Twin paradox: who decided who is the younger one

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ghwellsjr

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What do the diagonal red lines represent?
Interesting you should ask. The slanted lines are there to show the relative progress of each observer as they move along their respective 4th dimensions at the speed of light. When the traveling twin meets up with the black stay-at-home twin, the home twin is 13 years while the twin guy is 13 years. But wait--the home twin really couldn't be there could he -- if he has so far moved at the speed of light along his X4 to just his own proper time 10 yr point on X4???
Shouldn't one of these be 10 years? But I still can't make any sense out of what you are saying. Do you really think this kind of explanation helps someone learn Special Relativity?
If you think of observers moving at light speed along their 4th dimension, the home twin hasn't really arrived yet. But this is one reason why many physicists have concluded that objects in 4-dimensional space are 4-dimensional themselves. Thus the travel twin is meeting up with a different cross-section view of home twins 4-dimensional body. And the physical bodies really aren't doing any traveling anyway, because they are four-dimensional objects fixed --frozen in a 4-D universe.

It's ugly, I know. My subjective self does not want to accept that picture at all, but as I've said before, there is no alternative external objective picture out there that contradicts the 4-dimensional universe populated by 4-dimensional objects. I think many just abandon the idea of an external objective reality.
You don't seem to be a very good promoter of the Block Universe idea when you make comments like this. Why do you continue to promote it?
 

ghwellsjr

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Here's another example. The hyperbolic calibration curves were computed in MatLab. Notice that when the home twin is 20 yrs old his view of the universe includes the travel twin with his (red's) clock showing 10 yrs. Red (travel twin) is about 17.5 light-years away from home (along the black X1 axis, going about 0.875c) at this point. That's the usual time dilation.
Don't you mean length contraction?

And your example would fit better if you used a speed of 0.866c with a distance according to the traveling twin of 17.32 light-years away at his turn-around point. This would make the traveling twin 20 years old when he met up with his 40 year-old stay-at-home twin as you graphic depicts.

These discrepancies make it very difficult to follow what you are presenting.

But is this another example of a usual Twin Paradox explanation or of a Block Universe explanation?

If you want to put in diagonal lines to show how each twin watches the aging of the other twin, why don't you just do the normal pair of graphics like is shown half way down this page?

http://en.wikipedia.org/wiki/Twin_paradox
 
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Shouldn't one of these be 10 years?
You are right of course. It was a typo. The travel twin is 10 yrs and the home twin is 13 yrs when they meet.

But I still can't make any sense out of what you are saying.
It is simply the special relativity picture of the twin paradox. Sorry I was unable to communicate it in a way that you could comprehend the 4-dimensional description of Minkowski space-time.

Do you really think this kind of explanation helps someone learn Special Relativity?
It's always difficult for someone new to the subject to learn special relativity. Others have been doing a pretty good job of presenting the transformations and the frame-of-reference concepts, so I didn't think I could add much there. However, sometimes it is useful to visualize the paradoxes in the context of the 4-dimensional Minkowski space.

You don't seem to be a very good promoter of the Block Universe idea when you make comments like this. Why do you continue to promote it?
I'm not trying to promote Block Universe. I'm trying to Minkowski 4-D space-time in the context of external physical reality. The idea of an external physical reality is distasteful to some on a philosophical basis. I mean no offense to those and am quite sympathetic to the view. I just happen to be in the realist camp, but at the same time am frustrated with the implications of the 4-dimensional space of special relativity.

The concept of a 4-dimensional universe is very much a part of the subject of special relativity and should not be hidden from newcomers to the subject. Nor should space-time diagrams.

Your tone seems a little confrontational, and I have no interest in participating in confrontational discussions and don't think it's good for the forum.
 
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Don't you mean length contraction?
No. I was specifically referring to the time of 10yrs on the travel twin clock as seen in the the black's world at the black 20 yr point.

And your example would fit better if you used a speed of 0.866c with a distance according to the traveling twin of 17.32 light-years away at his turn-around point. This would make the traveling twin 20 years old when he met up with his 40 year-old stay-at-home twin as you graphic depicts.
That's why I said "approcximately 17.5 yrs. But, you're right. Thanks for giving us the exact numbers.

