Nonlinear perturbations of higher dimensional anti-de Sitter spacetime

Abstract

We study nonlinear gravitational perturbations of vacuum Einstein equations, with Λ < 0 in (n + 2) dimensions, with n > 2, the n = 2 case already having been done before. We follow the formalism by Ishibashi, Kodama and Seto to decompose the metric perturbations into tensor, vector and scalar sectors, and simplify the Einstein equations. We render the metric perturbations asymptotically anti-de Sitter by employing a suitable gauge choice for each of the sectors. Finally, we analyze the resonant structure of the perturbed equations at second order for the five dimensional case, by starting out with tensor-type perturbations at the linear level. We find evidence for resonances at second order.

Resonant Hamiltonian systems and weakly nonlinear dynamics in AdS spacetimes

“…Secular terms only appear at the second nontrivial order of the perturbation theory, and the corresponding resonant approximation will involve a cubic resonant system similar to (21). This vanishing of quadratic secular term, which is again a form of AdS selection rules, has been proved for the squashed sphere ansatz (40) in [29] and observed in practice in the numerical work for general perturbations outside spherical symmetry [64,67,70]. Similarly, there are indications that selection rules for cubic terms in the resonant approximation are in place, similar to the selection rules described for probe scalars under (18).…”

“…As one moves away from the ansatz (40), the situation becomes forbiddingly complicated. A number of partial treatments exist in the literature [62][63][64][65][66][67][68][69][70], but they mostly focus on evaluating the interaction coefficients involving specific mode quartets and analyzing their properties. These treatments typically rely on the Regge-Wheeler formalism [101,102] and the Ishibashi-Kodama construction [103].…”

“…Construction of time-periodic solutions in AdS [59][60][61] that has been approached both by perturbative methods (where one fine-tunes the initial conditions to eliminate the secular terms at higher and higher orders of the perturbative expansion) and numerically. (We will, however, briefly touch on related considerations [62][63][64][65][66][67][68][69][70] within the topic of gravitational dynamics in AdS outside spherical symmetry. )…”