**A. Compactness and relativistic motion**

The first theoretical clues to the necessity of relativistic
motion in GRBs arose from the Compactness problem
[353].
The conceptual argument is simple. GRBs show a
non thermal spectrum with a significant high energy tail (see
Section IIA1). On the other hand a naive
calculation
implies that the source is optically thick. The fluctuations on a
time scale *t*
imply that the source is smaller than
*c*
*t*. Given an observed flux *F*, a duration *T*, and an
distance *d* we can estimate the energy *E* at the source. For a
typical photon's energy
_{} this
yields a photon density
4 *d*^{2}
*F* / _{}
*c*^{3}
*t*^{2}.
Now, two -rays
can annihilate and produce e^{+}e^{-} pairs, if the
energy in their CM frame is larger than 2*m*_{e}
*c*^{2}. The optical depth for pair creation is:

(5) |

where, *f*_{e±} is a numerical factor
denoting the average
probability that photon will collide with another photon whose
energy is sufficient for pair creation. For typical values and
cosmological distances, the resulting optical depth is extremely large
_{e±}
~ 10^{15}
[306].
This is, of course, inconsistent with the non-thermal spectrum.

The compactness problem can be resolved if the emitting matter is
moving relativistically towards the observer. I denote the Lorentz
factor of the motion by
. Two corrections
appear in this
case. First, the observed photons are blue shifted and therefore,
their energy at the source frame is lower by a factor
.
Second, the implied size of a source moving towards us with a
Lorentz factor is
*c* *t*
^{2} (see
Section IVB below). The first effect
modifies *f*_{e±} by a factor
^{-2} where
is the photon's index
of the observed
-rays
(namely the number of observed photons per unit energy is proportional to
*E*^{-}).
The second effect modifies the density estimate by a factor
^{-4} and it
influences the optical depth as
^{-2}.
Together one finds that for
~ 2 one needs
100 to obtain an
optically thin source.

The requirement that the source would be optically thin can be
used to obtain direct limits from specific bursts on the minimal
Lorentz factor within those bursts
[15,
91,
199,
220,
306,
308,
442].
A complete calculation requires a detailed integration over
angular integrals and over the energy dependent pair production
cross section. The minimal Lorentz factor depends also on the
maximal photon energy, *E*_{max}, the upper energy cutoff of
the spectrum. Lithwick and Sari
[220]
provide a detailed comparison of the different calculations and point
our various flaws in some of the previous estimates. They find that:

(6) |

where the high end of the observed photon flux is given by
*E*^{-}
(photons per cm^{2} per sec per unit photon
energy). A lower limit on
is obtained by
equating Eq. 6 to unity.