Position space analysis of the AdS (in)stability problem
Fotios V. Dimitrakopoulos, Ben Freivogel, Matthew Lippert, I-Sheng Yang
PPreprint typeset in JHEP style - HYPER VERSION
Position space analysis of the AdS (in)stability problem
Fotios V. Dimitrakopoulos, Ben Freivogel, Matthew Lippert, and I-Sheng Yang
ITFA and GRAPPA, Universiteit van Amsterdam,Science Park 904, 1090 GL Amsterdam, Netherlands
Abstract:
We investigate whether arbitrarily small perturbations in global AdS spaceare generically unstable and collapse into black holes on the time scale set by gravi-tational interactions. We argue that current evidence, combined with our analysis,strongly suggests that a set of nonzero measure in the space of initial conditions doesnot collapse on this time scale. We perform an analysis in position space to study thispuzzle, and our formalism allows us to directly study the vanishing-amplitude limit. Weshow that gravitational self-interaction leads to tidal deformations which are equallylikely to focus or defocus energy, and we sketch the phase diagram accordingly. We alsoclarify the connection between gravitational evolution in global AdS and holographicthermalization. a r X i v : . [ h e p - t h ] D ec ontents
1. Introduction 12. Stability islands or instability corners? 63. Weak gravitational self-interaction in position space 8
4. Focusing and defocusing 15
5. Discussion 21
6. Summary 29A. Analytical details 30B. Numerical examples 34
B.1 The asymmetry-focusing correlation 34B.2 The effect of another object 37
1. Introduction
Recently, the gravitational (in)stability of classical gravity in asymptotically globalAdS spacetime has drawn a lot of attention and also generated a lot of confusion.Given a spherically symmetric perturbation of arbitrarily small initial amplitude (cid:15) , twodramatically different behaviors have been observed at the timescale ∼ (cid:15) − , the earliest– 1 –ime on which interactions can have a significant effect [1–13]. Sometimes a black holeforms around this time; sometimes a long-lived quasi-periodic behavior emerges andgravity does not become strong. This is a great puzzle concerning both the gravitationaldynamics in the bulk and the corresponding thermalization process in the holographicboundary theory.In this paper we will focus on the bulk perspective and on the simple case of afree massless scalar field coupled to gravity. We treat the system classically and imposespherical symmetry. In the limit of small amplitude (cid:15) , the energy density is proportionalto (cid:15) and controls the strength of gravitational effects. Therefore, behavior at the timescale (cid:15) − is sensitive to the leading-order effects of gravitational interactions. Oneframework to study this is to analyze the nonlinear couplings between the linearizedmodes induced by the gravitational interactions. Linearized modes in AdS space allhave frequencies which are integer multiples of the AdS scale. A mode that is initiallyunexcited can be resonantly driven by the excited modes, which allows for the possibilityof efficient transfer of energy. Such efficient energy transfer between modes genericallyleads to the breakdown of na¨ıve perturbation theory, since the true solution does notremain close to the solution in the non-interacting theory. This resonance effect wasargued to be the cause of an energy cascade—energy spreads out into more and highermodes—in order to explain black hole formation and the power-law spectrum observedduring such processes [1, 2, 4, 14]. It was also argued that since the AdS spectrum isresonant, such an instability should be the generic outcome of small perturbations.Counter-examples to the above claim in the form of the stable, quasi-periodicsolutions were initially viewed as being special. It was conjectured in [4] that thesestable solutions will shrink to a set of measure zero in the small (cid:15) limit, and the term“stability island” was used to describe their existence in the generically unstable seaof phase space. However, more recent evidence suggests that such a conclusion is toostrong. Numerical evidence suggests that, at finite (cid:15) , the stable and unstable solutionsboth have nonzero measure in the space of initial conditions [7, 9, 11]. One can thenapply a simple scaling argument, described in more detail in Sec. 2, to show that in the (cid:15) → (cid:15) → I.S. Yang thanks Jorge Santos for stressing this point during a discussion. Note that we are always discussing stability on the interaction time scale (cid:15) − in this paper. Thequestion of the behavior on longer time scales is a fascinating one that touches on issues of ergodicity,Arnold diffusion, and the KAM theorem. We do not know how to attack questions on these longertime scales analytically or numerically. – 2 –here are some important misconceptions and misunderstandings in the currentliterature regarding the status of the AdS (in)stability problem, due in part to threepoints of confusion, which we would like to clarify here. First of all, an energy cascadeis not identical to, nor does it guarantee black hole formation. This distinction has notbeen made clear enough. Both have been frequently used interchangeably and referredto as the “instability of AdS space.” Black hole formation requires energy to be focusedinto a small spatial region. According to the uncertainty principle, energy flowing tohigh momentum is certainly a necessary condition for that, but it is not sufficient. Itis entirely possible for even unboundedly high momentum modes to be populated, butfor the energy distribution to stay roughly spatially homogeneous.Therefore, in this paper, AdS instability strictly refers to black hole formation only. Because the AdS geometry changes dramatically in this case, such nomenclaturealigns with a more gravity point of view. This also allows us to study its implicationson the boundary CFT. When we refer to a solution or initial condition as stable orunstable, we will always be indicating whether it collapses to form black hole or not.The second point of confusion is the use of term “generic.” Numerical evidencesuggests that, at finite (cid:15) , the stable and unstable solutions both form sets of nonzeromeasure in the space of initial conditions [7, 9, 11]. We are interested in the (cid:15) → (cid:15) → From the hydrodynamic point of view, the existence of an energy cascade might be a suitabledefinition of instability. Indeed, this is the perspective taken by some authors, and we wish the readerto see the distinction clearly. Strictly speaking, numerical results only cover discrete choices of initial conditions. So, it istherefore impossible on numerical grounds alone to prove that any such set has either zero or nonzeromeasure. This fact holds equally for both stable and unstable solutions. Nevertheless, if either setreally were measure-zero, unless the numerical code secretly enforced extra symmetries, it wouldbe extremely unlikely to find such a result even once in simulations. Thus, despite the numericalcontroversy over some of the stable solutions [15], we still interpret the current evidence that stableand unstable solutions both have nonzero measures. – 3 –easure zero in the small- (cid:15) limit, which is certainly arguing for only (1). However, thenumerical evidence in [1] showing that black holes continue to form as (cid:15) is reduced isconsistent with both (1) and (3). Since these are three physically different cases, wethink such a clear distinction is needed.Finally, when addressing the question of instability, one needs to specify a timescale. In this paper, we will only discuss the time scale that goes to infinity as (cid:15) − inthe (cid:15) → Indeed a na¨ıve perturbation analysis shows that something interestingcan happen at this time scale. The physical question we will address is whether that“something interesting” is generically black hole formation? In the end, we will try torelate the answer to the boundary CFT: Does the boundary system thermalize at thistime scale?After making all these definitions clear, in Sec. 2 we briefly review the recentprogress on this topic. We then present a very simple scaling argument which showsthat possibility (1) defined above, “generic instability”, is the most unlikely given byexisting evidence. This directly argues against the “stability island” conjecture [4].The remaining question then is whether AdS space is generically stable (2) or mixed(3). In Sec. 3 we set up our perturbative method for studying gravitational self-interaction.This position-space approach is more directly relevant than the usual momentum-spaceanalysis to the question of whether or not black holes form. If energy gets focusedinto a smaller region, then the solution is evolving toward a black hole. If energy isdefocused into a larger region, then the solution is evolving away from a black hole.We explicitly demonstrate that in the (cid:15) → Behaviors at shorter time scales are somewhat trivial. For example, for a given fixed time, blackhole forming solutions disappear as (cid:15) →
0, so the system is generically stable, case (2). The behaviorat longer time scales is a very deep problem that touches on issues of ergodicity, Arnold diffusion, andthe KAM theorem. We do not know how to attack those questions analytically or numerically. I.S. Yang thanks Jorge Santos for stressing this point during a discussion. In principle, one can include all the information about relative phases in the spectral analysis toachieve the same result. Our position-space approach is simply more direct. In addition, it technicallycircumvents the subtlety that the gravitational interaction imposes significant additional constraintson possible resonances [16]. – 4 – arrower r=0r=0 wider
Figure 1: A thin shell that has higher energy density in its front will come out narrowerafter gravitational self-interactions as it bounces through r = 0. A shell with higherenergy density in its tail will come out wider after the bounce.will defocus. More generally, the leading-order dynamics of focusing and defocusingare related by time reversal, so a local maximum of energy density is also equally likelyto grow or diminish within the time scale (cid:46) (cid:15) − .In Sec. 5 we present the new intuition our method provides and propose a con-jecture on the structure of the phase space. Based on the symmetry between focusingand defocusing dynamics, the stable, quasi-periodic solutions can be understood astrajectories that alternate between the two. As a result, they may form quasi-closedloops in phase space. In fact, some unstable solutions are also known to exhibit thisalternating behavior while in the weak-gravity regime. Based on this understanding ofthe dynamics, we propose a conjecture on how to visualize the phase space of smallperturbations in AdS space.We also discuss how these gravitational calculations can shed light on the generalconcepts of thermalization in a closed system. In particular, contrary to conventionalwisdom, black hole formation at the (cid:15) − time scale is not necessarily the holographicdual of thermalization in the boundary field theory. If the thermal gas phase is the equi- The shell profile will change in other ways, but all changes are suppressed by (cid:15) . The focusing ordefocusing behavior will last for a time scale (cid:46) (cid:15) − , so it is the dynamics we are interested in here. – 5 –ibrium state, then black hole formation describes prethermalization that significantlydelays true thermalization [17, 18].In Sec. 6, we provide a quick summary of six major points of this paper. InAppendices A and B, we provide the computational details of our method and numericalexamples to demonstrate how the shape of the profile determines whether its energy isfocused or defocused.
