# Angular momentum, what is it, is it conserved, and how do we know?

Firstly, most definitions of angular momentum assume a point about which you define angular momentum. I realize that you can consider the angular momentum about any point, and have many angular momentums, and that if they are all useful and all conserved then maybe it doesn't matter if there are lots of different ones. But it does make me wonder how fundamental it is, or even whether it always makes sense, or whether the term has the same meaning in every branch of physics.

I'm used to momentum being frame dependent, but conserved. Its conservation just looks like three of the free components of $0 = T^{\mu \nu}_{,\nu}$ from Special Relativity.

I'm also familiar with the fact that in Newtonian Mechanics, forces that act in the direction between two particles conserve angular momentum about any point.

I'm also familiar with the fact that for any variable that the Lagrangian doesn't depend on, there is a corresponding momentum conjugate to that variable that is conserved (so if a variable deserves the name angular, then the corresponding conjugate momentum is both conserved and equally deserves to be called an angular momentum).

I've also seen conservation of angular momentum from Special Relativity in the form $0 = (x^{\alpha} T^{\mu \nu} - x^{\mu} T^{\alpha \nu})_{,\nu},$ where this is the divergence of a rank 3 tensor. Though to be fair, this is usually derived from the symmetry of the stress-energy tensor $T^{\mu \nu}$ which itself is usually derived from considerations that angular momentum doesn't go crazy at fine scales. So it does make me wonder which ideas come first, conservation of angular momentum or the symmetry of the stress-energy tensor.

Also, you can have angular momentum in electromagnetic fields (or in electromagnetic potentials, which depend on gauge this is even more than just depending on a point). An example is the angular momentum set up by having a linearly charged wire inside a solenoid inside a charged cylindrical shell. But if you turn the current off from the solenoid, it begins to rotate so it seems like mechanical angular momentum is only conserved when coupled to field (or potential) angular momentum.

And in quantum theories you have intrinsic spin angular momentum as well as the regular angular momentum, and the regular one again depends on your origin or potentially also on the electromagnetic potentials (but here I'm less bothered by the apparent gauge dependence).

With so many potentially (pardon the pun) different momentums, and so many derivations of special cases where some of them are conserved. The question is whether they are the same thing in these very different theories, and whether they are always conserved or just some of them or just in some situations, and in particular how this relates to General Relativity.

For instance, a charged rotating black hole is often parameterized by a total angular momentum per unit mass. So it's common in General Relativity to talk about a total angular momentum. But does a total angular momentum make sense in General Relativity and what if anything from other theories does it correspond to (field angular momentum, potential angular momentum, mechanical angular momentum, intrinsic spin, combination of all or just some of those)?

I'm not even aware of a general differential conservation law for angular momentum in General Relativity. And knowing whether it is stored in the fields or in mechanical angular momentums seems relevant to knowing how much is inside a region if we want to get a total for a region.

As an example of how common the assumption of conservation of angular momentum is, a quick google search came up with

http://csep10.phys.utk.edu/astr161/lect/solarsys/angmom.html

as the second link and it says "all experimental evidence indicates that angular momentum is rigorously conserved in our Universe." The course is an Astronomy course, not a Cosmology or General Relativity course, but I'd still expect instructors and students in an Astronomy course to have some awareness of General Relativity and so wonder which things continue to hold in General Relativity.

The first link from a quick google search is to the wikipedia article on angular momentum, which cites the Newtonian mechanical versions, the gauge dependent electromagnetic case, the quantum case, and no mention of General Relativity except to say that angular momentum is not, in general, conserved in General Relativity, but it is missing a citation for that. It also just assumes at the beginning that no net external torque means conservation of angular momentum, even though the proofs used requires that forces act along the line between the particles even for Newtonian Mechanics and the article itself admits that angular momentum (however it is defined) is not consider, in general, in General Relativity.

