Let Z stand for a set of scale-invariant variables of the problem in a first-order formulation. Z (r) is an element of the phase space, and Z (r, t) a solution. The self-similar solution is of the form . In the echoing region, where dominates, we linearize around it. As the background solution is -independent, , its linear perturbations can depend on only exponentially (with complex exponent ), that is
where the are free constants. To linear order, the solution in the echoing region is then of the form
The coefficients depend in a complicated way on the initial data, and hence on p . If is a critical solution, by definition there is exactly one with positive real part (in fact it is purely real), say . As from below and , all other perturbations vanish. In the following we consider this limit, and retain only the one growing perturbation. By definition the critical solution corresponds to , so we must have . Linearizing around , we obtain
This approximate solution explains why the solution is universal. It is now also clear why Eqn. (15) holds, that is why we see more of the universal solutions (in the DSS case, more ``echos'') as p is tuned closer to . The critical solution would be revealed up to the singularity if perfect fine-tuning of p was possible. A possible source of confusion is that the critical solution, because it is self-similar, is not asymptotically flat. Nevertheless, it can arise in a region up to finite radius as the limiting case of a family of asymptotically flat solutions. At large radius, it is matched to an asymptotically flat solution which is not universal but depends on the initial data (as does the place of matching).
The solution has the approximate form (33) over a range of . Now we extract Cauchy data at one particular value of within that range, namely defined by
where is some constant , so that at this the linear approximation is still valid. Note that depends on p . At sufficiently large , the linear perturbation has grown so much that the linear approximation breaks down. Later on a black hole forms. The crucial point is that we need not follow this evolution in detail, nor does it matter at what amplitude we consider the perturbation as becoming nonlinear. It is sufficient to note that the Cauchy data at depend on r only through the argument x, because by definition of we have
Going back to coordinates t and r we have
These intermediate data at depend on the initial data at t =0 only through the overall scale . The field equations themselves do not have an intrinsic scale. It follows that the solution based on the data at must be universal up to the overall scale. In suitable coordinates (for example the polar-radial coordinates of Choptuik) it is then of the form
for some function f that is universal for all 1-parameter families . This universal form of the solution applies for all , even after the approximation of linear perturbation theory around the critical solution breaks down. Because the black hole mass has dimension length, it must be proportional to , the only length scale in the solution. Therefore
and we have found the critical exponent .
When the critical solution is DSS, the scaling law is modified. This was predicted in , and predicted independently and verified in collapse simulations by Hod and Piran . On the straight line relating to , a periodic ``wiggle'' or ``fine structure'' of small amplitude is superimposed:
with . The periodic function f is again universal with respect to families of initial data, and there is only one parameter c that depends on the family of initial data, corresponding to a shift of the wiggly line in the direction .
It is easy to see that for near-critical solutions the maximal value of the scalar curvature, and similar quantities, scale just like the black hole mass, with a critical exponent . Technically, it is easier to measure the critical exponent and the fine-structure in the subcritical regime from the maximum curvature than from the black hole mass in the supercritical regime .
|Critical Phenomena in Gravitational Collapse
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