Null Killing vectors and geometry of null strings in Einstein spaces

Einstein complex spacetimes admitting null Killing or null homothetic Killing vectors are studied. Such vectors define totally null and geodesic 2-surfaces called the null strings or twistor surfaces. Geometric properties of these null strings are discussed. It is shown, that spaces considered are hyperheavenly spaces (HH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {HH}$$\end{document}-spaces) or, if one of the parts of the Weyl tensor vanishes, heavenly spaces (H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {H}$$\end{document}-spaces). The explicit complex metrics admitting null Killing vectors are found. Some Lorentzian and ultrahyperbolic slices of these metrics are discussed.


Introduction
The idea of using the complex numbers in analysis of the spacetime is almost so old, as theory of relativity. [Einstein himself used the imaginary time coordinate in special theory of relativity]. However, advanced complex methods in general theory of relativity have been discovered in the sixties. R. Penrose [1] and Newman [2] introduced the spinorial formalism, Debney et al. [3] and others developed the null tetrad formalism. Several important works were dedicated to the methods of finding new metrics of real spacetime from already known metrics by appropriate complex transformations [4,5]. These methods had their origins in the work [6] where searching for the solutions to the Maxwell equations were considered. Then Newman [7] showed that each asymptotically flat spacetime defines some 4-dimensional complex analytic A. Chudecki (B) Center of Mathematics and Physics, Technical University of Lodz, Al. Politechniki 11, 90-924 Lodz, Poland e-mail: adam.chudecki@p.lodz.pl differential manifold endowed with a holomorphic Riemannian metric. This metric satisfies the vacuum complex Einstein equations and the self-dual or anti-self-dual part of its conformal curvature tensor (Weyl tensor) vanishes. Such a space was called by Newman the heavenly space (H-space).
Plebański [8]showed that vacuum complex Einstein equations for a heavenly space can be reduced to a single second order nonlinear partial differential equation, heavenly equation (H-equation), for one holomorphic function. Besides very interesting mathematical properties, the solutions of heavenly equation have been considered as "basic bricks" which could be used to construct the solutions of Einstein equations by appropriate superpositions of two such solutions [9]. This approach, however, appeared to be limited. This is why the new methods of looking for the real solutions from the complex ones have been developed. One of them is the hyperheavenly space theory.
Hyperheavenly spaces (HH-spaces) was introduced in 1976 in famous work by Plebański and Robinson [10] as a natural generalization of the heavenly spaces. Hyperheavenly spaces with cosmological constant are complex spacetimes with algebraically degenerate self-dual or anti-self-dual part of the Weyl tensor satisfying the vacuum Einstein equations with cosmological constant. The transparent advantage of hyperheavenly spaces theory is the reduction of Einstein equations to one, nonlinear differential equation of the second order, i.e. hyperheavenly equation.
Moreover, it has been pointed out that all the real, algebraically special, vacuum, Lorentzian metrics are hidden inside the hyperheavenly metrics! It seemed, that finding new real vacuum solutions of Einstein field equations with the Lorentzian signature was only a matter of time. It was enough to solve the hyperheavenly equation and then to find Lorentzian slices of respective complex spacetimes. This research programme, often called the Plebański programme has its origin in the works by Trautman [6] and Newman et al. [4,5] (see also [11]). Unfortunately, obtaining the real Lorentzian slices appeared to be more difficult then anyone suspected.
In order to better understand the problem, the structure of hyperheavenly spaces together with their spinorial description have been investigated by Plebański, Finley III et al's. [12][13][14]. Believing that symmetry of the spacetime simplifies the problem, some authors have studied Killing symmetries in heavenly and hyperheavenly spaces [15][16][17][18]. But Lorentzian slices still remain elusive. Except some examples [14,[19][20][21][22] and discussions [23] no general techniques to find the Lorentzian (physical) slices have been presented. Exceptionally, some reality conditions have been analyzed [24], but practical applications of these ideas appeared to be problematic. Probably it was the reason why hyperheavenly machinery became less popular in nineties.
