Necessary conditions for an AdS-type instability

Abstract

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.

Fully resonant scalars on asymptotically AdS wormholes

“…These results extend also for the family of Quasitopological Gravities [42], [43], [44], [45], since the dynamics on the spherically symmetric dynamical scenarios with matter, can be obtained from the same formulae (50)- (52), by including extra terms in the polynomial.…”

“…In[52] it was noted that in a direct perturbative at order ε 4 the presence of a Gauss-Bonnet term cannot be scaled out.…”

Moving closer to the collapse of a massless scalar field in spherically symmetric anti–de Sitter spacetimes

“…Recent studies [41,42] indicate that some initial configurations never form an AH, but numerical studies can only provide evidence for these islands of stability. The properties and characteristics of these islands are being currently investigated combining non-linear perturbation methods and numerical simulations [43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58], and the structure of the spectrum of AdS perturbations appears to play an important role. Applications of these studies to the AdS/CFT correspondence can be found in [59,60].…”

Gravitational perturbations in a cavity: Nonlinearities

“…In all cases, there are subtleties involved in the imposition of Dirichlet wall boundary conditions at higher orders, and in the analysis of solutions to the perturbation equations satisfying these boundary conditions. We analyze these solutions at arbitrary order and apply the results in [6] to comment on the nonlinear stability of this system. Section II describes the methodology of the perturbation theory analysis at arbitrary order.…”

“…The modes can be obtained by solving these equations and it can be readily seen that at linear order, these modes reduce to pure gauge as expected. Section VII contains a summary and discussion of the nonlinear stability of this system, by using the equations obtained in the pa-per and nonlinear dynamics results in [6]. Appendices A and B have detailed computations for some of the equations in the rest of the paper.…”