Resonant Dynamics and the Instability of Anti–de Sitter Spacetime

Bizoń, Piotr; Maliborski, Maciej; Rostworowski, Andrzej

doi:10.1103/physrevlett.115.081103

Cited by 87 publications

(172 citation statements)

References 26 publications

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“…A careful analysis of this separation using rigorous renormalization group methods was done in [12]. The two-time framework has been used since in [13], [14]. This has uncovered quasi-periodic and recurrent behaviour in the gravity-scalar field system for times O(1/ǫ 2 ), ǫ being the field amplitude at initial time.…”

Section: Introductionmentioning

confidence: 99%

Necessary conditions for an AdS-type instability

Menon

Suneeta

2016

Phys. Rev. D

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In this work, we analyze the necessary conditions for a nonlinear AdS-type instability in the gravity-scalar field system. In particular, we discuss the necessary conditions for a cascade of energy to higher modes by applying results in KAM theory. Our analytical framework explains numerical observations of instability of spacetimes even when the spectrum is only asymptotically resonant, and the fact that a minimum field amplitude is needed to trigger it. Our framework can be applied for similar stability analyses of fields in (locally) asymptotically AdS spacetimes. We illustrate this with the example of the AdS soliton. Further, we conjecture on the possible reasons for quasiperiodic behaviour observed in perturbations of AdS spacetime. Certain properties of the eigenfunctions of the linear system dictate whether there will be localization in space leading to black hole formation. These properties are examined using the asymptotics of Jacobi polynomials. Finally we discuss recent results on the AdS instability in Einstein-Gauss-Bonnet gravity in light of our work.

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Section: Introductionmentioning

confidence: 99%

Necessary conditions for an AdS-type instability

Menon

Suneeta

2016

Phys. Rev. D

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“…If collapse is captured by these equations, then extrapolation becomes possible. A numerical study of the flow equations (truncated to 172 normal modes) for collapsing initial data in AdS 5 suggests (for the infinite system) a finite‐time oscillatory singularity. A fit to the ansatz

a_{n} (ɛ^{2} t) \sim n^{- γ (ɛ^{2} t)} e^{- ρ (ɛ^{2} t) n} (n ≫ 1),

motivated by the analyticity strip method, shows indeed that ρ tends to zero in finite time, while γ tends to 2, leaving a characteristic power‐law spectrum.…”

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confidence: 99%

“…Many other developments and open questions are worth mentioning: –Several of the results we have discussed have been generalized to massive scalar fields coupled to gravity , to scalar fields in Gauss‐Bonnet gravity and to a spherical cavity in Minkowski space . –It is interesting to study the stability of spacetimes with normal mode spectra that are asymptotically resonant . –It would be nice to know whether all stability islands are anchored on quasiperiodic solutions . –Is there a way to prove collapse for arbitrarily small initial data ? –Can one get an analytic handle on the turbulent regime leading to black hole formation? Is there a relation between the power spectra observed in singular blow‐ups of solutions to the effective flow equations in and approximate power spectra seen numerically in the strongly turbulent regime in ? …”

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confidence: 99%

AdS (in)stability: an analytic approach

Craps

Evnin

2016

Fortschritte der Physik

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“…We note that the TTF has been verified in Ref. [4] and even the authors of the Comment have now used the method [5].With regard to claim (3), we did not use the phrase "two-mode stability islands" in our Letter. In later work [6] we referred to "stability islands," but in that case we were referring to quasi-periodic solutions.…”

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confidence: 99%

“…We note that the TTF has been verified in Ref. [4] and even the authors of the Comment have now used the method [5].…”

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confidence: 99%

Balasubramanianet al.Reply:

et al. 2015

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In a recent Comment [1], Bizoń and Rostworowski present a few criticisms of our recent Letter [2]. In particular, they present three arguments: (1) that their own evolutions of the two-mode initial data we had studied collapse to a black hole around a time t ≈ 1080 whereas we found no collapse for times up to roughly t = 1500, (2) that our two-timescale framework (TTF) "cannot be even used to infer stability," and (3) that our "claims about stability islands" may not be correct.We have studied evolutions of this initial data with resolutions higher than we had originally. As displayed in Fig. 1, we do find that higher resolutions display higher concentrations of energy, but we nevertheless have still not observed collapse to a black hole. We note that our code demonstrates convergence to a unique solution even at the late times in question (i.e., around t ≈ 1080). We also emphasize that we have demonstrated that the solution to which our numerical results converge is in fact a solution of the scalar anti-de Sitter (AdS) system we seek to solve. This latter property (consistency) is demonstrated with tests that the constraint residuals and mass loss converge to zero (see our Supplementary Material). And so, while it is possible that the continuum solution does in fact collapse around t ≈ 1080 (and indeed, the work of [3] does see collapse in a code for which both convergence and consistency are verified), we cannot confirm this (in time for the publication of [1]) and hence their claim (1).However even if collapse does take place at a time t ≈ 1080, the main claims of our paper still stand. We could not know if collapse occurred after the time which we ran our code, and so collapse for some time soon after t = 1500 was always a possibility. And so if instead collapse occurs at t ≈ 1080, there is no change to our claims. Within that time one still finds both direct and inverse cascades which is now confirmed by the Comment.Another important point is that, were we to decrease the initial amplitude , we would certainly observe any possible collapse pushed to times later than t ≈ 1080.Regarding their claims (2) and (3), there seems to be some misunderstanding of what we tried to communicate in our Letter. Determining whether some scalar perturbation of AdS with arbitrarily small amplitude collapses to a black hole is not possible with numerical evolutions which require some finite , finite resolution, and finite evolution time t.More particularly regarding their claim (2), we agree that the TTF cannot be used to infer stability beyond times ∝ −2 . Indeed, we stressed that we have carried our TTF analysis to O( 3 ) so that one can only trust predictions within times scaling as −2 . However, for shorter times, we disagree with the claims of the Comment that our truncation to j max = 47 "does not suffice to capture the dynamics of the turbulent cascade" for 2-mode initial data. As is clear from Fig. 4 of [2], the vast majority of the energy remains in the lowest modes of the system during evolution, and the dynamics of ...

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Resonant Dynamics and the Instability of Anti–de Sitter Spacetime

Cited by 87 publications

References 26 publications

Necessary conditions for an AdS-type instability

Necessary conditions for an AdS-type instability

AdS (in)stability: an analytic approach

Balasubramanianet al.Reply:

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