# The twin paradox and other variants.

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If I look at two supernovas, both 1000 light years from me, in opposite directions, I can say with confidence that they were happening simultaneously 1000 years ago. Even though they couldn't see each other or interact.

But someone else, moving relative to you, may say that one happened before the other. And someone else could say that they happened the other way round.

(And that is key to understanding the Twin Paradox, although I don't feel able to give a good explanation myself!)

Edited by Strange
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[..]

If I look at two supernovas, both 1000 light years from me, in opposite directions, I can say with confidence that they were happening simultaneously 1000 years ago. Even though they couldn't see each other or interact.

However, don't let yourself be misled: considerations of distant simultaneity have no bearing on local simultaneity, of two clocks side by side (ideally at zero distance, although that's not exactly achievable). Your arguments concerning that outcome stand.

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But someone else, moving relative to you, may say that one happened before the other. And someone else could say that they happened the other way round.

(And that is key to understanding the Twin Paradox, although I don't feel able to give a good explanation myself!)

To help illustrate this we can look at the following animations:

Two observers, one standing along a railway track and the other riding a rail car. Two flashes go off leaving sources that are stationary to, and a equal distance from our track observer. (these are like the supernovae explosions.)

The track observer sees both flashes at the same time, and as we see in the animation, this is because both flashes went off at the same time. At the same moment as he sees the flashes, the rail car observer is passing him, so the light from the flashes hits him simultaneously also.

Note how the light from the flashes expand out as even circles from the emission points.

Now we consider the same events, but from the frame of the railway car. Just like above, the light from the flashes hits him at the same time as he and the track observer pass each other. Now speed of light must have a constant speed for him just as it did for the track observer, so the light must expand outwards from the points of emission as even circles centered on points that are stationary with respect to himself. This means that unlike the track observer, the red dots which represents the original sources of the flashes do not remain at the center of the flash. The movement of these sources after the emission of the flashes has no bearing on the light travel of the flashes.

Now the track observer could conclude that the flashes took place simultaneously because not only did he see the flashes simultaneously, but he knew that the sources were an equal distance from him when the flashes were emitted.

But the only moment at which the railway observer is exactly halfway between the initial sources is when he sees the flashes, and the actual emissions must have taken place before that, when he was closer to one source than he was the other.

If the light coming at him from one source isn't coming at him any faster or slower than the light coming from the other source, and he couldn't have been an equal distance from the sources when they were emitted, then the only way he could see the flashes reach him at the same time would be for the emission of the flashes to have occurred at different times and not simultaneously.

This is how events transpire according to the railway car frame.

Now it is important to note, that you don't have to actually be the track or railway observer to make these same conclusions, anyone located anywhere and at rest with respect to the tracks will determine the same thing regarding the simultaneity of the flashes as we see in the first animation, and anyone located anywhere and at rest with respect to the railway car will make the same determination as to the simultaneity of the flashes as shown in the second animation.

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Strange, Tim and Janus, I'm sure you're right. I posted that about the Supernovas without much thought, as I wasn't really trying to make a point about simultaneity, but about deducing things from past events. So, that post was full of crap.

But I'll try to make the point about deducing things about the past, from current evidence, in a hopefully better way.

Most people accept that the big bang happened. People look at the universe today, and come to that conclusion.

It's the same methodology for practically all of science. You look at today's evidence, and deduce the past.

And that's what I was trying to say about the twins in the twin paradox.

If at the end of the trip, twin B's clock reads two years, and he is clearly a two-year-old, while twin A's clock reads twenty years, and he is clearly a twenty-year-old, then you can safely say that as some point in the process, twin A's clock was ACTUALLY going faster than that of B, and he was ACTUALLY ageing faster than B.

So at that point, even though they were not undergoing acceleration, their cases were not interchangeable.

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Strange, Tim and Janus, I'm sure you're right. I posted that about the Supernovas without much thought, as I wasn't really trying to make a point about simultaneity, but about deducing things from past events. So, that post was full of crap.

But I'll try to make the point about deducing things about the past, from current evidence, in a hopefully better way.

Most people accept that the big bang happened. People look at the universe today, and come to that conclusion.

It's the same methodology for practically all of science. You look at today's evidence, and deduce the past.

And that's what I was trying to say about the twins in the twin paradox.

If at the end of the trip, twin B's clock reads two years, and he is clearly a two-year-old, while twin A's clock reads twenty years, and he is clearly a twenty-year-old, then you can safely say that as some point in the process, twin A's clock was ACTUALLY going faster than that of B, and he was ACTUALLY ageing faster than B.

