This question concerns a paradox I've come up with in GR. If somebody could tell me where it breaks down, I'd be much obliged. Thank you!(adsbygoogle = window.adsbygoogle || []).push({});

Assume there is spaceship A hovering above a blackhole, and spaceship B further away from a blackhole The blackhole, A, and B are all lined up. (The spaceships are not in orbits, but rather, they have super engines that counteract the gravitational pull of the blackhole such that they remain a constant distance from the blackhole, and each other. Think gravitational tugboat.)

Now, GR predicts that A and B see each other quite differently. Spaceship A should view B as blueshifted and aging quicker. Spaceship B would view A as being redshifted and aging slower.

Spaceship B fortunately comes equipped with a very long (highly durable) spool of thread. It's unspooled towards A and the physical length is recorded; let's say they are 9.5 x 10^15 meters apart, about light year.

However, there seems to be a paradox here. They are a constant distance away from each other, and light travels at c, so how is it they can be aging at different rates? They should all age at the same rate! I'll exploit this paradox to make it more clear:

We can construct a time clocking using A and B as mirrors, and the knowledge that they are 1 light year apart.

A sends out a message saying: "I am 0 years old."

B receives this message and says: "I am 0 years old."

A receives this and says: "I am 2 years old." Since, if you were onboard B, you'd have waited 1 year for the signal to reach A, and 1 year for it to return.

B receives this and says: "I am 2 years old." Since, if you were onboard A, you'd have waited 1 year for the signal to reach B, and 1 year for it to return.

... etc

Neither of the spaceships will see the other aging quicker... yet somehow GR says they will. I can only imagine this being true if light somehow travels faster in one direction than in the other.

Now, I asked this to a very bright professor and after a bit of thought, he told me the problem reduces to measuring distance between two accelerating bodies. He remarked that you cannot measure distance between two accelerating frames of reference, because they appear to be receding away from each other (or something like that). He said it would be impossible to hold a piece of string between the two accelerating bodies. Why the hell is that true?

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# Time Dilation Paradox (variant distance between two accelerating bodies)

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