A What assumptions underly the Lorentz transformation?

[otennert]

There are no closed timelike curves in Minkowski space!

Does flat spacetime with periodic boundary conditions count as somewhat Minkowski? If not, I give up.

 

[Orodruin]

Does flat spacetime with periodic boundary conditions count as somewhat Minkowski? If not, I give up.

No, that’s a different geometry. As evidenced by the existence of closed timelike curves.

 

[vanhees71]

It's a question of what we choose first. If I choose any $v$, then you let $c \to \infty$, then $\gamma(v) \to 1$. Fair enough.

But, if you choose $c$, then I can choose any $v < c$ and we have the full range of gamma factors and the same geometry. Still Minkowski and no nearer to Galilean. So, there is no convergence from Minkowski to Gallilean

The question is which of these is correct?

I'm still puzzled. The Lorentz boost with fixed $\vec{v}$ goes to a Galilei boost when taking $c \rightarrow \infty$. So on this very elementary level, there's not such a complicated issue in "deforming" the Poincare group to the Galilei group.

It gets more subtle in the context of quantum theory, where you deal with ray representations, and doing the "deformation" in the correct way there you end up not with the Galilei group but with a central extension of it (despite the fact that anyway you end up with the covering group too, enabling half-integer representations of the rotation group in both Poincare and Galilei groups). Proper unitary representations of the Galilei group don't lead to useful dynamics of the quantum theory (Enönü and Wigner).

 

[vanhees71]

Ok, so here's the thing:

If there is an invariant speed $V$, then spacetime is Minkowski regardless of what that invariant speed is. We all agree on this as far as I can tell. The question is what happens afterwards in the line of thought. On the one hand, yes, taking the limit $V\to \infty$ formally reduces the Lorentz transformations to the Galilei transformations as long as everything else is kept constant. However, the objection here is that calling this limit $V \to \infty$ is somewhat misleading from the Minkowski geometry point of view. The reason for this is that the natural thing to do would be to use the same units for time and space directions such that the metric diagonal becomes $\pm (1,-1,-1,-1)$ depending on convention. This is in complete analogy with using the same units for the x- and y-axes in Euclidean space. In this regard, the invariant speed $V$ is simply a scaling choice between time and space units and such a scaling choice can never affect the geometry. Therefore, letting $V\to \infty$ while keeping $v$ fixed geometrically corresponds to only allowing small rapidities, which would be more accurately referred to as $v/V \to 0$.

Of course you can NOT use "natural units" when you want to take the Newtonian limit, because if there is no "limiting speed", $V$, there simply is no "natural unit" for velocities/speeds and also no natural way to measure time intervals and distances in the same unit. So before you take the limit $V \rightarrow \infty$ you have to fix space and time units.

In Newtonian mechanics there is no such connection between space and time. The geometry is completely different: In Newtonian mechanics you have time and at each point in time a Euclidean space. If I remember the formalism right that's a kind of fiber bundle, while in special relativity you have a pseudo-Euclidean affine 4D manifold. I don't know, how to formally describe the limit $V \rightarrow \infty$ to deform the Minkowski spacetime to the Newtonian spacetime. I'm sure, there should be some literature about this somewhere.

Compare to the Euclidean case in two dimensions. Geometrically it would be natural to use the same units for both directions, but for some applications it may be of more relevance to use nautical miles in one direction and fathoms in another. The conversion factor between those units is roughly $K = 1012$ fathoms/NM. A rotation now takes the form
$$
x' = \Gamma ( x - ky/K^2), \qquad y' = \Gamma (y - kx),
$$
where $\Gamma = 1/\sqrt{1+(k/K)^2}$ and $k$ is a rotation parameter ($k = K \tan\theta$ where $\theta$ is the rotation angle). Formally, for $K\to \infty$ with everything else fixed, we would find
$$
x' = x, \qquad y' = y - kx.
$$
Geometrically, nothing changed, but our rotations of fixed $k$ became smaller and smaller rotations as $K$ increased and the formal limit no longer displays the same group structure as that of the full geometry. I think this is why many, including myself quite often, do not really like talking about the limit as $c \to \infty$, with $c$ being just an arbitrary unit conversion factor unrelated to the actual geometry.

