Wikipedia

User:Gregor4/sandbox

< User:Gregor4

Transverse Doppler effect

edit

Transverse Doppler effect

edit
 
Figure 5–3. Transverse Doppler effect: Variant scenarios

The transverse Doppler effect (TDE) is one of the novel predictions of special relativity. Assume that a source and a receiver are both approaching each other in uniform inertial motion along paths that do not collide.

At the beginning, when the observer approaches the light source,the observer sees a blueshift, and later, when the distance with the source increass, he sees a redshift. The transverse Doppler effect describes the situation when the light source and the observer close to each other. At the moment when the source is geometrically at its closest point to the observer, one may distinguish

  1. the light that arrives at the observer,
  2. the light that is emitted by the source, and
  3. the light that is at half distance between the source and observer.

The situation of case (1) is shown in Fig. 5-3(a) in the rest frame of the source. The frequency observed by the observer is blueshifted by the factor γ because of the time delation of the observer (as compared with the rest frame of the source). The dotted blue image of the source shown in the figure represents how the observer sees the source in his own rest frame.


This light was emitted earlier when the source was in front (on the right side) of the observer (as indicated in the figure).


The situation of case (2) is shown in Fig. 5-3(b) in the rest frame of the observer. This light is received later when the source is not any more at closest distance, but it appears to the receiver to be at closest distance. The observed frequency of this light is redshifted by the factor γ because of the time delation of the source (as compared with the rest frame of the observer). This situation was Einstein's original statement of the TDE [1]

In the situation of case (3), the light will be received by the observer without any frequency change.

Whether an experiment reports the TDE as being a redshift or blueshift depends on how the experiment is set up. Consider, for example, the various Mössbauer rotor experiments performed in the 1960s.[2][3][4] Some were performed with a rotating source while others were performed with a rotating receiver, as in Fig 5‑3(c) and (d). Fig 5‑3(c) and (b) are corresponding scenarios, as are Fig 5‑3(d) and (a).



Einstein's original statement of the TDE described the shift seen when the receiver is pointed to where the image of the source appears at its nearest point, as in Fig 5‑3a.[1] In this case, the actual position of the source has moved beyond the position of its received image. The source's clock is time dilated as measured in the frame of the receiver. When the source was at the position of its received image, it had no longitudinal velocity with respect to the receiver. Therefore, the receiver observes light from the source as being redshifted by the Lorentz factor.

The fact that the orbiting objects are in accelerated motion does not preclude relativistic analysis, since an inertial frame, called the momentarily comoving reference frame (MCRF) can always be found which is momentarily comoving with the accelerating particle. Only a clock's instantaneous speed is important when computing time dilation.[5] When analyzed using MCRFs, it is evident that Fig 5‑3c and Fig 5‑3a are precisely equivalent scenarios, as are Fig 5‑3d and Fig 5‑3b.



Classically, one might expect that if source and observer are moving transversely with respect to each other with no longitudinal component to their relative motions, that there should be no Doppler shift in the light arriving at the receiver. On the other hand, considering Fig. 5-3, it is clear that there is a blueshift for the received light at the beginning when the observer approaches the light source, and a redshift later when the observer reaches the right part of the figure and increases its distance from the source. Therefore there must be a moment while the observer passes the source when there is no frequency shift for the received light.

In order to consider the situation in more detail, Fig. 5-3 shows the light source and the observer when they are at closest distance. We can consider the Doppler effect for the light arriving at the observer at three different instants, as explained in more detail in Relativistic_Doppler_effect#Transverse_Doppler_effect:

  1. The light arriving at the observer at the moment of closest distance: This light is blueshifted by a factor γ since this light was emitted earlier when the source was in front of the observer (as indicated in the figure). The distance between the points of emission and observation decreases.
  2. The light which is emitted by the source when source and observer are at closest distance: This light is redshifted by a factor γ since this light is observed later (as indicated in the figure) when the source appears to be at closest distance to the observer. The distance between the points of emission and observation increases.
  3. The light which is at half distance between the source and observer when they are at closest distance: This light is received without Doppler shift.



 
Figure 5–3. Transverse Doppler effect: Diagram showing light source and observer located at closest distance.

The transverse Doppler effect is one of the predictions of the special theory of relativity.

