Does loop quantum $μ_o$ scheme permit a black hole formation?

Abstract: We explore the way different loop quantization prescriptions affect the formation of trapped surfaces in the gravitational collapse of a homogeneous dust cloud, with a particular emphasis on the so called $\mu_o$ scheme in which loop quantum cosmology was initially formulated. Its undesirable features in cosmological models led to the so-called improved dynamics or the $\bar \mu$ scheme. While the jury is still out on the right scheme for black hole spacetimes, we show that as far as the black hole formation is concerned the $\mu_o$ scheme has another, so far unknown, serious problem. We find that in the $\mu_o$ scheme no trapped surfaces would form for a non-singular collapse of a homogeneous dust cloud in the marginally bound case unless the minimum non-zero area of the loops over which holonomies are computed or the Barbero-Immirzi parameter decreases almost four times from its standard value. It turns out that the trapped surfaces in the $\mu_o$ scheme for the marginally bound case are also forbidden for any arbitrary matter content as long as the collapsing interior is isometric to a spatially flat Friedmann-Lema^itre-Robertson-Walker (FLRW) spacetime. We find that in contrast to the situation in the $\mu_o$ scheme, black holes can form in the $\bar \mu$ scheme, and also other lattice refinements with a mass gap determined by quantum geometry.