Help With the Twin Paradox: Why Does Time Dilation Apply?

[PAllen]

Note also there are many complications from bringing in GR when SR is not understood. Even that in GR, in general, a timelike geodesic need not even be a local maximum of proper time (it can be a saddle point).

 

[Orodruin]

Can you provide a reference to that example?

This is a textbook example, but too advanced for a B thread. Suffice to say, you can consider a twin in orbit of a planet and the other traveling radially away from it (but still gravitationally bound so that it gets back). You can set up the problem such that the geodesics cross again after a fixed number of orbits for the first twin. The twins will experience different proper time.

Using coordinates in which the home planet is at rest, twins Alice and Bob both leave at the same time and accelerate to the same speed, .5c. Bob travels outbound for one year at .5c, turns around and returns also at .5c, and once home waits for Alice. Alice travels outbound for two years, turns around, and is back home four years after their common departure. All of their speed changes are done with the same acceleration.

This is not an example in which both twins are in freefall.

 

[pixel]

This is a textbook example, but too advanced for a B thread. Suffice to say, you can consider a twin in orbit of a planet and the other traveling radially away from it (but still gravitationally bound so that it gets back). You can set up the problem such that the geodesics cross again after a fixed number of orbits for the first twin. The twins will experience different proper time.

To an observer on the planet, aren't both twins accelerating?

Using coordinates in which the home planet is at rest, twins Alice and Bob both leave at the same time and accelerate to the same speed, .5c. Bob travels outbound for one year at .5c, turns around and returns also at .5c, and once home waits for Alice. Alice travels outbound for two years, turns around, and is back home four years after their common departure. All of their speed changes are done with the same acceleration.

Again, acceleration is involved, albeit equal for the two twins.

 

[Nugatory]

To an observer on the planet, aren't both twins accelerating?

Yes, although it's probably more important that they are experiencing proper acceleration according to all observers everywhere.

Again, acceleration is involved, albeit equal for the two twins.

Right. They both experience equal acceleration, yet they experience different amounts of aging. That suggests that the acceleration is not causing the difference in aging.

 

[GeorgeDishman]

To remove the question of acceleration, you can use the version with triplets. Each is moving inertially throughout. The departing sibling synchronises his clock with stay-at-home as he passes home station and the third triplet who is already returning home sync's his clock as he passes the departing ship. The returning triplet's clock is then not sync'd with home station when he gets back.

 

[PAllen]

To remove the question of acceleration, you can use the version with triplets. Each is moving inertially throughout. The departing sibling synchronises his clock with stay-at-home as he passes home station and the third triplet who is already returning home sync's his clock as he passes the departing ship. The returning triplet's clock is then not sync'd with home station when he gets back.

I have never found this the slightest bit convincing. The path you are measuring still changes direction, which is acceleration. That you bring in an extra person to measure part of it changes nothing - the path is non-inertial. You have just removed the 'feel' of the acceleration by arranging that no one body actually follows the path measured.

It is just trivially true that in Minkowski spacetime two inertial paths have at most one intersection, so acceleration somewhere on either path is a necessary condition for two intersections. This fact need not be artificially denied in order to motivate that acceleration isn't the cause of differential aging. For that, just arrange both to accelerate the same degree (at different times) and still find differential aging.

 

[GeorgeDishman]

The path you are measuring still changes direction, which is acceleration.

The point is that none of the participants accelerates but there is a proper time difference because the lengths of the paths are different. What it emphasises is that the non-symetrical part of the effect is due not to acceleration but to a switch from one inertial frame to another. Making that switch causes all the event coordinates to be recalculated, and each of the triplets has a complete picture of the whole experiment in their own frame. The three pictures are different but consistent.

 

[PAllen]

The point is that none of the participants accelerates but there is a proper time difference because the lengths of the paths are different. What it emphasises is that the non-symetrical part of the effect is due not to acceleration but to a switch from one inertial frame to another. Making that switch causes all the event coordinates to be recalculated, and each of the triplets has a complete picture of the whole experiment in their own frame. The three pictures are different but consistent.

