Let's take a standard ACME Thought Experiment Division car with a max speed of a leisurely, constant 0.75c in a straight line.

So, for an external observer, the body of the car behaves like a typical Thought Experiment Car, it's shorter than at rest, the dashboard clock goes slower than the clock in the observer's pocket, and so on.

Thing is, the car uses oft-neglected in thought experiments, wheels.

The surface of the wheel in contact with the road surface remains at rest relative to the road (friction etc). But from the classical mechanics point of view, the upper surface of the wheel would be moving at 1.5c relative to the ground (and the external observer). Obviously this is impossible.

What gives? What shape will the wheels be to the external observer? Will they shatter as the shearing forces grow towards infinity? What shape will they have to the driver of the car?

Just to avoid trivial answers ( e.g. "wheels explode in all directions torn apart by centrifugal force") let's impose some limits:

The wheels are extremely (though, let's say, not infinitely) durable, and quite (though not infinitely) flexible and stiff: you need a very strong force to distort them - certainly more than the their centrifugal force alone. And once that force is exerted it doesn't instantly break them - they distort flexibly for an observable while before an eventual critical failure.

They move over the surface without slipping, and the car's motor provides just enough torque (if any) to keep the car moving at constant speed despite any flexibility/friction/other forces the wheels might exert on it; the car retains its 0.75c no matter what the cost to the wheel integrity or such.

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    $\begingroup$ This is just another way to see that there are no rigid bodies in relativity. Your "trivial" answer is the answer - any time you derive a contradiction from special relativity + rigid body (such as the wheel), the solution is that the body doesn't stay rigid, but is torn apart. $\endgroup$
    – ACuriousMind
    Jan 24, 2016 at 16:38
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    $\begingroup$ 1. Special relativity means flat metric, not inertial frame. You don't need general relativity unless you have gravity, so that edit was actually good. 2. I'm now confused what your actual question is. If you think one can predict how exactly the wheels will meet their demise, you can't. How exactly they bend or tear part is dependent on the specific wheel and its weak points, not on some general principle. $\endgroup$
    – ACuriousMind
    Jan 24, 2016 at 16:58
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    $\begingroup$ @ACuriousMind you cannot perform a Lorentz Transformation between non-inertial frames. The comments of OP's question is not a good place for this discussion --- a good reason why you should post your answers as answers, instead of as comments. $\endgroup$ Jan 24, 2016 at 17:07
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    $\begingroup$ @ACuriousMind: In GR acceleration is indistinguishable from gravity... though I might be wrong here. 2. This is not a material engineering question. Assume whatever construction is comfortable to you but tough enough that it can survive a couple microseconds needed to make an observation of what goes on with it, with the rim moving at 0.75c relative to the car. It can shatter and explode all you want if it exceeds 0.8c. $\endgroup$
    – SF.
    Jan 24, 2016 at 17:15
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    $\begingroup$ Possible duplicate of Rotate a long bar in space and get close to (or even beyond) the speed of light $c$ $\endgroup$ Jan 24, 2016 at 17:20


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