Terrell Revisited: The Invisibility of the Lorentz Contraction

[PAllen]

Yes, I know. That's one reason I want to do the math explicitly.
Yes, when the motion is "exactly perpendicular to the line of sight", as the page says. This may be the underlying issue here; we may be simply talking about different conditions, and assuming that the same answer has to apply to all of them, when really it's a question of transition between them.

Taking all of the figures on that page into account, it looks to me like what would be observed in a real case is this: when the ruler is very far away and is approaching, it appears longer than its rest length. As it nears the point of closest approach, its apparent length decreases; at some point before closest approach, its apparent length is exactly equal to its rest length; and at the point of closest approach, its apparent length is equal to its length-contracted length. Then, as it recedes, its apparent length gets shorter still.

I think this may resolve the issue I was thinking of: if the ruler's apparent length is decreasing as it nears the point of closest approach, then the effect I was talking about (which is really just the "approaching" effect described on the page you linked to) is approaching zero effect.

I have done the math on this, spefically for a ruler passing you by, at some distance away. What I find is that there is point of minimum apparent size which matches Lorentz contraction, and (at that moment) the ruler divisions toward the front are stretched, toward the back are compressed, and the visual center does not correspond to ruler rest frame center.

 

[Ken G]

I have always felt Terrell's title was simply wrong. I have never had the opportunity to read his whole paper. Penrose is much narrower in his claims: that s circle always looks like a circle. He never makes a claim that length contraction in general is invisible.

Well I think that settles it.

 

[JDoolin]

Just for the record, here is a hard to find Wikipedia page with similar animations, and here is the page it was linked from.

Yeah, it's pretty much the same except for the choice of where the circle comes through.

He chose to have the circle move right through the origin (the observer) so you get that lima-bean shape when the back end of the circle reaches the observer.

 

[A.T.]

I have done the math on this, spefically for a ruler passing you by, at some distance away. What I find is that there is point of minimum apparent size which matches Lorentz contraction, and (at that moment) the ruler divisions toward the front are stretched, toward the back are compressed,

Are you sure it's not the other way around? The back (still approaching) should look stretched, and the front (already receding) should look compressed. See Fig.1 here:
http://www.spacetimetravel.org/bewegung/bewegung3.html

 

[Ken G]

Wait-- I finally understand something I've been missing, and I think this justifies Terrell's title. It all has to do with what "visible" means, as well as what "over a small enough angle" means. Terrell is right, we just have to understand what he is saying. First of all, in my simple case of a rod passing the point of closest approach, it is clear that the rod appears length contracted-- but only if you know how far away it is! So I've been including things that Terrell does not-- I've been including knowledge of the distance to the "string" that the rod is sliding along, perhaps via binocular vision of our eyes. But Terrell is pointing out that you don't get that information in a photograph-- it can't tell you how far the image is, so its overall size is ambiguous, all you "see" is the shape. And shapes are preserved!

So if we look at the photograph of the rod whose tickmarks are distorted, the photograph cannot tell us that the rod is all at a fixed distance. It could be a rod that is much longer than that, and some parts of it are farther from us than others-- that's what he means by "invisible." You have to have other information, like that the rod is all the same distance from us, things a photograph cannot tell you. So it's a much more limited meaning of "visible" than what we have been using, but it is literally correct, because all you know from a photograph is the shapes of the images, and that's what is preserved. (Also, he puts in that the angular size needs to be small, so even comparisons of sizes from place to place in the photo are disallowed, you cannot compare one tickmark to another without violating his stated restriction, you can only look at things that are invariant of scale so can be as small as you like-- and again that is shape only.)

So I think the bottom line is, the statement really only ever was that "length contraction keeps shapes invariant", and since scales are a matter of interpretation in a photograph, the contraction is literally "invisible", though in most practical applications we would say that we are still "seeing" it.

 

[JDoolin]

I have always felt Terrell's title was simply wrong. I have never had the opportunity to read his whole paper. Penrose is much narrower in his claims: that a circle always looks like a circle. He never makes a claim that length contraction in general is invisible.

