Trying to understand time dilation better

[rede96]

I'm only an amateur enthusiast when it comes to physics and have no background in science at all. It's just something I enjoy thinking about.

One thing I am really finding difficult to get my head around is how time dilation works. So I've put a thought experiment below and would appreciate it if anyone would check my understanding.

There are two friends of the same age, on different space ships, both in uniform motion but traveling in different directions relative to each other. While communicating they discover that their paths will cross so decide to do a little experiment.

They agree that when they pass each other, ship A will send a single to ship B and then they will both synchronise their clocks to a pre agreed date and time. They also agree that they will send each other an update once a month at an exact agreed time using the newly synchronised clocks. The message will include the current time and date from the respective ships so they can see if their clocks remain in sync and at each message, report back any difference. They will do this for 2 years.

They also measure that relative to each other they are moving apart at 0.8c and neither ship goes through any accretion during the experiment.

As I understand it, Ship A would start to receive the messages each month from ship B with longer and longer delays. However Ship B would also be experiencing the same thing.

But what would be a constant is that the delay for each message in sequence would be the same for each ship.

So after 2 years, they stop sending the messages to each other (24 in total) and eventually they each receive the final message from the other ship. They then compare together the total delays for each month interval from the other ship's messages. What they both find is that although each ship thought the other ship's time was moving slower, because the delays were getting longer, the delays for each set of messages sent, match for each ship.

Baffled by this they decide to use their new quantum teleportation devices (which is possible at least in theory) and see who has aged more.

So my questions are:

1) If they both decide that at a given date, say after each experience 10 years after they initially passed and set their clocks, they teleport to a mid point between them, will they find that they have both aged at the same rate once together?

2) If the guy from ship A decided to teleport to Ship B, would he find that his friend on ship B was actually younger? (and the same if Ship B guy teleported to Ship A)

3) Assuming the above is correct (which I have no idea if it is) then what would happen if the guy from Ship A teleported to Ship B and sees his friend is younger. But then teleports back to his ship. Then the guy from Ship B teleports over to ship A. Would he find that miraculously his friend on Ship A is now younger them him again?

All really confusing :)

 

[Nugatory]

Baffled by this they decide to use their new quantum teleportation devices (which is possible at least in theory) and see who has aged more.

Quantum teleportation, as you're understanding it here, is not possible even in theory.

This isn't just a quibble that you can bypass by saying "Yes, but suppose it were possible... Then what would hapen?". The basic assumption you're making is that something can be moved from one point to another instantaneously, and that assumption contains a hidden internal contradiction.

 

[Ibix]

I may be wrong on this, but I get the impression that you believe that quantum teleportation proceeds faster than light. This is not the case, so there is no difference between your "exchanging light signals" and your "teleportation" cases.

The next thing to note is that there are two separate effects at work here. The first is simply the Doppler Shift. Since the ships are moving apart, each month's signal has further to go than the last one. For this reason, each traveler expects to receive the pulses from the other ship more than a month apart. The second thing to happen is time dilation. After correcting for the Doppler Shift, each traveler finds that the other one's pulses are being emitted at intervals of longer than one month, by a factor of $\gamma=1/\sqrt{1-v^2/c^2}$, which is 1.667 for you quoted v=0.8c.

Now to answer your questions:

1. Both travelers have done the same thing (or at least, a reflection of the other). They must end up the same age.

2. Presumably your teleporter re-constitutes the person at the other end exactly as they were when they left. In that case, the guy who teleported would be younger because he's been "frozen" in transit. However, this is not a relativistic effect, and the details of their relative ages depend on implementation details of your teleporter.

3. Again, the details of this one depend on the details of your teleporter and exactly what sequence people teleport, since the transit time is not negligible. Basically, both travelers are going to be spending years "frozen" in transit, and their relative ages depend upon how long they spend "frozen" which depends on how far apart their ships were when they beamed over.

 

[rede96]

I may be wrong on this, but I get the impression that you believe that quantum teleportation proceeds faster than light. This is not the case, so there is no difference between your "exchanging light signals" and your "teleportation" cases.

No, I understood that there needs to be clasical communication between the two. But am not too sure how long it would take for one person's information to be scanned and then replicated at the other side. So didn't think about him being 'frozen'.

The next thing to note is that there are two separate effects at work here. The first is simply the Doppler Shift. Since the ships are moving apart, each month's signal has further to go than the last one. For this reason, each traveler expects to receive the pulses from the other ship more than a month apart. The second thing to happen is time dilation. After correcting for the Doppler Shift, each traveler finds that the other one's pulses are being emitted at intervals of longer than one month, by a factor of $\gamma=1/\sqrt{1-v^2/c^2}$, which is 1.667 for you quoted v=0.8c.

Ok, thanks. I did ignore doppler shift as it was just the relative delay I was looking at but should have included it to make it complete.

Now to answer your questions:

1. Both travelers have done the same thing (or at least, a reflection of the other). They must end up the same age.

I agree with this but thought I couldn't make that statement in one frame or the other, as the outcome is relative. So the only way to know who has aged more would be for them to meet so they are in the same frame, but by doing that it may effect who is older than who.

2. Presumably your teleporter re-constitutes the person at the other end exactly as they were when they left. In that case, the guy who teleported would be younger because he's been "frozen" in transit. However, this is not a relativistic effect, and the details of their relative ages depend on implementation details of your teleporter.

