Noncommutative fluid dynamics in the Kähler parametrization

L. Holender, M. A. Santos, M. T. D. Orlando, and I. V. Vancea
Phys. Rev. D 84, 105024 – Published 16 November 2011

Abstract

In this paper, we propose a first-order action functional for a large class of systems that generalize the relativistic perfect fluids in the Kähler parametrization to noncommutative spacetimes. The noncommutative action is parametrized by two arbitrary functions K(z,z¯) and f(j2) that depend on the fluid potentials and represent the generalization of the Kähler potential of the complex surface parametrized by z and z¯, respectively, and the characteristic function of each model. We calculate the equations of motion for the fluid potentials and the energy-momentum tensor in the first order in the noncommutative parameter. The density current does not receive any noncommutative corrections and it is conserved under the action of the commutative generators Pμ but the energy-momentum tensor is not. Therefore, we determine the set of constraints under which the energy-momentum tensor is divergenceless. Another set of constraints on the fluid potentials is obtained from the requirement of the invariance of the action under the generalization of the volume preserving transformations of the noncommutative spacetime. We show that the proposed action describes noncommutative fluid models by casting the energy-momentum tensor in the familiar fluid form and identifying the corresponding energy and momentum densities. In the commutative limit, they are identical to the corresponding quantities of the relativistic perfect fluids. The energy-momentum tensor contains a dissipative term that is due to the noncommutative spacetime and vanishes in the commutative limit. Finally, we particularize the theory to the case when the complex fluid potentials are characterized by a function K(z,z¯) that is a deformation of the complex plane and show that this model has important common features with the commutative fluid such as infinitely many conserved currents and a conserved axial current that in the commutative case is associated to the topologically conserved linking number.

  • Received 7 September 2011

DOI:https://doi.org/10.1103/PhysRevD.84.105024

© 2011 American Physical Society

Authors & Affiliations

L. Holender1,2,*, M. A. Santos2,†, M. T. D. Orlando2,‡, and I. V. Vancea1,§

  • 1Grupo de Física Teórica e Matemática Física, Departamento de Física, Universidade Federal Rural do Rio de Janeiro (UFRRJ), Cx. Postal 23851, BR 465 Km 7, 23890-000 Seropédica-RJ, Brasil
  • 2Departamento de Física e Química, Universidade Federal do Espírito Santo (UFES), Avenida Fernando Ferarri S/N-Goiabeiras, 29060-900 Vitória-ES, Brasil

  • *holender@ufrrj.br
  • masantos@cce.ufes.br
  • mtdorlando@terra.com.br
  • §ionvancea@ufrrj.br

See Also

Quantization of the relativistic fluid in physical phase space on Kähler manifolds

L. Holender, M. A. Santos, and I. V. Vancea
Phys. Rev. D 77, 045024 (2008)

Noncommutative fluid dynamics in the Snyder space-time

M. C. B. Abdalla, L. Holender, M. A. Santos, and I. V. Vancea
Phys. Rev. D 86, 045019 (2012)

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Vol. 84, Iss. 10 — 15 November 2011

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