Doppler length change
Doppler length change
The existence of a visual, photographable Doppler-equivalent length-change effect in moving bodies is trivially simple to calculate, but was not fully appreciated and accepted into the literature on special relativity until after around 1959, when the publication of separate papers by James Terrell and Roger Penrose triggered a wave of followup papers in the American Journal of Physics, exploring how the effect manifested itself in the special theory. The effect is not limited to special relativity.
A general statement of the Doppler length-change effect is that an object that is seen to have its frequency changed by the ratio 1:x will also have its visual, photographable length changed by exactly the same ratio, 1:x. visible frequency change ratio = visible length change ratio
This ratio equivalence seems to hold as an exact relationship regardless of our choice of propagation model and shift equations.
Suppose that we stand on a railway embankment and watch a one-kilometre-long train passing us at a significant proportion of the speed of light. At the moment that the front of the train reached our location, the rear of the train, further away, is seen with a time-lag, which makes it appear to still be at a somewhat earlier location, i.e., somewhere further back along the track than the position that we know that it "really" occupies at that moment. The result is that the full length of the train, from front to back, is seen from our position to be occupying more than 1km of track. The approaching train appears, according to "still" and "movie" cameras, to be stretched.
If we continue to film or take photographs of the passing train, there will be another notable moment when the rear of the train reaches our location. Images taken at this moment will show the train's rear to be at its actual location with no discernable timelag, but now the distant front of the train will be seen with a timelag, further back along the track than its actual position. As a result, the train now appears to be occupying less than 1km of track – the receding train appears length-contracted.
If we calculate the result of differential timelag by assuming that the speed of light is globally fixed in the observer's frame, we obtain a photographable length-change of
(a) len'/len = c/(c+v)
, while assuming that light instead propagates at c with respect to the train gives a different prediction due to differential propagation timelag, of
(b) len'/len = (c-v)/c
, where v is recession velocity. In both cases, the ratio len'/len is idential to the corresponding Doppler prediction for the ratio freq'/freq .
In the case of special relativity, we can obtain its photographable predictions either by assuming a propagation-based effect of (a) supplemented by a Lorentz contraction, or by assuming (b), supplemented by a Lorentz elongation due to the contraction of the moving observer's own reference rulers. Regardless of approach, we get the same final SR prediction that the train appears to change in length by the ratio
(b) len'/len = SQRT[ (c-v) / (c+v) ]
, which is, once again, exactly the same ratio as the selected theory's predictions for frequency-change
Similar logic can be applied to calculating the apparent or photographable length of a passing train, to obtain a length-change ratio that matches a theory's Doppler shift / aberration redshift predictions. For a train passing our location, we take a wide-angle photograph (or multiple photographs) of the passing train, and read off the combined photographed lengths of its approaching and receding parts.
The complication with this calculation is that one has to choose the "correct" moment to take the photograph – we get different results if we photograph the train when its centre carriage is at our location (understood mid-point of the train), to if we take the picture when the receding and approaching parts appear to have the same length (apparent mid-point of the train).
As a result, although we can argue that the exact correspondence between visible frequency-change and visible length-change holds for all angles, the transverse case is more ambiguous (there is no ambiguity at all in the non-transverse case).
Interpretations and alternative derivations
We can predict this equivalence law by pointing out that in observerspace, the set of static photographs of the train could be interpreted as showing the train immersed in a gravitational field, giving length-contraction effects proportional to the gravitational time-dilation effect along the viewing path (consider the extreme case of a black hole, where a body free-falling through the horizon is seen to contract to zero length while the frequency of its emitted signals also goes to zero. Visual consistency within observerspace requires the length-change and frequency-change ratios to be identical.
This equivalence between the proportional change in visible length and visible frequency was not widely understood until the latter part of the 1950s, since the length-contraction effect under special relativity was widely mistaught as describing what an observer sees, rather than describing the difference between what they see according to SR and what they might expect to see under the theory, if they only assumed a globally-fixed lightspeed in their own frame.
The two most popular examples of misteaching were:
- Nobel-prizewinning physicist George Gamow's 1940 "Mr Tomkins in Wonderland" book. The book describes moving objects, approaching and receding, as being seen to be flattened in their direction of motion, and just to make the faulty prediction explicit, supplied nice drawings of the supposed effect (" A single cyclist was coming slowly down the street ... the bicycle and the young man on it were unbelievably flattened in the direction of the motion ... Everything that moves relative to me looks shorter to me ... !". The illustrations also show the street as seen by a cyclist, supposedly with a uniform contraction regardless of whether buildings were approaching or receding. This is not what special relativity (or apparently any other theory) predicts as a visual result.
- Jacob Bronowski's 1973 BBC/Time-Life television series "The Ascent of Man", part 7, "The Majestic Clockwork".
Bronowski's episode on physics faithfully reproduced the error in the Bronowski book using computer graphics, without any of the series fact-checkers realising that this was a bad prediction that had been dropped over ten years earlier. The series re-release digitally remastered and corrected the offending scene, but the mistake can still be found in the series' original accompanying book|As late as the 1990s it was still possible to find physics lecturers insisting that the faulty pre-1959 teachign was correct, "moving objects are always shorter under SR" "otherwise particle accelerators wouldn't work")
While Doppler length effects are simple, general and straightforward to visualise, they tend to be skipped over in modern textbooks, perhaps because they aren't part of the traditional SR narrative, perhaps because they highlight that the early SR narrative was actually wrong for several decades without it being "called", and perhaps because the implication of there being a more general rule at work here is that that spacetime has to (appear to) distort around moving bodies in a way that doesn't reduce politely to the simple length-contraction effects of special relativity. Following that path of enqiry tends to lead t a relativistic theory that diverges form the current system.
- George Gamow, Mr Tomkins in Wonderland (p2-3), Google Books (books.google.co.uk)
- The Ascent of Man (1974) (wikipedia.org)
It's interesting to note that as late as the 1940s, a world-class physicist such as Gamow could still fail to recognise that the predictions that he'd been taught to expect from special relativity didn't correspond to the theory's actual mathematics.