Confused about the logic of a Special Relativity problem

[Nantes]

Hi guys! I'm a pharmacist who has been trying to understand how time dilation and Lorentz contraction and etc. work, out of pure curiosity. I have been reading a course by Michael Fowler, which I link the 4th section of: http://galileoandeinstein.physics.virginia.edu/lectures/sreltwins.html

In this section, he presents the problem of twins, one of which travels to Alpha Centauri (which is 4 light-years away) and back at relativistic speeds, and comes back younger than the other twin. It starts from the "How to give twins very different birthdays" part, in case you guys want to read the whole problem.

I'm puzzled by this declaration in bold in "What does he see?":

"To him, her outward journey of 4 light years’ distance at a speed of 0.6c takes her 4/0.6 years, or 80 months. BUT he doesn’t see her turn around until 4 years later, because of the time light takes to get back to Earth from alpha-centauri! In other words, he will actually see her aging at half his rate for 80 + 48 = 128 months, during which time he will see 64 flashes.

When he sees his sister turn around, she is already more than half way back!"

I can understand that, because at the moment the female twin in the spaceship turns around, the visual information of her turning around will take 4 years to reach Earth, during which time she will have traveled 60% of the distance this light traveled (since her speed is 60% that of light) and thus be 60% of the way back home already at the moment her earthbound twin sees her turning around. But then, suppose that, from the moment he sees her turning around, he doesn't do anything else but watch her coming back. At some point, when he's still seeing her part way towards Earth, her real self will reach the planet (since she's ahead of the light by a good margin). So he'd be able to meet with his sister face to face, but if he looked through his telescope, he would still be seeing her as being inside the spaceship on her way back... but that sounds absurd! Where is the flaw in my reasoning?

Thanks!

 

[Orodruin]

At some point, when he's still seeing her part way towards Earth, her real self will reach the planet (since she's ahead of the light by a good margin).

No, she is never ahead of the light. The light she is emitting while traveling back also never catches up with the light that was emitted when she turned around (both travel at light speed!). However, the light that she is emitting on the way back is going to appear blue-shifted to the twin back on Earth and so what the twin on Earth actually sees is the processes for the traveling twin occurring faster. Note that this goes in the opposite direction of the time dilation (which makes the traveling twin's time go slower), but the time dilation is compensated and more by the fact that the source (in this case the traveling twin) is moving towards the observer (the twin back on Earth) so that the light needs less and less time to arrive.

When dealing with relativity you must separate what an observer actually sees from what is actually going on in terms of coordinate time and space coordinates.

 

[Nantes]

This "redshifting of the light meaning that the process is observed as occurring faster" boggles me. So if the traveling twin waved her hand when she was coming back, the earthbound twin would see this motion occurring faster, as if he were fast-forwarding a movie? But how, if the light she emitted at each picosecond as she moved her hand never catches up with the light emitted the picosecond before it? Each photon would never catch up with its immediate neighbor and thus not transmit information any faster. How is the image accelerated?

 

[Orodruin]

If the light caught up with the light in front of it it would not be a fast forward movie, it would be a "everything happening at once"-movie...

Do you understand the non-relativistic Doppler effect?

 

[Mister T]

But then, suppose that, from the moment he sees her turning around, he doesn't do anything else but watch her coming back. At some point, when he's still seeing her part way towards Earth, her real self will reach the planet (since she's ahead of the light by a good margin).

Never is she ahead of the light. When he sees her turn around it means the light has reached him. At that time she's still headed home, so she's behind the light. If he stands there and watches he'll see a tie between the light she emits as she arrives and her arrival.

Where is the flaw in my reasoning?

It appears you were mixing up the time she turned around with the (later) time that the turn-around signal arrived home.

 

[Ibix]

This "redshifting of the light meaning that the process is observed as occurring faster" boggles me.

