Spacetimes with all Penrose limits diagonalisable

TL;DR

This paper classifies space-time metrics where all Penrose limits are diagonalisable plane waves, revealing specific algebraic types and conditions on the Weyl spinor, including vacuum and nonvacuum solutions.

Contribution

It derives a conformally invariant differential condition on the Weyl spinor for all Penrose limits to be diagonalisable, identifying new solutions beyond known plane waves.

Findings

Vacuum examples include certain nonrotating type D metrics.

Nonvacuum solutions satisfying the condition are also found.

Weyl spinor must be proportional to a valence-4 Killing spinor with a real function.

Abstract

We consider the problem of finding all space-time metrics for which all plane-wave Penrose limits are diagonalisable plane waves. This requirement leads to a conformally invariant differential condition on the Weyl spinor which we analyse for different algebraic types in the Petrov-Pirani-Penrose classification. The only vacuum examples, apart from actual plane waves which are their own Penrose limit, are some of the nonrotating type D metrics, but some nonvacuum solutions are also identified. The condition requires the Weyl spinor, whenever it is nonzero, to be proportional to a valence-4 Killing spinor with a real function of proportionality.