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.

This post imported from StackExchange Physics at 2025-01-21 21:47 (UTC), posted by SE-user David Elm

Comments

See Center of mass (relativistic).

This post imported from StackExchange Physics at 2025-01-21 21:47 (UTC), posted by SE-user AccidentalFourierTransform