A Macroscopic Field Equation in Spacetime Algebra

Abstract

Geometric or Clifford algebra is a powerful, coordinate-free language for mathematical physics, offering more compact and insightful descriptions of natural laws than existing legacy formalisms in areas ranging from basic geometry and complex analysis to quantum mechanics and general relativity [1]–[5]. In its four-dimensional form employing the Minkowski metric, known as spacetime algebra, it is extraordinarily effective at describing electrodynamics in a relativistic framework, compressing Maxwell's equations into one simple expression that applies independent of any preferred inertial reference frame.However, as the practitioners of spacetime algebra have typically come from either a pure mathematics or theoretical physics background, only the microscopic form of Maxwell's equation, that which applies to fundamental particles in a vacuum, has received widespread attention. The conventional macroscopic simplification used by engineers, wherein the effects of bound charges and currents in a dielectric medium are subsumed into phenomenological constituent parameters—the relative permittivity and permeability—is frequently neglected.In this paper I describe a macroscopic form of Maxwell's equation in spacetime algebra where only free charges and currents appear explicitly.