In summary, the critical spacetimes that arise asymptotically in the fine-tuning of gravitational collapse to the black hole threshold have a curvature singularity that is visible at infinity with a finite redshift. The Cauchy horizon of the singularity is mildly singular (low differentiability), but the curvature is finite there. It is unclear at present if the singularity is timelike or if there exists a continuation beyond the Cauchy horizon with a regular center, so that the singularity is limited, loosely speaking, to a point. Further work should be able to clarify this. In any case, the singularity is naked and the critical solutions therefore provide counter-examples to any formulation of cosmic censorship which states only that naked singularities cannot arise from smooth initial data in reasonable matter models. The statement must be that there is no open ball of smooth initial data for naked singularities.

Recent analytic work by Christodoulou on the spherical scalar field [49] is not directly relevant to the smooth (analytic or $C^{\infty }$  ) initial data discussed here. Christodoulou considers a larger space of initial data that are not $C^1$  . He shows that for any data set $f_0$  in this class that forms a naked singularity there are data $f_1$  and $f_2$  such that the data sets $f_0+c_1{f}_1+c_2{f}_2$  do not contain a naked singularity, for any $c_1$  and $c_2$  except zero. Here $f_1$  is data of bounded variation, and $f_2$  is absolutely continuous data. Therefore, the set of naked singularity data is at least codimension two in the space of data of bounded variation, and of codimension at least one in the space of absolutely continuous data. The semi-numerical result of Gundlach claims that it is codimension exactly one in the set of smooth data. The result of Christodoulou holds for any $f_0$  , including initial data for the Choptuik solution. The apparent contradiction is resolved if one notes that the $f_1$  and $f_2$  of Christodoulou are not smooth in (at least) one point, namely where the initial data surface is intersected by the past light cone of the singularity in $f_0$  ^$\text{11}$  . The data $f_0+c_1{f}_1+c_2{f}_2$  are therefore not smooth.

^$\text{11}$  Roughly speaking, $f_1$  and $f_2$  start throwing scalar field matter into the naked singularity at the exact moment it is born, and therefore depend on $f_0$  .

4.6 Beyond spherical symmetry

Every aspect of the basic scenario: CSS and DSS, universality and scaling applies directly to a critical solution that is not spherically symmetric, but all the models we have described are spherically symmetric. There are only two exceptions to date: A numerical investigation of critical collapse in axisymmetric pure gravity [1] , and studies of the nonspherical perturbations the spherically symmetric perfect fluid [76] and scalar field [99] critical solutions. They correspond to two related questions: Are the critical phenomena in the known spherically symmetric examples destroyed already by small deviations from spherical symmetry? And: Are there critical phenomena in gravitational collapse far from spherical symmetry?

4.6.1 Axisymmetric gravitational waves

The paper of Abrahams and Evans [1] was the first paper on critical collapse to be published after Choptuik's PRL, but it remains the only one to investigate a nonspherically symmetric situation, and therefore also the only one to investigate critical phenomena in the collapse of gravitational waves in vacuum. Because of its importance, we summarize its contents here in some technical detail.

The physical situation under consideration is axisymmetric vacuum gravity. The numerical scheme uses a 3+1 split of the spacetime. The ansatz for the spacetime metric is

$$\begin{array}{ccc}d{s}^2=-{\alpha }^{2}d{t}^2+{\phi }^4[& & e^{2\eta /3}(dr+{\beta }^{r}dt{)}^2+r^2{e}^{2\eta /3}(d\theta +{\beta }^{\theta }dt{)}^2+\end{array}$$

$$\begin{array}{ccc}& & +e^{-4\eta /3}{r}^2{sin}^2\theta d{\phi }^2],\end{array}$$

(66)

parameterized by the lapse $\alpha$  , shift components ${\beta }^r$  and ${\beta }^{\theta }$  , and two independent coefficients $\phi$  and $\eta$  in the 3-metric. All are functions of $r$  , $t$  and $\theta$  . The fact that $d{r}^2$  and $r^{2}d{\theta }^2$  are multiplied by the same coefficient is called quasi-isotropic spatial gauge. The variables for a first-order-in-time version of the Einstein equations are completed by the three independent components of the extrinsic curvature, $K_{\theta }^r$  , $K_r^r$  , and $K_{\phi }^{\phi }$  . The ansatz limits gravitational waves to one “polarisation” out of two, so that there are as many physical degrees of freedom as in a single wave equation. In order to obtain initial data obeying the constraints, $\eta$  and $K_{\theta }^r$  are given as free data, while the remaining components of the initial data, namely $\phi$  , $K_r^r$  , and $K_{\phi }^{\phi }$  , are determined by solving the Hamiltonian constraint and the two independent components of the momentum constraint respectively. There are five initial data variables, and three gauge variables. Four of the five initial data variables, namely $\eta$  , $K_{\theta }^r$  , $K_r^r$  , and $K_{\phi }^{\phi }$  , are updated from one time step to the next via evolution equations.

As many variables as possible, namely $\phi$  and the three gauge variables $\alpha$  , ${\beta }^r$  and ${\beta }^{\theta }$  , are obtained at each new time step by solving elliptic equations. These elliptic equations are the Hamiltonian constraint for $\phi$  , the gauge condition of maximal slicing ( ${K_i}^i=0$  ) for $\alpha$  , and the gauge conditions $g_{\theta \theta }=r^2{g}_{rr}$  and $g_{r\theta }=0$  for ${\beta }^r$  and ${\beta }^{\theta }$  (quasi-isotropic gauge).

For definiteness, the two free functions, $\eta$  and $K_{\theta }^r$  , in the initial data were chosen to have the same functional form they would have in a linearized gravitational wave with pure $(l=2,m=0)$  angular dependence. Of course, depending on the overall amplitude of $\eta$  and $K_{\theta }^r$  , the other functions in the initial data will deviate more or less from their linearized values, as the nonlinear initial value problem is solved exactly. In axisymmetry, only one of the two degrees of freedom of gravitational waves exists. In order to keep their numerical grid as small as possible, Abrahams and Evans chose the pseudo-linear waves to be purely ingoing ^$\text{12}$  . This ansatz (pseudo-linear, ingoing, $l=2$  ), reduced the freedom in the initial data to one free function of advanced time, $I^{\left(2\right)}\left(v\right)$  ^$\text{13}$  . A suitably peaked function was chosen.

Limited numerical resolution (numerical grids are now two-dimensional, not one-dimensional as in spherical symmetry) 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 power-law scaling of the black hole mass, with a critical exponent $\gamma \simeq 0.36$  . Determining the black hole mass is not trivial, and was done from the apparent horizon surface area, and the frequencies of the lowest quasi-normal modes of the black hole. There was tentative evidence for scale echoing in the time evolution, with $\Delta \simeq 0.6$  , with about three echoes seen. This corresponds to a scale range of about one order of magnitude. By a lucky coincidence, $\Delta$  is much smaller than in all other examples, so that several echoes could be seen without adaptive mesh refinement. The paper states that the function $\eta$  has the echoing property $\eta (e^{\Delta }r,e^{\Delta }t)=\eta (r,t)$  . If the spacetime is DSS in the sense defined above, the same echoing property is expected to hold also for $\alpha$  , $\phi$  , ${\beta }^r$  and $r^{-1}{\beta }^{\theta }$  , as one sees by applying the coordinate transformation ( 12 ) to ( 66 ).

