Rumsey's Reaction Concept Generalized

TL;DR

This paper extends Rumsey's reaction concept from a scalar to a four-dimensional differential-form formalism, revealing it as a one-form with both scalar and vector components, enhancing the theoretical understanding of electromagnetic source interactions.

Contribution

It introduces a four-dimensional, coordinate-free differential-form representation of Rumsey's reaction concept, generalizing the original scalar form to include electromagnetic force terms.

Findings

Reaction concept represented as a one-form in four dimensions

Scalar reaction is the temporal component of the one-form

Spatial component corresponds to electromagnetic force terms

Abstract

The reaction concept, introduced by Rumsey in 1954, describes interaction between time-harmonic electromagnetic sources through the fields radiated by the sources. In the original form the concept was a scalar quantity defined by three-dimensional field and source vectors. In the present paper, the representation is extended to four dimensions applying differential-form formalism. It turns out that, in a coordinate-free form, the reaction concept must actually be a one-form, whose temporal component yields Rumsey's scalar reaction. The spatial one-form component corresponds to a three-dimensional Gibbsian-vector reaction which consists of electromagnetic force terms.