So at that point, even though they were not undergoing acceleration, their cases were not interchangeable.

There is no point in the trip where you can say that in an absolute sense that either clock was running slower than the other. There is a period where A and B do not measure( and by "measure" I mean determine the relative tick rate of the other clock at that moment) reciprocal time dilation in each other and that is when B undergoes the acceleration to return to A.

Assuming a non-instantaneous change of velocity for B at turn-around but rather a finite acceleration for a non-zero time:

While B is accelerating, A will measure B's clock as initially running slow, then speeding up as the difference in speed decreases until for a brief instant it runs at the same rate as his own, then slowing down again as B regains speed for the return leg and finally ending up running as slow as it was before the acceleration of B.

B, while accelerating will measure this: Just before acceleration, A's clock will be running slow. At the initiation of acceleration, A's clock will start running fast. This rate will increase to a maximum when B's acceleration brings B to rest with respect to A, and then decreases again until the acceleration phase ends, at which point, A's clock returns to running slow again.1

Thus according to A, the total time difference between the two clocks when they meet again will be due to B's Clock running slow by a constant rate on the outbound and return legs, with a period in between where B's clock ran slow by a varying rate.

According to B, the time difference is brought about by clock A running slow during the outbound and return legs with a period in between where A's clock ran fast by a varying rate. (B has to add the time elapsed for A's clock during the periods it ran slow to the time elapsed on it when it ran fast to get the correct total time elapsed for A's clock.)2

While A and B agree at the end that more total time elapsed for A than B, They will not agree on why things end up this way, and both "viewpoints" have equal validity as to be correct.

1 3 factors determine exactly how fast A's clock ticks according to B:

1) The instantaneous relative speed difference between A and B which adds a standard SR time dilation factor. This is at its greatest at the start and end of the acceleration phase and zero at the moment when B is briefly at rest with respect to A

2) The magnitude of the acceleration. This and the third factor below combine to produce a faster tick rate factor for A's clock as measured by B. The greater the acceleration, the larger the effect. Assumed to be a constant in this example.

3) The instantaneous distance and direction of A from B. This effect, combined with factor 2 above results in a increased time rate factor for clock A as the distance increases. (The direction is important as it determines whether or not this causes an increase or decrease in the measured clock's rate. Clocks in the direction of the acceleration will show an increase in rate and clock's in the opposite direction show a decrease in rate.)

This effect is at its least at the start and end of acceleration when A and B are closer together than they are at the midpoint of the acceleration while A and B are their furthest apart.

2 As can be seen, assuming finite acceleration for a non-zero time complicates the actual calculations a bit, which why most examples assume instantaneous or at least near instantaneous acceleration.(The down side of this is that it produces an apparent discontinuous "forward leap" of A's clock as measured by B)

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Janus, I think your analysis makes sense, and the acceleration phase is where the time difference becomes actual.

It's the claim that the acceleration isn't responsible for the time difference that seems illogical.

If you work out what twin B sees if his telescope is trained on A for the whole trip, ( and assuming he's already in motion at the start ), then he sees A's clock running slow over both travelling legs, so the only way that it can end up faster than his over the whole trip, is if it runs extremely fast during the turn-around.

If the difference in the two twins' clocks is 2yrs and 20 years at the end, then he needs to see A's clock spin through about 18 years during the turn-around, because when they are reunited, they will be able to look directly at each other's clocks, and A's will read 20 and B's will read 2.

Presumably, the reason that B will see such a dramatic speed-up on A's clock, is because he's in the equivalent of a gravity well, due to the large acceleration in the direction of A, which slows down his clock, making A's appear to him to be spinning very fast.

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Janus, I think your analysis makes sense, and the acceleration phase is where the time difference becomes actual.

It's the claim that the acceleration isn't responsible for the time difference that seems illogical.

It isn't. The time difference is "actual" during the entire scenario.

If we take your 2 yr to 20 year difference and assume a negligible time period for the acceleration, then the relative velocity needs to be about 0.995c. And thus according to A, B turns around when he gets ~ 10 light years away from him. However, due to length contraction, the same point that A measures as being 10 ly, B only measures as being 1 ly. Thus for him, the two legs of the trip only take ~1 year each at 0.995c And he returns only 2 years older. The acceleration has nothing to do this this. What the acceleration does it make it so that he will concur at the end of the trip that 20 yrs passed for A.

