Special Relativity

Length Contraction

The relative speed between two inertial reference frames, $v$, is the same in both frames. Think about the astronaut and astronomer in the light clock example from the previous chapter.

• The astronaut observes the astronomer moving at a constant speed, $v$.
• The astronomer observes the astronaut moving at a constant speed, $v$.
• The astronomer observes the elapsed time on the astronaut’s clock, $t^{\prime }$, to be dilated by the Lorentz factor, $\gamma$.

Remember the definition of velocity, $v=x/t$. Since we know that the time interval $t$ is dilated by the Lorentz factor, $\gamma$, then the distance traveled by the astronaut, $x^{\prime }$, must also be stretched by $\gamma$ to keep the velocity constant.

$$\begin{gathered} v=x/t,v=x^{\prime }/t^{\prime },t=\gamma t^{\prime } \\ v=\frac{x}{t}=\frac{x^{\prime }}{t^{\prime }},\gamma =t/t^{\prime } \end{gathered}$$ $$x^{\prime }=\frac{x{t}^{\prime }}{t}=\frac{x}{\gamma }$$ $$x^{\prime }=x\sqrt{1-(\frac{v}{c}{)}^2}$$

For any velocity $v$ less than the speed of light, $c$, the length contraction factor is less than 1, meaning the length of the object in the moving frame is shorter than the length of the object in the stationary frame.