General Relativity rocket puzzle

[PeterDonis]

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 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?

 

[PAllen]

In the first bullet (2) I assume you mean arrives at buoy #3 and is stopped, rather than buoy #2, as stated.

 

[PeterDonis]

In the first bullet (2) I assume you mean arrives at buoy #3 and is stopped, rather than buoy #2, as stated

You're right. Fixed now, thanks for catching that!

 

[PAllen]

Before giving a real answer, I’ll give a trivial answer. The flat spacetime case is a GR solution.

 

[PeterDonis]

The flat spacetime case is a GR solution.

Yes, it is. But there is also at least one GR solution in a curved spacetime. That's the one I'm looking for.

 

[PAllen]

Yes, it is. But there is also at least one GR solution in a curved spacetime. That's the one I'm looking for.

I know that’s what you are looking for, I just had to be facetious first.

 

[PeterDonis]

I know that’s what you are looking for, I just had to be facetious first.

Fair enough.

 

[PAllen]

I assume you mean that for the curved spacetime case, all 3 buoys must be accounted for? Just looking at #2 in relation to the rocket seems completely trivial to account for in SC geometry. Also, I assume you mean to consider only phase 2 of the flat spacetime case for a curved case, i.e. you are not looking for reversal of rocket proper acceleration?

 

[PeterDonis]

I assume you mean that for the curved spacetime case, all 3 buoys must be accounted for?

Yes. But see below.

I assume you mean to consider only phase 2 of the flat spacetime case for a curved case, i.e. you are not looking for reversal of rocket proper acceleration?

Yes. Which means that, really, buoy #1 is not important (though you can certainly include it). The important buoys are buoy #2, which is co-located with the rocket (and moving upward at very nearly the speed of light relative to it) at the start of phase 2, and buoy #3, which is co-located with the rocket (and at rest relative to it) at the end of phase 2.

 

[PAllen]

So it seems to me the biggest challenge here is define a reasonable meaning in curved spacetime for two world lines to being far apart and mutually at rest in curved spacetime. There is clearly no unique, objective way to do this, so it comes down to defending that some choice is reasonable for some specific manifold.

 

[PeterDonis]

the biggest challenge here is define a reasonable meaning in curved spacetime for two world lines to being far apart and mutually at rest in curved spacetime

For a general curved spacetime, that's true, but for this puzzle, you can assume that the massive body is static and spherically symmetric, which means there is a timelike Killing vector field that is hypersurface orthogonal, so there is an obvious choice for how to address the challenge you pose. I did not intend for that challenge to be a factor in the solution of the puzzle.

 

[PeterDonis]

for this puzzle, you can assume that the massive body is static and spherically symmetric

I've edited the OP to add this clarification.

 

[PAllen]

For a general curved spacetime, that's true, but for this puzzle, you can assume that the massive body is static and spherically symmetric, which means there is a timelike Killing vector field that is hypersurface orthogonal, so there is an obvious choice for how to address the challenge you pose. I did not intend for that challenge to be a factor in the solution of the puzzle.

But the buoys cannot be stationary, in this sense, as they must be geodesics, while the rocket is stationary (and stationary world lines are not geodesics). Note that in flat spacetime both the buoys and Rindler observers are integral curves of timelike killing fields, because there are many in flat spacetime. This cannot be the case in SC geometry, so it seems some plausible choice must be defended for the buoy world lines and the definition of distance between them.

 

[PAllen]

Is it possible that you do not mean to carry over to the curved case any requirement that the distance between the buoys is constant per each other?

Instead, you only care that after 10 years per the rocket, the distance per the rocket (using stationary congruence distance) between the buoys is a million light years, and that buoy #3 at this moment is coincident with the rocket and has the same 4 velocity?

 

[PeterDonis]

Is it possible that you do not mean to carry over to the curved case any requirement that the distance between the buoys is constant per each other?

No. The curved case must have every specific detail the same as the flat case. The whole point is to see how the equivalence principle is satisfied in the curved case, and that means finding a curved spacetime that has a region that can be considered flat at least to enough of an extent that allows you to set up the same scenario that is set up in the flat case.

 

[PAllen]

No. The curved case must have every specific detail the same as the flat case. The whole point is to see how the equivalence principle is satisfied in the curved case, and that means finding a curved spacetime that has a region that can be considered flat at least to enough of an extent that allows you to set up the same scenario that is set up in the flat case.

Well, then another silly solution is inside a 10 million light year spherical shell.

 

[PeterDonis]

another silly solution is inside a 10 million light year spherical shell

As you are no doubt aware, this is also not the solution I'm looking for. There really is an actual curved solution.

 

[PAllen]

No. The curved case must have every specific detail the same as the flat case. The whole point is to see how the equivalence principle is satisfied in the curved case, and that means finding a curved spacetime that has a region that can be considered flat at least to enough of an extent that allows you to set up the same scenario that is set up in the flat case.