It sounds as though Penrose and Terrell published two different results that were mistaken for independent confirmation of the same result. But actually, Penrose claimed that a specific geometric shape (the sphere) maintained a circular cross-section. While Terrell's claim was that there was no evidence of Lorentz Contraction at all.

Well I think that settles it.

Well, it settles it for you and me, but does Wikipedia go by references to threads on Physics Forums, or by articles in Physics Review?

I think it seems settled here, with PAllen, Ken G, Peter Donis, A. T, and myself all agreeing that you can see Lorentz Contraction. But is that enough to get it corrected on Wikipedia? This article has been a reference in the Physics Review since June 22, 1959, fifty-six years ago.

Would it be possible to get Physics Review to go back to that paper and analyze it again for its accuracy, and officially redact the verbal conclusion of the paper? Or does someone else need to write a paper which analyzes the problems of Terrell's paper, which then gets published as a redaction piece?

 

[JDoolin]

Terrell is right, we just have to understand what he is saying. First of all, in my simple case of a rod passing the point of closest approach, it is clear that the rod appears length contracted-- but only if you know how far away it is!

To ask your own question: "Are you giving him a pass for his misleading title?"

 

[Ken G]

You mean, are we giving him a pass on his misleading title? I would say yes we are-- but it's a legitimate "pass", length contraction is strictly invisible unless you can augment what you see with additional information that is simply not in that image itself, but is part of what a brain can legitimately infer. I think we'd have to get into the processing of mental images, and what constitutes a "literal" interpretation of seeing, versus what we really mean by what "seeing" is in practice. Of course if we do that, we can say we can't really see shape changes either, because we don't know the object isn't itself deforming...

 

[A.T.]

And shapes are preserved!

Not sure what you mean here, but in general the shapes will be distorted:
http://www.spacetimetravel.org/tompkins/tompkins.html

Even rods with some width look skewed (Fig 2):
http://www.spacetimetravel.org/bewegung/bewegung3.html

 

[JDoolin]

length contraction is strictly invisible unless you can augment what you see with additional information that is simply not in that image itself.

Hmmmmmmm. I think in physics we should use all of the data we have at our disposal, and seek generalities which apply whether we know all of the data or not. And we frequently use physics to infer information that does not appear in an image. To say "Lorentz Contraction is invisible because I cannot compare it to an image of a stopped object of the same size"

All you're saying is that you do not necessarily have enough information to confirm the Lorentz Contraction empirically. You're not really saying that the Lorentz Contraction isn't there.

 

[harrylin]

I did not follow the discussion, but here are my 2 cts:
What about a rod that passes at close distance a CCD array without optics, so that the rod blocks the light when passing? That's a very basic form of "seeing" and aberration cannot play a role.
More direct would be a rod with on one side illuminating LEDs in the same setup, and IMHO, also then aberration cannot prevent a length contracted picture.

 

[JDoolin]

and what constitutes a "literal" interpretation of seeing, versus what we really mean by what "seeing" is in practice.

I think, also, if someone is going to claim that Lorentz Contraction is invisible, then you should be using a "liberal" interpretation of the word "see."

If I tell someone "you cannot see a ghost." I am not saying "you cannot see a ghost if you turn your head away from it, are blindfolded, and are in a different room." The words imply "You cannot see a ghost, even under the best conditions, looking directly at it, with the best possible technology."

Rather than claim "Lorentz Contraction is Invisible" it would make more sense to seed to define what constitutes a literal interpretation of seeing... What we really mean by seeing in practice, and establishing a general rule for modeling what we see. And I think, unless you are specifically trying to define "seeing" in such a way to salvage Terrell's claim of the invisibility of Lorentz Contraction, you'll find that Terrell's claim is not accurate.

When physics says you "can" do something, you can be as explicit as you want about how to do it. But when physics says you "cannot" do something, then as soon as even one way of doing it is figured out, or even one circumstance is found where it can be done, you should redact the "cannot".

 

[A.T.]

