Bell's Spaceship Paradox and Length Contraction

[PeterDonis]

AIt seems the only source of asymmetry beside the acceleration, which can be obviated by swapping clocks with a rocket going in the other direction, is length contraction.

No, there is another asymmetry. Even if clocks are swapped at the turnaround point so no single observer actually feels any acceleration, it is still true that the "traveling twin" (which may be a pair of observers who synchronize clocks when they pass at the turnaround point) sees a change in Doppler shift in light signals from the stay-at-home twin right at the turnaround, but the stay-at-home twin doesn't see a change in Doppler shift in light signals from the traveling twin until much later (the time it takes for light to travel from the turnaround point back to the stay-at-home twin).

This is explained in the Usenet Physics FAQ on the twin paradox, in the section on the Doppler Shift Analysis. There have also been some previous PF threads that have gone into this in some detail.

 

[nosepot]

PeterDonis: Looking at wavefronts arriving in a Doppler example is just a time keeping exercise to show time dilation - it is not a resolution. Why is it that the traveling twin doesn't manage to get as many wavelengths out in his frame? The journey does not as far for him due to length contraction.

In fact the Doppler presentation is even more confusing, as we must also try to imagine the time delay for the light to travel between twins. It's a terrible example.

 

[ghwellsjr]

However, it adds absolutely nothing to the discussion except to reinforce what we have all been saying all along. I wonder why you find it more satisfying than our answers to you.

It may not add much for you, as you have a clearer understanding. My original question showed that I misunderstood that length contraction also involve an apparent dynamical shortening of objects when they are seen to accelerate. Bell's chapter was a sensible step by step argument towards why SR works. I've put it here for the benefit of anyone else who might be struggling with the same misconceptions I am. I'm for the most part converted - you should be pleased! :P

But there's still a serious problem saying that there is "an apparent dynamical shortening of objects when they are seen to accelerate." Think about this: a rod is accelerated along its long axis to some speed relative to its previous state. Now it is accelerated along its long axis in the opposite direction so that it is in its previous state. Has it experienced two Length Contractions so that it is shorter than it was originally?

 

[ghwellsjr]

PeterDonis: Looking at wavefronts arriving in a Doppler example is just a time keeping exercise to show time dilation - it is not a resolution. Why is it that the traveling twin doesn't manage to get as many wavelengths out in his frame? The journey does not as far for him due to length contraction.

In fact the Doppler presentation is even more confusing, as we must also try to imagine the time delay for the light to travel between twins. It's a terrible example.

The Doppler presentation is the raw data that any theory must explain. It doesn't specify time dilation or length contraction or time delay for light nor is it related to a specific frame. Those come into play when you invoke an explanation from Special Relativity and assign a specific Inertial Reference Frame (IRF). Then, depending on the IRF, you get all those particulars which are different in each IRF but which maintain the exact same Doppler presentation. The Doppler presentation is the problem that a theory must resolve.

 

[nosepot]

Has it experienced two Length Contractions so that it is shorter than it was originally?

This is a good point, because it highlights why SR is so confusing. The answer is both two contractions and two extensions. An object accelerating to join another frame would be seen to contract from the view of the frame it leaves, and uncontract from the view of the frame it joins. That's simple relativity, is it not?

If we may be so brave to consider an ether theory viewpoint, it could either contract or uncontract in an absolute way depending on whether it speeds up or slows down with respect to the ether, but we would have no way to know what it's really doing, as we don't know which observer is moving slower relative to the ether frame. Hence the result would be identical to the SR observations.

 

[nosepot]

The Doppler presentation is the raw data that any theory must explain. It doesn't specify time dilation or length contraction or time delay for light nor is it related to a specific frame.

That is wrong - of course it does. If you are using a spacetime diagram to figure out how many wavelengths each counts thne you are using both time dilation and length contraction. If you talking about the Earth frame only, then you are not resolving the paradox.

The resolution must be in understanding why the traveling twin does not manage to get as many wavelength out during his journey. If you do not consider the reduced distance the traveller thinks they must travel then you are falling into the same trap that gives rise to the paradox. Without the length contraction for the traveller (and we've ruled out acceleration as an issue), the experience of each twin is completely symmetric.

(Am I being trolled now? Touche.)

 

[nosepot]

Nobody biting. I see there was a civil war over this recently!

https://www.physicsforums.com/showthread.php?t=689621&highlight=twin+paradox

Told you the Doppler explanation was terrible.

 

[pervect]

The doppler explanation is actually very good at describing what one actually measures. It's based on measuring signal arrival times with "proper clocks", essentially. This is something one can actually, directly, measure, and can discuss.

There doesn't appear to be a good cure for erroneous mental constructs (erroneous in that they are incompatible with special relativity, which is itself consistent with experiment) that have nothing to do with what one measures, but rather how the measurements are put together into a view of the world.

If one restricts the notion of time to something that one can measure with a single clock and explicitly disallow the notion of "synchronizing" clocks, all the "paradoxes" disappear.

There isn't any fundamental reason why clocks that take different paths through space-time SHOULD agree when they meet up again, just as there isn't any reason why the odometers of two different cars who take different paths through space should agree when they both arrive at the same destination.

