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On the formation of trapped surfaces. (English) Zbl 1261.35139

Holden, Helge (ed.) et al., Nonlinear partial differential equations. The Abel symposium 2010. Proceedings of the Abel symposium, Oslo, Norway, September 28–October 2, 2010. Berlin: Springer (ISBN 978-3-642-25360-7/hbk; 978-3-642-25361-4/ebook). Abel Symposia 7, 181-206 (2012).
This paper presents an overview of recent results on formation of trapped surfaces upon evolution of suitable characteristic initial data for the vacuum Einstein field equations. The presence of trapped surfaces is sufficient to conclude that the maximal Cauchy development of the data is geodesically incomplete, via the Penrose singularity theorem. The spacetimes are also black holes, provided the weak cosmic censorship holds. Obtaining an open set of regular initial data leading to the formation of trapped surfaces (and hence black holes, under weak cosmic censorship) is an outstanding problem in General Relativity which was solved for the first time by Christodoulou. In this paper, the author gives a very clear explanation of the heuristic argument behind the validity of Christodoulou’s argument and describes the set of characteristic initial data, called short pulse data, for which trapped surfaces were proved to arise upon evolution. The size of the pulse was controlled by a smallness parameter and estimates of various quantities involving different powers of the parameter were necessary in order to make the heuristic argument rigorous. Obtaining these estimates was technically laborious. The author also reports on work by himself and I. Rodnianski [Acta Math. 208, No. 2, 211–333 (2012; Zbl 1246.83028)] where the arguments were substantially simplified and extended. This was accomplished by defining suitable scalings for each dynamical quantity and introducing scale invariant norms for all dynamical objects, except for two curvature components called “anomalous”. The nice scaling behaviour of the field equations allow for a substantial simplification of the derivation of the required estimates for proving existence of trapped surfaces during the evolution. The initial data for which this result was established extended the short pulse data analyzed by Christodoulou. The paper also discusses the results of another paper by the author and I. Rodnianski [Discrete Contin. Dyn. Syst. 28, No. 3, 1007–1031 (2010; Zbl 1193.35225)] where the scale invariant norms are further localized in the angular variables. This allows the authors to also localize the estimates and obtain sufficient conditions on the initial data making sure that a surface with negative outer null expansion on a prescribed angular section forms upon evolution.
For the entire collection see [Zbl 1231.35003].

MSC:

35Q75 PDEs in connection with relativity and gravitational theory
83C75 Space-time singularities, cosmic censorship, etc.
83C57 Black holes
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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