Is acceleration absolute or relative - revisited

[Dale]

]

Changes in different reference frames (example velocity) = relative
Same in different reference frames (example charge on an object) = invarient (absolute)

Yes, that is correct.

Velocity is relative. Are you able to say if acceleration (of the proper type) is relative or invariant (absolute) ?

Proper acceleration is invariant. That is precisely why it is of interest to physicists.

 

[PeroK]

Then returning to my original post, having determined that proper acceleration is invariant (and accepting the postulates of SR/GR), we can deduce there would be no observed difference whether you consider the bucket is rotating in a static universe, or a static bucket is in a rotating universe. Because the acceleration must be the same (equivalent) in both reference frames

No. If you get on a merry-go-round, you have an invariant proper centripetal acceleration and someone watching does not: you can tell you are "really" rotating; not the rest of the universe.

The alternative, Mach's principle, is not SR/GR. You need something like Brans-Dicke theory:

https://en.wikipedia.org/wiki/Brans–Dicke_theory

 

[Peter Leeves]

you can tell you are "really" rotating; not the rest of the universe.

I have to respectfully disagree. Not because I'm being obnoxious, but because logic tells me you cannot say with certainty you are really rotating. All the forces you experience would be identical if the merry-go-round was considered stationary and the universe rotated around it.

What you're effectively saying is if you were in an enclosed lift freefalling towards the earth, somehow you'd still "know" or be able to figure out that you weren't really in instellar space with no forces acting on you or the lift. Logic says there's no way to tell the difference, because they are directly equivalent. And it's the same with the rotating merry-go-round scenario. There's simply no way to tell, because all observations and measurments would be identical.

 

[hmmm27]

Then returning to my original post, having determined that proper acceleration is invariant (and accepting the postulates of SR/GR), we can deduce there would be no observed difference whether you consider the bucket is rotating in a static universe, or a static bucket is in a rotating universe. Because the acceleration must be the same (equivalent) in both reference frames

Well, the surface of the water is trough-shaped in the bucket-rotation model ("climbing the walls"), as opposed to dome-shaped in a stationary-on-Earth model, or actually flat if it's just hanging around in space, or under straight-line acceleration.

 

[PeroK]

All the forces you experience would be identical if the merry-go-round was considered stationary and the universe rotated around it.

Not in the theory of SR/GR it wouldn't. If you get on a trampoline, you feel the force of acceleration/deceleration. Someone not on the trampoline doesn't feel those forces in your reference frame.

You can consider the universe as bouncing up and down, but that doesn't create real forces. For example, if you construct something that breaks under a given acceleration and you jump up and down, it doesn't break. Whatever frame of reference you adopt.

Or, you have a bottle with separated fluids. You shake the bottle the fluids mix. You put the bottle on a table and you jump up and down, the fluids don't mix.

You know when the bottle is "really" being shaken. Being shaken up and down and fluids mixing is not relative; it's invariant.

 

[Peter Leeves]

Well, the surface of the water is trough-shaped in the bucket-rotation model ("climbing the walls"), as opposed to dome-shaped in a stationary-on-Earth model, or actually flat if it's just hanging around in space, or under straight-line acceleration.

I suspect you may not have read the entire thread. I'm not going to repeat large parts of it because it's already there. But I will just say there is no reason the water would be dome shaped as you suggest. If the universe rotated around the bucket the water would form an identical trough shape.

 

[PeroK]

If the universe rotated around the bucket the water would form an identical trough shape.

Can you justify that statement? Let's see the mathematics.

 

[Nugatory]

Then returning to my original post, having determined that proper acceleration is invariant (and accepting the postulates of SR/GR), we can deduce there would be no observed difference whether you consider the bucket is rotating in a static universe, or a static bucket is in a rotating universe. Because the acceleration must be the same (equivalent) in both reference frames

It is not necessarily the case that the two situations are equivalent. The GR calculation that predicts the curvature of the water surface is completely local: a bucket floating in space far from the rest of the universe is surrounded by a region of flat space; we apply a torque to it by firing up a motor attached to the bucket; we calculate the motion of the water relative to the rim of the bucket (strictly speaking, we calculate the invariant geodesic deviation of the water and the rim); we get the curved surface.

This is a different physical situation than spinning the rest of universe while not turning on the motor. There’s substantial intuitive appeal to the idea that the effect should be the same... but that’s not the same as a proof, let alone a proof backed up by observations.

