Understand Special Relativity and Time paradox

[PAllen]

Seeing as how one need only use the lines of simultaneity for the two inertial components of the outgoing twin's trip, your comments about non-inertial observers aren't relevant to this particular discussion. Maybe in a broader context, but not to what I was saying.

Actually, making turnaround instant doesn't make the twin world line inertial. The worldline as a whole is extremely non-inertial - proper acceleration is infinite (or undefined) at one point.

Note that for a sudden turnaround, say from going East to going West, if you inquire about events east of this non-inertial world line using your preferred simultaneity convention, you choose to accept the concept that an event you considered in your past a moment ago is now in your future. You can certainly do this; and I can say that's nonsensical, and I would not try to justify that to the neophyte who asked about the other side of the space time diagram.

 

[bobc2]

... I'm struggling to see how I could have been clearer, but I'll accept that we've been talking past each other if you say so...

LastOneStanding, I've been quite sympathetic with your comments. Here I've prepared a set of sketches to help anyone interested visualize what is going on in the 4-dimensional universe with the twins in the context of simultaneity. This focuses on the turn-around phase. As the traveling twin changes velocity, his X1 axis (representing hyperplane of simultaneity) continuously rotates such that the photon world line always bisects the angle between his blue X4 and blue X1 axes. These are what we refer to as a continuous sequence of boosts. The Lorentz transformations describe the boosts mathematicially. For anyone needing a little more background on the space-time sketches and the 4-dimensional universe concept, you can go to this earlier post that outlines the concept, beginning with post #19 (be sure you understand relativity of simultaneity--if not ask questions):

https://www.physicsforums.com/showthread.php?p=4138802#post4138802

In the sketch below, I've added in a Red guy here in order to make a point about the interesting sequence of simultaneous worlds for the traveling twin as he does his turna-round. So, this additional red guy is at rest along with the black guy. As the Blue guy does his turn-around, the sequence of Black clock readings (as presented in the sequence of Blue's 3-D world cross-sections of the 4-D universe) progress into the future whereas the Red clock readings proceed into the past. As Blue enters the turn-around, the Black clock begins with event "black E" and at the end of the turn-around the last Black clock reading corresponds to event "black h." But, although Red's clocks begin with event "red E" (same simultaneous plane as "Black E") and end with event "red h", we see the sequence of Red clock readings going into the past along Red's X4 axis world line. The E, a, b, c, ... h designations identify the discrete hyperplanes (planes of simultaneity) in the movement through 4-D Space-Time of Blue as he progresses along his world line.

 

[PAllen]

So, if I propose a "W" shaped twin scenario, with (in rest frame of inertial 'stay home' twin):

Home twin world line (t,x) from (0,0) to (0,10)

Travel twin world line (t,x) from (0,0) to (3,2) to (5, 2/3) to (7,2) to (0,10) // c=1 convention

This is perfectly easily explained using inertial frame, metric, or Doppler methods. Using 'objective' lines of simultaneity you are required to say travel twin must consider that home twin's clock went as follows (imagine slightly rounding each turnaround)

slow, fast, slow, backwards fast, slow, fast, slow

and you think this will help a new student understand and accept relativity? [edit: what you actually see or detect via exchanged signals, is slow, fast, slow, fast, none extreme, and consistent with the direct comparison at the end.]

 

[bobc2]

So, if I propose a "W" shaped twin scenario, with (in rest frame of inertial 'stay home' twin):

Home twin world line (t,x) from (0,0) to (0,10)

Travel twin world line (t,x) from (0,0) to (3,2) to (5, 2/3) to (7,2) to (0,10) // c=1 convention

This is perfectly easily explained using inertial frame, metric, or Doppler methods. Using 'objective' lines of simultaneity you are required to say travel twin must consider that home twin's clock went as follows (imagine slightly rounding each turnaround)

slow, fast, slow, backwards fast, slow, fast, slow

and you think this will help a new student understand and accept relativity? [edit: what you actually see or detect via exchanged signals, is slow, fast, slow, fast, none extreme, and consistent with the direct comparison at the end.]

Absolutely. The student first gets the hang of the basic picture for an observer moving at contstant velocity with respect to the reference "rest system." Then, the twin paradox. Then the triple paradox. Then your W pattern example. And if he figures that one out on his own he probably has a pretty good handle on what's going on with time dilation and hyperplanes of simultaneity.

I certainly have no fuss about doppler. Any special relativity course would not be complete without understanding that. But the real fundamental stuff of special relativity is intimately related to the time dilation, length contraction and hyperplanes of simultaneity as manifest in the Lorentz transformations and the space-time diagrams.

 

[PAllen]

Absolutely. The student first gets the hang of the basic picture for an observer moving at contstant velocity with respect to the reference "rest system." Then, the twin paradox. Then the triple paradox. Then your W pattern example. And if he figures that one out on his own he probably has a pretty good handle on what's going on with time dilation and hyperplanes of simultaneity.

