Aberration of Light in a Uniformly Moving Optical Medium Aleksandar Gjurchinovskia) Department of Physics, Faculty of Natural Sciences and Mathematics, Sts

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Aberration of light in a uniformly moving optical medium Aleksandar Gjurchinovskia) Department of Physics, Faculty of Natural Sciences and Mathematics, Sts. Cyril and Methodius University, P.O. Box 162, 1000 Skopje, Macedonia ͑Received 7 August 2003; accepted 19 December 2003͒ The transverse Fresnel–Fizeau light drag in the presence of a nondispersive homogeneous optical medium in uniform rectilinear motion is explained by using a simple Huygens’ construction. As a consequence of the motion of the medium, the wavefront of every individual secondary wavelet is an ellipse partially dragged by the moving medium. The resulting formula agrees with the experiment by Jones in the early 1970s and with Fresnel’s formula for transverse light drag. © 2004 American Association of Physics Teachers. ͓DOI: 10.1119/1.1648683͔

I. INTRODUCTION tical black holes would be observable only if the speed of rotation of the medium exceeded the speed of light in the The first step toward a theoretical understanding of the actual medium. A possible way to overcome this difficulty is effects on light propagation in moving media was made by 1 to utilize a so-called ‘‘slow light’’ medium, in which the Fresnel in his pioneering work in 1818. By considering light speed of the light is drastically reduced to several meters per as a mechanical wave propagating through the luminiferous second.10 These ‘‘slow light’’ materials also can be used for ether, Fresnel predicted that the velocity of light in a moving increasing the light drag considerably when the medium is medium would depend on the angle between the light ray moving at constant velocity. Furthermore, under certain con- and the direction of motion of the medium. According to ditions, a vortex flow in a fluid can cause other less dramatic, Fresnel, in the presence of a dispersionless medium in uni- but equally interesting phenomena, such as an optical analog form rectilinear motion at constant speed u, light propagat- 11 ing parallel to the motion of the medium would have a ve- of the Aharonov–Bohm effect. locity component Ϯu(1Ϫ1/n2) in addition to its phase The purpose of this paper is to provide additional insight into the motion of light in a moving medium. Although we velocity c/n which it would have if the medium were at rest. will restrict our attention to optical medium in uniform rec- The sign of this additional component depends on whether the light and the medium are moving in the same or opposite tilinear motion, our analysis should be extendible to more directions. The formula for the light velocity was verified complicated situations, such as the case of a nonuniformly experimentally by Fizeau,2 and since then the effect is known moving ‘‘slow light’’ material. The last statement can be as the longitudinal Fresnel–Fizeau light drag. Later investi- clarified by the fact that the curved space–time that the light gations showed that if the dispersion of the medium is taken ‘‘feels’’ while propagating through a nonuniformly moving into account, an extra term also must be considered.3 medium can be considered as being locally flat, consisting of Almost a century later, Fresnel’s naive approach was re- a large number of different local inertial reference frames vised, and the resulting formula was confirmed in the frame- related to each other in a way determined by the curvature work of Einstein’s theory of special relativity4 and the mod- tensor.11 ern formulation of electrodynamics of moving media.5 The We will consider a hypothetical case of a nondispersive theory of electrodynamics of moving media has certain ad- material medium, which should work well if the dispersion vantages over a purely special relativistic approach. It ap- in the real material is so small that correcting terms due to pears to be an irreplaceable mathematical tool for describing dispersion cannot be detected by the measuring equipment. the advancement of the wavefront of the light beam through We will argue that Huygens’ construction can still be used as a uniformly moving boundary separating two media, each of a ray-tracing tool in a dispersionless optical medium in uni- which is in uniform rectilinear motion at a different speed in form rectilinear motion if it is applied to the distorted sec- ͑ ͒ a different direction see Fig. 1 . This example is one for ondary wavelets. We will investigate a special case of refrac- which the standard Lorentz transformation procedure is not tion of light, a situation for which the plane surface of the applicable, because there is no reference frame in which the moving medium, on which the light is incident, is parallel to boundary and the optical media divided by it are all at rest at the motion of the medium. The law of refraction obtained in the same time. In other words, there exists no reference this manner will be used for describing an experiment by frame in which Snell’s law of refraction takes place, so there 12,13 is nothing to be Lorentz transformed.6 Jones, in which a light-beam probe is allowed to pass through a rotating disk made of glass, parallel to its axis of Of particular recent theoretical and experimental interest is 2 light propagation through a nonuniformly moving medium.7 rotation. The setup is an analogue to that of Fizeau, but here In this case, the light actually ‘‘sees’’ the moving medium as the incident light enters the medium perpendicularly to the an effective gravitational field. This optical analog of a direction of its motion. curved space-time could provide a laboratory test bed for Our procedure will replace the sophisticated mathematical general relativity. One could simply create an optical black apparatus of the electrodynamics of moving media by the hole8 in an ordinary glass of rotating water, and then, for simpler problem of using ordinary plane geometry and el- example, investigate what would be the equivalent of Hawk- ementary analysis to analyze the propagation of light in a ing radiation.9 However, precise calculations showed that op- uniformly moving optical medium.

