Generalized Siklos space-times

TL;DR

This paper introduces generalized Siklos space-times, extending the classical solutions by incorporating complex Dirac spinors satisfying a Killing-spinor-like equation, and analyzes their geometric and physical properties.

Contribution

It derives the local form of these generalized space-times and characterizes their matter content and gravitational wave degrees of freedom.

Findings

Killing function must be real in Lorentz signature

Dirac spinor is Majorana unless space-time is conformally flat

Matter sources include pure radiation and space-like perfect fluid

Abstract

Motivated by supersymmetry methods in general relativity, we study four-dimensional Lorentzian space-times with a complex Dirac spinor field satisfying a Killing-spinor-like equation where the Killing constant is promoted to a complex function. We call the resulting geometry a generalized Siklos space-time. After deriving a number of identities for complex spaces, we specialize to Lorentz signature, where we show that the Killing function must be real and that the corresponding Dirac spinor is Majorana (as long as the space-time is not conformally flat), and we obtain the local form of the metric. We show that the purely gravitational degrees of freedom correspond to waves, whereas the matter sources generically correspond, via Einstein's field equations, to a sum of pure radiation and a space-like perfect fluid. Consequently, we conclude that the physically relevant case is obtained when the Killing function is homogeneous on the wave surfaces.