Stability of AdS in Einstein Gauss Bonnet Gravity
SStability of AdS in Einstein Gauss Bonnet Gravity
Nils Deppe, ∗ Allison Kolly, † Andrew Frey, ‡ and Gabor Kunstatter § Physics Department, University of Winnipeg (Dated: November 11, 2018)Recently it has been argued that in Einstein gravity Anti-de Sitter spacetime is unstable againstthe formation of black holes for a large class of arbitrarily small perturbations. We examine theeffects of including a Gauss-Bonnet term. In five dimensions, spherically symmetric Einstein-Gauss-Bonnet gravity has two key features: Choptuik scaling exhibits a radius gap, and the mass functiongoes to a finite value as the horizon radius vanishes. These suggest that black holes will not formdynamically if the total mass/energy content of the spacetime is too small, thereby restoring thestability of AdS spacetime in this context. We support this claim with numerical simulations anduncover a rich structure in horizon radii and formation times as a function of perturbation amplitude.
INTRODUCTION
Anti-de Sitter (AdS) spacetime has been shown to beunstable against the formation of black holes for a largeclass of arbitrarily small perturbations, except for spe-cific initial data [1–9]. Given the interpretation of blackhole formation as thermalization in the AdS/conformalfield theory (CFT) duality, the questions of stability andturbulence of AdS are very important. The instability isapparently due to a subtle interplay of local non-lineardynamics and the non-local kinematical effect of the AdSreflecting boundary. An important question thereforeconcerns the dependence of the instability and turbulentbehaviour on the local dynamics. We investigate the ef-fects of higher curvature terms, which translate to finite N and ’t Hooft coupling corrections in the dual CFT.The most tractable higher curvature term is the Gauss-Bonnet (GB) term, since the equations of motion con-tain only second derivatives and are readily amenableto a Hamiltonian analysis. Since AdS /CFT is a pri-mary case of interest in the context of the AdS/CFTcorrespondence, we focus on 5D; the GB term, like othercurvature-squared terms, is dual to differing a and c cen-tral charges in the 4D CFT. As a result, the GB term iscommonly studied in the AdS/CFT context.On the gravity side, the GB term changes the local dy-namics in regions of high curvature and radically altersthe critical behaviour (Choptuik scaling) of microscopicblack hole (BH) formation [10, 11]. One interesting fea-ture of 5D Einstein-Gauss-Bonnet (EGB) gravity is thatthe horizon radius of a static spherically symmetric BHvanishes for a critical value of the ADM mass, so a BHcannot form dynamically for ADM mass less than thiscritical value. Such an algebraic mass gap is also presentin the 3D Einstein gravity case [12]; nonetheless, 5D EGBgravity differs in that the Riemann tensor is not deter-mined by the Ricci tensor (as opposed to 3D) and theGB term introduces a new length scale.Due to the reflecting boundary conditions at infinityin AdS spacetime, in the sub-critical region there aretwo possible endstates: a naked singularity or a quasi-periodic state in which the matter continues to bounce back and forth. It is important to determine which ofthese endstates is realized generically.Of potentially greater interest is whether the GB termstabilizes the spacetime above the algebraic threshold,given evidence [10] that some initial data with super-critical ADM mass still do not form black holes in asymp-totically flat spacetime, i.e. that there is a radius gap.This dynamical radius gap is expected to be a feature ofEGB in at least all odd dimensions [10] and may also bepresent in other higher curvature theories. We confirmthe presence of a radius gap and observe that in asymp-totically AdS spacetime it affects black hole formationeven at ADM mass far above the critical value.In the following we present 5D numerical simulationsconsistent with the conjecture that the stability of AdSin 5D EGB gravity is restored for arbitrarily small per-turbations. In the AdS/CFT correspondence, this wouldimply that low-energy perturbations of Yang-Mills theo-ries on S need not thermalize when finite N and ’t Hooftcoupling are taken into account. ACTION AND EQUATIONS OF MOTION
