The pioneering work of Choptuik on the spherical massless scalar field has been followed by a plethora of further investigations. These could be organised under many different criteria. We have chosen the following rough categories:

- Systems in which the field equations, when reduced to spherical symmetry, form a single wave-like equation, typically with explicit r-dependence in its coefficients. This includes Yang–Mills fields, sigma models, vector and spinor fields, scalar fields in 2+1 or in 4+1 and more spacetime dimensions, and scalar fields in a semi-classical approximation to quantum gravity.
- Perfect fluid matter, either in an asymptotically flat or a cosmological context. The linearised Euler equations are in fact wave-like, but the full non-linear equations admit shock heating and are therefore not even time-reversal symmetric.
- Collisionless matter described by the Vlasov equation is a partial differential equation on particle phase space as well as spacetime. Therefore even in spherical symmetry, the matter equation is a partial differential equation in 4 dimensions (rather than two). Intuitively speaking, there are infinitely more matter degrees of freedom than in the scalar field or in non-spherical vacuum gravity.
- Spherically symmetric nonlinear wave equations on 3+1 Minkowski spacetime, and other nonlinear partial differential equations which show a transition between singularity formation and dispersal.

Some of these examples were constructed because they may have intrinsic physical relevance (semiclassical gravity, primordial black holes), others as toy models for 3+1-dimensional gravity, and others mostly out of a purely mathematical interest. Table 1 gives an overview of these models.

4.1 Matter obeying wave equations

4.1.1 2-dimensional nonlinear model

4.1.2 Spherical Einstein–SU(2) sigma model

4.1.3 Einstein–Yang–Mills

4.1.4 Vacuum 4+1

4.1.5 Scalar field collapse in 2+1

4.1.6 Scalar field collapse in higher dimensions

4.1.7 Other systems obeying wave equations

4.2 Perfect fluid matter

4.2.1 Spherical symmetry

4.2.2 Nonspherical perturbations

4.2.3 Cosmological applications

4.3 Collisionless matter

4.4 Criticality in singularity formation without gravitational collapse

4.5 Analytic studies and toy models

4.5.1 Exact solutions of Einstein–Klein–Gordon

4.5.2 Toy models

4.6 Quantum effects

4.1.1 2-dimensional nonlinear model

4.1.2 Spherical Einstein–SU(2) sigma model

4.1.3 Einstein–Yang–Mills

4.1.4 Vacuum 4+1

4.1.5 Scalar field collapse in 2+1

4.1.6 Scalar field collapse in higher dimensions

4.1.7 Other systems obeying wave equations

4.2 Perfect fluid matter

4.2.1 Spherical symmetry

4.2.2 Nonspherical perturbations

4.2.3 Cosmological applications

4.3 Collisionless matter

4.4 Criticality in singularity formation without gravitational collapse

4.5 Analytic studies and toy models

4.5.1 Exact solutions of Einstein–Klein–Gordon

4.5.2 Toy models

4.6 Quantum effects

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