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1-1-2003 Analogue Of The iF zeau Effect In An Effective Optical Medium K. K. Nandi Department of Mathematics, University of North Bengal, Darjeeling WB 734430, India; CCAST (World Laboratory), P. O. Box 8730, Beijing 100080, China
Yuan-Zhong Zhang CCAST (World Laboratory), P. O. Box 8730, Beijing 100080, China; Institute of Theoretical Physics, Chinese Academy of Sciences, P. O. Box 2735, Beijing 100080, China
P M. Alsing Albuquerque High Performance Computing Center, University of New Mexico, Albuquerque, New Mexico 87131
James C. Evans University of Puget Sound, [email protected]
A. Bhadra High Energy and Cosmic Ray Research Center, University of North Bengal, Darjeeling, WB 734430, India
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Citation Nandi, K. K., Yz Zhang, Pm Alsing, James C. Evans, et al. 2003. "Analogue of the Fizeau effect in an effective optical medium." Physical Review D 67(2): 025002-025002.
This Article is brought to you for free and open access by the Faculty Scholarship at Sound Ideas. It has been accepted for inclusion in All Faculty Scholarship by an authorized administrator of Sound Ideas. For more information, please contact [email protected]. PHYSICAL REVIEW D 67, 025002 ͑2003͒
Analogue of the Fizeau effect in an effective optical medium
K. K. Nandi,1,2,* Yuan-Zhong Zhang,2,3,† P. M. Alsing,4,‡ J. C. Evans,5,§ and A. Bhadra6,ʈ 1Department of Mathematics, University of North Bengal, Darjeeling WB 734430, India 2CCAST (World Laboratory), P. O. Box 8730, Beijing 100080, China 3Institute of Theoretical Physics, Chinese Academy of Sciences, P. O. Box 2735, Beijing 100080, China 4Albuquerque High Performance Computing Center, University of New Mexico, Albuquerque, New Mexico 87131 5Department of Physics, University of Puget Sound, Tacoma, Washington 98416 6High Energy and Cosmic Ray Research Center, University of North Bengal, Darjeeling, WB 734430, India ͑Received 14 August 2002; published 8 January 2003͒ Using a new approach, we propose an analogue of the Fizeau effect for massive and massless particles in an effective optical medium derived from the static, spherically symmetric gravitational field. The medium is naturally perceived as a dispersive medium by matter de Broglie waves. Several Fresnel drag coefficients are worked out, with appropriate interpretations of the wavelengths used. In two cases, it turns out that the coefficients become independent of the wavelength even if the equivalent medium itself is dispersive. A few conceptual issues are also addressed in the process of derivation. It is shown that some of our results comple- ment recent work dealing with real fluid or optical black holes.
DOI: 10.1103/PhysRevD.67.025002 PACS number͑s͒: 03.65.Ta, 04.20.Ϫq, 42.15.Ϫi
I. INTRODUCTION AND REAPPRAISALS effective optical medium. In the process, we shall also see the extent to which the curved space analogy compares with The historical Fizeau effect for light in moving media has the results derived in a genuine gravitational field. For sim- been reconsidered by several authors ͓1–5͔ in recent times. plicity, we shall assume only uniform motion of our effective We shall consider it here in the context of static, isotropic medium resulting from the relative motion between the gravity. To our knowledge, such an investigation has not gravitating source and the observer. been undertaken before. We deemed it worthwhile to exam- An outline of the Fizeau effect is this. Consider a tube ine how an old effect would look in a new theoretical model through which a fluid with a refractive index n is flowing and what conceptual issues are involved. However, we must with velocity V. Then, let light pass through the tube parallel make clear at the outset that the only quantity to be borrowed to its axis. In the comoving frame of the water, the speed of ϭ from general relativity is the effective refractive index. The light is vЈ ( c0 /n), but in the frame in which the water rest of the analysis is special relativistic ͑see Sec. III͒. appears to be flowing, the speed of light has been found to be In the literature, generally, the Fizeau effect is considered in connection with its close relative, the analogue of the 2 vЈ c0 1 Aharonov-Bohm ͑AB͒ effect in a real material medium. Sev- vϭvЈϮͩ 1Ϫ ͪ VϩO͑V2͒Ϸ Ϯͩ 1Ϫ ͪ V, ͑1͒ 2 n 2 eral interesting results have followed from these analyses. c0 n For instance, Leonhardt and Piwnicki ͓6͔ showed that a non- uniformly moving medium appears to light as an effective where c0 is the speed of light in vacuum. The quantity (1 Ϫ gravitational field for which the curvature scalar is nonzero. Ϫn 2) is called the Fresnel drag as he was the first to pre- They also showed how light propagation at large distances dict it theoretically. Obviously, the resultant speed v does not around a vortex core shows the Aharonov-Bohm effect and conform to the Galilean law of addition of velocities vЈ at shorter distances resembles propagation around what are ϮV. The effect, after it was experimentally observed by termed optical black holes. Berry et al. ͓7͔ demonstrated the Fizeau in 1851, was regarded as an empirical fact awaiting a AB effect with water waves and Roux et al. ͓8͔ observed it correct theoretical interpretation. It came only after the ad- for acoustical waves in classical media. The curved space vent of Einstein’s special theory of relativity in 1905. It has analogy has been predicted for fluids and superfluids as well since been realized that the Fizeau effect symbolizes only a ͓9͔. The spirit of the present work, in some sense, is in a first order approximation of the exact one-dimensional spe- direction that is the reverse of the idea of the above curved cial relativistic velocity addition law ͑VAL͒ derived from space analogy. That is, our interest is to calculate the Fizeau Lorentz spacetime transformations. Originally, Fizeau did type effect for both massless and massive particles in a static, not consider dispersion but nowadays it is recognized that spherically symmetric gravity field but portraying it as an the effect also contains a term due to the effect of dispersion. In our investigation, we shall adopt an approach involving quantum mechanics and general and special relativity using *Email address: [email protected] the method of what is known as the optical-mechanical anal- †Email address: [email protected] ogy. The historical and fundamental role of the analogy in ‡ Email address: [email protected] the development of modern theoretical physics need not be § Email address: [email protected] emphasized. Apart from the crucial role it played in the de- ʈ Email address: [email protected] velopment of quantum mechanics, especially in the de Bro-