But is this another example of a usual Twin Paradox explanation or of a Block Universe explanation?
I don't care what you wish to call it. It's just an attempt to present the Twin Paradox using a space-time diagram and noticing obvious implications.

If you want to put in diagonal lines to show how each twin watches the aging of the other twin, why don't you just do the normal pair of graphics like is shown half way down this page?

http://en.wikipedia.org/wiki/Twin_paradox
You are missing the whole point of the diagonal lines. My lines definitely do not represent the instantaneous cross-section views of the 4-dimensional space as depicted in your link (I've done that in several other sketches). This was to emphasize the comparative progress through 4-dimensional space along their respective X4 coordinates. The twins each move at light speed.

The slanted lines call attention to the home twin arriving at his 10-yr point (proper time) in 4-dimensional space when the travel twin has arrived at his 10-yr point (lproper time) and it coincides with the home twin's future point of 13 years (they each travel a speed c along their respective X4 coordinate). This leads one to ponder how the stay home twin can be present to greet his travel twin if he has only moved to his own (black time coordinate) 10-yr point
 
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ghwellsjr

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It is simply the special relativity picture of the twin paradox. Sorry I was unable to communicate it in a way that you could comprehend the 4-dimensional description of Minkowski space-time.
I already comprehend the 4-dimensional description of Minkowski space-time. I didn't ask about that. I asked about the diagonal lines. I have never seen diagonal lines like that going at different angles on any Minkowski space-time diagram except yours. Can you point to an example of anyone else drawing them like that with an explanation of why they are slanted differently?
It's always difficult for someone new to the subject to learn special relativity. Others have been doing a pretty good job of presenting the transformations and the frame-of-reference concepts, so I didn't think I could add much there. However, sometimes it is useful to visualize the paradoxes in the context of the 4-dimensional Minkowski space.
It would be useful if the visualization would explain why there is no paradox rather than just add to the confusion. Please look at your answer in post #24 and tell me why I should have understood your explanation or taken it any more seriously than you seem to have.
I'm not trying to promote Block Universe. I'm trying to Minkowski 4-D space-time in the context of external physical reality. The idea of an external physical reality is distasteful to some on a philosophical basis. I mean no offense to those and am quite sympathetic to the view. I just happen to be in the realist camp, but at the same time am frustrated with the implications of the 4-dimensional space of special relativity.
Then why did you label your axes X1 and X4 instead of x and ct like everyone else does? Can you show me an example of a Minkowski space-time diagram not promoting Block Universe Theory that is labeled like you did yours?
The concept of a 4-dimensional universe is very much a part of the subject of special relativity and should not be hidden from newcomers to the subject. Nor should space-time diagrams.
Yes, 3 dimensions of space and one dimension of time is what makes 4-dimensional space-time, not 4 dimensions of space.
Your tone seems a little confrontational, and I have no interest in participating in confrontational discussions and don't think it's good for the forum.
I'm frustrated, just like you. When I'm trying to help someone understand what I'm saying, I welcome their confused questions and requests for clarification. I don't blame the questioner for being confrontational and threaten to end the conversation.

So please answer my questions above.
 
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I'm frustrated, just like you. When I'm trying to help someone understand what I'm saying, I welcome their confused questions and requests for clarification. I don't blame the questioner for being confrontational and threaten to end the conversation.

So please answer my questions above.
Thanks for your reply, ghwellsjr. I guess I was getting the wrong impression for a minute there. I understand now that you were responding out of frustraion. I was thinking that maybe you were not so much interested in following the logic of a 4-D spatial universe as you were as painting me as some kind of crackpot. Actually, I've always respected your posts. You always seem to present accurate information and have been quite helpful to the newcomers.

I'll be glad to respond to each of your questions. Right now I'm still at the office with a little deadline. I'll get back on as soon as I can this evening.
 

ghwellsjr

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You are missing the whole point of the diagonal lines. My lines definitely do not represent the instantaneous cross-section views of the 4-dimensional space as depicted in your link (I've done that in several other sketches). This was to emphasize the comparative progress through 4-dimensional space along their respective X4 coordinates. The twins each move at light speed.