If you work out what twin B sees if his telescope is trained on A for the whole trip, ( and assuming he's already in motion at the start ), then he sees A's clock running slow over both travelling legs, so the only way that it can end up faster than his over the whole trip, is if it runs extremely fast during the turn-around.

If the difference in the two twins' clocks is 2yrs and 20 years at the end, then he needs to see A's clock spin through about 18 years during the turn-around, because when they are reunited, they will be able to look directly at each other's clocks, and A's will read 20 and B's will read 2.

Using the same numbers as above, For B, between the time he leaves A and arrives at the turnaround point, A's clock runs slow and thus has only ticked 0.1 yrs. The same is true during the period after the turn around. Thus during acceleration, A's clock must advance 19.8 yrs according to B. the running slow of A's clock during the periods that B coasts is just as real as the running fast is during acceleration.

Presumably, the reason that B will see such a dramatic speed-up on A's clock, is because he's in the equivalent of a gravity well, due to the large acceleration in the direction of A, which slows down his clock, making A's appear to him to be spinning very fast.

A doesn't just "appear" to run fast, it runs fast. You can use the pseudo-gravity field viewpoint, but you have to do it properly. You have to keep in mind that even gravitational time dilation is not due to a difference in local gravity strength but a difference in gravitational potential. Thus if you are going to use this viewpoint you have to think of it this way:

The pseudo-gravity field is uniform and extends to infinity. In this viewpoint B is not accelerating towards A, but A is "falling" towards B in response to this field. A's clock ticks faster due to the fact that it is higher in the field. Saying that B runs slow would be incorrect as we are making our measurements from B.( According to B, there could be clocks lower than it in the field which run slower than it does.)1

The point being is that you cannot pick any one clock to be representative of the one "true" absolute time while other clocks tick at altered rates relative to the "true" rate. The "true" rate of time is whatever the local clock measures.

1This is true even for clocks sharing the same acceleration as B. Imagine B is sitting with his clock at the mid-point of his ship and there are two other clocks, one in the nose and one in the tail of the ship. During the acceleration, the nose clock will tick fast and the tail clock will tick slow according to B, even though both clocks are under the same acceleration as B is. Once the acceleration phase is over, these clock could be brought together with B's clock and the accumulated time difference between them would remain.

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I think, from looking at what B sees, I can now get an idea of what happens in the instance where B's clock reading is transferred to a third twin C.

If B was physically transferred with his clock reading, he would experience infinite acceleration for zero time.

So his original observation of seeing A's clock whizzing round fast through about nineteen years, would happen in an instant, in this case. He would see A's clock jump 19 years.

So that's where the instantaneous jump in time comes from. You are simulating an event that can't physically happen, so an instantaneous jump in time is not inconsistent with that.

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[...]

Now it is important to note, that you don't have to actually be the track or railway observer to make these same conclusions, anyone located anywhere and at rest with respect to the tracks will determine the same thing regarding the simultaneity of the flashes as we see in the first animation, and anyone located anywhere and at rest with respect to the railway car will make the same determination as to the simultaneity of the flashes as shown in the second animation.

Sorry for nitpicking here, but it may be important to note that the above is in fact wrong. It is correct for those who use as "rest system" a system in which they are in rest. And while that is the assumption in Einstein's illustrations, it is in common modern practice rarely the case; usually people in different states of motion all use GPS receivers that calculate the speed of light based on the same ECI system (both the train and the track are moving in it).

In other words, as a generalization one has to realize that it's not needed to be actually be on the track (as you said), nor does one have to be at rest with respect to the track in, order to determine the "track animation simultaneity" - one only has to use the track's reference frame as reference. As a matter of fact, that is already clear from the way you phrased your examples, as none of us were in that train or on that track.

[..] You look at today's evidence, and deduce the past.

And that's what I was trying to say about the twins in the twin paradox.

If at the end of the trip, twin B's clock reads two years, and he is clearly a two-year-old, while twin A's clock reads twenty years, and he is clearly a twenty-year-old, then you can safely say that as some point in the process, twin A's clock was ACTUALLY going faster than that of B, and he was ACTUALLY ageing faster than B.

So at that point, even though they were not undergoing acceleration, their cases were not interchangeable.

