# Moment of Inertia in SR & GR?

1. Feb 13, 2014

### bolbteppa

In classical mechanics you want to calculate the moment of inertia for hollow & solid:

lines, triangles, squares/rectangles, polygons, planes, pyramids, cubes/parallelepiped's, circles, ellipses, parabola's, hyperbola's, sphere's, ellipsoid's, paraboloid's, hyperboloid's, cones & cylinder's

setting them up in either scalar notation or in tensor notation (i.e. two ways of thinking for all those cases which requires two very error-prone constructions), which at least for me is an immense task I still haven't fully carried out

My question is, how does all this translate over the special &/or (?) general relativity? Do you have to re-do every one of those calculations from a more general standpoint or is it just that the density in the integral is usually veriable?

As a side note, is there an easier & more way to do all of the above? For instance, in calculus books they sometimes put MoI into 3 different chapters, leading to single, double & triple integral modelling on top of physical modelling (in physics books) or tensor modelling (in advanced physics books) which is really 5 f'ing ways to do about 30 calculations However, you can apparently sometimes use Stokes theorem
e.g. http://www.slideshare.net/corneliuso1/green-theorem (slide 26)
to show some of these models are exactly equivalent, but how do I deal with it all in general in a unified manner? Thanks

2. Feb 13, 2014

### WannabeNewton

Both the moment of inertia tensor and mass quadrupole moment tensor are defined in the rest space of the center of mass frame in the same way as in Newtonian mechanics. From there just convert Euclidean indices to Lorentz indices in the usual way, that's all there is to it.

3. Feb 13, 2014

### pervect

Staff Emeritus
The tensor might be defined in the same way, but I don't believe you can use it to calculate the angular momentum of a large system (assuming such a system has a conserved angular momentum).

I believe the idea that mass causes angular momentum has some of the same problems as the idea that mass causes gravity - it doesn't work in GR, where one needs to use the stress-energy tensor, and not "mass", as the source.

4. Feb 13, 2014

### WannabeNewton

Well it's essentially the same thing if we're considering fluids, at least in special relativity wherein the angular momentum is given by $S^{l} = \epsilon^{lrs}\int y^{r}T^{0s}d^{3}y$. Say the fluid is dust (e.g. a spinning thin shell of dust or spinning cylindrical shell of dust) and that we're in the center of mass frame so that $T^{0i} = \gamma^2\rho v^i$ relative to the center of mass and $y^i$ just becomes the displacement of each dust element from the center of mass. Then $S^{l} = \epsilon^{lrs}\int \gamma^2 y^r v^s dm = \epsilon^{lrs}\epsilon^{sij}\int \gamma^2 y^r y^j\omega^i dm$ or in more transparent notation $\vec{S} = \int \gamma^2 \vec{r}_{\text{CM}} \times (\vec{\omega}_{\text{CM}}\times \vec{r}_{\text{CM}})dm$. In the low velocity limit this is just $\vec{S} = I_{\text{CM}} \vec{\omega}_{\text{CM}}$ where $I_{\text{CM}}$ is the moment of inertia tensor in the center of mass frame.

I'm not sure what you mean by "mass causes angular momentum". Are you referring to, for example, the emergence of non-vanishing vorticity and orbital angular momentum for a family of static observers hovering outside of an axisymmetric stationary rotating source?

5. Feb 14, 2014

### pervect

Staff Emeritus
It's a bit vague, but first let me say that I'm saying mass doesn't cause angular momentum, not that it does :-). And that I'm talking about angular momentum in GR.

MTW, for instance, discusses angular momentum in terms of the behavior of one of the metric coeffficients (g_0j) as a function of r^3. (Not the most modern definition anymore).

In general the metric coefficients contain information not present in the stress-energy tensor, (for instance the contributions due to gravitational waves).

I believe one can get an answer for angular momentum in terms of some integral of the stress energy tensor T_ij for stationary space-times, but I'm not sure what it is offhand, Wald writes down the expression for energy in this form, but doesn't write one down for angular momentum.

6. Feb 14, 2014

### WannabeNewton

See exercise 6 of chapter 11 in Wald.