0556-2821/2003/67͑2͒/025002͑11͒/$20.0067 025002-1 ©2003 The American Physical Society NANDI et al. PHYSICAL REVIEW D 67, 025002 ͑2003͒ glie wave-particle duality, it provides an excellent tool that deeper understanding of the parameter A is still awaited.1 enables one to visualize the problems of geometrical optics Our basic strategy is to regard the gravity field as an ef- as problems of classical mechanics and vice versa. fective refractive optical medium imposed on a fictitious In a series of papers ͓10͔, it has been shown that the Minkowski space so that Lorentz transformations can be optical-mechanical analogy can be recast into a familiar form used to relate two relatively moving observers in that space. that allows one to envisage the mechanical particle equation ͑Note that we are not talking of a division of the metric as a geometrical optical ray equation and the latter as a New- tensor into two parts, but rather of a scalar field placed upon ͒ tonian Fϭma equation: a flat space. For more discussion, see Sec. III. This is just an intermediate exercise. The final outcome has to be translated back into actually observable quantities in a gravity field. The idea that a gravity field could be formally equivalent to d2rជ 1 ϭٌជ ͩ 2 2 ͪ ͑ ͒ a refractive medium with respect to optical propagation is n c0 , 2 dA2 2 not new. It goes all the way back to Eddington ͓12͔ who was the first to advance the expression of a gravitational refrac- tive index in an approximate form. It was used later, in vary- ing degrees, by several other researchers ͓13,14͔ in the in- dt dAϭ , ͑3͒ vestigation of specific problems. But none of the work really n2 focused on how the exact general relativistic equations of trajectories, frequency shifts, or Shapiro time delay for mass- less particles could be obtained in that equivalent medium. where rជϵ(x,y,z)or(r,,), ٌជ is the gradient operator, n is The motion of massive particles was not addressed at all. An ͓ ͔ the index of refraction, not necessarily constant, and A was extension of the work in Ref. 11 that also includes the massive particle motion now exists ͓15,16͔: A suitably modi- originally called the stepping parameter but could also be ϭ identified as the optical action and related to several other fied index of refraction together with the F ma formulation immediately reproduce all the desired exact equations in the physical parameters. Many illustrations in ordinary gradient static, isotropic gravity field. The method has been applied index optics demonstrated the validity of Eqs. ͑2͒ and ͑3͒ and very successfully to Friedmann cosmologies as well, which their usefulness as a heuristic tool. yielded some new interesting insights. All the above system- An interesting turn in the direction of investigation was atic developments amply indicate the usefulness of the con- ͓ ͔ signaled by the introduction of general relativity 11 . Exact cept of an effective gravitational index of refraction. By way equations for light propagation in the static, spherically sym- of a further extension, the index has been calculated also for metric field of Schwarzschild gravity do indeed follow from a more general class of rotating metrics ͓17͔. A new and Eqs. ͑2͒ and ͑3͒ when an appropriate gravitational index of significant development has come in the shape of the most refraction n(rជ) is employed. The analysis also brings forth recent formulation ͓18͔ of a single set of unified optical- the distinct but complementary roles played by the optical mechanical equations that allow easy introduction of quan- action A and coordinate time t. To see this, note that the first tum relations into the index. As a consequence, one then integral of Eq. ͑2͒ is finds that massive de Broglie waves necessarily perceive the gravity field as a dispersive optical medium. In this paper, our basic aim hinges around calculating the consequences arising out of this dispersion in the form of drជ ͯ ͯϭnc , ͑4͒ what may be termed the gravitational Fresnel drag, disper- dA 0
1An interesting identification of A is that it is simply the affine 2ϭ or equivalently, using Eq. ͑3͒, parameter lambda of the null geodesic in the optical metric ds Ϫ(c2/n2)dt2ϩdx2ϩdy2ϩdz2. To see this, note that the timelike Killing vector is Kaϭ(1,0,0,0), while it is a standard result that the inner product between the Killing vector and the tangent vector of a drជ c ͯ ͯϭ 0 ͑ ͒ geodesic is conserved, provided the tangent vector is normalized . 5 a bϭ dt n using an affine parameter. That implies K gabK const. That is, (Ϫc2/n2)(dt/d)ϭconst, or d is proportional to dt/n2. This im- plies that d is proportional to dA. Of course, this is still a formal However, the force laws have changed thereby. In Eq. ͑4͒, mathematical statement, but the fact that the parameter A is related in this way to a null affine parametrization in the sense of general the potential is 1 n2c2 while in Eq. ͑5͒ the potential is 2 0 relativity is quite promising. In this context, we should mention that Ϫ 1 2 2 ͑ ͒ ͑ ͒ 2 c0/n . On eliminating A from Eq. 4 or t from Eq. 5 , we gave, in Ref. ͓18͔, four more relations connecting A with other we therefore obtain two path equations for light on a plane, quantities—proper time, the integration measure in the WKB ex- but only the former, not the latter, gives the right answer. On pansions, Born and Wolf’s parameter , and phase speed times the other hand, Eq. ͑5͒ gives the correct equation for the group displacement. All these issues require a more detailed inves- Shapiro time delay ⌬t, while ⌬A from Eq. ͑4͒ does not. A tigation.
