Complex geometry and pre-metric electromagnetism

TL;DR

This paper investigates the deep connection between complex geometry and pre-metric electromagnetism, proposing a framework where spacetime metrics emerge from electromagnetic wave properties using complex structures on bivectors.

Contribution

It introduces a novel approach linking complex structures on bivectors to the emergence of spacetime metrics in electromagnetism, without assuming a metric a priori.

Findings

Complex structures on bivectors relate to 3+1 spacetime decompositions.

Scalar products on bivectors can induce Lorentzian metrics.

Electromagnetic waves influence the spacetime metric structure.

Abstract

The intimate link between complex geometry and the problem of the pre-metric formulation of electromagnetism is explored. In particular, the relationship between 3+1 decompositions of R4 and the decompositions of the vector space of bivectors over R4 into real and imaginary subspaces relative to a choice of complex structure is emphasized. The role of the various scalar products on the space of bivectors that are defined in terms of a volume element on R4 and a complex structure on the space of bivectors that makes it C-linear isomorphic to C3 is discussed in the context of formulation of a theory of electromagnetism in which the Lorentzian metric on spacetime follows as a consequence of the existence of electromagnetic waves, not a prior assumption.