Twin clocks-is it acceleration?

1. Feb 24, 2013

phyti

Acceleration is an equivalent answer to differential aging only in special cases such as the one restricting all acceleration to one twin.
View attachment 56070

Last edited: Mar 9, 2014
2. Feb 24, 2013

mathman

Doesn't sound right. If both are accelerating, but with different (vector) functions, the results will be different.

3. Feb 25, 2013

Staff: Mentor

Acceleration is only invoked to break the symmetry between the twins. In this case, it still breaks the symmetry since their acceleration profiles are different.

4. Feb 25, 2013

ghwellsjr

Not as I read it. The twins experience exactly the same accelerations, just at different times. Actually, the first twin has a long gap between his two accelerations while the second twin has no gap.

But I think it would have been easier to see this if extra time were inserted between both gaps. Then we would see the two twins starting out inertially and colocated (at rest with respect to each other), one of them accelerates (the document says decelerates, but it's the same thing) and then the two twins have a relative separation speed between them. After some time, the second twin accelerates in exactly the same way as the first one did, bringing them to rest with respect to each other but with a constant separation between them. Next the second twin accelerates again in the same direction, possibly with the same acceleration as before but this is not necessary. This causes the twins to have a relative closing speed between them that continually reduces their separation. Just before they make contact, the first twin accelerates in exactly the same way as the other twin's second acceleration, bringing them once more into mutual inertial rest and colocation but with different aging accumulated during their time of separation.

I think this is a brilliant variant of the Twin Paradox that shows that acceleration has nothing to do with the difference in aging and it also can't be used to determine which twin is younger, in other words, acceleration cannot be "invoked to break the symmetry between the twins". It also shows that it is not necessary for the twins to return to the same location that they started out in or even that they need to be at rest in the same frame they started out in. But it has the same "paradoxical" issue that while separated, in each twin's inertial rest frame(s), the other twin's time is dilated.

I just don't understand the U-frame/U-time explanation but like all SR scenarios, the standard explanations using inertial frames work just fine. Did you make the document, phyti? Do you understand the explanation?

5. Feb 25, 2013

Staff: Mentor

That is asymmetric. What is the symmetry operation which maps one twin's acceleration profile to the other?

According to SR the laws of physics are invariant under spatial rotations, spatial and temporal translations, and boosts. If you make an acceleration and time measurement profile that is symmetric under some combination of those symmetry operations then the twins will have the same elapsed age. Otherwise there is an asymmetry and they do not have the same age.

Last edited: Feb 25, 2013
6. Feb 25, 2013

A.T.

The age difference depends on acceleration and the spatial separation during that acceleration.
Different acceleration profiles break the symmetry.

7. Feb 25, 2013

ghwellsjr

Sorry, I didn't realize that you were including both intervals of acceleration plus the interval of inertial travel between them for the first twin as one acceleration profile. I assumed that the first twin's two intervals of acceleration were separate acceleration profiles.

The document looked to me like it was showing the two acceleration intervals of the first twin concatenated together to form the acceleration profile of the second twin. I broke it apart into two acceleration profiles and inserted a time gap between them (and added the same time gap to the first twin) to make this more obvious in my explanation.

So both twins experience the same two acceleration profiles but with different times between them. It's not the acceleration profiles alone that explain the difference in aging but rather we have to include the time interval between them to get the full story. I just thought this was a brilliant way to counter the argument that "it's the twin that accelerates that ends up younger" and the erroneous conclusion that it is only the acceleration that creates the time difference between them.

8. Feb 25, 2013

bobc2

phyti, too much focus is given to the acceleration. It's the path lengths through space-time that should be compared, regardless of whether one or both are accelerating and when the accelerations occur.

9. Feb 25, 2013

A.T.

An "acceleration profile" is the entire history of acceleration between the two meetings. Different timings means different profiles.
If you know the acceleration profiles between the two meetings, you can compute the age difference.

10. Feb 25, 2013

ghwellsjr

Well how am I supposed to know that except by making this mistake and getting corrected by someone else? Where is the official dictionary of approved relativity terms that I can reference to make sure I never make another mistake like this again?
True, given the official definition of "acceleration profile", did you get the impression that I didn't already know this?

11. Feb 25, 2013

phyti

The deceleration profiles are purposely made equivalent, the two for A connected together equal the one for B, in duration and curvature and tangent (speed) vectors. If you employed integration methods for the two complete curved profiles, they are equal. In the traditional simple case, B changes course 3 times, departs, reverses, and returns. He has 3 speed changes to 0 for A, thus his accumulated time is shorter, so the argument goes. Are you going to invoke asymmetry in this case in favor of A because he has 2 changes to 1 for B?
By eliminating the acceleration/deceleration issues, the focus is on the inertial portions, and specificaly on time lost as it relates to speed. The example also emphasizes the time lost is permanent, since there is no speed by which a clock can gain time.

