Another Twins Paradox question

[Al68]

No, it represents how much the Earth twin aged during the experiment in the earth-twin's frame.

You're right here. But isn't that the definition of aging? How much time passes for a person in their own frame. Someone does't grow old according to time elapsed in a different frame. Someone grows old at the rate time elapses in their own frame. I'm talking about real aging here.

Also, my evaluation here is the same as the first half of the Twins Paradox. Is the differential aging not real unless the twins reunite? Sure, it is more convenient to have the twins reunite for observational purposes, but observation (in the macroscopic world) does not cause reality. This differential aging is a real phenomenom that occurs. It is not a figment of observation, and therefore exists in reality, even if it is not convenient to measure.

Maybe I should state this point more clearly. Actual aging of a person between two events, as measured in their own frame, is equal to the time elapsed between those events, as measured in their own frame. Is this not a valid definition of actual aging?

Also, in your resonse to RandallB, you say that "the traveller disagrees that at the moment he left earth, the star-clock read t=0 years..." Why would the traveler ever conclude that the star-clock would not read exactly the same time as the Earth clock, if the Earth and star are in the same inertial frame. If you say that at any event is simultaneous with t=0 in a particular frame, then the event is simultaneous with t=0 for any location in that frame. What if I specified that the star-clock reads t=0 when the ship leaves earth? And all three clocks are synchronized at the start. Why didn't I think of that?

How about my example where neither twin even has a clock? That made things simple, I thought. Your objection was based on the fact that a clock on Earth would appear to run slow to the ship's twin. But, that assumes that they have clocks. I was referring to total elapsed time for each twin in their own respective frame, between the time the ship left Earth and the time it reached the star, again as observed in their own frame, respectively, and they don't need clocks to figure that out. As in the Twins Paradox, each twin will always see the other's clock run slow. That's different than the elapsed time recorded on a clock. As in the Twin's paradox, the total elapsed time recorded on a clock will not always be observed to be less by a different reference frame, like the rate of time passing is.

And pervect, if you're still around, I agree that we have been going around in circles here. But now you acknowedge my original point of trying to have the distance specified in the frame of the twin on the ship. Even though you think this is not possible. That's a lot better than discussing whether or nor rigid rods exist, which is pointless and my own fault, I admit.

Thanks,
Alan

 

[JesseM]

You're right here. But isn't that the definition of aging? How much time passes for a person in their own frame. Someone does't grow old according to time elapsed in a different frame. Someone grows old at the rate time elapses in their own frame. I'm talking about real aging here.

But you're not talking about an individual twin's age, you're comparing the ages of two different twins, and to do that you must pick a definition of simultaneity. It is of course really true that when the Earth twin's clock reads 11.55 years, he has aged 11.55 years since the traveling twin departed. It is not "really true" (in some objective frame-invariant sense) that when the earth-twin's clock reads 11.55 years, he is older than the traveling twin is at the same moment. In the traveling twin's frame, the event of the earth-twin's clock reading 11.55 years is simultaneous with the event of the traveling twin's clock reading 23.09 years. And when the traveling twin's clock reads 23.09 years, he has really aged by 23.09 years since departing the earth-twin. Likewise, in the traveling twin's frame, when his clock reads 5.77 years, that is simultaneous with the event of the earth-twin's clock reading 2.89 years. And when that event occurs, the earth-twin has really aged by 2.89 years. So why is it any less valid to take the traveling twin's definition of which events are simultaneous rather than the Earth twin's?

Also, my evaluation here is the same as the first half of the Twins Paradox. Is the differential aging not real unless the twins reunite?

No, it isn't, not if by "differential aging" you mean an objective truth about which one has aged less before either one accelerates. Of course there is a definite truth about who has aged less in any given frame--and in the frame where the traveling twin was at rest and the Earth was moving, it is the earth-twin who aged less during the outbound leg. Only if they depart and the reunite can there be an objective truth about which one aged more, and by how much.

Sure, it is more convenient to have the twins reunite for observational purposes, but observation (in the macroscopic world) does not cause reality.

This isn't about "observation causing reality", it's about some quantities not being objective features of reality at all, but instead depending on your choice of coordinate system. Velocity is an example of this. If two objects are moving apart, in one frame the first object may be moving faster than the second, while in another frame the second may be moving faster than the first--would you say there is an objective truth about which one is moving faster? Likewise, with two different spatial coordinate systems, in one coordinate system an object may be at position x=2 and in another it may be at position x=5...would you say there is an objective truth about what its "true" x-coordinate is? Assuming your answer to both these questions is "no", why is it so hard for you to accept that the question of which of two spatially separated events (like the earth-twin's clock reading 5.77 years vs. the traveling twin's clock reading 5.77 years) happened earlier and which happened later? In relativity this is every bit as coordinate-dependent as the question of which of two objects has a larger velocity, or which has a larger x-coordinate.

Maybe I should state this point more clearly. Actual aging of a person between two events, as measured in their own frame, is equal to the time elapsed between those events, as measured in their own frame. Is this not a valid definition of actual aging?

This definition is fine for inertial observers, although it's better to just use the proper time along their wordline between the two events, because that will work for non-inertial observers who don't have a single frame of their own. In any case, even for inertial observers this definition tells you nothing about which of two spatially separated observers has aged more without making some assumption about simultaneity.

Also, in your resonse to RandallB, you say that "the traveller disagrees that at the moment he left earth, the star-clock read t=0 years..." Why would the traveler ever conclude that the star-clock would not read exactly the same time as the Earth clock, if the Earth and star are in the same inertial frame.

This question makes me think you don't understand the way simultaneity works in relativity. Two clocks which are synchronized in their own mutual rest frame will always be out-of-sync in other frames--were you unaware of this? This is just a consequence of Einstein's clock synchronization convention. His idea was that each observer should define space and time coordinates of events using local readings on a network of clocks and measuring-rods which are at rest relative to himself, and with all the clocks "synchronized" using the assumption that light travels at a constant speed in all directions in that observer's rest frame. This necessarily implies that different frames must disagree about simultaneity, as I discussed on this thread:

Suppose I'm on a rocket, and I have two clocks, one at the front of the rocket and one at the back, that I want to synchronize. If I assume that light travels at the same speed in all directions in the rocket's rest frame, then I can set off a flash at the midpoint of the two clocks, and set them to both read the same time at the moment the light reaches them. But now imagine you are in another frame, one in which the rocket is moving forward with some positive velocity. In your frame, the back of the rocket will be moving towards the point in space where the flash was set off, while the front of the rocket will be moving away from it; therefore if you assume light travels at the same speed in both directions in your frame, you will naturally conclude the light must catch up with the clock at the back at an earlier time than it catches up with the clock at the front, since both were at equal distances from the midpoint of the rocket when the flash was set off there. This means that you will judge my two clocks to be out-of-sync if I use the above synchronization procedure, with the back clock ahead of the front clock in your frame.

More technically, you can show using the Lorentz transformation that if two events are simultaneous and 10 light years apart in one frame, then in another frame which is moving at 0.866c relative to the first, one of the events must have happened 8.66 years after the other. Are you familiar with how to use the Lorentz transformation?

If you say that at any event is simultaneous with t=0 in a particular frame, then the event is simultaneous with t=0 for any location in that frame. What if I specified that the star-clock reads t=0 when the ship leaves earth? And all three clocks are synchronized at the start. Why didn't I think of that?

They can only all be synchronized in one frame. In every other frame, the star-clock must either read an earlier or later time than the earth-clock at the moment the earth-clock reads t=0.

How about my example where neither twin even has a clock? That made things simple, I thought. Your objection was based on the fact that a clock on Earth would appear to run slow to the ship's twin. But, that assumes that they have clocks. I was referring to total elapsed time for each twin in their own respective frame, and they don't need clocks to figure that out.

But I don't disagree about the "total elapsed time for each twin in their own respective frame", I just disagree that this tells you anything about which twin aged more in an objective, frame-independent way. When 11.5 years have passed for the earth-twin, then in his frame this is the "same moment" that the traveling twin is reaching the star, but in other frames the traveling twin either has yet to reach the star or already reached it long ago.

To make sense of your claims about who has aged more, it would really help if you would answer this question:

Conceptually, it might also help to make the experiment more symmetrical. Suppose instead of the ship traveling to a star 10 light years from the earth, the ship is traveling along a measuring rod which has one end at the Earth and is at rest relative to earth, and is 10 light years long in the Earth's frame. Now suppose the ship itself is also attached to one end of a measuring rod that's at rest relative to the ship, and extends in the opposite direction as the Earth's measuring rod, and is 10 light years long in the ship's frame. So in this way, during a single trip we can be doing two separate but symmetrical experiments, one where we see how long it takes the ship to reach the far end of the Earth's measuring-rod, and another to see how long it takes the Earth to reach the far end of the ship's measuring rod. In this case the answers for the local readings will be the same--at the time the ship reaches the end of the Earth's measuring rod, the ship's own clock reads t = 5.77 years, and at the time the Earth reaches the end of the ship's measuring rod, the Earth's own clock reads t = 5.77 years. But then for each experiment, we could ask a question analogous to the one you ask in your experiment, namely:

1. How much time passes on Earth during the experiment of the ship traveling from one end of the Earth's measuring-rod to the other?

2. How much time passes on the ship during the experiment of the Earth traveling from one end of the ship's measuring-rod to the other?

How would you answer these questions?

If you used the same logic as you use above, it seems to me you'd have to conclude that the traveling twin aged less than the earth-twin in #1, and that the earth-twin aged less than the traveling twin in #2, despite the fact that both these experiments can be carried out in a single journey, with each twin moving alongside the other twin's measuring rod at the same time. Is that indeed what you'd say?

 

[Al68]

You're right here. But isn't that the definition of aging? How much time passes for a person in their own frame. Someone does't grow old according to time elapsed in a different frame. Someone grows old at the rate time elapses in their own frame. I'm talking about real aging here.

But you're not talking about an individual twin's age, you're comparing the ages of two different twins, and to do that you must pick a definition of simultaneity.

I'm not talking about either one. Of course you can never compare the ages of people in different rest frames at any given point in time. Not in any objective way, as you point out. My #1 and #2 statements earlier were not comparing their ages at any given point in time. I was stating how much each twin aged during the experiment.

It is of course really true that when the Earth twin's clock reads 11.55 years, he has aged 11.55 years since the traveling twin departed. It is not "really true" (in some objective frame-invariant sense) that when the earth-twin's clock reads 11.55 years, he is older than the traveling twin is at the same moment. In the traveling twin's frame, the event of the earth-twin's clock reading 11.55 years is simultaneous with the event of the traveling twin's clock reading 23.09 years. And when the traveling twin's clock reads 23.09 years, he has really aged by 23.09 years since departing the earth-twin. Likewise, in the traveling twin's frame, when his clock reads 5.77 years, that is simultaneous with the event of the earth-twin's clock reading 2.89 years. And when that event occurs, the earth-twin has really aged by 2.89 years. So why is it any less valid to take the traveling twin's definition of which events are simultaneous rather than the Earth twin's?.

I agree with all of your statements here. I wasn't choosing either twin's version of which events were simultaneous. I never said that the event of the Earth twin t=11.55 years was simultaneous with the event of the ship's twin's t=5.77 years. These two events are not simultaneous.

How about my example where neither twin even has a clock? That made things simple, I thought. Your objection was based on the fact that a clock on Earth would appear to run slow to the ship's twin. But, that assumes that they have clocks. I was referring to total elapsed time for each twin in their own respective frame, and they don't need clocks to figure that out.

But I don't disagree about the "total elapsed time for each twin in their own respective frame", I just disagree that this tells you anything about which twin aged more in an objective, frame-independent way.

