Why is charge/current density 4 vector a twisted differential 3-form?

[stevenb]

I thought I would post this question here rather than in the Classical Physics formum because I expect the GR experts might be better able to answer this.

I'm trying to get a phyisical/intuitive/geometrical explanation for why the charge-density/current 3-form (sometimes called a 4-vector and often referred to as J) is a twisted form.

By "twisted form" I mean a differential form that can be defined on a nonorientable manifold. It's clear to me why a 3-form is appropriate for J, but I can't seem to fathom why a twisted form is needed.

I think a key part of my question is that I don't really understand (aside from some non-intuitive mathematical statements) the important differences between a conventional differential form and a twisted differential form.

Any insight, even if incomplete, will be greatly appreciated.

 

[dx]

I think a key part of my question is that I don't really understand the important differences between a conventional differential form and a twisted differential form.

Consider a line segment. There are two ways one can orient this: along the segment and across the segment. For example, if you wanted to represent a segment of the world-line of a particle, then the first type of orientation is appropriate. On the other hand, imagine a circle drawn on a plane. A segment of this circle naturally has an orientation of the second type: it is oriented 'across' the segment, depending on which side of the circle is 'inside' and which is 'outside'

This is the main difference between differential forms and their twisted counterparts, i.e. the type of orientation.

The contour lines of a function have the 'across' orientation, and are represented by 1-forms. But if we wanted to represent coutour lines with an orientation along them instead of across, you would use a twisted 1-form.

Imagine 2+1 dimensional spacetime. I assume you're familiar with the usual picture of a 2-form in a three dimensional space. The 'tubes' or 'boxes' in the picture of this 2-form will have an orientation that is 'around' them, i.e. clockwise or anticlockwise. Of course, one can always convert from clockwise/anticlockwise to up/down using things like right-hand rules, but that is not the natural type of orientation of a current. For a twisted 2-form, on the other hand, the tubes or boxes will have the correct 'along' orientation. So, in 2+1 dimensional spacetime, current density is a twisted 2-form. Similarly, in 3+1 dimensions, it is a twisted 3-form.

 

[stevenb]

Thank you dx (I like your name by the way).

I appreciate your feedback on my question. I thought I wouldn't get any help on this.

I'm at work now, so I'll need to go through your explanation more carefully tonight. But, on quick review, I like that your description is an intuitive explanation which is really what I need right now. When I hit a mental road block like this I need to bounce between intuitive descriptions and formal math statements until it finally makes sense. I'll hit some books again with your comments in mind. I feel like I'm getting close now.