i , v j ), will, by the above calculation have zeros above the semi-diagonal. Thus its determinant is computed as the product of the terms on this semi-diagonal, namely det To see when this is nonzero we compute the factors. The first is ω( The second depends on the admissibility type, since dk ♭ (k, t) = g(∇ k k, t) − g(∇ t k, k) vanishes in the geodesic-admissible case, equals αg(k, t) in the pre-geodesic-admissible case, and equals −2g(∇ t k, k) = −d t (g(k, k)) in the Killing-admissible case. Combining these results shows that the determinant of [ω(v i , v j )] does not vanish when the stated inequalities hold, where for the first one we also use the fact that G and ℓ are nowhere vanishing.
We note that to prove this result we did not use all the assumptions concerning admissibility, but mainly just properties (10), (18) and (19)i).

Kähler metrics. We now examine under what conditions
on an admissible manifold, where ω is one of the symplectic forms (22), and J is the integrable almost complex structure of Theorem 1.
Theorem 3. On any admissible 4-manifold (M, g), a 2-tensor g K of the form (24) represents a Kähler metric in the region where where ι above is computed as in (6) via an ordered orthonormal frame of the form {x , y=Jx }.
Note that the second inequality in the geodesic-admissible case can be simplified to f ′ /ℓ > 0 in the Lorentzian case, using (13).
Proof. Since in the regions in question, by Proposition 4.1, ω is symplectic, while J, by Corollary 3.7, is integrable it remains to check that J is compatible with ω.
Consider first J-invariance of ω. For the first term in (23), as it vanishes when one of its arguments lies in H, it is enough to check it on vector fields in V, specifically on k,t, and clearly any 2-form is J-invariant on a pair of vector fields of the form a, Ja.
For the second term in (23), the same holds for the pair k, t and for an orthonormal frame {x , y = Jx } for H. It remains to check dk ♭ on a pair of vector fields, one in H and the other in V. If the latter vector field is k this will vanish in all three admissible cases, as we have seen in the proof of Proposition 4.1. If, on the other hand, the vector field in V is t, this term also vanishes x ]) = 0, the last equality being included in (17).
To complete the compatibility check of J with ω we need to find the region where J is ω-tame, yielding positive definiteness of g K . Since we have just shown that ω, and hence g K vanish on a pair of vector fields, one from H and the other from V, it remains simply to check when is positive, if a is a nonzero vector field in V, or in H. As g K also clearly vanishes on the pairs k, t and x ,y , it is enough to check positivity of (25) when a is one of the frame vector fields. But these calculations are equivalent to those made for ω(x , y ) and ω(k, t) in Proposition 4.1, and they yield positivity under the inequalities stated in the theorem.
Remark 4.2. Let J + := J, and denote by J − the almost complex structure of Remark 3.3, defined using the opposite orientation. As the proof of Theorem 3 shows that H and V are also g K -orthogonal, it follows that g K is almost hermitian also with respect to J − . By Remark 3.3, if k ± are shear-free, g K is in fact hermitian with respect to these two complex structures, which respect opposite orientations. In other words g K is ambihermitian, in the sense of [3]. In some cases, g K is also ambiKähler, meaning that there exists another metric in its conformal class which is Kähler with respect to this second complex structure. This happens, for example when g K is an SKR metric as in Section 8. A separate observation is that ω(·, J − ·) is a metric, not in the same conformal class, which differs from g K by having a different sign on its restriction to H. This can be seen directly, and also follows since ι switches sign when the ordered basis of H is flipped in the passage from J + to J − . As a result this metric has signature (2,2) in the domain of g K , while it is Kähler in the region where the expression defining g K has signature (2, 2).
Remark 4.3. It is instructive to consider to what degree Kähler metrics of this type are independent from the particular special cases of Theorem 1 that admissibility types represent. If the conditions of this theorem are satisfied, we can define ω and g K if additionally (19)i) holds, and one can ask whether the latter is Kähler. Going through the proofs of Proposition 4.1 and Theorem 3, we see that J-invariance of ω is guaranteed by only two more conditions. This is summarized in the following theorem.
Theorem 4. Let (M, g) be an oriented semi-Riemannian manifold such that (10), (11) and the conditions of Theorem 1 hold for vector fields k + = k and k − = t and the associated almost complex structure J. If both conditions (19) also hold and k is proper, then g K , given as in (24), defines a Kähler metric on any region where where ι is computed as in Theorem 3.
From this it is clear that the geodesic, pre-geodesic and Killing properties are not necessary for obtaining Kähler metrics. This will be utilized this later on, for example in Section 12.
We will often describe a Kähler metric provided by the above two theorems as induced by an admissible semi-Riemannian 4-manifold.

Sasaki metrics.
We end this section noting that under certain circumstances the Kähler metrics induced by admissible manifolds are cone metrics over Sasaki 3-manifolds. Proof. Recall that the symplectic form associated to g is and that in the geodesic-admissible case, only the first term on the right is nonzero on V. Evaluating the corresponding term of g K , we have, apart from the exponential coefficient, This can be verified by evaluating both sides on pairs taken from k, t and using the condition g(k, k) = −g(t, t) on these vector fields. Thus Applying the change of variable r = 2e τ /2 , we see that which has the form of a cone metric. Now the distribution (∇r) ⊥ = t ⊥ is spanned by H and a certain linear combination of k and t (with constant coefficients, since in the geodesic-admissible case the values of g V on these vectors are constant). As the Lie bracket of k, and of t, with vectors in H lies in H for an admissible manifold, the condition on [H, H] guarantees that t ⊥ is integrable. Thus in the region U where the Kähler metric g K is defined, h ι is a Sasaki metric lifted from a 3-manifold N such that U = N × R.

First-order properties of the induced Kähler metric
In this section we will show that many of the defining properties of admissible manifolds are also shared by their induced Kähler metric. Specifically, constancy of the metric value when evaluated on k, t, the mixed and the shear conditions and the fact that H is spacelike, all hold for the Kähler metric. In addition, under certain assumptions, the g-geodesic vector fields are also g Kgeodesic in the geodesic-admissible case, while k is g K -Killing in the Killingadmissible case. Finally, some conditions imply that t is g K -near-gradient, in the sense of (19)i). Additionally, we give a short description of how, in the geodesic-admissible case, completeness of the induced Kähler metric, together with certain bounds on metric-related quantities, implies completeness of the integral curves of the two vector fields. We also show that in the geodesic-admissible case, one of the vector fields is g K -conformal.
5.1. Admissibility properties of g K . We first record a number of basic facts.
Lemma 5.1. For a Kähler metric induced from an admissible semi-Riemannian metric, with ι computed as in Theorem 3.
Proof. The first equality was shown in the proof of Theorem 3. The second is obvious as ω is a 2-form. The third follows by comparing ) and −f ιg H (a, b), where a, b are taken from an ordered orthonormal frame x , y=Jx for H.
Remark 5.2. Note that as ω is a 2-form, we also have It follows that in one of our standard frames of the form {k, t, x , y }, the matrix of g K is diagonal, with at most two distinct diagonal values at each point.
From now on we will employ the notation τ c := τ − c, for a constant c.
Proposition 5.3. Let g be a geodesic-admissible semi-Riemannian metric with t a geodesic vector field and ℓ = 1. Assume g K is the Kähler metric induced according to Theorem 3 with f (τ ) = τ c . Then the associated vector fields k, t are geodesic and of constant length also with respect to g K .
Proof. First, k has constant g K -norm as g is geodesic-admissible, f ′ = ℓ = 1 and G is constant. Let x , y be a local orthonormal frame of g H , and ∇ K denote the Levi-Civita connection of g K . By the Koszul formula, the con- x ]) = 0, the last equality following as [k, x ] lies in H, since admissible metrics satisfy the hypotheses of Theorem 1. The same holds for x replaced by y , and clearly also As the matrix A of g V is nonsingular (see (10)), we have α = β = 0 and [k, t] is thus tangent to H. Applying this, we see by invoking the Koszul formula once more, this time for g K , that, as the matrix of g K V has constant entries, Thus g K vanishes on any pair consisting of ∇ K k k and any vector field in the frame {k, t, x , y }, so, using again that A is nonsingular, we see that ∇ K k k = 0. A similar argument shows that t has constant length and ∇ K t t = 0. The corresponding result in the Killing-admissible case requires more conditions.
Proposition 5.4. Let g be a Killing-admissible semi-Riemannian metric with k proper, ℓ, ι functions of τ , g(k, t) = 0 and [k, t] = 0, with g K a Kähler metric induced by g according to Theorem 3. Then the g-Killing field k is also g K -Killing.
