Wasn't that the super-duper intelligent guy who has a self-published book on Amazon whose mom was a scientist, who also couldn't figure out limits of infinite sums.

Evolving wrote:I think he's conjecturing that if Einstein had only thought about integrating the differences in time delays into his equations, that would have been sufficient to account for all the observations that he developed SR to explain.

Ah ok. After having slept I can see what you (and therefore him) mean. Time dilation has already been confirmed by experiment (flying a clock around the globe at high speed, IIRC), so Einstein obviously got that right with his equations.

A question I have is related to my skimming of that SA article yesterday. It mentioned that the travelling twin's clock would appear slow as it sped past the twin on Earth. I was too tired to concentrate properly on the article, and am still not awake enough yet to go read it now, so I'm unsure what effect it was talking about there. Does the clock appear slower to the Earth twin because of relativistic effects, or because of this time delay thing the OP is talking about? Or perhaps both (in which case it would be even slower?)?

When I was first taught SR, at a very elementary level, I felt very strongly that the twin paradox had been fudged. We were told: ah, SR only applies in inertial frames, and the travelling twin is accelerating; and that’s true, he is, at three points in the story, but that’s not really a satisfying explanation.

The article that Thommo linked to really is very accessible, and I don’t mind admitting that, having worked through the (very simple) calculations, that twin “paradox” has never been so clear to me. Of course it’s not a paradox at all: it’s a mere curiosity, and ought to be called the Twin Curiosity or the Twin Surprise.

I wrote down the following table. One of the neat things about it is that it shows that both twins see the other twin’s clock going at half the speed of their own, and yet when the travelling twin comes home he has aged less than the homebody. The experiment is asymmetrical.


t(homebody)______ t(traveller)_______event

0_________________0_______________traveller departs

10________________8_______________traveller arrives at star; sees homebody’s clock as showing
__________________________________t(homebody) = 4 = half of t(traveller)

10________________8_______________traveller immediately departs for home

16________________________________homebody sees traveller’s arrival at star; sees traveller’s clock as showing
__________________________________t(traveller) = 8 = half of t(homebody)

20________________16______________traveller returns

I wondered what we would get if, instead of sending the travelling twin to a star six light years away, we sent him in orbit around the Earth at the same speed and for the same length of time, having previously organised an international pact to clear all the space débris out of the way. Could we then ignore the time delay between event and observation?

No, we can’t (I then realised). First, we have to remind ourselves that SR indeed only applies to inertial frames, and in orbit, of course, our twin is constantly accelerating. That problem, however, is easy to overcome: we simply break down the orbit into a large number of straight lines, each of which represents an inertial frame, and at the limit, as the length of those lines approaches zero, we get our elliptical or circular orbit (repeated a very large number of times). In each of those inertial frames, the travelling twin is a finite distance away from his brother; not very far, but there is a time delay between when a twin (or his clock) does something and when his brother sees him (or it) doing it, we have to sum up all those time delays over all the little straight pieces of orbit, and I am pretty sure (without, however, actually having worked it out) that in aggregate we would get the same result as in the table above.

scott1328 wrote:Wasn't that the super-duper intelligent guy who has a self-published book on Amazon whose mom was a scientist, who also couldn't figure out limits of infinite sums.

Those were two different people, but yes, the poster I was referring to is the one with the book.

The table, by the way, illustrates what I was saying about first adjusting for differences in time delays, and then applying the equations of special relativity.

Look at lines one and four in the table, when t(homebody) is zero and 16.

At t(homebody) = t(traveller) = 0, there is no time delay between the traveller's clock showing zero and the homebody seeing it showing zero.

At t(homebody) = 16 there is a delay of six years, and so we have to adjust for that now different time delay. The homebody works out that, at the moment when the traveller's clock showed 8 (the event that he is now observing), his own clock showed 10. That difference between 10 and 8 is what the equations of SR predict.