It tells you a lot if we are talking about how much each twin aged during the experiment. It tells us nothing if we are trying to figure out how much each twin has aged at a particular moment in time, since that moment would not be simultaneous in both frames. Again, I am not claiming anything about how old either twin is at any particular time.

As far as the star clock, the ship's twin would agree that it read t=0 when he left earth, since he was in the same frame when they were synchronized. That doesn't mean that the clocks (on Earth and the star) will always read the same as seen by the ship. I didn't mean to imply that. I was just pointing out that all three clocks were at rest relative to each other when t=0. At least that's the way I read RandallB's post. And I did not mean to imply that I agreed with the rest of his post, either.

I just reread my earlier post, so I need to edit this part. I should have said that the ship would agree that the star-clock read t=0 the moment prior to leaving earth, but not the moment after. If you assume instaneous acceleration, the exact time the ship left Earth could be interpreted as either of these moments, since the ship is both at rest and in relative motion to Earth at t=0. So, you were right, since you were referring to t=0 after the ship was moving relative to earth. This t=0 will no longer be simultaneous at Earth and the star after the ship starts moving.

To make sense of your claims about who has aged more, it would really help if you would answer this question:
Quote:
Conceptually, it might also help to make the experiment more symmetrical. Suppose instead of the ship traveling to a star 10 light years from the earth, the ship is traveling along a measuring rod which has one end at the Earth and is at rest relative to earth, and is 10 light years long in the Earth's frame. Now suppose the ship itself is also attached to one end of a measuring rod that's at rest relative to the ship, and extends in the opposite direction as the Earth's measuring rod, and is 10 light years long in the ship's frame. So in this way, during a single trip we can be doing two separate but symmetrical experiments, one where we see how long it takes the ship to reach the far end of the Earth's measuring-rod, and another to see how long it takes the Earth to reach the far end of the ship's measuring rod. In this case the answers for the local readings will be the same--at the time the ship reaches the end of the Earth's measuring rod, the ship's own clock reads t = 5.77 years, and at the time the Earth reaches the end of the ship's measuring rod, the Earth's own clock reads t = 5.77 years. But then for each experiment, we could ask a question analogous to the one you ask in your experiment, namely:

1. How much time passes on Earth during the experiment of the ship traveling from one end of the Earth's measuring-rod to the other?

2. How much time passes on the ship during the experiment of the Earth traveling from one end of the ship's measuring-rod to the other?

How would you answer these questions?
If you used the same logic as you use above, it seems to me you'd have to conclude that the traveling twin aged less than the earth-twin in #1, and that the earth-twin aged less than the traveling twin in #2, despite the fact that both these experiments can be carried out in a single journey, with each twin moving alongside the other twin's measuring rod at the same time. Is that indeed what you'd say?

Yes, that is what I'd say. Except the part about "each twin moving alongside the other twin's measuring rod at the same time". These two experiments would not end simultaneously in either frame. So they wouldn't be "at the same time". Notice that the Earth twin would see the end of the ship's measuring rod pass Earth at t=5.77 years, and the ship reach the end of the Earth's measuring rod at t=11.5 years. And the ship's twin would see the ship reach the end of the Earth's rod at t=5.77 years, and the Earth reach the end of the ship's rod at t=11.5 years. In other words, your two experiments would not be simultaneous, but would be symmetrical.

And we have to notice that the conclusions you made (on my behalf), and that I agree with, are not contradictary at all. The events that define the ends of your two experiments are not simultaneous in either frame. Yes, the ship's twin will age less than Earth's twin during one of the experiments, and the Earth's twin will age less than the ship's twin during the other. You could not conclude that was contradictary unless you say the experiments are both symmetric and simultaneous. These two experiments would not occur simultaneously in any frame. And notice that the conclusions here are symmetrical.

And notice that I am not saying (in my experiment) that the event of t=5.77 on the ship clock is simultaneous with the event of t=11.5 years on the Earth clock. These clock readings represent how much each twin aged during the experiment, as observed by each twin. These clock readings do not represent represent moments in time that are simultaneous.

And I agree that both twins would have to come to relative rest to ever say one was older than the other at a particular time. Otherwise you could never define this particular time. But they would not have to come to rest to say that one aged more than the other during the experiment.

Thanks,
Alan

 

[RandallB]

They will both read the same time in the earth/star frame, of course. In this example T would be 11.55 years. But the two clocks will not read 11.55 years simultaneously in the traveller's own rest frame.

Well duh, that's the point of simultaneity – what I don’t get is why you refuse to or are unable to identify exactly WHERE and WHEN each of those two Earth-Star frame clocks can be seen in the traveler frame as 11.55.

No, because the traveller disagrees that at the moment he left earth, the star-clock read t=0 years. Again, in the traveller's frame the star-clock already read t = 8.66 years at the moment he left earth, and since it read 11.55 years at the moment he arrived at the star, he will say the star-clock only advanced forward by 11.55 - 8.66 = 2.89 years from the time he left Earth to the time he arrived, while his own clock advanced by 5.77 years between these events.

Balderdash, now that is gibberish and an unfounded arbitrary assumption within the stated problem. By what observation in this example did the traveler SEE the star-clock is at 8.6 the only clock to be seen in the earth-star frame by the traveler is the one there at Earth t=0 just like his t’=0. But what the traveler can do is send a light/radio signal back to all the other travelers behind him – stationary is his frame – to advise them of what t & t’ where when he passed (or Earth passed by him) both “0”.

None of this really helps me in understanding what you meant when you said "What is not simultaneous is the point and time Earth will observe locally going by in the traveling frame", though.

Let's see, how many different places can an Earth bound traveler see in a two reference frame problem? The Earth observer knows he is at location “0” and t= 11.55 in the Earth frame so what else is there to look at! HOW ABOUT THE TIME AND LOCATION OBSERVABLE IN THE OTHER FRAME – the traveler frame! At least put a clock over there, connected by a long measuring rod to the traveler if not another traveler – WHERE and WHEN is it. (Extended translation: What time does a traveler frame clock synchronized with the traveler within their frame read and how far away is it from the traveler as measured within the traveler frame viewable from Earth at Earth t=11.55). There is only room for one clock there, it must have a specific time and location in the traveler frame! So ALAN what is the time and location seen there? What conclusions will be forwarded up to the lead traveler based on same observation that can be seen from either frame?

 

[JesseM]

Well duh, that's the point of simultaneity – what I don’t get is why you refuse to or are unable to identify exactly WHERE and WHEN each of those two Earth-Star frame clocks can be seen in the traveler frame as 11.55.

What are you talking about? I didn't "refuse" to answer this question, you never asked it to me. The answer is that when the traveller's clock reads 11.55, since the traveller observes the Earth clock to be ticking at half the rate of his own clock, he will observe the earth-clock to read 5.77 years. And since the traveller observed the star-clock to start out at 8.66 years, and to be ticking at the same rate as the earth-clock, he will observe the star-clock to read 8.66 + 5.77 = 14.43 years.

No, because the traveller disagrees that at the moment he left earth, the star-clock read t=0 years. Again, in the traveller's frame the star-clock already read t = 8.66 years at the moment he left earth, and since it read 11.55 years at the moment he arrived at the star, he will say the star-clock only advanced forward by 11.55 - 8.66 = 2.89 years from the time he left Earth to the time he arrived, while his own clock advanced by 5.77 years between these events.

Balderdash, now that is gibberish and an unfounded arbitrary assumption within the stated problem. By what observation in this example did the traveler SEE the star-clock is at 8.6 the only clock to be seen in the earth-star frame by the traveler is the one there at Earth t=0 just like his t’=0.

I wasn't talking about what the traveller SEES if he looks around in his immediate vicinity--I try to avoid using the word "sees", instead using the word "observes" which in relativity is usually understood to just mean what is true about what's happening in that observer's rest frame (with 'frame' understood as a coordinate system filling all of space and time), not what he's actually seeing using light-signals...for example, the rate that I "see" a moving clock ticking is different from the rate I "observe" it ticking, because what I "see" is affected by doppler shift as well as time dilation. In this particular quote which you're responding to, I didn't use the word "see" or "observe", I just talked about what is true "in the traveller's frame" so there would be no ambiguity. And it is true that in the traveller's frame, the event of the star-clock reading 8.66 years happens at a time-coordinate of t' = 0 in terms of his frame's time-coordinate. Do you deny this? If so, try doing a Lorentz transformation starting from the coordinates of this event in the earth/star frame (ie x = 10 light years, t = 8.66 years).

But what the traveler can do is send a light/radio signal back to all the other travelers behind him – stationary is his frame – to advise them of what t & t’ where when he passed (or Earth passed by him) both “0”.

Well, likewise, if the traveler has a clock that is 5 light years ahead of him, at rest with respect to him and synchronized with his own clock according to the Einstein clock synchronization convention, then when that clock reads t'=0 years, it will be passing by the star and noting that the star's clock reads t=8.66 years. It could radio that observation to the traveller, and that's one way for the traveller to verify that the star-clock read 8.66 years at the moment he left in his own frame (but he could also figure that out without an actual physical clock in front of him, just by doing calculations based on when he observed the star-clock reading 8.66 years in his telescope, or based on backtracking from the time he observed when he passed the star clock and the rate he observed it ticking).

None of this really helps me in understanding what you meant when you said "What is not simultaneous is the point and time Earth will observe locally going by in the traveling frame", though.

Lets see, how many different places can an Earth bound traveler see in a two reference frame problem? The Earth observer knows he is at location “0” and t= 11.55 in the Earth frame so what else is there to look at! HOW ABOUT THE TIME AND LOCATION OBSERVABLE IN THE OTHER FRAME – the traveler frame! At least put a clock over there, connected by a long measuring rod to the traveler if not another traveler – WHERE and WHEN is it. (Extended translation: What time does a traveler frame clock synchronized with the traveler within their frame read and how far away is it from the traveler as measured within the traveler frame viewable from Earth at Earth t=11.55).

OK, I think I see what you're saying now, you're specifying that all statements about when things happen in a frame must be based on local measurements by clocks that are synchronized in that frame. If that's what you mean, it isn't obvious to me how that original sentence was supposed to imply that, but never mind. Anyway, in principle I agree all statements about times of distant events in a particular frame should be describable this way, although it isn't completely necessary since you can also figure out the times by calculation, based on backtracking from the time you received the light from an event.

As for the question of what a second clock synchronized with the traveller's will read when it passes by the Earth and the earth-clock reads t = 11.55 years, the answer is that it will read 23.09 years.

I am not really sure what the point of this discussion is--can you explain again what you were disagreeing with in the first post of mine that you responded to?

 

[JesseM]

I'm not talking about either one. Of course you can never compare the ages of people in different rest frames at any given point in time. Not in any objective way, as you point out. My #1 and #2 statements earlier were not comparing their ages at any given point in time. I was stating how much each twin aged during the experiment.

But to answer how much the Earth twin aged "during the experiment", you have to identify what point on the Earth twin's worldline corresponds to the end of the experiment, and you can't do that without picking a definition of simultaneity.

I think we may not be having any disagreement on substantive issues here, just over the meaning of the phrase "during the experiment". You seem to feel that the question "how much does the earth-twin age during the experiment" automatically implies we define the end of the experiment in terms of the definition of simultaneity in the earth-twin's own frame. I don't think so, I think it makes perfect sense to ask a question like "how much does the earth-twin age during the experiment in the traveller's frame"--would you say this question is incoherent or unnatural somehow?

