### 2.1 Universality

Consider GR as an infinite-dimensional continuous dynamical system. Points in the phase space are
initial data sets (3-metric, extrinsic curvature, and suitable matter variables, which together obey the
Einstein constraints). We evolve with the Einstein equations in a suitable gauge (see Section 2.5). Solution
curves of the dynamical system are spacetimes obeying the Einstein-matter equations, sliced by specific
Cauchy surfaces of constant time t.
An isolated system in GR can end up in qualitatively different stable end states. Two possibilities are
the formation of a single black hole in collapse, or complete dispersion of the mass-energy to infinity.
For a massless scalar field in spherical symmetry, these are the only possible end states (see
Section 3). Any point in phase space can be classified as ending up in one or the other type
of end state. The entire phase space therefore splits into two halves, separated by a “critical
surface”.

A phase space trajectory that starts on a critical surface by definition never leaves it. A critical surface
is therefore a dynamical system in its own right, with one dimension fewer than the full system. If it has an
attracting fixed point, such a point is called a critical point. It is an attractor of codimension one in the full
system, and the critical surface is its attracting manifold. The fact that the critical solution is an attractor
of codimension one is visible in its linear perturbations: It has an infinite number of decaying perturbation
modes spanning the tangent plane to the critical surface, and a single growing mode not tangential to
it.

As illustrated in Figures 1 and 2, any trajectory beginning near the critical surface, but not necessarily
near the critical point, moves almost parallel to the critical surface towards the critical point. Near the
critical point the evolution slows down, and eventually moves away from the critical point in
the direction of the growing mode. This is the origin of universality. All details of the initial
data have been forgotten, except for the distance from the black hole threshold. The closer
the initial phase point is to the critical surface, the more the solution curve approaches the
critical point, and the longer it will remain close to it. We should stress that this phase picture is
extremely simplified. Some of the problems associated with this simplification are discussed in
Section 2.5.