Critical Phenomena in Gravitational Collapse
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Abstract
As first discovered by Choptuik, the black hole threshold in the space of initial data for general relativity shows both surprising structure and surprising simplicity. Universality, powerlaw scaling of the black hole mass, and scale echoing have given rise to the term “critical phenomena”. They are explained by the existence of exact solutions which are attractors within the black hole threshold, that is, attractors of codimension one in phase space, and which are typically selfsimilar. Critical phenomena give a natural route from smooth initial data to arbitrarily large curvatures visible from infinity, and are therefore likely to be relevant for cosmic censorship, quantum gravity, astrophysics, and our general understanding of the dynamics of general relativity.
Keywords
Black Hole Scalar Field Spherical Symmetry Critical Phenomenon Apparent Horizon1 Introduction
1.1 Overview of the subject
 Near the threshold, black holes with arbitrarily small masses can be created, and the black hole mass scales aswhere p parameterises the initial data and black holes form for p > p_{*}.$$M \propto {(p  {p_\ast})^\gamma},$$(1)

The critical exponent γ is universal with respect to initial data, that is, independent of the particular 1parameter family, although it depends on the type of collapsing matter.

In the region of large curvature before black hole formation, the spacetime approaches a selfsimilar solution which is also universal with respect to initial data, the critical solution.
Critical phenomena were discovered by Choptuik [48] in numerical simulations of a spherical scalar field. They have been found in numerous other numerical and analytical studies in spherical symmetry, and a few in axisymmetry; in particular critical phenomena have been seen in the collapse of axisymmetric gravitational waves in vacuum [1]. Scaling laws similar to the one for black hole mass have been discovered for black hole charge and conjectured for black hole angular momentum.
It is still unclear how universal critical phenomena in collapse are with respect to matter types and beyond spherical symmetry, in particular for vacuum collapse. Progress is likely to be made over the next few years with better numerical simulations.
Critical phenomena can usefully be described in dynamical systems terms. A critical solution is then characterised as an attracting fixed point within a surface that divides two basins of attraction, a critical surface in phase space. Such a fixed point can be either a stationary spacetime, or one that is scaleinvariant and selfsimilar. The latter is relevant for the (“type II”) critical phenomena sketched above. We shall also see how the dynamical systems approach establishes a connection with critical phenomena in statistical mechanics.
Therefore, we could define the field of critical phenomena in gravitational collapse as the study of the boundaries among the basins of attraction of different end states of selfgravitating systems, such as black hole formation or dispersion. In our view, the main physical motivation for this study is that those critical solutions which are selfsimilar provide a way of achieving arbitrarily large spacetime curvature outside a black hole, and in the limit a naked singularity, by finetuning generic initial data for generic matter to the black hole threshold. Those solutions are therefore likely to be important for quantum gravity and cosmic censorship.
1.2 Plan of this review
Faced with more material to review, we have attempted to make this update shorter and more systematic than the original 1999 version [102]. We begin with the abstract theory in Section 2. Since 1999, progress on the theory side has mainly been made on the global spacetime structure of the critical solution and cosmic censorship in the spherical scalar field model. We have included this new material in an enlarged Section 3 on the spherical scalar field, although we hope it will turn out to be sufficiently generic to merit inclusion in Section 2. Nonspherical perturbations of the spherical scalar field are also discussed in Section 3.
In Section 4 we review the rich phenomenology that has been found in many other systems restricted to spherical symmetry. Numerical work in spherical symmetry has proliferated since 1999, but we have tried to keep this section as short as possible. There has been less progress in going beyond spherical symmetry than we anticipated in 1999, even though we continue to believe that important results await there. What is known today is summarised in Section 5.
The reader unfamiliar with the topic is advised to begin with either Sections 2.1, 2.2, and 2.3, which give the key theory of universality, selfsimilarity and scaling, or Sections 3.1 and 3.2, which describe the classic example, the massless scalar field.
This review is limited to numerical and theoretical work on phenomena at the threshold of black hole formation in 3 + 1dimensional general relativity. We report only briefly on work in higher and lower spacetime dimensions and nongravity systems that may be relevant as toy models for general relativity. We exclude other work on selfsimilarity in general relativity and work on critical phenomena in other areas of physics.
Other reviews on the subject are [129], [15], [96], [100], [50], [51], [33], [102], [156]. The 2002 review [105] by Gundlach gives more detailed explanations on some of the basic aspects of the theory. A review of the role of selfsimilarity in the formation of singularities in evolutionary PDEs in general is [71].
2 Theory

universality with respect to initial data;

scaleinvariance of the critical solution;

black hole mass scaling.
2.1 Universality
Consider GR as an infinitedimensional continuous dynamical system. Points in the phase space are initial data sets (3metric, extrinsic curvature, and suitable matter variables, which together obey the Einstein constraints). We evolve with the Einstein equations in a suitable gauge (see Section 2.5). Solution curves of the dynamical system are spacetimes obeying the Einsteinmatter equations, sliced by specific Cauchy surfaces of constant time t.
An isolated system in GR can end up in qualitatively different stable end states. Two possibilities are the formation of a single black hole in collapse, or complete dispersion of the massenergy to infinity. For a massless scalar field in spherical symmetry, these are the only possible end states (see Section 3). Any point in phase space can be classified as ending up in one or the other type of end state. The entire phase space therefore splits into two halves, separated by a “critical surface”.
A phase space trajectory that starts on a critical surface by definition never leaves it. A critical surface is therefore a dynamical system in its own right, with one dimension fewer than the full system. If it has an attracting fixed point, such a point is called a critical point. It is an attractor of codimension one in the full system, and the critical surface is its attracting manifold. The fact that the critical solution is an attractor of codimension one is visible in its linear perturbations: It has an infinite number of decaying perturbation modes spanning the tangent plane to the critical surface, and a single growing mode not tangential to it.
2.2 Selfsimilarity
Using the gauge freedom of general relativity, the lapse and shift in the ADM formalism can be chosen (nonuniquely) so that the coordinates become adapted coordinates if and when the solution becomes selfsimilar (see Section 2.5). τ is then both a time coordinate (in the usual sense that surfaces of constant time are Cauchy surfaces), and the logarithm of overall scale at constant x^{i}. The minus sign in Equation (3) and hence Equations (4) and (5), is a convention assuming that smaller scales are in the future. The time parameter used in Figure 2 is of this type.
2.3 Mass scaling
Let Z stand for a set of scaleinvariant variables of the problem, such as \({\tilde g_{\mu v}}\) and suitably rescaled matter variables. If the dynamics is scaleinvariant (this is the case exactly for example for the scalar field, and approximately for other systems, see Section 2.6), then Z(x) is an element of the phase space factored by overall scale, and Z(x, τ) a solution. Note that Z(x) is an initial data set for GR only up to scale. The overall scale is supplied by τ.
2.4 Type I
Type I critical phenomena occur when a mass scale in the field equations becomes dynamically relevant. (This scale does not necessarily set the mass of the critical solution absolutely: There could be a family of critical solutions selected by the initial conditions.) Conversely, as the type II power law is scaleinvariant, type II phenomena occur in situations where either the field equations do not contain a scale, or this scale is dynamically irrelevant. Many systems, such as the massive scalar field, show both type I and type II critical phenomena, in different regions of the space of initial data [34].
2.5 Coordinate choices for the dynamical systems picture
The time evolution of Cauchy data in GR can only be considered as a dynamical system if the ADM evolution equations are complemented by a prescription for the lapse and shift. To realise the phase space picture of Section 2.1, the critical solution must be a fixed point or limit cycle. We have seen how coordinates adapted to the selfsimilarity can be constructed, but is there a prescription of the lapse and shift for arbitrary initial data, such that, given initial data for the critical solution, the resulting time evolution actively drives the metric to a form (5) that explicitly displays the selfsimilarity?
Garfinkle and Gundlach [85] have suggested several combinations of lapse and shift conditions that leave CSS spacetimes invariant and turn the Choptuik DSS spacetime into a limit cycle (see [91, 81] for partial successes). Among these, the combination of maximal slicing with minimal strain shift has been suggested in a different context but for related reasons [188]. Maximal slicing requires the initial data slice to be maximal (\({K_a}^a = 0\)), but other prescriptions, such as freezing the trace of K together with minimal distortion, allow for an arbitrary initial slice with arbitrary spatial coordinates.
