Relativistic second gradient theory of continuous media

TL;DR

This paper extends Souriau's variational relativity framework to develop a second gradient theory of continuous media in General Relativity, identifying new invariants and connecting relativistic and classical continuum mechanics.

Contribution

It introduces a second order gradient extension of Souriau's relativistic continuum mechanics, revealing new invariants and clarifying the theoretical basis of higher gradient continuum theories.

Findings

Derived new higher order diffeomorphism invariants.

Connected relativistic invariants to classical continuum mechanics.

Showed some invariants converge to objective quantities in the Galilean limit.

Abstract

Variational Relativity is a framework developed by Souriau in the sixties to better formulate General Relativity and its classical limit\,: Classical Continuum Mechanics. It has been used, for instance, to formulate Hyperelasticity in General Relativity. In that case, two primary variables are involved, the universe (Lorentzian) metric $g$ and the matter field $\Psi$. A Lagrangian density depending on the 1-jet of these variables is then introduced which must satisfy the principle of General Covariance. Souriau proved in 1958 that under these hypotheses, the Lagrangian density depends only on the punctual value of the matter field $\Psi$ and of a secondary variable $\mathbf{K}$, the conformation, an invariant of the diffeomorphism group, which is the Relativistic analog of the inverse of the right Cauchy--Green tensor. In the present work, an extension of Souriau's results to a second order gradient theory in General Relativity is presented. Accordingly, new higher order diffeomorphisms invariants are found. Their classical limits are calculated, showing that the 3-dimensional Continuum Mechanics second gradient theory can be derived from such a relativistic theory. Some of these invariants converge to objective quantities in the Galilean limit, others to non-objective quantities. The present work contributes thus to clarify the theoretical foundation of higher gradient Continuum Mechanics theory.