Let the Earthbound twin be in Frame S' and the traveling twin be in Frame S. Further let the trip start at t=t'=0 and x=x'=0. Event 1: Ship leaves Earth. Event 2: Ship returns to Earth. Then the Lorentz Transformation relating the two times is: t'=(gamma)(t-vx/c^2) Now, for the traveling twin, there is no spatial separation between the two events, so x=0, giving us: t'=(gamma)t There is the formula. Tom
Tom2, You seem to have forgotten something. From the travelling twin's frame of reference: Event 1a: Earth leaves ship. Event 2a: Earth returns to ship. Unless you're suggesting that one frame of reference is more valid than another. Tom
The problem with the simple analysis I gave is that it runs roughshod over the point at which the ship changes direction (IOW, 'accelerates'). An intermediate event is overlooked, namely: Event 1.5: Ship reaches Alpha Centauri (or some other destination). What happens then? An acceleration. Which twin accelerates? The one on the ship. Now, this takes us into the whole worldline business. It turns out that the shortest path in Minkowski 4-space is the path of maximal aging. That path is taken by the twin who does not accelerate. I don't know how to put fancy equations in this forum, and I don't have my own webspace yet, but this problem is solved in a sample exercise on page 10 of the following book, and then it is really beaten to death in Section 4.5: edit: fixed the link http://www.physics.nyu.edu/hogg/sr/sr.pdf The formula for calculating the ages is, of course, the one I quoted. You just need the deeper analysis to resolve the apparent "paradox". Tom
Tom2, Are you saying that if it was theoretically possible to accelerate and decelerate the travelling twin instantaneously, the twins would be the same age at the end of the journey?
No, I'm saying that their ages cannot be the same. The thing that breaks the paradox is that there are not 2 frames involved, but 3. The ship is in 2 different frames during the process, and the Earth is in 1 (ignoring acceleration due to orbital and rotational motion). Tom
Tom2, But from the travelling twin's frame of reference, isn't the ship in one frame of reference while the Earth is in two?? I can only see two things in the twin paradox that aren't symmetrical: 1) The travelling twin feels the force of acceleration/deceleration while the Earth twin does not. 2) The Earth twin feels the force of the Earth's gravitational field, while the travelling twin does not. I can't see any other asymmetries.
No, he definitely knows that it is he who switched frames. That's the solution to the paradox. The Earth's gravity is not strong enough to affect the problem, so it is left out. If that is a thorn in your side, then leave the Earth out altogether. Just picture the "stay-at-home" twin on a nonaccelerating space station or something. Tom
Tom2, Are you claiming that the time dfference between the twins only occurs during the acceleration/deceleration part of the travelling twin's journey and not during the parts of the journey in which they are both moving at constant speeds (relative to each other)?
No, I am saying that the changing of frames resolves the paradox because it tells you unambiguously who follows the straight worldline, and who follows the bent worldline. Tom
Ahh Haa Lethe, I knew it all along.Please Register or Log in to view the hidden image! Please Register or Log in to view the hidden image! Please Register or Log in to view the hidden image!
Actually the time rate depends upon one's position within a gravitational field. Again, it is not the relative strength of the gravitational force the observers feel, but their positions(or gravitational potential) within the respective fields that counts. The Earth's gravity field falls off as you move away from it, this puts a limit on how much of a gravitational potential difference have with respect to it. ( for a person standing on the Surface of the Earth, that would be GMm/r. (with r being the radius of the Earth. ) This would be the potential difference between a mass at an inifinte distance from the Earth and one on the surface. For a one kg object, this works out to 61728912 joules. At the distance of the Moon it it is 1025278 joules, a difference of 60703733 joules. IOW, The vast majority of gravity potential difference happens while you are still quite close to Earth. Now let's consider what happens with the "traveling" twin while he is stopping and turning around at the end of his trip. Assuming that this starts at the half way point, he is 2.15 lys from Earth. He has to fire is engines away from the Earth. Meaning that in the "gravitational Field" he experiences, the Earth is "above" him. Now this Field is a Uniform one(It extends at uniforn Strength for an inifinte length both "above" and "below" the traveling twin. Thus from the "Traveling" twin's perspective, the Earth is 2.15 lys higher( and climbing) than him in a gravity field that maintains a constant strength throughout. IOW, the traveling twin will see himself much "deeper" in the Gravity field than the Earth. Because of this vast difference in gravity potential, He will see the Earth's time rate as traveling much faster than his. (a much greater difference in times rates than a person standing on the Surface of the Earth would see between himself and an object hanging 2.15 lys over his head. This is what breaks the symmetry.
Tom: Referring to your original post in this thread: Sections 1a and 1b are not symmetrical. Both observers will agree that the travelling twin's clock is running slower than the earth twin's clock in these sections of the journey. Moreover, they can both easily tell who is "really" accelerating, since one of them experiences forces and the other does not. Sections 2a and 2b are symmetrical. Both observers see each other's clock as running slower. Sections 3a and 3b are not symmetrical. Again, one observer actually experiences acceleration, whilst the other one does not. Same for 4a and 4b. Sections 5a and 5b are symmetrical. Sections 6a and 6b are not. The net result is that after the entire trip, the travelling twin is younger. They both agree on that.
1g James R, This has been my point and I see it again here. Due to the "Equivelence Principle", that is that gravity and accelerations are one and the same thing, both purportedly dilate time, why is it that it is claimed that if the twins rocket maintained a 1g acceleration that there would be any difference? 1g gravity and 1g acceleration. The use of the term "Turn Around" confuses me a bit in that I visualize a linear vector trip. That is +1g*t and -1g*2t and finally +1g*t. You are back where you started and have maintained the equivelent effects as the earth bound twin at 1g gravity.
Re: 1g Read my post above. It is not the strength of the G-force felt that effects time, but the relative positions in and nature of each "G" field. The Earth's G- field falls of by the square of distance. The twin's G-field (due to acceleration) is uniform and does not fall off in strength along the axis of acceleration.
Janus58, Welcome to sciforums. Do you agree that a 10 m/s/s acceleration caused by the Earth's gravitational field will have the same time dilation on the Earth twin as a 10 m/s/s acceleration caused by the travelling twin's rocket engine will have on the travelling twin?? Tom
Tom2, It seems that you have the same explanation for the twin paradox as does chroot and everneo. Unfortunately, I don't understand the "wordline" representation of the paradox. I also don't understand how the travelling twin will age slower than the Earth twin during the parts of the journey in which the travelling twin isn't accelerating. Maybe you, chroot, or everneo can explain it in some more detail. Tom
