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'Paradoxes' of Relativity Part 2A: The Twin Paradox

Posted December 13, 2006 11:00 PM by Jorrie
Pathfinder Tags: special relativity Twin Paradox

The "Twin Paradox" was born in 1911, after Paul Langevin restated Einstein's time dilation of traveling clocks as an interesting variant: one of two twin brothers undertakes a long, very fast space flight and on his return to Earth, finds his brother quite a bit older than himself.

The 'paradox' came in when skeptics argued that the 'away twin' could just as well have viewed himself as stationary, with the Earth and his twin brother speeding away from him and return to him later. In such a case, the Earth and the 'home twin' must be the younger ones, because they were in the 'moving frame of reference'. When compared to the original postulate, this is obviously paradoxical.

Relativists then say, "remember, the away twin had to suffer 'g-forces' to get up to speed, then again to turn around and head back and finally, again to land on Earth. So, the situation is not symmetrical".

Skeptics reply, "so what, we can make the journey arbitrarily long, so that the "g-forces" parts become negligible". Further, they say, "we are told that acceleration does not affect the rate of good clocks. So, the paradox stays".

Most relativists will then show how the problem can be broken up into three space-time intervals, bordered by four events, as shown below, from the home twin's point of view.

Figure 1

In the home twin's reference frame, the turning point is 3 light-years away and the away twin travels at 60% of c both ways. Hence, the trip takes 5 years each way and home arrival is 10 years after departure.

The space-time interval applicable to this scenario is expressed as follows:

ds2 = c2dt2 - dx2,

where dx is a space interval, dt a time interval and c=1, due the fact that distance is here expressed in light-years and time in years (light covers one light-year in one year).

Let's calculate the first space-time interval (between events 1 and 2) in the home twin's frame of reference:

ds2 = 52 - 32 = 16 square light-years.

Now, as has been confirmed theoretically and practically many times, space-time intervals are reference frame independent. In more technical terms, space-time intervals are invariant under coordinate transformations.

Therefore, the away twin must get the same interval than what the home twin calculated. Since the away twin was present at both events 1 and 2, his space interval dx' = 0. So his dt' must be 4 years. In symbolic form:

ds'2 = dt'2 - dx'2 = 16.

But dx' = 0, so dt'2 = 16.

This means that the away twin will reach the turn-around point after 4 years of his own elapsed time and not the 5 years that it will take in the home twin's reference frame. This underlines an important principle of special relativity: an inertial observer present at two events will always observe a shorter time interval between the events than what an observer not present at both events will observe.

If we ignore the short turnaround time and we assume that the inbound flight will take as long as the outbound flight, the away twin's clock and calendar will show that 8 years have elapsed when he arrives on Earth again. The home twin, however, would have recorded 10 years.

Engineers are not a very skeptical lot, but, after nodding their heads and saying, yep, that's what the theory and the calculations show, most will still not feel comfortable with this relativistic explanation.

In the part 2B of this mini-series, a more 'engineering-like' view, using relativistic Doppler shift, will be discussed. Watch this space!

You can learn more about space-time intervals in a free download from this page at the website/eBook Relativity 4 Engineers. (There's also a x-mas special on there...)

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#1

Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

12/14/2006 10:19 AM

Jorrie, interesting and refreshingly different from the usual time dilation and length contraction explanations.

I can see that it relates to the above concepts, but you somehow "gave preference" to the home twin in that you did the calculations in his reference frame only. Can you show us how to do the same from the away twin's reference frame?

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#2
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Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

12/15/2006 1:36 AM

Guest, you asked: "Can you show us how to do the same from the away twin's reference frame?"

Except for one little detail, it is also straightforward. The away twin, being present at both events 1 and 2 can directly measure the time interval of 4 years. With the zero space interval implied, it means that the space-time interval is also 4 years.

The home twin can now take the known 3 light-years space interval in his reference frame and calculate the time interval in his frame as 5 years, from sqrt[42 + 32].

The "one little detail" is that the home twin does not know the space interval (or distance) between events 1 and 2 from direct measurement. Or does he?

In the typical thought experiment, the home twin can put up a grid system with observers (or just sensors) at fixed distances so that he can detect exactly where the away twin does the turn-around thing.

For almost the same effort, the home twin can also equip the grid with synchronized clocks so that he can directly obtain space and time intervals.

