Understanding Special Relativity: Time Dilation and Space Contraction Explained

[JDoolin]

I don't know why you think any restrictions are necessary. I only mentioned an observer because I wanted to provide continuity with your earlier description but we don't need an observer so let's just eliminate him. We are not concerned with any observer trying to reconstruct what happened in his past. We are just doing this from the definition of a Frame of Reference.

Let's first consider a Frame of Reference in which the scenario you described earlier is stationary (but without the smoke):

If you had made an animation of this scenario, it would be just as you described it, an expanding circle of light simultaneously hitting all parts of the mirrored wall and collapsing back to the origin, repeating forever.

Now let's consider a new FoR that is moving at 0.3c with respect to the first FoR. Again, there is no observer in the scenario, just a light source and a circular wall of mirrors. Now your animation show exactly what is happening in this new FoR. Don't you agree?

I agree, but let me ask you another question: Can you construct an inertial reference frame without an origin? Can you describe the location and times of the events in an inertial reference frame without reference to some known and defined x = y = z = t = 0? And from that origin, for an inertial reference frame, there is a world-line extending through t along x=y=z=0.

You can get away without any observers in your reference frame, but you can't define a reference frame without placing an origin. And if you have to define an origin, doesn't it make sense to place it in a bird's-eye-view, where the effect we wish to describe is the most pronounced?

 

[JDoolin]

Now consider a stationary far away observer watching the room moving at 30% the speed of light. What would he see?

Are you serious? It's exactly the same scenario as your previous question. Exactly the same answer as post #34.

 

[ghwellsjr]

Terrell covers all these situations in the paper you linked to and in those cases where an object does not appear in its normal shape (distorted, in your words), it appears rotated rather than shortened to the degree required by Lorentz contraction. This is because of the fact that the image seen by the eye is not of the various parts of the object being viewed from the same distance to the eye and therefore not of the same time delay, and therefore not simultaneous, which is a requirement for measuring the correct Lorentz contraction.

I believe you have misunderstood Terrell's paper:

It appears to me that Terrell is specifying an angle

$$\theta' = \cos^{-1}(v/c)$$

and perhaps at that particular angle, for an approaching object, the Lorentz contraction and the lengthening effect due to the time-delay cancel out. But in general, except for that one special case, I would expect the appearance of the object is distorted the entire time.

He's not talking about an approaching object in your quote of the equation and he's not saying that something happens at just one particular angle. He's saying that in all cases, Lorentz contraction is never visible--note the title: Invisibility of Lorentz Contraction.

You might want to read this:

http://en.wikipedia.org/wiki/Penrose-Terrell_rotation

 

[ghwellsjr]

I don't know why you think any restrictions are necessary. I only mentioned an observer because I wanted to provide continuity with your earlier description but we don't need an observer so let's just eliminate him. We are not concerned with any observer trying to reconstruct what happened in his past. We are just doing this from the definition of a Frame of Reference.

Let's first consider a Frame of Reference in which the scenario you described earlier is stationary (but without the smoke):

If you had made an animation of this scenario, it would be just as you described it, an expanding circle of light simultaneously hitting all parts of the mirrored wall and collapsing back to the origin, repeating forever.

Now let's consider a new FoR that is moving at 0.3c with respect to the first FoR. Again, there is no observer in the scenario, just a light source and a circular wall of mirrors. Now your animation show exactly what is happening in this new FoR. Don't you agree?

I agree, but let me ask you another question: Can you construct an inertial reference frame without an origin? Can you describe the location and times of the events in an inertial reference frame without reference to some known and defined x = y = z = t = 0? And from that origin, for an inertial reference frame, there is a world-line extending through t along x=y=z=0.

You can get away without any observers in your reference frame, but you can't define a reference frame without placing an origin. And if you have to define an origin, doesn't it make sense to place it in a bird's-eye-view, where the effect we wish to describe is the most pronounced?