In a subsequent paper [2] , universality of the critical solution, echoing period and critical exponent was demonstrated through the evolution of a second family of initial data, one in which $\eta =0$  at the initial time. In this family, black hole masses down to $0.06$  of the ADM mass were achieved. Further work on critical collapse far away from spherical symmetry would be desirable, but appears to be held up by numerical difficulty.

^$\text{12}$  In nonlinear general relativity, no exact notion of ingoing and outgoing waves exists, but this ansatz means that the wave is initially ingoing in the low-amplitude limit.

^$\text{13}$  In the linear limit, everything would then remain a function of advanced time forever.

4.6.2 Perturbing around spherical symmetry

A different, and technically simpler, approach is to take a known critical solution in spherical symmetry, and perturb it using nonspherical perturbations. Addressing this perturbative question, Gundlach [76] has studied the generic nonspherical perturbations around the critical solution found by Evans and Coleman [53] for the $p=\frac{1}{3}\rho$  perfect fluid in spherical symmetry. There is exactly one spherical perturbation mode that grows towards the singularity (confirming the previous results [91, 98] ). There are no growing nonspherical modes at all. A corresponding result was established for nonspherical perturbations of the Choptuik solution for the massless scalar field [99] .

The main significance of this result, even though it is only perturbative, is to establish one critical solution that really has only one unstable perturbation mode within the full phase space.

As the critical solution itself has a naked singularity (see Section  4.5 ), this means that there is, for this matter model, a set of initial data of codimension one in the full phase space of general relativity that forms a naked singularity. This result also confirms the role of critical collapse as the most “natural” way of creating a naked singularity.

4.7 Black hole charge and angular momentum

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 one-parameter family of initial data with both electric charge and angular momentum (for suitable matter), and fine-tunes the parameter $p$  to the black hole threshold. Does the mass still show power-law scaling? What happens to the dimensionless ratios $L/M^2$  and $Q/M$  , with $L$  being the black hole angular momentum and $Q$  its electric charge?

Tentative answers to both questions have been given using perturbations around spherically symmetric uncharged collapse.

4.7.1 Charge

Gundlach and Martín-García [78] have studied scalar massless electrodynamics in spherical symmetry. Clearly, the real scalar field critical solution of Choptuik is a solution of this system too. Less obviously, it remains a critical solution within massless (and in fact, massive) scalar electrodynamics in the sense that it still has only one growing perturbation mode within the enlarged solution space. Some of its perturbations carry electric charge, but as they are all decaying, electric charge is a subdominant effect. The charge of the black hole in the critical limit is dominated by the most slowly decaying of the charged modes. From this analysis, a universal power-law scaling of the black hole charge

$$\begin{array}{c}Q\sim (p-p_{*}{)}^{\delta }\end{array}$$

(67)

was predicted. The predicted value $\delta \simeq 0.88$  of the critical exponent (in scalar electrodynamics) was subsequently verified in collapse simulations by Hod and Piran [86] . (The mass scales with $\gamma \simeq 0.37$  as for the uncharged scalar field.) General considerations using dimensional analysis led Gundlach and Martín-García to the general prediction that the two critical exponents are always related, for any matter model, by the inequality

$$\begin{array}{c}\delta \ge 2\gamma .\end{array}$$

(68)

This has not yet been verified in any other matter model.

4.7.2 Angular momentum

Gundlach's results on nonspherically symmetric perturbations around spherical critical collapse of a perfect fluid [76] allow for initial data, and therefore black holes, with infinitesimal angular momentum. All nonspherical perturbations decrease towards the singularity. The situation is therefore similar to scalar electrodynamics versus the real scalar field. The critical solution of the more special model (here, the strictly spherically symmetric fluid) is still a critical solution within the more general model (a slightly nonspherical and slowly rotating fluid). In particular, axial perturbations (also called odd-parity perturbations) with angular dependence $l=1$  (i.e. dipole) will determine the angular momentum of the black hole produced in slightly supercritical collapse.

Using a perturbation analysis similar to that of Gundlach and Martín-García [78] , Gundlach [74] (see correction in [77] ) has derived the angular momentum scaling law

$$\begin{array}{c}L\sim (p-p_{*}{)}^{\mu }.\end{array}$$

(69)

For the range $0.123<k<0.446$  of equations of state, the angular momentum exponent $\mu$  is related to the mass exponent $\gamma$  by

$$\begin{array}{c}\mu \left(k\right)=\frac{5(1+3k)}{3(1+k)}\gamma \left(k\right).\end{array}$$

(70)

In particular for $k=1/3$  , one gets $\mu =(5/2)\gamma \simeq 0.898$  . An angular momentum exponent $\mu \simeq 0.76$  was derived for the massless scalar field in [63] using second-order perturbation theory. Both results have not yet been tested against numerical collapse simulations.

5 Aspects of current research

Like the previous section, this one contains extensions of the basic ideas, but here we group together topics that are still under active investigation.

5.1 Phase diagrams

In analogy with critical phenomena in statistical mechanics, let us call a graph of the black hole threshold in the phase space of some self-gravitating system a phase diagram. The full phase space is infinite-dimensional, but one can plot a two-dimensional submanifold. In such a plot the black hole threshold is generically a line, analogous to the fluid/gas dividing line in the pressure/temperature plane.

Interesting phenomena can be expected in systems that admit more complicated phase diagrams.

The massive complex scalar field for example, admits stable stars as well as black holes and flat space as possible end states. There are three phase boundaries, and these should intersect somewhere. A generic two-parameter family of initial data is expected to intersect each boundary in a line, and the three lines should meet at a triple point.

Similarly, many systems admit both type I and type II phase transitions, for example the massive real scalar field, and the $SU\left(2\right)$  Yang-Mills field in spherical symmetry. In a two-dimensional family of initial data, these should again generically show up as lines, and generically these lines should intersect. Is the black hole mass at the intersection finite or zero? Is there a third line that begins where the type I and type II lines meet?

Choptuik, Hirschmann and Marsa [42] have investigated this for a specific two-parameter family of initial data for the spherical $SU\left(2\right)$  Yang-Mills field, using a numerical evolution code that can follow the time evolutions for long after a black hole has formed. As known previously, the type I phase transition is mediated by the static Bartnik-McKinnon solution, which has one growing perturbation mode. The type II transition is mediated by a DSS solution with one growing mode.

There is a third type of phase transition along a third line which meets the intersection of the type I and type II lines. On both sides of this “type III” phase transition the final state is a Schwarzschild black hole with zero Yang-Mills field strength, but the final state is distinguished by the value of the Yang-Mills gauge potential at infinity. (The system has two distinct vacuum states.) The critical solution is an unstable black hole with Yang-Mills hair, which collapses to a hairless Schwarzschild black hole with either vacuum state of the Yang-Mills field, depending on the sign of its one growing perturbation mode. The critical solution is not unique, but is a member of a 1-parameter family of hairy black holes parameterized by their mass. At the triple point the family ends in a zero mass black hole.

