Rigid Motions in Einstein Spaces

Abstract

The dyadic formulation of general relativity is used systematically to discuss rigid congruences in Einstein space-time. For space-time of uniform curvature, the quotient space metrics of rotating and accelerating rigid bodies are obtained. For Einstein space-time of nonuniform curvature, all irrotational, nonisometric, rigid motions are explicitly displayed. They have one degree of freedom,and occur only in degenerate static metrics of Class B. Rotating rigid congruences in Einstein space-timeof nonuniform curvature are shown to have no degrees of freedom. Their evolution is in fact found to be governed by a complete set of 14 first-order total differential equations, linear in the time derivatives of the dyadic variables. Such rotating motions are shown further to be constrained by a set of algebraic conditions, and the implication of this for the validity of the Herglotz-Noether theorem in Einstein space-time is discussed.