Scalar field critical collapse in 2+1 dimensions
Abstract
We carry out numerical experiments in the critical collapse of a spherically symmetric massless scalar field in 2+1 spacetime dimensions in the presence of a negative cosmological constant and compare them against a new theoretical model. We approximate the true critical solution as the Garfinkle solution, matched at the lightcone to a Vaidyalike solution, and corrected to leading order for the effect of . This approximation is only at the lightcone and has three growing modes. We conjecture that pointwise it is a good approximation to a yet unknown true critical solution that is analytic with only one growing mode (itself approximated by the top mode of our amended Garfinkle solution). With this conjecture, we predict a Ricciscaling exponent of and a massscaling exponent of , compatible with our numerical experiments.
Contents
 I Introduction
 II Numerical results

III Theory
 III.1 The Garfinkle solution
 III.2 Continuation beyond the lightcone
 III.3 The similarity coordinate
 III.4 Boundary conditions and gauge conditions
 III.5 Perturbations of the Garfinkle solution
 III.6 Perturbations of the null continuation
 III.7 Matching of perturbations on the lightcone
 III.8 corrections of the Garfinkle solution
 III.9 corrections of the null continuation
 III.10 Derivation of
 III.11 Derivation of
 III.12 An exact continuation of the Garfinkle solution beyond the lightcone with
 III.13 Construction of initial data for the amended Garfinkle solution
 IV Conclusions
I Introduction
i.1 Critical collapse
Starting with Choptuik’s investigation of scalar field collapse Choptuik1993 , and since then generalised to many other systems GundlachLRR , critical collapse is concerned with the threshold of black hole formation in the space of initial data. A practical way of investigating this threshold is to pick any oneparameter family of asymptotically flat initial data, with parameter , such that for the data form a black hole, and for they do not.
More specifically, “type II” critical collapse is concerned with the case where the black hole mass can be made arbitrarily small at the threshold. A necessary condition for this to happen is that the system of Einstein equations and matter evolution equations is scaleinvariant, or effectively scaleinvariant on sufficiently small length scales. As far as we know, exact scaleinvariance is also sufficient for the existence of type II critical collapse.
In type II critical collapse in spacetime dimensions, for (“subcritical” data), the maximum value of curvature (say the Ricci scalar) achieved on the spacetime scales as
(1) 
and for (“supercritical” data), the black hole mass scales as
(2) 
where in
(3) 
The relation (3) follows essentially from dimensional analysis, with the dimension (in gravitational units ) of mass (or energy). The exponent depends on the type of matter and spacetime dimension, but is universal for all 1parameter families of initial data.
In a small spacetime region just before the point of maximum curvature, or just before the formation of an apparent horizon, the spacetime and matter field are approximated by a “critical solution” which is again universal for a given system and spacetime dimension. The critical solution has three defining properties: it is regular, scaleinvariant (continuously selfsimilar, CSS) or scaleperiodic (discretely selfsimilar, DSS), and it has precisely one unstable mode. Continuous selfsimilarity means that there is a conformal Killing vector field such that . In coordinates adapted to CSS and spherical symmetry (but otherwise general), such that , this means that the metric takes the form
(4)  
where is an arbitrary length scale. This functional form of the metric is invariant under gauge transformations of the form
(5) 
[In DSS, in adapted coordinates, the metric takes the same form, with , , , (and , ) now depending periodically on with some scaleechoing period .]
The most general ansatz for a massless scalar field that is compatible, via the Einstein equations with , with continuous selfsimilarity of the metric is the Christodoulou ansatz Christodoulou
(6) 
for some constant . [For DSS, depends also on with period .] The constant does not depend on the choice of similarity coordinates. The spherical scalar field critical solution in higher dimensions is DSS with but, as we shall see later, in 2+1 dimensions it seems to be CSS with .
In a spherically symmetric critical solution, the regular centre corresponds to one value of . (for all ) represents a single spacetime point at the centre, the accumulation point, where the curvature blows up. Another value of corresponds to the past lightcone (or soundcone, for fluid matter) of the accumulation point, where the critical solution must also be regular. The critical solution can be continued in to the future lightcone of the accumulation point. Beyond the future lightcone, there is no unique continuation, but that part of the critical solution is not relevant for critical collapse.
If we choose to be timelike or null, we can interpret it both as a time coordinate on spacetime and as the logarithm of scale in renormalisation group theory. From selfsimilarity and the existence of precisely one unstable mode, using a little dynamical systems theory and dimensional analysis, one can then derive both universality and the above scaling relations. turns out to be the inverse Lyapunov exponent of the one unstable mode.
This scaling argument KoikeHaraAdachi1995 ; GundlachLRR goes roughly as follows: the closer to , the smaller the initial value of the one growing mode, the longer (larger ) the spacetime stays close to the critical solution. But larger also means scalar field variation on smaller length scales, and hence larger curvature, before the solution either starts dispersing or forms an apparent horizon.
