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

[Phrak]

I did. I pointed out the error in your argument, and I gave an argument to the contrary.

Referring you to a book is not an appeal to authority. If you get out of your chair, walk to the library, and pull the book off the shelf, you can read it for yourself and judge for yourself whether your believe the argument.

Telling you that this is a totally standard statement in every textbook is not an appeal to authority. It's just a warning that you're being foolish. Similarly, if you try to tell me that the Nile river is in South America, I will tell you that a vast number of sources of information contradict you. That's not an appeal to authority, it's an appeal to evidence. If you want to make an argument that the Nile is in South America, feel free to go ahead and knock yourself out, but it's going to take an extraordinarily strong argument. Don't expect anyone else to take you seriously during your process of constructing such an argument, especially if you make other elementary errors along the way.

One of the problems here is that you seem to have a poor grasp of extremely basic issues, such as Lorentz scalars and vectors and the tensor transformation laws, but rather than fixing your confusion about those issues you keep insisting on trying to talk about things at an even higher level of mathematical sophistication. If you don't understand that a vector can't be invariant, then you aren't going to understand all the more sophisticated formalism that you're trying to write down.

If you can't keep up with the conversation, it is better to stay silent. You have had nothing to offer from your original claim beyond repetitious claims to authority and insulting innuendos and implications about myself. Please stop.

 

[dextercioby]

It doesn't matter which maths one uses (and how fancy it is) in a flat spacetime without torsion, a classical result of electrodynamics is that the electric charge is globally and locally conserved.

There's a book by Brian Felsager which pretty much exhausts these issues in electrodynamics.

 

[atyy]

Yes, certainly. Differential forms and the exterior algebra are a very powerful tools. The old way results in some very suspicious combination of terms such as combining a volumetric density and a flux density into a single 4-vector. For instance, current density is a type(0,3) tensor in a spacelike hypersurface and current density is a type(0,2) tensor. These are combined into a type(1,0) tensor in spacetime. What's that all about?

In differential forms we get a far less peculiar development. Promoting these objects to four dimensions, they combine as a single type(0,3) tensor. From here, a dual tensor of type (0,1) is had on applying the Hodge duality operator--or through regrouping of terms, as I did above. Multiply by the metric tensor and the type(1,0) is had.

OK, let me see see if I can translate it into old fashioned language.

We start with Maxwell's equations (http://farside.ph.utexas.edu/teaching/em/lectures/node53.html , Eq 556 - 559).

We can integrate Eq 556-559 to get Maxwell's equations in integral form (Eq 560-563).

These imply charge conservation (http://farside.ph.utexas.edu/teaching/em/lectures/node46.html , Eq 406).

In Eq, 406 the charge density and current density are Lorentz covariant.

We pick a particular Lorentz inertial frame. In that frame we have a box of metal that is stationary at a particular coordinate time t. If we integrate Eq 406 over the box of metal (analagous to going from Eq 556-559 to Eq 560-563 ), are the resulting terms Lorentz covariant or Lorentz invariant?

 

[Phrak]

It doesn't matter which maths one uses (and how fancy it is) in a flat spacetime without torsion, a classical result of electrodynamics is that the electric charge is globally and locally conserved.

Yes. Better known as the charge continuity equation. It remains unchanged whether the connection is torsion free or not, as long as spacetime is orientable. In fact, charge continuity is independent of the torsion.

 

[Mentz114]

I apologise if this is not relevant to this thread, but I have a question.

I've been working with a spacetime that contains charges and can calculate the EM field tensor (F) from a potential. I can also find the current by contracting the covariant derivative of F

<br /> {F^{\mu\nu}}_{;\nu}=J^\mu<br />

then using J^0=\sigma u^0 I can get \sigma, the charge density. All this is in the holonomic basis.

I've also done the calculation in the local comoving frame basis, and I don't get the same result for the charge density. I'm surprised by this, but should I be ?

