Understanding Angular Momentum in Space | SR/GR and Centrifugal Force Explained

[Topher925]

I'm having trouble understanding angular momentum and how it relates to SR/GR. To keep it simple, if I'm in space and all matter and energy is removed (ie. reference frames) and I start rotating at a specific velocity will I feel my arms and legs being pulled out? If the only possible frame of reference is myself it will appear to me that I am not rotating at all. But my intuition tells me that any angular velocity will cause me to feel angular momentum as a centrifugal force.

Lets say that my intuition is correct and I do feel my arms and legs being pulled even though there is no other frame of reference. Why is this? Would this not constitute absolute space?

 

[haushofer]

I think this has everything to do with Mach's principle :P

If I remember correctly, Mach stated that these kinds of fictitious forces were the result of movement with respect to all the other matter in the universe. If you would remove this matter, you wouldn't feel a centrifugal force when rotating.

 

[Tomsk]

I read that rotation should always be taken as relative to the local gravitational field in GR.

I'm not sure how angular momentum transforms between frames in GR. The conserved relativistic angular momentum is $x_\mu T^\rho_\nu - x_\nu T^\rho_\mu$, but I think the 3-angular momentum must be removable by transforming to a rotating frame. This is similar to how a magnetic field can be removed by an appropriate transformation. But the norm of the Faraday tensor is invariant so you still see the same effects, you just say they are caused by an electric field. So I think similarly the norm of the relativistic angular momentum is invariant, so you still see the same thing happen, you just describe it in a different way.

 

[weaselman]

Mach's principle was before GR. In GR (and SR) the time-space is "absolute" in the sense that any accelerated motion is "absolute" - all (local) observers are supposed to agree on whether or not an object is moving with acceleration. This is because of the difference in the trajectories in the time-space. An inertial trajectory is always a "straight line" (a geodesic actually), and an accelerated one is always curved. It is this curvature that makes your hands pull out when you are spinning, and yes, it is "absolute" in the sense that you don't need another point of reference to tell whether your trajectory is curved or not.

 

[bcrowell]

I'm having trouble understanding angular momentum and how it relates to SR/GR. To keep it simple, if I'm in space and all matter and energy is removed (ie. reference frames) and I start rotating at a specific velocity will I feel my arms and legs being pulled out? If the only possible frame of reference is myself it will appear to me that I am not rotating at all. But my intuition tells me that any angular velocity will cause me to feel angular momentum as a centrifugal force.

Lets say that my intuition is correct and I do feel my arms and legs being pulled even though there is no other frame of reference. Why is this? Would this not constitute absolute space?

Einstein started worrying about this kind of thing ca. 1910, and for the next 50 years it was just an exercise in philosophizing. The classic paper that actually made some headway was this:

C. Brans and R. H. Dicke, Physical Review 124 (1961) 925

The paper is unfortunately not free online. There's a summary of the ideas here: http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.3

Brans and Dicke came up with a generalization of GR, with an adjustable parameter $\omega$. The limit of $\omega\rightarrow\infty$ gives GR, which predicts a totally non-Machian result for the thought experiment you've posed. Small values of $\omega$ give a Machian result. It then becomes an experimental task to measure $\omega$ and quantify how Machian our universe is. Brans and Dicke thought our universe might have $\omega\sim 1$. The latest experimental limit, however, is $\omega>40,000$, showing that our universe is highly non-Machian.

So the answer to your question is yes, and that's just the way the universe happens to have been designed.

 

[Topher925]

Interesting, so despite the name of general relativity, there is somewhat of an "absolute" space-time when it comes to acceleration. This raises a few questions on the working of mechanical gyroscopes but I think I will ponder that for a while before asking about it.

 

[bcrowell]

Interesting, so despite the name of general relativity, there is somewhat of an "absolute" space-time when it comes to acceleration.

I think it would be more accurate to say "when it comes to rotation."

 

[weaselman]

Not really. Any acceleration is absolute. Rotation is just one of many possibilities of accelerated motion.

 

[bcrowell]

Not really. Any acceleration is absolute. Rotation is just one of many possibilities of accelerated motion.

No, that's incorrect in GR. It's only correct in Newtonian mechanics. The whole point of the equivalence principle is that acceleration is not absolute in GR.

 

[shoehorn]

C. Brans and R. H. Dicke, Physical Review 124 (1961) 925

The paper is unfortunately not free online.

Carl has made lots of his papers available online at his Loyola page. In particular, lots of his scalar-tensor work, including the Brans-Dicke paper, can be found there.

 

[bcrowell]

Carl has made lots of his papers available online at his Loyola page. In particular, lots of his scalar-tensor work, including the Brans-Dicke paper, can be found there.

Aha! Thanks for the correction! Here's the URL: http://loyno.edu/~brans/ST-history/CHB-RHD.pdf

 

[yossell]

I'm having trouble understanding angular momentum and how it relates to SR/GR. To keep it simple, if I'm in space and all matter and energy is removed (ie. reference frames) and I start rotating at a specific velocity will I feel my arms and legs being pulled out? If the only possible frame of reference is myself it will appear to me that I am not rotating at all. But my intuition tells me that any angular velocity will cause me to feel angular momentum as a centrifugal force.

In SR, a rotating body isn't a frame of reference in Einstein's sense, it's not an inertial frame. In SR, in every inertial frame, you'll still be spinning and so there will be distortion.

Even in GR, there are solutions to GR where the whole universe has non-zero angular momentum, and these solutions are not equivalent to ones where it has zero angular momentum. Some take these solutions to cast doubt on the idea that GR, by itself, really implements Mach's principle. Though, like anything in GR, it's contentious.

http://en.wikipedia.org/wiki/Gödel_metric

(ouch - I don't know if this link works - the `o' in `Godel' has an accent and I don't know how the copying and pasting goes - google `godel universe' should give you the page.)

 

[weaselman]

No, that's incorrect in GR. It's only correct in Newtonian mechanics. The whole point of the equivalence principle is that acceleration is not absolute in GR.

What I mean is that as long as you are not moving on a geodesic, you will be experience "acceleration" or "gravity", and that notion is absolute in the sense that all observers will agree on it (except, finding a frame large enough to cover more than one observer is tricky in GR, but that's a different story

Another way to put it is that the property of a reference frame being inertial is absolute. You can always tell whether your frame is inertial or not as opposed to the inertial motion, which is always relative (i.e., indistinguishable from rest).