An what about the much more realistic smooth accelerations? You posit uncountably infinite inertial frame changes? In any case, you are measuring a path that if followed by one object would feel acceleration, and have simply measured this path using multiple devices to avoid any object following the path. It is analogous to claiming that if I use a series of straight ruler measurements to measure a bent path on a plane I have somehow removed that feature that a non-geodesic path in a plane must bend.

 

[GeorgeDishman]

It is analogous to claiming that if I use a series of straight ruler measurements to measure a bent path on a plane I have somehow removed that feature that a non-geodesic path in a plane must bend.

No, it is analogous to saying that if I lay down three rulers so that they cross, the sum of the lengths measured by two between crossing points isn't equal to the length measured by the third.

The Twins Paradox is a way to teach people how SR works, and the triplet version illustrates that acceleration need not be included, it is the change of frames that is the key point.

 

[Ibix]

I think the easiest thing to do is ask what explanation the listener would accept for why my legs are more tired than theirs if I walk two sides of a triangle and they walk the other one. Whatever answer they give can be applied to the twin paradox worldlines.

 

[Orodruin]

To an observer on the planet, aren't both twins accelerating?

This has nothing to do with coordinate acceleration. We are talking freefall observers, which means zero proper acceleration.

Yes, although it's probably more important that they are experiencing proper acceleration according to all observers everywhere.

The passage you quoted quoted my example, where none of the twins experience proper acceleration. You can lock them up in a black box without windows and both can tell the other that they did not accelerate at the end of the experiment.

No, it is analogous to saying that if I lay down three rulers so that they cross, the sum of the lengths measured by two between crossing points isn't equal to the length measured by the third

Yes it is. It is completely analogous. See my insight on the geometrical interpretation of the paradox.

 

[PAllen]

No, it is analogous to saying that if I lay down three rulers so that they cross, the sum of the lengths measured by two between crossing points isn't equal to the length measured by the third.

The Twins Paradox is a way to teach people how SR works, and the triplet version illustrates that acceleration need not be included, it is the change of frames that is the key point.

I think frame changing is itself a red herring because it is tortured to apply it do smoothly curved world lines or in GR.

I think it is important to emphasize that in SR an inertial path maximizes proper time and that any path elapsing less time than the inertial path between events must be non-inertial and feel acceleration if followed by a body. In GR, there are no correspondingly simple statements; best is that for some definition of sufficiently close events, there is a unique geodesic within a causal diamond that is proper time maximizing for all (timelike) paths within that causal diamond. Or the global statement that a global maximizing path [if such exists] must be a possibly nonunique geodesic, out of possibly many which are not maximal.

 

[GeorgeDishman]

Ultimately I think the use of teaching aids like the Twins Paradox is a matter of taste. This is my version, it was just an exercise in using an HTML5 canvas, I should label the birthday parties with the twins ages and clean up the description a bit, but it would be equally valid if I rewrote the text to describe it with triplets and no accelerations, and I could extend the lines beyond the crossing points to make the three inertial motions clear:

http://www.georgedishman.f2s.com/html5/twins.html

 

[PeterDonis]

What it emphasises is that the non-symetrical part of the effect is due not to acceleration but to a switch from one inertial frame to another.

But this isn't what causes the non-symmetrical effect. Switching inertial frames is not anything physical; it's just switching our description of what's happening. You can't cause something physical by just switching your description.

What you should be emphasizing is the geometry of the situation: we basically have the Minkowski spacetime version of the triangle inequality. The twist is that the direction of the inequality is reversed: the two sides of the triangle that make up the "traveling" portion are shorter than the third side, not longer as they would be in Euclidean space. But it's still geometry, and it can be described independently of any inertial frame.

