Uses of the Angular Momentum 4-Tensor

Question

The angular momentum 4-tensor has 6 independent components, three angular momentum components and three new guys. Some call these new guys the 'boosts', but since they are the conjugate momentum of the Lorentz Transformation (aka the actual boosts) I think 'boost momentum' is a better name.

These quantities are related to energy and momentum by (up to a sign, depending on the text)

$\vec K=\frac{E}{c} \vec x-ct \vec p$

While conservation of angular momentum is pretty useful for a lot of problems, conservation of the boost momentum seems to be a loose end.

It seems like conservation of the boost momentum only serves to ensure continuity of motion. A particle undergoing a discontinuous jump (hopping from one point to another in an unphysical way) could conserve energy, momentum, and angular momentum, but it wouldn't conserve boost momentum.

Are there physical problems where conservation of boost momentum could be as useful as conservation of angular momentum?

Edit: I don't think I unpacked the idea I was thinking about very well. Here's one thing I'm looking at with a lot more detail:

Starting with

$\vec K=\frac{E}{c} \vec x-ct \vec p$

We can use

$\vec p=\frac{E}{c^2} \vec v$

to get

$\vec K=\frac{E}{c}(\vec x-\vec vt)$

If our velocity is constant, then we have

$\vec x=\vec x_0+\vec v t$

and so

$\vec K=\frac{E}{c} \vec x_0$

What I'd like to do from here is get a sense about what $x_0$ means. We know it's a relativistic version of the terms we add up when finding the center of mass.

$\bar x=\frac{m_1 x_1+m_2 x_2+...}{m_1+m_2+...}$

And there are a couple of nice problems, like a guy walking on a board with frictionless wheels where the combined center of mass stays the same and so

$m_1 \Delta x_1=-m_1 \Delta x_2$

Which tells us that no matter how weird the motion of the guy walking on the board is, the center of mass stays put and the motion of the board is a scaled reflection of the guy's motion.

So here's what I'm getting at: What is the relativistic version of this like? What (if any) scenarios do we have where the conservation of boost momentum gives us a simple insight into a physical process? That's what I'm trying to get at.

The other item, the continuity of motion, is related to a discussion I saw on a great Feynman lecture, Lecture 3 - The Great Conservation Principles where he considers the idea that maybe an object could disappear in one place and appear in another at the same time. Feynman then shows how this couldn't work, since that would mean in one frame that object would be missing for a while and in another that object would appear at the new location before it disappeared in the first.

I haven't figured it out too much, but it seems to me that means there is a nice connection between the conjugate momentum of the Lorentz Transformation and continuity of motion. Also, where conservation of the momentum of a particle doesn't prevent it from disappearing in one place and appearing in another, conservation of boost momentum would, since it would change the $\vec x_0$ from above and nothing else.

Answer 1

Not really a full answer to your question, but I wanted to point out that the "conservation of boost momentum" is tightly connected with the equivalence between mass and energy; or more precisely to the equivalence between momentum and flux of energy.

Here's a summary in flat spacetime, for simplicity. Let's use coordinates $(x_0, x_i) \equiv (t, x_i)$ and a flat metric $(g_{\mu\nu}) = \operatorname{diag}(-c^2,1,1,1)$. The energy-momentum tensor at an event is $$ ({T^{\mu}}_{\nu}) = \begin{pmatrix} -\epsilon & p_j \\ -q^i & {\sigma^i}_j \end{pmatrix} $$ where $\epsilon$ is the (total) energy density, $q^i$ the energy flux density, $p_j$ the momentum density, and ${\sigma^i}_j$ the pressure tensor (negative of stress tensor).

