Critical behavior in collapse

2.2 Critical behavior in collapse

2.2.1 Gravitational collapse simulations

For a more detailed discussion of critical behavior see [131 ]. Since Gundlach’s Living Review article has appeared, the updates in this section will be restricted to results I find especially interesting.

Critical behavior was originally found by Choptuik [77 ] in a numerical study of the collapse of a spherically symmetric massless scalar field. For recent reviews see [126 , 129 ]. We note that this is the first completely new phenomenon in general relativity to be discovered by numerical simulation. In collapse of a scalar field, essentially two things can happen: Either a black hole (BH) forms or the scalar waves pass through each other and disperse. Choptuik discovered that for any 1-parameter set of initial data labeled by $p$ , there is a critical value $p ∗$ such that $p > p∗$ yields a BH. He found

where $MBH$ is the mass of the eventual BH. The constant $CF$ depends on the parameter of the initial data that is selected but $γ ≈ 0.37$ is the same for all choices. Furthermore, in terms of logarithmic variables $ρ = ln r + κ$ , $τ = ln (T∗ − T0) + κ 0$ ( $T0$ is the proper time of an observer at $r = 0$ , where $r$ is the radial coordinate, with $∗ T 0$ the finite proper time at which the critical evolution concludes, and $κ$ is a constant which scales $r$ ), the waveform $X$ repeats (echoes) at intervals of $Δ$ in $τ$ if $ρ$ is rescaled to $ρ − Δ$ , i.e. $X (ρ − Δ, τ − Δ) ≈ X (ρ,τ )$ . The scaling behavior (1

) demonstrates that the minimum BH mass (for bosons) is zero. The critical solution itself is a counter-example to cosmic censorship (since the formation of the zero mass BH causes high curvature regions to become visible at $r = ∞$ ). (See, e.g., the discussion in Hirschmann and Eardley [145

].) The numerical demonstration of this feature of the critical solution was provided by Hamadé and Stewart [135

]. This result caused Hawking to pay off a bet to Preskill and Thorne [61 , 170 ].

Figure 3: This figure is the final frame of an animation of Type II critical behavior in Einstein–Yang–Mills collapse. Note the echoing in the near-critical solution. For the entire movie and related references see [78 ].

Soon after this discovery, scaling and critical phenomena were found in a variety of contexts. Abrahams and Evans [1 ] discovered the same phenomenon in axisymmetric gravitational wave collapse with a different value of $Δ$ and, to within numerical error, the same value of $γ$ . (Note that the rescaling of $r$ with $Δ e ≈ 30$ required Choptuik to use adaptive mesh refinement (AMR) to distinguish subsequent echoes. Abrahams and Evans’ smaller $Δ$ ( $Δ e ≈ 1.8$ ) allowed them to see echoing with their 2 + 1 code without AMR.) Garfinkle [109 ] confirmed Choptuik’s results with a completely different algorithm that does not require AMR. He used Goldwirth and Piran’s [119 ] method of simulating Christodoulou’s [80 ] formulation of the spherically symmetric scalar field in null coordinates. This formulation allowed the grid to be automatically rescaled by choosing the edge of the grid to be the null ray that just hits the central observer at the end of the critical evolution. (Missing points of null rays that cross the central observer’s world line are replaced by interpolation between those that remain.) Hamadé and Stewart [135 ] have also repeated Choptuik’s calculation using null coordinates and AMR. They are able to achieve greater accuracy and find $γ ≈ 0.374$ .

