Length Contraction
The rest-frame length of an object is its proper length. A frame in which the object moves measures a shorter length along the direction of motion because endpoint positions must be recorded simultaneously in that measuring frame.
Why is a moving object's measured length shorter along its motion?
Length means the distance between two endpoint positions measured at the same time. But simultaneity is frame-dependent. Once different frames slice spacetime differently, they do not use the same pair of endpoint events to measure a moving object.
is proper length, measured in the object's own rest frame.
Compare one rod in two inertial frames.
The rest frame measures both endpoints simultaneously while the object itself is stationary, defining L₀.
Lengths perpendicular to the relative motion are unchanged by a Lorentz boost along the motion direction.
Length contraction is a statement about coordinate measurements made with synchronized clocks and simultaneous endpoint positions, not simply how an object looks after light-travel delays.
To measure a length, you must first decide what “at the same time” means.
That is why length contraction is not an independent trick. It is another consequence of Lorentz geometry and frame-dependent simultaneity.