4 More Spherical Symmetry
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
- 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
- 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
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.