Characteristics of the curvature effect of uniform jets or spherical   fireballs
  Qin et al. (2004) have derived a formula of count rates based on a model of highly symmetric expanding fireballs, where the Doppler effect is the key factor to be concerned. In this paper, we employ the formula to both the Qin model and the uniform jet model to study how the rising timescale, $\Delta\tau_{\theta,r}$, and the decay timescale, $\Delta\tau_{\theta,d}$, of a local pulse affect the light curve of GRBs. Our analysis shows that they do make contributions to both rising and decay portions of the light curve of GRBs. Associated with a local pulse with both rising and decay portions, the light curve of GRBs in the rising portion is expected to undergo a concave phase and then a convex one, which we call a "concave-to-convex" character, whereas that in the decay portion is expected to evolve an opposite process which is called a "convex-to-concave" character, regardless of being in Qin model or in the uniform jet model. These characteristics are independent of local pulse shapes and their rest frame radiation forms. The diversity of light curves are caused by different forms of local pulses and different ratios of $\Delta\tau_{\theta,r}$ to $\Delta\tau_{\theta,d}$. We study a sample of 86 GRB pulses detected by the BATSE instrument on board the Compton Gamma Ray Observatory and find that the "concave-to-convex" (expected in the rising portion of the light curve) and "convex-to-concave" (expected in the decay portion) characters, which result from the curvature effect of spherical surface (or the Doppler effect), do exist in the light curve of some GRBs. 
Rui-Jing Lu and Yi-Ping Qin