Energy Estimates and Gravitational Collapse

In this paper, we establish energy estimates for Einstein vacuum equations in order to prove the evolutionary formation of black holes. The novelty of the paper is that we can completely avoid using rotation vector fields to prove the global existence of solution. More precisely, we use only canonical null vector fields as commutators to derive energy estimates at the level of the first derivatives of null curvature components. The special cancelations from the null structure of non-linear interaction yield the desired estimates.