Then, hopefully, someone shows the poor student that if the traveler uses the light synchronization based simultaneity used to derive SR (i.e. re-deriving the case of a non-inertial observer rather than assuming it is correct to model non-inertial motion using formulas derived for inertial motion), they get no such nonsense. They relate points on the two world lines in a smooth, non-repeating way. Thus, there is no reason a rational traveler should ever believe the home clock is running backwards.

[edit: A little challenge: can you propose any experiment or observation that detects the purported backwards progress of the home clock, that the traveler could do? ]

 

[jaumzaum]

No, your brother will not receive many messages from you at the moment of his about-face. His about-face will not cause him to receive any messages from you. What's going to happen is that for the first half of his trip, he will receive messages from you at a slower rate than he sends them (1/R as I said in my first post to you), then for the second half of his trip he will receive messages from you at a faster rate than he sends them (R).

So for your example of your brother traveling at 0.5c, we can use the Relativistic Doppler formula to calculate what R is:

√((1+β)/(1-β)) = √((1+0.5)/(1-0.5)) = √((1.5)/(0.5)) = √3 = 1.732

And 1/R is the reciprocal, 0.57735.

This means that he will see your yearly messages coming to him slower than his during the first half of the trip. In fact it will take 1.732 years before he sees your first message.

And for the last half his trip, he will see your messages arriving more often than once per year according to his clock. It will only take 0.57735 years between each of your messages.

Now without knowing how long the trip will take, we can average these two numbers:

(1.732+0.57735)/2 = 2.30935/2 = 1.154675

This is the final ratio of your two clocks when he returns. How ever many years it took him according to his clock, yours will be 1.154675 times that amount.

So let's say your brother travels away at 0.5c for 13 years and then takes 13 years to get back at the same speed. Here is a spacetime diagram to show what is happening according to your rest frame. I show you as a thick blue stationary line with dots every year and your messages going out as thin blue lines traveling at c. I show your brother as a thick black line traveling at 0.5c with black dots every year and his messages coming back to you as thin black lines:

Now let's see how the previous calculations based on Relativistic Doppler fit in with this diagram. First off, I said that the rate at which your brother receives your yearly messages take 1.732 of his years. Can you see that on the graph? For example, at about his year 12, just before he turns around, he is just receiving the message you sent at your year 7. Can you follow that? If we divide 12 by 7 we get 1.714 which is about right. (We don't expect it to be exact because he didn't receive your message exactly at his year 12.)

Furthermore, if you look at your year 12, you can see that you are just receiving his message from year 7. It's symmetrical.

Now you should be able to see that after he turns around, he starts receiving your messages faster than one per year. In fact, from about his year 19 (you'll have to count his dots) to when you meet at his year 26, he will have received your messages from year 18 to 30. That is a ratio of the differences of (26-19)/(30-18) = 7/12 = 0.583, close enough to 0.577.

And in a similar manner, you can see that from his year 14 (just after he turns around) until you meet (12 years of messages from him), you will see them from your year 23 to your year 30 (7 years) and the reciprocal ratio applies.

Now that we can look at a spacetime diagram, we can see that the reason why the two of you age differently is because your brother sees these two ratios for half of his total trip time each but you see the smaller ratio for three-quarters of the time and the higher ratio for just one-quarter of the time. This means you are seeing him age less for a longer time while he sees you age less for half the time.

The last thing we want to notice is that ratio of your final age difference is 30/26 or 1.1538, very close to the actual 1.154675. (Again, these numbers are not exact because we're eyeballing them off the diagram.) This ratio is the famous value of gamma which is also the time dilation factor which shows in the diagram as the ratio of the coordinate time for your brother compared to his actual time on his clock. Can you see that?

Now I want to show you what the exact same information presented in the first diagram looks like in two more diagrams based on the IRF's in which your brother is at rest, first during his outbound leg and then during his inbound leg. First the outbound leg:

Notice how your brother's time is not dilated during the outbound leg (because he is at rest) but yours is. Note also that he has to travel at a higher speed than 0.5c (look up "veloctiy addition" in wikipedia to see that this higher speed is 0.8c) when he turns around and therefore now has more time dilation than you have. Nevertheless, all the signals between the two of you continue to travel at c and arrive at exactly the same times according to your own clocks as they did before. Does this all make sense to you?

Finally the diagram for the IRF in which your brother is at rest during the inbound leg:

This is very similar to the previous diagram so I won't go into any more explanation except that I want to point out that when your brother turns around, in no case does that have any bearing on what you see, until some time later and even then, each diagram shows accurately what you actually see and what your brother actually sees during the entire scenario.

Any questions?

