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.
Highlights
It’s well-known that, given a smooth 3-dimensional metric, Riemannian or Lorentzian, coordinates can locally be found in which the metric is diagonal
In a celebrated paper [18], Penrose showed that every space-time has a plane wave limit
This plane wave Penrose limit as originally defined, [18], entailed defining a coordinate system based on a choice of null-geodesic segment in a 4-dimensional space-time and taking a limit defined explicitly in these coordinates, which leads to a metric in the highly symmetric class of plane waves
Summary
It’s well-known that, given a smooth 3-dimensional metric, Riemannian or Lorentzian, coordinates can locally be found in which the metric is diagonal (see [3] for Riemannian, and [5] for Lorentzian). We consider the problem of characterising those space-times which do have all Penrose limits diagonalisable This turns out to be a strong condition on the Weyl curvature, if the extra condition of vacuum or Einstein is imposed in the space-time, when the only examples are plane waves themselves, the Lobachevski plane waves of Siklos [21] and some non-rotating type D solutions. The condition is seen to be conformally-invariant and any conformally-flat space-time necessarily has all its Penrose limits diagonalisable by the result in [23] The analysis of this condition, and in particular its association with Killing spinors, leads to our main result, Proposition 1. In an appendix we consider Penrose limits of the vacuum Kasner metric
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