On the critical behaviour of gapped gravitational collapse in confined spacetime
OOn the critical behavior of gapped gravitational collapse in confined spacetime
Rong-Gen Cai , , ∗ Li-Wei Ji , † and Run-Qiu Yang ‡ CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Quantum Universe Center, Korea Institute for Advanced Study, Seoul 130-722, Korea
The gravitational collapse of a massless scalar field enclosed with a perfectly reflecting wall in aspacetime with a cosmological constant Λ is investigated. The mass scaling for the gapped collapse M AH − M g ∝ ( (cid:15) c − (cid:15) ) ξ is confirmed and a new time scaling for the gapped collapse T AH − T g ∝ ( (cid:15) c − (cid:15) ) ζ is found. We find that both of these two critical exponents depend on the combination Λ R , where R is the radial position of the reflecting wall. Especially, we find an evolution of the critical exponent ξ from 0 .
37 in the confined asymptotic dS case with Λ R = 1 . . R → −∞ ), while the critical exponent ζ varies from 0 .
10 to 0 .
26, which shows the new criticalbehavior for the gapped collapse is essentially different from the one in the Choptuik’s case.
I. INTRODUCTION
In 1993, Choptuik [1] found an interesting so-calledtype II critical phenomenon in the gravitational col-lapse of a massless scalar field in spherically symmet-ric asymptotically flat spacetimes. He looked closely atthe threshold between black hole(BH) formation and dis-persion, and found that the masses of BHs formed fromsupercritical configurations follow a power-law behavior: M BH ∝ ( (cid:15) − (cid:15) ) γ . Here (cid:15) parameterizes a one-parameterfamily of initial data, (cid:15) corresponds to the threshold and γ is a universal exponent which is around 0 .
37. A fewyears later, Choptuik, Chmaj and Bizon found the typeI critical phenomenon in gravitational collapse of Yang-Mills field [2]. The BH formations here turned on at afinite mass (mass gap), unlike the type II case where theBH formation turned on at zero mass. The BH masses ofsupercritical solutions in this case do not follow a power-law scaling. Instead, the span of time which describeshow long the configuration stays in the vicinity of thecritical solution scales as: T ∝ ln( (cid:15) − (cid:15) ). For recentreviews on the critical phenomenon in gravitational col-lapse, please see [3, 4].When it comes to the asymptotically AdS spacetime,the situation is very different. Due to the confinementproperty of the timelike boundary of AdS, the subcriti-cal configurations can be reflected by the boundary andalso collapse into BHs. There are many thresholds (cid:15) n which divide the supercritical and subcritical configura-tions. The type II critical phenomenon was confirmedfor the supercritical configurations and the masses ofthe BHs follow a power-law behavior [5–8]. For thesubcritical configurations, a new power-law behavior: M AH − M g ∝ ( (cid:15) c − (cid:15) ) ξ was first observed recently in[9, 10], where ξ = 0 . M AH is the initial black hole ∗ [email protected] † [email protected] ‡ [email protected] mass at the subcritical solution and M g is the mass gap.Contrary to the cases discovered by Choptuik and hiscooperators which have been understood very well, thisnew gapped scaling behavior is still mysterious. For ex-ample, we still don’t know if there is any asymptotic scal-ing symmetry near the critical point as the case found inthe Choptuik’s Type II critical phenomena, and whetherit can appear in other matter fields and gravity theoriessuch as Yang-Mills field, Gauss-Bonnet theory and so on.Note that the critical exponent γ in the Choptuik’s scal-ing law is universal for the asymptotically flat and AdScases. It would be very interesting to see whether thecritical exponent ξ in the gapped scaling law is universalor not in confined spacetimes.In order to see whether the turbulent behavior is anexclusive domain of asymptotically AdS spacetime or atypical feature of “confined” Einstein’s gravity with re-flecting boundary