 

[Peter Leeves]

Not in the theory of SR/GR it wouldn't. If you get on a trampoline, you feel the force of acceleration/deceleration. Someone not on the trampoline doesn't feel those forces in your reference frame.

You can consider the universe as bouncing up and down, but that doesn't create real forces. For example, if you construct something that breaks under a given acceleration and you jump up and down, it doesn't break. Whatever frame of reference you adopt.

Or, you have a bottle with separated fluids. You shake the bottle the fluids mix. You put the bottle on a table and you jump up and down, the fluids don't mix.

You know when the bottle is "really" being shaken. Being shaken up and down and fluids mixing is not relative; it's invariant.

I'm entirely happy to be wrong. But I've yet to see a argument that convinces me I am. The trampoline (and you) would feel those exact same forces if you were static and the universe bounced up and down.

The reason the fluids don't mix is because the bottle moves with the rest of the universe. Only you remain static and detect the emerging gravitational field which then bounces up and down. The bottle see no emerging gravitational field and therefore remains static relative to the rest of the universe and is not therefore shaken up and down.

 

[PeroK]

The trampoline (and you) would feel those exact same forces if you were static and the universe bounced up and down.

It's up to you to produce the mathematical justification that this is the case. What does the universe bouncing up and down even mean? That's mysticism (*), not empirical physics.

Mysticism: vague speculation : a belief without sound basis

 

[Peter Leeves]

It's up to you to produce the mathematical justification that this is the case. What does the universe bouncing up and down even mean? That's mystisicm, not empirical physics.

Maths isn't necessarily required if you can sufficiently describe a premise and it is logically consistent.

The spinning bucket scenario (and the trampoline and the merry-go-round) are completely analagous to the oft-used example in Physics of man in a lift freefalling towards earth. He can't tell whether he's freefalling to Earth or in interstellar space with no forces acting on him - because the two scenarios are completely equivalent. I have merely turned that example from linear to rotational. It wasn't me that used bouncing up and down on a trampoline to try and prove me wrong, lol. But yes, it's EQUIVALENT to a bouncing universe. There's no way you can say one version is more real than the other. They are equivalent.

 

[PeroK]

It's completely analagous to the oft-used example of man in a lift freefalling towards earth. He can't tell whether he's freefalling to Earth or in interstellar space with no forces acting on him - because the two scenarios are completely equivalent. I have merely turned that example from linear to rotational. It wasn't me that used bouncing up and down on a trampoline to prove me wrong. But yes, it's EQUIVALENT to a bouncing universe.

That's nonsense, because the equivalence principle can be empirically tested. The bouncing universe is meaningless metaphysics.

 

[hmmm27]

I suspect you may not have read the entire thread. I'm not going to repeat large parts of it because it's already there.

Guilty : I saw your first post where you were going on about water climbing up the walls, texted a reply, then went away for awhile. When I came back I saw that y'all were going on about SR/GR. Read it, didn't really grok it.

But I will just say there is no reason the water would be dome shaped as you suggest. If the universe rotated around the bucket the water would form an identical trough shape.

By "dome shaped" I mean a spherical section (ie: dome) ; of course the bucket has to be big enough to be able to eyeball an horizon (because that's what it is) if an unaided human measurement is required. What shape is the surface of the ocean ? the surface of a lake ? ... the surface of a bucket of water sitting on the ground ?

Which pales in comparison to the gymnastics you're going to have to go through to justify "trough shaped" with a "rotating universe" model.

 

[Nugatory]

The trampoline (and you) would feel those exact same forces if you were static and the universe bounced up and down.

As with the rotating bucket, that is an unproven assertion. General relativity allows us to calculate the forces between the bouncer the and the trampoline, and the trajectory of the bouncer relative to the trampoline just by calculating their paths through the local and essentially flat spacetime around them. The results are the same even if the universe were empty aside from the the bouncer and the trampoline; the GR calculation just doesn’t care about the rest of the universe, only the local spacetime.

Sadly, we don’t have access to a spare empty universe in which we can set up the experiment. Thus, we have no way of knowing whether the universe really is as local and non-Machian as this naive application of GR suggests, or whether the local geodesic-following behavior of matter is related to the large-scale non-emptiness of the universe.

 

[Peter Leeves]

That's nonsense, because the equivalence principle can be empirically tested. The bouncing universe is meaningless metaphysics.