5.2 The renormalisation group as a time evolution

It has been pointed out by Argyres [3] , Koike, Hara and Adachi [91] and others that the time evolution near the critical solution can be considered as a renormalisation group flow on the space of initial data. The calculation of the critical exponent in Section  3.3 is in fact mathematically identical to that of the critical exponent governing the correlation length near the critical point in statistical mechanics [123] , if one identifies the time evolution in the time coordinate $\tau$  and spatial coordinate $x$  with the renormalisation group flow. But those coordinates were defined only on self-similar spacetimes plus linear perturbations. In order to obtain a full renormalisation group, one has to generalize them to arbitrary spacetimes, or, in other words, find a general prescription for the lapse and shift as functions of arbitrary Cauchy data.

For simple parabolic or hyperbolic differential equations, a discrete renormalisation (semi)group acting on their solutions has been defined in the following way [67, 24, 32, 33] : Evolve initial data over a certain finite time interval, then rescale the final data in a certain way. Solutions which are fixed points under this transformation are scale-invariant, and may be attractors. One nice distinctive feature of GR as opposed to these simple models is that one can use a shift freedom in GR (one that points inward towards an accumulation point) to incorporate the rescaling into the time evolution, and the lapse freedom to make each rescaling by a constant factor an evolution through a constant time ( $\tau$  , in our notation) interval.

The crucial distinctive feature of general relativity, however, is that a solution does not correspond to a unique trajectory in the space of initial data. This is because a spacetime can be sliced in different ways, and on each slice one can have different coordinate systems. Infinitesimally, this slicing and coordinate freedom is parameterized by the lapse and shift. In a relaxed notation, one can write the ADM equations as $(\dot{g},\dot{K})=$  functional $(g,K,\alpha ,\beta )$  , where $g$  is the 3-metric, $K$  the extrinsic curvature, $\alpha$  the lapse and $\beta$  the shift. The lapse and shift can be set freely, independently of the initial data. Of course they influence only the coordinates on the spacetime, not the spacetime itself, but the ADM equations are not yet a dynamical system. If we specify a prescription $(\alpha ,\beta )=$  functional $(g,K)$  , then substituting it into the ADM equations, we obtain $(\dot{g},\dot{K})=$  functional $(g,K)$  , which is an (infinite-dimensional) dynamical system. We are then faced with the general question: Given initial data in general relativity, is there a prescription for the lapse and shift, such that, if these are in fact data for a self-similar solution, the resulting time evolution actively drives the metric to the special form ( 28 ) that explicitly displays the self-similarity?

An algebraic prescription for the lapse suggested by Garfinkle [59] did not work, but maximal slicing with zero shift does work [64] if combined with a manual rescaling of space. Garfinkle and Gundlach [62] have suggested several combinations of lapse and shift conditions that not only leave CSS spacetimes invariant, but also turn the Choptuik DSS spacetime into a limit cycle. The combination of maximal slicing with minimal strain shift has the nice property that it also turns static spacetimes into fixed points (and probably periodic spacetimes into limit cycles). Maximal slicing requires the first slice to be maximal ( ${K_a}^a=0$  ), but other prescriptions 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 only for 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.

5.3 Analytic approaches

A number of authors have attempted to explain critical collapse with the help of analytic solutions. The one-parameter family of exact self-similar real massless scalar field solutions first discovered by Roberts [112] has already been presented in Section  4.5 . It has been discussed in the context of critical collapse in [18, 106] , and later [121, 26] . The original, analytic, Roberts solution is cut and pasted to obtain a new solution which has a regular center $r=0$  and which is asymptotically flat. Solutions from this family [see Eqns. ( 64 )] with $p>1$  can be considered as black holes, and to leading order around the critical value $p=1$  , their mass is $M\sim (p-p_{*}{)}^{1/2}$  . The pitfall in this approach is that only perturbations within the self-similar family are considered, so the formal critical exponent applies only to this one, very special, family of initial data. But the $p=1$  solution has many growing perturbations which are spherically symmetric (but not self-similar), and is therefore not a critical solution in the sense of being an attractor of codimension one. This was already clear because it did not appear in collapse simulations at the black hole threshold, but Frolov has calculated the perturbation spectrum analytically [55, 56] . The eigenvalues of spherically symmetric perturbations fill a sector of the complex plane, with $\text{Re}\lambda \le 1$  . All nonspherical perturbations decay. Other supposed critical exponents that have been derived analytically are usually valid only for a single, very special family of initial data also.

Other authors have employed analytic approximations to the actual Choptuik solution. Pullin [109] has suggested describing critical collapse approximately as a perturbation of the Schwarzschild spacetime. Price and Pullin [108] have approximated the Choptuik solution by two flat space solutions of the scalar wave equation that are matched at a “transition edge” at constant self-similarity 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 $\Delta$  . While the insights of this paper are qualitative, some of its ideas reappear in the construction [70] of the Choptuik solution as a 1+1 dimensional boundary value problem. Frolov [57] has suggested approximating the Choptuik solution as the Roberts solution plus its most rapidly growing (spherical) perturbation mode, pointing out that it oscillates in $\tau$  with a period $4.44$  , but ignoring the fact that it also grows exponentially. This is probably not a correct approach.

In summary, purely analytic approaches have so far remained unsuccessful in explaining critical collapse.

5.4 Astrophysical black holes

Any real world application of critical phenomena would require that critical phenomena are not an artifact of the simple matter models that have been studied so far, and that they are not an artifact of spherical symmetry. At present this seems a reasonable hypothesis.

Critical collapse still requires a kind of fine-tuning of initial data that does not seem to arise naturally in the astrophysical world. Niemeyer and Jedamzik [102] have suggested a scenario that gives rise to such fine-tuning. In the early universe, quantum fluctuations of the metric and matter can be important, for example providing the seeds of galaxy formation. If they are large enough, these fluctuations may even collapse immediately, giving rise to what is called primordial black holes. Large quantum fluctuations are exponentially more unlikely than small ones, $P\left(\delta \right)\sim exp-{\delta }^2$  , where $\delta$  is the density contrast of the fluctuation. One would therefore expect the spectrum of primordial black holes to be sharply peaked at the minimal $\delta$  that leads to black hole formation. That is the required fine-tuning. In the presence of fine-tuning, the black hole mass is much smaller than the initial mass of the collapsing object, here the density fluctuation. In consequence, the peak of the primordial black hole spectrum might be expected to be at exponentially smaller values of the black hole mass than expected naively. See also [103, 124] .

The primordial black holes work assumes that the critical phenomena will be of type II. If one could fine-tune the gravitational collapse of stars made of realistic matter (i.e. not scalar fields) it seems likely that type I critical phenomena could be observed, i.e. there would be a universal mass gap. Critical collapse is not likely to be relevant in the real universe (at least at the present epoch) as there is no mechanism for fine-tuning of initial data.

5.5 Critical collapse in semiclassical gravity

As we have seen in the last section, critical phenomena may provide a natural route from everyday scale down to much smaller scales, perhaps down to the Planck scale. Various authors have investigated the relationship of Choptuik's critical phenomena to quantum black holes. It is widely believed that black holes should emit thermal quantum radiation, from considerations of quantum field theory on a fixed Schwarzschild background on the one hand, and from the purely classical three laws of black hole mechanics on the other (see [120] for a review). But there is no complete model of the back-reaction of the radiation on the black hole, which should be shrinking.

In particular, it is unknown what happens at the endpoint of evaporation, when full quantum gravity should become important. It is debated in particular if the information that has fallen into the black hole is eventually recovered in the evaporation process or lost.

