Conservation of Newtonian Force and the Invariance of Maxwell's Equations

[davidge]

The electromagnetic wave equation being of the same form in all intertial frames is because Newton's force is a vector quantity? I mean, if the wave equation changes its form from a intertial frame to another one, would the electromagnetic force be different in the two frames?

I know that one of the Eintein's postulates is that laws of nature are invariant under change of intertial frames. Also, I know that one can argue that it should be this way independently from Newton's theory. (e.g. by guessing the Eintein's elevator).

But would conservation of Newtonian force be another way to say that the electromagnetic wave equation should be the same?

 

[PeterDonis]

The electromagnetic wave equation being of the same form in all intertial frames is because Newton's force is a vector quantity?

I'm not sure why you talk about "Newton's force" here, since Newtonian mechanics is not Lorentz invariant.

one of the Eintein's postulates is that laws of nature are invariant under change of intertial frames

Yes, but "change of inertial frames" here means Lorentz transformations. Newtonian mechanics is not invariant under Lorentz transformations. The equations of electromagnetism (Maxwell's Equations, from which the wave equation for electromagnetic waves can be derived) are, but there is no "Newtonian force" in those equations. There is an equation for the Lorentz force which is Lorentz covariant, but it's an equation for a 4-vector, not a 3-vector.

I know that one can argue that it should be this way independently from Newton's theory. (e.g. by guessing the Eintein's elevator)

Einstein's "elevator" thought experiment, if it's the one I think you're talking about, was not about Lorentz invariance but about the equivalence principle, which is a separate concept.

conservation of Newtonian force

Newtonian force is not conserved, so again I'm not sure what you're talking about here.

 

[davidge]

I'm not sure why you talk about "Newton's force" here, since Newtonian mechanics is not Lorentz invariant.

It's not Lorentz invariant, but it is invariant from one inertial frame to another, isn't?

was not about Lorentz invariance but about the equivalence principle, which is a separate concept

Oh, I thought these were the same thing.

Newtonian force is not conserved, so again I'm not sure what you're talking about here.

Isn't Newtonian force conserved under change of inertial frames?

 

[PeterDonis]

It's not Lorentz invariant, but it is invariant from one inertial frame to another, isn't?

"Lorentz invariant" and "invariant from one inertial frame to another" are the same thing.

Before special relativity was discovered, it was thought that "invariant from one inertial frame to another" meant the kind of invariance that Newtonian physics obeys, which is called Galilean invariance. But now we understand that that's not the case--the correct kind of invariance from one inertial frame to another is Lorentz invariance, not Galilean invariance. (If the velocity difference between the two frames is much smaller than the speed of light, the difference between the two kinds of invariance is not detectable, which is one reason why it took so long to figure out the difference.)

Isn't Newtonian force conserved under change of inertial frames?

No. See above.

 

[davidge]

Ah, ok. Thanks.

So what is the reason the electromagnetic wave equation should have the same form in all inertial frames?

 

[Nugatory]

So what is the reason the electromagnetic wave equation should have the same form in all inertial frames?

It's not just E&M; we expect all the laws of physics to have the same form in all inertial frames. We expect this because it is equivalent (with some slight hand-waving) to saying that we expect (intuitively, and also with abundant experimental support) that experiments in terrestrial physics labs will yield the same results in June and December, even though the lab is moving in a completely different direction.

The key question is how are inertial frames related to one another? For centuries it was assumed that they were related by the Galilean transformations, under which Newton's laws are invariant. When Maxwell discovered his laws of electromagnetism in 1861, it soon became apparent that they are not invariant under the Galilean transformations; therefore if the Galilean transformations were the correct way to relate inertial frames, then the laws of physics could not conform to our expectation that they all would be the same in all inertial frames. This was the great unsolved problem of physics in the second half of the 19th century, and it absorbed the energies of two entire generations of physicists.

Einstein's contribution was to demonstrate that if inertial frames were related by the Lorentz transformations instead of the Galilean ones, then Newtonian physics becomes not exactly correct but still a very good approximation, and everything can be made to work properly.

 

[davidge]

I understand what you say. My question actually is, what would happen specifically in the electromagnetic case if we allow the galelian transformation in the wave equation? How do we know that
$$- \frac{\partial^2}{\partial t^2}\phi + \frac{\partial^2}{\partial x^2}\phi + \frac{\partial^2}{\partial y^2}\phi + \frac{\partial^2}{\partial z^2}\phi = - \frac{\partial^2}{\partial t'^2}\phi' + \frac{\partial^2}{\partial x'^2}\phi' + \frac{\partial^2}{\partial y'^2}\phi' + \frac{\partial^2}{\partial z'^2}\phi'$$

 

[vanhees71]

The Maxwell equations are not Galilei invariant. I don't understand the debate in this thread to begin with :-(.

 

[PeterDonis]

what would happen specifically in the electromagnetic case if we allow the galelian transformation in the wave equation

What do you mean by "allow"? The transformation properties of the electromagnetic wave equation are easy to verify mathematically. Do the math and you will see that the equation is left invariant by Lorentz transformations, but not by Galilei transformations. There's nothing to "allow"; it's a mathematical fact.

 

[davidge]

the equation is left invariant by Lorentz transformations, but not by Galilei transformations

There's nothing to "allow"; it's a mathematical fact

Yes. I do know that. What I'm asking is how one knows that using galilean transformation gives a wave equation that doesn't describe the actual physics.

What do you mean by "allow"?

I mean what if we just use galilean transform for expressing the wave equation in another inertial frame

 

[Dale]

I mean what if we just use galilean transform for expressing the wave equation in another inertial frame

Why don't you try it out yourself. Write down a simple plane wave. Confirm that it satisfies all of Maxwell's equations in vacuum. Use the Galilean transformation. Check if it still satisfies Maxwell's equations in vacuum.

 

[davidge]

Why don't you try it out yourself. Write down a simple plane wave. Confirm that it satisfies all of Maxwell's equations in vacuum. Use the Galilean transformation. Check if it still satisfies Maxwell's equations in vacuum.

Ok. I will!