Abstract
We revisit the issues of non-linear AdS stability, its relation to growing (secular) terms in naive perturbation theory around the AdS background, and the need and possible strategies for resumming such terms. To this end, we review a powerful and elegant resummation method, which is mathematically identical to the standard renormalization group treatment of ultraviolet divergences in perturbative quantum field theory. We apply this method to non-linear gravitational perturbation theory in the AdS background at first non-trivial order and display the detailed structure of the emerging renormalization flow equations. We prove, in particular, that a majority of secular terms (and the corresponding terms in the renormalization flow equations) that could be present on general grounds given the spectrum of frequencies of linear AdS perturbations, do not in fact arise.
Highlights
According to the Poincare-Lindstedt method, the secular term can be absorbed in a small frequency shift, leading to a generalized asymptotic expansion that provides more accurate approximations for longer time intervals, x(t) = cos
We review a powerful and elegant resummation method, which is mathematically identical to the standard renormalization group treatment of ultraviolet divergences in perturbative quantum field theory
In particular, that a majority of secular terms that could be present on general grounds given the spectrum of frequencies of linear anti-de Sitter (AdS) perturbations, do not arise
Summary
When dealing with a system subject to a small perturbation, it is natural to describe its evolution by an asymptotic series in the perturbation magnitude, an approach familar under the name of perturbation theory. A problem of much greater phenomenological significance is to be able to trace the effect of a small perturbation over large times, when its impact on the evolution becomes appreciable despite its smallness This is precisely the regime when the so-called secular terms in perturbation theory come into play. The growing terms at higher orders in the naıve perturbation theory typically appear in realistic situations, and they have become known as ‘secular’ terms (from the Latin word for ‘century’, referring to terms that become significant when considering planetary perturbations over the course of centuries) Such terms need to be restructured by means of resummation, if one is aiming at a perturbative description of the large-time dynamics at all.
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