Symmetry in Travel and Time Dilation

[g_sanguinetti]

-Is it possible for two objects to have symmetrical geodesics in which each accelerate with equal magnitude but in opposite directions? If not why not?
-If so, how can both see each others clocks run slow? Would one have aged less than the other when then they reunite? Which? If not what happened to time dilation?
-Does relativity ever allow for relative velocities to produce the phenomenon of someone else's clock to run faster - for example on a return trip? Wouldn't that contradict special relativity?

Thanks,
- George -

 

[JesseM]

-Is it possible for two objects to have symmetrical geodesics in which each accelerate with equal magnitude but in opposite directions? If not why not?

Yes, it's possible.

If so, how can both see each others clocks run slow?

The usual time dilation equation of SR which says that a moving clock will run slow in your frame only works if you are an inertial observer (inertial = non-accelerating). In a non-inertial frame things are more complicated, if you're accelerating then a distant clock could be running faster than your own at times.

Would one have aged less than the other when then they reunite?

No, if their paths are symmetrical as seen in some inertial frame--if, for example, they both move away from the Earth at 0.6c for 1 year in the Earth's frame, then turn around and move back towards it at 0.6c for another year--then they will have aged the same amount when they reunite.

Does relativity ever allow for relative velocities to produce the phenomenon of someone else's clock to run faster - for example on a return trip? Wouldn't that contradict special relativity?

Again, the usual time dilation equation of special relativity only deals with the rate a clock is ticking as measured in an inertial reference frame. In such a frame, a moving clock will always be running slow--if its speed at a particular moment is v, its rate of ticking at that moment will be slow by a factor of $$\sqrt{1 - v^2/c^2}$$.

 

[g_sanguinetti]

Thanks for your reply JesseM.
Now the question is, when will the clocks be running faster?
If it is only during the periods of accelerations, then when the two objects are moving at constant velocity there should still be dilation in which each sees the other's clock moving slower, shouldn't there?
And if that is true, how can equal accelerations make up for the different dilations equally? Let me ask that again and elaborate: And if that is true, how can equal accelerations (for various different scenarios i.e., different trips, each involving the symmetrical geodesics of two objects, each having equal accelerations [between the objects within a trip and between all trip, i.e., accelerations in trip A = accelerations in trip B = accelerations in trip C = accelerations in…{The acceleration of each object within any particular trip is of course equal.}] and yet be of different lengths [length of trip A not equaling length of trip B not equaling the length of trip C not equaling the length of …{The length of the distance covered by each object in any particular trip is of course equal.} ] and so have different dilations.) make up for the different dilations equally? How can dilation during accelerations make up for the dilation for each possible different trip of different lengths and still have special relativity apply to the non-accelerating segments? The dilations in the direction of running faster would have to be different in trips of different lengths, even though the accelerations were of equal length, wouldn’t it? This cannot be possible, can it?

Thanks,

- George -

 

[pervect]

Thanks for your reply JesseM.
Now the question is, when will the clocks be running faster?

Because simultaneity is relative, the answer will depend on your choice of frame.

What everyone will agree on is that in flat space-time (no gravity) anyone following a geodesic path will have the maximum elapsed time on his clock.

Not everyone will explain things in the same manner, because they have differing notions of what "at the same time" means, hence there is no observer-independent way to compare clocks that are not at the same location.

 

[JesseM]

Thanks for your reply JesseM.
Now the question is, when will the clocks be running faster?

In what coordinate system are you asking the question? The normal way to analyze the situation in SR would be to take an inertial frame and analyze both paths from the perspective of that frame, in which case each clock's time dilation will just be a function of velocity in that frame, not acceleration. On the other hand, if you want to analyze things from the perspective of a non-inertial coordinate system where each twin is at rest throughout the entire trip (both before and after accelerating), it depends on the details of how you construct this non-inertial coordinate system (how you define simultaneity at different points in the trip, for example), there isn't really a single set way to do it. I suppose the most natural way would be to construct your non-inertial coordinate system so that its definition of simultaneity and distance at each moment always matches the definitions of the instantaneous inertial rest frame of the ship at that moment, and such that the time coordinate of events along the ship's worldline always matches the proper time of the ship (time according to the ship's own clock). If you do it this way, then in each ship's non-inertial coordinate system the second ship's clock will be running slow during the non-accelerating portions of the first ship's worldline, but will run fast--possibly extraordinarily fast, depending on how quick the acceleration is--during the accelerating portions.

If it is only during the periods of accelerations, then when the two objects are moving at constant velocity there should still be dilation in which each sees the other's clock moving slower, shouldn't there?

If you construct each ship's non-inertial coordinate system in the way I describe above then yes, though like I said you have no obligation to construct the coordinate system this way.

And if that is true, how can equal accelerations make up for the different dilations equally?

Why shouldn't they? In the first ship's own non-inertial coordinate system the second ship's clock will advance forward by some large amount during the period where the first ship is accelerating, and in the second ship's non-inertial coordinate system the first ship's clock will advance forward by exactly the same amount during the period where the second ship is accelerating.