 

[atyy]

OK, let me see see if I can translate it into old fashioned language.

We start with Maxwell's equations (http://farside.ph.utexas.edu/teaching/em/lectures/node53.html , Eq 556 - 559).

We can integrate Eq 556-559 to get Maxwell's equations in integral form (Eq 560-563).

These imply charge conservation (http://farside.ph.utexas.edu/teaching/em/lectures/node46.html , Eq 406).

In Eq, 406 the charge density and current density are Lorentz covariant.

We pick a particular Lorentz inertial frame. In that frame we have a box of metal that is stationary at a particular coordinate time t. If we integrate Eq 406 over the box of metal (analagous to going from Eq 556-559 to Eq 560-563 ), are the resulting terms Lorentz covariant or Lorentz invariant?

Hmmm, the charge defined this way is not obviously Lorentz invariant if, as Phrak said, stuff is flowing out of the box, since in a different frame, the walls of the box will enclose different stuff at a "particular point in time" because of the different definition of simultaneity.

I looked up Jackson's treatment and it is very simple.

He treats the case of a point particle.

The Lorentz force law is dp/dt=q(E+vXB).

q must be invariant if things are going to be ok.

Having defined q to be invariant, the corresponding charge and current densities for a particle with position r(t) in an inertial frame are qz(x-r(t)) and qv(t)z(x-r(t)), where z is the Dirac delta function.

 

[PAllen]

I'm not saying I agree with Pauli, but in his "Theory of Relativity" he states, or rather restates from a paper by Minkowski that charge is not invariant between coordinate systems. At what point did that become clarified or corrected in the literature?

What section is this? I have Pauli's 1921 "Theory of Relativity", and section 27, equations 197 - 200b, he derives that "The charge contained in given material volume element is an invariant".

 

[atyy]

What section is this? I have Pauli's 1921 "Theory of Relativity", and section 27, equations 197 - 200b, he derives that "The charge contained in given material volume element is an invariant".

Does he require that no stuff is flowing out of the volume?

 

[PAllen]

Does he require that no stuff is flowing out of the volume?

It doesn't seem so. He derives a law for transformation of charge density, and also for the volume of volume element, and shows that the product is invariant. This is after showing that (charge density, current density) form a 4-vector.

 

[atyy]

It doesn't seem so. He derives a law for transformation of charge density, and also for the volume of volume element, and shows that the product is invariant. This is after showing that (charge density, current density) form a 4-vector.

I see. That's what I said earlier in this thread, but then usually when we say charge, we (or at least I) usually mean the integral. What I don't understand is how the integral is going to be invariant once we include the limits.

 

[PAllen]

I see. That's what I said earlier in this thread, but then usually when we say charge, we (or at least I) usually mean the integral. What I don't understand is how the integral is going to be invariant once we include the limits.

In another old book I have, "Principles of Relativity Physics", by James L. Anderson (this book was praised in MTW for its day), the integral situation is addressed in section 8-4. I think when one says 'charge is invariant' it is sort of silly to talk about the case current flowing through a finite volume. What Anderson derives is that if current density 4 vector is nonzero in some bounded region, then for any hypersurface containing this region, and any coordinate system or frame of reference, the charge will be invariant.

 

[atyy]

In another old book I have, "Principles of Relativity Physics", by James L. Anderson (this book was praised in MTW for its day), the integral situation is addressed in section 8-4. I think when one says 'charge is invariant' it is sort of silly to talk about the case current flowing through a finite volume. What Anderson derives is that if current density 4 vector is nonzero in some bounded region, then for any hypersurface containing this region, and any coordinate system or frame of reference, the charge will be invariant.

Ok, that's seems to be in complete agreement with Phrak's point of view.

Wow, you have some good books (yes, I'd heard about Anderson's from MTW).

 

[Phrak]

Yes. Better known as the charge continuity equation. It remains unchanged whether the connection is torsion free or not, as long as spacetime is orientable. In fact, charge continuity is independent of the torsion.