For a spherically symmetric massless scalar field in the presence of a negative cosmological constant, critical collapse has been investigated in 3+1 dimensions BizonRostworowski2011 . In higher dimensions, critical collapse has been investigated in Birukouetal for , and in JalmuznaRostworowskiBizon2011 for . A cosmological constant (of either sign) obviously breaks scaleinvariance, but one would expect it to become negligible in regions of sufficiently large curvature, and hence in the regime where type II critical phenomena are seen. Indeed this seems to be the case in 3+1 and higher dimensions. A further effect of a negative cosmological constant is to replace asymptotic flatness with asymptotically antideSitter (adS) boundary conditions. The only boundary conditions for a massless scalar field compatible with the Einstein equations are totally reflecting. As a consequence, it appears that arbitrarily weak generic initial data collapse after sufficiently many reflections off the boundary. (But see MaliborskiRostworowski2013 for exceptions to this). However, at the thresholds , , for black hole formation after zero, one, two, and so on, reflections the same type II critical phenomena are seen as in asymptotically flat spacetime. Because of the reflecting boundaries, all the mass must fall into the black hole eventually, but the mass of the apparent horizon when it first forms does scale, with the same as the black hole mass in asymptotically flat spacetime.
i.2 2+1 dimensions
The situation is quite different in 2+1 dimensions. First, this is the critical dimension for the wave equation, meaning that the scalar field energy ( integrated over space dimensions) is dimensionless. Similarly, for gravity the black hole mass and the 2+1 dimensional equivalent of the Hawking mass are dimensionless. This already indicates that any mass scaling cannot be derived using the standard dimensional analysis argument. Secondly, in the absence of a cosmological constant there are no black hole solutions, and finite mass regular initial data cannot form an apparent horizon dynamically.
Standard gauge choices in spherical symmetry in spacetime dimensions are polarradial coordinates ,
(7) 
where the area radius is a coordinate, and double null coordinates ,
(8) 
where is a metric coefficient. With and , this can also be written as
(9) 
[In dimensions, the same coordinate choices exist, with replaced by the line element on the unit sphere.]
In 2+1 dimensions, the field equations
(10) 
for the metric (8) are
(11)  
(12)  
(13)  
(14)  
(15) 
These are the field equations that we will use in the theory Section III below.
In 2+1 dimensions, if , then from (13) . In a region containing a regular centre, one can then make the same gauge choice as in flat spacetime. But, always in 2+1 dimensions, the coefficients of the spherical wave equation (11) depend only on , not on , and so the matter evolution equation is not modified by curvature. This is one intuitive way of seeing why gravitational collapse cannot occur in 2+1 with .
However, in the presence of a negative cosmological constant black holes do exist in 2+1 spacetime dimension, and can be formed from regular data. These black holes are the BTZ solutions, which in polarradial coordinates are given by
(16) 
Although this looks similar to the SchwarzschildadS solution in higher dimensions, it is locally flat. This is because in 2+1 dimensions, the Ricci tensor determines the Weyl tensor, and so a vacuum region is not only Ricciflat but flat. The BTZ solution with is the dimensionsonal adS spacetime. All other BTZ solutions with have a naked conical singularity, while the BTZ solutions with are black hole solutions. This mass gap between the ground state and the smallest black hole is another feature of 2+1 dimensions. Regular initial data with cannot form a black hole (although they can develop arbitrarily large curvature BizonJalmuzna2013 .)
There seems to be a dilemma for type II critical collapse: in order to form a black hole at all, a cosmological constant is needed, but for curvature and mass scaling to occur, it must be dynamically negligible.
It is convenient to introduce the local mass function defined by
(17) 
This is the 2+1 dimensional equivalent of the Hawking mass for spherical symmetry in 3+1 dimensions, and has similar properties: it is constant in vacuum, while in the presence of matter it increases with on any spacelike surface in regions where . A spherically symmetric marginally outertrapped surface (MOTS) is given by , and so its mass is given by , as is the mass of the BTZ horizon.
i.3 Previous work
The first numerical simulations of critical collapse of a spherically symmmetric scalar field in 2+1 dimensions with a negative cosmological constant were carried out by Pretorius and Choptuik PretoriusChoptuik and Husain and Olivier HusainOlivier .
In order to avoid the complications associated with the reflecting boundary conditions, Pretorius and Choptuik, like others in 3+1 and higher dimensions after them, focused on the scaling of maximum curvature and the mass of the apparent horizon when it first appears. They found that for each of several oneparameter families of initial data they examined, there was a such that the maximum of the Ricci curvature scaled as (1) where . They also gave evidence for a universal CSS critical solution. They claimed also that the apparent horizon mass at first appearance scales as
(18) 
with , although their Figs. 4 and 5 correctly suggest a mass scaling exponent somewhere between 0 and 1. [We use the terminology FMOTS for for “first marginally outer trapped surface”, as the terminology “apparent horizon mass” is ambiguous in this context; see Sec. II.2.2 below.] Their theoretical argument for is that the dimensionless mass and area radius of an apparent horizon are related by , and should scale as suggested by its dimension. We shall correct this argument in Sec. III.11. Husain and Olivier found apparent horizon mass scaling with , consistent with our results, but their data are fairly far from criticality.
