Aims. We re-examine the possibility that astrophysical jet collimation may arise from the geometry of rotating black holes and the presence of high-energy particles resulting from a Penrose process, without the help of magnetic fields. Methods. Our analysis uses the Weyl coordinates, which are revealed better adapted to the desired shape of the jets. We numerically integrate the 2D-geodesics equations. Results. We give a detailed study of these geodesics and give several numerical examples. Among them are a set of perfectly collimated geodesics with asymptotes ρ = ρ 1 parallel to the z-axis, with ρ 1 only depending on the ratios, where a and M are the parameters of the Kerr black hole, E the particle energy and Q the Carter's constant.

We present the whole set of equations with regularity and matching conditions required for the description of physically meaningful static cylindrically symmmetric distributions of matter, smoothly matched to Levi-Civita vacuum spacetime. It is shown that the conformally flat solution with equal principal stresses represents an incompressible fluid. It is also proved that any conformally flat cylindrically symmetric static source cannot be matched through Darmois conditions to the Levi-Civita spacetime. Further evidence is given that when the Newtonian mass per unit length reaches 1/2 the spacetime has plane

Observations suggest that relativistic particles play a fundamental role in the dynamics of jets emerging from active galactic nuclei as well as in their interaction with the intracluster medium. However, no general consensus exists concerning the acceleration mechanism of those high energy particles. A gravitational acceleration mechanism is here proposed, in which particles leaving precise regions within the ergosphere of a rotating supermassive black hole produce a highly collimated flow. These particles follow unbound geodesics which are asymptotically parallel to the spin axis of the black hole and are characterized by the energy E, the Carter constant Q and zero angular momentum of the component L z . If environmental effects are neglected, the present model predicts at distances of about 140 kpc from the ergosphere the presence of electrons with energies around 9.4 GeV. The present mechanism can also accelerate protons up to the highest energies observed in cosmic rays by the present experiments.