I intended to say "In fact, charge continuity is independent of the connection."

 

[Phrak]

Hmmm, the charge defined this way is not obviously Lorentz invariant if, as Phrak said, stuff is flowing out of the box, since in a different frame, the walls of the box will enclose different stuff at a "particular point in time" because of the different definition of simultaneity.

You seem to have answered your own question well, as far as I can see. This is what Physics Monkey and I were discussing in posts #13, 15, 16 and 18, if you can read between the lines of mathematical incantations.

I looked up Jackson's treatment and it is very simple.

He treats the case of a point particle.

The Lorentz force law is dp/dt=q(E+vXB).

q must be invariant if things are going to be ok.

Having defined q to be invariant, the corresponding charge and current densities for a particle with position r(t) in an inertial frame are qz(x-r(t)) and qv(t)z(x-r(t)), where z is the Dirac delta function.

Can you post Jackson's argument from empirical evidence, if he has one? It looks as if this could be the oil drop experiment where the 'particle' is the drop of oil.

By the way I have Melvin Schwartz, Principles of Electrodynamics. He has a rather different argument for Lorentz invariant charge than the first guy I quoted, Allen Shadowitz.
I could post it if anyone is interested.

 

[Phrak]

I apologise if this is not relevant to this thread, but I have a question.

I've been working with a spacetime that contains charges and can calculate the EM field tensor (F) from a potential. I can also find the current by contracting the covariant derivative of F

<br /> {F^{\mu\nu}}_{;\nu}=J^\mu<br />

then using J^0=\sigma u^0 I can get \sigma, the charge density. All this is in the holonomic basis.

I've also done the calculation in the local comoving frame basis, and I don't get the same result for the charge density. I'm surprised by this, but should I be ?

I'm not familiar with your reference to holonomic basis. But maybe I should be; it could be something I should know! However, I am a little suspicious of you starting equation. Is this a textbook derived equation?

F is antisymmetric, so can you you can replace nabla with partial derivatives eliminating the connections?


However, presented in differential forms, j=d*F. It's just a different way to write 2 out of 4 of Maxwell's Equations. Same as your own, perhaps. So here goes.

F is the electromagnetic tensor with lower indeces. The shorthand is not so easy to expand in Latex, so I'll do it in parts.

G = *F = \frac{1}{2!}{\epsilon_{\mu\nu}}^{\sigma\rho}F_{\sigma\rho}

This defines the star operator acting on an a type(0,2) tensor. Epsilon is the completely antisymmetric tensor (in either curved or flat spacetime).

j = dG = 3\partial_{[\lambda} G_{\mu\nu]}

This defines 'd', the 'exterior derivative' acting on a type(0,2) tensor. The funny bracket placement is deliberate. It means dG is an antisymmetric tensor in all three indeces.

(dG)_{\lambda\mu\nu} = 3\left( \left[ (dG)_{\lambda\mu\nu} - (dG)_{\lambda\nu\mu} \right] + \left[ (dG)_{\nu\lambda\mu} - (dG)_{\nu\mu\lambda} \right] +\left[ (dG)_{\mu\nu\lambda} - (dG)_{\mu\lambda\nu} \right]\right)

j is a three index antisymmetric tensor and contains the charge and current densities. j has lower indeces. F, G, and j all have lower indeces and are all antisymmetric.

*j = \frac{1}{3!}{\epsilon_\pi}^{\lambda\mu\nu}j_{\lambda\mu\nu}

Here, the star operator acts on a type (0,3) tensor. The factor is 1/3! instead of 1/2! as it was the first time.

J_\pi = (*j)_\pi

Finally, and I'm running out of greek,

J^\zeta = g^{\zeta\pi}J_\pi

This should be equal to your vector for J. Is it? I don't know. If it isn't it cannot be right.

 

[PhilDSP]

What section is this? I have Pauli's 1921 "Theory of Relativity", and section 27, equations 197 - 200b, he derives that "The charge contained in given material volume element is an invariant".