On the grounds that should be dynamically negligible in critical collapse, Garfinkle Garfinkle looked for exactly CSS solutions for that are analytic between the two values of corresponding to the centre and to the past lightcone of the accumulation point (the standard procedure in higher dimensions). As we shall review in Sec. III.1, he found a family of these parameterised by . The solution is the FriedmannRobertsonWalker solution. In hindsight it is surprising that these solutions exist, as we have seen that with gravity does not affect the scalar field and so cannot regularise it, something that is essential for the existence of regular CSS solutions in higher dimensions. The Garfinkle solution is also in closed form, whereas critical solutions for spherical massless scalar field collapse in higher dimensions can only be constructed numerically (but see ReitererTrubowitz for an existence proof of the Choptuik critical solution in 3+1 dimensions).
Garfinkle Garfinkle noted that the solution showed good agreement with the numerical data of Pretorius and Choptuik inside the lightcone. However, the lightcone is also an apparent horizon, whereas the critical solution in higher dimensions has no trapped surfaces. Furthermore, the analytic continuation of the Garfinkle solution through the lightcone has a spacelike central curvature singularity, for all . This means that it is the CSS equivalent of a black hole, rather than a critical solution. (We will fix these problems in Secs. III.2, III.8, III.9 and III.12 below.)
Ignoring these obvious problems of the Garfinkle solution, Garfinkle and Gundlach GarfinkleGundlach computed its perturbation spectrum, by making the standard requirement that perturbations be analytic at both the centre and lightcone. As we shall review in Sec. III.5, they found that the Garfinkle solution with parameter has unstable modes. This then raised the problem that the Garfinkle solution does not fit the numerical data, while the Garfinkle solution, which does, has three growing modes. We have no theoretical solution for this problem, but we will show numerically in Sec. II.3 that our modified Garfinkle solution appears to have only one growing mode when evolved with .
Ii Numerical results
ii.1 Numerical method
We experimented with a time evolution code using polarradial coordinates, the standard coordinate choice for critical collapse in higher dimensions. However, as we want to continue the evolution after the time slicing crosses the apparent horizon, we have changed over to the numerical method of Pretorius and Chopuik PretoriusChoptuik .
The metric ansatz is essentially (9), but reparameterised as
(19) 
so that
(20) 
This brings the timelike infinity of asymptotically antide Sitter spacetimes to and the centre to . Note that the adS spacetime is given by . We refer the reader to PretoriusChoptuik for the field equations in these coordinates.
The metric effectively represents the metric in doublenull coordinates and (which go through apparent or event horizons), but the numerical algorithm evolves it on a grid in and , timestepping in .
Both and obey wave equations and are evolved from initial data at . The residual gauge freedom is and . We fix this in part by setting the initial data at . With and also set freely, the initial data for and are then determined by the Hamiltonian and momentum constraints. During the evolution, we impose the gauge fixing boundary conditions at the adS timelike infinity , and the regularity boundary conditions and at the centre .
The Hamiltonian and momentum constraints become singular on any time slice that contains a trapped surface, but we solve them only on the initial slice. The evolution equations remain regular at a trapped surface.
We choose units such that and , so that represents the regular centre and the adS boundary. We choose , and typically equally spaced grid points in .
For a geometric analysis of the results, and in particular for looking for selfsimilarity near the centre, more geometric coordinates fixed at the centre are helpful. This will be discussed in Sec. II.2.4 below.
ii.2 Evolution of finetuned generic initial data
In scalar field critical collapse in 3+1 and higher dimensions there is a clear distinction between two outcomes. Either the scalar field forms a black hole, and the remaining scalar field escapes to infinity, or the scalar field disperses, leaving behind flat spacetime. With a negative cosmological constant, there are the twin complications that a scalar wave that disperses initially can collapse after one or more reflections at the outer boundary, and that more scalar field can fall into an initially small black hole after reflection. However, locally in space and time there are still two distinct outcomes, at least as long as the initial data are on scales much smaller than the scale set by . From now on, the previously arbitray length scale in (4) is set by the cosmological constant for definiteness. Hence , from (4), indicates spacetime scales smaller than .
In 2+1 dimensions, the situation appears initially more confusing. The Ricci scalar at the centre either blows up while increasing monotonically, or it goes through one or more extrema before blowing up a short time later. Similarly, the mass of the first MOTS appearing anywhere on a time slice (what PretoriusChoptuik call the apparent horizon mass) behaves in a nonmonotonic way with .
We adopt the working definition of that for , monotically increases and blows up at finite , while for it goes through at least one maximum and minimum before blowup. We shall see that with this definition, controls all scaling phenomena. This is in itself an important observation, as it strongly indicates that the scaling is controlled by a single growing mode of a selfsimilar critical solution.
