Twin Paradox: Einstein's Explanation and Alternative Interpretations

[JesseM]

time on Earth does seem to the traveler to speed up while he is accelerating.

Unlike with inertial frames, there is no preferred way to define the coordinate system of an accelerating observer, so this really depends on a totally arbitrary choice of coordinate systems. You can find coordinate systems where the accelerating observer is at rest and time runs faster on the Earth clock, but you can also find coordinate systems where the accelerating observer is at rest and time runs slower on the Earth clock, or runs at exactly the same rate, or alternates running fast and slow, etc.

 

[granpa]

I'm just using the instantaneous frame of the traveler at each instant. I didnt think there was anything controversial about it.

 

[JesseM]

I'm just using the instantaneous frame of the traveler at each instant. I didnt think there was anything controversial about it.

In the instantaneous inertial frame of the traveler at each instant, the Earth-clock is always ticking slower at that instant in that frame, not faster. Only if you construct a non-inertial coordinate system which has the property that its definition of simultaneity at each point on the traveler's worldline matches the definition of simultaneity in the traveler's instantaneous inertial rest frame at that point, and the coordinate time along the traveler's worldline matches the traveler's proper time, can you say that the Earth's clock will be ticking faster in this non-inertial coordinate system.

 

[granpa]

its ticking slower but there is a change of simultaneity from frame to frame. I'm just taking that into account.

 

[phyti]

Hey cos;
I'm a salmon guy, not herring.
Anyway, I agree with you that acceleration implies a change in course, and in the simple twin problem there are only three parts to the trip. But as you add more accelerations for both, it's the total path that counts. At some point, words are not sufficient to explain these things.
Here is a geometrical explanation for the twin aging problem.
Time is not altered by motion, only the rate of activity and thus time measurement. When the twins separate and later rejoin, the same amount of time has elapsed for the universe, but the two clocks have sliced that time interval into different lengths.

The vertical line representing the height of the triangle, is the time for an object not moving, i.e., the minimum interval. Any motion to the right or left results in a longer/dilated interval. Divide the paths at the change of direction for A. For each segment, rotate the minimum interval to horizontal. Where the paths cross the horizontal,draw a vertical
line. Where the vertical intersects the arc, represents the amount of time for each twin for that part of the trip. The one who travels the greatest distance in a given time, will travel the fastest, and thus have more time dilation.

 

[cos]

That's what I said, the experiment was more complicated than Einstein's thought experiment because they had to take gravitational time dilation into account.

So when they took gravitational time dilation into account and eliminated that factor they ended up with Einstein’s prediction which didn't take it into account in the first place.

Was the eventual lag between the HKX traveled clocks after the gravitational time dilation factor was removed from the results any different from Einstein’s predicted lag, unaware of any gravitational time dilation factor?

And I'm certain that when he said "more slowly" he meant something like "more slowly on average over the course of a full rotation", or "more slowly at every instant in the rest frame of the sphere", not "more slowly at every instant in an objective frame-independent sense". For him to mean the last one would be a clear contradiction with his own theory.

You have your interpretation of what Einstein meant by “more slowly”; I have mine.

Every instant that an observer alongside a ‘polar’ clock looks at an ‘equatorial’ clock on a transparent massless sphere the size of the Earth he will see that compared with his observation made at a previous instant the equatorial clock will lag even further behind his own clock indicating to him that between those instances the equatorial clock has continuously ticked over at a slower rate than his own clock.

Er, why would you imagine I meant that? Of course I am talking about Einstein's thought experiment where the sphere is spinning. The point is that the sphere has a center, we can either pick an inertial frame where the position of the center of the sphere remains constant over time, or we can pick a frame where the center of the sphere is moving at some nonzero constant velocity.

Other than as an attempt to confuse the debate why do we even need to “pick a frame where the center of the sphere is moving at some nonzero constant velocity.”? The opinions expressed by an observer in that frame have no bearing whatsoever on the determinations made by an observer in the “inertial frame where the position of the center of the sphere remains constant over time.”

An observer standing next to clock B doesn't have their own inertial rest frame because they're not moving inertially. If we choose the inertial frame in which the center of the sphere is at rest, then in this frame B will be moving at constant speed so it's true that B will be ticking at a constant slowed-down rate, while A will be ticking at a normal rate.

On the basis that the observer at the pole determines that the clock at the equator (B) is “ticking at a constant slowed-down rate” isn’t he of the opinion that clock B is incurring time dilation?

Isn’t an observer accompanying clock B of the opinion that clock A is (or at least appears to be) ticking over at a faster rate than his own clock?

If so, he can either conclude that clock A is incurring time contraction OR that his clock is incurring time dilation (i.e. is “ticking at a constant slowed-down rate”)!

Assuming that he is aware of, and accepts, Einstein’s paragraph 3, chapter 4 comment - that the clock at the equator “goes more slowly” than (i.e. is “ticking at a constant slowed-down rate” compared to) the polar clock - might he not tend to take Einstein’s word for it thus determine that B is not ‘ticking over at a constant increased rate’ but realize that his clock is “ticking at a constant slowed-down rate”?

On the other hand, if we choose a different inertial frame in which the center of the sphere is moving inertially at some constant velocity...

On the other hand we could stick to the subject under discussion and not introduce extraneous materiel that obfuscates same.

Again, the formulas of special relativity are not concerned with visual appearances, but with the coordinates of events in inertial reference frames. As I said, a clock moving towards you would actually appear to be ticking faster than your own clock visually, but in your inertial rest frame it would still take a longer coordinate time between ticks than the coordinate time between ticks of your own clock, by an amount given by the time dilation formula.

Which is precisely why I usually stick the word ‘see’ in quotation marks and often follow it with parenthesised (‘determine’ or ‘calculate’)

I still have no idea what you mean by "time contraction".

On the basis of your word ‘still’ I assume that you previously had no idea what I meant by “time contraction” however I am not aware of any earlier comments of yours to that effect. If I had seen one I would have responded - if one clock (A) is ticking over at over at a slower rate than another clock (B) some people state that clock A is incurring time dilation (i.e. that it’s seconds are ‘compressed’ or ‘shorter’) on the other hand some people insist that clock A is not ticking over at a slower rate than B but that B is ticking over at a faster rate than A thus that clock B’s seconds are extended i.e. contracted!

On the basis that, as I understand it, the idea of time contraction was an anathema for Einstein I am of the opinion that he would not have accepted this concept.

Do you understand that in relativity there is no frame-independent truth about whether a clock is ticking slow or not, that we can only talk about its rate of ticking relative to some inertial coordinate system?

And in chapter 4 where Einstein wrote that a clock at the equator goes more slowly than a clock at one of the poles he was talking about it’s rate of ticking relative to the polar observer’s inertial coordinate system.

Whilst it is quite possible that Einstein might have been aware of the fact that the Earth is moving through space he was not positing what some purely hypothetical observer contained in another (‘different’) imaginary inertial reference frame would determine but his comment was strictly in relation to what is taking place in the Earth’s reference frame.

On the basis that, in relativity, we can only talk about some other clock’s rate of ticking relative to some inertial system then the claim that the traveling twin can (whilst he is accelerating following turn-around) ‘talk about’ the Earth clock ticking over at a faster rate than it was before he started moving appears, to me, to contradict relativity.

That’s what I’ve been saying!

Of course it's true that a clock can only tick slower than the coordinate time of an inertial frame, never faster,

So when the traveler is returning to Earth at uniform velocity he cannot, according to relativity, say or determine or calculate that the Earth clock is ticking faster than it was before he started moving ergo he can only conclude that his clock (as Einstein posited in paragraph 2, chapter 4 with respect to a clock moving in any polygonal line) is ticking over at a slower rate than it was before he started moving toward the Earth clock analogous to clock A in Einstein’s paragraph 1, chapter 4 STR.

but the clock is not ticking slow in any "objective" sense, and different inertial frames will disagree about which of two clocks is ticking slower (relative to their own coordinate time) at any given instant. And Einstein made clear that all inertial frames are equally valid, there is no reason to consider one frame's perspective to be more "true" than any other's.

Whilst “Einstein made clear that all inertial frames are equally valid..” in chapter 4 he referred to a single inertial reference frame in each paragraph of that presentation!

Although it is quite obvious that an observer (C) in a different inertial reference frame may have an entirely different perspective his opinion has nothing whatsoever to do with, and has no bearing on, determinations made by observers accompanying Einstein’s clocks A and B!

Only in one particular inertial frame. If an object is moving in a circle at constant speed in the inertial rest frame of the center of the circle, then in a different inertial frame where the center of the circle is moving at constant velocity, the speed of the object will be variable (and in this frame the path of the object will look like some type of cycloid rather than a circle). Again, in SR all inertial frames are equally valid.

Which, of course, has absolutely nothing whatsoever to do with the subject on hand!

In chapter 4 Einstein made no reference whatsoever to what an observer in a different inertial reference frame would conclude but referred to what takes place in the stationary system of the respective ‘at rest’ clocks.

The conclusion arrived at by a third observer moving past Einstein’s paragraph 1 “points A and B of K” or his paragraph 3 clock moving in a closed curve around another clock that has remained at rest or past a planet that has clocks at the equator and at one of the poles has absolutely nothing whatsoever to do with what the person accompanying the respective stationary clocks determines!

If the astronaut understands relativity at all, he knows that to talk about the rate a clock is ticking in any objective "physical" sense is totally meaningless, you can only talk about the rate a clock is ticking in one inertial coordinate system or another, and different coordinate systems give different (equally valid) answers. If you don't understand this, you really have missed one of the most basic ideas about relativity!

So if an astronaut ‘talks about’ or ‘determines’ or ‘predicts’ that the Earth clock is physically ticking over at a faster rate than it was before he started moving I presume that he has missed one of the most basic ideas about relativity?

