Lorentz contraction using odometers? In principle, would cars moving between a pair of points at different speeds show different odometer readings due to Length contraction? When we use odometers to measure length between two points, what can we say of length contraction?
 A: Yes, in theory the odometer will measure a contracted length. To show that rigorously is quite a complicated matter, because the rotating wheel of the car is no longer circular in the frame of the car or of the road.
Relative to the road, the bottom of the wheel is stationary, the centre of the wheel is moving at the speed of the car, and the top of the wheel is moving faster still (you have to use the relativistic velocity formula to work out the speed).
Relative to the car, the section of the bottom of the wheel that is in contact with the road is moving at the same speed as the road, and will therefore be length contracted by the same amount as the road.
Generally speaking, the circumference of the wheel is no longer 2pi times the radius in either the frame of the car or the frame of the road.
Of course, that is all hypothetical, since you could never accelerate the wheel of a car to produce a measurable relativistic effect.
If you want to get into the subject in more detail, google the Ehrenfest paradox.
A: You ask what can be said in principle about the measured distance in that situation, and the answer to that is "nothing". The principle of relativity only says that Lorentz-boosting the whole system can't change the physics. Lorentz-boosting a system consisting of a car driving on a road at 100 km/h gets you another system consisting of a car driving on a road at 100 km/h. The principle of relativity says nothing about the relationship between a car driving on a road at 100 km/h and a car driving on a road at 1,000,000,000 km/h. Those situations are not physically equivalent at all.
An odometer that naively counted the number of wheel rotations and multiplied by a constant would probably report different distances at different speeds, but the relationship depends on tricky details of the wheel-road interaction and can't be derived from universal principles. In a world where relativistic car speeds were possible, car odometers probably wouldn't be constructed in that naive way, because it would defeat the purpose of having an odometer. E.g., if you want to use the odometer reading to determine your current location on a road map, you need it to report proper road length. It is possible (not forbidden by the principle of relativity) to construct a device that measures proper road length, so they probably would.
