Plan of this review

1.2 Plan of this review

Faced with more material to review, we have attempted to make this update shorter and more systematic than the original 1999 version [102

]. We begin with the abstract theory in Section 2. Since 1999, progress on the theory side has mainly been made on the global spacetime structure of the critical solution and cosmic censorship in the spherical scalar field model. We have included this new material in an enlarged Section 3 on the spherical scalar field, although we hope it will turn out to be sufficiently generic to merit inclusion in Section 2. Nonspherical perturbations of the spherical scalar field are also discussed in Section 3.

In Section 4 we review the rich phenomenology that has been found in many other systems restricted to spherical symmetry. Numerical work in spherical symmetry has proliferated since 1999, but we have tried to keep this section as short as possible. There has been less progress in going beyond spherical symmetry than we anticipated in 1999, even though we continue to believe that important results await there. What is known today is summarised in Section 5.

The reader unfamiliar with the topic is advised to begin with either Sections 2.1, 2.2, and 2.3, which give the key theory of universality, self-similarity and scaling, or Sections 3.1 and 3.2, which describe the classic example, the massless scalar field.

This review is limited to numerical and theoretical work on phenomena at the threshold of black hole formation in 3+1-dimensional general relativity. We report only briefly on work in higher and lower spacetime dimensions and non-gravity systems that may be relevant as toy models for general relativity. We exclude other work on self-similarity in general relativity and work on critical phenomena in other areas of physics.

Other reviews on the subject are [129 ], [15 ], [96 ], [100 ], [50 ], [51 ], [33 ], [102 ], [156 ]. The 2002 review [105 ] by Gundlach gives more detailed explanations on some of the basic aspects of the theory. A review of the role of self-similarity in the formation of singularities in evolutionary PDEs in general is [71 ]. Update