Intertial Frame Dragging in an Acoustic Analogue spacetime
Chandrachur Chakraborty, Oindrila Ganguly, Parthasarathi Majumdar
IIntertial Frame Dragging in an Acoustic Analogue spacetime
Chandrachur Chakraborty,
1, 2, ∗ Oindrila Ganguly, † and Parthasarathi Majumdar ‡ Tata Institute of Fundamental Research, Mumbai 400005, India Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China Institute of Physics, Bhubaneswar 751005, India Ramakrishna Mission Vivekananda University, Belur Math 711202, India
We report an incipient exploration of the Lense-Thirring precession effect in a rotating acousticanalogue black hole spacetime. An exact formula is deduced for the precession frequency of agyroscope due to inertial frame dragging, close to the ergosphere of a ‘Draining Bathtub’ acousticspacetime which has been studied extensively for acoustic Hawking radiation of phonons and alsofor ‘superresonance’. The formula is verified by embedding the two dimensional spatial (acoustic)geometry into a three dimensional one where the similarity with standard Lense-Thirring precessionresults within a strong gravity framework is well known. Prospects of experimental detection of thisnew ‘fixed-metric’ effect in acoustic geometries, are briefly discussed.
I. INTRODUCTION
A stationary spacetime with angular momentum dragsalong with it local inertial frames, causing a test gyro-scope in the spacetime to precess with a precise frequencygiven as a function of the spacetime parameters. Thisis known in the gravity literature as the Lense-Thirringprecession [1]. Inertial frame dragging and the ensuinggyroscopic precession relative to a Copernican frame ofreference are consequences of the ‘fixed-metric’ geometryof a stationary spacetime. Until date, most observationsand measurements of Lense-Thirring effect have been car-ried out only for spacetimes with curvatures small enoughto justify the weak gravity approximation [2, 3]. It hasnot been possible so far to directly detect inertial framedragging in regions with very strong gravitational fieldsas near rotating black holes and neutron stars [4, 5].A whole new exciting arena for addressing strong grav-ity issues has opened up with the discovery by Unruh[6] that acoustic perturbations in the velocity poten-tial of a locally irrotational, barotropic, inviscid New-tonian fluid behave exactly as a minimally coupled mass-less scalar fields propagating in a curved (3 + 1) dimen-sional Lorentzian ‘acoustic analogue’ spacetime. Acous-tic analogue spacetimes have since been studied exten-sively to discern laboratory analogues of gravitational ef-fects which have so far evaded detection in real space-time. Unruh’s incipient prediction of Hawking radiationof phonons from acoustic black hole spacetimes have beenthe object of study for decades [7, 8], culminating in avery recent announcement [9] that correlation betweenHawking particles and their partners beyond the acous-tic horizon of analogue black holes, in accord with the-oretical predictions, have been observed in certain Bose-Einstein condensates.Following up on work on Hawking emission of virtualphonons, the acoustic analogue of superradiance named ∗ [email protected] † [email protected] ‡ [email protected] as ‘superresonance’ has been quantitatively worked out[10]. Follow-up work on acoustic superradiance has goneon over the last decade, with another exciting new an-nouncement made last winter of experimental observa-tion of this effect [11]. One of the key novel predictedfeatures of acoustic superresonance is the possibility thatthe amplification of the reflected wave may exhibit dis-creteness characteristic of quantum mechanical vortices.Observation of such discrete jumps in the superradiantscattering coefficient may be a signature of acoustic su-perradiance. Such theoretical predictions have now madeit possible to design in the laboratory simple and robustanalogue models of various classes of curved Lorentzianspacetimes in kinematical situations (where the acous-tic metric is non-dynamical). Notice that such discretejumps in the superradiant reflection coefficient is absentfor superradiance in physical spacetime.In this letter, we explore a new physical effect in acous-tic geometry - the Lense-Thirring precession of a test gy-roscope in a rotating acoustic black hole spacetime. Sincethe phenomenon is a familiar one in physical spacetime,and yet has never been observed except in its weak fieldversion, acoustic analogue spacetimes afford us an arenafor a more complete study where strong