---
abstract: 'We present an exact collapsing solution to 2+1 gravity with a negative cosmological constant minimally coupled to a massless scalar field, which exhibits physical properties making it a candidate critical solution. We discuss its global causal structure and its symmetries in relation with those of the corresponding continously self-similar solution derived in the $\Lambda=0$ case. Linear perturbations on this background lead to approximate black hole solutions. The critical exponent is found to be $\gamma = 2/5$.'
author:
- |
    Gérard Clément$^{a}$ [^1] and Alessandro Fabbri$^{b}$ [^2]\
    \
    [$^{(a)}$Laboratoire de Physique Théorique LAPTH (CNRS),]{}\
    [B.P.110, F-74941 Annecy-le-Vieux cedex, France]{}\
    [$^{(b)}$Dipartimento di Fisica dell’Università di Bologna]{}\
    [and INFN sezione di Bologna,]{}\
    [Via Irnerio 46, 40126 Bologna, Italy]{}
date: 'March 23, 2001'
title: |
    **Analytical treatment of critical collapse in 2+1 dimensional AdS spacetime:\
    a toy model**
---

Introduction
============

Since its discovery, the BTZ black hole solution \[1\] of 2+1 dimensional AdS gravity has attracted much interest because it represents a simplified context in which to study the classical and quantum properties of black holes. A line of approach which has been opened only recently [@CP; @HO; @gar; @burko] concerns black hole formation through collapse of matter configurations coupled to 2+1 gravity with a negative cosmological constant. As first discovered in four dimensions by Choptuik [@chop], collapsing configurations which lie at the threshold of black hole formation exhibit properties, such as universality, power-law scaling of the black hole mass, and continuous or discrete self-similarity, which are characteristic of critical phenomena [@gund]. In the case of a spherically symmetric massless, minimally coupled scalar field, a class of analytical continously self-similar (CSS) solutions was first given by Roberts [@rob; @brady; @oshiro]. These include critical solutions, lying at the threshold between black holes and naked singularities, and characterized by the presence of null central singularities. Linear perturbations of these solutions [@fro; @hay] lead to approximate black hole solutions with a spacelike central singularity.

Numerical simulations of circularly symmetric scalar field collapse in 2+1 dimensional AdS spacetime were recently performed by Pretorius and Choptuik [@CP] and Husain and Olivier [@HO]. Both groups observed critical collapse, which was determined in [@CP] to be continuously self-similar near $r=0$. In [@gar], Garfinkle has found a one-parameter family of exact CSS solutions of 2+1 gravity without cosmological constant, and argued that one of these solutions should give the behaviour of the full critical solution ($\Lambda\neq 0$) near the singularity.

The purpose of this paper is to present a new CSS solution to the field equations with $\L=0$ which can be extended to a threshold solution of the full $\L \neq 0$ equations. The new $\L
= 0$ solution is derived in Sect. 3. It presents a null central singularity and, besides being CSS, possesses four Killing vectors. In Sect. 4 we address the extension of this CSS solution to a quasi-CSS solution of the full $\L < 0$ problem, and show that the requirement of maximal symmetry selects a unique extension. This inherits the null central singularity of the $\L = 0$ solution, and has the correct AdS boundary at spatial infinity. Finally, we perform in Sect. 5 the linear perturbation analysis in this background, find that it does lead to black hole formation, and determine the critical exponent.

CSS solutions
=============

The Einstein equations for cosmological gravity coupled to a massless scalar field in (2+1) dimensions are G\_ - Łg\_ = T\_, with the stress-energy tensor for the scalar field T\_ = \_\_- 12 g\_ \^\_. The signature of the metric is (+ - -), and the cosmological constant $\L$ is negative for AdS spacetime, $\L = -l^{-2}$. Static solutions of these equations include the BTZ black hole solutions [@BTZ] with a vanishing scalar field $\phi = 0$, and singular solutions when a non-trivial scalar field is coupled with the positive sign for the gravitational constant $\k$ [@sigcosm].

We shall use for radial collapse the convenient parametrisation of the rotationally symmetric line element in terms of null coordinates $(u,v)$: ds\^2 = e\^[2]{}dudv - r\^2d\^2, with metric functions $\s(u,v)$ and $r(u,v)$. The corresponding Einstein equations and scalar field equation are & r\_[,uv]{} = r e\^[2]{}, &\
& 2\_[,uv]{} = e\^[2]{} - \_[,u]{}\_[,v]{}, &\
& 2\_[,u]{} r\_[,u]{} -r\_[,uu]{} = r\_[,u]{}\^2, &\
& 2\_[,v]{} r\_[,v]{} -r\_[,vv]{} = r\_[,v]{}\^2, &\
& 2r\_[,uv]{} + r\_[,u]{}\_[,v]{} + r\_[,v]{}\_[,u]{} = 0 & .

