What Causes Time Dilation in the Twin Paradox?

[neopolitan]

I think he might be referring to a function describing clock "A" in B's frame, which would have a discontinuity if we assume the rate of acceleration to be infinity, and the elapsed time on clock "B" to be zero during the turnaround. The Earth's clock would read 7.2 then 32.8 yrs, but not values in between. But this is just an artifact of treating the turnaround as instantaneous. If we say the turnaround is just "near" instantaneous, then the Earth's clock will advance from 7.2 to 32.8 very quickly according to clock "B", but it will read all times in between with no discontinuity.

Al

No, I am as against quick advancing of the clock (I called it scrolling) from 7.2 years to 32.8 years as I am against the clock being 7.2 years one moment and 32.8 years the next.

Acceleration qua acceleration has no effect on timing, only as a consequence of the altered relative speeds. (And simultaneity is only altered as a consequence of altered relative velocities.)

cheers,

neopolitan

 

[matheinste]

Hello neopolitan

As regards the turnaround, it need not be abrupt, it does not matter how long it takes. For the sake of my present post let us assume that it is takes a finite time. Now this finite time could be almost instantaneous, fairly fast………very slow, imperceptibly slow.. If the turnaround took a long time and was smooth and continuous and the clock showed a very slow advancement ( the one with B telling A’s time ), would you accept this. If you would, then let us crank the rate of turnaround up a notch and ask if you would accept this also ( assuming you accept the first degree of slowness ).
Obviously you see where I am going. If the discontinuity bothers you, by the way I think it makes for a bad scenario and would not have used it myself, let's just do away with it. If that was your worry would you accept scenarios with increasingly fast but smooth turnarounds and if not, at which point would they cease to be acceptable, assuming you accept accelerated rates of clock advancement of A’s clock from B’s point of view, at at all?

Matheinste.

 

[Fredrik]

• "B" calculates "A"'s age on an ongoing basis.
  
• Most significantly: "B" does not at any time assume that the prevailing inertial frame is eternally valid (due to awareness of accelerations).

And what would you say that A's age according to B is right after the turnaround? If it isn't "original age" + 32.8 years, then you have two major problems: 1) The time dilation formula says that A will only age another 7.2 years during the return trip, and he's going to have aged 40 years when B gets home, so your result contradicts time dilation. 2) Your result defines some other line than the red line to be B's simultaneity line, and that implies that B won't agree that the speed of light is =1 (i.e. =c).

 

[Antenna Guy]

As regards the turnaround, it need not be abrupt, it does not matter how long it takes.

If B's trajectory is circular in A's frame, the "turn-around" would take the entire trip to complete.

Regards,

Bill

 

[Dale]

This is exactly why I dislike this resolution of the twins paradox. It always leads inevitably to this discussion because simultaneity is fundamentally difficult to define in a non-inertial reference frame.

I much prefer the spacetime geometric resolution. It is simple, clear, and universal.

 

[atyy]

Is Michael Weiss claiming this question is not even meaningful?

"How much Terence ages during the turnaround is not something you can directly observe, according to SR. The Doppler Shift Analysis focuses on what Terence and Stella actually see through their telescopes, which avoids the difficulty."
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_distance.html

Edit: Is this at all related to how we observe distant galaxies, but we don't say anything about how old they are now, but we do at least colloquially say that this gives us information now about how the galaxies were then?

 

[Antenna Guy]

I much prefer the spacetime geometric resolution. It is simple, clear, and universal.

I agree.

The peculiar aspect of the circular trajectory is that the magnitude of B's velocity in A's frame could be constant for the entire trip (changing only in direction).

Regards,

Bill

 

[neopolitan]

Is Michael Weiss claiming this question is not even meaningful?

"How much Terence ages during the turnaround is not something you can directly observe, according to SR. The Doppler Shift Analysis focuses on what Terence and Stella actually see through their telescopes, which avoids the difficulty."
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_distance.html

Edit: Is this at all related to how we observe distant galaxies, but we don't say anything about how old they are now, but we do at least colloquially say that this gives us information now about how the galaxies were then?

Even more interesting are the two diagrams under this by the same author:

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html#gap

Pretty much the same thing that I have been talking about, so thanks atty :)

For the ever-snarky Fredrik, look at the angle of the red lines in the second of the two diagrams in Figure 2. Note that this angle is much more acute than the angles in the diagram posted by phyti and you made the comment about tachyons ...

cheers,

neopolitan

 

[Fredrik]

This is exactly why I dislike this resolution of the twins paradox. It always leads inevitably to this discussion because simultaneity is fundamentally difficult to define in a non-inertial reference frame.

I much prefer the spacetime geometric resolution. It is simple, clear, and universal.

I assume that you mean that the final ages of the two twins follows immediately from the fact that "what a clock measures" is the "proper time" (defined as an integral) along the curve in Minkowski space that represents the clock's motion

I agree that this is all you need to determine their final ages, but in my opinion, a "resolution" of the paradox should also explain what's wrong with the argument that lead to the conclusion that there's a paradox. The resolution discussed in this thread does that. It tells us that if you only take the time dilation into account, you get a paradox, but if you also take the "simultaneity shift" into account, you don't.

 

[atyy]

Your result defines some other line than the red line to be B's simultaneity line, and that implies that B won't agree that the speed of light is =1 (i.e. =c).

It always leads inevitably to this discussion because simultaneity is fundamentally difficult to define in a non-inertial reference frame.

Fredrik's comment about the speed of light is good. Is it possible to state, considering DaleSpam's comment, that a non-inertial frame only has a local plane of simultaneity? If so, how is "local" to be defined?

Edit: There seems to be no problem with the global planes of simultaneity, at least in this particular version of the twin paradox. However, we also don't require the speed of light to be constant in a non-inertial frame.

 

[Dale]

I agree that this is all you need to determine their final ages, but in my opinion, a "resolution" of the paradox should also explain what's wrong with the argument that lead to the conclusion that there's a paradox. The resolution discussed in this thread does that.

You can also explain what's wrong with the argument using the spacetime geometric approach.

The key error in the paradox is the incorrect assumption that the two paths are equally valid inertial reference frames. By the time the student is exposed to the twins paradox they have done a number of problems where they have to treat two different observers as equivalent even though they intuitively feel that one frame is "right". So, although they don't really feel that the two situations are equivalent they have learned to neglect their intuition and treat them as equivalent anyway.

With the spacetime geometric approach you can re-cast the inertial frames idea in terms of straight lines through spacetime. Then it becomes easy to see that different straight lines are equivalent, the difference being only a rotation. It is also easy to see that a bent line is not equivalent to a straight line, the bent line remains bent regardless of any rotation.

 

[RandallB]

assumptions ….
….
Most significantly: "B" does not at any time assume that the prevailing inertial frame is eternally valid (due to awareness of accelerations).

I do think that the last is important, and is possibly the sticking point

Equivalent to the what Einstein concluded about Simultaneity in SR.

