Textbooks say that for a distant external observer, an infalling object never reaches the horizon because dr/dt tends to 0 as r approaches 2M. But this conclusion assumes the object is a test particle with negligible mass. If we take the object's own mass m seriously, then the exterior spacetime is determined by the total mass M+m. The new horizon is at 2(M+m), so the old surface 2M is no longer the true coordinate singularity. That means the object's radial velocity does NOT vanish at 2M; it only vanishes asymptotically at the new horizon 2(M+m). So the coordinate time needed for the object to cross the original 2M surface is finite. My question: Does this show that the "eternal freezing" result is only valid for test particles, and that a real massive object can indeed enter the original horizon within finite external time? Am I missing something?