RELATIVISTIC DOPPLER EFFECT AND VELOCITY TRANSFORMATIONS

Transcription

1 Fundamental Journal of Modern Physics ISSN: Vol. 11, Issue 1, 018, Pages 1-1 This paper is aailable online at Published online December 11, 017 RELATIVISTIC DOPPLER EFFECT AND VELOCITY TRANSFORMATIONS Département de Physique Nucléaire et Corpusculaire Uniersité de Genèe 4, quai Ernest-Ansermet CH-111 Genèe 4 Switzerland john.field@cern.ch Abstract The relatiistic Doppler effect for a train of discrete signals is analysed both for acoustical pulses and pulsed particle beams. The effects of space-time geometry and time dilation are used to derie elocity transformations and the corresponding Doppler frequency shifts. It is shown that the conentional relatiistic elocity transformation is incompatible with the experimentally erified relatiistic Doppler shift, and the correct, corresponding, elocity transformation is gien. Some preious treatments of the problem are critically discussed. The spacetime relatiistic Doppler effect is also compared and contrasted with relatiistic energy momentum transformations and those of the de Broglie waes of quantum mechanics. The relatiistic Doppler effect for acoustical signals with parallel motion of source and obserer was analysed by Bachman [1] and subsequently generalised to the case of arbitrary directions of source and Keywords and phrases: special relatiity, Doppler effect, relatie elocity transformations. Receied October 17, 017; Accepted Noember 7, Fundamental Research and Deelopment International

2 signal elocities by Frankl [] and Bachman [3]. A more recent, pedagogical, article [4] presented an alternatie deriation of the general result. The aim of the present article is to calculate the Doppler effect for different types of signals: acoustic disturbances, massie particles or photons, paying special attention to the role of elocity transformations in the calculations. There are two distinct physical effects, one purely classical, the other relatiistic, that contribute to the Doppler effect, i.e., to the frequency transformation formula. The first, classical, one is the effect of source or obserer motion on the signal separation. The second, relatiistic, one is the time dilation effect that modifies the rate of clocks in the proper frames of the source or obserer relatie to that of a clock at rest in the frame where the signal elocity is specified. Velocity transformations, like the Doppler effect, are also sensitie to both these effects. A space-time analysis for the case of parallel motion of source and obserer is first presented, before critically reiewing the approaches used in preious deriations. Finally, the space-time Doppler effect is compared and contrasted with relatiistic energy-momentum transformations, as well as those of the corresponding waelengths and frequencies introduced in the wae-mechanical formulation of quantum mechanics. A source of frequency ν of acoustic or luminous signals: S 1, S, S3,..., is at rest in the inertial frame S. The signals moe isotropically with speed u in this frame. At epoch t 0 as recorded by a clock at rest in S, the signal detector D, moing with speed in S, detects the signal S 1 (see Figure 1a). At the same instant, a clock at rest in the proper frame S of D records an epoch t 0. In Figure 1b is shown the configuration of signals and detector at the later epoch t at which the signal S 3 is detected. The relatie phase of the signals S 3 and S 1 is:

3 RELATIVISTIC DOPPLER EFFECT AND 3 φ π( N 1) 4π, where N 3 is the number of detected signals at S S epoch t, The corresponding configuration in S at epoch t is shown in Figure 1c. Since the number of detected signals 1 is the same in the frames S and S, the corresponding phase difference in S, φ, between S 1 and S 3 must also be 4π. It then follows that the following relation holds between the space-time parameters specified in Figure 1: φ 4π π( u ) t λ φ πut. λ (1) Re-arranging, the elocity transformation formula between the frames S and S is obtained: ( ) t λ u u. () t λ With time dilation (TD): t γ t ( γ 1 1 β ; β c, where c is the speed of light in free space) and length contraction (LC): λ γ λ, it is found that: u γ ( u ) (TD and LC). (3) This relation is incompatible with the relatiistic parallel elocity addition relation: u u. (4) u 1 (3) and (4) are equialent only if u so that u 0, in contradiction with the initial hypothesis that u and are independent parameters. As c 1 This may be indicated by a isual counter connected to D. Properly taking into account the releant light propagation delays, different obserers, with arbitrary motions relatie to S, will agree on the reading of this counter at any epoch.

4 4 (a) (b) Figure 1. Spatial geometry in the frames S, S : (a), S: (b) and S : (c) of detection of a train of pulses S 1, S, S3,... moing at speed u in the frame S, by a detector D in parallel motion at speed in the same frame. (a): configurations in S, S at epoch t 0, S1 is detected. (b): configuration in S at epoch t at which S 3 is detected. (c): configuration in S at the epoch of detection of S 3. (c)