For the scalar field initial data we choose approximately ingoing (that is ) Gaussian or kink profiles located at with width . Their amplitude is a free parameter used for finetuning the initial data to the black hole threshold. Note that both chosen families of initial data are the same as considered in PretoriusChoptuik , which allows us to compare results. We find that for both these two families. All plots and numbers presented in the current Subsection II.2 use the Gaussian family, but we have checked that we obtain the same results for the kink data.
The absolute value of for any given oneparamter family is irrelevant and depends on the parameterisation. However, with of order one, is a meaningful measure of the amount of finetuning. For simplicity, we use the terminology “sub10” for initial data with and “super10” for . The best finetuning we have achieved is of the order of .
ii.2.1 Ricci scaling at the centre
As stated above, we define so that for , monotically increases until blowup, while for there is at least one maximum and minimum before blowup. For subcritical data further away from criticality than approximately sub15, the Ricci scalar at the centre goes through a second maximum and minimum before blowup. Fig. 1 illustrates this for representative values of . Going further away from criticality, the second minimum and eventual blowup moves to larger values of . For about sub8, the blowup moves to a time that indicates one reflection from the outer boundary; see Sec. II.2.3 below. Decreasing the amplitude further, below about sub7 we obtain initial data with mass below the threshold for black hole formation and these data cannot form a black hole. (While we therefore cannot observe mass scaling for these data, we still observe Ricci scaling.)
The scaling of the maxima and minima of the value of the Ricci scalar at the centre is shown in Fig. 2. For subcritical data, the first local maximum of scales as in (1) with , the same value, to within our numerical precision, as found by PretoriusChoptuik . We determine to high precision by fitting to the Ricci scaling law (1). The critical value defined in this way is consistent with the definition we have given before, but can be determined more accurately in practice.
The first minimum also scales, with . Further away from criticality than approximately sub10, the first minimum reaches a floor set by the cosmological constant, . Extrapolating beyond the limit of our finetuning, the scaling of the first maximum and first minimum would suggest that they merge at sub38. However, this extrapolation is probably incorrect, as by definition we would expect them to merge precisely at .
The value of the second maximum scales with , similar to the first maximum, and the second minimum with . At approximately sub15 its value agrees with the second maximum, and at this point the second maximum and minimum merge and disappear. The second miminum reaches the same floor as the first maximum, but only at sub2 and then at large , which is out of the range of critical phenomena at first implosion.
Fig. 3 shows the scaling of the locations, in proper time at the centre , of the first minimum, second maximum, and second minimum, all with respect to the first maximum, as well as the location of the first maximum with respect to the accumulation point . The scaling exponents are 1.2(2), 1.12(7), 1.2(8) and 1.4(3) respectively, see Fig. 3. (The reason that we do not use the accumulation point as our primary reference point is that its location is obtained by curvefitting, and is therefore less accurate than the location of the extrema with respect to each other.)
Checking pointwise convergence in of our time evolutions is difficult in the critical regime because of the sensitive dependence on initial data. At best we can compare scalar quantities such as at finetuning “sub” for the same at different numerical resolutions. (Note that itself is resolutiondependent). This works for and , but not for and . However, physical results such as Ricci and mass scaling should converge with resolution. In Fig. 4 we demonstrate that the first maximum of the Ricci scalar as a function of converges with resolution to better than fourth order from sub3 to sub22.
ii.2.2 Apparent horizon mass scaling
The scaling argument GundlachLRR only determines the size and hence mass of the black hole when it first forms, in a regime where the transition from the critical solution to black hole formation is still universal up to an overall scale. However, in asymptotically flat spacetimes and for massless scalar field matter, little additional mass falls inlater (when the scaling argument no longer holds), so one effectively has a scaling law for the asymptotic black hole mass. (In a cosmological context, there may be significant infall Hawke .) In 2+1 dimensions, the cosmological constant can never be neglected where collapse takes place, and so the local scaling argument breaks down already. Furthermore, for in any dimension all the mass eventually falls into the black hole because of reflecting boundary conditions. Therefore Pretorius and Choptuik focused on the mass at the first appearance (with respect to a given time slicing) of a marginally outer trapped surface (MOTS), which they call the apparent horizon mass. To explain the phenomenology we observe, we need to use a more explicit terminology, as follows.
We assume spherical symmetry. We shall use the term MOTS to denote any point where . We shall call the union of all MOTS the apparent horizon (AH), parameterised in coordinates as a curve . It bounds the region of outertrapped spherically symmetric surfaces (circles in 2+1). It is easy to see using the field equations that the AH is spacelike for (meaning that energy crosses the horizon) and outgoing null for . What Pretorius and Choptuik denoted by apparent horizon mass is the mass of the first appearance of a MOTS for a given time slicing, that is the absolute minimum of the AH curve with respect to the time coordinate . For clarity, we shall call this the first MOTS (FMOTS).