Proof. The proof consists of a systematic verification of the Killing field identity when a, b are taken from the usual frame vector fields k, t, x , y=Jx , with the latter two orthonormal for g H . In fact, this equation is additive in a and b and multiplying one of these vector fields by a function r changes both sides by (d k r)g(a, b). Thus it is enough to check its validity on these frame fields.
In fact, it is is easy to see that it is enough to check the cases where {a, b} is one of the pairs x x , kk, tt, kx , kt, tx , x y. As this is a repetitive task, we only show some of the cases where the extra conditions of the proposition are needed. First, for a = b = x , by Lemma 5.1, with prime denoting differentiation with respect to τ , we have x ], x ) = 0 because k is g-Killing and x has constant g-length. Thus Next we show that d k (g K (k, k)) = 0, and hence clearly equal to 2(g K ([k, k], k). First note that d k (g(k, k)) = 0 since k is g-Killing (using the analog of (26) for g). Then, computing as in Proposition 4.1, we have g(k, k)). Now as t has constant length, d k (f ′ G/ℓ) = (f ′ /ℓ) ′ g(k, ∇τ )G + (f ′ /ℓ)g(t, t)d k (g(k, k)) = 0, the last equality following as k is g-Killing and, as above, g(k, ∇τ ) = 0. Also, d k f = 0 as before, while d k d t (g(k, k)) = d t d k (g(k, k))+d [k,t] (g(k, k)) = 0 as k is g-Killing and [k, t] = 0. This completes the proof that d k (g K (k, k)) = 0. The other verifications of (26) are similar.
Similar calculations employing the Koszul formula show that in the Killingadmissible case, under the assumptions of this proposition, if also g(k, k) is a function of τ , then t is g K -pre-geodesic (not necessarily of constant length, i.e. not a geodesic). Examples where all these conditions are satisfied will be given in Section 8.
Next, we have Lemma 5.5. If g is an admissible semi-Riemannian metric with g(k, t) = 0 and g(k, k) a function of τ of (19)i), then t =l∇ K τ , with the g K -gradient of the functionl lying in V.
Proof. Let x , y be an orthonormal frame for g H . Since g K (∇ K τ, b) = dτ (b) = g(∇τ, b) = g(t, b)/ℓ and g K (t, b) are both zero for b = k, x , y , the only nonzero component of ∇ K τ in the usual frame, which is g K -orthogonal, is a multiple of t. The multiple itself can be found as follows. If p := g(k, k) and q := g(t, t), then the multiple is g But the result is a function of τ , whose g K -gradient we know to be parallel to t, which lies in V.
k]) = 0. Finally, we show that the shear condition (14)ii) holds also with the shears taken with respect to g K . In fact, this follows from the following relations between shears of k and t with respect to an admissible metric g and with respect to a Kähler metric g K it induces. Specifically, We prove the first relation only, as the second is similar. Let {x , y } be, as usual, a local orthonormal frame for g| H . From (5) we have . We now compute the corresponding shear coefficient for the Kähler metric g K and the g K -orthonormal basisx := x /s,ỹ := y /s, where s := −f ι k . The Koszul formula for g K gives, using the first relation in Lemma 5.1, since g K H = s 2 g H so that g K (y , y ) = s 2 g(y , y ) = s 2 g(x , x ) = g K (x , x ). A similar calculation shows σ K,k 2 = σ k 2 and thus the shear relation for k follows. Of course, there is a more conceptual argument for the validity of the shear condition (14)ii) with respect to the Kähler metric. Namely, the proof of Theorem 1 shows that if conditions (14)i) hold, then the Nijenhaus tensor vanishes if and only if the shear condition (14)ii) is satisfied. This vanishing, along with (14)i), are of course metric independent conditions, so it follows that, assuming (14)i), if J is integrable, condition (14)ii) must hold for any metric for which (10) and (11) hold. Now (11) certainly holds for the Riemannian metric g K . Given that, according to Remark 3.1, condition (10) follows automatically from the metric independent condition that k, t are linearly independent at every point. The mixed condition for g K can also be explained similarly. Generally speaking, the independence of i) and ii) of Theorem 1 from the particular metric employed forms the underlying reason for most of the results of this section. However, as some of the conditions analyzed in this section represent special circumstances in which i) holds, more assumptions are sometimes needed, as we have seen, to show that such circumstances also occur with g K .

5.2.
Completeness of geodesic vector fields. We mention here an outcome related to the assumption of completness of g K .
Proposition 5.7. Let g be a geodesic-admissible semi-Riemannian metric on a manifold M , inducing a Kähler metric g K as in Theorem 3 on all of M . If g K is complete, then any inextendible integral curve of the geodesic field k is defined on the whole real line if and only if f ′ G/ℓ is bounded below on M . If t is also geodesic, the same holds for its integral curves. If g is Lorentzian or one of these vector fields is null, the same holds if f ′ /ℓ is bounded above on M .
Proof. According to [8,Proposition 3.4], a geodesic will be extendible beyond a finite interval of its parameter domain if and only if in some Riemannian metric the length of its tangents is bounded. Given a geodesic which is an integral curve of k, its length in g K is given by the square root of the quantity g K (k, k) along it. In the geodesic case this quantity is −f ′ G/ℓ, which is of course positive (see Theorem 3), so we have the first conclusion if and only if it is bounded above. As g K (t, t) = g K (k, k), the same holds if t is geodesic. If either g is Lorentzian or one of the vector fields is null, then G is negative (see (13)), so the remaining claim follows.
Conversely, note that by the above-mentioned proposition in [8], if g is complete and, say, g K (t, t) is unbounded, then g K cannot be complete. A gravitational plane wave example for which this occurs will be described in Section 10.
5.3. g K -conformal fields in the geodesic-admissible case. We note here special properties of a normalization of the vector field k, one of which was already discovered in [1].
Proposition 5.8. Let (M, g) be geodesic-admissible with k null. Then the vector fieldk is Liouville with respect to the symplectic form (22). If k and t are also shearfree and commute, while f (τ ) = e τ and ℓ = 1, then k is a g K -conformal vector field (in fact homothetic).
Proof. The Liouville property was noted in [1]. In fact, we have seen that dk ♭ (k, ·) = 0 because k is geodesic.
The shear and commutation conditions guarantee first that k is holmorphic by Proposition 3.10. The conditions on f and ℓ then guarantee that f /(f ′ kτ ) = 1/g(k, t) is constant, so thatk is holomorphic as well. Being holomorphic and Liouville, it immediately follows that Lkg K = g K , and so k is similarly homothetic.
This proposition also applies to the gravitational plane wave example of Section 10.
6. Ricci and scalar curvatures of g K Suppose an admissible metric satisfies g(k, k) = −g(t, t) and g(t, k) = 0. Due to Lemma 5.1 and Remark 5.2, similar relations (up to sign) hold for any induced Kähler metric g K , and this lemma also shows that on H, g K is a function multiple of g. This verifies the claim made in the introduction, that after changing the sign of g(t, t) (a Wick rotation), the resulting metricg and g K become biconformal, i.e. their restrictions to V and H are each a function multiple of the other. This implies a fairly complicated relation between their connections and curvatures, which we will not pursue at this time. Instead we give more direct formulas for the Ricci and scalar curvature of g K based on their very definition via a symplectic form, which easily yield their volume form. These formulas hold with extra assumptions, most importantly that the metrics conform to a certain bundle structure.
The Ricci form of a Kähler metric is given via where µ is the volume form coefficient in a coordinate system and ν is the coefficient for the coordinate volume form in corresponding complex coordinates (cf. [32, §1.4.3]). We apply this general formula to the case of the Kähler metric g K associated to an admissible semi-Riemannian manifold, computed under certain assumptions detailed below. We give two formulas, one for the geodesic-admissible case, the other for the Killing-admissible one. Examples fulfilling these assumptions will be obtained in Section 8.
Proposition 6.1. Let (M, g) be geodesic-admissible with its standard complex structure J, inducing a Kähler metric g K as in Theorem 3. Assume that M is an open set in the total space of a holomoprhic line bundle over a Kähler surface (N, h) with holomorphic projection map π, and g H = π * h. Suppose k, t commute and are shear-free, ℓ is a function of τ , k is not null and k ♭ is locally a sum of an exact form and one vanishing on V.
