Causal Structure of a recent Loop Quantum Gravity Black Hole Collapse Model

TL;DR

This paper analyzes the causal structure of a recent loop quantum gravity black hole collapse model, revealing how matter and spacetime discontinuities influence horizons and proposing a continuous, time-like collapse scenario with an extended eternal black hole solution.

Contribution

It introduces a detailed causal analysis of the loop quantum gravity black hole collapse model and proposes a novel continuous matter collapse solution with a global extension.

Findings

Eternal metric has a causal horizon, but collapsing case does not due to matter travel through matter.

Discontinuities allow matter to travel super-luminally from vacuum perspective.

A continuous, time-like collapse solution with an extended eternal black hole is constructed.

Abstract

The causal structure of the recent loop quantum gravity black hole collapse model [1] is analysed. As the spacetime is only approximately diffeomorphism invariant up to powers of $\hbar$, it is not straight forwardly possible to find global conformally compactified coordinates and to construct the Penrose diagram. Therefore, radial in- and outgoing light rays are studied to extract the causal features and sketch a causal diagram. It was found that the eternal metric [2], 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 in- 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 travelling super-luminal along a space-like trajectory from the vacuum point of view, but remaining time-like 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 time-like evolution are studied. It was 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 time-like 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.