Ignore relativity entirely for the moment. Imagine a clock one light minute away and stationary with respect to you. At some point it reads 12.00. But, as you explained, you won't see that until 12.01 because it takes the light that bounced off the clock face at 12.00 one minute to reach you. Similarly, you'll see the clock reading 12.01 at 12.02, 12.02 at 12.03, and so on.

Now imagine the clock is coming towards you at half the speed of light. At 12.00 it's one light minute away from you - you see this at 12.01. At 12.01, it's moved half a light minute closer to you, so it's only half a light minute away and it only takes thirty seconds for the light from this to reach you. So you see the clock reading 12.01 at 12.01.30. Finally at 12.02 it's moved a whole light minute and is right on top of you - so there's no delay at all and you see 12.02 at 12.02. You've seen two whole minutes elapse on the clock in one minute.

But what happens next? At 12.03, the clock is now half a light minute away from you - so you won't see light from this until 12.03.30. At 12.04 the clock is a whole light minute away and you won't see it read 12.04 until 12.05. So it takes three minutes for you to see two minutes elapse on the clock.

This effect is called the Doppler effect, and is totally non-relativistic. Each successive tick of the clock happens one minute apart, but the lag in you seeing each one decreases as the clock approaches and increases as it goes away.

In relativity there is one additional complication: if you correct for the Doppler shift using the argument above, you'll find that the moving clock is ticking slower than a stationary clock. This is time dilation.

 

[Nantes]

Do you understand the non-relativistic Doppler effect?

I think so, such as the one for sound. When a moving body is emitting a sound, the sound waves get pressed closer together and thus the sound is perceived as being heightened in pitch. So you're saying light's photons are also closer together when parting from a moving object? I guess I never assumed this was possible, since it's not possible to accelerate light. I now realize however that the two concepts (waves being physically closer in space, and speed) are not related.

If he stands there and watches he'll see a tie between the light she emits as she arrives and her arrival.

Yes, I thought so too, as she gets really close to Earth, the light she emits reaches Earth almost instantly and her twin should be seeing that. But what happened to the light from before? Consider the following logic:

The distance between Earth and Alpha Centauri is X
The point where she turns around is Y

When light which parted from Y reaches Earth, the traveller is now at 0,4X (because she has transposed 60% of the way, or 0,6). The Earth twin keeps watching her. When he sees her at 0,6X (thus having transposed 40% of the way), the real her should be reaching Earth, because that's the distance that remained for her when Earth saw her at Y (but she was actually at 0,4X!).
Thus, the only way I can find this to reasonably make sense is if time dilation compensates for this, making the traveller slower and/or accelerating what Earth sees. But if that happened, it doesn't seem like she would be at 0,4X when Earth saw her at Y in the first place. Also, Lorentz contraction also applies, so it makes her path even shorter!

 

[Nantes]

This effect is called the Doppler effect, and is totally non-relativistic. Each successive tick of the clock happens one minute apart, but the lag in you seeing each one decreases as the clock approaches and increases as it goes away.

Right, that explains the time lag between you seeing the event and it actually occurring, but that shouldn't affect the light itself. Why does it turn bluer or redder in that case (a manifestation of its wavelength having been altered?). I believe it takes an unimaginably small amount of time for the photon to change direction as it bounces, and as light will always be faster than whatever it's bouncing off of, I don't see how the body being moving or stationary could affect this bouncing process in such a way as to change the light's wavelength.

 

[Nugatory]

I believe it takes an unimaginably small amount of time for the photon to change direction as it bounces, and as light will always be faster than whatever it's bouncing off of, I don't see how the body being moving or stationary could affect this bouncing process in such a way as to change the light's wavelength.

Light is not a stream of photons moving through space and reflection is not photons bouncing off the reflective surface. Light is an electromagnetic wave. (Photons don't show up until you get to quantum mechanics and start to look at the microscopic interactions between electromagnetic waves and subatomic particles).

