Spatial Doppler effect
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Relative motion between objects does not just affect the frequency of received light-signals (the usual Doppler effect on light) it also affects the apparent dimensions that differently-moving objects will see each other to have when using those light-signals. This has been referred to as the spatial analogue of the Doppler effect.
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[edit] Simple optical effects
[edit] Receding objects appear contracted
A photograph of an object moving directly away from an observer should show it to be compacted in its direction of motion. This is due to the different distances that the light signals have to travel from different parts of the object, the finite speed of light, and the distance that the object moves while these different signals are in flight
Illustration:
- A train 1 kilometre long is travelling at an appreciable fraction of the speed of light, and a trackside camera takes a photograph at the moment that the rear end of the train reaches the camera’s position. The resulting photograph will show the train’s rear to be at its “correct” position, but although the front of the train is known to be 1 km away at this moment, the light hitting the camera at that moment will show an older image of the front of the train, when it had not progressed quite so far along the track.
- The photograph will show the entire length of the train to occupy less than 1 km of track.
[edit] Approaching objects appear elongated
Similarly, differential signal time lags should make an object moving directly towards an observer appear to be elongated in its direction of motion.
Illustration:
- We repeat the earlier experiment, but instead photograph the train at the moment that the front of the train reaches the camera’s position. This time the front of the train is photographed with no timelag at its correct position alongside the camera, but the signal from the approaching rear of the train is out to date, and belongs to an earlier time when the train was further back along the track.
- The photograph shows the train to be elongated, and to be occupying more than 1km of track.
[edit] Strength of the effect
If we assumed that light propagates at fixed speed with respect to the observer, or propagates at fixed speed with respect to the emitter, we would expect to see the image of a receding object showing it to be contracted by the ratios
and
respectively (for approach velocities we can use negative velocity values).
These are the same ratios that we calculate for the apparent change in frequency of the object’s light (Doppler effect). This general correspondence between apparent change in depth and frequency also holds in special relativity.
[edit] Special relativity
Under special relativity, the apparent visible depth of a receding or approaching object again alters by the same ratio as the frequency of the object's lightsignals. In this case, the appropriate relationship is given by the special theory's relativistic Doppler equations, giving a prediction of
This relationship (see e.g.: Terrell 1959, Weinstein 1960, McGill 1968 ...) is the geometric mean of the two relationships mentioned in the last section.
[edit] Lorentz contraction
If we choose to interpret special relativity’s predictions by assuming a fixed lightspeed in the observer’s frame, we can rewrite the previous equation as
where the first part is the presumed propagation component, and the second part is the Lorentz-FitzGerald contraction effect.
[edit] History
The Lorentz contraction idea was popularised by Hendrik Antoon Lorentz, as a way of explaining how experiments to measure the Earth’s motion relative to an absolute aether had produced a “null” result. If the arms of a Michelson interferometer contracted with their motion through the aether by exactly this amount, the interferometer would not be able to identify the frame in which the speed of light was isotropic. Lorentz developed the idea into Lorentzian electrodynamics (1904), the forerunner to Einstein’s special theory of relativity (1905). Lorentz credited Fitzgerald as having suggested a similar ad hoc hypothesis some years earlier, and it is often referred to as Fitzgerald-Lorentz contraction.
[edit] Visibility of the effect
Early writings on special relativity did not always make it clear whether the Lorentz contraction effect was supposed to be a verifiable physical effect or something more interpretative, and as a result there was some confusion over whether a moving object should be seen to contract according to special relativity, as many descriptions seemed to suggest. Terrell and Penrose, working independently, decided that some researchers had been mistaken, and Terrell argued that an object passing the observer should be seen to be rotated rather than contracted (Terrell rotation). More recent texts tend to suggest that perhaps it may be more accurate to describe the appearance of passing object as being skewed rather than rotated.
[edit] Correct interpretation under special relativity
Objects can appear visibly longer or shorter overall depending on whether they are approaching or receding.
Special relativity predicts that the apparent depths of moving objects will be visibly shorter in their directions of motion than we might predict by just assuming a speed of light fixed in our own frame.
[edit] Gravity
Similar arguments may extend into the realm of gravitational effects: since a gravitational well’s internal light-distances are greater than its external dimensions (see: Shapiro effect), one could argue that when we see a gravitationally redshifted object, a projection of the object’s dimensions onto our own coordinate system should make the object appear contracted (and when we see a gravitationally blueshifted object it should appear elongated).
It is difficult to judge whether this should be classed as a “useful” visible effect since we cannot easily place a reference ruler alongside the object that is not inside the same gravitational field.
[edit] References and background reading
- Albert Einstein, "On the Electrodynamics of Moving Bodies" (1905),
- translated and reprinted in The Principle of Relativity (Dover, NY, 1952) pp.35-65.
- — gives the relativistic Doppler formula in a similar format to the third length change equation presented here
- James Terrell, "Invisibility of the Lorentz Contraction"
- Phys. Rev. 116 1041-1045 (1959).
- — says that Lorentz contraction under SR is not "visible": only an apparent Doppler expansion or contraction. This refers to the "relativistic Doppler" formula.
- Roger Penrose, "The Apparent Shape of a Relativistically Moving Sphere"
- Proc. Cambridge Phil. Soc. 55 137-139 (1959)
- V.F. Weisskopf, “The visual appearance of rapidly moving objects”
- Physics Today pp.24-27 (Sept 1960).
- Roy Weinstein, "Observation of Length by a Single Observer"
- Am.J.Phys. 28 607-610 (1960).
- — includes a useful graph of apparent length-change vs approach/recession velocity under special relativity
- Mary L. Boas, “Apparent shape of large objects at relativistic speeds,”
- Am. J. Phys. 29 283-286 (1961).
- Sten Yngström, “Observation of moving light-sources and objects,”
- Arkiv för Fysik 23 367-374 (1962).
- G.D. Scott and M.R. Viner, “The Geometrical Appearance of Large Objects Moving at Relativistic Speeds,”
- Am. J. Phys. 33 534-536 (1965).
- N. C. McGill, "The Apparent Shape of Rapidly Moving Objects in Special Relativity,"
- Contemp. Phys. 9 33-48 (1968).
- — presents the third relationship given here for apparent length-change, which corresponds to Einstein's 1905 "relativistic Doppler" equation for apparent frequency change, and refers to this as "the spatial analogue of the Doppler effect"
- James Terrell, "The Terrell Effect,"
- Am. J. Phys. 57 9-10 (1989)
- T.M. Kalotas and A.M. Lee, “A two-line derivation of the relativistic longitudinal Doppler formula”
- Am. J. Phys. 58 187-188 (1990)
- — presents the "geometric mean" relationship between special relativity and the two more basic propagation calculations for the Doppler effect.
- W. Mückenheim, "Is Lorentz Contraction Observable?"
- Annales de la Fondation Louis de Broglie 17 351-354 (1992)
