Boson Stars in AdS
aa r X i v : . [ g r- q c ] J un UWO-TH-13/5
Boson Stars in AdS
Alex Buchel, , , Steven L. Liebling , and Luis Lehner Perimeter Institute for Theoretical PhysicsWaterloo, Ontario N2J 2W9, Canada Department of Applied MathematicsUniversity of Western OntarioLondon, Ontario N6A 5B7, Canada Department of Physics, Long Island UniversityBrookville, NY 11548, U.S.A.
Abstract
We construct boson stars in global Anti de Sitter (AdS) space and study their stability.Linear perturbation results suggest that the ground state along with the first threeexcited state boson stars are stable. We evolve some of these solutions and study theirnonlinear stability in light of recent work [1] arguing that a weakly turbulent instabilitydrives scalar perturbations of AdS to black hole formation. However evolutions suggestthat boson stars are nonlinearly stable and immune to the instability for sufficientlysmall perturbation. Furthermore, these studies find other families of initial data whichsimilarly avoid the instability for sufficiently weak parameters. Heuristically, we arguethat initial data families with widely distributed mass-energy distort the spacetimesufficiently to oppose the coherent amplification favored by the instability. From thedual CFT perspective our findings suggest that there exist families of rather genericinitial conditions in strongly coupled CFT (with large number of degrees of freedom)that do not thermalize in the infinite future.September 25, 2018
Introduction
Understanding the gravitational behavior of spacetimes which asymptoticallybehave as anti deSitter (AdS) has, since holography [2], attracted significantinterest. The AdS/CFT correspondence conjecture implies that such under-standing is critically important for a plethora of phenomena described by fieldtheories. Remarkably however, relatively little is known about dynamicalscenarios, especially in comparison to spacetimes which are asymptoticallyflat (AF) or asymptotically deSitter (dS). An important reason behind thisdifference is that the boundary of AdS is in causal contact with the interiorof the spacetime, in stark contrast to the boundaries of AF and asymp-totically dS spacetimes. That the boundary affects the interior prevents astraightforward extension of standard singularity theorems to asymptoticallyAdS (aAdS) spacetimes [3].Significant advances have recently been achieved via different approaches,including strictly analytic [4, 5, 6, 7], perturbative [8, 9] and numerical [1,10, 11, 12, 13, 14, 15] (to name a few representative) efforts. Particularlyintriguing is the evidence first presented in [1] that pure AdS is unstable toscalar collapse to black hole (BH), regardless of how small a scalar perturba-tion is considered. The effect of the AdS boundary allows for the reflectionof scalar pulses. Hence, a weak scalar pulse bounces off the boundary atinfinity, returning to concentrate again at the origin. The weakly turbulentinstability of [1] results in the sharpening of the pulse so that, eventually, itachieves sufficient concentration to form a black hole. The black hole criticalbehavior discovered by Choptuik [16] in AF spacetimes appears here repeat-edly. In particular, after each reflection off the AdS boundary, there is yetanother threshold for prompt BH formation (before the next bounce).To explain this behavior, a study of the normal modes of scalar [1] andtensor [9] perturbations revealed a non-linear mode coupling at third order,shifting energy to higher frequencies. Consequently, this shift in frequencywould eventually lead to the formation of a black hole and so generic instabil-ity of AdS was conjectured. Interestingly though, further studies suggestedthe existence of non-linearly stable solutions [8, 13, 15, 17]. These obser-vations hint that a straightforward application of the resonance picture canonly partially capture the dynamical behavior.We consider the dynamics of boson stars in AdS, motivated towards bet-ter understanding of this instability and its implications for holographic sce-narios. Horizon formation in gravitational collapse from the holographicperspective implies thermalization of the dual conformal theory. When thegravitational problem is formulated in global d + 1 space-time dimensionalasymptotically AdS d +1 , the dynamics of the dual d space-time dimensional CF T d occurs on a S d − sphere. Furthermore, the holographic duality arisesas a correspondence between a String Theory and a CFT — it can be trun-2ated to a gravity/CFT duality when a CFT has a large number of stronglyinteracting degrees of freedom . A natural expectation from the field theoryperspective is that a large number of strongly interacting degrees of freedomin a finite volume thermalize from generic initial conditions; thus, one doesexpect no-threshold BH formation as advocated in [1]. Early work on bo-son stars in AdS [18] demonstrated