I One more talk about the independence of Einstein's SR axioms

[brotherbobby]

But the first postulate doesn't assert that the laws of Electrodynamics are actually Maxwell's equations.

True there - I understand this difficulty. I suppose I am going with Resnick here - Robert Resnick - Introduction to Special Relativity.

I will present his argument briefly. There are three issues at hand that rose. Galilean transformations, the relativity principle and Maxwell's equations. We can at best have any two of them satisfied. Most physicists dropped the relativity principle (for electrodynamics). They asserted a special frame (ether) in which Maxwell's equations are valid. Some, notably Hertz, dropped Maxwell's equations! More correctly, he tried to modify Maxwell's equations so that it would fit both Galilean transformations and the relativity principle for electrodynamics. And some, like Poincare and Einstein, adopted Maxwell's equations and felt that it is in fact the Galilean transformations that need to be altered. Lorentz role here is tricky. However, his belief in the "length contraction" violates the relativity principle. A bar moving relative to ether would be contracted in ether's frame, but a bar at rest in ether's frame would not be contracted for a moving frame, or so he felt. Hence it might be correct to say Lorentz too was part of the first group.

In order to solve our riddle here, we need to find out to what extent is Resnick justified. Was Einstein thinking about Maxwell's equations when he mentioned "electrodynamic and optical laws"? I argue that he did. For one, electrodynamics was only recently "tied" to optics back then - so he felt the need to take the trouble to put the two of them together. Today, we know that light waves and x-rays as electromagnetic waves so well that we wouldn't bother to do it. Hence, in mentioning both in the same vein, Einstein was making recourse to Hertz's experiment (e.m. waves). I do not think he wondered if some other equations of electrodynamics, other than those of Maxwell's, would turn out to be the true ones in the (pseudo) flat space of Minkowski.

 

[A.T.]

I suppose I am going with Resnick here...

Why not with Einstein himself?

The special relativity theory rests on the following postulate which also holds valid for the Galileo-Newtonian mechanics.

If a co-ordinate system K be so chosen that when referred to it the physical laws hold in their simplest forms, these laws would be also valid when referred to another system of co-ordinates K′ which is subjected to an uniform translational motion relative to K. We call this postulate "The Special Relativity Principle".
...
The Special Relativity Theory does not differ from the classical mechanics through the assumption of this postulate, but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the special relativity postulate, gives in a well-known way, the relativity of synchronism as well as the Lorentz transformation, with all the relations between moving rigid bodies and clocks.

Source

Was Einstein thinking about Maxwell's equations when he mentioned "electrodynamic and optical laws"? I argue that he did.

Instead of making up what Einstein was thinking about, why not read his own statement quoted above. It explicitly says that his first postulate by itself is consistent with Galileo-Newtonian mechanics.

 

[vanhees71]

The Galilean transformations do not leave Maxwell's equations covariant. The first postulate asserts that the laws of Mechanics and Electrodynamics must both be left covariant.

It's clear that the special principle of relativity together with the assumption that Maxwellian electromagnetism obeys it (together with some other symmetries about the space-time model) leads uniquely to Minkowski space as space-time model. Einstein only took the most simple assumption out of Maxwellian electromagnetism, namely that there is a fundamental constant with the dimension of speed, as a postulate.

The modern idea is to have the symmetry principles and then deduce the form of the physical laws from them, and if you only take the special principle of relativity together with the Euclidicity of space and homogeneity of time for all inertial observers you have two options as a space-time model, namely Galilei-Newton space-time or Einstein-Minkowski space-time. You can now build all kinds of mathematical models to describe the phenomena obeying the one or the other symmetry defining these space-time models. The only way to decide, which one is correct is empirical evidence, and it's well known that already "Newtonian electrodynamics" can be ruled out in favor of Maxwell's equations, which respects the Poincare symmetry of Minkowski space-time.

It's not so important which postulates you use to find the space-time model but which one describes nature best!

 

[Nugatory]

Somewhere further up in this thread, you said

I must confess I haven't read all the responses in this thread, but there is a basic thing in here.

If you still haven't, you might want to - there's an element of talking past one another in the last few posts in thread.