Evolving wrote:When I was first taught SR, at a very elementary level, I felt very strongly that the twin paradox had been fudged. We were told: ah, SR only applies in inertial frames, and the travelling twin is accelerating; and that’s true, he is, at three points in the story, but that’s not really a satisfying explanation.

Here is a very patient explanation

http://www.physicsguy.com/ftl/index.html

Parts I and II are about special relativity in detail, but in perhaps too much detail for anyone who doesn't wish to have a start on actually making calculations in relativistic kinematics. Part II in particular addresses the twin 'curiosity' in terms of the reference frames that the 'traveling' twin needs to recognize in making his computations of the rate of the clock used by the 'homebody'.

Part I includes a nice basic analysis of the slow rate of clocks that are moving in your reference frame.

In the final chapter, the author addresses himself to the irresolvable paradoxes of FTL travel in terms of known physics with rationalizations of how Star Trek avoided or evaded those problems.

I haven't had a chance to read your subsequent posts yet, Evolving. I decided to revisit that SA link from Thommo and see if I could digest that first. I didn't get very far, as one particular point doesn't seem sensible to me (although, I'm sure it actually is). This paragraph:

Both the traveler and homebody set their clocks at zero when the traveler leaves the earth for the star (event 1). When the traveler reaches the star (event 2) his clock reads eight years. (Click here for graph.) However, when the homebody sees the traveler reach the star, the homebody?s clock reads 16 years. Why 16 years? Because, to the homebody, the craft takes 10 years to make it to the star and the light six additional years to come back to the earth showing the traveler at the star. So to the homebody, the traveler?s clock appears to be running at half the speed of his clock (8/16.)?

The homebody wouldn't read the traveller's clock running at half the speed of his. His clock at 16 years isn't reading the time it took the traveller to reach the star. It's reading the time it took him to reach the star and then the light to get back to earth. The traveller's clock is only showing the time it took him to reach the star. So it would be 8/10 I would have thought.

Putting that aside, I can see how using the length/time-dilation calculations you wind up with one clock at 20 and one at 16 when the traveller arrives back home at Earth. But this doesn't really address the problem I've always had with the paradox. It's privileging one reference frame over the other. Why are the dilation calculations done for the traveller and not the homebody? The homebody could just as validly (I would have thought) done the calculations as he and the Earth and the star accelerate and move relative to the traveller. In that case the homebody would be younger on his and the Earth's return to the traveller.

Any signal, such as a transmission of a photo of the clock aboard the spaceship would contain the information of the clock at the time it was sent. The signal corresponding to the arrival of the traveler at the star would be sent at 10 years as measured by the homebody, but the signal would arrive at 16 years as measured by homebody. That signal arriving at that point would be of a clock that displayed a passage of 8 years.

Homebody could then calculate that this arrival occurred after 10 years of his own time, but what he would observe would be clock readings of 8 years and 16 years.

With regard to your second post - no the Earth does not accelerate relative to the traveler. The laws of physics are the same in all inertial frames, they are not the same in all accelerated frames (or more generally when changing from one IF to another). There is no relativity on that point, the traveler has accelerated and the homebody has not and they will both agree about that.

Bernoulli wrote:Putting that aside, I can see how using the length/time-dilation calculations you wind up with one clock at 20 and one at 16 when the traveller arrives back home at Earth. But this doesn't really address the problem I've always had with the paradox. It's privileging one reference frame over the other. Why are the dilation calculations done for the traveller and not the homebody? The homebody could just as validly (I would have thought) done the calculations as he and the Earth and the star accelerate and move relative to the traveller. In that case the homebody would be younger on his and the Earth's return to the traveller.

Because the homebody always stays in his frame of reference and never leaves it; the traveller changes his frame of reference (twice) and then comes back to the homebody's frame of reference. What effect that has, is described in the SA article and summarised in my table, that I extracted from the article.

We can construct a scenario in which no frame of reference is privileged by stopping our observation at line four of the table. At that point both twins see the other's clock as running half as fast as their own. The table describes it in one particular way, because we are assuming that the homebody's frame of reference is somehow the "correct" one; but we can equally validly describe it from the point of view of the other twin and call him the homebody and his frame of reference the stationary, correct one.