I want to point out that one reason I express skepticism about the definition of angular momentum in General Relativity is because it does not appear to be conserved. If you had a quantity with a differential conservation law, then there are grounds to say we have something legitimate. If you define something and then see that it isn't conserved, then it begs the question about whether a different definition would have been conserved. And it seems strange to have a non-conserved thing exist in a theory that is supposed to be an experimentally verified limit of a quantum theory where thing you are supposed to correspond to is generally assumed to be conserved. In particular if every star is made up of ions and fields and you expect angular momentum to be conserved by them, then a definition that is supposed to correspond to angular momentum but isn't a conserved thing should be suspect. But the tensors I see for angular momentum in Special Relativity seem to generalize mechanical momentum, so don't include any intrinsic spin. So when a parameter refers to the total angular momentum of a system in General Relativity, it's not clear what is does or is supposed to correspond to. And I don't know if we are happy with our definitions as they are (say from a Lagrangian formulation of General Relativity), or whether this is an area of active research that goes by a name I'm not familiar with.

In case this helps anyone answer, there are classical electromagnetic forces besides the Lorentz force law for objects with intrinsic magnetic moments, e.g. $\vec{F}=\vec{\nabla}(\vec{\mu}\cdot \vec{B})$. So the usual angular momentum tensors might simply be ignoring the stress-energy associated with this kind of work (which unfortunately is non conservative, so I'm not immediately aware of how to get a stress-energy tensor from those kinds of forces). I also am not looking just for a pointer to Einstein-Cartan theory (or any other alternative to General Relativity) without a discussion about whether current General Relativity is considered to be confused, wrong, or deficient or whether these are just two theories that appear to be equally valid and functional at this juncture in our studies of Physics.

P.S. I wanted my fifth tag to be correspondence-principle but can't make a new tag.

• Conservation of angular momentum about a point is just a consequence of rotational symmetry about that point, courtesy of Noether's theorem. That seems to me to be a reasonably fundamental property. Aug 25 '14 at 5:53
• You can pick an arbitary point in special relativity and newtonian mechanics, because both of these theories are rotationally invariant, and translationally invariant -- so you can move any origin to any other origin using the translation symmetry. If you have a set of physical laws with a special origin, then you can't do this, and angular momentum will only be "good" around your special point where the rotational symmetry works. Aug 25 '14 at 16:40
• The question is extremely long and not very focused, so it's hard to tell what you're asking. With respect to GR, the foundational status of angular momentum is exactly the same as the foundational status of energy-momentum. Both are locally conserved. Neither is globally conserved, because GR doesn't have global conservation laws. There's a good discussion of this in Hawking and Ellis.
– user4552
Aug 25 '14 at 17:01

Some bits and pieces on angular momentum:

Angular momentum is that which is conserved in rotationally invariant systems, just like energy is that which is conserved in time translation invariant systems and momentum is that which is conserved in space translation invariant system. This is the essence of Noether's theorem. The analogue in QFTs are Ward-Takahashi-(like) identities, essentially generalizing the Noether conservation laws as operator equations.

The normal approach to such conservation laws in GR is to consider Killing vectors, see also this old question - and so, if you have a Killing vector fields corresponding to "rotations", you have found what angular momentum is in that theory.

Asking what angular momentum is in a setting where it is not conserved by symmetries would be rather pointless, since it would then not tell you anything about the system as such, so I do not think you should be bothered that no one writes downs a "general" form for the angular momentum, because it would not be conserved in all theories, and as such not very useful.

Discussing angular momentum in a most general GR/QFT setting is quite challenging, and there are various expositions on this. One discussing matter fields minimally coupled to GR is here - look around eq. $(57)$ for the generalization of the conservation of total angular momentum - it is the joint conservation of the "ordinary" angular momentum obtained by the antisymmetriation of the energy-momentum tensor and "spin terms".

• Killing vectors in GR are not really a fundamental way of describing conservation laws, and I think what the OP wants is a description of the fundamentals. Quantities like energy-momentum and angular momentum are locally conserved in GR regardless of whether there is a Killing vector. The Killing vector just gives a conservation law for test particles.
– user4552
Aug 25 '14 at 16:59