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

[Phrak]

And if so, How?

From the post 15 and 16 of the thread http://www.physicsforums.com/showthread.php?t=474719

yes. Charge is the timelike component of the four-current. So it is relative the same way that the components of any four-vector is.

Wait, you mean the charge *density*, right? Charge is a Lorentz scalar; this is verified to extremely high precision because the hydrogen atom is electrically neutral.

But total charge and total current, Q and I, do form a 4-vector, don't they? There seem to be two ways to solve this, but I can't figure out which one is right.

Properly speaking, charge density in 3 space is a pseudo scalar and current density is a pseudo vector.

\rho = \rho_{ijk}\ dx^i dx^j dx^k \

j = j_{ij}\ dx^i dx^j \ *

To each of these, there is a corresponding dual.

\hat{\rho_n}= \epsilon^{ijk}\rho_{ijk}

\hat{j}_i = {\epsilon_i}^{jk} j_{jk}

Where j can be though of as current flux density, \hat{j} might be called the "current strength" or "current intensity". It would be nice to call \rho a current flux, but is doesn't sound very good in 3 dimensions of space unlike the case in spacetime.

Raising the index on j and combining to a 4-vector,

(\hat{\rho}, \hat{j}^i )

could form a proper Lorentz invariant 4-vector. Integrating over a 4-volume could then yield (Q, I), give or take a negative sign.

The second way would be the inverse sequence of operations above. Integrate rho and j over a 4-volume then raise the index on the spatial components.

* To be precise, these should be a directed volume and a directed area,
\rho = \rho_{ijk} \ dx^i \wedge dx^j \wedge dx^k
j = j_{ij} \ dx^i \wedge dx^j \ .

[bcrowell]

I think there's a pretty straightforward proof that (Q,I) can't be a four-vector. We know that Q is a Lorentz scalar, and therefore under a Lorentz transformation it has to stay the same. So (Q,I) would be a four-vector whose timelike component was invariant under Lorentz transformations. But that can't be right, because it's always possible to change the timelike component of a given four-vector by performing some Lorentz transformation.

[DaleSpam]

Charge is a scalar, so it cannot be a component of any four-vector.

I don't think that current is a component of any four-vector since it is equal to a scalar times a three-vector. But I am not certain about that.

[Phrak]

I think there's a pretty straightforward proof that (Q,I) can't be a four-vector. We know that Q is a Lorentz scalar, and therefore under a Lorentz transformation it has to stay the same. So (Q,I) would be a four-vector whose timelike component was invariant under Lorentz transformations. But that can't be right, because it's always possible to change the timelike component of a given four-vector by performing some Lorentz transformation.

Charge is a scalar, so it cannot be a component of any four-vector.

I don't think that current is a component of any four-vector since it is equal to a scalar times a three-vector. But I am not certain about that.

Well, velocity itself can be coerced into a 4-vector.

If not charge, then what quantity, X, satisfies (X,I) is a Lorentzian 4-vector?

I'm very shocked myself to find what looks like charge to be sitting in a 4-vector. So I still have some doubts. Charge is conserved according to the charge continuity equation directly derived from Maxwell, but this this is not the same as saying charge is a Lorentz scalar.

Charge Q satisfies

Q = \int_\Omega dx^4 \rho \ .

Charge density \rho is not a Lorentz scalar, so charge cannot be a Lorentz scalar, right?

[atyy]

Charge Q satisfies

Q = \int_\Omega dx^4 \rho \ .

Charge density \rho is not a Lorentz scalar, so charge cannot be a Lorentz scalar, right?

I'd do it over d3x, which should transform such that Q is the same (I haven't checked, but in principle ...).

[bcrowell]

Charge density \rho is not a Lorentz scalar, so charge cannot be a Lorentz scalar, right?

Charge is a Lorentz scalar, as demonstrated to extremely high precision by the fact that hydrogen atoms are neutral. Charge density is not a Lorentz scalar. Charge density is the timelike component of a four-vector made out of the charge density and the current density. You don't need to speculate about all this. You can find it in any standard relativity text.

[Phrak]

Charge is a Lorentz scalar, as demonstrated to extremely high precision by the fact that hydrogen atoms are neutral. Charge density is not a Lorentz scalar. Charge density is the timelike component of a four-vector made out of the charge density and the current density. You don't need to speculate about all this. You can find it in any standard relativity text.