It's no different than the typical animations of a moving one-dimensional light clock which show two moving mirrors with a photon or light flash bouncing along diagonals between them. Your animation is just a moving two-dimensional circular light clock. What's the difference? Why do you have to be so rigorous when nobody else is? People talk all the time about Frames of Reference in very general terms even to the point of saying "the rest frame of an observer" as if there is only one. They even draw sketches of scenarios all the time without worrying about those details. It really only matters if you want to do a rigorous Lorentz Transform on the scenario when you have to be very precise.

"And if you have to define an origin, doesn't it make sense to place it in a bird's-eye-view, where the effect we wish to describe is the most pronounced?"​

I don't understand this at all. Even if you defined an origin in your animation along with axes labels and a definition of time, where is the effect most pronounced and what has that got to do with a bird's-eye-view?

 

[JDoolin]

I'd still like some feedback from this last quote. Why don't you just describe your animation as what is happening in a Frame of Reference to your moving circular room?

My point was not to attack your explanation, but to defend my own. As long as you acknowledge that, yes, there are specific positions where you could place a camera where the Lorentz Contraction and relativity of simultaneity would be seen as shown, then you can explain it however you want.

I think it is less confusing to invoke an actual position from which you're watching the events unfold, and describe what you would see from that position.


[PLAIN]http://www.psychologytoday.com/files/u248/long_hallway_JeffK.jpg
I decided to throw in this image for good measure. What is "really happening in this frame of reference" is describable in four dimensions. But your perspective is still important. And in my animation, there is an implied perspective, which is straight-on. Just claiming that "this is what is really going on" implies that there is some way of visualizing things without a perspective. Philosophically, I don't really believe there is such a thing. A reference frame without an observer, is one that no one is imagining. It's an ambiguous and undefinable construct. If you're imagining a reference frame, you have to imagine it from a perspective.

 

[ghwellsjr]

First off, let me appologize for completely changing my previous post which you just quoted. I hadn't realized when I composed it that you had already provided the feedback I requested and I thought I could then respond to your feedback before you had a chance to see my unnecessary duplicate request. But I appreciate this further feedback which I will respond to now.

My point was not to attack your explanation, but to defend my own. As long as you acknowledge that, yes, there are specific positions where you could place a camera where the Lorentz Contraction and relativity of simultaneity would be seen as shown, then you can explain it however you want.

I'm not convinced that it's possible to place a camera anywhere that would show Lorentz Contraction and relativity of simultaneity, especially after you pointed out Terrell's article and he was just talking about the shape of a solid object. When it comes to tracking the expanding circle of light, I'm really not convinced that there is any way to take a picture of it that would look like your animation and that's not because of the smoke issue. To me, it's the same issue as two observers, moving with respect to each other "observing" the same expanding circle of light that started when they were colocated, who will both correctly determine that they continue to be in the center of that expanding circle, even though they are no longer colocated. So it seems to me that the camera is just another observer that would simply record an image of the expanding circle of light that showed the camera always in the center and so it would be incapable of showing the expanding circle of light in a different location from the collapsing circle of light or the point of reflection sweeping around the mirrored wall. But I could be wrong.

I think it is less confusing to invoke an actual position from which you're watching the events unfold, and describe what you would see from that position.

Well, as I pointed out in my first post on this subject, Frames of Reference are made for us, outside of the scenario, to actually observe. No one inside the scenario can possibly actually see what we illustrate in a FoR.

[PLAIN]http://www.psychologytoday.com/files/u248/long_hallway_JeffK.jpg
I decided to throw in this image for good measure. What is "really happening in this frame of reference" is describable in four dimensions. But your perspective is still important. And in my animation, there is an implied perspective, which is straight-on. Just claiming that "this is what is really going on" implies that there is some way of visualizing things without a perspective. Philosophically, I don't really believe there is such a thing. A reference frame without an observer, is one that no one is imagining. It's an ambiguous and undefinable construct. If you're imagining a reference frame, you have to imagine it from a perspective.

