Twin Paradox: Einstein's Explanation and Alternative Interpretations

[phyti]

cos;
from post 110

In that chapter (paragraph 1) Einstein wrote:- “If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by .5tv2/c2 ... t being the time occupied in the journey from A to B.”

In paragraph 3 Einstein refers to a clock that has remained at rest compared with an identical clock that has moved in a closed curve around that clock. I am of the opinion that he implied that clock B in paragraph 1 remains at rest ergo, in that chapter, Einstein does not allow that clock B “moves through spacetime”.

If someone were to draw a diagram of clocks A and B moving through spacetime they would be presenting that phenomenon from the point of view of another reference frame however Einstein specifically pointed out that the event is “viewed in the stationary system” (i.e. clock B’s reference frame).

Here is a space-time diagram from the B-frame.

Here is a quote from the Max Born book, page 257, which you have (A and B swapped).

"The paradoxical feature of this result lies in the circumstance that every internal process in the system A must take place more slowly than the same process in the system B."

Because A & B are synchonized initially, the only change is the motion of A. If Albert states there is a time difference when they meet, the time effect must be caused by the motion. He authored the theory, so he should know.

For those who do not understand, just because we can find a reference frame with a different perspective, does not mean this case is less real.

 

[JesseM]

cos;
from post 110Here is a space-time diagram from the B-frame.

Here is a quote from the Max Born book, page 257, which you have (A and B swapped).

"The paradoxical feature of this result lies in the circumstance that every internal process in the system A must take place more slowly than the same process in the system B."

I'd like to see the context of that quote, I wonder if Born specified that he was talking about what was happening in the rest frame of A. Looking at some other sample pages from the google book preview, I note that on p. 255 Born writes:

The same remarks apply to the relativity of time. An ideal clock has always one and the same rate of beating in the system of reference in which it is at rest. It indicates the "proper time" of the system of reference. Regarded from another system, however, it goes more slowly. In such a system a definite interval of the proper time seems longer. Here, too, it is meaningless to ask what is the "real" duration of an event.

Because A & B are synchonized initially,

Only relative to the time coordinates of the frame where they are initially at rest.

the only change is the motion of A. If Albert states there is a time difference when they meet, the time effect must be caused by the motion. He authored the theory, so he should know.

For those who do not understand, just because we can find a reference frame with a different perspective, does not mean this case is less real.

I didn't say any frame's perspective was "less real" than any others, just that no frame's perspective is more "physical" than any other's, so if different frames disagree about some question there can be no singe "true" physical answer to that question. Of course it is valid to say that in the rest frame of B, A is running slower than B after A accelerates...but do you claim that in this scenario A is "actually, physically" running slower than B as they approach one another?

 

[atyy]

"The paradoxical feature of this result lies in the circumstance that every internal process in the system A must take place more slowly than the same process in the system B."

Because A & B are synchonized initially, the only change is the motion of A. If Albert states there is a time difference when they meet, the time effect must be caused by the motion. He authored the theory, so he should know.

For those who do not understand, just because we can find a reference frame with a different perspective, does not mean this case is less real.

Time dilation and length contraction are said to be "real" by many standard books (Born is hardly the only one). On the other hand, it is also said that the only "real" things are frame independent quantities. I suspect it is two definitions of "real" here.

We could try "fast moving muons really decay more slowly, but they are not really moving fast". If the second half of that statement makes sense, then the first half cannot make sense. So maybe we try,"If a muon is found to be moving fast, it will decay more slowly", which makes perfect sense since the conditional is equivalent to specifying a reference frame. Born has a similar conditional "if the clocks are synchronized", which is again equivalent to a choice of reference frame. So if we define "velocity" or "synchronization" to be "real", then time dilation automatically becomes real.

It seems that the twin paradox can be stated with or without an initial synchronization. This is because there is no need to define simultaneity to mark the start and end of the story (unlike Einstein's Chap 4 or Born's clock, which requires a definition of simultaneity to mark the start of the story). Regardless of initial synchronization, the difference in accumulated proper time is real in an observer independent sense. Yet, the relative rate of their ageing is not real in an observer independent sense, since like velocity, the relative rate of ageing cannot even make sense without a definition of simultaneity, which is the choice of a reference frame. Does this make any sense?