You earlier seemed to make the argument that we should define things this way because "isn't that the definition of aging? How much time passes for a person in their own frame. Someone does't grow old according to time elapsed in a different frame. Someone grows old at the rate time elapses in their own frame. I'm talking about real aging here." I would basically agree with this, but for me the question of how much the earth-twin aged "during the experiment" is broken up into two steps:

1. Find the points on the earth-twin's worldline that correspond to the beginning and the end of the experiment

2. Find the proper time along the earth-twin's worldline between these two points

If the earth-twin was moving inertially, then #2 is equivalent to finding the amount of time between these two points in the earth-twin's own rest frame. So while I agree that you need to use the earth-twin's own frame to figure out how much he aged in #2, that doesn't imply that you also have to use the earth-twin's definition of simultaneity to figure out the point on his worldline that's "the end of the experiment" in #1. For example, if someone asked you "how much did the earth-twin age during the experiment in the traveller's frame", that means in #1 you'd use the traveller's definition of simultaneity to figure out which point on the earth-twin's worldline corresponds to the end of the experiment, but then in #2 you'd use the earth-twin's own frame to find out how much he aged between these two points. So your argument "Someone grows old at the rate time elapses in their own frame" is true but as long as we use the earth-twin's own frame in #2 I think it is being satisfied, regardless of whose frame we use to define "the end of the experiment" in #1.

It tells you a lot if we are talking about how much each twin aged during the experiment. It tells us nothing if we are trying to figure out how much each twin has aged at a particular moment in time, since that moment would not be simultaneous in both frames. Again, I am not claiming anything about how old either twin is at any particular time.

But how can you define "during the experiment" without specifying what moment in time on Earth corresponds to the end of the experiment?

As far as the star clock, the ship's twin would agree that it read t=0 when he left earth, since he was in the same frame when they were synchronized.

I thought you said earlier the traveling twin was just passing by the Earth at constant velocity on his way to the star, rather than accelerating from the earth. In post #33 you had said:

A spaceship plans to go to the nearest star system a distance of 10 light years away, at speed v = 0.866c. This spaceship first travels in the opposite direction from this star system, then turns around and passes Earth at speed v = 0.866c and both twins start their clocks at this time. Since they are not separated by any distance in the direction of travel, this will be t=0 for both twins. There will be no acceleration of the spaceship between Earth and the nearest star system.

In any case, a frame cannot accelerate, so I could rephrase that by saying "in the inertial frame where the traveller is at rest during his trip between Earth and the star, the star-clock read 8.66 years at the moment the earth-clock read 0 years".

Yes, that is what I'd say. Except the part about "each twin moving alongside the other twin's measuring rod at the same time". These two experiments would not end simultaneously in either frame.

I agree, I just meant that whichever frame you choose, there will be some duration of time between the event of the ship and Earth initially departing and the event of one of them reaching the end of the other one's measuring-rod, and for that duration they will both be moving alongside each other's measuring-rod "at the same time".

And we have to notice that the conclusions you made (on my behalf), and that I agree with, are not contradictary at all. The events that define the ends of your two experiments are not simultaneous in either frame. Yes, the ship's twin will age less than Earth's twin during one of the experiments, and the Earth's twin will age less than the ship's twin during the other. You could not conclude that was contradictary unless you say the experiments are both symmetric and simultaneous. These two experiments would not occur simultaneously in any frame. And notice that the conclusions here are symmetrical.

And notice that I am not saying (in my experiment) that the event of t=5.77 on the ship clock is simultaneous with the event of t=11.5 years on the Earth clock. These clock readings represent how much each twin aged during the experiment, as observed by each twin. These clock readings do not represent represent moments in time that are simultaneous.

And I agree that both twins would have to come to relative rest to ever say one was older than the other at a particular time. Otherwise you could never define this particular time. But they would not have to come to rest to say that one aged more than the other during the experiment.

OK, as long as you aren't claiming there's a single objective truth about who is aging more slowly in this type of experiment, then like I said I don't think we're really having any substantive disagreements over the physics of SR, just over semantic issues like what it means to ask how much the earth-twin aged "during the experiment".

 

[Al68]

Jesse,

First of all, you are right about the star-clock. There has been so much discussion of different experiments that I lost track of which one RandallB was talking about. He was talking about my example where the ship was already moving relative to earth. So that was my mistake.

Also, I agree that we are not disagreeing about any relevant facts about my example. We only disagree about how relevant each fact is.

Another way to point out the asymmetry in my experiment is this:

The ship's twin will say, "during the experiment, as measured in my frame, I aged 5.77 years and my brother aged 2.89 years."

The Earth twin will say, "during the experiment, as measured in my frame, I aged 11.5 years and my brother aged 5.77 years."

Each twin will agree that the other is telling the truth.

I think these statements incorporate your points and mine, and show that the experiment is still asymmetrical. We could add more true statements, and the asymmetry would still exist. My main point was that the experiment is asymmetrical, even without acceleration.

Do you agree with this?

Thanks,
Alan

 

[JesseM]

Jesse,

First of all, you are right about the star-clock. There has been so much discussion of different experiments that I lost track of which one RandallB was talking about. He was talking about my example where the ship was already moving relative to earth. So that was my mistake.

Also, I agree that we are not disagreeing about any relevant facts about my example. We only disagree about how relevant each fact is.

Another way to point out the asymmetry in my experiment is this:

The ship's twin will say, "during the experiment, as measured in my frame, I aged 5.77 years and my brother aged 2.89 years."

The Earth twin will say, "during the experiment, as measured in my frame, I aged 11.5 years and my brother aged 5.77 years."

Each twin will agree that the other is telling the truth.

I think these statements incorporate your points and mine, and show that the experiment is still asymmetrical. We could add more true statements, and the asymmetry would still exist. My main point was that the experiment is asymmetrical, even without acceleration.

Do you agree with this?

Thanks,
Alan

Yup, I agree it is asymmetrical in this sense. The asymmetry here can be understood as a consequence of the fact that the distance each sees the other move cannot be symmetrical, since the traveling twin sees the distance between the Earth and the star Lorentz-contracted. If instead each looked at the time for the Earth to pass by a buoy which was behind the ship and at rest relative to it, then the asymmetry would be in the opposite direction--the ship twin's value for how much he aged would be twice the Earth twin's value.

 

[Al68]

Yup, I agree it is asymmetrical in this sense. The asymmetry here can be understood as a consequence of the fact that the distance each sees the other move cannot be symmetrical, since the traveling twin sees the distance between the Earth and the star Lorentz-contracted. If instead each looked at the time for the Earth to pass by a buoy which was behind the ship and at rest relative to it, then the asymmetry would be in the opposite direction--the ship twin's value for how much he aged would be twice the Earth twin's value.

Thanks, Jesse. What you just stated was exactly what my initial point was. Maybe I should have worded it the way you did.

That being said, it's not even that important. If I had known it would take up 4 forum pages, I would have never brought it up.

I only wanted to make this point because this asymmetry is usually ignored in the Twins Paradox.

Thanks,
Alan

 

[RandallB]

The answer is that when the traveller's clock reads 11.55, since the traveller observes the Earth clock to be ticking at half the rate of his own clock, he will observe the earth-clock to read 5.77 years. And since the traveller observed the star-clock to start out at 8.66 years, and to be ticking at the same rate as the earth-clock, he will observe the star-clock to read 8.66 + 5.77 = 14.43 years.

I was specific in pointing the need for knowing WHERE and WHEN in the traveling frame for BOTH clocks in the Earth Star Frame. Again you only work the star clock. I’m only complaining about leaving out the easy details such as where and when in the traveling frame is the EARTH CLOCK at t = 11.55.

'Yada – Yada
Yada – Yada - Yada'
As for the question of what a second clock synchronized with the traveler’s will read when it passes by the Earth and the earth-clock reads t = 11.55 years, the answer is that it will read 23.09 years.

Well finally at least you identified the WHEN; but what about the WHERE in the traveler frame for this clock that reads 23.09 years. We know it’s at distance “0” in the Earth Frame.

Alan are you working these numbers, would this distance be the length of the measuring rod you were referring to earlier?

At least there is enough here to see that this means the Earth frame can clearly report that t’ = 23.09 years on the traveler clock near Earth is simultaneous with t’ = 5.77 years on the traveler clock near the star. OBVIOUSLY the traveling frame has screwed up synchronization – right?.

BUT Alan, work out the details on that rod length your were thinking about. Where is the Earth in relation to the traveler when t’= 5.77 years?? Once you know where the Earth is at you can have another observer in traveler frame there with a clock at t’=5.77 look over to the Earth to see WHEN that is in the Earth frame.
You shouldn’t be surprised to see that now it is the Earth frame that has lost touch with keeping things synchronized.
An “Asymmetrical disagreement” if you like.
BUT that is not the point.
And nether is which twin in the one way trip here is “really” younger or older!
Einstein’s point here is that what you perceive to be simultaneous between things separated by any distance, even just across the room from your is only within your own frame of reference and is not “real” and does not need to be “real”. IF you demand that some frame be a reference that is best described as a preferred frame of reference as in Lorentz Relativity "LR" not SR.

TO get a better ‘feel’ for these I recommend detailing out accurate “When” and “Where” information in every frame you use for everything used in an exsample. IMO using the ‘ability’ to SEE or OBSERVE clocks at great distances can to easily cause problems in understanding and doesn’t really make sense anyway. With any distance you must have a delayed observation, which is in effect local observer sending a report to you.

Keep working the details and it will get clearer.

 

[Al68]

Alan are you working these numbers, would this distance be the length of the measuring rod you were referring to earlier?

At least there is enough here to see that this means the Earth frame can clearly report that t’ = 23.09 years on the traveler clock near Earth is simultaneous with t’ = 5.77 years on the traveler clock near the star. OBVIOUSLY the traveling frame has screwed up synchronization – right?.

BUT Alan, work out the details on that rod length your were thinking about. Where is the Earth in relation to the traveler when t’= 5.77 years?? Once you know where the Earth is at you can have another observer in traveler frame there with a clock at t’=5.77 look over to the Earth to see WHEN that is in the Earth frame.
You shouldn’t be surprised to see that now it is the Earth frame that has lost touch with keeping things synchronized.
An “Asymmetrical disagreement” if you like.
BUT that is not the point.
And nether is which twin in the one way trip here is “really” younger or older!
Einstein’s point here is that what you perceive to be simultaneous between things separated by any distance, even just across the room from your is only within your own frame of reference and is not “real” and does not need to be “real”. IF you demand that some frame be a reference that is best described as a preferred frame of reference as in Lorentz Relativity "LR" not SR.

TO get a better ‘feel’ for these I recommend detailing out accurate “When” and “Where” information in every frame you use for everything used in an exsample. IMO using the ‘ability’ to SEE or OBSERVE clocks at great distances can to easily cause problems in understanding and doesn’t really make sense anyway. With any distance you must have a delayed observation, which is in effect local observer sending a report to you.

Keep working the details and it will get clearer.

Randall,

Although it looks like we've got my point resolved, you seem to be hinting at something else. Is there some other important point to be made here?

Just because I kept trying to redirect the discussion to my point doesn't mean that I'm not aware of other aspects of SR and the Twins Paradox that are more important.

I just didn't want to get sidetracked discussing points that have been addressed extensively in discussions of the Twins Paradox and SR. Are you referring to something else here?

Thanks,
Alan

 

[pervect]

I only wanted to make this point because this asymmetry is usually ignored in the Twins Paradox.

Alan

That's because there isn't any.

 

[Al68]

I only wanted to make this point because this asymmetry is usually ignored in the Twins Paradox.

Alan

That's because there isn't any.

pervect,

You do not think it's asymmetric that, in the Twins Paradox, one twin travels a longer distance than the other (as measureed in each twin's respective frame)?

Or do you just think that this point is irrelevant or unimportant?

Just curious.

Thanks,
Alan

 

[JesseM]

pervect,

You do not think it's asymmetric that, in the Twins Paradox, one twin travels a longer distance than the other (as measureed in each twin's respective frame)?