2. Stability islands or instability corners?
We are interested in the perturbative stability of global AdS space. We will work in(3 + 1) dimensions and employ the following metric for vacuum
AdS : ds AdS = − (cid:18) r R (cid:19) dt + dr r R + r d Ω (2.1)where R AdS is the AdS radius and d Ω is the round metric on S . Our perturbationswill take the form of a real, massless scalar field φ minimally coupled to Einstein gravitywith a negative cosmological constant: S = (cid:90) d x √ g (cid:18) π R + 6 R − ∂ µ φ∂ µ φ (cid:19) . (2.2)The Planck scale has been set to one. We will consider only spherically symmetricsolutions, so both the scalar φ and the metric functions g tt and g rr will only depend on t and r .Here we will review some existing evidence and argue that a careful interpretationstrongly supports the following conclusion for spherically symmetric perturbations ofa massless scalar field in AdS space: In the (cid:15) → limit, at the T ∼ (cid:15) − time scale, AdS space is either generically sta-ble, or that neither stable nor unstable perturbations are generic. The first part of our argument is based on ample numerical evidence at small butfinite (cid:15) . The initial conditions that lead to black hole formation (unstable) and thosethat lead to quasi-periodic solutions (stable) both form open sets in the phase space ofnonzero measure. Note that the phase space is infinite dimensional, so no numericalevidence can prove that any set is really open. Nevertheless, whatever extrapolationsare being made should be applied equally to both stable and unstable solutions, and the Our radial coordinate r is related to the radial coordinate x used in [1] by r = tan x . – 6 –xisting numerical evidence is quite sufficient to show that they are on equal footing.More specifically, numerical tests can scan a one-parameter family of initial conditions,corresponding to a line in phase space. It has been clearly demonstrated that for a fewsuch lines, the initial conditions that lead to stable and unstable solutions both formfinite segments [7, 9, 11]. We will pragmatically take this as evidence that both stableand unstable sets in phase space have nonzero measure at small but finite (cid:15) .In particular, within the set of stable solutions, one can identify a subset for which“gravity never becomes strong” during the ∼ (cid:15) − time scale; that is, ∃ φ ( (cid:15), r, t ) , such that (cid:16) ˙ φ + φ (cid:48) (cid:17) < δ (cid:28) ≤ t ≤ T ∼ (cid:15) − . (2.3)Our next step is to show that in the (cid:15) → (cid:15) . We will demonstrate that in the (cid:15) → R − , meaning that the field profile is exactly periodic in time.Heuristically, a spherical wavefront shrinks toward the origin r = 0, passes throughit, expands again to infinity, and finally bounces off the boundary back to the originalposition. It is natural to describe the dynamics as a function of the “number ofbounces” N = tπR AdS instead of the microscopic time t : φ ( r, N + 1) ≡ φ ( r, t + πR AdS ) = φ ( r, N ) ≡ φ ( r, t ) . (2.4)Now, introducing gravitational self-interaction, as long as the field amplitude (andtherefore the resulting back-reaction) is small, we have a small correction to the aboveexactly periodic solution, φ ( r, N + 1) − φ ( r, N ) = A [ φ, ˙ φ ] + O ( φ ) . (2.5)The functional A describes the small, leading-order changes to the profile, which wewill analyze further in the following sections. Here we only need to know that it scaleslike φ . It is convenient to introduce the rescaled field, ¯ φ ≡ φ/(cid:15) , whose evolution isgiven by ¯ φ ( r, N + 1) − ¯ φ ( r, N ) = A [ ¯ φ, ˙¯ φ ] (cid:15) + O ( (cid:15) ) , (2.6) The periodicity of geodesics in AdS is 2 πR AdS , and in that time they pass through the origintwice. However, a shell of massless scalar field with Dirichlet boundary conditions at the boundary isactually periodic in half that time, πR AdS , during which the wavefront passes through the origin onlyonce. – 7 –lthough the value of N is discrete, in the (cid:15) → d ¯ φd ( (cid:15) N ) = A . (2.7)Thus, the scaling behavior is exact:¯ φ (cid:15) ( r, N ) = ¯ φ (cid:15)α ( r, α N ) . (2.8)Reducing the amplitude of the fluctuation simply slows down the dynamics by α : if (cid:15) is reduced by a factor of α , it takes α more bounces to reach the same configuration.Therefore, if there is a stable solution at some finite (cid:15) and for a time T ∼ (cid:15) − duringwhich gravity never becomes strong, this must also be a stable solution at any smaller (cid:15) , all the way to the (cid:15) → Interestingly, this same argument is not applicable to unstable solutions. In orderto form a black hole, the scalar field profile must first evolve to have large energydensity somewhere, (cid:16) ˙ φ + φ (cid:48) (cid:17) ∼
1. In other words, gravity must become strong, atwhich point the higher order terms in Eq. (2.6) cannot be ignored. In those cases thescaling behavior is lost. A collapsing solution at some small but finite (cid:15) might escapethat fate if we reduce (cid:15) further [21].At this point, we are left with two possibilities: • Neither stable nor unstable perturbations are generic, since they both occupy setsof nonzero measure in the phase space. • AdS space is perturbatively stable generically, but there are special “instabilitycorners”, which shrink to measure zero in the limit (cid:15) →
3. Weak gravitational self-interaction in position space
We now present our approach to explicitly calculating the functional A in Eq. (2.5).Our result, a precise expression for A , is given in Eq. (3.21). Many of its properties will We should note that the expansion in powers of (cid:15) is most likely asymptotic [19], but its leading-order result has been accurate for many similar applications [20]. – 8 –elp us to better understand the dynamics and the possibility of instability corners.Our calculation will be in position space. The advantage for this approach is easilyseen if we first picture the evolution of a thin shell of total energy E ∼ (cid:15) , thickness w and initial size r , such that r (cid:29) w . This corresponds to an initial field profile that isroughly given by φ ( r, t ) | t ∼ t i ∼ − (cid:15) √ wr f (cid:18) − r − r + t − t i w (cid:19) . (3.1)We will take the profile f ( x ) to be some function that peaks at x = 0 and has compactsupport an order-one range around around this peak ( i.e. f ( x ) = 0 for | x | (cid:38) Notethat we have carefully chosen the dependence on w such that it does not affect the totalmass, which is controlled solely by (cid:15) . The small-perturbation limit then corresponds to (cid:15) → r = 0. In other cases, the perturbation originates from a quench in theboundary CFT and appears as a wavefront coming in from r = ∞ [22]. Rememberthat in the small- (cid:15) limit, the leading-order behavior is the same as in empty AdS space;that is, the radiation shell simply bounces back and forth between r = 0 and r = ∞ .Therefore, all of these initial positions of the shell are related by a shift in time on theorder of R AdS . Since we are interested in the outcome at longer time scales, they areall equivalent for our purposes.One advantage of our position-space approach is that we can choose an r whichimplements the following “two-region” approximation: • For r < r , we will ignore that the background is AdS space and consider onlythe back-reaction of the scalar field on Minkowski space. • For r > r , we will ignore the scalar field back-reaction and treat the geometryas empty AdS space.In order justify this simplification, we first recall the general form of the Schwarzschild-AdS metric: ds SAdS = − (cid:18) − M ( r ) r + r R (cid:19) dt + dr − M ( r ) r + r R + r d Ω (3.2) We choose the profile to have compact support only to make the subsequent calculations somewhatcleaner. The shell only needs a narrow, well-defined width. Alternately, one could take f to have, forexample, Gaussian tails without affecting the results. – 9 –here M ( r ) is the total mass located inside the sphere of radius r .For r < r we will ignore the r /R terms in g tt and g rr responsible for the AdSasymptotics and calculate M ( r ) due to the back-reaction of the radiation shell. Thiseffect is strongest when the shell is near the origin and its energy is concentrated in asmall region within r < w . We find that M ( r ) ∼ (cid:15) .At r = r , we will start including the AdS terms and ignoring the back-reactionterms, such that for r > r the metric is just that of empty AdS space. Na¨ıvely, thisis allowed if the metric at r is approximately that of Minkowski space; that is, thecorrections due to both AdS and back-reaction must be small: r R (cid:28) (cid:15) r (cid:28) . (3.3)However, we should really ask for a stronger condition.Our perturbative back-reaction calculation will be organized as an expansion inpowers of (cid:15) /w , and we will work up to some power n using the Minkowski background.In order to be able to trust our results up to that order, we cannot allow the transitionat r to have a competing effect, meaning r R (cid:28) (cid:18) (cid:15) w (cid:19) n and (cid:15) r (cid:28) (cid:18) (cid:15) w (cid:19) n . (3.4)For any R AdS , we can choose the shell small enough and thin enough to accommodatethe hierarchy of scales R AdS (cid:29) r (cid:29) w (cid:29) (cid:15) , (3.5)which satisfies Eq. (3.4) for any choice of n .The two-region approximation provides a very simple picture. In the (cid:15) → r < r . The propagation in the r > r region is just propagation in an empty AdS space; the shell simply travels out, reflectsoff the boundary, and repeats the gravitational evolution near the origin. Since theprofile is modified by a small fraction ∼ ( (cid:15) /w ) during each bounce, we expect on thetime scale ∼ (cid:15) − an order-one change to accumulate.For example, the self-interaction might make the shell thinner after each bounce,meaning that the gravitational effect becomes stronger, since more energy is squeezedinto a smaller region. If that behavior persists, then eventually the energy will be com-pressed during a bounce into a region near the origin smaller than its Schwarzschildradius. At this point, the weak-gravity approximation will break down, and it is verylikely that in the (cid:15) − time scale, the shell will evolve into a black hole. On the other– 10 –and, it is also possible that the shell becomes wider after each bounce, and energyis dispersed into a larger region. In this case, there is no particular reason why grav-itational effects would necessarily become strong and no indication that a black holewould form in the (cid:15) − time scale. The main goal here is to set up a calculation thatcan capture these two different behaviors.Before moving on, we need to address the applicability of the thin-shell approx-imation. A full dynamical picture should accommodate energy distributions of allthicknesses. However, when w ∼ R AdS , there is no clean way to separate the self-interaction from the effects of the AdS space. Nevertheless, our main interest is theinstability in AdS toward black hole formation. In the small- (cid:15) limit, the energy mustbecome concentrated into thin shells to even have a chance of eventually forming ablack hole. Note that not all of the energy needs to be in one thin shell. But, theevolution toward a black hole is determined by the shell with the highest radial energydensity, which is dominated by its self-interaction, so we can ignore the influence ofother energy distribution outside the shell.