that such stationary configurations arelinearly-stable. Thus, initializing a CFT state as a dual to an AdS bosonstar might result in a slow thermalization of the latter.In this paper we show that, rather, boson stars in AdS are non-linearly stable. This non-linear stability is unaffected by boson star perturbations, aslong as these perturbations are sufficiently small. Even more surprising, wefind that families of initial conditions with widely distributed mass-energyare non-linearly stable for sufficiently small mass as well. This suggests thatnon-linear stability is not a feature of states carrying a global charge (as dualto boson stars). As a result, it appears that there exist large sets of initialconfigurations in CFT which never thermalize in their evolution.The paper is organized as follows. In section 2 we setup the gravitationaldual of the generic, spatially-isotropic CF T , initial condition specified by apair of dimension-3 operators with a global U (1) symmetry: an Einstein grav-ity with a negative cosmological constant and a massless, minimally coupledbulk complex scalar field. In section 3 we construct stationary configurationsof the coupled scalar-gravity system, charged under the global U (1) symme-try — boson stars. We show that boson stars are stable under linearizedfluctuations. In section 4 we present results of fully-nonlinear simulations ofgenuine boson stars, perturbed boson stars, and U (1)-neutral initial configu-rations with widely distributed, bulk mass-energy. We conclude in section 5and outline future directions. In this section, following [13], we review the formulation of the problem ofthe gravitational collapse of a complex scalar in asymptotically anti-de-Sitterspace-time. We focus on asymptotically
AdS collapse, dual to CF T (wechoose d = 3 in the notation of [13]).The effective four-dimensional action is given by S = 116 πG Z M d ξ √− g ( R + 6 − ∂ µ φ∂ µ φ ∗ ) , (2.1) This can be quantified as a large central charge for even d , or more generically, thelarge number of excited degrees of freedom at thermal equilibrium. We set the radius of AdS to one. φ ≡ φ + i φ is a complex scalar field and M = ∂ M × I , ∂ M = R t × S , I = { x ∈ [0 , π } . (2.2)Adopting the line element as in [19], ds = 1cos x (cid:18) − Ae − δ dt + dx A + sin x d Ω (cid:19) , (2.3)where d Ω is the metric of unit radius S , and A ( x, t ) and δ ( x, t ) are scalarfunctions describing the metric. Rescaling the matter fields as in [13]ˆ φ i ≡ φ i cos x , (2.4)ˆΠ i ≡ e δ A ∂ t φ i cos x , (2.5)ˆΦ i ≡ ∂ x φ i cos x , (2.6)we find the following equations of motion (we drop the caret from here for-ward) ˙ φ i = Ae − δ Π i , ˙Φ i = 1cos x (cid:0) cos xAe − δ Π i (cid:1) ,x , ˙Π i = 1sin x (cid:18) sin x cos x Ae − δ Φ i (cid:19) ,x . (2.7) A ,x = 1 + 2 sin x sin x cos x (1 − A ) − sin x cos xA (cid:18) Φ i cos x + Π i (cid:19) ,δ ,x = − sin x cos x (cid:18) Φ i cos x + Π i (cid:19) , (2.8)together with one constraint equation A ,t + 2 sin x cos xA e − δ (Φ i Π i ) = 0 , (2.9)where a sum over i = { , } is implied. We are interested in studying thesolution to (2.7)-(2.9) subject to the boundary conditions:Regularity at the origin implies these quantities behave as φ i ( t, x ) = φ ( i )0 ( t ) + O ( x ) ,A ( t, x ) =1 + O ( x ) ,δ ( t, x ) = δ ( t ) + O ( x ) ; (2.10)4t the outer boundary x = π/ ρ ≡ π/ − x so that we have φ i ( t, ρ ) = φ ( i )3 ( t ) ρ + O ( ρ ) ,A ( t, ρ ) =1 − M sin ρ cos ρ + O ( ρ ) ,δ ( t, ρ ) =0 + O ( ρ ) . (2.11)The asymptotic behavior (2.11) determines the boundary CFT observ-ables: the expectation values of the stress-energy tensor T kl , and the opera-tors O ( i )3 , dual to φ i ,8 πG h T tt i = M , h T αβ i = g αβ h T tt i , πG d +1 hO ( i )3 i = 12 φ ( i )3 ( t ) , (2.12)where g αβ is a metric on a round S . Additionally note that the conserved U (1) charge is given by Q = 8 π Z π/ dx sin x cos x (Π (0 , x ) φ (0 , x ) − Π (0 , x ) φ (0 , x )) , (2.13)and that since ∂ t Q = 0, the integral in (2.13) can be evaluated at t = 0.The constraint (2.9) implies that M in (2.11) is time-independent, ensur-ing energy conservation ∂ t h T tt i = 0 . (2.14)It is convenient to introduce the mass aspect function M ( t, x ) as A ( t, x ) = 1 − M ( t, x ) cos x sin x . (2.15)Following (2.8) we find M ( t, x ) = Z x dz tan z cos z A ( t, z ) (cid:20) Φ i ( t, z )cos z + Π i ( t, z ) (cid:21) . (2.16)Comparing (2.16) and (2.11) we see that M = M ( t, x ) (cid:12)(cid:12)(cid:12)(cid:12) x = π . (2.17)5 Boson stars in