To study these issues, various 2-dimensional toy models of gravity coupled to scalar field matter have been suggested which are more or less directly linked to a spherically symmetric 4-dimensional situation (see [66] for a review). In two spacetime dimensions, the quantum expectation value of the matter stress tensor can be determined from the trace anomaly alone, together with the reasonable requirement that the quantum stress tensor is conserved. Furthermore, quantizing the matter scalar field(s) $f$  but leaving the metric classical can be formally justified in the limit of many such matter fields. The two-dimensional gravity used is not the two-dimensional version of Einstein gravity but of a scalar-tensor theory of gravity. $e^{\phi }$  , where $\phi$  is called the dilaton, in the 2-dimensional toy model plays essentially the role of $r$  in four spacetime dimensions. There seems to be no preferred 2-dimensional toy model, with arbitrariness both in the quantum stress tensor and in the choice of the classical part of the model. In order to obtain a resemblance of spherical symmetry, a reflecting boundary condition is imposed at a timelike curve in the 2-dimensional spacetime. This plays the role of the curve $r=0$  in a 2-dimensional reduction of the spherically symmetric 4-dimensional theory.

How does one expect a model of semiclassical gravity to behave when the initial data are fine-tuned to the black hole threshold? First of all, until the fine-tuning is taken so far that curvatures on the Planck scale are reached during the time evolution, universality and scaling should persist, simply because the theory must approximate classical general relativity. 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. This ansatz would recursively solve a semiclassical field equation, where powers of $e^{\tau }$  (in coordinates $x$  and $\tau$  ) signal the appearances of quantum terms. Note that this is exactly the ansatz ( 48 ), but with the opposite sign in the exponent, so that the higher order terms now become negligible as $\tau \to -\infty$  , that is away from the singularity on large scales. On the Planck scale itself, this ansatz would not converge, and self-similarity would break down.

Addressing the question from the side of classical general relativity, Chiba and Siino [34] write down a 2-dimensional toy model, and add a quantum stress tensor that is determined by the trace anomaly and stress-energy conservation. They note that the quantum stress tensor diverges at $r=0$  . Ayal and Piran [4] make an ad-hoc modification to these semiclassical equations. They modify the quantum stress tensor by a function which interpolates between 1 at large $r$  , and $r^2/L_p^2$  at small $r$  . They justify this modification by pointing out that the resulting violation of energy conservation takes place only at the Planck scale. It takes place, however, not only where the solution varies dynamically on the Planck scale, but at all times in a Planck-sized world tube around the center $r=0$  , even before the solution itself reaches the Planck scale dynamically.

This introduces a nongeometric, background structure, effect at the world-line $r=0$  . With this modification, Ayal and Piran obtain results in agreement with our expectations set out above. For far supercritical initial data, black hole formation and subsequent evaporation are observed. With fine-tuning, as long as the solution stays away from the Planck scale, critical solution phenomena including the Choptuik universal solution and critical exponent are observed ^$\text{14}$  . In an intermediate regime, the quantum effects increase the critical value of the parameters $p$  . This is interpreted as the initial data partly evaporating while they are trying to form a black hole.

Researchers coming from the quantum field theory side seem to favor a model (the RST model) in which ad hoc “counter terms” have been added to make it soluble. The matter is a conformally rather than minimally coupled scalar field. The field equations are trivial up to an ODE for a timelike curve on which reflecting boundary conditions are imposed. The world line of this “moving mirror” is not clearly related to $r$  in a 4-dimensional spherically symmetric model, but seems to correspond to a finite $r$  rather than $r=0$  . This may explain why the problem of a diverging quantum stress tensor is not encountered. Strominger and Thorlacius [115] find a critical exponent of $1/2$  , but their 2-dimensional situation differs from the 4-dimensional one in many aspects. Classically (without quantum terms) any ingoing matter pulse, however weak, forms a black hole. With the quantum terms, matter must be thrown in sufficiently rapidly to counteract evaporation in order to form a black hole. The initial data to be fine-tuned are replaced by the infalling energy flux. There is a threshold value of the energy flux for black hole formation, which is known in closed form; recall this is a soluble system. The mass of the black hole is defined as the total energy it absorbs during its lifetime. This black hole mass is given by

$$\begin{array}{c}M\simeq {\left(\frac{\delta }{\alpha }\right)}^{\frac{1}{2}},\end{array}$$

(71)

where $\delta$  is the difference between the peak value of the flux and the threshold value, and $\alpha$  is the quadratic order coefficient in a Taylor expansion in advanced time of the flux around its peak.

There is universality with respect to different shapes of the infalling flux in the sense that only the zeroth and second order Taylor coefficients matter.

Peleg, Bose and Parker [107, 17] study the so-called CGHS 2-dimensional model. This (nonsoluble) model does allow for a study of critical phenomena with quantum effects turned off. Again, numerical work is limited to integrating an ODE for the mirror world line. Numerically, the authors find black hole mass scaling with a critical exponent of $\gamma \simeq 0.53$  . They find that the critical solution and the critical exponent are universal with respect to families of initial data.

Turning on quantum effects, the scaling persists to a point, but the curve of $lnM$  versus $ln(p-p_{*})$  then turns smoothly over to a horizontal line. Surprisingly, the value of the mass gap is not universal but depends on the family of initial data. While this is the most “satisfactory” result among those discussed here from the classical point of view, one should keep in mind that all these results are based on mere toy models of quantum gravity.

Rather than using a consistent model of semiclassical gravity, Brady and Ottewill [23] calculate the quantum stress-energy tensor of a conformally coupled scalar field on the fixed background of the perfect fluid CSS critical solution and treat it as an additional perturbation, on top of the perturbations of the fluid-GR system itself. In doing this, they neglect the coupling between fluid and quantum scalar perturbations through the metric perturbations. From dimensional analysis, the quantum perturbation has a Lyapunov exponent $\lambda =2$  . If this is larger than the positive Lyapunov exponent ${\lambda }_0$  , it will become the dominant perturbation for sufficiently good fine-tuning, and therefore sufficiently good fine-tuning will reveal a mass gap. For a spherical perfect fluid with equation of state $p=k\rho$  , one finds that ${\lambda }_0>2$  for $k>0.53$  , and vice versa. If ${\lambda }_0>2$  , the semiclassical approximation breaks down for sufficiently good fine-tuning, and this calculation remains inconclusive.

^$\text{14}$  The exponent is measured as $0.409$  , indicating a limited accuracy of the numerical method.

6 Conclusions

We conclude with a summary of what is known today and an outlook what still needs to be investigated and understood.

6.1 Summary

When one fine-tunes a smooth one-parameter family of smooth, asymptotically flat initial data to get close enough to the black hole threshold, the details of the initial data are completely forgotten in a small spacetime region where the curvature is high, and all near-critical time evolutions converge to one universal solution there. (This region is limited both in space and time, and at late times the final state is either a black hole or empty space.) At the black hole threshold, there either is a universal minimum black hole mass (type I transition), or black hole formation starts at infinitesimal mass (type II transition). In a type I transition, the universal critical solution is time-independent, or periodic in time, and the closer the initial data are to the black hole threshold, the longer it persists. In a type II transition, the universal critical solution is scale-invariant or scale-periodic, and the closer the initial data are to the black hole threshold, the smaller the black hole mass, by the famous formula ( 1 ).

