4.3 Approximate self-similarity and universality 4 Extensions of the basic 4.1 Black hole thresholds with

4.2 CSS and DSS critical solutions

Critical solutions are continuously or discretely self-similar, and have exactly one growing perturbation mode. Other regular CSS or DSS solutions have more than one growing mode, and so will not appear as critical solution at the black hole threshold. An example for this is provided by the spherically symmetric massless complex scalar field. Hirschmann and Eardley [84Jump To The Next Citation Point In The Article] found a way of constructing a CSS scalar field solution by making the scalar field tex2html_wrap_inline2325 complex but limiting it to the ansatz


with tex2html_wrap_inline2739 a real constant and f real. The metric is then homothetic, while the scalar field shows a trivial kind of ``echoing'' in the complex phase. Later, they found that this solution has three modes with tex2html_wrap_inline2743  [83] and is therefore not the critical solution. On the other hand, Gundlach [74Jump To The Next Citation Point In The Article] examined complex scalar field perturbations around Choptuik's real scalar field critical solution and found that only one of them, purely real, has tex2html_wrap_inline2743, so that the real scalar field critical solution is a critical solution (up to an overall complex phase) also for the free complex scalar field. This had been seen already in collapse calculations [35].

As the symmetry of the critical solution, CSS or DSS, depends on the matter model, it is interesting to investigate critical behavior in parameterized families of matter models. Two such one-parameter families have been investigated. The first one is the spherical perfect fluid with equation of state tex2html_wrap_inline2505 for arbitrary k . Maison [98Jump To The Next Citation Point In The Article] constructed the regular CSS solutions and its linear perturbations for a large number of values of k . In each case, he found exactly one growing mode, and was therefore able to predict the critical exponent. (To my knowledge, these critical exponents have not yet been verified in collapse simulations.) As Ori and Piran before [104, 105], he claimed that there are no regular CSS solutions for k > 0.88. Recently, Neilsen and Choptuik [100, 101] have found CSS critical solutions for all values of k right up to 1, both in collapse simulations and by making a CSS ansatz. Interesting questions arise because the stiff (tex2html_wrap_inline2759) perfect fluid, limited to irrotational solutions, is equivalent to the massless scalar field, limited to solutions with timelike gradient, while the scalar field critical solution is actually DSS. These are currently being investigated [20].

The second one-parameter family of matter models was suggested by Hirschmann and Eardley [85], who looked for a natural way of introducing a nonlinear self-interaction for the (complex) scalar field without introducing a scale. (We discuss dimensionful coupling constants in the following sections.) They investigated the model described by the action


Note that tex2html_wrap_inline2325 is now complex, and the parameter tex2html_wrap_inline2449 is real and dimensionless. This is a 2-dimensional sigma model with a target space metric of constant curvature (namely tex2html_wrap_inline2449), minimally coupled to gravity. Moreover, for tex2html_wrap_inline2445 there are (nontrivial) field redefinitions which make this model equivalent to a real massless scalar field minimally coupled to Brans-Dicke gravity, with the Brans-Dicke coupling given by


In particular, tex2html_wrap_inline2441 (tex2html_wrap_inline2771) corresponds to an axion-dilaton system arising in string theory [51]. tex2html_wrap_inline2433 is the free complex scalar field coupled to Einstein gravity. Hirschmann and Eardley calculated a CSS solution and its perturbations, and concluded that it is the critical solution for tex2html_wrap_inline2775, but has three unstable modes for tex2html_wrap_inline2777 . For tex2html_wrap_inline2779, it acquires even more unstable modes. The positions of the mode frequencies tex2html_wrap_inline2573 in the complex plane vary continuously with tex2html_wrap_inline2449, and these are just values of tex2html_wrap_inline2449 where a complex conjugate pair of frequencies crosses the real axis. The results of Hirschmann and Eardley confirm and subsume collapse simulation results by Liebling and Choptuik [97] for the scalar-Brans-Dicke system, and collapse and perturbative results on the axion-dilaton system by Hamadé, Horne and Stewart [80]. Where the CSS solution fails to be the critical solution, a DSS solution takes over. In particular, for tex2html_wrap_inline2433, the free complex scalar field, the critical solution is just the real scalar field DSS solution of Choptuik.

Liebling [95] has found initial data sets that find the CSS solution for values of tex2html_wrap_inline2449 (for example tex2html_wrap_inline2433) where the true critical solution is DSS. The complex scalar field in these data sets is of the form tex2html_wrap_inline2793 times a slowly varying function of r, for arbitrary r, while its momentum tex2html_wrap_inline2799 is either zero or tex2html_wrap_inline2801 . Conversely, data sets that are purely real find the DSS solution even for values of tex2html_wrap_inline2449 where the true critical solution is the CSS solution, for example for tex2html_wrap_inline2441 . These two special families of initial data maximize and minimize the U (1) charge. Small deviations from these data find the sub-dominant ``critical'' solution for some time, then veer off and find the true critical solution. (Even later, of course, the critical solution is also abandoned in turn for dispersion or black hole formation.)

4.3 Approximate self-similarity and universality 4 Extensions of the basic 4.1 Black hole thresholds with

image Critical Phenomena in Gravitational Collapse
Carsten Gundlach
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