General study and basic properties of causal symmetries

Abstract

We fully develop the concept of causal symmetry introduced in our earlier publication (2003 Class. Quantum Grav. 20 L139). A causal symmetry is a transformation of a Lorentzian manifold (V, g) which maps every future-directed vector onto a future-directed vector. We prove that the set of all causal symmetries is not a group under the usual composition operation but a submonoid of the diffeomorphism group of V. Therefore, the infinitesimal generating vector fields of causal symmetries—causal-preserving vector fields—are associated with local one-parameter groups of transformations which are causal symmetries only for positive values of the parameter—one-parameter submonoids of causal symmetries. The pull-back of the metric under each causal symmetry results in a new rank-2 future tensor, and we prove that there is always a set of null directions canonical to the causal symmetry. As a result of this it makes sense to classify causal symmetries according to the number of independent canonical null directions. This classification is maintained at the infinitesimal level where we find the necessary and sufficient conditions for a vector field to be causal preserving. They involve the Lie derivatives of the metric tensor and of the canonical null directions. In addition, we prove the stability property of these equations under the repeated application of the Lie operator. Monotonicity properties, constants of motion and conserved currents can be defined or built using causal preserving vector fields. Some illustrative examples are presented.