Critical phenomena in gravitational collapse were first discovered by Choptuik [47, 48, 49] in the model of a spherically symmetric, massless scalar field minimally coupled to general relativity. The scalar field matter is both simple, and acts as a toy model in spherical symmetry for the effects of gravitational radiation. Given that it is still the best-studied model in spherical symmetry, we review it here as a case study. For other numerical work on this model, see [109, 80, 111, 77, 182, 212]. Important analytical studies of gravitational collapse in this model have been carried out by Christodoulou [57, 58, 59, 60, 61, 62, 63].

We first review the field equations and Choptuik’s observations at the black hole threshold, mainly as a concrete example for the general ideas discussed above. We then summarise more recent work on the global structure of Choptuik’s critical solution, which throws an interesting light on cosmic censorship. In particular, the exact critical solution contains a curvature singularity that is locally and globally naked, and any critical solution obtained in the limit of perfect fine-tuning of asymptotically flat initial data is at least locally naked. By perturbing around spherical symmetry, the stability of the Choptuik solution in the full phase space can be investigated, and the scaling of black hole angular momentum can be predicted. By embedding the real scalar field in scalar electrodynamics and perturbing around the Choptuik solution, the scaling of black hole charge can be predicted.

3.1 Field equations in spherical symmetry

3.2 The black hole threshold

3.3 Global structure of the critical solution

3.4 Near-critical spacetimes and naked singularities

3.5 Electric charge

3.6 Self-interaction potential

3.7 Nonspherical perturbations: Stability and angular momentum

3.2 The black hole threshold

3.3 Global structure of the critical solution

3.4 Near-critical spacetimes and naked singularities

3.5 Electric charge

3.6 Self-interaction potential

3.7 Nonspherical perturbations: Stability and angular momentum

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