You wrote, above, “Of course it's true that a clock can only tick slower than the coordinate time of an inertial frame, never faster.” It seems that you agree with me that the claim - that from the astronaut’s point of view his clock is not ticking over at a slower rate than it was before he started moving following turn-around but that it was the Earth clock that started ticking over at faster rate than it was before he started moving - is inappropriate.

Do you also agree with me that an astronaut, having come to stop at the end of his outward-bound trip is equivalent to Einstein’s paragraph 1, chapter 4 clock A at “points A and B of K” with the planet represented by ‘B’?

If so, do you also agree that his return trip is equivalent to Einstein’s paragraph 1, chapter 4 depiction of clock A moving to B’s location?

I assume that you would agree with Einstein’s comment that clock A (the astronaut’s clock) lags (even further) behind clock B (the Earth clock) when A arrives at B’s location?

Regardless of the fact that it may not actually be aware of the phenomenon - A can either conclude that it incurred time dilation (physically ticking over at a slower rate than it was before it started moving) or that B incurred time contraction and on the basis that “a clock can only tick slower than the coordinate time of an inertial frame, never faster.” does it not follow that A cannot say that B ticked over at a faster rate than it did before he started moving?

He can, of course, declare that B appeared to be ticking over at a faster rate than it did before he started moving however is he justified in insisting that it did, physically, tick over at a faster rate than it was before he started moving?

It would be very much appreciated if you did not refer to a conclusion arrived at by an observer in a different reference frame as, in my view, their opinion has absolutely nothing whatsoever to do with what A or B determine and can only obfuscate the discussion.

 

[cos]

Hey cos;
I'm a salmon guy, not herring.

Regardless of personal taste - there's something fishy.

Here is a geometrical explanation for the twin aging problem.

Thank for your presentation however I am not looking for an explanation of same but am seeking comments in relation to the claim that from the astronaut's point of view, his clock does not tick over at a slower rate than it did before he started moving but that the Earth clock physically ticked over at a faster rate than it did prior to his starting to move.

When the twins separate and later rejoin, the same amount of time has elapsed for the universe...

In accordance with the above claim, it is (from that astronaut's point of view) not only the Earth-bound twin that ages at a faster rate than the astronaut but also the entire universe.

 

[JesseM]

So when they took gravitational time dilation into account and eliminated that factor they ended up with Einstein’s prediction which didn't take it into account in the first place.

Was the eventual lag between the HKX traveled clocks after the gravitational time dilation factor was removed from the results any different from Einstein’s predicted lag, unaware of any gravitational time dilation factor?

I don't think you can break down the total time dilation into a linear sum of SR velocity-based time dilation and GR gravitational time dilation in the way you're suggesting. I think the only way to calculate the total time elapsed on the clocks is to do an integral over the path of each clock in curved spacetime; paths through the same regions of space at different speeds would have different times elapsed, so in that sense you're taking into account velocity-based time dilation, but I'm pretty sure you can't calculate what the velocity-based time dilation would be in flat spacetime and add it to a pure gravitational-based time dilation that ignores velocity to get the total time dilation.

And I'm certain that when he said "more slowly" he meant something like "more slowly on average over the course of a full rotation", or "more slowly at every instant in the rest frame of the sphere", not "more slowly at every instant in an objective frame-independent sense". For him to mean the last one would be a clear contradiction with his own theory.

You have your interpretation of what Einstein meant by “more slowly”; I have mine.

So are you saying you do think he meant "more slowly at every instant in an objective frame-independent sense"? If so, what you're saying goes against the basics of relativity, and is objectively wrong, it's not just a matter of opinion. On the other hand, if you don't mean to imply that one clock is going more slowly than another at every moment in a frame-independent sense, then please spell out explicitly what you do mean.

Every instant that an observer alongside a ‘polar’ clock looks at an ‘equatorial’ clock on a transparent massless sphere the size of the Earth he will see that compared with his observation made at a previous instant the equatorial clock will lag even further behind his own clock indicating to him that between those instances the equatorial clock has continuously ticked over at a slower rate than his own clock.

But as I said, visual appearances are completely different than time dilation. If you talk about what would be happening in the inertial rest frame of the observer at the pole, in this case it's true that the clock at the equator is ticking slower than his own clock. However, there are other equally valid inertial frames where at certain times the clock at the pole is ticking slower than the clock at the equator (because the clock at the pole has a higher speed at those times in that frame).

Other than as an attempt to confuse the debate why do we even need to “pick a frame where the center of the sphere is moving at some nonzero constant velocity.”? The opinions expressed by an observer in that frame have no bearing whatsoever on the determinations made by an observer in the “inertial frame where the position of the center of the sphere remains constant over time.”

Because I am making the point that there is no objective frame-independent truth about which of two clocks is ticking slower at any given moment, since all inertial frames are equally valid and there are some frames where the clock at the pole is ticking slower at some moments. That has been my point all along--do you agree with it or disagree? If you agree, but just want to talk about the frame-dependent facts about what is happening in the rest frame of the observer at the pole (in which case I certainly agree that the clock is ticking slower at every moment in that frame), then just say so! But that's the only point I was making all along, if you never disagreed with it than you could have said so earlier and we would have saved a lot of time.

On the basis that the observer at the pole determines that the clock at the equator (B) is “ticking at a constant slowed-down rate” isn’t he of the opinion that clock B is incurring time dilation?

Yes, in his own rest frame, but not in an objective frame-independent sense.

Isn’t an observer accompanying clock B of the opinion that clock A is (or at least appears to be) ticking over at a faster rate than his own clock?

Again, visual appearances are a totally different matter than time dilation in SR, as evidenced by the fact that a clock moving towards you will appear to be ticking faster than yours even though in your rest frame it is actually ticking slower. If the observer at the equator considers what is happening in the inertial rest frame where he is at rest at a particular moment (i.e. the frame of an inertial observer moving in a straight line whose instantaneous velocity is the same as the instantaneous velocity of the observer at the equator at that instant), then in that frame the clock at the pole is ticking slower at that moment, since in that frame the clock at the pole has a nonzero velocity while his instantaneous velocity is zero.

Assuming that he is aware of, and accepts, Einstein’s paragraph 3, chapter 4 comment - that the clock at the equator “goes more slowly” than (i.e. is “ticking at a constant slowed-down rate” compared to) the polar clock - might he not tend to take Einstein’s word for it thus determine that B is not ‘ticking over at a constant increased rate’ but realize that his clock is “ticking at a constant slowed-down rate”?

If he is only interested in the rate each clock is ticking in the rest frame of the observer at the pole (or any inertial observer at rest relative to the center of the sphere), then in this frame he'll certainly realize that his clock is ticking at a constant slowed-down rate. But if he understands relativity he knows this is a frame-dependent fact, not an objective physical fact, since he could equally well look at the problem from the perspective of a different inertial frame and get a different answer, and all inertial frames have equal validity in SR.

On the other hand, if we choose a different inertial frame in which the center of the sphere is moving inertially at some constant velocity...

On the other hand we could stick to the subject under discussion and not introduce extraneous materiel that obfuscates same.

It certainly is not extraneous to the subject of whether there is any objective physical truth about which of two clocks is ticking slower at a given moment, which is the one I have been focused on all along. If you have no objection to the idea that there is no objective answer to this question, only different frame-dependent answers, then you shouldn't have objected to my statement in my first post on the thread (post #18) where I said:

It is true that, no matter which frame you choose, the average rate of ticking on the clock of the traveling twin must be slower than the average rate on the clock of the Earth twin. But you can find inertial frames where the Earth twin's clock ticks slower than the traveling twin's clock during the trip away from the Earth, then the traveling twin's clock ticks slower than the Earth twin's on the return journey after the turnaround; you can also find frames where the opposite is true, and the traveling twin's clock is slower on the outbound trip but faster on the inbound leg. So, there is no objective truth about whose clock is ticking slower at any given moment, even if the average of the traveling twin's clock is always slower than the Earth twin's clock over the course of the whole trip.

Do you, in fact, have any objection to this statement?

Again, the formulas of special relativity are not concerned with visual appearances, but with the coordinates of events in inertial reference frames. As I said, a clock moving towards you would actually appear to be ticking faster than your own clock visually, but in your inertial rest frame it would still take a longer coordinate time between ticks than the coordinate time between ticks of your own clock, by an amount given by the time dilation formula.

Which is precisely why I usually stick the word ‘see’ in quotation marks and often follow it with parenthesised (‘determine’ or ‘calculate’)

Well, you didn't put "see" in quotation marks in your statement about the astronaut, nor did you give any indication that you meant "see" to stand for some set of calculations:

At any given instant he would see that the time indicated by that clock lapses even further behind his own time than the time indicated by that same clock at a previous instant indicating to him that clock A has continuously ticked over at slower rate than his own clock between those instances (observations) and, on that basis, it is (irrespective of the fact that he may be consciously unable to discern same) physically ticking over at a slower rate than his own clock in the one-tenth of a second that it takes for his cerebral processes to inform him that he is looking at that clock.

If you are saying the astronaut calculates that clock A is ticking slower, can you spell out what these calculations are? Is he calculating the rate that A is ticking in some inertial frame?

On the basis of your word ‘still’ I assume that you previously had no idea what I meant by “time contraction” however I am not aware of any earlier comments of yours to that effect.

I didn't respond to the specific phrase "time contraction", but I said that I didn't understand the distinction you were making between the idea that his clock was ticking slower than the other clock vs. the idea that the other clock was ticking faster than his clock, and I assumed that your distinction between "time dilation" and "time contraction" was basically the same thing.

If I had seen one I would have responded - if one clock (A) is ticking over at over at a slower rate than another clock (B) some people state that clock A is incurring time dilation (i.e. that it’s seconds are ‘compressed’ or ‘shorter’) on the other hand some people insist that clock A is not ticking over at a slower rate than B but that B is ticking over at a faster rate than A thus that clock B’s seconds are extended i.e. contracted!

