### 2.3 Mass scaling

Let stand for a set of scale-invariant variables of the problem, such as and suitably rescaled
matter variables. If the dynamics is scale-invariant (this is the case exactly for example for the scalar field,
and approximately for other systems, see Section 2.6), then is an element of the phase space
factored by overall scale, and a solution. Note that is an initial data set for GR only up to
scale. The overall scale is supplied by .
For simplicity, assume that the critical solution is CSS. It can then be written as . Its
linear perturbations can depend on only exponentially. To linear order, the solution near the critical
point must be of the form

The perturbation amplitudes depend on the initial data, and hence on . As is a critical
solution, by definition there is exactly one with positive real part (in fact it is purely real), say .
As , all other perturbations vanish. In the following we consider this limit, and retain only the one
growing perturbation.
From our phase space picture, the evolution ends at the critical solution for , so we must have
. Linearising in around , we obtain

For , but close to it, the solution has the approximate form (7) over a range of . Now we
extract Cauchy data at one particular -dependent value of within that range, namely defined
by

where is some constant such that at this the linear approximation is still valid. At
sufficiently large , the linear perturbation has grown so much that the linear approximation breaks
down, and for a black hole forms while for the solution disperses. The crucial point is
that we need not follow this evolution in detail, nor does the precise value of matter. It is sufficient to
note that the Cauchy data at are
Due to the funnelling effect of the critical solution, the data at is always the same, except for an overall
scale, which is given by . For example, the physical spacetime metric, with dimension is
given by , and similar scalings hold for the matter variables according to their dimension. In
particular, as is the only scale in the initial data (9), the mass of the final black hole must be
proportional to that scale. Therefore
and, comparing with Equation (1), we have found the critical exponent .
When the critical solution is DSS, a periodic or fine structure of small amplitude is superimposed on this
basic power law [98, 127]:

where has period and is universal, and only depends on the initial data. As the critical
solution is periodic in with period , the number of scaling “echos” is approximated by
Note that this holds for both supercritical and subcritical solutions.