I don't think that this is a good analogy, because here you stay within a fixed Euclidean geometry of the plane. There it indeed doesn't make sense to arbitrarily choose different units along different directions and then take a limit of the conversion factor.

This is a bit of a misdirected analogy as it involves a physical quantity going into a limit. The more relevant analogy would be letting $k_B$, which is effectively a conversion factor between units of temperature and units of energy, go to zero. Neither is really relevant of course, as the big question is typically how $k_B T$ relates to a typical difference in energies.

That's of course true. So it's a difference to have a concrete physical situation, of which you make some approximation in certain limits.

In our context an example is the above quoted work by LeBellac and Levy-Leblond on the different "Galilean limits" of Maxwell's equations. After reading this paper I guess even the most anti-relativity sceptic should get convinced that relativity is the right foundation of electromagnetic theory ;-)).

 

[PeroK]

I'm still puzzled. The Lorentz boost with fixed $\vec{v}$ goes to a Galilei boost when taking $c \rightarrow \infty$. So on this very elementary level, there's not such a complicated issue in "deforming" the Poincare group to the Galilei group.

It gets more subtle in the context of quantum theory, where you deal with ray representations, and doing the "deformation" in the correct way there you end up not with the Galilei group but with a central extension of it (despite the fact that anyway you end up with the covering group too, enabling half-integer representations of the rotation group in both Poincare and Galilei groups). Proper unitary representations of the Galilei group don't lead to useful dynamics of the quantum theory (Enönü and Wigner).

Here's an example. Let$$S_n = \{\frac 1 n, \frac 2 n, \dots \frac {n-1} n\}$$For any $k$ we have:$$\lim_{n \to \infty} \frac k n = 0$$But, it would be wrong to conclude that$$\lim_{n \to \infty} S_n = \{0\}$$The pointwise argument fails to capture the limiting behaviour of the set.

 

[Sagittarius A-Star]

Therefore, letting $V\to \infty$ while keeping $v$ fixed geometrically corresponds to only allowing small rapidities, which would be more accurately referred to as $v/V \to 0$.

Of course, you will not get exactly the GT, except for the special case of $v=0$, by calculating in case of $V=1$:
$\lim_{v\to0}\frac{x-vt}{(1-v^2)^{1/2}}=x$

If you are talking about "allowing small rapidities" for an approximately GT, then you must also demand "small enough x-coordinate", to limit the term for "relativity of simultaneity" ($vx/V^2$) in the transformation of time.

 

[vanhees71]

Here's an example. Let$$S_n = \{\frac 1 n, \frac 2 n, \dots \frac {n-1} n\}$$For any $k$ we have:$$\lim_{n \to \infty} \frac k n = 0$$But, it would be wrong to conclude that$$\lim_{n \to \infty} S_n = \{0\}$$The pointwise argument fails to capture the limiting behaviour of the set.

But nowhere does one make such a false conclusion when taking non-relativistic limits by doing an expansion in powers of $1/c$. I don't know, where you get this idea from.

 

[otennert]

Here's an example. Let$$S_n = \{\frac 1 n, \frac 2 n, \dots \frac {n-1} n\}$$For any $k$ we have:$$\lim_{n \to \infty} \frac k n = 0$$But, it would be wrong to conclude that$$\lim_{n \to \infty} S_n = \{0\}$$The pointwise argument fails to capture the limiting behaviour of the set.

OK, but your concrete series does not correctly reflect the LT. To remain with your analogy, you should rather consider:
$$L_n = \{\frac{0}{\sqrt{1-\frac{0^2}{n^2}}}, \frac{1}{\sqrt{1-\frac{1^2}{n^2}}},\frac{2}{\sqrt{1-\frac{2^2}{n^2}}},\ldots,\frac{n-1}{\sqrt{1-\frac{(n-1)^2}{n^2}}}\},$$

then

$$\lim_{n\to\infty} L_n = \{0,1,2,3,...\},$$

which in my view correctly represents the GT in this analogy.

