Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 967curves. And the no escape condition is the most general, that there shouldbe either a trapped surface or a closed space like three surface.The theorems predict singularities in two situations. One is in the futurein the gravitational collapse of stars and other massive bodies. Suchsingularities would be an end of time, at least for particles moving on the incompletegeodesics. The other situation in which singularities are predictedis in the past at the beginning of the present expansion of the universe.The prediction of singularities means that classical general relativity isnot a complete theory. Because the singular points have to be cut out ofthe space–time manifold one cannot define the field equations there andcannot predict what will come out of a singularity. With the singularityin the past the only way to deal with this problem seems to be to appealto quantum gravity. But the singularities that are predicted in the futureseem to have a property that Penrose has called, Cosmic Censorship. Thatis they conveniently occur in places like black holes that are hidden fromexternal observers. So any break down of predictability that may occurat these singularities will not affect what happens in the outside world, atleast not according to classical theory.Hawking Cosmic Censorship Hypothesis says: “Nature abhors a nakedsingularity” [Hawking and Israel (1979); Hawking and Penrose (1996)].However, there is unpredictability in the quantum theory. This is relatedto the fact that gravitational fields can have intrinsic entropy which is notjust the result of coarse graining. Gravitational entropy, and the fact thattime has a beginning and may have an end, are the two main themes ofHawking’s research, because they are the ways in which gravity is distinctlydifferent from other physical fields.The fact that gravity has a quantity that behaves like entropy was firstnoticed in the purely classical theory. It depends on Penrose’s CosmicCensorship Conjecture. This is unproved but is believed to be true forsuitably general initial data and state equations.One makes the approximation of treating the region around a collapsingstar as asymptotically flat. Then, as Penrose showed, one can conformallyembed the space–time manifold M in a manifold with boundary ¯M. Theboundary ∂M will be a null surface and will consist of two components,future and past null infinity, called I + and I − . One says that weak CosmicCensorship holds if two conditions are satisfied. First, it is assumed that thenull geodesic generators of I + are complete in a certain conformal metric.This implies that observers far from the collapse live to an old age and arenot wiped out by a thunderbolt singularity sent out from the collapsing