[alancalverd]

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

The cause is relative velocity.

But you can't have a relative velocity between initially synchronised twins unless one of them accelerates: twins are conceived with zero relative velocity and generally reach maturity with v << c. If one becomes a pilot or an astronaut, the result of the Haefle-Keating experiment tells you exactly how much their ages will differ after each tour of duty.

[Dimensional (OP)]

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.

I am not sure I am following.  An event is just a point.

Let me try to "prove" it to you geometrically.

Imagine a 2d spacetime diagram (as you know time is the y axis, and x is the spatial axis).  How can any 2 points horizontal with each other (in other words have the same time) have differential aging without a curve or bend?

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 I understand.  But we can say that cause should be the same for all observers as time passes (of course except in extreme cases in GR).

As for the "causes" like in the scenario you described, we should stay consistent with only fundamental physics/math terms if we can.

[Halc]

I am not sure I am following.  An event is just a point.
Let me try to "prove" it to you geometrically.

Prove what?

Imagine a 2d spacetime diagram (as you know time is the y axis, and x is the spatial axis)

x and t (ct techmically) axis since y is traditionally another spatial axis.

How can any 2 points horizontal with each other (in other words have the same time) have differential aging without a curve or bend?

Points in spacetime (events) don't age. If they did, they be a different point since they'd have a different t coordinate. Furthermore, two events at the same time are space-like separated. It is impossible to travel from one to the other since it would require you to do it in no time. All events at the beginning and end of scenarios like the twins scenario are time-like separated, meaning their coordinates differ more by ct than they do by x.  So if the two events (ct, x) are at (0, 0) and (2, 1), something can travel between those if it moves at 0.5c. If the second event is at (2, 0) then the thing can stay stationary and get there, path length t = 2. The path length to the (2, 1) event is √(2² - 1²) = √3, shorter than the path to the (0,0) event.

Yes I understand.  But we can say that cause should be the same for all observers as time passes (of course except in extreme cases in GR).

Sorry, but still have no idea what you mean by 'cause'. Differential aging is about geometry, not causation. Causation is not the same for everybody. I like vanilla for the flavor, the other guy picks it because he thinks the drips will not be as easily seen on his shirt. Same effect, different causes. This has nothing to do with Minkowskian geometry, where the mathematics accurately describes (doesn't cause) the ages the the twins at the reunion.

[Janus]

Imagine a 2d spacetime diagram (as you know time is the y axis, and x is the spatial axis).  How can any 2 points horizontal with each other (in other words have the same time) have differential aging without a curve or bend?

They would only be "at the same time" for one given inertial frame.
Draw your two dots on  sheet of paper, so that from your view, they are horizontal to each other.  In this view vertical is the time axis and horizontal is the space axis. 
Now rotate the paper.  The time and space axis do not turn with the paper, but stay with your view of vertical and horizontal. This is the equivalent of viewing the events from a different inertial frame.  The two dots are not horizontal to each other,and the events then represent do not happen at the same time for this inertial frame. 

[alancalverd]

instantaneous velocity change

I must take issue with anyone who uses this phrase!
For any body with nonzero mass, an instantaneous velocity change requires the input of a finite quantity of energy (½m(Δv)²) in zero time, i.e infinite power. This implies that the laws of physics have been suspended, including those of relativity and time dilation. The subsequent analysis is therefore invalid.
 

[Janus]

instantaneous velocity change

I must take issue with anyone who uses this phrase!
For any body with nonzero mass, an instantaneous velocity change requires the input of a finite quantity of energy (½m(Δv)²) in zero time, i.e infinite power. This implies that the laws of physics have been suspended, including those of relativity and time dilation. The subsequent analysis is therefore invalid.
 

The issue with considering "instantaneous velocity change"   is that the magnitude of the acceleration becomes undefined.
v =at, so to find acceleration needed for a known change of v over a known period of t, you use a= v/t. but if t is zero, this is division by zero which is undefined. (not infinite).

[alancalverd]

v =at, so to find acceleration needed for a known change of v over a known period of t, you use a= v/t. but if t is zero, this is division by zero which is undefined. (not infinite).

No. v = aΔt assuming constant a applied for a duration Δt or ∫adt if a is variable.
a = dv/dt at any time, including t = 0. 

[Dimensional (OP)]

I am not sure I am following.  An event is just a point.
Let me try to "prove" it to you geometrically.

Prove what?

Imagine a 2d spacetime diagram (as you know time is the y axis, and x is the spatial axis)

x and t (ct techmically) axis since y is traditionally another spatial axis.

How can any 2 points horizontal with each other (in other words have the same time) have differential aging without a curve or bend?