gravity aspectsof this precession might be revealed in the laboratory.To this end, we shall mostly restrict ourselves to a studyof frame dragging close to the ergosphere of a particu-lar fluid mechanical analogue model of a rotating blackhole, namely the so-called Draining Bathtub [7]. In termsof the fluid flow, the ergosphere characterises a transitionsurface from a sonic to a supersonic flow but sound wavesfrom inside it may still travel outwards, away from thesurface. Such a supersonic region can be considered asan acoustic analogue of the ergoregion of a rotating blackhole, characterised by the reversal in sign of the acousticspacetime metric component g . Within this ergoregion,a surface on which at every point the normal componentof fluid velocity is inward pointing and greater than thelocal speed of sound, exists. This surface then traps allacoustic disturbances within it like an outer trapped sur-face, thus acting as the sonic horizon of an acoustic blackhole. Dragging of local inertial frames occurs in the sta- a r X i v : . [ g r- q c ] S e p ionary spacetime outside the ergosphere bounding theergoregion.The derivation of the exact Lense-Thirring precessionfor four dimensional stationary spacetime has been dis-cussed earlier in [4] (following the textbook [12]). Thederivation of the precession frequency to the three di-mensional acoustic spacetime situation being discussedhere is new, since precession itself is not such an easyidea to visualise in two space dimensions where angularmomentum is a scalar quantity. However, we may re-call that most elementary treatments of the perihelionprecession of Mercury in general relativity is confined tothe equatorial plane, and is therefore two dimensional,and yet there has not been any difficulty with its mea-surement. With this in mind, an exact formula for theprecession frequency appropriate to the Draining Bath-tub black hole has been derived here. To ensure that theformula is correct, the three dimensional metric is nextembedded into a four dimensional one by simple exten-sion of the geometry along the extra spatial dimension.The known four dimensional formula for the precessionfrequency is then seen to be completely consistent withwhat emerges from the extended Draining Bathtub com-putation.The calculated Lense-Thirring precession frequency forthe Draining Bathtub acoustic spacetime is seen to en-hance rapidly as one approaches the ergosphere from afar.Very close to the ergosphere, it dominates the Kepler fre-quency (the angular velocity of a test particle/gyroscopewhich moves along a circular geodesic) corresponding toorbiting phonon geodesics, and this provides the clue forits observation. As such, inertial frame dragging in thistwo space dimensional acoustic spacetime will distort cir-cular geodesics encircling the acoustic black hole, mak-ing them encircle with a centre away from the vortexcentre. The Lense-Thirring precession would then resultin the centre of circular orbits to fluctuate periodicallyabout the centre of the vortex characterising the blackhole, with a frequency given by the calculated precessionfrequency. This frequency should exhibit the same en-hancement pattern as one approaches the ergosphere, inaccord with our theoretical result mentioned above.The paper is organised as follows : After a brief reviewof the Draining Bathtub acoustic spacetime, we presentthe derivation of our main formula for the Lense-Thirringprecession frequency. We then discuss the dependence ofthis frequency on the radial distance from the acousticblack hole centre, and exhibit its sharp enhancement asone approaches the ergosphere from outside. We thendiscuss ways that this effect, novel in acoustic gravity, canbe observed and the Lense-Thirring precession frequencymeasured in laboratory experiments. We then concludewith a few remarks on future outlook. II. ROTATING ACOUSTIC BLACK HOLE