The equivalence principle doesn't need to be empirically tested (IMHO). The man in an enclosed lift in freefalling towards Earth is logically consitant with a man in instellar space who isn't subject to any forces. I don't think it's necessary to put a man in an enclosed lift and drop him from a height. I'm comfortable that in freefall an accelerometer would read zero. I'm equally comfortable I don't need to put someone in interstella space with an accelerometer to know if no forces are applied it will also read zero.

 

[Peter Leeves]

Which pales in comparison to the gymnastics you're going to have to go through to validate "trough shaped" with a "rotating universe" model.

I can sense a certain reluctance in you to go back and read the whole thread ;)

We've clarified that proper acceleration is invariant (the same in all reference frames). I propose that the ref frame in which the universe rotates around a static bucket is directly equivalent to the ref frame where the bucket rotates in a static universe.

Many people seem to assert if the bucket isn't rotating then there's no way for the water to be forced out towards the circumerence and thus climb up the bucket. I think the water will be forced towards the circumference and here's why. When the bucket (and water) is released, the entire universe begins to accelerate rotationally. A new rotating gravitational field emerges which is only seen by the bucket/water system and is only present while acceleration/decelerating is occurring. This new gravitational field is rotating and due to the distortion of space-time (frame dragging) the water and the bucket are influenced in the very same way as if the bucket was spining and the universe was static. I note that the proper acceleration is invarient and remains identical no matter what reference frame it's in. That's it in a nutshell. It's important to note that the new rotating gravitational field is only felt by the static bucket/water (everything else is moving with the rotating universe), and only for the duration of the acceleration/deceleration. It's the rotational analogy of the man in the enclosed lift in freefall towards Earth being directly equivalent to a man in insterstella space with no forces acting on him.

 

[Nugatory]

The bouncing universe is meaningless metaphysics.

There are some subtleties here.

If by “bouncing universe” we mean a universe that is empty except for the trampoline and person, then it is clearly meaningless - there’s nothing to bounce.

If we mean a universe in which he have arranged to apply an oscillating force to every single particle in all the matter in the universe except the trampoline and the bouncer, that’s obviously not a realizable experiment but we might consider ourselves able in principle to calculate the effects. But there is a catch: there will be reaction forces so it’s not clear that a non-empty universe can be bounced in this sense.

So either way, considering a bouncing universe isn’t going to help us any more than considering the bucket and the rotating universe - and “meaningless metaphysics” is a pretty good two-word summary for the tl;dr crowd.

The only way I can see of settling the question of whether the universe is more Machian than GR suggests is to find an alternative to GR that: is Machian; agrees with all the experimentally confirmed predictions of GR; and makes some local prediction that disagrees with GR. Absent such a candidate theory, the discussion is somewhat sterile (and tends to provoke impatience and irritability in those of us who have been down this rabbit hole repeatedly)

 

[Nugatory]

I propose the water WILL be forced towards the circumference and here's why...

Please be mindful of the forum rule about personal theories.

 

[Peter Leeves]

But there is a catch: there will be reaction forces so it’s not clear that a non-empty universe can be bounced in this sense.

Negative. Only the static body (bucket/water) feels the influence of the emerging rotational gravitational field and only for the duration of the acceleration/deceleration. Everything else in the universe remains in sync and hence no reation forces.

It might be rotation that makes it more difficult to see. Let's change to linear.

Say you have a large spaceship (on the left) and a small spaceship (on the right) stationary with respect to each other in interstella space with no outside influences. The small ship fires it's rocket and accelerates away from the big ship. I hope we could all agree in the absence of any external reference points, it's equivalent to say the small ship fires it's rocket and the big rocket moves off to the left. I understand that it seems a bit nonsensical. But it's true to say it is equivalent and equally applicable from the perspective of the people in both spaceships.

It's intuitively easier to see things from the first perspective. It just seems to make more sense to say if the small ship fires his rocket then surely it's him that really moves. Maybe so. But that doesn't mean we aren't entitled to consider if there's an equivalent viewpoint. That is, the small ship fires his rocket and the big ship moves away. But that does leave the question, why would the big ship move away even though it's the small ship firing a rocket ? Well, the logic of equivalence says that the small ship must be firing his rocket to remain stationary. But why should he need to fire his rocket to stay still ? It can only be if firing the rocket generates a new gravitational field (linear this time) that is pushing the entire universe (including his chum in the big rocket) off to the left. This gravitational field only emerges for the duration of the rocket firing. Soon as the rocket stops, the gravitational field dies away. The influence of this temporary gravitational field only apply to the static small spaceship (everything else in the universe is being pushed to the left) and is the same magnitude precisely to the the first scenario (small ship firing rocket and moving to the right).