Thanks George! I mean, I really want to thank you, in all of the threads I've already posted here, I've never seen such a good and detailed answer as yours. I'm new in special relativity and you explained everything so carefully I could understood almost completely. I thought I had understood before, but I didn't know the explanation had nothing to do to what I suppose it was correct. There should be more guys like you here in PF. I want to thank everyone who answered this thread but the george's answer was phenomenal (at least for me). Now I think I'm startin g to understand special relativity.

And yes, I do have some questons, I would appreciate if you could help me again

How it would be the diagram if we take the referential frame as the whole trip of my brother (I mean, my brother is at rest in the whole time). I'm not being able to "close" the graphic. If my brother is a straight line, my lines can't be together as they are smaller than his. Is this right?

 

[ghwellsjr]

...I've prepared a set of sketches...

Bobc2, your Sketch III looks kind of like my first diagram in post #23 except that I have an instantaneous turn-around. The lines (or surfaces or volumes or whatever you want to call them) of simultaneity in my diagram are simply the horizontal grid lines (and the infinite number of horizontal lines in between them). In normal orthogonally drawn diagrams of IRF's, the issue of simultaneity is automatically handled by the grid lines and doesn't need any explanation. You don't show any equivalent grid lines either horizontally or vertically (or diagonally??) so it is very difficult for me to tell what is going on with your diagrams. I wonder if the OP is understanding them.

Secondly, I don't see any equivalent yearly messages traveling at c between the OP and his brother as he requested in post #13. Can you please put them in? And can you address his concern that a bunch of messages from him will arrive in a flood during his brother's about-face?

 

[bobc2]

Then, hopefully, someone shows the poor student that if the traveler uses the light synchronization based simultaneity used to derive SR (i.e. re-deriving the case of a non-inertial observer rather than assuming it is correct to model non-inertial motion using formulas derived for inertial motion), they get no such nonsense. They relate points on the two world lines in a smooth, non-repeating way. Thus, there is no reason a rational traveler should ever believe the home clock is running backwards.

It is certainly not nonsense. It makes sense logically. The approach depicted in my sketches follows that suggested by Einstein as a quite reasonable and logical approach. You are basicaly disregarding the fundamental concept of hyperplanes of simultaneity as directly expressed using the Lorentz transformations. The attempt to replace the direct Lorentz based relativity of simultaneity with the doppler approach is just an argument based on philosophical ideas. Further delving into the philosophical basis for selecting the doppler approach would not be appropriate given the forum's preference for avoiding those kinds of discussions.

[edit: A little challenge: can you propose any experiment or observation that detects the purported backwards progress of the home clock, that the traveler could do? ]

It is not difficult to derive the time dilation relationships and logic leading to the concept known as relativity of simultaneity. It is a well established concept in special relativity. If you are wanting to remove all derived concepts from theoretical physics, the relativity of simultaneity would perhaps be thrown out with all the rest. But, what a setback for physics that would be. You would be throwing out the concept of the speed of light, given its derivation based on the measurement of two different quantities from which the speed is derived. And of course throwing out the Lorentz transformations (derived) would pull the rug out from under relativity of simultaneity.

 

[PeterDonis]

The attempt to replace the direct Lorentz based relativity of simultaneity with the doppler approach is just an argument based on philosophical ideas.

I'm sure PAllen will respond to this too, but I have to disagree with this comment. The doppler effect is a direct observable; it's relativity of simultaneity, time dilation, length contraction, etc. that are derived concepts that can bring in "philosophical" issues. As PAllen commented in a prior post (I think it was in this thread, but there have been so many lately on this general theme that I've lost track), you can analyze a scenario like the "twin paradox" *entirely* in terms of the doppler-shifted light signals that each observer directly *observes* coming from the other; you don't *need* relativity of simultaneity, time dilation, length contraction *at all* to predict that the "traveling" twin will have aged less when the two reunite. So the doppler effect is the *last* thing that I would say is likely to bring "philosophical" baggage into a discussion about physics.

 

[Dale]

I think you lost track of how we even got here if you think I was at any point talking about SR in general.

I wasn't responding to the whole thread in general. I was responding only to the specific sentences I quoted. If you were not talking about SR in general in that quote, then it was very poorly worded.

 

[bobc2]

I'm sure PAllen will respond to this too, but I have to disagree with this comment. The doppler effect is a direct observable; it's relativity of simultaneity, time dilation, length contraction, etc. that are derived concepts that can bring in "philosophical" issues. As PAllen commented in a prior post (I think it was in this thread, but there have been so many lately on this general theme that I've lost track), you can analyze a scenario like the "twin paradox" *entirely* in terms of the doppler-shifted light signals that each observer directly *observes* coming from the other; you don't *need* relativity of simultaneity, time dilation, length contraction *at all* to predict that the "traveling" twin will have aged less when the two reunite. So the doppler effect is the *last* thing that I would say is likely to bring "philosophical" baggage into a discussion about physics.

