PAllen said:
[edit 2: OK, I see that if you allow the angular span of a tilted ruler to go to zero with distance, the ratio angles subtended by ruler lines goes to 1. However, if you fix the angular span of a tilted ruler (e.g. 2 degrees), then distance doesn't matter and the ratio front and back ruler lines remains constant. This is what I was actually modeling when comparing to rotation - all angles. I remain convinced the the rotation model is quite accurate for small, finite spans, e.g several degrees.]
This is the crux of the matter, it is what I find confusing about the language relating to "rotation." A rotation looks different at different angular sizes, because of how it makes some parts get closer, and other parts farther away. Is that being included, or just the first-order foreshortening? And what angular scales count as "sufficiently small"? Baez said:
"Well-known facts from complex analysis now tell us two things. First, circles go to circles under the pixel mapping, so a sphere will always photograph as a sphere. Second, shapes of objects are preserved in the infinitesimally small limit."
I interpreted that to mean the shapes are only preserved in the infinitesmally small limit, i.e., for the Lorentz contracted cross to look like a rotated cross, it has to be infinitesmally small, so this would not include how the forward tilted arm can look longer than the backward tilted arm on a large enough angular scale. You are saying I am overinterpreting Baez here, and what's more, your own investigation shows a connection between that longer forward arm, and what Lorentzian relativity actually does. So perhaps Baez missed that, or did not mean to imply what I thought he implied.
This is what Terrell says in his abstract:
"if the apparent directions of objects are plotted as points on a sphere surrounding the observer, the Lorentz transformation corresponds to a conformal transformation on the surface of this sphere. Thus, for sufficiently small subtended solid angle, an object will appear-- optically-- the same shape to all observers."
The answer must lie in the meaning of having a conformal transformation on the sphere of apparent directions. If we use
JDoolin's asterisk, instead of a cross, we can see that a rotation will foreshorten the angles of the diagonal arms, and it is clear that a conformal transformation will keep those angles fixed, so certainly Terrell is saying that the Lorentz contraction will foreshorten the angles in exactly the same way. But what about the contrast in the apparent lengths of the arms tilted toward us and the arms tilted away, is that contrast also preserved in the conformal transformation? You are saying that it does, and that seems to be the key issue. We do have one more clue from Terrell's abstract:
"Observers photographing the meter stick simultaneously from the same position will obtain precisely the same picture, except for a change in scale given by the Doppler shift ratio"
So the word "precisely" says a lot, but what is meant by this change in scale, and is that change in scale uniform or only locally determined? You are saying that it looks precisely like a rotation, including the
contrast between the fore and aft distortions, not just the first-order foreshortening effect.
A sphere with continents on it might be a good case to answer this. We all agree the sphere still looks like a sphere, and in some sense it looks rotated because we see different continents than we might have expected. But the key question that remains open is, do the continents in the apparent forward regions of the sphere appear larger than the continents in the most distant parts of the sphere, or is that element
not preserved in the conformal transformation between the moving and stationary cameras? I agree Terrell's key result is essentially that it is easier to predict what you will see for small shapes by using the comoving camera at the same place and time as the stationary camera, but what we are wondering about is over what angular scale, and what types of detail, we should expect the two photos to agree on. The mapping between the two cameras is conformal, but it is not the identity mapping, so can we conclude the continents will look the same size in both photos? Certainly distortions on the surfaces of
large spheres should look different between the two photos, but even large spheres will still look like spheres.