And it is exactly this distinction that I said I didn't understand, and still don't understand. For me, if A is ticking slower than B that is the same as saying that B is ticking faster than A, just like if a number N is larger than M that is the same as saying M is smaller than N. What we can say is that in inertial frames, clocks can only run slow relative to coordinate time in that frame, never faster, but I already brought this idea up and you didn't agree that this is what you were talking about.

On the basis that, as I understand it, the idea of time contraction was an anathema for Einstein I am of the opinion that he would not have accepted this concept.

Your use of the phrase "time contraction" still seems meaningless to me if it's supposed to be a comparison between two clocks. Again, it is true that in relativity a clock can only slow down relative to coordinate time in a particular inertial frame, never speed up relative to coordinate time, but apparently this isn't what you're talking about. Maybe what it comes down to is that you do think there is some objective frame-independent truth about whether a clock is running slow, and you're saying that a clock can never be running fast in this objective sense; but if so, that's the whole point I've been disputing, since I'm saying there is no objective answer to the question of how fast a clock is ticking, only an infinite number of different (and equally valid) frame-dependent answers.

And in chapter 4 where Einstein wrote that a clock at the equator goes more slowly than a clock at one of the poles he was talking about it’s rate of ticking relative to the polar observer’s inertial coordinate system.

I don't know if that's what he meant, but if you recall I did suggest that as at least one possibility:

And I'm certain that when he said "more slowly" he meant something like "more slowly on average over the course of a full rotation", or "more slowly at every instant in the rest frame of the sphere" [which of course is the same as the rest frame of the polar observer], not "more slowly at every instant in an objective frame-independent sense". For him to mean the last one would be a clear contradiction with his own theory.

So, I'd certainly have no objection to the idea that this might have been what I meant. The only point I have been making from the beginning is that there is no objective-frame dependent answer to the question of which of any two clocks is ticking slower at a given moment, but of course there is a single correct answer to the question of which clock is ticking slower at a given moment in a particular inertial frame. Again, do you have any disagreement with this point, or have you been arguing with me for no reason?

Whilst it is quite possible that Einstein might have been aware of the fact that the Earth is moving through space he was not positing what some purely hypothetical observer contained in another (‘different’) imaginary inertial reference frame would determine but his comment was strictly in relation to what is taking place in the Earth’s reference frame.

Reference frames are just coordinate systems, you don't need to have an actual physical observer at rest in a particular coordinate system in order to calculate the time-coordinates of events in that system, any more than you'd need an observer physically present at the center of the Earth in order to place the origin of your spatial axes there. No frame is any more or less "imaginary" than any other. But again, it might be true that Einstein was talking about what would be true in the inertial frame where the center of the Earth was at rest, it's not relevant to my main point which is that the only objective physical truths are the ones which are agreed on by all inertial frames.

On the basis that, in relativity, we can only talk about some other clock’s rate of ticking relative to some inertial system then the claim that the traveling twin can (whilst he is accelerating following turn-around) ‘talk about’ the Earth clock ticking over at a faster rate than it was before he started moving appears, to me, to contradict relativity.

That's not what I said, I just said that at any given moment, you can find a valid inertial frame where the Earth clock is ticking slower than the traveling twin's clock at that moment. Of course, since we're assuming the Earth is moving inertially, no matter what inertial frame you pick, the Earth will have a constant speed in that frame, so its clock will be ticking at an unchanging rate in that frame.

Of course it's true that a clock can only tick slower than the coordinate time of an inertial frame, never faster,

So when the traveler is returning to Earth at uniform velocity he cannot, according to relativity, say or determine or calculate that the Earth clock is ticking faster than it was before he started moving ergo he can only conclude that his clock (as Einstein posited in paragraph 2, chapter 4 with respect to a clock moving in any polygonal line) is ticking over at a slower rate than it was before he started moving toward the Earth clock analogous to clock A in Einstein’s paragraph 1, chapter 4 STR.

Regardless of what inertial frame you choose, the Earth's speed is constant in that frame, so the rate of ticking of the Earth clock doesn't change. However, in some frames there will be particular moments when the speed of the traveling twin is lower than that of the Earth, and therefore in such a frame the Earth-clock is ticking slower than his clock at those moments. Do you disagree with that? Note that in such a frame there will also be moments when the speed of the traveling twin is greater than the Earth's and his clock is therefore ticking slower than the Earth's, and it will always be true that when you look at the average rate his clock was ticking from the beginning to the end, it is less than the (constant) rate the Earth-clock was ticking, so he'll have aged less when he returns to Earth.

Whilst “Einstein made clear that all inertial frames are equally valid..” in chapter 4 he referred to a single inertial reference frame in each paragraph of that presentation!

He didn't explicitly refer to any particular inertial frame in the last paragraph of section 4, for example. If you believe he was implicitly talking about the inertial frame where the Earth was at rest, you could be right, I don't have any wish to argue this point. Again, my point all along has just been that there is never a frame-independent truth about which of two clocks is ticking slower at a particular moment.

Although it is quite obvious that an observer (C) in a different inertial reference frame may have an entirely different perspective his opinion has nothing whatsoever to do with, and has no bearing on, determinations made by observers accompanying Einstein’s clocks A and B!

And this doesn't conflict with my point. But again, I'll note that although it is a usual convention that an observer calculate things from the perspective of his own inertial rest frame, this is just a convention, any observer is free to use any coordinate system for the purpose of calculations, there isn't any physical reason he's forced to treat the frame where he's at rest as his own "perspective".

Only in one particular inertial frame. If an object is moving in a circle at constant speed in the inertial rest frame of the center of the circle, then in a different inertial frame where the center of the circle is moving at constant velocity, the speed of the object will be variable (and in this frame the path of the object will look like some type of cycloid rather than a circle). Again, in SR all inertial frames are equally valid

Which, of course, has absolutely nothing whatsoever to do with the subject on hand!

The reason we got into this long discussion was because you had some kind of objection to my original post on the thread, and the fact that different frames disagree on which of two clocks is ticking slower is certainly relevant to the point I made in that post. If you just didn't follow what I was arguing there, but now have no objection to the point that there is no objective frame-independent truth about which of two clocks is ticking slower at a given moment (regardless of whether you think this point is interesting or relevant to what you were talking about earlier), then we'll be in agreement and can drop the whole thing.

(continued in next post)

 

[JesseM]

(continued from previous post)

I don't know what you mean by this distinction--in any given frame, if clock A is ticking slower than clock B, then how is that different from saying that clock B is ticking faster than clock A in this frame?

As previously pointed out - the claim is that according to the astronaut the Earth clock is physically ticking over at a faster rate than it was before he commenced his trip and for the astronaut to be of the opinion that this is physically taking place he must also believe (predict, determine) that the Earth’s axial spin and orbit of the sun have physically increased.

If the astronaut understands relativity at all, he knows that to talk about the rate a clock is ticking in any objective "physical" sense is totally meaningless, you can only talk about the rate a clock is ticking in one inertial coordinate system or another, and different coordinate systems give different (equally valid) answers. If you don't understand this, you really have missed one of the most basic ideas about relativity!

So if an astronaut ‘talks about’ or ‘determines’ or ‘predicts’ that the Earth clock is physically ticking over at a faster rate than it was before he started moving I presume that he has missed one of the most basic ideas about relativity?

For a fact to be objectively "physical" it must be frame-independent. So yes, for the astronaut to say the Earth clock is "physically" ticking faster or slower than his is to miss a basic idea about relativity, although he can certainly say the Earth clock is ticking faster or slower than his clock in a particular frame, or that the Earth clock is ticking slow (never fast) relative to the coordinate time of a particular inertial frame.

You wrote, above, “Of course it's true that a clock can only tick slower than the coordinate time of an inertial frame, never faster.” It seems that you agree with me that the claim - that from the astronaut’s point of view his clock is not ticking over at a slower rate than it was before he started moving following turn-around but that it was the Earth clock that started ticking over at faster rate than it was before he started moving - is inappropriate.

You need to specify what you mean by the astronaut's "point of view". I would agree that if the astronaut picks some (arbitrary) inertial frame to call his "point of view", then the Earth clock will be ticking at a constant rate in that inertial frame (since it's moving at a constant speed in that frame).

Do you also agree with me that an astronaut, having come to stop at the end of his outward-bound trip is equivalent to Einstein’s paragraph 1, chapter 4 clock A at “points A and B of K” with the planet represented by ‘B’?

To make the situation completely equivalent, you'd need to have a clock at the planet B that was synchronized with the astronaut's clock the moment before he left Earth, in the rest frame of the Earth and the planet--in this case I'd agree the situations are equivalent. (by the way, it's not the first paragraph of section 4 where he talks about 'points A and B of K', it's actually the third-to-last).

If so, do you also agree that his return trip is equivalent to Einstein’s paragraph 1, chapter 4 depiction of clock A moving to B’s location?

On the return trip, for it to be fully equivalent you'd need to label the Earth as B rather than the distant planet as in the outward trip, and here you'd need the clock on Earth to be synchronized with the astronaut's clock the moment before he left the distant planet, in the rest frame of the Earth and the planet.

I assume that you would agree with Einstein’s comment that clock A (the astronaut’s clock) lags (even further) behind clock B (the Earth clock) when A arrives at B’s location?

In each case, if the clocks are initially synchronized in the way I describe, then yes.

Regardless of the fact that it may not actually be aware of the phenomenon - A can either conclude that it incurred time dilation (physically ticking over at a slower rate than it was before it started moving) or that B incurred time contraction

You leave out a possibility here, which is that B experienced time dilation, but B's clock was ahead of A's clock at the moment before A began to move towards it (instead of A's clock being synchronized with B's at this moment, as would be true according to the definition of simultaneity in the rest frame of B), so that even though B's clock was ticking slower than A's throughout the journey, B's clock is still ahead of A's when they meet thanks to that "head start". This is exactly what would be true in the inertial rest frame where A is at rest during the trip and B is moving towards him. If you exclusively want to talk about how things would work in the rest frame of B, that's fine, but if you just want to talk about a particular frame you should always spell this out explicitly.

and on the basis that “a clock can only tick slower than the coordinate time of an inertial frame, never faster.” does it not follow that A cannot say that B ticked over at a faster rate than it did before he started moving?

