Journal of High Energy Physics

, 2018:113 | Cite as

A nonrelativistic limit for AdS perturbations

  • Piotr BizońEmail author
  • Oleg Evnin
  • Filip Ficek
Open Access
Regular Article - Theoretical Physics


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 Λc2 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.


Classical Theories of Gravity AdS-CFT Correspondence Conformal and W Symmetry Integrable Field Theories 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Institute of PhysicsJagiellonian UniversityKrakówPoland
  2. 2.Department of Physics, Faculty of ScienceChulalongkorn UniversityBangkokThailand
  3. 3.Theoretische Natuurkunde, Vrije Universiteit Brussel and The International Solvay InstitutesBrusselsBelgium

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