All these coordinate conditions are elliptic equations that require boundary conditions, and will turn CSS spacetimes into fixed points (or DSS into limit cycles) only given correct boundary conditions. Roughly speaking, these boundary conditions require a guess of how far the slice is from the accumulation point t = t_{*}, and answers to this problem only exist in spherical symmetry. Appropriate boundary conditions are also needed if the dynamical system is extended to include the lapse and shift as evolved variables, turning the elliptic equations for the lapse and shift into hyperbolic or parabolic equations.
Turning a CSS or stationary spacetime into a fixed point of the dynamical system also requires an appropriate choice of the phase space variables Z(x^{i}). To capture CSS (or DSS) solutions, one needs scaleinvariant variables. Essentially, these can be constructed by dimensional analysis. The coordinates x^{i} and τ are dimensionless, le^{−τ} has dimension length, and g_{μν} has dimension l^{2}. The scaling for the ADM and any matter variables follows.
Even with a prescription for the lapse and shift in place, a given spacetime does not correspond to a unique trajectory in phase space. Rather, for each initial slice through the same spacetime one obtains a different slicing of the entire spacetime. A possibility for avoiding this ambiguity would be to restrict the phase space further, for example by restricting possible data sets to maximal or constant extrinsic curvature slices.
Another open problem is that in order to talk about attractors and repellers on the phase space we need to define a norm on a suitable function space which includes both asymptotically flat data and data for the exact critical solution. The norm itself must favour the central region and ignore what is further out and asymptotically flat if all black holes of the same mass are to be considered as the same end state.
2.6 Approximate selfsimilarity and universality classes
The field equations for the massless scalar field coupled to the Einstein equations are scalefree. Realistic matter models introduce length scales, and the field equations then do not allow for exactly selfsimilar solutions. They may however admit solutions which are CSS or DSS asymptotically on small spacetime scales as the dimensionful parameters become irrelevant, including type II critical solutions [49, 34, 52]. This can be explored by a formal expansion in powers of the small parameter Le^{−τ}, where L is a parameter with dimensions length in the evolution equations. The zeroth order of the expansion is the selfsimilar critical solution of the system with L = 0. A similar ansatz can be made for the linear perturbations of the resulting background. The zeroth order of the background expansion determines Δ exactly and independently of L, and the zeroth order term of the linear perturbation expansion determines the critical exponent 1/λ_{0} exactly, so that there is no need in practice to calculate any higher orders in to make predictions for type II critical phenomena where they occur. (With L ≠ = 0, the basin of attraction of the type II critical solution will depend on L, and type I critical phenomena may also occur; see Section 2.4.) A priori, there could also be more than one type II critical solution for L = 0, although this has not been observed.)
This procedure has been carried out for the EinsteinYangMills system [97] and for massless scalar electrodynamics [106]. Both systems have a single length scale 1/e (in geometric units c = G = 1), where e is the gauge coupling constant. All values of e can be said to form one universality class of field equations [112] represented by e = 0. This notion of universality classes is fundamentally the same as in statistical mechanics. Other examples include modifications to the perfect fluid equation of state (EOS) that do not affect the limit of high density [161]. A simple example is that any scalar field potential V(ϕ) becomes dynamically irrelevant compared to the kinetic energy ∇ϕ^{2} in a selfsimilar solution [49], so that all scalar fields with potentials are in the universality class of the free massless scalar field. Surprisingly, even two different models like the SU(2) YangMills and SU(2) Skyrme models in spherical symmetry are members of the same universality class [22].
If there are several scales L_{0}, L_{1}, L_{2} etc. present in the problem, a possible approach is to set the arbitrary scale in Equation (29) equal to one of them, say L_{0}, and define the dimensionless constants l_{i} = L_{i}/L_{0} from the others. The scope of the universality classes depends on where the l_{i} appear in the field equations. If a particular L_{i} appears in the field equations only in positive integer powers, the corresponding l_{i} appears only multiplied by e^{−τ}, and will be irrelevant in the scaling limit. All values of this l_{i} therefore belong to the same universality class. From the example above, adding a quartic selfinteraction λϕ^{4} to the massive scalar field gives rise to the dimensionless number λ/m^{2} but its value is an irrelevant (in the language of renormalisation group theory) parameter.
Contrary to the statement in [106], we conjecture that massive scalar electrodynamics, for any values of e and m, is in the universality class of the massless uncharged scalar field in a region of phase space where type II critical phenomena occur. Examples of dimensionless parameters which do change the universality class are the k of the perfect fluid, the κ of the 2dimensional sigma model or, probably, a conformal coupling of the scalar field [47] (the numerical evidence is weak but a dependence should be expected).
2.7 The analogy with critical phase transitions
Some basic aspects of critical phenomena in gravitational collapse, such as finetuning, universality, scaleinvariant physics, and critical exponents for dimensionful quantities, can also be identified in critical phase transitions in statistical mechanics (see [208] for an introductory textbook).
Phase transitions in thermodynamics are thresholds in the space of external forces f at which the macroscopic observables A, or one of their derivatives, change discontinuously. We consider two examples: the liquidgas transition in a fluid, and the ferromagnetic phase transition.
Quantities such as m or ρ_{liquid} − ρ_{gas} are called order parameters. In statistical mechanics, one distinguishes between firstorder phase transitions, where the order parameter changes discontinuously, and secondorder, or critical, ones, where it goes to zero continuously. One should think of a critical phase transition as the critical point where a line of firstorder phase transitions ends as the order parameter vanishes. This is already clear in the fluid example. In the ferromagnet example, at first one seems to have only the one parameter T to adjust. But in the presence of a very weak external field, the spontaneous magnetisation aligns itself with the external field B, while its strength is to leading order independent of B. The function m(B, T) therefore changes discontinuously at B = 0. The line B = 0 for T < T_{*} is therefore a line of first order phase transitions between directions (if we consider one spatial dimension only, between m up and m down). This line ends at the critical point (B = 0, T = T_{*}) where the order parameter m vanishes. The critical value B = 0 of B is determined by symmetry; by contrast p_{*} depends on microscopic properties of the material.
We have already stated that a critical phase transition involves scaleinvariant physics. In particular, the atomic scale, and any dimensionful parameters associated with that scale, must become irrelevant at the critical point. This is taken as the starting point for obtaining properties of the system at the critical point.
The behaviour of thermodynamical quantities at the critical point is in general not trivial to calculate. But the action of the renormalisation group on length scales is given by its definition. The blowup of the correlation length ξ at the critical point is therefore the easiest critical exponent to calculate. The same is true for the black hole mass, which is just a length scale. We can immediately reinterpret the mathematics of Section 2.3 as a calculation of the critical exponent for ξ, by substituting the correlation length ξ for the black hole mass M, T_{*} − T for p − p_{*}, and taking into account that the τevolution in critical collapse is towards smaller scales, while the renormalisation group flow goes towards larger scales: ξ therefore diverges at the critical point, while M vanishes.
In type II critical phenomena in gravitational collapse, we should think of the black hole mass as being controlled by the functions P and Q on phase space defined by Equation (27). Clearly, P is the equivalent of the reduced temperature T − T_{*}. Gundlach [104] has suggested that the angular momentum of the initial data can play the role of B, and the final black hole angular momentum the role of m. Like the magnetic field, angular momentum is a vector, with a critical value that must be zero because all other values break rotational symmetry.
3 The Scalar Field
Critical phenomena in gravitational collapse were first discovered by Choptuik [47, 48, 49] in the model of a spherically symmetric, massless scalar field ϕ minimally coupled to general relativity. The scalar field matter is both simple, and acts as a toy model in spherical symmetry for the effects of gravitational radiation. Given that it is still the beststudied model in spherical symmetry, we review it here as a case study. For other numerical work on this model, see [109, 80, 111, 77, 182, 212]. Important analytical studies of gravitational collapse in this model have been carried out by Christodoulou [57, 58, 59, 60, 61, 62, 63].
We first review the field equations and Choptuik’s observations at the black hole threshold, mainly as a concrete example for the general ideas discussed above. We then summarise more recent work on the global structure of Choptuik’s critical solution, which throws an interesting light on cosmic censorship. In particular, the exact critical solution contains a curvature singularity that is locally and globally naked, and any critical solution obtained in the limit of perfect finetuning of asymptotically flat initial data is at least locally naked. By perturbing around spherical symmetry, the stability of the Choptuik solution in the full phase space can be investigated, and the scaling of black hole angular momentum can be predicted. By embedding the real scalar field in scalar electrodynamics and perturbing around the Choptuik solution, the scaling of black hole charge can be predicted.