I'm pretty sure I have the last revision from the 50's. Before I read the OP I was comparing several texts on a similar subject and was surprised to read something to the effect that a charge can appear in one frame but not another, especially because the value of charge can't be reduced to dimensions of either time or length in any known way. I remember it being a footnote, but now after searching Pauli's text I can't find the note. It could well have been from another book. I'll search again.

 

[Mentz114]

I'm not familiar with your reference to holonomic basis. But maybe I should be; it could be something I should know! However, I am a little suspicious of you starting equation. Is this a textbook derived equation?

F is antisymmetric, so can you you can replace nabla with partial derivatives eliminating the connections?However, presented in differential forms, j=d*F. It's just a different way to write 2 out of 4 of Maxwell's Equations. Same as your own, perhaps. So here goes.

[...]

'Holonomic basis' is a gnomic way of referring to the global coordinate basis as opposed to a local frame basis.

That formula, I assumed was an accepted way of writing some of Maxwell's equations in curved spacetime. But it looks as if I can replace the covariant deriv with a partial, even in curved spacetime. The equation is refrerred to as the EM part of the Einstein-Maxwell equations, see for instance ( but beware, this is slightly off the beaten track and regarded as 'unphysical' by a lot of relativists). Equation (5) in here

http://arxiv.org/PS_cache/gr-qc/pdf/0201/0201073v1.pdf

Thanks a lot for your trouble, I'll check your derivation as well as I can.

Going back to my original question, I still am not sure whether a local comoving observer can 'see' a different charge density from the holonomic one.

Local observers can see (feel?) different tidal tensors for instance, but a charge density ?

 

[PAllen]

I have a thought on the issue of the boundary of region containing charges and currents, versus different observers. Note that when we say the proper length of a ruler is invariant we mean that everyone computes the same length for the spacelike path that *one* observer views as a ruler, but other observers view as a sequence of non-simultaneous events with spacelike relationship to each other. The integral over the path is invariant, but the interpretation of the path is frame dependent.

Similarly, the charge in a given 3-volume of a given spacelike hypersurface is invariant; whether the hypersurface is a simultaneity surface is frame dependent.

 

[Mentz114]

I have a thought on the issue of the boundary of region containing charges and currents, versus different observers. Note that when we say the proper length of a ruler is invariant we mean that everyone computes the same length for the spacelike path that *one* observer views as a ruler, but other observers view as a sequence of non-simultaneous events with spacelike relationship to each other. The integral over the path is invariant, but the interpretation of the path is frame dependent.

Similarly, the charge in a given 3-volume of a given spacelike hypersurface is invariant; whether the hypersurface is a simultaneity surface is frame dependent.

I think I can see what you're saying and it inclines me to the view that different observers can see (measure, experience?) different values for the same charge density and probably even for matter density. It seems to be obvious in SR where electric fields change under LTs.

I don't know why I've been having trouble with this ... must be the weather.

[Later]
My problem might sorted. When I go from the local frame to the coordinate frame I have to use the tensor density

<br /> \epsilon_0 D^{\mu\nu}=\sqrt{-g}F^{\mu\nu}<br />

instead of F. g is the determinant of g_{\mu\nu} and is part of the discrepancy I found in the charge density.

This article is useful.

http://en.wikipedia.org/wiki/Maxwell's_equations_in_curved_spacetime

 

[PhilDSP]

On page 103 Pauli (1958) states “Equations (274) also allow us to go over to the integral form. From transformation formulae (269a) it follows that the splitting of the current into a conduction and a convection current is not independent of the reference system. Even when there is no charge densíty and only a conduction current present in K', there will appear a charge density, and hence also a convection current, in K. The corresponding transformation formulae are obtained from (269a) and (275),”

\rho&#039; = \rho \sqrt(1 - \beta^2) - \frac{(1/c)(v \cdot J_c)}{\sqrt(1 - \beta^2)}

\rho = \rho&#039; + \frac{(1/c)(v \cdot J&#039;_c)}{\sqrt(1 - \beta^2)}
Pauli defers much basic analysis to that provided by Lorentz. Digging into what Lorentz had to say:

In "The Theory of Electrons" 1916, Lorentz gives a fairly extensive physical and mathematical rationale for the determination of charge density (occasionally pointing to some lack of rigor for relativistic velocities).

