4.3 Curvature blow up

The strong cosmic-censorship conjecture is of fundamental importance since it expresses the expectation that Einstein’s general theory of relativity is a deterministic theory (with some nongeneric exceptions). On the other hand, it is of a somewhat philosophical nature. However, it is strongly connected to a statement concerning the behavior of gravitational fields close to the singularity. Here, the existence of a singularity is equated with the existence of an incomplete causal geodesic, the motivation being the work of Hawking and Penrose resulting in the singularity theorems; see [614143] and [428960Jump To The Next Citation Point]. Even though the commonplace existence of singularities is established by the singularity theorems, their nature remains unclear; do, for example, the gravitational fields become arbitrarily strong in the vicinity of a singularity? It is natural to state the following conjecture:

Conjecture 2 For generic asymptotically-flat and generic spatially-compact initial data for Einstein’s vacuum equations, curvature blows up in the incomplete directions of causal geodesics in the MGHD.

In the statement of this conjecture, we shall in practice take curvature blow up to mean the unboundedness of the Kretschmann scalar,

R αβγδR αβγδ.

Of course, in the presence of matter one could also consider the contraction of the Ricci tensor with itself. Needless to say, there are many other possibilities. Again, the word “generic” has to be included. The reason is that the canonical counterexample to inextendibility of the MGHD in the cosmological vacuum case, Taub–NUT, is also a counterexample to curvature blow up in the incomplete directions of causal geodesics. Taub–NUT is causally geodesically-incomplete both to the future and to the past, but the spacetime can be extended and all curvature invariants consequently remain bounded along all geodesics; see [20Jump To The Next Citation Point] or [82Jump To The Next Citation Point]. The reason for wanting to prove Conjecture 2 is perhaps more clear than the reason for wanting to prove strong cosmic censorship; it would demonstrate the generic occurrence of singularities, not only in the sense of causal geodesic completeness, but in the sense of the gravitational fields becoming arbitrarily strong. Furthermore, if Conjecture 2 holds, then strong cosmic censorship follows (with the C2-concept of inextendibility).

Since it is not clear how to address Conjectures 1 and 2 in all generality, the work that has been carried out so far has been concerned with the analogous questions phrased in the context of special classes of spacetimes. Here, we shall be concerned with these questions phrased in the Gowdy class.

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