A more direct representation for complex relativity

TL;DR

This paper introduces a new, more direct way to represent complex relativity by using an almost-complex structure on the bundle of real 2-forms, simplifying the geometric framework of general relativity.

Contribution

It proposes an alternative to self-dual complex 2-forms by defining an almost-complex structure on real 2-forms, providing a more straightforward geometric representation.

Findings

Allows a complex orthogonal structure on 2-forms

Parallels the construction with self-dual forms

Supports Debever-Penrose classification

Abstract

An alternative to the representation of complex relativity by self-dual complex 2-forms on the spacetime manifold is presented by assuming that that the bundle of real 2-forms is given an almost-complex structure. From this, one can define a complex orthogonal structure on the bundle of 2-forms, which results in a more direct representation of the complex orthogonal group in three complex dimensions. The geometrical foundations of general relativity are then presented in terms of the bundle of oriented complex orthogonal 3-frames on the bundle of 2-forms in a manner that essentially parallels their construction in terms of self-dual complex 2-forms. It is shown that one can still discuss the Debever-Penrose classification of the Riemannian curvature tensor in terms of the representation presented here.