Let ω h = r dx ∧ dy be the Kähler form of h, expressed in local coordinates where z = x + iy is a holomorphic coordinate on N . Then the Ricci form and scalar curvature of g K are given, respectively, by where ω, * are the Kähler form and Hodge star operator of g K , respectively, and a is the coefficient of k in the expression for Jdτ as a linear combination of k and dτ .
Note that we could write the second term in (29) much more explicitly by replacing ω via (23), but we have chosen not to in order to keep the formula less cluttered. Also note that the assumption on k ♭ is fulfilled in the case it is a connection 1-form.
Proof. Given our assumptions on g H , the symplectic form ω takes the form where we dropped pull-backs by π from the notation. Thus the volume form is and µ = −f f ′ rι. Next, our assumptions, taken together with Proposition 3.10, mean that Ξ := k − it is a holomorphic vector field and ψ := k ♭ + it ♭ a holomorphic 1-form which evaluates to a nonzero constant on it. Employing ψ together with dz one computes the coordinate volume form coefficient to find, as t ♭ = ℓdτ is exact because ℓ is a function of τ , that up to a multiplicative constant, one can take ν = ℓ (note that the assumption on k ♭ is used in this step). Substituting in (28) gives the formula for the Ricci form.
To compute the scalar curvature, we first note that Jdτ = Jt ♭ /ℓ is a linear combination of k ♭ and dτ , as it vanishes on H. Substituting k and t in Jdτ = ak ♭ + bdτ and solving the linear system for a, b, one easily sees, as the metric g has constant values when evaluated on pairs from {k, t}, that b is constant, while a, which also depends on ℓ, is a function of τ . Thus Computing now ω ∧ ρ K , the contribution from the first two terms in the last line above combines to be Since the scalar curvature is s K = * (ω ∧ ρ K ), and * Vol = 1, the result follows.
A special case of this formula for the scalar curvature occurs when ℓ = 1 and f (τ ) = τ c , as then (a is constant and hence) the first term in (29) vanishes, giving Proposition 6.2. Let (M, g) be Killing-admissible with its standard complex structure J, inducing a Kähler metric g K as in Theorem 3. Suppose all assumptions of Proposition 6.1 hold (except geodesic-admissibility). If g(k, t) = 0, q := g(t, t) = 0 and p := g(k, k), ι are, in addition to ℓ, functions of τ . Then the Ricci form and scalar curvature of g K are given, respectively, by and * is the Hodge star operator of g K .
Proof. The symplectic form is now more complex. By comparing the values of dτ ∧ k ♭ and dk ♭ on the pair {k, t} (see the proof of Proposition 4.1), we have as d t p = p ′ d t τ = p ′ q/ℓ and G = pq because g(k, t) = 0. Thus the volume form is and µ = −f (f ′ + f p ′ /p)rι.
We turn to finding an appropriate complex coordinate chart. This time the form ψ = k ♭ /g(k, k) + it ♭ is constant on Ξ (here we are employing the vanishing of g(k, t)). Thus, the volume coefficient with respect to a holomorphic coordinate frame is ν = ℓ/p, so that the stated formula for the Ricci form is verified. Set P := − log(−f (f ′ +f p ′ /p)ιp/ℓ)/2, which is a function of τ . Writing Jdτ as a linear combination of k ♭ and t ♭ with coefficients a, b, this time as g(k, t) = 0 we see that a = −q/(pℓ), another function of τ , while b = 0 (as q = 0). Then d(P ′ Jdτ ) = d(P ′ ak ♭ ), so that Taking account of the volume form expression again, together with * Vol = 1, the stated formula for the scalar curvature of g K follows.
7. Obstructions to invertibility of the map g → g K In this section we outline various circumstances in which two different semi-Riemannian metrics induce the same Kähler metric. As the proofs involve arguments similar to ones we have already made in previous sections and the next one, we will be brief. Suppose on a given 4-manifold k, t are fixed pointwise linearly independent vector fields spanning V, and an almost complex structure J is given by its values on V as in subsection 3.1, and those on H = V ⊥ stipulated independently. Suppose functions ℓ, τ on this manifold are also given. Formula (22) for the symplectic form shows that if two semi-Riemannian metrics g,g whose restriction to H is hermitian with respect to J, satisfy k ♭ = k♭ and t = ℓ∇τ = ℓ∇τ , will yield the same symplectic form, and hence Kähler metric (for the same f ). But this condition implies g(k, k) =g(k, k) and g(k, t) =g(k, t), so that the only way g V andg V can differ is if g(t, t) =g(t, t). And this may occur even with the condition above on t. We do not discuss in this preliminary remark whether other admissibility conditions may hold for both metrics. Instead we turn to explicit ways to vary g that yield the same Kähler metric.

7.1.
Varying g H . Even if the two metrics agree on V, they may still differ on H while inducing the same Kähler metric. One situation where this occurs is in a biconformal change of the form for a nowhere vanishing function β.
Indeed, if x , y is the usual orthonormal frame for g H , then x /β, y /β is such a frame forg. Thus the twist of k with respect tog isι =g(k, [x /β, y /β]) = g(k, [x , y ])/β 2 = g(k, [x , y ])/β 2 = ι/β 2 because g andg agree on V. Thus, applying Lemma 5.1 we havẽ As ∇τ , the image of dτ under g −1 , is in V, it depends only on the inverse of g| V , so that under the biconformal change,∇τ = ∇τ , and thus both metrics share ℓ as well, so that glancing at the V-components of g K as given in the inequalities in Theorem 3,g K V = g K V as well, at least in the geodesic-admissible and Killing-admissible cases. One can check that the shear of k or t is invariant under the biconformal change. Also, if g is geodesic-admissible, and satisfies the conditions of Proposition 5.3, then k, t will also beg-geodesic and the mixed condition will be satisfied. So in factg will also be geodesic-admissible. In Section 8 Killing-admissible metrics will be produced from a special type of Kähler metric they induce. In that setting there will be a canonical choice for g H .

7.2.
Varying g V . We now discuss a complementary change of metric, where g H is fixed while g V varies.
Let g be a geodesic-admissible semi-Riemannian metric satisfying the hypotheses of Theorem 3, and again those of Proposition 5.3 as well. Letḡ = g + εdτ 2 for small ε > 0. The parameter ε is only introduced for the purpose of keeping the signatures of g andḡ equal. Note that for this metric change clearly k ♭ = k♭. We haveḡ(t, ·) = (1 + εdτ (t))dτ andḠ := detḡ V = G(1 + εdτ (t)), so that t is dual to the exterior derivative of a constant multiple of τ , and this constant is 1 if t is null. Now V is stillḡ-orthogonal to H. Choose as usual a local frame x , y , orthonormal with respect to one, and hence both ofḡ H and g H . Thenῑ =ḡ(k, [y , x ]) = ι + εdτ (k)dτ ([y , x ]) = ι, the last equality following as the Hessian of τ is symmetric.
The g-shear andḡ-shear of k and of t are also equal, and in factḡ is also geodesic-admissible. We show just one of the required properties. Using the Koszul formula one sees that Now dτ (k) = g(k, t) is constant so that this expression vanishes for a lying in H since [k, a] then lies in H; it clearly vanishes when a = k; and it vanishes for a = t because dτ (t) is constant while from the proof of Proposition 5.3, [k, t] lies in H. Thus∇ k k = 0 and k is geodesic. Also, using again that dτ (k) is constant, Thus, as the standard almost complex structure J is the same for g andḡ, the two induced Kähler metrics coincide:

Conformal change.
We comment here on a more common way to vary g that does not fix the induced Kähler metric, but one can still determine it after the change. Although a standard conformal changeĝ = β 2 g of an admissible metric g does not, in general, produce an admissible metric, it does preserve the twist and shear (cf. [7]). Thus integrability of J can be determined by checking the conditions of Theorem 1 forĝ. As t = ℓ∇τ = ℓβ −2∇ τ , we see that τ is recoverable from t,ĝ and β, and a similar claim follows for k ♭ , and hence for ω. As the twist is conformally invariant, it can be computed fromĝ and used to determine whether ω is symplectic, and similarly, to check that g K is Kähler. Section 11.4 will describe a pre-geodesic-admissible metric conformal to the Kerr metric.

Killing-admissible Lorentzian metrics inducing SKR metrics
In this section we begin the study of examples.
We first recall the description of a special type of Kähler metric, called SKR, also known as a metric admitting a special Kähler-Ricci potential. These include many conformally-Einstein Kähler metrics. We then produce via an explicit ansatz Lorentzian Killling-admissible metrics whose induced Kähler metric is SKR. In some cases, these Lorentzian metrics are also geodesic-admissible.
