Hermitian extension of the four-dimensional Hooke's law

TL;DR

This paper extends the classical four-dimensional Hooke's law into a Hermitian form using complex displacement vectors, unifying elastic, inertial, and electromagnetic phenomena within a relativistic framework.

Contribution

It introduces a Hermitian generalization of the four-dimensional Hooke's law, linking elasticity with electromagnetic field evolution through complex displacement vectors.

Findings

Hermitian extension unifies elastic and electromagnetic descriptions.

Real part describes elastic and inertial motion.

Imaginary part models electromagnetic field evolution.

Abstract

It has been shown recently that the classical law of elasticity, expressed in terms of the displacement three-vector and of the symmetric deformation three-tensor, can be extended to the four dimensions of special and of general relativity with a physically meaningful outcome. In fact, the resulting stress- momentum-energy tensor can provide a unified account of both the elastic and the inertial properties of uncharged matter. The extension of the displacement vector to the four dimensions of spacetime allows a further possibility. If the real displacement four-vector is complemented with an imaginary part, the resulting complex ``displacement'' four-vector allows for a complex, Hermitian generalisation of the four-dimensional Hooke's law. Let the complex, Hermitian ``stress-momentum-energy'' tensor density built in this way be subjected to the usual conservation condition. It turns out that, while the real part of the latter equation is able to account for the motion of electrically charged, elastic matter, the imaginary part of the same equation can describe the evolution of the electromagnetic field and of its sources. The Hermitian extension of Hooke's law is performed by availing of the postulate of ``transposition invariance'', introduced in 1945 by A. Einstein for finding the nonsymmetric generalisation of his theory of gravitation of 1915.