AdS There is an interesting class of stationary, perturbatively stable, fully-nonlinearsolutions to (2.7)-(2.11) with nonzero Q , referred to as boson stars [18, 20].Such solutions are characterized by a discrete integer n = 0 , , · · · , denot-ing the number of nodes of the complex scalar radial wave-function, anda continuous value of the global charge Q . In this section we discuss thenumerical construction of such solutions, their perturbative properties, lin-earized stability, and their relation (for small Q ) to linearized AdS massless,minimally coupled, scalar modes — the oscillons . Oscillons were reviewedin detail in [13]. These stationary solutions are uniquely characterized by anexcitation level j = { , , · · · } : e j ( x ) = d j cos x F (cid:18) − j, j ; 32 ; sin x (cid:19) ,w ( j ) = 3 + 2 j , d j = (cid:18) j + 1)( j + 2) π (cid:19) / , (3.1)where w ( j ) is an oscillon frequency and d j is a constant enforcing their or-thonormality, Z π/ dx e i ( x ) e j ( x ) tan x = δ ij . (3.2) Assuming a stationary solution in which the complex field varies harmonically φ ( x, t ) + iφ ( x, t ) = φ ( x )cos x e iωt , A ( t, x ) = a ( x ) , δ ( t, x ) = d ( x ) , (3.3)we find a system of ODEs from (2.7) and (2.8)0 = φ ′′ + (cid:18) x sin x + a ′ a − d ′ (cid:19) φ ′ + ω e d a − φ , d ′ + sin x cos x a − (cid:0) ( φ ′ ) a + φ ω e d (cid:1) , a ′ + 2 cos x − x sin x (1 − a ) + sin x cos x a − (cid:0) ( φ ′ ) a + φ ω e d (cid:1) . (3.4)The charge and the mass determined by these solutions are given by Q =8 π Z π/ dx ω sin xφ ( x ) e d ( x ) a ( x ) cos x ,M = Z π/ dx sin xa ( x ) cos x (cid:0) a ( x ) ( φ ′ ( x )) + e d ( x ) ω φ ( x ) (cid:1) . (3.5)6 physically relevant solution to equations 3.4 must satisfy:At the origin of AdS, i.e. , x → a =1 −
13 ( p h ) ω e d h x + O ( x ) ,d = d h −
12 ( p h ) ω e d h x + O ( x ) ,φ = p h − p h ω e d h x + O ( x ) . (3.6)Note that besides ω , the general solution is characterized by { p h , d h } . (3.7)and asymptotically (at the AdS boundary, i.e. , ρ → a =1 + a b ρ + O ( ρ ) ,d = 32 ( p b ) ρ + (cid:18)
34 ( p b ) −
14 ( p b ) ω (cid:19) ρ + O ( ρ ) ,φ = p b ρ + (cid:18) p b − p b ω (cid:19) ρ + O ( ρ ) . (3.8)Note that besides ω , the general solution is characterized by { p b , a b } . (3.9)From (3.7) and (3.9) we have precisely the correct number of coefficientsto find an isolated solution for a given ω . In this section we discuss analytic results for the spectrum of boson starsperturbatively in the amplitude φ . To this end we introduce φ = λ φ + O ( λ ) , a = 1 + λ a + O ( λ ) ,e d = 1 + λ d + O ( λ ) , ω = ω + ω λ + O ( λ ) , (3.10)where λ is an expansion parameter. Substituting (3.10) into (3.4) we findthat the equation for φ decouples to leading order0 = φ ′′ + 2sin x cos x φ ′ + ω φ . (3.11) As we discuss in section 3.2 solutions of a fixed charge are labeled by an integer,specifying the “level” of a boson star. φ as φ (cid:12)(cid:12)(cid:12)(cid:12) x → + = 1 , (3.12)the general solution of (3.11), subject to the boundary conditions (3.6) and(3.8), is given by φ ( j )1 = 1 d j e j ( x ) , ω ( j )0 = w ( j ) = 3 + 2 j , (3.13)where the integer j = 0 , , · · · parameterize the ’excitation level’ of theboson star. Notice that the asymptotic expansion for φ (in eqn 3.6) implies p h = λ . Given (3.13), we find from (3.5) M = λ π (3 + 2 j ) j + 1)( j + 2) + O ( λ ) , Q = λ π (3 + 2 j )2( j + 1)( j + 2) + O ( λ ) ,M = 3 + 2 j π Q + O ( Q ) = ω ( j )0 π Q + O ( Q ) . (3.14)Note from (3.14) that for a fixed charge Q , excited levels of boson stars aremore massive.It is straightforward to compute the leading-order background warp fac-tors { a , d } , as well as subleading frequency correction ω . In what followswe present explicit expressions for the first four levels of a boson star: j = 0 level, a (0)2 = 9 cos x x (cid:18)
14 sin(4 x ) − x (cid:19) , (3.15) d (0)2 = 32 cos x , (3.16) ω (0)2 = − j = 1 level, a (1)2 = 25 cos x
72 sin x (cid:18)
14 sin(4 x )(2 cos(4 x ) + 1) − x (cid:19) , (3.18) d (1)2 = 518 cos x (cid:0)
32 cos x −
40 cos x + 15 (cid:1) , (3.19) ω (1)2 = − ω − ω ( j )0 λ + O ( λ ) d h λ + O ( λ ) p b λ + O ( λ ) a b λ + O ( λ )0 − − π − − − π − − π − − − π Table 1: Condensate values (3.6) and (3.8) of boson stars, perturbatively in p h = λ . j = 2 level, a (2)2 = 49 cos x
144 sin x (cid:18)
18 sin(8 x )(2 cos(4 x ) + 1) − x (cid:19) , (3.21) d (2)2 = 790 cos x (cid:0)
960 cos x − x + 1904 cos x −
700 cos x + 105 (cid:1) , (3.22) w (2)2 = − j = 3 level, a (3)2 = 81 cos x
400 sin x (cid:18)
18 sin(8 x )(2 cos(8 x ) + 2 cos(4 x ) + 1) − x (cid:19) , (3.24) d (3)2 = 9350 cos x (28672 cos x − x + 130752 cos x − x + 33264 cos x − x + 525) , (3.25) ω (3)2 = − . (3.26)Further comparing the above with (3.6) and (3.8), we identify the condensatesperturbatively in p h = λ (see Table 1). 9 .1 0.2 0.3 0.40.850.900.951.00 PSfrag replacements p b a b p h d h ( ω ( j ) /ω ( j )0 ) PSfrag replacements p b a b p h d h ( ω ( j ) /ω ( j )0 ) - PSfrag replacements p b a b p h d h ( ω ( j ) /ω ( j )0 ) - - - - - PSfrag replacements p b a b p h d h ( ω ( j ) /ω ( j )0 ) Figure 1: (Colour online) Condensate values (3.6) and (3.8) for level j = { , , , } ( { dashed purple, dotted green,dot-dashed blue, solid orange } curves) boson stars. The red lines represent perturbative in p h approxi-mations, see Table 1. In the previous section we identified the first four levels of a boson starperturbatively in the amplitude p h . Here we report results for { ω, d h , p b , d b } ,as well as { M, Q } , for generic p h from the numerical solution of (3.4). Theseresults are collected in Figs. 1 and 2.The top-left panel of Fig. 1 presents the frequency ω ( j ) of a level- j bosonstar rescaled to that of a level- j oscillon frequency ω ( j )0 (see (3.13)). We usepurple/green/blue/orange color coding to denote j = 0 · · ·