I did not follow the discussion, but here are my 2 cts:
What about a rod that passes at close distance a CCD array without optics, so that the rod blocks the light when passing? That's a very basic form of "seeing" and aberration cannot play a role.
More direct would be a rod with on one side illuminating LEDs in the same setup, and IMHO, also then aberration cannot prevent a length contracted picture.

With or without optics, there is no aberration in the rest frame of the camera, just delayed signals coming from an outdated position.

 

[harrylin]

With or without optics, there is no aberration in the rest frame of the camera, just delayed signals coming from an outdated position.

Well, the delays are almost negligible in this case and anyway identical.

 

[Ken G]

I think, also, if someone is going to claim that Lorentz Contraction is invisible, then you should be using a "liberal" interpretation of the word "see."

Yes I agree with this, so I think we can find fault in the wording of that conclusion. But I used to think it was a scientifically flawed claim, whereas now I see it as more of a linguistic question. If someone says that "all we ever really see is shapes, all length scales are inferences of some kind", then one can support Terrell's conclusion using his argument. If we instead say "actually, what we mean by seeing involves a host of inferences, even the interpretation of shape requires that", then we can find fault in that wording.

If I tell someone "you cannot see a ghost." I am not saying "you cannot see a ghost if you turn your head away from it, are blindfolded, and are in a different room." The words imply "You cannot see a ghost, even under the best conditions, looking directly at it, with the best possible technology."

Yes, that's a particularly problematic element of the term "invisible." Normally, it means "cannot be seen at all", but Terrell is using it to mean "cannot be inferred from a purely literal analysis of an image." So the image is visible, but the attribute of being length contracted is not visible, but only if you hold that visibility requires no mental processing beyond what it takes to identify shapes! Which is a bit of a stretch, to say the least.

Rather than claim "Lorentz Contraction is Invisible" it would make more sense to seed to define what constitutes a literal interpretation of seeing... What we really mean by seeing in practice, and establishing a general rule for modeling what we see. And I think, unless you are specifically trying to define "seeing" in such a way to salvage Terrell's claim of the invisibility of Lorentz Contraction, you'll find that Terrell's claim is not accurate.

Yes, the conclusion is far better stated "because length contraction does not change the shapes of small things, and images are in some sense a cobbling together of small shapes, seeing it requires the processing of additional information, yet this is usually quite possible to do under practical conditions." In fact, if you think about it, you could use Terrell's argument to say that whether or not you are moving toward an object is also "invisible", because all that happens is the object appears bigger as you approach it-- none of the shapes change, so an image of the object doesn't tell you that you are approaching it. But we would not say that you cannot tell if you are approaching an object by looking at it, we would be very poor drivers!

When physics says you "can" do something, you can be as explicit as you want about how to do it. But when physics says you "cannot" do something, then as soon as even one way of doing it is figured out, or even one circumstance is found where it can be done, you should redact the "cannot".

Yes, "no-go" theorems need to be held to a high standard. If someone says "length contraction is invisible", this surely sounds like the claim that "you cannot tell if you are in a Galilean or Lorentzian universe just by looking", but in any practical situation that would not be true. If you know you have a rod sliding on a string at high speed, you can predict what that will look like in Galilean vs. Lorentzian universes, and even if the rod is so small that you cannot see the distortion in the tickmarks, the length of the rod at closest approach is still going to look different by the Lorentz factor in the two situations. If that doesn't mean "seeing length contraction", I don't know what does.

 

[JDoolin]

What about a rod that passes at close distance

The amount of distortion has more to do with the angular measure of the object than the distance. (The angular measure becomes LARGER when you're close.) If the object is close enough that the angular measure is greater than about 15 degrees, where you can no longer use the small angle approximation \theta \approx \sin \theta \approx \tan \theta I think you'll find that you would see significant differences in the compression of the front end of the object (which would be contracted) and the back end of the object (which would be stretched out.)

This phenomenon of having the front end contracted and the back end stretched out would always be present, but as long as you have the whole object fit within a "small angle" as it made its closest approach, I think the effect would be negligible.

I personally don't like the term "aberration" when it is applied to Special Relativity. Because it implies some kind of "illusory quality" to what is going on in "observer dependent measurements of distance" and it implies that there could or should be some "actual non observer dependent measurements of distance."