934 Am. J. Phys. 72 ͑7͒, July 2004 http://aapt.org/ajp © 2004 American Association of Physics Teachers 934 Fig. 1. The general case of light refraction between uniformly moving media. Observe that, in addition to the motion of the media, the boundary also Fig. 2. In the reference frame SЈ where the medium is at rest, the wavefront is in uniform rectilinear motion. of the elementary wavelet represents a circle with radius (c/n)tЈ.

II. HUYGENS’ CONSTRUCTION IN A MOVING of Eq. ͑2͒, we expect that the shape of the secondary wavelet MEDIUM will become substantially different if viewed from a frame of reference in which the medium is moving at constant To trace the path of an arbitrary light beam through a velocity.15 medium, a Huygens’ construction can be employed. It states The analytical description of the distorted elementary two- that every point that belongs to the primary wavefront at dimensional wavefront in the reference frame S in which the some time t0 serves as an elementary source of secondary wavelets which spread in all directions with the same fre- medium is moving at ﬁxed velocity u along the positive quency and velocity as the primary wavefront. The envelope direction of the X axis, can be done by applying the Lorentz of these wavelets is the wavefront of the light beam at a later transformations formulas, ϩ⌬ time t0 t. If the medium through which light propagates xϪut tϪ͑ux͒/c2 is made of an optically isotropic substance, then the light ray xЈϭ , yЈϭy, tЈϭ , ͑3͒ can be constructed as a line normal to every subsequent ͱ1Ϫ͑u/c͒2 ͱ1Ϫ͑u/c͒2 14 wavefront at all times. to the relation If the light propagates through a vacuum, the evolution of the wavefront of an arbitrary secondary wavelet is governed c 2 xЈ2ϩyЈ2ϭͩ ͪ tЈ2. ͑4͒ by n x2ϩy 2ϩz2ϭc2t2. ͑1͒ Equation ͑4͒ describes the evolution of the elementary light The wavefront of the elementary source is an expanding pulse observed in the reference frame SЈ where the medium sphere whose radius at time t is ct, where t is the time mea- is at rest ͑see Fig. 2͒. By making a transition from SЈ to S sured from the beginning of the emission of the wavelet, and using Eq. ͑3͒, we arrive at c is the speed of light in vacuum. If the propagation of light u2 1 ux 2 is observed from a frame of reference in which the optical ͑xϪut͒2ϩy 2ͩ 1Ϫ ͪ ϭ ͩ ctϪ ͪ . ͑5͒ c2 n2 c system is moving in a straight line with constant velocity, the shape of the wavelet remains unchanged because the equa- The wavefront of the wavelet as viewed from S is no longer tion of the elementary wavefront is invariant under Lorentz a circle, but an ellipse partially dragged by the moving me- transformations. dium ͑Fig. 3͒. As long as the speed of the medium u is However, a problem arises when we are dealing with light smaller than c/n, the wavefront will be incompletely propagating through a medium. In the presence of a homo- dragged and the origin of the elementary wavelet will remain geneous, isotropic, and nonconducting transparent medium inside the ellipse. Otherwise, if c/nϽuϽc, which is the case at rest, the evolution of the wavefront of the elementary of light propagation in a medium moving at superluminal source is described by velocity, the dragging is overwhelming and the ellipse no c 2 longer encloses the origin ͑Fig. 4͒. xЈ2ϩyЈ2ϩzЈ2ϭͩ ͪ tЈ2, ͑2͒ n By a careful examination of the plot shown in Fig. 3 of Eq. ͑5͒ for the case uϽc/n, we conclude that the speed of where n is the refractive index of the medium in its rest light in a uniformly moving optical medium depends on the frame. It can be easily demonstrated that Eq. ͑2͒ is not in- angle ␪ between the direction of propagation of the light and variant under Lorentz transformations due to the extra factor the velocity of the medium. In this way, a medium that is of n2 on the right-hand side. As a result of the noninvariance optically isotropic in its rest frame, possesses an optical an-

