I The wave equation interpretation of special relativity

[gentzen]

In most textbooks on special relativity or electrodynamics, it is mentioned sooner or later that the Lorentz transformations are symmetries of the wave equation (and of the vacuum Maxwell equations).
I no longer remember whether I ever worried about interpretation of special relativity. But this information really convinced me that there is nothing mysterious to worry about in special relativity. (I am no longer so sure about this specific argument today, because the wave equation only describes the massless case. So to do justice to special relativity, I would have to base my confidence on the Klein-Gordon equation instead. But I don't have the same confidence and intuitive understanding of that equation as for the wave equation.)

In this chapter we shall examine an easy way to explore how a disagreement about whose clocks are synchronized leads to all the relativistic effects we have found: the slowing down of moving clocks, the shrinking of moving sticks, the relativistic velocity addition law, the existence of an invariant velocity, and the invariance of the interval.
We shall do this by examining two frames of reference from the point of view of a third frame in which the first two move with the same speed, but in opposite directions. We take the third frame to be the proper frame of a space station. The first two frames are the proper frames of two trains of rockets: a gray train, moving to the left in the frame of the space station, and a white train, moving to the right in the frame of the space station, at the same speed that the gray train moves to the left.

The reason why I named this perspective the "wave equation interpretation" (and why Mermin's chapter 9 felt so natural to me) is as follows:
I do remember when I read about two different methods to simulate oblique incidence in the chapter on "Periodic Structures" in "Computational Electrodynamics the Finite-Difference Time-Domain Method" by Allen Taflove. One method used quasi-periodic boundary conditions where a phase shift is introduced between the values of a field at corresponding points on opposite boundaries. This was the method I had already implemented. The other methods was basically playing tricks with relativity of simultaneity, i.e. different grid cells stored the field at different times. It never said so explicitly, but this was the easiest way to make sense of the formulas.

 

[Dale]

@gentzen I have removed your rather lengthy complaint about moderation actions. Such complaints should be posted to the feedback forum. The technical forums are for discussion of the physics. Please respect that organization and keep the two separate.

That said, it is unclear what is your physics question or what physics topic you would like to discuss.

 

[robphy]

Mermin's chapter 9

Please give the full reference.
Mermin has two books on special relativity.

 

[robphy]

In most textbooks on special relativity or electrodynamics, it is mentioned sooner or later that the Lorentz transformations are symmetries of the wave equation (and of the vacuum Maxwell equations).
I no longer remember whether I ever worried about interpretation of special relativity. But this information really convinced me that there is nothing mysterious to worry about in special relativity. (I am no longer so sure about this specific argument today, because the wave equation only describes the massless case.

I think the point of mentioning the Lorentz Transformations
with reference to the wave equation arising from the Maxwell Equations
is that Galilean transformations don't preserve the wave equation.

So, Lorentz transformations are there in the context of the wave equation
to disprove Galilean transformations, not to prove special relativity.

As you say, the wave equation only describes the massless case.
Indeed, the wave equation is invariant under conformal transformations.

If I recall correctly, there's a quote by Finkelstein that
the conformal structure determines nine-tenths of the metric.
So, at this level, one doesn't have the square-intervals yet...
only the "signs of the square-intervals" (i.e. timelike vs spacelike vs lightlike).