Strong cosmic censorship

The conjecture is best understood by comparing the implications of Penrose's incompleteness theorem in the Schwarzschild and Kerr cases, respectively.

In the case of the Schwarzschild solution, thought of as the maximal vacuum Cauchy development of a two-ended Cauchy hypersurface, Penrose's theorem applies and the resulting incompleteness is in fact associated with blow up of curvature–for instance, the Kretschmann scalar blows up along all incomplete geodesics. The spacetime is inextendible as a spacetime with C^2 metric (in fact, as a spacetime with C^0 metric). One can attach a "singular boundary" to spacetime parametrised by suitable equivalence classes of such geodesics, and this boundary can be moreover thought of as "spacelike". 

In the case of the Kerr family for non-vanishing rotation parameter, however, the incompleteness associated to the above theorem is entirely global in origin. The maximal Cauchy development of two-ended initial data is in fact extendible to a larger spacetime with smooth metric, such that all incomplete geodesics pass into the extension. The boundary of the original manifold in the extension is known as a Cauchy horizon. What is going on here? The incompleteness imposed by Penrose's theorem is caused not by breakdown of regularity but by a failure of "global hyperbolicity": The above spacetime extensions are no longer uniquely determined by initial data, as they will necessarily contain past-directed causal geodesics which do not intersect the initial data hypersurface. 

The Kerr Cauchy horizon is associated to an infinite blue-shift, and the latter effect gave hope that the phenomenon of smooth Cauchy horizons is unstable. This motivated the so-called strong cosmic censorship conjecture, also originally formulated by Penrose, which in the language of the Cauchy problem can be formulated as follows:

ConjectureFor generic initial data for the vacuum equations or for suitable Einstein–matter systems, the maximal Cauchy development is inextendible.

In what sense inextendible? Here there are several choices in the formulation. We will return to this issue further down.  

The original underlying expectation connected with the above conjecture was that for generic perturbations of Kerr, the causal picture of spacetime would revert to that of Schwarzschild, i.e. with a spacelike "singularity", but where now the dynamics would be much more complicated as one approached this singular boundary, possibly exhibiting "chaotic" behaviour.

A hint that the basic assumption that the singularity should be spacelike might not in fact hold was given by the heuristic study of a series of toy models by William Hiscock, Werner Israel, Eric Poisson, and Amos Ori, followed by numerical study by Patrick Brady, which suggested a Cauchy horizon across which the metric is extendible, but the curvature blows up, in fact, the Christoffel symbols fail to be locally square integral. These are known as weak null singularities. Since in particular the so-called Hawking mass diverged on this singular boundary, the phenomenon was called mass inflation.

In a series of papers (see here and here) starting from my Ph.D. thesis (completed in 2001 under the direction of Demetrios Christodoulou), I proved mathematically that indeed a piece of weak null singularity emanating (in the topology of the well-known Penrose diagram representation) from timelike infinity occur for generic polynomially decaying data posed on a dynamic black hole event horizon settling down to Reissner-Nordstrom, in the context of the spherically symmetric Einstein–Maxwell–real scalar field system. The generic data can then be related to generic data posed on a spacelike hypersurface through my proof, jointly with Igor Rodnianski, that solutions arising from arbitrary asymptotically flat Cauchy data for this system indeed settle down to Reissner-Nordstrom with appropriate polynomial decay rates for the scalar field. See this link

Thus, for the simplest dynamic radiating spherically symmetric model where the question could be posed, the mass inflation scenario of Israel--Poisson and Ori was proven correct. 

The above works only considered the structure of the singular boundary in a neighbourhood of timelike infinity. In fact, in the case of small perturbations of two-ended Reissner–Nordstrom data in the context of the above system, I have more recently shown (see this link) that the entire boundary of the Cauchy development can be viewed as a bifurcate null hypersurface, in a spacetime extension with continuous metric to which all inextendible causal geodesic enter. This bifurcate null hypersurface is globally singular under suitable assumptions on the data. Thus, at least in the context of this spherically symmetric model, there are generic families of spacetimes whose singular boundary is no-where spacelike

There are good reasons to hope that the above model captures vacuum dynamics without symmetry. This suggests the following conjecture, which is essentially due to Ori:

ConjectureFor small perturbations of two-ended subextremal Kerr initial data, the maximal development is extendible as a continuous metric so that all incomplete geodesics pass into the extension. The boundary of the original spacetime in this extension is a bifurcate null hypersurface. For generic perturbations, however, any such continuous extension will fail to be C^2, in fact, will fail to have locally square integrable Christoffel symbols.

In view of the above conjecture, let us return to the issue of the proper statement of inextendibility in the formulation of the strong cosmic censorship conjecture.

The most convenient formulation, and one which is often used in the literature, is perhaps "inextendible as a C^2 metric". For then, a sufficient (but not necessary!) condition for verifying this is that some curvature component in a freely falling frame associated to each incomplete causal geodesic blows up. This formulation would indeed be compatible with the above conjecture, as failing to have locally square integrable Christoffel symbol implies that a metric is not C^2.

Experience from the analysis of partial differential equations, however, strongly suggests that this formulation is not appropriate, as the pointwise blowup of the second derivatives of the metric is no obstacle for existence of solutions, in fact, of "strong solutions" lying within the domain of well-posedness theory. Indeed, in view of a fundamental breakthrough of Sergiu Klainerman, Igor Rodnianski, and Jérémie Szeftel, the Einstein equations are well-posed in a suitable sense when the curvature is only assumed to be locally square integrable. See this link

The above conjecture concerning perturbations of Kerr, on the other hand, makes a stronger inextenbility statement, and this one is sufficient to rule out any reasonable notion of weak solution to the Einstein equations, as these would at the very least need to have locally square integrable Christoffel symbols. 

Thus a physically well motivated version of strong cosmic censorship which may still be true is

ConjectureFor generic initial data for the vacuum equations or for suitable Einstein–matter systems, the maximal Cauchy development is inextendible as a metric with locally square integrable Christoffel symbols.

This formulation is due to Demetrios Christodoulou and is discussed on page 13 of this link.

Ominously, in the cosmological version of this conjecture, this physically motivated formulation of inextendibility might not be true. See Section of 15 of this link.

A good introduction to spherically symmetric collapse, from which the reader can get a feeling of what the most general singular boundary of spacetime can look like for reasonable matter models, is contained in this paper of Jonathan Kommemi.