025002-2 ANALOGUE OF THE FIZEAU EFFECT IN AN EFFECTIVE... PHYSICAL REVIEW D 67, 025002 ͑2003͒ sion included. There are several spin-offs. It will be demon- if the action has a foot in the wave regime and a foot in the strated that, in the comoving frame, the expressions for the particle regime. The second step involves the introduction of Lagrangian and the dispersion relation are similar to those the Planck relation HЈϭបЈ and the de Broglie relation obtained by Leonhardt and Piwnicki ͓6͔ in the context of real pЈϭបkЈϭh/Ј, where hϭ2ប, in the expression for N.As media. These similarities provide a direct extension of these usual, HЈ and pЈ are the total energy and momentum, respec- expressions in a realistic gravity field. It will also be evident tively, and Ј (ϵ2Ј) and Ј are the coordinate frequency that the conditions for optical black holes ͓6,19,20͔ are natu- and the wavelength of the de Broglie waves. The physically rally met in the equivalent medium, irrespective of whether measurable corresponding proper quantities are ˜ ЈϭЈ/⍀ one considers light or massive de Broglie waves. and ˜ЈϭЈ/⌽, respectively. The third step finally gives the The paper is organized as follows. Section II contains a desired index of refraction N of the dispersive medium due brief survey of the basic equations that will be used through- to massive de Broglie waves: out the paper. Conceptual justifications for the adopted pro- cedure appear in Sec. III. In Secs. IV–VI, the gravitational 2 4⍀2͑ជ Ј͒ m c0 r Fresnel drags are calculated for different choices of the N͑rជ Ј,Ј͒ϭn͑rជ Ј͒ͱ1Ϫ , ͑11͒ wavelengths. Section VII contains a brief discussion of op- ប2Ј2 erational definitions. In Sec. VIII, we demonstrate how the results dealing with a real fluid medium compare with those where m is the rest mass of the test particle. One may also in a genuine gravity field considered in this paper. Finally, in rewrite N as Sec. IX, we summarize and add some remarks. 2 c0pЈ n vЈ Nϭ ϭ , ͑12͒ II. BASIC EQUATIONS HЈ c0
Consider a static, spherically symmetric, but not necessar- where vЈ is the ͑unobservable͒ coordinate speed of the clas- ily vacuum, solution of general relativity written in isotropic sical particle in the medium. It also follows that coordinates
c0 ds2ϭ⍀2͑rជ Ј͒c2dtЈ2Ϫ⌽Ϫ2͑rជ Ј͉͒drជ Ј͉2, ͑6͒ Јϭ2Ј, Јϭ . ͑13͒ 0 NЈ where ⍀ and ⌽ are the solutions of Einstein’s field equa- ͑ ͒ tions. Many metrics of physical interest can be put into this Using Eq. 13 , N can be rewritten in a more transparent isotropic form including the experimentally verified form: ជ Schwarzschild metric. The coordinate speed of light c(r Ј)is ͑ជ Ј͒ 2 n r determined by the condition that the geodesic be null (ds Nϭ , ͑14͒ ϭ ͱ ϩ ˜ ͒2 0): 1 ͑ Ј/ c ជ Ј ϭ ជ dr ជ ជ where c h/mc0 is the Compton wavelength of the particle. c͑r Ј͒ϭͯ ͯϭc ⌽͑r Ј͒⍀͑r Ј͒. ͑7͒ ϭ ϭ ͑ ͒ dt 0 Clearly, for light, m 0,N n, and one recovers Eqs. 2 and ͑3͒ from Eqs. ͑9͒ and ͑10͒, respectively. That is, light waves We take leave from the metric approach at this point and do not perceive the effective medium as dispersive. How- define the effective index of refraction for light in the gravi- ever, for mÞ0, dispersion seems inevitable, as evidenced tational field as from Eq. ͑11͒ or ͑14͒, if quantum relations are introduced. We shall require also the following. The mass shell con- n͑rជ Ј͒ϭ⌽Ϫ1⍀Ϫ1. ͑8͒ straint is given by ͓18͔
We shall omit further details here that can be found in c2ប2kЈ2 ប2Ј2ϭ 2 4⍀2ϩ 0 ͑ ͒ Ref. ͓18͔, and only state the results to be used in this paper. m c0 . 15 n2 The first step in the direction of introducing quantum me- chanics in a semiclassical way is to have a single refractive The phase velocity is index N and a single set of equations that should be valid for both massless and massive particles. The result is 2 HЈ Ј c0 c0 vЈϭ ϭ ϭ , vЈvЈϭ , ͑16͒ p p g 2 d2rជ Ј 1 pЈ kЈ N n ͒ ͑ ͒ ͑ ͪ ϭٌជ ͩ 2 2 N c0 light and particles , 9 dA2 2 giving the group velocity
ជ dr Ј d Ј c0N ͯ ͯϭ ͑ ͒ ͑ ͒ Јϭ ϭ ϭ Ј ͑ ͒ Nc light and particles , 10 vg v . 17 dA 0 dkЈ n2 where, once again, it is the same A, satisfying dAϭdt/n2, It should be mentioned that the validity of the expression that appears even for massive particle trajectories. It looks as ͑11͒ is established also by the WKB analysis of the massive
025002-3 NANDI et al. PHYSICAL REVIEW D 67, 025002 ͑2003͒ generally covariant Klein-Gordon equation ͓18͔. Moreover, velocities. Such types of natural constraints are unavailable the mass shell constraint ͑15͒ yields the exact Stodolsky in just any arbitrary medium consisting of solids or liquids. phase ͓21͔ in the case of the spin-1/2 Dirac equation in In these cases, dispersion is normally introduced by hand. curved spacetime ͓22͔. This last result is extremely interest- An important point should be noted here. In describing ing. the Fizeau experiment with an ordinary medium ͑such as With Eqs. ͑6͒–͑17͒ at hand, we are able to calculate the water͒, one takes the background spacetime to be flat. Such Fresnel drag factors under different scenarios, but, before Minkowski networks, composed of rods and clocks, are ac- this, we need to clear up a few relevant concepts. Note that tually unobservable in a gravity field due to the universality all the expressions