 

[cos]

As you consider the revised example, please remember that atyy is talking only about spatial distances, not time. So the duration and speed of the respective trips are explicitly not considered. This is purely a geometrical question.

Do you think that the distance between the ticks of Y's odometer was 2.3 times greater than the distance between the ticks of Z's odometer?

I’m having trouble getting my head around that depiction (I assume that it has some relationship to length contraction however on the basis that length contraction is ‘determined’ on the basis of relative speeds and, as you point out, the speeds of the respective trips are not taken into account) but - no - I do not think that the distance between the ticks of Y's odometer was 2.3 times greater than the distance between the ticks of Z's odometer.

The distance between the respective ‘ticks’ on both odometers is determined on the basis of the physical distance traveled by their vehicles i.e. on the number of rotations of their wheels due to their contact with the road and because the respective lengths of the amount of road traversed varies they attain different odometer readings.

If, in lieu of a direct trip home after turn-around, the astronaut takes a diversionary trip to Mars (off to one side of the direct route) but at the same speed as he would have been moving on his direct trip he will calculate that the distance he travels is greater than it would have been for the direct trip.

His ‘odometer’ will record more ticks on the stop-Mars-home journey than it would have for the direct trip however this does not mean that the distance between those ticks varies depending on which route he takes but merely that there are more of them as a result of the longer distance traveled on the stop-Mars-home journey than there would have been on the direct trip.

I’m sorry, but the answer to your question seems so obvious that perhaps I misinterpreted it.

 

[atyy]

Yet, the relative rate of their ageing is not real in an observer independent sense, since like velocity, the relative rate of ageing cannot even make sense without a definition of simultaneity, which is the choice of a reference frame. Does this make any sense?

Are there are cases where the relative rate of ageing makes sense in an observer independent way, but the accumulated proper time is observer dependent? How about two clocks which are stationary relative to each other in some inertial frame, but separated by some distance. Is it the case that their relative rates will be the same in all frames, but the elapsed proper time is frame dependent?

 

[atyy]

I do not think that the distance between the ticks of Y's odometer was 2.3 times greater than the distance between the ticks of Z's odometer.

Yes, that's what I also thought the answer should be. I'll let DaleSpam have the extended commentary since it was his version that clarified things nicely.

 

[cos]

Here is a quote from the Max Born book, page 257, which you have (A and B swapped).

"The paradoxical feature of this result lies in the circumstance that every internal process in the system A must take place more slowly than the same process in the system B."

Because A & B are synchronized initially, the only change is the motion of A. If Albert states there is a time difference when they meet, the time effect must be caused by the motion. He authored the theory, so he should know.

Thank’s for that support of my argument.

It was perhaps about 18 years ago that I read Born’s book in its entirety and it would have been then that I placed a bookmark in same which, to my surprize, now, I found to be located between pages 254 and 255.

To my shame I now find that my criticism of Born for depicting a sliced cucumber as an analogy for length contraction was (ignorantly, on my behalf) taken out of context thus was totally unwarranted.

Toward the conclusion of the previous paragraph (the penultimate sentence of same) on page 254 Born wrote:-

“Thus the contraction is only a consequence of our way of regarding things and is not a change of a physical reality.”

However, on page 255, he wrote:-

“In exactly the same way a rod in Einstein’s theory has various lengths according to the point of view of the observer. One of these lengths, the statical or proper length, is the greatest but this does not make it more real than the others.”

It seems, to me, that this comment conflicts with his abovementioned ensuing comment that an observation that a moving rod appears to contract in length “...is only a consequence of our way of regarding things and is not a change of a physical reality.”

(ref: my comment below in relation to my understanding of ‘reality’).

For those who do not understand, just because we can find a reference frame with a different perspective, does not mean this case is less real.

My interpretation of ‘reality’ is in relation to an event that takes place in an observer’s reference frame.

An observer in another reference frame may well have a different opinion of what is taking place in the first reference frame however I believe that he should be allowed to conclude that what appears to be taking place in the other reference frame is affected by their respective rates of travel.

Thus what appears to that observer to be taking place in another reference frame is only a consequence of his way of regarding things and is not a change of a physical reality in that other reference frame.