If you want A to make calculations from the perspective of some inertial frame (even though he will not remain at rest in this frame), then it's true that in any inertial frame B is moving at constant speed, and therefore B's clock was ticking at a constant rate. But depending on which frame is chosen, it may be that A's clock was ticking even slower than B during the approach.

It would be very much appreciated if you did not refer to a conclusion arrived at by an observer in a different reference frame as, in my view, their opinion has absolutely nothing whatsoever to do with what A or B determine and can only obfuscate the discussion.

Again, it's just a convention that inertial observers use their inertial rest frame to represent their "perspective", there is no physical reason that it should be harder to "determine" the coordinates of events in a frame where you are moving than it is to "determine" the coordinates of these events in your own rest frame. And A is not an inertial observer, he has different rest frames at different times, so I don't know what you mean by not using "different reference frames" here. Again, if we use the inertial frame where A is at rest during the journey, in this frame it is B's clock that is running slow while his is running normally, and the only reason B's clock is ahead when he reaches it is because it was already ahead at the start of the journey.

 

[phyti]

cos;

from the astronaut's point of view, his clock does not tick over at a slower
rate than it did before he started moving but that the Earth clock physically
ticked over at a faster rate than it did prior to his starting to move.

The dilation only applies to the ship and its contents, thus the pilot cannot
detect the difference within his frame. If he looks out, he should see a
polarized universe, events forward happening faster and events rearward
happening slower, a scenario which is not normal.
It should be obvious from studying the light clock, the rate can only
physically decrease. Observers will perceive the others clock rate increase
when converging, and decrease when diverging, but these are doppler effects.

When the twins separate and later rejoin, the same amount of time has elapsed
for the universe...

In accordance with the above claim, it is (from that astronaut's point of
view) not only the Earth-bound twin that ages at a faster rate than the
astronaut but also the entire universe.

This is true if you change 'faster' to 'slower', where 'slower' is 'measured
as slower'. All other clocks do not change rates, because the astronaut initiated the
motion, and the amount of energy used to move the ship would not move the rest
of the universe in the opposite direction, i.e., the situation is not symmetrical by reason of the conservation of energy.
The ship moving cannot alter the rules of physics in distant parts of the
universe, especially instantaneously. That's the nonsense!
It violates postulate (1) that the theory is built on.
And then there's this:
SR theory transforms coordinates not objects.
Would you please pass the cucumbers.

 

[cos]

(continued from previous post)Again, in relativity it is quite meaningless to talk about how fast any clock is ticking "physically" in a frame-independent sense.

So I assume that it is quite meaningless for someone to claim that the astronaut is of the opinion that the Earth clock is physically ticking over at a faster rate than it was before he started moving?

I am of the opinion that this claim is quite meaningless.

No matter what clock you are dealing with, different frames assign it different rates of ticking, and for any pair of clocks, different frames will disagree about whose rate of ticking is slower (since different frames will disagree about which clock's speed is greater).

I assume that by ‘different frames’ you are referring to frames other than those of the astronaut however I reiterate that I am not in the slightest bit interested in what ‘different frames’ agree or disagree about but purely and simply what the astronaut determines.

Einstein makes it quite clear that there is no reason to prefer one inertial frame's perspective over any other, and any relativity textbook you might care to look at should make this clear as well.

The claim by the astronaut that his clock is ticking over at its normal rate thus that the eventual time variation between his clock and his twin’s clock can only have occurred as a result of the Earth clock ticking over at a faster rate than it was before he started moving indicates that he is giving preference to his reference frame over that of the Earth.

On the basis that he determines that every other reference frame in the universe is moving relatively to him he is of the opinion that his is the only reference frame in the entire universe that is ‘at rest’.

"a velocity of close to the speed of light" relative to what?

Duh; relative to what it was before he started moving?

If the astronaut is moving at close to the speed of light in the rest frame of the Earth, then in the astronaut's own inertial rest frame the astronaut is at rest and the Earth has a velocity close to the speed of light.

A brilliant young student living in a small town is selected for astronaut training however as the aeroplane taking him on his very first flight lifts off he starts screaming “The sky is falling! The sky is falling!” What promised to be a very exciting career ends up in a rubber room.

An astronaut is in a ship prior to takeoff. He hears the command “ignition” and feels a tremendous force of acceleration pushing him back into his seat.

He keeps accelerating and having attained an instantaneous velocity of close to the speed of light sees the universe appearing to flash past his window in a blur.

He takes his foot off the gas pedal but according to the way of thinking that you presented he is then of the opinion that he is, at that very instant, no longer moving but that it is the universe that has incurred instantaneous acceleration and is now moving past him at close to the speed of light.

He carries out internal dynamic experiments involving “the phenomena of electrodynamics as well as of mechanics” but achieves no result that permits him to determine if his ship is moving with uniform velocity or if it is ‘at rest’ so by giving preference to his reference frame over that of the universe he determines that the universe must have incurred instantaneous acceleration and is now moving past him at that velocity.

It is my understanding that the Galilean principle of relativity showed that a state of rest or of uniform motion cannot be detected without reference to an outside point! i.e. without the astronaut being able to look out of his window.

On the basis that the astronaut is of the opinion that he has stopped moving and that it is the universe that is now moving past him he must also be of the opinion that it has taken a force of energy that was greater than the infinite mass of the universe to make it move!

Not only that, but the universe has instantaneously accelerated to near-light speed! What effect could that near-infinite force of acceleration have had on his twin?

If, at the very instant that he lifts his foot off the gas pedal, everything in the universe has been made to instantaneously accelerate to near-light speed why wasn’t his ship affected by that greater-than-infinite force?

Could it have been the infinitesimally weaker force generated by his ship’s engine which overcame that greater-than-infinite force of energy?

"According to his calculations"? If the astronaut is moving inertially, then in his own rest frame, it is the Earth that has the large velocity while he is at rest, and the time dilation formula must work the same way in every inertial frame according to the first postulate of relativity, so he must calculate that the Earth's clock is ticking 40,000 times slower than his own if he does the calculations relative to his own inertial rest frame, not 40,000 times faster.

The last few lines are my very argument!

The claim to which my posting refers is that the astronaut insists that his clock is not ticking over at the rate of 40,000 seconds for each Earth second but that the Earth clocks are ticking over at the rate of 40,000 seconds for each of his seconds!

You appear to have badly misunderstood the principle that the laws of physics work the same in all inertial reference frames, which was one of the two basic postulates of SR that Einstein put forward in his 1905 paper. Every frame must predict that clocks moving in that frame slow down, not speed up. Two observers moving inertially relative to one another will each calculate that the other one's clock is running slower than their own.

However, in his chapter 4 Einstein points out that, irrespective of the fact that clocks A and B are, effectively (ignoring the fact that in order to move to B’s location clock A must have incurred acceleration) “moving inertially relative to one another” an observer accompanying clock A, regardless of his calculations that B is running slower than his clock, finds upon his arrival at B’s location that B was not running slower than his own clock as predicted by his calculations but that his own clock was running slower than clock B (alternatively, according to some people, that B was running faster than his clock) resulting in A lagging behind B not B lagging behind A.

Your comment that “Two observers moving inertially relative to one another will each calculate that the other one's clock is running slower than their own.” is, of course, the origin of the clock paradox that Einstein sought to overcome with his 1918 article wherein he insisted that the only way those clocks can be accurately compared is if one of them is made to move to the other clock’s location which requires the former to undergo several periods of acceleration.

That relocated clock is, in my opinion, analogous to Einstein’s chapter 4 clock that follows a polygonal path.

It is interesting to note that when Galileo wrote his book ‘Two New Sciences’ he was already in trouble with authorities so he presented it as a hypothetical discussion between a teacher and two of his students.

Einstein similarly wrote his 1918 article in the form of a purely hypothetical conversation (this time between a relativist and a critic) perhaps in order to prevent criticism from his opponents, colleagues and authorities for his application of an aspect of general theory (gravitational acceleration) as a solution to a special theory related paradox.

He had already been criticised by his associates (particularly Max Abrahams) for suggesting, in the introduction to general theory, that the special theory law of the constancy of the speed of light required modification and, in his 1916 book ‘Relativity, the Special and General Theory’ that the law of the constancy of the speed of light was not valid.

And despite this seemingly counterintuitive result, all frames will nevertheless get identical predictions about all local events like what two clocks read at the moment they pass next to one another...

In chapter 4 Einstein makes no suggestion that clocks A and B are inertial reference frames that pass next to one another but points out that clock A is made to move toward, and come to a stop, alongside clock B as does his 1918 depicted clock!

As far as I am concerned, Einstein’s chapter 4 depictions (specifically his reference to a clock that is made to move in any polygonal path) is directly equivalent to his 1918 attempted negation of the twin paradox.

Relativity deals with plenty of instantaneous quantities such as instantaneous velocity, as do all dynamical theories of physics expressed using calculus.

The fact that the mathematical propositions of relativity and all dynamical theories of physics expressed using calculus deal with plenty of instantaneous quantities such as instantaneous velocity does NOT prove that they are reality!

“As far as the propositions of mathematics are certain, they do not refer to reality.” (A. Einstein)

So do you agree that if the astronaut is moving inertially, then in his inertial rest frame he is at rest while the planet is moving towards him at high speed, therefore in this frame his own clock is ticking at the normal rate while the planet's clock is ticking slower?

NO! I most certainly do NOT!