It will become much clearer in the next part of this mini-series, I hope!

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#3
In reply to #2

Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

12/15/2006 7:13 AM

Thanks Jorrie. It makes sense, but is it not better to just do what Einstein did - keep the away twin stationary, but let a 'uniform gravitational field' engulf the away twin during his turnaround acceleration. This slows the clock of the away twin sufficiently (during the turnaround) to more than counter the time dilation of the home twin.

In fact I believe the answer comes out the same as for any other explanation!

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#4
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Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

12/15/2006 9:47 AM

Hi Guest, you wrote: "... but is it not better to just do what Einstein did - keep the away twin stationary, but let a 'uniform gravitational field' engulf the away twin during his turnaround acceleration. This slows the clock of the away twin sufficiently (during the turnaround) to more than counter the time dilation of the home twin."

Einstein actually got away with this "blunder"! Gravitational fields (curved space-time) and "acceleration fields" are not equivalent. The twins paradox saga plays itself out in gravity-free space, so it is a pure special relativity case, including the acceleration phases. Acceleration in free space and gravitational time dilation have nothing to do with each other.

If you have lots of time, try and read the discussions on this Wikipedia talk section.

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#5
In reply to #4

Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

12/15/2006 10:12 PM

"TIME" is the rate of energy degradation (loss of energy) of the most elemental particle in the universe, (which makes up all of the universe). the faster the particle travels, the less energy it looses due to absorbtion of electromagnetic friction (energy). The slower it travels (throughtout the universe) the less energy it receives from electromagnetic friction, thus the more it degrades in energy and is negatively affected.

This is why "relativity" is. This is simply science at it's very most basic design. Try thinking like Einstien, SIMPLY!

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#6
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Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

12/17/2006 4:24 AM

Dear Guest(2), you wrote: '"TIME" is the rate of energy degradation (loss of energy) of the most elemental particle in the universe, (which makes up all of the universe).....'

Sorry, I do not know what you are talking about - do you?

It appears as if you have some theory of your own; if so, please start a new thread on that, because it will be slightly off-topic in this thread.

Jorrie

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#7
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Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

05/09/2007 9:01 PM

IF WE TAKE 2 clocks and send 1 on the journey above

and leave the other on earth when the clock returns

these clocks record time and date

will they read differently will 1 be 2 years behind the other

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#8

Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

10/08/2007 2:20 AM

From what I've read so far, including text books, there seems to be two basic ways of showing that the twin paradox is in fact not a paradox. We have the introduction of acceleration where the one twin turns around to go back and see his brother and we have the argument that there is not symmetry because for one twin he has the clock riding with him/her and the other does not. The latter is obviously by definition not symmetric. The introduction of acceleration seemingly resolves the problem, but to my mind it does not. If we just consider that the two are each moving relative to the other at high speeds (e.g., in opposite directions) we should be able to resolve the seeming(?) paradox without resorting to one or the other or both of the twins going thru an acceleration. I.e., resolve the paradox of each seeing the other's clock going slower than theirs without going any farther (not having them turn around to meet). The only thing I've been able to come up with is that for the situation to be understood in any meaningful way it does not suffice to say that one is moving at a speed relative to the other.. period. This statement implicitly implies that one knows their individual rates with respect to a third inertial frame. There is no way of understanding what one's motion with respect to the others means without the third system. So just saying twin1 is moving at rate 1 with respect to twin2 is meaningless unless one implicitly means that we know what the rates are of each in the third system. For simplicity let's assume that the third system is a frame fixed in the stars (or it could be the earth). Twin1 is moving at near light speed in an opposite direction to twin2 as measured in the star inertial frame. Likewise twin2 is moving at the same near light speed. Twin1 deduces that twin2's clock (light clock) is moving slower than his thru the usual analysis. And, similarly, Twin2 deduces that Twin1's clock is moving slower than his. Now a third observer in the star inertial frame sees that each is moving at near light speed and each will have a clock that runs slower than any clock in the star inertial frame and that they both are aging at a slower rate than anyone in the star system. He declares, "What paradox? There is none!" However, that still does not resolve the problem that Twin1 sees Twin2's clock running slower than his and vica versa. What is true in the star frame (each are aging slower than anyone in the star frame and their rates of aging are identical) is in the star frame. What about the two inertial frames that each twins exists in? Don't we still have a paradox? Isn't bringing in the third inertial frame similar to bringing in acceleration to resolve the question? I think the answer is no for the following reason. If the twins were moving at the same near light speeds in parallel and in the same direction, they would each see that the other is moving at the same rate as theirs. Only if they move in non parallel directions (e.g., towards each other) does the paradox arise. If one says that the clocks only run slower when they move in opposite directions I think all would agree that is nonsense. Motion in opposite directions to each other has no effect on time in each inertial frame. So does this resolve the seeming paradox? They still come to the conclusion of each other's clock running slower. I think the crucial point is as follows. Unless the twins consider what is happening with respect to a third inertial system (e.g. the star system) they have no idea whether one is moving and one is stationary (right there we have implicitly introduced the 3rd frame when we say stationary...stationary with respect to what?) or both are moving and so if they cannot describe what their motions are any conclusions drawn are without merit. And in order to know what their motions are we need the third inertial frame. Then they can make statements about their motions to each other in relation to this "fixed" frame. Then they will see that their conclusions about who's clock is slower must be made within the context of this 3rd frame. And, they will deduce that the only way they can tell who's clock is doing what is to measure the passage of time with respect to this fixed frame. Now they see that if their trajectories are at the same speed and in the same direction that their clocks are slower than the star frame, but the same as each others so that they each age at the same rate but slower than someone in the star system. And, when they travel in opposite directions where prior they deduced that each was aging slower than the other, they conclude that since they are each aging at the same slower rate than one in the star system they cannot each be aging slower than each other which of course is impossible and therefore the data they use and analyze only within their inertial frames is not the sufficient to draw any conclusions. And that is the bottom line. Analysis of time rates in inertial systems that are moving with respect to each other without reference to another fixed system is incorrect. A third system is necessary in order to determine just what speeds each are going. Is one stationary or not? You can't tell without the third system. If you don't have all the necessary data, you can't make any rational conclusions.