See Lorentz p. 304 - 307 Notes 53 & 54

"In the definition of a mean value \bar{\varphi} given in § 113, it was expressly stated that the space S was to be of spherical form. It is easily seen, however, that we may as well give it any shape we like, provided that it be infinitely small in the physical sense. The equation

\bar{\rho}S = \int{\rho dS}

may therefore be interpreted by saying that for any space of the said kind the effective charge (meaning by these words the product of \bar{\rho} and S) is equal to the total real charge."

Lorentz seems to be more or less equating charge density, carried to the limit of an infinitesimal area, to charge itself.

One problem with all of this is that Lorentz originally theorized that the electron (and presumably protons) suffered deformation in relation to incident radiation from moving bodies (Heavyside's idea and analysis), rather than the measure of space being modified. While he apparently partially accepted the alternate interpretation of Poincare and Einstein of the relationship of space to time becoming deformed rather than the electron, much of the original concept and mathematical expression remains today becoming freely mixed with space-time symmetry, Does this result in a conflagration of potentially antagonistic concepts?

Evidently the fundamental question is whether the shape or spatial extent of a single charge is invariant across inertial frames. And by extension, is the shape or spatial extent of charge distributions (such as atoms and molecules) invariant? If not, then charge is not invariant at the point or infinitesimal area being evaluated in different inertial frames, nicht wahr?

 

[PAllen]

On page 103 Pauli (1958) states “Equations (274) also allow us to go over to the integral form. From transformation formulae (269a) it follows that the splitting of the current into a conduction and a convection current is not independent of the reference system. Even when there is no charge densíty and only a conduction current present in K', there will appear a charge density, and hence also a convection current, in K. The corresponding transformation formulae are obtained from (269a) and (275),”

\rho&#039; = \rho \sqrt(1 - \beta^2) - \frac{(1/c)(v \cdot J_c)}{\sqrt(1 - \beta^2)}

\rho = \rho&#039; + \frac{(1/c)(v \cdot J&#039;_c)}{\sqrt(1 - \beta^2)}
Pauli defers much basic analysis to that provided by Lorentz. Digging into what Lorentz had to say:

In "The Theory of Electrons" 1916, Lorentz gives a fairly extensive physical and mathematical rationale for the determination of charge density (occasionally pointing to some lack of rigor for relativistic velocities).

See Lorentz p. 304 - 307 Notes 53 & 54

"In the definition of a mean value \bar{\varphi} given in § 113, it was expressly stated that the space S was to be of spherical form. It is easily seen, however, that we may as well give it any shape we like, provided that it be infinitely small in the physical sense. The equation

\bar{\rho}S = \int{\rho dS}

may therefore be interpreted by saying that for any space of the said kind the effective charge (meaning by these words the product of \bar{\rho} and S) is equal to the total real charge."

Lorentz seems to be more or less equating charge density, carried to the limit of an infinitesimal area, to charge itself.

One problem with all of this is that Lorentz originally theorized that the electron (and presumably protons) suffered deformation in relation to incident radiation from moving bodies, rather than the measure of space being modified (Heavyside's idea and analysis). While he apparently partially accepted the alternate interpretation of Poincare and Einstein of space-time becoming deformed rather than the electron, much of the original concept and mathematical expression remains today becoming freely mixed with space-time symmetry, Does this result in a conflagration of potentially antagonistic concepts?

