Self-similarity

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 $ξ$ , defined by [38

In coordinates $μ i x = (τ,x )$ adapted to the symmetry, so that

the metric coefficients are of the form

where the coordinate $τ$ is the negative logarithm of a spacetime scale, and the remaining three coordinates $xi$ can be thought of angles around the singular spacetime point $τ = ∞$ (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 $gμν(τ,xi)$ is periodic in $τ$ with period $Δ$ . 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). $τ$ 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 $xi$ . 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.