A Killing potential τ on a Kähler manifold (M, J, g K ) is, by definition, a smooth function τ such that J∇ K τ is a Killing vector field, where ∇ K denotes the Levi-Civita connection (or the gradient) with respect to the Kähler metric. We set v := ∇ K τ, u := Jv, V := span(v, u), H := V ⊥ .
This potential τ is called a special Kähler-Ricci potential, and g K an SKR metric, if τ is nonconstant, and at each regular point of τ , the nonzero tangent vectors in H are eigenvectors of both the Ricci endomorphism and the Hessian of τ . We will often denote such metrics g SKR . They include many Kähler conformally Einstein metrics in dimension four (and all of those, in higher dimensions).
Theorem 18.1 in [12] gives the local classification of SKR metrics. It states that for any SKR metric, every regular point of τ has a neighborhood U which is the domain of a biholomorphic isometry Ψ to an open set in a holomorphic line bundle over a Kähler manifold (N, h) with Kähler form ω h , equipped with the following metric, still denoted g SKR . There are in fact two metric forms, but we only give one, which we will call the irreducible form, as it describes metrics which are not Kähler local products. It is given as follows.
where τ c := τ − c as in Section 5.1 with c a constant, τ is the push-forward of the Killing potential under the above biholomorphism, a = 0 is a constant, Q is a function of τ which equals g SKR (v, v) = g SKR (u, u), π is the projection map from the line bundle to N ,û is the one-form having value a on u and zero on v and on lifts of vector fields on N , and V, H are also obtained via pushing-forward these distributions via the biholomorphism, with V being also the vertical distribution of the line bundle. In addition, H is the horizontal distribution for a Chern connection on the line bundle.
If M is compact and not biholomorphic to CP m , it follows from [13] that a biholomorphic isometry Ψ as above exists, with domain M , mapping onto a CP 1 -bundle over a Kähler manifold equipped with a canonical model metric. Furthermore, Ψ maps the non-critical set of τ onto the total space of a line bundle minus its zero section. Finally, in the irreducible case the model metric still has the form (34) on this subset of the total space.
We record some known relations for SKR metrics, all immediate from or appearing in [12], some of which will be employed below. In these, w, w ′ denote horizontal lifts of vector fields on the base manifold N .
Given an SKR metric in dimension four of the form (34), our first objective is to obtain a procedure for finding a Killing-admissible Lorentzian metric g, defined on an appropriate subset U , with an associated Kähler metric satisfying The construction is as follows. Set k := u, t := −v. Fix two of the three values of the metric on k, t as follows: g(k, t) := 0, while g(t, t) := q is defined to be an arbitrarily chosen negative constant. Choose p to be a function of a variable τ which is positive on {τ > c}. Set g(k, k) := p, with p now abusively denoting p • τ . Define g V by linear extension. Then define g H := π * h. Finally, declare g(V, H) = 0. Our theorem is then stated as follows.
Theorem 5. Let g SKR be an irreducible SKR metric on a complex manifold (M, J) of real dimension four with Killing potential τ , such that the abovementioned biholomorphic isometry Ψ has domain M . Then there exists a Killing-admissible Lorentzian metric on U := {dτ = 0} ∩ {τ > c}, which is isometric, via Ψ, to a metric g among those in the ansatz just described. The distinguished vector fields associated with g are k and t, and t = ℓ∇τ for ℓ := −q/Q. If for g, the function p is a positive constant, then this Killingadmissible metric is also geodesic-admissible.
Choosing f (τ ) := τ c /p, the metric g along with f induce a Kähler metric g K as in Theorem 3, whose isometric copy in M (also denoted g K ) is defined on U and satisfies If M is compact and not biholomorphic to CP 2 , then (after perhaps switching the sign of τ ) this admissible metric is in fact defined on the set U := {dτ = 0}, which is open and dense in M .
Proof. We identify U from now on with its image in the line bundle. Note first that on U the vector fields v, u have no zeros, hence the same holds for k, t, so that their assigned lengths via g are well-defined. As u = Jv, the fields k, t are linearly independent at each point of U .
On U , or even on {τ > c}, p is positive. Thus g(k, k) > 0, while g(t, t) = q < 0 and g(k, t) = 0. Hence g V is nondegenerate of index one. On the other hand g H = π * h is positive definite. Since g(H, V) = 0, g is a Lorentzian metric at each point of U . For this metric H is spacelike, and G = pq = 0 on U , so that conditions (10) and (11) hold.
We now prove that t is geodesic with respect to g. Let x be a horizontal lift of a vector field in N with respect to the projection π. Since t has constant g length and is g-orthogonal to H, the Koszul formula gives 2g (35)iii). Thus one can find a frame near each point of U such that g vanishes on the pairs consisting of ∇ t t and each frame vector field. Hence ∇ t t = 0.
Next we show that k is g-Killing. To show this we need to verify for any vector fields a, b. Just as in the proof of Proposition 5.4, it is enough to check that this relation holds on frame fields x , y = Jx , k, t, where x , y are horizontal lifts of an orthonormal frame for h. In fact, it is easy to see that it is enough to check the cases where {a, b} is one of the pairs x x , kk, tt, kx , kt, tx , x y. We check only a few of these, as the others are similar.
, as x has constant g-length while [k, x ] = [u, x ] = 0 by (35)v). Similarly, as p is a function of τ , denoting, as before, by a prime differentiation with respect to τ , we have The other verifications are similar.
Next, the condition that g(k, t) is constant is satisfied by the very definition of g. The mixed condition (21) holds because for any horizontal lift x of a vector field on N , [t, k])] = 0 via the Koszul formula, where the last equality holds because the Lie brackets in all three terms are zero.
As k is g-Killing, it is shear-free. Hence to verify ii) of (14) we have to show that t is shear-free. But if x , y are the usual horizontal lifts of ordered orthonormal vector fields on h, then 2σ t 1 = g(∇ y t, y ) − g(∇ x t, x ) = 0 by the Koszul formula, We now check that the near-gradient relation (19)i) holds. First, note that setting ℓ := −q/Q is well-defined on U as Q is nonzero there. Second, this ℓ is a function of τ , so as d x τ = g SKR (x , ∇ SKR τ ) = g SKR (x , v) = 0 for x lying in H, it follows that ℓ has a gradient lying in V. Now definet = t/ℓ (note that ℓ = 0). We wish to showt = ∇τ . The values of g on the pairing of either of these vector fields with x , y , k is zero, since, for example g(t, x ) = g(t, x )/ℓ = 0 = d x τ = g(x , ∇τ ), and similarly g(t, k) = 0, while we have seen that 0 = d k τ = g(k, ∇τ ). Their values when paired with t are compared as follows: This completes the proof that g is Killing-admissible. We need to check a number of extra relations, given in Theorem 3, to show g gives rise to a Kähler metric g K on U . First, k is proper, i.e. ∇k defines a map H → k ⊥ , since for We thus see that Also, applying (35)vi) and k = u, we see that for the usual horizontal lifts x , y =Jx of an ordered h-orthonormal frame, ι = g(k, [x , y ]) = −2π * ω h (x , y )p = −2π * h(y , y )p = −2p < 0 on U . Hence f ι < 0 on U . Thus, by Theorem 3, g induces a metric g K on U which is Kähler with respect to J.
We now show that g K = g SKR on U . In fact, g K (t, t) = g K (k, k) = −(f ′ G/ℓ + f d t p) = Q as we have just seen, i.e. this value is equal to g SKR (−v, −v) = g SKR (u, u), while g K (k, t) = 0 = g SKR (u, −v). Thus g K V = g SKR V . On the other hand, g K H = −f ιg H = −f (−2p)π * h = 2τ c π * h = 2|τ c |π * h = g SKR H on U . As g K (V, H) = 0 = g SKR (V, H), the first equality following from Lemma 5.1, the two Kähler metrics indeed coincide.
If p is constant so is g(k, k), so that k is Killing of constant length, hence also geodesic. In this case g is additionally geodesic-admissible.
Finally, if M is compact and not biholomorphic to CP 2 , we know that the noncritical set of τ is mapped via Ψ onto the line bundle minus its zero section, and we wish to show the former set is U . It was shown in [13] that for an SKR metric on a compact manifold, the range of τ is a closed interval and c is not in its interior. As choosing the sign of τ determines that of c, one can thus always arrange, when the manifold is compact, that τ > c. Then, because the τ -critical submanifolds are exactly the level sets of τ corresponding to the endpoints of the range of τ ( [13]), it follows that U ⊂ M is exactly the noncritical set of τ . As τ is a Killing potential, it is known that the latter set is open and dense in M (cf. [13]). This completes the proof.