Within five last years hyperheavenly spaces found their place in deep mathematical considerations. Their relation to Walker and Osserman geometry has been noticed in 2008. Some transparent results have been obtained with help of hyperheavenly formalism [25,26]. It appeared, that HH-spaces are the most natural tool in investigating real spaces of the neutral (ultrahyperbolic) signature (+ + −−). Moreover, a few works devoted to Killing symmetries in heavenly and hyperheavenly spaces appeared [27][28][29][30]. These papers generalized the previous ideas of Plebański, Finley III and Sonnleitner [16][17][18]. Between Killing vectors especially useful are these ones, which are tangent to self-dual null strings. The existence of such (null) Killing vectors simplifies the hyperheavenly equation, making it solvable in majority of cases.
The main aim of our work is to find all complex hyperheavenly and heavenly metrics admitting null homothetic and isometric Killing symmetry. Such metrics appear to be important in (++−−) real geometries. However, the existence of a null Killing vector appeared to be helpfull for finding the Lorentzian slices [28]. We develop this idea and examine all possible Lorentzian slices of the complex spacetimes admitting the null Killing vector.
It is well known [23] that if a complex spacetime admits any real Lorentzian slice then both self-dual and anti-self-dual part of the Weyl tensor must be of the same Petrov-Penrose type. So if this complex spacetime is a hyperheavenly space (with or without ) then by the Goldberg-Sachs theorem it admits both self-dual and antiself-dual congruences of null strings (i.e. totally null and geodesic 2-surfaces) which intersect each other and these intersections constitute the congruence of null geodesics. To assume the existence of null Killing vector field and identify this field with congruence of null geodesics seem to be the natural first steps in investigating Lorentzian slices.
The paper is organized as follows.
In Sect. 2 we investigate the general properties of Killing vectors, especially of null ones. Some useful theorems are given and relation between null Killing vector and null strings is pointed out. Then the detailed discussion on the possible Petrov-Penrose types admitting null Killing symmetry is presented. Section 3 is a concise summary of the properties of hyperheavenly spaces. The main goal of our work is to present explicit form of the metrics with null Killing symmetry. The results are gathered in Sects. 4 and 5. There are seven different hyperheavenly metrics with null isometric or homothetic Killing vector and five different heavenly metrics.
In (the most important) Sect. 6 we discuss the possible real slices of the metrics found in preceding sections. In order to find the Lorentzian slices we use the idea of complex coordinate transformations. We introduce the appropriate complex transformations of the complex metrics and then we replace all the holomorphic functions and coordinates by the real ones. After these steps we obtain, quite surprisingly, metrics with the Lorentzian signature. Our technique leads to a new method for obtaining the Lorentzian metrics. This method is different from the analysis of the reality conditions [24], superposition of two heavenly metrics [9] or complex transformation of the real metrics [6]. Finally Lorentzian slices of the type [II] and [D] are found. Then we discuss some metrics of the neutral signature. Two-sided Walker and globally Osserman spaces are obtained. The concluding remarks end our paper. Let M be a 4-dimensional complex analytic differentiable manifold endowed with a holomorphic metric ds 2 . Thus (M, ds 2 ) is a 4-dimensional holomorphic Riemannian manifold and one deals with complex relativity. The metric ds 2 on M can be written in terms of a null complex tetrad (e 1 , e 2 , e 3 , e 4 ) (where e i , i = 1, . . . , 4 are co-vectors) be the inverse basis of (complex) vectors. For our purposes it is useful to use the spinorial formalism. Thus we introduce the respective spinorial images of basis of co-vectors and vectors (2.2) We use the following rules of manipulating the spinorial indices where ∈ AB and ∈ȦḂ are the spinor Levi-Civita symbols Correspondence between the null tetrad formalism and spinorial formalism is realized with the use of the spin-tensor g a AḂ which is defined by the relation g AḂ = g AḂ a e a . It is easy to see that − 1 2 g a AḂ g b AḂ = δ a b and − 1 2 g a AḂ g aCḊ = δ A C δḂ D . The operators ∂ AḂ and ∇ AḂ are the spinorial images of operators ∂ a and ∇ a , respectively, given by In spinorial notation the metric can be written in the form [In complex relativity the 1-forms g AḂ are unrelated. In real relativity there are some constraints for the 1-forms g AḂ . For example, for the real spaces of the signature (+ + −−) there must be g AḂ = g AḂ and for the Lorentzian signature (+ + +−) there must be g AḂ = g BȦ where bar stands for the complex conjugation]. More complete treatment of the spinorial formalism in complex relativity see [31,32]. Now we recall some basic facts about the Killing vectors. The system of Killing equations are given by The Killing vector is said to be conformal, if χ = const, homothetic if χ = χ 0 = const = 0 and isometric if χ = 0. A conformal Killing vector K can be written as Components K a and K AḂ are related by Conformal Killing equations with conformal factor χ (2.7) in spinorial form read and this is equivalent to the following system of equations From (2.11a) and (2.11b) it follows that The integrability conditions of (2.12) in Einstein space (C ABĊḊ = 0, R = −4 ) consist of the following equations [Deep analysis of the Killing equations and their integrability conditions for Killing vectors and Killing tensor fields can be found in [15]].