It is of course undeniable that the average clock rate of B was greater than that of A, by mere definition. And this is agreed upon by any inertial reference system. Similarly, it's indeed safe to say - even undeniable by mathematical necessity - that at least for some part of the trajectory the clock rate of B must have been greater than that of A; however what part that is depends on which reference system you choose as "true". And since no inertial frame can be singled out as "true frame", we cannot determine such a thing.

I think, from looking at what B sees, I can now get an idea of what happens in the instance where B's clock reading is transferred to a third twin C.

If B was physically transferred with his clock reading, he would experience infinite acceleration for zero time.

So his original observation of seeing A's clock whizzing round fast through about nineteen years, would happen in an instant, in this case. He would see A's clock jump 19 years.

[..]

Not so. Twin C merely has a different estimation of distant time than twin B. If twin B chooses upon acceleration to switch reference system and adopts that of twin C, then he will himself make his estimation of A's "now" jump 19 years.

Alternatively he may choose to stick calculating with his earlier "rest" reference system. In that case he will conclude that due to his own speed which is now higher than that of A (he has to catch up with A!), his own clock ticks slower than that of A, such that on the average over the whole trip his clock will end up behind that of A.

Note also that B's act of suddenly turning around far away from twin A cannot have any effect on the clock rate of A; even the information from his turnaround will reach twin A, long after the turnaround with its imaginary clock jump of clock A.

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• 4 weeks later...

A doesn't just "appear" to run fast, it runs fast. You can use the pseudo-gravity field viewpoint, but you have to do it properly. You have to keep in mind that even gravitational time dilation is not due to a difference in local gravity strength but a difference in gravitational potential. Thus if you are going to use this viewpoint you have to think of it this way:

The pseudo-gravity field is uniform and extends to infinity. In this viewpoint B is not accelerating towards A, but A is "falling" towards B in response to this field. A's clock ticks faster due to the fact that it is higher in the field. Saying that B runs slow would be incorrect as we are making our measurements from B.( According to B, there could be clocks lower than it in the field which run slower than it does.)1

I think pseudo-gravity can't be used in this context.

Consider the following situation:

B (blue line) is moving away from A at 0.8c during 9 (of his) years, and then comes back at he same speed.

They both send a picture of their clocks every year. Red are A's signals, green are B's.

B will receive 3 pictures during the first part of the journey, and one every 4 months during the second part.

He won't see a sudden advance of A's time due to his own acceleration.

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I think pseudo-gravity can't be used in this context.

Consider the following situation:

B (blue line) is moving away from A at 0.8c during 9 (of his) years, and then comes back at he same speed.

They both send a picture of their clocks every year. Red are A's signals, green are B's.

B will receive 3 pictures during the first part of the journey, and one every 4 months during the second part.

He won't see a sudden advance of A's time due to his own acceleration.

It is important not to confuse what B visually "sees" with what "is happening right now" as far as A is concerned.

Here's a standard space time diagram for the trip in A's frame:

A is sending light signals to B at 3 year intervals (I chose 3 years because it keeps things from getting too crowded in later diagrams.)

The signals sent by A every 3 years from this point on arrive at B when it clock readings are only 1 year apart and B receives 27 years of signals during his return leg.

Now consider B's outbound leg as considered by B.

He again receives the light signal that left A when its clock read 3 years when his clock reads nine years. This is "old' information and does not represent the reading of A's clock at that moment. Instead, by following the orange "now" line, we see that A's clock actually reads 5.4 years when B's own clock reads 9 years. This is in line with time dilation which says that according to B, A's clock should tick 0.6 as fast as his own

Last, we have B's return leg as considered by B (after his has performed his acceleration)

If we assume a high acceleration for a negligible time period between the two legs, B's clock still reads 9 years, and he still is seeing the light signal that left A when it read 3 years. However, according to his orange "now" line, A's clock at that moment reads 24.6 years, it has jumped forward over 19 years. He doesn't visually see A's clock jump forward as he is seeing the same light as he was before the acceleration, but after the acceleration what that visual information means as far as what A's clock is reading "at that moment" has changed. The 24.6 year "now" reading on A's clock is in accord with time dilation again, as according to B, A's clock should tick off 5.4 years while his clock ticks off 9 years, and 30( the time A's clock reads when they meet up again) minus 5.4 is 24.6.

B still sees the rest of the signals coming from A as arriving 1 year apart for the rest of the trip until he meets up with A again and thus sees A's clock advance from 3 to 30 years in 9 years by his own clock. A light signal sent by B when its clock reads 9 years will arrive at A when its clock read's 27 years, and signals sent 3 years apart will arrive at A at 1 year intervals by its clock.

