A comment on AdS collapse of a scalar field in higher dimensions
aa r X i v : . [ g r- q c ] A ug A comment on AdS collapse of a scalar field in higher dimensions
Joanna Ja lmu˙zna, Andrzej Rostworowski, and Piotr Bizo´n Institute of Mathematics, Jagiellonian University, Krak´ow, Poland Institute of Physics, Jagiellonian University, Krak´ow, Poland (Dated: March 21, 2018)We point out that the weakly turbulent instability of anti-de Sitter space, recently found in [1]for four dimensional spherically symmetric Einstein-massless-scalar field equations with negativecosmological constant, is present in all dimensions d + 1 for d ≥
3, contrary to a claim made in [2].
In a recent paper [1] two of us reported on numeri-cal simulations which indicate that anti-de Sitter (AdS)space is unstable against the formation of a black holeunder arbitrarily small generic perturbations. This insta-bility was conjectured to be triggered by a resonant modemixing which moves energy from low to high frequencies.Although the simulations of [1] were done only in fourspacetime dimensions, it was clear from the nonlinearperturbation analysis given there that the same weaklyturbulent mechanism of generating instability operatesin all supercritical dimensions d + 1 for d ≥
3. Thus,we found it surprising that Garfinkle and Pando Zayas,who later looked at the same problem in 4 + 1 dimen-sions (apparently unaware of Ref.[1] as they did not citeit), wrote in [2]: ”We [...] establish that for small val-ues of the initial amplitude of the scalar field there is no[sic!] black hole formation, rather, the scalar field per-forms an oscillatory motion typical of geodesics in AdS.”The purpose of this comment is to show that, contraryto the above quoted claim,
AdS d +1 is unstable againstgravitational collapse for all d ≥
3. To this end, we firstgeneralize the formalism of [1] to higher dimensions andrecall the key argument for collapse of arbitrarily smallgeneric initial data. Second, using our code in d = 4 weevolve numerically the same initial data as those of [2]and show that, as expected, they collapse. Finally, wetry to identify a possible source of the error in [2].We parametrize the ( d +1)–dimensional asymptoticallyAdS metric by the ansatz ds = ℓ cos x (cid:0) − Ae − δ dt + A − dx + sin x d Ω d − (cid:1) , (1)where ℓ = − d ( d − / (2Λ), d Ω d − is the round metricon S d − , −∞ < t < ∞ , 0 ≤ x < π/
2, and A , δ arefunctions of ( t, x ). For this ansatz the evolution of a self-gravitating massless scalar field φ ( t, x ) is governed by thefollowing system (using units in which 8 πG = d − (cid:0) Ae − δ Π (cid:1) ′ , ˙Π = 1tan d − x (cid:0) tan d − x Ae − δ Φ (cid:1) ′ , (2) A ′ = d − x sin x cos x (1 − A ) − sin x cos x A (cid:0) Φ + Π (cid:1) , (3) δ ′ = − sin x cos x (cid:0) Φ + Π (cid:1) , (4) where · = ∂ t , ′ = ∂ x , Φ = φ ′ and Π = A − e δ ˙ φ . We wantto solve the system (2-4) for small smooth perturbationsof AdS solution φ = 0 , A = 1 , δ = 0. Smoothness at thecenter implies that near x = 0 φ ( t, x ) = f ( t ) + O ( x ) , δ ( t, x ) = O ( x ) ,A ( t, x ) = 1 + O ( x ) , (5)where we used normalization δ ( t,
0) = 0 so that t is theproper time at the center. Smoothness at spatial infinityand finiteness of the total mass M imply that near x = π/ ρ = π/ − x ) φ ( t, x ) = f ∞ ( t ) ρ d + O (cid:0) ρ d +2 (cid:1) , δ ( t, x ) = δ ∞ ( t ) + O (cid:0) ρ d (cid:1) ,A ( t, x ) = 1 − M ρ d + O (cid:0) ρ d +2 (cid:1) , (6)where the power series expansions are uniquely deter-mined by M and the functions f ∞ ( t ), δ ∞ ( t ) (which inturn are determined by the evolution of initial data). Onecan show that the initial-boundary value problem for thesystem (2-4) together with the regularity conditions (5)and (6) is locally well-posed.In [1] the instability of AdS was conjectured to resultfrom the resonant mode mixing which moves energy fromlow to high frequencies. It was argued that this process ofenergy concentration on increasingly small spatial scalesmust be eventually cut off by the formation of a blackhole. It is easy to see that the same mechanism is atwork for all d ≥
3. This follows from the fact that, usingthe PDE terminology, the system (2-4) is fully resonant.More precisely, the spectrum of the linear self-adjointoperator, which governs the evolution of linearized per-turbations of
AdS d +1 , L = − tan − d x ∂ x (cid:0) tan d − x ∂ x (cid:1) , is given by ω j = ( d + 2 j ) , ( j = 0 , , ... ). The keypoint is that the frequencies ω j are equally spaced, soalready at the third order of nonlinear perturbation anal-ysis one gets resonant terms for any frequency ω j suchthat j = j + j − j , where j k are indices of eigenmodespresent in the initial data. Some of these resonances leadto secular terms which signal the onset of instability attime t = O ( ε − ), where ε measures the size of initialdata.To substantiate this heuristic argument we solve thesystem (2-4) in d = 4 numerically, using the method of[1]. For easy comparison of results, we take the sameapproximately ingoing Gaussian initial data as Ref.[2]:Φ(0 , x ) = ∂ x φ (0 , x ) = Π(0 , x ), where φ (0 , x ) = ε √ (cid:18) − (tan x − r ) σ (cid:19) (7)with r = 4, σ = 1 .