The acoustic analogue of a rotating black hole space-time is best captured by a planar ‘Draining Bathtub’flow of an incompressible, barotropic, inviscid fluid withno global vortex present. The flow is characterized bythe velocity potential (cid:126)v b = − Ar ˆ r + Br ˆ φ . (1)Here, ( r, φ ) are plane polar coordinates while A, B areconstants. The constraints imposed on the fluid makethe background density ρ b position independent which inturn guarantees the constancy of background pressure p b and local speed of sound c s throughout the fluid. As asimplifying measure, we set c s = 1 and ignore an overallconstant factor of ρ b c s in the acoustic metric. The explicitform of the emerging acoustic black hole metric is, ds DB = − (cid:18) − A + B r (cid:19) dt + dr + r dφ + 2 Ar dr dt − B dφ dt . (2)It is clear that this curved analogue spacetime possessesisometries that correspond to time translations and rota-tions on the plane and hence, is not only stationary butalso axisymmetric. The radius of the ergosphere, r E , isdetermined by the vanishing of g : r E = A + B . The2-surface at r H = A acts as the future event horizon ofthe sonic black hole because just beyond it, the radialcomponent of (cid:126)v b exceeds the local speed of sound. Anylinearised fluctuation originating in the region boundedby the acoustic horizon is swept inward by the flow.In the recent literature [4], the general expression forthe angular velocity of precession of a test spin relativeto a Copernican frame of reference has been derived fora four dimensional stationary spacetime geometry. TheCopernican frame, with respect to which the gyroscopeprecession rate is given, is not a locally inertial frame.Freely-falling untorqued gyros cannot precess in such aframe. Rather, a Copernican frame [12] is a local or-thonormal tetrad at rest (so moving only in the “t” di-rection determined by the timelike Killing vector of thespacetime) and “locked” to the spatial part of whateversuch coordinate system is chosen, so that it is “at rest”with respect to the local inertial frames at infinity. Thatis why, a Copernican frame is also called “axes at rest”[12]. As a check on our calculations, it makes sense to em-bed the planar acoustic spacetime being considered here,into a three dimensional flow, by adding an extra spatialdimension and interpret the result as a superposition ofan ordinary vortex filament and a line sink: ds = − (cid:18) − A + B r (cid:19) dt + dr + r dφ + dz + 2 Ar dr dt − B dφ dt . (3)2
II. FRAME DRAGGING IN ACOUSTIC‘DRAINING BATHTUB’ GEOMETRY
Referring to our previous work [4] for a detailed deriva-tion, the Lense-Thirring precession ( ˜Ω of Eq.(13) of [4])in four dimensional spacetime can be expressed as˜Ω = 12 K ∗ ( ˜ K ∧ d ˜ K ) (4)where, K is the timelike Killing vector field correspond-ing to the stationarity of the spacetime, and ˜Ω is theone-form of Lense-Thirring precession vector (cid:126) Ω. Observethat in two dimensional space appropriate to the acousticspacetime under consideration, the Lense-Thirring pre-cession frequency Ω, now a spatial scalar, can be ex-pressed by the same formula Equation 4. The above ex-pression is clearly applicable in any (2 + 1) dimensional stationary spacetime.Specialising to a coordinate basis where K = ∂ , thecorresponding covector ˜ K takes the form:˜ K = g µ d ˜ x µ = g d ˜ x + g i d ˜ x i (5)It follows that, d ˜ K = g ,j d ˜ x j ∧ d ˜ x + g i,j d ˜ x j ∧ d ˜ x i , ( ˜ K ∧ d ˜ K ) = ( g g i,j − g i g ,j ) d ˜ x ∧ d ˜ x j ∧ d ˜ x i , ∗ ( ˜ K ∧ d ˜ K ) = ( g g i,j − g i g ,j ) ∗ ( d ˜ x ∧ d ˜ x j ∧ d ˜ x i )(6)Using ∗ ( d ˜ x ∧ d ˜ x j ∧ d ˜ x i ) = η ji = − √− g ε ji , we concludefrom Equation 6, ∗ ( ˜ K ∧ d ˜ K ) = ε ij √− g ( g g i,j − g i g ,j ) (7)Substituting the above in Equation 4, we end up withthe following expression for Ω (2+1) Ω (2+1) = 12 √− g (cid:15) ij g (cid:20) g i g (cid:21) ,j (8)To obtain the Lense-Thirring precession rate in DrainingBathtub case, we can easily apply Equation 8 where i, j take values 1 , DB (2+1) = − B r E r (cid:20) − r E r (cid:21) − (9)It is clear from Equation 9 that the the Lense-Thirringprecession is more pronounced closer to the ergosphere r → r E than away from it. This is precisely the sort ofqualitative behaviour of gyroscopic precession observedin earlier work [4] on physical stationary spacetimes. Inthe ‘weak field’, i.e., far away ( r >> r E ) from the ‘ergo-sphere’ | Ω DB (2+1) | decreases as 1 /r : | Ω DB (2+1) | (cid:39) B r E r , as r >> r E . (10) This is also similar to the weak-field approximation thathas been used for observation of the gyroscopic precessiondue to inertial frame dragging in the Earth’s gravitationalfield arising out of its diurnal rotation [13].As an extra technical verification that the formula inEquation 9 correctly gives the Lense-Thirring precessionfrequency, we use the embedding acoustic 3+1 dimen-sional geometry given in Equation 3, and use our fourdimensional formulae [4]. The Lense-Thirring precessionvelocity in any 4 dimensional stationary spacetime whenexpressed in a coordinate basis has the form (cid:126) Ω (3+1) = 12 ε ijl √− g (cid:20) g i,j (cid:18) ∂ l − g l g ∂ (cid:19) − g i g g ,j ∂ l (cid:21) (11)Substituting the metric components from Equation 3 intoEquation 11, we finally obtain the angular velocity withwhich a test gyroscope will precess relative to the frame { ∂ i } in the extended Draining Bathtub spacetime: (cid:126) Ω (3+1) = − B r E r (cid:20) − r E r (cid:21) − ∂ z (12)The formula matches Equation 9 exactly. With this ver-ification, we can now proceed to consider the physicalimplications of the gyroscopic precession formula Equa-tion 9, vis-a-vis laboratory detection of this effect. IV. COMPARISON OF LENSE-THIRRINGPRECESSION AND BACKGROUND FLOWROTATION
In this section, we present a comparison of the Lense-Thirring precession angular frequency Ω DB (2+1) for theDraining Bathtub flow in (2 + 1) dimensional acousticspacetime, with the angular velocity of the flow itself, Ω.Observe that the latter is given by,Ω( r ) = Br (13)On the other hand, the angular velocity of precessionoccuring due to dragging of local inertial frames has theform, | Ω DB (2+1) ( r ) | = (cid:34) B r E r (cid:18) − r E r (cid:19) − (cid:35) = (cid:34) Ω( r ) r E r (cid:18) − r E r (cid:19) − (cid:35) , (14)substituting the value of Ω( r ) = Br from Equation 13.For convenience, we introduce a dimensionless radial co-ordinate x ≡ r/r E such that the ergosphere at r = r E maps to x = 1 while r → ∞ corresponds to x → ∞ . Let3 c1 x ¯⌦ x c x c ¯⌦ = 1 x x c = p x c ⇡ . FIG. 1. Variation of ¯Ω with x showing the points x c and x c us also define ¯Ω ≡ | Ω DB (2+1) | / Ω which evidently is dimen-sionless also. Equation 14 then has a simpler appearance:¯Ω( x ) = 1 x − x = √
2, ¯Ω = 1. Thus, at thispoint, the angular velocity of precession becomes equalin magnitude to the angular velocity of rotation beforesurpassing it with approach towards the ergosphere ( x → , r → r E ). We denote the corresponding critical radius r = √ r E by r c and x c = r c /r E = √ r , unlike in aspinning black hole. Thus, it may also be useful to knowat what value of the radial coordinate the magnitude ofthe precession velocity, | Ω DB (2+1) | , becomes equal to the an-gular velocity of rotation of the ergosphere (Ω( r E ) = Br E ).This particular radius denoted by, say, r c ≈ . r E and hence, x c = r c /r E ≈ . resonance phenomenon, especially in thecontext of BEC systems, appears to be an interesting is-sue worthy of future investigation. Since this equalityoccurs outside the acoustic ergosphere, na¨ıvely, its con-nection to ‘superresonance’ is probably peripheral, if atall. However, there may be subtleties which escape us atthis point. V. OBSERVATIONAL PROSPECTS
With Bose-Einstein condensate systems (BEC) provid-ing the most promising experimental situations for morecomprehensive studies of superfluidity and other aspectsof low temperature physics, it is conceivable that they will also provide the arena for possible observation of in-ertial frame dragging in an acoustic geometry sensed byan acoustic perturbation on a rotating fluid. Startingwith the time-dependent Gross-Pitaevski equation [14]which follows from the microscopic Bogoliubov equationsfor the order parameter for a BEC system exhibiting off-diagonal long range order in the mean-field approxima-tion [15], the identification of the gradient of the phase ofthe order parameter with the velocity of the fluid leadsto the Euler hydrodynamic equations [8, 16, 17].Assuming that the velocity profile of the DrainingBathtub type is realisable experimentally for a BEC vor-tex system in the hydrodynamic limit, acoustic perturba-tions introduced into the system as virtual phonons willbe sensitive to such a flow as a rotating analogue blackhole, as discussed in the Introduction. Now, rotatingBose-Einstein condensates have been studied extensively[18], following earlier work on ‘BEC gyroscopes’ [19] (andreferences therein). In these works, the proposed gyro-scopes probe the rotating BEC and any precessional mo-tion that the entire BEC system might exhibit. Clearly,this is not adequate for our purpose. To actually observeinertial frame dragging in acoustic geometry, one mustconsider a gyroscope which rotates freely and probes thebehaviour of the virtual phonons. We assume that it isnot difficult to identify observationally part of the BECvortex for a Draining Bathtub flow as an ergosphere, al-beit a small portion of it, as the ergoregion of the acous-tic black hole, and an even smaller region as the acoustichorizon. The actual technical difficulties associated withmaking such identifications are admittedly beyond ourrange of expertise.
