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.
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.
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