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

In summary, the charge-density/current 3-form (sometimes called a 4-vector and often referred to as J) is a twisted form that is appropriate for J. 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. Spacetime is described by a twisted 2-form and a twisted 3-form.
  • #1
stevenb
701
7
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.
 
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  • #2
stevenb said:
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.
 
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  • #3
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.
 

1. What is a charge/current density 4-vector?

A charge/current density 4-vector is a mathematical concept used in the study of electromagnetism. It represents the charge density and current density at a given point in space and time, and is often used to describe the behavior of electric and magnetic fields.

2. Why is it important to understand the 4-vector in terms of twisted differential 3-form?

The 4-vector is important to understand in terms of twisted differential 3-form because it helps to fully describe and understand the behavior of electromagnetic fields. The 4-vector and 3-form are closely related and provide a comprehensive mathematical framework for analyzing and predicting the behavior of electric and magnetic fields.

3. How is the 4-vector related to special relativity?

The 4-vector is related to special relativity in that it allows for the transformation of physical quantities, such as charge and current density, between different inertial reference frames. This is essential for understanding the behavior of electromagnetic fields in different frames of reference.

4. Can you give an example of how the 4-vector is used in practical applications?

One practical application of the 4-vector is in the design and analysis of electrical circuits. By understanding the 4-vector and its relationship to twisted differential 3-form, scientists and engineers are able to accurately predict the behavior of electric and magnetic fields in a circuit and optimize its performance.

5. Is the 4-vector a universal concept or does it only apply to electromagnetism?

The 4-vector is a universal concept that is used in various branches of physics, including electromagnetism, special relativity, and quantum mechanics. While it is most commonly associated with electromagnetism, it also has applications in other areas of physics and mathematics.

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