My wife graduated from from Oberlin college in 1992, and as such she gets the Oberlin alumni magazine. The summer 2012 issue includes a one-page article entitled "The Entirety of Relativity", which I find to be a very unfortunate presentation of Relativity. (As a pedantic point, it's only talking about *Special* Relativity (SR), and doesn't address General Relativity (GR) at all, but that really is a pedantic point. When a physicist says "Relativity", she likely means GR (especially given that SR is a subset of GR, so nothing is lost), but when presented publicly we often use "Relativity" as a shorthand for SR.)

The basic problem with the article is that it presents the theory as if its nature were the way that SR has been taught to students for a long time. The article starts with three things that are correct as far as they go: moving clocks run slow, a moving rod is short, and moving clocks aren't synchronized. Where the article loses me, however, is on point number 4, "That's All There Is To It.".The brief text after this says that the first three points are the basis of relativity, and the rest of the article claims that all of SR is a consequence of these three points. This is at the very least a perverse way of describing the theory.

A *lot* of texts at both the high school and college level present Relativity by first presenting these three points. You're given formulae for each of these consequences; parts of them resemble each other, but they're each presented as if they were a fundamental formula that couldn't be derived from anything else, for you to memorize (or, in a more modern way of thinking about it, look up) and use. However, this is a back-assward way of presenting SR, and I would argue that stating that the rest of SR is a *consequence* of these three observations is not just back-assward, but in fact *wrong*.

In fact, these three points are themselves consequences of the theory of Relativity. The formulae for them can be derived from more fundamental considerations. They're no more fundamental than all of the various kinematic formulae you memorize or look up (such as *d*=½*a*^{2}) when you do a non-calculus Newtonian mechanics class; those kinematic formulae themselves are just results of the *definition* of velocity and acceleration as, respectively, the rate of change of position and the rate of change of velocity, together with Calculus. Those definitions are the fundamental thing, not all the various kinematic equations you learn to use if you take a non-Calculus physics class. I could *start* with *d*=½*a*^{2}, take a couple of derivatives, and say, "hey, acceleration is the rate-of-change of the rate-of-change of position, and that's a consequence of this kinematic equation". That would be back-assward and indeed wrong, and it's just as wrong to say that everything else in Relativity is a consequence of moving clocks running slow, separated moving clocks not being synchronized, and moving rods being short.

Special Relativity itself starts with just two very simple postulates— "simple" in the sense of "not complex", not in the sense of "easy to understand". Those postulates are:

- The laws of physics are the same for every freely-falling observer
- The speed of light is one of those laws of physics; every freely-falling observer will measure the speed of light in a vacuum to be 2.998×10
^{8}meters per second.

Everything else in SR— including moving clocks running slowly, separated moving clocks not being synchronized, and moving rods being short, *as well as other things* (such as the Doppler shift, focusing of light emitted by a moving object in the direction of motion, an apparent rotation of a moving object) are consequences of these two postulates.

I should note that both of these postulates do require more explanation to be truly precise. For the first postulate, you have to carefully define "freely-falling observer". You get it basically right if there are no net external forces *other than gravity* acting on that observer. (However, if you allow gravity to be around, things can get a little subtly complicated. It doesn't ruin the postulates, but you have to be careful in treating the consequences.) For the second postulate, in fact it's not the speed of *light* that's absolute, it's the speed of *any* object that both carries energy and is massless. Light just happens to be the thing that we think about the most that works like this, and thus we call the cosmic speed limit "the speed of light", even though we really ought to call it "the speed of spacetime" (at least in the context of Relativity).

One of the most interesting consequences of these two postulates it that you have to change the way you think about time. Most of us live our lives with a Galilean/Newtonian view of time: it's an absolute, that advances at the same rate and is the same for everybody. However, you can't maintain that idea *and* have the speed of the same bit of light be measured at the same rate by everybody regardless of how they're moving. Galileo and Newton would say that the latter is wrong; Einstein's postulate, from which all of Relativity springs, was that in fact it's this speed of massless objects that is absolute, and as such we just have to give up on the idea of absolute time. Some of the consequences of this are that separated moving clocks aren't synchronized and moving clocks run slow... as well as other things.

I'm fond of the way that Thomas Moore's *Six Ideas That Shaped Physics* presents Special Relativity. (This is the book series that I currently use when teaching introductory calculus-based physics.) His Book R of the series is written for college-level physics who have had Calculus (and indeed have had some Calculus-based Newtonian physics). It presents SR not in the old-fashioned and unfortunate pedagogical way that the Oberlin article does— by starting with the consequences such as time dilation and with their formulae, and only later getting to the fundamental structure of spacetime implied by Einsteins postulates— but rather by starting with the fundamental structure of spacetime implied by Einstein's postulates, and then developing the consequences out of that

Yes, it's easier to just learn the formulae and do calculations about time dilation and so forth, and presents fewer difficult abstract conceptual challenges to students coming across this for the first time. However, if you learn it this way, you're given a warped perspective of what the theory of Special Relativity really is. My beef with this Oberlin alumni article is that it presents Relativity as if *the theory itself* is based and structured in the way that it has often been taught.

While we learned SR ahead of classical electrodynamics in undergrad, it's worth noting that the Lorentz contraction comes out of Maxwell's equations as applied to moving charges. A point charge moving past an observer at high speed has equipotential surfaces that are flattened ellipses, and the math gives you exactly the familiar Lorentz contraction.

Sure-- and, indeed, the motivation that drove people along the lines of thinking that lead to Relativity was the inconsistency of Maxwell's Equation, this amazingly beautiful and elegant theory for electromagnetism (including light), with classical mechanics.

However, Special Relativity isn't about electromagnetism specifically. As such, yeah, you can get some of the results for SR specifically for electromagnetic structures purely from Maxwell's equations, that doesn't make SR something that's derived out of E&M. E&M naturally incorporates SR, but it took additional insights for physicists to realize that SR transcended E&M, and applied more generally.

I found your blog really helpful (a bit moreso than the notes you left on the article itself, though that amused me). To be fair to the Oberlin article author on one small point, though, he did say at the fine print at the bottom that he was talking specifically about special relativity.

You said

"Special Relativity itself starts with just two very simple postulates— "simple" in the sense of "not complex", not in the sense of "easy to understand". Those postulates are:

The laws of physics are the same for every freely-falling observer

The speed of light is one of those laws of physics; every freely-falling observer will measure the speed of light in a vacuum to be 2.998×108 meters per second.

Everything else in SR— including moving clocks running slowly, separated moving clocks not being synchronized, and moving rods being short, as well as other things (such as the Doppler shift, focusing of light emitted by a moving object in the direction of motion, an apparent rotation of a moving object) are consequences of these two postulates."

Actually this is confusing and not correct. The constancy of C is not an input, rather, a consequence of the argument that produces SR out of observations about the homogeneity and isotropy of space and time. You can read the argument here (mine, with some guest commentary, remembered from a paper I read as an undergraduate)..

http://membrane.com/sidd/wundrelat.txt

This has nothing to do with light as such, rather, symmetry arguments that are taken from experience. The real input is that we do not assume space and time to be independent of each other. It is really an argument about geometry, and of course the upshot of all of this is that space and time are part of a single geometry. You are right to be bothered about basing things on short rods and slow clocks, since these are only effects of perspective in the new geometry, not fundamental properties.

-drl

Of course I agree and there is no debate between us about the physics. However, saying that the speed of light is constant is a

postulatemakes it sound like "OK, one postulates this and derives the consequences but could have postulated something else" whereas we know that thispostulateis correct, which means it is not apostulatein the sense in which many people use the term.Hi Rob,

about the effort to discern what is fundamental in a theory:

As John Norton formulated it: "In any logical system, we have great freedom to exchange theorem and axiom without altering the system's content."

I do agree, of course, that things like time dilation and length contraction are better presented as physical consequences of underlying principles. But still there are several contenders for the title of 'underlying principle'.

If we want to choose a particular axiomatisation as more fundamental than others we need extra criteria.

I propose to examine newtonian physics and special relativity side by side, and choose axioms in such a way that they inherently fall in either of the two following categories:

- Principles that the two frameworks have in common

- Principles that are distinct (maybe even opposite) in the two frameworks.

Common:

First axiom:

Objects in motion (that do not interact with other objects) proceed to move along straight lines, covering equal distances in equal intervals of time.

Common:

Second axiom:

When objects interact then for each individual object F=m*a

(Of course F=m*a does not apply _globally_ in SR when considering interactions, but for each individual object it does apply for acceleration with respect to the instantaneously co-moving frame.)

Newtonian dynamics and Special relativity have the above in common. Now to the only difference between the two: what is asserted about simultaneity.

A procedure to find simultaneity can use the physics of mechanics or the physics of wave propagation.

- Mechanics:

A hub emits two particles in opposite directions; the hub transfers to both particles precisely the same amount of kinetic energy. Receiving stations record time of arrival, and they sent particles back as acknowledgement. With an appropriate protocol this will serve to define a self-consistent simultaneity.

- Propagation of waves in vacuum:

Undulations of a field (any field, including electromagnetic field) will propagate as waves. A hub emits pulses of waves in opposite directions. Receiving stations record time of arrival, and they sent pulses of waves back as acknowledgement, Etc, etc.

As we know, special relativity asserts that both types of physics will define _the same simultaneity_.

As we know, historically the newtonian expectation was that mechanics and wave propagation would _not_ indicate the same simultaneity.

Third axiom of Special Relativity:

For _any_ kind of physics the same simultaneity arises.

The third axiom of Special Relativity narrows things down to the Minkowski metric. In turn, the Minkowski metric implies that no physical entity can move with respect to any inertial coordinate system with a speed larger than c.

As you point out in the comments section:

Special Relativity isn't about electromagnetism specifically.

Einstein's 1905 axiomatisation was good for the specific purpose that it served in that article. However, by featuring light so prominently (the light postulate) there is a unintended suggestion that Special Relativity is all about electromagnetism. Avoiding that is better.

Once again the John Norton Quote: "In any logical system, we have great freedom to exchange theorem and axiom without altering the system's content."

You favor starting with an axiom of maximum speed (and the speed of light as one manifestation of that) and via that pathway narrow things down to the Minkowski metric. I favor the third axiom of Special Relativity, as given above, to arrive at the Minkowski metric.

It's tempting to _believe_ that one pathway is more fundamental than the other, but is it?

Cleon Teunissen