Ok, now you’ve added the weasel word “flat enough”. So now we have a notion of an exact Kvf defining distance per the rocket, and some approximate kvf defining mutual distance per the buoys.

Well, then just be sufficiently far from an ultra massive BH.

 

[PeterDonis]

now you’ve added the weasel word “flat enough”

That just means "local inertial frame". I'm not trying to make this difficult; I'm just describing the equivalence principle.

So now we have a notion of an exact Kvf defining distance per the rocket, and some approximate kvf defining mutual distance per the buoys.

Yes. Within the confines of the local inertial frame, these would correspond to the Rindler Kvf and the Minkowski (inertial frame) Kvf, respectively.

then just be sufficiently far from an ultra massive BH

Can you estimate how massive the BH would have to be? Also, can you calculate the proper acceleration the rocket has to have? (The second actually comes before the first, logically speaking.)

 

[PAllen]

The proper acceleration required I get as a little less than 1.5 g. This acceleration exists for any BH at some r. So then one needs a criterion for r and plus 1 million ly having minimal radial tidal gravity, and finding the minimum M such that r for 1.5 g meets this criterion. I might work on this soon, if no one else does.

 

[PeterDonis]

The proper acceleration required I get as a little less than 1.5 g.

Yes, that's what I get too. I suspect that that is what misled the poster of that old thread into thinking that a "planet" would have to have a local patch in the spacetime around it in which the scenario could be realized, which is of course not possible.

So then one needs a criterion for r and plus 1 million ly having minimal radial tidal gravity

There is also a criterion for r minus some distance having minimal radial tidal gravity.

 

[stevendaryl]

(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?

Just a comment: the equivalence principle only applies to situations where the variation of the apparent "acceleration due to gravity" can be ignored. If you pay attention to how $g$ varies with position, then you can distinguish between gravity due to massive bodies and artificial gravity due to accelerating in empty space. So taking that into mind, does your paradox still work if you assume that everything takes place in a small enough region that the variation of gravity is negligible?

 

[1977ub]

(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.

Leaves 1, passes 2 and on to 3 - or else leaves 1, turns around at 2 and back to 1 ? What am I missing ?

 

[Ibix]

Leaves 1, passes 2 and on to 3 - or else leaves 1, turns around at 2 and back to 1 ? What am I missing ?

That made me blink, too. Turnover might have been better wording than turnaround. The rocket accelerates from 1 to 2 then decelerates to 3.

 

[PeterDonis]

the equivalence principle only applies to situations where the variation of the apparent "acceleration due to gravity" can be ignored.

More precisely, where that variation cannot be distinguished from the variation it would have in flat spacetime in a family of Rindler observers.

does your paradox still work if you assume that everything takes place in a small enough region that the variation of gravity is negligible?

You're thinking of it backwards. The point of the puzzle is to find a curved spacetime in which there is a region large enough to fit the entire scenario as described in the flat spacetime version, and in which the variation of gravity is still indistinguishable from what it would be for a family of Rindler observers in flat spacetime (or, to put it in more conventional terms, tidal gravity must be negligible over the entire region, which must be large enough to fit the entire scenario as given in the flat spacetime version).

 

[PeterDonis]

Leaves 1, passes 2 and on to 3 - or else leaves 1, turns around at 2 and back to 1 ? What am I missing ?

Leaves 1, accelerates in the direction of 3 until it reaches 2, then turns around and decelerates until it reaches 3. When it passes 2 it's not at rest; it's traveling at almost the speed of light in the direction of 3 (its gamma factor, relative to the rest frame of the buoys, when it passes 2 is about a million).

 

[A.T.]

Now consider the following parallel scenario in the curved spacetime surrounding a static, spherically symmetric massive body

The point of the puzzle is to find a curved spacetime in which there is a region large enough to fit the entire scenario as described in the flat spacetime version, and in which the variation of gravity is still indistinguishable from what it would be for a family of Rindler observers in flat spacetime (or, to put it in more conventional terms, tidal gravity must be negligible over the entire region, which must be large enough to fit the entire scenario as given in the flat spacetime version).

Is the point to make the massive body very big, so its exterior tidal effects over the given region become negligible? Or are you looking for a mass distribution, within which the whole scenario takes place (like a nebula)?

 

[jartsa]

(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.

Well I guess the deceleration of buoy #2 must be very small. Obviously the buoy must lose kinetic energy as it climbs, but that loss of energy should correspond to a very small loss of speed. And that is the case if the buoy is hyper-relativistic.

Now this hyper-relativistic buoy behaves almost the same way as light - the coordinate speed of which increases as it climbs - halving of energy of light corresponds to doubling of coordinate speed of light.

 

[PeterDonis]

Is the point to make the massive body very big, so its exterior tidal effects over the given region become negligible?

Yes; the question is "what kind of massive body" and "how big"? @PAllen has already given one possible answer to the first question.

Or are you looking for a mass distribution, within which the whole scenario takes place (like a nebula)?