Classically, one might expect that if source and observer are moving transversely with respect to each other with no longitudinal component to their relative motions, that there should be no Doppler shift in the light arriving at the receiver. On the other hand, considering Fig. 5-3, it is clear that there is a blueshift for the received light at the beginning when the observer approaches the light source, and a redshift later when the observer reaches the right part of the figure and increases its distance from the source. Therefore there must be a moment while the observer passes the source when there is no frequency shift for the received light.

In order to consider the situation in more detail, Fig. 5-3 shows the light source and the observer when they are at closest distance. We can consider the Doppler effect for the light arriving at the observer at three different instants, as explained in more detail in Relativistic_Doppler_effect#Transverse_Doppler_effect:

  1. The light arriving at the observer at the moment of closest distance: This light is blueshifted by a factor γ since this light was emitted earlier when the source was in front of the observer (as indicated in the figure). The distance between the points of emission and observation decreases.
  2. The light which is emitted by the source when source and observer are at closest distance: This light is redshifted by a factor γ since this light is observed later (as indicated in the figure) when the source appears to be at closest distance to the observer. The distance between the points of emission and observation increases.
  3. The light which is at half distance between the source and observer when they are at closest distance: This light is received without Doppler shift.

Gregor4 (talk) 02:42, 9 November 2021 (UTC)


In the source frame, the light travels vertically from the source to the observer (the observer appears to be closest to the source), and

  1. In the observer frame, the light travels vertically from the source to the observer (the source appears to be closest to the observer).

Given that the momentum 4-vector of the light photons transforms in the Lorentz transformations like the space-time 4-vectors, one finds in situation (1) that the momentum 4-vector of the photos in the frame of the observer (a) has an energy component multiplied by the Lorentz factor γ, and (b) a momentum component in the opposite direction of the moving observer equal to γ v/c m (where m is the 3-momentum of the photon), which means that (a) there is a blueshift by a factor γ and (b) for the observer, the source appears to be located at the blue location shown in Fig. 5-3a.

In the situation (2), the energy component of the momentum 4-vector of the light photons emitted by the source are multiplied by the Lorentz factor γ, which means that the received light has a redshift of a factor γ. The observer, as seen by the source in its own frame appears to be located at the red spot in Fig. 5-3a. In the situation where the light travels vertically from the source to the observer in a frame that travels with speed v/2 in respect to the source, there is no frequency shift.

The explanation using time dilation arguments [6] as shown in Figs. 5-3 (b) and (c) is less convincing since (a) the Special Relativity does not deal with circular movements and (b) there is no way to distinguish between the scenarios of the figures (a) and (b). In Fig. 5-3b, the receiver observes light from the source as being blueshifted by a factor of γ. In Fig. 5-3c, the light is redshifted by the same factor.

  1. ^ a b Morin, David (2008). "Chapter 11: Relativity (Kinematics)" (PDF). Introduction to Classical Mechanics: With Problems and Solutions. Cambridge University Press. pp. 539–543. ISBN 978-1-139-46837-4. Archived from the original (PDF) on 4 April 2018.
  2. ^ Hay, H. J.; Schiffer, J. P.; Cranshaw, T. E.; Egelstaff, P. A. (1960). "Measurement of the Red Shift in an Accelerated System Using the Mössbauer Effect in 57Fe". Physical Review Letters. 4 (4): 165–166. Bibcode:1960PhRvL...4..165H. doi:10.1103/PhysRevLett.4.165.
  3. ^ Champeney, D. C.; Isaak, G. R.; Khan, A. M. (1965). "A time dilatation experiment based on the Mössbauer effect". Proceedings of the Physical Society. 85 (3): 583–593. Bibcode:1965PPS....85..583C. doi:10.1088/0370-1328/85/3/317.
  4. ^ Kündig, Walter (1963). "Measurement of the Transverse Doppler Effect in an Accelerated System". Physical Review. 129 (6): 2371–2375. Bibcode:1963PhRv..129.2371K. doi:10.1103/PhysRev.129.2371.
  5. ^ Misner, C. W., Thorne, K. S., and Wheeler, J. A (1973). Gravitation. Freeman. p. 163. ISBN 978-0716703440.{{cite book}}: CS1 maint: multiple names: authors list (link)
  6. ^ Cite error: The named reference Morin2007 was invoked but never defined (see the help page).
Retrieved from "https://en.wikipedia.org/w/index.php?title=User:Gregor4/sandbox&oldid=1056460866"