The influence of the jet opening angle on the appearance of relativistic jets

We reinvestigate the problem of the appearance of relativistic jets when geometrical opening is taken into account. We propose a new criterion to define apparent velocities and Doppler factors, which we think being determined by the brightest zone of the jet. We numerically compute the apparent velocity and the Doppler factor of a non homokinetic jet using different velocity profiles. We argue that if the motion is relativistic, the high superluminal velocities beta_{app} ~ gamma, expected in the case of an homokinetic jet, are only possible for geometrical collimation smaller than the relativistic beaming angle 1/gamma. This is relatively independent of the jet velocity profile. For jet collimation angles larger than 1/gamma, the apparent image of the jet will always be dominated by parts of the jet traveling directly towards the observer at lorentz factors < gamma resulting in maximal apparent velocities smaller than gamma}. Furthermore, getting rid of the homokinetic hypothesis yields a complex relation between the observing angle and the Doppler factor, resulting in important consequences for the numerical computation of AGN population and unification scheme model.Comment: Accepted in MNRAS, 12 pages and 9 Figure

2+12+1 Covariant Lattice Theory and t'Hooft's Formulation

We show that 't Hooft's representation of (2+1)-dimensional gravity in terms of flat polygonal tiles is closely related to a gauge-fixed version of the covariant Hamiltonian lattice theory. 't Hooft's gauge is remarkable in that it leads to a Hamiltonian which is a linear sum of vertex Hamiltonians, each of which is defined modulo $2 \pi$. A cyclic Hamiltonian implies that ``time'' is quantized. However, it turns out that this Hamiltonian is {\it constrained}. If one chooses an internal time and solves this constraint for the ``physical Hamiltonian'', the result is not a cyclic function. Even if one quantizes {\it a la Dirac}, the ``internal time'' observable does not acquire a discrete spectrum. We also show that in Euclidean 3-d lattice gravity, ``space'' can be either discrete or continuous depending on the choice of quantization. Finally, we propose a generalization of 't Hooft's gauge for Hamiltonian lattice formulations of topological gravity dimension 4.Comment: 10 pages of text. One figure available from J.A. Zapata upon reques