Twin paradox explained for laymen

[MikeLizzi]

Nobody is claiming that acceleration is not "relevant" simply because we idealize the turnaround to be instantaneous. In fact, the entire point of the other thread I linked to is that (as both my posts in the thread and the papers referenced show), even if we idealize the turnaround time as negligibly short for the traveling twin, we cannot ignore the time elapsed during the turnaround for the stay-at-home twin.
Nobody is claiming that the turnaround does not exist or can be neglected in all respects simply because we idealize it as being instantaneous. You are responding to a straw man.

I am beginning to think you are trolling.

So you think I am trolling. Whatever that means, I will stop posting. You must know that you and the other senior members of this forum have a communications problem with the sincere laymen who visit here. All we have is the basic SR for sophomore college. I took the course. In frustration, I bought 7 different textbooks entitled Modern Physics. Every one of them has a few chapters on SR and every one of them reads the same (I'm thinking plagiarism). Most people in this world are not capable of learning anything beyond that. Perhaps the Twins Paradox should not even be addressed at that level.

 

[PeterDonis]

You must know that you and the other senior members of this forum have a communications problem with the sincere laymen who visit here.

Consider this exchange:

Are you saying the traditional method for solving the twins paradox in textbooks is wrong?

What do you think the "traditional method" is? Can you give a reference?

No answer from you.

I don't see any communication problem on my side. I see one on yours.

 

[PeterDonis]

In frustration, I bought 7 different textbooks entitled Modern Physics.

Have you read the Usenet Physics FAQ article that was linked to earlier in this thread? I'll even link to it again (the table of contents this time):

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

 

[PeterDonis]

So you think I am trolling. Whatever that means

It means that you don't seem to be interested in actual constructive conversation, but simply in grinding whatever axe it is you have to grind.

 

[FactChecker]

The crucial point here is that you have to distinguish the frame-dependent concept of "time dilation", which by itself cannot be used to make accurate predictions about differential aging,
...
No. What appears is what I described as seen above. There is no direct observable that corresponds to the frame-dependent calculated time dilation. (There can't be, because frame-dependent quantities can never be direct observables.)

Suppose I am on Earth and I have communications with a friend far away who is not moving in my IRF. We have (EDIT) Einstein-synchronized our clocks and they are running at the same rate. Now suppose a high-speed traveler goes past me and continues inertially on toward my friend. The traveler is in an IRF with a high velocity compared to me and my friend. Suppose I record the traveler's clock time and my clock time when he passes me and my friend does the same. Is it not true that the traveler's clock times will indicate a shorter (EDIT)elapsed proper travel time between me and my friend than the difference between my friend's and my clock times?

 

[PeterDonis]

Is it not true that the traveler's clock times will indicate a shorter travel interval between me and my friend than the difference between my friend's and my clock times?

Yes.

 

[FactChecker]

Yes.

Ha! Sorry. Is that "Yes it is true." or "Yes it is not true."?
If it is not true, there must be something that I do not understand.
If it is true, isn't that observing time dilation that should be matched by slower aging?

 

[PeterDonis]

Is that "Yes it is true." or "Yes it is not true."?

This statement...

the traveler's clock times will indicate a shorter travel interval between me and my friend than the difference between my friend's and my clock times

...is true.

If it is true, isn't that observing time dilation

No. It could be taken to be observing a combination of time dilation and relativity of simultaneity (and, strictly speaking, length contraction). Or it could be taken as observing a consequence of Minkowski spacetime geometry. But it cannot be taken as an observation of time dilation alone.

 

[FactChecker]

No. It could be taken to be observing a combination of time dilation and relativity of simultaneity (and, strictly speaking, length contraction). Or it could be taken as observing a consequence of Minkowski spacetime geometry. But it cannot be taken as an observation of time dilation alone.

I don't see any issue of simultaneity (EDITED TO CLARIFY: here. In this scenario, there are no spacelike separated events that are hypothesized to be simultaneous.) There is never a question of whether two events, which are spatially separated are simultaneous. It seems to me that it is just comparing the elapsed time between the traveler being at your position and his being at your friend's position. It's just an elapsed time comparison. The issues of distance and simultaneity are not involved.

 

[PeterDonis]

There is never a question of whether two events, which are spatially separated are simultaneous

I don't know what you mean. Simultaneity is frame-dependent, so until you've picked a frame there most certainly is a question of whether two spacelike separated events are simultaneous. And once you've picked a frame the answer, whether it's yes or no, depends on the frame you picked; it's not an invariant.