I totally disagree with your assessment. Relativity of simultaneity is a well defined concept in special relativity. Everyone doing special relativity understands the motivation and significance of it. The concept is a direct outcome of the Lorentz transformations. Now, the doppler approach is based on the Lorentz transformations as well, but this approach just adds on another layer of complexity--transmitting and receiving light signals.

As I said before, it is very instructive to examine what observers actually measure--this should be, and typically is (doppler approach), included in any special relativity course. By the way, in the final analysis you will discover that doppler results are derived, resulting from measurements of more fundamental quanties than normally presented as "measurements."

But, the tendency to dismiss the concept of the hyperplanes of simultaneity based directly on the Lorentz transforms (removing the results of light travel delays, etc.) is probably motivated out of a philosophical preference for dismissing concepts that do not result directly from measurement. Theoretical physicists typically do not carry that philosophy to such an extreme as to lose concepts as important as time dilation, length contraction and relativity of simultaneity.

So, I think further discussion on this just spirals into philosophical arguments having to do basically with "...what are the real 3-D worlds that various observers live in, and can evidence of them be measured?" "...and to what extent can we rely on derivations based on measurements?"

 

[bobc2]

Bobc2, your Sketch III looks kind of like my first diagram in post #23 except that I have an instantaneous turn-around. The lines (or surfaces or volumes or whatever you want to call them) of simultaneity in my diagram are simply the horizontal grid lines (and the infinite number of horizontal lines in between them). In normal orthogonally drawn diagrams of IRF's, the issue of simultaneity is automatically handled by the grid lines and doesn't need any explanation. You don't show any equivalent grid lines either horizontally or vertically (or diagonally??) so it is very difficult for me to tell what is going on with your diagrams. I wonder if the OP is understanding them.

Secondly, I don't see any equivalent yearly messages traveling at c between the OP and his brother as he requested in post #13. Can you please put them in? And can you address his concern that a bunch of messages from him will arrive in a flood during his brother's about-face?

ghwellsjr, sorry I missed your post for a while there. I'll see if I can get back into this and respond to your questions sometime this weekend. Thanks.

 

[PAllen]

It is certainly not nonsense. It makes sense logically. The approach depicted in my sketches follows that suggested by Einstein as a quite reasonable and logical approach. You are basicaly disregarding the fundamental concept of hyperplanes of simultaneity as directly expressed using the Lorentz transformations. The attempt to replace the direct Lorentz based relativity of simultaneity with the doppler approach is just an argument based on philosophical ideas. Further delving into the philosophical basis for selecting the doppler approach would not be appropriate given the forum's preference for avoiding those kinds of discussions.

Einstein always analyzed non-inertial observers using a single inertial frame. He never proposed time going backwards as a reasonable interpretation, so far as I have seen. Using anyone inertial frame, you have one consistent (not self-intersecting) set simultaneity surfaces, and never any backwards time flow along any world line. Lorentz transforms apply between inertial frames. The γ factor for time dilation and length contraction is derived in the context of a single inertial frame. This is sufficient to compute and understand all observations.

The question of frames for non-inertial observers in SR is something I don't believe Einstein ever dealt with (he definitely dealt with their observations using the method above). You assume that it is valid and preferred to use formulas derived for inertial observers blindly for non-inertial observers. If, instead, one uses the procedure Einstein used for inertial frames for a non-inertial observer:

- simultaneity and synchronization defined by two way light signals
- distances computed via the metric along simultaneity surfaces defined as above

one comes up with a completely different description for the arbitrary traveling twin (one that has no clocks going backwards).

Now I don't claim this one is uniquely preferred either. I claim simultaneity is a non-observable convention. For inertial frames, there is a preferred convention in the sense that all reasonable procedures agree. For a non-inertial observer, the different, reasonable, ways to define simultaneity disagree, so the choice is arbitrary. Further, an observer trying to model a region of spacetime would want to use a valid coordinate system - blindly using inertial surfaces of simultaneity produces an invalid coordinate system for the 'W' twin (with turnarounds slightly rounded), because it multiply labels events - it assigns multiple time coordinates to certain events on the home world line.

A few more comments:

-The Lorentz transform is not an invariant, good everywhere, in SR; it is strictly a transform between inertial coordinates.
-The γ factor applies only in inertial coordinates, for all its uses.
- The invariants of SR, that hold in general coordinates that non-inertial observers might use are things proper times, and doppler and not simultaneity and time dilation. For coordinates suitable for a non-inertial observer, the metric expression is different than diag(1,-1,-1,-1) , and therefore time dilation is no longer given γ, because that form results from the metric in inertial coordinates. Simultaneity for any valid coordinate chart will never multiply label events (equivalently, will never reverse time on any world line).

 

[PeterDonis]

Relativity of simultaneity is a well defined concept in special relativity.

I didn't say it wasn't a well-defined concept; I said it was a *derived* concept, not a direct observable.