From the Einstein equations, the Ricci scalar is R = -6Ł+ 4e\^[-2]{}\_[,u]{}\_[,v]{}. It follows from (\[ric1\]) and (\[ein2\]) that the behavior of the solutions near the singularity is governed by the equations (\[ein1\])-(\[phi\]) with vanishing cosmological constant $\L = 0$ (see also [@burko]). Assuming $\L = 0$, Garfinkle has found [@gar] the following family of exact CSS solutions to these equations ds\^2 & = & -A()\^[c\^2]{}dudv - 14(v+u)\^2d\^2,\
& = & -2c( + ), depending on an arbitrary constant $c$ and a scale $A > 0$. In (\[gar1\]), $u$ is retarded time, and $-v$ is advanced time. These solutions are continuously self-similar with homothetic vector $(u\part_u + v\part_v)$. An equivalent form of these CSS solutions, obtained by making the transformation -u = (-|[u]{})\^[2q]{}, v = (|[v]{})\^[2q]{} (1/2q = 1 - c\^2) to the barred null coordinates $(\bar{u},\bar{v})$, is ds\^2 & = & -|[A]{}(|[v]{}\^q + (-|[u]{})\^q)\^[2(2q-1)/q]{}d|[u]{}d|[v]{} - 14(|[v]{}\^[2q]{} - (-|[u]{})\^[2q]{})\^2d\^2,\
& = & -2c(|[v]{}\^q + (-|[u]{})\^q). The corresponding Ricci scalar is R = (|[v]{}\^q + (-|[u]{})\^q)\^[2(1-3q)/q]{}(-|[u]{})\^[q-1]{} (|[v]{})\^[q-1]{}.

Garfinkle suggested that the line element (\[gar1\]) describes critical collapse in the sector $r = -(u+v)/2 \ge
0$, near the future point singularity $r = 0$ (where the Ricci scalar behaves, for $v \propto u$, as $u^{-2}$). The corresponding Penrose diagram (Fig. 1) is a triangle bounded by past null infinity $u \to -\infty$, the other null side $v = 0$, and the central regular timelike line $r = 0$. For $\k c^2 \ge 1$ ($q < 0$), the Ricci scalar R \~(|[v]{})\^[q-1]{} \~(v)\^[(q-1)/2q]{} is regular near $v = 0$, which moreover turns out to be at infinite geodesic distance. To show this, we consider the geodesic equation (e\^[2]{}) = -2rr\_[,u]{}\^2 = -2l\^2r\^[-3]{}r\_[,u]{} ($l$ constant) near $v = 0$, $u$ constant, which gives $v \propto (ls)^{4q}$ for $l \neq 0$, or $s^{2q}$ for $l=0$, so that in all cases the affine parameter $s \to \infty$ for $v \to 0$, and the spacetime is geodesically complete. For $\k c^2 < 1$ ($q
> 0$), we see from (\[ric2\]) that the null line $v = 0$ is a curvature singularity if $\k c^2 < 1/2$ ($q < 1$). If $1/2 \le \k c^2 < 1$ ($q \ge 1$), the surface $v = 0$ is regular. However, as discussed by Garfinkle, the metric (\[gar2\]) can be extended through this surface only for $q = n$, where $n$ is a positive integer. For $n$ even, the extended spacetime is made of two symmetrical triangles joined along the null side $\bar{v} = 0$, and has two coordinate singularities $r = 0$, one timelike ($\bar{u} - \bar{v} = 0$) and one spacelike ($\bar{u} + \bar{v} = 0$), but no curvature singularity. For $n$ odd, one of the $r = 0$ sides becomes a future spacelike curvature singularity ($e^{2\s} = 0$), similar to that of Brady’s supercritical solutions for scalar field collapse in (3+1) dimensions [@brady], except for the fact that in the present case the singularity is not hidden behind a spacelike apparent horizon (Fig. 2).

Let us point out that, besides the solutions (\[gar1\]), the system (\[ein1\])-(\[phi\]) also admits for $\L
= 0$ another family of CSS solutions ds\^2 & = & A()\^[c\^2]{}dudv - 14(v+u)\^2d\^2,\
& = & |[A]{}(|[v]{}\^q - (-|[u]{})\^q)\^[2(2q-1)/q]{}d|[u]{}d|[v]{} - 14(|[v]{}\^[2q]{} - (-|[u]{})\^[2q]{})\^2d\^2, with $\phi = -2c\ln(\sqrt{v} - \sqrt{-u})$, and we choose $A > 0$ and consider the sector $0 \le v \le -u$. These solutions have a future spacelike central ($r = 0$) curvature singularity at $(-\bar{u})^q =
\bar{v}^q$ (where the Ricci scalar (\[ric2\]) diverges) for all $q < 0$ or $q > 0$ (implying $q
> 1/2$). For $q < 0$, the Penrose diagram is a triangle bounded by past null infinities $\bar{u} \to
-\infty$ and $\bar{v} = 0$ (which is at infinite geodesic distance). For $q
> 0$, geodesics terminate at $\bar{v} = 0$, unless $q = n$ integer. For $n$ even, the extended spacetime has two central curvature singularities $r = 0$, one spacelike and the other timelike. The extended spacetime for $n$ odd is more realistic. In this case the extension from $\bar{v}
> 0$ to $\bar{v} < 0$ amounts to replacing (\[gar3\]) with $A > 0$ by the original Garfinkle solution (\[gar1\]) with $A > 0$, the resulting Penrose diagram being that of Fig. 2.

