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3.4 Weak null singularities and Price’s law

The results of this section concern spherically symmetric solutions but in order to explain their significance they need to be presented in context. A non-rotating uncharged black hole is represented by the Schwarzschild solution, which contains a singularity. At this singularity the Kretschmann scalar RabgdRabgd blows up uniformly and this represents an obstruction to extending the spacetime through the singularity, at least in a C2 manner.

A rotating uncharged black hole is represented by the Kerr solution in which the Schwarzschild singularity is replaced by a Cauchy horizon. This horizon marks a pathology of the global causal structure of the solution but locally the geometry can be extended smoothly through it. A similar situation is found in the non-rotating charged black hole which is represented by the Reissner-Nordström solution. These facts are worrying since they suggest that black holes may generally lead to causal pathologies. The rotating case is the more physically interesting one, but the charged case is a valuable model problem for the rotating case. Spherical symmetry leads to immense technical simplifications and so only that case will be discussed here. It is the only one where theorems on global existence and qualitative behaviour relevant to this problem are available.

It was early suggested that the Cauchy horizon of the Reissner-Nordström solution should be unstable and that a generic perturbation of the initial data would lead to its being replaced by a Schwarzschild-like singularity. This scenario turned out to be oversimplified. An alternative was suggested by Poisson and Israel [267]. In their picture a generic perturbation of the Reissner-Nordström data leads to the Cauchy horizon being replaced by what they call a weak null singularity. At this singularity the curvature blows up, but the metric can be extended through the singularity in a way which is continuous and non-degenerate. In this situation it is possible to make sense of the causal character of the singularity which turns out to be null. Furthermore, an important invariant, the Hawking mass, blows up at the singularity, a phenomenon known as mass inflation. All these conclusions were based on heuristic arguments which were later backed up by numerical results [185Jump To The Next Citation Point].

A mathematical understanding of these effects came with the work of Dafermos [122]. He showed how, starting from a characteristic initial value problem with data given on two null hypersurfaces, one of which is the event horizon, it is possible to prove that a weak null singularity forms and that there is mass inflation. He uses a model with an uncharged scalar field and a static charge and works entirely inside the black hole region.

Ideally one would wish to start with regular data on a standard Cauchy surface and control both formation of the black hole and the evolution in its interior. This requires using some kind of charged matter, e.g., a charged scalar field. This is what was done numerically in [185]. Analytically it remains out of reach at the moment.

In the original heuristic arguments it is important to make statements about the behaviour of the solution outside the black hole and what behaviour on the horizon results. Here there are classical heuristic results of Price [268] for a scalar field on a black hole background. He states that the scalar field falls off in a certain way along the horizon. Let us call this Price’s law. Now a form of Price’s law and its analogue for the coupled spherically symmetric Einstein-scalar field system have been proved by Dafermos and Rodnianski [126]. Thus we have come a long way towards an understanding of the problem discussed here. This has required the development of new mathematical techniques and these may one day turn out to be of importance in understanding the nonlinear stability of black holes.

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