But, you know, something about this is still bothering me and I just can't put my finger on it. What do you think? It may be just that we can't experience this phenomena. I'm not sure!

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#9
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Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

10/08/2007 3:04 PM

After posting my comments on the paradox it occurred to me that my conclusions were not as tight as I would have liked them to be. But after thinking about it I think I can now tighten things up as follows:

Two observers in two inertial frames cannot generally determine relative time passage without a third inertial frame for reference. In the same way that a train passenger cannot tell if he in his train is moving forward or backward when observing another train next to him without reference to some other object neither on his or the other train, it is the same with the passage of time. In each case the "other object" is another reference inertial frame. One could argue (incorrectly) that a third reference inertial frame should not be necessary, but the results of the examples we've seen show that this is not the case. One could sum up by saying that the paradox only exists when the analysis is done without all of the pertinent data and that data inlcudes a third reference inertial frame.

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#10
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Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

10/09/2007 1:54 AM

Hi jackpaul and welcome to the CR4 forum!

You wrote: "Two observers in two inertial frames cannot generally determine relative time passage without a third inertial frame for reference. "

No, it is not true that you necessarily need a third inertial reference frame in order to correctly analyze special relativity's time dilation. True, it can be done in any inertial frame of your choice, but the home-twin's frame is a good as any other one.

Even if there is no 'turn-around' by the away-twin in the exercise, one can place synchronized clocks in each of the two frames in such a way that they pass each other and can be directly compared.

It is true that if both twins are accelerating (i.e., they are both non-inertial), it is easier to analyze in a third inertial frame. But it is not an absolute requirement.

Jorrie

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#11
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Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

05/02/2008 7:17 PM

I hope I can be forgiven for introducing a different approach to this paradox. If desired I can of course start a new thread. Here goes...

Most of the back and forth argumentation on this stems from the assymmetric situation of both observers. Let us, instead, try to design an experiment which eliminates that complication so that we have a much clearer picture of what the paradox really implies.

Here is a suggestion for just such an experiment:

Suppose we have 2 identical, synchronized clocks (C1 and C2) accelerating away from each other, starting in an inertial frame O which we, the observers, occupy.

We (at point O) observe both clocks reach, say 0.5c within a very short time (compared with the whole journey).

After many years, both clocks decelerate very quickly (compared with the whole journey), come to rest and return along the same route.

When they eventually meet at their common point of origin (after a further, short, deceleration), they fire a light beam at us to confirm their time.