935 Am. J. Phys., Vol. 72, No. 7, July 2004 Aleksandar Gjurchinovski 935 Fig. 5. Huygens’ construction of the refracted wavefront when the medium on which the light is incident moves at constant velocity u to the right.

Fig. 3. As a consequence of the motion of the medium, the wavefront of the elementary wavelet is no longer a circle, but an ellipse partially dragged by homogeneous transparent media from the reference frame in the moving medium. The ﬁgure is a sketch of the situation when the speed which the entire system moves to the right at constant veloc- of the medium u is less than c/n. Note that the dragged ellipse possesses an ity u ͑Fig. 5͒. We take the medium in which the incident axial symmetry with respect to the X axis. wavefront propagates to be a vacuum, and the boundary between the vacuum and the material medium to coincide with isotropy when observed from the frame in which the medium the X axis. We denote by n the refractive index of the me- is in uniform rectilinear motion. The situation becomes more dium in its rest frame. The aim is to derive the formula that Ͻ Ͻ connects the angle of incidence ␣ to the angle of refraction interesting for the case c/n u c shown in Fig. 4, where ␤ the superluminal motion of the medium causes the existence . of a Mach cone, outside of which no light ever reaches. The wavefront AB of the incident plane-polarized light beam sweeps across the boundary starting at point A, caus- ing the atoms along the interface to radiate secondary wave- III. REFRACTION BETWEEN MOVING MEDIA lets. After the time t needed for the incident wavefront to The idea behind the analysis is to trace the advancement reach the ﬁnal point D, the wavefront of the secondary of the wavefront in a moving nondispersive medium by ap- wavelet originating from the elementary source at A is a plying the Huygens–Fresnel principle to the distorted sec- dragged ellipse whose shape is described by Eq. ͑5͒. The ondary wavelets. We will investigate the refraction of an ar- envelope of all the elementary wavelets whose sources lie bitrary light beam on a plane boundary separating two along AD is the distance CD, which is the wavefront of the

refracted light. The distance CD belongs to the line y t ϭ y t(x), which is a tangent line of all the elementary wave- fronts. Due to the apparent anisotropy of the moving medium, the line segments AC and CD are not orthogonal to each other, which is readily observable from Fig. 5. As a consequence, the ray and the normal to the wavefront are not identical in this case. The equation of the tangent line is dy y Ϫy ϭͩ ͪ ͑xϪx ͒. ͑6͒ t 0 dx 0 x0 ,y0 We have taken into account that the tangent line touches the elementary wavefront emanating from the initially disturbed point A at the point C (x0 ,y 0), which implies that the slope of the wavefront at the point C coincides with the slope of ϭ the tangent line y t y t(x). By taking the derivative with re- ͑ ͒ spect to x of Eq. 5 at the point (x0 ,y 0), we obtain 1 u2 utͩ 1Ϫ ͪ Ϫx ͩ 1Ϫ ͪ dy n2 0 n2c2 ͩ ͪ ϭ . ͑7͒ dx u2 Fig. 4. Illustration of the dragging effect when the optical medium is mov- x0 ,y0 y ͩ 1Ϫ ͪ ing at a superluminal velocity (c/nϽuϽc). 0 c2

