On the Curvature Effect of a Relativistic Spherical Shell

We consider a relativistic spherical shell and calculate its spectral flux as received by a distant observer. Using two different methods, we derive a simple analytical expression of the observed spectral flux and show that the well-known relation $\hat \alpha = 2+\hat \beta$ (between temporal index $\hat \alpha$ and spectral index $\hat \beta$) of the high-latitude emission is achieved naturally in our derivation but holds only when the shell moves with a constant Lorentz factor $\Gamma$. Presenting numerical models where the shell is under acceleration or deceleration, we show that the simple $\hat \alpha = 2+\hat \beta$ relation is indeed deviated as long as $\Gamma$ is not constant. For the models under acceleration, we find that the light curves produced purely by the high-latitude emission decay initially much steeper than the constant $\Gamma$ case and gradually resume the $\hat \alpha = 2+\hat \beta$ relation in about one and half orders of magnitude in observer time. For the models under deceleration, the trend is opposite. The light curves made purely by the high-latitude emission decay initially shallower than the constant $\Gamma$ case and gradually resume the relation $\hat \alpha = 2+\hat \beta$ in a similar order of magnitude in observer time. We also show that how fast the Lorentz factor $\Gamma$ of the shell increases or decreases is the main ingredient determining the initial steepness or shallowness of the light curves.