On the formation of trapped surfaces
S. Klainerman, I. Rodnianski
TL;DR
The paper addresses the long-standing problem of evolutionary formation of trapped surfaces in Einstein vacuum space-times by broadening Christodoulou’s initial data framework and weakening propagation requirements. It introduces scale-invariant, delta-coherent norms to organize nonlinear interactions and reduce derivative losses, enabling a near-universal, localized proof structure. The main contributions include a simplified semi-global existence result, refined control of curvature with only one derivative, and the demonstration that trapped surfaces (and in certain sectors, pre-scar surfaces) form under the new data conditions. This advances the mathematical understanding of gravitational collapse in general relativity and provides a framework amenable to further localization and generalization, potentially impacting global stability analyses and related geometric-analytic methods.
Abstract
In a recent important breakthrough D. Christodoulou has solved a long standing problem of General Relativity of evolutionary formation of trapped surfaces in the Einstein-vacuum space-times. He has identified an open set of regular initial conditions on an outgoing null hypersurface (both finite and at past null infinity) leading to a formation a trapped surface in the corresponding vacuum space-time to the future of the initial outgoing hypersurface and another incoming null hypersurface with the prescribed Minkowskian data. In this paper we give a simpler proof for a finite problem by enlarging the admissible set of initial conditions and, consistent with this, relaxing the corresponding propagation estimates just enough that a trapped surface still forms. We also reduce the number of derivatives needed in the argument from two derivatives of the curvature to just one. More importantly, the proof, which can be easily localized with respect to angular sectors, has the potential for further developments.