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The length contraction is no "illusion"; it is real in every way. Consider the "unrealistic" situation of a man driving a car of rest length 4 m wanting to get it into a 2 m garage.
He will drive at approximately 0.866 c in order to make = 2, so that his car contracts to 2 m. (It will be good to have a massive block of concrete at the end of the garage in order to ensure that there is no question that the car finally stops in the inertial frame of the garage, or vice versa.) Thus, the man drives his (now contracted) car into the garage and his gentle wife quickly closes the door!
When the car stops in the inertial frame of the garage, it is, in fact, "rotated in space-time" and will tend to obtain, if it can, its original length relative to the garage. Thus, if it survived, it must now either bend or burst the door.
At this moment a "paradox" might occur to the reader: What about the symmetry of this problem? Relative to the driver, will not the garage be only 1 m long?
Yes, of course!
How can a 4 m long car get into a 1 m long garage?
Let us consider the situation in the inertial frame of the car. The open garage now comes towards the car. Because of the concrete wall, the garage will keep on going even after the crash with the car, taking the front of the car with it. But the back of the car is still at rest; it cannot yet "know" that the front has crashed, because of the finite speed of propagation of information. Even if the "signal" (in this case the elastic shock wave) propagates along the car with the speed of light, that signal has 4 m to propagate against the garage front's 3 m, before reaching the back of the car. This race would be a dead heat if v were 0.75 c, but now v is approximately 0.866 c. Thus, the car more than just gets into the garage!