2.2.2 Critical solutions as an eigenvalue problem

Evans and Coleman [98 ] realized that self-similar rather than self-periodic collapse might be more tractable both numerically (since ODE’s rather than PDE’s are involved) and analytically. They discovered that a collapsing radiation fluid had that desirable property. (Note that self-similarity (homothetic motion) is incompatible with AF [94 , 108 ]. However, most of the action occurs in the center so that a match of the self-similar inner region to an outer AF one should always be possible.) In a series of papers, Hirschmann and Eardley [144 , 145 ] developed a (numerical) self-similar solution to the spherically symmetric complex scalar field equations. These are ODE’s with too many boundary conditions causing a solution to exist only for certain fixed values of $Δ$ . Numerical solution of this eigenvalue problem allows very accurate determination of $Δ$ . The self-similarity also allows accurate calculation of $γ$ as follows: The critical $p = p∗$ solution is unstable to a small change in $p$ . At any time $t$ (where $t < 0$ is increasing toward zero), the amplitude $a$ of the perturbation exhibits power law growth:

where $κ > 0$ . At any fixed $t$ , larger $a$ implies larger $MBH$ . Equivalently, any fixed amplitude $a = δ$ will be reached faster for larger eventual $MBH$ . Scaling arguments give the dependence of $MBH$ on the time at which any fixed amplitude is reached:

where

Thus

Therefore, one need only identify the growth rate of the unstable mode to obtain an accurate value of $γ = 1∕κ$ . It is not necessary to undertake the entire dynamical evolution or probe the space of initial data. (While this procedure allowed Hirschmann and Eardley to obtain $γ = 0.387106$ for the complex scalar field solution, they later found [146

] that, in this regime, the complex scalar field has 3 unstable modes. This means [128

, 131 ] that the Eardley–Hirschmann solution is not a critical solution. A perturbation analysis indicates that the critical solution for the complex scalar field is the discretely self-similar one found for the real scalar field [128

].) Koike et al. [176

] obtain $γ = 0.35580192$ for the Evans–Coleman solution. Although the similarities among the critical exponents $γ$ in the collapse computations suggested a universal value, Maison [183 ] used these same scaling-perturbation methods to show that $γ$ depends on the equation of state $p = kρ$ of the fluid in the Evans–Coleman solution. Gundlach [125 ] used a similar approach to locate Choptuik’s critical solution accurately. This is much harder due to its discrete self-similarity. He reformulates the model as nonlinear hyperbolic boundary value problem with eigenvalue $Δ$ and finds $Δ = 3.4439$ . As with the self-similar solutions described above, the critical solution is found directly without the need to perform a dynamical evolution or explore the space of initial data. Hara et al. extended the renormalization group approach of [176 ] and applied it to the continuously-self-similar case [136

]. (For an application of renormalization group methods to cosmology see [162 ].)

2.2.3 Recent results

Gundlach [128 ] completed his eigenvalue analysis of the Choptuik solution to find the growth rate of the unstable mode to be $γ = 0.374 ± 0.001$ . He also predicted a periodic “wiggle” in the Choptuik mass scaling relation. This was later observed numerically by Hod and Piran [152 ]. Self-similar critical behavior has been seen in string theory related axion-dilaton models [93 , 134 ] and in the nonlinear $σ$ -model [146 ]. Garfinkle and Duncan have shown that subcritical collapse of a spherically symmetric scalar field yields a scaling relation for the maximum curvature observed by the central observer with critical parameters that would be expected on the basis of those found for supercritical collapse [114 ].

Choptuik et al. [79 ] have generalized the original Einstein–scalar field calculations to the Einstein–Yang–Mills (EYM) (for $SU (2)$ ) case. Here something new was found. Two types of behavior appeared depending on the initial data. In Type I, BH formation had a non-zero mass threshold. The critical solution is a regular, unstable solution to the EYM equations found previously by Bartnik and McKinnon [14 ]. In Type II collapse, the minimum BH mass is zero with the critical solution similar to that of Choptuik (with a different $γ ≈ 0.20$ , $Δ ≈ 0.74$ ). Gundlach has also looked at this case with the same results [127 ]. The Type I behavior arises when the collapsing system has a metastable static solution in addition to the Choptuik critical one [132 ].

