Does the Loop Quantum μo Scheme Permit 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 particular emphasis on the so-called μo scheme in which loop quantum cosmology was initially formulated. Its undesirable features in cosmological models led to the so-called improved dynamics or the μ¯ scheme. While the jury is still out on the right scheme for black hole spacetimes, we show that as far as black hole formation is concerned, the μo scheme has another, so far unknown, serious problem. We found that in the μo scheme, no trapped surfaces would form for a nonsingular collapse of a homogeneous dust cloud in the marginally bound case unless the minimum nonzero 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 μo scheme for the marginally bound case are also forbidden for an arbitrary matter content as long as the collapsing interior is isometric to a spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. We found that in contrast to the situation in the μo scheme, black holes can form in the μ¯ scheme, as well as other lattice refinements with a mass gap determined by quantum geometry.