The first postulate says "all" laws of physics are covariant in inertial frames, "all" implying the laws of mechanics and electrodynamics.

Indeed it does. However, neither "postulate" really deserves that name. They aren't phrased with sufficient rigor, and @vanhees71 points out above that the original German could as reasonably have been translated to some other English word.

It's probably better to consider both postulates in historical context. Imagine yourself explaining special relativity to a group of physicists at the turn of the last century:

Folks, we all know that we've had a problem since 1865: Maxwell's electrodynamics only works covariantly with Galilean relativity if there is some luminous ether; but this approach is increasingly unsatisfactory and Michelson/Morley aren't helping any. I have a solution, if you'll just accept two things: Principle of relativity, which we already all agree about; and that we can go all in on that principle and apply it to E&M as well. Now let's see where this line of thought takes us

That's pretty close to what Einstein was doing in 1905. We wouldn't approach a modern audience with that argument; like you I tend to interpret the second postulate as just "And I really mean the first postulate, no exceptions for E&M" and prefer to incorporate the "no exceptions" bit into the first postulate. But that's largely a matter of personal taste in how to present the argument; an unnecessary postulate in a formal mathematical system is an abomination, but in a physical theory how we package the initial assumptions is less important.

 

[Ibix]

Einstein only took the most simple assumption out of Maxwellian electromagnetism, namely that there is a fundamental constant with the dimension of speed, as a postulate.

I like that way of putting it. In 1905, cutting edge physics assumed that Maxwell's equations were an approximation to, or special case of, an as-yet-undiscovered Galilean-covariant theory of electromagnetism. So, at that time, Einstein absolutely needs a separate postulate that brings something of Maxwell's electromagnetism into the argument.

 

[Dale]

The Galilean transformations do not leave Maxwell's equations covariant. The first postulate asserts that the laws of Mechanics and Electrodynamics must both be left covariant.

You are still assuming that Maxwell’s equations are the laws of electrodynamics. That is an additional assumption beyond the first postulate.

 

[vanhees71]

I like that way of putting it. In 1905, cutting edge physics assumed that Maxwell's equations were an approximation to, or special case of, an as-yet-undiscovered Galilean-covariant theory of electromagnetism. So, at that time, Einstein absolutely needs a separate postulate that brings something of Maxwell's electromagnetism into the argument.

At the end of the 19th century (beginning 20th century) there were also much more pragmatic approaches, and those lead finally to progress. I think among the most important contributions to the development of (S)RT was the research program by Helmholtz, who simply wrote down a set of equations with some free parameters which included all the theories on electromagnetism of the time (including action-at-a-distance theories a la Weber as well as Maxwell's). One of the most important outcomes was the discovery of em. waves by H. Hertz and finally the decision in favor of Maxwell's theory. I think there were not many physicists left around 1900 who thought there should be a Galilei invariant electrodynamics. Rather the majority believed in some kind of aether in the sense that there's a preferred inertial system defined by that inertial system, where the Maxwell equations are valid. The only trouble with that of course was that there was still no consistent formulation of the theory for moving media, and that's where Einstein ingeniously found the right "Copernicanian turn": One must give up the idea of a preferred frame of reference, because in fact not a single observation was known to really show the "asymmetries" in electromagnetic phenomena "usually assumed" and thus the necessity to change the space-time model and the fundamental equations of mechanics correspondingly rather than to assume a preferred frame of reference or even to give up Maxwell's empirically well established theory in favor of some Newtonian electrodynamics, and even more ingeniously he just picked the most simple and most general feature of the Maxwell equations, namely the above mentioned additional natural constant (additional as compared to Newton's mechanics, where the only natural constant known was the universal gravitational constant) of the dimensions of a speed and the fact that within Maxwell's electrodynamics it's the phase velocity of the electromagnetic waves in vacuo. Together with the empirical fact that light is also nothing else than such an em. wave, it's thus simply the speed of light in vacuum which must be invariant under transformations from one inertial frame to the other. As @Dale says the validity of Maxwell's equations (or more generally the existence of a fundamental speed as a natural constant) is in addition to the special principle of relativity, which indeed (together with the other symmetry assumptions, i.e., the Euclidicity of space for any inertial observer and the homogeneity of time) allows for both Galilei-Newton spacetime and Minkowski spacetime. The additional assumption Einstein has chosen was in a sense the minimal one, and he showed in his famous paper of 1905 that indeed this is sufficient for the Maxwell equations to be form-invariant under the now to be used transformations from one inertial frame to another, namely the Lorentz instead of the Galilei transformations.

