A nonrelativistic limit for AdS perturbations

Abstract

The familiar c → ∞ nonrelativistic limit converts the Klein-Gordon equation in Minkowski spacetime to the free Schrödinger equation, and the Einstein-massive-scalar system without a cosmological constant to the Schrödinger-Newton (SN) equation. In this paper, motivated by the problem of stability of Anti-de Sitter (AdS) spacetime, we examine how this limit is affected by the presence of a negative cosmological constant Λ. Assuming for consistency that the product Λc 2 tends to a negative constant as c → ∞, we show that the corresponding nonrelativistic limit is given by the SN system with an external harmonic potential which we call the Schrödinger-Newton-Hooke (SNH) system. We then derive the resonant approximation which captures the dynamics of small amplitude spherically symmetric solutions of the SNH system. This resonant system turns out to be much simpler than its general-relativistic version, which makes it amenable to analytic methods. Specifically, in four spatial dimensions, we show that the resonant system possesses a three-dimensional invariant subspace on which the dynamics is completely integrable and hence can be solved exactly. The evolution of the two-lowest-mode initial data (an extensively studied case for the original general-relativistic system), in particular, is described by this family of solutions.

Breathing Modes, Quartic Nonlinearities and Effective Resonant Systems

“…Indeed, if {H, ·} is a generator of a dynamical Lie group lying in the Cartan subalgebra and {B, ·} is a generator corresponding to a positive root, one gets a relation of the sort (2). In such situations, many breathing modes can be present on the same footing, corresponding to different generators of the dynamical symmetry group, as is indeed the case for the systems that motivate our current study [10][11][12][13][14][15][16]. Nonetheless, one can often construct consistent dynamical truncations of such systems to a subset of degrees of freedom, either at the level of the full system or at the level of the resonant approximation in the weakly nonlinear regime, so that only one breathing mode is relevant in each truncation, which is again what happens in [10][11][12][13][14][15][16].…”

“…which is familiar from [11,13,[15][16][17]. Note that, with S having dropped out, the resonant Hamiltonian enjoys two conservation laws…”

“…• Finally, there is a relativistic analog of the Pitaevskii-Rosch breathing mode that can be made manifest by considering the systems defined by (16) in three spatial dimensions and with the value of m 2 corresponding to a conformally coupled scalar [13]. In fact, it is more convenient to consider the same type of scalar field on a spatial 3-sphere, which is related to the above AdS setup by a conformal transformation.…”

“…The structures outlined above have been observed in the recent literature for a number of special cases, motivated by rather disparate topics in physics. Thus, the analysis of [10][11][12] is rooted in the Gross-Pitaevskii equation and the physics of Bose-Einstein condensates, while the analysis of [13][14][15][16] originates in studies of nonlinear dynamics in Anti-de Sitter spacetimes, which is of interest for mathematical general relativity and high-energy physics. In the case of Bose-Einstein condensates, the existence of breathing modes is well-known [1][2][3], though their relation to the weakly nonlinear solutions within the resonant approximation has not been duly appreciated in the literature.…”

Complex plane representations and stationary states in cubic and quintic resonant systems

“…(We note in passing that perhaps the most familiar spatially confined setting where a translationally invariant PDE is compactified on a torus does not possess the type of resonant spectrum of linearized perturbation frequencies underlying our studies.) Many of the cases listed above generate resonant systems possessing a rich algebraic structure, including special analytic solutions in the fully nonlinear regime and extra conserved quantities [10,13,[15][16][17]23]. This has led us to formulating conditions on the interaction coefficients C nmkl that guarantee such special properties, resulting in a large class of partially solvable resonant systems presented in [30].…”