In his book ‘An Introduction to the Special Theory of Relativity’ Professor Robert Katz wrote:-

“Is the moving rod really contracted in its direction of motion? Is time really dilated? These questions depend on what is meant by really. In physics what is real is identical with what is measured.”

In the preceding paragraph Katz wrote:-

“The way in which we set up to measure the length of a moving rod determines that we will measure a shorter length than the rod length.”

This gives me the impression that we could set up an experiment in such a way that we would not measure a shorter length than the rod length.

On the basis that ‘in physics what is real is identical with what is measured’ and, having set up the experiment in a different way, we measure the same (or even a greater) length than the rod length then that determination should be accepted as being reality - according to Katz.

When Albert Einstein wrote in the introduction to his general theory of relativity that the law of the constancy of the velocity of light ‘requires modification’ or in his book ‘Relativity’ that this law is not fully valid he was, apparently, castigated by his colleagues (particularly Max Abrahams) for ‘arguing against the validity of the [then] mainstream understanding of relativity’ however I believe that he had every right to do so!

If Einstein (or Max Born et al) were to have made a comment which, posted in this group, ‘argued against the validity of the mainstream understanding of relativity’ would it be censored?

When Einstein’s OEMB was published he was castigated by his colleagues for arguing against the validity of the mainstream understanding of Newtonian physics.

If anyone had been successful in having Einstein’s theory (or those of Lorentz and Fitzgerald et al) banned from dissemination we may not, now, be having these exchanges of opinions.

 

[JesseM]

My interpretation of ‘reality’ is in relation to an event that takes place in an observer’s reference frame.

So does that mean in your way of speaking, you allow different observers to have different "realities"? A reference frame is really just a coordinate system--even two observers at rest relative to one another may choose to place the origin of their coordinate systems in different places, and thus have different answers to the question of the x-coordinate assigned to a certain event, say. Would you say that for one observer, the "reality" is that the event occurred at x=6 meters and for another the "reality" is that it occurred at x=10 meters, simply because they choose to position the origins of their system differently? I would say that coordinate assignments are a matter of convention, not reality, and I think most physicists would speak the same way (I have seen many physicists talk about the 'simultaneity convention' of a particular coordinate system, for example), and would interpret a phrase like "objective truth" or "physical reality" in the context of relativity to mean a coordinate-invariant truth (see the discussion of the 'mainstream interpretation' of relativity on pages 22-25 here).

If Einstein (or Max Born et al) were to have made a comment which, posted in this group, ‘argued against the validity of the mainstream understanding of relativity’ would it be censored?

But as you said earlier, other statements of theirs make clear they agree there is no single absolute truth about the rate a clock is ticking. If someone comes to this group and posts an ambiguous comment, I'd ask for clarification about what they were saying. But you do not seem willing to answer my basic question about your argument, the same one I just asked phyti, concerning the thought-experiment in section 4 of Einstein's paper: do you claim that in this scenario A is "actually, physically" running slower than B as they approach one another?

Are you not answering me because you're ignoring me as some sort of "punishment" (note that this is a public forum rather than a private discussion, so I continue to respond to your posts in part so they don't mislead other readers), or do you not have a clear answer to this question in your own mind? If the issue is that you define words like "actually, physically" in a way that allows there to be multiple equally valid truths about what is "actually, physically" occurring, then as I said I think this would differ from the way virtually all physicists would use this type of language, but you are free to use these words any way you want, all you would need to do is clarify that you do define them such that there can be multiple equally valid truths and I would no longer see your posts as misleading.

 

[Dale]

I’m having trouble getting my head around that depiction (I assume that it has some relationship to length contraction however on the basis that length contraction is ‘determined’ on the basis of relative speeds and, as you point out, the speeds of the respective trips are not taken into account) but - no - I do not think that the distance between the ticks of Y's odometer was 2.3 times greater than the distance between the ticks of Z's odometer.

The distance between the respective ‘ticks’ on both odometers is determined on the basis of the physical distance traveled by their vehicles i.e. on the number of rotations of their wheels due to their contact with the road and because the respective lengths of the amount of road traversed varies they attain different odometer readings.

