Generalization of the Geroch-Held-Penrose formalism to higher dimensions
Mark Durkee, Vojtech Pravda, Alena Pravdova, Harvey S. Reall
TL;DR
The paper extends the GHP formalism to $d\ge4$ spacetimes, providing a covariant framework with derivatives that respect boosts and spins and yielding a streamlined set of curvature and Bianchi equations. It develops the GHP decomposition for Weyl and Ricci tensors, introduces covariant Maxwell equations for $(p+1)$-forms, and analyzes algebraic specialness, null rotations, and the priming operation to simplify computations. The authors apply the formalism to concrete settings, including codimension-2 hypersurfaces and the optics of Weyl-aligned null directions in Type N spacetimes, and establish results on algebraically special Maxwell fields in higher dimensions. Overall, the higher-dimensional GHP approach offers a compact, covariant toolkit for exploring perturbations and algebraic structure in gravity theories with $d>4$.
Abstract
Geroch, Held and Penrose invented a formalism for studying spacetimes admitting one or two preferred null directions. This approach is very useful for studying algebraically special spacetimes and their perturbations. In the present paper, the formalism is generalized to higher-dimensional spacetimes. This new formalism leads to equations that are considerably simpler than those of the higher-dimensional Newman-Penrose formalism employed previously. The dynamics of p-form test fields is analyzed using the new formalism and some results concerning algebraically special p-form fields are proved.