A conservation law formulation of nonlinear elasticity in general relativity

Details A conservation law formulation of nonlinear elasticity in general relativity A conservation law formulation of nonlinear elasticity in general relativity

2011-07-13

arxiv.org/abs/1107.2551v1

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

31 pages, 21 pdf figures

Scientific paper

We present a practical framework for ideal hyperelasticity in numerical relativity. For this purpose, we recast the formalism of Carter and Quintana as a set of Eulerian conservation laws in an arbitrary 3+1 split of spacetime. The resulting equations are presented as an extension of the standard Valencia formalism for a perfect fluid, with additional terms in the stress-energy tensor, plus a set of kinematic conservation laws that evolve a configuration gradient. We prove that the equations can be made symmetric hyperbolic by suitable constraint additions, at least in a neighbourhood of the unsheared state. We discuss the Newtonian limit of our formalism and its relation to a second formalism also used in Newtonian elasticity. We validate our framework by numerically solving a set of Riemann problems in Minkowski spacetime, as well as Newtonian ones from the literature.

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