[Paper Review] Electrodynamic forces in elastic matter

S. Antoci, L. Mihich|ArXiv.org|Dec 3, 1999

Magnetic and Electromagnetic Effects2 references3 citations

TL;DR

This paper proposes a macroscopic theory for elastic, isotropic matter under electromagnetic fields by generalizing Hooke's law to four dimensions using a spacetime displacement vector. It derives electrostrictive forces from a deformation-dependent electromagnetic Lagrangian, showing how realistic equations of motion can be constructed by linking electromagnetism and elasticity through relativistic principles.

ABSTRACT

A macroscopic theory for the dynamics of elastic, isotropic matter in presence of electromagnetic fields is proposed here. We avail of Gordon's general relativistic derivation of Abraham's electromagnetic energy tensor as starting point. The necessary description of the elastic and of the inertial behaviour of matter is provided through a four dimensional generalisation of Hooke's law, made possible by the introduction of a four dimensional ``displacement'' vector. As intimated by Nordstroem, the physical origin of electrostriction and of magnetostriction is attributed to the change in the constitutive equation of electromagnetism caused by the deformation of matter. The part of the electromagnetic Lagrangian that depends on that deformation is given explicitly for the case of an isotropic medium and the resulting expression of the electrostrictive force is derived, thus showing how more realistic equations of motion for matter subjected to electromagnetic fields can be constructed.

Motivation & Objective

• To develop a macroscopic theory describing the dynamics of elastic, isotropic matter under electromagnetic fields.
• To address the physical origin of electrostriction and magnetostriction by linking them to changes in the electromagnetic constitutive equations due to material deformation.
• To extend classical elasticity to four-dimensional spacetime using a generalized displacement vector.
• To derive a deformation-dependent electromagnetic Lagrangian for isotropic media, enabling accurate modeling of electrodynamic forces.
• To construct more realistic equations of motion for matter in electromagnetic fields by incorporating elastic and inertial responses via relativistic formalism.

Proposed method

• Adopt Gordon's general relativistic derivation of Abraham's electromagnetic energy tensor as a foundational framework.
• Introduce a four-dimensional 'displacement' vector to generalize Hooke's law into spacetime, describing elastic and inertial behavior of matter.
• Model the constitutive equation of electromagnetism as dependent on material deformation, following Nordström's insight.
• Explicitly construct the part of the electromagnetic Lagrangian that depends on deformation for isotropic media.
• Derive the resulting electrostrictive force from the modified Lagrangian using variational principles.
• Combine the deformed electromagnetic Lagrangian with the elastic energy term to yield a unified equation of motion for matter under EM fields.

Experimental results

Research questions

• RQ1 How can the dynamics of elastic matter be consistently coupled to electromagnetic fields in a relativistically invariant way?
• RQ2 What is the physical origin of electrostriction and magnetostriction in terms of material deformation altering electromagnetic constitutive relations?
• RQ3 How can Hooke's law be generalized to four-dimensional spacetime to describe elastic and inertial responses of matter?
• RQ4 What form does the electromagnetic Lagrangian take when it depends explicitly on the deformation of isotropic elastic media?
• RQ5 Can a unified equation of motion be derived that includes both electromagnetic and elastic forces in a consistent relativistic framework?

Key findings

• The paper successfully generalizes Hooke's law to four dimensions using a spacetime displacement vector, enabling a relativistic description of elastic and inertial behavior.
• Electrostriction and magnetostriction are attributed to changes in the electromagnetic constitutive equation due to material deformation, as suggested by Nordström.
• The deformation-dependent part of the electromagnetic Lagrangian is explicitly derived for isotropic media, forming the basis for force calculations.
• The resulting electrostrictive force is derived from the modified Lagrangian, providing a physically grounded mechanism for electromagnetic forces in elastic matter.
• The theory yields a consistent set of equations of motion that incorporate both electromagnetic and elastic forces through a unified variational principle.
• The approach provides a foundation for constructing more realistic models of matter dynamics in electromagnetic fields, particularly in condensed matter systems with strong coupling.