Exact versus Approximate Solutions in Gamma-Ray Burst Afterglows

We have recently obtained the exact analytic solutions for the relativistic equations relating the radial and time coordinates of a relativistic, thin uniform shell expanding in the interstellar medium in the fully radiative and fully adiabatic regimes. Here we reexamine the validity of the constant-index power-law relation between the Lorentz gamma factor and its radial coordinate, which is usually adopted in the gamma-ray burst (GRB) literature on the grounds of an "ultrarelativistic" approximation. Such expressions are found to be mathematically correct but only approximately valid in a very limited range of physical and astrophysical parameters and in an asymptotic regime that is reached for a very short time only, if at all, and they are shown to be nonapplicable to GRBs.