Length contraction is a phenomenon where objects appear shorter in the direction of motion when observed from a different inertial frame of reference. This is a relativistic effect that becomes significant at velocities approaching the speed of light.

Length contraction only occurs along the direction of motion, dimensions perpendicular to the motion remain unchanged.

Mathematical Description

The relationship between proper length and contracted length is given by using the Lorentz factor:

Reciprocity

An important aspect of length contraction is that each observer sees the other's measuring rod contracted.

This apparent paradox is resolved by recognizing that the measurement of length requires simultaneity, and simultaneity is relative in different frames.

Relationship to Time Dilation

Length contraction and time dilation are intimately connected through the Lorentz transformation. While time dilates by a factor of , length contracts by a factor of , ensuring that the speed of light remains constant in all frames.

Derivation

Time Dilation Based Derivation

Consider a moving clock moving at a velocity and a rod with rest length , the time dilation between the moving clock and the time the rod experiences is given by formula .

According to the postulates of special relativity the magnitude of the relative velocity of the clock will be the same in both the stationary clocks frame and the moving clocks frame.

Through knowing the velocity of the clock, we can calculate the length of the rod from the clocks frame, by determining the time it takes for the clock to travel from the start of the rod to the end.

We can determine the time at which the clock moves over the left side of the rod () and the time at which it moves over the right side () as follows:

Using the equations for time dilation we can convert these times into their respective times from the clocks frame ( and respectively):

Now that we have these times we can determine the length of the rod according to the clocks frame of reference.