Rheometric structure theory, convective differentiation and continuum electrodynamics

Abstract

Local dynamical theories of continuous media may be described in terms of what may be called rheometric structures, meaning local functional relations (equations of state) between relevant tensors on a three-dimen­sional (rheometric) base-space the points of which represent idealized particles of the medium. A general formalism of (relativistic) convective differentiation is developed here for the purpose of relating the (four­-dimensional) dynamical equations of motion to the (three-dimensional) rheometric structure. The formalism is illustrated by application to the case of an ideal electrodynamic medium, the energy-momentum and polarization tensors of which are functionally dependent only on the local values of the (relativistic) metric tensor g ab and the Maxwell field F ab . Self-consistency is shown to require a well-defined general form for the equation of state functions, and it is demonstrated that the resulting theory may be given a variational form by using as a Lagrangian E a D a — neϕ — ϵ where E a and D a are the rest-frame electric field and displacement, n is the conserved particle number density, e is the (fixed) charge per particle, ϕ = - u a A a , the rest-frame electric potential, and ϵ = T ab u a u b , the total rest frame energy density which implicitly contains electric and magnetic contri­butions (the Abraham-Minkowski controversy concerning their explicit form being an irrelevance), while u a and A a are the four-velocity and four-potential.