The slanted lines call attention to the home twin arriving at his 10-yr point (proper time) in 4-dimensional space when the travel twin has arrived at his 10-yr point (lproper time) and it coincides with the home twin's future point of 13 years (they each travel a speed c along their respective X4 coordinate). This leads one to ponder how the stay home twin can be present to greet his travel twin if he has only moved to his own (black time coordinate) 10-yr point
Yes, you are right, I am missing the whole point of your diagonal lines. And you are missing the whole point of the diagonal lines in the http://en.wikipedia.org/wiki/Twin_paradox" [Broken] They do not have anything to do with anything instantaneous or a cross-section view of 4-dimensional space (which is three dimensions of space and one dimension of time). The diagonal lines depict on a normal graph with t on the vertical axis and x on the horizontal axis the progress of signals (or images) emitted at the same repetitive interval by each twin as those signals travel at the speed of light from each twin to the other twin and very clearly show how each twin sees the aging of the other twin so that at the moment they reunite, they both agree that the traveling twin has accumulated less time. These diagrams depict Relativistic Doppler very clearly and without confusion or lingering questions.

All I have gotten out of your diagonal lines is that when the twins reunite, one of them still hasn't arrived yet. Don't you see that as pure nonsense? If not, I welcome further clarification.
 
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Yes, you are right, I am missing the whole point of your diagonal lines. And you are missing the whole point of the diagonal lines in the http://en.wikipedia.org/wiki/Twin_paradox" [Broken] They do not have anything to do with anything instantaneous or a cross-section view of 4-dimensional space (which is three dimensions of space and one dimension of time).
We had a misunderstanding. I was referring to the following space-time diagram in the wiki link:This shows the planes of simultanaeity for the travel twin, which are exactly the diagonal lines I mentioned in earlier posts (yes, they represent 3-D cross-sections of a 4-dimensional space, x2 and x3 being supressed).
TwinParadox_Wiki.jpg


All I have gotten out of your diagonal lines is that when the twins reunite, one of them still hasn't arrived yet. Don't you see that as pure nonsense? If not, I welcome further clarification.
It's not at all nonsense in the context I was presenting it. But I'm trying to understand where we miss each other in following the logic of my comments about the slanted lines. My slanted lines were not meant to represent some standard space-time configuration. And, again, they were certainly not intended to be the planes of simultaneity shown in the sketch above (we had already discussed those more than once in this thread). Again, I just added them to the space-time diagram so you could follow the synchronous proper times of the twins. Maybe you are not accepting the concept of observers moving along their own world lines at the speed of light. What do you make of the often mentioned comment in special relativity discussions, "The observer moves along his own 4th dimension at the speed of light?" And what do you make of x4 = ct?

When taken at face value, the synchronous proper times indicated with my slanted lines lead to nonsense without the full 4-dimensional space and 4-dimensional objects. That was what I was trying to emphasize by pointing out the absurdity you have without the 4-dimensional objects available over all world lines so that twins can share the same event even though their lapsed proper times are not the same.

Perhaps you have a better explanation as to how the twins can share a common event when their lapsed proper times (from the time of their previous shared event, i.e., the common coordinates origin) are not the same.
 
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I already comprehend the 4-dimensional description of Minkowski space-time. I didn't ask about that.
Then do you agree with the identification of the hyperbolic curves in my diagram as correctly representing a locus of points having the same lapsed proper times measured from the origin of both twins' coordinates? And do you agree that each observer moves along his world line at speed c? Do you agree that implies X4 = ct (distance traveled along the X4 axis)?

I asked about the diagonal lines. I have never seen diagonal lines like that going at different angles on any Minkowski space-time diagram except yours. Can you point to an example of anyone else drawing them like that with an explanation of why they are slanted differently?
I have never seen a diagram that includes those slanted lines. Again, it was not meant to represent any standard convention. I don't claim them to be any sort of standard space-time diagram convention. However, I thought it would help in the understanding of the use of the hyperbolic calibration curves.

