Gravitational non-commutativity and Gödel-like spacetimes

Abstract

We derive general conditions under which geodesics of stationary spacetimes resemble trajectories of charged particles in an electromagnetic field. For large curvatures (analogous to strong magnetic fields), the quantum mechanicical states of these particles are confined to gravitational analogs of {\it lowest Landau levels}. Furthermore, there is an effective non-commutativity between their spatial coordinates. We point out that the Som-Raychaudhuri and Gödel spacetime and its generalisations are precisely of the above type and compute the effective non-commutativities that they induce. We show that the non-commutativity for Gödel spacetime is identical to that on the fuzzy sphere. Finally, we show how the star product naturally emerges in Som-Raychaudhuri spacetimes.

Two sections added (Relation to the fuzzy sphere, Emergence of the star product). 10 pages, Revtex. To appear in General Relativity and Gravitation

References in corpus (8)

• String Theory and Noncommutative Geometry
• Noncommutative Field Theory
• Large N Limit of Non-Commutative Gauge Theories
• Physical Instances of Noncommuting Coordinates
• Spinning deformations of the D1-D5 system and a geometric resolution of Closed Timelike Curves
• Can branes travel beyond CTC horizon in Godel Universe?
• The Kerr-Newman-Godel Black Hole
• Observations on noncommuting coordinates and on fields depending on them

Cited by in corpus (18)

• Klein-Gordon Oscillator in Kaluza-Klein Theory
• The generalized Klein-Gordon oscillator in the background of cosmic string space-time with a linear potential in the Kaluza-Klein theory
• Linear confinement of a scalar particle in a Gödel-type spacetime
• Fermions in Gödel-type background space-times with torsion and the Landau quantization
• PDM Klein-Gordon oscillators in cosmic string spacetime in magnetic and Aharonov-Bohm flux fields within the Kaluza-Klein theory
• The generalized Klein-Gordon oscillator with position-dependent mass in a particular Gödel-type space-time
• Weyl fermions in a family of Gödel-type geometries with a topological defect
• Confined Klein-Gordon oscillators in Minkowski spacetime and a pseudo-Minkowski spacetime with a space-like dislocation: PDM KG-oscillators, isospectrality and invariance
• The Dirac Equation in (1+2)-dimensional Gürses space-time backgrounds
• PDM Klein-Gordon particles in Gödel-type Som-Raychaudhuri cosmic string spacetime background
• Relativistic and nonrelativistic Landau levels for the noncommutative quantum Hall effect with anomalous magnetic moment in a conical Gödel-type spacetime
• Ground state of a bosonic massive charged particle in the presence of external fields in a Gödel-type spacetime
• Quantum influence of topological defects on a relativistic scalar particle with Cornell-type potential in cosmic string space-time with a spacelike dislocation
• On the Klein-Gordon scalar field oscillators in a spacetime with spiral-like dislocations in external magnetic fields
• KG-oscillators in Som-Raychaudhuri rotating cosmic string spacetime in a mixed magnetic field
• Investigation of spin-$0$ massive charged particle subject to a homogeneous magnetic field with potentials in a topologically trivial flat class of Gödel-type space-time
• Weyl Fermions in a Linear Class of Gödel-Type Space-Time Backgrounds
• On the Klein-Gordon Gürses-oscillators and pseudo-Gürses-oscillators: vorticity-energy correlations and spacetime associated degeneracies