Now, the doppler approach is based on the Lorentz transformations as well, but this approach just adds on another layer of complexity--transmitting and receiving light signals.

You're missing the point. The doppler shift is a *direct observable*. You don't need to "interpret" anything. (Of course you may need to *calculate* what the observed doppler shift will be, but you can do that using the doppler formula knowing nothing more than the relative velocity of the emitter and the receiver; you don't need to know about simultaneity, or length contraction, or time dilation, or anything like that.)

in the final analysis you will discover that doppler results are derived, resulting from measurements of more fundamental quanties than normally presented as "measurements."

By that standard, the list of things which are "direct observables" is pretty small. And it certainly doesn't include relativity of simultaneity, time dilation, or length contraction, so I don't see how this helps you any.

But, the tendency to dismiss the concept of the hyperplanes of simultaneity based directly on the Lorentz transforms (removing the results of light travel delays, etc.) is probably motivated out of a philosophical preference for dismissing concepts that do not result directly from measurement.

Nobody that I can see is dismissing the concept of hyperplanes of simultaneity. If you want to use them, and it helps you to analyze scenarios, great, go for it. But when you start to make strong philosophical claims based on them, like saying that they require the block universe interpretation, then yes, you're going to get pushback from people, like me, who understand that hyperplanes of simultaneity, as a concept, just can't carry that much philosophical load.

Theoretical physicists typically do not carry that philosophy to such an extreme as to lose concepts as important as time dilation, length contraction and relativity of simultaneity.

Nobody is trying to discard the concepts. We just recognize their limitations.

So, I think further discussion on this just spirals into philosophical arguments having to do basically with "...what are the real 3-D worlds that various observers live in, and can evidence of them be measured?" "...and to what extent can we rely on derivations based on measurements?"

Well, you are one of the people who appears to care enough about these questions to bring them up in the first place, so it seems a bit odd for you to be (apparently) dismissing them as "philosophical".

 

[Dale]

In the sketch below, I've added in a Red guy here in order to make a point about the interesting sequence of simultaneous worlds for the traveling twin as he does his turna-round. So, this additional red guy is at rest along with the black guy. As the Blue guy does his turn-around, the sequence of Black clock readings (as presented in the sequence of Blue's 3-D world cross-sections of the 4-D universe) progress into the future whereas the Red clock readings proceed into the past.

I believe that I have told you before, but this is not correct. The naive simultaneity convention cannot map any coordinates at or to the right of the "crossover" point. One of the fundamental requirements of a coordinate system is that it must be one-to-one. This system violates that condition so you have to chop off the offending part of the coordinate system. The result is that it does not say that red's clock goes backwards, instead it says nothing about red's clock.

Your drawings are mathematically invalid if you extend them to or beyond the crossover point.

 

[Dale]

It is certainly not nonsense. It makes sense logically. The approach depicted in my sketches follows that suggested by Einstein as a quite reasonable and logical approach. You are basicaly disregarding the fundamental concept of hyperplanes of simultaneity as directly expressed using the Lorentz transformations.

I agree that the naive approach at defining a non-inertial frame is not nonsense in general, but only to the left of the crossover. At that point it violates one of the fundamental requirements of a coordinate system, and therefore it does become illogical. You can use this approach to make statements about the stay at home twin, but not the red guy. Your claims about the red guy's clock running backwards are therefore indeed nonsense.

Also, the Lorentz transform transforms between inertial frames, so it is reasonable to disregard it here where we are dealing with non inertial frames. And after you cobble together the various pieces the resulting transform is decidedly not the Lorentz transform anymore anyway.

 

[bobc2]

I have not presented the turn-around in the context of non-intertial frames. I followed Einstein's proposal to approach the turn-around problem as a sequence of inertial frames, each one individually amenable to the usual Lorentz transformation treatment. So, they are a sequence of incremental boosts. You can let the increments become as arbitrarily short as you please. Look again at my turn-around sketch showing a sequence of inertial frames.

It's a mistake to regard any part of the frame as nonsense just because it is non-intuitive with respect to one's personal philosophy of reality. The red guy in my earlier sketch is at rest in the black frame, and for him his clock his ticking forward in time. There is no going backward in time for the red guy. It's just that the sequence of red clock times as presented in blue's sequence of inertial frame increments appears as a sequence going backward in time--that does not mean that the red guy experiences time reversal. You can reject that on a philosophical basis if you wish, but it is simply a straight forward result of Lorentz transformations on individual frames.

[edit] As Vandam would say, "Einstein never needed the doppler approach to explain time dilation, length contraction and relativity of simultaneity." Vandam, where are you when we need you--you've been conspicuously absent of late.

