On the sideways expansion of relativistic non-spherical shocks and gamma-ray burst afterglows

Expansion of non-spherical relativistic blast waves is considered in the Kompaneets (the thin-shell) approximation. We find that relativistic motion effectively 'freezes out' the lateral dynamics of the shock front: only extremely strongly collimated shocks, with opening angles Delta theta <= 1/Gamma(2), show appreciable modification of profiles due to sideways expansion. For less collimated profiles, the propagation is nearly ballistic; the sideways expansion of relativistic shocks becomes important only when they become mildly relativistic. Even though the post-shock pressure is equilibrated downstream on the sound crossing time-scale, this does not mean that the shape of the shock changes on the post-shock sound crossing time-scale.