There are at most 14 independent real algebraic invariants of the Riemann tensor in a four-dimensional Lorentzian space. In the general case, these invariants can be written in terms of four different types of quantities: R , the real curvature scalar, two complex invariants I and J formed from the Weyl spinor, three real invariants I6, I7 and I8 formed from the trace-free Ricci spinor and three complex mixed invariants K, L and M. Carminati and McLenaghan [5] give some geometrical interpretations of the role played by the mixed invariants in Einstein-Maxwell and perfect fluid cases. They show that 16 invariants are needed to cover certain degenerate cases such as Einstein-Maxwell and perfect fluid and show that previously known sets fail to be complete in the perfect fluid case. In the general case, the invariants I and J essentially determine the components of the Weyl spinor in a canonical tetrad frame; likewise the invariants I6, I7 and I8 essentially determine the components of the trace-free Ricci spinor in a (in general different) canonical tetrad frame. These mixed invariants then give the orientation between the frames of these two spinors. The six real pieces of information in K, L and M are precisely the information needed to do this. A table is given of invariants which give a complete set for each Petrov type of the Weyl spinor 
$$\Psi _{A{\text{ }}B{\text{ }}C{\text{ }}D}$$

 and for each Segre type of the trace-free Ricci spinor 
$$\Phi _{A{\text{ }}B{\text{ }}\dot C{\text{ }}\dot D}$$

 This table involves 17 real invariants, including one real invariant and one complex invariant involving 
$$\Psi _{A{\text{ }}B{\text{ }}C{\text{ }}D}$$

 , 
$$\Psi _{\dot A{\text{ }}\dot B{\text{ }}\dot C{\text{ }}\dot D}$$

 and 
$$\Phi _{A{\text{ }}B{\text{ }}\dot C{\text{ }}\dot D}$$

 in some degenerate cases.

A Complete Set of Riemann Invariants

Killing equations for self‐dual, Euclidean Einstein spaces with nonvanishing cosmological term Λ are considered. It is shown that in this case one can reduce ten Killing equations to one master equation. Some important solutions of the master equation are found and one of these solutions is analyzed in detail as it leads to metrics defined by the field equation well known in Euclidean or complex relativity. Ernst potentials for the solutions considered are found.

Killing vector fields in self-dual, Euclidean Einstein spaces with Λ¬=;0

This is the first of a series of papers on complex spaces and their use in complex relativity. The basic aim is to develop the theory of complex relativity but only insofar as it helps in obtaining, and understanding, real solutions of Einstein's vacuum equations as slices of complex solutions. In this first paper, specific aims of the whole series are first presented. The basic equations and key entities, which are used later, are presented. The basic relativistic language used is that of Newman and Penrose. Included is a discussion of a number of important properties which arise in the development of the basic equations and key concepts, these properties being mainly ones which are not apparent in standard real formulations.

Complex relativity and real solutions. I: Introduction

In this paper an application of Newman-Janis algorithm in spherically symmetric metrics with the functions M(u,r) and e(u,r) has been discussed. After the transformation of the metric via this algorithm, these two functions M(u,r) and e(u,r) will be transformed to depend on the three variables u,r,θ. With these functions of three variables, all the Newman–Penrose (NP) spin coefficients, the Ricci as well as the Weyl scalars have been calculated from the Cartan’s structure equations. Using these NP quantities, we first give examples of rotating solutions of Einstein’s field equations like Kerr–Newman, rotating Vaidya solution and rotating Vaidya–Bonnor solution. It is found that the technique developed by Wang and Wu can be used to give further examples of embedded rotating solutions, that the rotating Kerr–Newman solution can be combined smoothly with the rotating Vaidya solution to derive the Kerr–Newman–Vaidya solution, and similarly, Kerr–Newman–Vaidya–Bonnor solution of the field equations. It has also shown that the embedded universes like Kerr–Newman de Sitter, rotating Vaidya–Bonnor–de Sitter, Kerr–Newman–Vaidya–de Sitter can be derived from the general solutions with Wang–Wu function. All rotating embedded solutions derived here can be written in Kerr–Schild forms, showing the extension of Xanthopoulos’s theorem. It is also found that all the rotating solutions admit non-perfect fluids.