Let me ask that again and elaborate: And if that is true, how can equal accelerations (for various different scenarios i.e., different trips, each involving the symmetrical geodesics of two objects, each having equal accelerations [between the objects within a trip and between all trip, i.e., accelerations in trip A = accelerations in trip B = accelerations in trip C = accelerations in…{The acceleration of each object within any particular trip is of course equal.}] and yet be of different lengths [length of trip A not equaling length of trip B not equaling the length of trip C not equaling the length of …{The length of the distance covered by each object in any particular trip is of course equal.} ] and so have different dilations.) make up for the different dilations equally?

Using the type of coordinate system I describe above, it would have to do with the fact that even though the accelerations are equal on different trips, if the inertial phases of the trips were different lengths than the distance to the second ship during the period when the first ship accelerates will be different on different trips (and vice versa), and the amount that the second ship's clock jumps forward in the first ship's non-inertial coordinate system is a function of distance, a bigger move forward at greater distances. You might get some idea of why this is the case if you look at this image from the too many explanations section of this twin paradox FAQ--the blue lines correspond to the non-inertial coordinate system's definition of simultaneity at different points on the ship's worldline, if you look at the region bounded by the lines from the beginning and end of the acceleration period you can see it's sort of wedge-shaped, so the farther a second worldline is from the first the larger the section of it that will be inside this wedge.

 

[g_sanguinetti]

You ask:
"Why shouldn't they? In the first ship's own non-inertial coordinate system the second ship's clock will advance forward by some large amount during the period where the first ship is accelerating, and in the second ship's non-inertial coordinate system the first ship's clock will advance forward by exactly the same amount during the period where the second ship is accelerating."

For the accelerations to make it come out so that each traveler is the same age upon coming back together, in each different symetrical trip of various lengths, each trip would have to have different time dilation effects during their acceleration periods; otherwise, one or the other set of travelers would expect some time dilation between them. But this cannot be since each acceleration is same for each trip. If in one trip time dilation was occurring for only a second (a meter) in the non-accelerated portion and in the other trip for decades (or a light-year) how can the same accelerations compensate for both when they advance the clocks the same amount in each trip of differing non-accelerated time dilation?

 

[g_sanguinetti]

So . . how can the same accelerations account / compensate for the different dilation effects / amounts for the different trips?

In one trip the time dilation due to relative velocities may be say "x" and in another, due to longer time spent in the trip, maybe "2x", but if each trip has the same accelerations how can the moving forward of the clocks during the accelerations always make up for the dilation in such a way as to have them came out equal?

Also:
Does anyone assert that time goes faster in the non-accelerated parts of the trip? If so doesn't that contradict relativity? How would the universe know that this particular relative velocity was part of particular trip?

 

[g_sanguinetti]

In what coordinate system are you asking the question? The normal way to analyze the situation in SR would be to take an inertial frame and analyze both paths from the perspective of that frame, in which case each clock's time dilation will just be a function of velocity in that frame, not acceleration. On the other hand, if you want to analyze things from the perspective of a non-inertial coordinate system where each twin is at rest throughout the entire trip (both before and after accelerating), it depends on the details of how you construct this non-inertial coordinate system (how you define simultaneity at different points in the trip, for example), there isn't really a single set way to do it. I suppose the most natural way would be to construct your non-inertial coordinate system so that its definition of simultaneity and distance at each moment always matches the definitions of the instantaneous inertial rest frame of the ship at that moment, and such that the time coordinate of events along the ship's worldline always matches the proper time of the ship (time according to the ship's own clock). If you do it this way, then in each ship's non-inertial coordinate system the second ship's clock will be running slow during the non-accelerating portions of the first ship's worldline, but will run fast--possibly extraordinarily fast, depending on how quick the acceleration is--during the accelerating portions. If you construct each ship's non-inertial coordinate system in the way I describe above then yes, though like I said you have no obligation to construct the coordinate system this way. Why shouldn't they? In the first ship's own non-inertial coordinate system the second ship's clock will advance forward by some large amount during the period where the first ship is accelerating, and in the second ship's non-inertial coordinate system the first ship's clock will advance forward by exactly the same amount during the period where the second ship is accelerating. Using the type of coordinate system I describe above, it would have to do with the fact that even though the accelerations are equal on different trips, if the inertial phases of the trips were different lengths than the distance to the second ship during the period when the first ship accelerates will be different on different trips (and vice versa), and the amount that the second ship's clock jumps forward in the first ship's non-inertial coordinate system is a function of distance, a bigger move forward at greater distances. You might get some idea of why this is the case if you look at this image from the too many explanations section of this twin paradox FAQ--the blue lines correspond to the non-inertial coordinate system's definition of simultaneity at different points on the ship's worldline, if you look at the region bounded by the lines from the beginning and end of the acceleration period you can see it's sort of wedge-shaped, so the farther a second worldline is from the first the larger the section of it that will be inside this wedge.

Also, let me direct you to this thread:
https://www.physicsforums.com/member.php?u=14944