This is exactly how this issue came up in the first place. Charge density combined with current density is not a 4-vector, it is a 3-form assignable to each point of a pseudo Riemann manifold. There is a complementary one-form obtained from acting the Levi-Civita tensor on the 3-form. If this one-form cannot be composed of Q and I, then what?

[Phrak]

I'd do it over d3x, which should transform such that Q is the same (I haven't checked, but in principle ...).

Yes, you're right. The correction doesn't seem to change expectations, but maybe...

[DaleSpam]

Well, velocity itself can be coerced into a 4-vector.AFAIK the three velocity is not the space like part of any four vector. It is limited to c, so it doesn't transform right. Gamma times v is, but not v.

[homology]

This is exactly how this issue came up in the first place. Charge density combined with current density is not a 4-vector, it is a 3-form assignable to each point of a pseudo Riemann manifold. There is a complementary one-form obtained from acting the Levi-Civita tensor on the 3-form. If this one-form cannot be composed of Q and I, then what?

I understand what you're saying, however bcrowell is correct, its pretty standard to define J^{\alpha}=(c\rho, \vec{J}). But now you've got me curious so I took out an old friend: "The Geometry of Physics" He agrees with everyone. He defines his current 4-vector but also ties it to an associated 3-form by contracting the 4-volume with the current vector. If you have a copy its section 7.2b on page 199 in my edition.

I'd say more but while I'm studying EM and relativity the course I'm in doesn't have this lovely geometric side :(

[Physics Monkey]

If you are interested in using a very geometric language, then a nice way to write Maxwell's equations is

d*F = J

where F = dA is the field strength (a 2-form) and J is a current 3-form. Local current conservation is written dJ = 0 .

Total charge is the Lorentz scalar obtained from the natural pairing between J and a spatial slice \Sigma as Q = \int_{\Sigma} J . I think this point of view is useful because it clarifies why there is no natural way to make an invariant current i.e. you're already using all of J just to make Q .

Hope this helps.

[Phrak]

AFAIK the three velocity is not the space like part of any four vector. It is limited to c, so it doesn't transform right. Gamma times v is, but not v.

Yes, good point. Raising velocity from a 3-vector to a 4-vector results in an unusual looking 4-vector. I'll have to see why that is. The energy momentum 4 vector should also be interesting to understand; it should have an equivalent energy density + momentum flux density counterpart.


I understand what you're saying, however bcrowell is correct, its pretty standard to define J^{\alpha}=(c\rho, \vec{J}). But now you've got me curious so I took out an old friend: "The Geometry of Physics" He agrees with everyone. He defines his current 4-vector but also ties it to an associated 3-form by contracting the 4-volume with the current vector. If you have a copy its section 7.2b on page 199 in my edition.

I'd say more but while I'm studying EM and relativity the course I'm in doesn't have this lovely geometric side :(

I don't have a copy, though it sounds like a good text to have. However, other than sign convention, I'm not sure there's any room left to define the 4-current once the 4-current density is defined, as Physics Monkey presents in the post directly following yours; that is J=dG. I expect to find out, one way or the other, though.


If you are interested in using a very geometric language, then a nice way to write Maxwell's equations is

d*F = J

where F = dA is the field strength (a 2-form) and J is a current 3-form. Local current conservation is written dJ = 0 .

I've spent tedious hours converting electromagnetism back and forth between differential forms and vector calculus. So, I'm OK with charge continuity.

Total charge is the Lorentz scalar obtained from the natural pairing between J

and a spatial slice \Sigma as Q = \int_{\Sigma} J . I think this point of view is useful because it clarifies why there is no natural way to make an invariant current i.e. you're already using all of J just to make Q .

Hope this helps.

Yes, thank you. I don't have a good grasp of how k-forms change between subspaces, let alone spacelike slices, but I'm getting there. Changing integrals to and from subspaces is a challenge.

Rephrased, my question is as follows: Given J=dF, define K=*J. Raise the index in K and separate into spacelike and timelike parts, or one scalar and a 3-vector. What are these two elements physically identified with?

[Physics Monkey]

Yes, thank you. I don't have a good grasp of how k-forms change between subspaces, let alone spacelike slices, but I'm getting there. Changing integrals to and from subspaces is a challenge.

Rephrased, my question is as follows: Given J=dF, define K=*J. Raise the index in K and separate into spacelike and timelike parts, or one scalar and a 3-vector. What are these two elements physically identified with?