It seems to me that it is just comparing the elapsed time between the traveler being at your position and his being at your friend's position.

Alex's elapsed time between the two events is directly observable since it's just the difference of the two readings on his clock.

There is no such thing as Bob's elapsed time between the two events, or Alice's elapsed time between the two events, because neither Bob or Alice is present at both events. Each of them is only present at one.

If you are going to call the difference between the reading on Bob's clock that Alex sees when he passes Bob, and the reading on Alice's clock that Alex sees when he passes Alice, the "elapsed time" for Alice or Bob, that statement requires a choice of simultaneity convention--namely, the simultaneity convention of the frame in which Bob and Alice are at rest. So any such statement is frame-dependent, not an invariant.

 

[PeroK]

Assume, the traveling twin just came back from Alpha Centauri and both twins are now sitting in a room on Earth with a constant distance of 1.5 meters from each other in their common restframe. Then their age difference is still frame-dependent.

Does the age difference in their common rest frame have no physical significance?

Let's focus on invariant quantities: quantities that are the same in all (inertial) reference frames. If, by an abstraction, we consider two clocks to be at the same spacetime location, then the difference in their readings is an invariant quantity. All reference frames will agree.

In the twin paradox, it's normally considered that when the traveller arrives back on Earth the difference in their ages can be measured and is a meaningful, invariant quantity.

Of course, you can argue that two clocks (or people) are never at exactly the same spacetime location. Einstein actually covers this in the 1905 paper in a footnote:

We shall not here discuss the inexactitude which lurks in the concept of simultaneity of two events at approximately the same place, which can only be removed by an abstraction.

One thing you could do is put some sort of bound on the lack of simultaneity between two events: two nearby clocks recording some time. Let's assume they are $1m$ apart and synchronised in their mutual rest frame. The maximum loss of synchronisation is $\pm \frac{1m}{c}$. Which can be taken to negligible in terms of the difference involving years.

Now, if we take the intermediate stop at Alpha Centauri, four light years from Earth. In the Earth/AC rest frame we have synchronised clocks at Earth and AC reading 5 years. And we have the traveller's clock at AC reading 3 years. Depending on the reference frame we have maximum loss of synchronisation of the Earth/AC clocks of $\pm 4$ years. This means that there is no absolute sense that the Earth twin is older than the traveller "when" the traveller reaches AC. When the traveller reaches Alpha Centauri, the Earth twin's age is between 1 and 9 years, depending on the reference frame.

This is where, again, the relativity of simultaneity is critical and we cannot do SR with a "time-dilation-is-all-there-is" approach.

Finally, to emphasise this point, let's analyse the outward leg in a frame in which the Earth and AC are moving at $0.9c$, with the Earth "leading". In this frame when the traveller accelerates away from Earth he/she slows down and is less time dilated for the entire outward journey. In this frame, the traveller is older than the Earth twin when he/she reaches AC. If it makes any sense to say that.

However, if you study the return leg in this frame as well, then the traveller is more time dilated on the return leg and when they return to Earth you get the same invariant ageing (elapsed proper time) of 10 years and 6 years respectively.

 

[SSequence]

I haven't read the thread, but here is something that helped me understand [or at least, be reasonably confident that many of the basic kinematical SR phenomenon are not that tricky to "explain" at least].

Few years ago I found a book that explains all the basic kinematics phenomenon using space-time diagrams [twin paradox was also included in later chapters]. I spent about two weeks studying it. The book didn't use any calculation or equations, but it was quite clear that all the equations [related to specific scenario] were not difficult to derive by "reading-off" from the space-time diagrams describing the scenario. I intended to make notes deriving all the equations but I didn't get around to doing that. I don't remember things fully well now since I didn't get back to reading about this topic again. But anyway, I found that way of understanding quite helpful since it made things clearer to me.

Another indirect effect, I think, of this was that it becomes clear [sub-conciously perhaps, since one is thinking of them in terms of points on space-time diagrams] that events which are separated only in time (and not in space) OR only in space (but not in time) for one intertial observer need not be for another intertial observer. Now I am sure that I lack good intuition on number of things. But to me being able to have a basic framework or principle way of explaining an arbitrary scenario makes things more comfortable (so to speak).

But anyway, my point is that this particular approach was easier for me than others. So perhaps, others might find it useful.