According to our approximation scheme, we can adopt the weak-gravity expansion inMinkowski space [20]: φ = (cid:15)φ + (cid:15) φ + ... (3.6) g µν = g µν + (cid:15) g µν + ... (3.7)At zeroth order in (cid:15) , the background is empty Minkowski space, ∼ O ( (cid:15) ) , ds = − dt + dr + r d Ω , (3.8)into which we put the initial shell profile. To first order, the equation of motion for φ is just that of a free field, ∼ O ( (cid:15) ) , ¨ φ − φ (cid:48)(cid:48) − r φ (cid:48) = 0 . (3.9)At the next order, gravity responds to the stress-energy tensor of the first-order profile.We therefore must solve the Einstein equation G µν = 8 πT µν to leading order in smallperturbations around empty Minkowski space. Spherical symmetry excludes dynamicaldegrees of freedom in the metric, so we only need to solve constraint equations. The tt and rr components suffice to provide the full answer, and the solution is parametrizedby two intuitive quantities: enclosed mass M and gravitational potential V . ds = − [1 + 2 (cid:15) V ( r, t )] dt + (cid:20) (cid:15) M ( r, t ) r (cid:21) dr + r d Ω . (3.10)– 11 –ote that we have also explicitly extracted the (cid:15) scaling from M and V , which aregiven in terms of the leading-order fields: ∼ O ( (cid:15) ) , M (cid:48) r = 8 π ˙ φ + φ (cid:48) , (3.11)2 r (cid:18) − Mr + V (cid:48) (cid:19) = 8 π ˙ φ + φ (cid:48) , (3.12)with boundary conditions M (0 , t ) = 0 and V ( ∞ , t ) = 0. Finally, the leading nontrivialdynamics comes at the next order—the change in geometry back-reacts on the fieldprofile. ∼ O ( (cid:15) ) , ¨ φ − φ (cid:48)(cid:48) − r φ (cid:48) = C (cid:18) ¨ φ + φ (cid:48)(cid:48) + 2 r φ (cid:48) (cid:19) + ˙ C ˙ φ + C (cid:48) φ (cid:48) . (3.13)Here we have abbreviated C = ( V − M/r ). We see that the field at this order obeys thesame wave equation as in the previous order with the addition of a nontrivial sourceterm.The radial wave equation can be rewritten as a (1 + 1)-dimensional wave equationby introducing u = rφ : r (cid:18) ¨ φ − φ (cid:48)(cid:48) − r φ (cid:48) (cid:19) = ¨ u − u (cid:48)(cid:48) . (3.14)This implies that the initial shell profile given in Eq. (3.1) is really just the left-movingpart of an exact, leading-order solution, rφ ( r, t ) = u ( r, t ) = √ w (cid:20) f (cid:18) r − tw (cid:19) − f (cid:18) − r − tw (cid:19)(cid:21) . (3.15)We remind the reader that in Eq. (3.6), the (cid:15) dependence has been extracted explicitlyfor φ , therefore also for u . We have taken the liberty to choose the initial time t i = − r to simplify this expression. This allows us to start this calculation once theshell enters the r < r region, and the center of the shell reflects off the origin at t = 0.Later, we will be interested in corrections to the profile at t f = r , when the shell isleaving the central Minkowski region.Rewriting the system in terms of the (1+1)-dimensional function u is essentiallyemploying a method of images; we extend the range of r into the unphysical r < r = 0 such that all physical quantitiesare finite and smooth, we require u ( r, t ) to be antisymmetric. Similarly, we can extendthe definition of M to negative r , M ( r, t ) = (cid:90) r d ˜ r ˙ φ + φ (cid:48) π ˜ r , (3.16)– 12 –hich is naturally an odd function of r . The same extrapolation shows that V is aneven function of r .In terms of these new variables, the problem of a shrinking shell has been mappedto the problem of two wavepackets colliding at r = t = 0. Note that this picture is morerealistic than it seems; antipodal points of the shell do indeed collide with each other.When the shell is far from the origin, even the leading-order radial energy densityis approximately equal to the naive definition of energy in this (1 + 1)-dimensionalsimplification: ρ = 4 πr ˙ φ + φ (cid:48) ≈ π ˙ u + u (cid:48) . (3.17)To leading order, the colliding shells simply pass through each other. Our goal is tosolve the next-order nontrivial effect of such a collision by solving Eq. (3.13), which interms of u is simply¨ u − u (cid:48)(cid:48) = C (¨ u + u (cid:48)(cid:48) ) + ˙ C ˙ u + C (cid:48) (cid:16) u (cid:48) − u r (cid:17) ≡ S ( r, t ) . (3.18)This description has a striking resemblance to soliton collisions [23, 24]. The key tothis type of problem is that, before solving the equations, we should already anticipatethe physical meaning of the answer. At t f = r , after the collision, the leading-ordersolution implies that an out-going shell of the opposite sign reaches exactly r = r . Ontop of that, we can organize the next-order correction into the following form: u + (cid:15) u = u − (cid:15) (cid:18) ∂u ∂r ∆ r + ∂u ∂w ∆ w + ... (cid:19) (3.19)We have again extracted the (cid:15) dependence explicitly. The shell is actually shifted by (cid:15) ∆ r from its expected position, its width has changed by (cid:15) ∆ w , and there will beother changes orthogonal to these.The function u at t f = r can be solved from Eq. (3.18) by integrating the retardedGreen’s function: u ( r, r ) = 12 (cid:90) r − r dt (cid:90) r + r − tr − r + t dr (cid:48) S ( r (cid:48) , t ) . (3.20)Note that the lower limit of r (cid:48) can be negative, which is allowed due to our method ofimages. The result, however, is the same if we replace the lower bound of the integrationrange by its absolute value. Note that the total energy E ≡ M ( ∞ , t ) = (cid:82) ∞ ρ ( r ) dr is in fact equal to the naive (1 + 1)-dimensional energy (cid:82) ∞−∞ π ˙ u + u (cid:48) dr. – 13 –ote that this u is only the difference between the incoming shell at t = − r andthe out-going shell at t = r , both at position r = r . Nevertheless, as we have arguedthat the propagation further to r = ∞ , the reflection, and the propagation back to r = r , can all be taken as trivial. This allows us to directly relate u to the functional A from Eq. (2.5), which gives the leading-order change due to one bounce. A (cid:20) u r , ˙ u r (cid:21) = − u (˜ r, t )˜ r , (3.21)where ˜ r = 2 r − r is the spatial reflection of r around r . The extra minus sign andchanging to this “flipped” position are due to the trivial propagation to and from r = ∞ .The full procedure to calculate u and extract physical information like ∆ r and ∆ w are tedious but straightforward. We will present the analytical and numerical detailsin Appendices A and B. Here we highlight two relevant features of the results:1. u has a ∼ log r contribution, which comes entirely from the position shift,∆ r = − (cid:82) u ∂ r u dr (cid:82) ( ∂ r u ) dr , (3.22)which has a clear physical meaning. The leading-order profile u follows the t = | r | trajectory, but the next-order correction to the metric modifies the nullgeodesics. The shell will therefore return to r = r not exactly when t = r .However, this shift is irrelevant to the pertinent question of whether energy getsfocused.
2. The change in the shell’s width is given by∆ w = (cid:82) u ∂ w u dr (cid:82) ( ∂ w u ) dr . (3.23)Since we have already scaled out the (cid:15) dependence, ∆ w only depends on the shapeof the shell (i.e. the function f one chooses in Eq. (3.15)), and it is independentof both (cid:15) and w .In particular, our main result is that ∆ w is just as likely to be positive as negative.Specifically, when we flip the profile of the incoming shell, f ( x ) → f ( − x ), then ∆ w →− ∆ w . As a special case, a symmetric profile with f ( x ) = f ( − x ) will result in no first This position shift is related to a shift in frequency in the momentum space analysis observed inother papers [11]. – 14 –rder ∆ w during one bounce. This demonstrates that the gravitational self-interactionin AdS is not biased toward focusing energy, and the collapse of small perturbationsinto black holes is probably not the generic behavior, at least not on time scales (cid:46) (cid:15) − .As a complementary calculation, we also investigate how the maximum radial en-ergy density ρ Max of the shell behaves under the same f ( x ) → f ( − x ) transformation.Like the width w , we find that if for a given profile ρ Max increases with each bounce,then for the flipped profile it decreases. This provides another indication that theweak-gravity dynamics are biased neither toward nor against focusing energy.In the next section, we will give general proofs of these statements. We will alsopresent numerical examples in Appendix B.