Do you agree that the slanted lines connect corresponding proper times from the home twin world line to the travel twin world line? I was simply attempting to call your attention to those synchronous proper times. Do you understand this now? I could explain further.

It would be useful if the visualization would explain why there is no paradox rather than just add to the confusion.
Again, the point of my calling attention to the synchronous proper times is to show that there is a paradox, which can be resolved if you apply the understanding of a 4-dimensional space with 4-dimensional objects. I can discuss that point more if it is not clear by now.

Please look at your answer in post #24 and tell me why I should have understood your explanation or taken it any more seriously than you seem to have.
I'm never sure whether my presentation of a concept is good or adequate. Evidently it was not adequate in your case. If you could ask a more specific question, maybe I could do a better job of explaining.

Then why did you label your axes X1 and X4 instead of x and ct like everyone else does?
It is not true that I'm the only physicist to label axes as x1, x2, x3, x4. Einstein has done that. I picked up the habit during a grad school special relativity course (my prof did it quite often-- sometimes ct and sometimes x4). Besides, the notation convention is a rather trivial issue. I like to emphasize the spatial character of the 4th dimension.

You could use a time convention with markers along the interstate from point A to point B if everyone drove at the same speed. Going 60 mph you could use one minute markers. It's not practical if everyone drives at different speeds. But, everyone drives at speed, c, along X4, and clocks are a very practical method of measurement. And you can always get the distance: X4 = ct. But, remember that a mechanical clock is just a 4-dimensional object with repetitive physical markers along X4. The 4-D clock and space are much more easily imagined than time as a physical dimension. And the idea of a "mixture of space and time" sounds appealing, but no one can really make sense out of such an idea (other than mathematically).

Can you show me an example of a Minkowski space-time diagram not promoting Block Universe Theory that is labeled like you did yours?
Why is that even relevant? If the problem as you see it is that a 4-dimensional external objective space is a concept out of mainline physics, then perhaps that should be the subject of another thread. We could start a thread, "Is the concept of a 4-dimensional external objective universe outside of mainline physics?" Many physicists pay no attention to that topic, because it has no effect on the way they are doing their physics. The 4-dimensional space is fully consistent with special relativity. If you do not agree with this, perhaps you could present information to the contrary.

Yes, 3 dimensions of space and one dimension of time is what makes 4-dimensional space-time, not 4 dimensions of space.
How do you know? Can you present the logic that would lead to such a conclusion?
 
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[..] there is no alternative external objective picture out there that contradicts the 4-dimensional universe populated by 4-dimensional objects. I think many just abandon the idea of an external objective reality.
As a matter of fact, I presented a well-known alternative objective picture in an earlier thread* and again referred to it in my post #19 here; it seemed to directly answer the OP's question. So, what did I miss (or what did you miss)?

*https://www.physicsforums.com/showthread.php?t=539826&page=2 ; see also p.3
 
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As a matter of fact, I presented a well-known alternative objective picture in an earlier thread* and again referred to it in my post #19 here; it seemed to directly answer the OP's question. So, what did I miss (or what did you miss)?

*https://www.physicsforums.com/showthread.php?t=539826&page=2 ; see also p.3
Thanks for jumping, harrylin. Your contribution is always good. I've read the Langevin paper again. It seems to me to be his summary of the special relativity theory with no new (for the time) interpretation. His summary seems to me to be consistent with Einstein's presentation of the theory. He refers to Minkowski's "world" and "world line" without any new embellishments.

I agree that his references to the 4-dimensional continuum, implied with his references to space and time, would be more consistent with ghwellsjr's emphasis of the 4-dimensional continuum as being three spatial dimensions and one time dimension. So, beyond that I don't see this as refuting a 4-dimensional space concept.