A new CSS solution for $\L = 0$
===============================

Among the one-parameter ($c$ or $q$) family of CSS solutions (\[gar1\]), the special solution, corresponding to $\k c^2 = 1$, ds\^2 = A( + )\^4 - 14(v+u)\^2d\^2, is singled out by the fact that the transformation (\[q\]) breaks down for this value. The transformation appropriate to this case, -u = 2e\^[-U]{}, v = 2e\^V = 2e\^[U-2T]{} (with $T \ge U$ for $u+v \le 0$) transforms the solution (\[gar4\]) to ds\^2 & = & e\^[-2U]{}\[-4A(1 + e\^[U-T]{})\^4dUdV - (1 - e\^[2(U-T)]{})\^2 d\^2\],\
& = & U - 2(1 + e\^[U-T]{}) (we use from now on units such that $\k = 1$, and have dropped an irrelevant additive constant from $\phi$).

Starting from this special CSS solution of the Garfinkle class, we now derive, by a limiting process, a new CSS solution which, as we shall see, exhibits a null singularity. We translate $T$ to $T-T_0$, and take the late-time limit $T_0 \to
-\infty$, leading to the new CSS solution (written for $A = -1/2$) ds\^2 = e\^[-2U]{}(2dUdV - d\^2), = U, with a very simple form which is reminiscent of the Hayward critical solution for scalar field collapse in 3+1 dimensions [@hay], ds\^2 = e\^[2]{}(2d\^2 - 2d\^2 - d\^2), = . The transformation |[u]{} = -e\^[-2U]{}, |[v]{} = V leads from (\[crit0\]) to the even more simple form of this solution ds\^2 = d|[u]{}d|[v]{} + |[u]{}d\^2, = -12(-|[u]{}), which is reminiscent of the other form of the Hayward solution ds\^2 = 2d|[u]{}d|[v]{} + |[u]{}|[v]{}d\^2, = -12(-|[u]{}/|[v]{}).

The solution (\[crit0\]) or (\[crit1\]) is continuously self-similar, with homothetic vector K = \_U = -2|[u]{}\_[|[u]{}]{}. It also has a high degree of symmetry, with 4 Killing vectors L\_1 & = & \_U + 2V\_V + \_,\
L\_2 & = & \_V + U\_,\
L\_3 & = & \_V,\
L\_4 & = & \_, generating the solvable Lie algebra & \[L\_1, L\_2\] = L\_4-L\_2, & \[L\_2,L\_3\] = 0,\
& \[L\_1, L\_3\] = -2L\_3, & \[L\_2,L\_4\] = -L\_3,\
& \[L\_1, L\_4\] = -L\_4, & \[L\_3,L\_4\] = 0.

The Ricci scalar (\[ric1\]) is identically zero for the solution (\[crit0\]), for which the sole nonvanishing Ricci tensor component is $R_{UU} = 1$. It follows that this metric is devoid of curvature singularity. However there is an obvious coordinate singularity at $U \to +\infty$, or $\bar{u} = 0$ (where $r=0$). To determine the nature of this singularity, we study geodesic motion in the spacetime (\[crit1\]). The geodesic equations are integrated by = , |[u]{} = l, + l = , where $\pi$ and $l$ are the constants of the motion associated with the Killing vectors $L_3$ and $L_4$, and the sign of $\varepsilon$ depends on that of $ds^2$ along the geodesic. The null line $\bar{u} = 0$ can be reached only by those geodesics with $\pi \neq 0$. Then, the third equation (\[geos\]) integrates to |[v]{} = |[u]{} - + [const.]{} = |[u]{} - (-|[u]{}) + [const.]{}. It follows that nonradial geodesics ($l \neq 0$) terminate at $\bar{u} = 0, \bar{v} \to +\infty$, while radial geodesics ($l =
0$), which behave as in cylindrical Minkowski space, can be continued through the null line $\bar{u} = 0$ to $\bar{u} \to
+\infty$ . So in this sense only the endpoint $\bar{v} \to
+\infty$ of the null line $\bar{u} = 0$ is singular. However formal analytic continuation of the metric (\[crit1\]) from $\bar{u} < 0$ to $\bar{u} > 0$ involves a change of signature from (+ - -) to (+ - +), leading to the appearance of closed timelike curves. So the null line $\bar{u} = 0$ corresponds to a singularity in the causal structure of the spacetime, analogous to the central singularity in the causal structure of the BTZ black holes [@BTZ]. The resulting Penrose diagram, reminiscent of that of the Hayward critical solution [@hay], is a diamond bound by three lines at null infinity ($\bar{v} = -\infty, \bar{u} = -\infty, \bar{v} =
+\infty$) and the null singularity $\bar{u} = 0$ (Fig. 3).

