2. Rolling Toward Relativity

Non-Relativistity

Before getting into the details, here are some general comments on the nature of the paradox and its treatment.

A basic function of SRT is to tell us the ‘where’ and ‘when’ of objects in one inertial reference frame as observed from a different inertial reference frame. If we have a train speeding past us at 86 mph, using grade school math we can calculate for any given instant where Seat 2A in Car 9 is, relative to our position. For a train going 86% lightspeed, using junior high math, we can do the same. If the method we use to calculate that position does not yield simple, rock-solid x,y coordinates, then we haven’t solved the problem.

A common characteristic of many papers on the paradox is the explicit or implicit assumption that SRT, and specifically Lorentz contraction, applies in the case of a rotating reference frame. It does not.

SRT is designed for inertial reference frames. It can cover straight-line acceleration, an example of ‘addition of velocities’ that Einstein addressed in his original paper. An object in an inertial frame cannot exceed the speed of light, even if enough power were available to propel it that fast. The simple reason is that the inertial frame itself may not exceed the speed of light. This is because at lightspeed, Lorentz contraction reduces all distances in the direction of travel in the frame to zero.

A rotating reference frame is not an inertial frame, and is not subject to the lightspeed limit. Every rotating frame, regardless of its spin rate, has a radius beyond which the frame itself exceeds ‘c’. For many SRT reasons, physical objects may not remain fixed in the frame beyond that radius, but the frame itself has no such limitation. The rotating frame is non-relativistic.

The question here is whether or not objects within that radius, maintaining fixed positions in the rotating frame, are subject to all the effects of SRT. We’ll briefly cover a few ingredients toward the answer, but bigger and better explanations of SRT are available on the web.

Lorentz and FitzGerald

It’s been known since Newton’s time that light has wavelike properties, and since waves must travel in a medium – sound through air, ocean waves through water – inquiries were made into the characteristics of light’s medium, named the ‘ether’ for convenience.

Michelson and Morley performed an experiment in 1887 that would have measured the motion of any ether, or our motion through the ether. They didn’t find a thing. More experiments indirectly determined that light always travels the same speed, regardless of the motion of the observer or source. This didn’t quite fit Newton’s mechanics. The Lorentz contraction hypothesis helped somewhat.

Lorentz contraction is more properly titled Lorentz-FitzGerald contraction, as Dutchman Hendrik Lorentz and Irishman George FitzGerald independently came up with the equation for transformation between reference frames.

The equation comes from a rearrangement of the Pythagorean theorem, and the c part makes the velocity a unitless fraction of lightspeed, much like Mach number in aviation.

The equation held that if length were variable, not absolute, that was how it would vary. Whether they believed it anything more than an ad hoc, empirical tool is doubtful, both because it’s a ridiculous notion to begin with, and because simply accepting the variability premise can quickly lead to the full set of SRT effects, and with a little more math, the equivalence of energy and matter, E=mc.

Einstein came up with the contraction hypothesis, too, but he did it the hard way, starting with Maxwell’s electrodynamic equations and moving electrons, and coming up with mass increase, time dilation, stellar aberration, addition of velocities, and a lot of other answers to questions that hadn’t been asked (and don’t apply to the Ehrenfest paradox). Here’s the paper
On The Electrodynamics Of Moving Bodies.

Let’s look at some SRT effects next.

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