Great to hear. Here are some additional details. Let me ignore signs, etc and just sketch the basic structure. The 3-form J looks like this
J = \rho \, dx dy dz + j_x \, dt dy dz + j_y \, dt dx dz + j_z \, dt dx dy and the dual is obtained by replacing the 3 d's with the fourth missing d as appropriate. \rho is the charge density and j_i is the current density in a particular frame. In principle, for a given hypersurface you need all four pieces of information to compute Q .

Here is a nice example. Let \Sigma_1 be the equal time surface in x,y,z,t coordinates above. Let \Sigma_2 be the hypersurface obtained from \Sigma_1 by "bumping" a little bit of \Sigma_1 forward in time. It helps to draw a picture.

Since \Sigma_2 is a continuous deformation of \Sigma_1 , the charges Q_1 and Q_2 are the same. Q_1 = \int_{\Sigma_1} J = \int_{\Sigma_2} J + \int_{M_{1 \cup 2}} dJ = Q_2 where M_{1 \cup 2} is a four dimensional subspace with boundary given by the differentce between \Sigma_1 and \Sigma_2 i.e. a little box.

More physically, integrating J over \Sigma_2 contains several terms: integrals over \rho at different times in different regions of space and integrals over the the "sides" of the box involving j_i . You'll notice that I need to integrate over dt dx dy, for example, to pick up a contribution from j_z . This is simply telling me that I need to figure out the total charge (integral over dt) that went through a given plane (integral over dx dy) in terms of the appropriate current density j_z . To summarize, I get the same answer for \Sigma_2 because even though I look at the charge density in one region (the bump) at a later time, I subtract off all the charge that entered that region via the terms involving j_i . However, it also shows that I really need all the components of J to make a lorentz invariant quantity.

I hope that wasn't too elementary.

[PhilDSP]

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?

[Phrak]

Great to hear. Here are some additional details. Let me ignore signs, etc and just sketch the basic structure. The 3-form J looks like this
J = \rho \, dx dy dz + j_x \, dt dy dz + j_y \, dt dx dz + j_z \, dt dx dy and the dual is obtained by replacing the 3 d's with the fourth missing d as appropriate. \rho is the charge density and j_i is the current density in a particular frame. In principle, for a given hypersurface you need all four pieces of information to compute Q .

Here is a nice example. Let \Sigma_1 be the equal time surface in x,y,z,t coordinates above. Let \Sigma_2 be the hypersurface obtained from \Sigma_1 by "bumping" a little bit of \Sigma_1 forward in time. It helps to draw a picture.

That helps a great deal. I need all the visual tools I can get, and simplifying the indexing to x,y,z,t helps put things in more manageable perspective

I'm not sure I agree with your answer, though. Integrating over spacial parts
Q = \int_\Sigma \rho_{xyz}dxdydy
seems to be what we measure, and call charge.

[Physics Monkey]

That helps a great deal. I need all the visual tools I can get, and simplifying the indexing to x,y,z,t helps put things in more manageable perspective

I'm not sure I agree with your answer, though. Integrating over spacial parts
Q = \int_\Sigma \rho_{xyz}dxdydy
seems to be what we measure, and call charge.

I agree, that is what we call the charge and is precisely what hypersurface \Sigma_1 directly computes. Following the bump construction I mentioned above, define B to be the bumped spatial region and A to the be the rest of space. Integrating over \Sigma_1 tells me that the total charge Q is (charge in A at time t0 + charge in B at time t0). This is how we normally evaluate the total charge. What happens with \Sigma_2 is simply that we write the total charge Q as (charge in A at time t0 + charge in B at time t1 > t0 - charge that flowed into B between t1 and t0). Either way we get the same total charge as a consequence of charge conservation.

[Phrak]

Charge is a Lorentz scalar, as demonstrated to extremely high precision by the fact that hydrogen atoms are neutral. Charge density is not a Lorentz scalar. Charge density is the timelike component of a four-vector made out of the charge density and the current density. You don't need to speculate about all this. You can find it in any standard relativity text.

This all gets better and better. I don't have a standard text on special relativity, but I do have Special Relativity by Albert Shadowitz:

The first question to arise is--do A and B [in relative motion] each measure the same total charge Q? [...] It turns of that for charge, however, there is excellent experimental evidence that the charge is, indeed, invariant: both A and B will measure the same value of Q regardless of the value of v. One such piece of evidence is provided by the electrical neutrality of both He and H. Each of these contain two electrons and two protons but the motions of the particles are quite different n the helium atom than in the hydrogen molecule.