Because light is a wave, the Doppler effect applies. If a wave is moving past me from left to right and you are moving from right to left relative to me, you will encounter more wave crests per second than I do so you will find that the wave has a higher frequency. The frequency and the wavelength are trivially related to the speed of the wave (speed equals frequency times wavelength) so the higher frequency implies a shorter wavelength, which is a blueshift.

 

[Mister T]

Moving at a speed of 0.6c, let's say that it's a 10 hour trip home for her, as measured in your frame of reference. It therefore takes light 6 hours to make the same trip. You arranged ahead of time that she would send you a flash of light at her turn-around point, when she begins her trip home.

You receive this flash at midnight, figure she must have sent it 6 hours ago, and so you figure she's already completed 6 hours of her trip, with only four hours to go. You expect her home at 4:00 am.

Now your puzzle is as you presented it to us. She has completed 60% of her journey, if she were to send another signal now, you'd get it in 2.4 hours, which would be 2:24 am. She's not scheduled to arrive for another 1.6 hours.

Suppose she knows you well enough to know exactly what you must be thinking and sends another signal when your clock reads 2:24. Since she's 1.6 hours out, the signal takes 0.96 hours to arrive.

Do you see that the time between the sendings of the light signals is not the same as the time between the receivings of the signals? This is what the others were explaining to you when they mentioned the Doppler effect.

Go to YouTube and search for "Paul Hewitt time dilation". The speed of the ship in that cartoon is 0.6c.

 

[Janus]

Right, that explains the time lag between you seeing the event and it actually occurring, but that shouldn't affect the light itself. Why does it turn bluer or redder in that case (a manifestation of its wavelength having been altered?). I believe it takes an unimaginably small amount of time for the photon to change direction as it bounces, and as light will always be faster than whatever it's bouncing off of, I don't see how the body being moving or stationary could affect this bouncing process in such a way as to change the light's wavelength.

Light being emitted from a source not moving with respect to the the observers (red and blue dots). It expands outward in circular waves with the distance between the waves being equal in all directions.

Light being emitted from a source moving wit respect to the observers. The light waves still expand outward as circles, but sine the source moves between the emission of one wave and the emission of the second wave, the center of each expanding wave is offset from the others, and the distance between the waves, thus the wavelength and frequency changing for the two observers. The red dot receives the waves at a lower frequency and the blue dot at a higher frequency.

 

[Orodruin]

When he sees her at 0,6X (thus having transposed 40% of the way), the real her should be reaching Earth, because that's the distance that remained for her when Earth saw her at Y (but she was actually at 0,4X!).

No, you are missing the fact that the closer she gets to Earth, the less time it takes for the light to reach Earth. The light will always be faster than her.

 

[Ibix]

I don't see how the body being moving or stationary could affect this bouncing process in such a way as to change the light's wavelength.

As others have said, think of light as a wave for now. The crest of one light wave is around 500nm ahead of the next one. In the time it takes light to move 500nm a ship doing 0.6c travels 300nm - so it's either crashed through that next crest already or is still running ahead of it. This is just like the clock example I gave earlier (with things happening early or late due to the motion of the ship), just on a shorter time scale.

The link to the quantum world is that wavelength is related to momentum and energy. Striking an object coming towards you is a higher energy process than striking one going away from you, so the energies (and hence wavelengths) of the "reflected" photons are different in those two circumstances. I've put scare quotes around "reflected" because it isn't a simple billiard-balls-bounce-off-the-cushion thing. I wouldn't worry about the details unless you want to go and take a quantum course.

 

[Nantes]

I've put scare quotes around "reflected" because it isn't a simple billiard-balls-bounce-off-the-cushion thing.

No, it never is xD

Thanks guys, I think I got it! Now, I've always had a nagging question: Since Lorentz contraction is an actual physical process that occurs and not an optical illusion that the person on another frame of reference (the stationary one) sees, isn't it dangerous for the traveling twin? With all her cells and organs being squashed and all...