Rotating metrics admitting non-perfect fluids

Der Lichtdruck auf Kugeln von beliebigem Material

This paper continues the examination of real metrics and their properties from the viewpoint of complex relativity as initiated by McIntosh and Hickman [1]. Tetrads of real metrics can be formally complexified by complex coordinate transformations and tetrad rotations and their properties investigated from the viewpoint of complex relativity. First, complex bivectors are examined and classified, partly by using the fundamental quadric surface of a metric in projective complex 3-space Pℂ3-an elegant but not well-known method of investigating the null structure of a metric. A generalization of the Mariot-Robinson theorem from real relativity is then given and related to various canonical forms of complex bivectors. The second part of the paper discusses four classes of complex metrics. Real metrics of the first class are ones with a null congruence whose wave surfaces have equal curvature. The second class, a subcase of the first one, is the main one; it contains integrable double Kerr-Schild metrics. Different, but equivalent, definitions of such metrics are given from various viewpoints. Two other subcasses are also discussed. The nonexpanding typs-D vacuum metric is considered and it is shown how complex transformations may be made to write it (and subcases) in double (or single) Kerr-Schild form.

Complex relativity and real solutions II: Classification of complex bivectors and metric classes

In this paper the theory of integrable double Kerr-Schild (IDKS) spaces is examined. The vacuum field equations are shown to reduce to the single equation of Plebanski and Robinson [20]. These metrics are given essentially in terms of one potentialH. First-order perturbations ofH lead to metric (gravitational) perturbations of vacuum algebraically degenerate spaces in a direct manner and give results in agreement with those of Cohen and Kegeles [6, 7, 8], Stewart [9], Teukolsky [5], Torres del Castillo [12, 13], and others. Higher-order perturbations ofH are also obtained with the view that, in the limit, these solutions should yield (new) exact vacuum solutions. The success of this construction lies in the (complex) geometric structure of IDKS spaces. This structure induces a natural splitting of the field equations which allows a potentialization of the perturbation (as well as the vacuum metric itself). It also allows massless spin 1/2 and 1 fields to be examined on the IDKS background in a similar manner.

Complex relativity and real solutions. IV. Perturbations of vacuum Kerr-Schild spaces

Einstein's vacuum field equations are integrated in complex relativity in a major subcase of the class whose Weyl tensor is of the type N⊗N, ie, when the left and right Weyl spinors Ψ and$\tilde \Psi $ are each of typeN The subcase is the complex equivalent of the real nontwisting case Five separate families of solutions are found Three of these are complexified versions of the two families of plane-fronted waves and the Robinson-Trautman real type-N metrics and two are complex solutions which do not have any real slices of Lorentz signature Before the equations are integrated, the relevant general theory and equations are developed in a tetrad frame which is well suited to the discussion of these and a wider class of complex solutions and is called aleft quarter flat frame The relationship between this frame and the coordinates used and some other frames and coordinates, including the complexified version of the frame often used for real type-N metrics, is discussed

Complex relativity and real solutions. III. Real type-N solutions from complexN ⊗N ones

One of the problems in dealing with double (and single) Kerr-Schild metrics is that there is a good deal of combined coordinate and tetrad freedom in writing any particular metric of this type in a standard form, and this freedom is not yet properly understood. This paper investigates part of this problem by examining the freedom in choosing the background flat metric for the standard form of a vacuum IDKS (integrable double Kerr-Schild) metric and by looking at the freedom in choosing the potentialH, which determines IDKS flat metrics in this standard form. Examples are given of different forms ofH which generate flat space. A slight generalization of the vacuum IDKS metric is also given in which thesurface equations andcurvature scalar are zero (i.e.,Φ00=Φ01=Φ11=Λ=0 in Newman-Penrose language) but the othercentral equations and theresidual equations are nonzero.

M. S. Hickman

Papers

Complex relativity and real solutions. I: Introduction

Complex relativity and real solutions II: Classification of complex bivectors and metric classes

Complex relativity and real solutions. IV. Perturbations of vacuum Kerr-Schild spaces

Complex relativity and real solutions. III. Real type-N solutions from complexN ⊗N ones

Complex Relativity and Real Solutions. V. The Flat Space Background