After one realizes that this (admittedly extreme) measure gets rid of the paradox, one can re-introduce clock synchronization as a concept that depends on "an observer" rather than as something that's defined in a universal matter, and realize that there are still no paradoxes.

 

[nosepot]

Suppose the traveling twin is in a windowless ship and after he embarks has no knowledge of the outside. How could he program his ship in advance to stop when he has reached his destination?

He would anticipate that time would slow down for him when he gets to speed, so he better decelerate a little sooner than he would have done if using classical physics to plot the course.

Once in the rocket he has no sense that time is slowed for him, and arrogantly reconsiders the correctness of relativity. He decides Newton was a legend and had it right all along and that he will continue on the classical physics course, when all of a sudden *boom* we've arrived, and he's burning up in the atmosphere of a Class M planet. "It wasn't as far away as I thought", he thinks!

So, am I still wrong that the traveling twin experiences length contraction of the distance between the origin and destination?

 

[PeterDonis]

If you are using a spacetime diagram to figure out how many wavelengths each counts thne you are using both time dilation and length contraction.

Really? How so? Where do either of those things appear in a spacetime diagram?

You appear to have things backwards. Length contraction and time dilation are not more fundamental than spacetime geometry. It's the other way around.

The resolution must be in understanding why the traveling twin does not manage to get as many wavelength out during his journey.

Because the length of the traveling twin's worldline is shorter than the length of the stay-at-home twin's worldline, between the two events that are common to both of them (the events at the start and end of the journey). It's just geometry.

Without the length contraction for the traveller (and we've ruled out acceleration as an issue), the experience of each twin is completely symmetric.

No, it isn't. The Doppler shift of the light signals is a direct observable, and it's different for the two twins. Once again, you have things backwards: you are trying to explain frame-independent, direct observables in terms of frame-dependent things like length contraction and time dilation. That doesn't work.

 

[PeterDonis]

am I still wrong that the traveling twin experiences length contraction of the distance between the origin and destination?

I don't think anyone said you were wrong to think that. All I'm saying is that the length contraction, which is frame-dependent, is the wrong thing to focus on if you are trying to explain frame-independent direct observables, like the difference in aging in the two twins when they meet up again, or the difference in observed Doppler shifts for the two twins during the journey. To explain those, you need to look for something frame-independent, like the geometry of spacetime.

 

[PeterDonis]

Suppose the traveling twin is in a windowless ship and after he embarks has no knowledge of the outside. How could he program his ship in advance to stop when he has reached his destination?

By using the equations of relativity. It's true that he won't get any external information that the equations are correct during the journey, but as you note, he certainly will at the end.

And note that when I say "the equations of relativity", I don't mean length contraction and time dilation. I mean simple spacetime geometry: he just has to compute the length of the worldline he intends to follow. Note, though, that "length" is spacetime length, *not* spatial length; he's not computing length contraction of the distance to the turnaround point.

Once in the rocket he has no sense that time is slowed for him, and arrogantly reconsiders the correctness of relativity. He decides Newton was a legend and had it right all along and that he will continue on the classical physics course, when all of a sudden *boom* we've arrived, and he's burning up in the atmosphere of a Class M planet. "It wasn't as far away as I thought", he thinks!

In other words, he runs an experiment that will give one result if relativity is correct, and a different result if Newtonian physics is correct. The actual result confirms relativity and falsifies Newtonian physics. What's the problem?

 

[ghwellsjr]

This is a good point, because it highlights why SR is so confusing. The answer is both two contractions and two extensions. An object accelerating to join another frame would be seen to contract from the view of the frame it leaves, and uncontract from the view of the frame it joins. That's simple relativity, is it not?

No, that's a confusing way to understand SR. There was a comment in both of the papers that you linked to earlier in this thread that stated that you should consider just one Inertial Reference Frame (IRF) when analyzing a scenario. You should follow that advice. So you should avoid statements like "an object accelerating to join another frame" or "the view of the frame it leaves". That's what creates confusion. Stick with one frame at a time. When you're all done, you can transform the entire scenario into any other IRF you want using the Lorentz Transformation process. It doesn't have to be one in which any object is at rest. Disassociate the idea that only the rest frame of an object is valid or that every object must have its own frame.

With this in mind, there is never any confusion about which objects are contracted or how they are contracted. If you detail their dimensions in one IRF, the Lorentz Transformation process will automatically apply the correct contraction. It's so simple, no confusion.

If we may be so brave to consider an ether theory viewpoint, it could either contract or uncontract in an absolute way depending on whether it speeds up or slows down with respect to the ether, but we would have no way to know what it's really doing, as we don't know which observer is moving slower relative to the ether frame. Hence the result would be identical to the SR observations.

If you can consider the single IRF viewpoint that an ether theory viewpoint presents, then just do the same thing with any single arbitrary IRF SR viewpoint.

 

[ghwellsjr]

Nobody biting. I see there was a civil war over this recently!

https://www.physicsforums.com/showthread.php?t=689621&highlight=twin+paradox

Told you the Doppler explanation was terrible.

