Relativity forbids rigid objects

Physicist Lubos Motl gives a relativity lesson:

It is clearly a totally rudimentary problem in special relativity. It has its own name and if you search for Ehrenfest paradox, you quickly find out that there's been a lot of debates in the history of physics – relatively to what one would expect for such a basic high school problem in classical physics. Born, Ehrenfest, Kaluza, von Laue, Langevin, Rosen, E ...

A rod can't be "unbendable" or "unsqueezable" or "unstretchable" because it would mean that there is something in the rod that guarantees its prescribed proper length at all times. ...

This non-existence of perfectly rigid rods in relativity should be totally obvious for rods. But it holds for disks, too. ...

At any rate, the non-existence of perfectly rigid bodies is undoubtedly a characteristic, almost defining, implication of relativity.

I am pretty amazed that even in 2015, 110 years after Einstein presented his relativity, this very simple point remains controversial. Well, I am convinced that at least since 1911, almost all good physicists have agreed what the correct answer basically is.

He is right. Part of the problem is that Einstein's famous 1905 relativity paper declared:

The theory to be developed is based — like all electrodynamics — on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters. ...

If a material point is at rest relatively to this system of co-ordinates, its position can be defined relatively thereto by the employment of rigid standards of measurement and the methods of Euclidean geometry, and can be expressed in Cartesian co-ordinates. ...

Let there be given a stationary rigid rod; and let its length be l as measured by a measuring-rod which is also stationary.

Einstein's whole presentation is in terms of rigid bodies. If there is no such thing as a rigid body, then it is hard to make any sense of that paper.

Motl is right, here. The whole discovery of special relativity was based on the insight by FitzGerald and Lorentz that the Michelson-Morley apparatus was not really rigid, but can contract as motion deforms the electromagnetic fields that hold the molecules together. Then Poincare and Minkowski had the insight that space and time were being deformed.

Poincare's 1905 paper defined distance in terms of how far light goes in a specified time. Minkowski made the non-Euclidean metric the fundamental entity. Einstein's use of rigid measuring rods does not make much sense, and apparently is still causing confusion today.

Update: I meant to also say this. The most important point about relativity is that it rejects action-at-a-distance. If you had a rigid object, you could push it at one end, and have an instantaneous effect at the other end. That is completely contrary to the whole spirit of relativity.