Yeah missed that one, thanks

3. Again, the details of this one depend on the details of your teleporter and exactly what sequence people teleport, since the transit time is not negligible. Basically, both travelers are going to be spending years "frozen" in transit, and their relative ages depend upon how long they spend "frozen" which depends on how far apart their ships were when they beamed over.

OK, but let's assume that after 2 years of travel at 0.8c the ships both stop moving apart so they are at rest wto each other and therefore keeping the symetery. Also let's assume that as they are now 1.6 light years apart, that the whole process for teleprotation, including sending the information takes 3 years.

If A goes to B, then B would expect A to be three years younger, as they only difference in aging came from being frozen.

Which means that if we adjust for teleportaion then of course they must have ages at the same rate during the 2 year trip.

And this is where I get confused with terminaolgy. As I understand it, A in his own frame of reference can't say that B aged the same during the 2 year period, as the results would prove otherwise. Neither can B in his own frame say the A has ages the same rate.

So as I understand it, the only way they can say with certanity that they have aged at the same rate is if they manipulate the process of getting together in just one frame so they have indeed aged the same. And as you said as long as the whole process from start to getting together again was symetrical, then they must age the same.

And if they process of getting back together isn't symetrical, then who ever broke the symetry by the biggest margin would have ages the less. (I think!)

Does that make sense?

 

[rede96]

Quantum teleportation, as you're understanding it here, is not possible even in theory.

Yeah, I just did a bad job of explaining it sorry. But at least in theory it is possible to use quantum entanglement to transport matter (or more replicate matter) from one place to another. They would have need to set all this up before the experiment obviously.

 

[Ibix]

I agree with this but thought I couldn't make that statement in one frame or the other, as the outcome is relative. So the only way to know who has aged more would be for them to meet so they are in the same frame, but by doing that it may effect who is older than who.

Because they are meeting, because they are in the same place, there is no ambiguity about the definition of "simultaneous", which there is for separate events. Since their world-lines were mirror images of one another in some frame, the elapsed proper time must be the same for both. So they will be the same age.

OK, but let's assume that after 2 years of travel at 0.8c the ships both stop moving apart so they are at rest wto each other and therefore keeping the symetery. Also let's assume that as they are now 1.6 light years apart, that the whole process for teleprotation, including sending the information takes 3 years.

OK - you're complicating things now. But this is enough to explain where you are going wrong: two years according to who? If both travelers fire their engines when their on board clocks read two years, both will say the other burned later because the other's clocks were slow. I suspect you are thinking in the frame in which the rockets are doing equal and opposite velocities - which is a good choice of frame - but note that two years in this frame is not two years according to the rocket's clocks.

And if they process of getting back together isn't symetrical, then who ever broke the symetry by the biggest margin would have ages the less. (I think!)

No. I'm not sure how you would measure distance from symmetry. What you do is calculate the proper time ("wrist-watch time") along the world line of each of the travelers and compare those at the instant they meet. If you see someone move a distance $\Delta x$ in a time $\Delta t$ at constant velocity, then the proper time (the elapsed time on their wrist-watch) is $\Delta\tau$:
$$(c\Delta\tau)^2=(c\Delta t)^2-(\Delta x)^2$$
Other observers will disagree on the $\Delta t$ due to time dilation and the$\Delta x$ due to length contraction, but they will all agree on $\Delta\tau$. As long as acceleration can be treated as instantaneous, all you need to do is calculate the $\Delta\tau$s between each acceleration and add them up.

This turns out to be closely analogous to saying that a zig-zag path is longer than a straight one. It is all best explained by a space-time diagram, which I suspect is being drawn by ghwellsjr as I type...

 

[rede96]

OK - you're complicating things now. But this is enough to explain where you are going wrong: two years according to who? If both travelers fire their engines when their on board clocks read two years, both will say the other burned later because the other's clocks were slow. I suspect you are thinking in the frame in which the rockets are doing equal and opposite velocities - which is a good choice of frame - but note that two years in this frame is not two years according to the rocket's clocks.

Yes, this is where I am getting confused. I wasn't thinking about what would happen if I looked at it from a third frame. Or any frame really. What I was thinking was more a prediction.

For example, I know Ship A and Ship B are both in inertial frames and are on a big X trajectory relative to me, as we can assume that I am at the mid point of the 'X'. I also know they are both equidistant from that mid point and traveling at the same speed relative to me. Hence how I know they will meet at that mid point. So I decided to make an experiment.

When they pass that mid point, I have asked them to synchronise their clocks, then agree that for each person, after 2 years have elapsed on their respective clocks, they will turn to ships to face each other and continue along the new trajectory until they meet the middle.

I tell as their paths are symmetrical wrt the mid-point where they synchronised their clocks, then when they meet, their clocks will have both elapsed the same amount of time.

Now I am assuming that if the above is correct, I can make that prediction in any frame of reference?

Is that correct?

 

[ghwellsjr]

I'm only an amateur enthusiast when it comes to physics and have no background in science at all. It's just something I enjoy thinking about.

One thing I am really finding difficult to get my head around is how time dilation works. So I've put a thought experiment below and would appreciate it if anyone would check my understanding.