Brady, Chambers and Gonçalves [71 , 52 ] conjectured that addition of a mass to the scalar field of the original model would break scale invariance and might yield a distinct critical behavior. They found numerically the same Type I and II “phases” seen in the EYM case. The Type II solution can be understood as perturbations of Choptuik’s original model with a small scalar field mass $μ$ . Here small means that $λμ < < 1$ where $λ$ is the spatial extent of the original nonzero field region. (The scalar field is well within the Compton wavelength corresponding to $μ$ .) On the other hand, $λ μ >> 1$ yields Type I behavior. The minimum mass critical solution is an unstable soliton of the type found by Seidel and Suen [229 ]. The massive scalar field can be treated as collapsing dust to yield a criterion for BH formation [120 ].

The Choptuik solution has also been found to be the critical solution for charged scalar fields [132 , 151 ]. As $p → p∗$ , $Q ∕M → 0$ for the black hole. $Q$ obeys a power law scaling. Numerical study of the critical collapse of collisionless matter (Einstein–Vlasov equations) has yielded a non-zero minimum BH mass [217 , 201 ]. Bizoń and Chmaj [47 ] have considered the critical collapse of skyrmions.

An astrophysical application of BH critical phenomena has been considered by Niemeyer and Jedamzik [199 ] and Yokoyama [252 ]. They consider its implications for primordial BH formation and suggest that it could be important.

2.2.4 Going further

The question is then why these critical phenomena should appear in so many collapsing gravitational systems. The discrete self-similarity of Choptuik’s solution may be understood as scaling of a limit cycle [136 ]. (The self-similarity of other systems may be understood as scaling of a limit point.) Garfinkle [110 ] originally conjectured that the scale invariance of Einstein’s equations might provide an underlying explanation for the self-similarity and discrete-self-similarity found in collapse and proposed a spacetime slicing which might manifestly show this. In fact, he later showed (with Meyer) [107 ] that, while the originally proposed slicing failed, a foliation based on maximal slicing did make the scaling manifest. These ideas formed the basis of a much more ambitious program by Garfinkle and Gundlach to use underlying actual or approximate symmetries to construct coordinate systems for numerical simulations [116 ].

An interesting “toy model” for general relativity in many contexts has been wave maps, also known as nonlinear $σ$ models. One of these contexts is critical collapse [146 ]. Recently and independently, Bizoń et al. [48 ] and Liebling et al. [180 ] evolved wave maps from the base space of 3 + 1 Minkowski space to the target space $3 S$ . They found critical behavior separating singular and nonsingular solutions. For some families of initial data, the critical solution is self-similar and is an intermediate attractor. For others, a static solution separates the singular and nonsingular solutions. However, the static solution has several unstable modes and is therefore not a critical solution in the usual sense. Bizoń and Tabor [49 ] have studied Yang–Mills fields in $D + 1$ dimensions and found that generic solutions with sufficiently large initial data blow up in a finite time and that the mechanism for blowup depends on $D$ . Husa et al. [158 ] then considered the collapse of $SU (2)$ nonlinear sigma models coupled to gravity and found a discretely self-similar critical solution for sufficiently large dimensionless coupling constant. They also observe that for sufficiently small coupling constant values, there is a continuously self-similar solution. Interestingly, there is an intermediate range of coupling constant where the discrete self-similarity is intermittent [238 ].

Until recently, only Abrahams and Evans [1 ] had ventured beyond spherical symmetry. The first additional departure has been made by Gundlach [130 ]. He considered spherical and non-spherical perturbations of $P = 13ρ$ perfect fluid collapse. Only the original (spherical) growing mode survived.

Recently, critical phenomena have been explored in 2 + 1 gravity. Pretorius and Choptuik [215 ] numerically evolved circularly symmetric scalar field collapse in 2 + 1 anti-de Sitter space. They found a continuously self-similar critical solution at the threshold for black hole formation. The BH’s which form have BTZ [8 ] exteriors with strong curvature, spacelike singularities in the interior. Remarkably, Garfinkle obtained an analytic critical solution by assuming continuous self-similarity which agrees quite well with the one obtained numerically [112 ].