The action for 5D EGB gravity with cosmological con-stant minimally coupled to a massless scalar is given by I = (cid:90) d x √− g (cid:26) − ∇ µ ψ ∇ µ ψ + 12 κ (cid:18) λ + R + λ (cid:2) R − R µν R µν + R µνρσ R µνρσ (cid:3)(cid:19)(cid:27) . (1)We will later rescale ψ to remove the Planck scale andnumerical factors from the equations of motion. As R →∞ , any static spherically-symmetric solution asymptotesto AdS with effective cosmological constant λ eff = (1 −√ − λλ ) / λ . It proves convenient to use coordinatesin which the AdS scale λ eff = 1.A Hamiltonian analysis of EGB (and more generalLovelock) gravity in the spherically symmetric contexthas been carried out in [13–16]; due to the Hamil-tonian constraint, the generalized Misner-Sharp mass a r X i v : . [ h e p - t h ] J a n function[17] M = R (cid:20) λ + (1 − R ,µ R ,µ ) R + λ R (1 − R ,µ R ,µ ) (cid:21) , (2)gives the energy due to matter within radius R andasymptotes to the ADM mass at R → ∞ [18]. In terms ofthe mass function, the horizon condition ( R ,µ R ,µ ) | R H =0 is M ( R H ) = 12 (cid:2) λR H + R H + λ (cid:3) , (3)which implies that R H → M ( R H ) → M crit ≡ λ / R H in the thirdterm of the mass function.To connect more readily to previous literature, we workin Schwarzschild-like coordinates with metric ds = R ,x (cid:18) − Ae − δ dt + A − dx + R R ,x d Ω (cid:19) (4)and spatial coordinate R = tan( x ). In future work, wewill consider AdS gravitational collapse in flat-slice coor-dinates, which are useful for studying scaling and singu-larity formation since they allow evolution past apparenthorizon formation.The resulting first order equations of motion areΦ ,t = (cid:0) Ae − δ Π (cid:1) ,x (5)Π ,t = 3sin( x ) cos( x ) Ae − δ Φ + (cid:0) Ae − δ Φ (cid:1) ,x (6) δ ,x = − cos( x ) sin ( x )(Π + Φ ) (cid:2) sin ( x ) − λ ( A − cos ( x )) (cid:3) (7) M ,x = A ( x )(Π + Φ ) (8) A = 1 + sin ( x )(1 − λ )2 λ × (cid:34) − (cid:115) M λ (1 − λ ) tan ( x ) (cid:35) . (9)Here, Φ = ψ ,x and Π is conjugate to ψ . In this parame-terization the horizon condition is A = 0.The boundary conditions at the origin are identicalto those in asymptotically flat spacetime and are well-known. At infinity, the boundary conditions areΦ = ρ (cid:0) Φ + Φ ρ + · · · (cid:1) , Π = ρ (Π + · · · ) , (10)where ρ = π/ − x .We solve the system (5-9) using the method of lines.[19]. We have verified that our code is consistently con-vergent, and that conserved quantities, such as the ADMmass, remain fully fifth order accurate throughout simu-lations. Additionally, we verify that altering parts of thealgorithm to higher and lower order methods providesthe expected convergence changes RESULTS
In all simulations we use Gaussian initial dataΦ = 0 , Π = 2 π (cid:15) exp (cid:32) − (cid:18) π tan( x ) σ (cid:19) (cid:33) , σ = 116 . (11)Fig. 1 shows the horizon radius vs amplitude for 5D Ein-stein gravity, indicating that our code gives the expectedresults for long times in this case. Specifically, we seeBH formation after the initial pulse bounces off the AdSboundary at infinity, possibly a large number of times.Since the coordinates break down at the horizon, the codesignals horizon formation when A ( x, t ) falls below 2 − k ,where k is the exponent in the number of grid points usedin the simulation, i.e. 2 + 1. FIG. 1. BH horizon radius on formation vs initial amplitudein Einstein gravity. Inset: horizon formation time vs ampli-tude.
The inset in Fig. 1 presents a plot of black hole for-mation time vs amplitude for 5D Einstein gravity. Itillustrates that BH formation occurs soon after an in-teger number of reflections from the AdS boundary (around-trip time from origin to boundary takes time π ).The formation time is approximately piecewise constant,which increases exponentially in each piece as the ampli-tude decreases.Fig. 2 shows the effect of introducing a non-zero GBparameter, λ = 0 . (cid:15) = 36 to 48 because BHformation for lower amplitudes required many reflectionsand requires more computation time. The lowest ampli-tude for which we successfully formed a black hole was (cid:15) = 36, which required 24 bounces.The inset of Fig. 2 illustrates the horizon formationtime vs amplitude for the same data. It shows that BH’sform directly for large amplitudes and transition to form-ing after one reflection off the boundary for amplitudes (cid:15) ∼ −
44. However, there is rich structure between (cid:15) ∼
44 and 45.3, where the horizon radius and formationtime vary unpredictably.
FIG. 2. BH horizon radius on formation vs initial amplitudein EGB gravity, λ = 0 . Fig. 3 shows the scaling plot as the critical amplitude (cid:15) = (cid:15) ∗ for BH formation is approached after zero andone bounce. Whereas in Einstein gravity these would FIG. 3. Scaling of horizon radius at formation after zero andone bounce for λ = 0 . . be straight lines[20] of slope γ = 0 . ± . x H ∼ .
014 in both cases, suggesting the existenceof a radius gap in agreement with [10].Another feature of both sets of data is a jump in hori-zon radius as the amplitude is lowered. This can be un-derstood by considering the horizon function, A ( x, t ). Inparticular, when the horizon radius gets small, A ( x, t )flattens out near horizon formation and additional min-ima (see Fig. 4) appear. The jump in horizon radiusoccurs as an outer minimum “overtakes” the inner onesin reaching the value that signals horizon formation inthe code first. This indicates that the scalar pulse formsmultiple thick shells interior to the outer minimum.To address the question of the endstate for ADM massbelow M crit , we simulated an amplitude (cid:15) = 20, where FIG. 4. Metric function A just prior to horizon formation for (cid:15) = 45 . (cid:15) crit = 21 .
