On the geometry of Petrov type II spacetimes

Abstract

In general, geometries of Petrov type II do not admit symmetries in terms of Killing vectors or spinors. We introduce a weaker form of Killing equations which do admit solutions. In particular, there is an analog of the Penrose-Walker Killing spinor. Some of its properties, including associated conservation laws, are discussed. Perturbations of Petrov type II Einstein geometries in terms of a complex scalar Debye potential yield complex solutions to the linearized Einstein equations. The complex linearized Weyl tensor is shown to be half Petrov type N. The remaining curvature component on the algebraically special side is reduced to a first order differential operator acting on the potential.

Parallel spinors, pp-waves, and gravitational perturbations

“…The operators Θ AA ′ and C AA ′ are well-defined in any signature 3 . Our interest in the (generalized) parallel spinor equations presented in table 1 is that they imply the existence of twistor surfaces, which are the basic object that give integration procedures.…”

“…This is conjectured to be true, locally, for perturbations of all algebraically special vacuum spaces, cf. the introduction in [3]; but, as far as we know, the problem has only been completely solved for the case of Minkowski space-time [4], [5], [6,Section 5.7]. Our main result is given in sections 3.…”