Null strings via null Killing vectors
The existence of a null Killing vector has a significant influence on the geometry of the space. To explain this we first note that the null Killing vector can be presented in the form where μ A and νḂ are some nonzero spinors. Moreover, it is well known that every spinor symmetric in all indices can be decomposed according to the formula A are some spinors. In particular, there exist spinors A A , B A , AȦ and BȦ such that We prove the following . Analogously we prove that lȦḂ = ν (Ȧ BḂ ) .
Before we formulate another theorem, we present some basic facts about the twodimensional holomorphic distributions and their integral manifolds, i.e. null strings.
Let us consider two-dimensional, holomorphic distribution D μ A = {μ A νḂ, μ A ρḂ} which is given by the Pfaff system Distribution D μ A is self-dual in the sense, that the 2-form "orthogonal" to the 2-plane belonging to D μ A is self-dual. Such a distribution is integrable in the Frobenius sense, if the spinor μ A satisfies the equations Integral manifolds of the congruence D μ A appear to be totally null and geodesic and they are called the null strings. They constitute the congruence of self-dual null strings. Analogously one can consider anti-self-dual two-dimensional distribution D νȦ = {μ A νḂ, σ A νḂ}, μ A σ A = 0 given by the Pfaff system Such distribution is integrable in the Frobenius sense if νḂνĊ ∇ AḂ νĊ = 0 and the integral manifolds are called anti-self-dual null strings. Null strings are the basic geometrical objects in the theory of heavenly and hyperheavenly spaces. Detailed analysis of the properties of the null strings can be found in [33,34]. Theorem 2.2 Let the null Killing vector K AḂ be of the form (2.15). Then the twodimensional self-dual holomorphic distribution {μ A νḂ, μ A ρḂ}, νḂρḂ = 0, is integrable and its integral manifolds constitute the congruence of self-dual null strings and the anti-self-dual distribution {μ A νḂ, σ A νḂ}, μ A σ A = 0, is also integrable and its integral manifolds constitute the congruence of anti-self-dual null strings. Moreover, both Weyl spinors C ABC D and CȦḂĊḊ are algebraically special with μ A and νḂ being the undotted and dotted, respectively, multiple Penrose spinors.
Proof Contracting (2.18) with μ A μ C and remembering that A A = μ A we get (2.21). This means that the spinor μ A defines a congruence of self-dual null strings in the sense that the 2-dimensional holomorphic distribution {μ A νḂ, μ A ρḂ}, νȦρȦ = 0 is integrable and its integrable manifolds constitute the congruence of self-dual null strings. From the complex Sachs-Goldberg theorem it follows [33], that C ABC D is algebraically special and μ A is multiple Penrose undotted spinor, i.e.
Analogously we prove that In particular, from Theorem 2.2 it follows that the integral curves of a null Killing vector are given by the intersection of self-dual and anti-self-dual congruences of null strings.
Note that where A and Ȧ describe the optic properties of the congruence of anti-self-dual and self-dual null strings, respectively. Indeed, if Ȧ = 0 then the self-dual null strings are parallely-propagated, if A = 0 then anti-self-dual null strings are parallely propagated. Therefore the concept of the expansion of the congruence of the null strings is rather different from the concept of expansion of geodesic congruence in the case of the Lorentzian spacetime. Inserting (2.25), A A = μ A and AȦ = νȦ into (2.18), after some straightforward calculations we obtain We prove another important theorem.

