Maximal acceleration,maximal angular velocity,and causal influence

The upper bounds to acceleration and angular velocity which are suggested by quantum gravitational effects are described, together with the relativistic bound to velocity, by means of a cone ⁺ in the Lie algebra of the Poincaré group. The connection between these bounds and the existence of a minimal measurable length (of the order of Planck's length) is illustrated by means of a simple model. The geometric properties of the cone ⁺ and of other related structures are examined in some detail. The new geometric background requires some modifications of the concepts of causal influence and of spacetime coincidence, which are analyzed and shown to lead to some nonlocal features of the theory. Due to the smallness of Planck's length, these modifications to the causal relations cannot be observed by means of available experimental methods, but they could have some influence on the structure of elementary particles and on the very early cosmology.