If, in lieu of a direct trip home after turn-around, the astronaut takes a diversionary trip to Mars (off to one side of the direct route) but at the same speed as he would have been moving on his direct trip he will calculate that the distance he travels is greater than it would have been for the direct trip.

His ‘odometer’ will record more ticks on the stop-Mars-home journey than it would have for the direct trip however this does not mean that the distance between those ticks varies depending on which route he takes but merely that there are more of them as a result of the longer distance traveled on the stop-Mars-home journey than there would have been on the direct trip.

I’m sorry, but the answer to your question seems so obvious that perhaps I misinterpreted it.

No, you didn't misinterpret it. It is obvious when we are dealing with space and normal Euclidean geometry. We have a very visceral understanding of geometric concepts like distance along a path, and we intuitively know that different paths through the same two points can have different lengths without paradox. We clearly understand that this difference in length is a geometric property of the paths themselves, not the odometers.

Now, special relativity can be most conveniently formulated mathematically using Minkowski geometry with one timelike dimension and three spacelike dimensions. Minkowski geometry is very analogous to the normal Euclidean geometry that we are so comfortable with. The only major difference is that distances are called intervals and are given by ds²=-dt²+dx²+dy²+dz².

In the Minkowski geometric formulation of special relativity a clock is nothing more than an odometer for measuring the interval along a timelike path (e.g. a clock measures one year per light-year). In this formulation, the time dilation and all other SR effects are immediately seen to be no more surprising or paradoxical than the different distances traveled by drivers Y and Z. In Minkowski spacetime clocks don't slow down in any physical sense, they simply take a different path through spacetime and the different interval is a property of that path rather than a property of the clock that measures that path.

 

[cos]

Are you not answering me because you're ignoring me...

You threatened to report me to the group moderators for having the temerity to express my opinions so irrespective of your repetitive taunts I see no point in providing you with potential ammunition.

----> CORRESPONDENCE TERMINATED <-----

 

[JesseM]

You threatened to report me to the group moderators for having the temerity to express my opinions so irrespective of your repetitive taunts I see no point in providing you with potential ammunition.

They are not "taunts", they are questions about what you are trying to argue. The point is that I don't know if your opinions actually are in conflict with SR because you aren't willing to answer these questions. As I said in my previous post, if you wish to clarify that when you said one clock was "actually, physically" running slower than the other, you did not mean this in the exclusive sense I had originally interpreted it (so that you allow for different observers to disagree about which of two clocks is 'actually, physically' running slower and do not claim that one observer is more correct than the other), then I would withdraw my objection that your statements conflict with relativity. On the other hand, if you think that there is a single, exclusive truth about which of two clocks is running slower, this does conflict with relativity, but if you said so I would not report you to the moderators if you were willing to engage in rational discussion about it. The stated purpose of this forum is to help people understand mainstream SR rather than to advocate alternative ideas, and I hope you respect that (there are plenty of other forums one can go to try to shoot down SR); but there is some leeway in that if people have doubts about some aspect of mainstream SR but are genuinely interested in discussing how advocates of the mainstream view would account for whatever seeming problems they have, rather than simply wanting to post polemics against the mainstream view, then I think that isn't a clear violation of the rules.

 

[cos]

In Minkowski spacetime clocks don't slow down in any physical sense, they simply take a different path through spacetime and the different interval is a property of that path rather than a property of the clock that measures that path.

My posting is not in relation to Minkowski spacetime but to Einstein's chapter 4 OEMB depiction wherein he wrote that a balance clock at the equator 'must go more slowly' than a similar clock at one of the poles.

It seems to me that although "In Minkowski spacetime clocks don't slow down in any physical sense..." equatorial clocks do, according to Einstein "go more slowly" than polar clocks.

 

[JesseM]

My posting is not in relation to Minkowski spacetime but to Einstein's chapter 4 OEMB depiction wherein he wrote that a balance clock at the equator 'must go more slowly' than a similar clock at one of the poles.

Minkowski spacetime just means that there is no spacetime curvature, it doesn't say anything about the shape of the paths. So, if you have a clock at the equator of a massless rotating sphere, you're still dealing with a situation in Minkowski spacetime, it's only if you assume the sphere's mass is curving spacetime (which Einstein wouldn't have been assuming in the 1905 paper since GR hadn't been invented) that you're no longer in Minkowski spacetime.