Extending the new solution to $\L \neq 0$
=========================================

In the preceding section we have found an exact solution for scalar field collapse with $\L = 0$, which presents a central null singularity. This property makes it a candidate threshold solution, lying at the boundary between naked singularities and black holes. However black holes exist only for $\L < 0$, so the solution (\[crit1\]) can only represent the behavior of the true threshold solution near the central singularity, where the cosmological constant can be neglected. This hypothetical $\L < 0$ solution cannot be self-similar, essentially because the scale is fixed preferentially by the cosmological constant [@CP]. So what we need is to find some other way to extend (\[crit1\]) to a solution of the full system of Einstein equations with $\L < 0$.

A first possible approach is to expand this solution in powers of $\L$, with the zeroth order given by the CSS solution (\[crit1\]). In the parametrisation (\[an\]), this zeroth order is (dropping the bars in (\[crit1\]) ) r\_0 = (-u)\^[1/2]{}, \_0 = 0, \_0 = -12 |u|. We look for an approximate solution to first order in $\L$ of the form r = (-u)\^[1/2]{} + Łr\_1, \_ = Ł\_1, = -12 |u| + Ł\_1 , \[lmk\]with the boundary condition that the fonctions $r_1$, $\sigma_1$ and $\phi_1$ vanish on the central singularity $u = 0$ . Eq (\[ein1\]) gives \[cmm\] r\_1 = (-u)\^[1/2]{}(13uv + f(u)), with $f(0) = 0$. Then, the linearized Eq. (\[ein4\]) gives 2r\_0\^[1/2]{}(r\_0\^[1/2]{}[\_1]{}\_[,v]{})\_[,u]{} = -[r\_1]{}\_[,v]{}[\_0]{}\_[,u]{} = 16(-u)\^[1/2]{}, which is solved by \[zzw\] \_1 = (1[15]{}uv + g(u)). The linearized Eq. (\[ein2\]) 2[\_1]{}\_[,uv]{} = 1 - [\_0]{}\_[,u]{}[\_1]{}\_[,v]{} = then gives \[nxn\] \_1 = 4[15]{}uv + h(u). Finally Eq. (\[ein2\]) leads to the relation between the arbitrary functions $f$, $g$, $h$ uf”(u) + f’(u) = g’(u) + h’(u).

Not only does this first order solution break the continuous self-similarity generated by (\[K\]), as expected, but it also breaks the isometry group generated by the Killings (\[kill\]) down to $U(1)$ (generated by $L_4 = \part_{\theta}$), except in the special case $f = g = h = 0$, where the Killing subalgebra $(L_1, L_4)$ remains. This suggests looking for an exact $\L < 0$ extension of the $\L = 0$ CSS solution of the form ds\^2 = e\^[2(x)]{}dudv + u\^2(x)d\^2, = -12|u| + (x), with $x = uv$. This will automatically preserve to all orders the Killing subalgebra $(L_1,
L_4)$. Inserting this ansatz into the field equations (\[ein1\])-(\[phi\]) leads to the system x” + 32’ & = & e\^[2]{},\
2(x” + ’) + ’(x’-12) & = & e\^[2]{},\
x\^2(-” + 2’’- ’\^2) + x(- ’ + (’+’)) & = & 0\
- ” + 2’’- ’\^2 & = & 0\
2x(’)’ + 52’ & = & 12’. ($' = d/dx$). The unique, maximally symmetric extension of the CSS solution (\[crit1\]) reducing to (\[crit1\]) near $u = 0$ is the solution of the system (\[einx1\])-(\[phix\]) with the boundary conditions (0) = 1, (0) = 0, (0) = 0.

The comparison of (\[einx3\]) and (\[einx4\]) yields = e\^[+]{}. The combination $(\ref{einx1}) + x(\ref{einx4})$ then gives, together with (\[rhospsi\]), x(2’\^2 + 2’’ - ’\^2) + 32(’ + ’) = e\^[2]{}. The third independent equation is for instance (\[einx2\]): 2(x” + ’) + ’(x’ - 12) = e\^[2]{}. Using these last two equations with the boundary conditions (\[boundx\]), one can in principle write down series expansions for $\s(x)$ and $\psi(x)$. Another simple relation, deriving from (\[einx4\]) and (\[rhospsi\]), is ” + ” - ’\^2 + 2’\^2 = 0.