 

[Orodruin]

See my insight on the geometrical interpretation of the paradox.

Dug up the link:
https://www.physicsforums.com/insights/geometrical-view-time-dilation-twin-paradox/

 

[GeorgeDishman]

But this isn't what causes the non-symmetrical effect. Switching inertial frames is not anything physical; it's just switching our description of what's happening. You can't cause something physical by just switching your description.

You have to be very careful expressing it that way though. A lot of newbies think that "something physical" means something like thermal expansion, i.e. that the clock is being affected in some mechanical way and not functioning correctly. The key point I would want to convey is that a properly functioning clock will tell you the length of its worldline measured along the path it took through spacetime.

What you should be emphasizing is the geometry of the situation: we basically have the Minkowski spacetime version of the triangle inequality. The twist is that the direction of the inequality is reversed: the two sides of the triangle that make up the "traveling" portion are shorter than the third side, not longer as they would be in Euclidean space. But it's still geometry, and it can be described independently of any inertial frame.

Absolutely, and that is precisely the intention of the interactive version I wrote. The aim is to convey that the underlying reality is unaffected by our choice of coordinates and moving the slider let's people see that it is nothing more than rotating or skewing the way we see that unchanging reality. That can be connected to the famous "Parable of the Surveyors" which explains it well. The twins and triplets versions are really not much different but using the triplets can solve a problem when someone thinks that acceleration is causing a physical effect in the sense of a "malfunctioning" of the traveller's clock (which I have encountered several times).

 

[GeorgeDishman]

Dug up the link:
https://www.physicsforums.com/insights/geometrical-view-time-dilation-twin-paradox/

Thanks, I had already dug around and found it. It's an excellent page and gives a lot more detail than my toy interactive version but what I was trying to convey is essentially the same message.

 

[robphy]

..and here is the low-tech brute-force diagram with clock diamonds
(once the clock diamonds for piecewise-inertial "outgoing Bob" and "incoming Bob" are determined then strung together)

from my Insight https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/
(Clock diamonds and causal diamonds encode the geometry of Minkowski spacetime.)
You can manually redraw the diagram in other reference frames.

 

[PeterDonis]

The key point I would want to convey is that a properly functioning clock will tell you the length of its worldline measured along the path it took through spacetime.

I agree. I'm simply pointing out that you can express this, as you just did, without ever mentioning inertial frames, much less talking about changing frames in mid-trip. So the talk about changing inertial frames just seems to me to be superfluous at best and confusing at worst.

 

[GeorgeDishman]

So the talk about changing inertial frames just seems to me to be superfluous at best and confusing at worst.

That is true, but simply saying "as the ships pass, the returning triplet synchronises his clock with the departing vessel" doesn't require discussion of frames at all. I'm all for keeping it simple.

 

[pixel]

To remove the question of acceleration, you can use the version with triplets. Each is moving inertially throughout. The departing sibling synchronises his clock with stay-at-home as he passes home station and the third triplet who is already returning home sync's his clock as he passes the departing ship. The returning triplet's clock is then not sync'd with home station when he gets back.

If the third triplet syncs his clock with one that is already out of sync with the stay-at-home's clock, it seems evident that then those two clocks would be out of sync when they pass each other. Where is the paradox?

 

[GeorgeDishman]

Where is the paradox?

When you know the answer and look at spacetime the right way, it is sometimes hard to see why anyone would think there was a paradox at all. It is the same argument as the twins, since the departing and returning twins both see stay-at-home's clock slowed relative to their own, and they sync'd theirs as they passed, the total time registered by stay-at-home should be less than the final time shown on the returning triplet's clock, but since stay-at-home saw both their clocks slowed, he thinks it is their cumulative time that should be less.

 

[dragoo]

There is no acceleration parameter In the equations. The acceleration is no matter.
"this frame switch is the reason for the aging difference, not the acceleration per se"
https://en.wikipedia.org/wiki/Twin_paradox