I'm not following this at all, sorry. A FoR is a perspective. And it's just one of an infinite number of equivalent FoRs. None of them is claiming that "this is what is really going on" if by that you mean what is really going on in nature. But within any FoR it's legitmate to say what is really going on in that FoR, by definition, I might add.

 

[JDoolin]

Terrell covers all these situations in the paper you linked to and in those cases where an object does not appear in its normal shape (distorted, in your words), it appears rotated rather than shortened to the degree required by Lorentz contraction. This is because of the fact that the image seen by the eye is not of the various parts of the object being viewed from the same distance to the eye and therefore not of the same time delay, and therefore not simultaneous, which is a requirement for measuring the correct Lorentz contraction.

I believe you have misunderstood Terrell's paper:

He's not talking about an approaching object in your quote of the equation and he's not saying that something happens at just one particular angle. He's saying that in all cases, Lorentz contraction is never visible--note the title: Invisibility of Lorentz Contraction.

You might want to read this:

http://en.wikipedia.org/wiki/Penrose-Terrell_rotation

I think this is the relevant question: Is the Lorentz contraction "invisible" as Terrell claims, or has James Terrell made a mistake which has gone unnoticed for decades?

I'll take some time to analyze Terrell's argument, and check whether my own methods (analyzing intersections of world-lines and light-cones) agree with his, (transformations of angles via an aberration equation) and if they don't agree, see if I can figure out why.

 

[formal]

That's right. Length contraction and time dilation are given by the more general Lorentz transform equations.

Is there any particular reason why transform must follow the curve of (co)secant?
Could you suggest a link?
Thanks

 

[JDoolin]

I think this is the relevant question: Is the Lorentz contraction "invisible" as Terrell claims, or has James Terrell made a mistake which has gone unnoticed for decades?

I'll take some time to analyze Terrell's argument, and check whether my own methods (analyzing intersections of world-lines and light-cones) agree with his, (transformations of angles via an aberration equation) and if they don't agree, see if I can figure out why.

(My method)

Consider a ruler lying in the y=1 plane and the z=0 plane. Consider marks on the ruler at points (-2.0, -1.9, -1.8, -1.7, ... 0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6). (You can imagine the ruler going on forever if you prefer.

Assume your position is x=0,y=0,z=0, and the time is now t=0. (This experiment will take a long time to discuss, but essentially takes zero time to perform.) Assume also that the ruler is aligned with its zero mark at x=0 (with you).

Now, you are observing several "events" on the ruler. Namely, light bounced off or emitted from the ruler sometime in the past, and you are now seeing those events which happened in the past. You can calculate when those events happened by the formula:

$$t=-\frac{\sqrt{x^2+y^2}}{c}$$

Now we consider another observer passing through the same location and time (0,0,0,0) but traveling at a speed of 0.8c. The two of you share past light-cones, so all of the events that you are observing, the other observer is observing at the same instant.

However, to find out where he is seeing these events, we must perform a lorentz transformation on each of them.

$$\begin{align*} t' &= \gamma t - \beta \gamma x\\ x'&=-\beta \gamma t + \gamma x \\ \end{align*}$$

When this is done, in particular, the ruler marks (-1.5, -1.4, -1.3, -1.2, -1.1, -1.0) are mapped to new positions:

-0.0963, -0.0394, 0.0202, 0.0827, 0.1488, 0.2190

We are particularly interested in the markers -1.4 and -1.2, which now appear at positions -.0394 and .0827. The uncontracted length of the ruler is $$(-1.2) - (-1.4) = 0.2$$, while the apparent length is $$.0827 - (-.0934)=.1221$$ The length contraction factor is $$.1221/.2=.6105$$

Which is roughly* the same as that which is expected by the lorentz contraction factor $$\sqrt{1-0.8^2}= 0.6$$

*If you wanted more fine detail, you should make more marks on the ruler around x=-1.33


Terrell derived an aberration equation from the Lorentz transformations, then uses the aberration equation to conclude that the Lorentz contraction effect "vanishes," but I find it suspicious, when by using the Lorentz transformations directly, I find that the Lorentz contraction is quite visible.