Thommo wrote:Any signal, such as a transmission of a photo of the clock aboard the spaceship would contain the information of the clock at the time it was sent. The signal corresponding to the arrival of the traveler at the star would be sent at 10 years as measured by the homebody, but the signal would arrive at 16 years as measured by homebody. That signal arriving at that point would be of a clock that displayed a passage of 8 years.

Homebody could then calculate that this arrival occurred after 10 years of his own time, but what he would observe would be clock readings of 8 years and 16 years.

No I understand that. The problem is with the wording in the article. He wouldn't read his clock running twice as slow as the travellers, because the two times - 8 years and 16 years - aren't measuring the same thing. 8 years is the measure of the time for the traveller to get from Earth to the star. 16 years is the measure of the traveller to get from Earth to the star plus the light to get from the star back to Earth.

With regard to your second post - no the Earth does not accelerate relative to the traveler. The laws of physics are the same in all inertial frames, they are not the same in all accelerated frames (or more generally when changing from one IF to another). There is no relativity on that point, the traveler has accelerated and the homebody has not and they will both agree about that.

Ah ok. Why is that the case? Intuitively they don't seem different; if you've ever sat on a train next to another train and watched as the other train accelerated away, for a moment it can be confusing as to whether the other train is accelerating away or you and your train are accelerating away.

Bernoulli wrote:Why is that the case? Intuitively they don't seem different; if you've ever sat on a train next to another train and watched as the other train accelerated away, for a moment it can be confusing as to whether the other train is accelerating away or you and your train are accelerating away.

Don't forget about the ground, which is the preferred reference frame for both trains, in a scenario where keeping the trains on time is considered 'important'.

It is manifestly untrue that you cannot detect acceleration in your own frame. Only you can't tell whether or not it's just gravity.

What is the reference frame in space? I'm assuming it's space-time. But how does one tell whether space time is moving relatively to one or not?

Bernoulli wrote:What is the reference frame in space? I'm assuming it's space-time. But how does one tell whether space time is moving relatively to one or not?

It is space-time, and you deal with events which help to determine your choices. You choose the reference frame that makes your calculations the easiest without letting them be incorrect.

Bernoulli wrote:
The homebody wouldn't read the traveller's clock running at half the speed of his. His clock at 16 years isn't reading the time it took the traveller to reach the star. It's reading the time it took him to reach the star and then the light to get back to earth. The traveller's clock is only showing the time it took him to reach the star. So it would be 8/10 I would have thought.

You say that because, understandably, you have decided to look at the events from the point of view of the homebody's frame of reference. But, as Thommo quite rightly pointed out, the only objectively verifiable facts are the observations. The idea of simultaneity, that the traveller arrives at the star when the homebody's clock shows t = 10, is an interpretation. In reality nobody can see those two allegedly simultaneous, events at the same time.

The traveller's observation and interpretation are equally valid at that point. If he takes his frame of reference as being stationary and the rest of the universe as moving with respect to him, then he thinks the homebody is experiencing time dilation, and so is the journey that the light has to take from the homebody's clock to him.

Bernoulli wrote:

Thommo wrote:Any signal, such as a transmission of a photo of the clock aboard the spaceship would contain the information of the clock at the time it was sent. The signal corresponding to the arrival of the traveler at the star would be sent at 10 years as measured by the homebody, but the signal would arrive at 16 years as measured by homebody. That signal arriving at that point would be of a clock that displayed a passage of 8 years.

Homebody could then calculate that this arrival occurred after 10 years of his own time, but what he would observe would be clock readings of 8 years and 16 years.

No I understand that. The problem is with the wording in the article. He wouldn't read his clock running twice as slow as the travellers, because the two times - 8 years and 16 years - aren't measuring the same thing. 8 years is the measure of the time for the traveller to get from Earth to the star. 16 years is the measure of the traveller to get from Earth to the star plus the light to get from the star back to Earth.