We are interested in the behavior of this extended solution in the sector $u < 0$, $v > 0$, i.e. $x < 0$. In this sector, Eqs. (\[einx1\]), (\[phix\]) and (\[einx2\]) can be integrated to (-x)\^[3/2]{}’ & = & \_x\^0 (-x)\^[1/2]{}e\^[2]{}dx,\
(-x)\^[5/4]{}’ & = & 14 \_x\^0 (-x)\^[1/4]{}’ dx,\
-x’ & = & 12 \_x\^0 ( e\^[2]{} + ’(12 - x’))dx . As long as $\rho > 0$, Eq. (\[sum1\]) (with $x < 0$, $\L < 0$) implies $\rho' < 0$, so that $\rho(x)$ decreases to 1 when $x$ increases to 0. It then follows from (\[sum2\]) that $\psi' <0$. Also, (\[sum2\]) can be integrated by parts to x’ = 14 - \_x\^0 (-x)\^[-3/4]{}dx, showing that $x\psi' < 1/4$. It then follows from (\[sum3\]) that $\s' < 0$. So, as $x$ decreases, the functions $\rho$ and $e^{2\s}$ increase and possibly go to infinity for a finite value $x = x_1$. If this is the case, the behavior of these functions near $x_1$ must be & = & \_1(1[|[x]{}]{} + 1[4x\_1]{} - + ... )\
e\^[2]{} & = & (1 + + ...)\
& = & \_1 + - (|[x]{}) + ... ($\bar{x} = x - x_1$).

These expectations are borne out by the actual numerical solution of the system x” + ’ &=& - e\^[2]{},\
-”+4’’ &=& ’\^2 +\^2’\^2, \[sist\] (this last equation comes from (\[einx4\]) where $\psi'$ is given by derivation of (\[rhospsi\]) ) where we have set $\L=-2$, with the boundary counditions $\rho(0)=1$, $\rho'(0)=-2/3$ (see eqs. (\[cmm\]) and (\[lmk\]) ), $\sigma(0)=0$. The plots of the functions $ \rho (x)$ , $\sigma (x)$ and $\psi' (x)$ are given in Figs. (4,5,6,). The value of $x_1$ is found to be approximately $-1.94$ (i.e. $\L x_1 = +3.88$).

The coordinate transformation[^3] u = Ł\^[-1]{}e\^[-|[U]{}]{}, v = e\^[|[V]{}]{} (|[U]{} = |[T]{} - |[R]{}, |[V]{} = |[T]{}+|[R]{}) leads to $x = \L^{-1} e^{2\bar{R}}$ and, on account of (\[anext\]) and (\[rhospsi\]), to the form of the metric ds\^2 = -Ł\^[-1]{}e\^[2((|[R]{})+|[R]{})]{}(d|[U]{}d|[V]{} - e\^[2(|[R]{}) -|[V]{}]{}d\^2). Near the spacelike boundary $\bar{R} = \bar{R}_1$ of the spacetime, the collapsing metric and scalar field behave, from (\[bound1\]), as ds\^2 -Ł\^[-1]{}(|[R]{}\_1-|[R]{})\^[-2]{}(d|[T]{}\^2-d|[R]{}\^2-e\^[|[T]{}\_1 -|[T]{}]{}d\^2), = \_1 + |[T]{}/2 ($\bar{R}-\bar{R}\simeq \bar{x}/2x_1$). This metric is asymptotically AdS, as may be shown by making the further coordinate transformation, |[R]{}-|[R]{}\_1 = -2/XT, |[T]{}-|[T]{}\_1 = 2(T/2), leading to ds\^2 -Ł\^[-1]{}(X\^2 dT\^2 - - X\^2d\^2 ), = \_1 + (T/2). The next-to-leading terms in the metric containing logarithms, this asymptotic behavior differs from that of BTZ black holes.

It follows from this discussion that the Penrose diagram of the $\L < 0$ threshold solution in the sector $v > 0$, $u < 0$ is a triangle bounded by the null line $v = 0$, the null causal singularity $u = 0$, and the spacelike AdS boundary $X \to \infty$. The null singularity $u = 0$ remains naked, i.e. is not hidden behind a trapping horizon, which would correspond to \_v r = -(-u)\^[3/2]{}’(x) = 0, because $\rho' <0$ (as discussed above) implies that the only solution of this equation is $u = 0$.

