Minisuperspace Stellar Collapse in Semi-Classical Gravity

We compute the Schr\"odinger and WKB propagators for the semi-classical collapse of a sphere of dust. This extends the work by Redmount and Suen (1993) from the free particle case to nontrivial gravity. In the Oppenheimer-Snyder model, a star can be idealised as a collapsing dust sphere of uniform density and zero pressure. The particles that make up the star have the attributes of classical dust where each particle is assumed to be infinitesimal in size and to interact only gravitationally with other matter. We include quantum mechanical effects, which lift some of these assumptions. This allows for the possibility that some configurations will not collapse to black holes. We find analytic, closed-form solutions for classical paths of a particle on the surface of a collapsing star in Schwarzschild and Kruskal geometries. Kruskal coordinates can be used to study the wavefunction inside the apparent horizon. The propagator is written in closed form in Schwarzschild coordinates in both the WKB and in the Schr\"odinger approximation. The waveform is computed and found to generally exhibit an ingoing and an outgoing component, where the former contains the probability of collapse, while the latter contains the probability that the star will disperse. In the limit when the mass of the black hole is zero, the WKB solution converges to the Schr\"odinger solution, which is the exact solution to the Wheeler-DeWitt equation and confirms the accuracy of the WKB approximation. Classically, time-like paths can only turn back in space, while space-like paths can turn in time. In Kruskal coordinates, we find space-like paths that turn back in time before entering $r=2M$. These calculations will pave the way for investigating the possibility of quantum collapse that does not lead to black hole formation as well as for exploring the nature of the wavefunction inside $r=2M$.