There is no "measuring the same thing" - the observers are in different places*. All measurements are relative. That is the fundamental point of special relativity.

What he sees is his clock running twice as fast as sequential images of clocks from the traveler that are being relayed to him. This is why the article chooses that word - "sees". As Evolving says, before the traveler gets to the point he turns around the two observers will not be "agreeing" because they don't "see" the same things - each will see the clock of the other running at half the speed of his own. He can then calculate (but you need to be careful about this, most people have an intuition of absolute simultaneity at distant locations that can cause them to "calculate" incorrectly) that the other brother has a clock running at 0.8 the speed of his own.

For the "like with like" comparison to be made the two observers, traveler and homebody, have to be back in the same place. But to do that one of them must undergo a change of inertial frame. That change will then alter the behaviour of the clocks that are "seen" after that point. Which one undergoes the change of IF determines which one is younger when they meet back up.

Bernoulli wrote:

With regard to your second post - no the Earth does not accelerate relative to the traveler. The laws of physics are the same in all inertial frames, they are not the same in all accelerated frames (or more generally when changing from one IF to another). There is no relativity on that point, the traveler has accelerated and the homebody has not and they will both agree about that.

Ah ok. Why is that the case? Intuitively they don't seem different; if you've ever sat on a train next to another train and watched as the other train accelerated away, for a moment it can be confusing as to whether the other train is accelerating away or you and your train are accelerating away.

Because the laws of physics don't add up the same, from the behaviour of clocks to the behaviour of forces. fictitious forces are one way of testing. If the train you're in suddenly slams you back against your seat or throws you forward you know about it even with your eyes shut in a dark tunnel. This is like measuring the Earth's rotation by use of Coriolis force, or the motion of a roundabout by centrifugal force.

The only time they seem similar is in cases of extremely small force so you simply don't notice. When you're talking about 2g acceleration for 200 days, you'd really notice. That's like spending the better part of a year at twice the gravity of Earth, instead of floating weightlessly in your space craft. It would be immediately obvious that you were accelerating without even any detailed experiment.

*Evolving puts this better:

Evolving wrote:The idea of simultaneity, that the traveller arrives at the star when the homebody's clock shows t = 10, is an interpretation. In reality nobody can see those two allegedly simultaneous, events at the same time.

The "allegedly" in that quote is really, really, really super important. There is no simultaneity of spatially separated events in relativistic theory. This is where people's intuition often leads them astray.

Cito di Pense wrote:

Bernoulli wrote:What is the reference frame in space? I'm assuming it's space-time. But how does one tell whether space time is moving relatively to one or not?

It is space-time, and you deal with events which help to determine your choices. You choose the reference frame that makes your calculations the easiest without letting them be incorrect.

That doesn't answer the question. There has to be a privileged frame for acceleration to be experienced by one body and not another body relative to it.

Evolving wrote:

Bernoulli wrote:
The homebody wouldn't read the traveller's clock running at half the speed of his. His clock at 16 years isn't reading the time it took the traveller to reach the star. It's reading the time it took him to reach the star and then the light to get back to earth. The traveller's clock is only showing the time it took him to reach the star. So it would be 8/10 I would have thought.

You say that because, understandably, you have decided to look at the events from the point of view of the homebody's frame of reference.

I didn't chose that. The author of the SA article chose that. It's his wording, and it's the wording that is inaccurate even in a hypothetical world where one person could read both clocks.

But, as Thommo quite rightly pointed out, the only objectively verifiable facts are the observations. The idea of simultaneity, that the traveller arrives at the star when the homebody's clock shows t = 10, is an interpretation. In reality nobody can see those two allegedly simultaneous, events at the same time.

Obviously. But to explain the procession of the paradox we need to assume the hypothetical situation that homebody can read the traveller's clock.

Thommo wrote:

Bernoulli wrote:

Thommo wrote:Any signal, such as a transmission of a photo of the clock aboard the spaceship would contain the information of the clock at the time it was sent. The signal corresponding to the arrival of the traveler at the star would be sent at 10 years as measured by the homebody, but the signal would arrive at 16 years as measured by homebody. That signal arriving at that point would be of a clock that displayed a passage of 8 years.