For the ingoing Gaussian data, the plot of shows powerlaw scaling down to a very small value of at , but then jumps to a larger value and varies only slowly with . This is shown in the upper plot of Fig. 6. This apparent jump is explained simply by the AH curve having two local minima for the range , which includes , see the lower plot. We shall refer to such a local minimum of as an earliest MOTS (EMOTS). It is helpful to consider the tracks of both EMOTS in the plane (Fig. 5), together with a plot of their masses against (Fig. 6).
At some very large value of (compare this to ) there is only one EMOTS, and it is located on the initial slice at some large and . As is decreased from , the EMOTS moves to smaller (on track that is approximately null) and smaller and hence . At (approximately sub19) the single EMOTS splits into two. To the limit of our finetuning of the initial data, the inner EMOTS approaches zero and as .
For , there is no inner EMOTS, and the outer EMOTS, whose mass does not scale, moves to larger and with decreasing on an approximately null track, until at sub10 it approaches the outer boundary. Presumably it will then move back in, but we have not followed this further. For the outer EMOTS appears first, so if one looks only for the first appearance of a MOTS, for any , its mass appears to jump at from the mass of the inner EMOTS to that of the outer EMOTS.
As far as our finetuning reaches, the mass of the inner EMOTS scales as (18) with , see Fig. 7. This value is roughly similar to the value of HusainOlivier (but different from the of PretoriusChoptuik ).
As is the same for both Ricci and mass scaling, to within our accuracy of finetuning (sub26 and super26), the exponents and must also be related. As we do not have the exact critical solution, we cannot give a complete derivation of this relation, but a tentative derivation of and based on an amended Garfinkle solution and approximate single growing mode is given below in Secs. III.10 and III.11.
ii.2.3 Second criticality
Decreasing further, we again find critical phenomena after reflection at the outer boundary. This means that we finetune the amplitude such that the initially ingoing Gaussian reflects off the center, moves towards the outer boundary, reflects off it and collapses while approaching the center for the second time. The bisection is again based on the behavior of the Ricci scalar at the centre. We find that there is a second critical amplitude such that the maxima and minima of the Ricci scalar for subcritical evolutions scale according to (1). This is demonstrated in Fig. 9. (We use to denote the critical amplitude after reflections, with our original .)
Note that is itself only sub8 with respect to , so that scaling maxima and minima are covered up by the initially ingoing part of the initial data. We were also unable to finetune as accurately as for the first criticality. At about sub20 relative to , the Ricci scalar still has maxima and minima, but their values fail to scale. We believe this is due to loss of numerical accuracy.
The accumulation point for immediate critical collapse, for the Gaussian initial data, was located at in coordinate time and in proper time at the centre. For critical phenomena after one reflection, for the same family of initial data, the corresponding values are and . Note that the two accumulation points are separated by , consistent with the intuitive picture of reflection at the outer boundary, but that they are separated in proper time only by . This is due to the fact that and jump down across the future lightcone of the first accumulation point to and then remain small. Hence after first nearcriticality, and correspond to much smaller physical scales than before, but by definition is still the outer boundary and the lightcrossing time is therefore still . See also Fig. 8 for an illustration of this memory effect in the sub10 evolution.
As a consequence of this separation of scales, the wave going back out in (first) nearsubcritical evolutions comes back in what is a very short time at the centre and interacts with the aftermath of first criticality. First and second criticality therefore overlap in time, and this may explain why they are also close in , in the sense that the scaling regimes overlap.
Furthermore, if we we compare the constant factors in front of the two Ricci scaling laws and , we find that . This may also be a consequence of the jump down in and .
For secondsupercritical data we also looked for evidence of mass scaling. The supercritical data with respect to can be also supercritical with respect to , and therefore to see second mass scaling one has to look at the proper range of amplitudes. We find some evidence that for second supercritical data the mass of an apparent horizon roughly behaves according to (2), but with a critical exponent , significally different from the found in first criticality. The evidence is presented in Fig. 10. We have no theoretical explanation of the discrepancy in the mass scaling exponent, but as the scaling appears to be very noisy anyway, the discrepancy may be just numerical error due to loss of resolution.
ii.2.4 Selfsimilarity inside the lightcone
We now examine the claim that a CSS critical solution is observed PretoriusChoptuik , and that inside the lightcone it agrees with the Garfinkle solution Garfinkle .
Recall that we denote by the proper time at the centre, starting with at . In a halfdiamond bounded on the left by , we can rescale and to new double null coordinates , so that both correspond to on the central worldline , which by ansatz is at and so is also at , and where denotes the accumulation point in central proper time. A plot of the contour lines of and in a nearcritical evolution, see Fig. 8 for sub10, shows that our numerical algorithm provides some automatic zooming in, which means we can resolve selfsimilarity over many foldings in scale without mesh refinement – to optimally resolve selfsimilarity, these lines should be equally spaced.
The first task is to find the accumulation point. With the scalar field at the centre in the Garfinkle solution given by , we make a linear fit
(21) 
for and . We can then compute
(22) 
from and . To see CSS, this needs to be done separately for each , but and depend only weakly on and have a limit as . We have fitted by the quadratic function . For subcritical evolutions of our Gaussian initial data, a least squares fit gives , , for the fitting interval . This range of is equivalent to . Hence we can strongly rule out any other than 4.
