Abstract. Introduction. The formulas of the Doppler effect

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1 1 Investigations on the Theory of the Transverse Doppler Effect Xing-Bin Huang College of Physics Science and Technology, Heilongjiang University, Harbin, Heilongjiang, China Abstract A widely accepted viewpoint asserts that the transverse Doppler effect for a plane electromagnetic wave is a red shift which is purely a relativistic effect caused by time dilation or slowing down of moving clocks. One bug of this assertion, however, is that little literature has proved its correctness in a rigorous way. This paper shows a further study in this assertion and finds that this seemingly authentic point of view is highly questionable because this red shift is not purely a relativistic effect, and the genuine transverse Doppler effect for a plane wave is a blue shift that means that a moving clock can also run faster than an identical clock at rest. This blue shift is in contradiction with time dilation, and might reflect the new features of special relativity. Introduction Shortly after the publication of Einstein s famous paper of On the Electrodynamics of Moving Bodies [1], Einstein proposed to experimentally detect time dilation by measuring the transverse Doppler shift of the light waves emitted from moving particles, and predicted the apparent frequency would be shifted to the red end [2]. From that time on, a widely accepted viewpoint is that the transverse Doppler red shift for a plane electromagnetic wave is an inevitable consequence directly caused by time dilation or the slowing down of moving clocks. Thus, the transverse Doppler red shift has to be purely a relativistic effect which means that the classical Doppler effect gives no such frequency shift when this transverse Doppler red shift occurs [3 8]. Up to now, various experiments for the transverse Doppler effects have been carried out, and these experiments have become one of the strongest proofs for supporting special relativity [9 15]. However, the author could not find out neither in published papers, nor in books on special relativity a rigorously theoretical proof which shows that the transverse Doppler red shift for a plane electromagnetic wave is purely a relativistic effect, and, as was unexpectedly found, this so-called red shift is not purely a relativistic effect, but is a combined effect that the genuine transverse Doppler effect for a plane wave which is a second-order blue shift is offset by a first-order red shift caused by the classical Doppler effect. This paper will first recall the formulas of the relativistic and classical Doppler Effect; secondly, the theories of the transverse Doppler effects will be investigated, our results show that, contrary to the accepted opinion, the genuine transverse Doppler effect for a plane wave is a blue shift that reveals that a moving clock can also run faster than an identical clock at rest; finally, the reasons which cause the blue shift will be discussed, and the main results and our own conclusions also be summarized. The formulas of the Doppler effect In 1905 [1], Einstein was the first person to have established the formulas of the relativistic Doppler shift and of the aberration of light by employing the model of a uniform monochromatic plane electromagnetic wave with an arbitrary direction of propagation. After that, some other rigorous derivations are now available in the literature on special relativity. Because these latter derivations and Einstein s derivation used the same model, their results are the same [3 8]. In order to recall the formulas of the relativistic Doppler effect and avoid ambiguities, we assume that the symbols are written in the same form.