Homebody could then calculate that this arrival occurred after 10 years of his own time, but what he would observe would be clock readings of 8 years and 16 years.

No I understand that. The problem is with the wording in the article. He wouldn't read his clock running twice as slow as the travellers, because the two times - 8 years and 16 years - aren't measuring the same thing. 8 years is the measure of the time for the traveller to get from Earth to the star. 16 years is the measure of the traveller to get from Earth to the star plus the light to get from the star back to Earth.

There is no "measuring the same thing" - the observers are in different places*. All measurements are relative. That is the fundamental point of special relativity.

What he sees is his clock running twice as fast as sequential images of clocks from the traveler that are being relayed to him. This is why the article chooses that word - "sees". As Evolving says, before the traveler gets to the point he turns around the two observers will not be "agreeing" because they don't "see" the same things - each will see the clock of the other running at half the speed of his own. He can then calculate (but you need to be careful about this, most people have an intuition of absolute simultaneity at distant locations that can cause them to "calculate" incorrectly) that the other brother has a clock running at 0.8 the speed of his own.

For the "like with like" comparison to be made the two observers, traveler and homebody, have to be back in the same place. But to do that one of them must undergo a change of inertial frame. That change will then alter the behaviour of the clocks that are "seen" after that point. Which one undergoes the change of IF determines which one is younger when they meet back up.

Bernoulli wrote:

With regard to your second post - no the Earth does not accelerate relative to the traveler. The laws of physics are the same in all inertial frames, they are not the same in all accelerated frames (or more generally when changing from one IF to another). There is no relativity on that point, the traveler has accelerated and the homebody has not and they will both agree about that.

Ah ok. Why is that the case? Intuitively they don't seem different; if you've ever sat on a train next to another train and watched as the other train accelerated away, for a moment it can be confusing as to whether the other train is accelerating away or you and your train are accelerating away.

Because the laws of physics don't add up the same, from the behaviour of clocks to the behaviour of forces. fictitious forces are one way of testing. If the train you're in suddenly slams you back against your seat or throws you forward you know about it even with your eyes shut in a dark tunnel. This is like measuring the Earth's rotation by use of Coriolis force, or the motion of a roundabout by centrifugal force.

The only time they seem similar is in cases of extremely small force so you simply don't notice. When you're talking about 2g acceleration for 200 days, you'd really notice. That's like spending the better part of a year at twice the gravity of Earth, instead of floating weightlessly in your space craft. It would be immediately obvious that you were accelerating without even any detailed experiment.

*Evolving puts this better:

Evolving wrote:The idea of simultaneity, that the traveller arrives at the star when the homebody's clock shows t = 10, is an interpretation. In reality nobody can see those two allegedly simultaneous, events at the same time.

The "allegedly" in that quote is really, really, really super important. There is no simultaneity of spatially separated events in relativistic theory. This is where people's intuition often leads them astray.

You are over-analysis this. The wording is assuming the hypothetical where the observers can see (and therefore measure) each other's clocks. The author saying that one clock is running twice as fast as the other is a measurement. But it's not running twice as fast. It's running 10/8 = 1.25 times as fast. If it was running twice as fast, then the age difference when the traveller returned would be greater than what it is.

You can only say how fast a clock is running once you have decided on your frame of reference. There is no absolute time, to which all specific frames of reference are subordinate.

Each of the two twins sees his own clock running twice as fast as the other twin's. Each of them thinks that 1.25 of that difference is due to time dilation experienced by the other twin, and that the remainder is due to the time taken for the signal to reach him.

Both are equivalent, and equally valid, if you don't want to privilege a particular frame of reference. In practice, of course, it feels a bit ridiculous to take the twin in the spaceship as being stationary and the rest of the universe as moving; but in principle there is no reason not to take that position, and the calculations work equally well - until one of the twins changes his frame of reference by accelerating.