 

[Mister T]

Each twin sees the other as being contracted. Each twin sees himself as not being contracted.

 

[Nantes]

Each twin sees the other as being contracted. Each twin sees himself as not being contracted.

But then isn't it an illusion seen by the traveling twin, since we know the stationary twin doesn't get contracted?

The traveling twin might not perceive herself as being contracted, but it's happening, right? Does it pose any danger to her physical integrity?

 

[Mister T]

But then isn't it an illusion seen by the traveling twin, since we know the stationary twin doesn't get contracted?

It's not an illusion. It's real. The stationary twin is contracted from the perspective of the traveling twin.

To someone passing by you right now at a speed of 087c, you are contracted to half your size in the direction of his motion. In the other two directions there is no contraction. You don't experience it because in your frame of reference you're at rest.

 

[Orodruin]

But then isn't it an illusion seen by the traveling twin, since we know the stationary twin doesn't get contracted?

The traveling twin might not perceive herself as being contracted, but it's happening, right? Does it pose any danger to her physical integrity?

No, you are currently length contracted by a large amount compared to many of the particles in the cosmic radiation. Do you feel any different?

The entire point is that both length contraction and time dilation are effects of how space and time are intertwined and behave in different frames. All inertial frames are equally valid in SR and whatever the physical process, a measurable outcome must be the same in all frames. This is one of the basic postulates in SR.

 

[Ibix]

Since Lorentz contraction is an actual physical process that occurs and not an optical illusion that the person on another frame of reference (the stationary one) sees, isn't it dangerous for the traveling twin? With all her cells and organs being squashed and all...

No. During the constant velocity phases of the journey, both twins are length contracted according to the other one. If length contraction were a problem, both would expect the other to be squished and themselves to be fine, which is contradictory.

One way to look at it is that length contraction happens at all scales. The traveller's cells are contracted by 50% (or whatever), but the atoms that make up the cells are also contracted by 50%. If you are being squashed with a heavy weight, it will hurt because there's nowhere for the matter inside you to go try to flow out sideways. But under length contraction, your atoms are 50% as long in the direction of motion and if you did not length contract on the macroscopic scale as well, there'd be extra space inside you - which would also be painful. Briefly.

That's not a particularly good explanation of length contraction (I'd suggest looking up the "block universe" for a better visualisation), but it does explain why length contraction doesn't hurt.

 

[Nantes]

I'm beginning to reach the conclusion that special relativity is probably out of my mind's league... the more you guys explain, the more questions pop up :-/

I'll just stop here lest my questions become infinite. Thank you very much everyone! I know these kinds of questions have probably been asked a hundred times before in this forum, and seeing you guys still take the time to explain it is heartwarming!

 

[peety]

I often find that confusion arises because relativistic problems have been mixed up with non-relativistic ones. It's easy to picture an approaching twin waving her hand once a minute. Each wave of the hand would take less time because she's getting nearer, so she seems to be waving more quickly than she is in reality. Strictly classical!

 

[Orodruin]

I often find that confusion arises because relativistic problems have been mixed up with non-relativistic ones. It's easy to picture an approaching twin waving her hand once a minute. Each wave of the hand would take less time because she's getting nearer, so she seems to be waving more quickly than she is in reality. Strictly classical!

But, as has already been mentioned in this thread, this is not the only effect going on at relativistic speeds. You have to add time dilation on top of that and unless you actually do the math it is not clear which will dominate (time dilation makes the waving look slower and the Doppler effect makes it look faster).

 

[Mister T]

I'm beginning to reach the conclusion that special relativity is probably out of my mind's league...

I don't think it's a case of it being out of your league. Everyone I know learns it the way you are learning it. You'll take a break, let the ideas soak in, and go back at it again. The notion that time and space are different from our current understanding of them is just too seductive of an intellectual concept to resist. There are far-reaching consequences to the notion that now here is not necessarily the same as now everywhere. It totally changes our understanding of the relationship between the past, present, and future.