Theorem 2.3 Assume that at least one of the spinors C ABC D or CȦḂĊḊ is nonzero. Then
Proof Assume that C ABC D = 0. Then from (2.14f) it follows that ∇Ȧ A χ is the quadruple Debever-Penrose spinor. However, as is well known, two quadruple DP spinors are necessarily linearly dependent so ∇ A1 χ has to be proportional to ∇ A2 χ or, equivalently Acting on (2.27) with ∇Ḃ B and using (2.14e) one quickly obtains Hence if = 0 then ∇Ḃ B χ = 0. Finally, using (2.14e) we get χ = 0 what proves (i). If = 0, then still ∇Ȧ N χ is a quadruple DP-spinor. However, we proved that μ N is a multiple DP-spinor, so with some χȦ. Inserting (2.29) into (2.14e) and contracting with μ B we arrive at the conclusion Ȧ χḂ = 0, so if we want to maintain possible conformal symmetries, the self-dual null string defined by the (conformal) Killing vector must be nonexpanding, Ȧ = 0. Consequently, lȦḂ = 0. Inserting this into (2.14b) and contracting with ∈ṘṠ we finally get χȦ = 0. From (2.29) it follows that ∇ NȦ χ = 0 and this proves (ii).
Summing up, null conformal symmetries can appear only in the Einstein spaces with C ABC D = 0 = CȦḂĊḊ, i.e. in the de-Sitter space (with = 0) or in Minkowski space (with = 0). We do not consider these spaces here.
The null Killing vector field defines congruence of null (complex) geodesics. The optical scalars of Killing vector field can be easily obtained. One gets Thus we conclude that null homothetic Killing field defines null geodesic congruence with nonzero expansion, twist and shear, while null isometric Killing field is nonexpanding, nontwisting and shearfree. It is worth to note, that optical scalars of the null (complex) geodesic congruence defined by K a are related to the expansions of the self-dual congruence of the null strings Ȧ and anti-self-dual congruence of the null strings A . This relation is given by the Eqs. (2.30a)-(2.30c) and the crucial condition (2.31f). More general analysis of the optical properties of the congruences of the null strings can be found in [34]. Gathering above considerations: we reduced the problem of null Killing vectors in Einstein space to the set of equations Inserting (2.31b) and (2.31f) into (2.12) we obtain The formula (2.32) proves, that first covariant derivative of the null Killing vector can be expressed only by the undotted and dotted multiple Penrose spinors and expansions of both self-dual and anti-self-dual congruences of null strings.
Algebraic degeneration conditions C ABC D μ A μ B μ C = 0 and CȦḂĊḊνȦνḂνĊ = 0 can be combined with (2.14a) and (2.14b). After some work we obtain where ,˙ , and˙ are defined by the relations We end this subsection by pointing out two relations essential in further analysis. Contracting (2.34a) with μ S μ T and using (2.31d) we obtain Analogously, contracting (2.34b) with νṠνṪ and using (2.31e) one gets Now we are ready to discuss the possible algebraic types admitting null Killing vector. Note, that we follow Petrov-Penrose algebraic classification of the Weyl spinors C ABC D and CȦḂĊḊ in complex spacetimes [32,8,13,14]. In real relativity with Lorentzian signature (+ + +−) the algebraic type of the undotted Weyl spinor is the same, as the type of dotted Weyl spinor. However, in complex specetimes spinors C ABC D and CȦḂĊḊ are unrelated. Hence, the "mixed" types (like can appear.

Null homothetic symmetries
Here we assume χ 0 = 0, what immediately gives = 0. Simple analysis of Eqs. (2.35)-(2.36) together with (2.31f) brings us to the conclusion that the only possibilities are • Ȧ = 0 (the congruence of self-dual null strings is nonexpanding), μ A A = 0 (the congruence of anti-self-dual null strings is necessarily expanding, more even, expansion A cannot be proportional to DP-spinor μ A ) • A = 0 (the congruence of anti-self-dual null strings is nonexpanding), νȦ Ȧ = 0 (the congruence of self-dual null strings is necessarily expanding, more even, expansion Ȧ cannot be proportional to DP-spinor νȦ) Of course, both possibilities constitute Eintein spaces with the same geometric properties. It is enough to consider only one of them with details, say we conclude that˙ =˙ = 0. Careful analysis of (2.34a) gives = 0. From (2.33a) and (2.33b) we obtain (the last formula is a consequence of (2.34a). The only possible anti-self-dual Petrov types are [N, −]. From (2.37a) we easily get that the only possible self-dual Petrov types are [III, −]; self-dual type [N] is not admitted. Indeed assume, that C ABC D is of the type [N], so C N RST μ N = 0 what gives = 0. Contracting (2.38) with T we obtain T ∇ RȦ T = 0, so T defines the congruence of self-dual null strings. But we proved earlier (see Theorem 2.2), that the self-dual null string is defined by μ T . The number of independent congruences of self-dual null strings is equal to the number of multiple undotted DP-spinors, so there are infinitely many independent congruences of self-dual null strings in the heavenly spaces, two for the self-dual type [D] and only one for the self-dual types [II, III, N]. But here we examine self-dual type [N], so there is only one congruence of the nullstrings. It means, that T must be proportional to μ T or T μ T = 0 ⇒ χ 0 = 0. This contradicts our assumption that χ 0 = 0 and proves that the only possible Petrov types which admit null homothetic symmetries are [III, −] ⊗ [N, −]. Then the congruence of self-dual null strings is nonexpanding and the congruence of anti-self-dual null strings must be expanding. All possible types via geometric properties of the congruence of null strings are presented in the table below: The upper index e means, that the corresponding congruence of null strings is expanding, index n -nonexpanding.

