The paradox of symmetrical time dilation

[spacecadet2563]

If someone stationary on Earth watches a vehicle leaving Earth very fast he sees that time is dilating for that vehicle. Any clocks he sees slow down. But since motion is relative the traveler sees the Earth race away from him at a high speed and the same thing happens. He sees the Earth clock slow down. This is a problem. Who is right? Saying both are is not good enough.

 

[Hill]

A sees B's clock slows down. B sees A's clock slows down. What is the problem?

 

[spacecadet2563]

A sees B's clock slows down. B sees A's clock slows down. What is the problem?

B is the traveler. He is supposed to be traveling into the Earth's future. From the Earth perspective. Apparently not from his. Which is it?

 

[Hill]

B is the traveler. He is supposed to be traveling into the Earth's future. From the Earth perspective. Apparently not from his. Which is it?

B ends up in the Earth's future after he turns around and gets back to Earth.

 

[spacecadet2563]

B ends up in the Earth's future after he turns around and gets back to Earth.

How? Why? He sees the Earth's clock moving slowly relative to his. The reverse would be true. Unless something happens when he turns around.

 

[Hill]

How? Why? He sees the Earth's clock moving slowly relative to his. The reverse would be true. Unless something happens when he turns around.

Exactly. Something happens during the turning around. B is not an inertial frame when that happens.

 

[PeroK]

How? Why? He sees the Earth's clock moving slowly relative to his. The reverse would be true. Unless something happens when he turns around.

Are you sure you are in the right frame of mind to learn relativity?

 

[Nugatory]

If someone stationary on Earth watches a vehicle leaving Earth very fast he sees that time is dilating for that vehicle. Any clocks he sees slow down. But since motion is relative the traveler sees the Earth race away from him at a high speed and the same thing happens. He sees the Earth clock slow down. This is a problem. Who is right? Saying both are is not good enough.

You have overlooked the relatively of simultaneity.

Say that observer A is located on earth, B on the ship, and both agree that the ship leaves earth at 12:00 noon. One hour later A looks at their clock and sees that it reads 1:00 PM, as we expect. A uses the Lorentz transformation to calculate what B's clock reads AT THE SAME TIME that their clock reads 1:00, or equivalently they are watching B's clock through a telescope and by allowing for light travel time they can see what B's clock read when light left B AT THE SAME TIME that A's clock reads 1:00 PM. Either way, they find that B's clock reads 12:30 AT THE SAME TIME that A's clocks reads 1:00. In other words, the events "A's clock reads 1:00" and "B's clock reads 12:30" are simultaneous when we use the frame in which A is at rest.
Clearly B's clock is running slow, by a factor of two.

But this is where the relativity of simultaneity comes in. Using the frame in which B is at rest, these two events do not happen at the same time. Instead, the event "A's clock reads 12:15" is simultaneous with the event "B's clock reads 12:30" so just as clearly A's clock is running slow by a factor of two.

And that’s how they’re both right.

As an aside, this setup is different from the twin paradox because the two clocks are not at the same place when we read them. Confusingly, people often use the term “time dilation” for both this case and the twin paradox even though they are completely different things.

 

[gmax137]

I think it helps to suspend your disbelief for as long as it takes you to learn how it works. Putting your mental energy into arguments showing why it can't be true is a waste of your time. Trust me, it will all make sense in the end.

PS, quantum mechanics is even worse.

 

[Dale]

Who is right?

They both are.

Saying both are is not good enough.

Sorry to hear that you feel the world is not good enough for you. It must be upsetting to feel that way.

B is the traveler. He is supposed to be traveling into the Earth's future.

A is also traveling into the Earth’s future. Neither A nor B can avoid traveling into the future.

 

[PeterDonis]

He is supposed to be traveling into the Earth's future. From the Earth perspective. Apparently not from his.

This is wrong. Both of them are traveling "into the future", and there is no way for either one of them to meet back up again with the other at a point in spacetime that is not to the future of the point in spacetime where they separated.

 

[Ibix]

If someone stationary on Earth watches a vehicle leaving Earth very fast he sees that time is dilating for that vehicle. Any clocks he sees slow down. But since motion is relative the traveler sees the Earth race away from him at a high speed and the same thing happens. He sees the Earth clock slow down.

Seen in terms of geometry, it is closely analogous to two cars driving at equal speeds $v$ down straight roads that make an angle $\theta$ to one another. Both cars will observe that the the other car is progressing at $v\cos\theta$, so both will see the other falling further behind. This is not a paradox, it's just that they have different ideas about what "forward" means.

Similarly, the two relativistic observers have different ideas about what "not moving in space, only advancing in time" means, and both say that the other is moving at an "angle" (called the rapidity) to them and hence their clocks are ticking slow.