3.1 Field equations in spherical symmetry
3.2 The black hole threshold
The free data for the system are the two functions Π(r, 0) and Φ(r, 0). Choptuik investigated several 1parameter families of such data by evolving the data for many different values of the parameter. Simple examples of such families are Π(r, 0) = 0 and a Gaussian for Φ(r, 0), with the parameter p taken to be either the amplitude of the Gaussian, with the width and centre fixed, or the width, with position and amplitude fixed, or the position, with width and amplitude fixed. For sufficiently small amplitude (or the peak sufficiently wide), the scalar field will disperse, and for sufficiently large amplitude it will form a black hole.
Generic 1parameter families behave in this way, but this is difficult to prove in generality. Christodoulou showed for the spherically symmetric scalar field system that data sufficiently weak in a welldefined way evolve to a Minkowskilike spacetime [58, 61], and that a class of sufficiently strong data forms a black hole [60].
Choptuik found that in all 1parameter families of initial data he investigated he could make arbitrarily small black holes by finetuning the parameter p close to the black hole threshold. An important fact is that there is nothing visibly special to the black hole threshold. One cannot tell that one given data set will form a black hole and another one infinitesimally close will not, short of evolving both for a sufficiently long time.
As p → p_{*} along the family, the spacetime varies on ever smaller scales. Choptuik developed numerical techniques that recursively refine the numerical grid in spacetime regions where details arise on scales too small to be resolved properly. In the end, he could determine p_{*} up to a relative precision of 10^{−15}, and make black holes as small as 10^{−6} times the ADM mass of the spacetime. The powerlaw scaling (10) was obeyed from those smallest masses up to black hole masses of, for some families, 0.9 of the ADM mass, that is, over six orders of magnitude [49]. There were no families of initial data which did not show the universal critical solution and critical exponent. Choptuik therefore conjectured that γ is the same for all 1parameter families of smooth, asymptotically flat initial data that depend smoothly on the parameter, and that the approximate scaling law holds ever better for arbitrarily small p − p_{*}.
3.3 Global structure of the critical solution
In adapted coordinates, the metric of the critical spacetime is of the form \({e^{ 2\tau}}\) times a regular metric. From this general form alone, one can conclude that τ= ∞ is a curvature singularity, where Riemann and Ricci invariants blow up like \({e^{4\tau}}\) (unless the spacetime is flat), and which is at finite proper time from regular points in its past. The Weyl tensor with index position C^{abcd} is conformally invariant, so that components with this index position remain finite as τ → ∞. This type of singularity is called “conformally compactifiable” [198] or “isotropic” [93]. For a classification of all possible global structures of spherically symmetric selfsimilar spacetimes see [107].
The global structure of the scalar field critical solution was determined accurately in [157] by assuming analyticity at the centre of spherical symmetry and at the past light cone of the singularity (the selfsimilarity horizon, or SSH). The critical solution is then analytic up to the future lightcone of the singularity (the Cauchy horizon, or CH). Global adapted coordinates x and τ can be chosen so that the regular centre r = 0, the SSH and the CH are all lines of constant x, and surfaces of constant τ are never tangent to x lines. (A global τ is no longer a global time coordinate.) This is illustrated in Figure 4.
As the CH itself is regular with smooth null data except for the singular point at its base, it is not intuitively clear why the continuation is not unique. A partial explanation is given in [157], where all DSS continuations are considered. Within a DSS ansatz, the solution just to the future of the CH has the same form as Equation (31). F_{reg}(τ) is the same on both sides, but F_{sing}(τ) can be chosen freely on the future side of the CH. Within the restriction to DSS this function can be taken to parameterise the information that comes out of the naked singularity.
There is precisely one choice of F_{sing}(τ) on the future side that gives a regular centre to the future of the CH, with the exception of the naked singularity itself, which is then a point. This continuation was calculated numerically, and is almost but not quite Minkowski in the sense that m/r remains small everywhere to the future of the SSH.
3.4 Nearcritical spacetimes and naked singularities
Choptuik’s results have an obvious bearing on the issue of cosmic censorship (see [203] for a general review of cosmic censorship). Roughly speaking, finetuning to the black hole threshold provides a set of data which is codimension one in the space of generic, smooth, asymptotically flat initial data, and whose time evolution contains at least the point singularity of the critical solution. The cosmic censorship hypothesis must therefore be formulated as “generic smooth initial data for reasonable matter do not form naked singularities”. Here we look at the relation between finetuning and naked singularities in more detail.
Christodoulou [63] proves rigorously that naked singularity formation is not generic, but in a rather larger function space, functions of bounded variation, than one would naturally consider. In particular, the instability of the naked singularity found by Christodoulou is not differentiable on the past light cone. This is unnatural in the context of critical collapse, where the naked singularity can arise from generic (up to finetuning) smooth initial data, and the intersection of the past light cone of the singularity with the initial data surface is as smooth as the initial data elsewhere. It is therefore not clear how this theorem relates to the numerical and analytical results strongly indicating that naked singularities are codimension1 generic within the space of smooth initial data.
3.5 Electric charge
Given the scaling power law for the black hole mass in critical collapse, one would like to know what happens if one takes a generic 1parameter family of initial data with both electric charge and angular momentum (for suitable matter), and finetunes the parameter p to the black hole threshold. A simple model for charged matter is a complex scalar field coupled to electromagnetism with the substitution ∇_{a} → ∇_{a} + ieA_{a}, or scalar electrodynamics. (Note that in geometric units, black hole charge Q has dimension of length, but the charge parameter e has dimension 1/length.)
3.6 Selfinteraction potential
An example of the richer phenomenology in the presence of a scale in the field equations is the spherical massive scalar field with a potential m^{2}ϕ^{2} [34] coupled to gravity: In one region of phase space, with characteristic scales smaller than 1/m, the black hole threshold is dominated by the Choptuik solution and type II critical phenomena occur. In another it is dominated by metastable oscillating boson stars (whose mass is of order 1/m in geometric units) and type I critical phenomena occur. (For the real scalar field, the type I critical solution is an (unstable) oscillating boson star [186] while for the complex scalar field it can be a static (unstable) boson star [120].)
When the scalar field with a potential is coupled to electromagnetism, type II criticality is still controlled by a solution which asymptotically resembles the uncharged Choptuik spacetime, but type I criticality is now controlled by charged boson stars [176]. There are indications that subcritical type I evolutions lead to slow, large amplitude oscillations of stable boson stars [141, 142, 176] and not to dispersion to infinity, as had been conjectured in [120]. Another interesting extension is the study of the dynamics of a real scalar field with a symmetric doublewell potential, in which the system displays type I criticality between the two possible vacua [128].
3.7 Nonspherical perturbations: Stability and angular momentum
Critical collapse is really relevant for cosmic censorship only if it is not restricted to spherical symmetry. MartínGarcía and Gundlach [154] have analysed all nonspherical perturbations of the scalar field critical solution by solving a linear eigenvalue problem with an ansatz of regularity at the centre and the SSH. They find that the only growing mode is the known spherical one, while all other spherical modes and all nonspherical modes decay. This strongly suggests that the critical solution is an attractor of codimension one not only in the space of spherically symmetric data but (modulo linearisation stability) of all data in a finite neighbourhood of spherical symmetry.
More recently, Choptuik and collaborators [54] have carried out axisymmetric time evolutions for the massless scalar field using adaptive mesh refinement. They find that in the limit of finetuning generic axisymmetric initial data the spherically symmetric critical solution is approached at first but then deviates from spherical symmetry and eventually develops two centres, each of which approaches the critical solution and bifurcates again in a universal way. This suggests that the critical solution has nonspherical growing perturbation modes, possibly a single l = 2 even parity mode (in axisymmetry, only m = 0 is allowed). There appears to be a conflict between the time evolution results [54] and the perturbative results [154], which needs to be resolved by more work (see Section 5.2).
Perturbing the scalar field around spherical symmetry, angular momentum comes in to second order in perturbation theory. All angular momentum perturbations were found to decay, and a critical exponent μ, ≃ 0.76 for the angular momentum was derived for the massless scalar field in [87]. This prediction has not yet been tested in nonlinear collapse simulations.