in this section refer to the comoving of gravitational interaction, or, putting it more technically, frame, that is, the frame fixed to the gravitating source. due to the principle of equivalence. There does not exist a Henceforth, in order to have conformity with notation in the unique division of the metric tensor into a background and a literature, all expressions in the comoving frame will be des- field part. We consider here a different kind of separation ignated by primes and those in the relatively moving lab according to which the gravity field is looked upon as analo- frame will be unprimed. gous to an optical medium imposed upon a flat background spacetime, the index N summarizing the nonlinearities of the III. CONCEPTUAL ISSUES gravity field, as it were. The important point is that the anal- ogy, although intended to be only of formal nature, may lead The following discussion is aimed at providing appropri- to results that could be testable by experiment ͑see Sec. IX ate interpretations of the quantities that appear in the various for a discussion͒. With this understanding, let us conceive of formulations of the Fizeau effect. There are two basic ingre- observers equipped with fictitious Minkowski networks and dients. The first is the VAL. In many works dealing with the apply, as an intermediate step, the full machinery of special effect, the one-dimensional VAL, which is valid for point relativity in what follows. particles, is also employed, implicitly or explicitly, for waves Thus, we take Eq. ͑19͒ in the form propagating with the phase speed c0 /n. The procedure is to use the one-dimensional Lorentz transformation equations in ⌬ϭ␥͑⌬Јϩ⌬ Ј ͒ ⌬ ϭ␥͑⌬ Јϩ ⌬Ј Ϫ2͒ k V , k k V c0 , the form ͑21͒ Ϫ Јϭ␥͑ϪkV͒, kЈϭ␥͑kϪVc 2͒, ͑18͒ Јϭ⌬Ј ⌬ Ј 0 which gives the VAL, denoting vg / k ,as ϭ␥͑Јϩ Ј ͒ ϭ␥͑ Јϩ Ј Ϫ2͒ k V , k k V c0 , ⌬ vЈϩV ϭ ϭ g ͑ ͒ vg . 22 ␥ϭ͑ Ϫ 2 2͒Ϫ1/2 ͑ ͒ ⌬k 1ϩVvЈ/c2 1 V /c0 , 19 g 0 and obtain a VAL as The second ingredient is the special relativistic Doppler shift in one dimension giving the frequency ͑or wavelength͒ Јϭ͑ Ϫ ͒͑ Ϫ 2͒ ϭ͑ Јϩ ͒͑ ϩ Ј 2͒ v p v p V 1 v pV/c0 , v p v p V 1 v pV/c0 , transformation between two frames in relative motion. Thus, ͑20͒ one takes Eq. ͑19͒ in the form ЈϭЈ Јϭ where v p /k c0 /n is the phase speed of light in the ϭ␥Ј͑1ϩkЈV/Ј͒, ͑23͒ ͑ ͒ ϭ primed comoving frame of the medium and v p /k is the phase speed in the ͑unprimed͒ lab frame in which the me- and specifies Ј/kЈ. At this point, let us note that Cook, dium appears to be moving with uniform relative velocity V. Fearn, and Milonni ͓2͔ considered two possibilities in the The phase speed, however, could well exceed c0 in many context of a Fizeau experiment with real media having re- physical configurations where nϽ1. fractive indices n. On the other hand, an a priori prescription that n be greater than unity ͑making c /nϽc ) somewhat diminishes 0 0 A. Case 1 the generality of the theory. However, this deficiency may ϭ ͑ ͒ not pose any realistic problem in a nondispersive medium. Take Ј/kЈ c0 in Eq. 23 . This case was considered by When dispersion is involved, the most appropriate quantity Synge ͓3͔. That is, take the usual Doppler shift formula, to use in the VAL is the group speed d/dk ͑which involves which, written in terms of the wavelength, is a knowledge of dn/d), which simply equals the velocity of 1ϪV/c the classical point particle, rather than the phase speed. As ϭЈͱ 0 ͑ ͒ ϩ . 24 stated before, the original Fizeau experiment did not consider 1 V/c0 any dispersion; the index n was taken to be a true constant, so that the group and the phase velocities coincided precisely The corresponding physical configuration consists of a block at c0 /n. In general, they are different for massive de Broglie of material moving with velocity V in an otherwise empty waves, as our later equations will reveal. In our calculation lab frame. The wavelength Ј of a light pulse measured by of the Fizeau effect, the mass shell constraint Eq. ͑15͒,or,by an observer stationed at the interface between the block and another name, the dispersion relation, plays a key role: It the empty space will appear to the lab observer as accord- provides well defined expressions for the group and phase ing to Eq. ͑24͒. Inside the block, however, Ј is assumed to
025002-4 ANALOGUE OF THE FIZEAU EFFECT IN AN EFFECTIVE... PHYSICAL REVIEW D 67, 025002 ͑2003͒
be a constant. The resulting Fresnel drag has been experi- dium. He/she measures the coordinate phase and group ve- ϭ Ј mentally confirmed to a very good accuracy by Sanders and locities of a massive de Broglie wave packet at r r0 as, Ezekiel ͓4͔. using Eq. ͑15͒, Unfortunately, it is difficult to conceive of a parallel con- figuration in our problem. The entire optical medium cannot Ј c0 be simply put inside a box with a certain boundary, nor need ϭvЈϭ , ͑27͒ Ј p ͑ Ј Ј͒ the wavelength Ј be constant throughout the medium. In- k N r0 , stead, it is easier to consider two relatively moving observers ⌬Ј c n2 associated with the background empty frame who may use ϭ Јϭ 0 Ͻ ¯ ϭ Ͼ ͑ ͒ ͑ ͒ vg c0 , N 1. 