Expanding Relativistic Shells and Gamma-Ray Burst Temporal Structure

Many models of cosmological gamma-ray bursts involve the sudden release of $\sim 10^{51}$ erg which produce shells which expand at relativistic speeds (Lorentz $\Gamma$ factors of $10^{2-3}$). We investigate the kinematic limits on the source size due to the observed time structure in three types of bursts: short spikes, FREDs (Fast Rise, Exponentail decay), and long complex bursts. The emitting shell keeps up with the photons it produces reducing apparent durations by $\Gamma^2$ so that source sizes can be very large ($2c\Gamma^2 T_{dur}). However, the thickness of the emitting region is not effected by the shell motion so it must always be small, $c\Delta T$ where $\Delta T$ is the subpeak time scale. We argue that one can only view the bulk motion head-on so it is inappropriate to treat GRBs as viewing the sides of a jet. Although photons come from a region within an angle $\Gamma^{-1}$, we show that the curvature of the shell within that angle creates delays comparable to those associated with the duration of the event. As a result, most bursts should be like FREDs with sharp rises related to how long the shell emits and power law decays related to how long the shell expanded before becoming gamma-ray active. Few bursts have the long decay phases required for large shells resulting in unacceptable high densities for ISM objects to cause the observed subpeaks. To be consistent with the observations, perhaps very thick shells (which act as parallel slabs) are required to avoid the effects of the curvature, or the duration is dictated by a central engine.