Evidently the fundamental question is whether the shape or spatial extent of a single charge is invariant across inertial frames. And by extension, is the shape or spatial extent of charge distributions (such as atoms and molecules) invariant? If not, then charge is not invariant at the point or infinitesimal area being evaluated in different inertial frames, nicht wahr?

I'm not sure this really disagrees with what Pauli derived in section 27 of the same book (where he showed that charge density times volume element is invariant, each changing in compensating way). Here his is not computing total charge, but the subdivision of current. As for the Lorentz analysis, I don't have access to see the whole, but have several books that rigorously derive charge invariance in various ways. Pauli's section 27 result alone implies what I claimed above about invariance of charge within a given volume of a spacelike hypersurface, which I claim is what is typically meant by an integrated invariant quantity. Anderson's book shows further, that if one computes total charge in a region such that there is no 4-current across its boundary, then the total contained charge will be the same for all such surfaces in all frames.

[Edit: If you go back to the text before eq. 272, on the prior page to your quote, he repeats the conclusion of section 27 (charge invariance), using it in the following derivations.]

 

[Phrak]

The more I delve into the standard accepted doctrine of classical electrodynamics, the more questionable it becomes. At key points I run into lack of mathematical rigor and questionable arguments.

However, I don't yet have a strong, rigorous reply.

One problem I am trying to resolve: If we accept current density as a 3-vector, can electromagnetism be CP invariant?

I am trying to argue that the so called "charge density vector" is not a vector. It can be physically represented as a type (0,2) tensor in 3-space without argument, but I think that arguing it to be a vector leads to the requirement that spacetime is not CP invariant.

 

[PAllen]

The more I delve into the standard accepted doctrine of classical electrodynamics, the more questionable it becomes. At key points I run into lack of mathematical rigor and questionable arguments.

I don't yet have a strong, rigorous reply.

Anderson's book supports this in a way. He notes that all treatments of point charges in classical EM are theoretically suspect hacks, only improved somewhat in QFT (this was in the early 60s, so newer reseults on the reliability of renormalization were not known). In the alternative, lumps of charged fluid have no theoretical anomalies, but are too cumbersome to derive many strong, rigourous results, especially since charged particles do not behave like lumps of fluid.

 

[atyy]

I haven't yet had the time to go back to the library and see if Jackson disucsses empirical evidence.

He does base his argument for charge invariance on the Lorentz force law for point particles.

But if Maxwell's equations + Lorentz force law are technically inconsistent, what is the replacement for the Lorentz force law?

Gralla et al have an interesting paper where the Lorentz force law is not assumed, only derived as an approximation. http://arxiv.org/abs/0905.2391

 

[PhilDSP]

In one sense, the Maxwell equations (plus the Lorentz force law which was included as a term within Maxwell's original equations) are immune or agnostic to whatever particular model of a charge is used. Or at least they give answers that are as accurate as the particular charge model is (point charge, spherical charge shell of an electron, deformed spherical electron, etc.,)

Doesn't the Dirac equation imply that the charge of an electron or components of it are in motion in at least 2 different ways and don't have a specific radius?

 

[atyy]

In another old book I have, "Principles of Relativity Physics", by James L. Anderson (this book was praised in MTW for its day), the integral situation is addressed in section 8-4. I think when one says 'charge is invariant' it is sort of silly to talk about the case current flowing through a finite volume. What Anderson derives is that if current density 4 vector is nonzero in some bounded region, then for any hypersurface containing this region, and any coordinate system or frame of reference, the charge will be invariant.

I have a thought on the issue of the boundary of region containing charges and currents, versus different observers. Note that when we say the proper length of a ruler is invariant we mean that everyone computes the same length for the spacelike path that *one* observer views as a ruler, but other observers view as a sequence of non-simultaneous events with spacelike relationship to each other. The integral over the path is invariant, but the interpretation of the path is frame dependent.

Similarly, the charge in a given 3-volume of a given spacelike hypersurface is invariant; whether the hypersurface is a simultaneity surface is frame dependent.