There was nothing in that thread that would account for your statement that the Doppler explanation was terrible. In fact it is elegant. Actually, as I stated before, it's the raw data that any other explanation must account for. And I did that in the four spacetime diagrams that I presented in post #52 at the top of page 4.

These four diagrams illustrate very clearly how Length Contraction is frame dependent:

In the first diagram depicting the rest frame of the stay-at-home blue twin, the black twin travels 9 light-months away and 9 light-months back.

In the second diagram depicting the rest frame of the black twin during the first "half" of the scenario, the blue twin moves away to a distance of 7.2 light-months and then the black twin has to travel 22.5 light-months to catch up to him.

In the third diagram depicting the rest frame of the black twin during the last "half" of the scenario, both twins are traveling but the black twin has traveled 22.5 light-months when he reaches his maximum distance of 7.2 light-months from the blue twin. Then he stops and waits for the blue twin to catch up to him.

In the fourth diagram depicting the non-inertial rest frame of the black twin, the blue twin travels away to a distance of 4.5 light-months, remains there for awhile and then returns.

In all of these diagrams, Length Contraction, Time Dilation and Simultaneity are different but the exact same Doppler information is maintained and the all-important speed of light is always c. Each twin always sees the other twin's clock running the same compared to his own throughout the scenario no matter what frame we use to describe the scenario. It's clearly presented. Why do you call it "terrible"?

 

[GrayGhost]

Looks like many folks have aleady posted on this, but if I may ...

Can someone please clarify for me whether length contraction in special relativity is considered a physical effect (a contraction of a cohesive material) or a kinematic effect (applied to the space the material occupies)? I've been thinking about Bell's Spaceship Paradox this week and realized that it stems from a discrepancy between these two different viewpoints.

Bodily length contraction is a real physical effect, however this is not to imply that the moving body ever changes in and of itself due to change in its own state of motion, or that it changes in and of itself simply because "a moving other" happens to gaze upon it. The body is measured its original proper length (call it L) by a ruler at rest with it, and it is measured a contracted length (L/γ) by a ruler in relative motion with it. If it's measurable, its real and physical. Both POVs are correct, even though they both beg to differ. They are allowed, and actually required to disagree of the body's length, when in relative motion with each other. It is true that kinematics (relative motion) produces the bodily length contraction, given the 2 relativity postulates true. Yet, it remains a real physical effect.

The spaceships are identically accelerated from rest to some speed. Therefore they will keep their separation, L, before and after acceleration (as observed in their original rest frame); although, each spaceship will be length contracted due to its speed relative to the rest frame.

Correct, however all the atoms of the string connecting them must also length contract, just as the rockets do. Per the launchpad, they are all always contracted by 1/γ because they each always possesses the very same velocity during their accelerations ... per the launchpad.

The paradox arises from the following. If the experiment is repeated with an inelastic string attached to the same point on each spaceship (say the back, near the rocket), then the entire connected setup can be considered as one large spaceship and so should under go length contraction as a whole, causing the string and hence the distance between the string attachment points to decrease. However, Bell poses the paradox in such a way that the string is too weak to draw the spaceships closer, and hence breaks.

correct. That is the scenario definition, and as such, we consider reality based upon that alone.

If length contraction is purely kinematic, then the string should feel no stress as the entire setup contracts; but then why are the spaceships not drawn closer when accelerated without a string present? A notion that resolves the issue is that the interatomic forces of the contracting string draw the spaceships closer as the string contracts, but I think this is at odds with standard interpretations of what length contraction means in special relativity (or is it?).

The scenario called for the ships to remain separated by their fixed original separation during their acceleration. So by scenario definition, the string (and each rocket) must length contract between rockets that never change in their separation per the launchpad POV, not per the string or either rocket. We could have considered an entirely different scenario if so desired ... Had the scenario instead said the rockets would not accelerate identically per the launchpad, then the rockets could be defined to accelerate in a way that their separation (per launchpad) always precisely matches the string's contracted length as they go (per launchpad). As such, no stress would be felt by the string (wrt relativistic effects), and so the rockets would always be they same separation "per the rockets and the string", as opposed to the launchpad POV. The taut string would never break.

I've seen some proposed solutions to this which move from the rest frame to the frame of the rockets, but this does not seem necessary, as the paradox occurs in the original rest frame, so it should be possible to resolve it without changing frames. Any ideas? Thanks.

The paradox is a "presumed" paradox, meaning it is a mistaken assumption that does not exist in reality. It's simply a misunderstanding of the scenario that gives rise to it. No matter what POV one considers, the outcome is the same. If the scenario setup requires the string to break in one POV, then it will break in all POVs. If the scenario setup instead required the string to NOT break in one POV, then it will NOT break in all POVs. So you are right in that it does not matter what frame one considers it from.

 

[GrayGhost]

nosepot,

You asked whether bodily length-contraction is a real physical effect.

