Curvature Collineations: A Fundamental Symmetry Property of the Space-Times of General Relativity Defined by the Vanishing Lie Derivative of the Riemann Curvature Tensor

Abstract

A Riemannian space Vn is said to admit a particular symmetry which we call a ``curvature collineation'' (CC) if there exists a vector ξi for which £ξRjkmi=0, where Rjkmi is the Riemann curvature tensor and £ξ denotes the Lie derivative. The investigation of this symmetry property of space-time is strongly motivated by the all-important role of the Riemannian curvature tensor in the theory of general relativity. For space-times with zero Ricci tensor, it follows that the more familiar symmetries such as projective and conformal collineations (including affine collineations, motions, conformal and homothetic motions) are subcases of CC. In a V4 with vanishing scalar curvature R, a covariant conservation law generator is obtained as a consequence of the existence of a CC. This generator is shown to be directly related to a generator obtained by means of a direct construction by Sachs for null electromagnetic radiation fields. For pure null-gravitational space-times (implying vanishing Ricci tensor) which admit CC, a similar covariant conservation law generator is shown to exist. In addition it is found that such space-times admit the more general generator (recently mentioned by Komar for the case of Killing vectors) of the form (−g Tijkmξiξjξk);m=0, involving the Bel-Robinson tensor Tijkm. Also it is found that the identity of Komar, [−g(ξi;j−ξj;i)];i;j=0, which serves as a covariant generator of field conservation laws in the theory of general relativity appears in a natural manner as an essentially trivial necessary condition for the existence of a CC in space-time. In addition it is shown that for a particular class of CC,£ξK is proportional to K, where K is the Riemannian curvature defined at a point in terms of two vectors, one of which is the CC vector. It is also shown that a space-time which admits certain types of CC also admits cubic first integrals for mass particles with geodesic trajectories. Finally, a class of null electromagnetic space-times is analyzed in detail to obtain the explicit CC vectors which they admit.