I am attaching a couple of spreadsheet files (Excel, or openoffice, or import them into google-docs), so you can see how I calculated things. But I think that Terrell's statement that the Lorentz Contraction is "invisible" is definitely wrong. At this point, it's just a matter of figuring out exactly what his mistake was.

 

[JDoolin]

Is there any particular reason why transform must follow the curve of (co)secant?
Could you suggest a link?
Thanks

The Lorentz Transformation has events move along arcs of $x^2 - (c t)^2 = constant$.

I'm not sure where you're getting a "(co)secant" from. These are hyperbolic arcs.

You might find https://www.physicsforums.com/showpost.php?p=3405177&postcount=20" helpful.

 

[ghwellsjr]

"Roughly"?? If you did it right you would get exactly the same number.

You have transformed a number of events from one frame to another frame and then used the distance difference in that second frame to "see" the contraction but this is the wrong way to do it. Your mistake is in this sentence:

The two of you share past light-cones, so all of the events that you are observing, the other observer is observing at the same instant.​

An event that is transformed from one frame to another frame is no longer "at the same instant" just like it is no longer at the same location. Comparing just the spatial distance between two events in one frame with the spatial distance between the same two events in another frame will not give you the lorentz contraction. What you have to do is calculate two events in the second frame that have the same time coordinates and then take the spatial difference between them. This means that you will have to interpolate between a pair of events. If you do that, you will get exactly the correct lorentz contraction.

But beyond that, when you use the Lorentz Transform two describe events in a new FoR, you are doing what I said will produce the correct interpretation of your animation. A FoR does not describe what a stationary observer in that FoR actually sees. It only describes what we can determine is happening according to the definition of that FoR. No observer that we define to be in the FoR can actually see anything happening remotely to their location, they can only see what is happening locally to them and then it doesn't matter what FoR we choose as they will all disclose the same images for every observer.

Terrell is not saying that the length contraction is not happening, he is just saying that in order to "observe" it (using his distinction between "observe" and "see"), the extra time delay caused by light having to travel extra distance must be taken into account so that the events at the source must all be "observed" at the same instant in time. This of course is impossible to do with any optical equipment, its something that take has to be done by calculation.

 

[formal]

I'm not sure where you're getting a "(co)secant" from.

I considered C = 1 the hypotenuse and v/C one cathetus I got
sq.rt 1- v^2/ C^2 = *(co)sine and ( * )^-1 = (co)secant.
Or does this transform apply only to mass increase?

 

[JDoolin]

"Roughly"?? If you did it right you would get exactly the same number.

We've already talked about this. I thought we had agreed that the effect will at a minimum at a certain angle, but would never competely go away.

You can get it as exact as you want by using events around the point t'=-1, x'=0

$$\begin{pmatrix} c t'\\ x' \end{pmatrix} = \Lambda ^{-1} \begin{pmatrix} c t\\ x \end{pmatrix}$$

$$= \begin{pmatrix} \gamma & \beta \gamma \\ \beta \gamma & \gamma \end{pmatrix} \begin{pmatrix} -1\\ 0 \end{pmatrix} = \begin{pmatrix} -\gamma\\ -\beta \gamma \end{pmatrix}$$

Since in this case,
$$\begin{matrix} \beta = 0.8 \\ \gamma = \frac{1}{\sqrt{1-.8^2}}=\frac{5}{3}\\ \beta \gamma = \frac{4}{3} \end{matrix}$$

You can use points on the ruler right around x=-4/3, labeled, for instance 1.333 and -1.334, (and the corresponding t values, which you can easily calculate, using $- (ct)^2 = x^2 + y^2$
) .

