Time Dilation Paradox (variant distance between two accelerating bodies)

[quertying]

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!

Assume there is spaceship A hovering above a black hole, and spaceship B further away from a black hole The black hole, 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 black hole such that they remain a constant distance from the black hole, 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?

 

[Ich]

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?

If both are aging at different rates, and if both say that c= measured distance / measured time, then c has not the same value for both. Everything consistent again.
c=const. is valid only if you don't mean such large distances that time dilation becomes important.

He said it would be impossible to hold a piece of string between the two accelerating bodies.

It isn't true generally. But you have to fine-tune the accelerations such that the distance between the bodies does not change. The interesting point (and what your professor apparently meant) is that the bodies have to go at different accelerations to make the distance constant. Has to do with time dilatation as well.

 

[quertying]

If both are aging at different rates, and if both say that c= measured distance / measured time, then c has not the same value for both. Everything consistent again.
c=const. is valid only if you don't mean such large distances that time dilation becomes important.

The measured distance, in this case, is constant... it's the length of string between the two spaceships. And the speed of light is constant as well -- from either reference frame its c. Thus, from either reference frame (spaceship A or B) it will take 2 years for the light to make a round trip, and both will thus age at the same rate.

It isn't true generally. But you have to fine-tune the accelerations such that the distance between the bodies does not change. The interesting point (and what your professor apparently meant) is that the bodies have to go at different accelerations to make the distance constant. Has to do with time dilatation as well.

Could you explain this in more detail? In the example given, would it be impossible to hold a string between the two spaceships? If so, why?

 

[Ich]

And the speed of light is constant as well -- from either reference frame its c.

No.

Could you explain this in more detail? In the example given, would it be impossible to hold a string between the two spaceships? If so, why?

In your example, it is possible to hold a string between the ships, because their distance is constant. Try http://en.wikipedia.org/wiki/Bell_spaceship_paradox" .

 

[JesseM]

The measured distance, in this case, is constant... it's the length of string between the two spaceships. And the speed of light is constant as well -- from either reference frame its c.

What reference frames are the ships using? For a nonrotating black hole it's most common to use Schwarzschild coordinates, you wouldn't have separate "frames" for observers at different distances in this case. And the speed of light is not constant in Schwarzschild coordinates, this is only supposed to be true in inertial coordinate systems (which are normally used in flat SR spacetime, although you can have a "locally inertial" coordinate system in an infinitesimally small region of curved spacetime thanks to the http://www.aei.mpg.de/einsteinOnline/en/spotlights/equivalence_principle/index.html ...any coordinate system large enough to cover observers at different radii from a black hole would not be inertial though).

 

[sylas]

The measured distance, in this case, is constant... it's the length of string between the two spaceships. And the speed of light is constant as well -- from either reference frame its c. Thus, from either reference frame (spaceship A or B) it will take 2 years for the light to make a round trip, and both will thus age at the same rate.

The measured distance is constant over time, for either ship; but it is not the same value for each ship.

The speed of light is "c" if measured locally; but it will not be "c" if measured as the time it takes for a light signal to travel between the two ships and back. The length of the string is also different. Values such as the speed of light between the two ships, and the distance between the two ships, depend on what co-ordinate system you use.

Could you explain this in more detail? In the example given, would it be impossible to hold a string between the two spaceships? If so, why?

It IS possible to hold a string between the two space ships in the problem you have described; but it's possible your professor was describing different cases. You would need to quote him exactly; the first post makes it not entirely clear what situation you described for the professor.

Cheers -- sylas

 

[Ich]

The speed of light is "c" if measured locally; but it will not be "c" if measured as the time it takes for a light signal to travel between the two ships and back. The length of the string is also different.

I can think of only one partly sensible definition of "distance" that depends on direction: radar distance. But there the speed of light would be "c", per definition.
Generally, in static spacetimes, one would choose a distance measure that gives one and only one value for any path in space. Like Schwarzschild r, or proper distance, or coordinate light travel time.

 

[sylas]

I can think of only one partly sensible definition of "distance" that depends on direction: radar distance. But there the speed of light would be "c", per definition.
Generally, in static spacetimes, one would choose a distance measure that gives one and only one value for any path in space. Like Schwarzschild r, or proper distance, or coordinate light travel time.

Thanks for the correction. That makes sense.

Cheers -- sylas

 

[Dale]

From the OP I have a hard time understanding what the paradox is supposed to be. Both observers agree that light follows null geodesics. Both observers agree on the proper time along each observer's worldline between any two events. Both observers agree on the proper length along any spacelike line you might choose to call the length of the string. Everything else would be a coordinate-dependent quantity that they are free to disagree on. So what is the paradox?