OOn the stability of anti-de Sitter spacetime
Nils Deppe
Cornell Center for Astrophysics and Planetary Science,Cornell University, Ithaca, New York 14853, USA ∗ (Dated: June 10, 2016)We present results from a detailed study of spherically symmetric Einstein-massless-scalar fielddynamics with a negative cosmological constant in four to nine spacetime dimensions. This study isthe first to examine dynamics in AdS beyond five dimensions and the gauge dependence of recentlyproposed perturbative methods. Using these perturbative methods, we provide evidence that theoscillatory divergence used to argue for instability of anti-de Sitter space by Bizo´n et al. is a gauge-dependent effect in five spacetime dimensions. Interestingly, we find that this behavior appears tobe gauge-independent in higher dimensions; however, understanding how this divergence dependson the initial data is more difficult. The results we present show that while much progress has beenmade in understanding the rich dynamics and stability of anti-de Sitter space, a clear route to theanswer of whether or not it is stable still eludes us. Introduction.
Stability of de Sitter and Minkowskispacetimes under small perturbations was establishedin 1986[1] and 1993[2]. Following the Anti-de Sitter(AdS)/Conformal Field Theory (CFT) conjecture[3], thequestion of the stability of AdS became more interest-ing. Using the AdS/CFT conjecture it is possible to ad-dress the important question of thermalization and equi-libration of strongly coupled CFTs, which is dual to thequestion of whether or not small perturbations of AdScollapse to a black hole. The stability of AdS against ar-bitrarily small scalar field perturbations was first studiednumerically in spherical symmetry by Bizo´n and Rost-worowski in 2011[5], where the authors suggested that alarge class of perturbations eventually collapse to form ablack hole even at arbitrarily small amplitude, (cid:15) . How-ever, in such simulations a finite (cid:15) must be used, leavingroom for doubt as to whether arbitrarily small pertur-bations do actually form a black hole[6]. The probingof small-amplitude perturbations is aided by the recentlyproposed renormalization flow equations (RFEs)[7–9] forwhich any behavior observed at amplitude (cid:15) and time t/(cid:15) is also present at an amplitude (cid:15) (cid:48) and time t/(cid:15) (cid:48) .This rescaling symmetry was used by Bizo´n et al. to ar-gue for the instability of AdS based on a divergence inthe RFE solution for specific initial data[10]. However,it is suspected that this divergence is a gauge-dependenteffect[11].In this paper, we address the AdS stability questionand the concerns of [11] by performing a detailed studyof the RFEs and the nonlinear Einstein equations. Ourstudy is the first to examine the gauge dependence ofthe RFEs and dynamics in AdS beyond five dimensions.Our numerical methods enable us to study the RFEs toa much higher accuracy than previous work, providingnew insight into when the RFEs are no longer valid and Novel results beyond spherical symmetry were recently presentedby Dias and Santos[4]. the reasons they fail. With a new understanding of theRFEs we revisit AdS , finding agreement with previouswork[7, 12] but strong contrast with what is observedin higher dimensions. Finally, we show that our resultsare largely robust against the choice of initial data andpresent evidence that the dynamics of AdS are moreintricate than in higher dimensions. Model.
