[Halc]

So in other layman words, is it integrating all the moments of its velocities throughout the acceleration (please excuse my layman wording, I hope the point of my question comes across.)?

Integrating the speeds, but yes. The direction doesn't matter when doing it this way, only the magnitude. Magnitude of velocity is speed.

One other question, when you said, "In the frame in which the two comparison events are at the same spatial location, the travelling twin is moving at a higher velocity than the Earth twin" do you mean at the moment he starts?

A frame is not a moment, it is (in this case) an inertial coordinate system that assigns spatial and temporal coordinates to all events in spacetime.

I ask because I wanted them both to start at rest and end at rest in the thought experiment.

At rest is relative to a frame. Yes, they're both at rest relative to each other at the the start, end, and also the middle of your new scenario. At all other times, they're not.

And most importantly, the temporal length of his worldline between those two events is shorter than that of the Earth twin. That worldline length defines the differential aging they experience.

Yes, I agree.  I am not arguing that the worldline should be longer than the Earth twin.

Shorter. There is less time along that path.

My argument is how did it become longer; acceleration, something else?

It's a path, a time-like line of adjacent events (points) in spacetime. It has an intrinsic geometric length, just like a space-like worldline line has an intrinsic spatial length. The path doesn't ever become anything since it is always there. The one twin just happens to follow this particular path.
So it's like asking why a curved line on paper between two points is longer than the straight one. It never became longer, but one might choose the longer path rather than the shorter one. Paper is Euclidean geometry where the shortest path is a straight line. You find the length of the hypotenuse of a triangle (x/y axis) by L=√(y²+x²). Minkowskian spacetime isn't Euclidean like that. The length of the hypotenuse of a timelike worldline is L=√(ct²-x²). Notice the minus sign in there, which means if you move through space in your coordinate system (wander away from the straight line), the length of the line gets shorter, not longer like it does on Euclidean paper.

The twins scenario can also be described (explained?) just by what each observer sees. The each see the clock of the other run slow as he recedes, but see it run faster as the approach each other. The symmetry is very nice in that instance, except the times of each phase are different, which explains the differential aging. But again, what anybody measures has nothing to do with causing something observed to age.

This is confusing to me when there is no acceleration.

There is acceleration in this case. It is the standard twins scenario, same story, different way of looking at it.

Let's say 2 clocks have always been moving at a constant velocity with respect to one another.  Each clock sees the other clock run faster as they approach each other.  When they meet, there will be no time dilation.

There is always dilation because each clock is moving in the inertial frame of the other. That's pretty much one of the three ways to define dilation (inertial frames, accelerating frames, and curved frames: gravity). So each clock will run slower relative to the frame of the other, but an observer watching the incoming clock will see it running faster, mostly due to Doppler effect, just like the siren of an approaching ambulance.

How can this be time dilation without acceleration?

Inertial dilation is all about speed and is not a function of acceleration at all. In the twins scenario, the acceleration is necessary for the twins to meet twice in Minkowskian (flat) spacetime. It can be done without acceleration, but doing so requires curved space, meaning it involves gravity.

For example, take two satellites in very eccentric orbits about Earth. Sans engines, both follow a geodesic (a straight line) through curved spacetime. They meet at apogee of one satelite and perigee of the other, meaning one has a shorter orbit and one much larger, such that the period of one is exactly twice the other so they meet repeatedly where clocks can be compared. The clock on the inner satellite will record less time than the outer one at each comparison event. Everything is weightless (not accelerating) the whole time, so no proper acceleration. Everything traces straight lines through curved spacetime, but the path lengths between successive intersection events are not the same.

[MikeFontenot]

This is confusing to me when there is no acceleration.  Let's say 2 clocks have always been moving at a constant velocity with respect to one another.  Each clock sees the other clock run faster as they approach each other.

In the above, I think you are describing what the home twin (she) looks like to the traveling twin (him) if she has been transmitting a TV image of herself for a long time.  He WILL see her ageing faster than himself on his TV monitor as he moves toward her.  But that is entirely different from the question: "How old is she right now", which is the really important question.  He knows that her image on his TV is out of date ... it doesn't show her age "NOW".

[Dimensional (OP)]

So in other layman words, is it integrating all the moments of its velocities throughout the acceleration (please excuse my layman wording, I hope the point of my question comes across.)?

Integrating the speeds, but yes. The direction doesn't matter when doing it this way, only the magnitude. Magnitude of velocity is speed.