In the following, we denote by the value of in the preferred doublenull coordinates . It is given in terms of the numerically evolved metric coefficient as
(23) 
Following GarfinkleGundlach , we then define similarity coordinates by
(24) 
for a positive integer. Hence the regular centre is given by and the lightcone by . We also define
(25)  
(26) 
where is a familydependent, dynamically irrelevant constant. The solution is then CSS if and only if , , , and are functions of only. These functions for the countable family of Garfinkle solution are reviewed in Sec. III.1.
Finally, we define
(27) 
where is the affine parameter along outgoing null geodesics, measured away from the centre, and normalised so that the inner product of with the 4velocity of the central observer is . With the centre at , this gives
(28) 
which we integrate along each line of constant . In particular,
(29) 
The rescaled
(30) 
is a function of only in CSS, and having tested this, we will later use it as the similarity coordinate in place of .
Fig. 11 shows a comparison of the mass function of the Garfinkle solution with against of the sub25 evolution. There is good agreement everywhere between the regular centre and the lightcone (), over the range , which means that the solution is CSS inside the lightcone over 8 foldings of scale, all of which are much smaller than the scale set by the cosmological constant. Fig. 12 shows a similar comparison of , against , where the constant depends on the family of initial data (but not on ) and has been determined by fitting. Figs. 13 and 14 show the corresponding tests for and .
Even though there is good numerical evidence that the critical solution inside the lightcone is the Garfinkle solution (up to small corrections in powers of ), we keep generic in the following for clarity of presentation.
ii.2.5 Outside the lightcone
Garfinkle Garfinkle compared his exact solution with the numerical evolutions of PretoriusChoptuik only inside the lightcone. Here we will go significantly beyond the lightcone. We will see that the analytic continuation of the Garfinkle solution is definitely ruled out, but that a different, , continuation proposed in Sec. III.2 below, and which we call the null continuation, appears to be at least a rough approximation to the true critical solution.
The best choice of data for this comparison would appear to be an evolution with the best available finetuning, as there we expect to see the critical solution most clearly. However, in nearcritical evolutions, even subcritical ones, the evolution ends in a central singularity very soon after the accumulation point of the CSS regime. This is different from critical collapse in 3+1 and higher dimensions, where subcritical evolutions go to essentially vacuum after the CSS regime (in the case , at least until the next reflection at the outer boundary). Hence we also consider the sub10 evolution, which corresponds to the closest we can get to critical initial data while still having a significant evolution in after the accumulation point of the CSS region. In sub10, we have access to large positive values of , but because and are very negative in this regime, this does not correspond to large values of the proper retarded time or area radius , and so we are not far away in this sense from the accumulation point. See again Fig. 8 in this context.
Recall that is normalised to proper time at the regular centre, so it is not defined outside the past of blowup at the centre. Moreover, even in subcritical evolutions, where blowup occurs significantly after the accumulation point, spacetime at the centre after the accumulation point is not expected to be selfsimilar. Hence we cannot use the similarity coordinate based on and outside the lightcone of the critical solution. We use instead. It is given in terms of for both the Garfinkle solution and its null continuation in Sec.III.3 below.
Fig. 15 shows contour lines of , and in the plane for the sub25 evolution with singularity excision. Near the center the contour lines of and are approximately parallel, as one would expect in a CSS spacetime. Near the lightcone, they are not even approximately parallel, and the contour line is not particularly close to the past lightcone of the accumulation point. (The contour line is precisely the past lightcone of the accumulation point by definition.) This disagreement is already visible in Fig. 14, but appears more clearly here because both and vary very slowly with respect to and near the lightcone. We believe that the origin of the discrepancy is that the true critical solution has a symmetry that is approximately CSS only inside the lightcone, but changes over to a different symmetry outside the lightcone in analytic manner; see Sec. III.12 below. Hence we expect some deviation from CSS already as we approach the lightcone from the inside.
Fig. 19 shows contour lines of , and for the sub10 evolution. The discrepancy between and is visible here, too. Sub25 gives us the larger range of (better finetuning), while sub10 gives us the larger range of (larger before the simulation stops). Overlaying the two sets of and contour lines in Fig. 20 shows that the contours are essentially the same, while the contours differ significantly for , as does the coordinate location of the accumulation point. Yet when we plot , and against , the two evolutions agree perfectly with the Garfinkle solution, and therefore each other, inside the lightcone.
By comparing , and with the nullcontinued Garfinkle solution, Figs. 1618 for sub25 and Figs. 2123 for sub10 also demonstrate that the analytic continuation is clearly ruled out, while the null continuation appears more plausible. In sub10, the strongest indication of this is that outside the lightcone, while the evidence from and is somewhat less clear.
