Two special classes of space-times admitting a non-null valence two Killing spinor

TL;DR

This paper investigates specific space-times with non-null Killing spinors, revealing that many are not explicitly known and depend on particular algebraic conditions, with some lacking Killing vectors and characterized by arbitrary functions.

Contribution

It identifies the assumptions behind standard constructions of such space-times and provides an exhaustive classification of cases where these assumptions fail.

Findings

Many space-times with non-null Killing spinors are not explicitly known.

Violations of key algebraic conditions lead to new classes of space-times.

Some space-times admit no Killing vectors and depend on arbitrary functions.

Abstract

Non-conformally flat space-times admitting a non-null Killing spinor of valence two are investigated in the Geroch-Held-Penrose formalism. Contrary to popular belief these space-times are not all explicitly known. It is shown that the standard construction hinges on the tacit assumption that certain integrability conditions hold, implying two algebraic relations, KS1 and KS2, for the spin coefficients and the components of the Ricci spinor. An exhaustive list of (conformal classes of) space-times, in which either KS1 or KS2 are violated, is presented. The resulting space-times are each other's Sachs transforms, in general admit no Killing vectors and are characterized by a single arbitrary function.