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).

 

[gentzen]

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

The reference given in the original post (before the edit) was:

Mermin in "It's About Time: Understanding Einstein's Relativity" has a chapter (9 Trains of Rockets) where it is shown that relativity of simultaneity is enough to fully explain both time dilation and length contraction.

 

[gentzen]

I have removed your rather lengthy complaint about moderation actions. ...
That said, it is unclear what is your physics question or what physics topic you would like to discuss.

It was not ment as a complaint, and especially the lengthy parts were not part of that reference to moderation actions.

My physics question (or physics topic) is the bold part below:

That is, “there is no mention in relativity of exactly how clocks slow, or why meter sticks shrink” (no “constructive efforts”), nonetheless the principles of special relativity are so compelling that “physicists always seem so sure about the particular theory of Special Relativity, when so many others have been superseded in the meantime

(Quoted from RUTA's Insights article)

I thought the mechanism would be "relativity of simultaneity," together with a definition of how to measure lengths based on measuring times (for why meter sticks shrink).

I think in modern times, the foundations of Special Relativity are well known to be the symmetries of an inertial reference frame and the POR. C is simply a constant that appears in the theory that needs to be fixed by experiment. From both theoretical and experimental considerations, it is the speed of light but is not an axiom at its foundations. See a paper I post a lot:
http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

Certainly a nice paper, but my goal was for RUTA to acknowledge that "relativity of simultaneity" is indeed a concrete mechanism in case of SR, actually explaining why meter sticks shrink.

Is there really nothing to be explained except for "relativity of simultaneity"? It might look like that to me from the perspective of the wave equation. But that equation leaves out mass, and hence probably also most of classical mechanics.

Also, what "Mermin's chapter 9" and my wave equation perspective have in common is that there is a preferred frame of reference. Even so it defines a reference time, in the end the different "observed times" for different observers are all that matters. The nice paper referenced by bhobba fails to provide the same conceptual clarity to me. (But of course, it simply has a different goal.)

 

[Ibix]

Also, what "Mermin's chapter 9" and my wave equation perspective have in common is that there is a preferred frame of reference.

I don't have Mermin so I can't comment on his view, but you certainly can add a preferred frame to SR. Essentially that is Lorentz Ether Theory, which is kind of what's left over after you remove all mechanical properties from the mechanical ether tested by (among others) Michelson and Morley - it id SR except that you additionally define one frame to be the "real" one. Which frame is the "real" one is undetectable.

It's not a terribly popular view these days because it seems like extraneous fluff. Things that are moving in this frame are "really" length contracted, while things that are at rest in this frame only appear length contracted according to some other frame due to the actual length contraction (etc) of the apparatus that is "really" moving. The fact that the same analysis can be performed using any frame as the "really at rest" frame is to be viewed as an interesting coincidence.

That doesn't necessarily mean it's wrong, of course, but it does mean it's got extra untestable assumptions, so "vanilla" SR with no preferred frame is preferred by most scientists.

 

[gentzen]

Essentially that is Lorentz Ether Theory, which is kind of what's left over after you remove all mechanical properties from the mechanical ether tested by (among others) Michelson and Morley - it id SR except that you additionally define one frame to be the "real" one. Which frame is the "real" one is undetectable.

I "claimed" that the wave equation perspective is different from Lorentz Ether Theory (original post, before the edit). In the Lorentz Ether Theory, you accept that moving things "really get length contracted". That would be something mysterious to worry about.

I don't have Mermin so I can't comment on his view

It is more his pedagogic attempt to "drive home" the explanatory power of "relativity of simultaneity" than "his view". It is possible that he doesn't have to worry about length contraction, because he only looks at the case where there is "exactly one speed":

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.

 

[Dale]

I thought the mechanism would be "relativity of simultaneity," together with a definition of how to measure lengths based on measuring times (for why meter sticks shrink).

One issue is that length contraction is not a shrinking, but a disagreement. In other words, length contraction occurs even if the length of the object is constant in every inertial reference frame. In such cases, there is no shrinking (a change in length over time), only a disagreement about the length. That disagreement is length contraction.

So, given that length contraction is a disagreement, what would you consider to be a “mechanism” for a disagreement? My wife and I disagree on how well done meat should be cooked. What would you consider to be a mechanism for that? Or, more relevant, in Newtonian mechanics two frames disagree on momentum and KE, what would you consider to be a mechanism for that?

Once you can describe what you mean by a “mechanism” for a disagreement then we can see if a mechanism can be found for length contraction.

 

[gentzen]

So, given that length contraction is a disagreement, what would you consider to be a “mechanism” for a disagreement?

A "good explanation" is what I mean by “mechanism”. Therefore, my question becomes whether the explanation from Mermin's chapter 9 is actually "good", especially whether it is better than Lorentz Ether Theory (LET). My guess is that the third frame (with its suitably chosen speed) is a good explanation for the disagreement (i.e. better than LET), but not a better explanation for general length contraction than LET.

Whether the wave equation is a "good explanation" is trickier. It is used as a "no need to worry" explanation, but it is hard to nail down how much it really explains. It cannot model slowly moving sticks directly. Whether the Klein-Gordon equation can model slowly moving sticks would have to be checked. Still, the fact that the Klein-Gordon equation is invariant under Lorentz transformations seems to be part of the explanation why special relativity just works.

 

[Dale]

A "good explanation" is what I mean by “mechanism”. Therefore, my question becomes whether the explanation from Mermin's chapter 9 is actually "good"

I would assume that every author who writes any explanation of length contraction considers their explanation to be a good one. Including Mermin. And I would further assume that a lot of students will be confused by any given explanation, and so probably not consider it to be good. Including Mermin.

So, any explanation will both be a mechanism and also not be a mechanism.

That said, a lot of scientists like geometric explanations. A geometric explanation for length contraction is fundamentally more difficult than for time dilation because geometrically time dilation involves worldlines and length contraction involves worldsheets. The relativity of simultaneity is important because it is the process of picking out a particular line in the worldsheet.