4. Focusing and defocusing
As a preliminary step in proving the statements of the previous section, we need todetermine how the first-order correction u responds to a spatial flip of the initial profile u such that f ( x ) → ˜ f ( x ) = f ( − x ). We find that u ( r, r ) → ˜ u ( r, r ) (cid:39) − u (˜ r, r ) , (4.1)where ˜ r = 2 r − r is again the spatial reflection of r around r . Note that this isan approximate statement; for a shell of width w , the error in Eq. (4.1) is of order w /r . As we argued in Sec. 3, in the (cid:15) → r to make this errorarbitrarily small.The quantities that enter the expression (A.1) for u are u and its derivatives and C and its derivatives. So, let us first see how these quantities transform under the flip.From Eq. (3.15), we can see that: u ( r, t ) → ˜ u ( r, t ) = − u ( r, − t ) . (4.2)Then, simply by differentiating the two sides of the equation (either with respect to r or t ), we obtain the same transformation behavior for the derivatives of u . Now, tosee how C transforms, all we need is to determine the transformation of M , defined inEq. (3.16): M ( r, t ) → ˜ M ( r, t ) = 2 π (cid:90) r dr (cid:48) (cid:32) ˙˜ u ( r (cid:48) , t ) + ˜ u (cid:48) ( r (cid:48) , t ) + (cid:18) ˜ u ( r (cid:48) , t ) r (cid:48) (cid:19) − u ( r (cid:48) , t ) ˜ u (cid:48) ( r (cid:48) , t ) r (cid:33) = 2 π (cid:90) r dr (cid:48) (cid:32) ˙ u ( r, − t ) + u (cid:48) ( r, − t ) + (cid:18) u ( r, − t ) r (cid:48) (cid:19) − u ( r (cid:48) , − t ) u (cid:48) ( r, − t ) r (cid:48) (cid:33) = M ( r, − t ) . (4.3)– 15 –ince V has the same behavior as M , then C ( r, t ) = V − Mr transforms under the flipas: C ( r, t ) → ˜ C ( r, t ) = C ( r, − t ) . (4.4)Again, a similar relation holds for the derivatives of C . Combining the above results,we see that the source term S ( r, t ), defined in Eq. (3.18), behaves as S ( r, t ) → ˜ S ( r, t ) = − S ( r, − t ) (4.5)under flipping of the initial profile. Also, by demanding regularity at the origin r = 0,the initial profile is antisymmetric in r , which in turn implies that M ( r, t ) is also anti-symmetric in r ; hence C ( r, t ) is symmetric. These properties imply the antisymmetryof S ( r, t ) in its first argument, S ( r, t ) = − S ( − r, t ).Now we are ready to prove Eq. (4.1), starting from the integral expression Eq. (A.1)for u . The integration regions are illustrated in Fig. 2.We first make an approximation to Eq. (A.1). The upper limit of the r (cid:48) integral is r + r − t . Instead, we will extend the region of integration up to r (cid:48) = ∞ . Because thewavepacket has compact support only over a region of width w , the error introduced bythis approximation comes just from the yellow shaded triangle in Fig. 2. The area ofthis added triangle is O ( w ) and, since C ( r, t ) ∼ r , the integrand is of order r . Hence,the error is suppressed by a factor of w r .A similar, and perhaps even more physical, approximation, albeit with more cum-bersome limits of integration, can be made by considering the area of integration de-noted by the red lines together with the orange line in Fig. 2. In that case, instead ofadding the extra contribution from the yellow triangle at the top, we would subtractthe area of the green triangle at the bottom. However, the results would be the same.After this approximation, we have: u ( r, r ) (cid:39) (cid:90) r − r dt (cid:90) ∞| r − r + t | dr (cid:48) S ( r (cid:48) , t ) . (4.6)Now, flipping the initial profile we get:˜ u ( r, r ) (cid:39) (cid:90) r − r dt (cid:90) ∞| r − r + t | dr (cid:48) ˜ S ( r (cid:48) , t ) . (4.7)Using the flipping property of S ( r, t ), as discussed above, we can write:˜ u ( r, r ) (cid:39) − (cid:90) r − r dt (cid:90) ∞| r − r + t | dr (cid:48) S ( r (cid:48) , − t ) . (4.8)– 16 – t ( r, r ) r t = r t = − r Figure 2: The areas of integration for the computation of u ( r, r ). The blue solid linerepresents the source at times t = ± r . The red lines (solid and dashed) indicate the ac-tual area of integration in Eq. (A.1) ( i.e. the integral (cid:82) r − r dt (cid:82) r + r − t | r − r + t | d ˜ r ), and the greenshaded region indicates where the integrand is nonzero. The solid red lines togetherwith the dashed green lines correspond to the region of integration (cid:82) r − r dt (cid:82) ∞| r − r + t | d ˜ r used in our approximate Eq. (4.6). The yellow triangle at the top shows the extranonzero contribution included in the second integral, which is suppressed by w r . Al-ternatively, the integral can be approximated by using the orange line instead of thehorizontal red line.We can now change the dummy integration variable t to − t and use the relation r =2 r − ˜ r to rewrite the lower integration limit, giving˜ u ( r, r ) (cid:39) − (cid:90) r − r dt (cid:90) ∞| (2 r − ˜ r ) − r + t | dr (cid:48) S ( r (cid:48) , t ) . (4.9)– 17 –omparing this expression to Eq. (4.6), we obtain˜ u ( r, r ) = − u (˜ r, r ) . (4.10)which is our desired result. In this subsection we will prove Eq. (3.23); that is, under a spatial flip f ( x ) → ˜ f ( x ) = f ( − x ), the leading-order correction to the width is antisymmetric:∆ w → ∆ ˜ w = − ∆ w (4.11)We assume the profile u has compact support within r − w/ < r < r + w/
2, andevaluate ∆ w at late time t f = r , well after the collision, at which point the left-movingand the right-moving wavepackets are far away from r = 0 and do not interfere witheach other. In that case, when computing ∆ w from Eq. (3.23), we can just integrateover the right-moving wavepacket; integrating over both wavepackets would just doubleboth the the numerator and denominator in Eq. (3.23), yielding the same result. Theexpression for the change in width of the flipped profile is then∆ ˜ w = (cid:82) r + w/ r − w/ dr ˜ u ( r, r ) ∂ w ˜ u ( r, r ) (cid:82) r + w/ r − w/ dr ( ∂ w ˜ u ) . (4.12)It is convenient to define y = r − r , such that the spatial flip of the initial profileis given by u ( r + y, r ) → ˜ u ( r + y, r ) = u ( r − y, r ) . (4.13)Starting with the numerator of Eq. (4.12) and using the properties of u and u underthe flip, we can write (cid:90) r + w/ r − w/ dr ˜ u ( r, r ) ∂ w ˜ u ( r, r ) = (cid:90) w/ − w/ dy ˜ u ( r + y, r ) ∂ w ˜ u ( r + y, r )= − (cid:90) w/ − w/ dy u ( r − y, r ) ∂ w u ( r − y, r )= − (cid:90) w/ − w/ dy u ( r + y, r ) ∂ w u ( r + y, r ) (4.14)In the third line, we changed the dummy integration variable from y to − y . Theflip therefore changes the sign of the numerator. Following these same steps with thedenominator of Eq. (4.12), we can see that it is invariant under the flip. Putting thesetwo statements together yields the desired result, ∆ ˜ w = − ∆ w .– 18 – .3 Energy density A similar argument holds for the leading-order change in the energy density ∆ ρ at time t f = r due one bounce through the origin. Specifically, for f ( x ) → ˜ f ( x ) = f ( − x ), wefind ∆ ˜ ρ ( r, r ) (cid:39) − ∆ ρ (˜ r, r ) . (4.15)where recall ˜ r = 2 r − r . The full radial energy density far from the origin is approx-imately the (1+1)-dimensional expression, as in Eq. (3.17). Expanding it to the nextorder, we find ρ + (cid:15) ∆ ρ = 4 π ˙ u + u (cid:48) π(cid:15) ( ˙ u ˙ u + u (cid:48) u (cid:48) ) . (4.16)We kept our principle of always extracting (cid:15) explicitly. The first term is the initialenergy density given in Eq. (3.17), which is actually (cid:15) − times the actual physicalenergy density. The second term is the leading change due to a single bounce.The formula (4.1) we found for the behavior of u under the flip holds at the specifictime t = r , and it is not straightforward to see that the same relation holds for ˙ u . Analternative way to proceed is to include the explicit expression for the derivatives of u at t f = r : u (cid:48) ( r, r ) = 12 (cid:90) r − r dt [ S ( r + r − t ) − S ( r − r + t )] (4.17)˙ u ( r, r ) = 12 (cid:90) r − r dt [ S ( r + r − t ) + S ( r − r + t )] . (4.18)Using these formulae and the expression for u Eq. (3.15), and omitting irrelevantconstants, we can write down the explicit expression for ∆ ρ .∆ ρ ( r, r ) = (cid:90) r − r dt (cid:104) f (cid:48) ( − r − r ) S ( r + r − t, t ) − f (cid:48) ( r − r ) S ( r − r + t, t ) (cid:105) (4.19)Since the function f ( x ) has compact support of width w around x = 0 and we considervalues of r on the order of r , then the first term in the integrand vanishes. We cannow determine how ∆ ρ behaves under f ( x ) → f ( − x ):∆ ˜ ρ ( r, r ) = − (cid:90) r − r dtf (cid:48) ( − r + r ) S ( r − r + t, − t ) . (4.20)Changing the dummy integration variable t to − t and substituting with r = 2 r − ˜ r ,we obtain ∆ ˜ ρ ( r, r ) = − (cid:90) r − r dtf (cid:48) (˜ r − r ) S ( − ˜ r + r − t, t ) . (4.21)– 19 –rom the antisymmetry of S in its first argument, we get∆ ˜ ρ ( r, r ) = + (cid:90) r − r dtf (cid:48) (˜ r − r ) S (˜ r − r + t, t )= − ∆ ρ (˜ r, t ) , (4.22)which is indeed Eq. (4.15).Eq. (4.15) relates the change in energy density at an arbitrary point r and itsimage ˜ r under the flip. However, we are particularly interested in how the change inthe maximum energy density is affected by the flip.The energy density at the position of the maximum, r Max , after one bounce, canbe expanded as ρ Max ≡ ρ ( r Max ) = ρ (0) ( r Max ) + (cid:15) ∆ ρ ( r Max ) . (4.23)It might be tempting to directly identify this with the change of maximum energydensity. However, we should remember that in addition, the location of the maximum r Max is also, in general, affected by the bounce, receiving corrections at the same order, r Max = r (0)Max + (cid:15) ∆ r Max . So, expanding to order (cid:15) , we find ρ (0)Max + (cid:15) ∆ ρ Max = ρ (0) (cid:16) r (0)Max (cid:17) + (cid:15) ρ (0) (cid:48) (cid:16) r (0)Max (cid:17) ∆ r Max + (cid:15) ∆ ρ (cid:16) r (0)Max (cid:17) . (4.24)However, r Max is an extremum of ρ (0) , and so ρ (0) (cid:48) (cid:16) r (0)Max (cid:17) = 0. To leading-order, thechange in the maximum is due only to a change in ρ and not a shift in the location ofthe maximum: ∆ ρ Max = ∆ ρ (cid:16) r (0)Max (cid:17) . (4.25)Now, under a flip, r (0)Max is mapped to ˜ r (0)Max = 2 r − r (0)Max , the location of the maximumof ˜ ρ ; that is, ∆ ˜ ρ Max = ∆ ˜ ρ (cid:16) ˜ r (0)Max (cid:17) . (4.26)From Eq. (4.15), we can see that∆ ˜ ρ Max = − ∆ ρ (cid:16) r (0)Max (cid:17) = − ∆ ρ Max . (4.27)Therefore, flipping the profile reverses the direction of the change in the maximumenergy density. This result nicely complements our result regarding the the change inwidth Eq. (4.11). Both of these results indicate that there is no bias in the weak-gravitydynamics toward either increasing or decreasing the energy concentration.