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Journal of Physical Studies 22(1), Article 1001 [6 pages] (2018)
DOI: https://doi.org/10.30970/jps.22.1001 THE MOTION OF A PARTICLE IN A GRAVITATIONAL FIELD IN A ROTATIONALLY-INVARIANT NONCOMMUTATIVE SPACE OF A CANONICAL TYPE AND THE WEAK EQUIVALENCE PRINCIPLEKh. P. Gnatenko, O. O. Morozko, Yu. S. Krynytskyi
Department for Theoretical Physics, Ivan Franko National University of Lviv, |
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A noncommutative space of a canonical type with preserved rotational symmetry is considered. The space is constructed involving additional coordinates, which build the tensor of noncommutativity and correspond to a rotationally-invariant system. The system is considered to be a harmonic oscillator with a high frequency. In the rotationally-invariant noncommutative space, the influence of noncommutativity on the classical and quantum equations of motion of a particle in the gravitational field is studied up to the second order in the parameter of noncommutativity. The weak equivalence principle is considered. We find that the noncommutativity of coordinates causes additional terms in the equation of motion, which depend on the mass. Therefore, the weak equivalence principle, also known as the principle of the universality of free fall or the Galilean equivalence principle, is not preserved in a noncommutative space with rotational symmetry. We show that in the case when the tensor of noncommutativity which corresponds to the motion of a particle in a noncommutative space is inversely proportional to its mass, the classical equations of motion in the gravitational field do not depend on the mass. Therefore, the weak equivalence principle is recovered in the rotationally-invariant noncommutative space. Also, in the case when the condition for the tensor of noncommutativity is satisfied, the dependence of the quantum equations of motion on the mass $m$ is represented by the ratio $\hbar/m$.
PACS number(s): 02.40.Gh, 04.20.Cv