On the alternative description of complex holomorphic and Lorentz geometries in four dimensions

TL;DR

This paper introduces an alternative approach to describing four-dimensional complex holomorphic and Lorentz geometries using transection fields and exterior forms, bypassing direct reliance on the metric tensor.

Contribution

It develops a novel framework for space-time geometry based on transection fields and exterior forms, providing a metric-independent description of curvature and metricity.

Findings

Provides a new method for describing space-time geometry without metric tensor references.

Defines a fiber bundle framework for globalizing local geometric constructions.

Offers a criterion to distinguish Lorentz geometry within this new framework.

Abstract

The equivalence of a conformal metric on 4-dimensional space-time and a local field of 3-dimensional subspaces of the space of 2-forms over space-time is discussed and the basic notion of transection is introduced. Corresponding relation is spread to the metric case in terms of notion of normalized ordered oriented transection field. As a result, one obtains a possibility to handle the metric geometry without any references to the metric tensor itself on a distinct base which nevertheless contains all the information on metricity. Moreover, the notion of space-time curvature is provided with its natural counterpart in the transection `language' in a form of curvature endomorphism as well. To globalize the local constructions introduced, a certain fiber bundle is defined whose sections are equivalent to normalized ordered oriented transection fields and locally to the metric tensor on space-time. The criterion distinguishing the Lorentz geometry is discussed. The resulting alternative method of the description of space-time metricity, dealing with exterior forms foliation alone, seems to be of a power compatible with one of the standard concept based on the metric tensor.