Ivancevic_Applied-Diff-Geom

966 Applied Differential Geometry: A Modern IntroductionFinally there is the generic energy condition, which says:(1) The strong energy condition holds.(2) Every time–like or null geodesic has a point where l [a R b]cd[e l f] l c l d ≠ 0.One normally thinks of a space–time singularity as a region in whichthe curvature becomes unboundedly large. However, the trouble with thisdefinition is that one could simply leave out the singular points and saythat the remaining manifold was the whole of space–time. It is thereforebetter to define space–time as the maximal manifold on which the metric issuitably smooth. One can then recognize the occurrence of singularities bythe existence of incomplete geodesics that cannot be extended to infinitevalues of the affine parameter.Hawking–Penrose Singularity TheoremsHawking–Penrose Singularity is defined as follows [Hawking and Israel(1979); Penrose (1989); Hawking and Penrose (1996)]:A space–time manifold is singular if it is time–like or null geodesicallyincomplete but cannot be embedded in a larger space–time manifold.This definition reflects the most objectionable feature of singularities,that there can be particles whose history has a beginning or end at a finitetime. There are examples in which geodesic incompleteness can occur withthe curvature remaining bounded, but it is thought that generically thecurvature will diverge along incomplete geodesics. This is important if oneis to appeal to quantum effects to solve the problems raised by singularitiesin classical general relativity.Singularity Theorems include:(1) Energy condition (i.e., weak (5.485), strong (5.486), or generic (5.12.3)).(2) Condition on global structure (e.g., there should not be any closedtime–like curves).(3) Gravity strong enough to trap a region (so that nothing could escape).The various singularity theorems show that space–time must be time likeor null geodesically incomplete if different combinations of the three kindsof conditions hold. One can weaken one condition if one assumes strongerversions of the other two. The Hawking–Penrose Singularity theorems havethe generic energy condition, the strongest of the three energy conditions.The global condition is fairly weak, that there should be no closed time like