– 20 – . Discussion In Sec. 3, we provided the recipe to compute the change in a field profile after onebounce, and it contained all the information about the functional A in Eq. (2.5). Inprinciple, one can add the resulting ¯ φ to the original ¯ φ to make a new initial condition,and calculate the result of the next bounce. Choosing a small (cid:15) and reiterating thisprocess ∼ (cid:15) − times is equivalent to solving Eq. (2.7). In principal, this will directlyreproduce the long term evolution. Unfortunately, there is one technical difficulty thatwe have not been able to overcome.Our method has one disadvantage: energy conservation is by definition an approx-imation. We basically “turn on” a self-gravitational potential when a shell shrinksbelow r , let energy flow between it and the field kinetic terms, then turn it off whenthe shell expands over r . The amount of potential energy we turn on and off differsby ∼ (cid:15) w/r . Although this is suppressed by an extra factor of w/r from the quanti-ties we care about in Sec. 4 and so it does not invalidate our results, it is technicallydifficult to control. We could na¨ıvely go to a larger r for better energy conservation,which would increase the integration range required to solve Eq. (3.18). However, morenumerical resources would then be necessary in order to proceed.Nevertheless, we might have learned enough about what happens in a single bounceto make a reliable extrapolation. We will attempt to do so by drawing a phase-spacediagram. Since there are no gravitational degrees of freedom within spherical symmetry,the phase space of perturbations is given by all possible scalar field profiles. Due toenergy conservation, we can focus on one fixed-energy, co-dimension-one surface in thisinfinite dimensional phase space. Within this surface, we can draw a two-dimensionalprojection (Fig. 3) and understand its structure based on our knowledge of the dynamicsof one bounce.One guiding principle of this diagram is that during one bounce, the profile changesby an infinitesimal amount ∼ (cid:15) , which is also an infinitesimal distance in the diagram.Within the weak-gravity time scale ∼ (cid:15) − , the evolution trajectory covers a finitedistance of the diagram. In this way, the diagram directly represents the dynamicalevolution in the rescaled time as given by Eq. (2.7).The horizontal axis of this two-dimensional diagram represents “how close is thisprofile to becoming a black hole”. More technically, it is quantified by the maximumradial energy density at r = 0 that is reached during one AdS time. In the small- (cid:15) limit, the profile is basically freely propagating, so this is a well-defined quantity.Heuristically, this maximum is reached when the highest “peak” goes through r = 0,and its value depends on the height of this peak, ρ Max .– 21 – ax non-thin-shell ρ strong gravity Figure 3: A two-dimensional projection within a constant-energy slice of the phasespace. The horizontal axis is the peak energy density, and the big, green arrows towardleft and right represent the focusing and defocusing flow due to gravitational self-interaction. Together with the upward and downward flows represented by the small,red arrows, the phase space has a circular flow pattern. The blue loop representsquasi-periodic solutions that stay within the center of this circular flow.Note that throughout this paper, we have been referring to ρ as the rescaled energydensity. In our conventions, the actual physical energy density is given by (cid:15) ρ . It is stillconvenient to consider the rescaled density here, since ρ Max quantifies how much higherthis peak is than the average, namely the relative concentration of energy. Its valueincreases toward the right-hand-side of Fig. 3. For any finite (cid:15) , there is a finite valueof ρ Max ∼ (cid:15) − that represents a density high enough to become a black hole. It can bedrawn as a vertical line. Some finite distance to the left of this line, we have anotherline signifying that the energy density is high enough to make gravity too strong to bedescribed by the weak-gravity expansion.To the left of this second line, gravity is weak enough that our analysis applies.Somewhere even further to the left, our approximation starts to fail for a differentreason: we can no longer describe this peak as an isolated thin shell satisfying thehierarchy in Eq. (3.5). In the small- (cid:15) limit, the region to apply our method alwaysexists. This left boundary is not a very clear line. Nevertheless, in this diagram we canroughly picture it as ρ Max R AdS ∼
1, that the maximum peak density is comparable tothe average density. Clearly, energy is too evenly distributed in the entire AdS spacethat nothing could be treated as an isolated thin shell.– 22 –he vertical axis of this diagram is not intended to represent any particular pa-rameter of the field profile. It is merely reflecting the fact we established in Sec. 4 thatfocusing and defocusing dynamics are equally likely in one bounce. This means one canalways find some parameter such that the middle region is divided into two halves: inthe upper half, the evolution makes the peak grow higher and moves closer to forminga black hole, and in the lower half, the peak gets lower and moves away from forminga black hole. In App. B, we give specific numerical examples and argue the parametercontrolling focusing and defocusing is the asymmetry of energy distribution: focusingoccurs when the shell is is denser in the leading edge, and defocusing when it is denserin the tail.In addition to focusing and defocusing which correspond to flowing horizontallyin Fig. 3, what tendencies to flow in the vertical direction can we identify? Generallyspeaking, when ρ Max is large, on the left side of Fig. 3, the system will tend to flowupwards. This is because a shrinking peak cannot remain the highest peak forever:a growing peak with smaller initial height will eventually take over. If that were theonly vertical motion, it would lead to only two possible trajectories: starting in theupper half, ρ Max would keep increasing and flow directly toward black hole collapse;alternatively, staring in the lower region, ρ Max would first decrease, then bounces backto become a black hole. This directly violates many numerical results, so there mustbe a downward flow somewhere in Fig. 3.The two numerical examples in App. B provide tentative evidence for a downwardflow. What we see is that given a symmetric profile on the boundary between the upperand lower region, after one bounce it picks up an asymmetry similar to the profiles inthe lower region—its energy becomes denser in the tail. Of course, we studied only twoone-parameter slices through an infinite dimensional phase space, so better numericaland/or analytical investigation is required to verify this. At the level of this paper,we will simply conjecture that such a downward flow exists, because the resultingcirculatory flow, shown in Fig. 3, explains existing numerical results very well: • The quasi-periodic solutions stay within the circular flow near the center, as inFig. 3. • The unstable solutions initially stay within the circular flow, but their radii varywildly and eventually these solutions enter the strong gravity regime, as in Fig. 4Note that the actual motion in the true phase space is still very complicated. Inthis is a two-dimensional projection, evolution trajectories are allowed to cross eachother. Nevertheless, this circular flow allows us to better visualize the dynamics in thephase space. – 23 – ax non-thin-shell strong gravity ρ Figure 4: The same circular flow in the phase space, but the blue trajectory now rep-resents an unstable solution. Though initially it follows the circular flow, it fluctuatesto larger radius, eventually enters the strong gravity region, and collapses into a blackhole.We can also repeat the argument in Sec. 2 in a more pictorial manner. As onereduces (cid:15) , most of this diagram does not change. Due to the scaling behavior, alltrajectories to the left of the strong gravity line remain the same, and so most of thestable solutions remain stable. The trajectories for unstable solutions must cross thestrong gravity line to form black holes, so they potentially can change.Actually, the location of the strong gravity line shifts to the right when (cid:15) decreases.As the total energy is reduced, it needs to be increasingly focused in order for gravityto become strong, and a collapsing solution must therefore to evolve further, across thenew weak-gravity regime. In the (cid:15) → ρ Max goes all the way to infinity.Although the trajectory for a black hole collapse appears divergent, it does notmean that we can immediately rule out such an evolution. In fact, this divergence isan artifact of the parameter choice, and we should appreciate that ρ Max = ∞ is not aninfinite distance away in phase space. Recall that ∆ w in one bounce is independentof w , so the change in width need not be a small fraction of the total. It is certainlypossible to have a profile such that after (cid:15) − bounces, ∆ w is negative and order one,leading to a diverging ρ Max .The real advantage of this picture is that it recasts the (cid:15) → ρ Max diverges. In-– 24 –erestingly, analyzing the regularity of
AdS perturbations at finite (cid:15) below the blackhole mass gap is similarly a question of determining whether the energy density di-verges. In that case, there is already strong evidence to support regularity [25,26]. Onemight hope to reproduce this AdS result in higher dimensions in order to confirm thatthe instability corners indeed shrink to measure zero. We should again caution thatspectral analysis can only provide necessary conditions; it can rule out an instability,but it cannot provide equally strong evidence to support one. If the power spectrumof perturbations agrees with a diverging ρ Max , a long-time evolution of Eq. (2.6) inposition space is still required for the final answer to the AdS stability problem.