4 More Spherical Symmetry

Systems in which the field equations, when reduced to spherical symmetry, form a single wavelike equation, typically with explicit rdependence in its coefficients. This includes YangMills fields, sigma models, vector and spinor fields, scalar fields in 2 + 1 or in 4 + 1 and more spacetime dimensions, and scalar fields in a semiclassical approximation to quantum gravity.

Perfect fluid matter, either in an asymptotically flat or a cosmological context. The linearised Euler equations are in fact wavelike, but the full nonlinear equations admit shock heating and are therefore not even timereversal symmetric.

Collisionless matter described by the Vlasov equation is a partial differential equation on particle phase space as well as spacetime. Therefore even in spherical symmetry, the matter equation is a partial differential equation in 4 dimensions (rather than two). Intuitively speaking, there are infinitely more matter degrees of freedom than in the scalar field or in nonspherical vacuum gravity.

Spherically symmetric nonlinear wave equations on 3 +1 Minkowski spacetime, and other nonlinear partial differential equations which show a transition between singularity formation and dispersal.
Critical collapse in spherical symmetry.
Matter  Type  Collapse simulations  Critical solution  Perturbations of critical solution 

Perfect fluid, p = kρ  II  
 in 4 + 1, 5 + 1, 6 + 1  II  CSS [7]  [7]  
Vlasov  I?  [155]  
Real scalar field:  
 massless, minimally coupled  II  DSS [95]  
 massive  I  [34]  oscillating [186]  
II  [49]  
 conformally coupled  II  [48]  DSS  
 4 + 1  II  [13]  
 5 + 1  II  [83]  
Massive complex scalar field  I, II  [120]  [186]  [120] 
Massless scalar electrodynamics  II  [126]  DSS [106]  [106] 
Massive vector field  II  [90]  DSS [90]  [90] 
Massless Dirac  II  [200]  CSS [200]  
Vacuum BransDicke  I  [53]  static [199]  [53] 
2d sigma model:  
 complex scalar (κ = 0)  II  [46]  DSS [98]  [98] 
 axiondilaton (κ = 1)  II  [110]  [110]  
 scalarBransDicke (κ > 0)  II  CSS, DSS  
 general κ including κ < 0  II  CSS, DSS [124]  [124]  
SU(2) YangMills  I  [52]  static [10]  [143] 
II  [52]  DSS [97]  [97]  
“III”  [56]  
SU(2) YangMillsHiggs  (idem)  [158]  (idem)  
SU(2) Skyrme model  I  [17]  static [17]  [17] 
II  [22]  DSS [22]  
SO(3) Mexican hat  II  [148]  DSS 
4.1 Matter obeying wave equations
4.1.1 2dimensional nonlinear σ model
Generalizing their previous results for κ = 0 [123, 122], HE constructed for each κ a CSS solution based on the ansatz ϕ(τ, x) = e^{iωτ} ϕ(x) for the critical scalar field. Studying its perturbations HE concluded that this solution is critical for κ > 0.0754, but has three unstable modes for κ < 0.0754 and even more for κ > −0.28. Below 0.0754 a DSS solution takes over, as shown in the simulations of Liebling and Choptuik [150], and HE conjectured that the transition is a Hopf bifurcation, such that the DSS cycle smoothly shrinks with growing κ, collapsing onto the CSS solution at the transition and then disappearing with a finite value of the echoing period Δ.
The close relation between the CSS and DSS critical solutions is also manifest in the construction of their global structures. In particular, the results of [122] and [69] for the CSS κ = 0 and κ = 1 solutions respectively show that the Cauchy horizon of the singularity is almost but not quite flat, exactly as was the case with the Choptuik DSS spacetime (see Section 3.3).
4.1.2 Spherical EinsteinSU(2) sigma model
This is an N = 3 sigma model, and it also displays a transition between CSS and DSS criticality, but this is a totally different type of transition, in particular showing a divergence in the echoing period Δ [131].
Aichelburg and collaborators [131, 144] have shown that for η ≳ 0.2 there is clear DSS type II criticality at the black hole threshold. The period Δ depends on η, monotonically decreasing towards an asymptotic value for η → ∞. Interesting new behavior occurs in the intermediate range 0.14 < η < 0.2 that lies between clear CSS and clear DSS. With decreasing η the overall DSS includes episodes of approximate CSS [197], of increasing length (measured in the logscale time τ). As η → η_{c} ≃ 0.170 from above the duration of the CSS epochs, and hence the overall DSS period Δ diverges. For 0.14 < η < 0.17 time evolutions of initial data near the black hole threshold no longer show overal DSS, but they still show CSS episodes. Black hole mass scaling is unclear in this regime.
It has been conjectured that this transition from CSS to DSS can be interpreted, in the language of the theory of dynamical systems, as the infinitedimensional analogue of a 3dimensional Shil’nikov bifurcation [145]. Highprecision numerics in [3] further supports this picture: For η > η_{c} a codimension1 CSS solution coexists in phase space with a codimension1 DSS attractor such that the (1dimensional) unstable manifold of the DSS solution lies on the stable manifold of the CSS solution. For η close to η_{c} the two solutions are close and the orbits around the DSS solution become slower because they spend more time in the neighbourhood of the CSS attractor. A linear stability analysis predicts a law \(\Delta \simeq  {2 \over \lambda}\log (\eta  {\eta _c}) + b\) for some constant b, where λ is the Lyapunov exponent of the CSS solution. For η = η_{c} both solutions touch and the DSS cycle dissapears.
4.1.3 EinsteinYangMills
Choptuik, Chmaj and Bizoń [52] have found both type I and type II critical collapse in the spherical EinsteinYangMills system with SU(2) gauge potential, restricting to the purely magnetic case, in which the matter is described by a single real scalar field. The situation is very similar to that of the massive scalar field, and now the critical solutions are the wellknown static n = 1 BartnikMcKinnon solution [10] for type I and a DSS solution (later constructed in [97]) for type II. In both cases the black holes produced in the supercritical regime are Schwarzschild black holes with zero YangMills field strength, but the final states (and the dynamics leading to them) can be distinguished by the value of the YangMills final gauge potential at infinity, which can take two values, corresponding to two distinct vacuum states.
Choptuik, Hirschmann and Marsa [56] have investigated the boundary in phase space between formation of those two types of black holes, using a code that can follow the time evolutions for long after the black hole has formed. This is a new “type III” phase transition whose critical solution is an unstable static black hole with YangMills hair [14, 202], which collapses to a hairless Schwarzschild black hole with either vacuum state of the YangMills field, depending on the sign of its one growing perturbation mode. This “coloured” black hole is actually a member of a 1parameter family parameterized by its apparent horizon radius and outside the horizon it approaches the corresponding BM solution. When the horizon radius approaches zero the three critical solutions meet at a “triple point”. What happens there deserves further investigation.
Millward and Hirschmann [158] have further coupled a Higgs field to the EinsteinYangMills system. New possible end states appear: regular static solutions, and stable hairy black holes (different from the coloured black holes referred to above). Again there are type I or type II critical phenomena depending on the initial conditions.
It is known that the spherical critical solutions within the magnetic ansatz become more unstable when other components of the gauge field are taken into account, and so they will not be critical in the general case.
4.1.4 Vacuum 4 + 1
In evolutions with the general ansatz where all L_{i} are different (triaxial solutions) [21], the U(1) symmetry is recovered dynamically in the approach to the critical surface. However, each biaxial solution, and in particular the critical solution, exists in three copies obtained by permutation of the σ_{i}. Therefore, in the triaxial case, the critical surface contains three critical solutions. The boundaries, within the critical surface, between their basins of attraction contain in turn codimensiontwo DSS attractors. It is conjectured that there is in fact a countable family of DSS solutions with η unstable modes.
Szybka and Chmaj [196] give numerical evidence that these boundaries within the critical surface are fractal (in contrast to the critical surface itself, which is smooth as in all other known systems.)
A similar ansatz can be made in other odd spacetime dimensions, and in 8 + 1 dimensions type II critical behaviour is again observed [19].
4.1.5 Scalar field collapse in 2 + 1
Spacetime in 2 + 1 dimensions is flat everywhere where there is no matter, so that gravity is not acting at a distance in the usual way. There are no gravitational waves, and black holes can only be formed in the presence of a negative cosmological constant (see [39] for a review).
Scalar field collapse in circular symmetry was investigated numerically by Pretorius and Choptuik [178], and Husain and Olivier [134]. In a regime where the cosmological constant is small compared to spacetime curvature they find type II critical phenomena with a universal CSS critical solution, and γ = 1.20 ± 0.05 [178]. The value γ ≃ 0.81 [134] appears to be less accurate.
