Causal structure of a recent loop quantum gravity black hole collapse model

The causal structure of the recent loop quantum gravity black hole collapse model J. G. Kelly et al. , Classical Quant. Grav. 38, 04LT01 (2021) is analyzed. As the spacetime is only approximately diffeomorphism invariant up to powers of, it is not straightforwardly possible to find global conformally compactified coordinates or to construct the Penrose diagram. Therefore, radial ingoing and outgoing light rays are studied to extract the causal features and sketch a causal diagram. It was found that the eternal metric J. G. Kelly et al. , Phys. Rev. D 102, 106024 (2020), which is the vacuum solution of the collapse model, has a causal horizon. However, in the collapsing case, light rays travel through matter to causally connect the regions inside and outside the horizon---the causal horizon is not present in the collapsing scenario. It is worked out that this is related to the shock wave and spacetime discontinuity, which allows matter traveling superluminally along a spacelike trajectory from the vacuum point of view, but remaining timelike from the matter perspective. The final causal diagram is a compact patch of a Reissner-Nordstr\"om causal diagram. Further, possibilities of a continuous matter collapse with only timelike evolution are studied. It is found that the time-reversed vacuum metric is also a solution of the dynamical equations, and a once continuously differentiable matching of the vacuum spacetime across the minimal radius is possible. This allows an everywhere continuous and timelike collapse process at the cost of an infinite extended causal diagram. This solution is part of an infinitely extended eternal black hole solution with a bounce, whose global extension is constructed. Due to the analysis of radial light rays, it is possible to sketch causal diagrams of these spacetimes.