28 Eq. 24 . We have to calculate how one observer translates ⌬kЈ N¯ ͑rЈ ,Ј͒ N the observations of another at a certain point when they hap- 0 pen to pass each other. This is done in Sec. IV. Јϭ Јϭ For a light pulse, v p vg c0 /n and these are independent of the wavelength Ј or wave number kЈ. The same holds for B. Case 2 vg as well. This implies that the trajectories of light rays do ϭ ͑ ͒ ͓ ͔ Take Ј/kЈ c0 /n in Eq. 19 for k. Cook et al. 2 pro- not depend on the wave properties of light. However, in gen- ЈÞ Ј ͑ ͒ ͑ ͒ vide the corresponding physical configuration in this case. eral, v p vg , as is evident from Eqs. 27 and 28 . According to Lerche ͓1͔, the lab observer can exercise two Consider another observer B moving in the same radial options. He/she either uses ͑i͒ a wavelength given by the direction with uniform velocity V with respect to A. Then, in ͑ ͒ ϭ ͑ ͒ Doppler formula 23 but with Ј/kЈ c0 /n or uses ii a the frame of B, identified as the lab observer, the entire me- ϭ vacuum wavelength 0 2 c0 / . The forms for the drag dium moves uniformly, that is, A becomes the comoving ob- coefficients will be different in the two cases. The parallel server. How will B translate the observations of A, when options in our case are the same, except that we have to use their origins coincide at rϭ0? To find out, note that the Ј N instead of n, so that the Doppler formula reads coordinate length r0 will appear to B as
Ϫ1 Ј V 2 ϭ ϩ ͑ ͒ V ␥ ͩ 1 ͪ . 25 ϭ Јͱ Ϫ ͑ ͒ Nc0 r0 r0 1 2 . 29 c0 We shall work out both the options in Sec. V. This particular Ј formula appears to be more consistent with our formulation Also, the velocity vg observed by A will appear to B as vg per se as we will be using our own definition of Ј/kЈ given given by the special relativistic VAL Eq. ͑22͒. We may ex- Ј Ј Ј in Eq. ͑27͒. We can also add a third possibility worked out in plicitly express vg in terms of (r0 , )as Sec. VI. This is a special feature of the gravitational case we are considering. c0 c0 vЈϭ ϭ . g ¯ ͑ Ј Ј͒ ͑ Ј͒ϫͱ ϩ͑Ј ͒2⌽Ϫ2͑ Ј͒ N r0 , n r0 1 / c r0 C. Case 3 ͑30͒ Consider a stationary observer ˜A at a point in the gravity When this expression for vЈ is plugged into the right hand ͑ ͒ ˜Ј g field measuring the proper or physical wavelength . He/ side of Eq. ͑22͒, one finds the answer to the question above: she also measures the proper velocity of light in his/her v (rЈ ,Ј) is the exact radial group velocity of the de Broglie ˜ g 0 neighborhood to be just c0. A freely falling observer B at waves to be observed by B. But B uses the Doppler shifted ˜ ˜ that point, having an instantaneous velocity V relative to A, wavelength instead of Ј. Then, to first order in (V/c0), would measure ˜ according to the options, which, to first we get from Eq. ͑24͒ order, are ЈϷ͑ ϩ ͒ϭϩ⌬ ЈϷ ͑ ͒ 1 V/c0 , r0 r0 . 31 ˜ Ϸ˜ ϩ˜ ͒ ˜ Ϸ˜ ϩ˜ ˜ ͒ ˜ ϭ ˜ Ј ͑1 V/c0 , Ј ͑1 VN/c0 , 0 2 c0 / . Considering the right hand side of Eq. ͑30͒ and writing the ͑26͒ ¯ Јϭ Ј ϵ¯ ϩ⌬ denominator as N(r0 r0 , ) N( ), we get from the Note that there is a difference between the present stationary Taylor expansion observer and the stationary observer associated with the background flat space of case 1: The group velocities of the ¯Nץ N¯ Vץ matter de Broglie waves measured by them are not the same N¯ ͑ϩ⌬͒ϷN¯ ͑͒ϩ⌬ ϭN¯ ͑͒ͩ 1ϩ ͪ . ץ ¯ ץ ͑see below͒. We now proceed to calculate the Fresnel drags c0N successively in all three cases using the same VAL Eq. ͑22͒, but different Doppler formula, Eqs. ͑24͒–͑26͒. From Eqs. ͑22͒ and ͑28͒, we get, using the above, a redefined index N¯ Ј such that IV. FRESNEL DRAG: CASE 1 c c 1 Suppose that an observer A, equipped with a Minkowski ͑Ј͒ϭ 0 ϭ 0 ϩͩ Ϫ ͪ ͑ ͒ vg 1 V. 32 network, is at rest at rϭ0 in a spherically symmetric me- N¯ Ј͑Ј͒ N¯ ͑Ј͒ N¯ 2͑Ј͒
025002-5 NANDI et al. PHYSICAL REVIEW D 67, 025002 ͑2003͒
¯ 2 Ј 2 In other words, in the approximation considered, N ( ) mc0 ϷN¯ 2() and we have Hϭ . ͑39͒ ͱ Ϫ Ј2 2 1 vg 2/c0 c 1 c ͑͒ϭ 0 ϩͩ Ϫ ͪ ϭ 0 ϩ Then one recovers the special relativistic mass shell condi- vg 1 V F1V, N¯ ͑ϩ⌬͒ N¯ 2͑͒ N¯ ͑͒ tion. It follows that, in this case, the drag measured by B in ͑33͒ terms of his/her wavelength is where ͒2 ͑ / c 1 F flatϭ ϫͫ 1Ϫ ͬ. ͑40͒ 1 ͓ ϩ͑ ͒2͔ ͓ ϩ͑ ͒2͔1/2 N¯ 1 / c 1 / cץ 1 ϵͩ Ϫ ͪ Ϫ ϫ ͑ ͒ F1 1 34 As one can see, Eqs. ͑37͒–͑40͒ are restatements of the wellץ N¯ 2͑͒ N¯ 2͑͒ known special relativistic expressions, only interpreted in a is the Fresnel drag we have been looking for. It can be easily different way. verified that the same F1 follows also from the ordinary ex- Ј Ј ͑ ͒ pansion of vg(r0 , ) in Eq. 22 in conjunction with Eqs. V. FRESNEL DRAG: CASE 2 ͑21͒ and ͑30͒ under the small velocity approximation, Eq. ͑31͒, but the steps as given above are the simplest. For light According to the first option ͑i͒ in Sec. III B the Doppler ͑ ͒ waves, N¯ →n, and one has shift is given by Eq. 25 . Thus, we have, to first order in (V/c0), nץ 1 ϵͩ Ϫ ͪ Ϫ ϫ ͑ ͒ V F1 1 . 35 ЈϷ͑ ϩ ͒ϭϩ⌬ ⌬ϭ ͑ ͒ 1 V/Nc0 , . 41ץ n2͑͒ n2͑͒ Nc0
Interestingly, although the dependence of n on is not ¯ Јϭ Ј ϵ¯ ϩ⌬ Then, writing again N(r0 r0 , ) N( ), we get known, the dispersion nonetheless follows here as an inher- from the Taylor expansion itance from Eq. ͑34͒. This is the formula proposed by Synge ͓3͔ and also experimentally tested ͓4͔ with n as the refractive ¯Nץ N¯ Vץ index of the block. N¯ ͑ϩ⌬͒ϷN¯ ͑͒ϩ⌬ ϭN¯ ͑͒ͩ 1ϩ ͪ . ץ ¯ ץ Using Eqs. ͑14͒ and ͑28͒, we can have the explicit expres- c0NN sion from Eq. ͑34͒ as ͑42͒