We consider a self-gravitating massless scalarfield in a spherically symmetric, asymptotically AdSspacetime in d spatial dimensions. The metric inSchwarzschild-like coordinates is ds = (cid:96) (cid:2) − Ae − δ dt + A − dx + sin (cid:0) x(cid:96) (cid:1) d Ω d − (cid:3) cos (cid:0) x(cid:96) (cid:1) , (1)where d Ω d − is the metric on S d − , x/(cid:96) ∈ [0 , π/ t/(cid:96) ∈ [0 , ∞ ). The areal radius is R ( x ) = (cid:96) tan( x/(cid:96) ), andwe henceforth work in units of the AdS scale (cid:96) (ie, (cid:96) = 1).The evolution of the scalar field ψ is governed by thenonlinear systemΦ ,t = (cid:0) Ae − δ Π (cid:1) ,x , Π ,t = ( Ae − δ tan d − x Φ) ,x tan d − x , (2)where Π = A − e δ ψ ,t is the conjugate momentum andΦ = ψ ,x is an auxiliary variable. The metric functionsare solved for from δ ,x = − sin x cos x (Π + Φ ) (3) A ,x = d − x sin x cos x (1 − A ) − sin x cos x (Φ + Π ) . (4)See [13, 14] for a detailed discussion of the code we use tosolve this system. For asymptotic flatness at the originwe require A ( x = 0 , t ) = 1. Two common gauge choicesare the interior time gauge (ITG), where δ ( x = 0 , t ) =0, and the boundary time gauge (BTG), where δ ( x = π/ , t ) = 0. We perform evolutions of the fully nonlineartheory in the ITG. a r X i v : . [ g r- q c ] J un We are particularly interested in perturbations aboutAdS ( d +1) whose evolution at zeroth order is governed byˆ L = − (tan − d x ) ∂ x (tan d − x∂ x ) (this can be seen by set-ting A = 1 , δ = 0 and Π = ψ ,t in Eq. (2)). The eigen-modes of ˆ L are given in terms of Jacobi polynomials, e j ( x ) = κ j cos d ( x ) P ( d/ − ,d/ j (cos 2 x ) (5)with eigenvalues ω j = d + 2 j and where κ j =2 (cid:112) j !( j + d − / Γ( j + d/ was presented in [12], while[10] investigated AdS . To study the RFEs, a “slowtime” τ = (cid:15) t is introduced, and dynamics on very shorttime scales can be thought of as being averaged over.The scalar field perturbation is expanded as ψ ( x, t ) = (cid:80) ∞ l =0 A l cos( ω l t + B l ) e l ( x ), where A l ( τ ) and B l ( τ ) aretime-dependent coefficients. The evolution of A l and B l is given by the RFEs[9] − dA l dt = (cid:88) i,j,ki + j = k + l { i,j }(cid:54) = { k,l } S ijkl ω l A i A j A k sin( B l + B k − B i − B j ) , (6) − dB l dt = (cid:88) i,j,ki + j = k + l { i,j }(cid:54) = { k,l } S ijkl ω l A l A i A j A k cos( B l + B k − B i − B j )+ T l ω l A l + (cid:88) ii (cid:54) = l R il ω l A i , (7)where { i, j } (cid:54) = { k, l } means both i and j are not equal to k or l , and the coefficients T l , R il and S ijkl are given byintegrals over the eigenmodes in appendix A of [9] andby recursion relations in [11]. The gauge dependence ofthe coefficients is discussed in [11].In our numerical computations we typically truncatethe RFEs (6-7) at l max = 399, giving a good balance be-tween computational cost and accuracy, and refer to thissystem as the truncated RFEs (TRFEs). We note thatthe evolutions dominate the computational cost, not theconstruction of T l , R il and S ijkl , which we have computedto l max > Results.
We present results from a detailed study ofthe TRFEs and fully nonlinear numerical evolutions infour to nine spacetime dimensions. For concreteness wefocus on two-mode initial data of the form ψ ( x,
0) = (cid:15) ( e ( x ) + κe ( x )) /d (8)but have also studied Gaussian initial data given byΠ( x,
0) = (cid:15) exp (cid:18) − tan ( x ) σ (cid:19) , ψ ( x,
0) = 0 . (9) l l og | A l | τ = . × − τ = . × − τ = . × − τ = . × − τ = . × − τ = . × − FIG. 1. The spectrum A l with l max = 399 for initial data (8)in AdS using the ITG. The spectrum becomes singular when τ (cid:63) = (cid:15) t (cid:63) ≈ . × − . In the evolutions presented here we choose κ = d/ ( d + 2),which has been studied extensively in AdS [7, 12, 14] andin AdS [10] using the ITG. A logarithmic divergence inthe time derivative of the phases, dB l /dt , was observedin [10]. This is consistent with an asymptotic analysisof the equations in the ITG; however, the terms leadingto the logarithmic divergence appear to be absent in theBTG[11]. We will address this in detail below.An interesting technique for analyzing solutions to theTRFEs is the analyticity-strip method[10, 15]. Thismethod involves fitting the spectrum A l to A l = C ( t ) l − γ ( t ) e − ρ ( t ) l (10)for l (cid:29)
1. The analyticity radius ρ ( t ) should be inter-preted as the distance between the real axis and the near-est singularity in the complex plane. When ρ becomeszero the TRFEs have evolved to a singular spectrum. Wedenote the time when the spectrum becomes singular by t (cid:63) (or τ (cid:63) in slow time) and in d > t is slightly larger than t (cid:63) . All fits usedata from simulations done with l max = 399 and omit thelowest and highest twenty modes to reduce errors fromtruncation. For concreteness we present results in AdS but observe qualitatively identical behavior for d > is shown inFig. 1. At τ = 1 . × − the spectrum is alreadysingular, so we show it only for completeness.In Fig. 2 we plot ρ ( t ) for both the ITG and BTG forAdS . We observe that the spectrum becomes singular atapproximately the same t (cid:63) in both the BTG and the ITG,independent of the dimension being studied, suggesting See Eq. (2.2) of [15] for more details.