P.S.
I wonder if there is a similar "easy" approach to understand the effect of relativity on dynamics, electromagnetism etc. [since I am completely uninformed about this].

And what about general relativity? Is there also an "easy" way to explain the principle idea behind it or not (similar to space-time diagrams in SR)? Anyway, I presume it would still require fair amount of mathematical background on certain specific topics (which I don't have I think).

 

[vanhees71]

I don't know, why some people cannot accept that the physical observables are always covariant quantities. Arbitrary spacetime coordinates have no other a-priori meaning than specifying the location of an event in spacetime. What is measurable is always a quantity which is independent of the choice of coordinates as well in SR as in GR, i.e., it's a tensor (including scalars and vectors).

The most simple example is time. It's defined for any "point-like observer" as his proper time along the (necessarily) time-like world line. That's it.

The twin paradox is thus simply comparing the proper times of two such observers moving along their time-like worldlines from two given points. By definition they synchronize their clocks at the starting point, i.e., setting $\tau_1=\tau_2=0$ at this point and then compare their clocks after meeting again at the final event, and there in general they realize that their clocks are no longer synchronized, i.e., $\tau_1 \neq \tau_2$.

Thinking in geometric terms, it's not surprising since the length of a curve between two points depends on the curve and is of course in general different for different curves.

This of course is based that you measure time with a clock that is not in any way influenced by (maybe very strong) inertial or gravitational forces. One such clock, where it has also been tested under extremely high accelerations, is the life time of elementary particles (like muons) or nuclei. There have been made high-accuracy life-time measurements in storage rings to the effect being in agreement with the clock paradox, i.e., that the lifetime is given by the time-dilation effect of their proper lifetime which is the time measured with a clock comoving with the particle (i.e., the time in the particle's rest frame).

http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html#Clock_Hypothesis

 

[A.T.]

Nobody is claiming that the turnaround does not exist or can be neglected in all respects simply because we idealize it as being instantaneous.

This is key, but often ignored, leading to misconceptions. In my opinion, the instantaneous turnaround idealization simplifies some of the math, but doesn't necessarily help with the intuitive understanding.

 

[vanhees71]

But the "sudden-turnaround approximation" is also simple to interpret drawing a Minkowski diagram. Many students seem to like Minkowski diagrams. So maybe it helps also here? See, e.g., this article in "The Physics Teacher":

https://doi.org/10.1119/1.4947152

 

[FactChecker]

I don't know what you mean. Simultaneity is frame-dependent, so until you've picked a frame there most certainly is a question of whether two spacelike separated events are simultaneous.

I agree. But in this scenario there are no spacelike separated events that are hypothesized to be simultaneous. So that issue is not relevant in my question. I have edited my post to clarify.

There is no such thing as Bob's elapsed time between the two events, or Alice's elapsed time between the two events, because neither Bob or Alice is present at both events. Each of them is only present at one.

The relatively traveling observer is present at both events. The "stationary" observer (I called him "you") is communicating with a friend in the same IRF who observes the event of the traveler passing. Your clock and your friend's clock have been Einstein synchronized.

 

[Sagittarius A-Star]

Potential Energy formula? What's the purpose for introducing the potential energy formula into this thread?

In this thread, a discussion is ongoing about the potential in a certain "hight" in an accelerated frame and it's relation to potential energy and clock tick-rate of a fast moving mass (for example a twin), relative to one at rest at "hight" = Zero.

Also, papers are discussed, that calculate the tick-rate of the "stationary" twin's watch in the accelerated rest frame of the "travelling" twin while "turnaround". In that frame, the tick-rate of the "stationary" twin's watch is much faster than that of the "travelling" twin's watch.

In accelerated frames, different formulas for (frame-dependent) tick-rates apply than in inertial frames. That is one way to prove the asymmetry of the twin scenario, but not the only one.

 

[jbriggs444]

I don't see any issue of simultaneity (EDITED TO CLARIFY: here. In this scenario, there are no spacelike separated events that are hypothesized to be simultaneous.) There is never a question of whether two events, which are spatially separated are simultaneous. It seems to me that it is just comparing the elapsed time between the traveler being at your position and his being at your friend's position. It's just an elapsed time comparison. The issues of distance and simultaneity are not involved.

This scenario was introduced as an example of time dilation. Time dilation involves a comparison between the coordinate time between two events and the elapsed proper time between the same two events. There are most certainly issues of simultaneity involved.

