Chapter 1 Relating material and space-time metrics within relativistic elasticity: a dynamical example EGLR Vaz, Irene Brito and J Carot Abstract Given a space-time and a continuous medium with elastic properties de- scribed by a 3-dimensional material space, one can ask whether they are compati- ble in the context of relativistic elasticity. Here a non-static, spherically symmetric spacetime metric is considered and we investigate the conditions for that metric to correspond to different 3-dimensional material metrics. 1.1 General results Let (M, g) be a spacetime. The material space X is a 3-dimensional manifold en- dowed with a Riemannian metric γ , the material metric; points in X can then be thought of as the particles of which the material is made of. Coordinates in M will be denoted as x a for a = 0, 1, 2, 3, and coordinates in X as y A , A = 1, 2, 3. The ma- terial metric γ is not a dynamical quantity of the theory and it roughly describes distances between neighboring particles in the relaxed state of the material. The spacetime configuration of the material is said to be completely specified whenever a submersion ψ : M → X is given; if one chooses coordinate charts in M and X as above, then y A = y A (x b ) and the physical laws describing the mechanical properties of the material can then be expressed in terms of a hyperbolic second order system of PDE. The differential map ψ ∗ : T p M → T ψ( p) X is then represented in E.G.L.R. Vaz Departamento de Matem´ atica para a Ciˆ encia e Tecnologia, Universidade do Minho 4800 058 Guimar˜ aes, Portugal, e-mail: evaz@mct.uminho.pt Irene Brito Departamento de Matem´ atica para a Ciˆ encia e Tecnologia, Universidade do Minho 4800 058 Guimar˜ aes, Portugal, e-mail: ireneb@mct.uminho.pt J Carot Departament de Fsica, Universitat de les Illes Balears, Cra Valldemossa pk 7.5, E-07122 Palma de Mallorca, Spain, e-mail: jcarot@uib.es 1