Simultaneity says that while have observers in Chicago and Dallas both watching their clocks turn to exactly 1 PM you can assume that they make these observations simultaneously “at the same time”. Because Simultaneity tells use that the same times separated by any distance in a common Reference Frame cannot in reality be assumed to happen Simultaneously.

It is a fundamental concluding principal of Special Relativity that the synchronized times in any Reference Frame regardless of what the observers think within that frame (Even us in our reference frame) cannot be used to measure when event are actually happen Simultaneously “at the same time” just because they are observed as happening at the same clock time defined in there common reference frame. Only the “Preferred Frame” can correctly define that and Special Relativity not only does not define a Preferred Frame it implies that there may not be a Preferred Frame.

Now thinking that through – and that may take a bit of thinking so take your time –
It means whatever reference frame B is in when changing speed the simultaneously time in that frame at the location for A cannot be known because as you say B cannot; “assume that the prevailing inertial frame is eternally valid” (or is THE Preferred Frame).
Therefore, if by definition (SR-Simultaneity) you cannot reliably define the correct simultaneous time on a B frame clock located near A, you cannot presume to define what time it is for A in that reference frame when B turns around. Not without ignoring the conclusions of SR Simultaneously or revising it to establish a Preferred Frame of Reference.

 

[neopolitan]

With the spacetime geometric approach you can re-cast the inertial frames idea in terms of straight lines through spacetime. Then it becomes easy to see that different straight lines are equivalent, the difference being only a rotation. It is also easy to see that a bent line is not equivalent to a straight line, the bent line remains bent regardless of any rotation.

Dale,

I imagine that here you are saying that the significant fact is that the "bent line" is bent and that that bend leads to a different path through spacetime which is, in total, equivalent to the straight one.

The result is as if "A" and "B" are given the same amount of spacetime dollars each, "A" converts all of it to time dollars (40 years worth), while "B" converts a share into space dollars (32 light years and 24 years, the equation for converting spacetime dollars into time and space dollars is the old equation s^2 = t^2 + x^2). In all, both have used 40 (light-)years worth of spacetime dollars

What is interesting to note is that the distance traveled in the equation above is the distance that "A" must calculate that "B" travels (40 years at 0.8c) rather than what "B" would calculate (12 years at 0.8c). This seems to indicate that the true distance traveled by "B" is 16 lightyears, not 9.6 lightyears. This does make sense since "B" is subject not only to dilated time, but also contracted space.

I am not sure that this is a standard interpretation though and I do see a problem. Say that "B" stops at the turnaround point and uses a laser range finder to determine the separation from "A". If "B" discovers that "A" is 16 lightyears distant (and obtains a statutory declation from "A" that "A" did not move away), then "B" will have cause to suspect FTL travel since "B" got that far away in only 12 years travelling.

What is the standard resolution to this? or is it just accepted that you can achieve greater distances than the inertial velocity maintained and time spent maintaining it would indicate?

Anyway, the important thing in my argument is that both "A" and "B" spend their spacetime fortune at the same rate and, in fact, everyone spends their spacetime dollars at the same rate. I think that both "A" and "B" will expect the other to spend their spacetimes dollars at the same rate and I don't think that "B" will expect "A" to blow a pile of them just because the spaceship changes direction

cheers,

neopolitan

 

[neopolitan]

Now thinking that through – and that may take a bit of thinking so take your time –
It means whatever reference frame B is in when changing speed the simultaneously time in that frame at the location for A cannot be known because as you say B cannot; “assume that the prevailing inertial frame is eternally valid” (or is THE Preferred Frame).
Therefore, if by definition (SR-Simultaneity) you cannot reliably define the correct simultaneous time on a B frame clock located near A, you cannot presume to define what time it is for A in that reference frame when B turns around. Not without ignoring the conclusions of SR Simultaneously or revising it to establish a Preferred Frame of Reference.

I think that "B" can make projections, based on the information to hand and predefined rules of the experiment.

I think that you have identified an assumption which I did not state, so I add it.

• "B" knows that under the terms of the experiment, "A" will not move and "A" will continue to transmit fairly (ie exactly 1 day between each daily ageing transmission).

It is this information which will allow "B" to make an accurate prediction - but only because future events have been predefined.

I do think that by using the spacetime geometry model, "B" can determine what the "A" clock will read. Indulge me and read the previous post where I raise the simile of spacetime dollars. I just need to clarify something.

When "A" and "B" are collocated and stationary wrt each other, I used the conceit of implying that they both spend their spacetime dollars as time dollars. However, this is not entirely true, since they both could be in motion relative to third observer and relative to that third observer, they are spending a combination of space and time dollars.

However, the exchanges are such that "A" can consider that all dollars are spent as time dollars and it is the deviation from this standard that "B" creates by taking the trip that matters, not the actual raw figures.

Why does this matter? Because the standard understanding phrasing of the twins paradox is that both "A" and "B" should expect the other to have faster clocks. However, this is not the case if "B" deviates from the standard distribution of spacetime dollar spending. If "B" can determine the new distribution of spacetime dollars, which is possible is the accelerations are known, then "B" will be able to work out what the "A" clock reads.

It is late here, I hope to sleep and read you tomorrow,

cheers,

neopolitan

 

[Dale]

I imagine that here you are saying that the significant fact is that the "bent line" is bent and that that bend leads to a different path through spacetime which is, in total, equivalent to the straight one.

I think you understand, but just to be absolutely clear I would definitely not say that e.g. a semi-circle is equivalent to a diameter. They pass through two of the same points, but otherwise they are completely different. Similarly with the two twins' paths through spacetime.

 

[RandallB]

I think that "B" can make projections,

Actually B cannot make projections, it can only use the projections as can be defined by the many different reference frames B uses as it changes speed and direction at the turnaround. And because of the differences between them you cannot take the projected time by anyone of those reference frames to be “correct” or “real” without violating the SR-Simultaneity Rule and establishing that one frame as preferred.

You will have the same problem in trying to decided which clock run slow or fast between B and little “b” following shortly behind B. They run at the same rate before the turn around, but after B turns around and before little “b” does they will pass next to each other. Measuring each other as the pass going in different directions which will they measure their time as slow, fast or the same as compared to the other. If you can prove they run at the same rate then A is in the preferred frame – Simultaneity says you will not be able to decide between the three options.

If you intend to do these problems using SR you cannot ignore the limitation imposed by the SR-Simultaneity Rule.

 

[matheinste]

Hello RandallB.

There is of course an underlying problem with simultaneity which it may be worth restating here. It has no naturally defined meaning, although we seem to have an ingrained intuitive idea of what we think it means. Einsteins definition of simultaneity is a conventional one, in the sense that it is man made. It does of course seem intuitively reasonable. We cannot describe it as right or wrong, it is merely a definition, in many cases a useful one otherwise it would not be used. Some feel understandably that it is misleading and that the whole idea of simultaneity is not relevant and unnecessary.

