Noncommutative fluid dynamics in the Kähler parametrization

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 ( √ −j 2 ) that depend on the fluid potentials and represent the generalization of the Kähler potential of the complex surface parametrized by z andz, 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 P µ 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. *

Noncommutative geometry and fluid dynamics

“…NC fluid dynamics, compatible with a Snyder form [21][22][23] of operatorial spacetime has been discussed in [15,16]. However, for the type of applications we have in mind, the simplest NC extension studied here induces very interesting and nontrivial effects in the fluid model and so other alternatives have not been considered, although these can be studied in our approach.…”

“…A form of "exotic plasma" fluid where the particle Lagrangian coordinates are endowed with an "exotic" algebra (that is, effectively a NC system), was studied in [54]. In another approach [15,16] NC effects, (of the same canonical form used here), were incorporated directly in the form of star products in the fluid action in Kahler parametrization to NC spacetimes. For completeness we show a few steps explicitly.…”

“…Incidentally we have put the 'time' in parentheses simply to indicate that conventionally the NC extension is restricted to the spatial sector only without affecting the time. Some relevant work in this context is [15][16][17].…”

“…See e.g. earlier reviews [18][19][20] for generic NC field theory construction and * -product formalism in [15,16] for NC fluid model. The other alternative that is especially suited for fluids is to start from the Lagrangian (discrete) fluid model, introduce the NC space effect and subsequently exploit the map that connects the Lagrangian framework to an Euler Hamiltonian (continuum) framework to induce NC effects in the fluid field theory.…”

Plane waves in noncommutative fluids

“…Motivated by recent works in which several noncommutative systems of charges were shown to display fluid properties at quantum scale [1,2,3,4,5,6,7] as well as cosmological scale [8], we have proposed the first field theoretical model of fluid in the canonical noncommutative space [9] and the noncommutative Lorentz covariant space [10], respectively [11,12,45]. The noncommutative fluid was constructed as a noncommutative field theory [13,14] in the realization method approach [15,16,17,18,19,20,21,22,23,24,25,26,27] by generalizing the first order action functional of the commutative perfect relativistic fluid [28,29,30,31].…”