Both types of behavior arise because there is a solution which is an intermediate attractor, or attractor of codimension one. Its basin of attraction is the black hole threshold itself, a hypersurface of codimension one that bisects phase space. Any time evolution that begins with initial data near the black hole threshold (but not necessarily close to the critical solution) first approaches the critical solution, then moves away from it along its one growing perturbation mode. At late times, the solution only remembers on which side of the black hole threshold the initial data were, and how far away from the threshold.

Our understanding of critical phenomena rests on this dynamical systems picture, but crucial details of the picture have not yet been defined rigorously. Nevertheless, it suggests semi-analytic perturbative calculations that have been successful in predicting the scaling of black hole mass and charge in critical collapse to high precision.

The importance of type II behavior lies in providing a natural route from large (the initial data) to arbitrarily small (the final black hole) scales, with possible applications to astrophysics and quantum gravity. Fine-tuning any one generic parameter in the initial data to the black hole threshold, for a number of matter models, without assuming any other symmetries, will do the trick.

Type II critical behavior also clarifies what version of cosmic censorship one can hope to prove.

At least in some matter models (scalar field, perfect fluid), fine-tuning any smooth one-parameter family of smooth, asymptotically flat initial data, without any symmetries, gives rise to a naked singularity. In this sense the set of initial data that form a naked singularity is codimension one in the full phase space of smooth asymptotically flat initial data for well-behaved matter. Any statement of cosmic censorship in the future can only exclude naked singularities arising from generic initial data.

Finally, critical phenomena are arguably the outstanding contribution of numerical relativity to knowledge in GR to date, and they continue to act as a motivation and a source of testbeds for numerical relativity.

6.2 Outlook

Clearly, more numerical work will be useful to further establish the generality of critical phenomena in gravitational collapse, or to find a counter-example instead. In particular, future research should include highly nonspherical situations, initial data with large angular momentum and/or electric charge, and matter models with a large number of internal degrees of freedom (for example, collisionless matter instead of a perfect fluid). Both going beyond spherical symmetry and including collisionless matter pose formidable numerical challenges.

The fundamental theoretical challenge is to explain why so many matter models admit a critical solution, that is, an attractor of codimension one at the black hole threshold. If the existence of a critical solution is really a generic feature, then there should be at least an intuitive argument, and perhaps a mathematical proof, for this important fact. On the other hand, the spherical Einstein-Vlasov system may already be providing a counter-example. A more thorough mathematical and numerical investigation of this system is therefore particularly urgent.

The critical spacetimes and their perturbations are well known only in the past light cone of the singularity. The Cauchy horizon and the naked singularity itself, as well as the possible continuations beyond the Cauchy horizon, of the critical spacetimes have not yet been investigated thoroughly. It is unknown if all possible continuations have a timelike naked singularity, and in what manner this singularity is avoided when one perturbs away from the black hole threshold.

An important mathematical challenge is to make the intuitive dynamical systems picture of critical collapse more rigorous, by providing a distance measure on the phase space, and a prescription for a flow on the phase space (equivalent to a prescription for the lapse and shift).

The latter problem is intimately related to the problem of finding good coordinate systems for the binary black hole problem.

On the phenomenological side, it is likely that the scope of critical collapse will be expanded to take into account new phenomena, such as multicritical solutions (with several growing perturbation modes), or critical solutions that are neither static, periodic, CSS or DSS. More complicated phase diagrams than the simple black hole-dispersion transition are already being examined, and the intersections of phase boundaries are of particular interest.

6.3 Thanks

A large number of people have contributed indirectly to this paper, but I would particularly like to thank Pat Brady, Matt Choptuik, David Garfinkle, José M. Martín-García, Alan Rendall and (last but not least) Bob Wald for stimulating discussions on many aspects of critical collapse.