Our plots of , and against or , at a range of fixed values of , show that inside the lightcone the deviations from the Garfinkle solution are very small. Such deviations are expected from a number of sources. In Sec. III.8 below we compute perturbative corrections to the Garfinkle solution for a nonvanishing . These are of order and hence very small. We also expect one growing perturbation, which is small by virtue of finetuning, infinitely many decaying perturbations, small by virtue of large , and numerical error. As these deviations from the Garfinkle solution are unlikely to cancel systematically, our plots indicate that they are all separately small.
Outside the lightcone, the deviations from our proposed null continuation of the Garfinkle solution are larger than inside the lightcone. It is clear that they cannot be mainly corrections, as they increase with , rather than depending on as . Rather, we believe that these deviations depend on the initial data in a manner that does not vanish in the finetuning limit. Mathematically, this may reflect that the discrete perturbation modes of the critical solution are not complete, or that a sum over those modes does not converge outside the lightcone. The latter could happen because individual modes that decay more rapidly with grow more rapidly as functions of outside the lightcone. (While we formally construct the discrete mode spectrum in Sec. III.5 below, we have only explicitly calculated the growing modes as functions of .) Yet another way of looking at this is to note that while demanding CSS and analyticity at the centre and the lightcone uniquely defines the countable family of Garfinkle solutions, the null data on the lightcone define a unique analytic continuation only if we demand CSS everywhere.
In Sec. III.12 below, we find an ODE system whose solution is an exact solution of the full field equations for finite outside the lightcone, and which can be matched at the lightcone to the Garfinkle solution and its first corrections as smoothly as the null continuation itself, namely . Hence, this is better than the null continuation plus corrections, but one may wonder how the two are related. As discussed in Sec. III.2 below, the bare null continuation has a null translation invariance in addition to spherical symmetry and CSS. Our exact outer solution has only one continuous symmetry of a hitherto unknown type: it acts as an isometry on the plane but a CSS on the orbits of spherical symmetry. However, if we expand this solution into a series in powers of , we obtain term by term the null continuation and its perturbative corrections, so we can also think of it as an approximate CSS symmetry. (Recall that with finite , exact CSS is impossible.) In the regime where we have plotted nearcritical solutions, even the firstorder corrections are very small compared to the zeroth order null continuation, so the deviations from the null continuation that we see cannot be caused mainly by these. For the same reason, we cannot distinguish the null continuation plus first correction from the exact solution of which it is the expansion.
ii.3 Evolving initial data for our amended Garfinkle solution
ii.3.1 Motivation and overview
Our working hypothesis, compatible with the numerical results presented so far, is that there is a true critical solution, which is asymptotically CSS, and which has one growing mode with . We have given strong numerical evidence that this critical solution is very well approximated by the Garfinkle solution inside the lightcone. We have also given, somewhat weaker, numerical evidence that outside the lightcone it is approximated not by the analytic continuation of the Garfinkle solution, but by what we have called its null extension.
The Garfinkle solution has a MOTS on its lightcone, and the null extension has a MOTS at every point. Therefore, on theoretical grounds, we need to add a correction to both, which removes the MOTSs. (These corrections are de facto so small, at least inside the lightcone, that we would have no reason to add them only to improve agreement with our numerical data.) We shall call this nullcontinued, corrected Garfinkle solution the “amended Garfinkle solution”.
Our amended Garfinkle solution still has two obvious shortcomings, namely that both it and its linear perturbations are not analytic but only at the lightcone, and that it has three growing modes. Analyticity at the lightcone is a natural requirement if the critical solution is required to arise from the evolution of generic initial data. Hence the nonanalyticity is not a mere technical shortcoming, and may well be related to the incorrect number of growing modes. Similarly, any universal critical solution can only have one growing mode.
In this Subsection we will give numerical evidence that, in some way that we do not yet understand theoretically, these twin problems seem to cancel each other out. We shall evolve initial data for our amended Garfinkle solution, matched outside its lightcone to asymptotically adS data, and add perturbations from one of five families: two that we consider as generic, and the three growing perturbation modes of the nullcontinued Garfinkle solution. We shall find that these data evolve in the expected CSS way, and that our amended Garfinkle solution with (approximately) zero perturbation is critical in all of these five families, showing scaling with and in each case, with no indication of any other growing mode. Hence we conclude that analyticity and the presence of the cosmological constant together somehow suppress the and growing modes, while the top mode survives.
ii.3.2 Data and results
The technical details of how we construct the initial data at for our amended Garfinkle solution are given in Sec. III.13 below. Here we need to say only that they are parameterised by , the value of at , which governs the magnitude of the corrections, the value of where the lightcone of the Garfinkle solution intersects , and the location and width of the switchover from Garfinkle data to vacuum.
We have chosen in order to make the correction small throughout the initial data, and , and in order to minimise spurious mass generated by the switching. With these parameters the total mass is . The correction to the initial data is small enough not to be visible in plots, and is of course expected to decay further as . Hence we can expect to compare the time evolution of these data against the nullcontinued Garfinkle solution within our plotting accuracy.
