# General Relativity Paradox - Different local times of two frames a constant distance apart

• Suppose there is a habitable star with a significantly large mass, and thus a huge gravitation field. It has a clock on it that ticks each local second. And it also has a mirror. This is Star A.
• Suppose there is another habitable star with a much smaller mass, also with a clock, called Star B.
• Finally, suppose that these two stars somehow maintain a fixed distance between them. (Eg: the two stars have perfectly calibrated rocket thrusters pointing toward one another).

Please correct me if I'm wrong: An observer on Star A looking through a telescope at clock B would see it ticking quickly. Likewise, an observer on Star B looking at the clock on Star A would see it ticking slowly.

Now, suppose a person on A sends a light pulse towards B and starts a clock. They measure it takes 10 seconds for it to come back.

Now, a person on B sends a pulse to A, and measures how long it takes to get back. Does it also take 10 seconds? If so, there's a pretty clear paradox. If not, how could it take light different times to travel the same distance?

Thanks

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Don't understand what you think is paradoxical here. You already established the two clocks do not tick at the same rate, therefore the observers measure different amounts of time. How is this in contradiction with anything? –  Muphrid Mar 15 '13 at 21:42
@Muphrid - If the signal takes the same amount of local time to bounce back and forth, A and B could synchronize their clocks with that signal. There would be no way for one clock to run faster than the other. Eg: There is no situation in which one observers clock runs fast enough to the point where it sends out two signals before the other observer reaches one. –  not-a-physicist Mar 18 '13 at 15:00
And in no way have I suggested that it does take the same amount of local time. –  Muphrid Mar 18 '13 at 15:33
Then the question is why A and B would measure different amounts of time for light taking a round trip across a fixed distance. –  not-a-physicist Mar 18 '13 at 16:21
@not-a-physicist you have to understand there is no such thing as a fixed distance. General relativity is a theory in which the very notion of distance and time is a dynamical variable that depends on the mass distribution of the universe. –  Mark Mitchison Mar 18 '13 at 16:23

## 1 Answer

Time is not the only measurement that is affected by a gravitational field. What makes you think that A and B measure the same distance between them? It helps to think about how you would actually measure the distance to a faraway object. If you are patient, you could do this measurement by bouncing a light beam off the object and seeing how long it takes to return, in exactly the manner you described already. (I have heard that this has actually been done to measure the distance to the Moon precisely.)

So let A bounce a light beam off B. It takes a time $2T$ to arrive back, therefore A thinks that B is at a distance $L = cT$, where $c$ is the speed of light.

Now let B do the same experiment. Of course, A measures the same time $2T$ for the beam to arrive and bounce back to B. But as you stated, if from A's point of view a time $2T$ has passed, from B's point of view a different time $2T^{\prime}$ has passed when the light returns. You know that the speed of light appears constant for all observers, so $B$ must measure a distance $L^{\prime} = cT^{\prime} \neq L$.

The physical meaning is that gravity is a warping of spacetime, not just space or time separately. The dilation of time measured by clocks due to any relativistic effect is exactly balanced by the contraction of metre rules (or yard sticks, or whatever) so that the speed of light is always constant.

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How could the stars' distance appear to be different to both A and B? What if A spools out a very stiff wire of some sort and measures it as it gets to B. Then once it gets there, B reels it in and measures it. Would it not match? Would this not be a valid way to measure distance? –  not-a-physicist Mar 18 '13 at 14:47
@not-a-physicist The short answer is no, they would not match, due to the different length contraction associated with the different gravitational fields at A and B. You seem quite happy to accept that time is different for different observers. The same is true for lengths –  Mark Mitchison Mar 18 '13 at 16:25