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 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.

Klein–Gordon oscillator in Kaluza–Klein theory

“…In the limit of = 0, a spectrum similar to the Landau levels is recovered. Note that this dynamics in the presence of the confining potential due to the Klein-Gordon oscillator reinforces the characteristics of the quantum dynamics observed in this Gödel-type spacetime [59][60][61], which are characterized by the similarity with the Landau problem for a charged particle on a surface exposed to a uniform magnetic field. Based on this analogy, Druker et al [59] have suggested the picture of a holographic description for a single chronologically safe region.…”

Linear confinement of a scalar particle in a Gödel-type spacetime

“…They analyzed the similarity between the spectrum of the energy of a scalar quantum particle in these class of Gödel-type spacetimes and the Landau levels in curved backgrounds. This similarity has also been observed by Das and Gegenberg [9] by studying the Klein-Gordon equation in the Som-Raychaudhuri spacetime (Gödel flat solution). They also compared with the Landau levels in the flat space.…”

“…This close relation can also be used in solving problems of the linear confinement of quarks in the presence of a topological defect, where a magnetic field can be introduced in a geometric way through the Som-Raychaudhuri spacetime metric. We claim that the results obtained here can be used to investigate heavy quarkonia subject to a strong magnetic field [45] in the presence of a topological defect, due to the similarity between the quantum dynamics in Som-Raychaudhuri spacetime and the Landau levels [8,9,16]. Models with confining potentials have been used to describe the spectrum of quarkoniatype systems.…”

“…This expression is analogous to the vector potential that yields a uniform magnetic field in the z-direction. An interesting property of this type of metric is that the geodesics for this spacetime are circles, which have a physical description similar to Larmor orbits for an electron that moves in a perpendicular magnetic field [8,9]. This analogy also arises from the point of view of quantum mechanics, where the quantum dynamics of scalar and spinorial quantum particles in this spacetime is an analogue of the Landau levels, as observed in Refs.…”

“…This analogy also arises from the point of view of quantum mechanics, where the quantum dynamics of scalar and spinorial quantum particles in this spacetime is an analogue of the Landau levels, as observed in Refs. [8][9][10]16,17]. We shall discuss this point below.…”