  We fully develop the concept of causal symmetry introduced in Class. Quant.
Grav. 20 (2003) 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 motions-- are associated to 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 a causal motion.
They involve the Lie derivatives of the metric tensor and of the canonical null
directions. In addition, we prove a 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 casual
motions. Some illustrative examples are presented.