3. As p h (andcorrespondingly the mass and the charge — see Fig. 2 ) of a boson stargrows, its frequency decreases. The remaining panels in Fig. 1 present thedependence of { d h , p b , d b } as a function of p h . Notice that p b saturates; thissaturation is the main obstacle in generating boson stars with ever increasingvalues of p h (or mass). The red curves indicate perturbative approximationsin p h as collected in Table 1, (Fig. 1), and perturbative approximation (3.14)in Q (right panel of Fig. 2).In the limit of vanishing charge Q , the level- j boson star radial profile φ ( j ) is a single level- j oscillon (see (3.13), (3.14)) : φ ( j ) ( x ) ∝ p Q e j ( x ) . (3.27)10 .1 0.2 0.3 0.40.050.100.150.200.25 PSfrag replacements
MQ p h PSfrag replacements
M Qp h Figure 2: (Colour online) Mass vs. p h (left panel) and vs. charge (right panel)for level j = { , , , } ( { dashed purple, dotted green,dot-dashed blue, solidorange } curves) boson stars. The solid red lines represent perturbative in Q approximations, see (3.14).For finite Q all the oscillons are excited. In Fig. 3 we present the spectraldecomposition in the oscillon basis of the most massive level- j boson starsthat we were able to construct c ( j ) i ≡ (cid:12)(cid:12)(cid:12)(cid:12) Z π/ dx φ ( j ) ( x ) e i ( x ) tan x (cid:12)(cid:12)(cid:12)(cid:12) . (3.28)Note that the maxima of c ( j ) i are achieved for i = j , much like in the small- Q limit. For all levels considered c ( j ) i approach a universal fall-off: c ( j ) i ∝ (1 + i ) − , i ≫ j , (3.29)represented by a dashed black curve in Fig. 3. In this section we explore the linearized stability of boson stars. Considerperturbations of stationary solutions (3.3) to leading order in ǫ : φ ( x, t ) + iφ ( x, t ) = cos − x (cid:18) φ ( x ) + ǫ ( f ( t, x ) − iφ ( x ) g ( t, x )) (cid:19) e iωt ,A ( t, x ) = a ( x ) + ǫ a ( t, x ) ,δ ( t, x ) = d ( x ) + ǫ δ ( t, x ) . (3.30)Further introducing f ( t, x ) = F ( x ) cos( χt ) , g ( t, x ) = − G ( x ) sin( χt ) , (3.31)11 æ æ æ æ æ æ æ æ æ æ æ ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ ò ò ò ò ò ò ò ò ò òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò à à à à à à à à à à à ààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààààà ì ì ì ì ì ì ì ììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììììì - - - - - PSfrag replacements ln( i + 1)ln c ( j ) i Figure 3: (Colour online) Spectral decomposition of level j = { , , , } ( { purple circles,green triangles,blue squares,orange diamonds } ) boson starsin oscillon basis, see (3.29). For large i all the curves approach a universalfall-off c ( j ) i ∝ (1 + i ) − (the black dashed curve).the equations for a ( t, x ) and δ ( t, x ) can be solved explicitly: a ( t, x ) = sin(2 x ) a ( x ) (cid:0) ω φ ( x ) G ′ ( x ) − φ ′ ( x ) F ( x ) (cid:1) cos( χt ) , (3.32) δ ( t, x ) = − e − d ( x ) cos( x ) φ ( x ) ω sin( x ) (cid:18) a ( x ) cos( x ) φ ( x ) G ′′ ( x ) sin( x ) + ( − x ) φ ( x ) e d ( x ) sin( x ) ω + a ( x )(2 a ( x ) cos( x ) φ ′ ( x ) sin( x )+ 2 a ( x ) cos( x ) φ ( x ) − a ( x ) φ ( x ) − x ) φ ( x ) + 3 φ ( x ))) G ′ ( x )+ cos( x ) φ ( x ) e d ( x ) G ( x ) χ sin( x ) − x ) F ( x )(cos( x ) φ ( x ) φ ′ ( x ) − cos( x ) φ ( x ) φ ′ ( x ) − sin( x )) ωe d ( x ) (cid:19) cos( χt ) . (3.33)where F ( x ) and G ( x ) satisfy a coupled system of equations0 = F ′′ + − x ) + 2 cos( x ) a + 3 − aa sin( x ) cos( x ) F ′ − φω G ′′ + 2 ωa sin( x ) cos( x ) (cid:18) x ) φ φ ′ cos( x ) + 2 cos( x ) φ − aφ cos( x ) − x ) aφ ′ cos( x ) − x ) φ φ ′ cos( x ) + aφ − φ (cid:19) G ′ − (cid:18) a cos( x ) ( φ ′ ) − e d χ − φ ′ ) a + 3 e d ω (cid:19) a − F , (3.34)12 = G ′′′ + (cid:18) − x ) φ + 6 aφ cos( x ) + 2 sin( x ) aφ ′ cos( x ) + 9 φ − aφ (cid:19)(cid:18) aφ sin( x ) cos( x ) (cid:19) − G ′′ + (cid:18) a φ + 4 a sin( x ) φ ′ φ cos( x ) − a sin( x ) φ ′ φ cos( x ) − a sin( x ) φ ′ φ cos( x ) + 6 a sin( x ) φ ′ φ cos( x )+ 2 e d aφ ω cos( x ) − e d a cos( x ) φ χ − e d aφ ω cos( x ) + e d a cos( x ) φ χ + 2 a ( φ ′ ) cos( x ) φ − a ( φ ′ ) cos( x ) φ + 3 a ( φ ′ ) cos( x ) φ + 9 e d cos( x ) φ ω + 6 e d cos( x ) φ ω − e d cos( x ) φ ω − a φ cos( x ) + 8 a φ cos( x ) + 9 aφ − a φ + 24 a cos( x ) φ − a cos( x ) φ − a cos( x ) φ + 4 a cos( x ) φ + 2 a ( φ ′ ) cos( x ) − a ( φ ′ ) cos( x ) (cid:19)(cid:18) a φ sin( x ) cos( x ) (cid:19) − G ′ + 2 e d ωa φ F ′ + (4 φ cos( x ) − a − φ ) e d