Effect of coordinate noncommutativity on the mass of a particle in a uniform field and the equivalence principle

Abstract

We consider the motion of a particle in a uniform field in noncommutative space which is rotationally invariant. On the basis of exact calculations it is shown that there is an effect of coordinate noncommutativity on the mass of a particle. A particular case of motion of a particle in a uniform gravitational field is considered and the equivalence principle is studied. We propose the way to solve the problem of violation of the equivalence principle in the rotationally invariant noncommutative space.

Rotationally invariant noncommutative phase space of canonical type with recovered weak equivalence principle

“…The influence of noncommutativity on the implementation of the equivalence principle was studied in the case of coordinates noncommutativity [18,19,[38][39][40][41], noncommutativity of coordinates and noncommutativity of momenta [42][43][44]. The authors of paper [43] concluded that the equivalence principle holds in the sense that an accelerated frame of reference is locally equivalent to a gravitational field, unless noncommutative parameters are anisotropic (η xy = η xz ).…”

“…The authors of paper [43] concluded that the equivalence principle holds in the sense that an accelerated frame of reference is locally equivalent to a gravitational field, unless noncommutative parameters are anisotropic (η xy = η xz ). In our previous papers we studied the possibility to recover the weak equivalence principle in a space with noncommutativity of coordinates [18,39,41], in four-dimensional noncommutative phase space of canonical type [44]. In the present paper we study the implementation of the weak equivalence principle in rotationally invariant noncommutative phase space (5)- (7).…”

“…In the limit η 2 → 0 one obtains x i (t) = δ 1,i gt 2 /2 + x 0i as it should be. From (39) we have that particles with different masses move on different trajectories in a uniform gravitational field in rotationally invariant noncommutative phase space. From this we can conclude that the weak equivalence principle also known as the uniqueness of the free-fall principle is violated.…”

“…The principle states that velocity and position of a point mass in a gravitational field are independent of mass, composition and structure and depend only on its initial position and velocity. We would like to stress that if condition (19) holds from ( 35), (39), one has…”

“…Note also that taking into account (39), the motion of the center of mass of a system of N noninteracting particles with masses m a in a uniform gravitational field is described by the following trajectory:…”