Terrell Revisited: The Invisibility of the Lorentz Contraction

[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 [tex]\theta \approx \sin \theta \approx \tan \theta[/tex] 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 [tex]\theta \approx \sin \theta \approx \tan \theta[/tex] 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?

[tex]\cos \theta_o=\frac{\cos \theta_s-\frac{v}{c}}{1-\frac{v}{c} \cos \theta_s}[/tex]

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 [itex]\theta_s[/itex] 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.

 

[PAllen]

How do you derive the aberration equation?

[tex]\cos \theta_o=\frac{\cos \theta_s-\frac{v}{c}}{1-\frac{v}{c} \cos \theta_s}[/tex]

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 [itex]\theta_s[/itex] 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.

The wikipedia description is poor. The 's' angle is measured in one reference frame, the 'o' angle is measured in the other. The discussion in the mathpages link is much clearer.

 

[harrylin]

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: [..]Now, if you're sophisticated in it enough that you've thought through all this, more power to you. [..]

Sorry in the past I was sophisticated enough to do that, but this time I imagined a simple set-up with identical emitter-receiver pairs that utilizes a basic physical principle - the laws of nature (including aberration) do not depend on position.
No math or drawings are needed (OK a mind sketch is useful) to know that if one LED shines at a certain angle, then an identical LED in an identical state must shine at the same angle, because the calculations and drawings are identical.
In the setup that I considered with identical LED's with matching CCD's, only anti-SR space anisotropy can provide a different outcome.

 

[JDoolin]

The wikipedia description is poor. The 's' angle is measured in one reference frame, the 'o' angle is measured in the other. The discussion in the mathpages link is much clearer.

Since that's a rather long page, I thought it might be helpful to focus in on what I think is the most relevant part.

I've lost my link to Terrell's paper, but I'm trying to imagine how I could use this equation to determine the shape of a relativistically passing object?

If I had an extended source which had a length L, along its velocity vector, then it is not an "object at point A" because a point cannot have a length L.

• The angle [itex]\alpha[/itex] at one end of length L, would be different from the angle [itex]\alpha[/itex] at the other end of length L.
• The angle [itex]\theta_s[/itex] would be different at the two ends of the object.
• The time [itex]t_1[/itex] would be different at the two ends of the object.

If there is enough L there to measure length contraction, you have an object that is not located wholly at the origin, so it would become difficult, if not impossible to use any form of the aberration equation derived from having the object at the origin.

 

[Ken G]

An important thing to notice about aberration is that it is an effect that appears at order v/c, so it is primarily a simple time-of-flight effect, similar to what happens when you have directional hearing of sound waves. All we are concerned with are Lorentzian effects, i.e., that which is different in Lorentzian relativity versus Galilean relativity. Has anyone tried to calculate what a moving "plus sign" would "look like" in Galilean relativity, and compare it? I'm sure things would look pretty weird in either relativity, but we can only claim length contraction is "invisible" if what we see looks the same in both forms of relativity.

This also raises the key question: what did Terrell actually show to be true? Baez thinks he showed something really interesting to be true, and he seemed to be saying that small shapes would look the same to two observers in relative motion, but that does not seem to be true for the plus sign photographed at the instant that it appears to be at the point of closest approach for the stationary observer, because certainly the observer moving with the plus sign will never see anything but a fully symmetric plus sign. So what did Terrell prove, and did both he and Baez draw erroneous conclusions from what was actually shown?

The one thing that gives me pause is that I can't help wondering if maybe the aberration that makes the plus sign appear to be somewhere other than where it actually was when it emitted that light, means that when the stationary observer sees the light emitted when the plus sign really was at closest approach, and also sees it skewed to be shorter along the direction along its motion, aberration will make it look like it is not yet at the point of closest approach-- so they might think "oh, it's skewed because I'm seeing it from an angle that is rotated by its lateral position." Then it wouldn't "look" length contracted, it would just look rotated in a perfectly normal way and nothing relativistic would be apparent (if it was small enough).

But that doesn't sound like what Baez is saying at all-- he is saying the two photographs taken through shutters at the same time and place would photograph the same shape, so would have to be a symmetric plus sign in both, and I just can't see how that could be true. But I hesitate to conclude that something Baez has thought about this much is wrong!

 

[JDoolin]

The wikipedia description is poor. The 's' angle is measured in one reference frame, the 'o' angle is measured in the other. The discussion in the mathpages link is much clearer.

You know? I was able to confirm the equations from the mathpages link, once I understood the definitions of all the variables. It's a fairly straightforward application of the Lorentz Transformation on the vector between two events.

[tex]\begin{pmatrix} t'\\ x'\\ y' \end{pmatrix} = \begin{pmatrix} -\beta \gamma &\gamma &0 \\ \gamma &-\beta \gamma &0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} t_1\\ t_1 \cos \alpha\\ t_1 \sin \alpha \end{pmatrix}[/tex]

Then the velocity angles can be calculated from x'/t', and y'/t'.

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:"

Although I said, before, that there is nothing wrong with the math--I should point out that it would have been incredibly difficult to guess the meaning of the 's' angle from the description given in the wikipedia article.

The angle between the source and the observer and the velocity vector "at the time the light is emitted" is NOT [itex]\theta_s[/itex].

[itex]\theta_s[/itex] is the angle between the observer and the source and the velocity vector "at the time the light is received by the observer" in the source's reference frame.