 

[PAllen]

Are you sure it's not the other way around? The back (still approaching) should look stretched, and the front (already receding) should look compressed. See Fig.1 here:
http://www.spacetimetravel.org/bewegung/bewegung3.html

You are right. I did that math a while ago, re-did it this morning. I had remembered it backwards.

 

[JDoolin]

Although, I suppose aberration wouldn't necessarily have to imply "illusionary" Rather, aberration would be that sort of funny-looking shape you get when you take an extended relativistically traveling object and find its intersection with the observer's past light-cone.

Whereas the non-aberration shape of the object would be the shape you get when you take an extended object and find its intersection with the observer's t=0 plane.

But the location of the events which produce the "aberration effect" are the real locations of events.

For example, if I see an object coming at me at 99% of the speed of light, it's image appears to be moving toward me superluminally. But the actual events I am seeing are at the distances they appear to be. That the object is coming toward me at superluminal speeds might be an illusion. But that the events occurred at the distances where they seem to have occurred is NOT an illusion.

That the object is stretched out might be an illusion, but that the observed events occurred where they appear to have occurred in my reference frame is NOT an illusion.

 

[PAllen]

I have several summary points of my own to make.

First, any common sense definition of 'seeing length contraction' means with knowledge of the object's rest characteristics. It is only relative to that there is any meaning to 'contraction'.

Second, there are obvious ways to directly measure/see any changes in cross section implied by the coordinate description. Simply have the object pass very close to a sheet of film, moving along it (not towards or away) and have a bright flash from very far away so you get as close as you want to a plane wave. Then circles becoming ovals, and every other aspect of the coordinate description will be visible. (You will have a negative image; or positive if old fashion film that you develop but don't print).

Third, the impact of light delays on idealized camera image formation has nothing to do with SR. However it combines with SR in such a way that with my common sense definition of 'see', length contraction is always visible (if it occurs, e.g. not for objects fully embedded in the plane perpendicular to their motion ). That is, if you establish what you would see from light delay under the assumption that the object didn't contract, and compare to what you would see given the contraction, they are different. You have thus seen (the effect of, and verified) length contraction.

Finally, I am posting the formula for the case of a ray traced image of a line of rest length L moving moving at v in the +x direction, along the line y=1, with angles measured down from the horizontal (e.g. on a approach, and angle might be -π/6, on recession -5π/6). I let c=1. I use a parmater α between 0 and 1 to reflect positions along the line in its rest frame. The sighting point is the origin. Then, to describe the range of angles seen at some time T, you simply solve (for each α):

cot(θ) = v csc(θ) + vT + αL/γ

The T corresponding to the symmetrically placed image that shows the exact same angular span (but not internal details) as a stationary ruler of length L/γ centered on the Y axis is:

T = -(L/2γ + v csc(θ_L))/v

where cot(θ_L) = -L/2γ

[Edit: It is not too hard to verify (formally) that you have stretching whenever cot(θ) < 0, and compression whenever cot(θ) > 0. ]

 

[harrylin]

The amount of distortion has more to do with the angular measure of the object than the distance. (The angular measure becomes LARGER when you're close.) If the object is close enough that the angular measure is greater than about 15 degrees, where you can no longer use the small angle approximation \theta \approx \sin \theta \approx \tan \theta I think you'll find that you would see significant differences in the compression of the front end of the object (which would be contracted) and the back end of the object (which would be stretched out.)

I can totally not follow that argument; in my analysis of SR space is homogeneous. The aberration of light from a LED with velocity v at x=x1 that shines towards a CCD element at x=x1 must be equal to the aberration of light from a LED with velocity v at position x=x2 that shines towards a CCD element at x=x2.

I personally don't like the term "aberration" when it is applied to Special Relativity. Because it implies some kind of "illusory quality" to what is going on in "observer dependent measurements of distance" and it implies that there could or should be some "actual non observer dependent measurements of distance."

You can call it angle of reception.

PS. I see that PAllen elaborates in post #49 the first argument I made in post #41.

