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Gravitomagnetism and Non-commutative Geometry

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Abstract

Similarity between the gravitoelectromagnetism and the electromagnetism is discussed. We show that the gravitomagnetic field (similar to the magnetic field) can be equivalent to the non-commutative effect of the momentum sector of the phase space when one maintains only the first order of the non-commutative parameters. This is performed through two approaches. In one approach, by employing the Feynman proof, the existence of a Lorentz-like force in the gravitoelectromagnetism is indicated. The appearance of such a force is subjected to the slow motion and the weak field approximations for stationary fields. The analogy between this Lorentz-like force and the motion equation of a test particle in a non-commutative space leads to the mentioned equivalency. In fact, this equivalency is achieved by the comparison of the two motion equations. In the other and quietly independent approach, we demonstrate that a gravitomagnetic background can be treated as a Dirac constraint. That is, the gravitoelectromagnetic field can be regarded as a constrained system from the sense of the Dirac theory. Indeed, the application of the Dirac formalism for the gravitoelectromagnetic field reveals that the phase space coordinates have non-commutative structure from the view of the Dirac bracket. Particularly, the gravitomagnetic field as a weak field induces the non-trivial Dirac bracket of the momentum sector which displays the non-commutativity.

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Notes

  1. However, there are some interpretation on the cosmological constant (associated with dark energy) that acts effectively as a repulsive (gravity) force, particularly on the large scale. In this issue, see, e.g., Refs. [14–16].

  2. This proof was never published by Feynman [19].

  3. The functions that appear in the definition of the NC product.

  4. That is, the ratio v 2/c 2 is ignored against the unity.

  5. This product is usually called the Weyl–Moyal product.

  6. The variables of the classical phase space obey the usual Poisson brackets, namely {x i ,x j }=0={p i ,p j } and {x i ,p j }=δ ij .

  7. In the case of classical mechanics, and when the equations of motion are canonical, the commutation relations can be considered as the usual Poisson brackets [19, 34].

  8. In the non-stationary case, the electric-like and the magnetic-like fields in (15) do not satisfy the two Maxwell-like equations [37].

  9. Here, the underlying manifold is the phase space.

  10. For a more complete discussion of this issue see, e.g., Refs. [40, 43, 44].

  11. The rank of a matrix is the number of its independent row (column).

  12. The full illustration of this example can be found in Ref. [45].

  13. Though, a considerable amount of charge q or a very small mass m can also make this large ratio, but we do not consider these cases.

  14. Obviously, the full conditions are the weak field and the slow motion approximations.

  15. For this reason, these relations are usually called the consistency conditions.

  16. The notation ε ij is the Levi–Civita symbol in two dimensions.

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Authors and Affiliations

  1. Department of Physics, University of Kurdistan, P.O. Box 66177-15175, Sanandaj, Iran

    Behrooz Malekolkalami

  2. Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran

    Mehrdad Farhoudi

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  1. Behrooz Malekolkalami
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Correspondence to Behrooz Malekolkalami.

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Malekolkalami, B., Farhoudi, M. Gravitomagnetism and Non-commutative Geometry. Int J Theor Phys 53, 815–829 (2014). https://doi.org/10.1007/s10773-013-1870-2

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