VII. Relativistic optics. Electromagnetic fields in inertial frames of reference. dt j ( ) ψ = 0. ri r j. Galilean transformation

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1 VII. Relatiisti optis eletromagneti fields in inertial frames of referene VII. Relatiisti optis Eletromagneti fields in inertial frames of referene Galilean transformation Before 1900 the spae and time oordinates (x, y, z, t) and (x, y, z, t ) of referene frames K and K moing with onstant speeds differing by were belieed to be related by Galilean transformation r = r t t = t (VII-1) proided the origins in spae and time are hosen suitably. Both the Newton equation of lassial mehanis and the Shrödinger equation of quantum mehanis an be shown to be inariant under Galilean transformation. For instane, if in K the Newton equations for a mehanial system onsisting of a group of partiles interating ia two-body potentials read as d ri m i = ( ) i Vij ri r j (VII-) dt j then in K the equation of motion has the same form d ri m i = ( ) i Vij ri rj (VII-3) dt j By ontrast, the equation goerning wae phenomena is not presered under the transformation (VII-1). For a field ψ(r,t ) satisfying the wae equation (VII-4) 1 ψ ψ = 0 t in K the Galilean transformation yields the equation 1 ψ ψ 1 ψ ( ψ ) = 0 (VII-5) t t in the referene frame K. For sound waes, the lak of inariane under transformation from one frame moing with a onstant eloity to another is aeptable gien the fat that propagation of these waes relies on a transmitting medium. The preferred referene frame K in whih (VII-4) is alid is obiously the frame in whih the propagation medium is at rest

2 VII. Relatiisti optis eletromagneti fields in inertial frames of referene Postulates of Einstein s speial theory of relatiity Analogously, a preferred referene frame for light propagation alls for a medium through whih light an propagate. This medium has been alled the ether and assumed to permeate all spae and to be of negligible density and to hae negligible interation with matter. Howeer, efforts to obsere the motion of laboratories on the Earth relatie to the rest frame of ether, for example, the famous Mihelson-Morley experiment, had failed. Fizeau s experiments on the eloity of light in moing fluids ould also be understood in terms of the ether hypothesis only by assuming that the ether was dragged along partially by the moing fluid, with the effetieness of the medium in dragging the ether related to its index of refration! These experiments made Einstein abandon the hypothesis of an ether. The only alternatie was to modify the transformation of the spae and time oordinates between uniformly moing referene frames so that Maxwell s equations beome inariant under the new transformation. This immediately implied that the laws of mehanis were in need of modifiations. To establish the new transformation, Einstein introdued two postulates. 1. Postulate of relatiity: The laws of nature and the results of all experiments performed in a uniformly moing frame of referene are independent of the translational motion of the system as a whole. These equialent oordinate systems are alled inertial referene frames.. Postulate of the onstany of the speed of light: The speed of light is onstant in eery frame of referene, independently of the motion of its soure. From these two postulates the rules of the new transformation of spae and time, named after Lorentz, an be deried in a straightforward manner 1. Before we address the deriation of the Lorentz transformation of oordinates, we summarize a few major experiments baking these postulates, performed long after Einstein proposed them. Experimental erifiation of Einstein s relatiity It is a general belief that the null result of the Mihelson-Morles experiment (1887) proided an unambiguous eidene for the seond postulate of the Speial Theory of Relatiity. This is not true. This experiment an also be explained without abandoning the onept of an ether by the hypothesis of the FitzGerald-Lorentz ontration. Compelling eidene ame long after the formulation of the theory by Einstein. One of the most beautiful lass of experiments draws on gamma-ray emission from nulei with an extremely narrow bandwidth resulting from the Mössbauer effet. Let s suppose that the inertial frames K and K are onneted with the Galilean oordinate transformation. The phase of a plane wae must be inariant quantity, the same in all oordinate frames: nr n φ=ω t =ω t r (VII-6) Expressing t and r in terms of t and r from (VII-1) leads to ωt (1 n ) ω n r =ω t ω n r (VII-7) 1 The general struture of the Lorentz transformation an be dedued from the first postulate alone, see e.g. N. D. Mermin, Relatiity without light, Am. J. Phys. 5, 119 (1984). A summary of aailable eidene is gien by Shankland et al., Re. Mod. Phys. 7, 167 (1955)

3 VII. Relatiisti optis eletromagneti fields in inertial frames of referene This equality must hold for alues of t and r. As a onsequene, the oeffiients of t, x, y, equal. From this requirement, we obtain the Doppler-shift formulas for Galilean relatiity: ω=ω (1 n ) = n z on both sides must be (VII-8a) (VII-8b) n = n (VII-8) The diretion of the wae etor (gien by n) appears to be inariant. Howeer, the diretion of energy flow hanges from one frame to the other. Suppose that inertial frame K is the preferential referene frame, in whih the ether is at rest, hene the light wae propagates at with both the wae etor and the diretion of energy flow (diretion of moement of the wae paket represented by the segments of a plane wae in Fig. VII-1) direted along n. Fig. VII-1 The diretion of energy flow is not parallel to n in K but parallel with the unit etor m = n n (VII-9) The experiments are performed in the laboratory, therefore it is expedient to be able to express the Doppler formulas (VII-8) in terms of m appropriate to the laboratory rather than n. To this end, we write n in terms of m. In the limit of 0 / << 1we an aording to Fig. VII- approximately write m 0 0 n 1 m + (VII-10) - 0 -

