On 3+1 decompositions with respect to an observer field via differential forms

TL;DR

This paper develops a general framework for 3+1 decompositions of differential forms on Lorentzian manifolds relative to any observer field, utilizing connection theory, with explicit formulas and applications to Maxwell equations.

Contribution

It introduces a unified approach to 3+1 decompositions using differential forms and connections, providing explicit formulas applicable to Maxwell equations.

Findings

Explicit formulas for 3+1 decompositions are derived.

The framework applies to Maxwell equations on Lorentzian manifolds.

Connections on principal bundles are used to facilitate the decomposition.

Abstract

3+1 decompositions of differential forms on a Lorentzian manifold (M,g;+ - - -) with respect to arbitrary observer field and the decomposition of the standard operations acting on them are studied, making use of the ideas of the theory of connections on principal bundles. Simple explicit general formulas are given as well as their application to the Maxwell equations.