Rigid Motion and Adapted Frames

Abstract

The aim here is to describe the rigid motion of a continuous medium in special and general relativity. Section 7.1 defines a rigid rod in special relativity, and Sect. 7.2 shows the link with the space coordinates of a certain kind of accelerating frame in flat spacetimes. Section 7.3 then sets up a notation for describing the arbitrary smooth motion of a continuous medium in general curved spacetimes, defining the proper metric of such a medium. Section 7.4 singles out rigid motions and shows that the rod in Sect. 7.1 undergoes rigid motion in the more generally defined sense. Section 7.5 defines a rate of strain tensor for a continuous medium in general relativity and reformulates the rigidity criterion. Section 7.6 aims to classify all possible rigid motions in special relativity, reemphasizing the link with semi-Euclidean frames adapted to accelerating observers in special relativity. Then, Sects. 7.7 and 7.8 describe rigid motion without rotation and rigid rotation, respectively. Along the way we introduce the notion of Fermi–Walker transport and discuss its relevance for rigid motions. Section 7.9 brings together all the above themes in an account of a recent generalization of the notion of uniform acceleration, thereby characterizing a wide class of rigid motions.