Effective quantum dust collapse via surface matching

Abstract

The fate of matter forming a black hole is still an open problem, although models of quantum gravity corrected black holes are available. In loop quantum gravity (LQG) models were presented, which resolve the classical singularity in the centre of the black hole by means of a black-to-white hole transition, but neglect the collapse process. The situation is similar in other quantum gravity approaches, where eternal non-singular models are available. In this paper, a strategy is presented to generalise these eternal models to dynamical collapse models by surface matching. Assuming (1) the validity of a static quantum black hole spacetime outside the collapsing matter, (2) homogeneity of the collapsing matter, and (3) differentiability at the surface of the matter fixes the dynamics of the spacetime uniquely. It is argued that these assumptions resemble a collapse of pressure-less dust and thus generalises the Oppenheimer–Snyder–Datt model, although no precise model of the matter has to be assumed. Hawking radiation is systematically neglected in this approach. The junction conditions and the spacetime dynamics are discussed generically for bouncing black hole spacetimes, as proposed by LQG, although the scheme is approach independent. Further, the equations are explicitly solved for the recent model Bodendorfer et al (2019 Class. Quantum Grav. 36 195015) and a global spacetime picture of the collapse is achieved. The causal structure is discussed in detail and the Penrose diagram is constructed. The trajectory of the collapsing matter is completely constructed from an inside and outside observer point of view. The general analysis shows that the matter is collapsing and re-expanding and crosses the Penrose diagram diagonally. This way the infinite tower of Penrose diagrams, as proposed by several LQG models, is generically not cut out. Questions about different timescales of the collapse for in- and outside observers can be answered.