 

[Ibix]

I'm beginning to reach the conclusion that special relativity is probably out of my mind's league... the more you guys explain, the more questions pop up :-/

Please keep asking questions - answering them is why we're here.

SR isn't, basically, incredibly complicated (although the implications can get quite complex). The reason it strains the brain it that it forces you to dump pre-conceptions that you probably developed as a toddler. It's not beyond you - it's just that there is an awful lot of un-learning to do before it becomes comfortable to think about it.

 

[Orodruin]

It's not beyond you - it's just that there is an awful lot of un-learning to do before it becomes comfortable to think about it.

Why come to mind this does?

 

[Ibix]

Why come to mind this does?

No idea of what you speak have I.

 

[Nantes]

Please keep asking questions - answering them is why we're here.

Since you insist... :)

This special on time by Brian Greene has a pretty interesting segment about there being different "slices" of space time. The segment only lasts about 5 mins. Click the video below, it should start around 18 mins in:


I'm intrigued by the whole thing about the alien moving towards Earth being able to see slices of the future. It would mean the future is already predetermined!

And even if somehow light from 200 years into the future were sent towards the alien (since apparently the effect is magnified with distance), it still would take 10 billion years to reach him. So he'd never be able to use that information to anything useful: in practice he'd be seeing 9 billion, 999 million, 999 thousand 800 years in the past. This is probably true for any distance as well: the future-seeing effect is always less than the time of its light to reach the distant observer, so in practice, he'll always be seeing the past (from Earth's point of view).

And let's exaggerate the magnitude of this effect for theory's sake: suppose an alien one light year away moving towards Earth could see a week into Earth's future. We could send him a message asking what's next's week's lottery numbers. Of course, we'd wait two years for the answer, so in practice the info would be useless. But if we took careful note of the time it takes for the message to come back to ensure the alien's not cheating (if he takes a week to reply, he could have received the lottery numbers normally and not from the future-seeing effect), as well as account for the position he'll actually be in when the message reaches him (since he's moving towards Earth at relativistic speeds) and all the time dilation effects, wouldn't we be able to prove that future is indeed predetermined?

I should probably read Greene's book, which should have this thing much better explained than the TV show.

 

[Orodruin]

This special on time by Brian Greene has a pretty interesting segment about there being different "slices" of space time. The segment only lasts about 5 mins:

But the video is 50 minutes long, it would help if you could refer to the actual segment - for example tell us at which time it starts.

I'm intrigued by the whole thing about the alien moving towards Earth being able to see slices of the future. All my knowledge so far tells me this is impossible. It would mean the future is already predetermined!

And even if somehow light from 200 years into the future were sent towards the alien (since apparently the effect is magnified with distance), it still would take 10 billion years to reach him.

I have not actually watched the video since I am not going to watch through 50 minutes of Brian Green, but there is a very common confusion about what "see" means in relativity. The alien will not actually see anything until the light reaches it. There is nothing in there which requires the world to be predetermined, the only relevant piece of information is that you can only affect events in your future light-cone and be affected by events from your past light-cone. This is saying nothing else than that you will not be able to influence what is going to happen in two days at a position which is 500 light-years away from you. It may not be "determined" but there is still nothing you can do about what is going to happen.

 

[Nantes]

But the video is 50 minutes long, it would help if you could refer to the actual segment - for example tell us at which time it star.

I did. Click the video thumbnail and should start in the appropriate timestamp. It does for me, at least, it's 18 minutes in.

The alien will not actually see anything until the light reaches it.

I know, which is why I said "So he'd never be able to use that information to anything useful: in practice he'd be seeing 9 billion, 999 million, 999 thousand 800 years in the past."

But what about the experiment about proving the future-seeing effect I described?