Looking for the critical solution in closed form, Garfinkle [82] found a countable family of exact spherically symmetric CSS solutions for a massless scalar field with Λ = 0, but his results remain inconclusive. The q = 4 solution appears to match the numerical evolutions inside the past light cone, but its past light cone is also an apparent horizon. The q = 4 solution has three growing modes although the top one would give γ = 8/7 ≃ 1.14 if only the other two could be ruled out [86]. An attempt at this [125] seems unmotivated. At the same time, it is possible to embed the Λ = 0 solutions into a family of Λ < 0 ones [64, 65, 43], which can be constructed along the lines of Section 2.6, so that Garfinkle’s solution could be the leading term in an expansion in e^{−τ} (−Λ)^{−1/2}.
4.1.6 Scalar field collapse in higher dimensions
Critical collapse of a massless scalar field in spherical symmetry in 5 + 1 spacetime dimensions was investigated in [83]. Results are similar to 3 + 1 dimensions, with a DSS critical solution and mass scaling with γ ≃ 0.424. Birukou et al [13, 133] have developed a code for arbitrary spacetime dimension. They confirm known results in 3 + 1 (γ − 0.36) and 5 + 1 (γ ≃ 0.44) dimensions, and investigate 4 + 1 dimensions. Without a cosmological constant they find mass scaling with γ ≃ 0.41 for one family of initial data and γ ≃ 0.52 for another. They see wiggles in the ln M versus ln(p − p*) plot that indicate a DSS critical solution, but have not investigated the critical solution directly. With a negative cosmological constant and the second family, they find γ = 0.49. Bland and Kunstatter [29] have made a more precise determination: γ = 0.4131 ± 0.0001. This was motivated by an attempt to explain this exponent using an holographic duality between the strong coupling regime of 4 + 1 gravity and the weak coupling regime of 3 + 1 QCD [5], which had predicted γ = 0.409552.
Kol [140] relates a solution that is related to the Choptuik solution to a variant of the critical solution in the blackstring black hole transition, and claims to obtain analytic estimates for γ and Δ. This has motivated a numerical determination of γ and Δ for the spherical massless scalar field in noninteger dimension up to 14 [190, 30].
4.1.7 Other systems obeying wave equations
Choptuik, Hirschmann and Liebling [53] have presented perturbative indications that the static solutions found by van Putten [199] in the vacuum BransDicke system are critical solutions. They have also performed full numerical simulations, but only starting from small deviations with respect to those solutions.
Ventrella and Choptuik [200] have performed numerical simulations of collapse of a massless Dirac field in a special state: an incoherent sum of two independent lefthanded zerospin fields having opposite orbital angular momentum. This is prepared so that the total distribution of energymomentum is spherically symmetric. The freedom in the system is then contained in a single complex scalar field obeying a modified linear wave equation in spherical symmetry. There are clear signs of CSS criticality in the metric variables, and the critical complex field exhibits a phase of the form e^{iωτ} for a definite ω (the Hirschmann and Eardly ansatz for the complex scalar field critical solutions), which can be considered as a trivial form of DSS.
Garfinkle, Mann and Vuille [90] have found coexistence of types I and II criticality in the spherical collapse of a massive vector field (the Proca system), the scenario being almost identical to that of a massive scalar field. In the selfsimilar phase the collapse amplifies the longitudinal mode of the Proca field with respect to its transverse modes, which become negligible, and the critical solution is simply the gradient of the Choptuik DSS spacetime.
Sarbach and Lehner [185] find type I critical behaviour in q + 3dimensional spacetimes with U(1) × SO(q + 1) symmetry in EinsteinMaxwell theory at the threshold between dispersion and formation of a black string.
4.2 Perfect fluid matter
4.2.1 Spherical symmetry
Evans and Coleman [72] performed the first simulations of critical collapse with a perfect fluid with EOS p = kρ (where ρ is the energy density and p the pressure) for k = 1/3 (radiation), and found a CSS critical solution with a massscaling critical exponent γ ≈ 0.36. Koike, Hara and Adachi [138] constructed that critical solution and its linear perturbations from a CSS ansatz as an eigenvalue problem, computing the critical exponent to high precision. Independently, Maison [153] constructed the regular CSS solutions and their linear perturbations for a large number of values of k, showing for the first time that the critical exponents were modeldependent. As Ori and Piran before [171, 172], he claimed that there are no regular CSS solutions for k > 0.89, but Neilsen and Choptuik [161, 162] have found CSS critical solutions for all values of k right up to 1, both in collapse simulations and by making a CSS ansatz. The difficulty comes from a change in character of the sonic point, which becomes a nodal point for k > 0.89, rather than a focal point, making the ODE problem associated with the CSS ansatz much more difficult to solve. Harada [114] has also found that the critical solution becomes unstable to a “kink” (discontinuous at the sonic point of the background solution) mode for k > 0.89, but because it is not smooth it does not seem to have any influence on the numerical simulations of collapse. On the other hand, the limit k → 1 leading to the stiff EOS p = ρ is singular in that during evolution the fluid 4velocity can become spacelike and the density ρ negative. The stiff fluid equations of motion are in fact equivalent to the masslessscalar field, but the critical solutions can differ, dependending on how one deals with the issue of negative density [35]. Summarizing, it is possible to construct the EvansColeman CSS critical (codimension1) solution for all values 0 < k < 1. This solution can be identified in the general classification of CSS perfectfluid solutions as the unique spacetime that is analytic at the center and at the sound cone, is ingoing near the center, and outgoing everywhere else [40, 41, 42]. There is even a Newtonian counterpart of the critical solution: the Hunter (a) solution [115]. ÁlvarezGaumé et al. [7] have calculated γ using perturbation theory for the spherically symmetric perfect fluid with P = kρ for k ≳ 1/4 in d =5, 6, 7 dimensions.
p = kρ is the only EOS compatible with exact CSS (homothetic) solutions for perfect fluid collapse [38] and therefore we might think that other equations of state would not display critical phenomena, at least of type II. Neilsen and Choptuik [161] have given evidence that for the ideal gas EOS \(p = k{\rho _0}\epsilon\) (where ρ_{0} is the rest mass density and ϵ is the internal energy per rest mass unit) the black hole threshold also contains a CSS attractor, and that it coincides with the CSS exact critical solution of the ultrarelativistic case with the same k. This is interpreted a posteriori as a sign that the critical CSS solution is highly ultrarelativistic, ρ = (1 + ϵ)ρ_{0} ≃ ϵρ_{0} ≫ ρ_{0}, and hence rest mass is irrelevant. Novak [167] has also shown in the case k = 1, or even with a more general tabulated EOS, that type II critical phenomena can be found by velocityinduced perturbations of static TOV solutions. A thorough and much more precise analysis by Noble and Choptuik [165, 166] of the possible collapse scenarios of the stiff k = 1 ideal gas has confirmed this surprising result, and again the critical solution (and hence the critical exponent) is that of the ultrarelativistic limit problem. Parametrizing, as usual, the TOV solutions by the central density ρ_{c}, they find that for lowdensity initial stars it is not possible to form a black hole by velocityinduced collapse; for intermediate initial values of ρ_{c}, it is possible to induce type II criticality for large enough velocity perturbations; for large initial central densities they always get type I criticality, as we might have anticipated.
Noble and Choptuik [165] have also investigated the evolution of a perfect fluid interacting with a massless scalar field indirectly through gravity. By tuning of the amplitude of the pulse it is possible to drive a fluid star to collapse. For massive stars type I criticality is found, in which the critical solution oscillates around a member of the unstable TOV branch. For less massive stars a large scalar amplitude is required to induce collapse, and the black hole threshold is always dominated by the scalar field DSS critical solution, with the fluid evolving passively.
4.2.2 Nonspherical perturbations
Ori and Piran [172] have pointed out that there exists a CSS perfect fluid solution for 0 < k < 0.036 generalizing the LarsonPenston solution of Newtonian fluid collapse, and which has a naked singularity for 0 < k < 0.0105. Harada and Maeda [113, 115] have shown that this solution has no growing perturbative modes in spherical symmetry and hence a naked singularity becomes a global attractor of the evolution for the latter range of k. This is also true in the limit k = 0, which can be considered as the Newtonian limit [116, 117]. Their result has been confirmed with very high precision numerics by Snajdr [189]. This seems to violate cosmic censorship, as generic spherical initial data would create a naked singularity. However, the exact result (41) holds for any regular CSS spherical perfect fluid solution, and so all such solutions with k < 1/9 have at least one unstable nonspherical perturbation. Therefore the naked singularity is unstable to infinitesimal perturbations with angular momentum when one lifts the restriction to spherical symmetry.