ωφ ′ (cid:18) a φ (cid:19) − F . (3.35)Notice that (3.34)-(3.35) are left invariant under the shift G → G + G , (3.36)where G is an arbitrary constant. From (3.33) is it clear that this constantis fixed uniquely requiring thatlim x → π/ δ ( t, x ) = 0 , (3.37) i.e. , we keep the time coordinate at the boundary fixed. We do not haveto worry about the shift symmetry (3.36), provided we rewrite (3.34)-(3.35)using dG ( x ) ≡ G ′ ( x ) . (3.38)Eqs. (3.34)-(3.35) must be solved subject to constraints that F ( x ) and G ( x ) φ ( x ) are regular for x ∈ [0 , π/ x → π/ i.e. , F ∝ ρ , G ∝ const , as ρ → . (3.39)The latter regularity condition implies that G can have a simple pole (or dG can have a double pole) precisely where φ ( x ) has a zero . These polesrepresent a technical difficulty in identifying the fluctuations about excitedboson stars — specifically, a straightforward shooting method: integrating Recall that excited level stationary boson stars are characterized by the number ofnodes in the radial profile φ ( x ), see (3.3). .Finally, since the system of equations (3.34)-(3.35) is linear, we can fur-ther fix the normalizable mode F :lim ρ → F ρ = 1 . (3.40)Given (3.39) and (3.40), we can specify the boundary conditions for { dG , F } :at the origin of AdS, i.e. , as x → dG =2 g h x + O ( x ) , F = f h + O ( x ) ; (3.41)at the AdS boundary, i.e. , as ρ → dG =2 g b ρ + O ( ρ ) , F = ρ + O ( ρ ) . (3.42)Note that along with χ , the physical solution { dG , F } is characterizedby { g h , f h , g b } — which is the correct number of coefficients necessary todetermine a unique (or isolated) solution for a pair of coupled second-orderODEs (3.34)-(3.35).Using the boundary conditions (3.41) and (3.42) it is easy to see that thecharge of a fluctuating boson star does not change to leading order in ǫ : δQ ∝ Z π/ dx ddx (cid:26) sin( x ) φ ( x ) a ( x ) G ′ ( x ) e d ( x ) cos( x ) (cid:27) = 0 . (3.43) We report here the results for solving (3.34)-(3.35) for light boson stars, i.e. ,perturbatively in λ , see section 3.2. In general, we search solution to aboveequations as a series dG ( j )1 ( x ) = λ dG ( j )1 , + λ dG ( j )1 , + O ( λ ) ,F ( j )1 = F ( j )1 . + λ F ( j )1 , + λ F ( j )1 , + O ( λ ) ,χ ( j ) = χ ( j )0 + λ χ ( j )1 + λ χ ( j )2 + O ( λ ) ; (3.44) We verified explicitly that while dG (1)1 to order O ( λ ) has a double pole at the locationof the zero of φ (1) (constructed perturbatively in λ to order O ( λ ) inclusive — see (3.10)),the full radial profile of physical fluctuations, G (1)1 φ (1) , is smooth for x ∈ [0 , π/ j is the excitation level of a boson star.We find : j = 0 level, χ (0)0 = 6 , χ (0)1 = − , χ (0)2 = 1215128 π − , (3.45) F (0)1 , = cos x , (3.46) F (0)1 , = cos( x ) x ) (5760 cos( x ) sin( x ) − x ) sin( x )+ 18738 cos( x ) sin( x ) + 7560 cos( x ) x − x ) sin( x ) − x cos( x ) + 1890 sin( x ) π − x sin( x )) − x ) (32 cos( x ) −
54 cos( x ) + 27) C , , (3.47) dG (0)1 , = − x ) cos( x )(5760 cos( x ) − x ) + 9900 cos( x ) − x ) cos( x )(8 cos( x ) − x ) − C , . (3.48)The integration constant C , is not fixed at order O ( λ ), but is uniquelydetermined at order O ( λ ): C , = 3163421120 . (3.49) j = 1 level, χ (1)0 = 10 , χ (1)1 = − . (3.50) j = 2 level, χ (2)0 = 14 , χ (2)1 = − . (3.51) j = 3 level, χ (3)0 = 18 , χ (3)1 = − . (3.52) j = 0 boson stars The spectrum of linearized fluctuations about j = 0 boson stars is presentedin Fig. 4. We find that over the whole range of charges Q we were able For excited levels we present only the coefficients { χ ( j ) i } , i = 0 , , j = 1 · · · This pattern extends to higher orders in λ : a solution at order λ n , { F , n , G , n − } ,is determined up to a constant C ,n , which is being uniquely fixed at order O ( λ n +1) ). .1 0.2 0.3 0.430323436 PSfrag replacements χ p h = λ Figure 4: (Colour online) Spectrum of linearized fluctuations about j = 0boson stars as a function of p h (black dots). The dashed orange/solid redcurves are successive approximations to χ = (cid:0) χ ( p h ≡ λ ) (cid:1) in λ , see (3.44).to construct j = 0 boson stars, the frequency of their fluctuations squared( χ ) is positive. This strongly suggests that the ground state boson stars areperturbatively stable.As discussed in the previous section (see (3.50)-(3.52)), excited bosonstars are perturbatively stable for small charge. Our numerical simulationssuggest that both the j = 0 and the excited boson stars are nonlinearlystable. We take the constructed boson star solutions described above { φ ( x ) , d ( x ) , a ( x ) , ω } and employ them to provide initial data for our dynamical studies via φ i = φ cos x δ i , (4.1)Φ i = φ ′ cos x δ i , (4.2)Π i = ωφe d a δ i , (4.3)where the metric functions are obtained by solving the constraints. We con-firm convergence of the obtained solutions (by monitoring the constraintresiduals, charge and mass conservation and self-convergence vs time) as res-olution is increased (see also [13]). 