Abstract:We revisit the issues of non-linear AdS stability, its relation to growing (secular) terms in naïve 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.

We continue our analytic investigations of non-linear spherically symmetric perturbations around the anti-de Sitter background in gravity-scalar field systems, and focus on conservation laws restricting the (perturbatively) slow drift of energy between the different normal modes due to non-linearities. We discover two conservation laws in addition to the energy conservation previously discussed in relation to AdS instability. A similar set of three conservation laws was previously noted for a self-interacting scalar field in a non-dynamical AdS background, and we highlight the similarities of this system to the fully dynamical case of gravitational instability. The nature of these conservation laws is best understood through an appeal to averaging methods which allow one to derive an effective Lagrangian or Hamiltonian description of the slow energy transfer between the normal modes. The conservation laws in question then follow from explicit symmetries of this averaged effective theory.

Using Wigner transforms of Green functions, we discuss non-equilibrium generalizations of spectral functions and occupation numbers. We develop methods for computing time-dependent spectral functions in conformal field theories holographically dual to thin-shell AdS-Vaidya spacetimes.