Conjugaisons finies et milieux continus en relativité générale. (Finite conjugacy and continuous media in general relativity) (Q1098341)

We present here the main elements of a mechanics of relativistic continua relying upon a concept of ``finite conjugacy'' between two relativistic motions described by two unit vector-fields u and u' defined on two different relativistic manifolds \({\mathcal M}\) and \({\mathcal M}'.\) This purely relativistic, global, and intrinsic theory leads, together with a new approach of the deformation tensors in relativity, to a differential system of equations for the conjugacies which is neither under-determined nor over-determined. A rough study of the propagation of the conjugacy-waves shows then that it is advisable to consider the notion of a finite conjugacy as a satisfying relativistic extension of the classical and tridimensional notion of a finite deformation in mechanics, and to identify the spatial conjugacy-waves obtained with the ordinary acoustic waves. Drastic particularizations of the space-times \({\mathcal M}\) and \({\mathcal M}'\), of the motions u and u', of the admissible types of conjugacies and of the elastic behaviour of the continua under study allow to recover, as very important but particular cases, the tridimensional non-relativistic theory of elasticity for finite deformations and nonlinear behaviour, as well as the main theories of relativistic elasticity already proposed e.g. by \textit{Y. Choquet-Bruhat} and the author [C. R. Acad. Sci., Paris, Ser. A 276, 1317-1320 (1973; Zbl 0286.73019)], \textit{C. B. Rayner} [Proc. R. Soc. Lond., Ser. A 272, 44-53 (1963; Zbl 0108.216)] \textit{B. Carter} and \textit{H. Quintana} [ibid. 331, 57-58 (1972; Zbl 0249.73093)].