1. 10 1. 15 1. 20 1. 25 1. 30 1. 35 τ × ρ ( τ ) ITGBTG
FIG. 2. ρ ( τ ) for the l max = 399 AdS evolution in the ITG andBTG. In both gauges the spectrum becomes singular at τ (cid:63) ≈ . × − , suggesting this behavior is gauge-independent. this behavior is gauge-independent. Interestingly, in ourstudy of d > below.We study dB l /dt for several different values of l upto l = 300 (far below l max to minimize mode truncationerrors) in the ITG and BTG to see if the logarithmicdivergence observed in [10] is gauge-dependent as sug-gested by the asymptotic analysis in [11]. In Fig. 3 weplot dB /dt for AdS and AdS in the ITG and BTG.To assess the presence of a logarithmic divergence we fit a ln(¯ t − t ) + b to dB l /dt . In AdS a logarithmic diver-gence is observed in the ITG and ¯ t is within ∼ .
3% of t (cid:63) . We estimate the error in ¯ t to be (cid:46) . t is approximately 2% larger than t (cid:63) . However, evolu-tions carried out beyond t (cid:63) no longer exhibit divergentbehavior in dB l /dt , making it difficult to interpret thesignificance of ¯ t in the BTG in AdS . This suggests thedivergence in dB l /dt coinciding with ρ going to zero is agauge-dependent effect in AdS .For d > ¯ t and t (cid:63) differ by less than 1%, in AdS by less than 0 . ∼ .
1% in AdS . While the divergent behaviorappears more prominently in the ITG, it is clearly alsopresent in the BTG (see Fig. 3). This behavior disagreeswith the asymptotic analysis of the equations discussed in[11], where it is suggested that the logarithmic behaviorarises from terms that are absent from the coefficients T l and R il in the BTG. The improving agreement between ¯ t and t (cid:63) with increasing dimension demonstrates that un-derstanding the origin of the divergence in dB l /dt is morecomplicated than initially thought and that attempting
100 200 300 400 500 t | d B / d t | ITG AdS ITG AdS BTG AdS BTG AdS FIG. 3. Evidence that the logarithmic divergence is a gauge-dependent effect in AdS . To assess the behavior we fit a ln(¯ t − t ) + b (solid lines) to the data. For d > t and t (cid:63) improves with increasing dimensionality. to connect the divergent behavior to critical phenomenaor thermalization would be a premature conclusion.In spatial dimensions d > ( x = 0 , t ), which is proportional to the Ricci scalar atthe origin, for several different values of l max and differ-ent values of (cid:15) for nonlinear evolutions in AdS . There isgood agreement between the nonlinear and TRFE solu-tions and the agreement improves with increasing dimen-sionality, at least for Π ( x = 0 , t ). This may be related tothe eigenmodes having larger values at x = 0 in higher di-mensions. For example, we find that in AdS e ( x = 0)is ∼ times larger than in AdS .