References

1. Abrahams, A.M., and Evans, C.R., “Critical behavior and scaling in vacuum axisymmetric gravitational collapse”, Phys. Rev. Lett., 70(20), 2980–2983, (May, 1993).
2. Abrahams, A.M., and Evans, C.R., “Universality in axisymmetric vacuum collapse”, Phys. Rev. D, 49, 3998–4003, (1994).
3. Argyres, P.C., Barvinski, A.O., Frolov, V.P., Giddings, S.B., Lowe, D.A., Strominger, A., and Thorlacius, L., “Quantum Aspects of Gravity”, (December, 1994). URL (cited on 10 December 1999): . ☻ open access ✓
4. Ayal, S., and Piran, T., “Spherical collapse of a mass-less scalar field with semi-classical corrections”, Phys. Rev. D, 56, 4768–4774, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
5. Barenblatt, G.I., Similarity, Self-Similarity, and Intermediate Asymptotics, (Consultants Bureau, New York, U.S.A; London, U.K., 1979).
6. Barenblatt, G.I., Dimensional Analysis, (Gordon and Breach, New York, U.S.A., 1987).
7. Barenblatt, G.I., and Zel'dovich, Y.B., “Self-similar solutions as intermediate asymptotics”, Annu. Rev. Fluid Mech., 4, 285, (1972).
8. Bartnik, R., and McKinnon, J., “Particlelike Solutions of the Einstein–Yang–Mills Equations”, Phys. Rev. Lett., 61, 141–144, (1988).
9. Bicknell, G.V., and Henriksen, R.N., “Self-similar growth of primordial black holes, I. Stiff equation of state”, Astrophys. J., 219, 1043, (1978).
10. Bizoń, P., “Colored black holes”, Phys. Rev. Lett., 64, 2844–2847, (1990).
11. Bizoń, P., “How to Make a Tiny Black Hole?”, Acta Cosm., 22, 81, (1996). Related online version (cited on 10 December 1999): . ☻ open access ✓
12. Bizoń, P., and Chmaj, T., “First order phase transitions in gravitational collapse”, Acta Phys. Pol. B, 29, 1071, (1998). Related online version (cited on 10 December 1999): . ☻ open access ✓
13. Bizoń, P., and Chmaj, T., “Formation and critical collapse of Skyrmions”, Phys. Rev. D, 58, 041501–1–4, (1998). Related online version (cited on 10 December 1999): . ☻ open access ✓
14. Bizoń, P., and Chmaj, T., “Remark on formation of colored black holes via fine-tuning”, Phys. Rev. D, 61, 067501–1–2, (June, 2000). Related online version (cited on 10 December 1999): . ☻ open access ✓
15. Bizoń, P., Chmaj, T., and Tabor, Z., “Equivalence of critical collapse of non-Abelian fields”, Phys. Rev. D, 59, 104003–1–3, (1999). Related online version (cited on 10 December 1999): . ☻ open access ✓
16. Bogoyavlenskiĭ, O.I., “General relativistic self-similar solutions with a spherical shock wave”, Sov. Phys. JETP, 46, 633, (1978).
17. Bose, S., Parker, L., and Peleg, Y., “Predictability and semiclassical approximation at the onset of black hole formation”, Phys. Rev. D, 54, 7490–7505, (1996). Related online version (cited on 10 December 1999): . ☻ open access ✓
18. Brady, P.R., “Analytic example of critical behaviour in scalar field collapse”, Class. Quantum Grav., 11, 1255–1260, (1994). Related online version (cited on 10 December 1999): . arXiv preprint title: Does scalar field collapse produce `zero mass' black holes? ☻ open access ✓
19. Brady, P.R., “Self-similar scalar field collapse: naked singularities and critical behavior”, Phys. Rev. D, 51, 4168–4176, (1995). Related online version (cited on 10 December 1999): . ☻ open access ✓
20. Brady, P.R., and Cai, M.J., “Critical phenomena in gravitational collapse”, in Piran, T., ed., The Eighth Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Gravitation and Relativistic Field Theories, Proceedings of the meeting held at the Hebrew University of Jerusalem, June 22–27, 1997, 689–704, (World Scientific, Singapore, 1999). Related online version (cited on 10 December 1999): . ☻ open access ✓
21. Brady, P.R., Chambers, C.M., and Gonçalves, S.M.C.V., “Phases of massive scalar field collapse”, Phys. Rev. D, 56, 6057–6061, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
22. Brady, P.R., Gundlach, C., and Neilsen, D.W., unknown format. in preparation.
23. Brady, P.R., and Ottewill, A.C., “Quantum corrections to critical phenomena in gravitational collapse”, Phys. Rev. D, 58, 024006–1–6, (1998). Related online version (cited on 10 December 1999): . ☻ open access ✓
24. Bricmont, J., and Kupiainen, A., “Renormalizing Partial Differential Equations”, in Rivasseau, V., ed., Constructive Physics, Proceedings of the Conference held at École Polytechnique, Palaiseau, France, 25–27 July 1994, vol. 446 of Lecture Notes in Physics, 83–115, (Springer, Berlin, Germany; New York, U.S.A., 1995). Related online version (cited on 10 December 1999): . ☻ open access ✓
25. Brodbeck, O., Heusler, M., Straumann, N., and Volkov, M.S., “Rotating Solitons and Nonrotating, Nonstatic Black Holes”, Phys. Rev. Lett., 79, 4310–4313, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
26. Burko, L.M., “Comment on the Roberts Solution for the Spherically-Symmetric Einstein-Scalar Field Equations”, Gen. Relativ. Gravit., 29, 259–262, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
27. Cahill, M.E., and Taub, A.H., “Spherically symmetric similarity solutions of the Einstein equations for a perfect fluid”, Commun. Math. Phys., 21, 1–40, (1969). Related online version (cited on 18 May 2005): . ☻ open access ✓
28. Carr, B.J., and Coley, A.A., “Self-similarity in general relativity”, Class. Quantum Grav., 16, R31–R71, (1999). Related online version (cited on 10 December 1999): . ☻ open access ✓
29. Carr, B.J., Coley, A.A., Goliath, M., Nilsson, U.S., and Uggla, C., “Critical phenomena and a new class of self-similar spherically symmetric perfect-fluid solutions”, Phys. Rev. D, 61, 081502–1–5, (2000). Related online version (cited on 10 December 1999): . ☻ open access ✓
30. Carr, B.J., and Henriksen, R.N., “On the nature of the self-similar critical solution”, unknown status. in preparation.
31. Carter, B., and Henriksen, R.N., “A covariant characterization of kinematic self-similarity”, Ann. Phys. (Paris) Suppl., 14, 47, (1989).
32. Chen, L.-Y., and Goldenfeld, N., “Numerical renormalization group calculations for similarity solutions and traveling waves”, Phys. Rev. E, 51, 5577–5581, (1995).
33. Chen, L.-Y., Goldenfeld, N., and Oono, Y., “Renormalization-group theory for the modified porous-medium equation”, Phys. Rev. D, 44, 6544–6550, (1991).
34. Chiba, T., and Siino, M., “Disappearance of black hole criticality in semiclassical general relativity”, Mod. Phys. Lett. A, 12, 709, (1997).
35. Choptuik, M.W., personal communication.
36. Choptuik, M.W., “ `Critical' behavior in massless scalar field collapse”, in d'Inverno, R.A., ed., Approaches to Numerical Relativity, Proceedings of the International Workshop on Numerical Relativity, Southampton, December 1991, 202, (Cambridge University Press, Cambridge, U.K.; New York, U.S.A., 1992).
37. Choptuik, M.W., “Universality and scaling in gravitational collapse of a massless scalar field”, Phys. Rev. Lett., 70, 9–12, (1993).
38. Choptuik, M.W., “Critical behavior in scalar field collapse”, in Hobill, D., Burd, A., and Coley, A., eds., Deterministic Chaos in General Relativity, Proceedings of a NATO Advanced Research Workshop on Deterministic Chaos in General Relativity, held July 25–30, 1993, in Kananaskis, Alberta, Canada, 155, (Plenum Press, New York, U.S.A., 1994).
39. Choptuik, M.W., “The (Unstable) Threshold of Black Hole Formation”, in Dadhich, N., and Narlikar, J.V., eds., Gravitation and Relativity: At the Turn of the Millenium, Proceedings of the 15th International Conference on General Relativity and Gravitation (GR-15), held at IUCAA, Pune, India, December 16–21, 1997, 67–85, (IUCAA, Pune, India, 1998). Related online version (cited on 10 December 1999): . ☻ open access ✓
