Singularities

Abstract This chapter presents a computable sufficient condition for the future causal completeness of a spacetime, and then a sufficient condition for its future or null incompleteness. It gives the fundamentals of the definitions pertinent to the study of incompleteness of spacetimes by the geometric methods introduced and developed by Penrose, Hawking, and their followers. It provides some elements of black hole theory and comments on Penrose's weak cosmic censorship conjecture, which says essentially that singularities developing from smooth initial data are hidden inside black holes. The conjecture is not easy to make mathematically precise without impoverishing its possible physical content. The chapter analyzes the study by Christodoulou of the singularities in spherically symmetric solutions of the Einstein-scalar equations. An up-to-date survey of results on the Belinskii, Khalatnikov, and Lifshitz (BKL) conjecture is presented. Finally, how the Fuchs theorem permits the analysis of some types of initial (Big Bang) singularities occurring in solutions of the Einstein equations, called asymptotically velocity term dominated (AVTD) behavior, is discussed.