˜ ͒2 The resultant group velocity as observed by B, who uses of 1 ͑ / c F ϭ1Ϫ Ϫ . Eq. ͑41͒,is 1 2 ͒ ϩ ˜ ͒2 ͒ ϩ ˜ ͒2 3/2 n ͑r0 ͓1 ͑ / c ͔ n͑r0 ͓1 ͑ / c ͔ ͑ ͒ 36 c c 1 ϭ 0 ϭ 0 ϩͩ Ϫ ͪ vg 1 V Note that, in the asymptotic region r→ϱ, or in the ab- N¯ Ј͑Ј͒ N¯ ͑Ј͒ N¯ 2͑Ј͒ sence of gravity, one has n(r)→1, ˜→Ј, so that, from Eq. ͑28͒, the group and phase velocities of de Broglie waves, as c0 1 ϭ ϩͩ 1Ϫ ͪ V measured by A, are, respectively, N¯ ͑ϩ⌬͒ N¯ 2͑͒
c0 vЈϭvЈϭ Ͻc , c0 g 2 1/2 0 ϭ ϩ ͑ ͒ ͓1ϩ͑Ј/ ͒ ͔ F2V, 43 c N¯ ͑͒ Ј 2 1/2 Јϭ ͫ ϩͩ ͪ ͬ Ͼ ͑ ͒ where v p c0 1 c0 , 37 c ¯Nץ 1 and thus one finds that matter de Broglie waves perceive F ϵͩ 1Ϫ ͪ Ϫ ϫ ͑44͒ ץ 2͑͒ ¯ 2͑͒ ¯ 2 even the flat space as a dispersive medium with an index of N NN refraction is the drag factor. For light waves, Nϭn, N¯ ϭn so that Ј 2 1/2 ¯ ϭͫ ϩͩ ͪ ͬ ͑ ͒ nץ N flat 1 . 38 1 c ϵͩ Ϫ ͪ Ϫ ϫ ͑ ͒ F2 1 . 45 ץ n2͑͒ n3͑͒ Ј ͑ ͒ One recognizes that it is this vg in Eq. 37 , together with ϭ Ј/ c mc0 /pЈ, that provides the energy transformation This formula was first given by McCrea ͓23͔. Writing explic- law: itly, we find, from Eq. ͑44͒,
025002-6 ANALOGUE OF THE FIZEAU EFFECT IN AN EFFECTIVE... PHYSICAL REVIEW D 67, 025002 ͑2003͒
1 so that ˜A measures, in his/her neighborhood, the proper F ϭ1Ϫ . ͑46͒ 2 2͑ ͒ phase and group velocities of the de Broglie waves which are n r0 connected by This coefficient comes out to be independent of . Accord- ˜ Ј d˜ Ј ing to the second option ͑ii͒, B uses a vacuum wavelength. In ˜ Ј˜ Јϭ ϭ 2 ͑ ͒ v pvg c0 , 54 this case, the calculations would proceed slightly differently. ˜kЈ dk˜Ј Consider Eq. ͑18͒ for Ј instead of Eq. ͑41͒. Then we have, to first order, where, using Eq. ͑27͒,
VN͑͒ ˜ Ј Ј ЈϷ Ϫ ͑͒ ϭϩ⌬ ⌬ϭϪ ˜ Јϭ ϭ ͩ ͪ ϭ ˜ ͑ Ј ˜Ј͒Ͼ 1 VN /c0 , . v p n c0N r0 , c0 , „ … c0 ˜kЈ kЈ ͑47͒ ˜ d Ј c0 Then, proceeding as before, ˜vЈϭ ϭ Ͻc , g ˜Ј ˜ ͑ Ј ˜Ј͒ 0 dk N r0, ¯ץ c0 1 N v ͑͒ϭ ϩͫͩ 1Ϫ ͪ ϩ ϫ ͬV. nץ g 2 ͑͒ N¯ ͑͒ N¯ ͑͒ N N˜ ϵ Ͼ1. ͑55͒ ͑48͒ N
Now B uses the vacuum wavelength as ϭ2c /, so that Note that these phase and group velocities are not the same 0 0 ͑ ͒ ͑ ͒ Eq. ͑48͒ gives as those measured by A, viz., Eqs. 27 and 28 , which high- lights the difference between the two observers. The observer ˜A measures the velocity of light as ˜vЈϭ˜vЈϭc since N c0 p g 0 v ͑ ͒ϭ ϩF V, ͑49͒ ϭ ˜ g 0 ¯ ͒ 3 n. Consider another observer B falling freely in the same N͑ 0 radial direction attaining an instantaneous speed ˜V at r ϭ Ј ˜ where r0 . Since the frame in which B is at rest is locally inertial in virtue of the principle of equivalence, the speed of light ¯ץ ͒ 1 0N͑ 0 N measured by ˜B will also be c and hence ˜A and ˜B can be F ϭͫͩ 1Ϫ ͪ Ϫ ϫ ͬ. ͑50͒ 0 ˜ Јץ ͒ ͒ ¯ 2 2 ¯ 3 N ͑ 0 N ͑ 0 0 connected by a Lorentz transformation. Then vg will appear ˜ ϭ ˜ to B at r r0 as vg given by the VAL For light waves, we get ˜ Јϩ˜ vg V v ϭ . ͑56͒˜ ץ 1 0 n g ϩ˜˜ Ј 2 F ϵͩ 1Ϫ ͪ Ϫ ϫ . ͑51͒ 1 Vvg/c0 ץ ͒ ͒ n͑ 2 3 n ͑ 0 0 0 Employing arguments similar to those in cases 1 and 2 we This is the expression given by Lerche ͓1͔ and Cook et al. can immediately write down the corresponding drag coeffi- ͓2͔ for a Fizeau experiment with water with index n. Writing cients explicitly, we find from Eq. ͑50͒ ͑a͒ ˜A measures ˜Ј and ˜B uses ˜ connected by ˜ЈϷ˜ (1 ϩ˜V/c ): ˜ ͒2 0 1 ͑ 0 / c F ϭ1Ϫ Ϫ ͑52͒ ˜Nץ ϩ͑˜ ͒2͔ ͓ ϩ͑˜ ͒2͔2 1 ˜ ͓͒ 2͑ 3 n r0 1 0 / c 1 0 / c ˜ ϵͩ Ϫ ͪ Ϫ ϫ F1 1 ˜ץ N˜ 2͑˜͒ N˜ 2͑˜͒ ˜ ϭ ⌽Ϫ1 ¯ where 0 0 . Thus, so far, corresponding to N and N, ˜ ͒2 we have three Fresnel coefficients F , F , and F depending ͑ / c 1 1 2 3 ϭ ϫͫ 1Ϫ ͬ. ͑57͒ on the VAL and the various Doppler shifted wavelengths ͓1ϩ͑˜/ ͒2͔ ͓1ϩ͑˜/ ͒2͔1/2 used by B, as considered in the literature. c c ͑b͒ ˜A measures ˜Ј and ˜B uses ˜ connected by ˜ЈϷ˜ (1 VI. FRESNEL DRAG: CASE 3 ϩ˜ ˜ VN/c0): Consider an observer ˜A at rest with respect to the gravi- ˜Nץ ϭ Ј 1 ˜ tating source at a coordinate radial distance r r0 . He/she ˜ ϵͩ Ϫ ͪ Ϫ ϫ ϭ ͑ ͒ F2 1 0. 58 ˜ץ will measure proper quantities. The mass shell condition is N˜ 2͑˜͒ N˜ ͑˜͒ given by ͑ ͒ ˜ ˜ ˜ ˜ ˜ c A measures and B uses 0 connected by 0 ប2˜ Ј2ϭm2c4ϩc2ប2˜kЈ2, ˜kЈϭ⌽kЈ, ͑53͒ ϭ ˜ 0 0 2 c0/ :
025002-7 NANDI et al. PHYSICAL REVIEW D 67, 025002 ͑2003͒
Ϸ ЈϷ N˜ r0 r l, where l is the physically measurable distanceץ ˜ 1 ˜ ϵͩ Ϫ ͪ Ϫ 0 ϫ 0 F3 1 from the center of the gravitating source to the field point. ˜ץ ͒ N˜ 2͑˜ ͒ N˜ 3͑˜ 0 0 0 Then ˜ ͒2 1 ͑ 0 / c ϭ1Ϫ ϫͫ 1ϩ ͬ, ͑59͒ 2M ͓ ϩ͑˜ ͒2͔ ͓ ϩ͑˜ ͒2͔ n͑r ͒Ϸn͑l͒Ϸ1ϩ . ͑64͒ 1 0/ c 1 0 / c 0 l