 

[Ken G]

PS. I see that PAllen elaborates in post #49 the first argument I made in post #41.

Yes that was the way I was thinking originally as well, that you could easily "see" that the rod was length contracted. But then I realized what Terrell meant, which is that a shortened rod still looks like a rod-- it's not distorted, so you only know it's contracted if you know how far away it is from the CCD. I agree with JDoolin that this does not constitute good use of the concept of "invisibility", because seeing always involves some inclusion of additional information to make sense of the image, I'm just saying that Terrell's meaning of invisibility is only about the non-distortion of small shapes. That's what I was struggling with before, I couldn't see how Terrell was missing such an obvious point, but now I see he just had an odd interpretation of the words.

 

[PAllen]

Yes that was the way I was thinking originally as well, that you could easily "see" that the rod was length contracted. But then I realized what Terrell meant, which is that a shortened rod still looks like a rod-- it's not distorted, so you only know it's contracted if you know how far away it is from the CCD. I agree with JDoolin that this does not constitute good use of the concept of "invisibility", because seeing always involves some inclusion of additional information to make sense of the image, I'm just saying that Terrell's meaning of invisibility is only about the non-distortion of small shapes. That's what I was struggling with before, I couldn't see how Terrell was missing such an obvious point, but now I see he just had an odd interpretation of the words.

I think there is more to it. Terrell was (I think) modeling an idealized camera, not a shadow cast image such as Harrylin and I mentioned. In the latter, shape change is trivially visible - a moving circle becomes an oval (as does a moving sphere).

Yet another point is that the effect of light delays on a idealized camera would distort shapes more if weren't for length contraction (a sphere would be elongated if it weren't for length contraction). Thus the absence of many types of shape distortion is direct evidence of length contraction!

Finally, other sources derive that shapes change anyway - a rectangle can become a curved parallelogram.

So I really think there is no substantive way in which the title of the paper is defensible.

 

[Ken G]

I think there is more to it. Terrell was (I think) modeling an idealized camera, not a shadow cast image such as Harrylin and I mentioned. In the latter, shape change is trivially visible - a moving circle becomes an oval (as does a moving sphere).

I'm not sure the moving sphere would look squashed, even in its shadow, since to make a shadow the sphere must scatter away the light, but light moving as the sphere goes by is going to scatter at multiple places around the sphere. A flat disk I can see, but then if you see a squashed flat disk, it can look rotated rather than squashed. But if you know it's all at the same distance, because you know something about the setup, you can include that knowledge in what you are calling the "image." I think Terrell's point is you will always need to include that knowledge, it's not in the "raw" image. But I admit I'm still unclear on just what the claim is.

Yet another point is that the effect of light delays on a idealized camera would distort shapes more if weren't for length contraction (a sphere would be elongated if it weren't for length contraction). Thus the absence of many types of shape distortion is direct evidence of length contraction!

But that's all right, Terrell knows you can infer length contraction from what you see, he is only claiming you can't "see it" without some analysis.

Finally, other sources derive that shapes change anyway - a rectangle can become a curved parallelogram.

So I really think there is no substantive way in which the title of the paper is defensible.

But if that's true, it's not just the title-- it's essentially every word in the abstract that is wrong. That requires a flaw in the mathematics, does it not?

 

[JDoolin]

I can totally not follow that argument; in my analysis of SR space is homogeneous. The aberration of light from a LED with velocity v at x=x1 that shines towards a CCD element at x=x1 must be equal to the aberration of light from a LED with velocity v at position x=x2 that shines towards a CCD element at x=x2.

You can call it angle of reception.

PS. I see that PAllen elaborates in post #49 the first argument I made in post #41.

Is it the angle of reception?

I may be misunderstanding the equation for aberration but look at the following diagram

Now there's nothing wrong with the math here, insofar as it goes:
"the source is moving with speed

at an angle

relative to the vector from the observer to the source at the time when the light is emitted. Then the following formula, which was derived by Einstein in 1905, describes the aberration of the light source, [PLAIN]http://upload.wikimedia.org/math/3/9/9/39994abba112928ccc9e9d70a502fb93.png, measured by the observer:"

I think that the hardest thing to do is to figure out what these angles mean verbally and intuitively. For instance the light that goes along that "measured observed angle" never actually hits the observer along the vector between the observer and the source. It's just where the light passes through the observers reference frame.