We observe that the time registered by both clocks is identical and, incidentally, slowed by the amount given by SR/GR due to the fact that the clocks were in motion with respect to our 'rest' frame. This equivalence in the duration measured by both clocks is a requirement of the symmetrical setup we have chosen and because we have remained at rest during the entire journey.

However, during the periods of inertial flight, each clock is entitled to consider itself at rest (SR postulate leaving GR aside for now) and consider the other as moving away at a speed given by the usual Lorentz transform. C1 will observe C2's clock as running slow. By symmetry, C2 will observe C1's clock running slowly by the same amount. When they therefore return to the point of origin and stop, each can claim that the other's clock has run more slowly. This is clearly impossible.

I have been unable to resolve this, rather troubling contradiction whilst upholding the postulates of SR or GR and would be most grateful for any insights you might offer.

Observations:

1. The reason for choosing the symmetrical setup, as already stated, was to eliminate the complicating factors which the classical clock/twin paradox suffers from (asymmetrical non-inertial frames). The GR and turn-around acceleration effects cancel by reason of symmetry in this case.

2. The periods of non-inertial motion in our setup should be chosen in such a way that they represent only a fraction of the whole journey time. This is easily achieved if we extend the journey to an arbitrarily long time.

3. The classical twin paradox admits complicating factors such as take-off, turn-around and GR effects (not to mention portable gravitational fields applied at the turn-around point for the travelling twin!). The symmetical setup proposed strips the problem down to its fundamentals I believe.

P.S. I have, in the meantime, located an interesting paper by Dr. L.J. Wang (physics researcher and lecturer) of the University of Tennessee Physics Dept. published in Physics and Modern Topics in Mechanical and Electical Engineering 1999 which details a very similar setup and concludes that it represents a logical contradiction inherent in SR. So it seems I am at least in good company with this *crank* question! Here is the url in case you have time to study it: http://www.utc.edu/Faculty/LingJun-Wang/Paradox.pdf

I used the word *crank* due to the fact that I was described as such by one other person I addressed the question to. He was not able or willing to study the problem, so I am rather worried now since I feel that the issue is being dodged...

Many thanks for your insights.

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#12
In reply to #11

Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

05/03/2008 2:06 PM

Hi Laharnaman, you wrote:

"C1 will observe C2's clock as running slow. By symmetry, C2 will observe C1's clock running slowly by the same amount. When they therefore return to the point of origin and stop, each can claim that the other's clock has run more slowly. This is clearly impossible."

Indeed it is impossible, simply because neither C2 nor C1 can observe the other's clock directly during the journey, except at the start and at the end. Sending time stamped light signals are not the same as 'observing the other clock'. Reading the other traveller's synchronized coordinate clocks are also not the same as reading the other one's clock. The idea that all clocks on the move relative to an observer run slow is not a SR postulate, as is easy to see from your example. One can only say that a clock ran slow if you compare the readings of ideal clocks directly, meaning they are brought to the same spot in space.

I haven't had time to read the reference yet, but my first reaction is that if it says what you attributed to it, it is badly flawed - there are no logical contradictions in SR. I will read it in due coarse and reply again. Thanks for bringing it to my attention.

Jorrie

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#13
In reply to #12

Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

05/03/2008 5:51 PM

Thank you so much for that prompt answer. To be clear, are we to understand that their clocks will show the same time when they come to rest and meet again, despite the long periods during which each was in relative motion to the other?

That would mean that neither had observed the other's clock moving slower. I thought that one of the main contentions of relativity was that clocks which move at speeds comparable to c wrt each other will observe time dilation. Further, the only means of observing a moving clock is via light signals emitted from the clock so I believe that approach would be justified. Regrettably, I am unable to see how this resolves the matter but would appreciate any further clarification you could find time to provide.

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#14
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Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

05/04/2008 1:35 AM

Hi again Laharnaman. You asked:

"To be clear, are we to understand that their clocks will show the same time when they come to rest and meet again, despite the long periods during which each was in relative motion to the other?"

Yep, this is what SR tells us.

Your problem may lurk in this statement:

"I thought that one of the main contentions of relativity was that clocks which move at speeds comparable to c wrt each other will observe time dilation."

The main contentions of SR are that the observed speed of light is isotropic in all inertial frames and hence there is no absolute motion detectable. Time dilation is only directly observable if there is an asymmetry in the experiment, e.g., one clock is moving purely inertially and the other one not.