Now, I could have just said that the moving space "that the moving pencil is at rest in" length-contracts, and so so too does the pencil ... because its atoms contract right along with the space they are at rest with. There is space between atoms, beyond atoms, and inside atoms. The size of the atoms are defined by a plot of locations in their spacetime system, and that spacetime system length-contracts when in motion, and so so too do the atoms and hence the body length. However, this response is somewhat insufficient IMO. Time desynchronisation cannot be ignored in any good explanation of the realness of bodily length-contraction, otherwise the meaning of it all is missed. Considering both time desynchronisation in unison with bodily length contraction allows one to undertand why the moving body REALLY IS length contracted per the stationary observer (because it's measurable, and the math requires it so), while at the same time possesses an understanding as to how the moving body is also always its original proper length per itself (because it's measureable per a ruler at rest with the body).

No body ever changes in and of itself simply because it changes in its own state of motion, or becomes gazed upon by relatively moving others. Relative velocity produces the relativistic effect of length contraction of moving bodies, as recorded by the inertial spectator. However, how does it contract if it also never changes in and of itself? ...

All relativistic effects arise and vanish in unison, when relative motion arises and vanishes. Bodily length-contraction is not the only relativistic effect. We may envision the moving body to have clocks affixed eveywhere inside synchronised with each other. Per the stationary observer, the body's fwd clocks will lag in time readout wrt aftward clocks. That's called time desycnrhonisation, another relativistic effect. Each atom along its length exists in a different time era of the moving body. The desynchronisation arose from v>0 just as length contraction did, and as it turns out, the length contraction cannot be fully explained without it.

In analogy, we might image a pencil at rest in a spacetime system S. An observer off the side of the pencil holds a ruler parallel to the pencil and records its length, L. Next, the pencil is rotated angularly thru (say) 60 deg, and the pencil appears shorter per the observer. He's told he cannot rotate his ruler to align it with the pencil-axis to verify if the pencil's length really changed or not. Deep down, from everyday experience, he knows the pencil length now spans a depth dimension it did not span before, and that the pencil still has its proper length of L. All micro wonder clocks affixed everywhere to the pencil always remain synchronised per both the pencil and the observer. However ...

In the case of relativity, the spacetime systems of the moving pencil and the observer (who measures it) possesses an angular orientation differential in the fused 4 dimensional spacetime continuum. That is, the 2 frames are angularly rotated wrt one another, called frame rotation, and that drives all the relativistic effects. The pencil & observer's systems are angularly rotated (in spacetime) wrt each other, because a relative velocity exists between them. Therefore, the observer cannot just turn his ruler to realign it parallel (in 4d) with the pencil, but instead must accelerate to the pencil's state of motion to become at rest with it, and then his ruler is again parallel with the ruler's length axis in 4d spacetime. OK, so the moving pencil is angularly rotated in 4d spacetime wrt the observer (and his ruler), and so the ruler measures a shorter pencil length, ie a length contracted body. Now then, is the ruler truly contracted in length, or is it an illusionary effect of sort? ...

The math requires the contracted length be real, just as time dilation, time desynchronisation, and all other relativistic effects are real. Light signals (theoretically) could verify this, technology permitting. If it's measureable, its a real physical effect. If the math requires it to exist as such, it must be real. As such, the moving contracted body length is a real physical effect ... even though said body never changes in and of itself as it goes, as it accelerates, or if moving others happen to gaze upon it. It's proper length never changes, and that's just as real and physical. Bottom line, it is not required that a moving contracted length be non-physical simply because it also "always holds itself" at its original proper length. All POVs are equally correct, and each their own measurements are equally real "to them". It's a (real physical) relativistic effect, not an illusionary (or optical) effect.

Hope that helps.

 

[ghwellsjr]

The body is measured its original proper length (call it L) by a ruler at rest with it, and it is measured a contracted length (L/γ) by a ruler in relative motion with it. If it's measurable, its real and physical. Both POVs are correct, even though they both beg to differ. They are allowed, and actually required to disagree of the body's length, when in relative motion with each other. It is true that kinematics (relative motion) produces the bodily length contraction, given the 2 relativity postulates true. Yet, it remains a real physical effect.

Length Contraction is a coordinate effect, meaning that it is different in each frame that is established. It cannot be measured with just a ruler, unless you know something that I don't know. What exactly did you have in mind?

 

[nosepot]

Hope that helps.

Yes, it does. Thank you for your reply.

What do you reckon about the part that length contraction plays in the explanation of the twin paradox?

 

[nosepot]

That is wrong - of course it does. If you are using a spacetime diagram to figure out how many wavelengths each counts thne you are using both time dilation and length contraction.

Really? How so? Where do either of those things appear in a spacetime diagram?

Ok, I understand where we have been disgreeing all this time. I'm seeing the length contraction and time dilation as the fundamental effect which the spacetime diagram explains. You see the spacetime as the fundmental of which length contraction and time dilation are a result.

Fair enough. I don't think it's right to say that we can show is more correct than the other though. If you ask in SR what's more fundamental, then it's obviously spacetime, by definition.

I think that's a philisophical point. However, you must understand that most people come to SR with a classical physics schooling. From a pedegogical viewpoint it would make sense to start with time dilation and length contraction for each frame and then convince that they are just artifacts of a greater spacetime union.

I don't think there's anything wrong with my view, I just couldn't officially call it a Special Relativity view.

 

[PeterDonis]

I'm seeing the length contraction and time dilation as the fundamental effect which the spacetime diagram explains. You see the spacetime as the fundmental of which length contraction and time dilation are a result.