You have transformed a number of events from one frame to another frame and then used the distance difference in that second frame to "see" the contraction but this is the wrong way to do it. Your mistake is in this sentence:

The two of you share past light-cones, so all of the events that you are observing, the other observer is observing at the same instant.​

An event that is transformed from one frame to another frame is no longer "at the same instant" just like it is no longer at the same location.

You are missing the point. Your statement is exactly correct.

However, I did not claim that the events were at the same location. I claimed that the light-cone contains the same events, but at different times and different locations. (Are you able to access the spreadsheet information?)

Maybe I should clarify that I am talking about the SURFACE of the light cone. I think the point you're missing is that the surface of the past light-cone with its point at (0,0,0,0) is the locus of events which can be detected at the event (0,0,0,0). This fact remains the same, regardless of the observer's reference frame.

No observer that we define to be in the FoR can actually see anything happening remotely to their location, they can only see what is happening locally to them and then it doesn't matter what FoR we choose as they will all disclose the same images for every observer.

I hope you can understand that the observer can see the locus of events on the surface of his or her past light-cone. The tip of the light-cone is the event where the information arrives. It is not a remote event, but THE local event. Naturally, it is also the one event which does not move when you perform a lorentz transformation. All of the other events move, but any event which is in the surface of the past light-cone stays in the surface of the past light-cone.

(If you have any doubt on this, think, how could it be otherwise? How could light that is arriving at an event (0,0,0,0) in one reference frame be NOT arriving at the event (0,0,0,0); the same exact event, in another reference frame? Also you can check the before and after transformation coordinates in my spreadsheet files, to check that indeed t' = -sqrt(x'^2+y'^2).)

 

[JDoolin]

I considered C = 1 the hypotenuse and v/C one cathetus I got
sq.rt 1- v^2/ C^2 = *(co)sine and ( * )^-1 = (co)secant.
Or does this transform apply only to mass increase?

Hmmm, maybe think of it with the slopes involved.

cosine(θ) = adjacent/hypotenuse = $$\frac{\Delta x}{\sqrt{\Delta x^2+\Delta y^2}} = \frac{1}{\sqrt{1 + \left( \frac {\Delta x}{\Delta y} \right )^2}}$$

where θ=arctan(y/x)

hyperbolic cosine (φ)= gamma = (time component)/(space-time-interval)

$$\frac{\Delta t}{\sqrt{\Delta t^2-\Delta x^2}} = \frac{1}{\sqrt{1 - \left( \frac {\Delta x}{\Delta t} \right )^2}}$$

where φ=arctanh(x/t)



P.S. sqrt(1-x^2) is only the cosine of an angle assuming that x^2 is the sine of an angle, right?

 

[formal]

Hallo Sir, thanks for your attention ( I was greatly impressed by your applet!)
I repeat : I was referring to relativistic-mass, and don't know for sure how this relates to time-dilation/space-contraction

I consider triangle AOB with angle λ at O: AO (hypothenuse) = 1 = C, AB = β = v/C = sin λ,BO = cos λ

If an electron has speed β = AB = 0.866 C, then, => λ = 60° , => ΒΟ = cos λ= 0.5, => AO/ BO = cosec λ = 2 =
mass doubles is. (m = M) : restmass+massincrease= 2

m + M = cosec λ = 2
$$\frac{\Delta t}{\sqrt{\Delta t^2-\Delta x^2}} = \frac{1}{\sqrt{1 - \left( \frac {\0.866 }{\1 } \right )^2}}$$

P.S. could you tell me what is space-contraction and time-dilation in this example (0.5-2)?

I'm following with interest your academical debate. I'd like to propose an interesting case that might help give insight to effects of SR:
consider a spaceship orbiting at r= C around the Earth at v = 0.866 C, communicating via radio in real time...
(...let's dispose of smokes and mirrors...).
If you are interested I'll give details. But probably you'll figure out the funny consequences by yourself.
(btw, you could detect space-contraction with your own eyes or measure it inside the ship in a simple manner, child's play.)

