  The detection of delayed emission at X-ray optical and radio wavelengths,
``after-glow'', suggests that the relativistic shell which emitted the initial
GRB due to internal shocks decelerates on encountering an external medium,
giving rise to the after-glow. At a very early stage (few seconds), the
observed bolometric luminosity increases as t^2. On longer time scales (more
than about 10s), the luminosity drops as t^{-1}. If the main burst is long
enough, an intermediate stage of constant luminosity will form. In this case,
the after-glow overlaps the main burst, otherwise there is a time separation
between the two. On the long time scale, the flow decelerate in a self similar
way, reaching non relativistic velocities after about 30days. Due to the
deceleration and the accumulation of ISM material, the relation between the
observed time, the shock radius, and its Lorentz factor, is given by
t=R/16\gamma^2c which is a factor of 8 different from the usual expression. The
majority of particles are those of the original ejecta (and not the ISM) up to
about 900s. These particles reach sub-relativistic velocities on a time scale
of about 2hours, well before the flow becomes sub-relativistic. Therefore the
ejecta particles are probably unimportant for most of the after-glow radiation.
We show that even though only a small fraction of the energy is given to the
electrons, most of the energy can be radiated over time. If this fraction is
greater than 10% radiation losses will significantly influence the
hydrodynamical evolution at early times (less than 1day).