 

[FactChecker]

This scenario was introduced as an example of time dilation. Time dilation involves a comparison between the coordinate time between two events and the elapsed proper time between the same two events. There are most certainly issues of simultaneity involved.

The scenario is that the traveler measures his elapsed proper time between his passing point A and B (both A and B in another IRF with Einstein synchronized clocks, $A_c$ at A and $B_c$ at B). The clock times, of $A_c$ at A and $B_c$ at B are recorded when the traveler passes. Then the traveler's elapsed proper time is compared to $A_c - B_c$. I do not see any issues of distance or simultaneity in this. Am I wrong?

 

[jbriggs444]

The scenario is that the traveler measures his elapsed proper time between his passing point A and B (both A and B in another IRF with Einstein synchronized clocks, $A_c$ at A and $B_c$ at B). The clock times, if $A_c$ at A and $B_c$ at B are recorded when the traveler passes. Then the traveler's elapsed proper time is compared to $A_c - B_c$. I do not see any issues of distance or simultaneity in this. Am I wrong?

You said "Einstein synchronized clocks". That is a simultaneity convention. It does not make the scenario ill-defined. But it does mean that it involves a simultaneity convention.

 

[FactChecker]

You said "Einstein synchronized clocks". That is a simultaneity convention. It does not make the scenario ill-defined. But it does mean that it involves a simultaneity convention.

Are you saying that it is not an accepted form of synchronization within a IRF?

 

[jbriggs444]

Are you saying that it is not an accepted form of synchronization within a IRF?

It is one possible standard for synchronized clocks. Not the only one.

 

[vanhees71]

Well, it's the usual way to synchronize clocks in a global IRF of special relativity, underlying also the construction of Minkowski diagrams. In this way the clocks of all observers being at rest wrt. this global IRF show the same proper time, which is chosen as the "coordinate time" using "Galilean coordinates". It's the most natural choice, I'd say.

 

[FactChecker]

It is one possible standard for synchronized clocks. Not the only one.

I guess then the question is whether the different synchronization methods would give answers so significantly different that they would change the answer to my scenario in Post #215.

 

[jbriggs444]

I guess then the question is whether the different synchronization methods would give answers so significantly different that they would change the answer to my scenario in Post #215.

Let us review #215... What claim is made in that post?

Suppose I am on Earth and I have communications with a friend far away who is not moving in my IRF. We have (EDIT) Einstein-synchronized our clocks and they are running at the same rate. Now suppose a high-speed traveler goes past me and continues inertially on toward my friend. The traveler is in an IRF with a high velocity compared to me and my friend. Suppose I record the traveler's clock time and my clock time when he passes me and my friend does the same. Is it not true that the traveler's clock times will indicate a shorter (EDIT)elapsed proper travel time between me and my friend than the difference between my friend's and my clock times?

You mention "the difference between my friend's and my clock times". You have invoked a synchronization convention to compute this difference. The convention that you have invoked is Einstein synchronization.

Yes, it is true that the traveler's clock times will indicate a shorter elapsed proper time between the two passing events than the difference between the times that you and your friend record for the same two passing events.

Now you ask whether a different synchronization method would give a different result. Yes, of course. If, for instance, you and your friend discarded Einstein synchronization and instead adopted a convention that your friend uses whatever clock reading he is currently receiving in a television picture of your clock then the difference between your reported passing times in this scenario may be quite different.

 

[vanhees71]

The important point is that if you discuss the twin paradox hat is compared are physical quantities, which are independent of any choice of coordinates or reference frames. It's the proper time of the two twins between to events, where they compare their clock readings. The proper time is uniquely defined as the time in an observer's rest frame, which is the frame that is distinguished from all others by the physical situation, concerning this specific "object", i.e., this specific observer in this case. That's why you can measure this time in the real world in a well-defined and unambigous way.

E.g., one real-world version, well known from many textbooks as an example, is the lifetime of muons being produced in the atmosphere. Way more muons are reaching the ground than you expect if using Newton's notion of time, and the reason is, as seen from the point of view of us, as earth-bound observers (for this purpose we can neglect gravity and the acceleration of the Earth and take the reference frame of an Earth-bound observer as an inertial reference frame with sufficien accuracy) it's the time dilation effect, i.e., the lifetime of the muon (as measured by its proper time) is time-dilated by $\gamma$ when measured in our time, i.e., $t=\gamma \tau$. Seen from the rest frame of the muon (which we can also consider with sufficient accuracy here as an inertial frame) of course the muon's life time is $\tau$, but when measuring the distance to the Earth a observer in the muon's restframe would measure a distance shorter by $1/\gamma$ as compared to the distance measured by the Earth-bound observer. Thus there is no paradox: If you calculate the probability for the muon reaching Earth from its point of creation comes out the same, no matter using which coordinates/frames are used to calculate it.