Just to clarify, I see spacetime as fundamental, not the diagram. The diagram is just a handy tool for helping us to visualize spacetime. And also, I agree that you can certainly use spacetime diagrams to illustrate length contraction and time dilation; but you don't *need* to do that to use spacetime diagrams to make correct predictions.

I don't think it's right to say that we can show is more correct than the other though.

I agree, because any viewpoint that allows you to make correct predictions must be considered correct from a scientific viewpoint.

However, the questions you have been asking are not about what is correct, but about how to explain "why" whatever is correct, is correct. That, IMO, is where the philosophy comes in, because whatever criteria you are using to determine what counts as a valid explanation for you, they seem philosophical to me. And to be fair, so are the criteria I'm using to determine that "spacetime geometry" counts as a valid explanation for me, and that length contraction and time dilation are derived effects.

 

[nosepot]

All I'm saying is that the length contraction, which is frame-dependent, is the wrong thing to focus on if you are trying to explain frame-independent direct observables, like the difference in aging in the two twins when they meet up again, or the difference in observed Doppler shifts for the two twins during the journey. To explain those, you need to look for something frame-independent, like the geometry of spacetime.

Do we really need to go spacetime to get an answer? As a less capable learner (which I must be) it is more natural to place myself sequentially in either frame and play each out with an understanding that frames moving relative to me will be contracted and they will experience time more slowly. I don't see what's wrong with that, even if you consider it a less sophisticated way to view the problem.

 

[nosepot]

However, the questions you have been asking are not about what is correct, but about how to explain "why" whatever is correct, is correct. That, IMO, is where the philosophy comes in, because whatever criteria you are using to determine what counts as a valid explanation for you, they seem philosophical to me. And to be fair, so are the criteria I'm using to determine that "spacetime geometry" counts as a valid explanation for me, and that length contraction and time dilation are derived effects.

Point taken. Thanks. Not sure that approach helps the uninitiated too much though in the beginning.

 

[nosepot]

There was nothing in that thread that would account for your statement that the Doppler explanation was terrible. In fact it is elegant. ... Why do you call it "terrible"?

Beauty is in the eye of the beholder. I would not consider that explanation anything less than bamboozling. I'm trying to appeal to the idea of giving a plain explanation for a lay person. I don't dispute your spacetime diagram will convey that to a skilled individual, but to a layman it looks like trickery. Hidden in there somewhere, is the notion that the distance between origin and destination is apparently shorter for the traveller, which would probably satisfy most who protest that a paradox exists.

The other thing I don't like about the Doppler version is that we must correct for the travel time of the light when solving from each inertial frame in turn to figure out how each appears to age (this is how most people would do it - sit themselves in the positions of each twin and start the experiment). The traveling twin is trying to observe that the Earth twin is aging slower, but on the outward journey the wavefronts trickle in and during the return trip he is bombarded. The same is true for the Earth twin. It's very confusing... but, each to their own.

I'm giving you insight into the brain of a Special Relativity retard. This should be gold! :)

 

[nosepot]

Also, I was wrong about the target planet "looking" immediately bigger from the point of view of the traveling twin. Some simple trigonometry shows a shortening of the ship would make image appear smaller in the front window. There is a cool video here (you've probably seen it) which shows an animation of the effect where the distance clouds recede at the start:

www.youtube.com/watch?v=JQnHTKZBTI4

Strange stuff.

 

[PeterDonis]

Do we really need to go spacetime to get an answer?

No. You can work things out in a particular frame, as ghwellsjr has pointed out. I am not saying that the spacetime method is the only way to get the right answer. I am only saying that, for me at least, the spacetime method presents a much cleaner conceptual picture.

Point taken. Thanks. Not sure that approach helps the uninitiated too much though in the beginning.

Not the way relativity is often taught, no. But that doesn't mean there aren't better ways to teach it (or at least to try).

The way relativity is often taught, IMO, wrongly tries to take slowly the process of dismantling one's natural intuitions about how space and time work. That's what leads to all the talk about length contraction, time dilation, relativity of simultaneity, etc. It tries to postpone the realization of how conceptually radical relativity actually is. But this often backfires, because the slow approach keeps up the hope in one's mind that there will be *some* intuitions that don't have to be unlearned. Then, when it turns out that those intuitions, too, are wrong, one hits a wall.

IMO it's better, at least if a person is serious about learning relativity (as opposed to just wanting some quick pop science "sound bites"), to dive right into the deep end of the pool and say, right up front, that *every* intuition you have about how space and time works is wrong: that *all* of the intuitive concepts you bring from classical physics have to be unlearned, that in order to really understand how relativity works, you have to build up a whole new set of concepts, things like spacetime and invariants. The shock of the cold water is drastic, but it can help you to get used to it faster.

 

[lugita15]

You didn't transform the laws of physics. I doubt that you could do that. I couldn't do it. I don't know how. I trust the experts that say that when they transform Maxwells's equations using the Lorentz Transformation, they come out the same.

You don't have to trust the experts, when you can have the experts explain it to you. See this excerpt from the Feynman Lectures, especially the last two pages. It shows how Lorentz got the Lorentz transformations from Maxwell's equations.

 

[ghwellsjr]

Beauty is in the eye of the beholder. I would not consider that explanation anything less than bamboozling. I'm trying to appeal to the idea of giving a plain explanation for a lay person. I don't dispute your spacetime diagram will convey that to a skilled individual, but to a layman it looks like trickery. Hidden in there somewhere, is the notion that the distance between origin and destination is apparently shorter for the traveller, which would probably satisfy most who protest that a paradox exists.

Didn't you read post #84 of this thread? I said in the first diagram of the other thread depicting the rest frame of the blue stay-at-home twin (who remains at the spatial origin) that the black twin's destination is 9 light-months away. I even put the 9 in bold so you wouldn't miss it. It's at the coordinate time of 15 months. Do you see it? Maybe it would help to know that distances between two of the thick lines are established along the horizontal grid lines.

Then I said in both of the next two diagrams, the maximum distance the blue twin gets away from the black twin is 7.2 light-months. In the second diagram this happens at 12 months and in the third diagram it happens at 25.5 months. Do you see those two distances?

And then in the fourth diagram the blue twin only gets 4.5 light-months away from the black twin. Can you see it?

Hopefully, my extra explanation will help you understand where those distances are. If not, please try to help me understand why you think they are still hidden. I'd like to be able to communicate this to a lay person.

The other thing I don't like about the Doppler version is that we must correct for the travel time of the light...

No you don't; at least you don't have to do anything special. As long as you use units like months and light-months, you just draw in the light signals along the 45-degree diagonals. It's especially easy on a computer--you just make sure there are no kinks in the lines. The only part that might be less than trivial is knowing where to place the dots for any observer that is moving but that is merely the Time Dilation factor (gamma) which I'm sure you already know about. You just place the dots higher up in Coordinate Time by the gamma factor. In this example, gamma is 1.25 at a speed of 0.6c so the first dot for the traveler is placed at the Coordinate Time of 1.25 months.

Or another way you can do it is to divide that Coordinate Time for each leg of travel by the gamma factor for that travel speed and then place the dots evenly spaced along the line. So for the first diagram, the traveling twin arrives at his destination at the Coordinate Time of 15 months so the Proper Time on his clock is 15/1.25=12 months. You can see that there are a total of 12 segments representing 12 months each along both of the black twin's travel legs.

If you had instead specified the time that the traveling twin took before he turned around according to the Proper Time on his clock, in this case 12 months, then you multiply that by the gamma factor to get the Coordinate Time. I presume you know what slope to draw the line for any particular speed.

when solving from each inertial frame in turn to figure out how each appears to age (this is how most people would do it - sit themselves in the positions of each twin and start the experiment).

But to solve for each additional inertial frame, you just decide on a speed that you want it to move relative to the original one and then you plug the coordinates of each worldline's endpoints into the Lorentz Transformation formulas and plot them on a new graph and connect them with appropriate colored lines. Then you calculate gamma for each line segment based on its speed or you can simply just place the same number of dots equally spaced on each line segment. Finally, you draw in new light signals along the diagonals, just like you did before.

The traveling twin is trying to observe that the Earth twin is aging slower,

But neither twin can actually observe the Time Dilation of the other twin. All they can do is observe the Doppler, which simply means they observe the progress of the other ones clock or they receive radio signals sent out at a predetermined and agreed on rate compared to their own. Then they can make some assumptions and construct a spacetime diagram after the scenario is all over but the specific diagram they make is dependent on the assumptions they make.

but on the outward journey the wavefronts trickle in and during the return trip he is bombarded.

True, that's exactly what he will observe, no matter which frame you chose to depict it in. Have you noticed that? This is a description of what actually, in reality, for real happens. Why shouldn't this be pointed out in any analysis of the Twin Paradox?

The same is true for the Earth twin.

Not true. The same does not happen for the Earth twin. The Earth twin does not see the change from trickle to bombarded until a long time after the half-way point of the time the traveler is gone. The faster the speed of the traveling, the more lopsided this change happens. And again, this is what actually happens.

It's very confusing... but, each to their own.

I'm giving you insight into the brain of a Special Relativity retard. This should be gold! :)

But it would give me great pleasure to know that you have learned and can even teach this to someone else. That's the stated purpose of this forum. Don't give up. Ask questions until the confusion evaporates. Remember, Einstein said it was a simple theory and he was a genius so he should know.

 

[PeterDonis]

The other thing I don't like about the Doppler version is that we must correct for the travel time of the light

No, that's exactly what you *don't* have to do with Doppler. Time dilation is where you have to correct for light travel time.

 

[ghwellsjr]

I think a good way to teach Special Relativity to a complete novice is to start with a laser rangefinder which can be purchased at any hardware store. Explain that it determines distance by sending out a short burst of laser light and measuring how long it takes for the reflection to get back. Then knowing the speed of light, it calculates the round trip distance the light had to take and divides by two. So we can easily determine the distance to any object that isn't moving with respect to us with this tool.