Now, if you're sophisticated in it enough that you've thought through all this, more power to you. But as for me, I find the idea of finding the location of the object according to the intersection of past-light-cones with the worldlines of the object much more intuitive.

Rather than figuring out where a particular aimed vector of light passes through your reference frame, it figures out the locus of events being seen from a particular point in space and time.

 

[PAllen]

I'm not sure the moving sphere would look squashed, even in its shadow, since to make a shadow the sphere must scatter away the light, but light moving as the sphere goes by is going to scatter at multiple places around the sphere. A flat disk I can see, but then if you see a squashed flat disk, it can look rotated rather than squashed. But if you know it's all at the same distance, because you know something about the setup, you can include that knowledge in what you are calling the "image." I think Terrell's point is you will always need to include that knowledge, it's not in the "raw" image. But I admit I'm still unclear on just what the claim is.

No scattering is needed for shadow casting. Imagine all light striking the body is absorbed. Then a moving sphere clearly casts an oval shadow. As for distance, you assume it is nearly touching the film for shadow casting. Terrell was simply not analyzing this scenario. I don't know why you are trying to defend a different case than Terrell analyzed. It really is trivial that shape change from length contraction is visible via shadow casting (given a perfect plane wave of near zero duration). It is a perfect measure of simultaneity for the frame generating the plane wave flash. In another frame, different elements of the flash are generated at different times, so the explanation of the shape distortion is frame dependent, but not the fact of the shape distortion.

But that's all right, Terrell knows you can infer length contraction from what you see, he is only claiming you can't "see it" without some analysis.

But if that's true, it's not just the title-- it's essentially every word in the abstract that is wrong. That requires a flaw in the mathematics, does it not?

Yes, it would, and on this I don't know for sure who is right. I have never done a complete ray tracing for a complex shape from first principles on my own. I do know there are many videos such as A.T. has linked that show even the same object changing shape as it approaches, passes, and recedes. Unless these are all wrong, then even a limited claim of shape preservation is wrong.

 

[PAllen]

On carefully reading Terrel's abstract I can see how my detailed analysis of the rod could be considered consistent with it. The increased space between ruler marks on the approaching part, and decrease on the receding part could be consistent with and interpretation of the ruler being rotated rather than contracted. However, as Penrose noted in his book, it would be easy to establish that this physically the wrong interpretation - imagine the rod as having wheels, moving on a stationary track. You would never see the wheels leave the track. Therefore, seeing this, you would be forced to interpret the image as contracted with stretching and compression of ruler lines.

As for the discrepancy between parts of the abstract and various ray traced videos, it is possible the video cases exceed his 'small subtended angle' restriction.

 

[PAllen]

No scattering is needed for shadow casting. Imagine all light striking the body is absorbed. Then a moving sphere clearly casts an oval shadow. As for distance, you assume it is nearly touching the film for shadow casting. Terrell was simply not analyzing this scenario. I don't know why you are trying to defend a different case than Terrell analyzed. It really is trivial that shape change from length contraction is visible via shadow casting (given a perfect plane wave of near zero duration). It is a perfect measure of simultaneity for the frame generating the plane wave flash. In another frame, different elements of the flash are generated at different times, so the explanation of the shape distortion is frame dependent, but not the fact of the shape distortion.

Actually, if you use my proposal from #49, the distant flash will produce what is interpreted as plane wave pulse in all frames. The simultaneity detection comes from the sheet of film. If the image interaction is simultaneous across the sheet in one one frame, it will not be simultaneous in a different frame, and that will explain the shape change per that frame.