For the sake of completeness, let us also discuss the behavior of the solution of the system (\[einx1\])-(\[phix\]) in the sector $x > 0$. In this case, one can write down integro-differential equations similar to (\[sum1\])-(\[sum3\]), from which one again derives that $\rho' < 0$, $\psi' < 0$ and $\s'< 0$. It follows that the metric function $e^{2\s}$ decreases as $x$ increases, eventually vanishing for a finite value $x = x_0$, corresponding to a spacelike curvature singularity (this has been confirmed numerically). The behavior of the solution near this singularity is found to be (x\_0-x), (x\_0-x), (x\_0-x) (= 3 - 1), and the coordinate transformation $u = e^U, v = e^V (x = e^{2T})$ leads to the form of the metric near the singularity ds\^2 (T\_0-T)\^[\^2]{}(dT\^2-dR\^2) +e\^[R\_0-R]{}(T\_0-T)\^2d\^2.

Perturbations
=============

To check whether the quasi-CSS solution (\[anext\]) of the full $\L \neq 0$ problem determined in the preceding section is indeed a threshold solution, we now study linear perturbations of this solution. Our treatment will follow the analysis of perturbations of critical solutions in the case of scalar field collapse in 3+1 dimensions [@fro; @hay].

The relevant time parameter in critical collapse being the retarded time $U = -(1/2)\ln(-u)$ (the “scaling variable” of [@fro]), we expand these perturbations in modes proportional to $e^{kU} = (-u)^{-k/2}$, with $k$ a complex constant. We recall that only the modes with $Re\ k>0$ grow as $U \to +\infty$ ($u \to
-0$) and lead to black hole formation, whereas those with $Re\ k<0$ decay and are irrelevant. The other relevant variable is the “spatial” coordinate $x =
uv$, and the perturbations are decomposed as r & = & (-u)\^[1/2]{}((x) + (-u)\^[-k/2]{}(x)),\
& = & -12|u| + (x) + (-u)\^[-k/2]{}(x),\
& = & (x) + (-u)\^[-k/2]{}(x). Then, the Einstein equations (\[ein1\])-(\[phi\]) are linearized in $\tilde{r}$, $\tilde{\phi}$, $\tilde{\sigma}$, using \_[,u]{} =-(-u)\^[[-k/2]{}-1]{}(x’-2), \_[,v]{} =-(-u)\^[[-k/2]{}+1]{}’ . The resulting equations are homogeneous in $u$, which drops out, and the linearized system reduces to & & x” + ([-k/2]{}+3/2)’ = e\^[2]{}( + 2),\
& & 2x” +([-k]{}+2)’ = Łe\^[2]{} - (2x’-1/2)’ + (k/2)’,\
& & -(-k+1)x’ + ((-k+1)x’-(k\^2-1)/4) + x’ - k(x’ + /2) =\
& & -(x’ - k(1/2-x’)) + (1/4-x’),\
& & 2(’’ + ’’) - ” = ’(2’ + ’),\
& & 2x” + (2x’ + (-k+5/2))’ - (k/2)’ + (2x’-1/2)’\
& & + (2x” + ([-k/2]{}+5/2)’) = 0.

What is the number of the independent constants for this system? The perturbed Klein-Gordon equation (\[pphi\]) is clearly redundant, while Eqs. (\[pein3\]) and (\[pein4\]) are constraints. So, as in the (3+1)-dimensional case [@fro; @hay], the order of the system is four, and the general solution depends on four integration constants. However, one of these four independent solutions corresponds to a gauge mode and is irrelevant. The parametrisation (\[anext\]) is invariant under infinitesimal coordinate transformations $v \to v + f(v)$. For $f(v) =
-\alpha v^{1+k/2}$, these lead to $x \to x - \alpha (-u)^{-k/2}(-x)^{1+k/2}$, giving rise to the gauge mode \_k(x) & = & (-x)\^[1+k/2]{}’(x),\
\_k(x) & = & (-x)\^[1+k/2]{}’(x),\
\_k(x) & = & , which solves identically the system (\[pein1\])-(\[pphi\]). So, up to gauge transformations, the general solution of this system depends only on three independent constants.

These will be determined, together with the possible values of $k$ (the eigenfrequencies) by enforcing appropriate and reasonable boundary conditions. We shall use here the “weak boundary conditions” of [@hay] on the boundaries $u = 0$ and $x = x_1$ ($X \to \infty$) \_[u 0]{}r\^[-1]{} 0, \_[x x\_1]{}r 0, together with the condition (0) = 0, which guarantees that the singularity of the perturbed solution starts smoothly from that of the unperturbed one. On the third boundary $v = 0$, we shall impose a stronger condition by requiring that the perturbations are analytic in $v$, in order for the perturbed solution to be extendible beyond $v=0$ to negative values of $v$ at finite $u$.