 

[JDoolin]

The one thing that gives me pause is that I can't help wondering if maybe the aberration that makes the plus sign appear to be somewhere other than where it actually was when it emitted that light, means that when the stationary observer sees the light emitted when the plus sign really was at closest approach, and also sees it skewed to be shorter along the direction along its motion, aberration will make it look like it is not yet at the point of closest approach-- so they might think "oh, it's skewed because I'm seeing it from an angle that is rotated by its lateral position." Then it wouldn't "look" length contracted, it would just look rotated in a perfectly normal way and nothing relativistic would be apparent (if it was small enough).

I have been thinking today about modeling an asterisk-shaped model. A set of eight or more tubes that would show the light paths as they came out of it, as well as the Lorentz contracted moving structure.

What I'd want to show is an animation of the paths of the light following the paths predicted by the aberration equation:

(Image from http://mathpages.com/rr/s2-05/2-05.htm )

But at the same time as it shows those paths of light, it should be showing overlying simply Lorentz contracted structure of the object.

 

[A.T.]

maybe the aberration that makes the plus sign appear to be somewhere other than where it actually was when it emitted that light

The camera always sees the light coming from where it was emitted, in the rest frame of the camera.

 

[Ken G]

I have been thinking today about modeling an asterisk-shaped model. A set of eight or more tubes that would show the light paths as they came out of it, as well as the Lorentz contracted moving structure.

What I'd want to show is an animation of the paths of the light following the paths predicted by the aberration equation:

View attachment 82988
(Image from http://mathpages.com/rr/s2-05/2-05.htm )

But at the same time as it shows those paths of light, it should be showing overlying simply Lorentz contracted structure of the object.

Can you contrast a similar picture for Galilean and Lorentzian relativity? I'm wondering if the Lorentz contraction cancels out the Lorentzian modification to the aberration equation.

 

[Ken G]

The camera always sees the light coming from where it was emitted, in the rest frame of the camera.

I'm not sure that's true, wouldn't the camera see the image in the same direction that a tube would need to be pointed to accept a stream of photons from the source, not along the path of any single one of those photons? In other words, imagine a helicopter flying along a straight path, firing straight-line bullets to try to hit a single point on ground (so they have to be aimed to account for the motion of the helicopter). It seems to me the stream of bullets will arrive, at any moment, along a line that does not track the actual trajectories of the individual bullets that are coming in. If we wanted to point a tube to accept those bullets, you would have to keep the tube rotating to track the incoming bullets, and at any instant the tube would not point along the trajectory of the bullets that are hitting the bottom of the tube at that moment. So I think if the bullets are photons, the eye will see the apparent image along the direction the tube is pointing instantaneously as the photons hit the bottom, not along the direction of motion of the photons. If one takes a wavefront picture, this must have to do with how the wavefronts are turned by the phase variations coming from the movement of the source, such that we cannot expect the arriving plane wave to be perpendicular to the line from the point where the light was emitted. Is that not what aberration is?

 

[JDoolin]

I'm not sure that's true, wouldn't the camera see the image in the same direction that a tube would need to be pointed to accept a stream of photons from the source, not along the path of any single one of those photons? In other words, imagine a helicopter flying along a straight path, firing straight-line bullets to try to hit a single point on ground (so they have to be aimed to account for the motion of the helicopter). It seems to me the stream of bullets will arrive, at any moment, along a line that does not track the actual trajectories of the individual bullets that are coming in.

The bullets would arrive along many lines, each tracking the actual trajectories of the individual bullets that are coming in.

If we wanted to point a tube to accept those bullets, you would have to keep the tube rotating to track the incoming bullets, and at any instant the tube would not point along the trajectory of the bullets that are hitting the bottom of the tube at that moment.

That's a good point. The tube would have to be rotating even as it was receiving the light. If the tube were narrow enough, and the passing object were moving fast enough, you'd have to rotate the tube so fast that the photons would hit the side of the tube before they made it into the camera.

So I think if the bullets are photons, the eye will see the apparent image along the direction the tube is pointing instantaneously as the photons hit the bottom, not along the direction of motion of the photons.

Check the thumbnail. If the top of the tube is rotating to stay aligned with the incoming "bullets" the bullet arriving at the bottom is not necessarily traveling along the direction the tube is oriented.

If one takes a wavefront picture, this must have to do with how the wavefronts are turned by the phase variations coming from the movement of the source, such that we cannot expect the arriving plane wave to be perpendicular to the line from the point where the light was emitted. Is that not what aberration is?

Well, would that be consistent with the derivation I copied from http://mathpages.com/rr/s2-05/2-05.htm in post number #63? The derivation there uses the Lorentz Transformation of two events in a pair of reference frames orthogonal to the relative velocity vector of a source and an observer.

What is the principle by which we know that the Lorentz Transformation works? It is the fact that the LT is the unique transformation that preserves light-cones, while serves as an approximation for Galilean Relativity at low velocities. But at what point during all that did anyone ever say "We cannot expect the arriving plane wave to be perpendicular to the line from the point where the light was emitted?" Never. Quite the contrary, the Lorentz Transformation absolutely preserve the principle where spherical wave-fronts create images of objects at their center. That seems to me, one of the many selling-points of having a transformation which preserves the light-cone.

Now, I don't know what other people have said about the aberration equation, but, according to the derivation, I would say, yes we CAN expect the arriving plane wave to be perpendicular to the line from the point where the light was emitted.[/QUOTE]