Logically, once you have a 4-dimensional continuum where you refer to the 4th dimension as time (provided the continuum is consistent with special relativity), you are automatically implying a 4th spatial dimension. This is manifestly so when looking at any space-time diagram representing a 4-dimensional continuum. And X4 is equivalent to ct. Different cross-section cuts through the 4-dimensional continuum (whether you use X4 or ct) do not separate out time from spatial dimensions.

This is obvious when you imagine extruding a 3-D space along the "time dimension." That is, an observer, when advancing along his world line, always observes a continuous sequence of 3-D spaces. Thus, it is seen that the world line at each successive point populates the sequence of 3-D spaces. But that means that as you move along the world line you move through space.

Einstein often spoke of the 4-dimensional continuum in the context of 3 spatial and 1 time (again, consistent with hgwellsjr's comments). He spoke of the problem of "Now" in special relativity and commented that physicists make no distinction between past, present and future. He seemed clear in his understanding of an external objective physical reality that embodies a 4-dimensional continuum. And he used

ds^2 = dX1^2 + dX2^2 + dX3^2 - dX4^2

as a 4-dimensional line element.
 
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You seem both to define it similarly to me?

So how about defining where you differ in your definitions? If I get you right Bob? You want SpaceTime to be a 'absolute' 4-dimensional, not differing 'time' from the other 'dimensions', as a unified 'jello' of sort? And that this 'jello' represent all observers?

Or do I get you wrong there?

And the other definition seems to be one in where 'time' has a unique flavour, although being a 'dimension' too? But both agree on that in 'reality' all 'dimensions' must be included for a time dilation, that is if I read you right?
 
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Thanks for jumping, harrylin. Your contribution is always good. I've read the Langevin paper again. It seems to me to be his summary of the special relativity theory with no new (for the time) interpretation. His summary seems to me to be consistent with Einstein's presentation of the theory. He refers to Minkowski's "world" and "world line" without any new embellishments.

I agree that his references to the 4-dimensional continuum, implied with his references to space and time, would be more consistent with ghwellsjr's emphasis of the 4-dimensional continuum as being three spatial dimensions and one time dimension. So, beyond that I don't see this as refuting a 4-dimensional space concept.
I think that the 4-dimensional space concept (in the way of a rather weird 4D physical space, not just mathematical space) is incompatible with Langevin's concept of a physical space: a stationary ether, of which we can detect the existence by a change of motion relative to it.
Logically, once you have a 4-dimensional continuum where you refer to the 4th dimension as time (provided the continuum is consistent with special relativity), you are automatically implying a 4th spatial dimension. [..]
Only if you interpret that 4th dimension as a spatial dimension; and clearly Langevin can not have meant that, for the reason that I just mentioned.
For example also temperature is a dimension, see http://en.wikipedia.org/wiki/Dimensional_analysis.
Einstein often spoke of the 4-dimensional continuum in the context of 3 spatial and 1 time (again, consistent with hgwellsjr's comments). He spoke of the problem of "Now" in special relativity and commented that physicists make no distinction between past, present and future. He seemed clear in his understanding of an external objective physical reality that embodies a 4-dimensional continuum.
Even Einstein stressed that (according to him, perhaps not Minkowski) space is a three-dimensional continuum, and that in contrast the "four dimensional space" of Minkowski is the "world" of events (brackets his).
- http://www.bartleby.com/173/17.html (that fits rather nicely with Langevin's speech; I wonder if he was influenced by it?).

Events take place in the physical world; the "4D continuum" of events is not the physical world itself. Another way to put it: "the map is not the territory".

As this is of course mostly* a matter of metaphysics, the point here is that the same mathematics has been interpreted or explained in very different ways, based on very different views of the physical world.

Harald

*only mostly: I don't think that you can walk in the negative time direction

PS I hope that the above also answers some of yoron's questions: there is no need to believe in a unified 'jello' of sorts!
 
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You seem both to define it similarly to me?

So how about defining where you differ in your definitions? If I get you right Bob? You want SpaceTime to be a 'absolute' 4-dimensional, not differing 'time' from the other 'dimensions', as a unified 'jello' of sort? And that this 'jello' represent all observers?

Or do I get you wrong there?