condition, Maliborski [11] investigatedthe collapse of a massless scalar field enclosed in a cavity.He observed a similar turbulent behavior and multiplecritical phenomena as in the asymptotically AdS case.The recent work [12] further pointed out that the newgapped critical relationship found in Refs. [9, 10] canalso appear in asymptotic flat space-time with reflectingwall. One of interesting results reported in Ref. [12] isthe critical exponent in the mass scaling law with a gapis 0.6, which is different from its value in the asymptoticAdS case reported by Refs. [9, 10]. This difference givesa very important signal that this new gapped critical be-havior has some essential difference compared with whatwe have known in the Choptuik’s type II critical behav-ior, as it has been proven that the critical exponent isindependent of the cosmological constant.To understand why such difference happens and whatare the roles of cosmological constant and reflecting wallin this difference, one can investigate such gapped criticalbehavior with a cosmological constant Λ and a reflectingwall located in the radius R . Such a setup was first pro-posed in Refs. [13–15] to investigate the role played bythe fully resonant spectrum of AdS in the turbulent in-stability. It is found that backgrounds with non-resonantfrequencies cannot cause collapse at arbitrarily small fre- a r X i v : . [ g r- q c ] S e p quencies [15]. In this paper, we study the model as thesame as the one in Refs. [13–15]. However, we focus onthe critical phenomena near the threshold of black holeformation, instead of turbulent instability of the space-time. One of our main purposes is to make a bridge tounderstand the difference between the results in the con-fined asymptotic flat case and those in the asymptoticAdS case. To be specific, our main motivation is to in-vestigate the influence of cosmological constant and theposition of wall on the exponent of mass scaling of thesubcritical configurations, which will be shown in thispaper that only the value of Λ R is relevant. Our nu-merical results show a clear change of critical exponent ξ from 0.37 in the asymptotic dS case with Λ R = 1 . R → −∞ foundin [9, 10]. This result is very different from the case intype II mass scaling of supercritical configurations whichhas already been proven to be independent of the cos-mological constant [16]. In addition, we also find a newtime scaling T AH − T g ∝ ( (cid:15) c − (cid:15) ) ζ for the forming timeof the gapped black hole, where the critical exponent isalso dependent of the value of Λ R .Our paper is organized as follows: In Sec.II, we presentthe equations of motion in double null coordinates andintroduce the algorithms briefly. In Sec.III, we displaythe results of our numerical simulations. We conclude inSec.IV. II. SET UP
We consider the gravitational collapse of a real scalarfield in a spherical cavity defined by setting a perfectlyreflecting mirror at some finite radial radius r = R . Thedynamics of the system is governed by the Einstein-scalarequations, G αβ + Λ g αβ = 8 πG (cid:20) ∇ α φ ∇ β φ − g αβ ( ∇ φ ) (cid:21) , (1) g αβ ∇ α ∇ β φ = 0 , (2)where G αβ is the Einstein tensor, Λ is the cosmologicalconstant, φ is the massless scalar field and G is Newtongravitational constant. In what follows we set G = 1 forconvenience.Following [12, 17], we take the metric ansatz in thedouble null coordinate, ds = − g r (cid:48) dudv + r d Ω , (3)where d Ω is the two-dimensional sphere metric, u is anoutgoing null coordinate and v is an ingoing null coordi-nate. We take u as the null time and v as the null spatialcoordinate. For any quantity f , an overdot ˙ f stands for ∂f∂u , while a prime f (cid:48) represents ∂f∂v . We introduce two auxiliary fields h and ¯ g such that, φ ≡ ¯ h = 1 r (cid:90) vv h r (cid:48) d ˜ v, (4)¯ g = 1 r (cid:90) vv (1 − Λ r ) g r (cid:48) d ˜ v, (5)where v is defined by r ( u, v ( u )) = 0. From the ( v, v )and ( u, v ) components of the Einstein equations (1), weget, g = exp (cid:20) π (cid:90) vv ( h − ¯ h ) r r (cid:48) d ˜ v (cid:21) , (6)˙ r = −