 

[Orodruin]

I did. Click the video thumbnail and should start in the appropriate timestamp. It does for me

It does not for me, it may be browser dependent. It is always good not to assume everyone's browser works the same and provide the information.

And let's exaggerate the magnitude of this effect for theory's sake: suppose an alien one light year away moving towards Earth could see a week into Earth's future.

I do not get it, you have said yourself that you agree that it cannot see into the future. The light needs to reach it.

 

[Orodruin]

I should probably read Greene's book, which should have this thing much better explained than the TV show.

Oh, and there is another thing you should think about before you go out to buy Greene's book: All popular science will only teach you about science, it will not teach you science. It can teach you about relativity, but it will give you the flashy version with the paradoxes and similar stuff, presenting the problems and solutions in a flashy way that is designed to make you go "whoaaaooooooo" rather than to learn anything about the theory itself. A lot of the misconceptions that people come here to ask question about are based on overthinking popularised science and extending the analogues that the popularisers make far beyond their region of validity. Bottom line is: You will not learn relativity by watching or reading Brian Greene. In order to do so, you need to pick up an actual course in relativity. A (free) resource such as @bcrowell 's Relativity for Poets would be a much better place to start than a popular book.

 

[Nantes]

I do not get it, you have said yourself that you agree that it cannot see into the future. The light needs to reach it.

First, thanks for the relativity for poets link!

Yes, from an utility standpoint he cannot. But let me put some dates to try and clarify what I mean by seeing into the "future" (dates are from Earth's perspective):

Message asking the alien what's next's weeks' lottery number (for, according to the video, his "now" slice of the Earth is one of the future, which we are pretending to be one week here) is sent on December 14th, 2015
Alien one light year away receives message on December 7th, 2016 our time (because he's seeing Earth one week into the future, thus he'll receive our message one week earlier than we would expect him to). The timestamp on the message still says December 14th, 2015
It will still take one week for him to be able to see December 21th 2015's lottery numbers, which is what we asked him to relay to us
He'll send us December 21th 2015's lottery numbers on December 14th, 2016
We'll receive the numbers on December 14th, 2017

Because we received not "next week's" lottery numbers, but rather the lottery numbers of 1 year, 11 months and 3 weeks ago (that is, December 21th 2015), in practice the numbers are not from the future, but in reality we'd be able to conclude the alien really was seeing the future at that time, as by all rights we should have received the message with the numbers of December 21th 2015's lottery on December 21th 2017 if the alien could not see into the future.

All that is, of course, assuming the alien is stationary (he'd have to be moving in reality) and completely disregarding relativistic effects, and both things would have to be accounted for/compensated for the real experiment.

 

[Mister T]

but in reality we'd be able to conclude the alien really was seeing the future at that time,

Our future, not the future. There is no such thing as a universal future. There are things that are in our future but in his past.

And he saw what would become our future, but he didn't see it until after it had already happened to us.

 

[Orodruin]

What you are seeing is really nothing else than the fact that "now" is not really well defined in relativity.

All that is, of course, assuming the alien is stationary (he'd have to be moving in reality) and completely disregarding relativistic effects, and both things would have to be accounted for/compensated for the real experiment.

This is also not really possible. You cannot cherrypick which relativistic effects you consider and you certainly cannot ignore that something is moving when it has to.

 

[Mister T]

This is also not really possible. You cannot cherrypick which relativistic effects you consider and you certainly cannot ignore that something is moving when it has to.

The video purposely chooses a scenario where the speed of the observer is nonrelativistic so that relative simultaneity is the only relativistic effect. The alien is many many light years away from our galaxy, if I recall correctly. I didn't watch the video recently, but I remember it. I thought that particular piece of it was well done.

In other words, the Lorentz transformation equation $\Delta t'=\gamma(\Delta t-\beta \Delta x)$ doesn't become $\Delta t'=\Delta t$ in the limit of low speed if $\Delta x$ is very very large.