Elasticity Theory in General Relativity

Abstract

The general relativistic theory of elasticity is reviewed from a Lagrangian, as opposed to Eulerian, perspective. The equations of motion and stress-energy-momentum tensor for a hyperelastic body are derived from the gauge-invariant action principle first considered by DeWitt. This action is a natural extension of the action for a single relativistic particle. The central object in the Lagrangian treatment is the Landau-Lifshitz radar metric, which is the relativistic version of the right Cauchy-Green deformation tensor. We also introduce relativistic definitions of the deformation gradient, Green strain, and first and second Piola-Kirchhoff stress tensors. A gauge-fixed description of relativistic hyperelasticity is also presented, and the nonrelativistic theory is derived in the limit as the speed of light becomes infinite.

12 pages, 2 figures, to be published in Classical and Quantum Gravity

Cited by in corpus (10)

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• Relativistic Elasticity II
• Relativistic Theory of Elastic Bodies in the Presence of Gravitational Waves
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• Extended body dynamics in general relativity: hyperelastic models
• Some geometrical aspects of gravitational waves using continuum mechanics analogy: State of the art and potential consequences
• Souriau's Relativistic general covariant formulation of hyperelasticity revisited
• Encounter between an extended hyperelastic body and a Schwarzschild black hole with quadrupole-order effects
• Nonlocal extension of causal thermodynamics of the isotropic cosmic fluid
• Compact elastic objects in general relativity