5. As in [2] we set the AdS radius ℓ = 1, hence their and our radial coordinates are relatedby r = tan x , while the time coordinates are identical.We denote their amplitude A by ε (to avoid conflict ofnotation); the factor 1 / √ πG = 3, while in [2] 8 πG = 1. Notethat the data (7) slightly violate the regularity condition(5) since Φ(0 ,
0) is not exactly zero, however an errorgenerated by this ’corner singularity’ is negligible. t H ε t H ε FIG. 1: Time of horizon formation t H vs amplitude for ini-tial data (7). The upper plot depicts the first ’step’ of the’staircase’ function t H ( ε ) corresponding to large data solu-tions which collapse during the first implosion. The horizonradius varies from x H = 0 . ε > . t H ( ε )corresponding to solutions which bounce several times fromthe AdS boundary before collapsing. For large ε , the solution quickly collapses (the forma-tion of an apparent horizon is detected by the metric
0 200 400 600 800 1000 1200 1400 Π ( t, ) t ε = 2 ⋅ -4 , t H =299.8 ε = 2 ⋅ -4 , t H =623.4 ε = 10 -4 , t H =1273.7 ⋅ -6 ⋅ -6 ⋅ -6 ⋅ -6 -5 ⋅ -5 ⋅ -5 ε - Π ( ε t, ) ε t ε = 2 ⋅ -4 ε = 2 ⋅ -4 ε = 10 -4 FIG. 2: Upper plot: the upper envelope of Π ( t,
0) for initialdata (7) with three relatively small amplitudes. After mak-ing 95 (for ε = 0 . ε = √ · . ε = 0 . ε − Π ( ε t,
0) seem to converge to a limiting curve. function A ( t, x ) touching zero at some x H ). As ε de-creases, the horizon radius takes the form of the rightcontinuous sawtooth curve x H ( ε ) (see Fig. 1 of [1]) withjumps at critical points ε n where lim ε → ε + n x H ( ε ) = 0 (theindex n counts the number of reflections from the AdSboundary before collapse). Accordingly, the time of hori-zon formation t H ( ε ) is a monotone decreasing piecewisecontinuous function with jumps at each ε n (see Fig. 1).For small initial data the weakly nonlinear perturba-tion analysis described in [1] predicts the onset of insta-bility at time t ∼ ε − . Numerics indicates that for suf-ficiently small ε this scaling holds approximately almostall the way to the collapse, that is t H ( ε ) ∼ ε − . Theevidence for this fact is shown in Fig. 2 which depicts theevolution of three solutions with small amplitudes differ-ing by a factor of √
2. Note that this scaling implies thatthe computational cost of numerical evolution increasesrapidly as ε decreases (since solutions have to be evolvedfor longer and longer times on finer and finer grids).Finally, let us make a few remarks about the paper [2].The content of that paper, when stripped of the standardpromotional material for the application of AdS/CFTcorrespondence to the quark-gluon plasma, boils downto two statements concerning the solutions of five di-mensional spherically symmetric Einstein-massless-scalarfield equations with negative cosmological constant: (i)large initial data lead to collapse and the larger the data,the shorter the time of horizon formation; (ii) small ini-tial data do not form black holes. The statement (i) istrue but trivial. The statement (ii), as we have shownabove, is false. We wonder what led the authors of [2] toreach this conclusion. The numerical method used in [2]is rather crude; it is based on the second order finite dif-ference code on a fixed nonuniform grid. The radial coor-dinate r is not compactified and the AdS timelike bound-ary at r = ∞ is mimicked by an artificial reflecting mirrorat r max = 10. Although this fact might make long-timequantitative results somewhat inaccurate (since the pulsegets reflected before reaching the true boundary), we donot think it had any effect on the claim (ii). We thinkthat the culprit was the lack of sufficient resolution. Inthe first (v1) version of [2], Fig. 2v1 shows three implo-sions for the solution with amplitude ε = 0 . x H = 1 . · − during the first implo-sion at time t H = 1 .
398 which means that the absence ofcollapse and the ”oscillatory motion” of Fig. 2v1 are nu-merical artifacts. It appears that the authors of [2] haverealized that their numerical simulations suffered frominsufficient spatial resolution because in the second (v2)version of [2] the number of grid points was increasedfrom 800 to 6400. We gather that this improvement ofresolution has helped to detect the collapse in Fig.2v1 during the first implosion because the new Fig. 2v2 de-picts four implosions of a ’non-collapsing’ solution withfive times smaller amplitude ε = 0 . x H ≈ . · − after time t H ≈ . +1 grid points just before collapse.Anyway, the lesson is that the long-time numerical simu-lations of asymptotically AdS spacetimes are challengingeven in spherical symmetry and one should be carefulin jumping to conclusions about the late time dynam-ics, especially without an analytic understanding of theproblem. Acknowledgments: