Bell's Spaceship Paradox and Length Contraction

[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.

 

[nosepot]

Didn't you read post #84 of this thread?

I did. The answer is there, but I'm asking you to consider if the protracted answer you have given me would connect with a reasonably smart 10 year old (about the level I'm putting myself)? Try again but don't say gamma, worldline, lorentz, transform, etc. Is there a more lay explanation?

I do now see the utility of Doppler over length contraction as an explanation though, as the traveller could spontaneously turn around at any time and come back... and GrayGhost's non-spacetime explanation blew my mind.

How about this to highlight the asymmetry?:

It's the traveller who decides when to turn around. The moment he does, he experiences the increased frequency of arrival of the signals from earth. However, the Earth twin must wait for information of this turn around to reach him at light speed, after which he will receive an increase in signal frequency from the traveller. Therefore the Earth twin counts less signals, because the traveller dictates proceedings.

 

[Dale]

Try again but don't say gamma, worldline, lorentz, transform, etc. Is there a more lay explanation?

The vocabulary is an important part of any field of study. If you want to learn relativity then you need to learn the vocabulary.

Would you try to learn to repair cars and ask the mechanic not to use words like "alternator" and "camshaft"?

It's the traveller who decides when to turn around. The moment he does, he experiences the increased frequency of arrival of the signals from earth. However, the Earth twin must wait for information of this turn around to reach him at light speed, after which he will receive an increase in signal frequency from the traveller..

That sounds good to me. The key point is the asymmetry.

 

[ghwellsjr]

How about this to highlight the asymmetry?:

It's the traveller who decides when to turn around. The moment he does, he experiences the increased frequency of arrival of the signals from earth. However, the Earth twin must wait for information of this turn around to reach him at light speed, after which he will receive an increase in signal frequency from the traveller. Therefore the Earth twin counts less signals, because the traveller dictates proceedings.

That's fine but what was wrong with wikipedia's similar statement that you linked to in post #69 where you summarized it with the word "argh!":

The asymmetry between the Earth and the spaceship is manifested in this diagram by the fact that more blue-shifted (fast aging) images are received by the ship. Put another way, the spaceship sees the image change from a red-shift (slower aging of the image) to a blue-shift (faster aging of the image) at the midpoint of its trip (at the turnaround, 5 years after departure); the Earth sees the image of the ship change from red-shift to blue shift after 9 years (almost at the end of the period that the ship is absent).

 

[ghwellsjr]

Didn't you read post #84 of this thread?

I did. The answer is there, but I'm asking you to consider if the protracted answer you have given me would connect with a reasonably smart 10 year old (about the level I'm putting myself)? Try again but don't say gamma, worldline, lorentz, transform, etc.

Don't you think a smart 10 year old would look up any word he wasn't familiar with or ask what it meant? Do you need more explanation for any of those words?

Is there a more lay explanation?

I gave a start in post #99. Did you read it? Did it make sense? I realize it's only a start but how was it for a start?

I do now see the utility of Doppler over length contraction as an explanation though, as the traveller could spontaneously turn around at any time and come back...

Good, that's what we want, understanding.

and GrayGhost's non-spacetime explanation blew my mind.

What does "blew my mind" mean?

 

[Samshorn]

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... 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.

Not true. Contrary to what your comments imply, the Lorentz invariance of physical objects and processes (including length contraction and time dilation) is neither conventional nor tautological. It has perfectly well-defined meaning, and can be tested empirically. I think your comments are unfortunate, because they may mislead noobies into thinking that relativity is simply a collection of circular definitions, and that it isn't possible to measure any frame-dependent quantity. What you're missing is the synchronization already entailed by the definition of inertial coordinate systems (meaning systems in which no fictitious forces appear in the statement of the laws of mechanics).

 

[GrayGhost]

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.

Well, the only problem with your assumption is that eyeballing a passing starship traveling at 86.6% light speed on passing with a stationary meter stick would be rather difficult. Yes?

Instead, you have applied a "measurement" that involves the prior adjustment of two clocks.

Agreed, however it also makes use of the original meter stick.

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?

All one does is synchronise the 2 clocks using the usual Einstein clock synchronisation method, before takeoff. The test only need be run once. Nothin to it.

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.

But you said to take the measurement multiple times with unadjusted clocks and reteeking the clocks until one force fits a 1/2 meter passing starship to be recorded, and I said to take the measurement (only once) with clocks adjusted prior per a pre-takeoff clock sync procedure and the recorded startship length should precisely match the LT predicted contracted-length. How then is your method equivalent to mine?

 

[Dale]

Not true. Contrary to what your comments imply, the Lorentz invariance of physical objects and processes (including length contraction and time dilation) is neither conventional nor tautological. It has perfectly well-defined meaning, and can be tested empirically. I think your comments are unfortunate, because they may mislead noobies into thinking that relativity is simply a collection of circular definitions, and that it isn't possible to measure any frame-dependent quantity.

I tend to agree with Samshorn here. ghwellsjr is 100% correct that the invariance of the one way speed of light is conventional. You can always define your space and time coordinates such that the speed of light is invariant. However, after you have fixed your convention you still have three free parameters that are not constrained by convention, but by the physics.

Your synchronization convention can reposition where those three free parameters show up, but it cannot get rid of them. Relativity predicts unique values for those three parameters and nature agrees: http://rmp.aps.org/abstract/RMP/v21/i3/p378_1

 

[Nugatory]

problem with your assumption is that eyeballing a passing starship traveling at 86.6% light speed on passing with a stationary meter stick would be rather difficult. Yes?

Define the length of the passing starship to be the distance between the point where its nose was and the point where its tail was, at the same time according to the observer measuring the length. This can be done by stationing observers all along the anticipated path of the ship, all at rest in the frame which we're measuring the length and all carrying synchronized clocks. Each one writes down on a piece of paper "At time X the nose passed me; at time Y the tail passed me". At our leisure and after the ship has passed, we examine all these pieces of paper; when we identify a pair of observers such that one of them saw the nose at time T and the other saw the tail at the same T, we use our meter stick to measure the distance between them.

That's the contracted length of the spaceship in the frame in which the observers are at rest, and it clearly depends on those all-important words "at the same time".

 

[GrayGhost]

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:

Your figure here looks OK ghwellsjr. However, it seems to focus only on the doppler and only during the inertial phases of twin B's flight, and does not present what happens during the virtually instant twin B acceleration at beginning, nor deceleration at the end. Those are more important to understanding my prior post.

 

[GrayGhost]

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

Well, we shall have to see about that then.

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, ...

Nope. I said 2 ly, not 4 ly. Before twin B's takeoff, the Earth and planet X are separated by 2 ly proper. That be per the earth/planetX rest frame. The dilated 4 ly sep is per twin B alone, and only after he completes his virtually instant proper acceleration from v=0 to v=0.866c ... and that's not the current separation between Earth and planet X per B (which is 1 ly), that's the separation between Earth and planet X wrt the 2 defined events of takeoff and turnabout (which is 4 ly).

, ... 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.

Well, I do realize what you are trying to say here. However, the problem is that I am doing and saying one thing to convey specific points, and you are suggesting I instead say and do something a little different which would convey differing points. I would like to stick to my intended points, thank you.

... , I have no idea what your point is about the Doppler shift but whatever it is, it doesn't change with the reference frame.

If the reference frame is defined as twin B's POV, then it changes during his accelerations. That was my point. Again, you changed the subject from B as reference to A/X as reference. I stated that the doppler freq of light emitted from planet X changes per B as B executes his virtually instant proper acceleration for takeoff. That's the only way B can know "that wrt the takeoff point as reference" planet X digresses into B's own past during B's own proper acceleration upon takeoff.

... 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.

Agreed.

... 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.

To understand the B POV, the light received from planet X is important. It's required to have a complete understanding of how (1) what exists in the B spacetime system in B's realtime (2) relates to what twin B actively experiences in realtime (via light signals during his own proper acceleration).

 

[Dale]

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.

The problem with this approach is that using this definition of simultaneity leads to a coordinate system which is mathematically invalid. A coordinate system must be 1 to 1, but this approach maps multiple coordinates to the same event.

 

[GrayGhost]

Define the length of the passing starship to be the distance between the point where its nose was and the point where its tail was, at the same time according to the observer measuring the length. This can be done by stationing observers all along the anticipated path of the ship, all at rest in the frame which we're measuring the length and all carrying synchronized clocks. Each one writes down on a piece of paper "At time X the nose passed me; at time Y the tail passed me". At our leisure and after the ship has passed, we examine all these pieces of paper; when we identify a pair of observers such that one of them saw the nose at time T and the other saw the tail at the same T, we use our meter stick to measure the distance between them.

That's the contracted length of the spaceship in the frame in which the observers are at rest, and it clearly depends on those all-important words "at the same time".

Agreed. That's really not any different from what I posted prior though. Where you use an infinite number of virtual super-observers to make the detection, I use 2 real detectors. Where you use wonder observers to document it, I used a wonder computer system with disk storage. I positioned my 2 detectors per a pre calculation using the LTs for a preplanned controlled flight test, and the results were as per predicted. I therefore need only run my wonder test once, and if the flight is properly controlled to the precision and accuracy required, then the flight test results match the prior LT prediction.

 

[GrayGhost]

The problem with this approach is that using this definition of simultaneity leads to a coordinate system which is mathematically invalid. A coordinate system must be 1 to 1, but this approach maps multiple coordinates to the same event.

Hello DaleSpam.

Hmmm. Well, I do understand your point wrt the impact of our usual defined rule for simultaneity per Einstein.

I see twin B's coordinate system as "always 1:1", per B himself. However, B will indeed map multiple coordinates to the same event. Personally, I'm OK with that though. I mean, no one said Minkowski's metric should not be allowed for relativistic rates simply because we used the euclidean metric for non-relativistic rates, yes? Given we are looking at a non-inertial B POV, why not allow the coordinate of a single event to change during B's own proper acceleration given relativistic rates exist?

See the animation about half way down this hyperlink's webpage, under the paragraph header "Visualizing the transformations in Minkowski space", on the right side, ...

http://en.wikipedia.org/wiki/Lorentz_transformation

It's clear from this animation that the coordinates for an event change in real time per B during B's own proper acceleration ... given a 1:1 relation of space-to-time is assumed to always exist "per B". Given such an assumption, bodies must whiz about the B spacetime system, even superluminally (B never "sees" this via light signals though, although doppler analysis of received light could prove it true) ... however not because they accelerated, but only because B's own sense of simultaneity dynamically rotates during his proper accelerations.

 

[Dale]

I see twin B's coordinate system as "always 1:1", per B himself.

No, even per B the coordinate system is not 1 to 1. It isn't a matter of perspective or opinion, it is an objective fact about the chosen coordinate system.

why not allow the coordinate of a single event to change during B's own proper acceleration given relativistic rates exist?

Because if a coordinate system is not 1 to 1 then you cannot do well defined coordinate transforms any more. If you cannot do coordinate transforms then you cannot use coordinate transforms to determine the laws of physics in that coordinate system and to transform results to other coordinate systems. So suddenly the coordinates become physically useless.

See the animation about half way down this hyperlink's webpage, under the paragraph header "Visualizing the transformations in Minkowski space", on the right side, ...

http://en.wikipedia.org/wiki/Lorentz_transformation

It's clear from this animation that the coordinates for an event change in real time per B during B's own proper acceleration ...

Thanks for the heads up. I fixed it.

 

[PAllen]

Hello DaleSpam.

Hmmm. Well, I do understand your point wrt the impact of our usual defined rule for simultaneity per Einstein.

Actually, Einstein never used this definition of simultaneity (simultaneity of a momentarily comoving inertial frame) for non-inertial observers. He analyzed non-inertial observers in SR using a single inertial frame; or using GR techniques with a well defined coordinate system. Applying the simultaneity procedure he actually used for inertial frames (radar simultaneity), to non-inertial observers produces a possible well defined set of coordinates; the simultaneity is quite different from that of momentarily com-moving inertial frames. This well defined set of coordinates (radar coordinates) does not have the anomalies of MCIF (or it wouldn't be a well defined coordinate system). However, locally (close to the non-inertial observer's world line) it approaches MCIF; thus it equally well describes local physics as MCIF. It disagrees more and more with MCIF simultaneity for non-inertial motion, the further away you get from the observer's world line.