Formation of Trapped Surfaces in Geodesic Foliation

We reprove the formation of trapped surfaces for the Einstein vacuum equation using the incoming geodesic foliation. All previous results, starting with the seminal work of Christodoulou, make use of the double null foliation. In the new proof, all Ricci coefficients are estimated by transport equations in the incoming null direction, and the incoming initial data is only used in the curvature (energy) estimates. The result is based on a version of the non-integrable PT frame, which was introduced in the proof of Kerr stability with small angular momentum.