12 ¯ g. (7)The evolution of h can be obtained from the Klein-Gordon equation (2),˙ h = 12 r ( g − ¯ g )( h − ¯ h ) . (8)The two boundaries of the computational domain arelocated at r ( u, v ) = 0 and r ( u, v ) = R , respectively. Inthe double null coordinates, these two boundaries are dy-namic and evolve with u . So we need to fix the compu-tational domain first by deleting extra grid points in thecenter and adding more points on the outer boundary[12], before we impose boundary conditions.Near r = 0, in order to overcome the inaccuracy due tothe explicit factor of 1 /r in the expressions for ¯ h, ¯ g and g , we expand h in a Taylor series in r [17], h = h + h r + h r + O ( r ) . (9)Then the expansions for the rest of the variables in r canbe obtained by substituting (9) into (4), (5) and (6). Thecoefficients of the expansions can be expressed in termsof h , h , h . All we need to do is to find out these threecoefficients. This is done by fitting the first three valuesof h to a second-order polynomial [16].We impose the Dirichlet boundary condition at r = R , φ ( u, v ) | r = R = 0 . (10)The value of φ at those added points can be given byfitting the last few values of φ and φ ( R ) to a fourth-order polynomial. The rest variables can be obtained bydefinition.The Misner-Sharp mass contained within the sphere ofradius r is defined as M ( u, r ) = r (cid:18) − ¯ gg − Λ r (cid:19) (11)To find the initial mass of the black hole, we need to findout where and when the black hole forms. This can beachieved by using apparent horizon(AH). One can easilyfind that the position of apparent horizon is given by¯ g/g = 0 [18]. Let r AH be the initial apparent horizonradius and r g is its value at the critical solution, from theEq. (11) we can easily see that δM AH = M AH − M g =(1 / − Λ r g / δr AH = (1 / − Λ r g / r AH − r g ), so themass and the radius of initial black hole have the samescaling behavior and share the same critical exponent. III. NUMERICAL RESULTS
We use a 4th-order Runge-Kutta scheme to solve thetime evolution equations (7), (8). At every time step,those points whose value of r is negative are removed andthose points whose value of r is smaller than R are added.Then the functions ¯ h, g, ¯ g are calculated in sequence using(4), (5), (6). A detailed description can be found in [12].We choose the Gaussian-type initial data for the scalarfield, φ ( r ) = (cid:15) exp (cid:20) − tan ( πrR − π ) σ (cid:21) (12)where (cid:15), σ are two parameters. For our simulations weset σ = 1 / R .Let’s consider a scaling transformation: u → uk , v → vk , r → rk , Λ → Λ k , R → Rk (13)This transformation will not change the forms of Eqs.(4)-(8) and will not change the solution of metric andscalar field. This scaling property means that for the twoparameters Λ and R , only the value of Λ R is relevant onthe system.This leads that we can fix the wall positionand only study the influence of the cosmological constanton the system. The AdS limit then can be obtained bytwo equivalent manners, one is fixing Λ to be any negativevalue and then taking R into ∞ , the other is fixing R andtaking Λ → −∞ .In addition, we will treat the cosmological constant as afree parameter in the model rather than constrain it to benegative. As the reflecting wall plays the role to confinethe energy, even for the asymptotic dS case, the bouncecan also appear and weak turbulence can still appear.When the amplitude is large enough, the wave packetcollapses to form an AH on its first implosion. Whenthe amplitude is smaller than some threshold (cid:15) , it is tooweak to form an AH on its first implosion. The wavepackage bounces, travels ”instantaneously” to the mirrorand is reflected back by the mirror. An AH might formon its second implosion. This scenario repeats again andagain as we decrease the amplitude continuously. Thecritical amplitude which separates those AHs formed ontheir ( n − n th implosions is denoted as (cid:15) n − .However, this picture breaks down when the cosmo-logical constant is too positive. In this case, the cosmo-logical constant is so large that the cosmological horizonwill be located inside the cutoff r = R , the bounced sub-critical wave package can’t touch the mirror and can’t - - Λ ϵ FIG. 1. Effect of cosmological constant Λ on the first criticalamplitude (cid:15) . be reflected back to start its second implosion. This islike the case that this mirror does not exist. In this case,there is no black hole formation. Therefore we will al-ways consider the case with the reflecting wall is insidethe cosmological horizon when the cosmological constantis positive.To study the influence of the rescaled radius Λ R onthe critical behavior, we change the cosmological con-stant Λ, with the cut off R fixed (we choose R = 1 for con-venience). Then the AdS limit corresponds to Λ → −∞ .Note the dS limit doesn’t need that Λ R → ∞ , as wehave discussed, because of the existence of the cosmo-logical horizon, it only needs that Λ R > Λ dS R > dS is the solution of ¯ g (0 , r ) /g (0 , r ) | r = R = 0 forgiven initial scalar field’s configuration. When (cid:15) → dS = 3 /R . For nonzero (cid:15) , Λ dS is less than 3 /R anddepends on the initial configuration. A. Critical amplitude
We first consider the effect of cosmological constant Λon the critical amplitudes (cid:15) n . For a fixed Λ, the criticalamplitudes (cid:15) n are found using bisection method. Theresult for (cid:15) between the first and second branches isshown in Fig. 1. As we can see, (cid:15) increases as we in-crease the cosmological constant Λ, which indicates thata positive Λ suppresses the collapse of the scalar field,while a negative Λ enhances it. This can be understoodsince a positive cosmological constant provides an addi-tional negative pressure, while a negative cosmologicalconstant does the opposite.When the magnitude of the cosmological constant isnear to zero, the dependence of the critical amplitude onΛ is almost linear, which is consistent with the result in[16]. B. Critical black hole
Mass gaps between the branches of collapsed scalarfields are also found in our model, which are first noticed M g ϵ (× - ) M a ss ( × - ) FIG. 2. Mass gap between the zero and first bounce brancheswhen Λ = 0. The inset shows the details in the red circle. by Santos-Oliv´an and Sopuerta [9, 10] in the asymptot-ically AdS case. The reason why there are such massgaps is that the subcritical configurations have to travelto the boundary and come back, suffering from a finitecompression, according to [9, 10]. Different cosmologicalconstant may have different effect on this compressionprocess.Since the critical amplitudes (cid:15) n for each Λ have al-ready been found in the previous section, we can run asimulation for each (cid:15) n to get the critical radius of theblack hole and calculate the corresponding Misner-Sharpmass. However, in these cases, the scalar field has alreadydeveloped very sharp feature before the bounce, whichcauses great numerical errors during evolution. Instead,we suppose the AH mass follows a power-law behavior ofthe type M AH − M n +1 g ∝ ( (cid:15) n − (cid:15) ) ξ , following [9]. We per-form a series of subcritical simulations near each criticalamplitude, and find the best fitting for M n +1 g and ξ (seeFig. 2). Interestingly, the collapse time near each criticalamplitude is also found to be well fitted by a power-lawof the type T AH − T n +1 g ∝ ( (cid:15) n − (cid:15) ) ζ . The time gaps T n +1 g and exponent ζ can be calculated in a similar way (seeFig. 3) Since the exponent ξ is universal (in the sensethat it is the same for all the mass gaps/branches [10]and it is also confirmed in our case), here we only focuson the critical behavior near the first mass gap M g forsimplicity.
1. Power-law of black-hole mass
Fig. 4 shows ln( m ) as a function of ln( a ), where a =( (cid:15) − (cid:15) ) /(cid:15) and m = M AH − M g . We vary the cosmologicalconstant Λ from − .