2 2 Let us consider K and K are two inertial reference frames, their coordinate axes are parallel to each other, and K moves with a constant velocity u relative to K in the positive direction of the x-axis, as viewed from K. For simplicity, let their origins O and O coincide at the initial time t = t = 0. In [1], Einstein assumed that in K, very far from O, let there be a source of light wave, which in a part of space containing O may be represented to a sufficient degree of approximation by the equations..., next, Einstein employed the phase invariance of a plane wave in the two inertial reference frames to obtain ν = ν γ(1 βcosα), (1) cos α = cosα β 1 βcosα. (2) Equations (1) and (2) are the formulas of the relativistic Doppler shift and of the aberration. Where u = cβ, γ = (1 β 2 ) 1/2 and c is the velocity of light in vacuum. The inverse transformation, from ν and α to ν and α, can be easily obtained from equations (1) and (2) by straightforward calculation. According to Einstein s statement, if an observer is moving with velocity u relative to an infinitely distant light source of frequency ν, in which way the connecting straight line source-observer makes the angle α with the velocity of the observer...the frequency ν of the light perceived by the observer is given by equation (1). The angle α is the angle between the wave-normal (the direction of the ray) in the moving system and the connecting line source-observer [1]. Despite Einstein s description about α is very uncertain, according to the full text of his article, we can easily conclude that α has to be the angle between the wave-normal (i.e. the direction of the wave vector of the plane wave in K ), as observed in K, and the direction of the velocity u (i.e. the positive direction of the x -axis). Similarly, α is the angle between the direction of the wave vector of the plane wave, as observed in K, and the positive direction of the x-axis. Ultimately, we should also particularly pay attention to the three directions employed by Einstein, i.e. the direction of the ray, the direction of the wave-normal, and the direction of the connecting line source-observer, the three directions and the direction of the wave vector of the plane wave are the same direction. Because Einstein assumed the light source to be infinitely distant from the observer, the light wave perceived by the observer is the plane wave. Thus, the α does not change with time despite the source and the observers either moving relative to each other. Furthermore, if a source is not very far from an observer, then the light waves perceived by the observer are spherical waves. But some authors employed the model of point-like light pulses [16] or spherical waves [17] to derive the formulas of the relativistic Doppler effect. In order to obtain the formulas (1) and (2), they have to use some additional approximation. Therefore, their derivations are not strictly valid or even wrong [18]. The other two innovative derivations based on single photon propagating are also noteworthy [18, 19]. On the surface, these derivations do not require the condition that the point light source must be far away from the point observer, and also obtain their expressions which are the same with Einstein s formulas (1) and (2) in form. However, in their expressions, the α is the angle between the propagating direction of a photon and the positive direction of the x -axis, thus, their α does not change over time. In short, no matter what kind of derivation is employed, we have to meet the model of a plane wave when we use expressions (1) and (2) to study on the transverse Doppler effect. In order to analyse the transverse Doppler effect, we recall the classical or nonrelativistic Doppler effect. According to the model of a plane wave assumed by Einstein, we use the Galilean transformation and the phase invariance for a plane wave among all inertial reference frames to strictly obtain the formulas of the classical or nonrelativistic Doppler effect for a plane wave given by [8] ν = ν(1 βcosα), (3) α = α, (4)

3 3 c = c ucosα. (5) Here c is the speed of light generated by the moving light source, as viewed from K, α and α are the angles between the positive direction of the common x-x axes and the directions of the wave vectors of a plane wave in K and K, respectively. From equation (4), we can obtain an important conclusion, that is, α must be an invariant, the same in all inertial reference frames [8]. Finally, in order to investigate the theory of the transverse Doppler effect, we should emphasize the special conditions that the classical Doppler effect for a plane wave vanishes. Obviously, from equations (3) and (4), only when both α and α are right angles, the frequency ν perceived by an observer and the proper frequency ν of a light source are the same in spite of being in relative motion between them. The theory of the transverse Doppler effect As you known, the classical or the nonrelativistic Doppler shift have to occur, only when the relative motion between a point observer and a point source has caused changes in the distance between them. In fact, the reason caused the classical Doppler shift is that each consecutive wave crest must spend slightly different time intervals from the source to reach the observer than the preceding crest due to the change of the distance between them. In other words, a moving light source causes a red shift for having