Null isometric symmetries
Here we assume χ 0 = 0. Analysis of Eqs. ( and the Eqs. (2.34a) and (2.34b) read Multiplying (2.42a) by˙ and (2.42b) by and adding both equations one arrives at the useful formula

Hyperheavenly spaces
The considerations from the previous section allow us to establish all possible algebraic types of the spaces which admit the null Killing symmetry. Main aim of the present paper is to find the explicit metrics with such symmetries. Due to Theorem 2.2, any null Killing vector defines the congruences of both self-dual and anti-self-dual null strings and implies the algebraic degeneration of both self-dual and anti-self-dual part of the Weyl curvature spinor. Let us remind the definition of hyperheavenly space [10,[12][13][14].

Definition 3.1
Hyperheavenly space (HH-space) with cosmological constant is a 4-dimensional complex analytic differential manifold endowed with a holomorphic Riemannian metric ds 2 satisfying the vacuum Einstein equations with cosmological constant and such that the self-dual or anti-self-dual part of the Weyl tensor is algebraically degenerate. These kind of spaces admits a congruence of totally null, self-dual (or anti-self-dual, respectively) surfaces.
A complex 4-dimensional space which admits the null Killing vector is equipped with both self-dual and anti-self-dual congruences of null strings. If, moreover, such a space is Einstein then by the Goldberg-Sachs theorem it is algebraically special for both sides and of course is a HH-space. Vacuum Einstein equations in HH-space can be reduced to one, nonlinear, partial differential equation of the second order, for one holomorphic function. This equation is called hyperheavenly equation.
The existence of the null strings allows us to introduce some useful tetrad and the coordinate system. The self-dual null string generated by the null Killing vector is given by the equation μ B μ C ∇Ȧ B μ C = 0 which is equivalent to the Pfaff system μ A g AḂ = 0. Choosing the spinorial basis in such a manner that μ A = (0, μ 2 ), μ 2 = 0 we arrive at the conclusion, that the congruence of self-dual null strings is defined by the Pfaff system e 1 = 0, e 3 = 0 ⇐⇒ g 2Ȧ = 0 (3.1) (the surface element of the null string is given by e 1 ∧ e 3 ). A null tetrad (e 1 , e 2 , e 3 , e 4 ) and a coordinate system (qȦ, pḂ) can be always chosen so that [13,10,14] where φ and QȦḂ = QḂȦ are holomorphic functions. Coordinates qȦ label the null strings and pȦ are coordinates on them. Dual basis is given by Of course, for consistency with (2.3), the rules to raise and lower spinor indices in spinorial differential operators read ∂Ȧ = ∂Ḃ ∈ȦḂ, ∂Ȧ =∈ḂȦ ∂Ḃ, ðȦ = ðḂ ∈ȦḂ, ðȦ =∈ḂȦ ðḂ, so The metric ds 2 is given by where FȦ, NȦ and γ are arbitrary functions of qĊ only (constant on each self-dual null string), is a cosmological constant and θ = θ( pȦ, qḂ) is the key function. The metric is defined by (3.6) with φ and QȦḂ given by (3.9).

