Complex conformal rescalings and complex Lorentz transformations
Making use of the group \(GL(2,{\mathbb{C}})\otimes G\tilde L(2,{\mathbb{C}})\) here the author examines in detail the effect of the elements \(\{1,\tilde A\}\) on spin-frames, tetrads, the derivative operator, various variables of the complexified Newman-Penrose formalism, twistors and massless free fields. The decomposition of a general element of \(GL(2,{\mathbb{C}})\otimes G\tilde L(2,{\mathbb{C}})\) into more basic ones such as standard conformal rescalings, complex null rotations, etc. is studied in the treatment of complex conformal rescalings and complex Lorentz transformations. It is proved that a general transformation can be decomposed into the following basic ones: 1. a pure spin transformation \(\{sI,s^{-1}I\}\), \(s\neq 0\); 2. one of a) a basic left conformal rescaling, b) a basic right conformal rescaling, c) a standard conformal rescaling; 3. a) a left null rotation \(\{A_ 1(t),I\}\) about \(\ell\), b) a right null rotation \(\{I,\tilde A_ 1(c)\}\) about \(\ell\); 4. a) a left null rotation \(\{A_ 3(b),I\}\) about n, b) a right null rotation \(\{I,\tilde A_ 3(b)\}\) about n; 5. a) a left boost-rotation \(\{A_ 2(a),I\}\), b) a right boost- rotation \(\{I,\tilde A_ 2(a)\}\); 6. a basic reflection. The transformation rules for various Newman-Penrose variables under these transformations are discussed in detail in this paper.
scientific article; zbMATH DE number 3981986