5 RELATIVISTIC DOPPLER EFFECT AND 5 discussed elsewhere [5, 6, 7], the putatie length contraction effect of special relatiity is spurious, resulting from an inconsistent definition of the spatial coordinates of eents in the frame S. Thus the correct relatiistic elocity transformation formula, gien by setting λ λ is: u γ ( u ) (TD no LC). (5) This formula has preiously appeared [8] in the literature in the context of the theoretical analysis of Sagnac interferometers. It is compatible with (4) only if: u c 1 1 β β ~ ( << c). (6) The relatiistic Doppler shift formula is deried by substitution in () of the relations connecting elocity, waelength and frequency in the frames S and S : u λν, u λ ν : λ ν ( ) t λ γ ν λν ν γ ν 1. (7) t λ λ u Note the cancellation of λ that renders the Doppler shift formula insensitie to the existence ( λ γ λ ) or absence ( λ λ) of LC. Using instead the conentional elocity transformation formula (4) to obtain the frequency transformation gies, instead of the usual relation (7): λ ν λν u 1 c ν λν u λ 1 c (8) or ν ( 1 u) ν u 1 c ( λ λ), (9)

6 6 ν ( 1 u) ν ( λ γλ), γ u 1 c (10) As will be discussed below, the relatiistic Doppler shift formula (7) can be deried from the standard elocity transformation (4) only by introducing a hypothetical formula (neither λ to λ. λ λ nor λ γ λ ) relating Elementary considerations of space-time geometry readily generalise the frequency transformation formula (7) to the case of a moing source. Suppose a source of proper frequency ν at rest in the frame S moes with speed w in the frame S in the same direction as the detector. Due to time dilation the obsered frequency of the source in S is: f ν γ w. If N signals are transmitted during the time interal t in S, then the spatial separation, λ, of successie signals is: where ( u w) λ t, (11) N N f t ( ν γ ) t. (1) w The frequency, (1) as: ν u λ of the signal train in S is then gien by (11) and ν ν γ [ 1 ( w u) ]. (13) w Combining (13) with (7) gies the relatiistic Doppler shift formula for arbitrary signal elocity with both source and detector in parallel motion: ν γ ν γ w u. u w (14) Seeral preious deriations of (14) may be found in the literature. Those

7 RELATIVISTIC DOPPLER EFFECT AND 7 presented in Refs. [, 10] based on space-time geometry in the rest frame of the medium and time dilation are similar to that presented aboe, but without any explicit consideration of elocity transformations. In Ref. [10] it is claimed to derie (14) by making use of the conentional elocity addition formula (4). Simplifying to the case of stationary source and moing detector and adopting the notation of the present paper, it is assumed, in this deriation, that the signal separation in the frame S is gien by the formula λ u f u, γ ν (15) where the time dilation relation gies the frequency of a clock at f νγ rest in S as obsered in S. Combining (4) and (15), and introducing the signal separation λ in the frame S gien by λ νu, leads to the expression: λ λ 1 [ 1 ( u c )]. γ (16) This corresponds neither to the correct relatiistic relation: λ λ nor LC: λ γ λ. Assuming the applicability of the standard elocity transformation (4): u λν( 1 u) u λ ν 1 ( u c ) 1 ( u c ). (17) So, diiding this equation through by λ and making use of (16) gies: λ 1 u ν ν γν( 1 u) λ 1 ( u c ) (18) Eq. (15) is (6) of Ref. [10] with u r u, r, s 0.

8 8 in agreement with (7). The relation (15) introduced in Ref. [10], though alid in the Galilean limit where γ 1, u u and λ λ does not hold in the relatiistic case where γ 1. This wrong ansatz allows to derie the correct frequency transformation relation (7) from the elocity transformation (4). Howeer, as shown in Eqs. (9) and (10) aboe, the latter transformation does not gie (7) when the constraints of space-time geometry on signal propagation, separately in the frames S and S, are properly taken into account. The article [4] has deried the frequency transformation formulas (7) and (13) by assuming that the space-time coordinates ( x, t), ( x, t ), ( x, t ) appearing in the inariant phase relation: φ πν( t x u) φ πν ( t x u ) φ πν ( t x u ) (19) are related by the Lorentz transformation equations. Considering the frames S and S, these were assumed to be: x γ ( x + t ), (0) t γ ( t + x c ). (1) Using them to eliminate ( x, t) from the first member of (19) and equating coefficients of t or x, respectiely, yields the relations: ν γν( 1 u), () u ν γ [ 1 ( u c )] (3) u so that the frequency transformation (7) is recoered in (). Taking the ratio of these equations and rearranging gies the standard elocity transformation formula (4). Howeer, the space-time coordinates in (19) are those of a wae-front or signal pulse in motion in both frames S and S, whereas in the Lorentz transformations (0) and (1) they are instead