I guess the frame invariant definition is the one that people have in mind when they say charge is invariant, since if stuff is flowing out, we can't even get a globally conserved charge (d(integral of charge density)/dt=0). So the statement must be that if there is a globally conserved charge, it is frame invariant.

 

[Phrak]

'Holonomic basis' is a gnomic way of referring to the global coordinate basis as opposed to a local frame basis.

Oh! Then maybe my assessment is wrong.

 

[Phrak]

I think I can see what you're saying and it inclines me to the view that different observers can see (measure, experience?) different values for the same charge density and probably even for matter density. It seems to be obvious in SR where electric fields change under LTs.

I don't know why I've been having trouble with this ... must be the weather.

[Later]
My problem might sorted. When I go from the local frame to the coordinate frame I have to use the tensor density

<br /> \epsilon_0 D^{\mu\nu}=\sqrt{-g}F^{\mu\nu}<br />

instead of F. g is the determinant of g_{\mu\nu} and is part of the discrepancy I found in the charge density.

This article is useful.

http://en.wikipedia.org/wiki/Maxwell's_equations_in_curved_spacetime

I scanned the article. I don't know why people keep harking back to vectors and tensors with upper indices in things concerning electromagnetism. I've been quite successful and learned many new things in adhering to antisymmetric tensors. What is true of antisymmetric tensors in Riemann normal coordinates is true under any general linear transformation of coordinates and is connection-free. A powerful statement. The only tensor density involved is the Levi-Civita tensor which reverts to the Levi-Civita tensor of Minkowski spacetime in Riemann normal coordinates. The derivative operator is just a vector. Derivatives act on tensors like tensors act on tensors. Everything looks like multiplication. Instead of jumbled hash that looks like a failed attempt to discover underlying reality, they are simply formed. All four of Maxwell's equations can be expressed, not in one or two equations but no equations at all by making assignments of tensor elements to physically measurable quantities.

This brings up another point. People are obsessed with making vectors of things; apparently uncomfortable with dual vectors, but feeling safe with vectors and even safer with scalars:-

It is wrong to say that charge density is a scalar in space. By "space" I mean a spacelike hypersurface of spacetime. It is a pseudoscalar which has very different index placements. If you want the scalar equivalent of charge density, you can define it, but it is just charge, Q. (People seem to be happy to say that the density of charge varies under a Lorentz transform, but militantly deny that its compliment does not. Stated in this way, how can this not seem foolish?) In the same manner, current density is not a vector; it is a pseudo vector; a 2-form in space. It has two lower indices.
Look closely at Maxwell's equations as commonly presented in the vector calculus. In the differential equations of Maxwell's equations, why are we adding vectors to pseudo vectors? In the integral Maxwell equations why are we taking a dot product with an area? And, yes, I know it's really the dot product of the differential area times it's surface normal. So what we are really doing is integrating a 2-form, E dx^i \wedge dx^j or B dx^i \wedge dx^j to get a 0-form; a scalar.

In any case, if there would be anyone here that could hint at a rigorous solution to the OP , this would first have to be recognized so that I would know how to properly promote k-forms in 3-space to (k+1)-forms in spacetime.

 

[Phrak]

In closing, for the old school, here was the common wisdom as given by Melvin Schwartz, Nobel prize winner in physics for the co-discovery of the mu neutrino, Principles of Electrodynamics, section 3-3, year 1972, where this wisdom apparenty propagated down to Jackson:

"Fortunately, when the laws of physics were first set down, this problem was averted through the Lorentz invariance of total charge."

And a half-page later, the Lortentz transform of charge density.

"
\rho = \frac{\rho_0}{\sqrt{1-v^2/c^2}} \;\; \; \; \; \; \; \;\; \; \; \; \; \;\;\; \; \; \; \; \;3-3-5
"

With Q = \int \rho[/tex], how is it that one is Lorentz invariant, and the other is not?

 

[Physics Monkey]

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.

Hope this helps.