  We present the results of theoretical analysis and numerical simulations of
the Weibel instability in two counter-streaming hot relativistic plasma flows,
e.g. flows of electron-proton plasma having rest-mass density $\rho \sim
10^{-4}\; \text{g}\, \text{cm}^{-3}$, Lorentz factors $\Gamma \sim 10$ and
proper temperature $T \sim 10^{13}\; \text{K}$. The instability growth rate and
the filament size at the linear stage are found analytically, and are in
qualitative agreement with results of three-dimensional particle-in-cell
simulations. In the simulations, incoherent synchrotron emission and pair
photoproduction in electromagnetic fields are taken into account. If the plasma
flows are dense, fast and/or hot enough, the overall energy of synchrotron
photons can be much larger than the energy of generated electromagnetic fields.
Furthermore, a sizable number of positrons can be produced due to the pair
photoproduction in the generated magnetic field. We propose a rough criterion
for judging copious pair production and synchrotron losses. By means of this
criterion we conclude that incoherent synchrotron emission and pair production
during the Weibel instability can have implications for the collapsar model of
gamma-ray bursts.