In the present scenario, while we may not be able, or may feel it unnecessary, to say what times are simultaneous on the clocks of A and B, we know that at separation the clocks read the same and when reunited A's clock is ahead of B's, this being the whole point of the apparent paradox. However, if both A and B ( A assumed stationary )move smoothly, although B must accelerate during turnaround, and assuming time itself is infinitely divisible,the readings on the clocks can be put in one to one correspondence, not necessarily linearly so. Therefore for each time on one clock there is a corresponding reading on the others clock. That fact of course says nothing about simultaneity but while we MAY not be able to determine what time on one clock coresponds to what reading on the other, such times do exist, no matter what the relative rate of one to the other when B is accelerating.

I do not know if this helps any of us in any way in the present discussion or if it agrees or disagrees in essence to what has already been said, but it may be food for thought.

Matheinste.

 

[Al68]

No, I am as against quick advancing of the clock (I called it scrolling) from 7.2 years to 32.8 years as I am against the clock being 7.2 years one moment and 32.8 years the next.

OK, I misunderstood. Would you agree that in the ship's frame(s), the coordinate distance of the ship from Earth increases (jumps) from 9.6 to 16 ly during the deceleration?

If so, it's a simple exercise to show that the Earth clock readings simultaneous (in Earth's frame) with the ship being at those two distances are 7.2 and 20 yrs, respectively.

Al

 

[RandallB]

Hello RandallB.

There is of course an underlying problem with simultaneity …
we seem to have an ingrained intuitive idea of what we think it means.
Einsteins definition of simultaneity is a conventional one, …
It does of course seem intuitively reasonable. …..

We cannot describe it as right or wrong, it is merely a definition,

Matheinste - I disagree

I don’t believe Einstein though he was merely stating an intuitive definition. He was pointing out what he felt had to be a fundamental principle about what we can know.
I never had an ingrained intuitive idea Einstein Simultaneity should be true.
I reason that while talking to someone on the phone a few hundred miles and synchronizing our watches that they would read the same time simultaneously.
But NO - Einstein Simultaneity demands that I not think that! That I must keep in mind our clocks only appear to be synchronized in our reference frame but in no other.
There is nothing conventional or intuitive about that
- which is why threads like this can run so long.

Even after understanding SR and the Twins my common sense would be willing to accept that only one of many reference frames is correct and all other frames must refer to that one “Preferred Frame” to translate their clocks so as to establish when things really do happen “at the same time”. Simple Lorentz Transforms against the preferred frame of an either would solve all our problems.

In the present scenario, B on the way out may be running faster than A as long as it running much slower on the back. Or slow on the way out and fast on the way back, it is just that B must spend much more “A Time” moving slow to cover the same distance traveled while moving fast. Or the third option is B could run slower than A in both directions.
Which of the three is correct?
If all reference frames acknowledge anyone of the three motions A, B outbound or B inbound as “The Preferred Frame” then you can determine exactly what clock A reads when B turns around.
But this problem is prevented from allowing that - boxed in by the simultaneity principle. Unless a way of getting outside that box while still using SR principles is defined. no frame can claim to “know” what clock A really reads when the turnaround happens..

Anyone that thinks they can say what the time is on clock A should also be able to give the time on clock A’. Where A’ is stationary wrt A but passing next to B at the moment of turnaround. Again Simultaneity states that no one can, thus there is no justification for time or distance “jumps” due to accereration.

 

[matheinste]

Hello RandallB

I believe we are in agreement.

When i say that some of us have an ingrained idea of what simultaneity means i did not mean that this idea is correct or justified. Also i did say SEEMS reasonable. Also saying that Einstein's definition is convential i did not mean conventional in the sense of "the accepted norm" but it the sense of not NATURALLY defined or dtermined by nature, but defined by convention and othe conventional definitions might serve as well.

I also, along with nearly everybody else do not believe in a single correct reference frame, although there may be preferred frames, preferred because they simplify the task in hand.

Matheinste.

 

[matheinste]

Hello again RandallB

As regards threads such as this dragging on:-

In non-relativistic physics and mathematics we have axioms or starting points and if we agree on these and use logic we reach conclusions. These conclusions appear in textbooks and are regarded as authoritative and generally accepted. In relativity we start with a set of axioms and IF we agree to accept these and apply logic we reach conclusions. These conclusions appear in textbooks and are very often NOT accepted as authoritative. Why is Relativity different. In the case of this “paradox” there appears to be an authoritative answer following logically from the axioms but many people are not accepting it.

I agree with the generally accepted answers but I am not capable of convincing others that they are correct. This may be my lack of skills but it may also be that these answers are wrong and that i misunderstand the problem. But the fact that many or most relativists accept the usual answers points to those answers being correct.

Matheinste

 

[RandallB]

When i say that some of us have an ingrained idea of what simultaneity means i did not mean that this idea is correct or justified. Also i did say SEEMS reasonable.

I’m not sure what you are saying here :
Do you mean miss-interpretations of Einstein Simultaneity may not be “correct or justified”?
OR that Einstein Simultaneity may not be “correct or justified”?

IMO the reason that threads like this run so long is not due to a miss-interpretations of Einstein Simultaneity – it is that the SR Axiom of Simultaneity is not applied at all.
Because that axiom makes it clear that no spatial separated events (Such as certain times on two clocks) can be defined as happening “at the time”. Only that one frame can predict that over time it will receive data that appears to indicate that they did – while all other frames will receive data that indicate they did not happen at the same time. And that is not enough to claim anyone frame as "correct".

Also saying that Einstein's definition is convential i did not mean conventional in the sense of "the accepted norm" but it the sense of not NATURALLY defined or determined by nature, but defined by convention and othe conventional definitions might serve as well.

I disagree – I think Simultaneity WAS NATURALLY defined by Einstein based on observations (M&M experiments) determined by nature.

 

[neopolitan]

Referring to posts from https://www.physicsforums.com/showpost.php?p=1887725&postcount=69" inclusive.

Specifically:

I don’t believe Einstein though he was merely stating an intuitive definition. He was pointing out what he felt had to be a fundamental principle about what we can know.

This is a problem statement in that, in context, it seems to deify the proponent of the theory. Irrespective of how right he is, this is not the way we should be doing science.

What Einstein thought is immaterial because, once he has given us the principle, it is fundamental or it isn't. His genius in coming up with the theory doesn't protect him from scrutiny or from error and he is actually known for having made blunders and acknowledging them.

As for the "SR-Simultaneity rule", a google search with that strict phase came up with one hit. This thread. That's a rule which either has another name or has only recently been created.

I would ask for the rule to be stated explicitly and referenced. Thanks.

-------------------------

The simultaneity thing is based on our definition of simultaneity, which due to relativity is fluid. But as has been pointed out in other threads, causality is not fluid. Cause leads to action, not action to cause - in whatever frame you wish to use.

