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 [38Jump To The Next Citation Point]:
ℒξgab = 2gab. (2 )
In coordinates μ i x = (τ,x ) adapted to the symmetry, so that
∂ ξ = − ---, (3 ) ∂τ
the metric coefficients are of the form
i −2τ i gμν(τ,x ) = e &tidle;gμν(x ), (4 )
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

gμν(τ,xi) = e−2τ&tidle;gμν(τ,xi), (5 )
such that gμν(τ,xi) is periodic in τ with period Δ. More formally, DSS can be defined as a discrete conformal isometry [95Jump To The Next Citation Point].

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 (3View Equation) and hence Equations (4View Equation) and (5View Equation), is a convention assuming that smaller scales are in the future. The time parameter used in Figure 2View Image is of this type.

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