Undergrad Time Paradox: A & B's Clocks Move Differently

[SaintRodriguez]

Two astronauts A and B are traveling at constant speed, one toward the other. From astronaut A's point of view, his partner B's clock is ticking at a slower rate than his. From astronaut B's point of view, it is his partner A's clock ticking at a slower rate. Does this set up a paradox? Because?

 

[Ibix]

Does this set up a paradox?

No.

Because?

The relativity of simultaneity. Set up the problem formally for one observer and use the Lorentz transforms to get the other observer's perspective and you'll see what's going on.

 

[Nugatory]

Some older posts about how this apparent paradox is caused by ignoring relativity of simultaneity:
https://www.physicsforums.com/threads/special-relativity-time-issues.846242/#post-5307390
https://www.physicsforums.com/threads/weird-time-dilation-question.883303/#post-5552719

 

[Dale]

Does this set up a paradox? Because?

No. Because it is not the right form for a paradox.

A paradox is something like $a<b$ and $b<a$. But that is not what we have here.

Let $\tau_A$ be the proper time of clock A and $t_A$ be the coordinate time in frame where A is at rest. Similarly for $\tau_B$ and $t_B$.

Then “From astronaut A's point of view, his partner B's clock is ticking at a slower rate than his” means $$\frac{d\tau_B}{dt_A}<1$$

And “From astronaut B's point of view, it is his partner A's clock ticking at a slower rate” means $$\frac{d\tau_A}{dt_B}<1$$

Both of these statements are perfectly compatible. They do not contradict each other. Hence it is not a paradox.

 

[Sagittarius A-Star]

Two astronauts A and B are traveling at constant speed, one toward the other. From astronaut A's point of view, his partner B's clock is ticking at a slower rate than his. From astronaut B's point of view, it is his partner A's clock ticking at a slower rate. Does this set up a paradox? Because?

Do you consider the following as a paradox? Because?

From astronaut A's point of view, his partner B moved via a greater spatial distance than himself, because A regards himself to be at rest. From astronaut B's point of view, it is his partner A's travel distance, which is the greater one.

 

[vanhees71]

Two astronauts A and B are traveling at constant speed, one toward the other. From astronaut A's point of view, his partner B's clock is ticking at a slower rate than his. From astronaut B's point of view, it is his partner A's clock ticking at a slower rate. Does this set up a paradox? Because?

It doesn't lead to a paradox, because the Lorentz transformation from one inertial frame of reference to another is a one-to-one map between coordinates, and the physical laws are covariant, looking the same when expressed in any inertial frame of reference.

 

[Hornbein]

It doesn't lead to a paradox, because the Lorentz transformation from one inertial frame of reference to another is a one-to-one map between coordinates, and the physical laws are covariant, looking the same when expressed in any inertial frame of reference.

Such a concise explanation.