16 erturbed, Genuine Boson Stars: We concentrate on studying thebehavior of these solutions when perturbed, and have considered variousforms of perturbation with qualitatively similar results. For concreteness, wehere present results obtained with Gaussian perturbations parametrized as G ( x ) = ǫe − ( r − R ) / ∆ and add it to the boson star solution via φ i = (cid:2) φ/ cos x + G ( x ) (cid:3) δ i , (4.4)Φ i = [ φ ′ / cos x + G ′ ( x )] δ i , (4.5)Π i = (cid:2) ωφ (cid:0) e d /a (cid:1) + G ′ (cid:3) δ i . (4.6)In analogy with previous studies, we set the amplitude of the Gaussian per-turbation to ǫ . We note that because the constraints are solved numericallyat the initial time for a ( x,
0) and δ ( x, | − A ( x, t ) | and δ (0 , t )) and they showedno signs of instability to BH formation. These observations prompted athorough study of the instability in AdS; independent work via perturbativestudies also pointed out that AdS should be stable for several families ofsolutions [8].A variety of boson star solutions were studied, including members oflevels 0, 1, and 3 (level 2 solutions presented regularity issues near the AdSboundary and we defer such analysis for future work). All examples appearedstable. Interestingly, in asymptotically flat scenarios, excited boson starsare generally unstable, radiating energy and settling into a ground statesolution [20]. In AdS, however, there is no way to rid itself of excess charge,which presumably explains their stability. Nevertheless, this property of AdSdoes not explain how the boson star can be immune to the weakly turbulentinstability. Below, we present an argument to this end, but first we discussthe behavior of a different family that lends support to our argument.17 ake Boson Stars: Of course numerical evolutions are limited to finitetimes, and so one cannot rule out that instability will manifest after the codehas been stopped or beyond the time for which one trusts the results. Tobetter assess the observed behavior, we compare these long-lived solutionsto a different family which can be considered “nearby” in some sense. Thisfamily, which we refer to as fake boson stars , represents purely real initialdata with the same mass and profile as their counterpart genuine boson stars.A fake counterpart of some boson star solution of (4.3) is achieved via thetransformation φ fake1 = φ BS1 , Π fake1 = Π BS2 , Π fake2 = 0 . (4.7)Remarkably, the evolution of this family also yields regular, long-livedsolutions for small perturbations that do not collapse to a black hole. Figs. 5and 6 illustrate the time of collapse as a function of ǫ both for genuine andfake boson stars. As indicated in the figures, successively higher resolutionslargely coincide with differences only apparent at the latest times. In allcases, the results indicate collapse times increasing quickly as ǫ decreaseswith no signs of collapse for smaller amplitudes of perturbation.Notice that these fake solutions are not stationary and have no charge,two seemingly essential features of genuine boson stars, and so their apparentimmunity to this weakly turbulent instability is surprising. This “stability”is apparently not tied to special features (e.g. charge or stationarity) butinstead suggests that the dynamics undergoes something akin to a frustratedresonance in which amplitudes increase at times but then disperse. In par-ticular, one essential aspect common to both genuine and fake boson starsappears to be their non-compact, long-wavelength nature. Because they haveenergy distributed throughout the domain, modes no longer propagate co-herently. Instead there is a continuing competition between dispersion andgravitational contraction; collapse to a black hole or not is then determinedby the outcome of this competition. Large σ : Admittedly, this argument is far from rigorous. But if it holds,then it would imply many other forms of stable initial data. In particular,perhaps other forms of initial data may be immune to this weakly-turbulentinstability when its extent is large. To explore this conjecture, we adopt thesame form of data considered in many previous studies of this instability (suchas those in [1, 13, 19, 21, 22]). We thus consider this family again, whichtakes the following form in our rescaled variablesΦ i (0 , x ) = 0 , Π i (0 , x ) = 2 ǫπ e − xπ σ cos − d x δ i . (4.8)To test the possibility of regular development of this data, we consideredthe time development of Eq. (4.8) with varying values of σ . The results are18igure 5: (Colour online) Collapse times for Gaussian perturbations of aground state boson star ( φ (0 ,
0) = 0 . ǫ evolutions show no sign of collapse.19igure 6: (Colour online) Collapse times for Gaussian perturbations of a firstexcited state boson star ( φ (0 ,
0) = − . ǫ for various values of σ . As is evident from the figures, for small values of σ ( σ < .
