Electromagnetic Field Theory in $(N+1)$ -Space-Time: A Modern Time-Domain Tensor/Array Introduction

Abstract

In this paper, a modern time-domain introduction is presented for electromagnetic field theory in ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> +1)-space-time. It uses a consistent tensor/array notation that accommodates the description of electromagnetic phenomena in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> -dimensional space (plus time), a requirement that turns up in present-day theoretical cosmology, where a unified theory of electromagnetic and gravitational phenomena is aimed at. The standard vectorial approach, adequate for describing electromagnetic phenomena in (3+1)-space-time, turns out to be not generalizable to ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> +1)-space-time for <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> >; 3 and the tensor/array approach that, in fact, has been introduced in Einstein's theory of relativity, proves, together with its accompanying notation, to furnish the appropriate tools. Furthermore, such an approach turns out to lead to considerable simplifications, such as the complete superfluousness of standard vector calculus and the standard condition on the right-handedness of the reference frames employed. Since the field equations do no more than interrelate (in a particular manner) changes of the field quantities in time to their changes in space, only elementary properties of (spatial and temporal) derivatives are needed to formulate the theory. The tensor/array notation furthermore furnishes indications about the structure of the field equations in any of the space-time discretization procedures for time-domain field computation. After discussing the field equations, the field/source compatibility relations and the constitutive relations, the field radiated by sources in an unbounded, homogeneous, isotropic, lossless medium is determined. All components of the radiated field are shown to be expressible as elementary operations acting on the scalar Green's function of the scalar wave equation in ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> +1) -space-time. Time-convolution and time-correlation reciprocity relations conclude the general theory. Finally, two items on field computation are touched upon: the space-time-integrated field equations method of computation and the time-domain Cartesian coordinate stretching method for constructing perfectly matched computational embeddings. The performance of these items is illustrated in a demonstrator showing the 1-D pulsed electric-current and magnetic-current sources excited wave propagation in a layered medium.