Gravitational non-commutativity and G\"odel-like spacetimes

TL;DR

This paper explores how certain stationary spacetimes exhibit non-commutative geometry similar to magnetic Landau levels, with specific focus on G"odel and Som-Raychaudhuri spacetimes, revealing a connection to fuzzy spheres.

Contribution

It establishes conditions for geodesics to resemble charged particle trajectories, links these to non-commutative geometry, and computes the induced non-commutativity in G"odel-like spacetimes.

Findings

G"odel spacetime induces non-commutativity identical to the fuzzy sphere.

Effective non-commutativity arises in G"odel and Som-Raychaudhuri spacetimes.

Star product naturally appears in Som-Raychaudhuri spacetime.

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 G\"odel 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 G\"odel spacetime is identical to that on the fuzzy sphere. Finally, we show how the star product naturally emerges in Som-Raychaudhuri spacetimes.