3 The Scalar Field
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.
