R. Desmorat - Sur la formulation covariante générale de l'hyperélasticité relativiste

Notes exposé

• Variational relativity geometric framework: Souriau (1958, 1960, 1964). This framework does not assume a priori the definition of a space-time.
• 4D formalism for continuum mechanics: perfect matter model of Souriau, inspired by Gauge Theory. Matter described by a compact orientable submanifold with border of dimension 3, called the body, the manifold which labels the particles.
• The matter studies in space time is sent into a 3D world.
• In classic 3D continuum mechanics: a configuration is an embedding. Different from the 4D formalism, where the configuration is an application from the 4D universe to the 3D body. $F=Tp:T\mathcal{B}\to T\mathcal{E}$.
• There is a relativistic version of mass conservation, doing the pullback of the mass mesure on the universe (4D), defining a vector field called by Souriau the current of matter.
• Conformation: cornerstone of the formulation of large-scale Relativistic hyperelasticity, defined by Souriau in 1958. Can be compared to the 3D definition of the Cauchy Green tensor.
• Fundamental postulate of general relativity: the laws of physics are independent of the choice of coordinates.
• General covariance: the Lagrangian is invariant by any local diffeomorphism of the Universe. It is important before the introduction of a Space-time.
• Another functional that is general covariant is the Lagrangian of the General Relativity of Hilbert-Einstein.
• The tensor impulse-energy can be obtained as the derivation of a Lagrangian of matter. Thanks to general covariance, we obtain that the divergence of the energy-impulsion tensor is equal to zero.
• Theorem by Souriau 1958: for a perfect matter (i.e. hyperlasticity), the Lagrangian density can be written as a function of the conformation tensor.
• Minkowski: general relativity mechanics.
• Then, they do orthogonal decompositions on the tensors they obtain, which gives the definition of some kind of a relativistic Cauchy stress tensor. Relativistic hyperelasticity
• Schwarzschild spacetime (1916) : Schwarzschild metric: static solution to Einstein vacuum equation. It can be expressed in cartesian isotropic coordinates. Then, each of the quantity obtained are looked from the point of view of the metric. This gives the definition of spatial velocity, generalised Lorentz factor, the conformation, and the stress-energy tensor. Each component of the stress-energy-momentum tensor give the energy density, the linear momentum. and the stress tensor. Galilean limit of relativistic hyperelasticity
• At the limit each of the components of the equations obtained before can correspond to the classical equations of continuum mechanics (conservation of mass, conservation of linear momentum)

Summary

Methodology

1. Choice of variables
2. Choice of jets of the
3. Invariance by diffeo
4. Introduce a direction of the observator
5. Introduce time
6. Passage to the limit to obtain the classical laws of continuum mechanics

Key points

• The conformation tensor plays a fundamental role
• The Newton-Cartan formulation ofGalilean relativity os obtained as the infinite light speed limit $c\to \infty$.