with 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  and is therefore not the critical solution. On the other hand, Gundlach  examined complex scalar field perturbations around Choptuik's real scalar field critical solution and found that only one of them, purely real, has , 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 .
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 for arbitrary k . Maison  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 () 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 .
The second one-parameter family of matter models was suggested by Hirschmann and Eardley , 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 is now complex, and the parameter is real and dimensionless. This is a 2-dimensional sigma model with a target space metric of constant curvature (namely ), minimally coupled to gravity. Moreover, for 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, () corresponds to an axion-dilaton system arising in string theory . 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 , but has three unstable modes for . For , it acquires even more unstable modes. The positions of the mode frequencies in the complex plane vary continuously with , and these are just values of 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  for the scalar-Brans-Dicke system, and collapse and perturbative results on the axion-dilaton system by Hamadé, Horne and Stewart . Where the CSS solution fails to be the critical solution, a DSS solution takes over. In particular, for , the free complex scalar field, the critical solution is just the real scalar field DSS solution of Choptuik.
Liebling  has found initial data sets that find the CSS solution for values of (for example ) where the true critical solution is DSS. The complex scalar field in these data sets is of the form times a slowly varying function of r, for arbitrary r, while its momentum is either zero or . Conversely, data sets that are purely real find the DSS solution even for values of where the true critical solution is the CSS solution, for example for . 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.)
|Critical Phenomena in Gravitational Collapse
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