 

[JDoolin]

I found a nice diagram of the aberration equation here: http://www.mathpages.com/rr/s2-05/2-05.htm which helps me make my point without a lot of math.

Even using the aberration equations, the Lorentz Contraction is visible, and for certain, it cannot be said that "all objects will appear normal" as Terrell claims.

In the attached diagram you can see that the length of the ruler swept out by angle A is approximately 5.5 units when in the original frame, but it is contracted to 3.6 units when the observer is going 50% of the speed of light, and to 3.1 units when the observer is going 90% of the speed of light.

The length of ruler swept out by angle B is 2.8 units, but contracted to 2.1 units when the observer is going 50% of the speed of light. However, when you go 90% of the speed of light, yes, there is ONE angle where the apparent length of the ruler is equal to the original length.

But Terrell's statements seem to indicate that he believes the "objects will appear normal" regardless of the angle viewed, which is simply not true.

 

[JDoolin]

I started another thread, entitled

Terrell Revisited: "The Invisibility of the Lorentz Contraction"

to summarize my argument, so far.

 

[ghwellsjr]

Terrell agrees that the length (along the direction of relative motion) of an object will appear "distorted". But he adds that the width would also appear "distorted" and the image of the object maintains the same ratio between between the length and the width as one taken by a camera not in relative motion to the object except for a magnification factor. Are you including that in your analysis?

He specifically states that the image of a moving sphere will always appear as a circle rather than an ellipsoid and he references another paper by Roger Penrose that proves the same thing. Do you doubt both of these papers with regard to the imaged shape of a moving sphere?

 

[JDoolin]

Terrell agrees that the length (along the direction of relative motion) of an object will appear "distorted". But he adds that the width would also appear "distorted" and the image of the object maintains the same ratio between between the length and the width as one taken by a camera not in relative motion to the object except for a magnification factor. Are you including that in your analysis?

He specifically states that the image of a moving sphere will always appear as a circle rather than an ellipsoid and he references another paper by Roger Penrose that proves the same thing. Do you doubt both of these papers with regard to the imaged shape of a moving sphere?

Hmmmm. I can imagine one way to check. In some ways it is a similar question to what I did here

(See https://www.physicsforums.com/showthread.php?t=432025&page=2"; the Mathematica source code is attached there.)

Except that we would only need the outside surface of the wheel, and not the spokes. (This would only do in two dimensions, but I think they are the dimensions length and width you are talking about.)

The method I would use would be again to find the intersection of the space-time surface representing the cross-section of the moving sphere, and the observer's past-light-cone*. Then find the x and y coordinates alone, and see if the shape is an oval or a sphere.

I strongly suspect that this intersection will be exactly circular only at specific locations (in fact, specific angles based on the speed of the oncoming object).

*Edit for clarification: The above animation does not use an intersection of a past light-cone, but the intersection of a plane. See the next post for further details on the idea of an intersection with a past-light-cone.

 

[JDoolin]

Terrell agrees that the length (along the direction of relative motion) of an object will appear "distorted". But he adds that the width would also appear "distorted" and the image of the object maintains the same ratio between between the length and the width as one taken by a camera not in relative motion to the object except for a magnification factor. Are you including that in your analysis?

He specifically states that the image of a moving sphere will always appear as a circle rather than an ellipsoid and he references another paper by Roger Penrose that proves the same thing. Do you doubt both of these papers with regard to the imaged shape of a moving sphere?

First decide whether or not you agree with me that the apparent position of object is the intersection of the observation's* past light-cone and the world-lines making up the object.

(I use the possessive form "observation's past-light-cone" here to mean the past-light-cone with the observation event at the tip.

As an example of one such possible configuration, see here:

In this case, though the time of the events varies somewhat unexpectedly with the position, if you viewed the diagram from directly above, you would only see a perfect circle. And your perception would be that you were standing on a circular object.