One motivation for studying the stability properties of AdS is to try to learn somethingabout the non-equilibrium dynamics of closed systems. This is due to the AdS/CFTcorrespondence, which relates this classical gravitational system to the dynamics of astrongly-coupled quantum system. Most investigations of holographic thermalizationstudy the Poincar´e patch of AdS, which has an infinite boundary (see, for example,[27–30]). In these cases, any nonzero energy density in the bulk will collapse into ablack hole, corresponding to thermalization on the boundary.Here, instead, we are considering global AdS which has a closed, spherical boundaryand therefore a very different thermalization behavior. Other studies of global AdS,such as [22], implicitly assume the connection between forming a black hole in thebulk and thermalization of the boundary system. Although that is valid in some cases,we would like to highlight other possibilities. What are the possible holographic dualdescriptions of the bulk story presented here?One caveat is that explicit examples of the AdS/CFT correspondence usually con-tain compact extra dimensions whose sizes are comparable to R AdS , for example in
AdS × S . In the (cid:15) → R AdS and is therefore too small to representthe most typical states; a ten-dimensional black hole of the same total energy, whichbreaks the symmetry of the S , has even higher entropy. Therefore, the gravitationalstability problem of AdS does not directly translate to the thermalization problem inthe boundary system. This might be an interesting direction for future work, but wewill set this concern aside for now. Let us take a very optimistic point of view thatthe AdS/CFT correspondence can work with extra dimensions arbitrarily smaller than R AdS , or even without them.After limiting our attention to the AdS space and treating our classical field theoryas a limit of a quantum gravity theory, the (cid:15) → (cid:15) → β ≡ R AdS R BH (cid:29) . (5.1)Namely, the Schwarzschild radius of the black hole made by collapsing the scalar fieldenergy is much smaller than the AdS size. On the other hand, the most straightforwardstandard for trusting classical gravity is γ ≡ R BH l Planck (cid:29) l Planck = √ G (cid:126) , which has been set to one inthe rest this paper. This condition implies that, at the very least, if a black hole forms,it could be described by classical gravity. For the limit we have been considering, both β and γ have been taken to infinity. We will see that whether the black hole or thethermal gas dominates depends on the details of how that limit is taken.The entropy of a black hole with energy E is S BH ∼ ( l Planck E ) ∼ (cid:18) R BH l Planck (cid:19) . (5.3)The entropy of a thermal gas in AdS space with the same total energy is S gas ∼ (cid:18) R BH R AdS l (cid:19) / . (5.4) Note that we are assuming here that the spacetime is effectively
AdS at distance scales of orderthe size of the black hole; in string constructions, such as AdS × S , black holes whose radius is smallcompared to the AdS radius would be eleven-dimensional rather than four-dimensional, leading todifferent formulas. – 26 –hus, black hole states dominate the micro-canonical ensemble when (cid:18) R BH l Planck (cid:19) > (cid:18) R AdS R BH (cid:19) / ; γ > β / . (5.5)This condition is equivalent to comparing the thermal wavelength λ th of the gas to theblack hole radius; the black hole dominates the ensemble if R BH > λ th . (5.6)We can see that whether the condition in Eq. (5.5) is satisfied depends on how thelimits of large β and large γ are taken. A classical and small- (cid:15) limit does not restrictthe system to being dominated by either the thermal gas or black hole states.Note that whether, and for how long, classical evolution is a good approximationdepends on more details of the state. For example, even if a black hole forms which isclassical according to Eq. (5.2), the process by which it formed might not be. A simplerule of thumb for the validity of the classical limit is that the occupation numbers inthe modes of interest have to be large. If the system is in a state where the energy isroughly equipartitioned between a number of modes up to some maximum frequency ω max , we require energy per mode (cid:29) ω max . (5.7)The thermal gas states can never satisfy this condition because modes with frequencyof order the temperature have occupation numbers of order one, yet contribute a sig-nificant fraction of the entropy of the gas. Independent of the limiting procedure andwhich states dominate, the thermal gas final state is never compatible with a classicaldescription. On the other hand, a spherically symmetric collapse into a black hole can oftenbe completely classical. Such a process only needs to excite the radial modes from thelongest wavelength ∼ R AdS to the shortest wavelength ∼ R BH , thusnumber of modes = R AdS R BH . (5.8)The condition on occupation numbers, Eq. (5.7), becomes β (cid:28) γ . (5.9) Nevertheless, from the position-space viewpoint, classical evolution may still describe the “processof approaching” a thermal gas state, at least distinguishing it from approaching a black hole. In thelater case energy becomes more concentrated, but in the former case it does not. – 27 –omparing this to Eq. (5.5), we see that Eq. (5.9) is always true when the black holedominates the ensemble, but it can still be true even if thermal gas dominates. Thus,the specific stability problem within classical gravity investigated in this paper, namelya spherically symmetric collapse into a black hole, is a valid dual to some boundarysystem, independent of whether such a process is equivalent to a efficient thermalizationor not. Furthermore, when the thermal gas dominates, if a black hole really forms in thetime scale we investigated, T weak gravity = R AdS R AdS R BH ∝ (cid:15) − , (5.10)it could represent a significant delay to thermalization. In order to confirm this, weneed to compare the na¨ıve thermalization time T weak gravity to the black hole lifetime,given by the the evaporation time scale, T evaporation = R l . (5.11)When T evaporation > T weak gravity , which requires β < γ , (5.12)the system thermalizes only after forming a long-lived black hole, which eventuallyevaporates. This process of thermalization via a quasi-stationary thermal-like state isknown as prethermalization and has been observed in finite-sized, isolated quantumsystems [17, 18]. Note that Eq. (5.12) is compatible with thermal gas domination anda classical collapse.To summarize, spherically symmetric black hole formation within T weak gravity canhave two different holographic interpretations: • When γ > β , it represents efficient thermalization of the boundary system. • When β < γ < γ < β , it represents prethermalization, which delays truethermalization (to thermal gas ) at a time scale (cid:38) T evaporation > T weak gravity . Note that spherical symmetry is very important here. Without it, the number of modes wouldhave been (cid:16) R AdS R BH (cid:17) , and with that many modes, the black hole collapse would have failed to remainclassical in the thermal gas-dominated regime. In this paper, “efficient” means that thermalization happens in the shortest time scale allowed bythe dynamics, ∼ (cid:15) − . One should not confuse this with, for example, the much faster thermalizationin the Poincar´e patch of AdS, where, within one AdS time, perturbations cross the horizon, form aplanar black hole, and appear to thermalize. Note that since thermal gas is never classical, we do not know exactly when will it really form. Weonly know that within T evaporation , the systems was too busy forming a black hole and then remainingas one, so it cannot reach the thermal gas state yet. – 28 –or the remaining possibility, when γ < β but β > γ , the implication of black holeformation is inconclusive from a thermalization point of view. Black holes decay toofast to be quasi-stationary intermediate states, but their evaporation cannot guaranteereaching the thermal gas state either.