Now we consider how to measure the distance to an object that is moving directly toward or away from us. Since the distance is changing, we have to establish when we want to apply the measured distance. We could apply it when the light was sent, or when it was received, or any other time in between but it would seem reasonable to apply it at the average time between the two extremes. And that's what we do in Special Relativity. Not only that, but we establish that the time the laser light reflected off the object is the same as the average time we calculated, no matter how fast or slow or even stopped the object is with respect to us.

Finally, we want to measure the speed of an object moving directly toward or away from us. That's simple, we just take two readings of distance applied as determined in the previous paragraph and from that we can determine the average speed during those two measurements.

Seems like a good way to start for me.

 

[GrayGhost]

Length Contraction is a coordinate effect, meaning that it is different in each frame that is established. It cannot be measured with just a ruler, unless you know something that I don't know. What exactly did you have in mind?

Agreed, a coordinate effect.

Far as what I had in mind, there are a number of ways to look at it. For one, a spherical 1 meter starship measures 1 meter long using its own ruler, before it leaves the space station. That ruler is then left behind with the station capt, as the starship departs, turns about, and does a flyby of the station at 0.866c inertial. That starship capt never discerns any change in his own starship's size or proportion as he goes, based on observations made within his own ship. On flyby, the space station capt uses the same ruler to assist measuring the starship on flyby. It is found to measure 1/2 meter.

 

[ghwellsjr]

Agreed, a coordinate effect.

Far as what I had in mind, there are a number of ways to look at it. For one, a spherical 1 meter starship measures 1 meter long using its own ruler, before it leaves the space station. That ruler is then left behind with the station capt, as the starship departs, turns about, and does a flyby of the station at 0.866c inertial. That starship capt never discerns any change in his own starship's size or proportion as he goes, based on observations made within his own ship. On flyby, the space station capt uses the same ruler to assist measuring the starship on flyby. It is found to measure 1/2 meter.

Tell me exactly what he does with the ruler to measure the starship flying by. I understand how to measure the starship when the ruler is at rest with it but you haven't explained how to do it when they are in relative motion. Remember what you said and what I am asking about:

The body is measured its original proper length (call it L) by a ruler at rest with it, and it is measured a contracted length (L/γ) by a ruler in relative motion with it. If it's measurable, its real and physical.

 

[GrayGhost]

Tell me exactly what he does with the ruler to measure the starship flying by. I understand how to measure the starship when the ruler is at rest with it but you haven't explained how to do it when they are in relative motion. Remember what you said and what I am asking about:

Imagine 2 lasers systems are setup at the space station to radiate orthogonally (along z) wrt the pre-planned axis-of-motion (x) of the starship. Each transmits a steady laser beam to a receiver a short distance away, and a processing system stores the data and analyses it using an incredibly fast wonder computer. The lasers are separated by precisely 1/2 meter, using the legendary 1 meter ruler specified in my prior post. When the starship passes by the space station at 0.866c, it momentarily breaks the 1st (aft) laser beam and then subsequently the 2nd (fwd) laser beam. The data shows that the 1st (aft) laser beam's continuity re-established at the moment the 2nd (fwd) laser beam's continuity broke, revealing the starship was precisely 0.5 meter long at v=0.866c (as predicted by the LTs). That same ruler measured the starship at 1 meter length before takeoff from the starship.

 

[GrayGhost]

Yes, it does. Thank you for your reply.

What do you reckon about the part that length contraction plays in the explanation of the twin paradox?

I reckon it plays a critical part of the explanation, however its only the half of it.

This is not an easy topic to discuss without spacetime figures, because of the dynamics. Let's assume twin B does a virtually instant acceleration to full speed (0.866c) at takeoff. Just before takeoff, planet X (the turn about point) is 2 ly away. Just after takeoff when at 0.866c, planet X is then at 1 ly range, so the range to planet X became contracted by 50% per B. Here's the rub ...

although planet X is then 1 ly distant per B, the separation between Earth and planet X (relative to takeoff) is not 1 ly, but rather 4 ly. That is, it's a 200% dilated length wrt its proper sep of 2 ly. One might ask, what the hell you talkin about? I might answer, "I have no idea and should probably have not forgotten to take my meds this morning". However as it turns out, that's not my answer at all ...

It dilates to 4 ly because the location of planet X relative to B's takeoff exists in B's past, not B's NOW. His NOW said it was 2 ly distant before takeoff at v=0, and his NOW says it's 1 ly distant after takeoff at steady v=0.866c. However, after takeoff, and after rapidly attaining 0.866c, planet X had to exist far in B's own past (not his NOW) at a range of 4 ly ... for X to then be at 1 ly range NOW (per B). It's as though the rapid acceleration of B caused the location of X to fast forward 75% along its own worldline, from 4 ly to 1 ly range. In fact, as though is not accurate. That's what Einstein's theory requires, per B. The location of planet X "at takeoff" goes from 2 ly just before the virtually instant acceleration, to 4 ly just after the acceleration, because planet X's position at takeoff shifts from B's NOW to a point in B's PAST. That shifting into the past makes the interval longer, and a longer duration requires a longer earth-planet X separation. That's how is must exist in B's spacetime system, yet B can only be aware of this by accounting for the noted doppler shift in received light from X, as per SR's doppler formula. The doppler shift would prove planet X reshifted within B's spacetime system, but not because X accelerated ... but rather only because Bs own POV rotated during his proper acceleration.