3) the collapse time increases monotonically as ǫ decreases; howeverfor larger values of σ the collapse time increases abruptly as ǫ is decreased.Notice that this abrupt growth in collapse time behaves quite similarly tothat seen for boson stars and fake stars, suggesting that for sufficiently large σ , the behavior would be regular. Furthermore, an analysis of the Fourierpower spectra of cases below σ ≈ . σ ≈ . σ initial data is immune to the weakly turbulent instabilityis consistent with the argument that widely distributed mass energy pre-vents the coherent amplification typical of the instability. It is interestingto consider what would happen in the semilinear wave equation on a fixedAdS background as studied in [22]. That model shows many of the samecharacteristics as the gravitating scalar collapse, but the nonlinear potentialplays the role of the attractive, focusing effect that gravity plays here. How-ever, numerical evidence from that model suggests that there is no large- σ effect, lending support to the idea that the distributed mass-energy affectsthe spacetime in a way that disturbs the coherent amplification.A change in behavior such as this merits a closer examination of the“transition region” between apparent stability and black hole collapse. Fig. 9illustrates this region 0 . ≤ σ ≤ . ǫ is decreased.From Figs. 7 and 9, it is clear that for σ ≥ . ǫ min belowwhich initial data does not form a black hole. The idea of this function ǫ min ( σ ) is similar to ǫ min ( x max ) studied in [22]. Preliminary study of ǫ min ( σ )shows it to be a roughly exponentially decreasing function (after the apparentdiscontinuity at σ ≈ . ǫ min = 0 → . ǫ min is demonstrated in Fig. 11. In particular, for this weak initial data when themetric is frozen at its initial profile, the evolution demonstrates dispersion.It is instructive to study the spectral decomposition of the initial data(4.8) in the oscillon basis for different σ . To relate with the analysis in Fig. 7,we keep σǫ = 1 fixed. For a select set, i.e. , σ = { . , . , . , · · · . } , wecompute the spectral coefficients c i ( σ ), see (2.5), c i = (cid:12)(cid:12)(cid:12)(cid:12) ω ( i ) Z π/ dx tan x A (0 , x ) e − δ (0 ,x ) Π (0 , x ) e i ( x ) (cid:12)(cid:12)(cid:12)(cid:12) , (4.9)where A (0 , x ) and δ (0 , x ) are obtained from integrating (2.8) with initialdata (4.8). The resulting spectral decompositions are collected in Fig. 10.21omparing with the spectral decomposition of boson stars (see Fig. 3), here,the large- j decay of the spectra is approximately exponential, instead of apower-law as in (3.29). The spectral profile achieves a minimum around σ ∼ . − .
4, which is roughly the critical value of σ separating the stableand unstable regions in the parameter space of the initial data (4.8). Restricted domain:
The last nonlinear effect presented in this sectionconcerns evolutions conducted in a restricted domain. In particular, an artifi-cial, reflecting boundary condition is applied at some x max < π/
2, restrictingthe propagating pulse to some subdomain of AdS. The motivation for this isto study whether this turbulent instability is itself just the manifestation ofthe nonlinear attraction of gravity occurring in a bounded domain, or insteadsome particular property of the full AdS (and hence would be destroyed bythis restriction).As found in [13], the imposition of such a reflecting boundary conditiondoes not eliminate black hole formation after multiple bounces. However, aminimum value of ǫ was found, below which no such black hole formationoccurred. In the semilinear model, it was found that the boundary conditionresulted in dispersion not seen with the full AdS domain.Here, we revisit this problem, showing the customary time of collapse plotin Fig. 12. As mentioned, there is some similarity with the large- σ effect inthe existence of some ǫ min . However, in contrast, one does observe the “stair-step” decrease in collapse time for ǫ increasing above ǫ min , characteristic ofsuccessive bounces.It is also possible that such restricted-domain evolutions result from twoeffects: (i) the imposition of the reflecting wall introduces dispersion as in thesemilinear model, and (ii) as the domain shrinks, there may be some effectdue to the fact that, for fixed σ , the fractional support of the initial data isincreasing. We have constructed boson star solutions in global AdS and shown thatthey are stable at linear order. Numerical studies of their dynamics stronglysuggest that these boson stars, both ground state and the first few excitedstates, are non-linearly stable. Along with solutions presented in [15], there isnow considerable evidence that the instability of AdS to scalar perturbationsreported in [1] is limited in scope; that there exist non-trivial, dynamicalexamples of stable solutions in AdS. These results are consistent with theperturbative arguments of [8] for the stability of boson stars, geons andsolitons.Comparison of the lifetimes of perturbed boson stars with other, non-stationary solutions, our fake boson stars, reveals an even wider class of22igure 7: (Colour online) Collapse times for initial data of the form Eq. (4.8)with varying width values, σ . Because changes to σ affect the amount ofmass, the natural parameter against which to plot is σǫ not just ǫ (also seeFig. 8 for this data plotted versus ǫ ). Note that for σ . . ǫ . In contrastfor σ & .