And the other definition seems to be one in where 'time' has a unique flavour, although being a 'dimension' too? But both agree on that in 'reality' all 'dimensions' must be included for a time dilation, that is if I read you right?
Yoron, I think you are pretty much on target with your summary here. And you can see from ghwellsjr and harrylin posts that my interpretation of the 4-dimensional continuum seems to be in the minority.

I recall one physicist writing on this subject fairly recently commenting that most physicists now embrace the "block universe" concept. He provided no basis for making that statement, and I'm doubtful it is true. The UK physicist, Cox, who does a lot of popular physics presentations with books and videos, championed the block universe concept on BBC. It was available on YouTube for a while until BBC had it blocked. In Brian Greene's video he illustrates the concept of observers having different cross-section views of the 4-dimensional continuum by slicing a loaf of bread in different directions. There of course is nothing original with me about this subject, and you can find many physicists accept the concept if you just do some googling on "block universe" and "block time". I don't believe it is an "out-of-the-mainstream" subject.
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I hope the airing of these views has been fruitful for visitors to the thread. I'll have to mull over harrylin's post. He always does a good job with his posts. Maybe there is more to say--and maybe we've all pretty much presented our views. Perhaps others will be motivated to research this question.
 
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Yoron, I think you are pretty much on target with your summary here. And you can see from ghwellsjr and harrylin posts that my interpretation of the 4-dimensional continuum seems to be in the minority.
I recall one physicist writing on this subject fairly recently commenting that most physicists now embrace the "block universe" concept. He provided no basis for making that statement, and I'm doubtful it is true. The UK physicist, Cox, who does a lot of popular physics presentations with books and videos, championed the block universe concept on BBC. It was available on YouTube for a while until BBC had it blocked. In Brian Greene's video he illustrates the concept of observers having different cross-section views of the 4-dimensional continuum by slicing a loaf of bread in different directions. There of course is nothing original with me about this subject, and you can find many physicists accept the concept if you just do some googling on "block universe" and "block time". I don't believe it is an "out-of-the-mainstream" subject.
It is often claimed that the interpretation that you presented here - although you indicated that you do not like it very much - is the majority interpretation. It might be interesting to do a poll here, similar to the one about interpretations of QM; but popularity is not all-important.

My point was that an alternative interpretation has been known from the start, and again other interpretations exist (e.g. "physical relativity" as expressed by Harvey R. Brown). It's not necessary to get hung up on what may be the most popular interpretation, especially for those for who that interpretation doesn't make much sense.
I hope the airing of these views has been fruitful for visitors to the thread. I'll have to mull over harrylin's post. He always does a good job with his posts. Maybe there is more to say--and maybe we've all pretty much presented our views. Perhaps others will be motivated to research this question.
Thanks! Let's hope that it was useful for someone. And perhaps Passiday, who started this topic, has a comment.
 

ghwellsjr

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I stopped reading Brian Greene's book fabric of spacetime because of him laying the block universe idea so heavily as the way to interpret 4D. I wasn't up for the same force feeding of strings.

A second book I got called Relativity a brief introduction by Russell Stannard (a more [STRIKE]scientific approach to explaing[/STRIKE] accurate explaination of SR/GR) also proposes the block universe as a better interpretaion of 4D. Superior to Greene though, Stannard does mention that this is not the interpretation of most physicists, and also lists oddities of both conceptions.

As I slowly understand spacetime diagrams better, I see the perspective of Greene and Stannard more clearly, but think it's taking the idea of spacetime diagrams a little too far (visualizing 4Ds at right angles to each other, and coming up with block universe or whatever).
 
Guys, I am overwhelmed by your feedback. I asked what I thought a pretty basic question, but got a very elaborate explanations, with accompanying heated discussion. I bow in front of your commitment.