936 Am. J. Phys., Vol. 72, No. 7, July 2004 Aleksandar Gjurchinovski 936 Then, the equation of the tangent line can be written as 1 u2 utͩ 1Ϫ ͪ Ϫx ͩ 1Ϫ ͪ n2 0 n2c2 y Ϫy ϭ ͑xϪx ͒. ͑8͒ t 0 u2 0 y ͩ 1Ϫ ͪ 0 c2 The tangent line intersects the interface at the point D (d,0). Therefore, the point (d,0) satisfies Eq. ͑8͒. Then, by setting ϭ ϭ ͑ ͒ x d and y t 0 in Eq. 8 , we have 1 u2 utͩ 1Ϫ ͪ Ϫx ͩ 1Ϫ ͪ n2 0 n2c2 Ϫy ϭ ͑dϪx ͒. ͑9͒ 0 u2 0 y ͩ 1Ϫ ͪ 0 c2 The point of tangency C belongs to the elementary wavefront emitted from A, which means that the coordinate point ͑ ͒ (x0 ,y 0) satisfies Eq. 5 , that is, 2 2 u 1 ux0 Fig. 6. Modified version of the famous Fizeau’s experiment. ͑x Ϫut͒2ϩy 2ͩ 1Ϫ ͪ ϭ ͩ ctϪ ͪ . ͑10͒ 0 0 c2 n2 c ͑ ͒ ͑ ͒ By solving Eqs. 9 and 10 for x0 and y 0 , we obtain If we let uӶc, so that u/cϷ0, we see that Eq. ͑15͒ reduces n2u2 ͑n2Ϫ1͒utdϩc2t2ͩ 1Ϫ ͪ to the usual Snell’s law of refraction c2 x ϭ , ͑11͒ sin ␣ϭn sin ␤. ͑16͒ 0 u2 n2dͩ 1Ϫ ͪ Ϫut͑n2Ϫ1͒ n2c2 It is possible to derive Eq. ͑15͒ by using Lorentz transformations ͑see the Appendix͒. and u2 ud 2 ctͱ1Ϫ ͱn2͑dϪut͒2Ϫͩ ctϪ ͪ c2 c IV. ABERRATION OF LIGHT IN A MOVING y ϭ . ͑12͒ 0 u2 MEDIUM 2 ͩ Ϫ ͪ Ϫ ͑ 2Ϫ ͒ n d 1 2 2 ut n 1 n c In the following we consider a modified version of Fizeau’s historic experiment.2 Unlike the original experi- Here we take only the positive solution in y 0 , which is the one that corresponds to the actual situation. It is evident from ment, we take the incident light to be perpendicular to the ␤ϭ ͑ ͒ direction of the motion of the fluid ͑see Fig. 6͒. As a result of Fig. 5 that tan x0 /y0 . By taking into account Eqs. 11 and ͑12͒, we have the dragging effect, the boundary rays of the incident wavefront will undergo a continual deflection toward the right as u d n2u2 the wave propagates through the moving water. ͑ 2Ϫ ͒ ϩͩ Ϫ ͪ n 1 1 2 By tracing the path of the light wave through the optical c ct c air–water–air assemblage, we find that the outgoing light tan ␤ϭ . d u 2 u d 2 u2 will emerge parallel to the incoming light and will be dis- ͱn2ͩ Ϫ ͪ Ϫͩ 1Ϫ ͪ ͱ1Ϫ placed to the right by the distance q. From Fig. 6, we have ct c c ct c2 ͑13͒ q tan ␤ϭ , ͑17͒ From the triangle ABD, we have D d 1 where D is the thickness of the layer. The refraction of the ϭ . ͑14͒ light beam from the layer is in agreement with Eq. ͑15͒ if we ct sin ␣ take the incident angle to be zero (␣ϭ0). Thus, We substitute Eq. ͑14͒ into Eq. ͑13͒ and find the law of refraction of the wave 1 u ͩ nϪ ͪ u n2u2 n c ͑n2Ϫ1͒ ϩͩ 1Ϫ ͪ sin ␣ tan ␤ϭ , ͑18͒ c c2 u2 u2 ␤ϭ ͱ Ϫ ͱ Ϫ tan . 1 2 2 1 2 u 2 u 2 u2 n c c ͱn2ͩ 1Ϫ sin ␣ ͪ Ϫͩ sin ␣Ϫ ͪ ͱ1Ϫ c c c2 where n is the refractive index of the fluid at rest. We sub- ͑15͒ stitute Eq. ͑17͒ into Eq. ͑18͒ to find