The whole point of the signals in my earlier posts was to introduce, as stated before, real universe thinking. With a signal, you have a causal chain - there is a sequence of signals which are transmitted by "A", each signal is causally linked to the next (you can surely understand that ageing 2 days is a result of having aged 1 day, rather than a cause). There is also a causal link between the transmission and reception. There is a causal link between "B"'s initial acceleration and the outwards cruising speed. There is a causal link between the accelerations at the turnaround point and the homeward cruising speed.

It is by arraying all these causal links, and using spacetime geometry, that "B" can calculate the time shown on "A"'s clock.

We've shown that we can work out what time will be on "A"'s clock when "B" turns around (at the middle of the turnaround, when both are stationary with respect to each other), 20 years. We can also work out from the information to hand (ie, that we know that "B" accelerated, rather than "A"), that "A" will be the twin who will experience more time.

Why will we become more stupid if we take the role of "B" (and I stress, we as "B" are aware of the accelerations)?

Why would we ignore the evidence provided by the reception of signals from "A"?

This just smacks of orthodoxy and orthodoxy is like death to critical thinking.

cheers,

neopolitan

BTW ...

When I get time I will put together something on that "spacetime dollars" allusion. I am pretty sure that if you do it in such a way that "A"'s frame is not privileged, ie we do not assume that "A" is stationary, you can still work out that "A" will experience more time than "B" and the figures will work out perfectly well.

However, in the meantime, you could attach a grid to "A" and make "B" travel within that grid, much as I think granpa referred to. A grid with co-stationary clocks which are synchronised in their rest frame and distance markings. Using that grid you can work out which events are synchronous with which events. Now even if that grid is sort of a privileged frame, the removal of the grid will not remove the underlying physics and destroy our knowledge of what happened when, according to each frame.

If you do the thinking, with such a grid in mind, you can calculate that there is no sudden change in "A"'s clock according to "B".

I strongly suspect that too many people here have not had to do some of the reasonably basic applied mathematics which is behind discrete element modelling. If they had, they would be less likely to ignore the fact that the twin's universe has to make as much sense in the micro scale as in the macro scale.

 

[matheinste]

Hello RandallB

By our ingrained intuitive sense of simultaneity i meant the pre-Einstein idea of absolute simultaneity. Obviously when it is defined or described as per Einstein then his definition becomes reasonable and obvious and i accept it without argument. What i was saying is that there are schools of thought, i suppose more philosophical ones, which would argue that non-local simultaneity is meaningless and for the teaching of SR a hindrance.

By conventional i meant "requiring definition". We do not feel to need to define "two events being colocated in the same inertial coordinate frame" but it was thought necessary to define "two events being simultaneous in the same inertial coordinate frame" . Of course none of these are my original thoughts but are borrowed from others.

Matheinste.

 

[Fredrik]

As for the "SR-Simultaneity rule", a google search with that strict phase came up with one hit. This thread. That's a rule which either has another name or has only recently been created.

I would ask for the rule to be stated explicitly and referenced. Thanks.

I explained simultaneity in SR earlier in this thread.

The whole point of the signals in my earlier posts was to introduce, as stated before, real universe thinking.

Some people might say that simultaneity is actually a relevant concept in "real world thinking".

We've shown that we can work out what time will be on "A"'s clock when "B" turns around (at the middle of the turnaround, when both are stationary with respect to each other), 20 years.

Why aren't you answering my question about how much B should say that A has aged right after the turnaround? Do you understand what I said about how answers other than "32.8 years" are problematic?

Why will we become more stupid if we take the role of "B" (and I stress, we as "B" are aware of the accelerations)?

Stupid? I don't know what you're talking about but this seems like a good time to remind you that the reason we have to consider B's point of view is that the twin paradox is all about explaining why B gets the wrong result for A's age if he just uses the time dilation formula. (The "simultaneity shift" explains why B gets the wrong result and also what the correct result is. Can your argument based on signals from Earth do that?).

Why would we ignore the evidence provided by the reception of signals from "A"?

You don't have to ignore it (or anything else), but you have to interpret it right.

This just smacks of orthodoxy and orthodoxy is like death to critical thinking.

You sound more and more like someone who just refuses to learn the theory he's criticizing.

However, in the meantime, you could attach a grid to "A" and make "B" travel within that grid, much as I think granpa referred to. A grid with co-stationary clocks which are synchronised in their rest frame and distance markings. Using that grid you can work out which events are synchronous with which events. Now even if that grid is sort of a privileged frame, the removal of the grid will not remove the underlying physics and destroy our knowledge of what happened when, according to each frame.

If you do the thinking, with such a grid in mind, you can calculate that there is no sudden change in "A"'s clock according to "B".

First you make A's frame a preferred frame, contradicting the principles on which SR was built, and then you describe something using this preferred frame and claim that the description is according to B!? Don't you see how wrong this is? You can't accuse others of being to "orthodox" when you're holding on to a pre-relativistic view of spacetime.

 

[RandallB]

… deify the proponent of the theory. Irrespective of how right he is, this is not the way we should be doing science.

As for the "SR-Simultaneity rule", a google search with that strict phase came up with one hit. This thread. That's a rule which either has another name or has only recently been created.

I would ask for the rule to be stated explicitly and referenced.

That doesn’t make sense:
acknowledging priority on SR to Einstein does not deify him
Why use that for a google search when the scientific term is simply “Simultaneity”

For references look under Wiki : http://en.wikipedia.org/wiki/Relativity_of_simultaneity

Wiki “The relativity of simultaneity is the concept that simultaneity is not absolute, but dependent on the observer.”

The “Not Absolute” part means that any reference frame can establish a grid of Synchronized Clocks but no reference frame can assume that those clocks are marking time increments simultaneously – to do so would establish an absolute simultaneity reference as a benchmark other frames would have to respect.

IMO the only way to gain a reference to an Absolute frame of simultaneity would be to recover the Newtonian concept of Absolute Space and Absolute Time and that is not current conventional thinking by any means.

WRT this twin problem, it means any frame of reference can use the information given at tiime = 0 including the turnaround point defined in one frame; to accurately predict when and where B will turn around in all frames. But no two frames will agree on both where and when A is in any of those frame simultaneous with the times and locations they all agree upon for the B turnaround.

…. non-local simultaneity is meaningless and for the teaching of SR a hindrance.

Quite the contrary; non-local simultaneity, or that Relativity can be considered “Non-Local” is not unreasonable at all; just more advanced than most people get in there understanding.
Not directly hindrance to teaching of SR or GR, but such advanced work could be confusing vs, QM use of “non-local” when those are two separate major branches of science (Cosmic vs Micro) still considered incompatible with each other. Thus I think you will find that rather than use Local Vs. Non-Local in SR or GR it is more common or conventional to use ‘Dependent Background’ vs. ‘Independent Background’.
IMO the case for GR being Background Independent (ie. non-local) grows most directly from the simultaneity principle.
But that is more advanced than this thread calls for; so those that want more on Background Independence ref: Smolin & Perimeter Institute for his book and current papers.

