Confused with a certain aspect of special relativity

[aguycalledwil]

I am confused with a certain aspect of special relativity. Here is my reasoning...

Someone in a rocket, traveling at 0.99c ages slower and thus would be younger after returning to Earth than his twin brother. I understand the 'light clock' explanation of this.
Here's where I'm getting confused.. Surely, the Earth would also be traveling at 0.99c, relative to the rocket. Does that not suggest that a clock on Earth would go slower than that of the rocket's, relative to the rocket. So to the rocket, the Earth is going slow in time. To the earth, the rocket is going slow in time. When the rocket comes back, what dictates that the brother in the rocket would have aged less?
Regards..

 

[HallsofIvy]

I am confused with a certain aspect of special relativity. Here is my reasoning...

Someone in a rocket, traveling at 0.99c ages slower and thus would be younger after returning to Earth than his twin brother. I understand the 'light clock' explanation of this.
Here's where I'm getting confused.. Surely, the Earth would also be traveling at 0.99c, relative to the rocket. Does that not suggest that a clock on Earth would go slower than that of the rocket's, relative to the rocket. So to the rocket, the Earth is going slow in time. To the earth, the rocket is going slow in time. When the rocket comes back, what dictates that the brother in the rocket would have aged less?
Regards..

Yes, as long as twin A is moving at 0.99c relative to twin B, twin B sees twin A aging more slowly. But you are right that twin B is moving at 0.99c relative to twin A so twin A see twin B aging more slowly. That's why it is called a paradox! There are two parts to the resolution of this. One is that twin B stays in Earth's gravity well while twin A does not and the Earth's gravity plays a part in the aging. You could, of course, avoid that by postulating that the two twins were born an raised on a spaceship without the deep gravity well. But in order to be able to say that the two twins are of the same or different ages, rather than just saying one appears to age faster or slower than the other, you must have them together in the same reference frame, motionless relative to one another. That means that one or the other must accelerate to a different speed, then decelerate in order to come back to the initial reference frame.

Assuming that one twin remains in the same initial frame, the other must accelerate and decelerate and it is that that causes the "asymmetry". The twin who accelerates and decelerates ages slower than the twin who does not.

Of course, you could avoid that asymmetry by postulating that both twins accelerate, in opposite directions, away from the initial frame of reference, the decelerate back again. In that case, the calculations say the twins would be of exactly the same age when they are together again.

Just to add to the complication, you could also assume that one twin remains in the original reference frame while the other moves along a "closed null geodesic". Assuming there is a null geodesic requires some pretty complicated geometry to begin with but then the "moving" twin would be able to move at a constant speed yet comeback to his original position. That is way above my head!

 

[jtbell]

When the rocket comes back, what dictates that the brother in the rocket would have aged less?

The fact that the brother in the rocket "turns around" (accelerates) at the midpoint of his trip, whereas the brother on Earth doesn't do anything special. The traveling brother doesn't remain in the same inertlal reference frame throughout the trip, but the earthbound brother does.

Assume that the two brothers synchronize their wristwatches before the traveling brother leaves. Just before the traveling brother "turns around", at that moment in his current reference frame the stay-at-home brother's wristwatch reads less time. Just after the traveling brother "turns around" (assume he does it very quickly so we can ignore the amount of time that passes on his own wristwatch), in his new reference the stay-at-home brother's wristwatch reading has "jumped" ahead by a large amount. From the traveling brother's point of view, this "jump" at the turnaround causes the total elapsed time on his brother's watch to be larger than on his own watch, when he returns to earth, even though his brother's watch runs slower than his own watch while he is going out and coming back.