Now swap "big spaceship" for "universe" and rotation instead of linear. You have the Newton's bucket scenario.

I'm not saying this is correct. I'm only saying I can follow the reasoning and is appears to be logically consistent. I guess I'm hoping someone will either agree, or I'd be equally happy to hear an explanation that genuinely says nope, that's wrong. And here's the reason why it can't be right.

 

[Dale]

The equivalence principle doesn't need to be empirically tested (IMHO)

Nevertheless, it has been tested extensively.

Then returning to my original post, having determined that proper acceleration is invariant (and accepting the postulates of SR/GR), we can deduce there would be no observed difference whether you consider the bucket is rotating in a static universe, or a static bucket is in a rotating universe. Because the acceleration must be the same (equivalent) in both reference frames

So, the invariant fact is that if the accelerometers detect (invariant) acceleration then the surface will be curved, and if the accelerometers do not detect acceleration then the surface will be flat. That is invariant and is true in any coordinate system. So proper acceleration is not relative.

Focusing on the scenario where the surface is curved, you can describe that in inertial coordinates or in (non inertial) coordinates where the bucket is stationary. In the inertial coordinates the bucket is undergoing coordinate acceleration (the universe is not) and in the co-moving coordinates the bucket is not undergoing coordinate acceleration (the universe is). So coordinate acceleration is relative.

Many people seem to assert if the bucket isn't rotating then there's no way for the water to be forced out towards the circumerence and thus climb up the bucket.

Here you are using the word “rotating” without specifying if you are talking about “proper” or “coordinate”. That is likely the source of the confusion.

 

[hmmm27]

I can sense a certain reluctance in you to go back and read the whole thread ;)

Not even slightly tempted - I hit up Wikipedia for "frame dragging", that was it ; maybe a few years from now.

We've clarified It was clarified for me that proper acceleration is invariant (the same in all reference frames). I propose It seems to me that the ref frame in which the universe rotates around a static bucket is directly equivalent to the ref frame where the bucket rotates in a static universe.

"The universe" is a bit big for me... how about we hollow out a small chamber in the center of the Earth and place a blob of water in the middle.

Spin the (spherical) blob and it flattens due to centrifugal force ; spin the Earth (lots) and the blob elongates (a tiny bit) thanks to frame-dragging. So, not the same... at least for that reasonably well defined scenario.

This is the part where Dale comes in and tells me that's not how frame-dragging works , and I have to go back and rethink it for awhile.

 

[Dale]

Spin the (spherical) blob and it flattens due to centrifugal force ; spin the Earth (lots) and the blob elongates (a tiny bit) thanks to frame-dragging. So, not the same... at least for that reasonably well defined scenario.

This is the part where Dale comes in and tells me that's not how frame-dragging works , and I have to go back and rethink it for awhile.

On the contrary, this is an excellent example. I really liked it.

For @Peter Leeves when @hmmm27 says “spin” above they are referring to the invariant situation where an accelerometer attached to the spinning object detects the rotation. In other words, proper rotation.

 

[reinhard55]

What happens if you have two buckets rotating in different directions? Or, a rotating bucket on a rotating Earth?

Which problem do you see then?

 

[PeroK]

Which problem do you see then?

I think I've had enough of metaphysics for now.

 

[PeterDonis]

I propose that the ref frame in which the universe rotates around a static bucket is directly equivalent to the ref frame where the bucket rotates in a static universe.

You can make this claim, but you have to be very careful about what it is and is not asserting.

Your claim is asserting that all coordinate charts are equivalent in GR. You can pick any coordinates you want to describe physics. In particular, you can pick coordinates in which the universe is at rest (actually our universe is expanding and parts of it are moving relative to other parts, but we'll ignore those complications for this discussion, they don't change the main point) and the bucket is rotating, or coordinates in which the bucket is at rest and the universe is rotating. Both coordinate charts will let you compute whatever physical quantities you like, and both will give the same answers for all invariants, such as the proper acceleration of a particular small parcel of water in the bucket or the shape of the water's surface.