I do not believe that an intelligent astronaut would adopt a Henny Penny attitude.

I believe that an intelligent astronaut, having accelerated to a relativistic velocity and having taken his foot off the gas pedal, would realize that he is still moving - either away from, or toward, the planet.

I believe that an intelligent astronaut would be of the opinion that there is no such thing as a force of energy that is greater than infinite that could cause the Earth, and the entire universe, to suddenly - instantaneously - start moving at close to the speed of light when he takes his foot off the gas pedal.

Conversely - I assume that somebody would be able to provide a mathematical proposition which ‘proves’ the idea of a force that is greater than infinite and that the Earth, along with the universe, could physically cope with being instantaneously accelerated to near-light speed.

Having initially set out to establish the infallible nature of mathematics Bertrand Russell was reluctantly forced to admit that “Mathematics may very well be a subject in which we never know what we are talking about nor that what we are saying is true.”

Furthermore, as I have pointed out several times, the claim to which I refer is not that the astronaut believes that “his own clock is ticking at the normal rate while the planet’s clock is ticking slower” but insists that his own clock is ticking at the normal rate while the planet’s clock is ticking FASTER!

He claims that this is the reason why his clock lags behind the Earth clock upon his return.

The astronaut, being of the opinion that, during his return journey, the Earth clock is ticking slower than his own clock would be surprised to find, upon his return the the planet, that his clock lags behind the Earth clock whereas, according to his calculations, it should be the Earth clock that (having, as you point out, ticked over at a slower rate than his clock) should lag behind his clock - but it doesn’t!

This is why the claim is that he finds that his clock lags behind the Earth clock due to the ‘fact’ that it is the Earth clock that has ticked over at a faster rate than his clock - not at a slower rate.

 

[cos]

I don't think you can break down the total time dilation into a linear sum of SR velocity-based time dilation and GR gravitational time dilation in the way you're suggesting.

I didn’t suggest that - Clifford Will did! Take up your argument with him.

I think the only way to calculate the total time elapsed on the clocks is to do an integral over the path of each clock in curved spacetime;

I’m not suggesting that we “calculate the total time elapsed on the clocks.”

Einstein pointed out that the total time elapsed is not ‘calculated’ but is observed!

So are you saying you do think he meant "more slowly at every instant in an objective frame-independent sense"? If so, what you're saying goes against the basics of relativity, and is objectively wrong, it's not just a matter of opinion. On the other hand, if you don't mean to imply that one clock is going more slowly than another at every moment in a frame-independent sense, then please spell out explicitly what you do mean.

What I mean is that in my opinion when Einstein wrote that a clock at the equator “must go more slowly” than a clock at one of the poles he meant that the equatorial clock will continuously tick over at a slower rate than the polar clock.

But as I said, visual appearances are completely different than time dilation. If you talk about what would be happening in the inertial rest frame of the observer at the pole, in this case it's true that the clock at the equator is ticking slower than his own clock. However, there are other equally valid inertial frames where at certain times the clock at the pole is ticking slower than the clock at the equator (because the clock at the pole has a higher speed at those times in that frame).

On the basis that despite my requests you continue to waste my time talking about other inertial frames that have absolutely no relevance whatsoever to the matter on hand this discussion is terminated.

 

[matheinste]

cos .

Quote:-

--I believe that an intelligent astronaut, having accelerated to a relativistic velocity and having taken his foot off the gas pedal, would realize that he is still moving - either away from, or toward, the planet.----

Yes,he can look and see the Earth moving relative to him (or vice versa ), but he cannot determine if he is "moving" in an absolute sense. For all he knows he and the Earth may have initially been "moving" and his acceleration may have brought him to "rest" while the Earth carries on "moving".

This is absolutely fundamental in SR.

Matheinste
-

 

[JesseM]

So I assume that it is quite meaningless for someone to claim that the astronaut is of the opinion that the Earth clock is physically ticking over at a faster rate than it was before he started moving?

I already said that this was not what I was claiming.

I assume that by ‘different frames’ you are referring to frames other than those of the astronaut however I reiterate that I am not in the slightest bit interested in what ‘different frames’ agree or disagree about but purely and simply what the astronaut determines.

The astronaut can "determine" the coordinates of events in any frame he wants, there is no physical reason he must use his own rest frame.

Einstein makes it quite clear that there is no reason to prefer one inertial frame's perspective over any other, and any relativity textbook you might care to look at should make this clear as well.

The claim by the astronaut that his clock is ticking over at its normal rate thus that the eventual time variation between his clock and his twin’s clock can only have occurred as a result of the Earth clock ticking over at a faster rate than it was before he started moving

An inertial frame is a system of coordinates that covers all of spacetime, and where any fixed coordinate position has been moving inertially. There is no inertial coordinate system where the "Earth clock is ticking over at a faster rate than it was before he started moving". If the astronaut chooses to use a coordinate system where he is at rest as the Earth approaches him, then in this coordinate system the astronaut was not at rest before he accelerated, only after accelerating. And in this inertial coordinate system the Earth was always moving inertially, it never changed velocities.

indicates that he is giving preference to his reference frame over that of the Earth.

"Preference" means saying that statements made in one coordinate system are somehow more physically correct than statements made in another coordinate system. I never said that the astronaut should say anything like that about statements in his frame. The astronaut isn't saying that his own clock is ticking at a normal rate while the Earth's clock is ticking slower in any objective, frame-independent sense; he's just saying it's what's happening in one particular inertial frame, the frame where he is at rest during the journey (but not before). The astronaut would certainly acknowledge that in the frame where the Earth is at rest, it is his clock that's ticking slower than the Earth's clock. So, he is not "preferencing" either frame, he's just acknowledging that in relativity neither statement is more physically correct than the other, they are both just the perspectives of two different equally valid inertial reference frames.

On the basis that he determines that every other reference frame in the universe is moving relatively to him he is of the opinion that his is the only reference frame in the entire universe that is ‘at rest’.

Not in any objective sense, just from the perspective of that frame. Do you acknowledge that in relativity there can be no single "real truth" about what object is at rest, that different frames treat different objects as being at rest and it's understood that all statements about rest are frame-dependent?

A brilliant young student living in a small town is selected for astronaut training however as the aeroplane taking him on his very first flight lifts off he starts screaming “The sky is falling! The sky is falling!” What promised to be a very exciting career ends up in a rubber room.

If both the Earth and the aeroplane are moving inertially, in relativity there is no objective truth about whether the ground is at rest and the aeroplane is moving up or the the aeroplane is at rest and the ground is moving down. But since you seemed to be confused about this point earlier, this does not mean there is any valid inertial frame where the ground was at rest while the aeroplane was on the ground, then suddenly started moving downward. An inertial frame is moving at the same velocity for all time, so in the frame where the aeroplane was at rest before taking off and the ground was moving downward, it must have been true that the ground was always moving downward at the same constant velocity (again, assuming for the sake of the argument that the Earth is moving inertially), and the aeroplane was originally moving downward along with it until it accelerated and came to rest.

An astronaut is in a ship prior to takeoff. He hears the command “ignition” and feels a tremendous force of acceleration pushing him back into his seat.

He keeps accelerating and having attained an instantaneous velocity of close to the speed of light sees the universe appearing to flash past his window in a blur.

He takes his foot off the gas pedal but according to the way of thinking that you presented he is then of the opinion that he is, at that very instant, no longer moving but that it is the universe that has incurred instantaneous acceleration and is now moving past him at close to the speed of light.

Again you are repeating the same confusion as above. Inertial frames move at constant speed for all time, so if the stars around him are moving inertially, there is no valid inertial frame where the stars were originally at rest and then suddenly accelerated to relativistic speed. But there is a valid inertial frame where these stars have been moving at the same relativistic speed for all time, and the astronaut was originally moving along with them, but then when he accelerated he came to rest in this frame. This frame is just as good as the frame where the astronaut was originally at rest and then moving at relativistic speed after acceleration, the first postulate of relativity says that we have no grounds for saying either perspective is more physically "true" than the other.

He carries out internal dynamic experiments involving “the phenomena of electrodynamics as well as of mechanics” but achieves no result that permits him to determine if his ship is moving with uniform velocity or if it is ‘at rest’ so by giving preference to his reference frame over that of the universe he determines that the universe must have incurred instantaneous acceleration and is now moving past him at that velocity.

Nope, this is just confused reasoning. If he felt the G-forces when accelerating, then all inertial frames agree it was him that accelerated, not the stars around him. They disagree about his original velocity before accelerating, and whether his speed increased or decreased after the acceleration, but they all agree that the inertial stars maintained a constant velocity throughout. If you try to construct a non-inertial frame where the astronaut was at rest both before and after accelerating, the first postulate does not apply to this frame, and indeed he could do experiments during the accelerating phase which would have different results than they would when he was moving inertially, so the laws of physics don't appear to work the same way in this non-inertial frame.

It is my understanding that the Galilean principle of relativity showed that a state of rest or of uniform motion cannot be detected without reference to an outside point! i.e. without the astronaut being able to look out of his window.

Your understanding is incorrect, the principle of relativity says that the whole idea of distinguishing "rest" from "uniform motion" is meaningless. Different inertial frames disagree about which objects are at rest and which are in motion, and all inertial frames are on equal footing as far as relativity is concerned.

"According to his calculations"? If the astronaut is moving inertially, then in his own rest frame, it is the Earth that has the large velocity while he is at rest, and the time dilation formula must work the same way in every inertial frame according to the first postulate of relativity, so he must calculate that the Earth's clock is ticking 40,000 times slower than his own if he does the calculations relative to his own inertial rest frame, not 40,000 times faster.

The last few lines are my very argument!

The claim to which my posting refers is that the astronaut insists that his clock is not ticking over at the rate of 40,000 seconds for each Earth second but that the Earth clocks are ticking over at the rate of 40,000 seconds for each of his seconds!

