Spinor calculus on 5-dimensional spacetimes

TL;DR

This paper extends Penrose's spinor calculus from 4-dimensional Lorentzian geometry to 5-dimensional Lorentzian geometry, developing new tools and formalism for higher-dimensional spacetime analysis.

Contribution

It introduces a 5-dimensional spinor calculus, including covariant derivatives and curvature spinors, and generalizes the Newman-Penrose formalism to 5D.

Findings

Development of 5D spin covariant derivative

Detailed study of curvature spinors in 5D

Extension of Newman-Penrose formalism to 5D

Abstract

Penrose's spinor calculus of 4-dimensional Lorentzian geometry is extended to the case of 5-dimensional Lorentzian geometry. Such fruitful ideas in Penrose's spinor calculus as the spin covariant derivative, the curvature spinors or the definition of the spin coefficients on a spin frame can be carried over to the spinor calculus in 5-dimensional Lorentzian geometry. The algebraic and differential properties of the curvature spinors are studied in detail and as an application we extend the well-known 4-dimensional Newman-Penrose formalism to a 5-dimensional spacetime.