Do Total Current and Total Charge form a Lorentz Covariant Vector.

[Phrak]

Hi Phrak,

I was under the impression that you were comfortable with the charge 3-form language we were using earlier. There the total charge is simply the integral of the 3-form over all space at fixed time. Thus the total charge is an invariant geometric object and we don't need to say anything about transformation laws, etc. Also, as long as the charge is contained in a finite size region (just to avoid tricky business at infinity), one can evaluate the charge using any space-like hypersurface with the same asymptotics because of current conservation.

Here is another point of view. Total charge merely counts the total number of electrons minus the total number of positrons etc. These are discrete quantities which cannot continuously vary. Note that this is unlike the charge density which involves a choice of length and can be varied continuously.

And another. Take the 3-form and convert it to a 1-form and then raise the index to produce a vector. Take a system with only charge density and no currents and boost to a new frame. You will find that the charge density has changed according to Schwartz's formula (there will also now be currents). It is thus a special case of a more general transformation rule. However, the integration measure has also changed because you've "mixed up" space and time and hence the integral you wrote for the lorentz invariant total charge actually changes in two compensating ways.

Thanks for your help. I did find what I was looking for. The 1-form itself is the generally covariant field of charge and current, having nice properties as well. It canonically obeys PT symmetry. By the definition of charge density it also obeys CPT symmetry. However, raising the index to a vector ruins all this. Vectors are evil.

 

[Phrak]

The attached file should explain how charge and current combine in a Lorentz and diffeomorphism invariant way, sufficiently concise to be disprovable---well, maybe not as concise as I wished, but I want to get this out of the way and move on, unless anyone has anything of interest to add.

Any disproofs?

It is of note that the question of charge invariance over a volume is irrelevant. See the variables q_R and i_R.

For others, William L. Burke, Applied Differential Geometry, has an introduction to electromagnetism in the language of differential forms, though it is not sufficiently well developed to cover 4-current invariance as I do, in shorthand, in the attached.

 

[Physics Monkey]

Sorry, Phrak, but I find your note quite confusing. In 2.1 and 2.2 seem ok. 2.3 and 2.4 are already confusing. You seem to be suggesting that beyond J there is another 3-form rho that is just the charge density, but the charge density is already in J. The charge contained in a spatial volume V is simply integral of J over that volume. Then you introduce little j in 2.5 again with no obvious relation to anything else. In 2.8 and 2.9 you seem to be acknowledging that J, rho, and j are all related, but in 2.9 you're mixing 2-forms and 3-forms. In 2.10 you've gone back to only 3-forms, and your I=*J is just the usual current vector (once you raise the index) whose time component is charge density and whose three spatial components are current densities. You have the units wrong in the paragraph between 2.10 and 2.11. I has units of charge density or units of current density (which are the same when speed of light = 1), not units of charge and/or current. In 2.13 it looks like you almost have it right, but you again seem to be confusing charge density and current density with charge and current. To have a non-infinitesimal current at a single point in spacetime is highly singular, instead one should have some smooth current density at each point.

I'm sure some of this is just presentational, but I can't help but feel you're making something relatively simple overly complicated. And I think you understand the simple thing. 2.1 and 2.2 look fine. The total charge is the integral of J over a given spatial volume. There is no lorentz invariant notion of total current. How could there be since it involves time?

 

[Phrak]

I'm shocked.

2.1 defines J as the exterior derivative of the Hodge dual of F. We don't yet know where it's parts make contact with experimentally measurable things or things already defined in terms of vector calculus.

2.2 is not a derivation, but simply expands J into its components and bases on the r.h.s.

2.3 separates J into it's space³ and time-space² parts.

2.4 rho is shown in expanded form on the r.h.s with components and bases.

2.5 is Gauss' law. [Oops. 2.5 is the definition of nonrelativistic total charge in terms of density.]

2.6 is current density expanded on the r.h.s.

2.7 defines total current as the integral of charge density taken over an area.

2.8 is equation 2.3 restated for convenience.

2.9 rho and j are substituted into J. It says that rho is the space³ part of J is what we define as charge density. rho_ijk=J_ijk.
It also says that the time-space² part is what we define as current density; j_ij=J_0ijdt.

It is the most subtle part. I'm glad you criticized this one. It seems to mix apples and oranges. To get a better handle on this look at what is commonly done to combine energy and momentum into a 4-vector.

E⁽⁴⁾ = (E,p)

The shorthand notation obscures what's really going on. Somehow we've got to take a scalar and a 3-vector and combine them into a 4-vector. (I'm not saying the E is a scalar and p a 3-vector, but just presenting this equation as a problem example).

How do you do it?

 

[Phrak]

Physics Monkey,

You're right. It's an embarrassing mess. I first took your criticism for misunderstanding. Sorry. So if you could please ignore the above...

Thanks for looking it over. If I'm much more satisfied with it, maybe I'll post it again.

By the way, it's not actually total charge and total current that I intended to coherently unify, but charge and current. I simply didn't know any better how to express it at the time of post #1.

 

[Physics Monkey]

Physics Monkey,

You're right. It's an embarrassing mess. I first took your criticism for misunderstanding. Sorry. So if you could please ignore the above...

Thanks for looking it over. If I'm much more satisfied with it, maybe I'll post it again.

By the way, it's not actually total charge and total current that I intended to coherently unify, but charge and current. I simply didn't know any better how to express it at the time of post #1.