There are two friends of the same age, on different space ships, both in uniform motion but traveling in different directions relative to each other. While communicating they discover that their paths will cross so decide to do a little experiment.

They agree that when they pass each other, ship A will send a single to ship B and then they will both synchronise their clocks to a pre agreed date and time. They also agree that they will send each other an update once a month at an exact agreed time using the newly synchronised clocks. The message will include the current time and date from the respective ships so they can see if their clocks remain in sync and at each message, report back any difference. They will do this for 2 years.

This is an excellent idea and I think I can show you how it is all they need to do to establish Time Dilation with the simple application of Einstein's second postulate.

They also measure that relative to each other they are moving apart at 0.8c and neither ship goes through any accretion during the experiment.

I will also show you how they can each measure the speed of the other.

As I understand it, Ship A would start to receive the messages each month from ship B with longer and longer delays. However Ship B would also be experiencing the same thing.

This is a very important feature of Einstein's first postulate; however slow ship A sees ship B's date/time going, ship B will see ship A's date/time going slow by the same factor.

But what would be a constant is that the delay for each message in sequence would be the same for each ship.

So after 2 years, they stop sending the messages to each other (24 in total) and eventually they each receive the final message from the other ship. They then compare together the total delays for each month interval from the other ship's messages. What they both find is that although each ship thought the other ship's time was moving slower, because the delays were getting longer, the delays for each set of messages sent, match for each ship.

Baffled by this they decide to use their new quantum teleportation devices (which is possible at least in theory) and see who has aged more.

I don't think they or anybody else should be baffled that a symmetrical situation should be, well, symmetrical. There's no need to propose any device as you suggest, they already have everything they need to establish Time Dilation.

First, let's suppose that they set their calendar/clocks to zero as they passed each other (just to make our calculations simpler). Then let's just suppose that as they are sending and receiving their messages, they each see the other ones signals coming in at one-half the rate that they are each sending them. At the end of the first month ship A will send a message to ship B saying that it is the end of A's first month. Ship B will get this message at the end of his second month so he will send a message back to A saying that at B's date/time of 2 months he got A's message sent at A's date/time of one month. Ship A will get this message at his date/time of 4 months.

So what does ship A do with this information? He applies Einstein's second postulate that his first signal took the same amount of time to get to B as B's response took to get back to him. Since the total time from sending to receiving was 3 months (4-1), he divides that time by 2 and gets 1.5 months as the time after he sent his signal, or 2.5 months as the time on his calendar/clock that ship B received the signal. Furthermore, ship A also establishes the distance away that ship B was as how far light travels in 1.5 months which is simply 1.5 light-months.

Since Time Dilation is the ratio of Coordinate Time (the same as the time on a stationary clock) to the time on a moving clock, ship A has just established that ship B's calendar/clock is dilated by a factor of 2.5/2 or 1.25.

Also, ship A establishes that since ship B was 1.5 light-months away at ship A's time of 2.5 months, then ship B is traveling away from ship A at 1.5/2.5 or 0.6c while ship A was stationary.

It should be clear that ship B could have also done the same thing and established that it was at rest while ship A was traveling away in the opposite direction at 0.6 c and that it was ship A's calendar/clock that was time dilated by a factor of 1.25.

So my questions are:

1) If they both decide that at a given date, say after each experience 10 years after they initially passed and set their clocks, they teleport to a mid point between them, will they find that they have both aged at the same rate once together?

2) If the guy from ship A decided to teleport to Ship B, would he find that his friend on ship B was actually younger? (and the same if Ship B guy teleported to Ship A)

3) Assuming the above is correct (which I have no idea if it is) then what would happen if the guy from Ship A teleported to Ship B and sees his friend is younger. But then teleports back to his ship. Then the guy from Ship B teleports over to ship A. Would he find that miraculously his friend on Ship A is now younger them him again?

You're right that teleporting would be miraculous so there is no way to answer your questions.

All really confusing :)

I hope you're still not confused.

You may have noted that the example that I used was not the example you described because you specified a speed of 0.8c and mine turned out to be 0.6c. But if you understood my explanation, I'd like you to work through another one where they each see the other ship's calendar/clock running at one third of there own and see if that turns out to be your example at 0.8c. Can you do that or do you need more help?

 

[rede96]

I hope you're still not confused.

You may have noted that the example that I used was not the example you described because you specified a speed of 0.8c and mine turned out to be 0.6c. But if you understood my explanation, I'd like you to work through another one where they each see the other ship's calendar/clock running at one third of there own and see if that turns out to be your example at 0.8c. Can you do that or do you need more help?

Thanks, that does help to understand a bit of how to calculate the time dilation.

But I'm still not clear on one point.

If the two guys were in a parallel path then they would in effect be at rest wrt each other. So if one of the guys decided to change course and go see the other person and traveled at 0.6c for a distance of x, when he arrived he would be the younger of the pair due to the effects mentioned. There is no ambiguity as the two people are together.

However if the guy instead turns to travel in the opposite direction away from his friend, but still travels the same distance x at 0.6c, it is valid for either pair to state that the other's clock has moved slower. We can't say the guy who traveled in the opposite direction is younger when he stops traveling as the two guys aren't together.

So it seems direction has a bearing on outcome?

 

[ghwellsjr]

Thanks, that does help to understand a bit of how to calculate the time dilation.

You're welcome.

But I'm still not clear on one point.

If the two guys were in a parallel path then they would in effect be at rest wrt each other. So if one of the guys decided to change course and go see the other person and traveled at 0.6c for a distance of x, when he arrived he would be the younger of the pair due to the effects mentioned. There is no ambiguity as the two people are together.

Don't confuse Time Dilation with differential aging. You can only compare the amount of aging between two guys if they first start out together, then separate, then rejoin. So in this case, there is ambiguity.

However if the guy instead turns to travel in the opposite direction away from his friend, but still travels the same distance x at 0.6c, it is valid for either pair to state that the other's clock has moved slower. We can't say the guy who traveled in the opposite direction is younger when he stops traveling as the two guys aren't together.

So it seems direction has a bearing on outcome?

Direction has no bearing on Time Dilation (which is what you are asking about) and you don't need a new scenario to get a change in direction. Your original scenario will work just fine. The two guys start out approaching each other and then after they pass, they are receding from each other.

You said that they were communicating before they passed each other so they could have easily agreed to send the same kind of signals and done the same kind of calculations that they did afterwards. The only difference is that they will each see the other's signals coming in at twice the rate of their own signals going out, instead of half the rate which is true after they pass.

Unfortunately, it gets a little hard to conceptualize how this works but let's pretend that while they are approaching, they simply keep track of the times on their own calendar/clocks and then they make an adjustment to accommodate setting their calendar/clocks to zero when they pass. This makes all their times have negative values prior to them passing.

So 4 months prior to passing (we'll call it -4 months), ship A sends a message to ship B who gets it at -2 months and sends a message back which ship A gets at -1 month. Ship A does similar calculations as described in my previous post and comes up with the same Time Dilation of 1.25.

Does this make sense to you?

 

[rede96]

You said that they were communicating before they passed each other so they could have easily agreed to send the same kind of signals and done the same kind of calculations that they did afterwards. The only difference is that they will each see the other's signals coming in at twice the rate of their own signals going out, instead of half the rate which is true after they pass.

Unfortunately, it gets a little hard to conceptualize how this works but let's pretend that while they are approaching, they simply keep track of the times on their own calendar/clocks and then they make an adjustment to accommodate setting their calendar/clocks to zero when they pass. This makes all their times have negative values prior to them passing.

So 4 months prior to passing (we'll call it -4 months), ship A sends a message to ship B who gets it at -2 months and sends a message back which ship A gets at -1 month. Ship A does similar calculations as described in my previous post and comes up with the same Time Dilation of 1.25.

Does this make sense to you?

Yes, I think so. What I understand by that is that it doesn't matter if they are traveling towards or away from each other, the time dilation factor will be the same. If course the time for the messages will be different due to the different distances.

You're welcome.

Don't confuse Time Dilation with differential aging. You can only compare the amount of aging between two guys if they first start out together, then separate, then rejoin. So in this case, there is ambiguity.

This is the bit that is confusing me.

1) Imagine A and B start off from the same point together with clocks set to zero. They head off in opposite directions, traveling at the same speed away (away from the point of origin). When their respect clocks reach 4 months, the stop so they are at rest rst the origin and hence each other.

When they stop I ask them to send their clock time back to me at the point of origin. I notice I receive their information on clock times both at the same time and that both their clocks say the same amount of time has lapsed (4 Months) However the time past for me will be longer.

But I can concluded that as 4 months have elapsed on both A and B's clocks during that time, and as they were the same age when the started off, they will be the same age now. I don't need to bring them back together to know that. (As I understand it) Although I could bring them back at the same speed and would see they have aged the same.


2) Now imagine a similar situation where A and B set off from the same point with their clocks set to zero. But this time A takes 'W' trajectory away from the point of origin and B takes a 'Z' trajectory. I arrange the trip so the total distance traveled in the W and Z trajectories is 4 light months AND that they travel at the same speed. I have also worked out that relative to me the end point of each trajectory is 2 light months (or some other equal distance) away from me at the point of origin.

I tell them to set off on their journeys and at the end of their journeys (4 months on their clocks) stop so they are at rest wrt to me at the point of origin and send back their clock times.

Now this is the part I get confused over. As A and B had both traveled the same distance in their respective frame and finish the same distance away from me at the point of origin, will I receive their signals at the same time as with the first experiment above?

Or would I receive them at different times because their journeys were not symmetrical and thus I would get the 'W' journey first as it had the most changes in direction?

Also, in the first experiment there seems to be a contradiction as during the travel time both A and B could say that the other's clock was running slower but when they finished their journey, the are both the same age.

 

[Bill_K]

Also, in the first experiment there seems to be a contradiction as during the travel time both A and B could say that the other's clock was running slower but when they finished their journey, the are both the same age.

Simultaneity is the answer. A key part of SR is the fact that which events an observer regards as simultaneous will change if his rest frame changes. While A and B are in relative motion, A regards B's clocks as running slow, and says for example that "now", B is younger than A, celebrating his 25th birthday. When they come to relative rest, A's definition of "now" has changed, and A says that B is now the same age as him, age 30.

 

[Dale]

But I can concluded that as 4 months have elapsed on both A and B's clocks during that time, and as they were the same age when the started off, they will be the same age now. I don't need to bring them back together to know that. (As I understand it) Although I could bring them back at the same speed and would see they have aged the same.

In your rest frame, this is true. This is not true in other frames. For instance, in a frame where you are moving to the left (in the direction of A) the distance that the light from A travels is less than the distance that the light from B travels. Therefore, the fact that they arrived at you at the same time implies that A is younger than B. This makes sense because in this frame A traveled faster than B.

The problem with comparing the ages of distant objects is not that it cannot be done. Clearly it can. The problem is just that the result depends on the reference frame used for the comparison.

 

[rede96]

Simultaneity is the answer. A key part of SR is the fact that which events an observer regards as simultaneous will change if his rest frame changes. While A and B are in relative motion, A regards B's clocks as running slow, and says for example that "now", B is younger than A, celebrating his 25th birthday. When they come to relative rest, A's definition of "now" has changed, and A says that B is now the same age as him, age 30.

Thanks for that. I think I get it but not sure I fully understand. Does this mean that in A's reference frame when he wants to compare ages 'at the same time' that what A thinks is a simultaneous point for A and B to compare ages, is in fact off by 5 years? (in the example you gave) So while A and B are in relative motion, 'now' for both is relative. But when they are at rest rst each other, 'now' can be simultaneous?

 

[rede96]

In your rest frame, this is true. This is not true in other frames. For instance, in a frame where you are moving to the left (in the direction of A) the distance that the light from A travels is less than the distance that the light from B travels. Therefore, the fact that they arrived at you at the same time implies that A is younger than B. This makes sense because in this frame A traveled faster than B.

The problem with comparing the ages of distant objects is not that it cannot be done. Clearly it can. The problem is just that the result depends on the reference frame used for the comparison.

Ok, thanks. I think what I was trying to understand was if the space-time interval (which I read was invariant) is the same for both A and B for their journey then would their clocks have to read the same at the end of the journey? Then anyone in any frame if they knew the space-time interval would know that A and B's clocks would read the same relative to A and B, even though they might not agree with A and B's clocks reading the same from their frame.

I may be over complicating things, sorry if I am. But my thinking was that if I know the space-time interval (or their journey through 4d space time just in case I am not using the right terminology) is the same, then I know the net effects of any difference in time from their journey would be the same.

 

[Dale]

Ok, thanks. I think what I was trying to understand was if the space-time interval (which I read was invariant) is the same for both A and B for their journey then would their clocks have to read the same at the end of the journey? Then anyone in any frame if they knew the space-time interval would know that A and B's clocks would read the same relative to A and B, even though they might not agree with A and B's clocks reading the same from their frame.

I may be over complicating things, sorry if I am. But my thinking was that if I know the space-time interval (or their journey through 4d space time just in case I am not using the right terminology) is the same, then I know the net effects of any difference in time from their journey would be the same.

The spacetime interval is the same for both journeys, and therefore in all frames their clocks do read the same at the end of their journeys.

But the journeys only end at the same time in one frame. In other frames they end at different times and therefore they are not the same age at anyone point in time.

 

[rede96]

But the journeys only end at the same time in one frame. In other frames they end at different times and therefore they are not the same age at anyone point in time.

Ah ok. got it thanks.

 

[ghwellsjr]

Don't confuse Time Dilation with differential aging. You can only compare the amount of aging between two guys if they first start out together, then separate, then rejoin. So in this case, there is ambiguity.

This is the bit that is confusing me.

1) Imagine A and B start off from the same point together with clocks set to zero. They head off in opposite directions, traveling at the same speed away (away from the point of origin). When their respect clocks reach 4 months, the stop so they are at rest rst the origin and hence each other.

When they stop I ask them to send their clock time back to me at the point of origin. I notice I receive their information on clock times both at the same time and that both their clocks say the same amount of time has lapsed (4 Months) However the time past for me will be longer.

But I can concluded that as 4 months have elapsed on both A and B's clocks during that time, and as they were the same age when the started off, they will be the same age now. I don't need to bring them back together to know that.

As has been pointed out by others, what you are calling "now" is only true in your rest Inertial Reference Frame (IRF) which we can illustrate in an example where the two travelers went away at 0.333c. Here is a spacetime diagram showing your scenario. You are shown as the thick black line, A is shown as the thick blue line and B is the thick red line:

Note that your clock has advanced to 5.67 months when you receive the two messages from A and B that they have reached the 4-month mark for them to stop. And by the time you get their messages, their clocks have advanced to 5.4 months.

We can use the Lorentz Transformation process to see what this scenario looks like in A's rest frame during his "travel time":

Note that you still receive the messages at your time of 5.67 months, even though it took different amounts of time for the messages to get to you and A and B are not the same age when this happens.

(As I understand it) Although I could bring them back at the same speed and would see they have aged the same.

Yes, if you bring them back at the same speed, they will continue to age at the same rate. Here is a diagram showing this:

You have aged more than the travelers (8.48 months) but they have aged the same (8 months).

But now this is true in all other IRF's. Here's the same one we did before:

2) Now imagine a similar situation where A and B set off from the same point with their clocks set to zero. But this time A takes 'W' trajectory away from the point of origin and B takes a 'Z' trajectory. I arrange the trip so the total distance traveled in the W and Z trajectories is 4 light months AND that they travel at the same speed. I have also worked out that relative to me the end point of each trajectory is 2 light months (or some other equal distance) away from me at the point of origin.

I tell them to set off on their journeys and at the end of their journeys (4 months on their clocks) stop so they are at rest wrt to me at the point of origin and send back their clock times.

Now this is the part I get confused over. As A and B had both traveled the same distance in their respective frame and finish the same distance away from me at the point of origin, will I receive their signals at the same time as with the first experiment above?

Yes, here is a diagram to depict what I think you described:

Or would I receive them at different times because their journeys were not symmetrical and thus I would get the 'W' journey first as it had the most changes in direction?

Symmetry doesn't matter. Remember in post #10 I said that direction doesn't matter? The only thing that matters is speed according to an IRF. As long as both travelers are always going at the same speed, then they will be subject to the same Time Dilation and end up with the same aging when they rejoin you.

 

[ghwellsjr]

1) Imagine A and B start off from the same point together with clocks set to zero. They head off in opposite directions, traveling at the same speed away (away from the point of origin). When their respect clocks reach 4 months, the stop so they are at rest rst the origin and hence each other.
...
Also, in the first experiment there seems to be a contradiction as during the travel time both A and B could say that the other's clock was running slower but when they finished their journey, the are both the same age.

If you apply the technique I described for you in post #8, sending messages back and forth, then A can make a non-inertial rest frame for himself and it will show you how even though A determines that B's clock was running slower than his at the beginning, it can still catch up at the end. To make an accurate diagram, A will have to send out continual messages and get continual responses back from B but I will only show those where B's clock increments one-month intervals. We will also have A communicating with you. Here's the signals going between A and B:

And here's the signals going between A and you:

Remember what A does with the information:

So what does ship A do with this information? He applies Einstein's second postulate that his first signal took the same amount of time to get to B as B's response took to get back to him. Since the total time from sending to receiving was 3 months (4-1), he divides that time by 2 and gets 1.5 months as the time after he sent his signal, or 2.5 months as the time on his calendar/clock that ship B received the signal. Furthermore, ship A also establishes the distance away that ship B was as how far light travels in 1.5 months which is simply 1.5 light-months.

This allows A to construct this diagram:

As you can see, even though both you and B are Time Dilated at the beginning, you eventually surpass A while B just catches up.

 

[rede96]

As has been pointed out by others, what you are calling "now" is only true in your rest Inertial Reference Frame (IRF) which we can illustrate in an example where the two travelers went away at 0.333c. Here is a spacetime diagram showing your scenario. You are shown as the thick black line, A is shown as the thick blue line and B is the thick red line:

Note that your clock has advanced to 5.67 months when you receive the two messages from A and B that they have reached the 4-month mark for them to stop. And by the time you get their messages, their clocks have advanced to 5.4 months.

Thanks for the space time diagrams, they really help make it clear.

Symmetry doesn't matter. Remember in post #10 I said that direction doesn't matter? The only thing that matters is speed according to an IRF. As long as both travelers are always going at the same speed, then they will be subject to the same Time Dilation and end up with the same aging when they rejoin you.

I understand now that direction doesn't matter. But I thought change in direction did? Due to acceleration? So although the distance and speed covered by the 'W' path and the 'Z' path is the same, as the 'W' path has changed direction more than the 'Z' path doesn't that mean that the two will age differently when the come to rest?

Although as both come to rest the same distance from the origin where the set their clocks, then I guess the total acceleration (change in direction) would be the same as the angles would have to add up the same?

Does that make sense?

 

[ghwellsjr]

Thanks for the space time diagrams, they really help make it clear.

You're welcome. I'm glad they helped.

I understand now that direction doesn't matter. But I thought change in direction did? Due to acceleration? So although the distance and speed covered by the 'W' path and the 'Z' path is the same, as the 'W' path has changed direction more than the 'Z' path doesn't that mean that the two will age differently when the come to rest?

Although as both come to rest the same distance from the origin where the set their clocks, then I guess the total acceleration (change in direction) would be the same as the angles would have to add up the same?

Does that make sense?

No, only the speed according to an IRF matters to the Time Dilation which means the rate that a clock ticks. Acceleration only matters if it results in a speed change, not a direction change.

 

[phyti]

But I can concluded that as 4 months have elapsed on both A and B's clocks during that time, and as they were the same age when the started off, they will be the same age now. I don't need to bring them back together to know that.

Your conclusion would be based on historical data, and an assumption that nothing has happened since that time. You cannot 'know' the state of a remote system 'now'.
As a counterexample, one of them could have died.

 

[rede96]

Your conclusion would be based on historical data, and an assumption that nothing has happened since that time. You cannot 'know' the state of a remote system 'now'.
As a counterexample, one of them could have died.

Yes, conclusions are based on historical data, that is that when A and B have finished their respective journeys, are at rest wrt me and have contacted me with their data, then I work out my conclusions.

No one died or was hurt in the making of this thought experiement :)

My 'now' is always based in my IFR and when the other ships have come to rest wrt me. What happens during the journey isn't really of concern, only as a matter of interest.

 

[Sugdub]

... No, only the speed according to an IRF matters to the Time Dilation which means the rate that a clock ticks. Acceleration only matters if it results in a speed change, not a direction change.

Can we say that “only the average speed according to an IRF matters to the Time Dilation which determines the average value of the period assigned, in this IRF, to a relatively moving clock” (I assume all velocities are alongside the x-axis and that both the initial and final meeting points are collocated)?

Then the Differential Aging of both clocks at their second meeting point would depend on the difference between the average speed, across its journey, of each clock in respect to the same IRF (this difference being independent on the selected IRF, irrespective of the maximum speed reached by any of the clocks, of the number of direction changes alongside the x axis, of the magnitude of their accelerations and finally not directly dependent on their relative speed in respect to each other).

 

[Nugatory]

Can we say that “only the average speed according to an IRF matters to the Time Dilation which determines the average value of the period assigned, in this IRF, to a relatively moving clock” (I assume all velocities are alongside the x-axis and that both the initial and final meeting points are collocated)?

No. Try calculating a few examples and you'll see that approach won't work.

 

[ghwellsjr]

Can we say that “only the average speed according to an IRF matters to the Time Dilation which determines the average value of the period assigned, in this IRF, to a relatively moving clock” (I assume all velocities are alongside the x-axis and that both the initial and final meeting points are collocated)?

Then the Differential Aging of both clocks at their second meeting point would depend on the difference between the average speed, across its journey, of each clock in respect to the same IRF (this difference being independent on the selected IRF, irrespective of the maximum speed reached by any of the clocks, of the number of direction changes alongside the x axis, of the magnitude of their accelerations and finally not directly dependent on their relative speed in respect to each other).

I'm not even sure how to apply your calculation. Let's take this diagram from post #18 and determine the difference in aging between the black and the red clocks:

The average speed of the black clock is easy--it's 0.333c--but how do you average the speed of the red clock? At first its speed is 0.6c and then it's 0c. According to the time on the clock it spends and equal time at both speeds so the average speed would 0.3c. Or do you take a weighted average according to the Coordinate Time where it spends 5/9 of its time at 0.6c and 4/9 at 0c so the weighted average would be 0.333c, the same as the black clock so I guess that won't work if we're trying to find a difference.

But if we use the non-weighted average we have the 0.333c for the black clock and 0.3c for the red clock.

Now look at the other diagram for the same situation where the black clock has an average speed of 0c and the red clock has an average speed of 0.333c:

I don't see how your scheme can work since in the first diagram the black clock has an average speed of 0.333c while the red clock is lower and in the second diagram the red clock has an average speed of 0.333c while the black clock is lower. We need to get the same answer in both diagrams.

Or maybe I haven't understood your scheme?

 

[phyti]

...they will be the same age now.
I don't need to bring them back together to know that.

The clocks are frequencies, and while in motion each observer records a perceived frequency resulting from doppler effects, i.e. relative motion of source and detector. This does not provide info about 'aging'.

You do have to bring two moving clocks together to compare accumulated time from a previous union.

The only way you can 'know' if they are the same age 'now', is wait until a later 'now'.

 

[Dale]

Can we say that “only the average speed according to an IRF matters to the Time Dilation which determines the average value of the period assigned, in this IRF, to a relatively moving clock”

So, the aging of a particle on a worldline in an inertial reference frame in units where c=1 is the proper time:
$$\tau= \int \sqrt{1-v(t)^2} dt$$

The average speed is:
$$\frac{1}{T}\int \left|v(t)\right| dt$$

I don't see any way to pull the latter out of the former. Even a series expansion does not give a v(t) term.

 

[Sugdub]

No. Try calculating a few examples and you'll see that approach won't work.

...I don't see any way to pull the latter out of the former...

Thanks for your explanations.

...I don't see how your scheme can work ...Or maybe I haven't understood your scheme?

The issue I intended to debate is that physicists seem to indicate that the relative motion between two clocks is responsible for (or causes) their differential aging. But the reciprocity of the relative motion prevents deciding which clock will age less unless some additional hypothesis gets injected. Also the symmetrical twins case leads to their equal aging in spite of their relative motion (as per the red and blue clocks in your second diagram).

So how can we better qualify what causes (the physical conditions leading to) a different aging? What do we actually need to know in order to calculate the time gap?

Let's assume that we specify the relative speed of each clock in respect to an IRF for which the initial and final meeting points are collocated, in such a way that their respective journeys are different but not symmetrical (so let's imagine a third diagram based on your second one, with the red and blue clocks meeting again at the same location following non-symmetrical journeys). On that basis, one can determine the period assigned to each moving clock (red or blue) as compared to the period of a third identical clock (black) assumed to remain at rest as per the aforementioned IRF, for each segment of their journey. So the differential aging between each of the moving clocks (red or blue) and the rest clock (black) can be determined by cumulating the contributions from all segments of their respective journey. Previously I've suggested that somehow the average speed between the red and black clocks (respectively the blue and black clocks), as calculated in the aforementioned IRF, “determines” their differential aging. Thanks to the explanations I received, I now understand this was not correct. However, since time dilation increases with the relative speed, I guess that a larger average red-black speed is likely to induce a larger red-black differential aging at the end of the journey. Anyway the difference between the red-black and the blue-black cumulated values gives the differential aging between the red and the blue clocks.