Or do you just think that this point is irrelevant or unimportant?

Just curious.

Thanks,
Alan

I think this point depends on what you and pervect mean by "asymmetry". Usual when physicists talk about symmetries they are talking about the laws of physics, and there is no asymmetry in how the laws of physics work in the two frames. But if you like you can say there is an "asymmetry" in the details of how this particular physical setup appears in the two frames; you don't even really have to get into the times to see this, you can just note that in one frame there's two parts of the system moving and the one in the middle at rest, while in the other frame two parts are at rest and one is moving between them (this 'asymmetry' would be present in the Newtonian version of the problem too).

 

[RandallB]

Randall
Although it looks like we've got my point resolved, you seem to be hinting at something else. Is there some other important point to be made here?
Are you referring to something else here?

My main concern was that IMO you were correct in looking for additional details related to the twins such as what amounted to the when and where the Earth was when the star saw the traveler arrive. When you were told that some of that information was arbitrary or unknowable I just disagreed and didn’t want you to be sidetracked from following a productive oath of your own choice. For any location and time in a reference frame there is one and only one time and place for it in another defined reference frame. There is no reason you cannot determine that kind helpful information exactly and there is nothing arbitrary in it.

As to “something else here” – that depends on where you want to go beyond the twins.
You can already see from looking at simultaneous events in the earth-star frame you find from complete SR detail it reveals t’= 5.77 and t’= 23.09 are. A paradox you can duplicate from the t’ frame when you looking at the earth-star t frame.
I don’t think your asymmetry point is the key.
Rather you might ask the question, can you trust your frame to correctly tell you if in reality two events separated by distance are simultaneous or not??

Based on what you’ve learned from the Twins, IMO I think you must say NO, and this was Einstein’s point.
Thus moving beyond the twins you might ask: ‘Can I define a "preferred frame" where I can determine if two separated events are REALLY simultaneous?’

Myself, I’m committed to SR and would say NO again, as it would take something very solid for me to reconsider LR “Lorentz Relativity” and the preferred reference frame used there.
So this could be “something else” for you to consider / work on / think about; beyond the twins, maybe open another post after you consider and research it a bit if you like.

You seem to have the twins issue fairly well in hand at this point,
Look on Twins/SR as a tool.
RB

 

[Al68]

RandallB,

OK, I thought you might be hinting at something else. I had other reasons to pursue my asymmetry point. Which is why I kind of ignored the other points presented. I just wanted to return to your points because it seemed like you were hinting at something else that hasn't been mentioned.

Thanks,
Alan

 

[Physical_Anarchist]

The reason why many people can't understand the Twins Paradox is simply because it doesn't make sense.
Never fear, I am here to cut through all the cr4p and tell it like it is.

No, I'm no physicist or anything, but even I can see that there's no logic to the explanations offered here or anywhere. Take this one for instance:
http://sciam.com/print_version.cfm?articleID=000BA7D8-2FB2-1E6D-A98A809EC5880105
It's clear from the word go that if you use your conclusion in the proof, you will at the end get the conclusion you were looking for. Let me sum up this especially pathetic attempt:
1.The star is 6 light-years away.
2.The trip takes 10 years (to the one staying at home).
3.The trip feels only like 8 years, because of length contraction.
4.Length contraction = Time Dilation
5.We're supposedly trying to prove time dilation!
6.Return trip, 10 years.
7.Again, feels like 8 years for some reason.
8. 8 + 8 = 16 < 20 = 10 + 10
9.Throw in useless Doppler shifts to confuse tired brains...
10.Conclude that you've proven your point.

Seriously, though, let's define the problem before attemping to solve it shall we?
I'm no expert, so feel free to correct me here but the root of the problem arises from some strange property of light: it passes you by at the constant speed of 299 792 458 m/s. Even when you travel at 100 000 000 m/s relative to your friend, the light passes him by at 299 792 458 m/s and it also passes you by at 299 792 458 m/s. The answer then is time dilation. It allows you to become slower even while you're traveling fast, so that you can see light pass you by at the same rate as before. Notice, however, that there is no acceleration in the problem, which means there shouldn't be in the paradox. Time dilation is a function of speed here, not acceleration.
How do we then define the Twin Paradox without needlessly confusing the issue with accelerations? There's a number of ways we can do that.
1.Suppose that the twins are both astronauts. They each embark on a spaceship. They accelerate at the same rate, for the same predetermined length of time. Afterwards, one of them immediately engages his thrusters in reverse, in order to decelerate, and then to accelerate in the opposite direction, and finally decelerate again in order to stop at the point of origin. Meanwhile, the second one has stopped accelerating, so he is cruising at a uniform velocity. One year later, he does the same decelarating and accelerating maneouvres his bro did earlier. One year later, on final approach, he decelerates the same way his twin did and also ends up at the point of origin. Now they are together again, and the only difference is that one has been static for 2 years, while his bro has been in motion at a constant rate of speed. SR tells us that Time Dilation did occur in this scenario and that consequently one of them will be younger than the other. I challenge anyone to prove such a thing happened. (Edit: I guess they could even have started out in opposite directions...)
2.Another way we can frame the question is by having an alien with an atomick clock, onboard a spaceship in motion coincidentally synchronise it with an atomic clock situated somewhere along his flighpath. I am not suggesting causality, so there shouldn't be a problem with the fact that the 2 atomic clocks were reset at the same exact instant. The question then becomes: when the alien gets a snapshot of the second clock while flying by it, will the clocks still be synchronised?

 

[pervect]

The reason why many people can't understand the Twins Paradox is simply because it doesn't make sense.
Never fear, I am here to cut through all the cr4p and tell it like it is.

No, I'm no physicist or anything, but even I can see that there's no logic to the explanations offered here or anywhere.

Take this one for instance:
http://sciam.com/print_version.cfm?articleID=000BA7D8-2FB2-1E6D-A98A809EC5880105
It's clear from the word go that if you use your conclusion in the proof, you will at the end get the conclusion you were looking for.

You should re-read the forum guidelines you signed, espeically the ones about "unfounded attacks on mainstream science will not be tolerated".

But a few things should be obvious even to a non-physicist like yourself.

1) The article does not attempt to "prove" relativity. The article only attempts to explain it.

2) The proof of relativity is not deductive. The proof of relativity is in how well it's theoretical results are confirmed by experiment.

 

[Physical_Anarchist]

What makes my "attack" unfounded? And how is it an attack on mainstream science? I can see it as an attack on lazy defining of a problem and an attack on equally lazy explanations of a theoretical phenomena.
I've also seen many posts from some users who "wish to argue Relativity's validity, or advertise their own personal theories", that didn't earn a warning about forum guidelines. What's so much worse about mine? Does it make too much sense? Scared? Go ahead and delete it if you must... If you want to stifle intelligent debate...

 

[JesseM]

I'm no expert, so feel free to correct me here but the root of the problem arises from some strange property of light: it passes you by at the constant speed of 299 792 458 m/s. Even when you travel at 100 000 000 m/s relative to your friend, the light passes him by at 299 792 458 m/s and it also passes you by at 299 792 458 m/s. The answer then is time dilation.

Not alone, no. The fact that light is measured to travel at the same speed by all observers is a consequence of how Einstein proposed that each observer should define their own coordinate system--using a network of rulers and clocks which are at rest with respect to themselves, with the clocks synchronized using the "Einstein clock synchronization convention", which is based on each observer assuming that light travels at the same speed in all directions in their coordinate system (so two clocks in an observer's system are defined to be synchronized if, when you set off a flash of light at their midpoint, they both read the same time at the moment the light from the flash reaches each one). The rationale for defining each observer's coordinate system this way becomes clear in retrospect, when you see that it is only when you define your coordinates this way that the laws of physics will be observed to work the same way in each observer's coordinate system (this is because the laws of physics have a property known as 'Lorentz-symmetry', the name based on the fact that the different coordinate systems described above will be related to each other by a set of equations known as the 'Lorentz transformation'). You're still free to define your coordinate systems in a different way, but then the laws of physics would take a different form in different observer's frames.

The Einstein clock synchronization is enough to insure that each observer will measure light to travel at a constant speed in all directions, as opposed to faster in some directions than others. But to explain why the magnitude of this constant speed is the same for different observers, you also have to know that each observer will measure the rulers of those moving at velocity v relative to him to shrink by a factor of \sqrt{1 - v^2/c^2} and the ticks of clocks expand by a factor of 1/\sqrt{1 - v^2/c^2} (as long as the laws governing the rulers and the clocks have the property of Lorentz-symmetry, it's guaranteed this will happen). So, the fact that all observers measure the speed of light to be constant in all directions and the same from one observer's frame to another's is really a consequence of three things combined: time dilation, length contraction and "the relativity of simultaneity" (meaning that different observers will disagree whether a given pair of events happened 'at the same time' or not) which is a consequence of Einstein's clock synchronization convention. I posted a simple numerical example of how these three factors interact to insure a constant speed of light in this thread, if you're interested.

It allows you to become slower even while you're traveling fast, so that you can see light pass you by at the same rate as before. Notice, however, that there is no acceleration in the problem, which means there shouldn't be in the paradox. Time dilation is a function of speed here, not acceleration.

Yes, in an inertial frame time dilation is always a function of speed--if a clock is traveling at velocity v at a given moment, its rate of ticking will always be \sqrt{1 - v^2/c^2} times the rate of ticking of clocks at rest in that frame at that moment.

How do we then define the Twin Paradox without needlessly confusing the issue with accelerations? There's a number of ways we can do that.
1.Suppose that the twins are both astronauts. They each embark on a spaceship. They accelerate at the same rate, for the same predetermined length of time.

It is usually convenient in statements of the twin paradox to just assume the acceleration period is instantaneously brief, so that the twin switches from one velocity to another instantaneously.

Afterwards, one of them immediately engages his thrusters in reverse, in order to decelerate, and then to accelerate in the opposite direction, and finally decelerate again in order to stop at the point of origin. Meanwhile, the second one has stopped accelerating, so he is cruising at a uniform velocity. One year later, he does the same decelarating and accelerating maneouvres his bro did earlier. One year later, on final approach, he decelerates the same way his twin did and also ends up at the point of origin. Now they are together again, and the only difference is that one has been static for 2 years, while his bro has been in motion at a constant rate of speed. SR tells us that Time Dilation did occur in this scenario and that consequently one of them will be younger than the other. I challenge anyone to prove such a thing happened.

Simple, just analyze the problem from the point of view of the inertial frame of the spot where they both departed and later reunited (we can assume it's the earth, say). In this frame, one twin spent only a brief time moving at high velocity (suppose he instantaneously accelerated to 0.8c moving away from the earth, then after 0.01 years instaneously accelerated to 0.8c moving back towards it, then after another 0.1 years he reached Earth again and instantaneously accelerated so he was at rest on earth), while the other spent a whole year moving at high velocity. The first twin's clock was only ticking slow in this frame during the time he was moving relative to the earth, while the other twin's clock was ticking slow during the entire year, so the second twin's clock will have elapsed less time. Using the Lorentz transform, we could analyze this same situation from the point of view of any other inertial frame, and we'd always get the same answer to what the two clocks read when they reunited--I could show you the math if you want.

2.Another way we can frame the question is by having an alien with an atomick clock, onboard a spaceship in motion coincidentally synchronise it with an atomic clock situated somewhere along his flighpath. I am not suggesting causality, so there shouldn't be a problem with the fact that the 2 atomic clocks were reset at the same exact instant. The question then becomes: when the alien gets a snapshot of the second clock while flying by it, will the clocks still be synchronised?