Okay, thanks, I did not know this.

My argument is how did it become longer; acceleration, something else?

It's a path, a time-like line of adjacent events (points) in spacetime. It has an intrinsic geometric length, just like a space-like worldline line has an intrinsic spatial length. The path doesn't ever become anything since it is always there. The one twin just happens to follow this particular path.
So it's like asking why a curved line on paper between two points is longer than the straight one. It never became longer, but one might choose the longer path rather than the shorter one. Paper is Euclidean geometry where the shortest path is a straight line. You find the length of the hypotenuse of a triangle (x/y axis) by L=√(y²+x²). Minkowskian spacetime isn't Euclidean like that. The length of the hypotenuse of a timelike worldline is L=√(ct²-x²). Notice the minus sign in there, which means if you move through space in your coordinate system (wander away from the straight line), the length of the line gets shorter, not longer like it does on Euclidean paper.

Ok, that's a good point.  I have already learnt about Minkowski metric vs Euclidean metric, as well as how to make basic spacetime diagrams.  But I am definitely fuzzy about accelerated worldlines vs nonaccelerated worldlines and using the Minkowski metric to show the difference, and then what that would mean for time dilation.

The twins scenario can also be described (explained?) just by what each observer sees. The each see the clock of the other run slow as he recedes, but see it run faster as the approach each other. The symmetry is very nice in that instance, except the times of each phase are different, which explains the differential aging. But again, what anybody measures has nothing to do with causing something observed to age.

This is confusing to me when there is no acceleration.

There is acceleration in this case. It is the standard twins scenario, same story, different way of looking at it.[/quote]

I meant that for other examples when acceleration does not happen I get confused as to how time dilation makes sense.  Then I started with an example that does not have acceleration.

Let's say 2 clocks have always been moving at a constant velocity with respect to one another.  Each clock sees the other clock run faster as they approach each other.  When they meet, there will be no time dilation.

There is always dilation because each clock is moving in the inertial frame of the other. That's pretty much one of the three ways to define dilation (inertial frames, accelerating frames, and curved frames: gravity). So each clock will run slower relative to the frame of the other, but an observer watching the incoming clock will see it running faster, mostly due to Doppler effect, just like the siren of an approaching ambulance.

I am not sure I understand exactly what you are saying about scenarios without acceleration (and gravity for this matter).  Are you saying that there is time dilation for each clock relative to the other in my example with the clocks?

How can this be time dilation without acceleration?

Inertial dilation is all about speed and is not a function of acceleration at all. In the twins scenario, the acceleration is necessary for the twins to meet twice in Minkowskian (flat) spacetime. It can be done without acceleration, but doing so requires curved space, meaning it involves gravity.

I meant for my example with the two clocks.

For example, take two satellites in very eccentric orbits about Earth. Sans engines, both follow a geodesic (a straight line) through curved spacetime. They meet at apogee of one satelite and perigee of the other, meaning one has a shorter orbit and one much larger, such that the period of one is exactly twice the other so they meet repeatedly where clocks can be compared. The clock on the inner satellite will record less time than the outer one at each comparison event. Everything is weightless (not accelerating) the whole time, so no proper acceleration. Everything traces straight lines through curved spacetime, but the path lengths between successive intersection events are not the same.

Yes, I know a little about this and that time dilation also comes from gravitational potential.  So I am saying that acceleration is the only reason for time dilation.  I am more interested in what exactly causes time dilation without gravity, and without acceleration for this matter.

[alancalverd]

  I am more interested in what exactly causes time dilation without gravity, and without acceleration for this matter.

It is not possible to experience time dilation without an energy change. Gravitational potential offers one form, acceleration another.

If A is moving relative to B, their identical clocks cannot synchronise, due to time dilation.

If A and B are synchronised they can only acquire a relative velocity if one of them accelerates.

Einstein recognised that a change in gravitational potential is equivalent to an acceleration. This makes cosmology rather more interesting as it implies that two widely separated clocks might appear to a third party to be synchronised and fixed in space, but one could then drift if it was approached by a large mass.

It is important to approach physics from the standpoint of relativistic mechanics rather than simple newtonian analysis. The relativistic equations neatly degenerate to classical mechanics whenever v<<c but you can't generate relativistic corrections if you begin with the degenerate form. And the relativistic results are confirmed by experiment.

[Dimensional (OP)]

This is confusing to me when there is no acceleration.  Let's say 2 clocks have always been moving at a constant velocity with respect to one another.  Each clock sees the other clock run faster as they approach each other.

In the above, I think you are describing what the home twin (she) looks like to the traveling twin (him) if she has been transmitting a TV image of herself for a long time.  He WILL see her ageing faster than himself on his TV monitor as he moves toward her.  But that is entirely different from the question: "How old is she right now", which is the really important question.  He knows that her image on his TV is out of date ... it doesn't show her age "NOW".

I was just trying to think of an example without acceleration.  But I just realized that my example does not make any sense because the clocks were never synchronized in the first place. 

So the new example is just 2 clocks, no twins, that have always been travelling toward each other since infinity.  When they meet they synchronize, and then continue along their paths in opposite directions.  I believe one will "see" (let's leave doppler effects and photons out of it if we can) the other clock as running slower.  I don't know if that counts as time dilation.

[alancalverd]

I think the light is beginning to dawn!

Be careful not to misuse "synchronise". As I pointed out in reply #11, the colloquial use of the word means to set them to both read t =  zero right now, on the presumption that v<<c so for all practical purposes you can guarantee simultaneity when t = x, the time you want something to happen. Perfectly adequate for making a rendezvous by ship or car, and indeed the basis of timekeeping for trade ever since Stonehenge  was built. But not for satellite navigation or cosmic red shift.

The whole point of specifying twins (or other identical clocks) is that initially each sees the other as having the same tick rate, so they have to observe one another for a finite elapsed time (at least one tick) to establish true synchronicity. But if they have a nonzero relative velocity, they cannot see each other as having the same tick rate.

[Dimensional (OP)]

I think the light is beginning to dawn!

Be careful not to misuse "synchronise". As I pointed out in reply #11, the colloquial use of the word means to set them to both read t =  zero right now, on the presumption that v<<c so for all practical purposes you can guarantee simultaneity when t = x, the time you want something to happen. Perfectly adequate for making a rendezvous by ship or car, and indeed the basis of timekeeping for trade ever since Stonehenge  was built. But not for satellite navigation or cosmic red shift.

The whole point of specifying twins (or other identical clocks) is that initially each sees the other as having the same tick rate, so they have to observe one another for a finite elapsed time (at least one tick) to establish true synchronicity. But if they have a nonzero relative velocity, they cannot see each other as having the same tick rate.

Yeah, good point!

[Halc]

I meant that for other examples when acceleration does not happen I get confused as to how time dilation makes sense.  Then I started with an example that does not have acceleration.

It was a good example, one that illustrated dilation without the differential aging. Dilation from inertial motion is a function of speed. Differential aging (from any kind of situation) is a function of path lengths. The twin paradox is meant to illustrate the latter.

I am not sure I understand exactly what you are saying about scenarios without acceleration (and gravity for this matter).  Are you saying that there is time dilation for each clock relative to the other in my example with the clocks?

Relative to the various inertial frames, yes. In one frame, the first clock runs faster and relative to another frame the second clock runs faster. Relative to some select frames, both tick at the same rate. It's all an abstract consequence of frame choice.
Differential aging is not a function of frame choice since the same answer is found regardless of choice of coordinate system with which to describe the situation.

How can this be time dilation without acceleration?

I meant for my example with the two clocks.

Your example was a perfect one of dilation. Relative a frame where one of the clocks is stationary, the other clock is dilated (runs slower).

Yes, I know a little about this and that time dilation also comes from gravitational potential.

Yes, because curved spacetime alters path lengths, and sometimes allows multiple straight lines to meet more than once just like straight train tracks will cross in two places on Earth. The surface of Earth is 2D but positively curved so parallel lines meet after a while.

So I am saying that acceleration is the only reason for time dilation.

Then you're not getting it. I gave several examples of the dilated clock being the non-accelerated one, or the less accelerated one. Without gravity, clocks can have dilation without acceleration (such as in your example of passing clocks) but since they never meet more than once, there is no pair of paths between the same two events that can be compared. Without resorting to gravity there is no way to make the clocks meet twice without one or both of them accelerating (or by staying in each other's presence the entire way).

It is not possible to experience time dilation without an energy change.

Time dilation, like length contraction, isn't something that is experienced. They are coordinate effects, and per the first postulate (Galilean relativity), physics is the same relative to any inertial frame, so by that postulate, dilation is not something that can be measured or felt.
Change of potential also cannot be felt, but proper acceleration very much can.