Note also that as all assumptions of Proposition 5.4 hold, it provides an alternative route for showing that k is g K -Killing, independent from the recognition that g K = g SKR . It is worth mentioning that if in the ansatz above we make g(t, t) := q a positive constant, while g(k, k) := p a function of τ which is negative on {τ > c}, the proof of Theorem 5 goes through without change, except that ι will now be positive and f negative on that set, hence we will still have f ı < 0. Thus we can choose our Lorentzian metrics inducing SKR metrics to have a timelike Killing field. Another point that needs emphasizing is that in the case where g SKR is defined on a compact manifold (not biholomorphic to CP 2 ), the metric g, given via the ansatz on an open dense set U , does not extend to M . The reason is that M \ U consists of zeros of t and k, but g(t, t) is a negative constant on U , so cannot become zero smoothly on M \ U . This, of course, goes along with the fact that the expression (22) for the exact Kähler form cannot hold on the entire compact manifold, even though the Kähler form does extend smoothly.

Kähler metrics induced by Lorentzian warped products
In this section we construct Kähler 4-manifolds from pre/geodesic-admissible Lorentzian 4-manifolds, in the sense of Definition 3.6, which are warped prodcuts.
9.1. The construction. We begin with the case when the Lorentzian 4manifold is a warped product (R × w N, −dt 2 + w 2ḡ ), with (N,ḡ) a Riemannian 3-manifold. In what follows, ∇ will denote the Levi-Civita connection of the warped product, and ∇ that of (N,ḡ). Theorem 6. Let (N,ḡ) be a Riemannian 3-manifold with a unit length vector fieldk, whose flow is geodesic, shear-free, and has a nowhere vanishing twist function. Suppose w(t) is a smooth positive function on R satisfying w ′ /w > −1. Then (R × w N, g, k, ∇t) is pre/geodesic-admissible with respect to a chosen orientation, where g is the Lorentzian warped product and k := ∂ t +k/w. The metric g then induces a Kähler metric on R × N .
Note that in the expression for k, the notationk refers to the obvious lift of this vector field from N to R × N .
Proof. We verify the pre/geodesic-admissible properties of Definition 3.6. To begin with our vector field k is clearly g-null, and has pre-geodesic flow since Here we have used the fact that ∇kk has R-component nor ∇kk = − g(k,k) w ∇w = ww ′ ∂ t and N -tangential component tan ∇kk = ∇kk = 0 (see, e.g., [25, Proposition 35, p. 206]). Next, setting t := ∇t = −∂ t , the vector fields k and t are pointwise linearly independent and satisfy g(k, t) = k(t) = 1. Furthermore, t clearly has constant length and geodesic flow (in the notation of Definition 3.6, ℓ = 1). We now establish relations between the shears and twist functions of k andk. In what follows, the shear coefficients and twist function (up to sign) ofk will be denoted byσ 1 ,σ 2 andῑ, respectively, while those of k will be denoted by σ 1 , σ 2 and ι. Let {x ,ȳ } be an ordered localḡ-orthonormal frame ofk ⊥ḡ ⊂ T N , whose ordering is chosen so thatῑ is negative. Lifting these vector fields trivially to R× N , the g-orthonormal vector fields x :=x /w and y :=ȳ/w comprise a basis for the two-dimensional (spacelike) distribution H = span(k, t) ⊥g . Furthermore, where we have used the fact that nor ∇ȳk = − g(ȳ ,k) w ∇w = 0. Likewise, Now we can compare the shears.
Sincek is shear-free in (N,ḡ), It follows that k must be shear-free with respect to g, because It is likewise verified that t is also shear-free, since ∇ x t = −(w ′ /w)x and ∇ y t = −(w ′ /w)y . Thus, the almost complex structure J defined by Finally, because ∇ k t = w ′ w 2k and ∇ t k = 0, condition (20) is satisfied, hence the distribution V = span(k, t) satisfies the mixed condition. This concludes the verification that (R × w N, g, k, t) is pre/geodesic-admissible. From Corollary 3.7, we see that J is integrable. To complete the proof, recall the definition of ι in (6). Observe that the twist function (up to sign) ι of k in (R × w N, g) is related to that ofk in (N,ḡ) by Then, in accord with Theorem 3, g K = d(e t k ♭ )(·, J·) will be a Kähler metric.
Indeed, by Lemma 2.2, this result generalizes as follows: Corollary 9.1. Let (N,ḡ) be a Riemannian 3-manifold and X a unit length vector field whose flow is complete, geodesic, and shear-free. If Ricḡ(X, X) > 0, then R × N admits a Kähler metric as in Theorem 6.
We now present some concrete realizations of Theorem 6.

9.2.
A Kähler metric via the direct product on R×S 3 . Our first example falls properly into the geodesic-admissible category (k in Theorem 6 will have geodesic flow). Letḡ denote the round metric on the 3-sphere S 3 ; i.e., the metric induced on S 3 from the flat metric on R 4 . On S 3 there is a well-known unit length Killing vector fieldk whose flow is tangent to the Hopf fibration.
In coordinates (x 1 , y 1 , x 2 , y 2 ) ∈ R 4 , it is the restriction to S 3 of the following vector field on R 4 : On, say, the upper hemisphere (y 2 > 0) of S 3 ,k takes the form By Lemma 2.3 the flow ofk is geodesic and shear-free in (S 3 ,ḡ). Its twist function ι satisfies ι 2 = 2Ricḡ(k,k) = 2. We now apply Theorem 6 with w(t) = 1, k = ∂ t +k, and t = ∇t, to conclude that (R × S 3 , −dt 2 ⊕ḡ, k, t) is geodesic-admissible and induces a Kähler metric on R × S 3 . 9.3. A Kähler metric on de Sitter spacetime. Four-dimensional de Sitter spacetime is the warped product (R × w S 3 r , −dt 2 + w 2ḡ ), where w(t) := r 2 cosh 2 (t/r) and r > 0 denotes the radius of S 3 r . For r ≥ 2, w ′ /w > −1 on the entire manifold (if r < 2, then on an open subset); thus with null vector field k = ∂ t +k/w and t = ∇t, Theorem 6 applies. Note that de Sitter spacetime is both globally hyperbolic and geodesically complete (see [5, p. 183-4]). 9.4. Kähler metrics on R × R 3 . Using Theorem 6 once again, a family of Kähler manifolds will now be constructed on R 4 out of the following distinguished class of Lorentzian 4-manifolds: Definition 9.2. A four-dimensional standard pp-wave is the Lorentzian manifold (R 4 , g) with coordinates (u, v, x, y) and with g given by If H is quadratic in x and y, then (R 4 , g) is called a plane wave.
Standard pp-waves originated in gravitational physics and have been intensely studied therein; see, e.g., [22] and [30], as well as [16] and [5,Chapter 13]. With respect to {∂ u , ∂ v , ∂ x , ∂ y } the matrices of g and its inverse are given by Note that ∂ v = ∇u is a parallel null vector field. Furthermore, for any choice of the function H(u, x, y), the corresponding pp-wave is scalar flat, and the only nonvanishing component of its Ricci tensor is where H xx and H yy denote the second partial derivatives of H with respect to x and y, respectively. Now let k, h : R 4 −→ R be two smooth functions independent of v and consider the following null vector field z on (R 4 , g): Observe that g(z , z ) = 0, that ∂ v and z are pointwise linearly independent and g(∂ v , z ) = −1, and finally that the pair of vector fields are orthonormal and span the distribution H = span(∂ v , z ) ⊥ , which is spacelike (in the literature, such bundles are usually referred to as "screen distributions"; see, e.g., [22]). Observe also that the twist function of z does not, in general, vanish, as ι z is Because ∂ v = ∇u, the hypersurfaces S u := {u = const.} are integral submanifolds of the orthogonal complement ∂ ⊥ v ⊂ T R 4 ; because ∂ v is null, it is tangent to these submanifolds. Now fix any u 0 and consider the hypersurface S u 0 ∼ = R 3 with global coordinates {v, x, y}. Set and define a Riemannian metricḡ on S u 0 , by giving its orthonormal coframe {k♭, x♭, y♭}, which isḡ-dual to {k, x , y }. It is given bȳ Observe that {k,x ,ȳ } is a global orthonormal frame for (R 3 ,ḡ). We then have the following: Proposition 9.3. On (R 3 ,ḡ) withḡ given by (45), the vector field k given by (44) is a unit length Killing vector field. If k y − h x is nowhere vanishing and w is a smooth positive function satisfying w ′ /w > −1, then (R × w R 3 , −dt 2 + w 2ḡ , k, ∇t) is pre/geodesic-admissible and induces a Kähler metric on R 4 .
