### 2.2 Self-similarity

Fixed points of dynamical systems often have additional symmetries. In the case of type II critical
phenomena, the critical point is a spacetime that is self-similar, or scale-invariant. These symmetries can be
discrete or continuous. The critical solution of a spherically symmetric perfect fluid (see Section 4.2), has
continuous self-similarity (CSS). A CSS spacetime is one that admits a homothetic vector field , defined
by [38]:
In coordinates adapted to the symmetry, so that
the metric coefficients are of the form
where the coordinate is the negative logarithm of a spacetime scale, and the remaining three
coordinates can be thought of angles around the singular spacetime point (see
Section 3.3).
The critical solution of other systems, in particular the spherical scalar field (see Section 3) and
axisymmetric gravitational waves (see Section 5.2), show discrete self-similarity (DSS). The simplest way of
defining DSS is in adapted coordinates, where

such that is periodic in with period . More formally, DSS can be defined as a discrete
conformal isometry [95].
Using the gauge freedom of general relativity, the lapse and shift in the ADM formalism can be
chosen (non-uniquely) so that the coordinates become adapted coordinates if and when the
solution becomes self-similar (see Section 2.5). is then both a time coordinate (in the usual
sense that surfaces of constant time are Cauchy surfaces), and the logarithm of overall scale at
constant . The minus sign in Equation (3) and hence Equations (4) and (5), is a convention
assuming that smaller scales are in the future. The time parameter used in Figure 2 is of this
type.