However, if we were to take that space-time cyinder, and lorentz transform it (basically skew it; i.e. lean it over) then pass the resulting structure through all different portions of the past-light-cone, and view the intersection from directly above, would you expect that the result would be a circle every time?

Or would you expect ovals of different shapes, depending on where the LT-transformed cylinder passed through the past-light-cone?

To me there is no doubt. You would have different shapes, depending on where the leaning cylinder is placed. If any paper says the ratio of length and width is the same, the author should revisit the question and consider the points I am making.

 

[ghwellsjr]

I have looked up the papers that were cited in the wikipedia article on the Terrell Effect along with the original paper you linked to.

If you look at the footnote on the second page of Terrell's paper concerning the paper that Roger Penrose wrote at about the same time, you will see the clues that explain what's going on here. Penrose proved that a moving spherical object will always appear as a sphere rather than an ellipsoid, that is, the outline of the image will be a circle. If there are any surface features on the sphere, they can be distorted. A sphere is the one object that always looks the same no matter how it is rotated.

Terrell, on the other hand, is discussing a different situation. He is talking about the image of an object that is far away so that it "subtends a very small angle" to the eye or camera. Although he states this in his paper, the way he states some of his conclusions makes it sound like he is talking about the more general situation, that is, those that apply for a sphere.

As a result, the wikipedia article is quite misleading in that it combines the two different effects that Penrose and Terrell separately identified as if they were the same issue. It could be vastly improved if this distinction were made clear instead of leaving it up to the reader to look up the papers in its bibliography to figure this out.

So there aren't really any mistakes in Terrell's paper and it has been extensively debated to point out the confusion surrounding it. To tell you the truth, I don't understand his paper or your analyses and I have no reason to disagree with either of you. I think you are doing some marvelous work in this area and I'm even more impressed with the animations that you are producing now than I was with your first one.

You asked me:

First decide whether or not you agree with me that the apparent position of object is the intersection of the observation's* past light-cone and the world-lines making up the object.​

Here's what's confusing to me about this:

1) I thought we were trying to determine the apparent visual shape of an object, not it's position, unless maybe you are talking about all the postitions on an object.

2) Isn't a past light-cone just a subset of the complete history of the light coming to the eye? Wouldn't it illustrate just what a single circular pattern of optical sensors would detect in a single plane? I'm guessing you are using the term in a broader three-dimensional sense which cannot be illustrated on a 2D picture, correct?

3) It seems to me that you have to do more than just establish an intersection of an object's world line with the light cone. Don't you have to perform a complicated algorithm to detect the first occurence of an intersection (looking back from the peak of the light-cone) to establish which surface of a three-dimensional object will be the one that is viewed? If you are doing this just for a two-dimensional object like you originally started with then this won't matter but now you are discussing other more complicated objects.

But getting back to your original animation, I would like to see you do that again with your new technique and show what it would look like from a reasonably close distance and then from a very far distance. I am now more inclined to agree that your original animation is a close approximation to what an observer would see,

 

[JDoolin]

I have looked up the papers that were cited in the wikipedia article on the Terrell Effect along with the original paper you linked to.

If you look at the footnote on the second page of Terrell's paper concerning the paper that Roger Penrose wrote at about the same time, you will see the clues that explain what's going on here. Penrose proved that a moving spherical object will always appear as a sphere rather than an ellipsoid, that is, the outline of the image will be a circle. If there are any surface features on the sphere, they can be distorted. A sphere is the one object that always looks the same no matter how it is rotated.

Terrell, on the other hand, is discussing a different situation. He is talking about the image of an object that is far away so that it "subtends a very small angle" to the eye or camera. Although he states this in his paper, the way he states some of his conclusions makes it sound like he is talking about the more general situation, that is, those that apply for a sphere.

As a result, the wikipedia article is quite misleading in that it combines the two different effects that Penrose and Terrell separately identified as if they were the same issue. It could be vastly improved if this distinction were made clear instead of leaving it up to the reader to look up the papers in its bibliography to figure this out.