If B were to perform a more gradual acceleration during turn around, rather than a sudden jump in A's clock, it would work out to be a period where the average tick rate of A's clock occurred at a higher rate than B's clock . (in terms of "now" time.)

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I agree with that.
The point is that the sentence (in post #32)

A doesn't just "appear" to run fast, it runs fast.

is confusing.

It would be true if A felt in a real gravitational field. Light signals sent during that period would also prove that A's time really accelerated.
It's not the case with the pseudo-gravity. At the end of the trip, B must accept that, as the light signals show, his calculation of A's time change was not in accordance with reality.

Another confusing sentence is:

The pseudo-gravity field is uniform and extends to infinity.

If the pseudo-gravity is supposed to have any effect, the field can't fill the entire space instantaneously as this would contradict the theory. In our example, the effect would reach A when he aged 27 years.
But instead, pseudo-gravity has no effect at all.

IMO, that leaves us with no explanation of the twin paradox from the viewpoint of the traveller in terms of Relativity Theory.

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I agree with that.

The point is that the sentence (in post #32)

A doesn't just "appear" to run fast, it runs fast.

is confusing.

It would be true if A felt in a real gravitational field. Light signals sent during that period would also prove that A's time really accelerated.

It's not the case with the pseudo-gravity. At the end of the trip, B must accept that, as the light signals show, his calculation of A's time change was not in accordance with reality.

Another confusing sentence is:

The pseudo-gravity field is uniform and extends to infinity.

If the pseudo-gravity is supposed to have any effect, the field can't fill the entire space instantaneously as this would contradict the theory. In our example, the effect would reach A when he aged 27 years.

But instead, pseudo-gravity has no effect at all.

IMO, that leaves us with no explanation of the twin paradox from the viewpoint of the traveller in terms of Relativity Theory.

Good points (+1). The confusing statements are based on the incorrect statement "Saying that B runs slow would be incorrect as we are making our measurements from B." That corresponds to taking a self-centered point of view as "absolute" so that nothing can happen to the observer. Such an approach leads to contradictions or "multiple realities" as well as magical "instantaneous action at a distance".

Instead, one has to conclude that, taken over the whole interval, B runs slow; that is the only logically reasonable conclusion. That the use of pseudo gravitational fields is not justifiable was already explained in "Resolution of the clock paradox", Builder 1957, Austr. J.P.

However we do have, as you know, other explanations of the twin paradox, which we compared at great length several months ago.

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However we do have, as you know, other explanations of the twin paradox, which we compared at great length several months ago.

Sure we have, but how compatible are they with Einstein's theory, as they are based on the idea of an absolute frame of reference, and on the fact that the two observers are not fully equivalent, even during the inertial periods?

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Sure we have, but how compatible are they with Einstein's theory, as they are based on the idea of an absolute frame of reference, and on the fact that the two observers are not fully equivalent, even during the inertial periods?

SR is not based on such things. It is purposefully based on generalizations of observed phenomena, so that it does not depend on underlying physical models. Therefore, any explanation or interpretation that is fully compatible with the predicted phenomena, is also compatible with SR.

By the way, although Minkowski Spacetime is a theory of an "absolute world" (in contrast with Einstein's original concept), I think that it is debatable to call that an "absolute frame of reference".

Edited by Tim88
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The primary difference between Lorentz and Minkowskii is that the Minkowskii metric is a classical field treatment.

The (at rest frame) shouldn't be construed as an absolute frame or a preferred frame.

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SR is not based on such things. It is purposefully based on generalizations of observed phenomena, so that it does not depend on underlying physical models. Therefore, any explanation or interpretation that is fully compatible with the predicted phenomena, is also compatible with SR.

By the way, although Minkowski Spacetime is a theory of an "absolute world" (in contrast with Einstein's original concept), I think that it is debatable to call that an "absolute frame of reference".

Maybe I was unclear. I didn't mean that SR was based on an absolute frame of reference. The explanations of the twin paradox are.

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Maybe I was unclear. I didn't mean that SR was based on an absolute frame of reference. The explanations of the twin paradox are.

Indeed, on that explanation we also elaborated at great length. It must have been me who was unclear, when I remarked

"However we do have, as you know, other explanations of the twin paradox, which we compared at great length several months ago."

With "other explanations" I meant the Lorentzian and the Minkowskian interpretations.

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