0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 1. 2 τ × l og ( Π / † ) l max = 99 ITG l max = 199 ITG l max = 399 ITG l max = 399 BTG † = 0 . † = 0 . † = 0 . FIG. 4. The upper envelope of Π ( x = 0 , t ) for two-modeequal-energy data, Eq. (8), for evolutions in AdS . Plottedare solutions to the TRFEs in the ITG (dash lines) and theBTG (dashed-dotted lines) for several different values of l max ,and nonlinear evolutions for (cid:15) = 0 . , . , . (cid:15) and larger l max . We now turn to the case of two-mode equal-energydata, Eq. (8), in AdS . This case has been studied ex-tensively using numerical relativity and the TRFEs[5, 7,14, 16, 17]. It was suggested in [12] that this solution t l og ( Π ) l max = 99 l max = 199 l max = 299 l max = 395 † = 0 . FIG. 5. The upper envelope of Π ( x = 0 , t ) for two-mode data(8) for evolutions in AdS . Plotted are solutions to the TRFEs(dash lines) for different values of l max , and the nonlinearevolution (solid line). The TRFE solutions with l max = 299and 395 differ only marginally until the third increase in Π . orbits a quasi-periodic solution of the same temperature.For these evolutions we use l max = 99 , , , and 395to test convergence and to understand how the analyt-icity radius depends on mode truncation. We compareΠ ( x = 0 , t ) with the nonlinear evolution in Fig. 5. The299 and 395 mode evolutions are almost indistinguish-able until the third increase in Π . This suggests thatthe agreement between the TRFE and nonlinear solu-tions would improve if a lower amplitude nonlinear evo-lution were studied, similar to what is observed in Fig. 4for AdS . Unfortunately, such an evolution requires aprohibitive amount of computational resources.To gain insight into the correct interpretation of thespectrum becoming singular, we plot the analyticity ra-dius using different l max in Fig. 6. We observe that ρ crosses zero several times. However, as the numberof modes is increased ρ crosses zero fewer times, un-like in AdS where increasing l max does not qualitativelychange the behavior of ρ . This suggests that if moremodes were used in AdS , ρ would not cross zero un-til at least a fourth energy cascade occurs. It is inter-esting to note that the spectrum becomes singular nearthe end of the inverse cascade when the energy startsto transfer to higher modes again. For l max = 99 thisoccurs at t ≈ , , l max = 395 onlyat t ≈ , l max . This l max -dependent flattening out ofthe spectrum further demonstrates that for equal-energytwo-mode data in AdS , ρ crossing zero is an artifact ofmode truncation and should only be interpreted as an in-dication that an insufficient number of modes was used.The flattening of the top 30% of modes is in contrastwith what is observed in AdS , where the entire spec-
200 400 600 800 1000 1200 1400 t ρ ( t ) l max = 99 l max = 199 l max = 299 l max = 395 FIG. 6. Analyticity radius ρ for two-mode equal-energy datain AdS using different l max . The minimum of ρ increaseswith l max suggesting the evolutions are still suffering frommode truncation to some extent. trum flattens out (see Fig. 2).While it may be tempting to speculate that the two-mode data in AdS is stable and that the theorems of [18]for behavior of stable solutions should be applied for alltimes, we believe that this would be premature in lightof our new understanding of when the perturbation the-ory suffers from mode truncation. However, given theevidence we have presented and that our perturbativeresults are not (severely) suffering from mode truncationuntil the third energy cascade, the theorems of [18] pro-vide strong evidence that small amplitude equal-energytwo-mode initial data is stable at least until τ ≈ . d = 4 we find that for l = 300¯ t is approximately 1.3 times larger in the BTG than inthe ITG. We are also unable to find agreement between¯ t and t (cid:63) in both gauges. However, increasing l max from299 to 399 increases t (cid:63) by ∼
1% so it is possible that us-ing l max (cid:29)