4.2.3 Cosmological applications
In the early universe, quantum fluctuations of the metric and matter can be important, for example providing the seeds of galaxy formation. Large enough fluctuations will collapse to form primordial black holes. As large quantum fluctuations are exponentially more unlikely than small ones, \(P(\delta) \sim {e^{ {\delta ^2}}}\), where δ is the density contrast of the fluctuation, one would expect the spectrum of primordial black holes to be sharply peaked at the minimal δ that leads to black hole formation, giving rise to critical phenomena [163]. See also [94, 209].
An approximation to primordial black hole formation is a spherically symmetric distribution of a radiation gas (p = ρ/3) with cosmological rather than asymptotically flat boundary conditions. In [163, 164] type II critical phenomena were found, which would imply that the mass of primordial black holes formed are much smaller than the naively expected value of the mass contained within the Hubble horizon at the time of collapse. The boundary conditions and initial data were refined in [119, 160], and a minimum black hole mass of ∼ 10^{−4} of the horizon mass was found, due to matter accreting onto the black hole after strong shock formation. However, when the initial data are constructed more realistically from only the growing cosmological perturbation mode, no minimum mass is found [177, 159].
4.3 Collisionless matter
A cloud of collisionless particles can be described by the Vlasov equation, i.e., the Boltzmann equation without collision term. This matter model differs from field theories by having a much larger number of matter degrees of freedom: The matter content is described by a statistical distribution f(x^{μ}, p_{ν}) on the point particle phase space, instead of a finite number of fields ϕ(x^{μ}). When restricted to spherical symmetry, individual particles move tangentially as well as radially, and so individually have angular momentum, but the stressenergy tensor averages out to a spherically symmetric one, with zero total angular momentum. The distribution f is then a function f(r, t, p^{r}, L^{2}) of radius, time, radial momentum and (conserved) angular momentum.
Several numerical simulations of critical collapse of collisionless matter in spherical symmetry have been published to date, and remarkably no type II scaling phenomena has been discovered. Indications of type I scaling have been found, but these do not quite fit the standard picture of critical collapse. Rein et al. [183] find that black hole formation turns on with a mass gap that is a large part of the ADM mass of the initial data, and this gap depends on the initial matter condition. No critical behavior of either type I or type II was observed. Olabarrieta and Choptuik [168] find evidence of a metastable static solution at the black hole threshold, with type I scaling of its life time as in Equation (13). However, the critical exponent depends weakly on the family of initial data, ranging from 5.0 to 5.9, with a quoted uncertainty of 0.2. Furthermore, the matter distribution does not appear to be universal, while the metric seems to be universal up to an overall rescaling, so that there appears to be no universal critical solution. More precise computations by Stevenson and Choptuik [192], using finite volume HRSC methods, have confirmed the existence of static intermediate solutions and nonuniversal scaling with exponents ranging now from 5.27 to 11.65.
MartínGarcía and Gundlach [155] have constructed a family of CSS spherically symmetric solutions for massless particles that is generic by function counting. There are infinitely many solutions with different matter configurations but the same stressenergy tensor and spacetime metric, due to the existence of an exact symmetry: Two massless particles with energymomentum p^{μ} in the solution can be replaced by one particle with 2p^{μ}. A similar result holds for the perturbations. As the growth exponent λ of a perturbation mode can be determined from the metric alone, this means that there are infinitely many perturbation modes with the same λ. If there is one growing perturbative mode, there are infinitely many. Therefore a candidate critical solution (either static or CSS) cannot be isolated or have only one growing mode. This argument rules out the existence of both type I and type II critical phenomena (in their standard form, i.e., including universality) for massless particles in the complete system, but some partial form of criticality could still be found by restricting to sections of phase space in which that symmetry is broken, for example by prescribing a fixed form for the dependence of the distribution function f on angular momentum L, as those numerical simulations have done.
A recent investigation of Andréasson and Rein [8] with massive particles has confirmed again the existence of a mass gap and the existence of metastable static solutions at the black hole threshold, though there is no estimation of the scaling of their lifetimes. More interestingly, they show that the subcritical regime can lead to either dispersion or an oscillating steady state depending on the binding energy of the system. They also conclude, based on perturbative arguments, that there cannot be an isolated universal critical solution.
More numerical work is still required, but current evidence suggests that there are no type II critical phenomena, and that there is a continuum of critical solutions in type I critical phenomena and hence only limited universality.
4.4 Criticality in singularity formation without gravitational collapse
It is well known that the YangMills field does not form singularities from smooth initial conditions in 3 + 1 dimensions [70], but Bizoń and Tabor [26] have shown singularity formation in 4 + 1 (the critical dimension for this system from the point of view of energy scaling arguments) and 5 + 1 (the first supercritical dimension). In 5 + 1 there is the countable family W_{n} of CSS solutions with n unstable modes, such that W_{1} acts as a critical solution separating singularity (W_{0}) formation from dispersal to infinity. In 4 + 1 there are no selfsimilar solutions and the formation of singularities seems to proceed through adiabatic shrinking of a static solution.
Completely parallel results can be found for wave maps, for which the critical dimension is 2 + 1. For the wave map from 3 + 1 Minkowski to the 3sphere, Bizoń [16] has shown that there is a countable family of regular (before the CH) CSS solutions labeled by a nodal number ≥ 0, such that each solution has n unstable modes. Simulations of collapse in spherical symmetry [23, 151] and in 3 dimensions [149] show that n = 0 is a global attractor and the n = 1 solution is the critical solution (see also [67, 68] for computations of the largest perturbationeigenvalues of W_{0} and W_{1}). Again, for the wave map from 2 + 1 Minkowski to the 2sphere generic singularity formation proceeds through adiabatic shrinking of a static solution [24].
These results have led to the suggestion in [26] that criticality (in the sense of the existence of a codimension1 solution separating evolution towards qualitatively different end states) could be a generic and robust feature of evolutionary PDE systems in supercritical dimensions, and not an effect particular of gravity.
Garfinkle and Isenberg [88] examine the threshold between the round end state and pinching off in Ricci flow for a familiy of spherically symmetric geometries on S^{3}. They have found intermediate approach to a special “javelin” geometry, but have not investigated whether this is universal. See also [89] and [135].
A scaling of the shape of the event horizon at the moment of merger in binary black hole mergers is noted in [44], but this is really a kinematic effect.
4.5 Analytic studies and toy models
4.5.1 Exact solutions of EinsteinKleinGordon
A number of authors have explored the possibility of finding critical phenomena with CSS (rather than DSS) massless scalar critical solutions. The Roberts 1parameter family of 3 + 1 solutions [184] has been analyzed along this line in [173, 32, 206, 137]. This family contains black holes whose masses (with a suitable matching to an asymptotically flat solution) scale as (p − 1)^{1/2} for p ≳ 1, but such a special family of solutions has no direct relevance for collapse from generic data. Its generalization to other dimensions has been considered in [75]. A fully analytic construction of all (spherical and nonspherical) linear perturbations of the Roberts solution by Frolov [73, 74] has shown that there is a continuum of unstable spherical modes filling a sector of the complex plane with Re λ ≥ 1, so that it cannot be a critical solution. Interestingly, all nonspherical perturbations decay.
Frolov [76] has also suggested that the critical (p = 1) Roberts solution, which has an outgoing null singularity, plus its most rapidly growing (spherical) perturbation mode would evolve into the Choptuik solution, which would inherit the oscillation in τ with a period 4.44 of that mode.
A similar transition within a single 1parameter family of solutions has been pointed out in [170] for the Wyman solution [207].
Hayward [121, 66] and Clement and Fabbri [64, 65] have also proposed critical solutions with a null singularity, and have attempted to construct black hole solutions from their linear perturbations. This is probably irrelevant to critical collapse, as the critical spacetime does not have an outgoing null singularity. Rather, the singularity is naked but first appears in a point. The future light cone of that point is not a null singularity but a CH with finite curvature.
Other authors have attempted analytic approximations to the Choptuik solution. Pullin [181] has suggested describing critical collapse approximately as a perturbation of the Schwarzschild spacetime. Price and Pullin [180] have approximated the Choptuik solution by two flat space solutions of the scalar wave equation that are matched at a “transition edge” at constant selfsimilarity 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 Δ.