Proof. We first show that k is a unit length Killing vector field, via Lemma 2.3. That k has unit length with respect toḡ is clear, sinceḡ(k, k) = −g(z , k) = 1.
So, too, does its shear: 2σ 1 =ḡ(∇ȳ k,ȳ ) −ḡ(∇x k,x ) = 0, and the Koszul formula on σ 2 yields Being unit length, geodesic, divergence-free, and shear-free, it follows by Lemma 2.3 that k is a unit length Killing vector field on the Riemannian 3-manifold (R 3 ,ḡ). Finally, we show that the twist function of k is nowhere vanishing, provided that k x = h y at any point. This follows from our particular choice of z , namely, that its twist function is nowhere vanishing: Applying now the contents of Theorem 6, the proof is complete.
Two further remarks regarding Proposition 9.3. First, the construction of Riemannian metrics on hyperplanes in R 4 via a pp-wave metric and one of its null vector fields, as we have done above, is derived from a well-known construction; see, e.g., [22]. Second, we record here that the scalar curvature of our Riemannian 3-manifold (R 3 ,ḡ) is strictly negative and directly proportional to the twist function (46) of k: This calculation was done with the aid of the software Mathematica.

A Kähler metric on a geodesic-admissible plane wave
In Proposition 9.3 we used three-dimensional submanifolds of pp-waves to construct Kähler metrics on four-dimensional Lorentzian warped products which were in general pre-geodesic-admissible. Key to this construction was the null vector field given by (41), which was used to define the Riemannian metric (45). In this section we will construct a Kähler metric which will be induced directly from a geodesic-admissible 4-dimensional pp-wave (in fact, a plane wave; recall Definition 9.2), without first constructing a Riemannian metric on a hypersurface. Thus, let (R 4 , g) be a standard pp-wave as in Definition 9.2, with g given by (40) and with the function H(u, x, y) to be determined. Recall that ∂ v = ∇u is a parallel null vector field; here it will play the role of ∇τ for us. Consider once again the null vector field given by (41): Recall that ∂ v and z are linearly independent and g(∂ v , z ) = −1, and that the spacelike distribution H = span(∂ v , z ) ⊥ is spanned by the orthonormal pair x := k∂ v + ∂ x , y := h∂ v + ∂ y . Important for our discussion below are covariant derivatives involving z , x , and y, so let us record them here. First, the only nonzero covariant derivatives of the coordinate basis {∂ v , ∂ u , ∂ x , ∂ y } are Using these, the following covariant derivatives are straightforwardly computed: With respect to the frame {∂ v , x , y , z }, the metric and its inverse are Now we observe that g(∇ z z , ∂ v ) = g(∇ z z , z ) = 0, so that z is geodesic if and only if the following two conditions are satisfied: Meanwhile, the twist function (up to sign) and shear coefficients of z are where the shear was written in complex notation. To construct a Kähler metric in accord with Theorem 3, we will use the null vector fields k + = z and k − = ∂ v = ∇u. Our almost complex structure J will be We now find conditions on k, h, H so that (R 4 , g, z , ∂u) is geodesic-admissible, under the assumption that k, h, H are all functions of x, y only. To begin with, observe that because ∂ v = ∇u is parallel, it is geodesic and shear-free. A glance at the shear condition says that we will therefore need z to be shear-free as well. This happens if and only if the function k + ih is holomorphic, which is equivalent to k being harmonic, In summary, for any smooth function k : R 4 −→ R independent of u, v and harmonic in x, y, with h : R 4 −→ R defined as above, a smooth function H : R 4 −→ R independent of u, v exists making z geodesic if and only if k satisfies (51). In the following proposition, we show that k, h can also be chosen to ensure that the twist function ι of z , Proof. That z is geodesic and shear-free is easily verified, as are (18) and (19) in Remark 3.5. Finally, (20) follows easily because ∇ ∂v z = ∇ z ∂ v = 0. Thus (R 4 , g, z , ∇u) is geodesic-admissible. Now set f = e u and note that ι = k y − h x = −2 < 0. That g K = d(e u z ♭ )(·, J·) is Kähler now follows from the geodesicadmissible case of Theorem 3: with G = −1 and ℓ = 1, as required.
We make three further remarks regarding Proposition 10.1. First, z is a conformal Killing vector field with respect to g K , in fact a homothetic one: This follows from Proposition 5.8. In fact z is null and geodesic in the Lorentzian manifold (R 4 , g), so z /z (u) = −z is a Liouville vector field of the symplectic form ω := d(e u z ♭ ): L −z ω = ω. As z and ∂ v also commute and are shear-free, while the latter is gradient, z is homothetic with respect to the Kähler metric associated to ω. Note that, by contrast, ∂ v is a Killing vector field with respect to g K : L ∂v g K = 0. The second remark concerns the completeness of the plane wave metric g in Proposition 10.1: One fundamental property of plane waves is that they are geodesically complete; see, e.g., [29]. As a consequence, if g R is any complete Riemannian metric on R 4 , then in particular g R (∂ v , ∂ v ) must be bounded because ∂ v has geodesic flow in (R 4 , g); see [8]. But g K (∂ v , ∂ v ) = e u is unbounded, therefore g K cannot be complete. The final remark concerns the Ricci curvature of the induced Kähler manifold (R 4 , g K ), with g K = d(e u z ♭ )(·, J·). Computed with the aid of Mathematica, we note here that the Ricci curvature is nonpositive. In the ordered frame {∂ v , x , y , z }, one has: The relationship between (especially) such spacetimes and Kähler geometry was considered in [15]. There, Flaherty described the complex structure we employ for NUT spacetimes (with Schwarzschild spacetime as a special case), but not the one we find for the Kerr metric. One reason for that is that he employed only the frames, called null tetrads, that appeared in Kinnersley's work [21,20], for which the criteria for integrability simplifies. The Kerr almost complex structure we find can be shown to be integrable either via Theorem 2, applied to a certain null tetrad, or via the more stringent sufficient criteria for integrability given in Theorem 1, applied to a natural frame that, while violating one of the conditions to be a null tetrad, contains the two shear-free geodesic null vector fields of the Kerr metric (see Proposition 11.2). Note that in Kinnersley's null tetrads one of the two null vector fields is only pre-geodesic.
Flaherty did not construct Kähler metrics out of the genuine complex structures he found, since he did not regard the latter as useful to Physics. He was not satisfied with their split-adjoint character mentioned in the introduction, and showed that they are not invariant under certain Lorentz transformations.
Consequently, he developed an ingenious alternative modification of the complex analytic treatment of spacetimes of type D. He extended the definition of an almost complex structure to allow it to take values in the complexified tangent bundle, and defined such a structure using null tetrads. He then showed that for this modified version integrabiliy holds in the Ricci flat case if and only if the spacetime is of type D. From this he shows that any such spacetime is conformal to a modified-Kähler metric, meaning a metric that is Kähler with respect to the modified almost complex structure; but the conformal factor may be complex, so that this modified-Kähler metric may be complex, in the sense of being defined on the complexified tangent bundle. When they are in fact real, these modified-Kähler metrics, which have, of course, Lorentzian signature and are in fact of type D themselves, turn out to be of a rather restrictive type, as they are all local products. But complex modified-Kähler metrics have a more varied structure and were related to other "complex spacetimes" that were studied at the time.
In comparison, here we stick with genuine almost complex structures and Kähler metrics associated to Kerr, conformally Kerr and NUT spacetimes. The cost of this choice is that the Lorentzian metric and its associated Kähler metric are not simply conformal to each other, but are related in more complex manner.
11.1. Kähler metrics for Lorentzian metrics of Petrov type D. As is well-known, by the Goldberg-Sachs Theorem [18] (see also [26,Chapter 5]), a Lorentzian metric of Petrov type D admits two null geodesic vector fields which are both shear-free. Even when these satisfy the conditions of Theorem 1, neither one may be gradient, or even near-gradient in the sense of (19)(i). In this case they do not give rise to a geodesic-admissible manifold, and one cannot form with them Kähler metrics with symplectic form (22). However, if one takes k + to be one of these geodesic fields, and k − a pre/geodesic field which is a function multiple of the second, it is still possible to form very similar Kähler metrics, with τ being replaced by a function that is not directly associated with k − .