Expanding hyperheavenly spaces
If the congruence of self-dual null strings is expanding, we obtain where NȦ and γ are arbitrary functions of qĊ only (constant on each self-dual null string). Instead of the ( pȦ, qḂ)-coordinate system , another one, namely (φ, η, w, t) is universally used with the operators In (φ, η, w, t)-language, the hyperheavenly equation reads where 2NȦ =: ν KȦ + JȦ. The metric (3.6) takes now the form We do not present here the curvature formulas and connection forms. The reader is referred to [12,25,26]. We note only that the expansion of the congruence of the selfdual null strings is proportional to nonzero spinor JȦ, namely Ȧ = − √ 2φ −1 μ 2 JȦ.  [29]. Killing vector has the form and The key function and the curvature read where F = F(x, w, t) and γ = γ (w, t) are arbitrary functions of their arguments such that F x x x = 0 and γ t = 0. Inserting the key function W (4.3) into hyperheavenly equation we get where w = τ w, t = τ t and γ = τ −2 γ . The general solution of the Eq. (4.5) is not known. The metric has the form  [29] for details) one gets C (1) = −2φ 7 f tt (4.8) where f = f (w, t) is an arbitrary function. The metric is Formally it is enough to set C (1) = 0 ⇐⇒ γ t = 0 in subsection 4.1. However, the Eq. (4.5) is still hard to solve. It appears to be much more convenient to attack the problem from the opposite side and consider the space of the type [III] n ⊗ [−] e . As a starting point we take the nonexpanding hyperheavenly spaces and we set CȦḂĊḊ = 0. It allows us to take the general key function as a third-order polynomial in pȦ coordinates (see [27,28] for details). The metric appears to be two-sided Walker [25]. Using the general results from [28] and [25] (especially Theorem 5.1 from [25]) we find the form of the Killing vector K = 2χ 0 pȦ ∂ ∂ pȦ (4.10) and the spinors l AB and lȦḂ The key function and the curvature read where X = X (qṀ ) and Y = Y (qṀ ) are arbitrary functions. The metric is (4.14) [Note, that the conditions CȦḂĊḊ = 0 = assure the existence of infinitely many congruences of nonexpanding anti-self-dual null strings, that is why the space considered belongs to the two-sided Walker class. However, the congruence of anti-self-dual null strings generated by the null Killing vector is expanding.]  [18] (with = 0) and then in [29] (with = 0). The Killing vector reads K = ∂ ∂η (5.1) and

Metrics admitting null isometric symmetries
The key function and the curvature are The metric reads After inserting the key function (5.3) into the hyperheavenly equation we get In order to maintain the type [II] ⊗ [II] we have to assume W t = 0 and W φφφφ = 0. The general solution of the Eq. (5.6) is not known. Its reduction to canonical form is realized by the transformation Considering the key function W as a function of the variables (z, s, w) we obtain the equation Multiplying Eq. (5.8) by φ φ 3 + 6μ 0 −2 one can bring it to the form From (5.9) we infer the existence of the potential = (z, s, w) such that From the (5.10b) one can calculate W Inserting it into (5.10a) we arrive to the equation Of course, φ has to be considered as a function of coordinates s and z, φ = φ(z, s), according to (5.7). The integral in (5.7) can be calculated in all subcases, leading to the condition z = z(s, φ). Unfortunately, the inverse function φ = φ(z, s) is an elementary function only if = 0. However, if = 0 more efficient transformation can be proposed.

Special case = 0
In what follows we assume = 0 and we introduce the transformation (φ, η, w, t) → (x, y, u, v): where is a function of the variables (x, y, u). The hyperheavenly Eq. (5.6) takes the form differentiating Eq. (5.16) twice by ∂ x + i∂ y and using definition of M, after some algebraic work we obtain x M x x + x M yy + M x = 0 (5.17) Equation (5.17) is equivalent to the Euler-Poisson-Darboux equation (EPD equation) and its solutions have been discussed in literature [35]. The form of the metric (5.14) and the Eq. (5.17) are especially useful in obtaining the Lorentzian slices (see Sect. 6).