75 to 1 .
75 with the position of themirror fixed ( R = 1). As we can see, the power-lawscaling of masse is clear for each Λ. We show the resultsfor the fittings in Table I.In [16], Hod and Piran showed that the exponent γ ofChoptuik’s mass scaling: M AH ∝ ( (cid:15) − (cid:15) ) γ for supercrit-ical configurations is immune to the existence of cosmo- T g ϵ (× - ) T i m e FIG. 3. Time gap between the zero and first bounce brancheswhen Λ = 0. The inset shows the details in the red circle. ◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦Λ = ξ = - - - - - - ( a ) l n ( m ) ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦Λ = ξ = - - - - - - - ( a ) l n ( m ) ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦Λ =- ξ = - - - - - - - ( a ) l n ( m ) ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦Λ =- ξ = - - - - - - ( a ) l n ( m ) FIG. 4. The black hole mass scaling: ln( m ) vs ln( a ) for dif-ferent Λ, where m = M AH − M g and a = ( (cid:15) − (cid:15) ) /(cid:15) . Thepoints are well fitted by a straight line whose slope ξ increasesas we decrease the cosmological constant Λ. logical constant Λ. It can be understood as follows. Thecritical solution which determines γ shows its structureon smaller and smaller spatial (and temporal) scales as itevolves. And the effect of cosmological constant Λ on thecritical solution becomes less and less important and ul-timately disappears when a naked singularity forms. Sothe exponent γ is expected to be independent of Λ. How-ever, the situation is different when it comes to the massscaling on the left side of the mass gaps. The criticalsolution which determines the exponent ξ collapses intoa black hole of a finite size, instead of a naked singular-ity. So the cosmological constant Λ always has a finitecontribution. We expect the exponent ξ to be differentwhen Λ is different, which is conformed in Table I.The exponent ξ grows as we decrease the cosmologicalconstant Λ. We expect it to approach the exponent inthe AdS limit which is around 0 .
70 [10], as Λ goes to −∞ .As we can see in Table I, the precision in the simulationis not so good when the magnitude of Λ is large. Thisis due to the fast adding or deleting grid points in thesecases, which causes extra numerical errors.The mass gap M g corresponds to the mass of critical Λ Mass gap ( M g × ) Exponent( ξ )1.75 6.8 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± M AH − M g ∝ ( (cid:15) − (cid:15) ) ξ . black holes into which the first subcritical configurationcollapses. We plot the mass gaps M g for different Λin Fig. 5(a).It shows a very interesting behavior. Weobserve a nearly linear increasing behavior of M g as weincrease Λ in the region Λ ∈ [ − . , . M g decreases and it starts toshrink around Λ = 1 . M g when Λ ∈ [ − . , . M g may bedecreased when cosmological constant is increased. Toseparate the effects coming from the total mass and seethis effect indeed happens clearly, we consider the massratio ρ ≡ M g /M total , which describes the ratio of blackhole mass to the total mass. The dependence of ρ on Λis displayed Fig. 5(c). The result is consistent with theargument we just give above: ρ is negatively correlatedto Λ. So the behavior of mass gap is just the competitionof this two effects and there is indeed an inflexion pointshown in Fig. 5(a).Fig. 5(c) shows a monotonous dependence of Λ. Twoextremal case that Λ → −∞ and Λ → Λ dS are very inter-esting, though our numerical solver can’t directly explorethem. For the former case, the system should recover tothe asymptotic AdS case, where the value of M g /M total isa finite nonzero value. For the latter one, when Λ > Λ dS ,cosmological horizon is now located inside the mirror, so ( a )- - - Λ M g ( b )- - - Λ M t o t a l ( c )- - - Λ ρ FIG. 5. Masses for different cosmological constant Λ. Top:Mass gap M g , Middle: Total mass of the configuration whichcollapses into a black hole with masse M g , Bottom: The massratio ρ = M g /M total the energy will be absorbed into cosmological horizon andthe subcritical configuration can not be reflected on theboundary to start its second implosion. When Λ → Λ dS ,as the wall is near to the cosmological horizon, one canexpect the most part of the energy will be holden by thereflecting wall and the initial black hole mass will con-tain a little part of the total energy. Then we have avery interesting conjecture that ρ will approach to somefinite value ρ as Λ goes to −∞ and ρ → → Λ dS .
