Covariant theory of conductivity in ideal fluid or solid media

Abstract

After a preparatory account of the established theory of non-conducting perfect fluid media, with emphasis on the important but traditionally neglected concept of the 4-momentum 1-form associated with each chemically independent constituent, it is shown how to generalise the theory to allow for conductivity by extending the variational formalism in terms of independent displacements of the world-lines.

Attention is concentrated initially on the simplest possible conducting model, in which appart from a single conserved particle current the only other constituent is the entropy-current whose flow world-lines are displaced independently of those of the conserved particles in the variational formulation, resistive dissipation being included by allowing the variationally defined force density acting between the particle and entropy currents to be non-zero. The model so obtained is fully determined by the specification of the resistivity coefficient and the traditional thermodynamic variables of the corresponding non-conducting thermal equilibrium state if it is restricted by postulating that it satisfies a “regularity ansatz” to the effect that the separate 4-momenta associated with the (non-conserved) entropy and the (conserved) particles are respectively directed allong the corresponding flow directions. It is shown that this regularity ansatz is consistent with good hyperbolic causal behaviour, unlike a previous ansatz proposed by Landau and Lifshitz, which is interpretable as a degeneracy requirement to the effect that the separate 4-momenta have the same direction as each other, and which results in (inevitably superluminal) parabolic behaviour. Another ansatz, proposed much earlier by Eckart, is shown to be effectively equivalent to the mixed-up requirement that the 4-momentum associated with the entropy to be directed not along its own flow direction but along that of the particles, and (as recently shown by Hiscock and Lindblom) results in even worse (quasi-elliptic) behavior.

After this analysis of the simplest possible well behaved thermally conducting model, it is shown how the principles by which it was constructed can be extended to allow for multiple (including electrically charged) currents, in solid as well as fluid media.