 

[ghwellsjr]

Tell me exactly what he does with the ruler to measure the starship flying by. I understand how to measure the starship when the ruler is at rest with it but you haven't explained how to do it when they are in relative motion. Remember what you said and what I am asking about:

Imagine 2 lasers systems are setup at the space station to radiate orthogonally (along z) wrt the pre-planned axis-of-motion (x) of the starship. Each transmits a steady laser beam to a receiver a short distance away, and a processing system stores the data and analyses it using an incredibly fast wonder computer. The lasers are separated by precisely 1/2 meter, using the legendary 1 meter ruler specified in my prior post. When the starship passes by the space station at 0.866c, it momentarily breaks the 1st (aft) laser beam and then subsequently the 2nd (fwd) laser beam. The data shows that the 1st (aft) laser beam's continuity re-established at the moment the 2nd (fwd) laser beam's continuity broke, revealing the starship was precisely 0.5 meter long at v=0.866c (as predicted by the LTs). That same ruler measured the starship at 1 meter length before takeoff from the starship.

Oh, I thought you meant that he was going to measure the passing starship with just his ruler like he did when they were at rest together.

Instead, you have applied a "measurement" that involves the prior adjustment of two clocks.
And how do you do that? Well, one way is to just start with the clocks unadjusted and then do your measurement of the starship. Chances are that it comes out to be some value other than 1/2 meter. So you tweak one of the clocks in the correct direction to get it closer to 1/2 meter. Keep repeating until it comes out exactly 1/2 meter. Do you call this a measurement?

Lest you think that I'm playing foul, realize that any other way of adjusting those two clocks is exactly equivalent to the method I just described.

 

[ghwellsjr]

I reckon it plays a critical part of the explanation, however its only the half of it.

This is not an easy topic to discuss without spacetime figures, because of the dynamics. Let's assume twin B does a virtually instant acceleration to full speed (0.866c) at takeoff. Just before takeoff, planet X (the turn about point) is 2 ly away.

You're right. You need some spacetime figures. Here's one for your scenario in the earth/planet's mutual rest frame. Earth is in red, planet X is in black and twin B is in blue. The dots represent one-month intervals of Proper Time for each observer/object. I have drawn in some thin black Doppler signal lines:

Just after takeoff when at 0.866c, planet X is then at 1 ly range, so the range to planet X became contracted by 50% per B.

Here's the same scenario transformed to the rest frame of twin B for his outbound trip:

Here's the rub ...

although planet X is then 1 ly distant per B, the separation between Earth and planet X (relative to takeoff) is not 1 ly, but rather 4 ly. That is, it's a 200% dilated length wrt its proper sep of 2 ly. One might ask, what the hell you talkin about? I might answer, "I have no idea and should probably have not forgotten to take my meds this morning".

I agree with you: you don't know what you're talking about.

However as it turns out, that's not my answer at all ...

It dilates to 4 ly because the location of planet X relative to B's takeoff exists in B's past, not B's NOW. His NOW said it was 2 ly distant before takeoff at v=0, and his NOW says it's 1 ly distant after takeoff at steady v=0.866c. However, after takeoff, and after rapidly attaining 0.866c, planet X had to exist far in B's own past (not his NOW) at a range of 4 ly ... for X to then be at 1 ly range NOW (per B). It's as though the rapid acceleration of B caused the location of X to fast forward 75% along its own worldline, from 4 ly to 1 ly range. In fact, as though is not accurate. That's what Einstein's theory requires, per B. The location of planet X "at takeoff" goes from 2 ly just before the virtually instant acceleration, to 4 ly just after the acceleration, because planet X's position at takeoff shifts from B's NOW to a point in B's PAST. That shifting into the past makes the interval longer, and a longer duration requires a longer earth-planet X separation.

While it's true that the initial events for Earth and planet X which were simultaneous in their mutual rest frame, have a Coordinate Distance for those same two events in the twin's initial rest frame that are separated by about 4 light-years, those two events are not simultaneous in this second frame and so you can't compare their separation with those two events. Instead, you have to pick two events on their respective worldlines that are simultaneous, for example, at the Coordinate Time of 0 where the black line for planet X is at the Coordinate Location of 12 light-months or 1 light-year. If you want to go into planet X's distant past, you have to also go into Earth's distant past (which I didn't draw) and you would see that their separation is a constant 12 light-months.

That's how is must exist in B's spacetime system, yet B can only be aware of this by accounting for the noted doppler shift in received light from X, as per SR's doppler formula. The doppler shift would prove planet X reshifted within B's spacetime system, but not because X accelerated ... but rather only because Bs own POV rotated during his proper acceleration.

I have no idea what your point is about the Doppler shift but whatever it is, it doesn't change with the reference frame. I have drawn some signal lines from planet X to twin B along 45-degree diagonals. The Doppler ratio is about 3.73 which means that 15 of planet X's months are seen by twin B in 4 of his months. After twin B reverses direction, it's the other way around, twin B sees 3.75 of planet X's months in 14 of his own months.

I think involving Doppler from planet X only confuses things. The Doppler that matters is between the two twins, which I didn't draw in.