Relative velocity time dilation in opposite orbits.?

Question

Relative velocity time dilation
From the local frame of reference (the blue clock), relatively accelerated clocks move more slowly.
When two observers are in relative uniform motion and uninfluenced by any gravitational mass, the point of view of each will be that the other's (moving) clock is ticking at a slower rate than the local clock. The faster the relative velocity, the greater the magnitude of time dilation. This case is sometimes called special relativistic time dilation.
For instance, two rocket ships (A and B) speeding past one another in space would experience time dilation. If they somehow had a clear view into each other's ships, each crew would see the others' clocks and movement as going too slowly. That is, inside the frame of reference of Ship A, everything is moving normally, but everything over on Ship B appears to be moving slower (and vice versa).
From a local perspective, time registered by clocks that are at rest with respect to the local frame of reference (and far from any gravitational mass) always appears to pass at the same rate. In other words, if a new ship, Ship C, travels alongside Ship A, it is "at rest" relative to Ship A. From the point of view of Ship A, new Ship C's time would appear normal too.[4]
A question arises: If Ship A and Ship B both think each other's time is moving slower, who will have aged more if they decided to meet up? With a more sophisticated understanding of relative velocity time dilation, this seeming twin paradox turns out not to be a paradox at all (the resolution of the paradox involves a jump in time, as a result of the accelerated observer turning around). Similarly, understanding the twin paradox would help explain why astronauts on the ISS age slower (e.g. 0.007 seconds behind for every 6 months) even though they are experiencing relative velocity time dilation. 
http://en.wikipedia.org/wiki/Time_dilati…

Thought experiment :  the above ships, side by side, simultaneously leave a base in relatively open space, their mission is to orbit the same distant star, taking observations of each other's clocks at various points along the way, then return to base. As they approach the star, a pre- determined way point is reached, at which, they must adjust their trajectories so that their orbits are in opposite directions and then simultaneously return to waypoint and so on to base.
( just so as not to get distracted by gravitational time dilation, let's remove the star and just follow the same route )

Am I right in thinking the paradox returns, or am I missing something? 
Thinkonit.

permeative pedagogy · Accepted Answer

I'm not 100% sure about the thought experiment you've set up. It sounds as if ship A and B fly out to a predetermined position and both fly off in different directions while still being capable of observing each other. If that is indeed the idea you meant to convey, then it is exactly the situation described in the Wikipedia article. In essence, this "paradox" arises in any situation when two objects are traveling at different velocites. So because your ships are traveling away from each other, they must be traveling at velocities away from each other and thus they would observe a time dilation on the other ship.

If you meant something else with your question, please clarify and I can try to help further.

EDIT: It seems that I correctly interpreted your question. In that case, neither of them is older. I think the solution to this paradox comes from the fact that time dilation only occurs in inertial reference frames. Keep in mind that this subject is very tricky and and I don't have a full grasp on it, but here is my interpretation. Obviously they cannot both have aged less because that makes no sense (hence the paradox). The key though is that they can only truly discover who has aged less by meeting up and comparing their calculations in the same inertial reference frame. They cannot do so while apart. BUT here's the rub, they can only ever get into the same inertial reference frame by accelerating. The problem already states that they are traveling in difference velocities in different directions. They must necessarily remove themselves from an inertial reference frame (by accelerating) to get back to each other in the same reference frame. Doing so cancels out any affects they may have received from time dilations and in doing so, they both observe, when they are back together again, that they are still the same age. That might not be the best answer out there, but its as good as I can describe it. You might just want to read around. I'm 100% certain someone before you has already asked this exact question and probably gotten a better answer.

? · Answer

It's a bit awkward to compare your thought experiment with the twin paradox. In the twin paradox we speak of Special Relativity's treatment of Inertial Frames. Strictly speaking, an orbit is not inertial since it is always accelerating. We may treat this problem, though, using Special Relativity. If we perform an integral of infinitesimal time dilations Δt = ∫γ(v(t'))dt' we can, in principle, work out each twin's observation of the other's clock. I'm not going to do a mathematical treatment here because I don't know how long it would take. Also it is only important that such a treatement is possible.
 
Because the motion is completely symmetric, I suspect that the answer to the paradox lies (as for the twin paradox) in the acceleration at the end when both ships must be brought into the same inertial frame to 'meet up'. 
I would expect that (since both ships have to accelerate into the same non-orbiting inertial frame) each twin aboard would see the other as being younger until the moment they both accelerate into the non-rotating frame. As they accelerate they would have to accept 'our' reality in which both are the same age as each other but considerably younger than a hypothetical triplet in our frame of reference. Hence; no paradox.
Let me just qualify that with an "I think". Since I haven't done the maths I wouldn't like to proclaim anything definite, but I *am* confident the result will turn out as expected.

It's a potentially interesting maths problem though!

Anonymous · Answer

paradox means apparent impossibility or contradiction

there is real experimental evidence for relativistic effects
relativity means " true from your point of view"  the closest to ' Real" is rest measurements made by observers in zero velocity with respect to each other

in a symmetrical situation the results should be the same but when they are back together they are no longer in relative motion

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Relative velocity time dilation in opposite orbits.?

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