4 VII. Relatiisti optis eletromagneti fields in inertial frames of referene where 0 is the eloity of the laboratory relatie to the ether rest frame. Fig. VII- A plane wae haing a frequeny ω in the ether rest frame will hae a frequeny ω =ω 0 1 n 0 (VII-11a) in the laboratory frame and a frequeny ω=ω 1 1 n 1 (VII-11b) in a frame K1 moing with a eloity 1 = u1 + 0 relatie to the ether rest frame. The frequenies ω1 and ω0 are onneted by u ω ω m + (VII-1) 1 0 We obtained (VII-1) from (VII-11) by eliminating n, whih an not be measured, by means of (VII-10). Equation (VII-1) is a onsequene of the assumed alidity of the wae equation (VII-4) in the ether rest frame and that of Galilean oordinate transformation onneting inertial frames. Owing to the appearane of 0 it is obiously able to proe or disproe the existene of a preferred frame of referene and the alidity of Galilean relatiity. Differenes between ω1 and ω0 an be measured aurately if the soure and detetors hae a narrow bandwidth. These requirements an be best met by using two Mössbauer systems of idential ω0, one emitter and one absorber. Suppose they moe with u1 and u in the laboratory. Eq. (VII-1) then implies for the differene frequeny between the emitter and the absorber

5 VII. Relatiisti optis eletromagneti fields in inertial frames of referene ω ω = ( u1 u) m+ ω (VII-13) 0 In the first resonant absorption experiment of this type 3 the emitter and the absorber were loated on the opposite ends of a rod of length R rotated about its entre with an angular eloity Ω as depited in Fig. VII-3. For this speifi ase (u1 - u)m= 0 and the frational frequeny differene is predited as ω ω ω 1 0 = ΩR ( ) sinω t (VII-14) 0 Fig. VII-3 In the experiment of Champeney, Isaak and Khan the Mössbauer line was at a photon energy of 14.4 kev with a frational bandwidth of x10-1, the emitter and the absorber foils were separated by R = 8m and the highest rotation speed lose to 8000 s -1. At a rotational speed of ~6000 s -1 Eq. (VII-14) yields a differene frequeny equal to the Mössbauer line width for an ether drift eloity of ~00m/s. Champeney and o performed the measurement in 4-hour yles and searhed for different ether drift eloities relatie to their laboratory due to the earth s rotation. Their experiment yielded a maximum ether drift speed omponent orthogonal to the earth s axis of rotation of < 5 m/s. An improed experiment in 1970 set the limit of 5 m/s. Clearly, the existene of a preferential referene frame and with that the idea of the ether must be abandoned, the first postulate of the speial theory of relatiity has been onfirmed. The seond postulate, the onstany of the speed of light irrespetie of the motion of its soure ould be unquestionably erified also only many years after postulating this law, in a beautiful experiment 4 performed at CERN, Genea, in The speed of 6 GeV gamma-ray photons produed in the deay of energeti pions flying with a speed of = was measured by time of flight. Within the experimental error the speed of the photons emitted by the fast moing partiles was found to be equal to, written as = + k, the experiment yielded k = (0±1.3)x10-4. Other experiments 5 onfirmed this result and proided eidene for the onstany of the light speed oer its frequeny, up to photon energies of 7 GeV 6, establishing onlusiely the alidity of the seond postulate of speial relatiity. Clearly, the onstany of the speed of light, independently of the motion of the soure, destroys the onept of time as a uniersal ariable independent of the spatial oordinates. Experiments taught us that light propagates in the 3 D. C. Champeney, G. R. Isaak, A. M. Khan, Phys. Lett. 7, 41 (1963); G. R. Isaak, Phys. Bull. 1, 55 (1970). 4 T. Alager, J. M. Bailey, F. J. M. Farley, J. Kjellman, and I. Wallin, Phys. Lett. 1, 60 (1964). 5 G. R. Kalbfleish, N. Baggett, E. C. Fowler, Phys. Re. Lett. 43, 1361 (1979). 6 B. C. Brown et al. Phys. Re. Lett. 30, 763 (1973)