9 RELATIVISTIC DOPPLER EFFECT AND 9 those of a massie object at rest in the frame S with x 0 and equation of motion: x t in the frame S. This is a consequence of the ery definition of the frames S and S and the operational meaning of the coordinate symbols in the Lorentz transformations [5]. The assumption that the coordinates in (19), (0) and (1) are one and the same is a logically untenable assumption, inalidating the deriation of () and (3). The deriation of Doppler shift formulas presented in Ref. [4] assuming phase inariance and the applicability of the Lorentz transformations (0) and (1) is an acoustic generalisation of that presented for light waes in Einstein s original special relatiity paper. The latter contains the same logical flaw just described in the calculation in Ref. [4]. It is of interest to compare the analysis of the Doppler effect in terms of space-time geometry (Figure 1) and time dilation, as just presented, with one based on relatiistic kinematics and de Broglie waes, of waelength λ h p. The acoustic signal of speed u of the preious analysis is replaced either by a photon of speed c or massie particle of speed uˆ pc E where the relatiistic energy, E, and momentum, p, in the frame S, of a particle of rest mass m are defined as: E γuˆ mc, p γuˆ muˆ. (4) The putatie transformation laws of energy and momentum, between the frames S and S preiously introduced: [ E pˆ ], p γ [ p Eˆ c ] E γˆ ˆ (5) together with the definitions (4) gie the relatiistic elocity transformation equations: γ uˆ γˆ γuˆ ( 1 βˆ βuˆ ), (6) γ uˆ uˆ γˆ γuˆ ( uˆ ˆ ). (7)

10 10 The equation obtained by taking the ratio of (7) to (6): uˆ ˆ uˆ (8) 1 βˆ βuˆ is just the standard parallel elocity addition relation (4) 3. The difference between the elocity transformation formula (5), deried from considerations of space-time geometry, and (8), deried from relatiistic kinematics, implies that the elocity u in the configuration shown in Figure 1 and the elocity u ˆ deried from the energy and momentum of a particle according to Eqs. (5) cannot be the same physical quantity. This requires a reinterpretation of the kinematical transformation equations in (5). E and p cannot be the energy and momentum, as obsered in the frame S, of a particle with energy, momentum E, p in the frame S. If this were the case, then u in Figure 1 would be gien by (8) instead of (5) which, gien the geometry of Figure 1, is impossible. Also the Doppler frequency transformation would be gien by the Eq. (9) instead of the experimentally confirmed formula (7). The transformations (5) must be considered, not as a relation between quantities describing the same system, as obsered in two different inertial frames, but rather as a mapping in the same frame between two possible kinematical configurations of the same object. This is analogous to a three-space rotation of a ector, within a fixed coordinate system, which conseres its length. The corresponding consered quantity for the transformations (5) is the mass, m, of the object. The correctness of the space-time elocity transformation formula (5), which predicts c γ ( c ) instead of c c according to the kinematical formula (8) is ouchsafed by the existence of the Sagnac effect [11, 8]. This was confirmed for light signals near the surface of the 3 Eqs. (6), (7) and (8) are algebraically equialent. If any one of them is postulated, the other two may be deried by purely algebraic manipulation.

11 RELATIVISTIC DOPPLER EFFECT AND 11 Earth, where the elocity is gien by the rotation of the Earth, in the experiment of Michelson and Gale [1] in 195. The elocity-dependent de Broglie waelength, λ û, associated with the massie particle is: h h λ uˆ. (9) p γuˆ um ˆ Combining (9) with (7) gies the waelength ratio: λuˆ λuˆ γuˆ uˆ γuˆ uˆ γˆ 1 ( 1 ˆ uˆ ). (30) The frequency ν û associated with de Broglie waes is gien by a generalisation of the Planck-Einstein for photons E hν as: E γuˆ mc ν uˆ. (31) h h The waelength of (9) and the frequency of (31) correspond to a superluminal phase elocity : c uφ λuˆ νuˆ (3) uˆ while the transformation law of the de Broglie frequency is, using Eq. (6): ν uˆ γuˆ νuˆ γuˆ γˆ ( 1 βˆ βuˆ ). (33) This is consistent with the Doppler formula (7) only for the case of photons or highly relatiistic massie particles for which u ˆ c and β u ˆ 1. The general discrepancy between (7) and (33) for the de Broglie waes of a massie particle demonstrates the breakdown of any analogy between the latter and acoustic phenomena. The identity between the space-time Doppler shift formula (7) and Eq. (33) for the case u ˆ c-

12 1 essentially the energy transformation formula of (5) for a photon, a completely different physical problem - must be considered as fortuitous. Further discussion of the ontology of de Broglie waes and the related concept of wae-particle duality may be found in Refs. [13, 14]. References [1] R. A. Bachman, Am. J. Phys. 50(9) (198), 816. [] Frankl, Am. J. Phys. 5() (1984), 171. [3] R. A. Bachman, Am. J. Phys. 54(8) (1986), 717. [4] S. P. Drake and A. Puris, Am. J. Phys. 8(1) (014), 5. [5] J. H. Field, Fundamental J. Modern Physics () (011), 139; arxi pre-print: [6] J. H. Field, Fundamental J. Modern Physics 4(1-) (01), 1; arxi pre-print: [7] J. H. Field, Fundamental J. Modern Physics 6(1-) (013), 1; arxi pre-print: [8] E. J. Post, Re. Mod. Phys. 39() (1967), [9] Ref. [1], Section IV. [10] Ref. [1], Section III. [11] G. Sagnac, Compt. Rend. 157 (1913), ; [1] A. A. Michelson and H. G. Gale, Astrophys. J. 61 (195), [13] J. H. Field, Eur. J. Phys. 33 (011), 63-87; arxi pre-print: [14] J. H. Field, Eur. J. Phys. 34 (013), ; arxi pre-print: