# Twin paradox again (yes again)

There is a simple truth to this world. Some people are dumb. I am one of them. I'm sorry to ask this but yet again but I just don't get it. Here is the latest explanation of the twin paradox that I have read.

http://mentock.home.mindspring.com/twins.htm" [Broken]

I understand the three reference frames concept. I just don't understand why "Bob" in this case must be the one who switches reference frames. Let me quote one section

Now, since special relativity lets us use either rest frame, we assume Bob is the at-home twin. Ann speeds away at 3/5c. No problem so far. But after 4 years of waiting, Bob must change his inertial frame. If we allow Ann to return, we've only restated the problem with the names switched. (emphasis mine)

EXACTLY EXACTLY EXACTLY. We just restate the problem with the names reversed. But why can't Bob do that. Why can't Bob says, "Heck No! I'm not switching reference frames! I'm not chasing after the wench at 15/17 the speed of light. I'm sitting right here in my rest frame and letting HER turn around and come back at 3/5c. And when she does, I'LL be younger. As far as I'm concerned, in Bob's eyes Ann darted off at 3/5c and darted back at 3/5c and he just sat there the whole time. The only difference is that he also saw the earth dart away and dart back at the same speed. She can switch reference frames all she wants but Bob is keeping his.

If we CAN"T let Ann switch reference frames but that would just restate the problem, then there is some sense in which Bob really must be the one moving or must be the one who turned around. He is always the one getting older!

What am I missing? Will this misery never end!!!

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JesseM
EXACTLY EXACTLY EXACTLY. We just restate the problem with the names reversed. But why can't Bob do that. Why can't Bob says, "Heck No! I'm not switching reference frames!
Bob can't say that because when Bob accelerates, he'll feel G-forces which tell him he's not moving inertially. If Ann was the one who turned around, Ann would be the one to feel the G-forces, while Bob would be weightless from beginning to end.
OS Richert said:
If we CAN"T let Ann switch reference frames but that would just restate the problem, then there is some sense in which Bob really must be the one moving or must be the one who turned around.
You can certainly let Ann switch reference frames--she just has to accelerate in the direction of Bob, so that after the acceleration the distance between her and Bob is decreasing rather than increasing. In this case it will be Ann, not Bob, who is younger when they reunite (assuming Bob did not also accelerate). But again, this will be different than the case where Bob turns around because Ann will feel the G-forces during acceleration while Bob will be weightless throughout.

jtbell
Mentor
Reference frames in general are a mathematical concept. Inertial reference frames also have physical significance. Newton's First Law of Motion is basically a statement that inertial reference frames exist, and have special physical status.

Bob can't say that because when Bob accelerates, he'll feel G-forces which tell him he's not moving inertially.

Okay, so even though we made this acceleration instintaneous, it is still the determing factor. It is not a question of relative motion but whoever actually experiences a force to change the relative motion between them, this one will be younger

On first glance, it would appear that one is going 3/5c in one direction and 3/5c in the other direction, so that the difference between the two frames is 6/5c! Faster than light? No, special relativity does not add speeds this way. The actual difference is only 15/17c, fast but not faster than light.

Does SR include math on how to add speeds? Obviously I havn't read the SR but I would like too. Where can I find Einsteins original formulation and math and whatever he publiced when he announced SR?

The change in direction and the resulting accelleration is relative to the mass of the entire universe not just between the twins so which one ages is the one accelerating. If two cars are traveling side by side and one steps on the gas and speeds away it's that cars occupants that feel the accelleration. Motion is relative but not accelleration. Remember that inertia and gravity are one and the same thing. -Robert.

JesseM
Okay, so even though we made this acceleration instintaneous, it is still the determing factor. It is not a question of relative motion but whoever actually experiences a force to change the relative motion between them, this one will be younger
Basically this is right, although keep in mind that if Bob is moving away from Earth at 0.8c, then briefly accelerates towards the Earth, then immediately accelerates away again until he's once again moving away from Earth at 0.8c, and then later Ann accelerated towards Bob and eventually catches up with him, then Ann will still be younger, and by almost exactly the same amount as if Bob had not accelerated at all.

