Timelike vs. null vs. spacelike four-current

[djy]

I've looked in several sources but haven't seen this addressed specifically.

If you have a ball of static electric charge, then the four-current is timelike. If the ball is moving, then you can transform to its rest frame, in which the four-current's spacelike components are all zero.

If you have a wire carrying current and it has no net charge density, then the four-current is spacelike. If the whole wire is moving, one can transform into its rest frame, in which the four-current's timelike component is zero.

It seems that one can always decompose a spacelike vector into two timelike vectors: for example, the four-current of the moving electrons and the four-current of the stationary protons.

As for null four-current, I wonder if there is any special significance, other than the fact that it is null in all reference frames? (Even by itself, this seems pretty intriguing.)

 

[Phrak]

I have no idea. I'll add it to my list of "100 or so physics questions to ponder." It's good enough that you're talking about massive stuff with null vectors.

Maybe I do have some ideas...

1. If we fail to attach mass to the divergence of the electric field (charge densty), then charge density will have a second order solutions that propagates at a velocity of c. But you've come up with this quantity that already has null nature, and doesn't require such a nonphysical charge attribute. Given all that, I'm curious as to what obtains when we put waves of null charge/current density in motion.

2. Extending your idea a little further, can we also consider null vectors in the energy and momentum of a combination of massive particles, so that we can also consider waves in this vector. Do these wave propagate at c?

3. How would light interact with null, or near null, electric current density?

 

[djy]

For #2, I don't think it's possible, because while electric charge can be positive or negative, mass can never be negative.

 

[yossell]

In trying to understand your question, I got stuck on your paragraph 4. In what sense can you decompose a spacelike vector into two timelike ones? I thought decomposing a vector meant taking its projections along the axes of a frame - but a spacelike vector cannot be decomposed into two timelike ones in this sense.

 

[djy]

In trying to understand your question, I got stuck on your paragraph 4. In what sense can you decompose a spacelike vector into two timelike ones? I thought decomposing a vector meant taking its projections along the axes of a frame - but a spacelike vector cannot be decomposed into two timelike ones in this sense.

Maybe 'decompose' is the wrong word. I mean that if you have a spacelike vector ${\bf j}$ then you can find two timelike vectors ${\bf j}_1, {\bf j}_2$ such that ${\bf j}_1 + {\bf j}_2 = {\bf j}$, as long as one is allowed to point "backward" in time--which is exactly what happens with negative charge density.

 

[yossell]

Ah - I think I see now. But then, in this sense, aren't null vectors also the sum of two timelike vectors, one forward and one backward in time?

 

[djy]

Yes, that too.