The evidence is misapplied. Shadowitz is wrong. It might be simplest to consider a single atom within a region V. As long as the entire system of charges is within the region, the total charge stays constant. If any charge leaves the region, there is an accompanying current leaving the box.

For a system of identically charged particles, the charge covaries with the number count. The number density and the number count are not Lorentz invariant. The number flux and number count together, both form a Lorentz invariant vector. This fact should be present in a sufficiently enlighten book on relativity, I would think. I don't have such a text, so I'm just guessing.

[Phrak]

I agree, that is what we call the charge and is precisely what hypersurface \Sigma_1 directly computes. Following the bump construction I mentioned above, define B to be the bumped spatial region and A to the be the rest of space. Integrating over \Sigma_1 tells me that the total charge Q is (charge in A at time t0 + charge in B at time t0). This is how we normally evaluate the total charge. What happens with \Sigma_2 is simply that we write the total charge Q as (charge in A at time t0 + charge in B at time t1 > t0 - charge that flowed into B between t1 and t0). Either way we get the same total charge as a consequence of charge conservation.

OK. This is good. Thanks for providing me some motivation in resolving this. The experimental setup is a bump function within a spacetime 4-volume. The bump function intersects the bounding hyersurfaces \Sigma_1 and \Sigma_2 but not the other 3-surface boundaries. Thinking of our bump function as a charged particle, the charge is the same on \Sigma_1 and \Sigma_2 as long as we don't let the particle out of the 3-volume we have it in. Otherwise there is a current flux out of the box. So we can see that this particular bump function is a special case, right?

In make sense of this, I may have found a remarkably easy way to simplify manipulation of k-forms, so I'll attempt to use it. It also has the very useful property of making the signs and the permutation count evident.

Again, using simplified notation and orthonormal coordinates in dx, dy, dz, and d(ct), partition the n-form J, where n=4 dimensions into spatial and temporal parts. (It makes sense to define J as an n-form, as you will see.)

J = J d(ct)^dx^dy^dz. The components have units of Q·D-4 and the bases have units of D4. So, overall, the tensor has units of charge. I’ve dropped the subscripts for clarity.

Grouping cJ dtdxdydz, the way we are used to, for charge density and current density:

J = 6(cJdt)dxdydz + 6c(Jdtdxdy)dz + 6c(Jdtdydz)dx + 6c(Jdtdzdx)dy

Note that in regrouping, the terms in parenthesis constitute the tensor components. The terms standing to the right of the parenthesis are the basis covectors.

(1/6)J = ρ dxdydz + cjzdz + cjydy + cjxdx

Grouping, instead, for total charge and total current:

J = -6(cJdxdydz)dt + 6c(Jdtdxdy)dz + 6c(Jdtdydz)dx + 6c(Jdtdzdx)dy

(1/6)J = -Qdt + cIzdz + cIxdx + cIydy

(-Q, cI) is a covector.
(Q, cI) is a vector.
Q is the total charge.
I is the total current.
c(Jdxdydzdt) is the Lorentz invariant charge pseudo scalar of charge and current.

A little work and we should have a properly formulated Lorentz invariant Kirchoff current law.

Notice in replacing units of charge with units of energy we would end up with a covector of (-E, cp), and should eventually discover that -J2 = m2c4= E2-c2p2. What do you think?

[bcrowell]

The evidence is misapplied. Shadowitz is wrong. It might be simplest to consider a single atom within a region V. As long as the entire system of charges is within the region, the total charge stays constant. If any charge leaves the region, there is an accompanying current leaving the box.

For a system of identically charged particles, the charge covaries with the number count. The number density and the number count are not Lorentz invariant. The number flux and number count together, both form a Lorentz invariant vector. This fact should be present in a sufficiently enlighten book on relativity, I would think. I don't have such a text, so I'm just guessing.

As far as I know there is no controversy on this point in the literature. Shadowitz is right, and charge is a Lorentz invariant.

[Phrak]

As far as I know there is no controversy on this point in the literature. Shadowitz is right, and charge is a Lorentz invariant.

Shadowitz jumped to a conclusion without a supporting argument. Do you have a valid argument to supply?