Just to be clear, are there 2 different clocks in the alien's flightpath, as well as a third atomic clock on the alien's ship? And you're saying the alien's clock reads the same time as the first clock in his path at the moment he passes it, and then you want to know what will happen as he passes the second clock in his path and compares it with his own clock? In this case the answer will depend on which frame the two clocks were synchronized, because again, the "relativity of simultaneity" means that different frames disagree on whether two events (such as two different clocks ticking 12 o clock) happened at the same time or different times. If the two clocks are at rest with respect to each other and synchronized in their own rest frame, then in the alien's rest frame the first clock he passes will be ahead of the second one by a constant amount, and this explains why, even though both clocks are running slower than his, his clock still reads a smaller time than that of the second clock he passes (in the clocks' own frame, this is because the alien's clock was running slow). Again, I could show you a numerical example to explain why both frames make the same prediction about what the clocks read at the moment they pass despite disagreeing about which clock was running slow and whether or not the two clocks in his path were synchronized.

 

[JesseM]

What makes my "attack" unfounded? And how is it an attack on mainstream science? I can see it as an attack on lazy defining of a problem and an attack on equally lazy explanations of a theoretical phenomena.
I've also seen many posts from some users who "wish to argue Relativity's validity, or advertise their own personal theories", that didn't earn a warning about forum guidelines. What's so much worse about mine? Does it make too much sense? Scared? Go ahead and delete it if you must... If you want to stifle intelligent debate...

My understanding of the rules is that you are free to say that certain aspects of relativity don't make sense to you, and ask questions about how relativity would explain things, as you did in that long post. What's not allowed is just making definite assertions that relativity is wrong without any room for further discussion or calls for explanation.

 

[Physical_Anarchist]

Thank you for your replies, JesseM. I also read the other thread you referenced. I understand how, using the Lorentz transformation, one can demonstrate that the Twin in the spaceship ages less.
The problem I have with the whole concept, however, is that, by showing how one twin ages less, the paradox is trying to demonstrate that time dilates when traveling at speeds close to c. Why then is it OK to use a formula that assumes time dilation within our demonstration? It becomes a circular argument.
It's exactly as if I were trying to demonstrate that 5=7 by assuming that 1=3 and demonstrating that 1+4=3+4.

 

[RandallB]

Never fear, I am here to ……… tell it like it is.

Seriously, though, let's define the problem
How do we then define ……without needlessly confusing accelerations?

They accelerate at the same rate, …..
Afterwards, one of engages his thrusters in reverse, to decelerate,
and then to accelerate
and finally decelerate again
Meanwhile, the second one has stopped accelerating,
later, he does the same decelarating and accelerating
later, he decelerates the way his twin did and also ends at the point of origin.
I challenge anyone to prove …………….

What makes my "attack" unfounded?

OH I fear – If that was to remove acceleration confusion – I fear what you might say when you get to something not so simple as SR like GR or QM – I fear a great deal!

Your "attack" is unfounded, because it is irrationality incomplete.

In the Twins Paradox there are no acceleration issues to deal with.
It is easy to eliminate acceleration calculations in relativity.
Just use infinite instantaneous accelerations that take zero time to make transfers.
All reference frames will agree that the time elapse for any object going though such acceleration will slow to zero.
But since it also takes zero time in all reference frames there is also no argument as to when and where it started or ended as measured in any frame from any frame.
They will all agree.

If you can not handle that simple assumption, just use high speed snap shots of clocks with a fresh stop watch attached to each photo image. Then you can track total time for a clock and its images without anything actually having to accelerate anything at all. Just keep track of exactly where and when in each reference frame each image was taken and recorded.

Both methods will give the same agreement with SR, which is the Twins issue is not a paradox at all.
Do the work and you will know it like it is.
Just be sure to be absolutely complete and detailed about the where and when of each event in all three reference frames.

 

[Physical_Anarchist]

While true that I didn't eliminate acceleration from the sequence of events, I did eliminate it from the equation by having both twins experience the same amount of it.
You say that "In the Twins Paradox there are no acceleration issues to deal with."
JesseM also said we can make the acceleration instantaneous in order to eliminate it from the equation.
Why then is it that in explanations of the Twin Paradox, acceleration always rears its ugly head, by claiming that it can't really be instantaneous after all and some (or all) of the discrepancy between the clocks happens there, or that we have to only consider the worldline of the one who remained in an uniform inertial frame?
Well, the way I arranged the problem, these strategies can no longer be used to avoid the issue.
Thinking of the problem as I stated it, where both experience the same accelerations, the difference between them is only the speed that they are experiencing. Since that is relative, how can you tell which one is moving and thus remaining younger?

 

[RandallB]

Why then is it that in explanations of the Twin Paradox, acceleration always rears its ugly head, by claiming that it can't really be instantaneous after all and some (or all) of the discrepancy between the clocks happens there, or that we have to only consider the worldline of the one who remained in an uniform inertial frame?

Acceleration is not always used to explain the differences in the twin’s ages. As you've already noted, JesseM certainly didn’t use that.
And as you said, when acceleration on the traveling twin is used as the reason for the difference in ages, it is ugly because it is wrong.

 

[pervect]

What makes my "attack" unfounded? And how is it an attack on mainstream science? I can see it as an attack on lazy defining of a problem and an attack on equally lazy explanations of a theoretical phenomena.

Your attack is unfounded because you set up a "straw man" which you then proceed to be demolish.

This is a rather shabby form of debating practice.

The general purpose of this forum is to answer questions that people have about relativity. You are not "asking questions", you are playing silly little debating games.

I don't really believe you for a second when you say that you are attacking the "lazy writers" of that article and not attacking relativity. But I'll pretend that I do, for the sake of politeness.

In that case, I will simply say that it is not the fault of the article that it does not address your particular questions. It was not intended to provide a "proof" of relativity.

So, now let us pretend that you politely asked us - if this article doesn't provide a "proof" of relativity, and that it is not necessarily a bad article for omitting such a "proof", for it never intended to provide such a "proof", where do I find an article that does?

We will then politely answer you that that science does not provide such proofs. Back in the days of Aristotle, it was thought that "man's mind could elucidate all the laws of the universe, by thought alone, without recourse to experimentation"

Nowadays, we know better. Or at least most of us do. If we take your post at face value, you apparently do not know better.

So now we will politely attempt to explain to you that the scientific method is based on doing experiments - not on "proof".

This is really basic stuff. I'll conclude to a link to the wikipeda with some basic introductory info on the scientific method:

http://en.wikipedia.org/wiki/Scientific_method

a link to Aristotle's view on science

http://en.wikipedia.org/wiki/Aristotle#Science

and a suggestion that if you want to debate the foundations of science that you try the philosphy forum and not the relativity forum.

Now, if you ever manage to progress to the question: "What sort of experimental evidence makes us believe in relativity" this would be a resonably good forum to ask such a question. Of course we'd have to believe that you were actually interested in the answer...

 

[JesseM]

Thank you for your replies, JesseM. I also read the other thread you referenced. I understand how, using the Lorentz transformation, one can demonstrate that the Twin in the spaceship ages less.
The problem I have with the whole concept, however, is that, by showing how one twin ages less, the paradox is trying to demonstrate that time dilates when traveling at speeds close to c. Why then is it OK to use a formula that assumes time dilation within our demonstration? It becomes a circular argument.

I think you're misunderstanding the point of the twin paradox. The original idea of the "paradox" was to show that there was an internal logical inconsistency in the theory of relativity, so that even if you started out assuming the laws of relativity were correct, you would get inconsistent predictions if you analyzed the same situation from the point of view of different reference frames. The basic idea of the paradox is something like "from the Earth twin's point of view, the twin in the rocket is the one moving so his clock will be running slower, therefore he'll have aged less when they reunite; but you could equally well look at things from the point of view of the twin in the rocket, who sees the Earth moving, therefore he should predict the Earth twin will have aged less." The flaw in this argument is that the standard rules of time dilation only work in inertial frames, and the rocket twin does not stick to a single inertial frame (this is true regardless of whether he changes velocities instantaneously or if the acceleration is spread out over a finite period of time). As long as you analyze the paths of both twins from the point of view of an inertial frame, you will always get the same answer to how much each twin will have aged along their entire path, even if you use a frame where the Earth is moving and the twin on the rocket is at rest during one leg of the journey (but in such a frame, the twin on the rocket will have to move even faster than the Earth on the other leg of the journey in order for them to reunite).

If you are looking for actual experimental evidence of time dilation, rather than just arguments for why the theory of relativity is internally consistent, that's a separate subject. There's certainly plenty of experimental evidence, like the longer decay time of particles moving at very high velocities, or the fact that the GPS satellite system is designed to factor time dilation into all its calculations and would not work correctly if time dilation did not exist.

 

[Physical_Anarchist]

First of all, perv, I did not set up a "straw man". And stop defending that one specific article. It was merely an example. I read multiple articles on the twins paradox, some of which I was directed to from numerous threads on the subject in this forum. There was one that had "faraday" and, I believe, the university of toronto in the url, and another one linked to from the end of that one, for instance. They all claim to be the resolution of the paradox, and none did so satisfactorily in my view. That is why I registered here. Having read a few threads around here, I thought this would be a place where I could possibly get some clarifications. I am not looking for proof of relativity, experimental or otherwise. I merely wanted to analyse the twins paradox.
JesseM: you said: "The flaw in this argument is that the standard rules of time dilation only work in inertial frames, and the rocket twin does not stick to a single inertial frame". In my version of the twin paradox, I had both twins accelerating the same way. That leaves us then only the parts of the trip where each twin's speed is uniform to consider for comparison. I have yet to see a resolution of that scenario that doesn't use the conclusion as an assumption in the process. (I still believe that if I ask "Why is the sky blue?", "Because blue is the color of the sky" is not a complete and satisfactory answer. Circular reasoning is just not my thing... Shoot me!)

 

[clj4]

I have yet to see a resolution of that scenario that doesn't use the conclusion as an assumption in the process. (I still believe that if I ask "Why is the sky blue?", "Because blue is the color of the sky" is not a complete and satisfactory answer. Circular reasoning is just not my thing... Shoot me!)

Read this. If you are still confused, ask me questions:

http://sheol.org/throopw/sr-ticks-n-bricks.html

 

[JesseM]

JesseM: you said: "The flaw in this argument is that the standard rules of time dilation only work in inertial frames, and the rocket twin does not stick to a single inertial frame". In my version of the twin paradox, I had both twins accelerating the same way.

They didn't accelerate in the same way though. One accelerated twice between the time they began to move apart and the time they reunited--first to turn around after having left the earth, then to match his speed to the Earth to wait there for the other twin to return. The second twin, who spent longer away from the earth, only accelerated once, to turn around (you don't have to have him accelerate again once he reaches earth, you can just have the two compare clocks at the moment they pass at constant velocity).

I like to think of the twin paradox in terms of the "paths through spacetime" explanation. If you draw two points on a piece of paper, and one straight-line path between them and another with a bend in it, then you will always find that the straight-line path is shorter. Similarly, if you draw a spacetime diagram for the twins (with just one spatial dimension and one time dimension for convenience), they are taking two different paths between two points in spacetime (the point where they leave each other and the point where they reunite), and the way the time elapsed on a path is calculated in relativity insures that a straight path through spacetime will always have a greater proper time than any non-straight path between the same two points. So while the duration of the acceleration is not really important, the fact that acceleration leads to a bend in a twin's path through spacetime insures that it will have a smaller proper time. In your example where both twins accelerate, it's as if I had drawn two non-straight paths between the same two points in space, one consisting of two straight line segments joined at an angle, and one consisting of three straight line segments joined at an angle. Here, which path is longer really depends on the shape of the paths. Similarly, in your example it also depends on the shape of the path--it would actually be possible for the twin who spends most of his time on Earth to nevertheless be younger when they reunite, if he had been traveling at a much greater velocity relative to the Earth during his trip away and back. But if you specify they were both traveling at the same speed relative to the Earth during their trip, he will always have aged more. Similary, if you specify that the three-line-segment path through space has one segment that is parallel to a straight line between the points (analogous to the section of the twin's path through spacetime spent at rest on earth), and the other two segments are at exactly the same angle relative to this straight path as the two segments of the second path (analogous to the fact that in your example both twins have the same velocity during the inbound and outbound legs of their trip), then the path with three segments will always be longer than the path with two segments. If my descriptions are unclear I could provide a diagram as well.

