4 Extensions of the basic 3 The basic scenario3.2 Scale-invariance and self-similarity

3.3 Black hole mass scaling 

The following calculation of the critical exponent from the linear perturbations of the critical solution by dimensional analysis was suggested by Evans and Coleman [53Jump To The Next Citation Point In The Article] and carried out by Koike, Hara and Adachi [91Jump To The Next Citation Point In The Article] and Maison [98Jump To The Next Citation Point In The Article]. It was generalized to the discretely self-similar (DSS) case by Gundlach [74Jump To The Next Citation Point In The Article]. For simplicity of notation we consider again the spherically symmetric CSS case. The DSS case is discussed in [74Jump To The Next Citation Point In The Article].

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 tex2html_wrap_inline2563 . In the echoing region, where tex2html_wrap_inline2565 dominates, we linearize around it. As the background solution is tex2html_wrap_inline2337 -independent, tex2html_wrap_inline2569, its linear perturbations can depend on tex2html_wrap_inline2337 only exponentially (with complex exponent tex2html_wrap_inline2573), that is


where the tex2html_wrap_inline2575 are free constants. To linear order, the solution in the echoing region is then of the form


The coefficients tex2html_wrap_inline2575 depend in a complicated way on the initial data, and hence on p . If tex2html_wrap_inline2565 is a critical solution, by definition there is exactly one tex2html_wrap_inline2583 with positive real part (in fact it is purely real), say tex2html_wrap_inline2585 . As tex2html_wrap_inline2587 from below and tex2html_wrap_inline2589, 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 tex2html_wrap_inline2215, so we must have tex2html_wrap_inline2593 . Linearizing around tex2html_wrap_inline2221, we obtain


This approximate solution explains why the solution tex2html_wrap_inline2565 is universal. It is now also clear why Eqn. (15Popup Equation) holds, that is why we see more of the universal solutions (in the DSS case, more ``echos'') as p is tuned closer to tex2html_wrap_inline2221 . The critical solution would be revealed up to the singularity tex2html_wrap_inline2603 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 (33Popup Equation) over a range of tex2html_wrap_inline2337 . Now we extract Cauchy data at one particular value of tex2html_wrap_inline2337 within that range, namely tex2html_wrap_inline2611 defined by


where tex2html_wrap_inline2613 is some constant tex2html_wrap_inline2615, so that at this tex2html_wrap_inline2337 the linear approximation is still valid. Note that tex2html_wrap_inline2611 depends on p . At sufficiently large tex2html_wrap_inline2337, 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 tex2html_wrap_inline2613 we consider the perturbation as becoming nonlinear. It is sufficient to note that the Cauchy data at tex2html_wrap_inline2627 depend on r only through the argument x, because by definition of tex2html_wrap_inline2611 we have


Going back to coordinates t and r we have


These intermediate data at tex2html_wrap_inline2639 depend on the initial data at t =0 only through the overall scale tex2html_wrap_inline2643 . The field equations themselves do not have an intrinsic scale. It follows that the solution based on the data at tex2html_wrap_inline2307 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 [83Jump To The Next Citation Point In The Article]. This universal form of the solution applies for all tex2html_wrap_inline2649, 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 tex2html_wrap_inline2643, the only length scale in the solution. Therefore


and we have found the critical exponent tex2html_wrap_inline2653 .

When the critical solution is DSS, the scaling law is modified. This was predicted in [74Jump To The Next Citation Point In The Article], and predicted independently and verified in collapse simulations by Hod and Piran [87]. On the straight line relating tex2html_wrap_inline2655 to tex2html_wrap_inline2657, a periodic ``wiggle'' or ``fine structure'' of small amplitude is superimposed:


with tex2html_wrap_inline2659 . 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 tex2html_wrap_inline2657 direction Popup Footnote .

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 tex2html_wrap_inline2669 . 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 [61].

4 Extensions of the basic 3 The basic scenario3.2 Scale-invariance and self-similarity

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
© Max-Planck-Gesellschaft. ISSN 1433-8351
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