Numerical approach to the black-to-white hole transition

We outline an algorithm to compute numerically the black-to-white hole transition amplitude, using the loop quantum gravity covariant formulation and the Lorentzian Engle-Pereira-Rovelli-Livine model. We apply the algorithm to calculate the crossing time of the transition in the deep quantum regime, comparing our result with previous analytical estimates of the same physical observable in the semiclassical limit. Furthermore, we show how to evaluate the crossing time analytically using an alternative approach with respect to the one currently present in the literature. This method requires much easier calculations and emphasizes that the crossing time does not depend on the extrinsic geometry of the transition.