(I still believe that if I ask "Why is the sky blue?", "Because blue is the color of the sky" is not a complete and satisfactory answer. Circular reasoning is just not my thing... Shoot me!)

You're equivocating on what kind of question you're asking though. If your question is about the internal logic of why relativity predicts that one twin will be younger, then in answering it we will take for granted the rules of relativity, and explain why the rules lead to these predictions. But if you're asking for experimental evidence that the rules of relativity are actually the ones that are seen in the real world, that's a totally separate question, the answer would involve various pieces of evidence for these rules such as the increased decay time of fast-moving particles or the workings of the GPS satellite system. If you want experimental evidence, than don't ask theoretical questions about the twin paradox, and if you want theoretical explanations of why relativity predicts one twin will be younger, then don't complain about "circular reasoning" when we assume the laws of relativity in our answer. Either one is inconsistent and illogical on your part.

 

[Al68]

Since this thread is still going, I have another question.

I personally don't like the Twins Paradox explanation with instantaneous turnaround. Or the two ship explanation where the incoming ship passes the outgoing ship at the distant star system. Considering this as two independent frames illustrates the math of time dilation, but, observers in the two ship frames will disagree with each other about the moment of departure from Earth and the moment of arrival back to earth. For example, with the two different ships passing each other in opposite directions at the star system 10 light years from Earth (earth frame), v = 0.866c, from Earth's frame the incoming ship arrived 23.1 years after the outgoing ship left. If you add up the times for the one way trip of each ship, we get 11.55 years. But, if we assume that Earth sends out a signal when the outgoing ship leaves earth, and the incoming ship received this signal, the incoming ship will calculate that he reaches Earth 46.2 years after the outgoing ship left earth. And similarly, if Earth sends out a signal when the incoming ship arrives at earth, and the outgoing ship eventually receives it, an observer on the outgoing ship will calculate that the incoming ship arrived at Earth 46.2 years after he left Earth (I hope I got this math right).

So, from the point of view of either ship frame, although it only took 5.77 years for them to get from Earth to the star system or vice versa, the total proper time (for either ship frame) between the event of the outgoing ship leaving Earth and the event of the incoming ship arriving at Earth is 46.2 years. Is this correct?

And this explanation also sidesteps some of the questions that arise from the Twins Paradox. With real acceleration involved, when the ship arrives at the star system, it will decelerate and at some point be at rest (at least momentarily) relative to the star system. And then will the distance between Earth and the star system "stretch back out" (as observed by the ship)? Does the ship observer observe Earth to "move" 5 light years farther away in a short period of time (v>c)? And then get 5 light years closer during the brief acceleration when the ship leaves the star system?

Is there a good explanation of the Twins Paradox available on the internet that addresses these kinds of questions?

I would like to find a good, comprehensive explanation to read before I ask a lot more questions.

And I hope nobody interprets this as a challenge to SR. Of course SR is mainstream science, and we have plenty of experimental evidence. But obviously this specific thought experiment has never been tested, and won't be in the foreseeable future. So it has to be resolved deductively, while assuming SR to be correct. But it's explanations are different and vary even in accepted textbooks. Of course they assume SR to be correct, since they are supposed to be explanations of how SR resolves the Twins Paradox. But, as far as I can tell, there is not universal agreement by mainstream sources about the details of this issue. And some details are not addressed at all.

Thanks,
Alan

 

[clj4]

Since this thread is still going, I have another question.

I personally don't like the Twins Paradox explanation with instantaneous turnaround. Or the two ship explanation where the incoming ship passes the outgoing ship at the distant star system. Considering this as two independent frames illustrates the math of time dilation, but, observers in the two ship frames will disagree with each other about the moment of departure from Earth and the moment of arrival back to earth. For example, with the two different ships passing each other in opposite directions at the star system 10 light years from Earth (earth frame), v = 0.866c, from Earth's frame the incoming ship arrived 23.1 years after the outgoing ship left. If you add up the times for the one way trip of each ship, we get 11.55 years. But, if we assume that Earth sends out a signal when the outgoing ship leaves earth, and the incoming ship received this signal, the incoming ship will calculate that he reaches Earth 46.2 years after the outgoing ship left earth. And similarly, if Earth sends out a signal when the incoming ship arrives at earth, and the outgoing ship eventually receives it, an observer on the outgoing ship will calculate that the incoming ship arrived at Earth 46.2 years after he left Earth (I hope I got this math right).

So, from the point of view of either ship frame, although it only took 5.77 years for them to get from Earth to the star system or vice versa, the total proper time (for either ship frame) between the event of the outgoing ship leaving Earth and the event of the incoming ship arriving at Earth is 46.2 years. Is this correct?

And this explanation also sidesteps some of the questions that arise from the Twins Paradox. With real acceleration involved, when the ship arrives at the star system, it will decelerate and at some point be at rest (at least momentarily) relative to the star system. And then will the distance between Earth and the star system "stretch back out" (as observed by the ship)? Does the ship observer observe Earth to "move" 5 light years farther away in a short period of time (v>c)? And then get 5 light years closer during the brief acceleration when the ship leaves the star system?

Is there a good explanation of the Twins Paradox available on the internet that addresses these kinds of questions?

I would like to find a good, comprehensive explanation to read before I ask a lot more questions.

And I hope nobody interprets this as a challenge to SR. Of course SR is mainstream science, and we have plenty of experimental evidence. But obviously this specific thought experiment has never been tested, and won't be in the foreseeable future. So it has to be resolved deductively, while assuming SR to be correct. But it's explanations are different and vary even in accepted textbooks. Of course they assume SR to be correct, since they are supposed to be explanations of how SR resolves the Twins Paradox. But, as far as I can tell, there is not universal agreement by mainstream sources about the details of this issue. And some details are not addressed at all.

Thanks,
Alan

The twins paradox has not been tested per se but there are plenty of other practical situations that received theoretical and experimental attention. Since you want something that you can read off the net, the best that comes to mind is the SR AND GR corrections that need to be applied prior to the launch of the GPS satellites. There may be more but this one is one of the best. See here:

http://relativity.livingreviews.org/open?pubNo=lrr-2003-1&page=node5.html

The other one that comes to mind is the Haefele - Keating experiment .You'll need to look up their paper.

 

[jtbell]

With real acceleration involved, when the ship arrives at the star system, it will decelerate and at some point be at rest (at least momentarily) relative to the star system. And then will the distance between Earth and the star system "stretch back out" (as observed by the ship)?

Yes. The key concept here is that of instantaneously co-moving inertial reference frames. During the time period (as measured by the ship's clock, say) that the ship is accelerating or decelerating, it is not stationary in any single inertial reference frame. Nevertheless, at any point in time according to the ship's clock, it is instantaneously stationary in an inertial reference frame which is moving along with the ship. At that point in (ship) time, the distance between the Earth and the star system is contracted according to the relative speed of that instantaneously co-moving inertial reference frame with respect to the inertial reference frame in which the Earth and star system are stationary.

Loosely speaking, we can say that the ship "passes through" a continuous series of instantaneously co-moving inertial reference frames, with infinitesimal relative velocities between each pair of successive frames.

Does the ship observer observe Earth to "move" 5 light years farther away in a short period of time (v>c)?

Yes, but you shoudn't think of this as a "genuine" v > c. The Earth's apparent superluminal velocity comes about because the ship is not moving inertially. The v <= c restriction applies to velocities of objects observed in a single inertial reference frame. (There's probably a more precise way to state this, but I can't think of it off the top of my head.)

obviously this specific thought experiment has never been tested, and won't be in the foreseeable future. So it has to be resolved deductively, while assuming SR to be correct. But it's explanations are different and vary even in accepted textbooks.

Ever hear the saying, "There's more than one way to skin a cat?"

 

[robphy]

But obviously this specific thought experiment has never been tested, and won't be in the foreseeable future. So it has to be resolved deductively, while assuming SR to be correct.

Of course, one way to test the twin paradox is to travel to a distant planet [insert: and back] at speeds close to the speed of light... then compare wristwatches. However, with an accurate enough clock, you don't need to go that far or that fast. Consider the clocks described here http://www.newscientist.com/article.ns?id=dn7397 "The first atomic clocks could pin this down to an accuracy of 1 part in 10^10. Today's caesium clocks can measure time to an accuracy of 1 in 10^15, or 1 second in about 30 million years." You can figure out the order of magnitude of v that corresponds to a gamma of (say) 10^(-15). I would think that such an experiment is possible in the forseeable future.

But it's explanations are different and vary even in accepted textbooks. Of course they assume SR to be correct, since they are supposed to be explanations of how SR resolves the Twins Paradox. But, as far as I can tell, there is not universal agreement by mainstream sources about the details of this issue. And some details are not addressed at all.

In my opinion, a "standard, mainstream textbook", especially one written by a non-relativist, is generally not the best place to find a definitive statement about "resolving the twin paradox", together with the various issues that may be raised. Such a textbook's explanation is usually based (read as "limited") by what material has been presented thus far in that textbook.

The variety of explanations arise from the many symmetries of Minkowski spacetime (see #5 in https://www.physicsforums.com/showthread.php?t=118994 ). In my opinion, the best explanations are the ones that use the fewest number of those symmetries... because they focus on the key physical idea: the proper-time [arc-length in spacetime] between two events is longest for the inertial observer.

As I have often said on this topic,
here's one of my favorite papers on the clock paradox:
http://links.jstor.org/sici?sici=0002-9890(195901)66%3A1%3C1%3ATCPIRT%3E2.0.CO%3B2-L
"The Clock Paradox in Relativity Theory"
Alfred Schild
The American Mathematical Monthly, Vol. 66, No. 1. (Jan., 1959), pp.1-18.
This addresses many of the approaches that have been suggested.

 

[RandallB]

I personally don't like the Twins Paradox explanation with instantaneous turnaround. Or the two ship explanation where the incoming ship passes the outgoing ship at the distant star system. Considering this as two independent frames illustrates the math of time dilation, but...

But nothing,
I thought you were getting the Twins but from the above I can see your still in the weeds.
You cannot do the twins where one returns the start with just two reference frames,
the traveler cannot get back to the other twin without a third frame.
And again if you don’t like transferring people at infinite accelerations – use the third returning frame to hold a clock and camera to take a photo back to earth.
Just collect ALL the data from all three frames with each photo to analyze what has happened.
Be sure Earth collects photos that include the WHERE and WHEN of all three reference frames as can be seen locally at Earth in all three frames for every event. Including: Three photos taken at Earth when that location is simultaneous with the traveling twin reaching the star for each of the three frames – That means three different photos of three unique events that hold 18 different pieces of information about Where and When those three events took place at earth.
Then do the same for star, based on a) when the Twin leaves Earth and b) when the photo of the twin and the star is brought back to Earth by someone in that third frame. That will be 36 pieces of information.
All these photos can be collected in one place after the fact for your review by whatever accelerations or data transfer is OK by you.
The conclusions you draw from this mathematical exercise using SR rules will correlate to the same kind of results that are always seen in experiments that confirm SR.

Take your time don’t lose track of a frame or locations and distances in it.

 

[JesseM]

With real acceleration involved, when the ship arrives at the star system, it will decelerate and at some point be at rest (at least momentarily) relative to the star system. And then will the distance between Earth and the star system "stretch back out" (as observed by the ship)?

