Length contraction

To study the relativity of the length, we will analyze the following situation: we will imagine the same wagon of the previous item with the same conditions already established. As shown in the following figure, the wagon will go through a tunnel. We will adopt two benchmarks to measure the length of the wagon. Are they:

  • R: reference at rest in relation to the body, whose length will be measured - the tunnel. For this frame, the length of the tunnel is l.
  • R ': mobile reference to the tunnel. In this frame, the length of the tunnel is l '.

In frame R, the length of the tunnel is l. Thus, as the wagon passes through the tunnel completely, the frame R sees it travel a distance l during a time interval Δt. Therefore, regarding the referential R, we have:

For the referential R ', the tunnel has length l' and moves to the left with speed v, as shown below.

In this way, R 'sees the tunnel pass entirely through it traveling a distance l' during a interval Δt '. Like this:


We can rewrite:

Substituting the above result in the wagon length expression:


We obtain:


The result is less than 1, so l 'is less than l. Therefore:

For a frame R, which is at rest in relation to a body, that body has length l. For a frame R 'that moves in relation to the same body, the length is l', being l 'smaller than l. We call this phenomenon the length contraction. Remember that the contraction occurs only in the direction of movement.