Twisting, type-N vacuum gravitational fields with symmetries

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The Einstein field equations for twisting, type-N fields in empty space possessing two noncommuting Killing vectors are reduced to a single second-order ordinary differential equation for a complex function. Alternative forms of this basic equation are also presented; in particular, an appropriate Legendre transform provides a partial linearization, leading to a single real, nonlinear, third-order ordinary differential equation.
© 1988 The American Physical Society. The present work was supported in part by the National Science Foundation Grant No. AST 8514911, and by the U.S. National Aeronautics and Space Administration Astronomy/Relativity Branch. I would like to thank F. B. Estabrook, for his hospitality at the Jet Propulsion Laboratory, for discussions, and for providing computing facilities; K. S. Thorne, for his hospitality at Caltech, for pointing out the interest of exact solutions in relation with numerical treatments, and for the opportunity to discuss matters within his research group; and H. D. Wahlquist, for conversations. The detailed form of Eq. (27) was found by using the symbolic manipulation program sMP. I thank Xiao-He Zhang for his help in getting me started in sMP. Support by the Del Amo (Spain) Foundation is gratefully acknowledged.
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