6. Summary • By combining existing numerical data with our analysis, we have argued that fora massless scalar field in AdS space, in the small-amplitude (cid:15) → T ∼ (cid:15) − form an open set. Thisimproves similar observations in finite- (cid:15) numerical simulations [7, 9] and arguesagainst the conjecture that the weakly turbulent instability occurs in all but aset of measure zero in the space of initial conditions [1, 4, 14]. • One important difference between our approach and previous work is that weanalyzed the problem in position space. We pointed out that only position spaceproperties can provide necessary and sufficient conditions for the collapse into ablack hole. Any analysis of the power spectrum can at most provide necessaryconditions for black hole formation. • In the position space analysis, we exploited the small-amplitude (cid:15) → r = 0. This argument requires a hierarchy of scales given inEq. (3.5), which is difficult to reach in realistic numerical simulations. • We showed that gravitational self-interaction near r = 0 obeys an exact antisym-metry under time reversal. As a result, it is equally likely for interactions to focusor defocus energy. This equality is consistent with existing numerical results. We remind the readers that gravity can be effectively repulsive: tidal forces tendto pull things apart. The possible defocusing of a radiation shell is due to suchtidal effect. • By making use of scaling symmetry, we simplified the stability problem in the (cid:15) → More specifically, one could take any numerical simulation and pause it at a moment when gravityis still weak. If one keeps the field profile but reverses the time derivative at this moment, thesimulation will literally evolve backward toward the original initial profile, up to the numerical errorand higher-order effects (which are small if the hierarchy of scales in Eq. (3.5) is satisfied during theprocess). – 29 –volution was recast as a simple, first-order differential equation. We hope thatthis point of view, combined with the other techniques in this work and theexisting literature, will allow a rigorous analysis of stability in the vanishingamplitude limit. • Even if black holes do form in the (cid:15) − time scale, we point out that it does notalways represent efficient thermalization of the boundary theory via AdS/CFTduality. In some cases, black hole formation describes prethermalization, andactual thermalization is delayed until this black hole evaporates. Acknowledgments
We thank Jan de Boer, Avery Broderick, Alex Buchel, Stephen Green, Diego Hofman,Matthew Johnson, Luis Lehner, Vladimir Rosenhaus, Andrzej Rostworowski, JorgeSantos and Ferdinand Verhulst for useful discussions. This work is part of the ∆-ITPconsortium and also supported in part by the Foundation for Fundamental Researchon Matter (FOM), both are parts of the Netherlands Organisation for Scientific Re-search (NWO) that is funded by the Dutch Ministry of Education, Culture and Science(OCW). F.D. is supported by GRAPPA PhD Fellowship. M.L. and I.S.Y. are sup-ported in part by funding from the European Research Council under the EuropeanUnion’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreementno. 268088-EMERGRAV.
A. Analytical details
In this appendix, we will clarify some analytical details omitted in Sec. 3. There weshowed how to reach a simple differential equation for u , Eq. (3.18), which can besolved simply by integrating the Green’s function: u ( r, t f ) = 12 (cid:90) r − r dt (cid:90) r + r − tr − r + t dr (cid:48) S ( r (cid:48) , t )= 12 (cid:90) r − r dt (cid:90) r + r − tr − r + t dr (cid:48) (cid:16) C (¨ u + u (cid:48)(cid:48) ) + ˙ C ˙ u + C (cid:48) (cid:16) u (cid:48) − u r (cid:48) (cid:17)(cid:17) . (A.1)Here we should be careful about our method of images. A physical solution φ is onlygiven by a u that is an odd function of r , and it is not obvious that the u given bythe above integral will have this property. Another potentially worrisome observationis that the lower limit of the r (cid:48) integral can be negative for some positive r , but a– 30 –hysical answer should only invoke an integration over the physical space r > C = V − M/r is naturally defined.In this case, these concerns about the method of images can be easily resolved. Asexplained in Sec. 3, we can generalize the definition of V and M to include the r < M is an odd function of r and V is even. Together with thefact that u is odd, we see that the integrand in Eq. (A.1) is odd. Any integrationover negative r is canceled by an equal region with positive r , so effectively the lowerlimit of the r (cid:48) integral is | r − r + t | . Eq. (A.1) is effectively only integrating over thephysical range. It is, however, more convenient to keep working in this form and avoidthe confusion of taking an absolute value. An odd integrand here also guarantees that u is an odd function which leads to a physical φ .The form of Eq. (A.1) clearly suggests some integrations by parts. u ( r, t f ) = − (cid:90) r − r dt (cid:90) r + r − tr − r + t dr (cid:48) C (cid:48) u r (cid:48) + 12 (cid:90) r − r dt (cid:90) r + r − tr − r + t dr (cid:48) ( Cu (cid:48)(cid:48) + C (cid:48) u (cid:48) )+ 12 (cid:90) r +2 r r − r dr (cid:48) (cid:90) r −| r − r (cid:48) |− r dt ( C ¨ u + ˙ C ˙ u ) (A.2)= − (cid:90) r − r dt (cid:90) r + r − tr − r + t dr (cid:48) C (cid:48) u r (cid:48) + 12 (cid:90) r − r dt C (cid:104) ( r + r − t ) , t (cid:105) u (cid:48) (cid:104) ( r + r − t ) , t (cid:105) − (cid:90) r − r dt C (cid:104) ( r − r + t ) , t (cid:105) u (cid:48) (cid:104) ( r − r + t ) , t (cid:105) + 12 (cid:90) rr − r dr (cid:48) C (cid:104) r (cid:48) , ( r − r + r (cid:48) ) (cid:105) ˙ u (cid:104) r (cid:48) , ( r − r + r (cid:48) ) (cid:105) + 12 (cid:90) r +2 r r dr (cid:48) C (cid:104) r (cid:48) , ( r − r (cid:48) + r ) (cid:105) ˙ u (cid:104) r (cid:48) , ( r − r (cid:48) + r ) (cid:105) − (cid:90) r +2 r r − r dr (cid:48) C ( r (cid:48) , − r ) ˙ u ( r (cid:48) , − r )= 12 (cid:90) r − r dt C (cid:104) ( r − r + t ) , t (cid:105) (cid:16) ˙ u (cid:104) ( r − r + t ) , t (cid:105) − u (cid:48) (cid:104) ( r − r + t ) , t (cid:105)(cid:17) + 12 (cid:90) r − r dt C (cid:104) ( r + r − t ) , t (cid:105) (cid:16) ˙ u (cid:104) ( r + r − t ) , t (cid:105) + u (cid:48) (cid:104) ( r + r − t ) , t (cid:105)(cid:17) − (cid:90) r − r dt (cid:90) r + r − tr − r + t dr (cid:48) C (cid:48) u r (cid:48) − (cid:90) r +2 r r − r dr (cid:48) C ( r (cid:48) , − r ) ˙ u ( r (cid:48) , − r ) . In the above equation, we first isolated two terms which should be integrated by parts,and for one of them we interchange the order of integration so it can be done with– 31 –espect to t instead of r (cid:48) . The integration by parts produces five boundary terms asline integral along five segments which we explicitly write down. Finally, two pairsof segments can combine with each other and be expressed as time integrals. Wecollect the remaining space integral and the only non-boundary term which cannot beintegrated by parts in the end.Note that up to this step, we have not used any approximations. We did not evenuse the property that C is sourced by φ . In other words, this expression could describethe change in the field profile under the influence of any other spherically symmetricgravitational effects, either apart from or on top of its self-interaction.Our next step is to plug in Eq. (3.15) and use our assumption that it represents athin shell: we assume that u is only nonzero within two narrow packages around r = t and r = − t . This significantly simplifies Eq. (A.2) to u ( r, t f ) = − (cid:15) √ w f (cid:48) (cid:18) r − r w (cid:19) (cid:90) r − r dt C (cid:104) ( r − r + t ) , t (cid:105) (A.3)+ (cid:15) √ w f (cid:48) (cid:18) − r − r w (cid:19) (cid:90) r − r dt C (cid:104) ( r + r − t ) , t (cid:105) − (cid:90) r − r dt (cid:90) r + r − tr − r + t dr (cid:48) C (cid:48) u r − (cid:90) r +2 r r − r dr (cid:48) C ( r (cid:48) , − r ) ˙ u ( r (cid:48) , − r ) . Note that here the f (cid:48) means a derivative with respect to the variable of f instead of a r derivative, which should always be clear from the context.Since in the end, we are only interested in the physical range r >
0, we canactually drop the second term because the profile f is zero there. This starts to takethe promised form of Eq. (3.19), and we can almost identify∆ r = (cid:90) r − r dt C (cid:104) | r − r + t | , t (cid:105) . (A.4)Note that we have added an absolute value to the first variable in C . This makes nodifference since it is even, but it helps to emphasize the fact that the integral can bestrictly limited to the physical r > (cid:18) Mr (cid:19) | dr | = (1 + V ) dt . (A.5)Thus an incoming null ray starting from r = r and t i = − r does not exactly returnto r = r at t f = r ; the amount it misses is exactly given by Eq. (A.4). The leading-order correction due to gravity, of course, includes the fact that geodesics are changed,– 32 –nd the shell simply follows the new geodesic. A geometric calculation is enough todetermine how much a localized object appears to be shifted from the position predictedby the zeroth-order theory.For any finite-sized source, the gravitational potential at large r is proportionalto 1 /r , so the integral in Eq. (A.4) actually had a piece proportional to log r . Sincewe are have taken r to be large, one might have worried that such a term would theruin perturbation expansion. However, such the log r term is totally expected fromthe change of geodesics and does not interfere with our goal of computing the changein width or other changes.One last concern about the position shift in Eq. (A.4) is that it is a function of r . This turns out not to be a problem either, since the r -dependent part of ∆ r is notproportional to log r . We can see this by taking a derivative with respect to r : ∂ r ∆ r = ∂ r (cid:18)(cid:90) r − r − r dt C (cid:104) ( r − r − t ) , t (cid:105) + (cid:90) r r − r dt C (cid:104) ( t − r + r ) , t (cid:105)(cid:19) = (cid:18) − (cid:90) r − r − r dt C (cid:48) (cid:104) ( r − r − t ) , t (cid:105) + (cid:90) r r − r dt C (cid:48) (cid:104) ( t − r + r ) , t (cid:105)(cid:19) . (A.6)According to the Einstein’s equation, we have C (cid:48) = V (cid:48) − M (cid:48) r + Mr = 2 Mr + r T rr − T tt ) . (A.7)This means that as long as we restrict the matter sources to (1) finite-sized sourcesthat vanish beyond some fixed r and/or (2) radiation in the radial direction, T rr = T tt ,then the r dependence of ∆ r will not have a log r (or any other large r ) dependence.Furthermore, there is no small- r divergence either, since the two terms in Eq. (A.6) takesopposite signs and cancel each other near r = 0. Pictorially, this means that differentinfinitesimal