A nonrelativistic limit for AdS perturbations

Abstract

The familiar $c\to \infty$ nonrelativistic limit converts the Klein-Gordon equation in Minkowski spacetime to the free Schroedinger equation, and the Einstein-massive-scalar system without a cosmological constant to the Schroedinger-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 $\Lambda$. Assuming for consistency that the product $\Lambda c^2$ tends to a negative constant as $c\to \infty$, we show that the corresponding nonrelativistic limit is given by the SN system with an external harmonic potential which we call the Schrodinger-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 analytically. 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.

arXiv commentv3: slightly expanded published version

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• Time-periodic quantum states of weakly interacting bosons in a harmonic trap
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• Complex plane representations and stationary states in cubic and quintic resonant systems
• Turbulent cascades in a truncation of the cubic Szego equation and related systems
• Obstruction to ergodicity in nonlinear Schrödinger equations with resonant potentials
• Time-periodicities in holographic CFTs
• Dark energy effects in the Schrödinger-Newton approach
• Exact solutions for a coherent phenomenon of condensation in conservative Hamiltonian systems
• Schrödinger-Newton-Hooke system in higher dimensions. Part I: Stationary states
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