It is shown that the curved spacetime induced in a thin film of superfluid
$^{3}$He-A by the presence of symmetric vortices with the unbroken symmetry
phase, admits the existence of closed timelike curves through which only
superfluid clusters formed by anti-$^{3}$He atoms can travel and violate
causality.Comment: 7 pages, LaTex, to appear in Phys. Lett. 

Buerle

Carmen L. Sigüenza

Grebenev

Gödel

Harari

Hendry

Leggett

Pedro F. González-Díaz

Ruutu

Turok

Unruh

Volovik

Zurek

English

arXiv

It is shown that the curved spacetime induced in a thin film of superfluid 3He-A by the presence of symmetric vortices with the unbroken symmetry phase, admits the existence of closed timelike curves through which only superfluid clusters formed by anti-3He atoms can travel and violate causality. © 1998 Elsevier Science B.V.The authors acknowledge DGICYT by support under Research Project No. PB94-0107.Peer Reviewe

González-Díaz, Pedro F.

Sigüenza, Carmen L.

Digital.CSIC

arXiv:cond-mat/9809140v1  9 Sep 1998Closed timelike curves in superfluid 3HePedro F. Gonza´lez-Dı´az and Carmen L. SigenzaInstituto de Matema´ticas y F´ısica FundamentalConsejo Superior de Investigaciones Cient´ıficasSerrano 121, 28006 Madrid (SPAIN)June 15, 1998AbtractIt is shown that the curved spacetime induced in a thin film of superfluid 3He-A bythe presence of symmetric vortices with the unbroken symmetry phase, admitsthe existence of closed timelike curves through which only superfluid clustersformed by anti-3He atoms can travel and violate causality.PACS numbers: 04.20.Gz , 67.57.FgKeywords: Rotating vortex, Spacetime metric, Closed timelike curves,Corresponding Author: P.F. Gonzlez-Daz, Instituto de Matemticas y Fsica Fun-damental, C.S.I.C., Serrano 121, 28006-Madrid, SpainTelephon: (34-1)5616800 Fax: (34-1)5645557E-Mail: iodpf21@cc.csic.es1One of the most appealing recent developments in physics is the discoveryof a common boundary between condensed-matter systems at low temperaturesand some cosmological and gravitational systems [1,2]. It has been theoreticallyestablished [3] and experimentally checked [4] that the vortices which form when3He-A undergoes the phase transition to its superfluid phase [5] can be regardedas the condensed matter analogs of the so called cosmic strings (i.e. the gauge-theory topological defects which are thought to have been created through phasetransitions in the early stages of the evolution of the universe and that could wellhave been the seeds for presently observed galaxies [6]). On the other hand, ithas also been pointed out [7] that event horizons and ergoregions similar to thoseoccurring in black and white holes with angular momentum can be found as wellin the curved spacetime of planar solitons moving in superfluid 3He-A.Most symmetric vortices in 3He-A can be created in thin films where the unitvector ℓˆ defining the direction of the gap nodes in momentum space is fixed alongthe normal to the film, and the superfluid circulates around the vortex axis witha velocity given by [8]vs =h¯2mr, (1)with m the mass of the 3He atoms and r the distance to the vortex axis. Inthis case, close to their two zeros, the energy spectrum of the Bogoliubov-Nambufermion quasiparticles becomes that of a charged, massless relativistic particlepropagating in a curved spacetime whose cylindrically symmetric metric can bewritten as [9]:ds2 =(c2⊥ − v2s)dt2+(h¯24m2c2⊥− r2)(1−v2sc2⊥)−1dφ2−c2⊥dz2c2‖−dr2+h¯mdtdφ, (2)where c⊥ =△pFand c‖ = vF =kFmare the speed of light transverse to and along2ℓˆ, respectively, with kF = PF ℓˆ the Fermi momentum and △ << kFvF the gapamplitude. In metric (2) the true singularity at the horizon vs = c⊥ marks theonset of the ergoregion with vs > c⊥ and occurs at exactly the radius of the vortexcore:rc =h¯2mc⊥. (3)In this letter we shall consider the possible formation of closed timelike curves(CTCs) in the spacetime decribed by metric (2). It will be seen that ordinary 3Heatoms in the superfluid phase cannot enter any of such CTCs and that it is onlyfor 3He anti-atoms, which are made of the antiparticles to the particles that formup an 3He atom, that the spacetime described by metric (2) can display CTCs.We also discuss the possibility to make observable the effects that such CTCsmight have in the neighbourhood of the transition temperature to the superfluidphase.We first note that the line element (2) becomes the spacetime metric of aspinning cosmic string [10] at points sufficiently far from the vortex axis, r >> rc,only if we interpret the quantity h¯/8mG as the internal angular momentum Jper unit length in the resulting spinning string and adscribe to this a zero massper unit length, µ = 0. Far from the vortex axis, metric (2) reduces tods2 = c2⊥dt2 +(h¯24m2c2⊥− r2)dφ2 −c2⊥dz2c2‖− dr2 +h¯mdtdφ. (4)Much as for the cylindrically symmetric metric that describes the spacetime ofa homogeneously rotating Gdel universe [11], at first glance, one could expectthat CTCs would be formed in some regions of the spacetime described by metric(4). However, in order for the Killing vector ∂φ with closed orbits to be timelikeeverywhere and so allow for CTCs, the coordinate r must be constrained to the3range r < rc, a condition that clearly contradicts the approximation where metric(4) is valid.For the more general metric (2) one would in principle not expect CTCs toappear, too, neither outside nor inside the horizon at rc, the reason for the lattercase being that the condition r < rc necessarily implies