Derivation of the symmetric stress-energymomentum tensor in exterior algebra

Colombaro, Ivano; Font Segura, Josep; Martinez, Alfonso

doi:http://dx.doi.org/10.1088/1742-6596/2090/1/012050

Derivation of the symmetric stress-energymomentum tensor in exterior algebra

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Colombaro I, Font-Segura J, Martinez A. Derivation of the symmetric stress-energymomentum tensor in exterior algebra. J Phys Conf Ser. 2021;2090:012050. DOI: 10.1088/1742-6596/2090/1/012050

Abstract

We present a derivation of a manifestly symmetric form of the stress-energy-momentum using the mathematical tools of exterior algebra and exterior calculus, bypassing the standard symmetrizations of the canonical tensor. In a generalized flat space-time with arbitrary time and space dimensions, the tensor is found by evaluating the invariance of the action to infinitesimal space-time translations, using Lagrangian densities that are linear combinations of dot products of multivector fields. An interesting coordinate-free expression is provided for the divergence of the tensor, in terms of the interior and exterior derivatives of the multivector fields that form the Lagrangian density. A generalized Leibniz rule, applied to the variation of action, allows to obtain a conservation law for the derived stress-energy-momentum tensor. We finally show an application to the generalized theory of electromagnetism.