Either one can be true depending on which inertial frame you use--in the rest frame of the Earth, 40,000 seconds go by on the Earth clock for every one second that goes by on the astronaut's clock, while in the inertial frame where the astronaut is at rest during the journey, 40,000 seconds go by on the astronaut's clock for every one second that goes by on the Earth's clock. As always, there is no reason to say one inertial frame's perspective is more physically true than another's.

However, in his chapter 4 Einstein points out that, irrespective of the fact that clocks A and B are, effectively (ignoring the fact that in order to move to B’s location clock A must have incurred acceleration) “moving inertially relative to one another” an observer accompanying clock A, regardless of his calculations that B is running slower than his clock, finds upon his arrival at B’s location that B was not running slower than his own clock as predicted by his calculations but that his own clock was running slower than clock B (alternatively, according to some people, that B was running faster than his clock) resulting in A lagging behind B not B lagging behind A.

Einstein only says that clock A will be behind clock B when they meet, but this does not prove that clock A was running slower. Do you understand what is meant by the phrase "relativity of simultaneity"? Do you understand that if two clocks are synchronized in their own rest frame, that means that in other inertial frames they'll be out-of-sync? There are perfectly valid frames where clock B was running slower, but in these frames B's time was already well ahead of A's time at the moment A accelerated, so because of this "head start" B is still ahead of A's when A meets it, despite the fact that B was ticking slower. If I place a clock 1 mile away from you that reads 5 PM when you start to walk towards it, and that clock is also running at half the correct rate, and if your clock reads 3 PM at the moment you start walking towards it, and your clock is ticking at the normal rate, then if it takes you half an hour to reach the other clock, your clock will read 3:30 and the other clock will read 5:15. In this example it's clear that despite the fact that the other clock was running slower than yours, it's still ahead of yours when you meet because it was already ahead by a lot when you started out.

Your comment that “Two observers moving inertially relative to one another will each calculate that the other one's clock is running slower than their own.” is, of course, the origin of the clock paradox that Einstein sought to overcome with his 1918 article wherein he insisted that the only way those clocks can be accurately compared is if one of them is made to move to the other clock’s location which requires the former to undergo several periods of acceleration.

No, you missed the point of the thought-experiment, Einstein certainly wasn't saying that bringing the two clocks together would settle the question of which was "really" ticking slower during the initial phase when they were moving apart inertially, that would contradict the first postulate of relativity. It is true that the fact that one clock has to accelerate explains why one clock has elapsed less time in total when they reunite, but different frames can disagree on which clock was ticking faster during the initial phase when they were moving apart, yet they'll still all make the same prediction that the clock that accelerates will have elapsed less time in total when they reunite.

In chapter 4 Einstein makes no suggestion that clocks A and B are inertial reference frames that pass next to one another but points out that clock A is made to move toward, and come to a stop, alongside clock B as does his 1918 depicted clock!

I don't know what "no suggestion that clocks A and B are inertial reference frames" means. A clock is a physical object, an inertial reference frame is a coordinate system.

The fact that the mathematical propositions of relativity and all dynamical theories of physics expressed using calculus deal with plenty of instantaneous quantities such as instantaneous velocity does NOT prove that they are reality!

First of all, we're dealing with thought-experiments here. Second of all, if space and time are continuous, and an object has a definite position x(t) at all times, the instantaneous velocity is just defined as dx/dt. If you're arguing that maybe space and time are not infinitely divisible, feel free to just talk about the nearly instantaneous velocity defined as (change in position)/(change in coordinate time) for the smallest possible time-interval you're willing to accept, and likewise we can talk about nearly instantaneous rate of clock ticking defined as (time elapsed on clock)/(change in coordinate time).

So do you agree that if the astronaut is moving inertially, then in his inertial rest frame he is at rest while the planet is moving towards him at high speed, therefore in this frame his own clock is ticking at the normal rate while the planet's clock is ticking slower?

NO! I most certainly do NOT!

I do not believe that an intelligent astronaut would adopt a Henny Penny attitude.

I believe that an intelligent astronaut, having accelerated to a relativistic velocity and having taken his foot off the gas pedal, would realize that he is still moving - either away from, or toward, the planet.

There is no such thing as absolute motion in relativity. If you want to argue that relativity is wrong and that there is some objective truth about whether an object is moving or at rest, this is not the forum to do so--see the thread IMPORTANT! Read before posting

I believe that an intelligent astronaut would be of the opinion that there is no such thing as a force of energy that is greater than infinite that could cause the Earth, and the entire universe, to suddenly - instantaneously - start moving at close to the speed of light when he takes his foot off the gas pedal.

Again, this is a misunderstanding of yours. There is no inertial frame where the astronaut is at rest both before and after the acceleration, I was talking only about a frame where the astronaut was at rest after accelerating, while the planet he took off from was moving inertially at the same speed both before and after.

Furthermore, as I have pointed out several times, the claim to which I refer is not that the astronaut believes that “his own clock is ticking at the normal rate while the planet’s clock is ticking slower” but insists that his own clock is ticking at the normal rate while the planet’s clock is ticking FASTER!

There is no frame where the rate the planet's clock is ticking changes when the astronaut accelerates. It is true, though, that in the planet's own rest frame the astronaut's clock is ticking slower than the planet's clock after the astronaut accelerates, which is the same as saying that in this frame the planet's clock is ticking faster than the astronaut's after the astronaut accelerates, though this is not to say the planet's clock is ticking faster than it was before the astronaut took off in this frame.

The astronaut, being of the opinion that, during his return journey, the Earth clock is ticking slower than his own clock would be surprised to find, upon his return the the planet, that his clock lags behind the Earth clock whereas, according to his calculations, it should be the Earth clock that (having, as you point out, ticked over at a slower rate than his clock) should lag behind his clock - but it doesn’t!

If the astronaut calculates things from the perspective of the inertial frame where he is at rest during the return trip, then again this is a matter of the relativity of simultaneity--in this frame the Earth's clock is already well ahead of his own at the moment the astronaut begins the journey, so even though the Earth's clock is ticking slower than his throughout the journey, it will still be ahead of his clock when he reaches the amount. If you factor in both the relativity of simultaneity and the time dilation factor, you'll get exactly the same prediction in this frame about how much the Earth clock is ahead as you would if you calculated things in the Earth's rest frame.

 

[JesseM]

On the basis that despite my requests you continue to waste my time talking about other inertial frames that have absolutely no relevance whatsoever to the matter on hand this discussion is terminated.

I continue to bring it up because you are explicitly contradicting basic principles of SR by saying there is a definite truth about which of two clocks is ticking slower in thought-experiments like the one where the astronaut travels from a distant planet to Earth, and my point is that different inertial frames disagree about which of two clocks is ticking slower, yet they all make the same predictions about physical events (like what two clocks read when you bring them together) and it's a basic principle of SR that all inertial frames are equally valid. If you continue to talk as though there is an objective truth about which clock is ticking slower without being willing to listen to counterarguments explaining your error (and these counterarguments necessarily require you to look at the same situation in multiple inertial frames), then you are violating the rules of this forum as laid out in the IMPORTANT! Read before posting thread.

 

[cos]

cos; The dilation only applies to the ship and its contents, thus the pilot cannot detect the difference within his frame.

In paragraph 1, chapter 4, Einstein wrote that clock A - moved to clock B’s location - will lag behind B. In paragraph 2 Einstein extended this event to clock A moving in any polygonal line then, in paragraph 3, he again extended the phenomenon to a clock that is moving in a closed curve around another clock then that a clock at the equator “must go more slowly” than a clock at one of the poles.

It is my understanding that by the term “must go more slowly” Einstein was suggesting that clock A ticked over at a slower rate than (i.e. incurred time dilation relatively to) clock B.

I believe, although he did not state this, that Einstein was of the opinion that clock A (in paragraph 1) “must go more slowly” (i.e. tick over at a slower rate than) clock B.

An observer accompanying clock A who has read and fully accepts Einstein’s paragraph 1, chapter 4, could take Einstein’s word for it and realize that his clock is ‘going more slowly’ (ticking over at a slower rate) than B irrespective of the fact that his clock (A) appears to have remained unchanged.

Although applicable to general theory - a person at sea level should determine that an identical clock at that location is ticking over at the same rate as his own clock. If he moves to the top of a nearby mountain he could insist (on the basis that a clock at that location is also ticking over at the same rate as his clock) that time does not vary depending on a clock’s location in a gravitational tidal area (as GR shows) on the basis that both clocks tick over at the same rate as his clock OR he could apply his knowledge of the Wallops Island experiment and realize that his watch has also been affected by its location in a different gravitational tidal area irrespective of the fact that it’s rate of operation appears to have remained unchanged.

When Hafele and Keating conducted the first leg of their experiment they could either assume (having returned to Washington to find, as Einstein predicted, that their clocks lagged behind the clocks that had remained ‘at rest’ and having taken into account and eliminating any variations due to gravitational time distortion effects) that the laboratory clocks physically ticked over at a faster rate than they did prior to the flight commencing OR that their clocks physically ticked over at a slower rate than they did prior to the commencement of the flight as per Einstein’s chapter 4.

I am of the opinion that Hafele and Keating (et al) would have accepted the latter explanation irrespective of the fact that, during the flight, their clocks appeared to them to be ticking over at an unchanged rate in the same way that an observer at the top of a mountain could insist that his clock is ticking over at an unchanged rate compared to when he was at sea-level.

Confucius wrote:-

“Knowledge is one-dimensional, the proper application of knowledge is multi-dimensional. Only the extremely wise, and the exceptionally foolish, are not prepared to change.”

Assuming that the mountain-ascending observer is aware of, and fully accepts, the results of the Wallops Island experiment he could (and should) apply that knowledge thus realize that although his clock appears to have remained unaffected it is actually, physically, ticking over at a faster rate than it was when he was at sea-level.