937 Am. J. Phys., Vol. 72, No. 7, July 2004 Aleksandar Gjurchinovski 937 Fig. 8. Refraction of light from a material medium observed in the reference frame SЈ where the medium is at rest.

Fig. 7. Simplified presentation of the experiment performed by Jones ͑Refs. ͒ modification is in perfect agreement with the experiment. 12 and 13 . The formula that replaces Eq. ͑20͒ for a dispersive material is17,18 Du 1 Du 1 ͩ Ϫ ͪ qϭ ͩ n Ϫ ͪ ͑21͒ n g c n c n p ϭ ͑ ͒ q . 19 to first order in u/c. Here, n is the phase refractive index of u2 u2 p ͱ Ϫ ͱ Ϫ the medium, which is a function of the frequency f of the 1 2 2 1 2 n c c light, and ng is its group refractive index related to n p by ϭ ϩ Ϸ For uӶc, we can expand the right-hand side of Eq. ͑19͒ in ng n p fdnp /df. For a low dispersive medium, ng n p ϭn, Eq. ͑21͒ reduces to Eq. ͑20͒. powers of u/c and neglect all the second and higher order ͑ ͒ terms to obtain The derivation of Eq. 21 by a procedure similar to the one used here is an open question, and we leave it as a Du 1 subject for future study. We expect that the method of analy- qϷ ͩ nϪ ͪ . ͑20͒ c n sis in this paper can be applied to more complicated configu- rations, such as the one shown in Fig. 1, where in addition to ͑ ͒ Equation 20 is identical to the Fresnel formula for the the motion of the media, the boundary between them also is transverse aberration of light.1 The result was experimentally 12,13 moving at constant velocity. We believe that the method also confirmed by Jones for white light passing through a could lead to a simpler and more intuitive approach to the rapidly rotating disk made of a low-dispersive glass, parallel problem of light propagation through a nonuniformly mov- to its axis of rotation ͑Fig. 7͒. In this case, q is the tangential ing material medium. displacement of the light beam due to the dragging effect caused by the motion of the disk, u is the tangential speed of the rotating disk in the region of passage, D is the thickness APPENDIX of the disk, and n is the refractive index of the glass. In reference frame SЈ where the whole system is at rest ͑Fig. 8͒, Snell’s law of refraction is V. CONCLUDING REMARKS sin ␣Јϭn sin ␤Ј, ͑A1͒ We have presented an alternative approach to the propagation of light in a nondispersive optical material in uniform and the velocity components of the propagating light beam rectilinear motion. We have limited our discussion to a spe- are cific geometry of light refraction, where we considered the vЈ ϭc sin ␣Ј, ͑A2͒ surface of the material, on which the light is incident, to be x1 parallel to the motion of the material. This sort of arrange- vЈ ϭc cos ␣Ј, ͑A3͒ ment gave us an opportunity to compare our derivation with y1 the results of the experiment by Jones to measure the trans- c Ј ϭ ␤Ј ͑ ͒ verse Fresnel–Fizeau light drag effect. Our conclusions are vx sin , A4 in agreement with the result of the experiment, and matches 2 n the one derived by Fresnel for transverse light drag. c Ј ϭ ␤Ј ͑ ͒ However, when the rotating disk used in the experiment is vy cos . A5 made of a highly dispersive glass material, a significant en- 2 n hancement of the drag effect was noticed.16 In this case, the In the reference frame S where the medium is in uniform value of the displacement q of the beam was found to be rectilinear motion to the right ͑Fig. 9͒, the components of the different from the one predicted by Eq. ͑20͒. It has been velocity of the light beam are shown17,18 that the original Fresnel formula is modified by v ϭc sin ␣, ͑A6͒ the presence of an additional term due to dispersion, and this x1