If A is moving relative to B, their identical clocks cannot synchronise, due to time dilation.

In standard physics texts, there is frequent mention of syncing passing clocks. Any clock can be zeroed or set to the reading of the other when in its presence, and it is not a misuse of the term 'synchronise' to do this. It does not require v<<c. There's no need for a tick to pass in order to set a clock to a specific time. With the twins, it's a calendar, hardly a stopwatch. Yes, if they're moving relative to each other, they cannot stay in sync except in some select frames in which they have the same speed.
This kind of sync just means the two clocks read the same value at that event. There's no implication that they're expected to subsequently tick at the same rate. The operation doesn't require a selection of frame

The other kind of sync is for clocks that are stationary relative to each other and not in each other's presence. This requires a selection of frame and a sync convention to get them to read the same value relative to the selected frame. You seem to only be aware of this sort of sync and not the other.

Einstein recognised that a change in gravitational potential is equivalent to an acceleration.

Actually, it was the lack of change of gravitational potential that is equivalent to acceleration. For instance, I cannot locally tell if I'm accelerating in a box in space or if the box is sitting on Earth at constant potential. Satellites on the other hand are often continuously changing potential with eccentric orbits, and yet they're in freefall, not accelerating. It is equivalent to no acceleration in flat spacetime.

This makes cosmology rather more interesting as it implies that two widely separated clocks might appear to a third party to be synchronised and fixed in space, but one could then drift if it was approached by a large mass.

Yes, and if this happens, the clocks will acquire a velocity relative to each other, further mucking up the sync well after the big object has gone by.

I was just trying to think of an example without acceleration.  But I just realized that my example does not make any sense because the clocks were never synchronized in the first place.

No, they were probably never synchronized in the first place, but they have calculators and could have computed when they'll pass and set the clocks to happen to read the same value when they do. One way to do this is to sync them to a frame in which the clocks tick at the same rate. No need to wait for them to meet. It can be done well ahead of time.

I believe one will "see" (let's leave doppler effects and photons out of it if we can) the other clock as running slower.

Scare quotes are appropriate here. They don't actually see this, but they compute it relative to their choice of frame. Remember, dilation is mostly abstract, not physical. Differential aging is physical, but there is none of that in this example.

I don't know if that counts as time dilation.

It exactly counts as dilation.

[alancalverd]

Time dilation, like length contraction, isn't something that is experienced.

https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment suggests otherwise.

There's no need for a tick to pass in order to set a clock to a specific time.

Indeed, but that isn't what is implied by "synchronicity" when considering time dilation. It's all about tick rate, and consequently elapsed time as measured by two clocks moving relative to one another.

[Dimensional (OP)]

I meant that for other examples when acceleration does not happen I get confused as to how time dilation makes sense.  Then I started with an example that does not have acceleration.

It was a good example, one that illustrated dilation without the differential aging. Dilation from inertial motion is a function of speed. Differential aging (from any kind of situation) is a function of path lengths. The twin paradox is meant to illustrate the latter.

Here is an example of how my brain gets scrambled when thinking about this scenario.

Imagine that someone was watching the clocks come together.  It is a perfectly symmetrical situation.  The observer is between the clocks at their meeting point.  The clocks are on a collision course as they have the same velocity relative to the observer in the middle.  The clocks are synchronized. 

If I am correct, each clock will see the other ticking more slowly.  How does this get resolved as they come together? 

I am not sure I understand exactly what you are saying about scenarios without acceleration (and gravity for this matter).  Are you saying that there is time dilation for each clock relative to the other in my example with the clocks?

Relative to the various inertial frames, yes. In one frame, the first clock runs faster and relative to another frame the second clock runs faster. Relative to some select frames, both tick at the same rate. It's all an abstract consequence of frame choice.
Differential aging is not a function of frame choice since the same answer is found regardless of choice of coordinate system with which to describe the situation.

Ok, that is my understanding too. 

So I am saying that acceleration is the only reason for time dilation.

Then you're not getting it. I gave several examples of the dilated clock being the non-accelerated one, or the less accelerated one. Without gravity, clocks can have dilation without acceleration (such as in your example of passing clocks) but since they never meet more than once, there is no pair of paths between the same two events that can be compared. Without resorting to gravity there is no way to make the clocks meet twice without one or both of them accelerating (or by staying in each other's presence the entire way).

Oh how frustrating, I meant to put, "So I am not saying ...".  Sorry.

I don't know if that counts as time dilation.

It exactly counts as dilation.

I pretty much agree.  I am still a little foggy how the dilation no longer becomes dilation as the two clocks approach each other from the example above, but I will wait for your response.

Getting back to the twins, I will try to use what I have learnt from this discussion (I have double checked much of what you said, and everything seems to be correct, just for my own peace of mind).

I have learnt that when there is velocity/speed involved, the result is that both twins seem to undergo time dilation relative to the other (just like in the clock example).  But this is telling me that the speed/velocity is not the reason for the change in aging.

And regarding the path lengths, if I look at a simple spacetime diagram of the "instant-acceleration-version" of the twin paradox, I see that the stationary twin's worldline/path is straight up, and the travelling twin's makes a shape like this > only stretched out much more.  And the only logical thing that I can think of is that the difference is that the twin had to change directions.  So I am saying that the path length is what it is because the twin changes directions.  Now the question is, does "change directions" absolutely imply acceleration?  If so, then I see no alternative than to say that the change in age is caused by acceleration.

[alancalverd]

Here is an example of how my brain gets scrambled when thinking about this scenario.

Imagine that someone was watching the clocks come together.  It is a perfectly symmetrical situation.  The observer is between the clocks at their meeting point.  The clocks are on a collision course as they have the same velocity relative to the observer in the middle.  The clocks are synchronized. 

If I am correct, each clock will see the other ticking more slowly.  How does this get resolved as they come together? 

Oddly, I was sitting in a doctor's waiting  room thinking about this at the time you were writing it!.

From the central observer's point of view the two clocks may indeed be synchronised with each other as each is approaching with velocity v,  but each sees the other as advancing at 2v, therefore speeded up (or blue shifted) relative to his own.

[alancalverd]

Now the question is, does "change directions" absolutely imply acceleration? 

Acceleration is a change of velocity, whether speed, direction, or both.Hence the occasionally quoted "instantaneous relative velocity" is meaningless.

[Halc]

Imagine that someone was watching the clocks come together.  It is a perfectly symmetrical situation.  The observer is between the clocks at their meeting point.  The clocks are on a collision course as they have the same velocity relative to the observer in the middle.  The clocks are synchronized.

They would tick at the same rate in the inertial frame in which that middle observer is stationary. They'd be synchronized only if they happen to read the same value simultaneously in that frame.

If I am correct, each clock will see the other ticking more slowly.  How does this get resolved as they come together?

It doesn't need resolution. Observers with both clocks will always compute (not see) the other ticking slower relative to their own frame. This doesn't change when they come together and part company again.

I am still a little foggy how the dilation no longer becomes dilation as the two clocks approach each other from the example above, but I will wait for your response.

It never stops being dilation. The other clock is always moving at some speed relative to you, so it is dilated relative to you, before, during, and after it passes by.

I have learnt that when there is velocity/speed involved, the result is that both twins seem to undergo time dilation relative to the other (just like in the clock example).  But this is telling me that the speed/velocity is not the reason for the change in aging.

Differential again and time dilation are different things. The former is due to physically different path lengths. The latter due to abstract coordinate choices and speed relative to those choices.

And regarding the path lengths, if I look at a simple spacetime diagram of the "instant-acceleration-version" of the twin paradox, I see that the stationary twin's worldline/path is straight up, and the travelling twin's makes a shape like this > only stretched out much more.

That's right. It's because paper is Euclidean and cannot correctly represent spacetime which is not. The mathematics says the > path is shorter (because of the -x² instead of the +x² you get with Euclidean geometery), and the mathematics is what counts.

So I am saying that the path length is what it is because the twin changes directions.

In this case, but not necessarily so. I can have a twin with more direction changes and still have him come out older than another with less.
 

Now the question is, does "change directions" absolutely imply acceleration?

To change your direction of nonzero motion, yes, that's acceleration. To just face/point a different way is to change direction without implying any acceleration, but I don't think you're talking about that.

So I am saying that acceleration is the only reason for time dilation.

Then you're not getting it.

Oh how frustrating, I meant to put, "So I am not saying ...".  Sorry.

But here you are asserting the same thing again:

If so, then I see no alternative than to say that the change in age is caused by acceleration.

As I said before, I don't think I can help you further. I gave several examples contradicting this suggestion, and it seems pointless to post in a topic with so many obfuscating and often outright wrong replies being given.

[Dimensional (OP)]

Imagine that someone was watching the clocks come together.  It is a perfectly symmetrical situation.  The observer is between the clocks at their meeting point.  The clocks are on a collision course as they have the same velocity relative to the observer in the middle.  The clocks are synchronized.

They would tick at the same rate in the inertial frame in which that middle observer is stationary. They'd be synchronized only if they happen to read the same value simultaneously in that frame.

I agree.  Let's say that is what happened.

If I am correct, each clock will see the other ticking more slowly.  How does this get resolved as they come together?

It doesn't need resolution. Observers with both clocks will always compute (not see) the other ticking slower relative to their own frame. This doesn't change when they come together and part company again.

But when they meet, don't they have to have the same time if they synchronized correctly?  In other words, the middle observer should always "see" their clocks as the same until they meet, right?

I am still a little foggy how the dilation no longer becomes dilation as the two clocks approach each other from the example above, but I will wait for your response.

It never stops being dilation. The other clock is always moving at some speed relative to you, so it is dilated relative to you, before, during, and after it passes by.

What do you mean by "you" in the scenario?  In the scenario there are the 2 clocks closing in on each other and an observer in the middle, which is the "you"?

I have learnt that when there is velocity/speed involved, the result is that both twins seem to undergo time dilation relative to the other (just like in the clock example).  But this is telling me that the speed/velocity is not the reason for the change in aging.

Differential again and time dilation are different things. The former is due to physically different path lengths. The latter due to abstract coordinate choices and speed relative to those choices.

Yes, I agree.  But I am not sure if this response agrees or not with what I said.

And regarding the path lengths, if I look at a simple spacetime diagram of the "instant-acceleration-version" of the twin paradox, I see that the stationary twin's worldline/path is straight up, and the travelling twin's makes a shape like this > only stretched out much more.

That's right. It's because paper is Euclidean and cannot correctly represent spacetime which is not. The mathematics says the > path is shorter (because of the -x² instead of the +x² you get with Euclidean geometery), and the mathematics is what counts.

Yes, that I understand.  Whatever other-worldly shape it actually is, it is not the straight vertical line that the other twin has, which seems to be due to the turn around.  Do you agree?

So I am saying that the path length is what it is because the twin changes directions.

In this case, but not necessarily so. I can have a twin with more direction changes and still have him come out older than another with less.

Okay, you agree, so for the sake of my side of the argument, let us just stick with this case in this part of the post and really try to nail down the exact "cause" of the differential aging.
 

Now the question is, does "change directions" absolutely imply acceleration?

To change your direction of nonzero motion, yes, that's acceleration. To just face/point a different way is to change direction without implying any acceleration, but I don't think you're talking about that.

Okay, that's what I was thinking too.

So I am saying that acceleration is the only reason for time dilation.

Then you're not getting it.

Oh how frustrating, I meant to put, "So I am not saying ...".  Sorry.

But here you are asserting the same thing again:

No, it is not the same thing.  The adjusted statement with the word "not" is very much different than the statement that I made by mistake.   

If so, then I see no alternative than to say that the change in age is caused by acceleration.

As I said before, I don't think I can help you further. I gave several examples contradicting this suggestion, and it seems pointless to post in a topic with so many obfuscating and often outright wrong replies being given.

Well I am sorry you feel that way.  I was trying to create a chain of logic that throughout my post that results in what I said. 
Please tell me what I said that was wrong, and I will address it directly.  And please tell me the contradictions that I did not address.  I really want to understand this topic once and for all.

[MikeFontenot]

Can you please tell me why I am doing this?

So that you'll understand the twin paradox.

[Halc]

I agree.  Let's say that is what happened.

OK. We'll have clock coming from the left, one in the middle, and one coming from the right. We'll name the frames and the observers L, M, and R. Clocks L and R have been,
The L and M clocks, both equally dilated in the M frame, are synced relative to that M frame, so they'll always say the same thing at all times. They will still run slow since they're moving, just equally slow.

But when they meet, don't they have to have the same time if they synchronized correctly?
  In other words, the middle observer should always "see" their clocks as the same until they meet, right?

Yes to all.

I am still a little foggy how the dilation no longer becomes dilation as the two clocks approach each other from the example above, but I will wait for your response.

It never stops being dilation. The other clock is always moving at some speed relative to you, so it is dilated relative to you, before, during, and after it passes by.

What do you mean by "you" in the scenario?[/quote]You at say the L clock. Relative to frame L, the R clock is moving always, so it will always run slower than the clock by you. It passing you doesn't change that.  It was an answer to the statement: "If I am correct, each clock will see the other ticking more slowly.".

I have learnt that when there is velocity/speed involved, the result is that both twins seem to undergo time dilation relative to the other (just like in the clock example).  But this is telling me that the speed/velocity is not the reason for the change in aging.

Differential again and time dilation are different things. The former is due to physically different path lengths. The latter due to abstract coordinate choices and speed relative to those choices.

Yes, I agree.  But I am not sure if this response agrees or not with what I said.[/quote]For the twins scenario, you can always compute it purely in terms of speeds. Pick any inertial frame, but stick with it for all calculations. No matter the frame chosen, the result (the differential ages) will always be the same. That's computing the differential aging via speed computations. It's quite simple and doesn't involve complicated Lorentz transforms.

Whatever other-worldly shape it actually is, it is not the straight vertical line that the other twin has, which seems to be due to the turn around.  Do you agree?

The bend to the line causes it to not be a straight line, yes. In this case, it's the only way they're going to meet again. You can construct scenarios where both lines bend, or with lots of bends, or with smooth curves instead of these brutal instant speed changes.

Okay, you agree, so for the sake of my side of the argument, let us just stick with this case in this part of the post and really try to nail down the exact "cause" of the differential aging.

You can say that the bends (the accelerations) cause the path length to change, and the shorter path length results in less duration than the straight path. You seem to really need accelerations to be a cause even though there's no function for acceleration to duration. It's not like you need to pass a college test. Sabine was wrong because people will generalize what she says and imply that clocks on Mercury run faster than on Earth because the proper acceleration is less on Mercury, but clocks there actually run slower. That's why I'm resisting coupling acceleration with 'causes the age difference.

[Dimensional (OP)]

For the twins scenario, you can always compute it purely in terms of speeds. Pick any inertial frame, but stick with it for all calculations. No matter the frame chosen, the result (the differential ages) will always be the same. That's computing the differential aging via speed computations. It's quite simple and doesn't involve complicated Lorentz transforms.

What formula are you referring to?

Sometimes math does not tell the whole story even though it is correct.  For example, it may take 10 minutes for me to walk to a park and back.  If I walked at constant speed, the math says that it takes 5 minutes to get there and 5 to get back, which gives the just of the story.  But it did not mention anything about an acceleration at the turn-around.  It's not really the part of the description that people want to know about.

You can say that the bends (the accelerations) cause the path length to change, and the shorter path length results in less duration than the straight path.

So if acceleration can cause shorter temporal path lengths, and shorter path lengths can cause differential aging, then doesn't this mean that acceleration can cause differential aging?

You seem to really need accelerations to be a cause even though there's no function for acceleration to duration. It's not like you need to pass a college test.

I don't need the cause to be acceleration; I just want to know what the cause is.  I am only leaning towards acceleration because at the moment it makes the most sense to me.

Sabine was wrong because people will generalize what she says and imply that clocks on Mercury run faster than on Earth because the proper acceleration is less on Mercury, but clocks there actually run slower. That's why I'm resisting coupling acceleration with 'causes the age difference.

Yeah, I agree that she was definitely wrong about that.

[Halc]

For the twins scenario, you can always compute it purely in terms of speeds. Pick any inertial frame, but stick with it for all calculations. No matter the frame chosen, the result (the differential ages) will always be the same. That's computing the differential aging via speed computations. It's quite simple and doesn't involve complicated Lorentz transforms.

What formula are you referring to?

I didn't mention a formula, but it was given before by ES:
Δτ = 

For the very simple case of choosing Earth frame and having the traveler always traveling at some constant speed, this works out to Δτ =
You only have to do the tedious integration if the speed varies along the way, such as with any actual rocket.

So if acceleration can cause shorter temporal path lengths, and shorter path lengths can cause differential aging, then doesn't this mean that acceleration can cause differential aging?

Yes, you can say that, so long as you don't generalize 'can cause' to 'causes' since the following statements about differential aging are false:
1) The twin that has accelerated (or accelerated more) will be found younger.
2) If a differential age shows one twin to be younger, that twin must have accelerated.
Meanwhile, both those statements would be true if we substituted the bit about path lengths instead of the accelerations. It does happen to be true that in the typical twin scenario, the twin that has accelerated will be found younger.

