2.3 Mass scaling

Let Z stand for a set of scale-invariant variables of the problem, such as &tidle;g μν 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 Z(x) is an element of the phase space factored by overall scale, and Z (x,τ) a solution. Note that Z (x) is an initial data set for GR only up to scale. The overall scale is supplied by τ.

For simplicity, assume that the critical solution is CSS. It can then be written as Z (x,τ) = Z ∗(x ). Its linear perturbations can depend on τ only exponentially. To linear order, the solution near the critical point must be of the form

∑∞ Z (x, τ) ≃ Z (x) + C (p)eλiτZ (x ). (6 ) ∗ i i i=1
The perturbation amplitudes Ci depend on the initial data, and hence on p. As Z ∗ is a critical solution, by definition there is exactly one λi with positive real part (in fact it is purely real), say λ0. As τ → ∞, 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 p = p∗, so we must have C0 (p∗) = 0. Linearising in p around p∗, we obtain

dC0- λ0τ τli→m∞ Z(x,τ ) ≃ Z∗(x) + dp (p − p∗)e Z0 (x). (7 )

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

dC0 ----(p − p∗)e−λ0τ∗ ≡ 𝜖, (8 ) dp
where 𝜖 is some constant ≪ 1 such that at this τ the linear approximation is still valid. At sufficiently large τ, the linear perturbation has grown so much that the linear approximation breaks down, and for C0 > 0 a black hole forms while for C0 < 0 the solution disperses. The crucial point is that we need not follow this evolution in detail, nor does the precise value of 𝜖 matter. It is sufficient to note that the Cauchy data at τ = τ∗ are
Z (x,τ∗) ≃ Z∗(x) + 𝜖Z0(x). (9 )
Due to the funnelling effect of the critical solution, the data at τ∗ is always the same, except for an overall scale, which is given by e−τ∗. For example, the physical spacetime metric, with dimension (length)2 is given by gμν = e−2τ&tidle;gμν, and similar scalings hold for the matter variables according to their dimension. In particular, as e−τ∗ is the only scale in the initial data (9View Equation), the mass of the final black hole must be proportional to that scale. Therefore
−τ∗ 1∕λ0 M ∝ e ∝ (p − p∗) , (10 )
and, comparing with Equation (1View Equation), we have found the critical exponent γ = 1∕λ0.

When the critical solution is DSS, a periodic or fine structure of small amplitude is superimposed on this basic power law [98Jump To The Next Citation Point127]:

ln M = γ ln (p − p ∗) + c + f (γ ln(p − p∗) + c) , (11 )
where f(z) has period Δ and is universal, and only c depends on the initial data. As the critical solution is periodic in τ with period Δ, the number N of scaling “echos” is approximated by
−1 N ≃ Δ γ ln |p − p∗| + constant. (12 )
Note that this holds for both supercritical and subcritical solutions.
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