receded or a blue shift for having approached. However, in some special cases, although there is in relative motion between the light source and the observer, the distance between them does not change with time, so that the classical Doppler effect, i.e. the geometric longitudinal or first order Doppler effect, vanishes. Thus, the classical Doppler shift is zero, but the relativistic Doppler shift is not zero. Therefore, when the classical Doppler shift is zero, a purely relativistic effect should alone remain [18, 19], and the remained effect was generally known as the transverse Doppler effect which is a second order effect. To obtain the transverse Doppler effect, the key point is to find out the condition that does not cause the classical Doppler shift. Next, we consider two special cases for spherical waves and a plane wave. For spherical waves: If a point light source is placed at the center of a rotating disk and a point observer at the rim of the disk, then the distance between them does not change. This configuration thus yields purely a relativistic effect which is a blue shift caused by clock paradox that can be deduced either by special relativity [20] or by the general relativity [21]. The blue transverse Doppler effect had been verified experimentally by Hay et al. [10] and Kündig [11] based on the Mössbauer effect. Conversely, if the observer is placed at the center of the rotating disk and the light source is placed at the rim of the disk, which is a no inertial reference frame, we would obtain the transverse Doppler red shift, i.e. the red transverse Doppler effect. Therefore, there are two transverse Doppler effect for spherical waves: when the light source is at rest with respect to the accelerating system, the transverse Doppler effect for spherical waves is red shift; when the observer is at rest with respect to the accelerating system, the transverse Doppler effect for spherical waves is blue shift. However, we should notice that both the blue and red transverse Doppler effects for the spherical waves cannot be interpreted by using expressions (1) and (2), because expressions (1) and (2) are established by employing a plane wave. In other words, for a plane wave, there is only a transverse Doppler effect that is purely a relativistic effect, i.e. it is one of both the blue and red transverse Doppler effect for plane waves. Next, we will focus on the case of a plane electromagnetic wave. For a plane wave: In order to let an observer receive a plane wave emitted by a point light source, we have to suppose that the point light source is infinite away from the observer. Thus, the formulas (3) and (4) are rigorously valid, so that we can obtain the very important condition for the classical Doppler zero shift, i.e. ν = ν, given by α = α = π/2. (6)

4 4 From equation (6), we can calculate a purely relativistic Doppler effect for a plane wave which should be the transverse Doppler effect for a plane wave. However, we have to notice that equation (6) is only the classical conditions that the classical Doppler effect vanishes. In other words, we thus have to determine α and α of formulas of the relativistic Doppler effect to calculate the purely relativistic Doppler effect, because they are generally not the same, according to equation (2). In our assuming, because the light source is at rest in K, α is the proper angle generated by the stationary light source and α is the apparent angle generated by the moving light source. Thus, to measure α of the stationary light source, an observer has to be at rest with respect to the light source so that his measured result does not depend on the theory of relativistic or classical. Thus, α angle of formula (3) of the classical Doppler effect and α angle of formula (1) of the relativistic Doppler effect are the same on the same light source, because the source is at rest in K. Therefore, α = π/2 of equation (6) is the condition shared by the formulas (1) and (3) to deal with the same source. If substituting α = π/2 into equation (2), then cosα = β. Hence, α = π/2 and cosα = β are the relativistic conditions that the classical Doppler effect vanishes. In other words, to meet the classical Doppler zero shift, α = π/2 and cosα = β must be satisfied, as viewed from theory of special relativity. We should also stress that α is not the condition shared by formulas (1) and (3) to deal with the same light source due to α generated by the moving source. Now, we can employ formula (1) and α = π/2 to calculate the purely relativistic effect which is named as the transverse Doppler effect, because the classical Doppler shift is zero. Putting α = π/2 into equation (1), we can obtain the genuine transverse Doppler shift for a plane wave, that is, ν = γν. (7) So far, we have shown that equation (7) is the transverse Doppler effect for a plane wave which is a blue shift, and the blue shift is not compatible with time dilation. However, from the widely accepted viewpoint, the transverse Doppler effect for a plane wave is a red shift which is also purely a relativistic effect caused by time dilation or the slowing down of moving clocks [3 7]. Strangely enough, a strictly theoretical proof for this red shift is absent. In fact, we can prove that this widely accepted viewpoint is incorrect. To obtain the red transverse Doppler Effect for a plane wave, in general, let α = π/2, which is equal to letting cosα = β, from