 

[Sagittarius A-Star]

Of course you can NOT use "natural units" when you want to take the Newtonian limit, because if there is no "limiting speed", $V$, there simply is no "natural unit" for velocities/speeds and also no natural way to measure time intervals and distances in the same unit. So before you take the limit $V \rightarrow \infty$ you have to fix space and time units.

It is not important, if you choose a unit system with $V:=1$ or with $V :=3\cdot 10^8 \frac{m}{s}$. You generally cannot use a constant as input to a limit calculation.

As example, take the SI unit system, according to which $c=3 \cdot 10^8 m/s$. It makes no sense to write:
$\require{color}\lim_{3 \cdot 10^8 \frac{m}{s} \rightarrow \infty} {\frac{1}{\sqrt{1-v^2/c^2}} (t-\color{red}\frac{vx}{c^2}\color{black})}$

 

[vanhees71]

I'm really confused now about these arguments. Of course you can make an expansion in powers of $c$, and it doesn't make sense to make an expansion in powers of $1$.

E.g., take the equation of motion of a particle in an electromagnetic field. The most simple way to get the non-relativistic limit is to formulate it in terms of the coordinate time
$$m \frac{\mathrm{d}}{\mathrm{d} t} (\gamma \dot{\vec{x}})=q (\vec{E} + \vec{v}/c \times \vec{B}).$$
Then you expand in powers of $1/c$:
$$m \ddot{\vec{x}} +\mathcal{O}(1/c^2) = q(\vec{E} +\vec{v}/c \times \vec{B})=q \vec{E} +\mathcal{O}(1/c).$$
Depending on, whether you take everything into account up to powers of $1/c$ or only $(1/c)^0$ you get the Newtonian equation of motion either including the magnetic force or not.

 

[PeroK]

I'm really confused now about these arguments. Of course you can make an expansion in powers of $c$, and it doesn't make sense to make an expansion in powers of $1$.

E.g., take the equation of motion of a particle in an electromagnetic field. The most simple way to get the non-relativistic limit is to formulate it in terms of the coordinate time
$$m \frac{\mathrm{d}}{\mathrm{d} t} (\gamma \dot{\vec{x}})=q (\vec{E} + \vec{v}/c \times \vec{B}).$$
Then you expand in powers of $1/c$:
$$m \ddot{\vec{x}} +\mathcal{O}(1/c^2) = q(\vec{E} +\vec{v}/c \times \vec{B})=q \vec{E} +\mathcal{O}(1/c).$$
Depending on, whether you take everything into account up to powers of $1/c$ or only $(1/c)^0$ you get the Newtonian equation of motion either including the magnetic force or not.

That's only valid for $v <<c$. So, you must bound $v$ by some finite $c_0$ as you let $c \to \infty$.

The problem is that you do only an invalid pointwise convergence, then claim a universal convergence for a set of all $v < c$.

The example I gave uses the same idea and leads to an immediate and obvious contradiction.

 

[vanhees71]

That's only valid for $v <<c$. So, you must bound $v$ by some finite $c_0$ as you let $c \to \infty$.

The problem is that you do only an invalid pointwise convergence, then claim a universal convergence for a set of all $v < c$.

The example I gave uses the same idea and leads to an immediate and obvious contradiction.

That's of course true. So in this case it would indeed be better to do an expansion in powers of $\beta=|\vec{v}|/c$. That also tells you, where the approximation becomes invalid, namely when $\beta \gtrsim 1$.

 

[Orodruin]

I don't think that this is a good analogy, because here you stay within a fixed Euclidean geometry of the plane. There it indeed doesn't make sense to arbitrarily choose different units along different directions and then take a limit of the conversion factor.

It is not only a good analogy, it is an exact analogy. Just as I am stating within a fixed Euclidean geometry, Minkowski space describes the same geometry regardless of the value of c and the geometrically natural thing to do is to use the same units for all directions. This is the point.

If you are talking about "allowing small rapidities" for an approximately GT, then you must also demand "small enough x-coordinate", to limit the term for "relativity of simultaneity" ($vx/V^2$) in the transformation of time.

Sure, that is also true regardless of how you take the limit. See the discussion using $\pi_1$ and $\pi_2$ below.

But nowhere does one make such a false conclusion when taking non-relativistic limits by doing an expansion in powers of $1/c$. I don't know, where you get this idea from.

1/c is generally a bad expansion parameter because it is dimensional. When controlling limits of the behavior of physical relationships, the natural tjing to do is to look at dimensionless

Of course you can make an expansion in powers of c, and it doesn't make sense to make an expansion in powers of 1.

See above. The physical limits are those of dimensionless quantities. For dimensional quantities you can always readapt your units so that the numerical values become order one. In other words, you need a reference of some sort to be able to meaningfully say that something approaches zero or infinity.

In the case of the Lorentz transformations, the meaningful dimensionless parameters can be taken to be $\pi_1 = v/c$, $\pi_2 = x/ct$, and $\pi_3 = t’/t$ (for the time transformation). The limit colloquially referred to as ”$c\to\infty$” corresponds to both of those parameters approaching zero at the same rate.

The Lorentz transformations generally take the form
$$
\pi_3 = f(\pi_1,\pi_2)
$$
based on the Buckingham pi-theorem. In particular, we would have
$$
\pi_3 = \gamma(\pi_1) (1 - \pi_1\pi_2).
$$
Similarly, for $\pi_4 = x’/x$,
$$
\pi_4 = \gamma(\pi_1) (1 - \pi_1/\pi_2).
$$
With $\pi_1$ and $\pi_2$ going to zero at the same rate, it is clear that the limit becomes $\pi_3 \to 1$ and $\pi_4 \to 1 - \pi_1/\pi_2$.
(To make the transition to the GT more apparent, we could instead pick $\pi_2 = x/vt$.)

 

[otennert]

The Galilean limit often can be derived as the limit $\beta=v/c \rightarrow 0$, but also a bit depends on the theory you are looking at. E.g., for mechanics the limit $\beta \rightarrow 0$ is indeed usually getting you to the non-relativistic approximation.

For the Maxwell equations it's another business, and you have to distinguish between different "Galilean limits" ("electric" and "magnetic" ones):

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

Generally, I also do not understand what @PeroK is after.

I have read the paper

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

in a first round now, but am struggling with some basic prerequisites, or maybe motivations for their undertaking (by the way, I am happy to spin this off to a separate thread, if generally preferred):

For example, the authors make a distinction between "mostly-timelike" and "mostly-spacelike" 4-vectors. This is the first statement I am struggling with: "mostly-timelike" suggests essentially the space components are extremely small, and vice-versa for "mostly-spacelike". But a relativistic 4-vector to start with is either timelike or spacelike, invariantly so, although of course by LT, the value of the space and time components each change. In a Galilean limit, so that we are looking at a Galilei spacetime, there is no causal structure, hence no "timelike" and "spacelike" at all!

OK, but let's move on. Accordingly, 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 $c$ is still in both expressions.

But then in §§2.2/2.3 they look at the electromagnetic 4-current $j_\mu= (c\rho,\mathbf{j})$ and look at the 2 cases, essentially, $j_\mu$ either "mostly timelike" or "mostly spacelike". But the 4-current is a timelike 4-vector to start with! How and by what limiting process will that get transmuted into a "mostly-spacelike" 4-vector, on order to justify §2.3?

The authors criticize on p.234 the result referenced from Landau-Lifshitz, which "coincides neither with (2.7) nor with (2.14)", and remark that this does not correspond to any kind of Galilean limit. I have not studied those parts of Landau-Lifshitz yet (they seem to refer to some old edition, where the section numbering was different -- §3.10 in the most current Landau-Lifshitz editions does not exist).

Although the calculations are quite scarce, the math seems to be right, as it is not overall complicated, but I am also beginning to fail to see the point in the overall investigation, and I am struggling with the assumptions.

 

[otennert]

#1 Of course you can NOT use "natural units" when you want to take the Newtonian limit, because if there is no "limiting speed", $V$, there simply is no "natural unit" for velocities/speeds and also no natural way to measure time intervals and distances in the same unit. So before you take the limit $V \rightarrow \infty$ you have to fix space and time units.

#2 In Newtonian mechanics there is no such connection between space and time. The geometry is completely different: In Newtonian mechanics you have time and at each point in time a Euclidean space. If I remember the formalism right that's a kind of fiber bundle, while in special relativity you have a pseudo-Euclidean affine 4D manifold. I don't know, how to formally describe the limit $V \rightarrow \infty$ to deform the Minkowski spacetime to the Newtonian spacetime. I'm sure, there should be some literature about this somewhere.#3 In our context an example is the above quoted work by LeBellac and Levy-Leblond on the different "Galilean limits" of Maxwell's equations. After reading this paper I guess even the most anti-relativity sceptic should get convinced that relativity is the right foundation of electromagnetic theory ;-)).

#1:

I do agree with this statement. And now one specific paper comes back to my mind:

J.-M. Lévy-Leblond. “Nonrelativistic particles and wave equations”. In: Commun.
Math. Phys. 6 (1967), pp. 286–311

There, Levy-Leblond uses $c=1$ and -- let's skip the details -- concludes that by "linearizing" the Schrödinger equation as Dirac did with the Klein--Gordon equation, you end up with the Pauli equation, via a 4-spinor equation!

Now I don't want to open up another can of worms here, but essentially what Levy-Leblond arrived at in this paper is the "non-relativistic" limit of the Dirac equation in 4-spinor notation which can be found in any textbook, and there were no $c$'s around because they have been eliminated by using natural units from the beginning. What IMHO the author failed to see in his approach is that -- by linearizing the Schroedinger equation using by the way the good old Dirac matrices which are dimensionless (!) -- you have to include a constant with the dimensions of "velocity" that simply does not exist in non-relativistic physics. You can't have a velocity scale in Galilei physics. By using $c$ explicitly in the formulae, the validity of the whole approach would have been made explicitly doubtful from the outset.

By the way: His main motivation was to show that spin by itself is not a relativistic effect (which is valid, because it isn't), but the method he chose to show this fails completely.

Addendum: I would like to take back my respective statement in posting #68 on using natural units as just one of many options.

#2:

https://www.physicsforums.com/threads/special-relativity-and-fiber-bundle.1016699/post-6650311

#3:

Oh yes. I am really struggling to interpret the paper.

 

[PeterDonis]

I have read the paper

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

in a first round now, but am struggling with some basic prerequisites, or maybe motivations for their undertaking (by the way, I am happy to spin this off to a separate thread, if generally preferred):

Please do. This paper is not about taking a "Galilean limit" of standard SR. It is about a proposed alternative "Galilean" theory of electromagnetism, one which, to say the least, has not gotten any traction. Discussion of it definitely belongs in a separate thread (and to be honest, there isn't a lot to discuss given what I've said just above).

 

[PeterDonis]

For the Maxwell equations it's another business, and you have to distinguish between different "Galilean limits" ("electric" and "magnetic" ones):

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

Please note that, as I said in response to @otennert just now, this paper is not (just) about taking "Galilean limits" of standard electrodynamics. It is proposing an alternative "Galilean" theory of electrodynamics.

 

[robphy]

Please do. This paper is not about taking a "Galilean limit" of standard SR. It is about a proposed alternative "Galilean" theory of electromagnetism, one which, to say the least, has not gotten any traction. Discussion of it definitely belongs in a separate thread (and to be honest, there isn't a lot to discuss given what I've said just above).

Inspired by this article, there is an AJP article that proposes a new storyline to teaching relativity:

https://aapt.scitation.org/doi/10.1119/1.12239
“If Maxwell had worked between Ampère and Faraday: An historical fable with a pedagogical moral”
American Journal of Physics 48, 5 (1980);
https://doi.org/10.1119/1.12239
Max Jammer & John Stachel

(Further discussion along these lines should probably be moved to a new thread.)