Although an observer accompanying Einstein’s chapter 4 clock A could be of the opinion that his clock’s rate of operation remains unchanged he could also (assuming that he has read and accepted Einstein’s chapter 4 thought experiment as well as Einstein’s 1918 negation of the twin paradox and the reports pertaining to the HKX) realize that his clock is actually, physically, ‘going more slowly’ (ticking over at a slower rate) than it was before he started moving.

You may, perhaps, have missed my original posting which included the reason for that submission:-

“I have read several interpretations of the twin paradox one of which insists that the traveler’s clock does not (as Einstein expressed it in chapter 4) ‘go more slowly’ than the Earth clock but that the Earth clock, from the traveler’s point of view, ticks over at a faster rate than his own clock but only during the astronaut’s period of acceleration following turn-around however it is my understanding that the concept of time contraction was, for Einstein, an anathema.”

If he looks out, he should see a polarized universe, events forward happening faster and events rearward happening slower, a scenario which is not normal.

This, of course, is a result of Doppler shift which some postings regarding the twin paradox dismiss retaining the relativistic time dilation effect however I fully appreciate that during acceleration following turn-around an astronaut would observe that the Earth clock appears to be ticking over at a proportionally increasing faster rate however for him to insist that what he sees is reality - that the Earth clock is physically ticking over at a faster rate than it was before he started accelerating is, in my opinion, asinine!

It is to be noted that this ludicrous claim insists that the variations in the rate of operation of the Earth clock only takes place as the astronaut accelerates however the Doppler shift that the astronaut observes at the instant that he starts to take his foot off the gas pedal is precisely the same as it is when he has removed his foot.

In correspondence at that time (about the mid-90s) the author of that claim responded to my question about what happens when the astronaut stops accelerating and is moving with uniform velocity insisting that the astronaut’s clock is then ticking over at the same rate as the Earth clock! He also insisted that there is no variation in the rates of operation of those clocks as the astronaut moves away from (or, with uniform velocity) toward the planet and continued to insist that the total amount of the eventual discrepancy between the two clocks all takes place during acceleration following turn around.

When I asked him what would happen if the astronaut is transporting a light clock and, in lieu of landing back on the planet, traveled past same and his response was that the light clock would not necessarily be ticking over at a slower rate than an Earth-bound light clock.

One of my reasons for submitting the original posting was to see if anyone in this group agreed with such nonsense.

Of course there is the remote possibility that the author of that nonsense and his supporting cronies may have realized their error as the result of my arguments and have dropped the idea - but I doubt it.

All other clocks do not change rates, because the astronaut initiated the motion

That’s what I said; thank you for agreeing with me.

 

[JesseM]

Although an observer accompanying Einstein’s chapter 4 clock A could be of the opinion that his clock’s rate of operation remains unchanged he could also (assuming that he has read and accepted Einstein’s chapter 4 thought experiment as well as Einstein’s 1918 negation of the twin paradox and the reports pertaining to the HKX) realize that his clock is actually, physically, ‘going more slowly’ (ticking over at a slower rate) than it was before he started moving.

As I said, this claim that any clock is "actually, physically" going more slowly than another contradicts relativity (because all inertial frames are equally valid, and different inertial frames disagree about which clock is ticking more slowly), and if you aren't willing to listen to people explain your error, you're violating the rules of the forum, which is not meant to be a place for people to promote ideas that go against mainstream physics. Here again is my explanation of why you are free to use an inertial frame where A is at rest and B is moving, in which case B is ticking more slowly than A, but B will still be ahead when they meet because of the relativity of simultaneity:

Einstein only says that clock A will be behind clock B when they meet, but this does not prove that clock A was running slower. Do you understand what is meant by the phrase "relativity of simultaneity"? Do you understand that if two clocks are synchronized in their own rest frame, that means that in other inertial frames they'll be out-of-sync? There are perfectly valid frames where clock B was running slower, but in these frames B's time was already well ahead of A's time at the moment A accelerated, so because of this "head start" B is still ahead of A's when A meets it, despite the fact that B was ticking slower. If I place a clock 1 mile away from you that reads 5 PM when you start to walk towards it, and that clock is also running at half the correct rate, and if your clock reads 3 PM at the moment you start walking towards it, and your clock is ticking at the normal rate, then if it takes you half an hour to reach the other clock, your clock will read 3:30 and the other clock will read 5:15. In this example it's clear that despite the fact that the other clock was running slower than yours, it's still ahead of yours when you meet because it was already ahead by a lot when you started out.

If you continue to repeat these sorts of claims that one clock is "physically" ticking slower, and ignore explanations of why this is incorrect according to SR, I'll report your posts to the moderators.

 

[cos]

cos .

Quote (cos):-

--I believe that an intelligent astronaut, having accelerated to a relativistic velocity and having taken his foot off the gas pedal, would realize that he is still moving - either away from, or toward, the planet.----(unquote)

Yes,he can look and see the Earth moving relative to him (or vice versa ), but he cannot determine if he is "moving" in an absolute sense. For all he knows he and the Earth may have initially been "moving" and his acceleration may have brought him to "rest" while the Earth carries on "moving".

This is absolutely fundamental in SR.

Matheinste
-

"...he and the Earth may initially been 'moving'" relatively to what? The CBR? Newton's absolute rest?

On the basis that "This is absolutely fundamental in SR" what does SR suggest the Earth could have been moving relatively to?

I see absolutely no difference whatsoever to the astronaut (having come to a stop) moving toward the Earth and Einstein's paragraph 1, chapter 4, reference to clock A being made to move to clock B's location.

Is it possible that Einstein's clocks A and B could also have initially been moving relatively to some form of fundamental reference frame thus that whilst clock A is moving toward B with uniform velocity it would have been 'at rest' relatively to that FRF thus that it would not, as Einstein posited, eventually be found to lag behind clock B on the basis that it was clock B that "carried on moving" thus that B would be found to lag behind A?

Are you of the opinion that Einstein was wrong in suggesting that A will lag behind B on the basis that they could both have initially been moving thus that A came to 'rest'?

 

[JesseM]

"...he and the Earth may initially been 'moving'" relatively to what? The CBR? Newton's absolute rest?

Pretty sure that was matheinste's point, that there can be no evidence for any absolute notion of motion in relativity, and similarly no evidence for any objective truth about which of two clocks is ticking more slowly in an absolute sense.

On the basis that "This is absolutely fundamental in SR" what does SR suggest the Earth could have been moving relatively to?

What's fundamental is that there is no absolute truth about how fast an object is moving; your comment that the astronaut "would realize that he is still moving" seemed to say otherwise (it is perfectly valid to use an inertial frame where the astronaut comes to rest after finishing his acceleration, so in this frame he is not moving after that, and this inertial frame's perspective is no more or less valid than any other's).

Are you of the opinion that Einstein was wrong in suggesting that A will lag behind B on the basis that they could both have initially been moving thus that A came to 'rest'?

As I keep saying, if A and B initially had synchronized clocks in their mutual rest frame before A accelerated, then in a different frame where A was at rest after accelerating, B's clock was already significantly ahead of A's before A accelerated, so even though B ticks more slowly in this frame as it approaches A, it will still be ahead of A when they meet. This inertial frame's perspective is no more or less valid than any other's, therefore it is just as valid to say B was running slower than A as they approached each other as it is to say A was running slower than B as they approached each other.

 

[atyy]

I see absolutely no difference whatsoever to the astronaut (having come to a stop) moving toward the Earth and Einstein's paragraph 1, chapter 4, reference to clock A being made to move to clock B's location.

Referring to your first post, clocks B and A are specified to be "synchronous", a technical term. Is the astronaut, having come to a stop, also "synchronous" with the non-rotating non-gravitating Earth? (Non-gravitating since only special relativity is being discussed, and non-rotating to save the Earth twin from being spun off into space without gravity.)

 

[matheinste]

Hello cos

Quote:-

----"...he and the Earth may initially been 'moving'" relatively to what? The CBR? Newton's absolute rest?----

Relative to any non-accelerating object.

Matheinste

 

[cos]

Referring to your first post, clocks B and A are specified to be "synchronous", a technical term. Is the astronaut, having come to a stop, also "synchronous" with the non-rotating non-gravitating Earth? (Non-gravitating since only special relativity is being discussed, and non-rotating to save the Earth twin from being spun off into space without gravity.)

Let us imagine that the astronaut's outward journey is directly away from the South Pole and, having come to a stop, he is now looking back at a very large clock at that location which is mounted on a platform that allows it to remain stationary from the traveler's point of view (i.e. it is not spinning around with the planet).

According to my interpretation of Einstein's chapter 4, paragraph 1, his clock (Einstein's clock A) will then lag behind the Earth clock (Einstein's clock B) by .5tv²/c2. When he determines a lag created by the time that it takes that light to reach him as well as any gravitational time dilation created by the Earth's mass he can calculate the exact amount of that lag however his clock will then be ticking over at the same rate as the Earth clock on the basis that they are Einstein's paragraph 1, chapter 4, 'points A and B of K.

He then adjusts his clock so that it reads the same time as the Earth clock so yes, the astronaut's clock is (temporarily) synchronous with the Earth clock and whilst the Earth clock's rate of operation is affected by it's location in a gravitational tidal area the astronaut attains Einstein's chapter 4 (purely hypothetical) instantaneous velocity (of near-light speed for the astronaut) ergo his clock is then 'going more slowly' than the Earth clock by a factor of .5tv²/c2 and he will arrive back on the planet with his clock lagging behind the Earth clock by the same amount as it did at the end of his outward-bound trip.

Alternatively, if the astronaut (in a suitably equipped ship), accelerates at perhaps 100g his clock will very soon be 'going more slowly' than the gravitationally affected Earth clock.

