5.4 Astrophysical black holes5 Aspects of current research5.2 The renormalisation group as

5.3 Analytic approaches

A number of authors have attempted to explain critical collapse with the help of analytic solutions. The one-parameter family of exact self-similar real massless scalar field solutions first discovered by Roberts [112Popup Equation] has already been presented in Section  4.5 . It has been discussed in the context of critical collapse in [18, 106], and later [121, 26]. The original, analytic, Roberts solution is cut and pasted to obtain a new solution which has a regular center r =0 and which is asymptotically flat. Solutions from this family [see Eqns. (64Popup Equation)] with p >1 can be considered as black holes, and to leading order around the critical value p =1, their mass is tex2html_wrap_inline3449 . The pitfall in this approach is that only perturbations within the self-similar family are considered, so the formal critical exponent applies only to this one, very special, family of initial data. But the p =1 solution has many growing perturbations which are spherically symmetric (but not self-similar), and is therefore not a critical solution in the sense of being an attractor of codimension one. This was already clear because it did not appear in collapse simulations at the black hole threshold, but Frolov has calculated the perturbation spectrum analytically [56, 57]. The eigenvalues of spherically symmetric perturbations fill a sector of the complex plane, with tex2html_wrap_inline3453 . All nonspherical perturbations decay. Other supposed critical exponents that have been derived analytically are usually valid only for a single, very special family of initial data also.

Other authors have employed analytic approximations to the actual Choptuik solution. Pullin [109] has suggested describing critical collapse approximately as a perturbation of the Schwarzschild spacetime. Price and Pullin [108] have approximated the Choptuik solution by two flat space solutions of the scalar wave equation that are matched at a ``transition edge'' at constant self-similarity coordinate x . The nonlinearity of the gravitational field comes in through the matching procedure, and its details are claimed to provide an estimate of the echoing period tex2html_wrap_inline2377 . While the insights of this paper are qualitative, some of its ideas reappear in the construction [71] of the Choptuik solution as a 1+1 dimensional boundary value problem. Frolov [55] has suggested approximating the Choptuik solution as the Roberts solution plus its most rapidly growing (spherical) perturbation mode, pointing out that it oscillates in tex2html_wrap_inline2337 with a period 4.44, but ignoring the fact that it also grows exponentially. This is probably not a correct approach.

In summary, purely analytic approaches have so far remained unsuccessful in explaining critical collapse.



5.4 Astrophysical black holes5 Aspects of current research5.2 The renormalisation group as

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
http://www.livingreviews.org/lrr-1999-4
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