5. Length Contraction

Definitions

Proper Length

• $l_0$ is the proper length or the length of the stationary observer

Observer at constant $v$

• $l$ is the length measured in the frame of the observer

 

Derivations

Consider a rod at rest in frame $S'$

its endpoints have fixed coordinates

$$x_1' = 0, \quad x_2' = l_0$$

the proper length is measured at the same time in the rest frame

$$l_0 = x_2' - x_1'$$

now look at the same rod from frame $S$, where the rod moves with speed $v$

to measure length in $S$, you must record the positions of both ends simultaneously in $S$

$$t_1 = t_2$$

the measured length is

$$l = x_2 - x_1$$

use the Lorentz transformation for space

$$x' = \gamma(x - vt)$$

apply this to each endpoint

$$x_2' - x_1' = \gamma[(x_2 - vt_2) - (x_1 - vt_1)]$$

since the measurement in $S$ is simultaneous, $t_1 = t_2$, so the time terms cancel

$$x_2' - x_1' = \gamma(x_2 - x_1)$$

identify the lengths

$$l_0 = \gamma l$$

rearranging

$$l = \frac{l_0}{\gamma}$$

since $\gamma > 1$, then $l < l_0$

therefore the length measured by the moving observer is contracted

 

Useful Equations

$$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}, \quad l = \frac{l_0}{\gamma}$$