6.2 Outlook6 Conclusions6 Conclusions

6.1 Summary

When one fine-tunes a smooth one-parameter family of smooth, asymptotically flat initial data to get close enough to the black hole threshold, the details of the initial data are completely forgotten in a small spacetime region where the curvature is high, and all near-critical time evolutions converge to one universal solution there. (This region is limited both in space and time, and at late times the final state is either a black hole or empty space.) At the black hole threshold, there either is a universal minimum black hole mass (type I transition), or black hole formation starts at infinitesimal mass (type II transition). In a type I transition, the universal critical solution is time-independent, or periodic in time, and the closer the initial data are to the black hole threshold, the longer it persists. In a type II transition, the universal critical solution is scale-invariant or scale-periodic, and the closer the initial data are to the black hole threshold, the smaller the black hole mass, by the famous formula (1Popup Equation).

Both types of behavior arise because there is a solution which is an intermediate attractor, or attractor of codimension one. Its basin of attraction is the black hole threshold itself, a hypersurface of codimension one that bisects phase space. Any time evolution that begins with initial data near the black hole threshold (but not necessarily close to the critical solution) first approaches the critical solution, then moves away from it along its one growing perturbation mode. At late times, the solution only remembers on which side of the black hole threshold the initial data were, and how far away from the threshold.

Our understanding of critical phenomena rests on this dynamical systems picture, but crucial details of the picture have not yet been defined rigorously. Nevertheless, it suggests semi-analytic perturbative calculations that have been successful in predicting the scaling of black hole mass and charge in critical collapse to high precision.

The importance of type II behavior lies in providing a natural route from large (the initial data) to arbitrarily small (the final black hole) scales, with possible applications to astrophysics and quantum gravity. Fine-tuning any one generic parameter in the initial data to the black hole threshold, for a number of matter models, without assuming any other symmetries, will do the trick.

Type II critical behavior also clarifies what version of cosmic censorship one can hope to prove. At least in some matter models (scalar field, perfect fluid), fine-tuning any smooth one-parameter family of smooth, asymptotically flat initial data, without any symmetries, gives rise to a naked singularity. In this sense the set of initial data that form a naked singularity is codimension one in the full phase space of smooth asymptotically flat initial data for well-behaved matter. Any statement of cosmic censorship in the future can only exclude naked singularities arising from generic initial data.

Finally, critical phenomena are arguably the outstanding contribution of numerical relativity to knowledge in GR to date, and they continue to act as a motivation and a source of testbeds for numerical relativity.

6.2 Outlook6 Conclusions6 Conclusions

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
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