 

[Dale]

Again, all of this unnecessary confusion about simultaneity etc. goes away with the spacetime geometric approach. I see nothing to recommend any other approach over it.

 

[matheinste]

Hello RandalB.

As i said, the comments on non local simultaneity being meaningless were other peoples thoughts and not my own feelings. I think we are having problems communicating so i think i will call it a day. The fault is probably mine. Thanks for your time.

Matheinste.

 

[neopolitan]

I explained simultaneity in SR earlier in this thread.

I am after the "SR-Simultaneity rule", explicited stated. There is a world of difference between "this is how we define simultaneity" and "this is the simultaneity rule". The assumption seems to be that we cannot trangress the simultaneity rule, rather than we use use the simultaneity rule to do something.

Feel free to either state the SR-Simultaneity rule, explicitly, or advise that the rule is like Fleming's left hand rule, a useful device rather than a restriction.

Some people might say that simultaneity is actually a relevant concept in "real world thinking".

Sure it is, but it isn't a very useful concept if it comes saddled with a rule which seems to say "you can't work things out". Compare this to spacetime geometry which actually does allow you to wortk things out. I know which I find more attractive.

Why aren't you answering my question about how much B should say that A has aged right after the turnaround? Do you understand what I said about how answers other than "32.8 years" are problematic?

Instantaneously after "B" has turned around (ie no time elapsed in "B"'s frame, but "B" is now pointing in "A"'s direction and has the beginnings of a velocity, "B" should say that "A" has aged 20 years. Just before "B" turns around (ie no time elapses between this moment and the turnaround, while "B" still has a velocity away from "A"), "B" should say that "A" has aged 20 years. At the turnaround, which we are taking to be instantaneous and ignoring that that is not actually possible, "B" should say that "A" has aged 20 years.

This is all in accordance with the assumptions which I have previously posted.

Stupid? I don't know what you're talking about but this seems like a good time to remind you that the reason we have to consider B's point of view is that the twin paradox is all about explaining why B gets the wrong result for A's age if he just uses the time dilation formula. (The "simultaneity shift" explains why B gets the wrong result and also what the correct result is. Can your argument based on signals from Earth do that?).

"B" will only get the wrong age for "A" if "B" ignores the accelerations, the consequences of which are different inertial frames. The twin paradox is based on the incorrect assumption that we can treat "A" and "B" the same. The incorrectness of this assumption is highlighted if you look closely at the geometry involved.

I am not totally convinced that you have to even consider simultaneity at all in order to prove that the twin paradox is a false paradox. The only reason I am bringing it up is because I disagree very strongly that "B" is forced to calculate that "A" ages 25.6 years during the turn around.

You sound more and more like someone who just refuses to learn the theory he's criticizing.

Well, that is your opinion. Do you agree that orthodoxy is counterproductive (and that that would be equally so if I took the side of an alternative orthodoxy and refused to budge)?

However, in the meantime, you could attach a grid to "A" and make "B" travel within that grid, much as I think granpa referred to. A grid with co-stationary clocks which are synchronised in their rest frame and distance markings. Using that grid you can work out which events are synchronous with which events. Now even if that grid is sort of a privileged frame, the removal of the grid will not remove the underlying physics and destroy our knowledge of what happened when, according to each frame.

If you do the thinking, with such a grid in mind, you can calculate that there is no sudden change in "A"'s clock according to "B".

First you make A's frame a preferred frame, contradicting the principles on which SR was built, and then you describe something using this preferred frame and claim that the description is according to B!? Don't you see how wrong this is? You can't accuse others of being to "orthodox" when you're holding on to a pre-relativistic view of spacetime.

I did say that the grid is a sort of privileged frame, and from what we know of the scenario, it is most logical to attach the grid to "A"'s frame.

I am not sure you even tried to think about the scenario I presented. Possibly because you think that SR is built on the principle that there is no preferred frame.

May I remind you of the postulates of SR?

1. First postulate (principle of relativity)

The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.

2. Second postulate (invariance of c)

Light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body.

http://en.wikipedia.org/wiki/Postulates_of_special_relativity"

I do believe that no preferred frame is a consequence of SR, not a foundation. I don't state this lightly and present Einstein as witness for the defence:

"Physical space and the aether are only different terms for the same thing: fields are physical states of space. If no particular state of motion can be ascribed to the aether, there do not seem to be any grounds for introducing it as an entity of a special sort alongside space."

http://en.wikipedia.org/wiki/Relati...ativity_of_simultaneity_and_.22local_time.22"

He does not say it is impossible, he says it is unnecessary. Being unnecessary does prove anything impossible, or Laplace would have destroyed all religions with a simple phrase.

Note, I am not saying that a preferred frame is likely, necessary, attractive or whatever. I am just saying that the absence of one is categorically not a postulate or foundation of SR.

cheers,

neopolitan

 

[neopolitan]

That doesn’t make sense:
acknowledging priority on SR to Einstein does not deify him
Why use that for a google search when the scientific term is simply “Simultaneity”

For references look under Wiki : http://en.wikipedia.org/wiki/Relativity_of_simultaneity
The “Not Absolute” part means that any reference frame can establish a grid of Synchronized Clocks but no reference frame can assume that those clocks are marking time increments simultaneously – to do so would establish an absolute simultaneity reference as a benchmark other frames would have to respect.

IMO the only way to gain a reference to an Absolute frame of simultaneity would be to recover the Newtonian concept of Absolute Space and Absolute Time and that is not current conventional thinking by any means.

WRT this twin problem, it means any frame of reference can use the information given at tiime = 0 including the turnaround point defined in one frame; to accurately predict when and where B will turn around in all frames. But no two frames will agree on both where and when A is in any of those frame simultaneous with the times and locations they all agree upon for the B turnaround.

For most of this, check the post I addressed to Fredrik. I don't think simultaneity is required at all to prove that the twin paradox is not a real paradox. It seems to confuse more than elucidate. My primary concern is merely that I think that, in a real universe, "B" will not assume or calculate that "A" ages 25.6 years at turnaround. "B" has sufficient information and knowledge to not make that mistake, if he uses the spacetime geometric approach, rather than trying to work out which events at "A" are simultaneous with events in the spaceship and refusing to ignore the accelerations which will alter "B"'s spacetime trajectory.

I stress that there is no reference to an "SR-Simultaneity rule", and even the link you gave me, thanks, only refers to Poincare's "new rule". I rephrase the question I posed to Fredrik, is this rule supposed to be descriptive or prescriptive? Is it how you can determine simultaneity or is it how you must determine simultaneity?

cheers,

neopolitan

PS Oh and about the deifying Einstein thing, do you agree that now that the theory is out there, what Einstein thought is immaterial other than for historical purposes? Einstein's thoughts about how to work out simultaneity don't actually affect whether you can work out simultaneity or not, do they?

I was just pointing out that you should stick to the facts of the case rather than ruminating on what Einstein thought, because you could be accused (in all likelihood unfairly) of appeals to authority.