 

[PAllen]

I have not presented the turn-around in the context of non-intertial frames. I followed Einstein's proposal to approach the turn-around problem as a sequence of inertial frames, each one individually amenable to the usual Lorentz transformation treatment. So, they are a sequence of incremental boosts. You can let the increments become as arbitrarily short as you please. Look again at my turn-around sketch showing a sequence of inertial frames.

It's a mistake to regard any part of the frame as nonsense just because it is non-intuitive with respect to one's personal philosophy of reality. The red guy in my earlier sketch is at rest in the black frame, and for him his clock his ticking forward in time. There is no going backward in time for the red guy. It's just that the sequence of red clock times as presented in blue's sequence of inertial frame increments appears as a sequence going backward in time--that does not mean that the red guy experiences time reversal. You can reject that on a philosophical basis if you wish, but it is simply a straight forward result of Lorentz transformations on individual frames.

[edit] As Vandam would say, "Einstein never needed the doppler approach to explain time dilation, length contraction and relativity of simultaneity." Vandam, where are you when we need you--you've been conspicuously absent of late.

It is certainly true that Einstein didn't use Doppler analysis. It is also true that he never used your approach, in papers I can find. He used analysis in one inertial frame, in his 1905 paper and others that I checked. None of his writing that I could find use or 'lines of simultaneity' in the way you do. Certainly, many presentations do use this approach, but I find it at odds with Einstein's philosophy. By mandating that lines of simultaneity are physical and real, you have the view that simultaneity may be observer dependent, but is absolute for any given observer. The thrust of Einstein's writing is much more in the spirit of: "So we see that we cannot attach any absolute signification to the concept of simultaneity" taken fully to heart.

[edit: Einstein also used accelerated motion as a bridge to GR. But with more searching, I still find no example of Einstein using lines of simultaneity or continuously shifting inertial frames. Do you know of any such example?]

 

[PeterDonis]

The thrust of Einstein's writing is much more in the spirit of: "So we see that we cannot attach any absolute signification to the concept of simultaneity" taken fully to heart.

Exactly. Einstein's reasoning was along the lines of:

(1) All "real physical things" are frame-independent;

(2) Simultaneity is frame-dependent;

(3) Therefore, simultaneity is not a "real physical thing".

 

[Dale]

I have not presented the turn-around in the context of non-intertial frames. I followed Einstein's proposal to approach the turn-around problem as a sequence of inertial frames, each one individually amenable to the usual Lorentz transformation treatment.

Yes, you did, as soon as you started talking about the red guy's clock or time running backwards. That does not happen in any of the sequence of inertial frames and the Lorentz transform is not compatible with that happening.

As soon as you were talking about time running backwards you were connecting pieces of the various inertial frames into a single non-inertial frame, and you were doing so in a way that is mathematically wrong.

Either way you cannot make the conclusion that the red guy's clock runs backwards. Either you are considering inertial frames in which his time is well defined and his clock runs forwards or you are considering a non-inertial frame where his time is not even defined.

Furthermore, I don't believe that Einstein ever did what you suggest he did. Can you please provide a reference? The references I am aware of from Einstein are much more along the lines of the "one inertial frame" approach.

It's a mistake to regard any part of the frame as nonsense just because it is non-intuitive with respect to one's personal philosophy of reality.

It is also a mistake to continue to make mathematically wrong statements when the error has already been explained to you.

 

[bobc2]

I get the impression that the concept of the sequence of inertial frames to represent the turnaround is not being understood. In the sketches below we show how you might represent a turnaround with more and more inertial frames. The number of inertial frames increases in each sketch, a) through e). A procedure such as one of these could be implemented with a rocket if you perform a sequence decelerations followed by a sequence of accelerations back toward the start point once the turnaround point is reached. Between each rocket impulse you would coast in an inertial frame during which the Lorentz transformations apply. You could break the turn-around incrementally into as many inertial frames as desired. Should there be any limit in smallness and number of the worldline increments used in the sequence of inertial frames? Of course not.

So, it is incorrect to regard this scenario as requiring a non-inertial analysis. This is nothing more than a generalization of the accepted practice of using a single turnaround point when analyzing the final age differences of the twins.

Further, the full extent of the Lorentz frames (including positive and negative directions) should be recognized. Again, that does not mean that the red observer's clock runs backwards (see previous post sketch), even though red worldline clock time are presented in the sequence of traveling twin's inertial frames as a sequence of clock times going backwards in time. That certainly does not mean that either the twin or the Red observer are actually moving backward in time. Both observers are moving along their respective worldlines at the speed of light.

 

[PeterDonis]

I get the impression that the concept of the sequence of inertial frames to represent the turnaround is not being understood.

I understand it just fine. I can't speak for DaleSpam and PAllen and others, but I suspect they understand it just fine as well. But you appear to be claiming that your analysis tells us something about "reality"--that there is some "real meaning" to the fact that the times assigned to events on red's worldline "appear to go backwards" in the succession of frames that blue uses. That's the claim that we are disagreeing with.