6 VII. Relatiisti optis eletromagneti fields in inertial frames of referene same way in any inertial frame, onsequently Maxwell s equations must be inariant under the transformation onneting two inertial frames. Galilean oordinate transformation must be modified to meet this requirement. Lorentz transformation deriation from the requirement of the relatiisti inariane of Maxwell s equations A simple Gedankenexperiment reeals that the onstany of the speed of light indeed requires us to abandon the onept of absolute time. Imagine a light lok made up of a photon bouning bak and forth between two parallel mirrors (Fig. VII-4a). The lok tiks eah time the photon ompletes a round-trip journey. Fig. VII-4a Let s now imagine that the lok is put on a spae shuttle and is moed with respet to us with a high, onstant speed. Whilst a passenger of this spae shuttle still sees the photon moing perpendiularly to the mirror surfaes in Fig. VII-4a, from our perspetie, the photon in the sliding lok must trael at an angle, on a diagonal path (Fig. VII-4b). Fig. VII-4b This path is learly longer and therefore beause the photon s speed is the same in both frames we find that, from our perspetie the moing lok tiks less frequently. From our perspetie the passage of time is slowed down in a frame moing with respet to us. The hange in the passage of time is learly inonsistent with Galilean relatiity and alls for a modified oordinate transformation. In order to find this new transformation, let us onsider two inertial frames of referene K (t, x, y, z) and K (t, x, y, z ). The oordinate axes in the two frames are parallel and oriented so that K is moing in the positie z diretion with speed as iewed from K (Fig. VII-5)

7 VII. Relatiisti optis eletromagneti fields in inertial frames of referene Fig. VII-5 We derie the proper oordinate transformation from the requirement of the inariane of Maxwell s equations under the new transformation. Sine there is no relatie motion along the x and y axes, we hae x = x and y = y. Consequently, we an fous on the transformation of the z and t oordinates. To this end, we onsider a plane eletromagneti wae propagating along the z diretion, with its eletri and magneti fields polarized along the x and y diretion, respetiely: 1 Er (,) t = exex ( z,) t = exe0 e.. 1 Br (,) t = eyby( z,) t = eyb0 e.. ikz ( ωt) + (VII-15a) ikz ( ωt) + (VII-15b) The fields must obey the first and seond Maxwell s equations (IV-1 and IV-) in both oordinate systems, hene E B x y By 1 E = ; = z t z t E B B 1 E z t z x y y x = ; = t x (VII-16,17) (VII-16,17 ) The onnetion between the oordinates z, t and z, t must be linear and express that the origin of K, whih is defined by z = 0 moes with along the z axis of K. These onditions are fulfilled by z =γ( z t) (VII-18a) t = at+ bz (VII-18b)

8 VII. Relatiisti optis eletromagneti fields in inertial frames of referene These relations are indeed linear. It is impliit in (VII-18) that the origins of the spatial oordinates in K and K are oinident at t = t = 0. (VII-18a) reeals that z = 0 for z = t, ensuring that the origin of K moes with with respet to K. Inerting (VII- 18) yields z a γ = ( z + t ) aγ+ bγ a (VII-19a) t γ b = t aγ+ bγ aγ+ bγ z (VII-19b) We now require in the spirit of the first postulate that the form of (VII-19) is idential to that of (VII-18) exept for a hange in sign in front of, refleting that from the perspetie of an obserer residing in K, K moes with in the opposite (i.e. negatie z ) diretion. This requirement implies a aγ+ bγ γ = 1 a By introduing the notation = γ a =γ γ 1 b = (VII-0) aγ+ bγ = 1 γ γ 1 α= γ (VII-1) the new oordinate transformation fulfilling the requirements of the first postulate takes the form z =γ( z t) z =γ ( z + t ) t =γ( t α z) t =γ ( t +αz ) (VII-, ) If we now wish to transform Maxwell s equations from K to K we will hae to make use of z z t t =γ ; = γ ; =γ ; = αγ z t t z (VII-3) whih results in

9 VII. Relatiisti optis eletromagneti fields in inertial frames of referene =γ αγ z z t = γ γ t t z (VII-4) Substituting the differential operations (VII-4) into (VII-16) and (VII-17) leads to ( γex γ By) = ( γby αγe ) z t γ 1 ( γby E ) ( x = γe x αγ z t x (VII-16 ) B y) (VII-17 ) These equations take the same form as the original equations in frame K if and only if the expressions in the parentheses an be identified as the transformed field quantities E =γ( E B ) = γ( E α B ) x x y x By =γ( By α Ex) = γ( By E ) x y (VII-5) whih ditates α= (VII-6) From the requirement of the inariane of Maxwell s equations under the new transformation, we hae now obtained the last unknown parameter of the transformation. Substituting (VII-6) into (VII-1) yields 1 1 γ= = ; β= 1 β 1 (VII-7) With this the Lorentz transformation takes the form

10 VII. Relatiisti optis eletromagneti fields in inertial frames of referene z =γ( z t) z=γ ( z + t ) t =γ( t z) t =γ ( t + z ) (VII-8) From our analysis we hae also obtained the transformation rules for the eletromagneti fields Ex =γ( Ex By) ; By =γ By E x (VII-9a) The other transerse omponents, E y and polarisation of the plane wae ansatz (VII-15): B x, an be obtained by applying the same proedure after hanging the Ey =γ ( Ey + Bx) ; By =γ( By E ) x (VII-9b) The transformation laws of the z-omponent of the fields will be deried later