Inertially moving clocks maximizes the time interval between events. All non-inertial clocks record less time between events than inertial clocks. This also works for GR, where a clock sitting stationary in a gravitational field is not inertial and hence records less time than a distant static clock, where the gravitational field is vanishing. Static and stationary here mean with respect to the gravitational field, not in an absolute sense.

Hope this helps!

Jorrie

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#15
In reply to #14

Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

05/04/2008 6:20 AM

I think the best way to show my appreciation is to buy your book!I can say, however, that I'm surprised to hear that time dilation is only directly observable if one clock is moving inertially wrt to the other.

I've read again and again that (and this is just one example):

"In special relativity, clocks that are moving with respect to an inertial system of observation are measured to be running slower. This effect is described precisely by the Lorentz transformation." Wikipedia.

Perhaps we need to distinguish between inferred and observed time dilation? The observers can surely infer that the other's clock is slower even if they cannot directly observe it--that is after all what SR postulates?

Here is a simplified version of the experiment which should remove all non-inertial considerations as well as the direct 'unobservability' of time dilation between inertial frames of reference.The whole issue of acceleration can be taken out of the equation by re-designing the experiment as follows.

1. Let C1 and C2 be separated by a very large distance d from O and speeding towards each other at constant velocity and ourselves at O. We can ignore their past history.

2. The clocks of C1 and C2 are then synchronized from O when both are at a distance d from O. Synchronization is achieved via a light impulse from us (this method is allowed in SR and in fact suggested by Einstein and Poincare I understand).

3. When they eventually pass each other at O they both fire two impulses towards us to indicate:

3.1.The time as measured by their own clock.

3.2.The time as measured by the other clock which they briefly are able to observe at O(observed or inferred!).

Conclusions:

A. The times shown by C1 and C2 must be identical according to us by symmetry.

B. Nevertheless, using SR constraints alone, both clocks will have registered with us via the 2nd impulse a time which is different from the time registered by their first impulse.

C. A. contradicts B and we therefore have a logical inconsistency.

I'll read take time to read your book and, perhaps, return to your site in a month or two.Thank you so much Jorrie for your patience and willingness to address this issue.

Bob Kelly

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#16

Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

07/24/2010 11:23 AM

Though I do not have the time at the moment to explore this subject further now, I am glad to find this thread and topic here.

A number of years ago I posted the topic: 'Are Black Holes the Birth places of New Universes?' here in CR4. Though it was more of a scientific orientated topic than an engineering one, it sparked some interesting and lively discussions.

Currently the theory that our universe may in fact be within a black hole has recently surfaced. Though perhaps 'submerged' might be a more appropriate term in this instance (< 8)

The apparent motion that our entire universe is all being pulled towards one point is also very intriguing.

A number of years ago I encountered a short blurb about a book written by a layman/amateur cosmologist with the last name of 'Wright' that was published some 50 ~ 75 years ago with the title of 'Gravity is a Push' and from what I gleamed about it Mr Wright was putting forth and postulating the theory that Gravity is a repulsive force as opposed to a force of attraction. Given the current speculative hypotheses of 'Dark Matter &/or Energy' that is seeking to understand and explain the surprising new information and data that has come to light with the significantly enhanced sensing gathering capabilities in recent years; I wonder what, if any connection there are between these ideas and theories?

Addressing the 'Twins Paradox' that has been tested and proven to be accurate, it means that space travel is a 'one way trip' in that anyone venturing forth in space travel at ever increasing relativistic speeds that if they subsequently return to their point of origin, while they may have aged a significant portion of a 'normal human lifespan'. The matter from which they have departed from will have aged millions or even billions of years. Until such time that we may come to discover, understand and utilize 'warp speed' capacities, any such journeys will most definitely be 'one way trips' as their can be no possibility of 'going home.' At least not in the sense that anything including inert matter and living organisms will be in any way as they were for the Space and 'Time' travelers as it was when they departed and embarked on such journeys.

Very limited time permitting, I hope to delve into and explore such matters further along with the denizens of these cosmic realms and haunts.

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#17

Re: 'Paradoxes' of Relativity Part 2A: The Twin Paradox

07/25/2010 7:25 AM

errata addendum;

The book: Gravity is a Push was published much later than I originally thought it to be with the first edition by Walter C Wright being released from Carlton Press in 1979

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