Spaces of the type [D] e ⊗ [D] e
From the previous subsection one can easily obtain the general metric for the type [D] e ⊗ [D] e with . Self-dual type [D] we get after setting C (1) w). General solution of the hyperheavenly Eq. (5.6) reads W = admits, together with null isometric Killing vector ∂ η , three other isometric Killing vectors ∂ w , ∂ t and w∂ w −η∂ η and -if = 0 -one homothetic Killing vector 2 3 χ 0 (2t∂ t − φ∂ φ + η∂ η ). If = 0 the metric (5.18) can be easily transformed to coordinate system (x, y, u, v) defined by (5.13). According to (5.15) function M = 0 and the metric reads and The key function and the curvature where f = f (z, w) and g = g(w, t) are arbitrary functions of their variables. The metric reads In this case we deal with the hyperheavenly spaces of types [III,N] n ⊗ [N] e with nonexpanding congruence of self-dual null strings defined by the null Killing vector; the congruence of anti-self-dual null strings is still expanding. The respective metrics have been discussed in [28]. The Killing vector has the form The key function and the curvature read 3 (5.26) (1) δ2Ȧδ2Ḃδ2Ċ δ2Ḋ ,Ċ (1) = 2 p2 S(q2, q1 p2 p2) p2 p2 p2 p2 (5.27) where N = N (qṀ ) and S = S(q2, q1 p2 p2) are arbitrary functions and F 0 is a constant. The metric takes the form necessarily must be zero and the hyperheavenly metrics are characterized by • type [III] n ⊗[N] e : F 0 = 0, F 0 can be re-gauged to 1 without any loss of generality, The heavenly reductions of the metric obtained above are especially interesting, because they provide two different null string geometries. One can get the metrics of the type In order to obtain the heavenly metrics of the types [III,N] n ⊗ [−] e one must set S(q2, q1 p2 p2) = f (q2) q1 p2 p2 where f is an arbitrary function of the variable q2.
(It seems, that this is too strong condition and it is enough to set ( p2 S) p2 p2 p2 p2 = 0, but there is unused gauge freedom, which allows to simplify the function S). Heavenly metrics of the type [−] n ⊗ [N] e can be obtained by setting F 0 = 0 = N (once again condition N = 0 is stronger then necessary C (1) = 0, but N = 0 can be obtained by using gauge freedom). Finally The key function and the curvature are (5.32) where N = N (qṀ ) and A = A( p2, q2) are arbitrary functions. The metric reads There are two different heavenly degenerations but they lead to the equivalent heavens.
To get the heavenly space of the type [−] n ⊗ [N] n it is enough to set N q1q1 = 0, then by using gauge freedom one can gauge N away. The heavenly space of the type [N] n ⊗ [−] n can be obtained by setting the function A as a third-order polynomial in p2 (in fact, taking into considerations the remaining gauge freedom, it is enough to set A = f (q2) p2 p2 p2). Gathering, we arrive at the cases The metrics presented in Sects. 4 and 5 are holomorphic. It is an easy matter to carry over all the results to the case of real spaces of the signature (+ + −−). To this end, instead of the holomorphic objects (spinors, null strings, tetrads, coordinates, etc.) we simply deal with the real smooth objects. Real spaces of the neutral signature play an important role in Walker and Osserman geometry [36][37][38]. Recently, it has been recognized that the hyperheavenly formalism allows to obtain transparent results in Walker and Osserman geometry. For example, a new class of metrics admitting selfdual and anti-self-dual, parallely propagated null strings (two-sided Walker spaces) has been found in [25]. These spaces have a natural generalization when only one of the families of null strings is parallely propagated (sesqui-Walker spaces). Such spaces have been defined and investigated in [38]. Probably the most distinguished success of the hyperheavenly methods in Osserman geometry was finding all algebraically degenerate metrics of the globally Osserman space which do not have the Walker property, i.e. they do not admit any parallely propagated null strings [26]. Some of the metrics presented in Sects. 4

Real Lorentzian slices
Of course, the most interesting from the physical point of view are Lorentzian slices. There are still no general techniques of obtaining such slices, except some notes on their properties [23]. However, in some special cases they can be obtained quite easily.
It is well known, that there are only two subcases of the Einstein spaces with Lorentzian signature and null Killing vector (compare [39]). One of them is pp-wave solution. The real metric of the pp-wave solution can be obtained from the complex metric (5.33) of the type [N] n ⊗ [N] n . Detailed discussion of this case can be found in [28]. We only mention, that in this particular case it is enough to consider the necessary condition of existing Lorentzian slice, namely CȦḂĊḊ =C ABC D , which gives noẇ C (1) =C (1) .
