An exterior-algebraic derivation of the symmetric stress–energy–momentum tensor in flat space–time

Abstract

This paper characterizes the symmetric rank-2 stress-energy-momentum tensor associated with fields whose Lagrangian densities are expressed as the dot product of two multivector fields, e. g., scalar or gauge fields, in flat space-time. The tensor is derived by a direct application of exterior-algebraic methods to deal with the invariance of the action to infinitesimal space-time translations; this direct derivation avoids the use of the canonical tensor. Formulas for the tensor components and for the tensor itself are derived for generic values of the multivector grade $s$ and of the number of time and space dimensions, $k$ and $n$, respectively. A simple, coordinate-free, closed-form expression for the interior derivative (divergence) of the symmetric stress-energy-momentum tensor is also obtained: this expression provides a natural generalization of the Lorentz force that appears in the context of the electromagnetic theory. Applications of the formulas derived in this paper to the cases of generalized electromagnetism, Proca action, Yang-Mills fields and conformal invariance are briefly discussed.