And imagine how awful it would be in discussions about SR were based on WDET (What Did Einstein Think?)

 

[Fredrik]

Again, all of this unnecessary confusion about simultaneity etc. goes away with the spacetime geometric approach. I see nothing to recommend any other approach over it.

I still don't really agree with that. It's the best and easiest way to find the final ages of the twins, but I still don't see how it explains why it's wrong to just use the time dilation formula on the two straight pieces of B's path.

The post you made after the last time I said that doesn't really address that issue. You just pointed out that the paths aren't equivalent. That, plus the fact that the approach you prefer tells us the correct final ages, is all that we need to conclude that it's wrong to use the time dilation formula like that, but it doesn't tell us why it's wrong.

That's why I still prefer the standard resolution and keep posting it in these threads. That doesn't mean I think the other approach is irrelevant or anything like that. I think every student should learn both.

One more comment that isn't really important: The name "spacetime geometric approach" is a bit inappropriate. The standard resolution is geometric too. It just doesn't mention the metric explicitly. So maybe "the coordinate independent approach" would be a better name.

 

[atyy]

The post you made after the last time I said that doesn't really address that issue. You just pointed out that the paths aren't equivalent. That, plus the fact that the approach you prefer tells us the correct final ages, is all that we need to conclude that it's wrong to use the time dilation formula like that, but it doesn't tell us why it's wrong.

I have no problems with your solution, and its clear from your picture that A ages normally according to himself. But I also like neapolitan's point that everyone ages according to his own (proper) time.

Anyway, the "look at the picture and see the bend" resolution becomes intuitive for me once I'm told that the time dilation formula (in its simplest form) only works with inertial frames, which are the "preferred frames" of special relativity. There is no single preferred frame only because there are lots of preferred frames, and we have to prefer all of them equally. So A is clearly a preferred frame, and B is clearly not a preferred frame! Of course, since A and B are truly different, we could call B the preferred frame. The point is that if A and B were both inertial frames, then we could not prefer them unequally. But A is inertial, and B is not, so we can prefer one of them unequally.

 

[granpa]

I don't think that neopolitan is so much talking about a preferred frame as a convenient frame. if everyone who blasts off in a rocket ship inevitably returns back to Earth then Earth frame is a very convenient frame to calculate things in. a grid of mile markers all synchronized with Earth would indeed be very convenient.

but if everyone on Earth blasted off and went to andromeda galaxy and stayed there then we would all need a new frame.

anyway the whole point of the twins paradox is that the 2 frames arent equal. the accelerating frame is not equal to the stationary frame so in that sense the Earth frame is more 'convenient' than the rockets frame.

 

[Dale]

The post you made after the last time I said that doesn't really address that issue. You just pointed out that the paths aren't equivalent. That, plus the fact that the approach you prefer tells us the correct final ages, is all that we need to conclude that it's wrong to use the time dilation formula like that, but it doesn't tell us why it's wrong.

Sorry, I guess that I thought that was clear. There are three "natural" inertial reference frames in the standard twin paradox. If you draw the spacetime diagram in any of the three frames you see that the path of the traveling twin is bent. You can also use the time dilation formula in any of those three inertial frames to determine the ages as you would expect. The use of the time dilation formula on the straight pieces of the traveler's path is not incorrect, the treatment of the traveler's rest frame as an inertial frame is the incorrect part. This makes it clear IMO.

The standard resolution does not introduce the Minkowski geometry of spacetime, so it is not a spacetime geometric approach.

 

[Fredrik]

Sorry, I guess that I thought that was clear. There are three "natural" inertial reference frames in the standard twin paradox. If you draw the spacetime diagram in any of the three frames you see that the path of the traveling twin is bent. You can also use the time dilation formula in any of those three inertial frames to determine the ages as you would expect. The use of the time dilation formula on the straight pieces of the traveler's path is not incorrect, the treatment of the traveler's rest frame as an inertial frame is the incorrect part. This makes it clear IMO.

In my opinion, this only proves that something funny happens at the turnaround event. It doesn't say what that funny thing is. I'll be more specific:

You can also use the time dilation formula in any of those three inertial frames to determine the ages as you would expect. The use of the time dilation formula on the straight pieces of the traveler's path is not incorrect, the treatment of the traveler's rest frame as an inertial frame is the incorrect part.

So let's do that. First half of the trip: A's aging rate is 60% of B's, so B ages 12 years and A 7.2 years. Second half of the trip: Same thing. Add up the results: B has aged 24 years, and A 14.4 years.

Note that I didn't have to assume that "B's point of view" is an inertial frame to get this incorrect result. The incorrect part is the assumption that A has aged 7.2 years at the beginning of the second half of the trip, when in fact he has aged 32.8 years (in the inertial frame associated with the return trip).

Yes, you can figure that out using your approach too, but you have to find the correct result first and work backwards from there. First you have to note that the correct final ages are A=40, B=24. Then you note that A only ages 7.2 years on the second half of the trip, and that this implies that he must have aged 32.8 years at the start of the second half. Then you note that he had aged 7.2 years at the end of the first half, and that this implies that his age somehow "jumped" 25.6 years at the turnaround. But we still can't see why his age made a jump. The only way to do that is to look at the simultaneity lines of the relevant inertial frames, but now we're back in the realm of the "standard" resolution.

 

[Dale]

So let's do that. First half of the trip: A's aging rate is 60% of B's, so B ages 12 years and A 7.2 years. Second half of the trip: Same thing. Add up the results: B has aged 24 years, and A 14.4 years.

Note that I didn't have to assume that "B's point of view" is an inertial frame to get this incorrect result.

This is wrong, you did, in fact, consider B's rest frame to be an inertial frame.

There is only one inertial frame where A's aging rate is 60% of B's during the first leg of the trip. In that frame, the second leg of the trip is not the same thing at all, which is quite obvious in the spacetime geometric approach. During the second leg of the trip B is moving at about .98 c in the inertial frame, so B's time dilation is much greater than A's which correctly accounts for the difference in age at the reunion event.

There is no discontinuity in anyone's age, and you don't have to "work backwards" from the answer. I can work the example quantitatively in all three inertial frames if you wish.

 

[RandallB]

For most of this, check the post I addressed to Fredrik. I don't think simultaneity is required at all to prove that the twin paradox is not a real paradox. It seems to confuse more than elucidate. My primary concern is merely that I think that, in a real universe, "B" will not assume or calculate that "A" ages 25.6 years at turnaround. "B" has sufficient information and knowledge to not make that mistake, if he uses the spacetime geometric approach, rather than trying to work out which events at "A" are simultaneous with events in the spaceship and refusing to ignore the accelerations which will alter "B"'s spacetime trajectory.

I stress that there is no reference to an "SR-Simultaneity rule", and even the link you gave me,

I gave you the references you asked for (ten of them) at the bottom of that Wiki link. Obviously you have not looked at any of them. And as I already said; look for “Simultaneity” to learn more about it, not some phrase "SR-Simultaneity rule" – your insisting on that is just being belligerent and if you keep that attitude up we have nothing further to discuss.