My argument is simple enough. If you have a charge in any state of motion in a bound region of space, the charge doesn't change upon its state of motion. This is the same evidence supplied by the equivalence of charge of He and H2. The conclusion made is that charge, so defined as charge density times volume, is Lorentz invariant. But upon what grounds? If there is anything in the literature that can validly support this claim better than hand waving arguments, I'd be wildly surprised.

If you can find any valid errors in my argument I would like to hear it.

Now, you should be interested to know that there is a Lorentz invariant charge; a vector quantity. Using more intuitive units than I did in my previous post,

Q = (QR, -cI)

This obtains Q2 = QR2 - c2I2

Q is the invariant charge. QR is the relativistic charge. In the claim made by Shadowitz, it is QR that is invariant.

This is exactly the same derivation to obtain m2c4 = mR2c4 - c2p2 from energy rather than charge, where we would never claim that relativistic mass were Lorentz invariant.

[bcrowell]

Shadowitz jumped to a conclusion without a supporting argument. Do you have a valid argument to supply?
Yes, see #6.

My argument is simple enough. If you have a charge in any state of motion in a bound region of space, the charge doesn't change upon its state of motion. This is the same evidence supplied by the equivalence of charge of He and H2. The conclusion made is that charge, so defined as charge density times volume, is Lorentz invariant.
Yes, except that there is no reason to define charge as charge density times volume. If a hydrogen atom accelerates in an electric field, it's charged. If it doesn't, it's not.


If you can find any valid errors in my argument I would like to hear it.
The argument you gave in #17 doesn't make sense.

For a system of identically charged particles, the charge covaries with the number count. The number density and the number count are not Lorentz invariant.
It's true that the number density is frame-dependent, but that doesn't connect in any logical way to anything else you've said.


Now, you should be interested to know that there is a Lorentz invariant charge; a vector quantity.
There is no such thing as a Lorentz-invariant vector. A Lorentz-invariant quantity is a rank-0 tensor. A vector is a rank-1 tensor. Rank-1 tensors transform according to the tensor transformation law, so they can't be invariant.

For an example of a standard textbook presentation of the fact that charge is Lorentz invariant, see Jackson, Classical Electrodynamics, 3rd ed., section 11.9, "Invariance of Electric Charge; Covariance of Electrodynamics." Jackson is the standard graduate text on E&M. Of course it's possible that Jackson has made a mistake of truly epic proportions, that he repeated it in all three editions of his book, and that nobody ever corrected it -- but if you want to convince us of that, it's going to take an extraordinary effort. Another standard textbook discussion of this is given in Purcell, Electricity and Magnetism.

[Phrak]

Yes, see #6.


Yes, except that there is no reason to define charge as charge density times volume. If a hydrogen atom accelerates in an electric field, it's charged. If it doesn't, it's not.


The argument you gave in #17 doesn't make sense.


It's true that the number density is frame-dependent, but that doesn't connect in any logical way to anything else you've said.


There is no such thing as a Lorentz-invariant vector. A Lorentz-invariant quantity is a rank-0 tensor. A vector is a rank-1 tensor. Rank-1 tensors transform according to the tensor transformation law, so they can't be invariant.

For an example of a standard textbook presentation of the fact that charge is Lorentz invariant, see Jackson, Classical Electrodynamics, 3rd ed., section 11.9, "Invariance of Electric Charge; Covariance of Electrodynamics." Jackson is the standard graduate text on E&M. Of course it's possible that Jackson has made a mistake of truly epic proportions, that he repeated it in all three editions of his book, and that nobody ever corrected it -- but if you want to convince us of that, it's going to take an extraordinary effort. Another standard textbook discussion of this is given in Purcell, Electricity and Magnetism.

Good. Can you post it in any form?

[atyy]

Notice in replacing units of charge with units of energy we would end up with a covector of (-E, cp), and should eventually discover that -J2 = m2c4= E2-c2p2. What do you think?

To run the same argument for mass, you should start from the energy density, just like you started from the charge density. The energy density is some component of the stress-energy tensor.

[bcrowell]

Good. Can you post it in any form?

No. It's copyrighted. This is what libraries are for.

[Phrak]

You would do better as an authority to do more than appeal to authority.

I challenge you to obtain the relativistic charge-current equation from the relativistic Maxwell equations.

