# Near light angular momentum

1. Oct 11, 2007

### BulletRide

After reading a chapter on the conservation of angular momentum, I have had a radical idea growing in my mind ever since I finished reading the material. To cut to the chase, the law states that the angular momentum of a rotating object will remain constant unless an outside torque acts on the object. Since angular momentum = I x w, an objects rotational speed will increase as the rotational inertia decreases - the same principal behind how an ice skater increases how fast she spins by pulling in her arms. Radically thinking, if a spinning space station consisting of simply a wire anchored on both ends and had a length a of one hundred kilometers, or so, were to be accelerated to 75% the speed of light at the outermost point, and then reeled in at a constant rate towards the axis of rotation; what would happen once the rotational velocity began to reach the speed of light? Since, the law of conservation of angluar momentum indirectly states that an object's rotational speed will increase as its rotational intertia decreases, one would suspect that, at only 75%c at the edge, as the object began to be reeled in closer and closer, it would theoretically exceed the speed of light. Since this is an impossibility, what would happen as it approached the speed of light? Any aid on this problem would be greatly appriciated!

2. Oct 11, 2007

### pervect

Staff Emeritus
Relativistic angular momentum can be thought of as $\vec{p} \times \vec{r}$, or the 4-vector equivalent. Here $\vec{p}$ is the momentum, and $\vec{r}$ is the radial vector.

(There's also a representation as a bi-vector which is a bit more elegant if you happen to be familiar with clifford algebra. However, you can make do fine with the 3-vector or the 4-vector form for this problem. So just ignore this if you're not familiar with Clifford algebra).

So you can see immediately that if you halve r, you double the
momentum p, but nothing ever exceeds the speed of light, because the relativistic formula for the momentum p is
$$\vec{p} = \frac{m \vec{v}}{\sqrt{1-(|v|/c)^2}}$$

and p goes to infinity as v->c.

Last edited: Oct 11, 2007
3. Oct 11, 2007

### BulletRide

So, it would be impossible to bring the momentum up to a velocity greater than c, no matter how much the objects are "reeled" in? Does this also imply that the law of conservation of angular momentum only applies to objects of relatively low velocity (compared to 1c)?

4. Oct 11, 2007

### pervect

Staff Emeritus
Angular momentum is still conserved in special relativity, you just have to use the relativistic formula for angular momentum, not the Newtonian formula.

This basically involves using the correct relativistic formula for linear momentum, as I described earlier.

You might also want to check out http://panda.unm.edu/Courses/finley/P495/TermPapers/relangmom.pdf [Broken]

though it may be a bit advanced.

Last edited by a moderator: May 3, 2017
5. Oct 13, 2007

### BulletRide

Ah, I see. Thanks!