You think you can “prove that the twin paradox is not a real paradox” without applying Einstein Simultaneity.
Then show us the resolution to the problem under discussion:
When reaching the turnaround point B can directly observe clocks at the turn around point for the A reference frame, the return trip Frame and any other inertial frame, including those inertial speeds that may be crossed by short interval acceleration B may use to change inertial frame. None of these clock times (including B Proper time) change if B stops on the A frame or transfers to the Return Trip frame.

So the paradox here is what “really” is B Proper Time at the moment of turnaround compared to A Proper Time. – Is it More(fast), Less(slow), or the Same? In a normal real Reality in cannot actually BE all three it must be one of those.

IMO you cannot resolve this paradox without applying the rules of Simultaneity.
So if as you claim this “paradox is not a real paradox” show us the resolution and the rule(s) used to solve it. Please include an appropriate reference for rule(s) used.

 

[neopolitan]

Yes, you can figure that out using your approach too, but you have to find the correct result first and work backwards from there. First you have to note that the correct final ages are A=40, B=24. Then you note that A only ages 7.2 years on the second half of the trip, and that this implies that he must have aged 32.8 years at the start of the second half. Then you note that he had aged 7.2 years at the end of the first half, and that this implies that his age somehow "jumped" 25.6 years at the turnaround. But we still can't see why his age made a jump. The only way to do that is to look at the simultaneity lines of the relevant inertial frames, but now we're back in the realm of the "standard" resolution.

I hesitate at the keyboard before typing this, but ... this is if and only if "B" ignores the accelerations.

"B" knows that by not accelerating, "A" will take a route through spacetime which has a higher time component than anyone who is initially stationary wrt to "A", accelerates to travel away and then accelerates back towards "A".

"B" can therefore work out the correct result from first principles, without having rely on simultaneity at all.

One thing that is a little confusing about all of the bent path diagrams I have seen is that they are all vertical, implying that "A" is absolutely stationary. This just isn't the case, "A" is relatively stationary, in that "A"'s inertial frame does not change.

What you can do, however, is swing "A"'s path through spacetime around in any direction, subject to the limitation that "A"'s inertial velocity must be subluminal (ie "A" must have valid trajectory through spacetime), and the bent line will still trace a path that amounts to 24 years shipboard time, according to "B" and 32 light years traversed, according to "A".

A third observer, just watching "A" and "B" may report that "A" has a total path through spacetime of 40 units, but it won't be 40 and zero, it may be 28.28 and 28.28 (and this nonzero spatial component will result in a nonvertical path through spacetime for "A").

The bent line representing "B"'s journey, according to this third observer, will also amount to 40 units of spacetime, the distribution will pretty much rely on how "B" moves relative to "A"'s motion (according to the third observer).

Lets use that third observer to prove that the twin paradox is a false paradox.

We have two extreme options for how "B" can move relative to "A"'s motion, according to our third observer - orthogonally or parallel.

If the motion is orthogonal, according to the third observer, then "B"'s path will amount to additional velocity, all the way. This orthogonal motion by "B" is equivalent to "A" having been absolutely stationary (in other words, stationary according to our third observer).

If the motion is parallel, then "B"'s path will amount to reduced velocity (or overshoot) in one direction and increased velocity in the other direction - according to our third observer.

There is no subluminal velocity which "B" can maintain which will not result "B" maintaining, overall, a greater speed than "A" - (remember that I stated that "A"'s inertial velocity is limited to subluminal ... if this restriction is removed then all bets are off).

Any combination of the two will also result in a higher speed for "B" relative to "A" over the entirety of "B"'s journey.

Now note that I did not put any restrictions on the third observer. That is because the third observer can be anyone, with any valid trajectory through spacetime.

If the fact that "B" must have a greater speed than "A", over the entirety of the journey is true for ANYONE, it must be true for EVERYONE, including "A" and even, with the application of a bit of postmodern common sense, "B".

Twin paradox shown to be a false paradox, with no requirement to refer to simultaneity.

cheers,

neopolitan

(And if you feel like saying "You just can't do that!" ... come prepared with an argument as to why I can't. If I can do that, please weigh in and give a little support. Thanks.)

 

[neopolitan]

I gave you the references you asked for (ten of them) at the bottom of that Wiki link. Obviously you have not looked at any of them. And as I already said; look for “Simultaneity” to learn more about it, not some phrase "SR-Simultaneity rule" – your insisting on that is just being belligerent and if you keep that attitude up we have nothing further to discuss.

You are making the assumption that because I do not come to the same conclusions as you that I don't know or understand anything about SR and simultaneity. Is that not a little arrogant?

Please, can you at least answer one of my questions?

Is the SR-Simultaneity rule that you referred to descriptive or prescriptive? From all my readings it is descriptive, but you seem to be using it prescriptively. Are you, or is that a misreading on my part?

IMO you cannot resolve this paradox without applying the rules of Simultaneity.
So if as you claim this “paradox is not a real paradox” show us the resolution and the rule(s) used to solve it. Please include an appropriate reference for rule(s) used.

Done. I don't have "rules". There are merely logical steps. If I have made a huge misstep, I am sure you will be able to point it out to me. Please, just go for the substantial errors, it is late again where I am so typos may have got past me. If there is a step which you don't understand, please ask for clarification.

cheers,

neopolitan

 

[RandallB]

Done. I don't have "rules". There are merely logical steps.

NOT done:
the explanation you gave Fredrik did use the rules of Simultaneity I gave in post #76 – You just don’t understand Simultaneity to see that.
You may call that descriptive or prescriptive I don’t care.
Your logical steps did not address the question I asked:
Prescribe or describe your resolution to the part of the twin paradox I stated:

What “really” is B Proper Time at the moment of turnaround compared to A Proper Time. – Is it More(fast), Less(slow), or the Same?

And you ARE ALLOWED to say "You just can't do that!" because you cannot.
Just explain why no one can answer that question, including Minkowski spacetime geometry

Unless of course you think you can pick one of the three More, Less, or the Same.
Just show the "logical steps" that let you pick and define the correct answer.

 

[matheinste]

Hello neopolitan.

Quote:-

---One thing that is a little confusing about all of the bent path diagrams I have seen is that they are all vertical, implying that "A" is absolutely stationary. This just isn't the case, "A" is relatively stationary, in that "A"'s inertial frame does not change.----

The fact that A's path is vertical i.e. along the time axis means that A is stationary with respect to the coordinates of the spacetime diagram. In other words the analysis is taking place from the point of view of A. This is basic.

Matheinste.

 

[Fredrik]

"B" knows that by not accelerating, "A" will take a route through spacetime which has a higher time component than anyone who is initially stationary wrt to "A", accelerates to travel away and then accelerates back towards "A".

"B" can therefore work out the correct result from first principles, without having rely on simultaneity at all.