[Phrak]

To run the same argument for mass, you should start from the energy density, just like you started from the charge density. The energy density is some component of the stress-energy tensor.

Yes, it is, thank you. It is also the component of a generally covariant type(0,3) tensor.

[bcrowell]

You would do better as an authority to do more than appeal to authority.

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.

[Antiphon]

As long as we're hand-waiving (Shadowitz that is) I suggest that the neutrality of the H atom means nothing in this context. Consider the fact that the proton and electron have wildly different motion yet neither radiates. Should we conclude that radiation is independent of motion?

This has to resolved rigorously and this thread is on a good track. My intuition (about people, not physics) tells me that indeed if charge were not a relativistic invariant someone would have rushed to invalidate the textbooks and make a name for themselves. Now let's get to the bottom of it by derivation.

[atyy]

Yes, it is, thank you. It is also the component of a generally covariant type(0,3) tensor.

Do you get the rest mass to be an invariant if you run your argument for charge density (and current density) on the energy density (and energy flux)? (Sorry, for being so elliptic, I'm not familiar with your notation, just vaguely remembered having some trouble myself in old-fashioned notation, and thought this might be similar.)

[Phrak]

Do you get the rest mass to be an invariant if you run your argument for charge density (and current density) on the energy density (and energy flux)? (Sorry, for being so elliptic, I'm not familiar with your notation, just vaguely remembered having some trouble myself in old-fashioned notation, and thought this might be similar.)

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 in to 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.

[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 in to 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


{F^{\mu\nu}}_{;\nu}=J^\mu


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


{F^{\mu\nu}}_{;\nu}=J^\mu


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_{\si gma\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_{\lamb da\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


\epsilon_0 D^{\mu\nu}=\sqrt{-g}F^{\mu\nu}


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%27s_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' = \rho \sqrt(1 - \beta^2) - \frac{(1/c)(v \cdot J_c)}{\sqrt(1 - \beta^2)}

\rho = \rho' + \frac{(1/c)(v \cdot J'_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' = \rho \sqrt(1 - \beta^2) - \frac{(1/c)(v \cdot J_c)}{\sqrt(1 - \beta^2)}

\rho = \rho' + \frac{(1/c)(v \cdot J'_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


\epsilon_0 D^{\mu\nu}=\sqrt{-g}F^{\mu\nu}


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%27s_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 [itex]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.

[PAllen]

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

To answer just this question, Pauli showed that the dv transforms inversely to charge density, making the integral invariant.

[atyy]

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.

You've probably said it here, but I'm not familiar with the new language. What is the counterpart in that language that (charge density).dV is frame invariant?

In understand roughly enough what the counterparts are in your post to the other statements that PAllen gives in posts #41 and #48.

[pervect]

I very much doubt there is any fundamental problem with the usual formalism of treating current density and charge density as a four vector. Everything transforms as a tensor.

What's probably true is that treating current and charge (rather than current density and charge density) as a four vector is allowed if - and only if - a system is isolated.

This is rather similar to the way momentum and energy work.

It's fairly well known that the energy-momentum of an object with a volume greater than zero is not in general covariant, this is mentioned for instance in http://arxiv.org/abs/physics/0505004.

However, an isolated object does have an covariant energy-momentum 4-vector, as mentioned in basic SR books, for instance Taylor & Wheeler. The confusion sneaks in if one forgets the conditions mentioned in said basic textbooks that the object be isolated.

The situation with charge is similar, IMO.

[Phrak]

I very much doubt there is any fundamental problem with the usual formalism of treating current density and charge density as a four vector. Everything transforms as a tensor.

It's not CPT invariant, is it?

[PAllen]

It's not CPT invariant, is it?

Can you explain this? It's a 4-vecor in classical Maxwell theory in SR. I would have thought not beint CPT invariant is impossible for such an object (but I admit my limited expertise, would welcome an explanation).

[atyy]

Invariance under charge conjugation is a property of Maxwell's equations (reverse charge, reverse fields).

The analogous situation eg. time reversal invariance for Newtonian gravity is applied to the equations of motion (reverse t, reverse p), not to an object like p.

[PAllen]

I very much doubt there is any fundamental problem with the usual formalism of treating current density and charge density as a four vector. Everything transforms as a tensor.

What's probably true is that treating current and charge (rather than current density and charge density) as a four vector is allowed if - and only if - a system is isolated.

This is rather similar to the way momentum and energy work.