I can't envision any situation where the length of the angle subtended would matter. The objects approaching will look elongated, and the objects receding will look contracted. There is only one angle, (θ'=arccos(v/c), if Terrell's paper is in any way correct) where an approaching obect would appear to be at its normal uncontracted length.

So there aren't really any mistakes in Terrell's paper and it has been extensively debated to point out the confusion surrounding it. To tell you the truth, I don't understand his paper or your analyses and I have no reason to disagree with either of you. I think you are doing some marvelous work in this area and I'm even more impressed with the animations that you are producing now than I was with your first one.

Well, Terrell's paper is just amiguous, circular, and strange. He says at one point "a linear object which was oriented in the θ=0 angle at the earlier time when light left it, will appear contracted by the rotation just to the extent of the Lorentz contraction. This does not constitute proof of the visibility of the contraction, as this relation does not hold for other orietations, angles of observation, and shapes, and since the appearance of the object is normal at all time."

How can he claim that the object "will appear contracted" but "this does not constitute a proof of the visibility of the contraction." Secondly, shouldn't he leave out the phrase "since the appearance of the object is normal at all time," being as that is what he is trying to prove?

Furthermore for some reason he picks out angles where $\cos(\theta - \theta') = \sqrt{1-v^2/c^2}$, and doesn't do much of an analysis anywhere else.

You asked me:

First decide whether or not you agree with me that the apparent position of object is the intersection of the observation's* past light-cone and the world-lines making up the object.​

Here's what's confusing to me about this:

1) I thought we were trying to determine the apparent visual shape of an object, not it's position, unless maybe you are talking about all the postitions on an object.

The apparent visual shape of the object consists of the locus of events that happened to the object which are currently arriving at the eye.

This is the intersection of two space-time shapes: (1) the locus of events which are currently arrivng at the eye; the past light-cone, and (2) the locus of events which have and will happen happening to the object; all of its particles world-lines.

2) Isn't a past light-cone just a subset of the complete history of the light coming to the eye?

The INTERNAL portion of the light-cone is just a complete history of th light coming to the eye, but the SURFACE of the light-cone is the set of events which is arriving at the tip of the light cone at a single instant.

Wouldn't it illustrate just what a single circular pattern of optical sensors would detect in a single plane?

The TIP of the lightcone represents a SINGLE optical sensor at a single instant of time. (Even smaller really since it's a geometric point, while a single sensor is extended.) You could treat it, for instance, as the pinhole of a pinhole camera.

I'm guessing you are using the term in a broader three-dimensional sense which cannot be illustrated on a 2D picture, correct?

I am leaving out the z-component, and referring to the shape of the circle in the xy plane as the object moves by in the x direction.

I assumed that the "length" and "width" to which you were referring were in those planes. However, my argument would not change (except for further complication) if you were referring to "length" and "width" as the two directions perpendicular to your line of sight.

The object will look elongated as it approaches and contracted as it recedes, and right in the middle it will have the standard "Lorentz contracted" size.

3) It seems to me that you have to do more than just establish an intersection of an object's world line with the light cone. Don't you have to perform a complicated algorithm to detect the first occurence of an intersection (looking back from the peak of the light-cone) to establish which surface of a three-dimensional object will be the one that is viewed?

Quite right. That would be what I mean by the "surface" of the light-cone. ;)

If you are doing this just for a two-dimensional object like you originally started with then this won't matter but now you are discussing other more complicated objects.

I think there should not be too much difficulty with the x and y-axis cross-section in the z=0 plane. For the full sphere, I may have to think further on cross-sections of the sphere in the z≠0 planes.

But getting back to your original animation, I would like to see you do that again with your new technique and show what it would look like from a reasonably close distance and then from a very far distance. I am now more inclined to agree that your original animation is a close approximation to what an observer would see,

Thank you. I will see whether I can get Mathematica to work. (I have not been inspired to overcome the "password expired" message that I receive when I try to run it on my laptop these days.)