938 Am. J. Phys., Vol. 72, No. 7, July 2004 Aleksandar Gjurchinovski 938 vЈ ϩu x2 v ϭ , ͑A15͒ x2 uvЈ x2 1ϩ c2

u2 ͱ1Ϫ c2 ϭ Ј ͑ ͒ vy v . A16 2 y2 uvЈ x2 1ϩ c2

By eliminating v␤ from Eqs. ͑A8͒ and ͑A9͒, and using Eqs. ͑A15͒ and ͑A16͒, we have Fig. 9. Refraction observed in the reference frame S where the medium is in v vЈ ϩu uniform rectilinear motion. x2 x2 tan ␤ϭ ϭ . ͑A17͒ v 2 y2 u vЈ ͱ1Ϫ y2 c2 ϭ ␣ ͑ ͒ vy c cos , A7 1 Finally, we substitute Eqs. ͑A13͒ and ͑A14͒ into Eq. ͑A17͒ v ϭv␤ sin ␤, ͑A8͒ and obtain x2 u n2u2 v ϭv␤ cos ␤, ͑A9͒ 2 y2 ͑n Ϫ1͒ ϩͩ 1Ϫ ͪ sin ␣ c c2 where v␤ refers to the velocity of the light beam in the mov- tan ␤ϭ . ing medium in the direction determined by the angle ␤. The u 2 u 2 u2 ͱ 2ͩ Ϫ ␣ ͪ Ϫͩ ␣Ϫ ͪ ͱ Ϫ X components of the velocity of the incident light in S and n 1 sin sin 1 2 SЈ are connected to each other by c c c ͑A18͒ Ϫ vx u Ј ϭ 1 ͑ ͒ a͒ vx . A10 Electronic mail: [email protected] 1 uv 1 ͑ ͒ x1 A. Fresnel, Ann. Chim. Phys. 9,57–66 1818 . 1Ϫ 2 ͑ ͒ c2 H. Fizeau, Ann. Chim. Phys. 57, 385–404 1859 . 3I. Lerche, ‘‘The Fizeau effect: Theory, experiment, and Zeeman’s measure- If we use Eqs. ͑A2͒ and ͑A6͒, we have ments,’’ Am. J. Phys. 45, 1154–1163 ͑1977͒. 4A. Einstein, ‘‘Zur Elektrodynamik bewegter Ko¨rper,’’Ann. Phys. ͑Leipzig͒ u 17, 891–921 ͑1905͒; reprinted in Einstein’s Miraculous Year: Five Papers sin ␣Ϫ That Changed the Face of Physics, edited by John Stachel ͑Princeton U.P., c Princeton, 1998͒. sin ␣Јϭ . ͑A11͒ 5 u L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media 1Ϫ sin ␣ ͑Pergamon, Oxford, 1984͒; P. Penﬁeld and H. A. Haus, Electrodynamics of c Moving Media, Research Monograph No. 40 ͑MIT, Cambridge, MA, ͑ ͒ 1967͒. By applying Eq. A1 , we have 6See, for example, B. M. Bolotovskii and S. N. Stolyarov, ‘‘Wave reﬂection u from a moving mirror and related problems,’’ Usp. Fiz. Nauk 159,155– sin ␣Ϫ 180 ͑1989͒. 1 c 7U. Leonhardt and P. Piwnicki, ‘‘Optics of nonuniformly moving media,’’ sin ␤Јϭ . ͑A12͒ Phys. Rev. A 60, 4301–4312 ͑1999͒. n u 8 1Ϫ sin ␣ For a thorough mathematical discussion and description of optical black c holes, the reader is referred to V. A. De Lorenci, R. Klippert, and Yu. N. Obukhov, ‘‘Optical black holes in moving dielectrics,’’ Phys. Rev. D 68, Then, by using Eq. ͑A12͒, Eqs. ͑A4͒ and ͑A5͒ become 061502͑R͒-1–4 ͑2003͒; I. Brevik and G. Halnes, ‘‘Light rays at optical black holes in moving media,’’ ibid. 65, 024005-1–12 ͑2002͒. u 9 sin ␣Ϫ U. Leonhardt, ‘‘A laboratory analogue of the event horizon using slow c c light in an atomic medium,’’ Nature ͑London͒ 415, 406–409 ͑2002͒; Ј ϭ ͑ ͒ vx 2 , A13 ‘‘Theory of a slow light catastrophe’’ Phys. Rev. A 65, 043818-1–15 2 n u ͑2002͒. 1Ϫ sin ␣ 10 c L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, ‘‘Light speed reduction to 17 meters per second in an ultracold atomic gas,’’ Nature and ͑London͒ 397, 594–598 ͑1999͒. 11U. Leonhardt and P. Piwnicki, ‘‘Slow light in moving media,’’ J. Mod. Opt. c 1 u 2 u 2 48, 977–988 ͑2001͒. Ј ϭ ͱ 2ͩ Ϫ ␣ͪ Ϫͩ Ϫ ␣ͪ 12 v n 1 sin sin . R. V. Jones, ‘‘Aberration of light in a moving medium,’’ J. Phys. A 4, y2 n2 u c c 1Ϫ sin ␣ L1–L3 ͑1971͒. c 13R. V. Jones, ‘‘Fresnel aether drag in a transversely moving medium,’’ Proc. R. Soc. London, Ser. A 328, 337–352 ͑1972͒. ͑A14͒ 14However, when the medium is exhibiting optical anisotropy, the light ray is generally not perpendicular to the wavefront of the light. See, for ex- The components of the velocity of the refracted light beam in ample, M. Born and E. Wolf, Principles of Optics ͑Cambridge U.P., Cam- both reference frames are related by bridge, 1999͒, 7th ed.