˜ ϭ ⌽Ϫ1 ͑ ͒ where 0 0 . We also see that the radial proper veloc- With these inputs, Eq. 36 provides the theoretically pre- ˜ ϭ Ј dicted value of F1 after the known value of the Compton ity of the classical point particle as measured by A at r r0 is given by wavelength is plugged in. ˜ Interesting results are obtained in the case of F2 and F2. dlЈ drЈ One finds that F2 does not involve the wavelength at all. ˜vЈ ϭ ϭn ϭnvЈ . ͑60͒ This means that a Fizeau type experiment either with light or prop Ј Ј coord d dt with de Broglie waves will yield the same drag factor, if Eq. ͑23͒ is followed in conjunction with Eq. ͑27͒. In this case, it Јϭ⌽Ϫ1 Ј Јϭ⍀ Ј Ј Using the definitions dl dr , d dt , vcoord appears that the wavelength dependence introduced by the ϭ 2 ˜ Ј ϭ˜ Ј Nc0 /n , we find that v prop vg . For light, of course, group velocity is undone by the Doppler shift. A similar Ј ϭ ˜ Ј ϭ˜ Јϭ ˜ vcoord c0 /n and v prop vg c0. The last result is also con- thing occurs also in the case of F2 which is identically zero. 2ϭ 2 Ј2Ϫ Ј2ϭ sistent with the fact that ds c0d dl 0 gives ϭ ˜ ϭ ˜ VIII. COMPARISON WITH REAL MEDIUM dlЈ/d Ј c0. For light waves, we find N 1, so that F1 ϭ˜ ϭ˜ ϭ F2 F3 0. These indicate only the special relativistic in- Starting from the wave equation in a nonuniformly mov- variance of light speed, no matter what wavelength ˜B uses. ing fluid with refractive index n, Leonhardt and Piwnicki ͓6͔ For de Broglie waves, the difference among the drag coeffi- derived the Lagrangian and the Hamiltonian for a light ray as cients is evident from Eqs. ͑57͒–͑59͒. observed by a lab observer. From the action principle, they arrived at a completely geometrical picture of ray optics in a VII. OPERATIONAL DEFINITIONS moving medium. Light rays are geodesic lines with respect to Gordon’s metric, which in the comoving frame reads In order to operationally realize the value of F1 in a gravi- 2 tational field, consider a simple thought experiment. Let c0 ds2ϭ dtЈ2Ϫ͉drជ Ј͉2. ͑65͒ there be a source in free space that produces de Broglie 2 waves with wavelength Ј. Then is known via Eq. ͑31͒; n this is the wavelength measured by B. Let A take this source The Lagrangian, Eq. ͑49͒ of Ref. ͓6͔, that they derived for a to any point inside the refractive medium. Then, A will mea- light particle in the lab frame is sure the same Ј as ˜ЈϭЈ⌽Ϫ1 and B will find ˜ ϭ⌽Ϫ1 2 . The only other quantity is the coordinate distance 1 uជ vជ r appearing in the refractive indices n(r ) and ⌽(r ). The LϭϪmc ͱc2Ϫv2ϩͩ Ϫ1 ͪ ␥2ͩ c Ϫ • ͪ , ͑66͒ 0 0 0 0 0 2 0 c expression for the index is supplied by the metric functions. n 0 ϭ For instance, in the Reissner-Nordstrom field, with G c0 ͓ϵ Ј ϭ where u is the fluid velocity in the lab frame, v (v 1, we have ϩ ϩ Ј 2 ͔ u)/(1 v u/c0) is the velocity of the light particle con- Ϫ ␥2ϭ ͑ 2Ϫ 2͒ 2 ͑ 2Ϫ 2͒ 2 ceived of as having a fictitious mass m, and (1 M Q M M Q 2 2 Ϫ1 ជ ⍀2͑r͒ϭͫ 1Ϫ ͬ ͫ 1ϩ ϩ ͬ , Ϫu /c ) . In the comoving frame of the fluid element, u 2 2 0 4r r 4r ϭ0 so that ͑61͒ 1 vЈ2n2 2 2 2 ϭϪ 2ϫ ϫͱ Ϫ ͑ ͒ M ͑M ϪQ ͒ L mc0 1 . 67 ⌽Ϫ2͑ ͒ϭͫ ϩ ϩ ͬ ͑ ͒ n c2 r 1 , 62 0 r 4r2 Consider the Lagrangian for a massive particle in the comov- where M and Q are the mass and the electric charge. For the ing frame, derived in our Ref. ͓18͔, viz., Schwarzschild field, we have Qϭ0, so that Ј2 2 1/2 2 v n 3 LϭϪmc ⍀ͫ 1Ϫ ͬ , ͑68͒ ͑1ϩM/2r͒ 0 2 n͑r͒ϭ . ͑63͒ c0 ͑1ϪM/2r͒ where vЈ is the classical particle coordinate speed. Now note If the relative velocity V between A and B is small, V that the metric ͑65͒ with n as the real medium index can be Ӷ Ϸ Ј ͑ ͒ ͑ ͒ ⌽ c0, we can take r0 r0 from Eq. 29 . If we consider that obtained formally from Eq. 6 above simply by putting both the observers are in a weak gravity field, we can take ϭ1 and ⍀ϭ1/n. Clearly, the n in Eq. ͑68͒ has a different
025002-8 ANALOGUE OF THE FIZEAU EFFECT IN AN EFFECTIVE... PHYSICAL REVIEW D 67, 025002 ͑2003͒ origin: it derives from general relativity. Using this value of ϭ Ϫជ where k ( /c0 , k) is the wave four-vector. There are ⍀ in Eq. ͑68͒, one finds that it is exactly the same as Eq. several other ways in which Eq. ͑74͒ could be obtained, ei- ͑67͒. ther by the usual Legendre transformations from Eq. ͑72͒ or (xץ/Sץ)(xץ/Sץ)The dispersion relation for light in the comoving frame by the Hamilton-Jacobi equation g ជ ϭ ជ ͑ ͒ ͓ ͔ ϭ 2 4 ϭ⌽2ϫ͓ϩ 2Ϫ ͔ (u 0) following from Eq. 33 of Ref. 6 is m c0 with g (n 1)V V .Wedonot do it here. Ј2Ϫ 2 Ј2ϩ͑ 2Ϫ ͒Ј2ϭ ͑ ͒ c0k n 1 0. 69 A further interesting result holds as a corollary to Sec. IV: For light waves in flat space, vЈϭvЈϭc ϭv , as expected. ͑ ͒ p g 0 g This is precisely the same as that following from Eq. 15 It should be noted that the Minkowski observers A and B can ϭ with m 0 for light. also be located in the asymptotic region and the entire analy- ͑ ͒ Interestingly, taking a cue from Eq. 66 , we may proceed sis would remain the same. From the asymptotic vantage to write down the Lagrangian of the classical particle in the point, these observers can see that, near the horizon, n→ϱ, lab frame as follows. Our metric Eq. ͑6͒ in the comoving Ј Ј→ and thus v p ,vg 0, for both light and matter de Broglie frame can be written down as waves. It is exactly here that we find that the conditions for optical black holes required by Leonhardt and Piwnicki ͓19͔ 1 ͓ ͔ 2ϭ⌽Ϫ2ϫͫ 2 Ј2Ϫ ជ Ј2ϩͩ Ϫ ͪ 2 Ј2ͬ ͑ ͒ and Hau et al. 20 are provided most naturally, that is, ex- ds c0dt dr 1 c0dt . 70 n2 tremely low group velocity or high refractive index. In this respect, optical and gravitational black holes indeed look ϭ To go to the lab frame, we effect a Lorentz transformation. similar. Also, vg V, implying that, while A sees everything Note that there is a Lorentz invariant term in the parentheses standing still at the horizon, B sees them moving away at the and hence only the last term needs to be transformed. Thus, speed V because of B’s own relative motion. This is what we in the lab frame the metric is should really expect.