So IMHO we now have a more realistic specification for what “causes” a different aging. It does not refer any longer to the relative motion between the red and blue clocks, but it spells out the relative motion between the (black) rest clock and each of the moving (red and blue) clocks. However it refers to a peculiar IRF and I guess it can't be extended to any IRF. May be there is a better way to solve this issue...

 

[Nugatory]

The issue I intended to debate is that physicists seem to indicate that the relative motion between two clocks is responsible for (or causes) their differential aging.

No, we are saying no such thing, and you will continue to be confused until you understand the difference between differential aging and time dilation. Motion does cause time dilation, but it doesn't cause differential aging.

Differential aging: two different world lines connecting two points in spacetime can have two different lengths, just as two different roads between the same two places on Earth will, in general, have different lengths.

Time dilation: the instantaneous rate of change of the $t$ coordinate as a function of proper time along a world line depends on whose $t$ coordinate you're using. Choose a coordinate system in which the $x$ coordinate isn't changing along the world line (that is, one in which an object following that world line is at rest) and the rate of change of the $t$ coordinate will be greater than if you choose any other coordinate system.

Those may not be the definitions you're most accustomed to hearing; that's because I'm carefully not mentioning relative motion at all.

 

[ghwellsjr]

Thanks for your explanations.

You're welcome.

...
So IMHO we now have a more realistic specification for what “causes” a different aging. It does not refer any longer to the relative motion between the red and blue clocks, but it spells out the relative motion between the (black) rest clock and each of the moving (red and blue) clocks. However it refers to a peculiar IRF and I guess it can't be extended to any IRF. May be there is a better way to solve this issue...

DaleSpam already gave you the complete answer in post #28. Instead of concerning yourself with the black clock which happens to be at rest in a particular IRF, you want to use the Coordinate Time, represented by "t" in his equation. (They may be the same but you still don't want to restrict yourself to any particular clock at rest in an IRF.) In my diagrams, I use a simplified version of his equation which simply means that for the increment of the Coordinate Time grid lines, I determine (or specify) the speed of an object and calculate how much Time Dilation there is for that increment of time and that's how far apart I spread the dots.

Once I build a complete scenario in the defining IRF, I can use the Lorentz Transformation process to get to any other IRF moving at any speed within the range of +/- c. That is one reason why you don't want to conceptualize Time Dilation as dependent on or in relation to any other actual clock. It's related to the Coordinate Time of an IRF which may or may not have any clock at rest in it and even if it did, that clock would be subject to the same determination of Time Dilation as any other clock. For example, in the second and fourth diagrams on post #18, there is no clock at rest in those IRF's, and yet the determination of Time Dilation for all the clocks still works.

 

[rede96]

So just to check my understanding...

My twin and I are the same age. We both have very fast star ships and are challenged to a race to see who's star ship is the quickest. The race starts and ends on Earth and the first one back wins.

We both set off in our separate star ships to a star that is 10 ly's away. When we get there we must turn around and head back to earth.

I have a very special star ship that allows me to travel very close to the speed of light, so the time I actually experience on my journey is arbitrarily small, say just 1 day. However my twin is only traveling at say 0.5c so the time he experiences is longer.

When I arrive back on Earth I find that people I knew are nearly 20 years older then when I left. And, to my surprise I find my twin is already back on Earth and older than I am. So he won the race? But how can that be if I was traveling almost twice as fast?

Am I missing something or is that just how space-time works?

 

[ghwellsjr]

So just to check my understanding...

My twin and I are the same age. We both have very fast star ships and are challenged to a race to see who's star ship is the quickest. The race starts and ends on Earth and the first one back wins.

We both set off in our separate star ships to a star that is 10 ly's away. When we get there we must turn around and head back to earth.

I have a very special star ship that allows me to travel very close to the speed of light, so the time I actually experience on my journey is arbitrarily small, say just 1 day. However my twin is only traveling at say 0.5c so the time he experiences is longer.

When I arrive back on Earth I find that people I knew are nearly 20 years older then when I left. And, to my surprise I find my twin is already back on Earth and older than I am. So he won the race? But how can that be if I was traveling almost twice as fast?

Am I missing something or is that just how space-time works?

I'd be surprised too if your twin got back before you. Why do you think this? You are correct about your trip but your twin will take twice as long meaning he won't get back until 40 years after he left and he will have aged 34.6 years.

 

[rede96]

I'd be surprised too if your twin got back before you. Why do you think this? You are correct about your trip but your twin will take twice as long meaning he won't get back until 40 years after he left and he will have aged 34.6 years.

Ah right, of course. Thanks. And I wasn't thinking, but it has been a long day. :)

Cheers.

 

[Dale]

So how can we better qualify what causes (the physical conditions leading to) a different aging? What do we actually need to know in order to calculate the time gap?

The cause of differential aging is simply that different paths through spacetime have different intervals. This is no different nor any more surprising than the fact that different paths in space have different lengths. To determine the time elapsed on a timelike path through spacetime we calculate:

$$\tau=\int \sqrt{-g_{\mu\nu} dx^{\mu}dx^{\nu}}$$

This formula works for any timelike path, inertial or non-inertial, through any spacetime, curved or flat, using any coordinate system. It is the completely general explanation for differential aging. In an inertial frame in flat spacetime it simplifies to the equation I gave in 28.