Yes, that's the point of the approach that DaleSpam is advocating. The correct final ages follow immediately from the postulate that what a physical clock measures is the integral of \sqrt{-g_{\mu\nu} dx^\mu dx^\nu} along the curve in Minkowski space that represents the clock's motion.

One thing that is a little confusing about all of the bent path diagrams I have seen is that they are all vertical, implying that "A" is absolutely stationary. This just isn't the case, "A" is relatively stationary, in that "A"'s inertial frame does not change.

It doesn't imply that A is absolutely stationary. Any object that moves with a constant velocity is stationary in some inertial frame, and in this case it's simply convenient to draw the diagram using the inertial frame in which A is stationary.

What you can do, however, is swing "A"'s path through spacetime around in any direction,

I didn't read all the details of your argument, but a quick glance was enough for me to see that it's pointless. It's obvious that a third frame isn't going to tell you anything that you can't see by considering A's rest frame, and if you're not going to talk about simultaneity, your best option is to use DaleSpam's approach anyway.

 

[Fredrik]

This is wrong, you did, in fact, consider B's rest frame to be an inertial frame.

No, I didn't. I just used one frame for the first half and another for the second half.

you don't have to "work backwards" from the answer. I can work the example quantitatively in all three inertial frames if you wish.

You still seem to be missing the point. Maybe I'm missing your point too, but you're definitely missing mine. I can do those calculations too, obviously, but our task isn't to find multiple ways to calculate the correct final ages. It's to explain what's wrong with the incorrect solution. Your approach can only tell us that there's something wrong with it (by telling us the correct final answer), but it doesn't tell us why it's wrong. OK, it points out that there's a bend in B's path, but it doesn't explain why that matters. And you are going to have to work backwards from the answer to find the magnitude of the error (i.e. 25.6 years), unless you do what I did, which is to consider the simultaneity lines of the two inertial frames associated with B's motion.

 

[RandallB]

No, I didn't. I just used one frame for the first half and another for the second half.

But nothing is SR grants you permission to do that.
SR requires that you solve any problem like this wrt just one reference frame. Not two of your choice.
Pick the right two and you can make almost any weird thing appear in that approch including Backwards Causality.

That includes the “two straight pieces of B's path” if by that you mean the outbound reference frame and returning reference frame—those are two different frames.

SR requires you pick one of those frames or the A frame or any other fourth frame your care to make up. But your must work the problem all the way though using just one frame and translate all time and locations for other frames based on the 0 starting position and Lorentz transforms from the one reference frame. No matter what frame you pick you will get the same result on B returning to A.

 

[Dale]

No, I didn't. I just used one frame for the first half and another for the second half.

That is practically the definition of a non-inertial reference frame!

our task isn't to find multiple ways to calculate the correct final ages. It's to explain what's wrong with the incorrect solution. Your approach can only tell us that there's something wrong with it (by telling us the correct final answer), but it doesn't tell us why it's wrong. OK, it points out that there's a bend in B's path, but it doesn't explain why that matters.

Don't you see? The fact that there is a bend in B's path is the reason why the other approach is wrong. In Minkowski geometry the longest timelike interval between two points is a straight line, so a path with a bend in it is always shorter than a straight path. So the existence of a bend in a path is an important geometrical feature. The other approach is wrong geometrically because it straightens out a bent path and bends a straight path. In physical terms it is a non-inertial reference frame.

 

[neopolitan]

An early morning postscript to my post from last night.

The third observer will not work out that "A" and "B" have cashed in 40 units of spacetime, because according this observer, "A" will have converted a certain amount of spacetime into spatial translation. Therefore the amount of spacetime cashed in, according to the third observer, will be a lower value given by

sqrt ('time elapsed according to "A"' squared plus 'distance traveled according to the third observer')

and this value will be the same as the time elapsed according to the third observer.

However, the third observer will still work out that "A" and "B" cash in the same amount of spacetime overall, and "A" will have cashed in more as time, due to the fact that "A"'s single inertial path is the shortest between the start and finish points.

And, yes, this is the same solution as DaleSpam's but just worded differently.

As for Randall and asking a simultaneity based question about the relationship between "A" Proper time and "B" Proper time at the turnaround point ... is that really the crux of the twin paradox as generally stated?

I thought it was that one twin ends up older than the other twin. If the crux of the twin paradox is that you run into problems with working out Proper times at the turnaround, I would take this as a reason to avoid using simultaneity altogether when it comes to solving these sorts of problems and use the approach advocated by DaleSpam (and, to a more clumsy extent, myself). After all, Dale and I don't need to know Proper time to show that there really isn't a problem. You just shift the problem.

cheers,

neopolitan

 

[atyy]

If the crux of the twin paradox is that you run into problems with working out Proper times at the turnaround

Everyone ages according to the integral of his own proper time, under the assumption that he is an ideal clock whose time dilation depends on velocity and not acceleration. We know this is true for individual particles, and even for atomic clocks.

In an inertial frame, the observer A at spatial coordinate x=0 has proper time s_A which is the global time coordinate t of the frame ds_A²=dt². For any other observer B, accelerating or not, ds_B²=dt²-dx².

If we take the "point of view" of B, we get new global space and time coordinates p,q. By definition of taking his point of view, ds_B²=dq². Since this is not an inertial frame, the proper time of any other observer A is generally not ds_A²=dq²-dp², unless B is an inertial observer. So we can get the twin paradox even by integrating the proper time simply by using the wrong proper time formula.

In the derivation of the time dilation formula, there is a critical part where the proper time of one observer is identified with the global time axis of an inertial frame, and the proper time of the other observer is identified with the global time axis of another inertial frame. The final part of the derivation comes by noting that two inertial frames are related by a Lorentz transformation. So the presentation of the twin paradox using the time dilation formula is about proper times anyway.

It is tempting for me to want something to happen to B's clock at the bend, since surely it is his acceleration that is absolute. If B uses an ideal clock or an atomic clock, it will not be affected by acceleration. The most common non-ideal clock is the pendulum, but that will not work in outer space with no gravity (SR has trouble with gravity anyway). I guess I should make B use a quantum gravity clock?

 

[Antenna Guy]

It is tempting for me to want something to happen to B's clock at the bend, since surely it is his acceleration that is absolute.

Surely contracted meters per dilated seconds squared cannot be considered "absolute".

Regards,

Bill

 

[atyy]

Surely contracted meters per dilated seconds squared cannot be considered "absolute".

I just mean B will know that he is not an inertial observer.

 

[Al68]

Everyone ages according to the integral of his own proper time, under the assumption that he is an ideal clock whose time dilation depends on velocity and not acceleration. We know this is true for individual particles, and even for atomic clocks.

Do we? Or are you just talking about SR time dilation, not GR time dilation? It seems to me that if we're talking about an accelerated observer, we have to consider GR time dilation as well (equivalence principle).

Of course this would lead to crazy conclusions like Earth's clock running faster than the ship's clock (from the ship's POV) during the acceleration. Oh, wait...