In his seminal work [52], Choptuik considers a real, massless scalar field and numerically evolves various initial configurations, finding either dispersion or black-hole formation. By parameterizing these initial configurations, say by the amplitude of an initial pulse , and by tuning this parameter, he was able to study the threshold for black-hole formation at which he found fascinating black-hole critical behavior. In particular, his numerical work suggested that continued tuning could produce as small a black hole as one wished. This behavior is analogous to a phase transition in which the black-hole mass serves as an order parameter. Similar to phase transitions, one can categorize two types of transition that distinguish between whether the black-hole mass varies continuously (Type II) or discontinuously (Type I). For Choptuik’s work with a massless field, the transition is therefore of Type II because the black-hole mass varies from zero continuously to infinitesimal values.

Subsequent work has since established that this critical behavior can be considered as occurring in the neighborhood of a separatrix between the basins of attraction of the two end states. For , the system is precisely critical and remains on the (unstable) separatrix. Similarly other models find such threshold behavior occurring between a stationary state and black-hole formation. Critical behavior about stationary solutions necessarily involve black-hole formation “turning-on” at finite mass, and is therefore categorized as Type I critical behavior.

The critical surface, therefore, appears as a co-dimension 1 surface, which evolutions increasingly approach as one tunes the parameter . The distance from criticality serves as a measure of the extent to which a particular initial configuration has excited the unstable mode that drives solutions away from this surface. For Type II critical behavior, the mass of the resulting black-hole mass scales as a power law in this distance, whereas for Type I critical behavior, it is the survival time of the critical solution that scales as a power law. See [100] for a recent review.

We have seen that boson stars represent stationary solutions of Einstein’s equations and, thus, one would correctly guess that they may occur within Type I black-hole critical behavior. To look for such behavior, Hawley and Choptuik [109] begin their evolutions with boson-star solutions and then perturb them both dynamically and gravitationally. They, therefore, included in their evolutionary system a distinct, free, massless, real scalar field which couples to the boson star purely through its gravity.

The initial data, therefore, consisted of a boson star surrounded by a distant, surrounding shell of real scalar field parametrized by the amplitude of the shell. For small perturbations, the boson star oscillated about an unstable boson star before settling into a low mass, stable solution (see Figure 17). For large perturbations, the real scalar field serves to compress the initial star and, after a period of oscillating about an unstable boson star, the complex field collapses to a black hole. By tuning the initial perturbation, they find a longer and longer lived unstable boson star, which serves as the critical solution (see Figure 11). The survival time obeys a power law in terms of the distance from criticality

where is a real constant that depends on the characteristic instability rate of the particular unstable boson star approached in the critical regime.One can also consider these BSs in axisymmetry in which non-spherically symmetric modes could potentially become important. A first step in this direction studied spherically symmetric BSs within conformally flat gravity (which does not allow for gravitational waves) in axisymmetry [187]. Later, better resolution using adaptive mesh refinement within full general relativity was achieved by [141, 142], which upheld the results found within spherical symmetry. This work thus suggests that there are either no additional, unstable, axisymmetric modes or that such unstable modes are extremely slowly growing.

A very different type of critical behavior was also investigated by Lai [141]. By boosting identical boson stars toward each other and adjusting their initial momenta, he was able to tune to the threshold for black-hole formation. At the threshold, he found that the time till black-hole formation scaled consistent with Type I critical behavior and conjectured that the critical solution was itself an unstable boson star. This is one of the few fully nonlinear critical searches in less symmetry than spherical symmetry, and the first of Type I behavior in less symmetry. A related study colliding neutron stars instead of boson stars similarly finds Type I critical behavior [124] and subsequently confirmed by [127].

The gauged stars discussed in Section 3.9 also serve as critical solutions in spherical symmetry [53, 54, 166].

Living Rev. Relativity 15, (2012), 6
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