400 would resolve the discrepancies. Similarto the two-mode data, agreement of ¯ t in the BTG andITG and between ¯ t and t (cid:63) improves with increasing di-mensionality. Nevertheless, even for AdS we do not findagreement of ¯ t in the BTG and ITG or between ¯ t and t (cid:63) . Interestingly, in AdS the analyticity radius does notcross zero, at least up to l max = 399, even though thisdata collapses in nonlinear evolutions. However, increas-ing l max still decreases ρ , so the possibility of ρ crossingzero with sufficiently many modes is not ruled out. Thissuggests that at least in AdS , and possibly all dimen-sions, the TRFE solution spectrum not becoming singu-lar is insufficient to guarantee that the nonlinear evolu-tion will never collapse to a black hole. Conclusions . In summary, our study is the first to ex-amine the gauge dependence of the RFEs and dynamicsin AdS beyond five dimensions. Our numerical methodsallow us to test the RFEs to a much higher accuracythan previous studies, providing new insight into whenthe equations no longer accurately approximate the Ein-stein equations. We provide evidence that the oscillatorysingularity of the RFEs used to argue for the instabil-ity of AdS in [10] is a gauge-dependent effect in fivedimensions and that this behavior is independent of ini-tial data. However, the oscillatory singularity appears tobe gauge-independent for dimensions greater than five,which is in disagreement with an analysis of the RFEsin the ultraviolet limit[11]. Interestingly, we find that inhigher dimensions the singular behavior of the RFEs oc-curs at the same time that a black hole forms in the fullnonlinear theory. In agreement with [12], we do not ob-serve an oscillatory singularity in AdS . Additionally, wefind that at least in AdS , the spectrum becoming sin-gular is not indicative of black hole formation in the fulltheory but rather that the truncated RFEs are no longervalid. While our results aid in understanding the valid-ity and behavior of the RFEs, they also show that eventhough much progress has been made in understandingthe (in)stability of AdS, a clear route to answering thequestion of stability of AdS still eludes us. Acknowledgements ∗ [email protected] [1] H. Friedrich, Journal of Geometry and Physics , 101(1986).[2] D. Christodoulou and S. Klainerman, The Global Nonlin-ear Stability of the Minkowski Space (PMS-41) , PrincetonLegacy Library (Princeton University Press, 2014).[3] J. M. Maldacena, Int. J. Theor. Phys. , 1113 (1999),[Adv. Theor. Math. Phys.2,231(1998)], arXiv:hep-th/9711200 [hep-th].[4] O. J. C. Dias and J. E. Santos, (2016), arXiv:1602.03890[hep-th].[5] P. Bizo´n and A. Rostworowski, Phys. Rev. Lett. ,031102 (2011), arXiv:1104.3702 [gr-qc].[6] F. V. Dimitrakopoulos, B. Freivogel, M. Lippert, and I.-S. Yang, JHEP , 077 (2015), arXiv:1410.1880 [hep-th].[7] V. Balasubramanian, A. Buchel, S. R. Green, L. Lehner,and S. L. Liebling, Phys. Rev. Lett. , 071601 (2014),arXiv:1403.6471 [hep-th].[8] B. Craps, O. Evnin, and J. Vanhoof, JHEP , 048(2014), arXiv:1407.6273 [gr-qc].[9] B. Craps, O. Evnin, and J. Vanhoof, JHEP , 108(2015), arXiv:1412.3249 [gr-qc].[10] P. Bizo´n, M. Maliborski, and A. Rostworowski, Phys.Rev. Lett. , 081103 (2015), arXiv:1506.03519 [gr-qc].[11] B. Craps, O. Evnin, and J. Vanhoof, JHEP , 079(2015), arXiv:1508.04943 [gr-qc].[12] S. R. Green, A. Maillard, L. Lehner, and S. L. Liebling,Phys. Rev. D92 , 084001 (2015), arXiv:1507.08261 [gr-qc].[13] N. Deppe, A. Kolly, A. Frey, and G. Kunstatter, Phys.Rev. Lett. , 071102 (2015), arXiv:1410.1869 [hep-th].[14] N. Deppe and A. R. Frey, JHEP , 004 (2015),arXiv:1508.02709 [hep-th].[15] C. Sulem, P.-L. Sulem, and H. Frisch, Journal of Com-putational Physics , 138 (1983).[16] P. Bizo´n and A. Rostworowski, Phys. Rev. Lett. ,049101 (2015), arXiv:1410.2631 [gr-qc].[17] V. Balasubramanian, A. Buchel, S. R. Green, L. Lehner,and S. L. Liebling, Phys. Rev. Lett. , 049102 (2015),arXiv:1506.07907 [gr-qc].[18] F. Dimitrakopoulos and I.-S. Yang, Phys. Rev. D92 ,083013 (2015), arXiv:1507.02684 [hep-th].[19] J. Ja(cid:32)lmu˙zna, A. Rostworowski, and P. Bizo´n, Phys. Rev.
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