Personally, I think the best way to think of it is in terms of the geometry of the two worldlines. As an analogy, if you draw two paths connecting the same two points on an ordinary sheet of paper, one a straight line and the other with one or more bends in it, the straight line will always be the shortest one. You could even calculate the lengths of each path using different coordinate systems analogous to reference frames--draw an x and y axis, and then calculate how much the length of each path increases with each increment of the y axis (if a path is parallel to the y-axis from y=4 to y=6, then the path length will only have increased by 2 over that increment, but if the path has a slope of 2 from y=4 to y=6, meaning that it increments 1 in the x direction for every increment of 2 in the y-direction, then from y=4 to y=6 the path length will have increased by $$\sqrt{1^2 + 2^2} = \sqrt{5}$$ over the same interval). But even if you choose a number of different orientations for the y and x axis and recalculate the path length in each coordinate system using this same method, you'll always find that the straight-line path is shorter than a bent path between the same two points. In the same way, the geometry of spacetime is such that a straight worldline always has a greater proper time than a "bent" one (any worldline involving acceleration). And the case where Bob accelerates twice, returning to his initial velocity after the second acceleration, is analogous to a line that has one bend that sends it in a different direction, but then shortly after that another bend which returns it to its original direction--this path's length will only be slightly different from the length of a straight line path between the same two points.

JesseM
The change in direction and the resulting accelleration is relative to the mass of the entire universe
That would be true if [URL [Broken] principle[/url] were valid, but neither special relativity nor general relativity is truly "Machian" in this sense--the theories would allow you to have a totally empty universe apart from two test particles of infinitesimal mass, and it would still be true that whichever particle accelerates will have elapsed less time.

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JesseM
Does SR include math on how to add speeds?
Yes, see here.
Antman said:
Obviously I havn't read the SR but I would like too. Where can I find Einsteins original formulation and math and whatever he publiced when he announced SR?
You can find it here, but I wouldn't recommend this as a starting point for learning SR, more modern presentations would have proofs that are easier to follow along with useful aides that were developed later like spacetime diagrams, and more explanation of the background of Einstein's thinking like the problems physicists at the time were having understanding Maxwell's laws.

jtbell
Mentor
In general, the original presentation of any well-established theory is not the best place to start learning the theory from scratch, although it may be interesting and worthwhile to study it after you're already fairly fluent in that theory.

After all, how many people learn classical mechanics from scratch using Newton's "Principia Mathematica?" :uhh:

Thanks for the answers and th link! What is really interesting is what information Einstein had at the time when he thought up his special relativity theory and to follow the road his mind took from previous knowledge to SR.

The problem this thread discusses seems flaud in that way it says the twin paradox has nothing to do with acceleration.

JesseM
The problem this thread discusses seems flaud in that way it says the twin paradox has nothing to do with acceleration.
Who on this thread has said it has nothing to do with acceleration? I didn't say that.

The twin "paradox" has everything to do with acceleration.

Note though, that it is, in principle, possible to have a time interval differential for inertial twins in curved spacetime. Spacetime curvature can separate them and bring them back together, their elapsed time could be different even while both stay inertial the whole time.

But this is impossible in flat spacetime.

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Who on this thread has said it has nothing to do with acceleration? I didn't say that.

Sorry, noone in this thread said that. It was the problem in the link that did.

The confusion arises not because there are two equally valid inertial rest frames, but (here's the tricky part) because there are three. A lot of explanations of the twin paradox have claimed that it is necessary to include a treatment of accelerations, or involve General Relativity. Not so.

It seems everyone here agrees acceleration is what makes travelling twin younger but the problemformulation in the link says it does not have to do with accelleration. Or am I missreading totally?

sylas
It is not that acceleration "makes the twin younger". To think in such terms goes right back to the notion of some absolute reference from against which you decide who is younger. It is rather that acceration shows that you are not in an inertial reference frame.

