Geometric Representations of Whitney Forms and their Generalization to Minkowski Spacetime

TL;DR

This paper introduces two equivalent coordinate-independent formulas for Whitney forms, generalizes them to Minkowski spacetime, and demonstrates their application in covariant electromagnetism and wave simulations.

Contribution

It presents new coordinate-free representations of Whitney forms, extends their applicability to Minkowski spacetime, and explores their duality and geometric interpretation.

Findings

Formulas valid in any coordinates for Whitney forms.

Generalization of Whitney forms to Minkowski spacetime.

Successful simulation of a wave equation in 1+1 dimensions.

Abstract

In this work, we present two alternative yet equivalent representation formulae for Whitney forms that are valid for any choice of coordinates, and generalizes the original characterization of Whitney forms in Whitney (1957) that requires the use of barycentric coordinates. In addition, we demonstrate that these formulae appropriately generalize the notion of Whitney forms and barycentric coordinates to Minkowski spacetime, and naturally to any other flat pseudo-Riemannian manifold. These alternate forms are related to each other through a duality between the exterior algebras on vectors and covectors. In addition, these two formulae have a geometrically intuitive interpretation which provide interesting insights into their structure. Furthermore, we obtain an explicit characterization of the Hodge dual of the space of Whitney forms on Minkowski spacetime, and this opens the door to treating the theory of classical electromagnetism in a fully covariant fashion through the combination of multisymplectic variational integration and spacetime finite-element exterior calculus (FEEC) techniques. We conclude the paper with the results from an $\mathbb{R}^{1+1}$ wave equation simulation, as a proof-of-concept of the spacetime formulation described herein.