Ricci Collineations of Spacetimes

Abstract

In accord with the early observations of Noether, the existence of certain geometric symmetry properties described by continuous groups of motions or homotheties or Ricci collineations lead to conservation laws in the form of first integrals (i.e. constants of the motion) of a dynamical system. Indeed, the fundamental importance of group of motions in spacetime and their relation to the conservation laws of energy, linear momentum, and angular momentum for particles and fields is well known. [1]. Besides, it is very difficult to find exact solutions of Einstein’s field equations (EFEs), as they consist of ten highly non-linear, second order partial differential equations for ten functions of four variables, with inhomogeneous terms, depending upon the distribution of matter-energy. An alternative approach is to specify the symmetry in the physical problem and require the geometry to accommodate it. Requiring a high level of symmetry would not be of much use as it would restrict the problems that could be solved by this approach. The procedure adapted is to be require a minimal symmetry group and find all spacetimes, with their associated isometries, homotheties or Ricci collineations possessing that or higher symmetries [7-11]. For this approach to be effective we need to extend the procedure to smaller symmetry groups and deal with more of them. The original attempts to classify spacetimes by isometries were incomplete [3-4]. The first attempt at a complete classification was for spherically symmetric spacetimes according to their isometries and metrics [7-8]. Later extended to plane symmetric static[9], cylindrically symmetric static [10] and hyperbolically symmetric static spacetimes. The project under consideration is related to finding the classification of certain spacetimes according to their Ricci collineations. These spacetime include: the spherically symmetric spacetimes admitting SO (3) as the non-maxiamal isometry group [20] the hyperbolically symmetric static spacetimes [21] and the cylindrically symmetric spacetimes. Besides it, the general hyperbolically symmetric [11, 22] and the general plane symmetric spacetimes [25-26] are classified according to their isometries and metrics. The plane symmetric static [23-24] and the cylindrically symmetric static [27] manifolds are classified according to their homotheties.