Proposition 11.1. Let g be a Petrov type D metric on an oriented 4-manifold M , with null shear-free vector fields k ± , with k + geodesic, k − pre/geodesic and g(k + , k − ) constant. Assume also that the conditions in i) of Theorem 1 hold for k ± . Suppose u is any smooth function on M with gradient in V = span(k + , k − ), which is not constant along integral curves of k + . Then for any smooth realvalued function f defined on the range of u, and J determined by k ± as in Section 3, Thus as in Proposition 4.1, non-degeneracy of ω will hold if ω(x , y ) and ω(k + , k − ) do not vanish, for the usual x , y . Computing, we see that because d k + u and g(k + , k − ) are nonvanishing, this holds exactly when f f ′ ι is nonzero. J-invariance of ω is checked as in Theorem 3, where the case where one frame vector field is in H and the other in V requires both the constancy of g(k + , k − ) and the vanishing of du on H. Tameness of J is also checked as in Theorem 3.
In the next section we will need to apply a generalization of this proposition, with weaker assumptions, as g(k + , k − ) is not assumed constant and a multiple of k ♭ + is required to form the symplectic form. Proposition 11.2. Let g be a Petrov type D metric on an oriented 4-manifold, with null shear-free vector fields k ± , with k + geodesic, k − pre/geodesic and g(k + , k − ) < 0. Assume also that the conditions in i) of Theorem 1 hold for k ± . Set p := 1/ −g(k + , k − ). Suppose u is a smooth function on M and f a smooth positive function defined on the range of u, for which f (u)/p has a gradient in V = span(k + , k − ). If J is the complex structure determined by k ± as in Section 3, then g K = d(f (u)pk ♭ + )(·, J·) is Kähler in any region where ι k + < 0 and d k + (log(f (u)p)) < 0, with ι k + computed as in Theorem 3.
Note that if p = 1 above, this becomes essentially Proposition 11.1.
Proof. Settingk ± := pk ± , we note that g(k + ,k − ) = −1 whilek + is a null pre-geodesic vector field with ∇k +k + = (d k + p)k + , as k + is geodesic. Finallỹ k ± define the same almost complex structure as k ± . It follows from the null pre-geodesic character ofk + that ω := d(f (u)k ♭ + ) still vanishes on the pairk + , x , with x in H. Similarly, we also see that ω(k + ,k − ) = −(f ′ du(k + ) + f d k+ p), as can be verified by breaking ω into two terms as usual and using g(k + ,k − ) = −1. Next, for the usual orthonormal frame x , y on H we have ω(x , y ) = f dk ♭ + (x , y ) = −f pι k + . Thus ω will be symplectic if f (f ′ du(k + ) + f d k + p)ι k + = 0. For J-invariance of ω, again the crucial test is when one vector field lies in V and the other in H. Since we know it vanishes for the null pre-geodesic vector fieldk + and a vector field in H, we need to show it also fork − and such a vector field. As in Theorem 3, we calculate, noting that g([k − , x ], k + ) = 0 as usual, and relying on the constancy of g(k + ,k − ): Vanishing of this follows if d x (log f (u)) = d x (log p). Vanishing for any x with values in H will thus hold if log(f (u)/p), or equivalently f (u)/p, has a vertical gradient. Tameness of J in the region specified by the inequalities in the theorem follows from the above calculations of ω(x , y ) (note that f , p are positive) and ω(k + ,k − ), where for the latter we note that −( Note thatk ± satisfy the conditions of Theorem 2, rather than Theorem 1.
Remark 11.3. Note that Remark 3.3 applies here, while Remark 4.2 follows analogously, so that g K of Proposition 11.2 is ambihermitian. and Σ is a totally geodesic hypersurface called the "equatorial plane", which contains a cylinder called the "ring singularity", where the Kerr metric g is singular. We construct a Kähler metric g K on M ′ , in the case where the Kerr spacetime is "rapidly rotating", a status determined by an inequality between the two constant parameters in g. A similar Kähler metric can be constructed on M ′′ if the complex structure we choose is changed by a minus sign on a distribution denoted H and described below.
We observe in passing that while the limiting case a = 0, in which one recovers the Schwarzschild spacetime, is a warped product with fiber a 2-sphere, the Kerr metric itself is not warped in this manner. Kerr spacetime is "rapidly rotating" if a > m, where the parameter m corresponds to the mass, and a to the angular momentum per unit mass of the spherical body or black hole being modeled by the Kerr metric. In particular, note that if a > m then ∆ has no real roots. Our choice of the rapidly rotating version is made for convenience only, to simplify the singular domain of the metric.
On the other hand, the set of points where ρ = 0, namely {(t, 0, π/2, ϕ)}, constitutes a genuine curvature singularity, called the ring singularity, which is topologically Σ := R × S 1 . After a simple analytic extension to include {θ = 0}, {θ = π}, Kerr spacetime will be defined on R 2 × S 2 − Σ, endowed with an extension of the metric above. We refer the reader to [26] for more on the geometry of the Kerr metric.
The Kerr metric is the most famous example of a Ricci-flat Lorentzian 4manifold of Petrov Type D (as defined in [27]). As mentioned earlier, by the Goldberg-Sachs Theorem [18] (see also [26,Chapter 5]), this means that it has precisely two geodesic and shear-free null vector fields, which we denote by k ± .
In the standard coordinate frame {∂ t , ∂ r , ∂ ϑ , ∂ ϕ }, they are given by (see [26, p. 79ff.]), which are everywhere linearly independent. We now define an almost complex structure J in the usual way, by defining On the spacelike distribution H = span(k + , k − ) ⊥ , we take the orthonormal pair and complete J by defining (The notations E 2 , E 3 conform to the notation found in [26], which we reference below.) As mentioned in subsection 11.1, our setup is not admissible in any of the three senses of Definition 3.6: neither k ± can be expressed as proportional to a gradient, nor is g(k + , k − ) constant. To verify integrability of J, we will therefore use Proposition 11.2. We already mentioned that k ± are null, shear-free and geodesic. Observe next that conditions (i) of Theorem 1 hold because as can be easily verified using [26, p. 95-6]. 4 Next, one easily calculates that g(k + , k − ) = −2ρ 2 /∆ < 0, so that Note that on p. 95 of [26], there are two occurrences of "r/ √ ε∆(E0 ± E1)" which should in fact be "ρ/ √ ε∆(E0 ± E1)," with ρ given by (53). Also, on p. 96 of [26] the Lie bracket [E2, E3] should equal "−r √ ε∆ ρ −3 E3," not "−r √ ε∆ ρE3." Note that p is a function of only r and ϑ. Let u = e h(r) p, with h to be determined later, and f (u) = u, which is positive on the (positive) range of u. The conditions of Proposition 11.2 we want satisfied can be written in the form Now the inverse matrix to that of g has the nonzero components at the same entries as those of g, and g rr = 1/ρ 2 > 0. Thus ∇h(r) = g rr h r ∂ r , which lies in V . Thus the first condition in (57) is automatically satisfied, since log f (u) = log p + h(r). The second condition is just = (2g rr (log ρ) r − g rr (log ∆) r )g rr = g rr g rr (r/ρ 2 − 2(r − m)/∆).
As 1/ρ 2 > 1/(r 2 + a 2 + 1), this will be satisfied at any point of M − Σ if h(r) = h a,m (r) is given by for any fixed constant r 0 . The function ι, which is, up to sign, the twist function of k + , is given (cf. [26]) in the ordered basis E 2 , E 3 , by Thus ι vanishes only on the totally geodesic hypersurface P given by ϑ = π/2, known as the "equatorial plane" (a plane is gotten by also fixing a value of t). Hence ι will be negative "below" it, so that the Kähler metric g K = d(e h(r) p 2 k ♭ + )) is defined on the set M ′ := {(t, r, ϑ, ϕ) ∈ R 2 × S 2 | π/2 < ϑ < π}.
Note that Dixon describes in [14] a Kähler metric which is obtained by a type of Wick rotation of the Kerr metric, also not defined on the whole spacetime, and whose Kähler form was given earlier in [2]. He also showed this metric is in fact ambitoric, and studied its domain and asymptotic behaviour near the singular sets, including the non-rapdily rotating case.
11.3. A Kähler metric on NUT spacetime. As noted above, Kerr spacetime is an example of a Ricci-flat Lorentzian 4-manifold of Petrov type D.