It seems to me that although "In Minkowski spacetime clocks don't slow down in any physical sense..." equatorial clocks do, according to Einstein "go more slowly" than polar clocks.

The odometer analogy works in this case too. In spacetime, the worldline of the clock on the equator would look like a type of helix, as depicted in this diagram with the number of spatial dimensions reduced to 2, and the worldline of the clock at the pole would be a straight path through spacetime going through the center of the helix. So if we instead imagine submarines moving through a 3D volume of water, one going in a straight line and the other taking a helix-shaped path through space that wraps around the straight line, then if each submarine has some type of 3D odometer keeping track of the mileage of its path through the water, of course the helix-shaped path will accumulate greater mileage, but that's a function of the shape of its path through space rather than indicating its miles are a different length than the miles of the submarine going along a straight path.

 

[Dale]

My posting is not in relation to Minkowski spacetime but to Einstein's chapter 4 OEMB depiction wherein he wrote that a balance clock at the equator 'must go more slowly' than a similar clock at one of the poles.

It seems to me that although "In Minkowski spacetime clocks don't slow down in any physical sense..." equatorial clocks do, according to Einstein "go more slowly" than polar clocks.

One particular section of one particular work is a rather narrow focus. I am trying to provide you with a much more general conceptual tool.

In addition to being more intuitive than other approaches, the spacetime geometric approach is universally applicable. It applies equally well for any arbitrary set of clocks traveling on any arbitrary set of paths. Once you have the appropriate metric for a given coordinate system it even applies in non-Cartesian cases and in the curved spacetimes of general relativity. In short, it always works.

JesseM gave a brief description of how the geometric approach would work for the polar/equatorial case. If you have any questions I would be glad to address them.

 

[phyti]

I'd like to see the context of that quote, I wonder if Born specified that he was talking about what was happening in the rest frame of A. Looking at some other sample pages from the google book preview, I note that on p. 255 Born writes:


Only relative to the time coordinates of the frame where they are initially at rest.

I didn't say any frame's perspective was "less real" than any others, just that no frame's perspective is more "physical" than any other's, so if different frames disagree about some question there can be no singe "true" physical answer to that question. Of course it is valid to say that in the rest frame of B, A is running slower than B after A accelerates...but do you claim that in this scenario A is "actually, physically" running slower than B as they approach one another?

Einstein states when the clocks are togther, A lags behind B. The began in synch. Only B moved. The trip is over. How did the A clock get behind the B clock, without slowing down?
We are not talking Harry Potter physics!

 

[phyti]

to cos; post 127

It seems, to me, that this comment conflicts with his abovementioned ensuing
comment that an observation that a moving rod appears to contract in length
“...is only a consequence of our way of regarding things and is not a change
of a physical reality.”

He is talking about physical events and perception of the events, and assumes the reader can tell the difference from the context. He is saying time dilation is real/physical, an altered state of the moving system, and length contraction is a measurement/perception process. A biological system is just atoms and photon exchanges, thus the astronaut does not perceive the time dilation effect, he is part of it. The measurement process involves two spatial endpoints of an object, but each signal originated at a different time.
You are probably aware that some authors know the subject, but are not adept at communicating ideas, i.e., cucumbers.

My interpretation of ‘reality’ is in relation to an event that takes place in
an observer’s reference frame.

Your reality then is perception, which is sufficient for everyday life, unless you are an astronaut!

The saying 'things are not as they appear', requires a distinction between events (reality) and perception (awareness of events).
Subjective simultaneity is a sphere of light containing a composition of signals from various parts of the universe, converging on the viewer. I doubt you would conclude that a distant event happened simultaneously with something happening next to you, even though the light signals are coincident. Perception is 'now', events are always in the past.

I do not agree with Robert Katz. Suppose upon measuring a fast moving space probe from earth, you accept your length measurement of 20 ft. as 'real'. You build a 21 ft container for it, send it on a shuttle with meets the probe in orbit, and find it's really 24ft.
Which is real? Again, perception or reality?

If Einstein (or Max Born et al) were to have made a comment which, posted in
this group, ‘argued against the validity of the mainstream understanding of
relativity’ would it be censored?