40. Choptuik, M.W., Chmaj, T., and Bizoń, P., “Critical Behavior in Gravitational Collapse of a Yang–Mills Field”, Phys. Rev. Lett., 77, 424–427, (1996). Related online version (cited on 10 December 1999): . ☻ open access ✓
41. Choptuik, M.W., Hirschmann, E.W., and Liebling, S.L., “Instability of an `Approximate black hole”', Phys. Rev. D, 55, 6014–6018, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
42. Choptuik, M.W., Hirschmann, E.W., and Marsa, R.L., “New critical behavior in Einstein–Yang–Mills collapse”, Phys. Rev. D, 60, 124011–1–9, (1999). URL (cited on 10 December 1999): . ☻ open access ✓
43. Christodoulou, D., “Violation of cosmic censorship in the gravitational collapse of a dust cloud”, Commun. Math. Phys., 93, 171–195, (1984). Related online version (cited on 18 May 2005): . ☻ open access ✓
44. Christodoulou, D., “The problem of a self-gravitating scalar field”, Commun. Math. Phys., 105(3), 337–361, (1986). Related online version (cited on 18 May 2005): . ☻ open access ✓
45. Christodoulou, D., “A mathematical theory of gravitational collapse”, Commun. Math. Phys., 109(4), 613–647, (1987).
46. Christodoulou, D., “The formation of black holes and singularities in spherically symmetric gravitational collapse”, Commun. Pure Appl. Math., 44, 339–373, (1991).
47. Christodoulou, D., “Bounded variation solutions of the spherically symmetric Einstein-scalar field equations”, Commun. Pure Appl. Math., 46, 1131–1220, (1993).
48. Christodoulou, D., “Examples of naked singularity formation in the gravitational collapse of a scalar field”, Ann. Math., 140, 607–653, (1994).
49. Christodoulou, D., “The instability of naked singularities in the gravitational collapse of a scalar field”, Ann. Math. (2), 149, 183–217, (1999).
50. Coley, A.A., “Kinematic self-similarity”, Class. Quantum Grav., 14, 87–118, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
51. Eardley, D.M., Hirschmann, E.W., and Horne, J.H., “ $S$  duality at the black hole threshold in gravitational collapse”, Phys. Rev. D, 52, 5397–5401, (1995). Related online version (cited on 10 December 1999): . ☻ open access ✓
52. Evans, C.R., personal communication.
53. Evans, C.R., and Coleman, J.S., “Critical phenomena and self-similarity in the gravitational collapse of radiation fluid”, Phys. Rev. Lett., 72, 1782–1785, (1994).
54. Foglizzo, T., and Henriksen, R.N., “General relativistic collapse of homothetic ideal gas spheres and planes”, Phys. Rev. D, 48, 4645, (1992).
55. Frolov, A.V., “Perturbations and critical behavior in the self-similar gravitational collapse of a massless scalar field”, Phys. Rev. D, 56, 6433–6438, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
56. Frolov, A.V., “Critical collapse beyond spherical symmetry: General perturbations of the Roberts solution”, Phys. Rev. D, 59, 104011–1–7, (1999). Related online version (cited on 10 December 1999): . ☻ open access ✓
57. Frolov, A.V., “Continuous self-similarity breaking in critical collapse”, Phys. Rev. D, 61, 084006–1–14, (2000). Related online version (cited on 10 December 1999): . ☻ open access ✓
58. Garfinkel, D., “Choptuik scaling in null coordinates”, Phys. Rev. D, 51(10), 5558–5561, (1995). Related online version (cited on 10 December 1999): . ☻ open access ✓
59. Garfinkel, D., “Choptuik scaling and the scale invariance of Einstein's equation”, Phys. Rev. D, 56, 3169–3173, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
60. Garfinkel, D., Cutler, C., and Duncan, G.C., “Choptuik scaling in six dimensions”, Phys. Rev. D, 60, 104007–1–5, (1999). Related online version (cited on 10 December 1999): . ☻ open access ✓
61. Garfinkel, D., and Duncan, G.C., “Scaling of curvature in subcritical gravitational collapse”, Phys. Rev. D, 58, 064024–1–4, (1998). Related online version (cited on 10 December 1999): . ☻ open access ✓
62. Garfinkel, D., and Gundlach, C., “Symmetry-seeking spacetime coordinates”, Class. Quantum Grav., 16, 4111–4123, (1999). Related online version (cited on 10 December 1999): . ☻ open access ✓
63. Garfinkel, D., Gundlach, C., and Martín-García, J.M., “Angular momentum near the black hole threshold in scalar field collapse”, Phys. Rev. D, 59, 104012–1–5, (1999). Related online version (cited on 10 December 1999): . ☻ open access ✓
64. Garfinkel, D., and Meyer, K., “Scale invariance and critical gravitational collapse”, Phys. Rev. D, 59, 064003–1–5, (1999). Related online version (cited on 10 December 1999): . ☻ open access ✓
65. Gibbons, G., personal communication.
66. Giddings, S.B., “Quantum mechanics of black holes”, (December, 1994). URL (cited on 10 December 1999): . ☻ open access ✓
67. Goldenfeld, N., Lectures on Phase Transitions and the Renormalization Group, vol. 85 of Frontiers in Physics, (Addison-Wesley, Reading, U.S.A., 1992).
68. Goldwirth, D.S., and Piran, T., “Gravitational collapse of massless scalar field and cosmic censorship”, Phys. Rev. D, 36(12), 3575–3581, (1987).
69. Goode, S.W., Coley, A.A., and Wainwright, J., “The isotropic singularity in cosmology”, Class. Quantum Grav., 9, 445–455, (1992).
70. Gundlach, C., “The Choptuik Spacetime as an Eigenvalue Problem”, Phys. Rev. Lett., 75, 3214–3217, (1995). Related online version (cited on 10 December 1999): . ☻ open access ✓
71. Gundlach, C., “Critical phenomena in gravitational collapse”, in Chruściel, P.T., ed., Mathematics of Gravitation, Part I: Lorentzian Geometry and Einstein Equations, Proceedings of the Workshop on Mathematical Aspects of Theories of Gravitation, held in Warsaw, Poland, February 29 March 30, 1996, vol. 41 of Banach Center Publications, 143–152, (Polish Academy of Sciences, Institute of Mathematics, Warsaw, Poland, 1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
72. Gundlach, C., “Echoing and scaling in Einstein–Yang–Mills critical collapse”, Phys. Rev. D, 55, 6002–6013, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
73. Gundlach, C., “Understanding critical collapse of a scalar field”, Phys. Rev. D, 55, 695–713, (1997).
74. Gundlach, C., “Angular momentum at the black hole threshold”, Phys. Rev. D, 57, 7080–7083, (1998). Related online version (cited on 10 December 1999): . ☻ open access ✓
75. Gundlach, C., “Critical phenomena in gravitational collapse”, Adv. Theor. Math. Phys., 2, 1–49, (1998). Related online version (cited on 10 December 1999): . ☻ open access ✓
76. Gundlach, C., “Nonspherical perturbations of critical collapse and cosmic censorship”, Phys. Rev. D, 57, 7075–7079, (1998). Related online version (cited on 10 December 1999): . ☻ open access ✓
77. Gundlach, C., “Critical gravitational collapse of a perfect fluid: Nonspherical perturbations”, Phys. Rev. D, 65, 084021–1–22, (2002). URL (cited on 10 December 1999): . ☻ open access ✓
78. Gundlach, C., and Martín-García, J.M., “Charge scaling and universality in critical collapse”, Phys. Rev. D, 54, 7353–7360, (1996). Related online version (cited on 10 December 1999): . ☻ open access ✓
79. Gundlach, C., Price, R.H., and Pullin, J., “Late-time behavior of stellar collapse and explosions. II. Nonlinear evolution”, Phys. Rev. D, 49, 890–899, (1994). Related online version (cited on 10 December 1999): . ☻ open access ✓
80. Hamadé, R.S., Horne, J.H., and Stewart, J.M., “Continuous self-similarity and $S$  -duality”, Class. Quantum Grav., 13, 2241–2253, (1996). Related online version (cited on 10 December 1999): . ☻ open access ✓
81. Hamadé, R.S., and Stewart, J.M., “The spherically symmetric collapse of a massless scalar field”, Class. Quantum Grav., 13, 497–512, (1996). Related online version (cited on 10 December 1999): . ☻ open access ✓
82. Hara, T., Koike, T., and Adachi, S., “Renormalization group and critical behavior in gravitational collapse”, (July, 1996). URL (cited on 10 December 1999): . ☻ open access ✓
