Non-Riemannian vortex geometry of rotational viscous fluids and breaking of the acoustic Lorentz invariance
Introduction
The acoustic metric in fluids has been proposed by Unruh [1] in 1981 with the purpose of investigating more realistic Hawking effect and sonic spectrum of temperature, which allowed him to proposed the concept of sonic black hole or dumb hole. More recently Garcia de Andrade [2] has extended this concept to allow for acoustic torsion, with the very strong constraint of irrotational perturbations in fluids originally already in rotation to compare the Euler fluid equations with the Riemann–Cartan (RC) wave equation to obtain the so-called analog gravity models [3] endowed with torsion. Actually the idea of applying non-Riemannian geometry in hydrodynamics is not new, but stands from Kazuo Kondo in 1947 [4] when he used the idea of non-holonomicity to investigate hydrodynamical turbo-machines, what is new here is that torsion here is used as analogous model for rotation, distinct from Kondo case where torsion appears, for example, in the Boltzmann–Hammel equation [4]. Here the dynamical equations are given by the Navier–Stokes [5] and conservation of mass density of the fluid equations which do not contain Cartan torsion whatsoever. In other words, the dynamics of the effective gravity here is not given by the Einstein equations, let alone by the Einstein–Cartan equations [6]. Torsion only appears when we compare the fluid equations with the wave equation in the RC real spacetime. The effective or sonic torsion appears then, in terms of the parameters of the fluid such as rotation and fluid velocity. In this Letter we consider the generalization of the previous paper [2] to the case where viscosity appears together with rotation. One of the main advantages to include viscosity is not only to make the fluid more real (apart from superfluids which has the advantage of not possessing viscosity at all) is, as shown by Visser [7] to be able to deal with interesting physical phenomena as acoustic Lorentz violation. One of main differences between this previous work and the present one is that here we consider that viscosity is present partly in the acoustic torsion and in the Visser approach viscosity is present only in the acoustic metric. To take this advantage in the present work we also show that the acoustic torsion is able to induce acoustic Lorentz violation in strong gravitational analogy with recent work by Kostelecky [8] where he shows that explicitly Lorentz violation is found to be incompatible with generic RC geometries, but spontaneous Lorentz breaking avoids this problem. By considering viscosity and fluid vorticity we also pave the way to built a non-Riemannian approach to turbulent flow. The Letter is organized as follows. In Section 2 we briefly review the RC wave equation of massless fields while in Section 3 we present the new material concerning the non-Riemannian geometry of viscous rotational fluids and its vorticity perturbation, followed by the comparison with the RC wave equation and consequent derivation of the acoustic torsion. Section 4 presents the breaking of acoustic Lorentz symmetry through acoustic torsion and viscosity and vorticity of the classical fluid. Section 5 contains also new material and we provide an example of teleparallel gravity torsion loop which metric can be mapped with the Lense–Thirring (LT) metric as long as the LT rotation can be associated to torsion. Since we recently show that the Letelier teleparallel loops can be associated to the metric of superfluid 4He we may consider that this example is similar to the Volovik [9] example of associating the LT metric to be an analog model for the superfluid model with rotation. The basic difference with Volovik's example is that his case does not contain torsion and that his metric represents the motion of phonons around the vortex in 4He superfluid. In Section 6 conclusions are presented. In this Letter we also respond to some criticism concerning our previous work [2] where irrotational perturbations would have been contaminated by the background vorticity and this would have spoiled our acoustic torsion model. This problem here does not appear since the background flow is already rotational.
Section snippets
Wave equation in Riemann–Cartan spacetime
In general relativistic analogue models the fluid equations are expressed in terms of the Riemannian wave equation for a scalar field Ψ in the form where represents the Riemannian D'Lambertian operator given by In this case where g represents the determinant of the effective Lorentzian metric which components are , , , others zero. Throughout this Letter c represents the speed of sound, which quanta are
Dynamics of analog models in rotational viscous fluids
Consider the dynamics of the non-relativistic fluids by expressing the conservation of mass equation and the Navier–Stokes equation by where ν is the viscosity coefficient, ϕ the potential energy of the fluid and p is the pressure. Here and represents the vorticity of the fluid and the vector velocity in terms of the stream function [9] . Performing the perturbation to these equations according to the perturbed flow
Acoustic Lorentz symmetry breaking by acoustic torsion
Visser [7] has been able to incorporate viscosity of the flow into the acoustic metric by coupling the term to the flat part of the metric. In this section we shall discuss the possibility of considering the acoustic torsion inducing acoustic Lorentz violation, taking the advantage of the presence of acoustic torsion in our formulation of the non-Riemannian geometry of viscous flow endowed with vorticity [2]. Let us consider the viscous wave equation endowed with vorticity
Teleparallel loops and the Lense–Thirring metric in superfluid 4He
Recently we have shown [2] that the Letelier metric representing a teleparallel torsion loop [11] is given by the metric where is an arbitrary vector and , could be mapped to the acoustic metric of Unruh describing superfluid 4He. This metric can be written in terms of the basis 1-form , where , as where , , and . Let us now consider that torsion form is given by
Conclusions
We have shown that the LT metric can be considered an acoustic superfluid metric where sonic torsion is given by the LT angular velocity. This is actually a simple example of the use of acoustic torsion in classical and quantum fluids. Future prospects include the investigation of a sonic black hole endowed with acoustic torsion and possible contributions to the acoustic Hawking effect. The acoustic Lorentz breaking due to acoustic torsion analog of vorticity and viscosity is explicitly
Acknowledgments
I would like to dedicate this Letter to the memories of Professors Kazuo Kondo and Jeeva Anandan who taught me so much about torsion and inspired me through their works and talks. I am also very much indebted to Professor W. Unruh, Dr. C. Furtado, Dr. S. Bergliaffa and Professor P.S. Letelier for discussions on the subject of this Letter. Special thanks go to Professor V. Pagneux for some discussions on acoustic perturbations. I also thank Larry Smalley teacher and friend who told me for the
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