# Microscopic, and Tensoral Poynting Th., Vector, & Lorentz F.

1. Aug 13, 2015

### stedwards

It's a long title. I ran out of space.

It should be What are the microscopic, tensoral formulations of the Poynting theorem, Poynting vector and Lorentz force?

(If these cannot be stated in a diffeomorphism invariant way, what theoretical good are they?)

Oddly, I came up empty on an internet search for "covariant poynting theorem".

2. Aug 13, 2015

### ShayanJ

Read Special Relativity in General Frames by Eric Gourgoulhon.

3. Aug 14, 2015

### stedwards

Thanks, but the special theory is not generally covariant, and I don't have the text.

4. Aug 14, 2015

### ShayanJ

Its not about special or general relativity. Its about how the theory is presented. In the usual way that special relativity is presented, equations aren't generally covariant but the way this book presents them, they are.
I think you are confusing general covariance with the presence of curvature. You can have general covariance in a flat space-time, which is what the above book presents. So I guess you want this equations in the presence of space-time curvature. Then I should say that although the above book is about SR, it usually does the calculations in a general manner without assuming flat space-time, so it will be very useful even if it doesn't give you what you want exactly.
And I forgot to say, actually your question tricked me. The usual Poynting theorem is w.r.t. a reference frame because it chooses a particular slicing of space-time. Such a theorem can't be covariant. What you want is $\nabla \cdot (T^{mat}+T^{em})=0$, where $\nabla\cdot S$ is the covariant divergence of $S$ and $T^{mat}$ and $T^{em}$ are the SEM tensor of matter and EM field.

5. Aug 16, 2015

### stedwards

The poytning theorem without interpretation is directly desendendent from maxwells equations, and should not be in contradiction to the covariant derivative of the electromagnetic stress tensor, so there should be some logical development from maxwell to the stress tensor motivating the em stress tensor. Can you motivate the em stress tensor?

I was really hoping to see some parallel tensoral development for the pointing theorem, for one. You know, take the three dimensional nonsense in div, grad, dot, curl, ETC, and come up with something useful.

I have some old notes i've just reviewed (they're garbage), though two things are not so bad: The poynting theorem is the time-like part of a more general set of 4 equations directly decendent from maxwell's equations , and Hodge-wedge permutations of this set are repeated 3 other times.

Of this total of 16 identities, ponyting's theorem is 1/16.

Last edited: Aug 16, 2015
6. Aug 16, 2015

### ShayanJ

See here!

I'm not sure what you want!

7. Aug 17, 2015

### stedwards

I want it all and I want it now.

Seriously though, after little bit of review, this is really too much for a forum and I need to do some research. I do appreciate the hints.

Do you know an integral form of the Poynting vector equation?

8. Aug 18, 2015

### ShayanJ

See here![/PLAIN] [Broken]

Last edited by a moderator: May 7, 2017
9. Aug 18, 2015

### stedwards

Thanks. The integral form rounds things out. My job is to replace the hodge-podge development in vector calculus, at home in 3 dimensions + time, with a tensor equivalent development, and a few other things.

One thing to notice is that the left hand side of the Poynting vector (where charge and current are placed on the right), is $E \cdot (d*F) \pm B \cdot (dF)$.

F is the electromagnetic tensor, $*$ is the Hodge duality operator, and $d$ is the exterior derivative. I need to replace $E$ and $B$ with the electromagnetic tensor and resolve the sign issue.

Last edited: Aug 18, 2015
10. Aug 19, 2015

### vanhees71

Just look for Noether's theorem and electromagnetism. This should give you the derivation of the conservation laws from Hamilton's principle for classical field theory and Poincare invariance. If you can't find this, be patient, because I'm just writing on an SRT FAQ article, but just on the side as a hobby, and that's while it's going slow.

11. Aug 19, 2015