#### PeterDonis

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A while back I posted a black hole horizon puzzle. This is another puzzle in the same general spirit. It is based on the scenario described in this old PF thread:

https://www.physicsforums.com/threads/a-flaw-of-general-relativity.115418/

Needless to say, as was the case with my previous puzzle, this puzzle does not actually show a flaw in GR (as the OP of the old thread incorrectly claims). But it does provide a scenario that can be made into an interesting puzzle--at least interesting enough to me for me to do the work of putting this post together.

As I did with my previous puzzle, I'm going to restate things in my own words to make a clear statement of the scenario and the key question of the puzzle. We start with a scenario in flat spacetime:

(1) We have three buoys, floating in space, at rest relative to each other, and one million light years apart in the inertial frame in which they are all at rest. (If you like, you can think of one buoy as floating in space somewhere near Earth, one as floating in space somewhere in the Andromeda galaxy, and one as floating in space halfway between them; but in the flat spacetime version of this scenario, the Earth, and in fact the entire Milky Way galaxy, and the Andromeda galaxy, and anything else, have zero mass and zero gravity; they're just imaginary markers.)

(2) A rocket ship launches from buoy #1 at some acceleration; it passes buoy #2 after ten years of proper time according to the ship's clocks, and at that instant it turns around and has the same acceleration in the opposite direction; in another ten proper years it arrives at buoy #3 and is stopped.

(3) Consider the motion of buoy #2 according to the crew of the rocket during the second half of the trip. From their standpoint, there is a uniform gravitational field, in which the rocket is at rest due to its proper acceleration, and the buoy is free-falling upward in that field (and "decelerating due to gravity" because of that). So in the rocket's (non-inertial) frame, we have the buoy covering one million proper light-years in ten years of proper time.

I emphasize at this point that, so far, everything is straightforward SR. Even the part about "one million proper light years in ten proper years", i.e., a "proper speed" of 100,000 times the speed of light for the buoy, is fine, because the rocket frame is non-inertial and speeds don't have to be limited to c in a non-inertial frame. (For concreteness, this non-inertial frame is assumed to be Rindler coordinates; I'll leave as an exercise for the reader the verification that the second half of the trip can in fact be modeled using Rindler coordinates with appropriate numbers. One hint, though: in the standard spacetime diagram of Rindler coordinates, things look simplest if coordinate time ##t = 0## is the

Now consider the following parallel scenario in the curved spacetime surrounding a static, spherically symmetric massive body (but I'll be deliberately non-committal for now about just what kind of massive body or what its characteristics are):

(1) We have three buoys, all free-falling upward in the body's gravitational field, at rest relative to each other, and one million light years spacing between them.

(2) A rocket ship is hovering at rest in the planet's gravitational field, such that, at some instant, buoy #2 is just rising past it.

(3) Consider the motion of buoy #2 according to the crew of the rocket after that instant. They are at rest in a gravitational field and buoy #2 is free-falling upward in that field. But the equivalence principle says that this works the same as the rocket accelerating in flat spacetime and the buoys being at rest in an inertial frame in flat spacetime. So, according to the equivalence principle, it should be possible for the buoy to free-fall upward a million light-years from the massive body, in ten years of the ship's proper time.

The OP of the old thread I linked to above claims that, in fact, GR does not allow #3 just above to occur, which, if true, would mean there must be a problem with GR, since GR is supposed to uphold the EP. That claim is not correct, as I said above.

But the puzzle I want to pose here is simple: given that GR does, in fact, allow #3 just above to occur,

https://www.physicsforums.com/threads/a-flaw-of-general-relativity.115418/

Needless to say, as was the case with my previous puzzle, this puzzle does not actually show a flaw in GR (as the OP of the old thread incorrectly claims). But it does provide a scenario that can be made into an interesting puzzle--at least interesting enough to me for me to do the work of putting this post together.

As I did with my previous puzzle, I'm going to restate things in my own words to make a clear statement of the scenario and the key question of the puzzle. We start with a scenario in flat spacetime:

(1) We have three buoys, floating in space, at rest relative to each other, and one million light years apart in the inertial frame in which they are all at rest. (If you like, you can think of one buoy as floating in space somewhere near Earth, one as floating in space somewhere in the Andromeda galaxy, and one as floating in space halfway between them; but in the flat spacetime version of this scenario, the Earth, and in fact the entire Milky Way galaxy, and the Andromeda galaxy, and anything else, have zero mass and zero gravity; they're just imaginary markers.)

(2) A rocket ship launches from buoy #1 at some acceleration; it passes buoy #2 after ten years of proper time according to the ship's clocks, and at that instant it turns around and has the same acceleration in the opposite direction; in another ten proper years it arrives at buoy #3 and is stopped.

(3) Consider the motion of buoy #2 according to the crew of the rocket during the second half of the trip. From their standpoint, there is a uniform gravitational field, in which the rocket is at rest due to its proper acceleration, and the buoy is free-falling upward in that field (and "decelerating due to gravity" because of that). So in the rocket's (non-inertial) frame, we have the buoy covering one million proper light-years in ten years of proper time.

I emphasize at this point that, so far, everything is straightforward SR. Even the part about "one million proper light years in ten proper years", i.e., a "proper speed" of 100,000 times the speed of light for the buoy, is fine, because the rocket frame is non-inertial and speeds don't have to be limited to c in a non-inertial frame. (For concreteness, this non-inertial frame is assumed to be Rindler coordinates; I'll leave as an exercise for the reader the verification that the second half of the trip can in fact be modeled using Rindler coordinates with appropriate numbers. One hint, though: in the standard spacetime diagram of Rindler coordinates, things look simplest if coordinate time ##t = 0## is the

*end*of the second half of the trip.)Now consider the following parallel scenario in the curved spacetime surrounding a static, spherically symmetric massive body (but I'll be deliberately non-committal for now about just what kind of massive body or what its characteristics are):

(1) We have three buoys, all free-falling upward in the body's gravitational field, at rest relative to each other, and one million light years spacing between them.

(2) A rocket ship is hovering at rest in the planet's gravitational field, such that, at some instant, buoy #2 is just rising past it.

(3) Consider the motion of buoy #2 according to the crew of the rocket after that instant. They are at rest in a gravitational field and buoy #2 is free-falling upward in that field. But the equivalence principle says that this works the same as the rocket accelerating in flat spacetime and the buoys being at rest in an inertial frame in flat spacetime. So, according to the equivalence principle, it should be possible for the buoy to free-fall upward a million light-years from the massive body, in ten years of the ship's proper time.

The OP of the old thread I linked to above claims that, in fact, GR does not allow #3 just above to occur, which, if true, would mean there must be a problem with GR, since GR is supposed to uphold the EP. That claim is not correct, as I said above.

But the puzzle I want to pose here is simple: given that GR does, in fact, allow #3 just above to occur,

*how*can that happen?
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