Yes. The key concept here is that of instantaneously co-moving inertial reference frames. During the time period (as measured by the ship's clock, say) that the ship is accelerating or decelerating, it is not stationary in any single inertial reference frame. Nevertheless, at any point in time according to the ship's clock, it is instantaneously stationary in an inertial reference frame which is moving along with the ship. At that point in (ship) time, the distance between the Earth and the star system is contracted according to the relative speed of that instantaneously co-moving inertial reference frame with respect to the inertial reference frame in which the Earth and star system are stationary.

I agree with this description of what happens if you measure the distance in a series of instantaneously co-moving inertial frames, but I think it's misleading terminology to say this is what will be "observed by the ship", period. The series of co-moving inertial frames do not together define a single well-behave non-inertial coordinate system for an accelerating observer, because the same event could happen simultaneously with more than one point on the observer's worldline. And when dealing with non-inertial coordinate systems, there is no reason to see one choice as more physical than another, so you could equally well invent a very different non-inertial coordinate system in which the accelerating observer is at rest but the distance at any given moment does not correspond to the distance in the instaneous inertial frame at that moment. I think the word "observed" should only be used without qualification when talking about inertial observers, where there is a single well-known convention for how to define the coordinate system that constitutes their "rest frame", while it shouldn't be used for non-inertial observers, at least not unless you define in advance what coordinate system they are using to make "observations", with it being understood that this choice of coordinate system is a somewhat arbitrary one.

 

[pervect]

There is a shortish summary of many of the varioius methods of addressing the twin paradox at

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

including an addendum about "too many explanations", which has a lot of very useful diagrams.

There are two general subsets of the many approaches that are worth some attention.

The first approach considers only what the two space-ships actually see. By this I mean the signals that they actually receive from each other. This is the "doppler approach". See figure 2 in http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html for instance.

A second subset of explanations focuses on which events different obsevers think of as being "simultaneous".

The most basic thing one must understand with the later approach is that simultaneity is relative. One can actually draw "lines of simultaneity" on a space-time diagram that represent different observer's concepts of simultaneity.

As we have mentioned in another thread, the slope of a line of simultaneity for an inertial obsever is always c^2 / v, also written as c/\beta where \beta=v/c.It is probably better NOT to get too mired in the working out of "what events are simultaneous to other events according to which observer" but it seems that some people just can't help it.

See figures 3 and 4 in http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_vase.html

for diagrams which show the lines of simultaneity.

 

[Physical_Anarchist]

They didn't accelerate in the same way though. One accelerated twice between the time they began to move apart and the time they reunited--first to turn around after having left the earth, then to match his speed to the Earth to wait there for the other twin to return. The second twin, who spent longer away from the earth, only accelerated once, to turn around (you don't have to have him accelerate again once he reaches earth, you can just have the two compare clocks at the moment they pass at constant velocity).

I don't get it. You change the way I formulated the question, just so you can say they didn't accelerate in the same way? To have them experience the exact same accelerations overall is precisely why I had the second one decelerate before re-joining the point of origin.

Let me illustrate:
1.@@@@@@DDDDDD-@-@-@-@-@-@DDDDDD////////////////////////
2.@@@@@@////////////DDDDDD-@-@-@-@-@-@////////////DDDDDD

Legend: @=accelerate, D=decelerate, /=one month.
Since deceleration is merely acceleration in the opposite direction and wee're disregarding accelerations, since they are equivalent overall, we only have to compare 24 months at "low speed" to the 2 segments of 12 months at "high speed". Relativity tells us that the one at low speed could actually be the one moving at high speed. None has a more legitimate claim than the other. To understand this, imagine that their point of origin is actually moving, without them realizing it. Their trip, that they imagined as in the above illustration, actually could look like this:

1.DDDDDD@@@@@@DDDDDD-@-@-@-@-@-@///////////////////////
2.DDDDDD////////////@@@@@@DDDDDD////////////-@-@-@-@-@-@
This illustration is just as legitimate as the first one for the purposes of determining speed in a relative context.

As for your other argument, I'll always complain about circular resoning, because circular reasoning is simply bad logic. It should always be possible to work out the theory using logic.

Al68 said: "Of course they assume SR to be correct, since they are supposed to be explanations of how SR resolves the Twins Paradox." SR creates the Twins Paradox. That's why it's a paradox.

clj4: I read that thing, but it was really late at night and I'm still confused. I'll give it another try.

 

[clj4]

clj4: I read that thing, but it was really late at night and I'm still confused. I'll give it another try.

Yes, read again.

 

[JesseM]

I don't get it. You change the way I formulated the question, just so you can say they didn't accelerate in the same way? To have them experience the exact same accelerations overall is precisely why I had the second one decelerate before re-joining the point of origin.

My point is that as long as you're assuming instantaneous acceleration, accelerations at the start or end of each one's path (with the start being where they depart at a common time and place, the end being where they reunite at a common time and place) don't affect the length of the path in between those points, so they're irrelevant. And if the acceleration is brief but not instantaneous, the difference between accelerating right near an endpoint or not accelerating will be very small, it won't substantially change the answer to which twin is older when they reunite.

Did you read everything I wrote about the "paths through spacetime" way of thinking about the problem, and did you understand why, in your example, their two paths will be quite different, regardless of whether the second one accelerates when he reunites with his twin who's already at rest on earth? I'll try to render a diagram here if it helps:../\
./..\___
*...*
.\.../
..\.../
...\../
...\/

Here position is the vertical axis, and time is the horizontal axis (ignore the rows of dots, they're just there to keep everything spaced right since the forum automatically deletes multiple spaces in a row...if the diagram is unclear I can redraw it as a nicer-looking image file on request). The *'s are the endpoints of the path, the top path involves first moving away from Earth (the part of the path slanted like /), then moving back towards (the part of the path slanted like \), then resting on Earth while waiting for the other twin to return (the flat part of the path which looks like ___ ), while the bottom just involves moving away from the Earth (\) and returning (/)between the two endpoints. If you understand the diagram, it should be obvious that it doesn't matter if the twin on the bottom path accelerates to come to rest on Earth right as he is about to reach the endpoint or not, it will have no significant effect on the overall length of the path between the two endpoints, and that's all that's important.

As for your other argument, I'll always complain about circular resoning, because circular reasoning is simply bad logic.

You should review the meaning of the term "circular reasoning", because proving that there is no logical inconsistency in a theory using the axioms of the theory itself is definitely not circular reasoning. Of course this can't prove whether or not the theory is true in the real world, only whether it contains any internal inconsistencies.

It should always be possible to work out the theory using logic.

Complete nonsense. There is not a single scientific theory that can be proven using only "logic" without any need for observation. Both Newtonian mechanics and relativity are internally consistent, for example, it is only experimental tests which can tell you which is actually true in the real world.

 

[Physical_Anarchist]

I don't know why a position diagram is relevant. A speed graph would be more appropriate and I think it would look like this:
.../\
../...\
/...\...__________
...\.../
....\../
.....\/

versus
..._____
.../...\
../....\
/....\
......\.../
.....\.../
......\_____/

 

[JesseM]

I don't know why a position diagram is relevant. A speed graph would be more appropriate and I think it would look like this:
.../\
../...\
/...\...__________
...\.../
....\../
.....\/

versus
..._____
.../...\
../....\
/....\
......\.../
.....\.../
......\_____/

Either a speed graph or a position graph can be used to determine the total spacetime "length" of each path (ie the proper time along each path). Similarly, if you have two paths drawn on an ordinary piece of graph paper with x and y coordinate axes drawn on, then if you can describe the paths in terms of the y-position as a function of x-position, like y(x), or if you can describe the slope of each path at every point along the x-axis with the function S(x), then you can use either one to find the total length of the path between the two points.

You can see in your diagram above that the speed graphs for the two paths look quite different, and that your addition of that third acceleration at the very tail end of the last graph didn't make their overall shapes the same (also, if that acceleration was very brief compared to the overall time spend in space, the last upward section should be much shorter along the t-axis, but maybe they weren't meant to be to scale). In terms of figuring out the proper time, if you know the functions for speed as a function of time v(t) in a particular inertial reference frame, and you know the times of the two endpoints t_0 and t_1 in that frame, then to find the total proper time you'd integrate \int_{t_0}^{t_1} \sqrt{1 - v(t)^2/c^2} \, dt. The value of this integral for the second path will be only barely changed if you add a brief acceleration immediately before the time t_0 when they reunite.

So again, what's important is the precise nature of the position vs. time or speed vs. time functions for each one. The "whichever twin accelerates will have aged less" is not meant to be a general answer that covers all cases, it's only meant to cover the case where one twin moves inertially between the two endpoints while the other does not (as long as this is true, then no matter what specific position vs. time or speed vs. time function you pick for the second twin, you will find that his proper time is less). But in the case where both twins accelerate, you obviously can't apply this rule, you have to consider the two paths in a more specific way.

All this is directly analogous to the example of two paths drawn on an ordinary 2D piece of paper; if one is straight while the other has a bend in it, you can say "whichever path has the bend will be longer", but this is not a general rule that would cover all cases, in an example where neither path is straight you have to consider the specific shape of each path.

edit: I just noticed something about your graphs--is there a reason that the changes in speed in the first graph are sharp, while the changes in speed in the second graph have those flat intervals? If the flat intervals are meant to be extended periods of time at rest relative to the earth, so that both ships spend the exact same amount of total time at rest relative to the Earth from beginning to end, and also the same amount of time moving away from Earth at speed v and the same amount of time moving towards it at speed v, then in that case they will be the same age when they reunite. If this is what you meant all along and I misunderstood, then sorry for the confusion.

edit 2: OK, another thing I missed, but shouldn't constant speed always be a flat section of the graph? On a graph of speed vs. time, a non-flat slope would be a period of acceleration (speed changing at a constant rate), is that what you meant the sloped parts of the graph to represent? I thought in your example the idea was that each ship spent most of the time moving at constant velocity, with only brief periods of acceleration.

 

[Al68]

Does the ship observer observe Earth to "move" 5 light years farther away in a short period of time (v>c)?

Yes, but you shoudn't think of this as a "genuine" v > c. The Earth's apparent superluminal velocity comes about because the ship is not moving inertially. The v <= c restriction applies to velocities of objects observed in a single inertial reference frame. (There's probably a more precise way to state this, but I can't think of it off the top of my head.)

Why would we not call this v > c genuine? If we are calling the v < c restriction for the ship's velocity (on the way to star system) relative to Earth's position genuine. If we do not call Earth's position of 5 light years away from the star system "genuine", then our velocity of v = 0.866c would not be "genuine". Are you just saying this v > c is not genuine because this is not a restriction for accelerating observers?

Also, I have read all of the referenced explanations on the internet (except the one on jstor, I don't have access), and none of them address the questions I have. That's probably because I've read them all before, and no longer have the questions that they do address.

Also, it's my understanding that Einstein initially thought he should be able to consider the ship at rest with the Earth and star system moving back and forth relative to the ship, and still resolve the Twins Paradox in SR. But he gave up on this and tried to resolve it with GR. And physicists generally consider his GR resolution erroneous. Is this correct? I think this is what wikipedia says, also.

Thanks,
Alan

 

[pervect]

Why would we not call this v > c genuine? If we are calling the v < c restriction for the ship's velocity (on the way to star system) relative to Earth's position genuine. If we do not call Earth's position of 5 light years away from the star system "genuine", then our velocity of v = 0.866c would not be "genuine". Are you just saying this v > c is not genuine because this is not a restriction for accelerating observers?