4.5.2 Toy models
Birmingham and Sen [12] considered the formation of a black hole from the collision of two point particles of equal mass in 2 + 1 gravity. Peleg and Steif [175] have investigated the collapse of a dust ring. In both cases the mass of the black holes is a known function of the parameters of the initial condition, giving a “critical exponent” 1/2, but no underlying selfsimilar solution is involved.
Mahajan et al. [152] expand the initial data for Einstein clusters in powers of the radius and, assuming that there are no shell crossings, find a mass scaling exponent of 3/2 for two such expansion coefficients. Universality is not demonstrated, and so the connection with a CSS solution discussed by Harada and Mahajan [118] is unclear.
Frolov [78], and Frolov, Larsen and Christensen [79] consider a stationary 2 + 1dimensional NambuGoto membrane held fixed at infinity in a stationary 3 + 1dimensional black hole background spacetime. The induced 2 + 1 metric on the membrane can have wormhole, black hole, or Minkowski topology. The critical solution between Minkowski and black hole topology has 2 + 1 CSS. The mass of the apparent horizon of induced black hole metrics scales with γ= 2/3, superimposed with a wiggle of period \(3\pi/\sqrt 7\) in ln p. The mass scaling is universal with respect to different background black hole metrics, as they can be approximated by Rindler space in the mass scaling limit.
Horowitz and Hubeny [130], and Birmingham [11] have attempted to calculate the critical exponent in toy models from the adSCFT correspondence. ÁlvarezGaumhé et al. have attempted to use the adSCFT correspondence for calculating γ from the QCD side for the spherical massless scalar field in 5 dimensions [5], and the spherical perfect fluid with P = kρ for k = 1/(d − 1) in d = 5, 6, 7 dimensions [7].
ÁlvarezGaumé et al. [6] have calculated critical exponents for the formation of an apparent horizon in the collision of two gravitational shock waves.
Burko [37] considers the transition between existence and nonexistence of a null branch of the singularity inside a spherically symmetric charged black hole with massless scalar field matter thrown in.
Wang [205] has constructed homothetic cylindrically symmetric solutions of 3 + 1 EinsteinKleinGordon and studied their cylindrically symmetric perturbations. It is not clear how these are related to a critical surface in phase space.
4.6 Quantum effects
Type II critical phenomena provide a relatively natural way of producing arbitrarily high curvatures, where quantum gravity effects should become important, from generic initial data. Approaching the Planck scale from above, one would expect to be able to write down a critical solution that is the classical critical solution asymptotically at large scales, as an expansion in inverse powers of the Planck length (see Section 2.6).
Black hole evolution in semiclassical gravity has been investigated in 1 + 1 dimensional models which serve as toy models for spherical symmetry (see [92] for a review). The black hole threshold in such models has been investigated in [45, 9, 194, 137, 210, 174, 31]. In some of these models, the critical exponent is 1/2 for kinematical reasons.
In [36] a 3 + 1dimensional but perturbative approach is taken. The quantum effects then give an additional unstable mode with λ = 2. If this is larger than the positive Lyapunov exponent λ_{0}, it will become the dominant perturbation for sufficiently good finetuning, and therefore sufficiently good finetuning will reveal a mass gap. The mass gap is found also in numerical evolutions of a spherical scalar field in 3 + 1 dimensions with the semiclassical equations obtained in the framework of “singularity resolution” in loop quantum gravity [132, 211].
5 Beyond Spherical Symmetry

Weak gravitational waves in vacuum general relativity can focus and collapse. The black hole threshold in this process shows what in critical phenomena in gravitational collapse is intrinsic to gravity rather than the matter model.

Black holes are characterised by charge and angular momentum as well as mass. Angular momentum is the more interesting of the two because it is again independent of matter, but cannot be studied in spherical symmetry.

Angular momentum resists collapse, but angular momentum in the initial data is needed to make a black hole with angular momentum. Therefore it is an interesting question to ask what happens to the dimensionless ratio J/M^{2} at the black hole threshold.
5.1 Perturbative approach to angular momentum
We have already mentioned that when angular momentum is small, a critical exponent for J can be derived in perturbation theory. This has been done for the perfect fluid (see Section 4.2) in firstorder perturbation theory and for the massless scalar field (see Section 3.7), where secondorder perturbation theory in the scalar field is necessary to obtain an angular momentum perturbation in the stressenergy tensor [87]. However, neither of the predicted angular momentum scaling laws has been verified in numerical evolutions.
For a perfect fluid with EOS p = kρ with 0 < k < 1/9, precisely one mode that carries angular momentum is unstable, and this mode and the known spherical mode are the only two unstable modes of the spherical critical solution. (Note by comparison that from dimensional analysis one would not expect an uncharged critical solution to have a growing perturbation mode carrying charge.) The presence of two growing modes of the critical solution is expected to give rise to interesting phenomena [104]. Near the critical solution, the two growing modes compete. J and M of the final black hole are expected to depend on the distance to the black hole threshold and the angular momentum of the initial data through universal functions of one variable that are similar to “universal scaling functions” in statistical mechanics (see also the end of Section 2.7). While they have not yet been computed, these functions can in principle be determined from time evolutions of a single 2parameter family of initial data, and then determine J and M for all initial data near the black hole threshold and with small angular momentum. They would extend the simple powerlaw scalings of J and M into a region of initial data space with larger angular momentum.
5.2 Axisymmetric vacuum gravity
In order to keep their numerical grid as small as possible, Abrahams and Evans chose their initial data to be mostly ingoing. The two free functions \(K_\theta ^r\) in the initial data were chosen to have the same functional form they would have in a linearised gravitational wave with pure l = 2, m = 0 angular dependence. This ansatz reduced the freedom in the initial data to one free function of advanced time. A specific peaked function was chosen, and only the overall amplitude was varied.
Limited numerical resolution allowed Abrahams and Evans to find black holes with masses only down to 0.2 of the ADM mass. Even this far from criticality, they found powerlaw scaling of the black hole mass, with a critical exponent γ ≃ 0.36. The black hole mass was determined from the apparent horizon surface area, and the frequencies of the lowest quasinormal modes of the black hole. There was tentative evidence for scale echoing in the time evolution, with Δ ≃ 0.6, with about three echos seen. Here η has the echoing property η(e^{Δ}r, e^{Δ}t) = η(r, t), and the same echoing property is expected to hold also for α, ϕ, β^{r}, and r^{−1}β^{θ}. In a subsequent paper [2], some evidence for universality of the critical solution, echoing period and critical exponent was given in the evolution of a second family of initial data, one in which η= 0 at the initial time. In this family, black hole masses down to 0.06 of the ADM mass were achieved.
It is striking that at 14 years later, these results have not yet been independently verified. An attempt with a 3dimensional numerical relativity code (but axisymmetric initial data), using free evolution in the BSSN formulation with maximal slicing and zero shift, to repeat the results of Abrahams and Evans was not successful [4]. The reason could be a combination of rather lower resolution than that of Abrahams and Evans, growing constraint violations, and an inappropriate choice of gauge. (It is now known that BSSN with maximal slicing and zero shift is an illposed system [108].)
5.3 Scalar field
In 2003, Choptuik, Hirschmann, Liebling and Pretorius reported on numerical evolutions at the black hole threshold of an axisymmetric massless scalar field [54]. In axisymmetry with scalar field matter there is no angular momentum and only one polarisation of gravitational waves. The slicing condition is maximal slicing and the spatial gauge, in cylindrical coordinates, is g_{zz} = g_{ρρ}, g_{zρ} = 0, similar to the gauge used by Abrahams and Evans. The Hamiltonian constraint is solved at every time step, and the time derivatives of the spatial gauge conditions are substituted into the momentum constraints to obtain secondorder elliptic equations for the two shift components. Thus the evolution is partially constrained. Adaptive meshrefinement was used in the numerical time evolution.
The initial data were either timesymmetric or approximately ingoing, with the scalar field either symmetric or antisymmetric in z. In the symmetric case, even strongly nonspherical data were attracted to the known spherical critical solution for the massless scalar field. Scaling with the known γ was observed in the Ricci scalar. However, with sufficiently good finetuning to the black hole threshold, the approximately spherical region that approaches the critical solution suffers an l = 2 (and by ansatz m = 0) instability and splits into two new spherical regions which again approach the critical solution. The spatial separation of the two new centres is related to the smallest length scale that developed prior to the branching. There is evidence that with increasing finetuning each of these centres splits again. The antisymmetric initial data cannot approach a single spherical critical solution, but the solution splits initially into two approximately spherical regions where the critical solution is approached (up to an overall sign in the scalar field). The separation of these initial two centres is determined by the initial data, but there is evidence that they in turn split.