Your claim is not , however, asserting that the spacetime geometry changes when you change coordinates. And the fact that the universe is "static" and the bucket is not can be expressed as invariant properties of the spacetime geometry and particular families of worldlines within it. For example (since this is an "I" level thread, some technical jargon is not inappropriate), the family of worldlines describing the motion of objects "at rest relative to the universe" will be integral curves of a timelike Killing vector field that is hypersurface orthogonal (which is what "static" translates to in more technical GR language); whereas the family of worldlines describing the motion of the bucket will be integral curves of a timelike Killing vector field (assuming the bucket's angular velocity of rotation relative to the universe is constant) that is not hypersurface orthogonal (in more technical jargon, the bucket's motion will be stationary but not static).

So in fact the real answer to your original question is that there is indeed an invariant sense in which the universe is not rotating (it is static) and the bucket is (it is stationary but not static), and that invariant difference between them is the correct underlying explanation of why the water in the bucket experiences nonzero proper acceleration and why its surface has the shape it has.

 

[Peter Leeves]

Your claim is not , however, asserting that the spacetime geometry changes when you change coordinates.

For any imprecision on my part, I apologise. I agree with your statement above. I hope I was saying that one scenario (rotating bucket) is equivalent to the other (rotating universe) and that both had equal validity and gave the same observational results (due to the acceleration being proper and therefore identical impact in both reference frames).

 

[Peter Leeves]

Thanks for your indicated edits. I prefer your version to my own too.

Spin the (spherical) blob and it flattens due to centrifugal force ; spin the Earth (lots) and the blob elongates (a tiny bit) thanks to frame-dragging. So, not the same... at least for that reasonably well defined scenario.

In the politiest possible way, can I ask why you think spinning the blob flattens it due to centrifugal force, but the Earth spinning would elongate the blob ? The system is symmetrical through all 360° about the the spin axis. I can see no reason for elongation, but I can see reason for identical flattening ? It certainly appears (at first sight anyway) that both scenarios could yield identical results.

 

[Peter Leeves]

This is all about Mach's principle. The question is: does the (inertial) mass m of the water depend on all the other mass M of the universe? Mach believed so; he believed that, whatever m(M) is, the inertial property of it should vanish if M vanishes. I.e. water in the bucket shouldn't become concave in an otherwise empty universe. Newton would disagree; he defined acceleration w.r.t. space.

And Einstein...well, Einstein believed Mach's principle was true, but his own theory of General Relativity is not fully Machian. Yes, inertial properties of a test mass are defined by other masses; they curve spacetime. But in an empty universe a particle can still undergo inertial forces because Minkowski spactime solves the field equations of an empty universe.

In the end, this remains an open question, mostly because neither Mach, Einstein or other people can tell you exactly what m(M) is. Thats's also why nowadays most physicists lost their interest in the topic.

Good grief, I just re-read your post which I failed to understand entirely yesterday, and found that now I can actually follow it all ! Thank you for this post.

 

[Peter Leeves]

This is a different physical situation than spinning the rest of universe while not turning on the motor.

It's more accurate (and makes comprehension easier) when you realize that it's firing of the motor that causes the universe to spin and the new gravitational field to be generated. You then appreciate that when the motor stops, the gravitational field dies away. Also that the gravitational field essentially just produces an equivalent force to the motor, and identical observations.

There’s substantial intuitive appeal to the idea that the effect should be the same ... but that’s not the same as a proof, let alone a proof backed up by observations.

I can't dispute your point at all. All I can do it try and apply logic and reason as best I can. No one seems to have an issue with the equivalence principle (man in lift freefalling towards Earth = man in interstellar space with no external influences). I see the rotating bucket / rotating universe as circular version.

I put this thread in this forum in the hope that someone can either shoot it down (making me happy because my understanding has increased), or confirm it (making me equally happy because my understanding has increased). Even if I come away with no resolution I'm still learning an awful lot, believe me (which has certainly increased my understanding).

 

[PeterDonis]

For any imprecision on my part, I apologise

It's not a matter of imprecision; perhaps my choice of words was a little misleading. I was making the point that spacetime geometry is actually the underlying cause of the shape of the water in the bucket; spacetime geometry is the invariant thing that tells us that it "really" is the bucket that is rotating, not the universe.