relativistic formula (2), then substitute cosα = β into formula (1), we obtain so-called the transverse Doppler red shift, that is, ν = ν 1 β 2. (8) However, equation (8) is not purely a relativistic effect, because the classical Doppler effect does not vanish. According to the foregoing analysis, because the light source is at rest in K, α = π/2 is not the condition shared by relativistic formula (1) and classical formula (3) to deal with the same light source. Thus, we cannot directly employ α = π/2 and formulas (3) and (4) to calculate the classical Doppler shift for the same light source. Therefore, the classical Doppler shift does not vanish when equation (8) is named as the transverse Doppler effect. Next, let us employ the length contraction as an example to illustrate further this problem. Consider the proper length of a moving rod which is at rest in K is 2, and its apparent length is 1 due to the length contraction in terms of special relativity. Our question is that in the classical case, what are its proper and apparent lengths, respectively? The answer is obviously 2, of course, not 1. Similarly, when we analyse the two angles for the above relativistic red transverse Doppler effect, cosα = β, which is observed in K, is the proper angle because the light source is at rest in K, and α = π/2, which is observed in K, is the apparent angle because the light source is in motion with respect to K. Thus, in the classical case, this proper and apparent both angles have to be the same with the relativistic proper angle, namely, the classical angles are that cosα = cosα = β when the relativistic proper angle is cosα = β

5 5 and apparent angle is α = π/2. Therefore, the classical Doppler effect does not vanish for the widely accepted red transverse Doppler effect. In other words, if the known light source of a plane wave in K is moving with respect to observers, then when the relativistic observer concludes α = π/2 (i.e. cosα = β), the classical observer has to conclude cosα = cosα = β. Thus, in this case, we cannot employ α = π/2 to calculate directly classical Doppler shift. We have to employ cosα = β and classical formulas (3) and (4) to calculate the classical Doppler shift. We thus have ν = ν(1 β 2 ). (9) Equation (9) is a red shift caused by the classical Doppler effect. Therefore, our an important conclusion is that equation (8) is not purely a relativistic effect, and the red shift of equation (8) is the combined effect that the blue shift of equation (7) is overcome first order red shift of equation (9). In short, we have shown that the transverse Doppler effect for a plane wave is blue shift, not so-called red shift, and only the blue shift is purely a relativistic effect. The transverse Doppler blue shift for a plane wave means that a moving clock can also run faster than an identical clock at rest. Thus, this blue shift is in contradiction with time dilation. Results and Discussion It is well known that time dilation is that a point clock moving past the identical two synchronized point clocks at speed u runs more slowly than the two clocks by a factor of 1/γ. In other words, the moving point clock runs the time interval, i.e. proper time, which occurs at the same point in space, but the two synchronized point clocks to measure this proper time are at two different points in space [22]. However, the transverse Doppler effect for a plane wave means that we use a point clock (i.e. a point observer) to measure the proper time of another identical moving point clock (i.e. a point light source). In fact, if a moving point light source is placed infinitely away from a point observer and the classical Doppler effect vanishes, then the time intervals of two arbitrary consecutive light wave crests emitted from the point light source to the point observer have to be the same. Thus, the transverse Doppler effect is comparing the two proper times of a moving point clock and a point clock at rest. Therefore, we have no reason to expect that the transverse Doppler effect for a plane wave have to be equivalent to time dilation. Consider a plane electromagnetic wave is propagating along the positive direction of the y-axis emitted by a point light source at rest in K, as viewed from K. According to Huygens principle, every point of a wave front passing the plane (x, z) at any instant t can be viewed the source of secondary wavelets. Thus, we can imagine that the plane (x, z) is equivalent to a planar light source which emits the plane wave with the proper period T, and the plane (x, z) is thus equivalent to a moving infinite planar light clock, as seen from K. Therefore, when a point clock placed on O measures the proper period T, the result has to satisfy that a moving infinite clock can also run faster than an identical clock at rest, according to Lorentz time transformation [23]. Contrary to the widely accepted viewpoint, the transverse Doppler effect for a plane wave is blue shift which is the only purely relativistic effect. This blue shift raised herein should be a very important conclusion directly caused by special relativity, which means that a moving clock runs slow or fast depends on the method measured theoretically. This blue shift seems to reveal a new feature or a logical question of special relativity, and may inspire some physicists to research the new physics hidden by special relativity.

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