It's fairly well known that the energy-momentum of an object with a volume greater than zero is not in general covariant, this is mentioned for instance in http://arxiv.org/abs/physics/0505004.

However, an isolated object does have an covariant energy-momentum 4-vector, as mentioned in basic SR books, for instance Taylor & Wheeler. The confusion sneaks in if one forgets the conditions mentioned in said basic textbooks that the object be isolated.

The situation with charge is similar, IMO.

Is it really similar? The energy, by itself, whether of a particle or finite system, is frame dependent. The charge of any isolated object (point, or finite) is invariant. That's been the whole point of the discussion (going back to Dalespam's first response). Thus, even in this limited sense, charge (rather than charge density) cannot be made part of a 4 vector. Then, we've also been trying to clarify what exactly is meant by charge invariance in non-trivial cirumstances.

[Phrak]

Invariance under charge conjugation is a property of Maxwell's equations (reverse charge, reverse fields).

The analogous situation eg. time reversal invariance for Newtonian gravity is applied to the equations of motion (reverse t, reverse p), not to an object like p.

I think my use of the word invariance was incorrect--or too trite to be correct, by the way.

This could be an entire topic of it's own right in the domain of relativistic field theory that I'd considering opening in a thread of it's own. I'm not sure how to put it though.

There should be a concrete way to express it in concise mathematical terms. The even vs. odd number of inversions of spacetime coordinates might best be expressed as even and odd permutations of the indices of the Levi-Civita tensor of 4 dimensions. Adding charge as a dimension, CPT might be associated with a 5 dimensional Levi-Civita Symbol. How this n=5 LCS would be used to examine various n=4 tensors, however, could be bit challenging. But the n=4 LCS might not be so difficult for the case of electromagnetism expressed in proper 4 dimensional equations. Any ideas on how to present this?

[Phrak]

I very much doubt there is any fundamental problem with the usual formalism of treating current density and charge density as a four vector. Everything transforms as a tensor.

What's probably true is that treating current and charge (rather than current density and charge density) as a four vector is allowed if - and only if - a system is isolated.

This is rather similar to the way momentum and energy work.

It's fairly well known that the energy-momentum of an object with a volume greater than zero is not in general covariant, this is mentioned for instance in http://arxiv.org/abs/physics/0505004.

However, an isolated object does have an covariant energy-momentum 4-vector, as mentioned in basic SR books, for instance Taylor & Wheeler. The confusion sneaks in if one forgets the conditions mentioned in said basic textbooks that the object be isolated.

The situation with charge is similar, IMO.

I had time to look over your post and give it better consideration. Only a few hours earlier I lately came to the realization that there is really only one way to simply combine charge and current into a generally covariant form. This form is a 1-form and has not been discussed here but by myself, but only the 3-forms have been discussed. These entities, having charge and current density, are then integrated to indirectly obtain a relationship between charge and current.

Looking closely at the charge-current 1-form, it cannot be covariantly integrated to obtain total charge and total current. This comes from integrating the 3-form of charge and current density. The 1-form expresses "charge strength" and "current strength" at a point, if you will forgive my terminology.

Now, 1) I've been very careful to ensure all expressions are generally covariant and 2) have noticed the exact same relationships exists between energy and momentum: The energy momentum 1-form I obtain applies to to each point on the spacetime manifold and forms a field, but is not integrated over a system to obtain a generally covariant combination of total energy and total momentum. Integrating over a subspace breaks it.

So today, I'm not suprised at all to read

"It's fairly well known that the energy-momentum of an object with a volume greater than zero is not in general covariant,..."


as you stated.

[Phrak]

Can you explain this? It's a 4-vecor in classical Maxwell theory in SR. I would have thought not beint CPT invariant is impossible for such an object (but I admit my limited expertise, would welcome an explanation).

The usual presentations of Maxwell's equations, it's many varieties, are not caste in 4-vectors but given in terms of the vector calculus. The possible elements are vectors, pseudo vectors, scalars and pseudo scalars. Depending on how carefully the elements are defined, the set of equations may or may not have various symmetries.

[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 qR and iR.

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. rhoijk=Jijk.
It also says that the time-space2 part is what we define as current density; jij=J0ijdt.

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?title=Four-vector&oldid=442771887#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?title=Four-vector&oldid=442771887#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%27s_equations#Differential_geometric_formu lations


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.