83. Hirschmann, E.W., and Eardley, D.M., “Critical exponents and stability at the black hole threshold for a complex scalar field”, Phys. Rev. D, 52, 5850–5856, (1995). Related online version (cited on 10 December 1999): . ☻ open access ✓
84. Hirschmann, E.W., and Eardley, D.M., “Universal scaling and echoing in gravitational collapse of a complex scalar field”, Phys. Rev. D, 51, 4198–4207, (1995). Related online version (cited on 10 December 1999): . ☻ open access ✓
85. Hirschmann, E.W., and Eardley, D.M., “Criticality and bifurcation in the gravitational collapse of a self-coupled scalar field”, Phys. Rev. D, 56, 4696–4705, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
86. Hod, S., and Piran, T., “Critical behaviour and universality in gravitational collapse of a charged scalar field”, Phys. Rev. D, 55, 3485–3496, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
87. Hod, S., and Piran, T., “Fine-structure of Choptuik's mass-scaling relation”, Phys. Rev. D, 55, 440–442, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
88. Horne, J., personal communication.
89. Horne, J., “Critical behavior in black hole collapse”, Matters of Gravity, 7, 14, (1996). Related online version (cited on 10 December 1999): . ☻ open access ✓
90. Jhingan, S., Joshi, P.S., and Singh, T.P., “The final fate of spherical inhomogeneous dust collapse II: Initial data and causal structure of singularity”, Class. Quantum Grav., 13, 3057–3068, (1996). Related online version (cited on 10 December 1999): . ☻ open access ✓
91. Koike, T., Hara, T., and Adachi, S., “Critical Behavior in Gravitational Collapse of Radiation Fluid: A Renormalization Group (Linear Perturbation) Analysis”, Phys. Rev. Lett., 74, 5170–5173, (1995). Related online version (cited on 10 December 1999): . ☻ open access ✓
92. Koike, T., Hara, T., and Adachi, S., “Critical behavior in gravitational collapse of a perfect fluid”, Phys. Rev. D, 59, 104008–1–9, (1999).
93. Lake, K., and Zannias, T., “Naked singularities in self-similar gravitational collapse”, Phys. Rev. D, 41, 3866–3868, (1990).
94. Lavrelashvili, G., and Maison, D., “A remark on the instability of the Bartnik–McKinnon solutions”, Phys. Lett. B, 343, 214–217, (1995).
95. Liebling, S.L., “Multiply unstable black hole critical solutions”, Phys. Rev. D, 58, 084015–1–8, (1998). Related online version (cited on 10 December 1999): . ☻ open access ✓
96. Liebling, S.L., “Critical phenomena inside global monopoles”, Phys. Rev. D, 60, 061502–1–5, (1999). Related online version (cited on 10 December 1999): . ☻ open access ✓
97. Liebling, S.L., and Choptuik, M.W., “Black Hole Criticality in the Brans–Dicke Model”, Phys. Rev. Lett., 77, 1424–1427, (1996). Related online version (cited on 10 December 1999): . ☻ open access ✓
98. Maison, D., “Non-universality of critical behaviour in spherically symmetric gravitational collapse”, Phys. Lett. B, 366, 82–84, (1996). Related online version (cited on 10 December 1999): . ☻ open access ✓
99. Martín-García, J.M., and Gundlach, C., “All nonspherical perturbations of the Choptuik spacetime decay”, Phys. Rev. D, 59, 064031–1–19, (1999).
100. Neilsen, D.W., and Choptuik, M.W., “Critical phenomena in perfect fluids”, Class. Quantum Grav., 17, 761–782, (2000). Related online version (cited on 10 December 1999): . ☻ open access ✓
101. Neilsen, D.W., and Choptuik, M.W., “Ultrarelativistic fluid dynamics”, Class. Quantum Grav., 17, 733–759, (2000). Related online version (cited on 10 December 1999): . ☻ open access ✓
102. Niemeyer, J.C., and Jedamzik, K., “Near-Critical Gravitational Collapse and the Initial Mass Function of Primordial Black Holes”, Phys. Rev. Lett., 80, 5481–5484, (1998).
103. Niemeyer, J.C., and Jedamzik, K., “Dynamics of primordial black hole formation”, Phys. Rev. D, 59, 124013–1–8, (1999). Related online version (cited on 10 December 1999): . ☻ open access ✓
104. Ori, A., and Piran, T., “Naked singularities in self-similar spherical gravitational collapse”, Phys. Rev. Lett., 59, 2137–2140, (1987).
105. Ori, A., and Piran, T., “Naked singularities and other features of self-similar general-relativistic gravitational collapse”, Phys. Rev. D, 42, 1068–1090, (1990).
106. Oshiro, Y., Nakamura, K., and Tomimatsu, A., “Critical behavior of black hole formation in a scalar wave Collapse”, Prog. Theor. Phys., 91, 1265–1270, (1994). Related online version (cited on 10 December 1999): . ☻ open access ✓
107. Peleg, Y., Bose, S., and Parker, L., “Choptuik scaling and quantum effects in 2D dilaton gravity”, Phys. Rev. D, 55, 4525–4528, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
108. Price, R.H., and Pullin, J., “Analytic approximations to the spacetime of a critical gravitational collapse”, Phys. Rev. D, 54, 3792–3799, (1996). Related online version (cited on 10 December 1999): . ☻ open access ✓
109. Pullin, J., “Is there a connection between no-hair behavior and universality in gravitational collapse?”, Phys. Lett. A, 204, 7–10, (1995). Related online version (cited on 10 December 1999): . ☻ open access ✓
110. Rein, G., Rendall, A.D., and Schaeffer, J., “Critical collapse of collisionless matter: A numerical investigation”, Phys. Rev. D, 58, 044007–1–8, (1998). Related online version (cited on 10 December 1999): . ☻ open access ✓
111. Rendall, A.D., personal communication.
112. Roberts, M.D., “Scalar field counterexamples to the cosmic censorship hypothesis”, Gen. Relativ. Gravit., 21, 907–939, (1989).
113. Seidel, E., and Suen, W.-M., “Oscillating soliton stars”, Phys. Rev. Lett., 66, 1659–1662, (1991).
114. Straumann, N., and Zhou, Z.-H., “Instability of a colored black hole solution”, Phys. Lett. B, 243, 33–35, (1990).
115. Strominger, A., and Thorlacius, L., “Universality and scaling at the onset of quantum black hole formation”, Phys. Rev. Lett., 72, 1584–1587, (1994). Related online version (cited on 10 December 1999): . ☻ open access ✓
116. Tod, P., personal communication.
117. Volkov, M.S., Brodbeck, O., Lavrelashvili, G., and Straumann, N., “The number of sphaleron instabilities of the Bartnik–McKinnon solitons and”, Phys. Lett. B, 349, 438–442, (1995). Related online version (cited on 10 December 1999): . ☻ open access ✓
118. Volkov, M.S., and Gal'tsov, D.V., “Non-Abelian Einstein–Yang–Mills Black Holes”, J. Exp. Theor. Phys. Lett., 50, 346–350, (1989).
119. Wald, R.M., “Gravitational Collapse and Cosmic Censorship”, (1997). URL (cited on 10 December 1999): . ☻ open access ✓
120. Wald, R.M., “Black Holes and Thermodynamics”, in Wald, R.M., ed., Black Holes and Relativistic Stars, Proceedings of the Symposium dedicated to the memory of Subrahmanyan Chandrasekhar, held in Chicago, December 14–15, 1996, 155–176, (University of Chicago Press, Chicago, U.S.A.; London, U.K., 1998). Related online version (cited on 10 December 1999): . ☻ open access ✓
121. Wang, A., and de Oliveira, H.P., “Critical phenomena of collapsing massless scalar wave packets”, Phys. Rev. D, 56, 753–761, (1997). Related online version (cited on 10 December 1999): . ☻ open access ✓
122. Wilson, K.G., “Problems in physics with many scales of length”, Sci. Am., 241, 140–157, (August, 1979).
123. Yeomans, J.M., Statistical Mechanics of Phase Transitions, Oxford Science Publications, (Clarendon Press; Oxford University Press, Oxford, U.K.; New York, U.S.A., 1992).
124. Yokoyama, J., “Cosmological constraints on primordial black holes produced in the near-critical gravitational collapse”, Phys. Rev. D, 58, 107502, (1998). Related online version (cited on 10 December 1999): . ☻ open access ✓

Note: The reference version of this article is published by Living Reviews in Relativity