Effective connections and fields associated with shear-free null congruences

TL;DR

This paper investigates a special class of shear-free null congruences linked to Weyl tensor principal null directions, revealing their association with an effective affine connection containing nonmetricity and torsion, and introduces related invariant operators.

Contribution

It introduces a new subclass of shear-free null congruences with unique geometric properties and associates a Maxwell-like field with quantized charges, along with novel invariant differential operators.

Findings

Special shear-free null congruences are parallel with respect to an effective affine connection.

A Maxwell-like field associated with these congruences exhibits self-quantized electric charge.

Two invariant operators nullify the principal spinor of these congruences.

Abstract

A special subclass of shear-free null congruences (SFC) is studied, with tangent vector field being a repeated principal null direction of the Weyl tensor. We demonstrate that this field is parallel with respect to an effective affine connection which contains the Weyl nonmetricity and the skew symmetric torsion. On the other hand, a Maxwell-like field can be directly associated with any special SFC, and the electric charge for bounded singularities of this field turns to be ``self-quantized''. Two invariant differential operators are introduced which can be thought of as spinor analogues of the Beltrami operators and both nullify the principal spinor of any special SFC.