First, we consider the region $x\to 0$ where, according to Eqs. (\[zwx\]), (\[cmm\]), (\[zzw\]) and (\[nxn\]), \[wxc\] 1 + x,  e\^[2]{}1+x,   x. Let us assume a power-law behavior r(x) \~a (-x)\^[p]{} where $p$ is a constant to be determined. Then Eqs. (\[pein1\]), (\[pein2\]) and (\[pein4\]) can be approximated near $x = 0$ as & & x” + ([-k/2]{}+3/2)’ Ł,\
& & x” + ([-k/2]{}+1)’ 14’\
& & 2’’ - ” 2’’. Eliminating the functions $\tilde{\s}$ and $\tilde{\phi}$ between these three equations and using Eq. (\[wxc\]), we obtain the fourth-order equation 4x\^2”” + (-4k + 13)x”’ + (k/2-1)(2k - 5)” 0, which implies the power-law behavior (\[pow\]) with the exponent $p$ constrained by p(p-1)(p-k/2-3/4)(p-k/2-1)=0. Obviously the root $p = k/2 + 1$ corresponds to the gauge mode (\[gauge\]) and must be discarded as irrelevant. As a consequence the general solution near $x = 0$ can be given in terms of three independent constants as & &r(x) \~A +B(-x) +ŁC(-x)\^[3/4 +k/2]{},\
& &(x) \~- 2 + Ł\^[-1]{}2 - (k + )(-x)\^[-1/4 +k/2]{},\
& & (x) \~2 - Ł\^[-1]{}2 + (k + )(-x)\^[-1/4 +k/2]{}.\[pti\] Let us note that this solution remains valid in the limit $\L \to 0$, leading to the limiting solution $\tilde{r} \sim A +
B(-x)$ (with $B = 0$ for $k \neq 3$), which could also be obtained directly by solving the equation $\tilde{r}'' = 0$ which results from (\[pein4\]) in the limit $\L
\to 0$, together with the stronger condition (from Eq. (\[pein1\])) $(k-3)\tilde{r}' = 0$.

Now we enforce the boundary conditions at $x = 0$. For $k > 0$, $\tilde{r}$ is dominated by its first constant term in (\[rti\]), so that the condition (\[bound0\]) can only be satisfied for $u \to 0$ if A = 0. Then, for $k
> 1/2$, $\tilde{r}$ is dominated by its second term $-Bx$, leading to a perturbation $(-u)^{1/2-k/2}\tilde{r}(x)$ which blows up as $u \to 0$ and violates (\[bound\]) unless \[rang\] k 3. Then we impose the condition of analyticity in $v$ at fixed $u$. This is satisfied if \[unn\] k = 2n - 3/2, where $n$ is a positive integer. Combining eqs. (\[rang\]) and (\[unn\]) we find that $k$ has only two positive eigenvalues k = 1/2, k = 5/2. However, in the above analysis we have disregarded the fact that $k = 1/2$ is a double root of the secular equation (\[sec\]). For $k = 1/2$ the correct behavior of the general solution near $x = 0$ is & &r(x) \~A +B(-x) +ŁC(-x)|x|,\
& &(x) \~- 2 - Ł\^[-1]{} 4 - 4 - 4 |x|\
& & (x) \~4 + Ł\^[-1]{} 4 + 4 + 4 |x| , which satisfies the condition of analyticity only if $C = 0$.

At the AdS boundary ($x\to x_1$) the leading behaviour of the background is, from Eqs. (\[bound1\]), \[cdfg\] ,  e\^[2]{}(),  \_1. We again assume a power-law behavior \~b|[x]{}\^q ($\bar{x} = x-x_1$). Then Eq. (\[pein2\]), where $\tilde{\phi}$ can be neglected, gives q(q-1) = 2, i.e. $q = -1$ or $q = 2$. Then, Eq. (\[pein1\]) reduces near $\bar{x}
= 0$ to ” - 2|[x]{}\^[-2]{} 4b\_1|[x]{}\^[q-3]{}. If $q = -1$, the behavior of the solution is governed by the right-hand side, i.e. $\tilde{r} \propto \bar{x}^{-2}$, which violates the boundary condition (\[bound\]) for $x \to x_1$. So the behavior $\tilde{\s} \sim b\bar{x}^{-1}$ must be excluded, which fixes another integration constant $D=0$ (where $D$ is a linear combination of $B$ and $C$). Then, the generic behavior of the solution of Eq. (\[asr\]) with $q = 2$ is governed by that for the homogeneous equation, i.e. r \~. This is consistent with the boundary condition (\[bound\]), and is an admissible small perturbation if its amplitude is small enough, $E \ll \rho_1$.

For $k = 1/2$, we have seen that two of the three integration constants in (\[rtii\])-(\[ptii\]) are fixed ($A = C = 0$) by condition (\[bound0\]) and the analyticity condition, while the weak boundary condition at the AdS boundary fixes a third constant $D=0$. However this is impossible, as the perturbation amplitude must remain as a free parameter. So the mode $k = 1/2$ cannot satisfy all our boundary conditions, and we are left with a single eigenmode, k = 5/2, completely determined up to an arbitrary amplitude by the two conditions $A=D=0$.