A version of your depiction is that Einstein's paragraph 1, chapter 4, clocks A and B are twin astronaut's each in identical ships that, unlike Einstein's clocks A and B, are initially stationary alongside, and synchronous with, each other whereupon A moves in Einstein's paragraph 2 polygonal path (i.e. away from then back to B's location).

At the end of his 'outward-bound' trip A's clock, although lagging behind, is ticking over at the same rate as B's clock so A adjusts his clock in accordance with the calculated lag factor and they are once again synchronous.

 

[atyy]

Let us imagine that the astronaut's outward journey is directly away from the South Pole and, having come to a stop, he is now looking back at a very large clock at that location which is mounted on a platform that allows it to remain stationary from the traveler's point of view (i.e. it is not spinning around with the planet).

According to my interpretation of Einstein's chapter 4, paragraph 1, his clock (Einstein's clock A) will then lag behind the Earth clock (Einstein's clock B) by .5tv²/c2. When he determines a lag created by the time that it takes that light to reach him as well as any gravitational time dilation created by the Earth's mass he can calculate the exact amount of that lag however his clock will then be ticking over at the same rate as the Earth clock on the basis that they are Einstein's paragraph 1, chapter 4, 'points A and B of K.

He then adjusts his clock so that it reads the same time as the Earth clock so yes, the astronaut's clock is (temporarily) synchronous with the Earth clock and whilst the Earth clock's rate of operation is affected by it's location in a gravitational tidal area the astronaut attains Einstein's chapter 4 (purely hypothetical) instantaneous velocity (of near-light speed for the astronaut) ergo his clock is then 'going more slowly' than the Earth clock by a factor of .5tv²/c2 and he will arrive back on the planet with his clock lagging behind the Earth clock by the same amount as it did at the end of his outward-bound trip.

Alternatively, if the astronaut (in a suitably equipped ship), accelerates at perhaps 100g his clock will very soon be 'going more slowly' than the gravitationally affected Earth clock.

A version of your depiction is that Einstein's paragraph 1, chapter 4, clocks A and B are twin astronaut's each in identical ships that, unlike Einstein's clocks A and B, are initially stationary alongside, and synchronous with, each other whereupon A moves in Einstein's paragraph 2 polygonal path (i.e. away from then back to B's location).

At the end of his 'outward-bound' trip A's clock, although lagging behind, is ticking over at the same rate as B's clock so A adjusts his clock in accordance with the calculated lag factor and they are once again synchronous.

So which twin is older when they reunite?

 

[phyti]

JesseM;

As I said, this claim that any clock is "actually, physically" going more slowly than another contradicts relativity

Then what was measured in the prolonged half life of muons?

 

[JesseM]

Then what was measured in the prolonged half life of muons?

The muons have a slowed rate of decay in our frame where they are moving at relativistic speed, but they aren't slowed down in any objective, frame-independent sense. You can analyze the behavior of muons perfectly well in a frame where the muons are at rest and the Earth is moving at relativistic speed, and you get the exact same prediction about the point on Earth where they decay. Have a look at this page which analyzes muons created in the upper atmosphere in both the Earth frame and the muon frame; in the Earth frame, the muons are able to make it to the surface because their decay is slowed down, while in the muon frame, they decay at the normal rate but they are able to make it to the surface because the distance from the upper atmosphere to the surface is shrunk due to Lorentz contraction.

If you don't understand that any situation in special relativity can be analyzed in any inertial frame using precisely the same laws of physics (so in each frame you assume clocks moving faster in that frame are slowed down by a greater amount) and you'll always get all the same predictions about local physical events (like whether a muon reaches the surface, or what two clocks read at the moment they pass next to each other), you've missed one of the most central conceptual ideas of SR--this is the meaning of the first postulate.

 

[phyti]

JesseM;

in the Earth frame, the muons are able to make it to the surface because their decay is slowed down,

Here you gave a good answer.

while in the muon frame, they decay at the normal rate but they are able to make it to the surface because the distance from the upper atmosphere to the surface is shrunk due to Lorentz contraction.

Here you'll have to explain what physical process shrinks space!
To my knowledge, it has never been experimentally verified.

 

[JesseM]

Here you'll have to explain what physical process shrinks space!

What do you mean by "shrinks space"? It's just a fact about the coordinate systems given by the Lorentz transformation that objects have a shorter coordinate length in a frame where they're moving than in their rest frame, this has nothing to do with physics, it would be true even if you used these coordinate systems in a universe governed by Newtonian laws, for example. Where physics enters into it is that the equations of all known fundamental laws have the property of being invariant under the Lorentz transform, which means they'll look the same in all the different inertial coordinate systems given by the Lorentz transform--this means that if two observers construct physical rulers at rest in their own frame using identical procedures, then the coordinate distance between ends of these identically-constructed rulers will be identical in the rulers' rest frame (this would not be true in a universe with Newtonian laws), and from there it's just a property of the coordinate transformation that if you have two objects moving relative to one another that have the same coordinate length in their own rest frame, in each one's own rest frame the other one will have a shorter coordinate length.

To my knowledge, it has never been experimentally verified.

It's just a mathematical matter to check if some equations of physics are Lorentz-invariant, if they are that means they will be the same in different frames given by the Lorentz transform, and everything I said above will apply. So, to the extent that the equations we have for quantum field theory are Lorentz-invariant and all the tests have supported the idea that systems obey these equations, this is a form of indirect evidence for Lorentz contraction. The page on evidence for special relativity lists another form of evidence for length contraction:

A current-carrying wire is observed to be electrically neutral in its rest frame, and a nearby charged particle at rest in that frame is unaffected by the current. A nearby charged particle that is moving parallel to the wire, however, is subject to a magnetic force that is related to its speed relative to the wire. If one considers the situation in the rest frame of a charge moving with the drift velocity of the electrons in the wire, the force is purely electrostatic due to the different length contractions of the positive and negative charges in the wire (the former are fixed relative to the wire, while the latter are mobile with drift velocities of a few mm per second). This approach gives the correct quantitative value of the magnetic force in the wire frame. This is discussed in more detail in: Purcel, Electricity and Magnetism. It is rather remarkable that relativistic effects for such a tiny velocity explain the enormous magnetic forces we observe.

 

[cos]

Hello cos

Quote:-

----"...he and the Earth may initially been 'moving'" relatively to what? The CBR? Newton's absolute rest?----

Relative to any non-accelerating object.

Matheinste

So when he is, as you expressed it, at ‘rest’ this only applies to the reference frame of a non-accelerating object relatively to which the Earth was moving at the velocity that the astronaut attains at the end of his period of acceleration?

It seems that you are depicting a reference frame (an object) relatively to which the Earth was moving at near-light speed. The astronaut aims his ship at that object and accelerates until he attains a velocity whereby he is moving away from the Earth at near-light speed and is then at rest in that object’s reference frame.

So the only reference frames relatively to which he could be ‘at rest’ at the end of that period of acceleration are those that, before he started accelerating, the Earth was moving relatively to at near-light speed and only in the opposite direction to the astronaut’s planned route not to any (i.e. all) non-accelerating object’s reference frame?

So your response “Relative to any non-accelerating object.” combined with your comment that “his acceleration may have brought him to ‘rest’” does not, as you point out, apply to any ‘non-accelerating object’ but only to a non-accelerating object that was located in the direction of the astronaut’s route and only relatively to which the Earth was moving at near-light speed.

Having accelerated to near-light speed away from the Earth the astronaut could be ‘at rest’ in the reference frame of another (or even several) non-accelerating objects however he is also moving relatively to the ‘non-accelerating’ reference frame that is the planet Earth.

On the basis of your comment that the Earth could have been moving “relative to any non-accelerating object” is it possible that Einstein's clocks A and B could also have initially been moving relatively to some non-accelerating object thus that whilst clock A is moving toward B with uniform velocity it would have been 'at rest' relatively to that non- accelerating object thus that it would not, as Einstein posited, eventually be found to lag behind clock B on the basis that it was clock B that "carried on moving" thus that B would be found to lag behind A?

Are you of the opinion that Einstein was wrong in suggesting that A will lag behind B on the basis that they could both have initially been moving relatively to any non-accelerating object thus that A came to 'rest' in that object’s reference frame whilst B kept moving?

I am of the opinion that Einstein’s paragraph 1, chapter 4 depiction of clocks A and B was in relation to the fact that clock A was moving relatively to the non-accelerating object clock B and that Einstein’s conclusion that A lags behind B is from the point of view of clock B’s reference frame not from the point of view of any other non-accelerating object.

 

[cos]

So which twin is older when they reunite?

At the end of his outward-bound trip the astronaut's clock lags behind his Earth-bound twin's clock and when he returns to the planet his clock lags even further behind his twin's clock in accordance with Einstein's paragraph 1, chapter 4 depiction ergo, according to that depiction, the astronaut will have aged at a slower rate than his twin thus the Earth-bound twin will be the elder.

 

[JesseM]

On the basis of your comment that the Earth could have been moving “relative to any non-accelerating object” is it possible that Einstein's clocks A and B could also have initially been moving relatively to some non-accelerating object thus that whilst clock A is moving toward B with uniform velocity it would have been 'at rest' relatively to that non- accelerating object thus that it would not, as Einstein posited, eventually be found to lag behind clock B on the basis that it was clock B that "carried on moving" thus that B would be found to lag behind A?

As always, the relativity of simultaneity explains why, even if you look at a frame where A is at rest after accelerating and B is in motion, A's time will still be behind B's when they meet even though B was ticking slower as they approached each other in this frame. Einstein specified that A and B were synchronized in the frame where they were both at rest before A accelerated, which means that in the other frame where A is at rest after it accelerated, B's clock-reading was already well ahead of A's clock-reading before A accelerated.

(by the way, it's pretty hypocritical that you stopped responding to my posts because you didn't like my discussing other reference frames, and yet here you are asking matheinste about a different reference frame where A is at rest after accelerating, which is precisely what I had been talking about in relation to this problem)