Well, as I said above, I think you've already got a roughly correct idea, there's just some baggage attached. Except for where you mixed 3-forms and 2-forms, I mostly found your note confusing because of a proliferation of symbols that all turned out to be related.

What is unified is charge density and current density. These are encoded in the 3-form J, or the dual one form, or the index raised vector, etc. J satisfies $$J = d*F$$ and $$dJ = 0$$ and you can integrate it over a spatial slice to get the total charge. That's basically it, and it seems to me that you've almost got it. Just dispense with the baggage

 

[Phrak]

What is unified is charge density and current density. These are encoded in the 3-form J, or the dual one form, or the index raised vector, etc.

I understand what you say, though I don't see how you do this. How would you unify the charge density and current density in three dimensions into J, a 3-form in four dimensions?

 

[Phrak]

I've come back to this problem and resolved it.

Charge density and current density do not form a vector. This would be an abuse of the physical units involved. (In this, I am challenging my own eletrodynamics text by Schwart, and it seems others by Griffith and Jackson if I've interpreted the comments, here, correctly.)

These current and charge densities, together, comprise a spacetime density, $J_{\mu\nu\theta} dx^\mu dx^\nu dx^\theta$

Now, it's easiest to transform everything in Minkowski coordinates for clarity, and deal with general covariance later.

Under this condition a covector $J_\sigma$ can be obtained from the density.
$$J_\sigma = {\epsilon_\sigma}^{\mu\nu\theta} J_{\mu\nu\theta}$$

In Minkowski coordinates elements of epsilon are just ones and zeros and negative ones. In generized coordinates this changes, but doesn't effect the overall argument.

The vector $J^\pi$ is obtained by applying the metric.

$J^\pi = g^{\pi\sigma}J_\sigma$

$J^\pi$ is an honest to God vector and transforms as a vector under a Lorentz transformation as it should.

The problem is, that in the general literature, it is identified as a 4 charge/current density. It is not.

The densities are clearly defined in terms of a 4-tensor with three lower indices. The units properly associates with the elements of $J^\sigma$ are:

$J^0 = J^0 [QT]$
$J^i = J^i [QD]$

$$J[Q] = J^0[QT] \partial_t[T^{-1}] + J^i[QD] \partial_i [D^{-1}]$$

The total vector has units of charge.

The units of $J^{\pi}$ would be "charge/current intensities" or "charge/current strengths". There is no such animal as a vector comprised of densities.

Does anyone have a problem with this?

(There is a problem in the community of tucking Units under the carpet such as setting c=1. Setting c=1 is not so bad, but an increasing tendency to use more shorthand obfuscates the physics underlying the mathematical formalism. It might be really cool and demonstrate one's sophistication, but can also lead to misconceptions in problems such as this where units values are valuable analytical tools.)

 

[pervect]

Charge density and current density form a well-known four vector, i.e.

http://en.wikipedia.org/w/index.php...#Examples_of_four-vectors_in_electromagnetism

It's a special case of the more general number-flux four vector

http://web.mit.edu/edbert/GR/gr2b.pdf

But I believe the answer to the original question regarding total current and total charge is "no". You could probably come up with a four-vector whose norm was the total charge with some work, though.

 

[Phrak]

Charge density and current density form a well-known four vector, i.e.

http://en.wikipedia.org/w/index.php...#Examples_of_four-vectors_in_electromagnetism

It's a special case of the more general number-flux four vector

http://web.mit.edu/edbert/GR/gr2b.pdf

Well, yes, it does transform as a vector (or, at least sticking to orthonormal coordinates it does). Actually it would be a pseudovector. But there's more than one tensor containing as elements $c \rho$ and $j^i$.

See page 32, of this paper, for instance: http://www.math.purdue.edu/~dvb/preprints/diffforms.pdf"
or the section Differential Geometric Formulations within the Wikipedia article "Maxwell's Equations".
http://en.wikipedia.org/wiki/Maxwell's_equations#Differential_geometric_formulations"

But I believe the answer to the original question regarding total current and total charge is "no". You could probably come up with a four-vector whose norm was the total charge with some work, though.

That could be... I may review the OP and other posts. $P^\mu = (E,p^i)$ is often given as elements of a vector where mass is the conserved charge--or invariant norm of the vector. It may be appropriate to apply this only to pointlike particles or extended systems where spacetime is not curved. In the same sense, with the same restrictions, I should also expect there to be a vector quantity corresponding to electric charge as the norm, as you also seem to be saying.

 

[pervect]

Are pseudovectors three-forms?

It seems to me this might relate to signed vs. unsigned volume elements, I'm pretty sure a signed volume element can naturally be represented by a three-form.

Usually I ignore the sign issues, but one of these days I want to find out the mathematically pure way of dealig with them.

 

[Phrak]

Are pseudovectors three-forms?

It seems to me this might relate to signed vs. unsigned volume elements, I'm pretty sure a signed volume element can naturally be represented by a three-form.

Usually I ignore the sign issues, but one of these days I want to find out the mathematically pure way of dealig with them.

I highly recommend learning about the mathematical machinary. Sean Carroll introduced it in his Lecture Notes on General Relativity, chapter 2. http://preposterousuniverse.com/grnotes/"

I expect that much of the mathematics used in relativity theory will be replaced by "differential forms" as more of it filters down from the mathematician.

A signed 3-volume element can be represented by the alternating bases dx^dy^dz. The wedge (^)is a particual kind of multiplication where interchange of any two operands changes the sign of the product.
A^B=-B^A.