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

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General Relativity

Scientific paper

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, i.e., when the left and right Weyl spinors Ψ andtilde Ψ 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.

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