I don't need the cause to be acceleration; I just want to know what the cause is.

You've been told that repeatedly, and you keep pushing back to accelerations. You don't accept the answers, which is admittedly hard to do when you have all these people giving different stories and all insisting on being the right one. The physicsforums members are more consistently knowledgeable and will downvote users that give wrong answers.

It's not much of a claim, but I know my relativity better than anyone on this forum with the possible exception of Janus. The guys on physicsforums are a different league of expert and can get into the exact language required, and tensor calculus, and all that.

Yeah, I agree that she was definitely wrong about that.

Which is horrible since she's very respected and shouldn't make such obvious mistakes like that. Does she invite peer review of her vids before publishing them?

[Dimensional (OP)]

For the twins scenario, you can always compute it purely in terms of speeds. Pick any inertial frame, but stick with it for all calculations. No matter the frame chosen, the result (the differential ages) will always be the same. That's computing the differential aging via speed computations. It's quite simple and doesn't involve complicated Lorentz transforms.

What formula are you referring to?

I didn't mention a formula, but it was given before by ES:
Δτ = 

For the very simple case of choosing Earth frame and having the traveler always traveling at some constant speed, this works out to Δτ =
You only have to do the tedious integration if the speed varies along the way, such as with any actual rocket.

Isn't this just for time dilation?  What about the differential aging because that is what this is really about.

Math works, but it also needs proper context.

So if acceleration can cause shorter temporal path lengths, and shorter path lengths can cause differential aging, then doesn't this mean that acceleration can cause differential aging?

Yes, you can say that, so long as you don't generalize 'can cause' to 'causes' since the following statements about differential aging are false:
1) The twin that has accelerated (or accelerated more) will be found younger.
2) If a differential age shows one twin to be younger, that twin must have accelerated.
Meanwhile, both those statements would be true if we substituted the bit about path lengths instead of the accelerations. It does happen to be true that in the typical twin scenario, the twin that has accelerated will be found younger.

I totally agree.

I don't need the cause to be acceleration; I just want to know what the cause is.

You've been told that repeatedly, and you keep pushing back to accelerations. You don't accept the answers,

That is because I have not been given a convincing argument.  At this point in the discussion, please give your argument/s for how acceleration is not necessary in the case where the twin instantly accelerates in the turnaround.

It's not much of a claim, but I know my relativity better than anyone on this forum with the possible exception of Janus. The guys on physicsforums are a different league of expert and can get into the exact language required, and tensor calculus, and all that.

Yes, but I think this topic is a little more controversial.  Even though Sabine was incorrect about generalizing acceleration as a cause of time dilation the way she did, I find it impossible to pick apart her reasoning in the case of the turnaround in the twin paradox.

[Halc]

Isn't this just for time dilation?  What about the differential aging because that is what this is really about.

Yes, it is a dilation computation. If there are two paths delimited by the same two events, then it is also using those dilation equations to compute a differential age. If the delimiting events are not the same, then the calculation is still valid but it isn't a differential aging situation.

At this point in the discussion, please give your argument/s for how acceleration is not necessary in the case where the twin instantly accelerates in the turnaround.

Don't understand. You seem to be asking how acceleration is not necessary in a scenario with acceleration. If there wasn't acceleration, it would be a different scenario.
So tell me, using a simple geometric example. You have a paper with 2 dots on it. You draw several lines with meandering paths between the dots. What would you consider to be a convincing argument about what 'causes' one line to be longer than another? Maybe it's the amount of ink that causes the longer lines. It isn't the number of turns taken or how sharp or gradual those turns are. The turns are equivalent to accelerations. I personally don't see it as a causal situation at all. Some lines are just longer than others. A cause might be that you had an argument with your wife this morning and took out the frustration by scribbling one of the lines furiously. So as for the 'cause' of the twin scenario, it was the one twin's decision to make this trip that makes him younger than his sibling. See what I mean about 'cause' being sort of open to interpretation?

Yes, but I think this topic is a little more controversial.

No, it's actually kindergarten phase, a most simple situation that has been beaten to death over the decades and is only controversial to a novice because it grinds with the sort of Newtonian physics that seem more intuitive.

Even though Sabine was incorrect about generalizing acceleration as a cause of time dilation the way she did, I find it impossible to pick apart her reasoning in the case of the turnaround in the twin paradox.

Agree. Her mistake was generalizing it, asserting that all dilation and differential aging is due to acceleration, which contradicts the mathematics.