Figure 5: Schematic solution of the shock heating problem in spherical geometry. The initial state consists of a spherically symmetric flow of cold (p = 0) gas of unit rest mass density having a highly relativistic inflow velocity everywhere. A shock is generated at the center of the sphere, which propagates upstream with constant speed. The post-shock state is constant and at rest. The pre-shock state, where the flow is self-similar, has a density which varies as  = (1 + t/r)² with time t and radius r.