Peak Luminosity–Spectral Lag Relation Caused by the Viewing Angle of the Collimated Gamma-Ray Bursts

We compute the kinematical dependence of the peak luminosity, the pulse width, and the spectral lag of the peak luminosity on the viewing angle θ v of a jet. For appropriate model parameters we obtain a peak luminosity-spectral lag relation similar to the observed one including gamma-ray burst (GRB) 980425. A bright (dim) peak with short (long) spectral lag corresponds to a jet with small (large) viewing angle. This suggests that the viewing angle of the jet might cause various relations in GRBs such as the peak luminosity-variability relation and the luminosity-width relation. Our model also suggests that X-ray-rich GRBs (or X-ray flushes or fast X-ray transients) are typical GRBs observed from large θ v with large spectral lag and low variability.