segments within the wavepacket “shift” differently from one another bysome finite amount.In the last line of Eq. (A.3), the first term is also finite for the same reason asEq. (A.7), and the second term is finite because u has compact support. Theseterms should be combined and understood as some perturbative deformations of thewavepacket profile. They are cleanly separated from the ∆ r ∼ log r overall shift, whichis uniquely defined by a projection:∆ r = − (cid:82) u ∂ r u dr (cid:82) ( ∂ r u ) dr . (A.8)We can simply remove this shift mode from Eq. (3.19) and study the other deformations.A more physical way to understand the removal of this shift is letting the wavepacket– 33 –volve an extra time ∆ t = ∆ r such that it really reaches position r ; then it will befair to compare with the zeroth-order profile at the same position.In order to eventually form a black hole, we need the energy density to becomelarge. Since the total energy is conserved, the most trivial way to increase the energydensity is to narrow the width of the profile. The leading-order change in width canbe extracted from u by the following projection:∆ w = (cid:82) u ∂ w u dr (cid:82) ( ∂ w u ) dr . (A.9)Note that the (cid:15) dependence was already scaled out in Eq. (3.6). Interestingly, (cid:15) has theunit of length in our conventions, and the physical change in width is (cid:15) ∆ w . Therefore,∆ w is dimensionless. The width w is the only other dimensionful quantity that canpotentially affect ∆ w in the leading order ( r affects only the subleading error), and sothere is no way it can enter the expression for ∆ w .What we really wish to determine the sign of ∆ w . ∆ w < w > ∂ w u actually measures the scaling the profile around some center. If that center is notthe center of mass, then this scaling not only changes the width but also shifts theposition. A simple projection will be contaminated by the large ∼ log r contributionfrom the position shift. We will avoid this by always defining the profile f ( x ) to haveits center of mass at x = 0. This means that, on top of the normalization (cid:90) f (cid:48) ( x ) dx = 1 , (A.10)we also demand that (cid:90) xf (cid:48) ( x ) dx = 0 . (A.11)This guarantees that the scaling mode ∂ w u is orthogonal to the shift mode ∂ r u . B. Numerical examples
B.1 The asymmetry-focusing correlation
In this appendix, we numerically evaluate the change to a thin-shell profile after onebounce. Our example will be the following two one-parameter families of profiles. g a ( x ) = (1 + ax ± x ) e − x , (B.1)– 34 –igure 5: The left panel shows the energy density of the profiles with the “+” signdefined in Eq. (B.1), and the right panel shows the profiles with the “ − ” sign. Theblue curves are the symmetric, a = 0 profiles. The red (dashed) curves are for a = 0 . a . N a = (cid:115)(cid:90) g (cid:48) a dx , (B.2) X a = N − a (cid:90) xg (cid:48) a dx , (B.3) f a ( x ) ≡ N − a g a ( x + X a ) . (B.4)They are symmetric when a = 0, and varying a scans through two different directionsof asymmetry. Note that the quadratic term is necessary. Without it, a small a simplymeans an overall shift in position and the profile will be still symmetric to leadingorder. Our definition of f a ( x ) shifts the center of mass back to x = 0 and preservesonly the asymmetry generated by a .We plot some representative profiles in Fig. 5. Note that for both families, we have f a ( x ) = f − a ( − x ). Scanning through positive and negative values of a can confirm ouranalytical proof in Sec. 4 that flipping the profiles leads to opposite behaviors withinone bounce. It will also provide a better understanding about what physical quantityreally affects whether a profile becomes focused or not.There are infinite ways to be asymmetric, and our parameter a certainly is notthe unique parameter to quantify the asymmetry. It also has no reason to be theasymmetry directly responsible for focusing or defocusing the profile. However, for anyfamily of profiles centered around a symmetric one, we can define a natural parameterto quantify the asymmetry, at least for small values of a . Here is how it goes. First ofall, g a has a center of mass shifted by X a from g by the definition in Eq. (B.3). On theother hand, one can also naturally define the shift by a projection to the zero mode,– 35 –igure 6: The left panel shows the asymmetry parameter defined as the differencebetween the center-of-mass shift and the field-profile shift. The blue curve is for the“+” family and the red curve for the “ − ” family. The right panel shows the changeis width after one bounce, which qualitatively agrees with the asymmetry parameter.These are done with r = 60 and w = 1. Recall that the physical change in width isactually (cid:15) ∆ w .which is exactly the way we defined ∆ r in Eq. (A.8).¯ X a = N − (cid:90) [ g ( x ) − g a ( x )] g (cid:48) ( x ) dx . (B.5)These two shifts already disagree at linear order in a , therefore the amount of theirdisagreement, ∆ X a = ( X a − ¯ X a ), seems to be a reasonable way to quantify the asym-metry.Given these profiles, we solve Eq. (3.10) for M and V , and then we can integrateEq. (A.1). When we scan the parameter a from − . .
5, the change in width ∆ w defined in Eq. (A.9) follows a pattern closely resembling the behavior of this asymmetryparameter, ∆ X a . We compare them side-by-side in Fig. 6. Note that they are notidentical. For example, the relative slopes between the two families near a = 0 are notthe same. Thus, although we see a rough correlation between them, we cannot claimthat our asymmetry parameter directly controls focusing or widening in one bounce.In our conventions, the profiles are moving toward the right in Fig. 5. If we comparetheir shapes in Fig. 5 to their behaviors in Fig. 6, we get the following impression: • When the wavepacket is denser in the front, we get ∆ w <
0. The shell getsfocused into a smaller region, and gravitational effect will become stronger in thenext bounce. • When the wavepacket is denser in the tail, we get ∆ w >
0. It profile gets widerafter one bounce, and gravitational effect will become weaker in the next bounce.– 36 –igure 7: Results with a = 0 and varying r from 30 to 90 in steps of 10. The left panelshows ∆ w , which indeed goes to zero as 1 /r . The right panel shows the position shift∆ r defined in Eq. (A.8) which has the correct log r dependence.Since the family of profiles with “+” sign is much more sensitive to the parameter a ,we will use it to test other behaviors later in App. B.2.In Fig. 6, one might notice that for the a = 0 symmetric profiles, the changesin width are not exactly zero as we argued earlier. This deviation is not physicalbut simply an artifact of our approximation. That is because although the physicalsolution is symmetric in time, our technical choice breaks that symmetry by a smallamount. We have set the correction to the field profile at the initial time to zero, u ( r, − r ) = 0. This is a small error since the first-order correction to the metric inEq. (3.10) would have already modified the free field profile at that time, by a smallfraction ∼ (cid:15) V ∼ (cid:15) /r .We test this explanation by varying r and verifying that ∆ w goes to zero in theexpected way; see Fig. 7. We also verified that the position shift indeed has an r -dependent shift ∆ r ∝ log r , as discussed in App. A.Finally, with a symmetric profile, one can ask for a prediction for the next bounce.What we observed in these examples is that a symmetric profile will pick up a ∆ X a > w >
0, namely their energy become defocused. We stress again that this is not aproof, but merely two examples we observed. A more thorough investigation is requiredto support the generic downward flow we conjectured in Sec.5.
B.2 The effect of another object
Black hole formation does not always involve all of the energy becoming concentratedinto a thin shell. For example, an initially smooth field profile might start to developone or more sharp peaks. It is possible for the energy density in these peaks to becomelarge enough to induce strong gravity and black hole collapse before or even withoutthe average density of the entire profile ever becoming large.– 37 –n this section, we will present some evidence that sharp peaks evolve similarlyto thin shells. In the perturbative regime, one is free to separate the matter intocomponents in many ways. In particular, we can treat a smooth field profile with asharp peak as a thin shell propagating in the background of some additional diffusesource of gravity. Our approach is convenient since Eq. (A.1) and further analysis aboutit do not rely on the specific metric ansatz Eq. (3.10). As long as the additional sourceare also spherically symmetric, we can simply repeat the calculation in the previousappendix.We will start by considering a simple situation in which the additional mattersources are static. In addition to the thin shell with total mass 4 π(cid:15) , we have M ( r ) = 10 (cid:15) tanh (cid:18) rw star (cid:19) , (B.6) P r ≈ . (B.7)This is a star of roughly constant density, radius w star and total mass 10 (cid:15) . It doesnot interact with the massless field which forms the shell in any other way other thangravitationally, so it simply enters by altering the metric in Eq. (3.10). We assume thestar is stable and supports itself by a negligible amount of radial pressure (but it canhave angular pressure), so it does not add any extra term to modify T rr .According to the momentum space analysis, including this additional gravitationalsource breaks the AdS resonance structure and should interfere with black hole for-mation [12]. We show that such an interpretation is not necessary to understand thedynamics of thin shells in one bounce. Remember that for a symmetric shell profile,we argued that there cannot be a change in width due to the time-reversal symmetry.Adding an extra, static source does not break that symmetry, so symmetric profilesagain cannot change in size. And, it is straightforward to verify that asymmetry stillfocuses or defocuses in qualitatively the same way as before.In Fig. 8, we demonstrate that whether the shell gets thinner or thicker has thesame dependence on the asymmetry induced by the parameter a . Its magnitude doeshave an interesting dependence on the additional source. In the first set of data we fixthe size of the star to be equal to the shell. The change in width turns out to growlinearly with the additional mass. On the other hand if we keep the same density andincrease the size of the star, the change in width is not strongly affected. References [1] P. Bizon and A. Rostworowski, “On weakly turbulent instability of anti-de Sitterspace,”
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