CR Structures and Twisting Vacuum Spacetimes with Two Killing Vectors and Cosmological Constant: Type II and More Special

Abstract

Based on the CR formalism of algebraically special spacetimes by Hill, Lewandowski and Nurowski, we derive a nonlinear system of two real ODEs, of which the general solution determines a twisting type II (or more special) vacuum spacetime with two Killing vectors (commuting or not) and at most seven real parameters in addition to the cosmological constant Lambda. To demonstrate a broad range of interesting spacetimes that these ODEs can capture, special solutions of various Petrov types are presented and described as they appear in this approach. They include Kerr-NUT, Kerr and Debney/Demia\'{n}ski's type II, Lun's type II and III (subclasses of Held-Robinson), MacCallum and Siklos' type III (Lambda0) we found in an earlier paper, along with a new class of type II solutions as a nontrivial limit of Kerr and Debney's type II solutions. Also, we discuss a situation in which the two ODEs can be reduced to one. However, constructing the general solution still remains an open problem.Comment: 22 page