A new class of non-aligned Einstein-Maxwell solutions with a geodesic, shearfree and non-expanding multiple Debever-Penrose vector

TL;DR

This paper introduces a new class of Einstein-Maxwell solutions with a geodesic, shearfree, non-expanding multiple Debever-Penrose vector, specifically addressing cases where the null vector k is twisting but non-expanding, expanding previous classifications.

Contribution

It presents a novel family of solutions where the null vector k is twisting but non-expanding, filling a gap in the classification of algebraically special Einstein-Maxwell fields with zero cosmological constant.

Findings

Identifies solutions with twisting but non-expanding null vectors

Clarifies the role of the Newman-Penrose coefficient π in these solutions

Expands the classification of algebraically special Einstein-Maxwell fields

Abstract

In a recent study [NVdB2017] of algebraically special Einstein-Maxwell fields it was shown that, for non-zero cosmological constant, non-aligned solutions cannot have a geodesic and shearfree multiple Debever-Penrose vector k. When $\Lambda=0$ such solutions do exist and can be classified, after fixing the null-tetrad such that $\Psi_0 = \Psi_1 = \Phi_1 = 0$ and $\Phi_0 = 1$, according to whether the Newman-Penrose coefficient $\pi$ is 0 or not. The family $\pi = 0$ contains the Griffiths solutions (Griffiths 1986), with as sub-families the Cahen-Spelkens, Cahen-Leroy and Szekeres metrics. It was claimed in [Griffiths 1986] and repeated in [NVdB2017] that for $\pi = 0$ both null-rays k and l are non-twisting: while it is certainly true that $\mu \Im (\rho) = 0$, the case $\mu = 0$ appears to have been overlooked. A family of solutions is presented in which k is twisting but non-expanding.