2. Power-law of collapse time
The step-like increase of collapse time is common whenone considers the gravitational collapse of some matterfields in bounded domains [7, 8, 11]. We confirm thisstructure of collapse time in our simulations. What ismore interesting is that when we make a more close look ◦◦◦◦◦◦◦◦◦ ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦Λ = ζ = - - - - - - - - ( a ) l n ( t ) ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦Λ = ζ = - - - - - - - - ( a ) l n ( t ) ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦Λ =- ζ = - - - - - - - - ( a ) l n ( t ) ◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦◦Λ =- ζ = - - - - - - - - ( a ) l n ( t ) FIG. 6. The time scaling: ln( t ) vs ln( a ) for different Λ, where t = T AH − T g and a = ( (cid:15) − (cid:15) ) /(cid:15) . The points are well fittedby a straight line whose slope ζ increases as we decrease thecosmological constant Λ.Λ Time gap ( T g ) Exponent( ζ )1.75 3.620 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± T AH − T g ∝ ( (cid:15) − (cid:15) ) ζ . at the collapse time near the critical amplitude, we find apower-law behavior: T AH − T g ∝ ( (cid:15) − (cid:15) ) ζ , similar to thepower-law relation of the black hole mass. Fig. 6 showsln( t ) as a function of ln( a ), where t = T AH − T g . Thepoints are well fitted by a straight line whose slope is ζ for each Λ. The fitting results are displayed in Table II.The exponent ζ also increases as we decrease the cos-mological constant Λ. Since the collapse time is less sen-sitive to the numerical errors than the collapse mass ofthe black hole, the precision of the parameters displayedin Table II is better than the one displayed in Table I.Fig. 7 shows the time gap of critical collapse T g for dif-ferent Λ. It grows monotonously as we increase Λ, whichis consistent with the argument we gave in the last sec-tion: a negative (positive) Λ introduces an enhancement(suppression) to the formation of black hole.As the same as what we have analyzed, we also conjec-ture that T g will tend to a finite value when Λ R → ∞ and tend to infinite when Λ → Λ dS . However, our numer-ical solver can’t give a clear evidence for this conjecture. - - - Λ T g FIG. 7. Critical time gap versus different Λ
It is worth further studying in the future.
IV. CONCLUSION
Though the confined asymptotic flat space-time andasymptotic AdS space-time share many same propertiesin black hole forming, such as weak turbulence, multipletype II critical gravitational collapses and so on, an in-teresting but still poor understood difference among thecritical exponents at the new gapped critical point hasbeen pointed out recently. The main aim of this paperis to try make some attempts to understand why sucha difference happens and what the roles of cosmologicalconstant and reflecting wall are in this difference.Though the cosmological constant and the position ofreflecting wall are both involved, as there is a scalingsymmetry, only the value of Λ R is relevant. Our nu-merical results showed a clear evolution of the criticalexponent ξ when we change the value of Λ R . This isa very interesting result and gives out a clear piece ofevidence to show that there are some unknown essentialdifferences compared with the Choptuik’s type II massscaling of