Physical black holes: spacetime and matter near trapping surfaces

Within a semiclassical framework, we investigate spherically symmetric solutions of the Einstein equations that (i) develop a trapped region within a finite time as measured by distant observers, and (ii) remain sufficiently regular at the apparent horizon. These requirements characterize the self-consistent approach, and it imposes specific constraints on the limiting form of the energy momentum tensor (EMT). Previous works argued that only certain EMT classes admit viable dynamical solutions, while discarding others (see, e.g., Terno 2020) based on reasoning that is not entirely robust. Here, we explicitly demonstrate that the discarded classes are indeed unphysical. In addition, we also introduce a new class of static solutions that are regular at the horizon and allow finite-time horizon crossing according to a distant observer.