Wedging spacetime principal null directions

TL;DR

This paper explores the algebraic and differential properties of wedge products of principal null directions in various spacetimes, extending previous geometrical results to include algebraically special cases with multiple null directions.

Contribution

It introduces a unified framework for analyzing principal null directions using wedge products, extending prior results to algebraically special spacetimes with multiple null directions.

Findings

Derived algebraic relations for wedge products in type I spacetimes

Extended geometrical analysis to algebraically special spacetimes

Illustrated methods with vacuum and nonvacuum spacetime examples

Abstract

Taking wedge products of the $p$ distinct principal null directions associated with the eigen-bivectors of the Weyl tensor associated with the Petrov classification, when linearly independent, one is able to express them in terms of the eigenvalues governing this decomposition. We study here algebraic and differential properties of such $p$-forms by completing previous geometrical results concerning type I spacetimes and extending that analysis to algebraically special spacetimes with at least 2 distinct principal null directions. A number of vacuum and nonvacuum spacetimes are examined to illustrate the general treatment.