Generalized Maxwell equations for exterior-algebra multivectors in (k, n) space-time dimensions

Abstract

This paper presents an exterior-algebra generalization of electromagnetic fields and source currents as multivectors of grades r and r−1, respectively, in a flat space-time with n space and k time dimensions. Formulas for the Maxwell equations and the Lorentz force for arbitrary values of r, n, and k are postulated in terms of interior and exterior derivatives, in a form that closely resembles their vector-calculus analogues. These formulas lead to solutions in terms of potentials of grade r−1, and to conservation laws in terms of a stress–energy–momentum tensor of rank 2 for any values of r, n, and k, for which a simple explicit formula is given. As an application, an expression for the flux of the stress–energy–momentum tensor across an (n+k−1)-dimensional slice of space-time is given in terms of the Fourier transform of the potentials. The abstraction of Maxwell equations with exterior calculus combines the simplicity and intuitiveness of vector calculus, as the formulas admit explicit expressions, with the power of tensors and differential forms, as the formulas can be given for any values of r, n, and k.