The geodesics on the (1 + 3)-dimensional de Sitter spacetime are considered studying how their parameters are determined by the conserved quantities in the conformal Euclidean, Friedmann-LemaˆıtreLemaˆıtre-Robertson-Walker, de Sitter-Painlevé and static local charts with Carte-sian space coordinates. Moreover, it is shown that there exist a special static chart in which the geodesics are genuine hyperbolas whose asymptotes are given by the conserved momentum and the associated dual momentum.

As quotient spaces, Minkowski and de Sitter are fundamental, non-gravitational spacetimes for the construction of physical theories. When general relativity is constructed on a de Sitter spacetime, the usual Riemannian structure is replaced by a more general structure called de Sitter-Cartan geometry. In the contraction limit of an infinite cosmological term, the de Sitter-Cartan spacetime reduces to a singular, flat, conformal invariant four-dimensional cone spacetime, in which our ordinary notions of time interval and space distance are absent. It is shown that such spacetime satisfies all properties, including the Weyl curvature hypothesis, necessary to play the role of the bridging spacetime connecting two aeons in Penrose's conformal cyclic cosmology.Comment: 15 pages. V2: presentation changes aiming at clarifying the text, matches published versio

Abstract. Minkowski spacetime is transitive under ordinary translations, a transformation that do not have matrix representations. The de Sitter spacetime, on the other hand, is transitive under a combination of translations and proper conformal transformations, which do have a matrix representation. Such matrix, however, is not by itself a de Sitter generator: it gives rise to a conformal re-scaling of the metric, a transformation not belonging to the de Sitter group, and in general not associated with diffeomorphisms in spacetime. When dealing with variational principles and Noether's theorem in de Sitter spacetime, therefore, it turns out necessary to regularise the transformations in order to eliminate the conformal re-scaling of the metric.

Extending the results of our previous work we construct an uniparametric class of action principles for complex scalar fields with the property that their energy momentum tensor and the electric current vanish for certain massive configurations. These stealth fields do not curve the spacetime and they do not source electromagnetic fields in spite their non-trivial degrees of freedom and U p1q gauge invariance. We shall also show that the presence of these stealth fields can affect the strength of the gravity-matter and radiation-matter coupling of other massive configurations. Indeed, the energy momentum tensor of other massive (non-stealth) configurations and the electric current can be rescaled (with a stealth-mass depending factor) with respect to the predictions of the standard complex scalar field model. Hence we argue that stealth fields could be detected indirectly by means of their effects on standard configurations of matter.

We construct a generic class of models for complex scalar fields — minimally coupled to gravity and electromagnetism — with the property that their energy-momentum tensor and the electric current vanish for certain massive configurations. These are electromagnetically and gravitationally stealth fields. We shall see that the latter configurations can affect, in addition, the strength of the gravity-matter and electromagnetic-matter couplings of other (non-stealth) modes present in the system, which turn out to be equivalent to the re-scaling the electric charge and the Newton constant (with a stealth-mass depending factor).