939 Am. J. Phys., Vol. 72, No. 7, July 2004 Aleksandar Gjurchinovski 939 15G. P. Sastry and T. R. Ravury, ‘‘Modeling some two-dimensional relativ- 17M. A. Player, ‘‘Dispersion and the transverse aether drag,’’ Proc. R. Soc. istic phenomena using an educational interactive graphics software,’’ Am. London, Ser. A 345, 343–344 ͑1975͒. J. Phys. 58, 1066–1073 ͑1990͒. 18G. L. Rogers, ‘‘The presence of a dispersion term in the transverse Fresnel 16R. V. Jones, ‘‘Aether drag in a transversely moving medium,’’ Proc. R. aether drag experiment,’’ Proc. R. Soc. London, Ser. A 345, 345–349 Soc. London, Ser. A 345, 351–364 ͑1975͒. ͑1975͒.

VISUALIZING PHYSICS I found out that the main ability to have was visual, and also an almost tactile, way to imagine the physical situations, rather than a merely logical picture of the problems. The feeling for problems in physics is quite different from purely theoretical mathematical thinking. It is hard to describe the kind of imagination that enables one to guess at or gauge the behavior of physical phenomena. Very few mathematicians seem to possess it to any great degree. Johnny, ͓von Neumann͔ for example, did not have to any extent the intuitive common sense and ‘‘gut’’ feeling or penchant for guessing what happens in given physical situations. His memory was mainly auditory, rather than visual.

Stanislaw M. Ulam, Adventures of a Mathematician ͑Charles Scribner’s Sons, 1983͒. Reprinted in The World Treasury of Physics, Astronomy, and Mathematics ͑Little, Brown and Company, Boston, MA, 1991͒, p. 710.

940 Am. J. Phys., Vol. 72, No. 7, July 2004 Aleksandar Gjurchinovski 940