I hope I was saying that one scenario (rotating bucket) is equivalent to the other (rotating universe)

And my point is that this is only true as a statement about choices of coordinates for the same spacetime geometry; it is not true as a statement about invariants. As far as invariants are concerned, "rotating bucket in non-rotating universe" is not the same as "rotating universe with non-rotating bucket"; the latter would be a different spacetime geometry from the former (and the former is the spacetime geometry you have been describing).

 

[hmmm27]

heheh... I can't for the life of me now figure out why I thought/wrote the "elongated" bit, though it made sense at the time. Wait... almost had it... darn, slipped away again. Apparently @Dale and @PeroK understood : maybe they can help.

As for the other, more mundane scenario : the center of the Earth's gravitational force is the center of the Earth, no matter where your object is. Equilibrium for an object at the center of the Earth is a spherical shape, which a liquid easily accommodates. If you spin the sphere, the bits closest to the axis experience a slight outward force ; the bits furthest away experience a strong outward force. The outward force is balanced by the gravitational inward force. So, the sphere widens radially near the axial plane ("equator") and shrinks along the axis, which maintains the volume. You might even get a donut-shape out of the deal, since gravity's net force is zero at the center.

That is not going to happen if you spin the Earth, instead. Gravity (imparted by mass) is very much not the same thing as centripetal/fugal force, though a very limited amount of results are similar.

 

[Peter Leeves]

That is not going to happen if you spin the Earth, instead. Gravity (imparted by mass) is very much not the same thing as centripetal/fugal force, though a very limited amount of results are similar.

I would respectfully argue this: "The simplest way to state the equivalence principle is this: inertial mass and gravitational mass are the same thing." I would also add that Centripetal/fugal inertia/force is merely the rotational equivalent of linear inertia/force.

 

[PeterDonis]

No one seems to have an issue with the equivalence principle (man in lift freefalling towards Earth = man in interstellar space with no external influences). I see the rotating bucket / rotating universe as circular version.

No, this is not correct. The equivalence principle is local; it only covers a small patch of spacetime, not the entire universe. And it is not about the global equivalence of different choices of coordinates on the same spacetime geometry; it is about the local equivalence of the same state of motion (i.e., same proper acceleration--zero for free fall, or some fixed nonzero proper acceleration) in different global spacetime geometries (e.g., flat spacetime vs. the curved spacetime geometry around the Earth).

The global equivalence of different choices of coordinates on the same spacetime geometry is called "general covariance", not the equivalence principle. So the bucket thought experiment is an illustration of general covariance, not the EP.

 

[Nugatory]

It's more accurate (and makes comprehension easier) when you realize that it's firing of the motor that causes the universe to spin.

Now wait a moment... are you suggesting that when I switch on a motor to rotate something (like, for example a shaft with a bucket on the end) the motor is actually applying torque to all the rest of the universe causing it to rotate, while the bucket remains still?

You may be thinking of a different question, "does there exist a coordinate system in which the angular velocity of the bucket does not change while the angular velocity of the rest of the universe does?". The answer to that question is clearly yes (and in many problems, such as navigating on the surface of the earth, this coordinate system is more often used). However, the fact that we can use coordinates in which the bucket is rotating or coordiates in which the universe is rotating to describe the same relative motion does not mean that the two situations are otherwise equivalent. In one case, the water in the bucket is following its inertial geodesic path and in the other it is not because of the torque applied by the motor, and an accelerometer measuring proper acceeration will show the difference.

 

[hmmm27]

I would respectfully argue this: "The simplest way to state the equivalence principle is this: inertial mass and gravitational mass are the same thing." Centripetal/fugal inertia/force is merely the rotational equivalent of linear inertia/force.

Yes, and an aircraft driven by a propeller is equivalent to one driven by a rocket motor in that they both produce thrust which allows a properly built craft to fly around. They aren't the same, though and there are very good reasons why there are very few rocket-powered airplanes, and absolutely no propeller-driven spaceplanes.

I like the second sentence, though : no mention of gravity. Take a chair, duct-tape a rocket to the back and light it up. Assuming it flies straight, there's your linear acceleration. Impart the right amount of rotation to the chair, and you can end up with a circular spiral motion that uses the rocket thrust to impart centrifugal force.

[edit:]Impart *exactly* the right amount of rotation, calculated based on the thrust of the rocket, and you end up with the rocket facing directly outwards, always pushing the chair towards the center of a circle, but not getting any closer, ie: exactly the same as swinging the chair around with a rope.