The times assigned by blue in the succession of frames you describe are just coordinate times; they don't have any physical meaning by themselves. If you try to translate them into statements that have physical meaning, i.e., into statements about invariants, you will find, of course, that the invariants regarding time, such as red's own proper time, "move forward" as expected. I say "of course" because you already know this, since you said so in a previous post.

 

[Dale]

The times assigned by blue in the succession of frames you describe are just coordinate times; they don't have any physical meaning by themselves.

I actually disagree with this slightly. They are not even coordinate times (on the red worldline) since they violate the one-to-one requirement of coordinate charts.

 

[Dale]

Should there be any limit in smallness and number of the worldline increments used in the sequence of inertial frames? Of course not.

So, it is incorrect to regard this scenario as requiring a non-inertial analysis. This is nothing more than a generalization of the accepted practice of using a single turnaround point when analyzing the final age differences of the twins.

You seem to think that a worldline single turnaround point is inertial. This is incorrect. Even one "increment" representing a sharp instantaneous acceleration is enough to make the traveling twin and his frame be non-inertial.

As long as the worldline is bent at all in an inertial frame then it is non-inertial. It does not matter if it is a single sharp bend, a gradual continuous bend, or a series of small sharp bends. The answer to your question "how many increments before it is declared to be non-inertial" is 1.

 

[PeterDonis]

They are not even coordinate times (on the red worldline) since they violate the one-to-one requirement of coordinate charts.

The time assigned by each individual frame in the succession of blue's frames is a coordinate time; the time assignments of each individual frame are one-to-one. The problem only arises if you try to put together a single non-inertial frame whose coordinate assignments along a given worldline (such as red's) agree with those of the succession of inertial frames; as you point out, you can't do that globally without violating the one-to-one requirement.

It looks to me like bobc2 didn't actually intend to construct a single non-inertial frame in this way, but to me that's really a side issue; even if one is careful *not* to make any claims about a single non-inertial frame, it's still true that you can't get anything physically meaningful just by looking at coordinate times of events along red's worldline in the succession of blue's inertial frames.

 

[bobc2]

The time assigned by each individual frame in the succession of blue's frames is a coordinate time; the time assignments of each individual frame are one-to-one. The problem only arises if you try to put together a single non-inertial frame whose coordinate assignments along a given worldline (such as red's) agree with those of the succession of inertial frames; as you point out, you can't do that globally without violating the one-to-one requirement.

It looks to me like bobc2 didn't actually intend to construct a single non-inertial frame in this way...

You are exactly correct. That is what I've been trying to get across. Regarding the collection of individual inertial frames as representing a single non-inertial frame is definitely not the way to understand this.

...but to me that's really a side issue; even if one is careful *not* to make any claims about a single non-inertial frame, it's still true that you can't get anything physically meaningful just by looking at coordinate times of events along red's worldline in the succession of blue's inertial frames.

That is a philosophically based idea. Einstein cautioned against these kinds of ideas which trap you into solipsism.

You are saying that there is no reality to be associated with the hyperplanes of simultaneity for a given Lorentz frame. I don't think our monitor will want us to continue a discussion along those lines. The monitor may allow you to define for us how you would describe or define criteria for identifying the real world of existence in the context of physical theory--I'm not sure.

 

[Dale]

The time assigned by each individual frame in the succession of blue's frames is a coordinate time; the time assignments of each individual frame are one-to-one. The problem only arises if you try to put together a single non-inertial frame whose coordinate assignments along a given worldline (such as red's) agree with those of the succession of inertial frames; as you point out, you can't do that globally without violating the one-to-one requirement.

Right, and this is exactly what bobc2 did in his post 32. In that post he is explicitly NOT talking about a sequence of individual 4D inertial frames, but a sequence of instantaneous "3D worlds". This is the same as defining a simultaneity convention for a single non-inertial 4D coordinate system.

 

[Dale]

You are exactly correct. That is what I've been trying to get across. Regarding the collection of individual inertial frames as representing a single non-inertial frame is definitely not the way to understand this.

I already addressed this in post 50, but that is exactly what you were doing in claiming that the red guy's time was going backwards. If you talk about a sequence of inertial frames then his clock goes forwards at all times and in all frames. If you talk about a sequence of "3D worlds" then you are talking about a 4D non-inertial frame, and mathematically that frame cannot cover the red guy's worldline.

That is a philosophically based idea. Einstein cautioned against these kinds of ideas which trap you into solipsism.

Reference please?

 

[PeterDonis]

Regarding the collection of individual inertial frames as representing a single non-inertial frame is definitely not the way to understand this.

Then, as DaleSpam pointed out, you can't make any claims about red's time "running backwards". Just noticing that the coordinate times in a succession of different inertial frames "run backwards" doesn't say anything about red's time "running backwards".

That is a philosophically based idea. Einstein cautioned against these kinds of ideas which trap you into solipsism.