 

[Dale]

I think among the most important contributions to the development of (S)RT was the research program by Helmholtz, who simply wrote down a set of equations with some free parameters which included all the theories on electromagnetism of the time (including action-at-a-distance theories a la Weber as well as Maxwell's).

Oh, I didn't know about that. Do you have a reference handy? That seems like it was probably one of the first "test theories" ever. I think that test theories are in some ways as important as the actual theories they test, but certainly they get less attention.

 

[vanhees71]

If I'd always knew, where I've read things... By a bit of googling, I found this paper, which contains a summary of Helmholtz's theory on electrodynamics:

https://doi.org/10.1086/350399

Another good book on the history of electrodynamics, containing a long appendix on Helmholtz's electrodynamics and its relation to the various theories discussed from around 1870 on is

O. Darrigol, Electrodynamics from Ampere to Einstein, Oxford University Press (2000)

 

[Sagittarius A-Star]

That seems like it was probably one of the first "test theories" ever.

Tests seem to decide about $k$ in:
https://puhep1.princeton.edu/~mcdonald/examples/velocity.pdf

 

[Sagittarius A-Star]

Ah, yes, you're right.

I found another paper on this topic:

C.4.2 Lorentz Force

Another felicitous result is that the low-velocity transform of the electric field from lab frame to a moving frame has the same form as the lab-frame Lorentz force (per unit charge), $F/q=E+v/c×B$, for arbitrary velocity. Of course, neither Maxwell (1873) nor Lorentz (1892) were aware of this happy result of the Lorentz transformation of 4-force (Minkowski force).

Source (see page 44):
https://physics.princeton.edu//~mcdonald/examples/maxwell_rel.pdf

via:
https://physics.princeton.edu//~mcdonald/examples/

The electromagnetic field (or wave) cannot be measured directly, but only indirectly via the acceleration of a charged particle due to the force of the local electric field in it's rest frame.

 

[Sagittarius A-Star]

At Mathpages, the Lorentz transformation of the electromagnetic field is derived from the Lorentz force and the relativistic transformation for forces:

Force Laws and Maxwell's Equations
...
Returning again to equation (1a), we see that in the absence of a gravitational field the force on a particle with q = m = 1 and velocity v at a point in space where the electric and magnetic field vectors are E and B is given by

$\ \ \ \ \ \mathbf f = \mathbf E + \mathbf v \times \mathbf B$

...
Thus if the particle is stationary with respect to the original x,y,z,t coordinates, the force on the particle has the components

$\ \ \ \ \ f_x = E_x \ \ \ \ \ f_y = E_y \ \ \ \ \ f_z = E_z$

...
Therefore, from equation (9), we see that the transformed components of the total electromagnetic force are
$\ \ \ \ \ f_{x'} = f_x \ \ \ \ \ f_{y'} = \sqrt{1-v^2} f_y \ \ \ \ \ f_{z'} = \sqrt{1-v^2} f_z \ \ \ \ \ \$ (10)

...
Just as the Lorentz transformation for space and time intervals shows that those intervals are the components of a unified space-time interval, these transformation equations show that the electric and magnetic fields are components of a unified electro-magnetic field. The decomposition of the electromagnetic field into electric and magnetic components depends on the frame of reference.
...
we see that Maxwell's equations are invariant under Lorentz transformations. Moreover, any physical force consistent with special relativity must transform in accord with (10), because otherwise a comparison of the forces in different frames of reference would give different results.

Source:
https://www.mathpages.com/rr/s2-02/2-02.htm

Another related paper:

A WAY TO DISCOVER MAXWELL’S EQUATIONS THEORETICALLY

Source:
https://arxiv.org/pdf/1309.6531.pdf