I kind of dislike the oversimplification in science, you know, when talking about not-that-trivial things. I think, most of people when read about the twin paradox, take it at face value and then they think they've got the relativity. But I've got confused about the complex trajectories of both twins, in the field of gravity of the Earth and the Sun. The most important point I take from this discussion is that the frame of reference is irrelevant (the aging difference holds disregarding the FoR), and what serves here as the definition of the speed is "that what happens to an object that just felt acceleration". However, the fact that there would be an acceleration to be felt even in totally empty universe, that escapes my deep understanding, feels like mystery. But I am happy to learn that Einstein and Mach were not that clear on that subject either :)
 
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However, the fact that there would be an acceleration to be felt even in totally empty universe, that escapes my deep understanding, feels like mystery.

Why would feeling acceleration in "totally empty universe" be mysterious?
 
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I asked what I thought a pretty basic question, but got a very elaborate explanations, with accompanying heated discussion.
By the way, I very much liked your thread title. For me it calls to mind this mental image of Einstein and his contemporaries in a secret meeting behind closed doors in a room filled with cigar smoke taking a vote to decide which twin is younger.
 

tom.stoer

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To define the 'lengths' of the curves you said "Here the "length" and therefore the proper time is calculated according to the strange 4-dim. relativistic Pythagoras t² - x²." The variables "t" and "x" in that expression are not arbitrary, they must be inertial coordinates, ...
Of course not!

The definiton of the line integral is completely independent from any specific reference frame. If you want to calculate something you can use a coordinate system i.e. you introduce t and x. But the geometric property 'the length of a timelike curve C in a pseudo-Riemannian manifold' does not depend on a coordinate system. Neither does the geometric property 'this timelike curve is longer than that timelike curve' depend on a coordinate system.
 
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Why in your first example do you call the hyperbola curve "Proper Distance" and in the second one "Proper Time"?
The upper and lower cones (inside the cones) are normally referred to as "timelike" and the outside of the cone is referred to as "spacelike." Points on the cone sufrace are "lightlike."

The term "Proper Time" normally refers to measurements along world lines inside the light cone. I usually stick to that convention unless I am trying to emphasize the X4 distance along the 4th dimension in accordance with X4 = ct. Actually, some authors use units of spatial distance for Proper Time (usually with the greek tau symbol).
 
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Then why did you label your axes X1 and X4 instead of x and ct like everyone else does? Can you show me an example of a Minkowski space-time diagram not promoting Block Universe Theory that is labeled like you did yours?
Here is a copy of a space-time diagram from the text book "The Geometry of Spacetime" by Gregory L. Naber. He uses X1, X2, X3, and X4 notation throughout the book (along with ct and tau at times).

The author never mentions block time, nor can you find the term in the index. He does not engage in any interpretations of spacetime--just deals with the math. He takes a fairly formal approach to the subject.
Space-Time_Naber-1.jpg
 

ghwellsjr

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We had a misunderstanding. I was referring to the following space-time diagram in the wiki link:This shows the planes of simultanaeity for the travel twin, which are exactly the diagonal lines I mentioned in earlier posts (yes, they represent 3-D cross-sections of a 4-dimensional space, x2 and x3 being supressed).
TwinParadox_Wiki.jpg
I can see lines of simultaneity in the drawing and you say they represent 3-D cross-sections of a 4-D space, so why do you call them "planes of simultaneity" instead of "volumes of simultaneity"?
 

ghwellsjr

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...look closely and you will see that I included that hyberbolic calibration curves so that we could make comparisons between the two coordinate systems. I used the 5 year calibration curves for each of the two inertial start events for the traveling twin.
I'm not understanding what these calibration curves are for or even how you derived them. On the bottom one, you show it going through the 5 year Proper Time for the traveling twin and through the 5 year Proper Time of the stationary twin.

Later you said:
My slanted lines were not meant to represent some standard space-time configuration. And, again, they were certainly not intended to be the planes of simultaneity shown in the sketch above (we had already discussed those more than once in this thread). Again, I just added them to the space-time diagram so you could follow the synchronous proper times of the twins.
Are you saying that you have a calibration scheme that allows you to determine that the twins have synchronized clocks or that it provides some means by which to "calibrate" them? It seems rather trivial to me if all you want to do is draw a line from a particular Proper Time for one twin to the same Proper Time for the other twin but then I don't understand why you would need a calibration curve.
 

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