On the Growing Problem in Conventional and Relativistic Elasticity

In this paper, the principal aspects of the surface growth theory are considered and their application for the relativistic case is discussed. It is shown that such aspects can be reproduced within relativity without significant modifications. The only issue arises during this, is what are the properties of the space, that contains shapes of the body. It is suggested in the work that such the space should be space-like hypersurface of the space-time. Deformation is formalized as diffeomorphism between such hypersurfaces and strain measures are obtained by pulling the metric back or pushing it forward from one hypersurface to another. The evolution of material structure during growth is modeled by a motion of phase boundary along material manifold. The shapes are images of embeddings from such manifold to the leafs of foliation of space-time into space-like hypersurfaces. The reasonings are provided within the centrally symmetric case, that represent the accretion process of compact stars.