86 corresponds to M crit . Without the GBterm this amplitude results in black hole formation afterthree bounces. In the present case the simulation wascontinued to t = 200, corresponding to over 60 bounces,with no horizon formation. The dynamics of the pulse asit bounces back and forth is quite intricate [22].Comparison to Einstein gravity is instructive. Fig. 5graphs Π at the origin, which is proportional to the traceof the stress tensor, for (cid:15) = 12 . . Fig. 6 graphs Π for (cid:15) = 20 in EGB gravity. In con-trast, the pattern is irregular, and there is no apparenttendency to focus. FIG. 5. Π ( x = 0 , t ) in Einstein gravity for (cid:15) = 12 . From the inset in Fig. 6 one can see that there aremultiple peaks of Π ( x = 0). This agrees with our ob-servations from animations that the GB term causes theoriginal pulse to break up into multiple smaller pulses,which then propagate through the spacetime. The GBterm causes delays in the implosions resulting in a slightlydifferent phase for the different pulses. We have observedthat BH’s form when a sufficient number of these pulsesare within the horizon radius at the same time. Interest-ingly, this does not necessarily translate into the curva-ture being large at the origin.Additionally, the energy spectrum of the (cid:15) = 20 pulseshows no evidence of a turbulent cascade of energy tohigher frequencies as time passes [22]. This providessome support to the notion that the system settles intoa smooth quasi-periodic state, however more simulationsare necessary to draw a definitive conclusion. FIG. 6. Π ( x = 0 , t ) in EGB gravity for (cid:15) = 20. Inset:zoomed to show peaks with different relative phases. The above results are in stark contrast with what isseen in the 3D case where an algebraic mass gap is alsopresent. In 3D Einstein gravity, there is no lower boundon the BH radius [23], whereas the BH radius is boundedbelow in the present case. This behavior seems closelyrelated to the complex structure seen in Fig. 2. Further,the energy spectra for sub-critical collapse in 3D does notshare this characteristic behaviour [24, 25].
CONCLUSIONS
We have presented the results of numerical simulationsof spherically symmetric massless scalar field collapse in5D AdS EGB gravity. Our data are consistent with theconjecture that stability against small perturbations isrestored. Some speculations are perhaps in order: Af-ter each bounce from the boundary, the Einstein termfocuses the pulse of matter as it implodes at the origin.On the other hand, the observed dynamical radius gapleads to a defocusing effect that resists BH formation atsmall horizon radii and allows the matter to travel tothe boundary multiple times before BH formation. Thedefocusing effect is evident in the out-of-phase peaks inΠ( x = 0) seen in Fig. 6 as well as the flattened formof the horizon function (Fig. 4) in EGB gravity. Thisdefocusing in turn affects the time it takes for the pulse to disperse from the origin. Furthermore, extreme sen-sitivity of the outcome (BH formation vs dispersion) toinitial conditions is a hallmark of critical collapse. Thissensitivity along with altered dispersal timescales leadsto the complex structure seen in Fig. 2. One can spec-ulate further that the map from amplitude to horizonformation time may evince a fractal structure due to theinterplay between Einstein and GB dynamics at the ori-gin. In any case, the data clearly suggest that the GBcorrections to short distance dynamics inhibit the for-mation of black holes and that stability may indeed berestored. Of course, it is much more difficult to provestability, if indeed that is the case, than instability. Weplan a detailed study of these issues in future work.There are in principle an infinite number of possiblehigher curvature deformations to Einstein gravity. It isimportant to ask whether the qualitative features we ob-serve persist in the more general class of deformations.In brief, the suppression of black hole formation in EGBis a consequence of the dynamical radius gap, which is in-dicative of a non-zero mass critical solution. These arewell-known to occur when a new length scale becomesrelevant to the dynamics, as invariably happens in grav-itational collapse with higher curvature deformations.Thus, we expect the BH suppression to be generic insuch theories. Moreover, the sensitivity to initial data ofcritical collapse in combination with a radius gap shouldgenerically lead to complex structure in pulse waveformsand BH formation time in higher-curvature gravities.In conclusion, our analysis shows that BH formationinstabilities in AdS are highly sensitive to small scaledynamics of gravity. Moreover, our results imply finite N ∗ [email protected]; Current address: Physics Dept., Cor-nell University † [email protected] ‡ [email protected] § [email protected][1] P. Bizo´n and A. Rostworowski, Phys.Rev.Lett. ,031102 (2011), arXiv:1104.3702 [gr-qc].[2] J. Ja(cid:32)lmu˙zna, A. Rostworowski, and P. Bizo´n, Phys.Rev. D84 , 085021 (2011), arXiv:1108.4539 [gr-qc]. [3] A. Buchel, L. Lehner, and S. L. Liebling, Phys.Rev.
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