A. Intrinsic Spin of Phonons
If phonons representing acoustic perturbations insidethe fluid have an instrinsic spin, in addition to their or-bital angular momentum , which is free to precess aroundthe rotation axis of the background flow, one can studysuch a precession as a gyroscopic precession, as a func-tion of its distance from the ergosphere. We have madeprecise predictions above in Equation 14 for the mannerin which the precession frequency must increase as thedistance of the phonon from the ergosphere decreases. Ifthe observed behaviour matches with this prediction, itis very likely that one is observing the effect of inertialframe dragging in the acoustic spacetime.The notion of an intrinsic phonon spin has been pro-posed fairly recently by Zhang and Niu [20] for spin relax-ation in ionic crystals exposed to uniform magnetic fields,based on the Raman spin-phonon interaction which islinear in the phonon momentum. Such a momentum de-pendent interaction was first proposed by Ray and Ray[21] for paramagnetic spin relaxation for Van Vleck twophonon processes, and isotropy (conservation of angularmomentum) appears to require existence of this inter-action in an essential way. Of course, the importance4f such processes depends crucially on the symmetry as-pects of the crystal lattice under consideration. It hasbeen argued [20] that the intrinsic phonon spin is an in-herently quantum phenomenon, and it is shown that thespin vanishes in the classical limit. A very clear and ped-agogical explanation of this notion of phonon spin hasbeen presented by Garanin and Chudnovsky [22] where ithas been shown that small radius circular shear deforma-tion around the equilibrium points of the elastic mediummay produce a non-conserved angular momentum sincethe stress tensor is clearly non-symmetric in this case.To recover rotational invariance, one may either assumeanharmonicity of the elastic medium arising from non-linearity, or the effect of quantum spin relaxation in thelattice in interaction with phonons. The latter has beenargued to induce an intrinsic spin angular momentum forphonons.Now, the crucial issue here is: whether BEC systemsmay model such paramagnetic crystals so that the phe-nomenon of phonon spin becomes observable in the hy-drodynamic approximation in the mean field limit of suchcondensates. Indeed, the well-known correspondencebetween Bose-Einstein Condensates and the Bardeen-Cooper-Schrieffer model of superconductivity, and sim-ilar other correspondences have opened up possibilitiesof studying phenomena in condensed matter physics farmore cleanly within BEC systems confined to certaintypes of optical lattices [23]. We have not been able todiscern in the latest literature on BECs as to whetherintrinsic phonon spin has indeed been observed in BECsused to model paramagnetic crystals. There is no doubt,however, that within ionic crystals, the phenomenon isnow an accepted paradigm, and so it could be a matterof time before it is realised in BEC systems.There is another possible scenario which can be usefulfor observation of the phenomenon proposed here, involv-ing spinor condensates [14], [18]. Such condensates arelikely to have spin wave (magnon) excitations, with pos-sible spin-dependent coupling to phonons. It is not un-likely that because of such interactions, the inertial framedragging in spinor condensates will result in a flip of thespin wave excitation of the condensate. This spin-flip is adiscrete process and ought to be observable in quantumresonance experiments. Once again, there could be a flip-ping of the magnon spin with a frequency given in termsof the Lense-Thirring frequency. Even though such a sce-nario is speculative at this stage, since phonon-magnoninteractions in spinor condensates are yet to be observed,the one feature that is remarkable is that it may be a dis-crete process because of quantum dynamics of the spinor BEC. It is similar to the discrete amplification predictedfor superresonance in [10] for BECs or liquid helium. Wehasten to add that both these phenomena involving dis-crete transitions are speculative and have not been ob-served yet - it is possible that in the linear perturba-tion theory that we have employed to discern the acous-tic spacetime geometry, such discrete transitions cannotpossibly be incorporated.There is also the issue of using normal fluids like wa-ter for observation of the effect of frame dragging in theacoustic geometry. Water has a small viscosity whichleads to Lorentz violation [7] in that the acoustic metricmay not be extractable in a closed form [24]. Neverthe-less, it is still possible to demonstrate acoustic superra-diance for such fluids, provided the viscosity is withincertain limits which also delineates the range of super-radiant scattering. The question of how inertial framedragging may be detected in such fluids is not completelyunderstood yet and will be hopefully addressed in a fu-ture publication.
VI. CONCLUSIONS
Acoustic analogue gravity appears to be richer in termsof observational prospects for non-dynamic spacetimesthan actual physical non-dynamic spacetimes. This is sobecause cold atom condensates present a unique arena forobservation of phenomena like Hawking radiation, super-radiance and Lense-Thirring precession. These conden-sate systems involve discreteness because of their inher-ent quantum mechanical nature; such discreteness hasobservational prospects for phenomena in acoustic ana-logue black holes, which one can by no means expectfor physical continuum spacetime. It is therefore of ut-most importance to investigate thoroughly as to whetherthe effects discussed in the paper have precise signaturesstemming from the unique quantum properties of coldatom condensates.
ACKNOWLEDGEMENTS
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