Most people don't like change, in any aspect of life unless there is an immediately obvious benefit. Otherwise they have to learn new things and readjust their thinking.
Sometimes it's plain ignorance, eg., the LHC experiments at CERN. We could go way back, when it was blasphemy to suggest the Earth was not the center of the ego-centric universe. History is full of this.

Thanks for a sensible discussion.

 

[JesseM]

Einstein states when the clocks are togther, A lags behind B. The began in synch. Only B moved. The trip is over. How did the A clock get behind the B clock, without slowing down?
We are not talking Harry Potter physics!

Einstein only says they "began in sync" in the frame where B is at rest (his words are 'If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous'). Do you understand that because of the relativity of simultaneity, if we choose a different inertial frame where A is at rest after it accelerates, the two clocks were not in syc to begin with in this frame, instead B was already significantly ahead of A at the moment before A accelerated? So, even though B was running slower than A after A accelerated, B is still ahead when A reaches it because of this "head start". This follows directly from the Lorentz transformation, and I gave a numerical example and showed how the numbers work out in post #31 of this thread.

 

[Dale]

Einstein states when the clocks are togther, A lags behind B. The began in synch. Only B moved. The trip is over. How did the A clock get behind the B clock, without slowing down?

By taking a longer path through spacetime.

 

[JesseM]

By taking a longer path through spacetime.

In this case Einstein was talking about two clocks that started out a certain distance apart, rather than a twin paradox scenario where they start and finish at the same position, so I don't know if this answer is applicable...you'd have to pick points on each clock's worldline to be the "starting points" of each clock's path through spacetime (the ending point being when they meet of course), but the choice is somewhat arbitrary, and which one has a longer path through spacetime depends on what starting points you choose.

edit: I suppose you might have meant something like "longer path through spacetime from the moment each clock was set to zero to the moment they met", in which case your statement would make sense since they'd been synchronized in B's rest frame before A accelerated.

 

[Fredrik]

Einstein states when the clocks are togther, A lags behind B. The began in synch. Only B moved. The trip is over. How did the A clock get behind the B clock, without slowing down?

(I wrote this before JM and DS posted their replies).

I think the point was that a statement like "A is actually, physically running slower than B" doesn't make sense, not that it's wrong. The "actually, physically" only makes sense if we're talking about proper time, which is coordinate independent and only defined along a path. The "running slower" only makes sense if we specify one specific event on each of the world lines, and use the co-moving inertial frames (or two other coordinate systems which we must specify) to compare the clocks' ticking rates in those two frames, at those two events.

A doesn't "slow down". Both A and B do what they're supposed to, which is to measure a property of their respective world lines. It's the world lines that are different, not the clocks.

 

[Dale]

edit: I suppose you might have meant something like "longer path through spacetime from the moment each clock was set to zero to the moment they met", in which case your statement would make sense since they'd been synchronized in B's rest frame before A accelerated.

Yes, exactly. From the point on each worldline where they were synchronized to the reunion event there is a shorter spacetime interval for one clock than the other.

This is no more surprising than the fact that someone driving from Boston to New York shows a smaller odometer reading than someone driving from Atlanta to New York even if they both reset their odometers at the beginning of their respective trips. You could even find two starting cities with the same lattitude or longitude to make the analogy more exact, but I am too lazy.

A doesn't "slow down". Both A and B do what they're supposed to, which is to measure a property of their respective world lines. It's the world lines that are different, not the clocks.

Well said.

 

[phyti]

... In Minkowski spacetime clocks don't slow down in any physical sense, they simply take a different path through spacetime and the different interval is a property of that path rather than a property of the clock that measures that path.

The slowdown is built in, aka, the calibration curve.

 

[JesseM]

The slowdown is built in, aka, the calibration curve.

What do you mean by "calibration curve"? The definition here seems to have nothing to do with relativity, but maybe you're using the term differently.

 

[matheinste]

Hello phyti

Are you referring to the calibration of axes using the invariant hyperbolae.

Matheinste

 

[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.

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.

My interpretation of ‘reality’ is in relation to an event that takes place in an observer’s reference frame.

It seems to me that although "In Minkowski spacetime clocks don't slow down in any physical sense..." equatorial clocks do, according to Einstein "go more slowly" than polar clocks.

