  Many cosmological models of gamma-ray bursts (GRBs) assume that a single
relativistic shell carries kinetic energy away from the source and later
converts it into gamma rays, perhaps by interactions with the interstellar
medium or by shocks within the shell. We investigate methods of achieving the
observed variability using symmetry. Since our model emphasizes geometric and
statistical considerations, rather than the detailed physics of the shell, it
is applicable to any theory that relies on relativistic shells. In order to
achieve the observed variability in GRBs using a single-shell model, we have
found that the gamma-ray emission must occur from small patches on the shell
(entities). We have derived two significant characteristics of the time
histories expected from such a model. First, the average number of entities
contributing to the signal remains constant throughout much of the time
history, although the overall photon flux decreases due to relativistic
effects. Second, the "peaks" in a time history can be ascribed to Poisson
variations in the actual number of entities contributing to the signal at any
given time. Taken together, these properties imply that the relative variations
in a GRB (i.e. the heights of the peaks relative to the average envelope of the
signal) should remain constant throughout a time history.
