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Goedel-type Universes and the Landau Problem

Nadav Drukker, Bartomeu Fiol, Joan Simón

TL;DR

We address the question of holography in Gödel-type universes by establishing a precise analogy with the Landau problem on constant-curvature surfaces $S^2$, \mathbb{R}^2$, and $H_2$. The main approach combines classical geodesic analysis and Klein-Gordon mode solutions to reveal Landau-like spectra and localized wavefunctions, with a focus on the chronologically safe region inside a domain wall. The key contributions are (i) showing geodesics project to Larmor-like circles, (ii) deriving Landau-level–type spectra for KG fields with curvature- and sign-dependent degeneracies, and (iii) proposing a holographic description for a single safe region via a domain wall and a noncommutative cylinder whose boundary degrees of freedom scale linearly with the cavity radius. This framework provides a concrete, quasi–top-down path toward holography in spacetimes with closed timelike curves and connects to quantum Hall physics and matrix-model realizations of holographic screens, offering a platform to explore observer-dependent holography beyond AdS/CFT.

Abstract

We point out a close relation between a family of Goedel-type solutions of 3+1 General Relativity and the Landau problem in S^2, R^2 and H_2; in particular, the classical geodesics correspond to Larmor orbits in the Landau problem. We discuss the extent of this relation, by analyzing the solutions of the Klein-Gordon equation in these backgrounds. For the R^2 case, this relation was independently noticed in hep-th/0306148. Guided by the analogy with the Landau problem, we speculate on the possible holographic description of a single chronologically safe region.

Goedel-type Universes and the Landau Problem

TL;DR

We address the question of holography in Gödel-type universes by establishing a precise analogy with the Landau problem on constant-curvature surfaces , \mathbb{R}^2H_2$. The main approach combines classical geodesic analysis and Klein-Gordon mode solutions to reveal Landau-like spectra and localized wavefunctions, with a focus on the chronologically safe region inside a domain wall. The key contributions are (i) showing geodesics project to Larmor-like circles, (ii) deriving Landau-level–type spectra for KG fields with curvature- and sign-dependent degeneracies, and (iii) proposing a holographic description for a single safe region via a domain wall and a noncommutative cylinder whose boundary degrees of freedom scale linearly with the cavity radius. This framework provides a concrete, quasi–top-down path toward holography in spacetimes with closed timelike curves and connects to quantum Hall physics and matrix-model realizations of holographic screens, offering a platform to explore observer-dependent holography beyond AdS/CFT.

Abstract

We point out a close relation between a family of Goedel-type solutions of 3+1 General Relativity and the Landau problem in S^2, R^2 and H_2; in particular, the classical geodesics correspond to Larmor orbits in the Landau problem. We discuss the extent of this relation, by analyzing the solutions of the Klein-Gordon equation in these backgrounds. For the R^2 case, this relation was independently noticed in hep-th/0306148. Guided by the analogy with the Landau problem, we speculate on the possible holographic description of a single chronologically safe region.

Paper Structure

This paper contains 16 sections, 61 equations.

Table of Contents

  1. Introduction.
  2. A family of Gödel-type solutions
  3. Sphere $(l^2<0)$:
  4. Flat plane $(l^2=0)$:
  5. Hyperbolic plane $(l^2>0)$:
  6. Isometries
  7. Geodesics
  8. Flat plane $(l^2=0)$:
  9. Hyperbolic plane $(l^2> 0)$:
  10. The Klein-Gordon equation
  11. Flat plane $(l^2=0)$:
  12. Sphere $(l^2<0)$:
  13. Hyperbolic plane $(l^2>0)$:
  14. Holographic screens.
  15. Discussion
  16. ...and 1 more sections