3, there appears to exist a threshold ǫ ∗ below which collapse doesnot occur. For initial data above the transition, σ > .
3, evolutions withsmaller ǫ than shown reached at least t ≈ σ increases and the behavior of the time of collapse, t c ,changes dramatically from the “usual” stair-step to something else entirelywith a very sharp transition. Note that the spacing between runs is verynon-uniform and that tuning is necessary to resolve the transition.24igure 9: (Colour online) Collapse times for a range of σ that demonstratesthe transition from turbulent BH formation to frustrated resonance. For σ = 0 .
3, collapse appears inevitable for any value of ǫ in contrast to theresults for σ = 0 .
4. Interestingly, a “bump” appears for these values of σ in which the collapse times demonstrate a lack of monotonicity. Beginningwith σ = 0 . ǫ ≈
8, one sees a small bumpthat, as one looks to higher σ , sharpens and occurs at smaller ǫ values.25 ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òà à à à à à à à à à à à à à à à à à à à à à à à à à à à à à àò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ ƒ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ - - - - PSfrag replacements i ln( c i ( σ ))Figure 10: (Colour online) Spectral decomposition of the initial data (4.8)with σ = { . , . , . , . , . , . , . , . } (yellow circles, blue squares,cyan triangles, magenta squares, black triangles, red stars, green squares,orange circles) and σǫ = 1 in the oscillon basis. Spectral coefficients c i ( σ ) (see(4.9)) start relatively large at high oscillon numbers for σ = (circles); theydecrease for σ = 0 . σ = 0 . σ = 0 . σ = 0 . σ = 0 . σ = 0 . σ = 0 . σ ∼ . − . σ ' . σ , indicating that gravity plays a key role in thedynamics. In particular, a heuristic argument suggests that the widely dis-tributed mass-energy distorts the space sufficiently to introduce dispersionand thereby oppose the concentrating effect of the instability.The picture that emerges is a phase space with (at least) two regions.One region is subject to the weakly turbulent instability and therefore col-26igure 11: (Colour online) Demonstration of dispersion introduced by widelydistributed mass-energy. Shown is the behavior of Π at the origin during theevolution of ǫ = 1, σ = 1 initial data. By keeping A = 1 and δ = 0, we evolvethe scalar field in a pure AdS background, and this results in a periodicsolution (dashed, red line). Instead, by solving for the initial metric A ( x, δ ( x, < t < π anddisplay it shifted in time (dot-dashed black); that it overlays the pure AdSsolution shows that scalar solutions on a background of AdS are periodic.This figure is similar to Fig. 5 of [22] which shows the dispersion introducedby a restricted domain. 27igure 12: (Colour online) Collapse times of the initial data in Eq. (4.8) with σ = 1 /
16 with an artificial, reflecting wall at a various positions, x max . Alsoshown (solid black) are the results for the full domain with no reflecting wall.Note that the finite domain evolutions demonstrate a threshold ǫ min belowwhich collapse does not occur. 28apses for any initial “amplitude.” The other region can avoid the turbulentinstability and contains oscillons, boson stars, geons, and similar symmetricsolutions. However, it appears this second “stable” region contains a widerclass of solutions with no periodicity or stationarity, namely fake boson stars,large- σ and similarly distributed families of initial data. To be clear, this sec-ond region need not be strictly stable and can certainly possess collapsingsolutions. What is important is that in this second region, one can choosea sufficiently small amplitude such that the weakly turbulent instability isavoided.We do not know at this stage what is the precise criterion that separatesthe parameter region of scalar field initial data resulting in BH formationfrom the region of non-linear stability. It can be argued that such a criterionis encoded in the spectral decomposition of the initial data in the oscillonbasis. Indeed, consider the region (assuming it exists ) where collapse occursfor arbitrarily small amplitude of the scalar field. In this limit, the full initialdata is the oscillon spectrum, as the backreaction can be safely ignored. Inthis paper we presented strong evidence for initial configurations that do notcollapse in the limit of vanishingly small amplitude. Thus, the distinctionbetween stable and non-stable configurations (at least for small amplitudes)must be hidden in the initial scalar field spectral data. We have seen thatinitial profiles of [1] (4.8) become stable for small ǫ as σ increases; the latterincrease results in softening the decay of the asymptotic oscillon spectralcoefficients (see Fig. 10). Likewise, boson stars have an asymptotic power-lawoscillon spectral decomposition, in contrast to the exponential-decay profilefor initial data (4.8) (see Fig. 3).It is important to further investigate the nonlinear stability of AdS. Theissue has profound implications for a dual boundary conformal field theory,as it identifies CFT initial configurations that fail to thermalize. There isno obvious symmetry criterion “protecting” such configurations. A possiblefuture direction is to investigate the collapse of initial configurations specifiedby their oscillon spectral decompositions with various trial profiles c i . Theprimary goal, of course, is the identification of the stability criteria — itis possible that the latter can be established from analysis of the weaklynonlinear regime only. Finally, it is interesting to analyze the stability ofmore general configurations — such as boson stars with local bulk charge, asrecently discussed in [23, 24, 25]. Acknowledgments:
It is a pleasure to thank Oscar Dias, Chad Hanna,Gary Horowitz, Pavel Kovtun, Robert Myers, Andrzej Rostworowski andJorge Santos for interesting and helpful discussions. This work was sup-ported by the NSF (PHY-0969827 to Long Island University) and NSERC Numerical analysis of [1] strongly supports that this is the case.
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