I believe these statements are correct.

The equatorial clock is a clock in Minkowski space - that is just the technical name for the spacetime that Einstein discusses in Chap 4. So both are correct descriptions of the same physical situation.

The initial clock synchronization involves equalizing the zero-settings of the clocks and the rate of the clocks.

Once a reference frame is stated to be "real", the traveling clock is the one that "really" slows - in the sense that it lags.

This is exactly equivalent to saying that the Minkowski rate of the clock is unchanged. To resynchronize the clock, one only has to readjust its zero-setting (as you said in the above quotes), and not its rate (just like resetting an odometer).

 

[Fredrik]

The slowdown is built in, aka, the calibration curve.

I don't know what you mean by calibration curve, but for a clock to even have a ticking rate, we have to be talking about a specific point on its world line, and a specific coordinate system. This means that you can only compare the ticking rates of two different clocks if you specify the points on their world lines where you will compare, and the two coordinate systems you will use for the comparison. Since you can get any result you want to just by choosing appropriate coordinate systems, this comparison doesn't make much sense unless there's a situation where there's a "natural" choice of an event and a coordinate system to associate with the second clock once you have chose an event and a coordinate system to associate with the first clock.

The only one I can think of is an event where the two world lines intersect. We use one coordinate system: the co-moving inertial frame of one of the clocks. The ticking rates can be defined as "change of proper time"/"change of coordinate time". (This gives one of the clocks a ticking rate of 1, but the other can have any value).

 

[Fredrik]

Well said.

Of course it is. It's almost exactly the same thing you said earlier. (I didn't see that until now).

 

[phyti]

JesseM;
from your post 31, with [corrections]

A now ticks at the normal rate in this frame since it's at rest. Since the initial distance between them is 12 light-seconds [20 ls] in this frame, it will take 12/0.8c = 15 [20/.8 = 25] seconds for B to catch up with A. During this time A will advance forward by 15 [25] seconds but B will only advance forward by 15*0.6 = 9 [25*.6 = 15] seconds.

The example cited by cos would be trivial if it just meant one clock lagging behind the other, i.e., you could set either ahead and prove any scenario. Einstein intended to demonstrate the connection between time dilation and motion. Your example just mirrors the original setup, with A and B swapped. As the diagram shows, the longer leg has less proper time.

 

[cos]

One particular section of one particular work is a rather narrow focus. I am trying to provide you with a much more general conceptual tool.

'One particular work' - Einstein's special theory - is my specific interest and whilst I appreciate that you are, perhaps, providing a more general conceptual tool in relation to the twin paradox it is not the 'twin paradox' per se in which I am particularly interested but my argument relates to the claim that the astronaut is of the opinion that his clock does not 'go more slowly' than it did before he started moving but that the Earth clock physically 'goes faster' than it did when he had come to a stop at the end of his outward-bound trip.

One particular section (chapter 4) of one particular work (special theory) is the crux of the argument from my point of view.

I am of the opinion that having read and accepted special theory including chapter 4 as well as Einstein's 1918 article - an astronaut should be capable of realizing that although his clock appears to be ticking over at its standard rate as he returns to Earth it is, according to Einstein 'going more slowly' than it was before he started moving.

As I have pointed out, a person who moves to the top of a mountain could be of the opinion that a clock at that location is ticking over at the same rate as a clock at sea-level because they are both ticking over at the same rate as his own clock which is, in his opinion, ticking over at its normal rate when he is at both locations.

Alternatively he could (as Confucius suggested) apply his (assumed) knowledge of the Wallops Island experiment and realize that his clock is ticking over at different rates depending on his location in a gravitational field.

As the astronaut starts to accelerate for the return trip he will see his clock continuously appearing to be ticking over at a constant rate however, due to the fact that his velocity (v) is increasing his clock will, in accordance with Einstein's chapter 4 equation (.5tv²/c²), be 'going more slowly' than it was at a lesser instantaneous velocity.

Although Einstein's chapter 4 is 'one particular section of one particular work' I fail to see any reason why it should be ignored or not taken into consideration.