JesseM
Sorry, noone in this thread said that. It was the problem in the link that did.
The confusion arises not because there are two equally valid inertial rest frames, but (here's the tricky part) because there are three. A lot of explanations of the twin paradox have claimed that it is necessary to include a treatment of accelerations, or involve General Relativity. Not so.
It seems everyone here agrees acceleration is what makes travelling twin younger but the problemformulation in the link says it does not have to do with accelleration. Or am I missreading totally?
I don't think that quote is saying it has nothing to do with acceleration, just that you don't have to have a "treatment" of either how things look in an accelerating frame of reference, or of how the ship's clock is dilating during the acceleration phase; instead you can just make the acceleration instantaneous, and then add the time elapsed on the outbound leg of the trip and the time elapsed on the inbound leg (both of which are inertial) to find the total time elapsed on the travelling twin's clock between leaving Earth and returning.

pervect
Staff Emeritus
If you consider the space-time diagram of a twins paradox type experiment, you can see that the twins form a triangle.

Acceleration means that the triangle isn't perfect, that the points of the triangle are actually slightly rounded by the acceleration.

However, this can be made to be a very small effect, by taking the limit of high accelerations.

What the twin paradox says is very similar to the geometrical theorem that the shortest distance between two points is a straight line, or eqivalently, the "triangle inequality" that says that the sum of the length of two sides of a triangle is always longer than the remaining third side.

In the case of the twins paradox, though, the statement becomes that the observer following a geodesic path (analogous to the straight line, especially in the flat geometry of SR) has the longest proper time. The time is the longest, but the distance in the Euclidean geometrical analogy is the shortest. This has to do with the difference between Euclidean geometry with it's ++++ metric signature, and the Lorentzian geomery of SR with it's -+++ signature.

Acceleration (the curvature at the tips of the triangle) really has very little to do with this geometrical result.

That is why people say that the twin paradox isn't about the rounding of the corners due to the acceleration - it's due to the angle between the observers. This angle on a space-time diagram is simply a change of velocity. What is important is not the rate at which the velocity changes (the accleration) , but what is often called "delta-V", the change in velocity.

The geometrical analogy of the "twins pardox" would be the "triangle paradox". The triangle paradox would say:

If I go directly from A to B, I travel the shortest distance. And if I go from A to C to B, I travel a longer distance. It's "paradoxical" (?) that when I go from A to C to B, that this distance is longer than going straight from A to B. Furthermore, if I look at the distance from A to B, that's shorter than going from A to C to B. Now AB is the shortest distance!

Of course people gemerally don't get confused by the "triangle paradox" - it's really not that much more difficult not to get confused by the "twin paradox" either.

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It's not that accelleration does not occur, it's that it's value is not important but the fact that it, the accelleration, results in the creation of another inertial frame and a time differance. So do you need the actual rate of acceleration to find the time difference value? -Robert.

In the case of the twins paradox, though, the statement becomes that the observer following a geodesic path (analogous to the straight line, especially in the flat geometry of SR) has the longest proper time. The time is the longest, but the distance in the Euclidean geometrical analogy is the shortest. This has to do with the difference between Euclidean geometry with it's ++++ metric signature, and the Lorentzian geomery of SR with it's -+++ signature.
The longest or, more accurately, the extremal path in spacetime is a geodesic and records the largest proper time.

In flat spacetime extremal length of a world line is an indicator of straightness.

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The best guess I have is that acceleration only matters in that it is necessary in order for Bob to change inertial frames. The actual period in acceleration can be made arbitrarily small and does not factor into the actual time calculation. My original concern was that since we were making acceleration instantaneous, that for both Bob and Anne the experience truly would be identical; while they are sitting still, the other party blasts off, then returns. But in reality, one of them DID have to change frames in order for this to happen, and in this example it was Bob who was assigned to truly change frames. And he changed Frames because he actually experienced acceleration; even though we made it so fast that the time period he was actually accelerating did not factor into the calculation.

After the time deration of the accelleration periods, there is no time dillation since linier motion does not produce durrable time dillation just as in SR. So it seems that the accelleration values must be required to determine the actual time difference the twins will experiance. -Robert.

JesseM