1 2 Ϫ2 IX. SUMMARY AND CONCLUDING REMARKS ds ϭgdx dx ϭ⌽ ϫͫ ϩͩ Ϫ1 ͪ VVͬdx dx , 2 n The present investigation is inspired by recent discoveries ͑ ͒ 71 and analyses of light propagation in Bose-Einstein conden- sates ͓19,20͔. The extremely low velocity of light in such ϭ͓ 2 Ϫ Ϫ Ϫ ͔ ϭ␥ Ϫ ជ ␥ϭ where c0 , 1, 1, 1 , V (1, V/c0), (1 condensates lead to the possibility of creating optical ana- Ϫ 2 2 Ϫ1/2 ជ V /c0) , and V is the velocity of our medium in the lab logues of astrophysical black holes in the laboratory. In order frame. In the comoving frame, Vϭ(1,0,0,0). The action is to theoretically model this possibility, Leonhardt and Pi- given by wnicki ͓6͔ proceeded from the moving optical medium to an effective gravity field with a scalar curvature RÞ0 in which dx dx light propagation is shown to mimic that around a vortex ϭϪ ͱ ϭ S mc ͵ g dt ͵ Ldt. 0 dt dt core or optical black hole. It has been shown recently that the propagation of photons in a nonlinear dielectric medium can Defining vϭdx/dtϭ(1,vជ ), we can find the Lagrangian also be described as a motion in an effective spacetime ge- for a particle in the lab frame, ometry and several interesting results have followed thereby ͓24͔. Our approach here has been in the exact reverse direc- 2 tion: We proceed from the gravity field and arrive at an ef- 1 Vជ vជ LϭϪmc ⌽Ϫ1ϫͱc2Ϫv2ϩͩ Ϫ1 ͪ ␥2ͩ c Ϫ • ͪ . fective optical refractive medium and examine the theoretical 0 0 2 0 n c0 consequences. The motion of this medium is caused by the ͑72͒ relative motion between the observer B and the gravitating source. The dispersion relation ͑or the Hamiltonian͒ in the lab frame We must mention that works based on the above men- can also be obtained by a Lorentz transformation on the mass tioned analogies provide some curious theoretical insights shell equation ͑15͒ in the comoving frame, rewritten as both in real media and in the gravitational field, as a result of wisdom borrowed from one field and implanted into the 2 4 2⍀2 m c0n other. This has been the basic philosophy of the present pa- Ј2Ϫc2kЈ2ϩ͑n2Ϫ1͒Ј2ϭ . ͑73͒ 0 ប2 per. Many more interesting results are known apart from the possibility of optical black holes stated above. For instance, Note that the right hand side is a Lorentz scalar and the left an analysis in acoustic theory leads to the remarkable result Ј2Ϫ 2 Ј2 that the Hawking radiation in black hole physics is not of hand side has a Lorentz invariant part c0k . The re- ͑ ͓ ͔͒ maining part can be transformed to give dynamical but of kinematical origin Visser, Ref. 9 . Con- versely, a gravitational refractive index approach, similar in ˇ 2 4 2⍀2 spirit to ours, has yielded the possibility of Cerenkov radia- m c0n ͓ ͔ 2Ϫc2k2ϩ͑n2Ϫ1͒␥2ϫ͑Ϫkជ Vជ ͒2ϭ , tion in the outskirts of a wormhole throat 25–28 .Inthe 0 • ប2 present paper, we envisaged a nontrivial dispersive Fresnel ͑74͒ drag coefficient in a gravity field. We must emphasize that
025002-9 NANDI et al. PHYSICAL REVIEW D 67, 025002 ͑2003͒ these results are only of pedagogic interest at present. A fur- Moreover, we can find a direct extension of the expressions ther confirmation or otherwise of these results would estab- to a genuine gravity field ͑Sec. VIII͒. The resulting Lagrang- lish the extent to which these analogies can actually be relied ian and Hamiltonian describe the trajectories of a particle as upon. viewed from the lab frame, say, a rocket. It also appears that We saw above how dispersion effects, for both massless the nomenclature ‘‘optical black holes’’ is quite apt as the and massive particles, appear naturally as a consequence of conditions required for their creation are most naturally met the systematic development of an effective medium approach near the gravitational horizon. This gives an indication that to the gravitational field. Various expressions for the drag the behavior of the real optical medium should mimic that of coefficients result due to the use of VAL and different wave- our equivalent refractive medium around a coordinate singu- ͑ ͓ ͔ lengths used by the observer B. See Refs. 1–3 for more larity. A favorable situation is attained if light perceives the detailed arguments on the question of the use of the appro- ͒ highly refractive real optical medium as dispersionless priate wavelength. It is demonstrated that F2 is independent which, in our effective medium, is actually the case. Leon- of even in a dispersive medium for massive particles and hardt and Piwnicki ͓6͔ also make a similar statement in the ˜ that F2 is identically zero. These results may have interesting context of their vortex analysis. It is interesting to note that implications for both optical and general relativity black an index of the form nϭC/r, where C is a constant, when holes. put in Eq. ͑4͒ yields orbits that resemble those around an It does not seem easy to simulate real experiments, with optical vortex core ͓10͔. A similar investigation with a dif- our type of unbounded medium, that parallel those dealing ferent form of index has also been reported recently ͓29͔. with ordinary media like solids, fluids, or superfluids. For this reason, we limited ourselves only to theoretical calcula- tions of the drag coefficients, and the expressions may be ACKNOWLEDGMENTS useful in the study of the passage of light and cosmic par- ticles in astrophysical media, since what we actually see One of us ͑K.K.N.͒ wishes to thank the Director, Profes- from the moving Earth is not what was originally sent from sor Ouyang Zhong Can, for providing hospitality and excel- the source. This work is underway. lent working conditions at ITP, CAS. The work was in part We saw that the present analysis naturally complements supported by TWAS-UNESCO of ICTP, Italy, and also by the curved space analogy of a moving medium. Some of the NNSFC under Grants No. 10175070 and 10047004, as well key expressions in the comoving frame are indeed the same. as by NKBRSF G19990754.
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