Here's a quote from Baez's et al paper on GR ("The Meaning of Einstein's equations" which helps explain this point.

http://math.ucr.edu/home/baez/einstein/node2.html

(also available at http://arxiv.org/abs/gr-qc/0103044 in pdf if you don't like the chopped-up version)

Before stating Einstein's equation, we need a little preparation. We assume the reader is somewhat familiar with special relativity -- otherwise general relativity will be too hard. But there are some big differences between special and general relativity, which can cause immense confusion if neglected.

In special relativity, we cannot talk about absolute velocities, but only relative velocities. For example, we cannot sensibly ask if a particle is at rest, only whether it is at rest relative to another. The reason is that in this theory, velocities are described as vectors in 4-dimensional spacetime. Switching to a different inertial coordinate system can change which way these vectors point relative to our coordinate axes, but not whether two of them point the same way.

In general relativity, we cannot even talk about relative velocities, except for two particles at the same point of spacetime -- that is, at the same place at the same instant. The reason is that in general relativity, we take very seriously the notion that a vector is a little arrow sitting at a particular point in spacetime. To compare vectors at different points of spacetime, we must carry one over to the other. The process of carrying a vector along a path without turning or stretching it is called `parallel transport'. When spacetime is curved, the result of parallel transport from one point to another depends on the path taken! In fact, this is the very definition of what it means for spacetime to be curved. Thus it is ambiguous to ask whether two particles have the same velocity vector unless they are at the same point of spacetime.

This is the issue that you are running into with apparently FTL velocities. You are using a non-inertial coordinate system, and expecting it to act like an inertial coordinate system.

Note that because the underlying problem is in flat space-time, one actually CAN talk about the relative velocities of two particles. But in order to do so and get the right answer, one must restrict oneself to inertial coordinate systems.

Note that even in flat space-time, if the velocity between two objects is changing (because one of them is accelerating), the velocity of a distant object "at the same time" is ambiguous, because "at the same time" is an ambiguous concept in SR.

Thus the main problem is in your expectations. You are applying concepts which work in inertial coordinates and expecting them to apply in generalized coordinates.

You might also notice (or maybe you haven't) that the velocity of light is not constant in your accelerated coordinate system. Thus when you say that the distant object is moving "faster than light", it is actually not moving faster than light moves at that particular location. In your non-inertial coordinate system, light does not have a constant coordinate velocity, and in the region where the Earth appears to be moving faster than 'c', light appears to be moving even faster than the moving Earth.

Also, I have read all of the referenced explanations on the internet (except the one on jstor, I don't have access), and none of them address the questions I have. That's probably because I've read them all before, and no longer have the questions that they do address.

I also do not have access to the Jstor article.

As far as I can tell, you are trying to run before you can walk. It is possible to understand and work with non-inertial coordinates in relativity, but it requires some sophisticated mathematical techniques, such as the process of "parallel transport" that Baez alludes to.

It is both easier and more productive (IMO) to start to learn about relativity in a coordinate independent manner. This means learning about 4-vectors, and space-time diagrams. You need to have a firm grasp on SR, especially on the relativity of simultaneity (which you apparently still are struggling with from what I can infer from your remarks) before you can go on to handle GR and arbitrary coordinate systems.

You might also give some thought to the philosophical idea that coordinate systems are not the fundamental basis of reality.

Rather than treat coordinates as the basis of reality, think of the arrival of signals, and the readings of clocks, as being the fundamental basis - after all, that is actually what you can observe. You do not directly perceive the coordinates of some distant object, you percieve signals from that object.

The "coordinate" of a distant object are just something that you compute. Coordinates are supposed to be a convenience to make your life easier (and not a millstone around your neck that drags you into confusion). What you actually physically observe are signals emitted from and sent to said object (such as radar signals, or observations you make with a telescope).

The Doppler explanation of relativity, for instance, tells you all about how to compute the arrival time of such signals.

If you get into a muddle, think not about coordinates, but think instead about physical signals - when they were sent (and by whose clock that time deterimnation was made!), and when they arrived. Think about things that you actually could directly observe (i.e. NOT coordinates, which are things that you compute, not observe).

Also, it's my understanding that Einstein initially thought he should be able to consider the ship at rest with the Earth and star system moving back and forth relative to the ship, and still resolve the Twins Paradox in SR. But he gave up on this and tried to resolve it with GR. And physicists generally consider his GR resolution erroneous. Is this correct? I think this is what wikipedia says, also.

Thanks,
Alan

I can't make heads or tails of this remark. If you could provide a specific quote from the Wikipedia I might be able to say more.

 

[Al68]

pervect,

I was only objecting to the use of the phrase "not genuine". And I note that using observation as a basis for reality is objected to by some on this forum. I have my own reasons for asking these questions. I don't have a problem with apparent FTL velocities. I was just remarking that observations should not be referred to as "not genuine". And I would think that, after your comments here, that you would agree.

And here is a link to that wikipedia article: http://en.wikipedia.org/wiki/Twin_paradox#Resolution_of_the_Paradox_in_General_Relativity

It is part of their Twins Paradox article.

Thanks,
Alan

 

[pervect]

I might quibble with the exact way Jesse worded his remark, but I agree with the spirit. There may be some way to explain it more concisely than the rather long quote I gave, but I'm not sure how to do it :-(.

As far as the Wikipedia goes, I agree that there are some issues with the "gravitational time dilation" explanation of the twin paradox, but those problems don't actually come up in the twin paradox itself in my opinion.

My personal opinion is that the explanation IS good enough to explain the twin paradox, but has troubles further down the road.

I suspect you are traveling down that road right now, so I'll give you some idea of where I see the roadblock occurring. The roadblock is that "the" coordinate system of an acclerated observer does not cover all of space and time. It's only good locally.

This is covered in various textbooks, unfortunately I've never seen a textbook that covers this well that does not use tensors.

The problem can be illustrated with a simple diagram without the math, though. Basically, if you draw the lines of simultaneity for "the" coordiante system of an accelerated observer, they eventually cross.

Example: accelerated observers follow a "hyperbolic" motion, whose equations are just:

x^2 - t^2 = constant

If you draw the "lines of simultaneity" for such an accelrated motion, all of them cross at the origin of the coordinate system.

(I've got a picture of this somewhere, can't find the post though).

The fact that all of the coordinate lines of simultaneity cross leads to a coordinate system that is good only in the region before the lines cross. After the crossing occurs, one point has multiple coordinates - the origin, for instance, has an infinite number of "time" coordinates. This is very bad behavior, it does not meet the standards that every point must have one and only one set of coordinates.

The fact is closely related to the existence of the "Rindler horizon" for an accelerated observer. Another reason the acclerated observer does not have a coordinate system that covers all of space-time is that he cannot see all of space-time. An observer who accelerates away from Earth at 1 light year/year^2 will never actually see any event on Earth that occurs at a time later than 1 year, unless he stops accelerating, for example. All events more than 1 light year behind the accelerating observer are "behind" his Rindler horizon.

Detailed treatments of this do exist (my favorite is in MTW's "Gravitation")- unfortunately, as I 've said, most of them use tensor notation.

 

[JesseM]

I was only objecting to the use of the phrase "not genuine". And I note that using observation as a basis for reality is objected to by some on this forum.

If you're referring to me, my objection was not to "using observation as a basis for reality", but rather to the notion that the v>c thing constitutes an "observation" in the first place. In an inertial frame, what you "observe" is not what you actually see at the time, but what you reconstruct in retrospect based on the assumption that light always moves at c in your frame. For example, if I look through my telescope in 2006 and see the image of an explosion 10 light-years away as measured in my frame, and then in 2016 I look through my telescope again and see the image of an explosion 20 light-years away in my frame (which is the same one as before since I'm moving inertially), I can do a calculation of speed/distance for each and say that I "observed" these explosions to happen simultaneously in my frame, even though I certainly didn't see them happen simultaneously.

Now, have you thought about what type of calculation an accelerating observer would have to make to say in retrospect that he "observed" the Earth moving faster than c? For one thing, it would involve using a different set of rulers to measure distance at each point during his acceleration. And if you want things to work out so that he always "observes" an event's distance at a given moment to be identical with the distance in his instantaneous inertial rest frame, then he can't just take the time he actually sees the event and divide the distance in his inertial rest frame at the moment he saw it by c like in the inertial case, because his inertial rest frame at the moment he sees the event will be different from the inertial rest frame he was in at earlier moments when the light was on its way. I'd suggest that you take a shot at figuring out just how he could calculate in retrospect when he "observed" different events based on the moments he actually sees the light from them, and then perhaps you will change your mind about whether this highly abstract (and not very well-motivated physically, unlike the inertial case) calculation really deserves the commonsense word "observation".

 

[jtbell]

Does the ship observer observe Earth to "move" 5 light years farther away in a short period of time (v>c)?

Yes, but you shoudn't think of this as a "genuine" v > c.

Why would we not call this v > c genuine?

I shouldn't have said "not genuine." It has loaded connotations. A better statement is simply that coordinate velocities (and other physical quantities based on coordinate measurements) behave differently in non-inertial coordinate systems than in inertial ones. Restrictions that apply in inertial coordinate systems don't necessarily apply in non-inertial ones.

Consider a different, but somewhat similar example. Suppose you are studying the behavior of a distant star in a coordinate system that has your head as its origin, and its x-axis sticking out straight ahead from your nose. Turn your head. The coordinate system is now rotating. In that coordinate system, the star now has a huge velocity, many times the speed of light, perpendicular to the x-axis.

 

[Al68]

Jesse,

I was not referring to you. And I wasn't referring to a direct observation of a measurement of Earth's velocity relative to the ship, while the ship decelerated. Just the observation by the ship's observer that the Earth's apparent position relative to the ship changed by a few light years in a matter of days, according to the ship's clock. I was only suggesting that the observation of Earth at 5 light years from the star system prior to deceleration, then the observation of Earth at 10 light years from the star system after deceleration should be considered genuine. After all, if we cannot call this length expansion genuine, how could we say the ship ended up 10 light years from earth, while its clock only showed 5.77 years since it left earth?

Thanks,
Alan

 

[JesseM]

Jesse,

I was not referring to you. And I wasn't referring to a direct observation of a measurement of Earth's velocity relative to the ship, while the ship decelerated. Just the observation by the ship's observer that the Earth's apparent position relative to the ship changed by a few light years in a matter of days, according to the ship's clock.

But again, this won't be a straightforward "observation". If he looks through his telescope, he won't see the position of the Earth change by a few light years in a matter of days (or if he does, it'll only be in the sense that he has switched which set of rulers he is using to define distance, the apparent size of the Earth as seen in his telescope won't change significantly). Like I said above, even in SR, "observation" involves a process of calculating dates of events in retrospect, long after they were observed. And the calculation for an accelerated observer needed to insure that what he "observes" matches his instantaneous inertial reference frame at each moment would be very complicated and not too physically well-motivated. Again, I invite you to figure out what he will actually see through his telescope during the acceleration, and then to figure out exactly what sorts of calculations he'd have to do on the dates he saw things to get the dates he "observed" them to work out the way you want them to.

I was only suggesting that the observation of Earth at 5 light years from the star system prior to deceleration, then the observation of Earth at 10 light years from the star system after deceleration should be considered genuine.

Why? Again, please try to figure out what measurements and calculations he'd have to do to get this "observation", and explain why this series of complicated calculations should be considered more "genuine" than some different set of calculations corresponding to a different non-inertial coordinate system.

After all, if we cannot call this length expansion genuine, how could we say the ship ended up 10 light years from earth, while its clock only showed 5.77 years since it left earth?

This comment is only true in the Earth's inertial reference frame, it's not an objective coordinate-independent statement about reality, any more than the statement "the velocity of the Earth is zero". Even in other inertial frames, it is not true that the ship "ended up" 10 light years from earth. And there's certainly no reason to think this would have to be true in a non-inertial coordinate system.