Another Special Relativity Paradox inspired by Herbert Dingle's Paradox

Question

Suppose spaceships A and B with clocks A and B move along the same straight line at uniform speeds differing by 161 miles per second. At the moment when B passes A, their clocks both read noon. Shortly after they meet, at 2 o'clock according to A's clock, spaceship B passes space station C.

Furthermore, suppose that spaceship B has a computer monitor that is programmed to display the time according to clock B plus the word "yes" if spaceship B has already passed space station C or the word "no" if spaceship B has not yet passed space station C (according to B's perspective). Assume that spaceship A can see this computer monitor at all times from where it is.

Then according to the special theory of relativity, from A's point of reference, B's computer monitor will change from "no" to "yes" when as soon as it displays 1 o'clock.

But by symmetry, from B's point of reference, B's computer monitor will change from "no" to "yes" when as soon as it displays 2 o'clock.

Did I make a mistake, or is it a consequence of special relativity that two people can perceive completely different realities from completely different perspectives, A seeing the monitor change from "no" to "yes" at 1 o'clock and B seeing the monitor change from "no" to "yes" at 2 o'clock?

Answer 1

I think you're overlooking the fact that the ordering of distant events is ambiguous in special relativity. In general, observers will disagree on whether distant events are simultaneous. To recap the events you are considering:

Event AB: A and B pass one another. Clocks A and B both read noon.

Event BC: B passes C. B's clock reads 1 o'clock.

Anyone (including A) observing event BC will observe that B passes C and that B's clock reads 1 o'clock. However the observer's clock need not read 1 o'clock, even if it was synchronized with B earlier, because the rate at which B's clock ticks depends on its speed relative to the observer.

In regards to symmetry, B sees the same thing when observing A: A's clock ticks more slowly than B's clock. Both A and B think that the other's clock is slow. This is not a paradox because A and B can never meet again to compare clocks "in person" unless one of them accelerates - in which case the accelerated clock is the slower of the two when they meet again.

Answer 2

They sure can.

Every moving frame has its own time perception, and if different people are in differently moving frames they perceive different time(s?).

That might even result in exterior event sequences being switched - for example, star A becomes a supernova before star B for one frame, and the other way around for another frame.

Don't try to apply common sense to relativity, it leads you wrong. Common sense developed in a non-relativistic environment.

Answer 3

No, they can't, not as you've formulated the problem. Spaceship B and clock B are at the same point in spacetime, so all frames will agree whether or not two events at the spaceship and the clock are simultaneous.

Both A and B will observe B's computer monitor changing from "no" to "yes" when clock B reads 1:00. I don't understand your symmetry argument for why B would see it change at 2:00.

Answer 4

The symmetry argument is invalid. Clock B shows 1:00 in all frames of reference when it reaches space station C, so there is a mistake in the question.