All such 4-manifolds have been classified in [20,21]; there are fourteen such metrics, each of which contains between one and four independent parameters. Among these are the three NUT spacetimes (Newman-Unti-Tamburino), which were invented as generalizations of the Schwarzschild spacetime, and are also limiting cases of the Kerr-NUT spacetimes. The NUT spacetime we consider is given in [20] in local coordinates {u, r, x, y} by g uu = −|ρ| 2 (r 2 − 2mr − l 2 ) , g ur = −1 , g ry = 2l cos x, g uy = 2|ρ| 2 l cos x(r 2 − 2mr − l 2 ) , g xx = r 2 + l 2 , g yy = −|ρ| 2 (r 2 − 2mr − l 2 )(4l 2 cos 2 x) + (r 2 + l 2 ) sin 2 x, all other components being zero, with ρ := − 1 r + il and l a positive constant (the metric (60) is equation (3.47) in [21] with a = 0; note that [21,20] work in metric index 3, so the metric above is minus that which appears in [21]). NUT spacetime is Ricci-flat and topologically an open subset of R 2 × S 2 , with x, y serving the roles of that ϑ, ϕ played in the case of the Kerr spacetime, respectively. Its two shear-free null vector fields are k + := ∂ r , which is geodesic, and which is in fact properly pre-geodesic (i.e., ∇ k − k − = αk − for a function α(r, x) that is not identically zero). We now construct a Kähler metric via Proposition 11.1. However, instead of verifying condition (i) of Theorem 1 via some orthonormal pair E 2 , E 3 in H = span(k + , k − ) ⊥ (as in (55)), we shall first of all combine them to form the following complex-valued vector field, thereby defining the null tetrad {k + , m , m , k − }. Null tetrads constitute the frames of interest in the Newman-Penrose formalism [23]; we adopt this here because this is the formalism in which all the metrics and their quantities of interest in [21] are expressed. Here m is the complex-conjugate of m , which for Kerr-NUT spacetime, as derived in [21], takes the form m := −ρ √ 2 2il cot x ∂ u + ∂ x + i csc x ∂ y .
Our almost complex structure J now takes the form Jk + := k − , Jk − := −k + , Jm := im , Jm := −i m (the latter two equalities follow from defining JE 2 := E 3 and JE 3 := −E 2 ). We now wish to invoke Proposition 11.1, with u = −r and f (u) = e u = e −r . To verify that its conditions hold, note first that d k+ (−r) = −1 and ∇u = −∂ u + r 2 −2mr+l 2 r 2 +l 2 ∂ r ∈ span(k + , k − ). Next we show that conditions (i) of Theorem 1 hold. To begin with, Thus [k ± , m] are multiples of m, and hence [k ± , E i ] lie in H, for i = 2, 3, thus verifying conditions (i) of Theorem 1. Finally, according to the Newman-Penrose formalism, the twist of k + is determined from ρ as follows and hence g K = d(e −r k ♭ + )(·, J·) is Kähler on the entire domain of NUT spacetime lying in S 2 × R 2 . This is in contrast to the Kähler metric induced from Kerr spacetime that we constructed earlier, which was not defined on or above the equatorial plane ϑ = π/2. 11.4. A conformally Kerr metric and its induced Kähler metric. Our final Petrov type D example is in fact pre-geodesic-admissible. The Lorentzian metric will be conformal to Kerr spacetime, and the almost complex structure will be defined with respect to the frame of the form {k + , E 2 , E 3 , ∇r} instead of the frame {k + , E 2 , E 3 , k − }. Let g denote the metric of the maximally extended "rapidly rotating" Kerr spacetime given by (52) above, and recalling (53), definẽ This conformal factor is chosen so as to ensure that the gradient ∇r = ρ 2 ∆ ∇r ∆ ρ 2 ∂r = ∂ r hasg-unit length and thereforeg-geodesic flow: g( ∇r, ∇r) = ∆ ρ 2 g rr = 1.
It is well defined as k, ∇r are everywhere linearly independent. We now show thatg is pre-geodesic-admissible (recall Definition 3.6) and admits a Kähler metric in accord with Theorem 3. Indeed, observe that k is pre-geodesic via (63), t := ∇r is a geodesic gradient, because | ∇r|g = 1, andg(k, t) = k(r) = 1 is constant. We are thus left with verifying the shear conditionJ ∇ o k = ∇ oJ k and that V satisfies the mixed condition. For the former, we first recall that the shear, and in particular its vanishing, are conformally invariant notions, so vanishing automatically holds for k. For ∇r = ∂ r , on the other hand, we use the following facts. The E i 's are part of the well-known Boyer-Lindquist frame for the Kerr metric, in which one of the frame vector fields is a function multiple of ∂ r . Covariant derivatives and bracket relations for these fields are known and we use them to compute the shear and various quantities associated with theẼ i 's. In particular, straightforward computations of this type, using [26, p. 95-6], show that g( ∇ E 2 ∂ r , E 2 ) =g( ∇ E 3 ∂ r , E 3 ) ,g( ∇ E 2 ∂ r , E 3 ) =g( ∇ E 3 ∂ r , E 2 ) = 0, so that the (relative) shearσ ∂r =σ ∂r 1 + iσ ∂r 2 of ∂ r is identically zero: Similarly, one shows so thatg([∂ r , ·], k) H = 0, or equivalently, V is satisfies the mixed condition. Thus g is pre-geodesic-admissible and soJ is integrable. We now find a Kähler metric g K = d(f (r)k ♭ )(·,J ·) on the region M ′ \ {r = 0}, where M ′ given in (59).
First, the twist function of k is conformally invariant. Thus the corresponding functionι, when computed in the ordered basis E 2 , J E 2 , is given by (58), and in particular is negative on M ′ . As for this particular pre-geodesic-admissible example we have τ = r, we now set f to be the positive function of r given by f = e −h(r) , for h(r) = h a,m (r) := r r 0 q(x) dx with r 0 a constant, where q(r) = 2((r − m)/(r 2 − 2mr + a 2 ) − 2/r).
Note that q(r) is only defined for r = 0.

A Lie group example with vector fields which are not shear-free
Consider the 4-dimensional solvable real Lie algebra g with ordered basis k, t, x , y defined by the following Lie bracket relations, where we list only the non-zero ones: [k, x ] = y, [t, y ] = y , [t, k] = k, [x , y ] = y + rk, with r a nonzero real constant. Four dimensional solvable Lie algebras have been classified using symbolic software, cf. [11] (note that we are choosing a slightly different form for the bracket relations, isomorphic to one of the canonical normal forms given there). By Lie's third fundamental theorem, there exists a Lie group G whose Lie algebra is g, and one can take it to be simply connected, as we do. The left invariant vector fields of G give a realization of this Lie algebra, and we will denote the left invariant vector fields corresponding to the above generators by the same letters. Define a Lorentzian metric g on G by choosing an inner product on g making the above four vectors orthonormal, with g(k, k) = −g(t, t) = 1, and then extending it to the tangent bundle of G as a left-invariant metric. Then one easily checks that so by (5), relations (65) imply σ k 1 = −σ t 2 = 0, σ k 2 = σ t 1 = −1 = 0, so that the shear operators of k and t are nonzero, and for J defined as in Section 3 (with orientation compatible with the choice y := Jx ), the conditions of Theorem 1 hold, and thus J is integrable. Next, the twist function of k is |ι| = |g(k, [x , y ])| = |r| = 0, so that the sign of ι is fixed by the choice of r. Since k is proper and g(k, t) is constant, Theorem 4 implies that we can define a Kähler metric on a region of G to be determined, if t is near-gradient with respect to g, in the sense of (19)i). By (64), the orthogonal complement of t is integrable. If the constant length vector field t is also geodesic, then one can easily see that t ♭ is closed, hence t is locally gradient, and by simply connectedness of G, in fact globally a gradient. So it remains to show that t is geodesic.
As a left invariant vector field, ∇ t t = −ad * t (t), (see [9, Proposition 3.18]), where ad * t denotes the metric adjoint of the differential at t of the adjoint representation. As this differential is given by the Lie bracket with t, it follows from (64), that g(t, [t, ·]) vanishes identically on vector fields of G, so that ∇ t t = 0. Thus t = ∇τ for a function τ on G. One can check easily that k is not geodesic, pre-geodesic or Killing, so that g is not admissible. Thus by Theorem 4, G admits Kähler metrics of the form (24), with Kähler form (22), on the region of G in which f ι < 0 and f ′ G/ℓ − f (g(∇ k k, t) − d t (g(k, k))/2) = −f ′ − f < 0. Thus chosing f (τ ) = e τ and r < 0, the induced Kähler metric g K will be defined on the entire Lie group G.