Complex relativity and real solutions. V. The flat space background
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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.
KeywordsStandard Form Differential Geometry Good Deal Real Solution Complex Relativity
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