vs > c⊥, with vs = c⊥at r = rc, and hence the Killing vector ∂φ keeps being spacelike even on theregion inside the vortex. Let us consider however the line L0 defined by L0 :t = +βφ, r = R, z = 0. For that line the gφφ-component of the metric tensorbecomes:c−2⊥ gφφ =h¯24m2c2⊥− r2 +h¯βmc2⊥(c2⊥ − v2s). (5)Thus, ∂φ can still be timelike, provided thatrc < r = R < r0 =√h¯βm, (6)withβ >h¯4mc2⊥> 0. (7)We note now that any 3He atoms propagating along any line satisfying condi-tion (6), (7) forward in time can also be described as the corresponding anti-3Heatoms (i.e. atoms that are made of the antiparticles to the particles which an3He atom is made of) moving backward in time on that line. Thus, one canalways consider another line in the spacetime described by metric (2), defined asL1 : t = −αφ, r = R, z = 0, along which anti-3He atoms (with negative mass -m)propagate backward in time. If we set α and β to be both positive and satisfyingα > β, then line L1 will also be timelike everywhere and, since an initial pointQ(φ = 0) and a final point P (φ = 2π) on L1 will also be on L0, where P pre-cedes Q, one can readily deduce that, relative to a given observer that evolves4forward in time, CTCs are actually allowed to occur in the exterior superfluidregion r > rc of the spacetime described by metric (2), provided that superfluid3He can travel through such CTCs only in the form of anti-atoms.Nevertheless, the probability for these CTCs to exist and carry with super-fluid clusters of atoms through them would not depend on the presence of somenonzero proportion of anti-3He or any pair creation process originally in the sam-ple. Relative to a given observer evolving forward in time, even when no initialtrace of anti-3He existed, the above CTCs had to exist and be operative, sinceordinary 3He atoms traveling through them would behave like their correspondinganti-atoms to the given observer.Finally, we briefly comment on the possibility of an experimental verificationof these CTCs. This would naturally reduce to the question, how could superfluidanti-3He be detected in an experiment with normal superfluid 3He?. In order totry to answer this question, let us suppose that anti-3He is traveling through aCTC into the past. Of course, it can do so by time amounts which depend onthe value of β and, therefore, such a traveling can be so large as for making anobserver able to discriminate the presence of superfluid anti-3He in supercoolingexperiments on thin films containing ultrapure (without any trace of 4He) 3He,before this has reached the superfluid transition temperature Tc. If superfluidanti-3He can travel into the past the way we have discussed in this letter, onecould expect that before reaching Tc clusters of some tens of anti-3He atoms ableto show superfluid behaviour would momentary appear still in the viscous 3Hephase, even though the sample did not contain any trace of 4He.A way to try to detect these superfluid anti-3He clusters would require dis-solving a given proportion of a suitable small molecule A in the liquid helium,5checking whether the added molecules are able to freely rotate in short time in-tervals by the use of ultrafast laser spectroscopy, in an experiment similar to thatrecently performed by Grebenev, Toennies and Vilesov to determine the minimumnumber of helium atoms needed for superfluidity [12]. However, expectation todetect CTCs in this way would be very small because viscous 3He should attractthe molecules much more strongly than anti-3He would do. This expectation verymuch increased if, instead of molecule A, we were able to use the correspondinganti-molecule around which anti-3He atoms could much easierly cluster.Although superfluid liquid helium acts as a true vacuum with respect to itsviscous phase, experiments like the one discussed above would also be useful tocheck violation of causality because of the presence of an observer who couldalways decide to stop and invert the supercooling process before it reached thetransition temperature Tc, while still detecting causally-disconnected clusters ofsuperfluid anti-3He in the viscous liquid.Acknowledgements. The authors acknowledge DGICYT by support underResearch Project No. PB94-0107, and A. Ferrera, L.J. Garay and G.A. MenaMarugn for useful comments.6References1 W.H. Zurek, Phys. Rep. 276 (1997) 177.2 G.E. Volovik and T. Vachaspati, Int. J. Mod. Phys. B10, (1996) 471.W.H. Zurek, Nature 317 (1985) 505.4 P.C. Hendry, N.S. Lawson, R.A.M. Lee, P.V.E. McClintock and C.H.D. Williams,Nature 386 (1994) 315; C. Buerle, Yu.M. Bunkov, S.N. Fisher, H. Godfrinand G.R. Pickett, Nature 382 (1996) 332; V.M.H. Ruutu, V.B. Eltsov, A.J.Gill, T.W.B. Kibble, M. Krusius, Yu.G. Makhlin, B. Placais, G.E. Volovikand W. Su, Nature 382 (1996) 334.5 A.J. Leggett and S. Yip, in: Helium Three, eds. W.P. Halperin and L.P.Pitaevskii (Elsevier Science Publishers, B.V., 1990).6 N. Turok and R.H. Brandenberger, Phys. Rev. D33 (1986) 2175.7 T.A. Jacobson and G.E. Volovik, Event horizons and ergoregions in 3He, cond-mat/9801308; W.G. Unruh, Phys. Rev. Lett. 46 (1981) 1351; D51 (1995)2827.8 G.E. Volovik, Superfluid 3He, Particle Physics and Cosmology, cond-mat/9711031.9 G.E. Volovik, Pis’ma ZhETF 62 (1995) 58 [Engl. transl.: JETP Lett. 62 (1995)65].10 D. Harari and A.P. Polychronakos, Phys. Rev. D38 (1988) 3320.11 K. Gdel, Rev. Mod. Phys. 21 (1949) 447.12 S. Grebenev, J.P. Toennies and A. Vilesov, Science 279 (1998) 2083.7

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