Causal self-consistency of the Blandford--McKee self-similar solution
The Blandford--McKee (BM) solution describes the ultra-relativistic self-similar flow behind a strong spherical blast wave propagating into an external density profile $\propto r^{-k}$, where $k<4$ and $r$ is the distance from the center, but it applies only to the hot shell adjacent to the shock, sufficiently deep behind it, the fluid leaves the BM regime. Since the similarity profiles are determined solely by the shock conditions, with no conditions imposed at the inner end, the validity of the solution near the shock depends on whether the flow beyond the BM regime can imprint on this hot shell. For shallow density profiles, $k<k_g\simeq 2.062$, the characteristic structure of the BM solution already prevents forward-going acoustic information from the non-BM interior from reaching the shock during the ultra-relativistic stage. For steeper profiles, $k_g<k<4$, the hot equation of state fails while the flow is still relativistic and shock-connected. There, we derive a new self-similar solution for the cooling relativistic flow, which has its own similarity scale, equations, and characteristic structure. It overlaps with the hot BM solution toward the shock, and reaches a sonic point beyond this overlap. This critical point separates the shock-connected BM-plus-cooling composite from the deeper downstream region, so acoustic signals generated beyond it cannot propagate toward the shock. The mechanism is explicit at $k=7/2$, for which the cooling solution is obtained analytically. This causal structure is therefore what enables the BM solution, which remains self-consistent near the shock even though it does not globally describe the full downstream flow.