4.1. Relativistic Motion

Relativistic effects can fool us and, when ignored, lead to wrong conclusions. This happened thirty years ago when rapid variability implied "impossible" temperatures in extra-galactic radio sources. This puzzle was resolved when it was suggested [214, 215] that these objects reveal ultra-relativistic expansion. This was confirmed later by VLBA measurements of superluminal jets with Lorentz factors of order two to ten. This also happened in the present case. Consider a source of radiation that is moving towards an observer at rest with a relativistic velocity characterized by a Lorentz factor, = 1 / (1 - v² / c²)^1/2 >> 1. Photons with an observed energy h_obs have been blue shifted and their energy at the source was h _obs / . Since the energy at the source is lower fewer photons have sufficient energy to produce pairs. Now the observed fraction f_p, of photons that could produce pairs is not equal to the fraction of photons that could produce pairs at the source. The latter is smaller by a factor ^-2 (where is the high energy spectral index) than the observed fraction. At the same time, relativistic effects allow the radius from which the radiation is emitted, R_e < ² cT to be larger than the original estimate, R_e < c T, by a factor of ². We have

(10)

where the relativistic limit on R_e was included in the second line. The compactness problem can be resolved if the source is moving relativistically towards us with a Lorentz factor > 10^13/(4+2) 10². A more detailed discussion [210, 211] gives comparable limits on . Such extreme-relativistic motion is larger than the relativistic motion observed in any other celestial source. Extragalactic super-luminal jets, for example, have Lorentz factors of ~ 10, while the known galactic relativistic jets [216] have Lorentz factors of ~ 2 or less.

The potential of relativistic motion to resolve the compactness problem was realized in the eighties by Goodman [217], Paczynski [53] and Krolik and Pier [218]. There was, however, a difference between the first two approaches and the last one. Goodman [217] and Paczynski [53] considered relativistic motion in the dynamical context of fireballs, in which the relativistic motion is an integral part of the dynamics of the system. Krolik and Pier [218] considered, on the other hand, a kinematical solution, in which the source moves relativistically and this motion is not necessarily related to the mechanism that produces the burst.

Is a purely kinematic scenario feasible? In this scenario the source moves relativistically as a whole. The radiation is beamed with an opening angle of ^-1. The total energy emitted in the source frame is smaller by a factor ^-3 than the isotropic estimate given in Eq. (8). The total energy required, however, is at least (M c² + 4 F D² / ³) , where M is the rest mass of the source (the energy would be larger by an additional amount E_th if an internal energy, E_th, remains in the source after the burst has been emitted). For most scenarios that one can imagine Mc² >> (4 / ²) F D². The kinetic energy is much larger than the observed energy of the burst and the process is extremely (energetically) wasteful. Generally, the total energy required is so large that the model becomes infeasible.

The kinetic energy could be comparable to the observed energy if it also powers the observed burst. This is the most energetically-economical situation. It is also the most conceptually-economical situation, since in this case the -ray emission and the relativistic motion of the source are related and are not two independent phenomena. This will be the case if GRBs result from the slowing down of ultra relativistic matter. This idea was suggested by Mészáros, and Rees [27, 219] in the context of the slowing down of fireball accelerated material [220] by the ISM and by Narayan, et al. [28] and independently by Rees and Mészáros [29] and Paczynski and Xu [30] in the context of self interaction and internal shocks within the fireball. It is remarkable that in both cases the introduction of energy conversion was motivated by the need to resolve the "Baryonic Contamination" problem (which we discuss in the next section). If the fireball contains even a small amount of baryons all its energy will eventually be converted to kinetic energy of those baryons. A mechanism was needed to recover this energy back to radiation. However, it is clear now that the idea is much more general and it is an essential part of any GRB model regardless of the nature of the relativistic energy flow and of the specific way it slows down.

Assuming that GRBs result from the slowing down of a relativistic bulk motion of massive particles, the rest mass of the ultra-relativistic particles is:

(11)

where _c is the conversion efficiency and is the opening angle of the emitted radiation. We see that the allowed mass is very small. Even though a way was found to convert back the kinetic energy of the baryons to radiation (via relativistic shocks) there is still a "baryonic contamination" problem. Too much baryonic mass will slow down the flow and it won't be relativistic.