Motions in relativistic spaces

Abstract

A definition of rigidity for rectilinear motion in Minkowski space was first proposed by Born [I]. Herglotz [2] and Noether [3] independently formulated a condition of rigidity for general rigid motion in Minkowski space. Salzman and Taub [4] examined the concept of rigidity in curved spaces. More recently rigid motion has been investigated by Rayner [5], Pirani and Williams [6] and Boyer [7]. The rigidity condition used is that the distance between every neighboring pair of particles, measured orthogonal to the world line of either of them, remains constant along the world lines. Pirani and Williams [6] constructed integrability conditions for the equations of rigid motion in a gravitational field and gave a new proof for the Herglotz-Noether theorem that in the absence of a gravitational field every rotating rigid motion is isometric. Boyer [7] has obtained sufficient, but not necessary, conditions for the validity of the Herglotz-Noether theorem in curved manifolds. Irrotational motions in Relativistic Spaces have been investigated by Edelen [8]. Here such motions are classified as either strongly irrotational or Fermi irrotational. Necessary and sufficient conditions that a space allow such motions are found. Yano [9] used the Lie derivative as a tool for the investigation of isometric, conformal, and homothetic motions in a Riemannian Space. Here these motions are studied further, in Relativistic spaces, and their relation to rigid motion investigated.