The corresponding perturbed metric function $r$ behaves near $x = 0$ as r (-u)\^[1/2]{}\[1 + Łx - (-u)\^[-5/4]{}Bx\]. For $B < 0$, the central singularity $r = 0$ is approximately given by (-u)\^[1/4]{} -Bv. Our boundary conditions guarantee that it starts at $u=v=0$ (as for the unperturbed solution) and then becomes spacelike in the region $v>0$. This singularity is hidden behind a trapping horizon (defined by Eq. (\[trap\])) which, near $x = 0$, is null, (-u)\^[5/4]{}= (a null trapping horizon was also found in [@hay]). Let us point out the crucial role played by the cosmological constant $\Lambda$ in the formation of this trapping horizon. For $\L = 0$, $\rho(x) = 1$, while, as discussed after Eq. (\[pti\]), the perturbation $\tilde{r}$ with the boundary condition (\[bound0\]) vanishes for $\L = 0$, so that the perturbed radial function $r$ is (as in [@gar]) identical to the CSS one, and the trapping horizon does not exist. Near the AdS boundary $x \to x_1$, it follows from (\[cdfg\]) and (\[asol\]) that both the central singularity and the trapping horizon are tangent to the null line (-u)\^[5/4]{} = -E ()\^[-1/2]{}.

Thus, perturbations of the quasi-CSS solution lead to black hole formation, showing that this solution is indeed a threshold solution, and is a candidate to describe critical collapse. Near-critical collapse is characterized by a critical exponent $\gamma$, defined by the scaling relation $Q
\propto |p - p^*|^{s\gamma}$, for a quantity $Q$ with dimension $s$ depending on a parameter $p$ (with $p = p^*$ for the critical solution). Choosing for $Q$ the radius $r_{AH}$ of the apparent horizon, and identifying $p - p^*$ with the perturbation amplitude $B$, we obtain from (\[trap1\]) r\_[AH]{} ()\^[2/5]{}, leading to the value of the critical exponent $\gamma = 2/5$, in agreement with the renormalization group argument [@kha] leading to $\gamma = 1/k$.

Conclusion
==========

We have discussed in detail the causal structure of the Garfinkle CSS solutions (\[gar1\]) to the $\L= 0$ Einstein-scalar field equations. From a special solution of this class, we have derived by a limiting process a new CSS solution, which we have extended to a unique solution of the full $\L < 0$ equations, describing collapse of the scalar field onto a null central singularity. This is not a curvature singularity (all the curvature invariants remain finite), but a singularity in the causal structure similar to that of the BTZ black hole. Finally, we have analyzed linear perturbations of the $\L < 0$ solution, found a single eigenmode $k = 5/2$, checked that this mode does indeed give rise to black holes, and determined the critical exponent $\gamma = 2/5$.

For comparison, Choptuik and Pretorius [@CP] derived, by analysing the observed scaling behavior of the maximum scalar curvature, the value $1.15 < \gamma < 1.25$ for the critical exponent. This value is different from the value $\gamma\sim 0.81$ obtained in the numerical analysis of Husain and Olivier [@HO] from the scaling behavior of the apparent horizon radius. Our value $\gamma = 0.4$, while significantly smaller than these two conflicting estimates, is of the order of the theoretical value $\gamma = 1/2$ derived either from the analysis of dust-ring collapse [@ps], of black hole formation from point particle collisions [@bir], or of the $J = 0$ to $J \neq 0$ transition of the BTZ black hole [@kg].

It is worth mentioning here that, even though they were obtained for a vanishing cosmological constant and thus solve the $\L\neq 0$ equations only near the singularity, the Garfinkle CSS solutions are, for the particular value (chosen in order to better fit the numerical curves) $c = (7/8)^{1/2} \simeq 0.935$, in good agreement [@gar] with the numerical results of [@CP] at an intermediate time. The fact that this value is close to 1 suggests that the $c = 1$ CSS solution (\[gar5\]) approximately describes near-critical collapse at intermediate times. If this the case, then it would not be surprising if its late-time limit, our new CSS solution Eq. (\[crit0\]), gives a good description of exactly critical collapse near the singularity. A fuller understanding of the relationship between the numerically observed near-critical collapse and these various $\L = 0$ CSS solutions could be achieved by extending them to $\L < 0$, as done in the present work for the special solution (\[crit1\]).

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[^1]: Email: gclement@lapp.in2p3.fr

[^2]: Email: fabbria@bo.infn.it

[^3]: We have taken care that in (\[anext\]) $u$ has the dimension of a length squared while $v$ is dimensionless.
