# Spinning objects in GR

1. Jul 31, 2006

### MeJennifer

In GR, can a spinning object be traveling on a geodesic?

2. Jul 31, 2006

### cesiumfrog

Yes.

A ball falls the same way regardless of how it may be spinning (provided we can neglect friction, frame-drag, etc).

Just as an observer feels whether he is accelerating, it is possible to feel whether or not one is spinning. In general relativity this basically means that, as the spinning (but otherwise unaccelerated) observer traverses some geodesic path, the observer will simply note his/her "spacelike" laboratory coordinate axes are not being parrallel transported - they are twisting.

3. Jul 31, 2006

### Stingray

No. Something called the Papapetrou equation governs the motion of spinning test particles. They do not generally move on geodesics in the absence of external forces.

4. Aug 1, 2006

### MeJennifer

What force governs this behavior?

5. Aug 1, 2006

### Stingray

The momenta evolve according to
$$\frac{\delta P^{a}}{ds} = - \frac{1}{2} u^{b} S^{cd} R^{a}{}_{bcd}$$

$$\frac{\delta S^{ab}}{ds} = 2 P^{[a} U^{b]}$$
where the linear momentum is
$$P^{a} = m u^{a} - u_{b} \frac{\delta S^{ab}}{ds}$$
So the "force" comes from the spin coupling to the Riemann tensor.

6. Aug 1, 2006

### pervect

Staff Emeritus
Another, less technical (I hope) way of explaining the "extra" forces on a spinning particle.

The difference in motion between a spinning test particle and a non-spinning one can be ascribed to the "gravitomagnetic" effects.

One can divide the tensor into two parts - the magnetic part, and the electric part. The "extra" force on a spinning particle is due to the magnetic part of the tensor.

One can also draw an anology to the electromagnetic case. A spinning charge will have a magnetic moment, meaning it acts like a tiny bar magnet. This will cause the spinning charge to interact with magnetic fields even when it is stationary. The spinning mass does something rather similar.

7. Aug 1, 2006

### cesiumfrog

My mistake (Papapetrou is indeed mentioned in MWT's Gravitation for example), although I understand the deviation from geodesic motion is generally "negligible" and proportional to the mass of the text particle.