Points in spacetime (events) don't age. If they did, they be a different point since they'd have a different t coordinate. Furthermore, two events at the same time are space-like separated. It is impossible to travel from one to the other since it would require you to do it in no time. All events at the beginning and end of scenarios like the twins scenario are time-like separated, meaning their coordinates differ more by ct than they do by x.  So if the two events (ct, x) are at (0, 0) and (2, 1), something can travel between those if it moves at 0.5c. If the second event is at (2, 0) then the thing can stay stationary and get there, path length t = 2. The path length to the (2, 1) event is √(2² - 1²) = √3, shorter than the path to the (0,0) event.

Yes I understand.  But we can say that cause should be the same for all observers as time passes (of course except in extreme cases in GR).

Sorry, but still have no idea what you mean by 'cause'. Differential aging is about geometry, not causation. Causation is not the same for everybody. I like vanilla for the flavor, the other guy picks it because he thinks the drips will not be as easily seen on his shirt. Same effect, different causes. This has nothing to do with Minkowskian geometry, where the mathematics accurately describes (doesn't cause) the ages the the twins at the reunion.

Ok, I understand.

After thinking about all of this for the past few days, I think I know why this topic is so unclear (at least for me). 

In physics, we observe a quality of something, for example; mass, redness, or gravity.  Then we try to quantify, relate, reduce them down to their simplest forms, predict, formulize, etc.  Well with this situation, a velocity and a "turnaround" seems to be the necessary, and I think the minimum, that makes differential aging happen.  The turnaround by itself (without acceleration if that even makes any sense) is not really quantifiable, so I don't think it can be a function of anything.

I am really trying to figure this out, but it seems like this is a different kind of physics problem than we are used to.

What do you think is minimally necessary for a twin to have differential aging, without gravity?

[pzkpfw]

It may be simpler to not have a twin - I think it's easier to add a triplet.

Alice sits on her deck chair.
Bob zooms past Alice, away from Earth.
Bob later meets Carol, zooming towards Earth.
Carol eventually passes Alice too.

Nobody changes speed or direction at any time. No acceleration by anyone.

They all have perfect second-per-second timers (stopwatches).

When Bob passes Alice, they both start their timers.
When Bob passes Carol, he stops his timer, she starts hers.
When Carol passes Alice, they both stop their timers.

Some time later (experiment over) they get together and compare timers. Without relativity you'd expect Bob timer + Carol timer = Alice timer.
But they find Bob timer + Carol timer < Alice timer

(Essentially this is the instantaneous acceleration version of the twins paradox, but without the spherical cow.)

((In terms of your diagram in post #1, Alice's timer shows 10, Bob's and Carol's both show 4, adding to 8.))

Very very informally: when two observers are in relative motion, for both of them the others' time is slower. The "turnaround" has a sort of effect of picking up one of those "slower times" and bringing it to the other.

[Zer0]

This " Twin Paradox " is mostly always being discussed somewhere or the other...

Perhaps it would be Best to just put a lifetime Space Travel Ban on any twins/triplets/quadruplets etc..

That would surely solve this problem once n for all & put an end to all this fuss!

[Peter11]

I think a lot of confussion comes from trying to solve it using general relitivity when special relitivity solves it all.There is no need to use general relitivity.

[alancalverd]

For a fixed finite velocity change delta_v in a finite time delta_t, the energy required to make that happen is independent of delta_t ... i.e., it takes no more energy to change his velocity quickly (including instantaneously) than to change it slowly!

Which is why I discussed power, not energy.

[MikeFontenot]

For a fixed finite velocity change delta_v in a finite time delta_t, the energy required to make that happen is independent of delta_t ... i.e., it takes no more energy to change his velocity quickly (including instantaneously) than to change it slowly!

Which is why I discussed power, not energy.

The power DOES go to infinity as delta_t goes to zero.  But that is not important.  It just means that, if you were doing this experiment in a spaceship, doing a sequence of experiments with different amounts of thrust from your engine, but always making the velocity change be the same each time, you would find that no matter how quickly you make the velocity change, the amount of fuel you have to burn is the same in each experiment.  As the time to make the velocity change decreases, you need to squirt more fuel into the engine per unit time (to get more thrust), but the total amount of fuel burned stays the same in each experiment.

You can never actually DO the experiment with the duration of thrust being infinitesimal, and the thrust being infinite.  That limit is just a mathematical process that describes where your results are converging to as you keep repeating the experiments.  You can't ever actually have an infinite acceleration with zero duration, but that limiting value is nevertheless a useful concept, and your experimental results can get arbitrarily close to the computed limit.

If you understand what Dirac did when he invented his "Dirac delta function", that's really all that's happening here.  In the limit, our rocket power is described by a Dirac delta function, which gives a finite and constant result (the energy supplied) when integrated with respect to time.

[alancalverd]

The power DOES go to infinity as delta_t goes to zero.  But that is not important. 

I rather think that it might be just a tad significant. We don't like singularities in physics, particularly those that prevent us from conducting an experiment.