12. Feb 25, 2013

phyti

This is my own work. After the continuous rehashing of the 'twin thing', I thought there should be a simpler way, possibly without math, to demonstrate physically why there is differential aging. The U-frame represents a universal time standard, and the red portions represent what's missing after subtracting the dilated time for A and B.
You have a good understanding of the post, thank you.

13. Feb 25, 2013

stevendaryl

Staff Emeritus
I find the talk about acceleration "breaking the symmetry" in the twin paradox to be a little misleading. On the one hand, it is true that without invoking weird topologies, it is impossible for the twins to get back together without one or the other accelerating. It's also true, if they do get back together, that if one accelerated and the other didn't, then the one accelerated will be the youngest. But it's not correct, in my opinion, to conclude from this that acceleration somehow is responsible for the difference in ages, as if acceleration causes people to age slower. It doesn't. If you accelerate away from Earth, turn around and accelerate back, turn around and accelerate away, etc. you can arrange that you're always feeling acceleration. But if you never accelerate in one direction long enough to acquire relativistic velocity, then the acceleration isn't going to make you younger than the stay-at-home twin. The formula for elapsed time depends only on velocity, not acceleration.

Here's an analogy that I think is pretty good.

Suppose you hop in a car, and set your trip meter to 0, and travel from New York City to Chicago. What does your odometer read when you get to Chicago? The least it can read is 1270 km. But depending on the path you took, the reading can be different from 1270.

Suppose you hop in a rocket, and set your clock to 0, and travel from New York City in 2013 to Chicago in 2023. The most your clock can read is 10 years. But depending on the path you took, the reading can be different from 10.

The big difference between the two cases is that in the Euclidean geometry governing odometer readings, the straight-line path has the least length, while in the Minkowsky geometry governing clocks, the straight-line path has the greatest elapsed time. These facts follow from the definitions of length:

For Euclidean paths: $L = \int \sqrt{1+s^2} dx$ where $s$ is the slope of the path ($s = dy/dx$).

For Minkowsky paths: $\tau = \int \sqrt{1-\frac{v^2}{c^2}} dt$ where $v$ is the velocity of the path ($v = dx/dt$).

14. Feb 25, 2013

Staff: Mentor

It doesn't matter if you call it acceleration or deceleration. They are not symmetric.

In physics, "symmetry" means that something is unchanged under some mathematical operation. In the case of SR, the symmetry operations are spatial and temporal translations, rotations, and boosts. There is no way to turn a(t) for one twin into a(t) for the other twin through any combination of translations, rotations, and boosts. They are not symmetric.

15. Feb 25, 2013

ghwellsjr

I think we agree that the terms "acceleration" and "deceleration" are equivalent, at least I pointed that out in post #4 which phyti quoted and agreed with.
So are you saying that if we start with two colocated twins at rest with each other and they both identically accelerate in opposite directions, coast for the same length of time and then accelerate identically but in opposite directions so that they approach each other and then finally accelerate in opposite directions to come to mutual colocated rest, then we cannot say that symmetry applies here and we cannot conclude that they are the same age? The mathematical operation in this case is multiplying the acceleration profile of one twin by -1 to get the acceleration profile of the other twin.

16. Feb 25, 2013

A.T.

Velocity is frame dependent. The age difference after reunion and proper acceleration aren't.

You are missing the point of the twin paradox.

The question is not: "How can I explain the age difference geometrically in one inertial frame?"

The question is : "Why can't I use the same explanation in both twin's rest frames to get the opposite result?"

The answer to the question is : "Difference in proper acceleration profiles"

17. Feb 25, 2013

Staff: Mentor

That is a rotation by 180º, which is a symmetry of SR. We can conclude that they are the same age.

18. Feb 25, 2013

stevendaryl

Staff Emeritus
The integral $\tau = \int \sqrt{1-\frac{v^2}{c^2}} dt$ is frame-independent. $v$ and $t$ are both frame-dependent, but the combination
$\sqrt{1-\frac{v^2}{c^2}} dt$ is frame-independent.

It's exactly analogous to the Euclidean formula for the length of a curve:

$L = \int \sqrt{1-s^2} dx$

where $s = \dfrac{dy}{dx}$ is the slope of the curve at point $x$. Of course, slope is relative to a coordinate system, but that integral is the same for any (cartesian) coordinate system.

Last edited: Feb 25, 2013
19. Feb 25, 2013

PAllen

Well, if you are allowing non-inertial coordinates you need the metric contracted with coordinate velocities inside the square root. Obviously your main point is correct.

20. Feb 25, 2013

stevendaryl

Staff Emeritus
I think that the answer "different acceleration profiles" for why one twin ages more than the other is misleading. It certainly is true that in flat Minkowsky spacetime, there is no way for two twins to depart and reunite unless one of them accelerates. But using acceleration as the cause of the difference doesn't generalize to other kinds of topologies of spacetime, and doesn't generalize to General Relativity.

In a cylindrical spacetime, you can have twins start off together, depart and reunite later without either twin accelerating. I guess you could say that such an example goes beyond SR, but it doesn't involve any gravity, so it's not very GR-ish, either. It's just geometry.