 

[Ken G]

On carefully reading Terrel's abstract I can see how my detailed analysis of the rod could be considered consistent with it. The increased space between ruler marks on the approaching part, and decrease on the receding part could be consistent with and interpretation of the ruler being rotated rather than contracted. However, as Penrose noted in his book, it would be easy to establish that this physically the wrong interpretation - imagine the rod as having wheels, moving on a stationary track. You would never see the wheels leave the track. Therefore, seeing this, you would be forced to interpret the image as contracted with stretching and compression of ruler lines.

Terrell might say you are not allowed to compare the ruler lines, as then the object is not "small" in the way Terrell means. He is apparently arguing that if you allow yourself to compare different places in the image, you must make additional assumptions about what you are looking at in order to "connect the dots", and that could subject you to illusions that don't count as "seeing." This is the tricky part of his language. Terrell certainly knows that if we are allowed to include analytical details about the situation, especially time of flight information, we can correctly infer there is length contraction, that's how length contraction was discovered. So he is using a very restricted idea of what things "look like"-- he is comparing photographs made by two observers in relative motion, and saying the shapes of small things in photographs taken at the same time and place look the same. So he must say that your shadow analysis, done close to the film, subtends a solid angle that is too large to count for what he is talking about. In some sense he seems to be claiming that a shadow analysis is not what things look like, it is an analytical tool for saying what they are actually doing-- akin to using time-of-flight corrections to do the same thing.

So I think it all comes down to what is meant by saying a shape "looks no different". Maybe the explanation by Baez in the link PeterDonis provided will shed light on this:
"Now let's consider the object: say, a galaxy. In passing from his snapshot to hers, the image of the galaxy slides up the sphere, keeping the same face to us. In this sense, it has rotated. Its apparent size will also change, but not its shape (to a first approximation)."

But the more I think about what Baez is saying there, I just don't get it. Surely a camera moving at the same velocity as a "plus sign" of rods will see the symmetric plus sign, and the camera that sees the plus sign as moving can take an image of something apparently at closest approach, which will look distorted. A distorted image looks different, no matter which images you choose to match up to make the comparison. It doesn't seem to matter if you can attribute the distortion to rotation or length contraction, Baez claimed the images will have the same shape, and I don't see how that could be.

 

[JDoolin]

How do you derive the aberration equation?

\cos \theta_o=\frac{\cos \theta_s-\frac{v}{c}}{1-\frac{v}{c} \cos \theta_s}

You'll see I posted a quote from the wikipedia article about it from above... But the more I think about it, I start to think this might be the source of the problem in Terrell's paper.

From Wikipedia: ""the source is moving with speed

at an angle

relative to the vector from the observer to the source at the time when the light is emitted. Then the following formula, which was derived by Einstein in 1905, describes the aberration of the light source, [PLAIN]http://upload.wikimedia.org/math/3/9/9/39994abba112928ccc9e9d70a502fb93.png, measured by the observer:"

Now my reading of this is that the light is emitted along a "tube" that is aimed directly toward the observer in the reference frame of the observer when the source is at the given point.

The trouble is that if the "tube" is aimed directly toward the observer, in the reference frame of the observer, you're looking at the situation Post-Lorentz-Contraction. That is \theta_s is not the angle of the tube in the source's reference frame, but the angle of the tube in the observer's reference frame. So this equation is not relating a difference between appearances in the source's reference frame and the observer's reference frame.

Rather, it is relating a difference between two different angles measured in the observer's reference frame.

If I were to try to confirm this, I would probably try to set up a diagram similar to the one I gave in post 54, and do some vector and trigonometric calculations, dividing the velocities into well-chosen x and y components, setting the final speed of the photon through the moving tube at c, and see if I could reproduce the aberration equation from scratch.

My point is, I don't think you would find any evidence of Lorentz Contraction in the aberration equation, because the aberration equation may simply be figuring out the direction at which rays travel from already lorentz-contracted tubes.

 

[m4r35n357]

My point is, I don't think you would find any evidence of Lorentz Contraction in the aberration equation, because the aberration equation may simply be figuring out the direction at which rays travel from already lorentz-contracted tubes.

That aberration formula is just one of three basic definitions, using cos, sin and tan. It just happens that gamma cancels out in the cos definition. Reference. See for example equation (2) which contains gamma explicitly.