Holonomy and projective symmetry in general relativity
A study of the holonomy group of space-time is undertaken and related to the Segre and Petrov types of then Weyl and E-tensors respectively. Attention is then focused on the E-tensor, and a theorem is proved which states that any space-time M can be disjointly decomposed into open sets on which the Segre type of the E-tensor is constant, the union of which if dense in M. This theorem is then applied to prove a similar theorem for the Ricci tensor using the principal null directions of the E-tensor. Finally, a study of proper projective symmetry in null and non-null Einstein-Maxwell and static, spherically symmetric space-times is performed. A theorem is proved which states that no proper projective symmetry is possible in any null Einstein-Maxwell space-times. This result is then extended to the non-null case under some general restrictions. The static, spherically symmetric space-times are then considered, and those admitting proper projective symmetry are completely determined. The proper projective vector fields are also explicitly calculated.