No; the flat spacetime scenario takes place in vacuum, so the curved spacetime version does as well.

 

[PeterDonis]

I guess the deceleration of buoy #2 must be very small.

You don't have to guess; you can calculate the coordinate deceleration (note that the proper deceleration of all the buoys is zero; they are in free fall) in the flat spacetime case, and then just apply the EP to say that it is the same in the curved spacetime case.

Obviously the buoy must lose kinetic energy as it climbs, but that loss of energy should correspond to a very small loss of speed.

It will initially, but not through the entire scenario; at the end of the scenario, all of the buoys are at rest relative to the rocket.

halving of energy of light corresponds to doubling of coordinate speed of light.

I don't know where you're getting this from, but it's wrong. In any case, the buoys do not behave like light; they can't, since they're moving on timelike worldlines and light moves on null worldlines.

 

[PAllen]

Hopefully my rough math is good (only order of magnitude calculations are sensible, to me, for a scenario like this).

First, a general comment. If one requires and local inertial frame (LIF) in curved spacetime to have a flatness criteria which prevents build up of large effects over all of its dimensions, then the degree of flatness required (as a local measure) increases with increased LIF size. In particular, I find that doubling the LIF size doubles the required r coordinate for e.g. the center of the LIF. However, keeping the proper acceleration of a stationary observer the same when doubling r, requires quadrupling the the mass. But Schwarzschild radius is directly proportional to mass. So there must come a time when the scenario is no longer achievable, as you increase the LIF size (far any reasonably chosen flatness criterions) - because you will eventually be unable to have a stationary observer.

For the numbers I ran for this case, it doesn't work already with an, IMO, inadequate definition of flatness. What I picked, for simplicity, is simply that over the whole range of r (of rocket holding stationary world line) - million light years, to r + million light years, that change in proper acceleration of a stationary observer is no more than 1%. This is sort of ok from a spatial point of view, but not temporal. With this criteria, by the end of the process under discussion, the relative speed of the buoys will be large, due to how long this small acceleration difference will have acted. If one tightens the criteria a lot, it will be unachievable for the reason above.

So, I get that we need r of 400 million ly to achieve the modest flatness criterion, and then to have 1.5g proper acceleration of a stationary observer at this distance requires nearly 2*10²⁸ solar masses. This will have a Schwarzschild radius of about 6*10¹⁵ ly. Which of course means, you are out of luck.

So, assuming there is a valid solution, it appears it is not just 'getting far enough from a big BH', it must be something else.

 

[PeterDonis]

the degree of flatness required (as a local measure) increases with increased LIF size.

This is true, but "degree of flatness" is simply "range over which the Riemann tensor components are acceptably small". That's not the same as "change in proper acceleration of a stationary observer". See below.

What I picked, for simplicity, is simply that over the whole rang of r (of rocket holding stationary world line) - million light years, to r + million light years, that change in proper acceleration of a stationary observer is no more than 1%.

This can't be a correct criterion of flatness; it's much too strong, since even the flat spacetime scenario does not satisfy it. (Just compute the change in proper acceleration between two Rindler observers, one in the rocket as given in the scenario and one a million light-years above him, who is co-located with buoy #2 at the end of the scenario.)

Also, remember that the coordinate $r$ is not the same as radial distance, and since we must allow for the possibility that the LIF in question is close to the black hole's horizon, it might not even be approximately the same.

 

[1977ub]

Leaves 1, accelerates in the direction of 3 until it reaches 2, then turns around and decelerates until it reaches 3. When it passes 2 it's not at rest; it's traveling at almost the speed of light in the direction of 3 (its gamma factor, relative to the rest frame of the buoys, when it passes 2 is about a million).

I am simply not familiar with this usage of the phrase "turn around".

 

[PeterDonis]

I am simply not familiar with this usage of the phrase "turn around".

I was describing what the ship has to do to switch from accelerating to decelerating: it has to turn around from its nose pointing towards buoy #3 to its tail pointing towards buoy #3. But I can see how the phrase could be ambiguous. I have seen this usage in plenty of sci-fi stories (for example, Larry Niven's stories involving ramscoop ships), but I admit I have not seen it in the scientific literature.

 

[jartsa]

You don't have to guess; you can calculate the coordinate deceleration (note that the proper deceleration of all the buoys is zero; they are in free fall) in the flat spacetime case, and then just apply the EP to say that it is the same in the curved spacetime case.

Oh yes ... when the buoy sees the rocket passing by, the rocket's coordinate acceleration is its proper acceleration / gamma^3, according to the buoy. And the rocket crew says the same thing about the buoy.

And the gamma is 1600000 according to the relativistic rocket formula, if the proper acceleration is 1.5 g. So I guessed right, the coordinate acceleration of the buoy is very small at that point.

The speed of the rocket stays close to c most of the million light years distance, according to the buoy, so the rocket crew says the same thing about the buoy. So the crew could say that the buoy moves approximately like light, except for just a very short time at the end of the journey.

http://www.math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html