Light curve of pulses of gamma-ray bursts affected by the Doppler effect of fireballs

Light curves of pulses of gamma-ray bursts (GRBs) affected by the Doppler effect of fireballs are investigated. Formulas of the count rate in terms of the integral of time are presented. Applying a rest frame $\delta $ function form of pulses we obtain the light curve of pulses, ignoring the effect of the rest frame pulse width, for expanding fireballs. From the light curve we find that the time scale of a pulse would be significantly affected by the Lorentz factor. If the Lorentz factor $\Gamma $ is very large, the observed width of the light curve of a very narrow pulse would become much smaller than the time scale of the real size of the fireball, following the law of $\Gamma ^{-2}$. This explains why for many sources very short time scale of pulses is observed. In addition, light curves of rest frame rectangle pulses as well as other forms of rest frame pulses which last an interval of time are also studied. These light curves reveal the well known structure of a fast rise and an exponential decay (FRED) observed in many GRBs. We find that, the character of FRED is a consequence of the expanding motion of fireballs. The main difference of the observed profiles of pulses is due to their rest frame widths. The interval between the beginning and the peak of pulses depends not only on the Lorentz factor but also on the width of the rest frame pulse as well