Follow Up on M. LeBellac & J.M. Levy-Leblond's "Galilean Electromagnetism

[otennert]

TL;DR Summary

This is meant to be a follow-up thread to https://www.physicsforums.com/threads/what-assumptions-underly-the-lorentz-transformation.1015982/post-6657920, just in case, anyone would like to discuss. As agreed, it should be spun off from the original thread.

In https://www.physicsforums.com/threa...e-lorentz-transformation.1015982/post-6657920 a discussion evolved from the basic assumptions of the Lorentz transformations, to a paper

M. LeBellac, J. M. Levy-Leblond, Galilean electromagnetism, Nuovo Cim. 14B, 217 (1973)

which is outside the scope of the original discussion, and moreover may be considered "controversial". This thread is just to take over from there.

 

[otennert]

In eqs (2.1) and (2.2.) they define the 2 different Galilean limits. First of all, at this point, they are not Galilean limits at all, because is still in both expressions.

But then in §§2.2/2.3 they look at the electromagnetic 4-current and look at the 2 cases, essentially, either "mostly timelike" or "mostly spacelike". But the 4-current is a timelike 4-vector to start with!

How do you come to that conclusion? There are both types of currents in nature: A "convection current", i.e., the current due to a single moving charge is of course timelike. In continuum-mechanical notation it's given by $$j^{\mu}=q n c u^{\mu},$$
where $q$ is the charge of the particles making up the fluid, $n$ the particle density as measured in the rest frame of the fluid cell (a scalar), and $u^{\mu}$ the normalized four-velocity (with $u_{\mu} u^{\mu}=1$, using the (1,-1,-1,-1) signature).

Then there are conduction-current densities in wires, which are space-like. The charge density is close to 0 since there is the positive ion lattice in addition to the negative conduction electrons making up the current.

But even if the charge is close to zero, so is the current, as the total charge enters as an overall factor. How can lead to a spacelike 4-current? If there are 2 4-currents, each caused by opposite charges, still the 4-current of each is timelike, and they add up to a total time-like 4-current.

 

[Dale]

But the 4-current is a timelike 4-vector to start with!

The 4-current is usually spacelike in ordinary macroscopic circuits

FYI, thank you for “forking” this to a different thread! As a mentor that is very appreciated

 

[otennert]

The 4-current is usually spacelike in ordinary macroscopic circuits

FYI, thank you for “forking” this to a different thread! As a mentor that is very appreciated

Of course, as requested!

But could you be more explicit regarding spacelike 4-currents please? When you refer to macroscopic circuits, we are still talking about electromagnetism which is relativistic by nature...?

Is there some reference you could point to?

 

[Dale]

We might be better using:

https://arxiv.org/abs/1012.1068

Or

https://arxiv.org/abs/1303.5608v2

 

[Dale]

But could you be more explicit regarding spacelike 4-currents please? When you refer to macroscopic circuits, we are still talking about electromagnetism which is relativistic by nature...?

Is there some reference you could point to?

Here is the Wikipedia link https://en.m.wikipedia.org/wiki/Four-current which is consistent with the peer reviewed literature in this case.

The four current is defined as $J^\mu = (c\rho,\vec j)= (c\rho,j_x,j_y,j_z)$ in a local inertial frame. So in an uncharged current carrying wire $\rho=0$ and $\vec j \ne0$ so $J$ is spacelike.

Yes, this is all electromagnetism which is relativistic. I am just referring to ordinary everyday circuits that you might find around your house. Usually they have spacelike four-currents

 

[otennert]

We might be better using:

https://arxiv.org/abs/1012.1068

Or

https://arxiv.org/abs/1303.5608v2

Thanks...however, before I read these papers, I would first like to understand the concept of a spacelike 4-current. This seems to be the root cause of my confusion.

Moreover, both papers just build up on the paper under consideration from Le Bellac and Levy-Leblond, which I am still struggling to understand, and also the first of the papers above do not even get the name "Levy-Leblond" right, which may be forgiveable, but bears witness of bad reviewing...

 

[DrGreg]

If there are 2 4-currents, each caused by opposite charges, still the 4-current of each is timelike, and they add up to a total time-like 4-current.

For a current in a wire, in the rest frame of the wire, the electron 4-current has a negative temporal component and non-zero spatial component. The wire 4-current (i.e. due to the positive ions at rest in the wire) has a positive temporal component, and zero spatial component. Add them together and the temporal component is small (often zero) and the spatial component is much larger.

 

[Dale]

I would first like to understand the concept of a spacelike 4-current. This seems to be the root cause of my confusion.

It just means a current in a wire with little or no net charge. See my previous post for details

 

[otennert]

Here is the Wikipedia link https://en.m.wikipedia.org/wiki/Four-current which is consistent with the peer reviewed literature in this case.

The four current is defined as $J^\mu = (c\rho,\vec j)= (c\rho,j_x,j_y,j_z)$ in a local inertial frame. So in an uncharged current carrying wire $\rho=0$ and $\vec j \ne0$ so $J$ is spacelike.

So $J^\mu = (c\rho,\vec j)$ is of course the correct definition. So let's have a look at a normal conductor: a straight wire, for all practical purposes of infinite length. There is an electrical current in that wire, i.e. the electrons are moving a speed $v$, the positively charged ions are at rest.

I think there is some subtlety now because of Lorentz contraction: the 4-current of one positively charged ion at rest is $(c\rho,\vec 0)$. The 4-current of one negatively charged electron moving at speed $v$ is $\gamma(v)(-c\rho,-\rho\vec v)$, with $\gamma(v)$ being the Lorentz factor.

Adding both currents yields

$$J_{tot}^\mu = (c\rho(1-\gamma),-\gamma\rho\vec v),$$

and thus after a little algebra, if I have made no mistakes,

$$J_{tot}^2 = 2c^2\rho^2(1-\gamma(v))<0.$$

Alright, I agree: Obviously the total 4-current does not need to be timelike at all, it can be spacelike and obviously, for most electrical circuits mostly is, and even null! I admit this had never occurred to me, because I never spent any thoughts on this!

Everyday is a school day...

Which means I need to give the paper under consideration a 2nd thought...

 

[vanhees71]

In eqs (2.1) and (2.2.) they define the 2 different Galilean limits. First of all, at this point, they are not Galilean limits at all, because is still in both expressions.

But then in §§2.2/2.3 they look at the electromagnetic 4-current and look at the 2 cases, essentially, either "mostly timelike" or "mostly spacelike". But the 4-current is a timelike 4-vector to start with!But even if the charge is close to zero, so is the current, as the total charge enters as an overall factor. How can lead to a spacelike 4-current? If there are 2 4-currents, each caused by opposite charges, still the 4-current of each is timelike, and they add up to a total time-like 4-current.

Here's a complete relativistic analysis of the DC in a wire:

https://itp.uni-frankfurt.de/~hees/pf-faq/relativistic-dc.pdf

As you see in Eq. (7) the total charge density (conduction electrons + ions) is $\rho_{\text{tot}}=\rho_{\text{cond}} \beta^2$, where $\beta=v/c$, where $v$ is the speed of the conduction electrons. Since in normal house-hold currents $v$ is of the order 1mm/s, the total charge density is close to 0, and the four-current thus "very spacelike".

Of course, in practice, this is a perfect example, where the magnetic case of the Galilean approximation of electrodynamics is almost exact.

I don't think that LeBellac's and Levy-Leblond's paper is in any way controversial. It's just a pretty concise analysis, what possible Galilean approximations of Maxwell's theory (which is of course a relativistic theory) exist. It also clearly states that a Galilean electrodynamics is not satisfactory as a fundamental theory but that it's valid under certain conditions. One example, where we use both the "electric" and the "magnetic" Galilean limits, is the quasistationary approximation within standard AC circuit theory.

In fact almost all textbooks on electrodynamics use such non-relativistic approximations, unfortunately often not stating this clearly. E.g., the usual version of Ohm's Law, $\vec{j}=\sigma \vec{E}$ is a "non-relativistic"/Galilean approximation.

 

[otennert]

Here's a complete relativistic analysis of the DC in a wire:

https://itp.uni-frankfurt.de/~hees/pf-faq/relativistic-dc.pdf

Excellent, I will take some time to study it. Anyway, meanwhile I have identified the mistake in my reasoning...simple things never thought of before...

What I missed to see is that adding timelike 4-vectors with different causal directions (some are future-pointing, some are past-pointing), do in general not lead to another timelike vector. But this indefiniteness enters due to the various signs of the different charges.

 

[vanhees71]

It's amazing, how this simple example of the straight DC in a wire is messed up in the literature... They simply forget about the "self-induced Hall effect" (see the alternative derivation in Sect. 3), which is included in the correct form of Ohm's Law by taking into account also the magnetic part of the Lorentz force.

 

[Delta2]

is messed up in the literature

The so called "professional scientific literature" is just wrong sometimes, though I have to admit that probably this is the exception and not the rule.

 

[vanhees71]

I'd also say that overall peer-reviewing works pretty well.

 

[Sagittarius A-Star]

I'd also say that overall peer-reviewing works pretty well.

There exists also a comment:

Comment on ‘‘In what frame is a current‐carrying conductor neutral?’’ [Am. J. Phys. 53, 1165 (1985)] A. Hernández, M. A. Valle, and J. M. Aguirregabiria

Source:
https://aapt.scitation.org/doi/10.1119/1.15392

The related original paper from Peters is referred in:

Here's a complete relativistic analysis of the DC in a wire:
https://itp.uni-frankfurt.de/~hees/pf-faq/relativistic-dc.pdf

 

[vanhees71]

That's of course not contradicting the paper and my summary of it. Everything is manifestly Lorentz covariant. The point is to take the right relativistic version of Ohm's Law, which properly takes into account the Hall effect (what Hernandez et all call "reflection"). That's in the last section of my writeup, where I simply solve the problem in the rest frame of the wire using the correct version of Ohm's Law, which is in accordance with Peters's approach arguing in the rest frame of the conduction electrons.

 

[Sagittarius A-Star]

That's of course not contradicting the paper and my summary of it. Everything is manifestly Lorentz covariant. The point is to take the right relativistic version of Ohm's Law, which properly takes into account the Hall effect (what Hernandez et all call "reflection"). That's in the last section of my writeup, where I simply solve the problem in the rest frame of the wire using the correct version of Ohm's Law, which is in accordance with Peters's approach arguing in the rest frame of the conduction electrons.

I just found a counter-reply:

Reply to ‘‘Comment on ‘In what frame is a current‐carrying conductor neutral?’ ’’

Source:
https://aapt.scitation.org/doi/abs/10.1119/1.15393?journalCode=ajp