 

[otennert]

Inspired by this article, there is an AJP article that proposes a new storyline to teaching relativity:

https://aapt.scitation.org/doi/10.1119/1.12239
“If Maxwell had worked between Ampère and Faraday: An historical fable with a pedagogical moral”
American Journal of Physics 48, 5 (1980);
https://doi.org/10.1119/1.12239
Max Jammer & John Stachel

(Further discussion along these lines should probably be moved to a new thread.)

No need from my side at this point. I actually share @PeterDonis's assessment.

 

[vanhees71]

I have read the paper

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

in a first round now, but am struggling with some basic prerequisites, or maybe motivations for their undertaking (by the way, I am happy to spin this off to a separate thread, if generally preferred):

For example, the authors make a distinction between "mostly-timelike" and "mostly-spacelike" 4-vectors. This is the first statement I am struggling with: "mostly-timelike" suggests essentially the space components are extremely small, and vice-versa for "mostly-spacelike". But a relativistic 4-vector to start with is either timelike or spacelike, invariantly so, although of course by LT, the value of the space and time components each change. In a Galilean limit, so that we are looking at a Galilei spacetime, there is no causal structure, hence no "timelike" and "spacelike" at all!

Sure, but of course the Newtonian limit (for a mechanical situation) can be valid only in an inertial frame, where the bodies move with velocities much smaller than the speed of light, i.e., you'll get a good approximation to the relativistic dynamics only in such reference frames.

OK, but let's move on. Accordingly, 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 $c$ is still in both expressions.

But then in §§2.2/2.3 they look at the electromagnetic 4-current $j_\mu= (c\rho,\mathbf{j})$ and look at the 2 cases, essentially, $j_\mu$ 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.

How and by what limiting process will that get transmuted into a "mostly-spacelike" 4-vector, on order to justify §2.3?

The authors criticize on p.234 the result referenced from Landau-Lifshitz, which "coincides neither with (2.7) nor with (2.14)", and remark that this does not correspond to any kind of Galilean limit. I have not studied those parts of Landau-Lifshitz yet (they seem to refer to some old edition, where the section numbering was different -- §3.10 in the most current Landau-Lifshitz editions does not exist).

It's Eq. (2.23) on p. 224, and in my edition of Landau-Lifshitz's vol. 2 it's in Paragraph 24. That's indeed an approximation of the transformation law derived as an expansion in powers of $1/c$, but indeed this doesn't lead to a transformation group and in this sense is not a consistent Galilean theory. Of course the paper also demonstrates that there is indeed no Galilean electrodynamics which is consistent with the phenomenology anyway.

Although the calculations are quite scarce, the math seems to be right, as it is not overall complicated, but I am also beginning to fail to see the point in the overall investigation, and I am struggling with the assumptions.

Well, the question, whether there is a consistent Galilean electrodynamics is of some academic interest, but as the paper shows, it fails to describe the electromagnetic phenomenology right although of course there are good approximations to certain "non-relativistic" situations, e.g., the quasistationary approximations used to derive AC circuit theory.

 

[otennert]

Sure, but of course the Newtonian limit (for a mechanical situation) can be valid only in an inertial frame, where the bodies move with velocities much smaller than the speed of light, i.e., you'll get a good approximation to the relativistic dynamics only in such reference frames.

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.

It's Eq. (2.23) on p. 224, and in my edition of Landau-Lifshitz's vol. 2 it's in Paragraph 24. That's indeed an approximation of the transformation law derived as an expansion in powers of $1/c$, but indeed this doesn't lead to a transformation group and in this sense is not a consistent Galilean theory. Of course the paper also demonstrates that there is indeed no Galilean electrodynamics which is consistent with the phenomenology anyway.

Well, the question, whether there is a consistent Galilean electrodynamics is of some academic interest, but as the paper shows, it fails to describe the electromagnetic phenomenology right although of course there are good approximations to certain "non-relativistic" situations, e.g., the quasistationary approximations used to derive AC circuit theory.

I've opened another thread here.