Twin Paradox - Take 2

[Mowgli]

This was posted under the thread "Twin Paradox" earlier.

 

The Twin Paradox is usually explained away by arguing that the traveling twin feels the motion because of his acceleration/deceleration, and therefore ages slower.

 

But what will happen if the twins both accelerate symmetrically? That is, they start from rest from one space point with synchronized clocks, and get back to the same space point at rest by accelerating away from each other for some time and decelerating on the way back. By the symmetry of the problem, it seems that when the two clocks are together at the end of the journey, at the same point, and at rest with respect to each other, they have to agree.

 

Then again, during the whole journey, each clock is in motion (accelerated or not) with respect to the other one. In SR, every clock that is in motion with respect to an observer's clock is supposed run slower. Or, the observer's clock is always the fastest. So, for each twin, the other clock must be running slower. However, when they come back together at the end of the journey, they have to agree. This can happen only if each twin sees the other's clock running faster at some point during the journey. What does SR say will happen in this imaginary journey?

 

(Note that the acceleration of each twin can be made constant. Have the twins cross each other at a high speed at a constant linear deceleration. They will cross again each other at the same speed after sometime. During the crossings, their clocks can be compared.)

[Sisyphus]

 

(Note that the acceleration of each twin can be made constant. Have the twins cross each other at a high speed at a constant linear deceleration. They will cross again each other at the same speed after sometime. During the crossings, their clocks can be compared.)

 

I don't understand this part. You can't compare clocks that are in different inertial frames, because the observers in each frame won't agree. But why is that necessary? I thought you wanted to bring them back to being at rest with respect to one another?

[Janus]

This was posted under the thread "Twin Paradox" earlier.

 

The Twin Paradox is usually explained away by arguing that the traveling twin feels the motion because of his acceleration/deceleration, and therefore ages slower.

 

But what will happen if the twins both accelerate symmetrically? That is, they start from rest from one space point with synchronized clocks, and get back to the same space point at rest by accelerating away from each other for some time and decelerating on the way back. By the symmetry of the problem, it seems that when the two clocks are together at the end of the journey, at the same point, and at rest with respect to each other, they have to agree.

 

Then again, during the whole journey, each clock is in motion (accelerated or not) with respect to the other one. In SR, every clock that is in motion with respect to an observer's clock is supposed run slower. Or, the observer's clock is always the fastest. So, for each twin, the other clock must be running slower. However, when they come back together at the end of the journey, they have to agree. This can happen only if each twin sees the other's clock running faster at some point during the journey. What does SR say will happen in this imaginary journey?

Acceleration has an effect on what each twin sees.

After factoring out time dilation due to velocity:

If your accelerate away from a clock, you see it running slower.

If you accelerate towards a clock, you see it running faster.

How fast or how slow depends on the magnitude of the acceleration, and the distance to the clock as measured along the line of acceleration.

(Note, the actual distance between the observer and the clock does not have to be increasing or decreasing. If you have two clocks accelerating in the same direction at the same rate, the leading clock will run faster than the trailing clock, according to both clocks.)

 

If you factor in this effect, it turns out, that each twin will see the other's clock as running fast during a portion of the trip, and this will cancel out that portion when he saw it running slow, so that, at the end, each twin will have aged the same.

[Mowgli]

Acceleration has an effect on what each twin sees.

After factoring out time dilation due to velocity:

If your accelerate away from a clock, you see it running slower.

If you accelerate towards a clock, you see it running faster.

How fast or how slow depends on the magnitude of the acceleration, and the distance to the clock as measured along the line of acceleration.

 

Is this an SR or GR effect?

 

Note that the acceleration of each twin can be made constant. Have the twins cross each other at a high speed at a constant linear deceleration. They will cross again each other at the same speed after sometime. During the crossings, their clocks can be compared. Thus you can keep the accelerations of the clocks constant and always directed towards each other. Would that mean that they always run faster (time contraction) rather than slower?

[Janus]

Is this an SR or GR effect?

Well, that can depend on who you ask. Some have the position that SR only holds in intertial reference frames, and since it deals with acceleration, it is GR

Other would say that it is a result The Realtivity of Simultaneity and thus an SR effect.

Note that the acceleration of each twin can be made constant. Have the twins cross each other at a high speed at a constant linear deceleration. They will cross again each other at the same speed after sometime. During the crossings, their clocks can be compared. Thus you can keep the accelerations of the clocks constant and always directed towards each other. Would that mean that they always run faster (time contraction) rather than slower?

 

At the moment of the first passing the distance between the two twins is small so the acceleration effect will be small. The velocity difference will be large, so each twin would see the other's clock running slow. As time goes by, the distance will increase, and the velocity difference will decrease, each twin sees the other's clock as running fast.

 

When the two twins reach the turnaround point (that instant when their velocities stasrt to change direction) The total time each twin will have seen the other's clock gain and lose due to running fast and slow will cancel out, and the accumulated time on both of their clocks will be the same.

 

The two twins start to come back together and the reverse of the outbound leg happens. At the instant they pass each other, their clocks will once again agree. (but only for that instant, for each twin will see the other's as runing slow at this point.)