In the first oneparameter family of deformations of these data, we multiply and by a factor of . We find that the critical value is . As expected, this is small. Moreover, we find good agreement with the Garfinkle solution inside the lightcone from onwards, as there is no transition from generic initial data to the Garfinkle solution.
We find that , and outside the lightcone, as they would be in the null continuation. This is demonstrated for the sub8 evolution in Figs. 2426. (We do not know why the deviation in is relatively much larger). We have chosen sub8 because it is in the middle of the range of where we see convergence of the Ricci scaling, and hence trust our evolution.
With the same fitting procedure we used above for Gaussian initial data, a fit of for subcritical timesGarfinkle data gives , and for the fitting interval . This is equivalent to . The values of constants and are much smaller than for Gaussian initial data, meaning that depends only very weakly on . Clearly, the formal fitting error is an overoptimistic estimate of the error in , but the evidence strongly suggests again.
In a second 1parameter family of initial data, we take our best approximation to the critical point of the first family and add a Gaussian (centre , width ) in and with overall amplitude . The critical value for this family is . We would expect this to be very small, as the element of this family is already our best approximation to the critical point of the first family.
We have created three other 1parameter families of initial data by adding one of the , , and growing perturbations of the nullextended Garfinkle solutions to the best finetuned data of the first family. We add the perturbations for both and and their derivatives, with , and then solve the (nonlinear) constraints for and . The critical values are , and .
All five families show similar subcritical powerlaw scaling of the values and proper time locations of the extrema of the Ricci scalar at the centre. This is demonstrated in Figs. 2729 for the first family. The value of obatained for this family is compatible, within our numerical accuracy, with the theoretical value and slightly different from the result obtained for Gaussian initial data. However, the discrepancy is not significant if we take into account that the actual deviation of our data from a straight line is not random but smooth (i.e. systematic). By eye, a straight line with slope seems to be as good a fit as the leastsquares straight line. That the deviation from is smaller than for the Gaussian initial data might be explained by the fact that for our approximate critical solution we start much closer to the true critical solution, and less finetuning is needed to observe the scaling.
We have also looked at mass scaling for the supercritical evolutions. At low finetuning, for example from super5 to super15 for the family of initial data, we find a MOTS present already in the initial data. (To be precise, our initial data constraint solver fudges the MOTS, but it then appears on the first time step.) At larger finetuning, say for super17 to super27 for this family, the EMOTS occurs at some . We find that the mass of the MOTS in the initial data, or the EMOTS forming later, lie on a single curve with . In Sec. III.11, where we derive , we explain why it also applies for the MOTS in the initial data in this case. The mass scaling for Garfinkle family of initial data is presented in Fig. 30.
The timesGarfinkle family of initial data shows firstorder convergence in the intervals [sub4, sub14] (see Fig. 29) and [super3, super14]. For finetuned Garfinkle initial data plus a Gaussian perturbation we also see firstorder convergence in the intervals [sub5, sub22] and [super5, super19]. For subcritical evolutions of perturbations of the Garfinkle data we observe secondorder convergence in the interval [sub5, sub22] and first orderconvergence for supercritical data in the interval [super3, super19]. For perturbations we again have firstorder convergence in the intervals [sub6, sub21] and [super2, super20]. perturbations show secondorder convergence for subcritical evolutions in the interval [sub7, sub15] followed by firstorder convergence in [sub15, sub25], while for supercritical evolutions the convergence is first order in the interval [super3,super25]. The results obtained in fitting and to the power laws for all families of initial data studied are given in Table 1.
Family of initial data  fitting interval  fitting interval  
Gaussian  [26,5]  [26,17]  
[11,4]  [14,3]  
[18,10]  [19,9]  
[18,10]  [19,8]  
[20,10]  [20,11]  
[25,13]  [25,10]  
Theoretical values 
Iii Theory
iii.1 The Garfinkle solution
This Subsection is based on Garfinkle and is included here for completeness. In the metric coefficients and and coordinates defined in (2325), the general metric becomes
(31) 
where we have defined the shorthand by
(32) 
In order to eliminate from the field equations (10), we define the positive dimensionless parameter from the dimensionful parameter by
(33) 
The Garfinkle solution Garfinkle of the field equations (10) with , denoted by the subscript , is then given by
(34)  
(35)  
(36) 
The mass function is given by
(37) 
which takes value at the centre, as necessary for regularity, and at the lightcone.
So far, the Garfinkle solution is analytic at the centre for any and any positive integer , but it is generically singular at the lightcone , because is either zero or infinite there. However, with
(38) 
the overall power of in cancels and the solution is also analytic at the lightcone , with
(39) 
Hence, the Garfinkle solution is analytic at the centre and lightcone if and only if , thus restricting the possible values of .
In order to give an analytic form of the metric also in double null coordinates, we rescale to something that is proportional to , namely Garfinkle
(40) 
The metric then becomes
(41) 
Rescaling also ,
(42) 
the metric becomes
(43) 
where
(44) 
The Garfinkle solution then takes the more symmetric form CCF