All this is consistent with the assumption that the spherical critical solution has, besides the known one spherical unstable mode, precisely one further l = 2 unstable mode. (Without the restriction to axisymmetry, if such a mode exists, it would be 5fold degenerate with m = −2, …, 2.) This contradicts the calculation of the perturbation spectrum in [154]. Choptuik and coworkers do not state with certainty that the mode they see in numerical evolutions is a continuum mode, although they have no indication that it is a numerical artifact. The growth rate of the putative mode is measured to be λ ≃ 0.1 −0.4, which should be compared with the growth rate λ ≃ 2.7 of the spherical mode and the relatively small decay rate of λ ≃ −0.02 claimed in [154] for the least damped mode, which is also an l = 2 mode. We observe that the range of τ in Figures 6 and 7 of [54] is about 10, and over this range the plot of the amplitude of the l = 2 perturbation against the logscale coordinate τ seems equally consistent with linear growth in τ as with exponential growth. (In the notation of [54], τ denotes proper time and τ_{*} the accumulation point of echos, so that the logscale coordinate τ used in this review corresponds to − ln(τ − τ_{*}) in the notation of [54].)
For all initial data in numerical evolutions, a critical solution is approached that is discretely selfsimilar with logscale period Δ ≃ 0.42. (By ansatz this solution is axisymmetric but spherical symmetry is ruled out and so the critical solution cannot be the Choptuik solution.) The same critical solution is approached in particular for initial data with \(\Pi = 0\) and hence no angular momentum, and initial data where \(\Pi = \bar \Phi\) and hence with large angular momentum. A scaling exponent of γ ≃ 0.11 is observed in the Ricci scalar in subcritical evolutions. The critical solution is purely real (up to an initial datadependent constant phase) and hence has no angular momentum. Only m = 1 was investigated, but it is plausible that a different critical solution exists for each integer m.
Far from the black hole threshold J ∼ M^{2} in the final black hole, but nearer the black hole threshold, J ∼ M^{6}, where J and M are measured on the apparent horizon when it first forms. J/M^{2} → 0 is compatible with a nonrotating critical solution.
In the absence of angular momentum, the wave equations for the real and imaginary part of Φ decouple. Assuming that the background critical solution is purely real with δ = 0 and A ∼ 1, and angular momentum is provided by a perturbation with δ∼ e^{λτ}, one would expect ρ ∼ ΔA^{2} ∼ e^{2τ} and j ∼ A∇δ∼ e^{(1+λ)τ}. Integrating over a region of size e^{−3τ} when the black hole forms, we find \(M \sim {e^{ {\tau _\ast}}}\) and \(J \sim {e^{( 2 + \lambda){\tau _\ast}}}\). Then J ∼ M^{6} would imply λ = −4.
Olabarrieta et al. [169] study a similar system in spherical symmetry, by arranging 2l + 1 scalar fields given by ϕ_{lm} = ψ(r, t)Y_{lm}(θ, φ) for m = −l, …l with the same ψ(r, t) for all values of m so that the total stressenergy tensor and the spacetime are spherically symmetric. Note that not the ϕ_{lm} are added but their stressenergies, and each value of l describes a different matter content.ψ then sees a centrifugal barrier in its evolution equation, but there is no angular momentum in the spacetime. DSS critical behaviour is found, and the logarithmic echoing period Δ and mass scaling exponent γ both decrease approximately exponentially with l. It is observed empirically that the radius r_{0} of maximum compactness during evolution (a measure of the scale of the initial data) and the accumulation time of echos T* (measured from the initial data) obey T* ≃ r_{0}/(4.35Δ) for all l and families of initial data.
Lai [141] has studied type I critical phenomena for boson (massive complex scalar field) stars in axisymmetry, the first study of type I in axisymmetry. He finds that the subcritical end state is a boson star with a large amplitude fundamental mode oscillation.
5.4 Neutron star collision in axisymmetry
A first investigation of type I critical collapse in an astrophysically motivated scenario was carried out in [136, 204]. The matter is a perfect fluid with “Gamma law” EOS P = (γ − 1)ρϵ, where Γ ≃ 2 is a constant, P is the pressure, ρ the rest mass density, and ϵ the internal energy per rest mass (so that ρ(1 + ϵ) is the total energy density). The initial data are constructed with P = kρ^{Γ} for a constant k, which corresponds to the “cold” (constant entropy) limit of the Gamma law EOS. The initial data correspond to two identical stars which have fallen from infinity. (The evolution starts at finite distance, with an initial velocity calculated in the first postNewtonian approximation). The entire solution is axisymmetric with an additional reflection symmetry that maps one star to the other.
The parameter of the initial data that is varied is the mass of the two stars. Supercritical data form a single black hole, while subcritical data form a single star. The diagnostics given are plots against time of the lapse and the Ricci scalar at the symmetry centre of the spacetime, and of outgoing l = 2 gravitational waves. Collapse of the lapse and blowup of the Ricci scalar are taken as indications of black hole formation. The limited numerical evidence is compatible with type I critical phenomena, with the putative critical solution showing oscillations with the same period in the lapse and Ricci scalar. For the critical solution to be exactly timeperiodic, it would have to be spherical in order to not lose energy through gravitational waves, and there is some evidence that indeed it does not radiate gravitational waves.
Other 1parameter families of initial data were obtained by fixing the mass of the stars in the first family very near its critical value and then varying one of the following: the initial separation, the initial speed, and Γ. For all approximately the same scaling law for the survival time of the critical solution was found. However, as all these initial data are very close together, this only confirms the validity of the general perturbation theory explanation of critical phenomena, but does not provide evidence of universality.
A key question that has not been answered is how shock heating affects the standard critical collapse scenario. A priori the existence of a universal spherical critical solution is unlikely in the presence of shock heating as there is no dynamical mechanism to arrive at a universal spherical temperature profile, in contrast to nondissipative matter models or vacuum gravity where all equations are wavelike.
5.5 Black hole collisions
The standard dynamical systems picture of critical collapse, with the critical solution an attractor in the threshold hypersurface, appears to be consistent with these observation. Pretorius compares his data with the unstable circular geodesics in the spacetime of the hypothetical rotating black hole that would result if merger occurred promptly. These give orbital periods, and their linear perturbations give a critical exponent, in rough agreement with the numerical values for the full black hole collision.
Pretorius does not comment on the nature of the critical solution, but because of the mass loss through gravitational radiation it cannot be strictly stationary. The mass loss would be compatible with selfsimilarity, with a helical homothetic vector field, but exact selfsimilarity would be compatible with the presence of black holes only in the infinite boost limit. Therefore the critical solution is likely to be more complicated, and can perhaps be written as an expansion with an exactly stationary or homothetic spacetime as the leading term.
Pretorius also speculates that these phenomena generalise to generic initial data with unequal masses and black hole spins which are not aligned. This seems uncertain, given the claim by Levin [146] that the threshold of immediate merger is fractal if the spins are not aligned, and that the system is therefore chaotic. However, Levin’s analysis is based on a 2nd order postNewtonian approximation to general relativity, in which there is no radiation reaction, while the rapid energy loss observed here may suppress chaos. Nevertheless, the phase space is much bigger when the orbit is not confined to an orbital plane, and so the critical solution observed here may not be an attractor in the full critical surface.
Sperhake et al. [191] push the numerics towards higher energies emitted and conjecture that the merger of nonspinning black holes can in principle yield a black hole arbitrarily close to extremal Kerr. See also [187].
Notes
Acknowledgements
We would like to thank David Garfinkle for a critical reading of the manuscript and the Louisiana State University for hospitality while this work was begun. JMM was supported by the I3P framework of CSIC and the European Social Fund, and by the Spanish MEC Project FIS200505736C0302.
For the 2010 revision, JMM was supported by the French A.N.R. (Agence Nationale de la Recherche) through Grant BLAN071_201699 entitled “LISA Science”, and CG was supported by the A.N.R. Grant 062134423 entitled “Mathematical Methods in General Relativity”. CG would also like to thank the IAP and LUTH for hospitality.
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