# Interesting applications of relativistic angular momentum?

1. Dec 20, 2015

### bcrowell

Staff Emeritus
I've been working on my SR book http://www.lightandmatter.com/sr/ and its presentation of a suite of related mathematical tools such as the Levi-Civita tensor. This is aimed at the upper-division undergraduate level, although there are optional sections at the end of some of the chapters that go into topics that would normally be seen in a graduate course.

Currently I have a treatment of the Levi-Civita tensor and volume forms in an optional section at the end of ch. 7. Ch. 8 is on rotation and currently makes no mention of angular momentum. I would like to add a section on angular momentum and make some use of the tools previously developed in ch. 7. At this stage the reader does not yet know about the stress-energy tensor, which is discussed in ch 9.

Does anyone know of an interesting, simple, concrete, real-world application of relativistic angular momentum that can be done with these tools? When I think of relativistic objects that have angular momentum, I think of neutron stars, nuclei, hadrons, and atoms. Examples involving neutron stars seem not doable at this stage because they involve gravity, and this is an SR book. Nuclei, hadrons, and atoms seem like they would lead too far into quantum mechanics, which I don't assume my readers know much about beyond what's described in a freshman survey course.

Any suggestions? Maybe there would be something interesting from relativistic heavy ion physics? Or small corrections in high-precision experiments, such as tests of Lorentz invariance?

2. Dec 20, 2015

### Staff: Mentor

There is also hidden angular momentum, but you may need the stress energy tensor.

3. Dec 20, 2015

### Staff: Mentor

4. Dec 21, 2015

### sweet springs

5. Dec 21, 2015

### bcrowell

Staff Emeritus
From the tension that holds the disk together, right? But it might actually be an interesting example to work out for later in the book, after they know about stress-energy.

Yeah, but I don't think you can do anything quantitative with that without GR.

6. Dec 21, 2015

### bcrowell

Staff Emeritus
Following up on the relativistic heavy ion idea, I found this by googling: https://books.google.com/books?id=h...ivistic heavy ion" "angular momentum"&f=false . So for a simple plug-in example, we could have something like the following. RHIC collides beams of gold nuclei at 8.86 GeV/nucleon. If a gold nucleus is approximately a sphere with radius 6x10^-15 m, find the maximum angular momentum, in units of $\hbar$, about the center of mass for a glancing collision. The answer turns out to be about 10^5 $\hbar$.

Another possibility might be something to do with helical motion of an electron in a magnetic field.

7. Dec 21, 2015

### pervect

Staff Emeritus
For the lever paradox, see for instance http://arxiv.org/abs/1206.4487. It's related to the Trouton-Nobel experiment. There's been a few threads on it in PF. This meets a lot of your goals (it involves special relativity and angular momentum), but it may be hard to explain simply especially without using the stress-energy tensor, so it might not be what you're looking for. But it's close enough it may be worth checking out anyway.

8. Dec 21, 2015

### bcrowell

Staff Emeritus
I ended up figuring out a pretty fun example that gets some mileage out of relativistic angular momentum without requiring any knowledge of the stress-energy tensor or the Levi-Civita symbol. It turns out that if you add relativistic corrections to the Bohr model, you end up with quite an accurate equation for the energy spectrum. If you consider the states that have maximum angular momentum for a given n (which makes sense if the Bohr model is about circular orbits), you actually get the same result as the exact solution of the Dirac equation. You can also do hydrogenlike atoms with Z>1, and you get an unphysical result for $Z\gtrsim 1/\alpha$ that has a nice interpretation in terms of sparking the vacuum. I have this written up now in the book, along with a discussion of angular momentum as a rank-2 tensor. I added the RHIC example as a homework problem.

Stuff like the lever paradox, Mansuripur paradox, or a rotating disk would also be interesting to do, but in the organization of my book they would have to come later, after I've introduced the stress-energy tensor. At that point in the book it would also be nice to add a proof that the divergence-free property of the stress-energy tensor automatically implies conservation of angular momentum.

9. Dec 26, 2015

### PFfan01

In the book by Landau and Lifshitz [L.D. Landau and E.M. Lifshitz, "The classical theory of fields," (Butterworth-Heinemann, New York, 1975), pp. 83-84], there is a proof, but according to the article [Can. J. Phys. 93: 1470–1476 (2015), footnote 1; dx.doi.org/10.1139/cjp-2015-0198], this proof is questionable.

10. Dec 30, 2015

### vanhees71

Well, I'd call it relativistic angular momentum or simply angular momentum. There is nothing hidden about it. Everything that's called "hidden momentum" or "hidden angular momentum" is straightforwardly taken into account with the use of the correct energy-momentum tensor. I never understood, why one should treat this subject as something so confusing.

Usually apparent paradoxes appear (usually in classical electrodynamics), because at one place the authors make a non-relativistic approximation, usually in the mechanical part of the corresponding master equations.

I think the most illuminating treatment of these issues is to make simple (classical) fully relativistic toy models for the "matter part" of the em. problems associated with "hidden conserved quantities". The most natural way are simple continuum-mechanics models like to treat the electrons in a metal semi-classically as an ideal fluid or solid bodies with an ansatz for the mechanical part of the energy-momentum tensor.

A straightforward application of relativistic angular momentum for the case of the em. field is to construct plane-wave packets in a well-defined polarization state (helicity) or even higher total angular momentum (of course, in relativistic physics there's no clear way to split the total angular momentum into orbital and spin parts, particularly for the electromagnetic field).

11. Dec 30, 2015

### vanhees71

I don't understand, why one would avoid the energy-momentum tensor but introduce the angular-momentum tensor, and how do you do that? The natural way to introduce the conserved quantities is via Noether's theorem applied to the Poincare group, and the Mansuripur paradox is an infamous example for much ado about something that has been clarified very early in history (partially even in prehistory) of the development of special relativity by Poincare and von Laue.

12. Dec 30, 2015

### vanhees71

I don't find the time to read this paper (which you can find in the arXiv as a preprint for free legal download; the Canadian Journal is not available in my university; perhaps for some good reason...), but I'm pretty sure there must be something wrong with this paper, because the energy-momentum and angular-momentum balance equations are a straightforward application of Gauss's Integral Theorem in 4 dimensions. Of course, you have to assume appropriate boundary conditions. The most simple and most realistic case are distributions of energy-momentum-stress and angular-momentum that are compact in the spacelike directions, i.e., a kind of tube in the Minkowski space-time diagram. Now you use a four-volume with only time-like boundaries and apply Gauss's Integral Theorem to show that for a tensor field of this kind
$$T_{\text{tot}}^{\nu \rho \sigma\ldots}=\int_{V^{(3)}} \mathrm{d}^3 x \Theta^{0 \nu \rho \sigma \ldots}$$
is a Minkowski tensor and that it is conserved in any inertial reference frame, provided that
$$\partial_{\mu} \Theta^{\mu \nu \rho \sigma\ldots}=0,$$
and this is an important constraint, because neglecting it, leads to all kinds of paradoxes like "hidden momentum" and other unnecessary confusion.

13. Dec 30, 2015

### bcrowell

Staff Emeritus
I have a discussion of Thomas precession in section 8.3.3. BTW, note that this is a semi-necro-thread.