If Einstein's chapter 4 depiction is 'wrong' or if it argues against mainstream physics community interpretations of relativity then perhaps it should be expunged or publicly denounced as members of the, then, scientific community (representatives of The Church) would have liked to do with Galileo's 'Two New Sciences' or as members of the, then, physics community would have liked to do with Einstein's general theory comment that the law of the constancy of the velocity of light required modification and his comment in 'Relativity' that the same law is not fully valid - that it can only be applied in conditions that do not, to the best of extant scientific knowledge, exist (i.e. zero gravity).

Ignoring, or in our discussions not taking into account, 'one particular section of one particular work' will not make it go away.

If other sections of a particular work (special theory) contradict that particular section then it seems that special theory must contain an inconsistency but that's not what I'm saying!

What I'm saying is that if an observer is located in an inertial reference frame there is no internal dynamic experiment that he can carry out to determine if he is moving with uniform velocity or if he is 'at rest' and that the same thing applies to all inertial reference frames thus that no reference frame takes precedence over any other reference frame however, in chapter 4 (as well as in his 1918 article), Einstein points out that if a clock is made to move it will 'go more slowly than' an identical clock that remained at rest in the original reference frame. On the basis that a clock is made to move it must undergo acceleration.

According to Nikolai Rudakov in his book 'Fiction Stranger Than Truth' - "Very few relativists have actually adopted Einstein's [1918] explanation [of the twin paradox]. Not many authors mention the 1918 dialogue, and some who do imply that Einstein may have been wrong."

It is my opinion that Einstein's chapter 4 depictions are analogous (albeit sans reference to acceleration) to his 1918 article thus that some people may similarly insist that chapter 4 was wrong thus should not be referred to on the (assumed) basis that it argues against mainstream physics community interpretations of relativity however it is Einstein's relativity - not interpretations of same - to which I refer and it does contain chapter 4.

 

[JesseM]

JesseM;
from your post 31, with [corrections]

A now ticks at the normal rate in this frame since it's at rest. Since the initial distance between them is 12 light-seconds [20 ls] in this frame, it will take 12/0.8c = 15 [20/.8 = 25] seconds for B to catch up with A. During this time A will advance forward by 15 [25] seconds but B will only advance forward by 15*0.6 = 9 [25*.6 = 15] seconds.

What is the basis for your "corrections"? I specified that the initial distance between A and B was 20 light-seconds in the rest frame of B. Do you disagree that, according to relativity, this means that if we analyze the same situation from the perspective of the frame where A is at rest after accelerating (a frame moving at 0.6c relative to B's rest frame), the initial distance between A and B must have been 12 light-seconds in this frame, not 20 light-seconds?

The example cited by cos would be trivial if it just meant one clock lagging behind the other, i.e., you could set either ahead and prove any scenario.

No, you couldn't "set either ahead" by any amount you like, because Einstein specified that the clocks were initially synchronized in the rest frame of B. According to relativity, this uniquely determines the amount that A and B were initially out-of-sync in any other frame moving with some specified velocity relative to the rest frame of B. If A and B were initially 20 light-seconds apart and synchronized in the frame of B, then in a frame moving at 0.6c relative to this frame, moving in the direction from A to B, B must have been initially ahead of A by precisely 12 seconds. Do you disagree? Would you like me to demonstrate this using the Lorentz transformation?

Einstein intended to demonstrate the connection between time dilation and motion. Your example just mirrors the original setup, with A and B swapped.

It's not a different example, it's the same scenario viewed from the perspective of a different frame (you can see that I didn't just swap A and B by noting that in both Einstein's description and my description, A was the clock that accelerated, and in my description A and B were both originally in motion rather than both originally at rest as in Einstein's description). Do you understand that the Lorentz transformation shows us how to describe the same physical situation from the perspective of different reference frames?

As the diagram shows, the longer leg has less proper time.

The time and distance intervals in your diagram are incorrect--see my question about your "corrections" above--but the shape of the two worldlines is correct. I don't know what "tB" is supposed to represent, and I don't know what segment of each worldline you're talking about when you talk about the amount of proper time on each (obviously the endpoint is when the two clocks meet, but what's the starting point on each worldline? Is it the event of each clock reading t=0? If so, note that the starting point along the B worldline should be lower on the page than the starting point on the A worldline, since the B clock is ahead in this frame).