Except the pp-wave solution, null Killing vector is admitted by the Lorentzian, Einstein spaces of the type [II] and [D]. It means, that desired Lorentzian slice is hidden in the hyperheavenly metric (5.5) with the key function (5.3) and with curvature given by (5.4). Unfortunatelly, in this case the conditions CȦḂĊḊ =C ABC D are not straightforward and technique which succeeded in pp-wave case, failed.
However, one can consider the (complex) transformation (5.13) which brings the metric to the form (5.14). The metric (5.14) depends on one (complex) function M = M(x, y, u) which satisfies the Eq. (5.17). Treating now the coordinates (x, y, u, v) as real coordinates and the function M as a real smooth function we find that the metric (5.14) automatically becomes real and has the Lorentzian signature. The vacuum Einstein equations have been reduced to the Eq. (5.17). Exactly the same form of the metric with the same equation describing real vacuum Lorentzian types [II] and [D] admitting a null isometric Killing vector can be found in [39]. Summing up, the metric (5.14) with real coordinates is another example of Lorentzian slice of the complex space.
Why does this technique succeed? The first reason is, probably, the explicit use of the imaginary unit in transformation (5.13). It plays no role if we consider (5.13) as complex transformation and the coordinates (x, y, u, v) as complex. But if (x, y, u, v) are real, this step changes automatically the signature of the metric making (5.14) Lorentzian. The second reason is that the metric (5.14) does not depend directly on the key function W but on the function M. The relation between this two functions is given by (5.15) and it contains the imaginary unit. However, differentiating twice the hyperheavenly Eq. (5.16) one can bring it to the form (5.17) which is free of imaginary unit. Finally, we are left with the real metric and the real equation. It is enough to accomplish the construction of Lorentzian spaces of the type [II] and [D] admitting null Killing vector.
It is worth to note that this construction works only in the vacuum case. If cosmological constant = 0 we have not been able to find the Lorentzian slices.
We conclude also that no Einstein spaces with null homothetic symmetries admit Lorentzian slices.

Concluding remarks
In this paper the null Killing vectors (isometric and homothetic) in complex spacetime have been considered. The relation between the existence of null Killing vector and geometry of null strings has been studied in Sect. 2. Because of the existence of null strings the most natural apparatus in investigating null Killing vectors appeared to be the one provided by the theory of hyperheavenly and heavenly spaces. After short summary of the structure of hyperheavenly spaces (Sect. 3), we have been able to present all possible metrics admitting null Killing vector. Only two of them i.e.
• the metric (4.6) of the type [N] e ⊗ [III] n with null homothetic Killing vector • the metric (5.5) of the types [II] e ⊗[II] e with = 0 and with null isometric Killing vector have not been solved completely. In (4.6) the functions F = F(x, w, t) and γ = γ (w, t) satisfy the Eq. (4.5). No solution with F x x x = 0 and γ t = 0 have been found. However, the geometry of this space is so interesting and the type [N] e ⊗ [III] n so rare, that we are going to study the Eq. with = 0 has been reduced to the Eq. (5.12), but this reduction has obvious disadvantages. Like in the previous case, we will deal with this equation soon.
The transparent results are the metrics (4.9) and (4.14) which constitute all heavens with null homothetic symmetry. These cases have been considered in [40] but without giving any explicit form of the metric. We were able to integrate the problem completely.
Perhaps, the most interesting from the physical point of view is searching for examples of real Lorentzian slices of the complex metrics. The first such an example has been presented in [28]. Here we have been able to find the Lorentzian slices of the types [II] e ⊗ [II] e and [D] e ⊗ [D] e with = 0. They are given by the metric (5.14) which depends on one function M of three variables satisfying the Eq. (5.17). Such a metric has been presented earlier (see [41], or in a concise form [39]).
Both these examples gave some valuable hints about obtaining Lorentzian slices. In the first of them the condition CȦḂĊḊ =C ABC D has been successfully used, in the second one a reasonable using of imaginary unit appears to be essential. Nonetheless, the case with nonzero cosmological constant is still unsolved. Taking into considerations the optical properties of the congruences of null geodesics defined by the null isometric Killing vector (2.30a) -(2.30c) we conclude that all such slices must belong to the Kundt class ( [39], xxxi). The vacuum Einstein field equations with cosmological constant for the Kundt class have been gathered in [39]  We hope, that further investigations on the structure of complex spacetimes allow us to find some effective and more general techniques of obtaining real Lorentzian slices.
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