supercritical configurations which has alreadybeen proved to be independent of the cosmological con-stant in Ref. [16]. In addition, we also found a new timescaling T AH − T g ∝ ( (cid:15) c − (cid:15) ) ζ for the forming time of thegapped black hole, where the critical exponent is alsodependent of the value of Λ R .Note that the Choptuik’s type II critical behaviorcoming from an emergent discrete (or continuous) self-similarity near in the critical region and the critical ex-ponent are related to the Lyapunov’s index [19, 20]. Theself-similarity induces a conformal symmetry. The massand the charge of scalar and the value of cosmologicalconstant are all irrelevant operators in such a conformaltransformation. It is interesting to see whether there isany such a discrete (or continuous) self-similarity in thenew gapped critical solution. However, based on the thispaper, it becomes clear that if such a self-similarity doseexist, the cosmological constant must be a relevant op-erator. We hope to report further study on this issue inthe future. ACKNOWLEDGMENTS
This work was finalized during a visit of R.G. Cai as avisiting professor to the Yukawa Institute for TheoreticalPhysics, Kyoto University, the warm hospitality extendedto him is greatly appreciated. This work was supportedin part by the National Natural Science Foundation ofChina under Grants No.11375247 and No.11435006, andin part by a key project of CAS, Grant No.QYZDJ-SSW-SYS006. [1] M. W. Choptuik, Phys. Rev. Lett. , 9 (1993).[2] M. W. Choptuik, T. Chmaj, and P. Bizon, Phys. Rev.Lett. , 424 (1996), arXiv:gr-qc/9603051 [gr-qc].[3] C. Gundlach, Phys. Rept. , 339 (2003), arXiv:gr-qc/0210101 [gr-qc].[4] C. Gundlach and J. M. Martin-Garcia, Living Rev. Rel. , 5 (2007), arXiv:0711.4620 [gr-qc].[5] V. Husain and M. Olivier, Class. Quant. Grav. , L1(2001), arXiv:gr-qc/0008060 [gr-qc].[6] F. Pretorius and M. W. Choptuik, Phys. Rev. D62 ,124012 (2000), arXiv:gr-qc/0007008 [gr-qc].[7] P. Bizon and A. Rostworowski, Phys. Rev. Lett. ,031102 (2011), arXiv:1104.3702 [gr-qc].[8] J. Jalmuzna, A. Rostworowski, and P. Bizon, Phys. Rev.
D84 , 085021 (2011), arXiv:1108.4539 [gr-qc].[9] D. Santos-Olivn and C. F. Sopuerta, Phys. Rev. Lett. , 041101 (2016), arXiv:1511.04344 [gr-qc].[10] D. Santos-Olivn and C. F. Sopuerta, Phys. Rev.
D93 ,104002 (2016), arXiv:1603.03613 [gr-qc].[11] M. Maliborski, Phys. Rev. Lett. , 221101 (2012),arXiv:1208.2934 [gr-qc].[12] R.-G. Cai and R.-Q. Yang, (2016), arXiv:1602.00112 [gr- qc].[13] A. Buchel, L. Lehner, and S. L. Liebling, Phys. Rev.
D86 , 123011 (2012), arXiv:1210.0890 [gr-qc].[14] A. Buchel, S. L. Liebling, and L. Lehner, Phys. Rev.
D87 , 123006 (2013), arXiv:1304.4166 [gr-qc].[15] H. Okawa, J. C. Lopes, and V. Cardoso, (2015),arXiv:1504.05203 [gr-qc].[16] S. Hod and T. Piran, Phys. Rev.
D55 , 3485 (1997),arXiv:gr-qc/9606093 [gr-qc].[17] D. Garfinkle, Phys. Rev.
D51 , 5558 (1995), arXiv:gr-qc/9412008 [gr-qc].[18] Because of the decomposition of the ( u, v ) componentof metric in Eq. 3, ¯ g and g are both singular at AH.This leads that ¯ g/g cannot really reach to zero. We set athreshold value and suppose that AH appears when it isless than this threshold.[19] C. Gundlach, Phys. Rev. D55 , 695 (1997), arXiv:gr-qc/9604019 [gr-qc].[20] T. Koike, T. Hara, and S. Adachi, Phys. Rev. Lett.74