 

[FactChecker]

Now you ask whether a different synchronization method would give a different result. Yes, of course. If, for instance, you and your friend discarded Einstein synchronization and instead adopted a convention that your friend uses whatever clock reading he is currently receiving in a television picture of your clock then the difference between your reported passing times in this scenario may be quite different.

I should have been more clear. I am wondering about synchronization methods that are considered at least as valid for SR as Einstein-synchronization.

 

[jbriggs444]

I should have been more clear. I am wondering about synchronization methods that are considered at least as valid for SR as Einstein-synchronization.

@Dale is fond of radar coordinates. Special relativity is agnostic about coordinates. Use whatever convention you like. The math is easier with some conventions than with others.

To be clear I do agree that Special Relativity as it is taught in schools is all about inertial frames of reference using Einstein synchronization and transformations between those frames.

 

[Dale]

@Dale is fond of radar coordinates.

FYI, one of the reasons I like radar coordinates is that they preserve the second postulate, even for non-inertial frames.

 

[FactChecker]

@Dale is fond of radar coordinates.

Ha! I looked that up. I will have to leave that for another lifetime. (A lifetime where I am smarter than I am now.)

 

[Dale]

Ha! I looked that up. I will have to leave that for another lifetime. (A lifetime where I am smarter than I am now.)

It isn't that difficult. You send a radar pulse to an event and collect the radar echo from that event. The radar coordinates of the event are $t=(t_{echo}+t_{pulse})/2$ and $r=(t_{echo}-t_{pulse})c/2$.

 

[PeterDonis]

in this scenario there are no spacelike separated events that are hypothesized to be simultaneous

There are if you are going to make any claims about time dilation. If the only events you consider are "Alex passes Bob" and "Alex passes Alice", then you can make no statements at all about time dilation. You have to consider events like "the reading of Alice's clock at the same time Alex passes Bob" and "the reading of Bob's clock at the same time Alex passes Alice" to make any statements at all about time dilation.

The "stationary" observer (I called him "you") is communicating with a friend in the same IRF who observes the event of the traveler passing.

And this communication (between Alice and Bob) will involve events which are spacelike separated from the first two events, and any claims about time dilation will require a simultaneity convention.

 

[PeterDonis]

Am I wrong?

Yes. You have multiple people now trying to explain to you why. I strongly suggest that you listen to them. It is getting to the point where we are repeating the same explanations over and over.

 

[FactChecker]

I realize now that I have confused a few aspects of the Twins Paradox. Here is a top-level summary of what I think has been said here. I hope that I do not butcher some people's inputs because there are a great many details that I am not qualified to understand or explain.
1) The correct answer to the Twins Paradox can be calculated using only SR and the IRF of the non-traveling twin.
2) Within SR, there is no real symmetry in the twins' situations because the traveling twin can detect that he does not remain in an IRF. So one can not use his non-inertial reference frame and SR to calculate the correct answer.
3) In order to calculate the correct answer using the traveling twin's non-inertial reference frame, GR is required. Two approaches for that are to use pseudo-gravitational potential or to use relativistic Lagrangian dynamics. These approaches are taken in the reference given by @Sagittarius A-Star in Post #17. Both approaches give the same answer as the one calculated with SR using the IRF of the non-traveling twin.

I hope that this is a good representation of the situation. Thanks to all for clarifying it for me.

 

[PeterDonis]

1) The correct answer to the Twins Paradox can be calculated using only SR and the IRF of the non-traveling twin.

Yes.

2) Within SR, there is no real symmetry in the twins' situations because the traveling twin can detect that he does not remain in an IRF.

Yes.

ne can not use his non-inertial reference frame and SR to calculate the correct answer.

No.

3) In order to calculate the correct answer using the traveling twin's non-inertial reference frame, GR is required.

No.

Two approaches for that are to use pseudo-gravitational potential or to use relativistic Lagrangian dynamics. These approaches are taken in the reference given by @Sagittarius A-Star in Post #17.

These aren't two different approaches.

give the same answer as the one calculated with SR using the IRF of the non-traveling twin

Yes.

See my corresponding post in the other thread in which you made an almost identical post for the details behind the above responses.