The energy-stress tensor obeys these ten equations (because of Einstein's equations): $$ \begin{align} &\partial_{\mu} {T^{\mu}}_0 = 0 &\text{or}\quad&-\partial_t\epsilon - \partial_i q^i = 0 \tag{1}\label{E} \\ &\partial_{\mu} {T^{\mu}}_j = 0 &\text{or}\quad&\partial_t p_j + \partial_i {\sigma^i}_j = 0 \tag{2}\label{P} \\ &T^{i0} = T^{0i} &\text{or}\quad& q^i/c^2 = p^i \tag{3}\label{B} \\ &T^{ij} = T^{ji} &\text{or}\quad& \sigma^{ij} = \sigma^{ji} \tag{4}\label{A} \end{align} $$

Equation \eqref{E} is the balance of energy, \eqref{P} is the balance of momentum, \eqref{A} is the symmetry of the stress tensor, which is equivalent to the balance of angular momentum. Equation \eqref{B} is the generalization of "$E=mc^2$", saying that a flux of energy is equivalent to a flux of momentum, besides a $c^2$ conversion factor.

There's a clear difference between the set of equations \eqref{E}--\eqref{P}, and \eqref{B}--\eqref{A}. The first is local, while the latter is pointwise, that is, it doesn't even depend on the values of the energy-momentum tensor in a spacetime neighbourhood of the event.

But we can rewrite the system above as an equivalent system of equations, all of which are local: $$ \begin{align} &\partial_{\mu} {T^{\mu}}_0 = 0 \tag{1} \\ &\partial_{\mu} {T^{\mu}}_j = 0 \tag{2} \\ &\partial_{\mu} ({T^{\mu}}_0\, x^k/c + c\,{T^{\mu}}_k\, x^0) = 0 \tag{3'}\label{B2} \\ &\partial_{\mu} ({T^{\mu}}_j\, x^k - {T^{\mu}}_k\, x^j) = 0 \tag{4'}\label{A2} \end{align} $$ The last two correspond to $$ \begin{align} & \partial_t(-x^k\,\epsilon/c + c\,t\,p_k) +\partial_i(-x^k\,q^i/c + c\,t\,{\sigma^i}_k) = 0 \\ &\partial_t(x^k\,p_i - x^i\,p_k) +\partial_i(x^k\,{\sigma^i}_j - x^j\,{\sigma^i}_k) = 0 \end{align} $$ that is, the balance of "boost momentum" and the balance of angular momentum.

It's easy to see that \eqref{E}--\eqref{B} imply \eqref{E}--\eqref{B2}: $$ \begin{aligned} &\partial_{\mu} ({T^{\mu}}_0\, x^k/c + c\,{T^{\mu}}_k\, x^0) \\ &=\partial_{\mu}{T^{\mu}}_0\, x^k/c + {T^{\mu}}_0\, \partial_{\mu}x^k/c +c\,\partial_{\mu}{T^{\mu}}_k\,x^0 +c\,{T^{\mu}}_k\,\partial_{\mu}x^0 \\ &\stackrel{\eqref{E},\eqref{P}}{=} {T^{\mu}}_0\, {\delta_{\mu}}^k/c +c\,{T^{\mu}}_k\,{\delta_{\mu}}^0 \\ &={T^k}_0/c + c\,{T^0}_k \\ &\stackrel{\eqref{B}}{=} 0 \end{aligned} $$ and vice versa.

So we have a situation analogous to the one in Newtonian mechanics: there, balance of angular momentum can equivalently be written as a balance equation or as a symmetry condition for the stress tensor. In relativity, balance of "boost momentum" can be written as a balance equation or as the equivalence between momentum and energy flux.

All this can be extended to general relativity, where some source terms generally appear in non-flat spacetime (that's why I'm speaking of "balances" rather than "conservations").

Some interesting references about this are:

• Eckart 1940: The thermodynamics of irreversible processes. III. Relativistic theory of the simple fluid
• Hawking, Ellis 1973: The Large Scale Structure of Space-Time, §3.2, around eqn. 3.2.

Your final comments seem to be related to somewhat analogous situations where light enters and exit a medium, so that the medium must move as light "slows" down upon entering it, because of conservation of boost momentum. There's an interesting discussion about this in

• Pfeifer & al. (2007) Momentum of an electromagnetic wave in dielectric media; see for instance §§ VIII.B and IX.

Very recently some interesting discussions about this topic have appeared in

• Gralla & al 2024: Frames and Slicings for Angular Momentum in Post-Minkowski Scattering
• Gralla, Lobo 2024: Electromagnetic Scoot