Einstein did no such thing. Arguments from authority don't count anyway, but I believe Einstein would have agreed that coordinate times (and indeed coordinates in general) are not "physically real". See below.

The monitor may allow you to define for us how you would describe or define criteria for identifying the real world of existence in the context of physical theory--I'm not sure.

Read my post #49. I gave there a perfectly good definition of what counts as "real"--invariants, things that are not frame-dependent. As far as I know, as I said in that post, Einstein would have agreed with such a definition. Simultaneity is frame-dependent, hence it does not count as "real" by that definition. The same goes for coordinate times, and for your "3-D worlds". They are fine as logical constructions, or as elements in a model that helps you to understand things; but that's all.

 

[laurub]

No, your brother will not receive many messages from you at the moment of his about-face. His about-face will not cause him to receive any messages from you. What's going to happen is that for the first half of his trip, he will receive messages from you at a slower rate than he sends them (1/R as I said in my first post to you), then for the second half of his trip he will receive messages from you at a faster rate than he sends them (R).

So for your example of your brother traveling at 0.5c, we can use the Relativistic Doppler formula to calculate what R is:

√((1+β)/(1-β)) = √((1+0.5)/(1-0.5)) = √((1.5)/(0.5)) = √3 = 1.732

And 1/R is the reciprocal, 0.57735.

This means that he will see your yearly messages coming to him slower than his during the first half of the trip. In fact it will take 1.732 years before he sees your first message.

And for the last half his trip, he will see your messages arriving more often than once per year according to his clock. It will only take 0.57735 years between each of your messages.

Now without knowing how long the trip will take, we can average these two numbers:

(1.732+0.57735)/2 = 2.30935/2 = 1.154675

This is the final ratio of your two clocks when he returns. How ever many years it took him according to his clock, yours will be 1.154675 times that amount.

So let's say your brother travels away at 0.5c for 13 years and then takes 13 years to get back at the same speed. Here is a spacetime diagram to show what is happening according to your rest frame. I show you as a thick blue stationary line with dots every year and your messages going out as thin blue lines traveling at c. I show your brother as a thick black line traveling at 0.5c with black dots every year and his messages coming back to you as thin black lines:

Now let's see how the previous calculations based on Relativistic Doppler fit in with this diagram. First off, I said that the rate at which your brother receives your yearly messages take 1.732 of his years. Can you see that on the graph? For example, at about his year 12, just before he turns around, he is just receiving the message you sent at your year 7. Can you follow that? If we divide 12 by 7 we get 1.714 which is about right. (We don't expect it to be exact because he didn't receive your message exactly at his year 12.)

Furthermore, if you look at your year 12, you can see that you are just receiving his message from year 7. It's symmetrical.

Now you should be able to see that after he turns around, he starts receiving your messages faster than one per year. In fact, from about his year 19 (you'll have to count his dots) to when you meet at his year 26, he will have received your messages from year 18 to 30. That is a ratio of the differences of (26-19)/(30-18) = 7/12 = 0.583, close enough to 0.577.

And in a similar manner, you can see that from his year 14 (just after he turns around) until you meet (12 years of messages from him), you will see them from your year 23 to your year 30 (7 years) and the reciprocal ratio applies.

Now that we can look at a spacetime diagram, we can see that the reason why the two of you age differently is because your brother sees these two ratios for half of his total trip time each but you see the smaller ratio for three-quarters of the time and the higher ratio for just one-quarter of the time. This means you are seeing him age less for a longer time while he sees you age less for half the time.

The last thing we want to notice is that ratio of your final age difference is 30/26 or 1.1538, very close to the actual 1.154675. (Again, these numbers are not exact because we're eyeballing them off the diagram.) This ratio is the famous value of gamma which is also the time dilation factor which shows in the diagram as the ratio of the coordinate time for your brother compared to his actual time on his clock. Can you see that?

Now I want to show you what the exact same information presented in the first diagram looks like in two more diagrams based on the IRF's in which your brother is at rest, first during his outbound leg and then during his inbound leg. First the outbound leg:

Notice how your brother's time is not dilated during the outbound leg (because he is at rest) but yours is. Note also that he has to travel at a higher speed than 0.5c (look up "veloctiy addition" in wikipedia to see that this higher speed is 0.8c) when he turns around and therefore now has more time dilation than you have. Nevertheless, all the signals between the two of you continue to travel at c and arrive at exactly the same times according to your own clocks as they did before. Does this all make sense to you?

Finally the diagram for the IRF in which your brother is at rest during the inbound leg:

This is very similar to the previous diagram so I won't go into any more explanation except that I want to point out that when your brother turns around, in no case does that have any bearing on what you see, until some time later and even then, each diagram shows accurately what you actually see and what your brother actually sees during the entire scenario.

Any questions?

Awesome diagrams and explanation! thank you for this!