Invariance of c

Einstein’s relativity assumes that the speed of light $c$ is the same to all observers. This assumption leads to the fundamental effects of special relativity.

Moving clocks run slower

Consider an observer in a train $\mathcal O_T$ and an observer standing at the trian station $\mathcal O_S$ . The train moves with speed $v$ relative to the station. Inside, there is a light bulb and a mirror (crudely depicted below).

How long does it take for the light to bounce off the mirror and reach the bulb again?

According to $\O_T$ , the light just moves vertically, so $\Delta t_T = h/c$ .
According to $\O_S$ , the light travels distance $h$ vertically, but also distance $v\Delta t_S$ horizontally. This gives the equation $h^2 v^2\Delta t_S^2 = c^2 \Delta t_S^2$ , so

( Delta tS = frac{Delta tT}{sqrt{1 - v^2/c^2}}. )

More time passes according to $\O_S$ !

We define $\gamma := \big(1 - v^2/c^2\big)^{-1/2}$ since it appears often.

Moving rulers are shortened

Now consider a light bulb at the back of the train, with a mirror at the front.

How far does the light travel in each reference frame?

In $\O_T$ it’s simple, $\Delta x_T = c \Delta t_T / 2$ .

In $\O_S$ , we consider the trip in two parts:

\begin{align*} c \Delta t_{S1} &= \Delta x_S + v \Delta t_{S1}, \quad \Delta t_{S1} = \frac{\Delta x_S}{c - v} \\ c \Delta t_{S2} &= \Delta x_S - v \Delta t_{S2}, \quad \Delta t_{S2} = \frac{\Delta x_S}{c + v} \\ \Delta t_S &= \frac{2\Delta x_S}{c (1-v^2/c^2)}. \end{align*}

We then eliminate the $\Delta t$ terms using the equation found above, $\Delta t_S = \gamma \Delta t_T$ .

\begin{align*} \gamma \Delta t_T &= 2 \gamma^2 \frac{\Delta x_S}{c} \\ \Delta x_S &= \frac{\Delta x_T}{\gamma}. \end{align*}

We see the distance interval measured on the station is shorter than the distance measured on the train. In other words, from the station, the train appears to be squished in the direction of motion. From the train, the station (and the rest of the world) appears compressed in the direction of motion.

Note

It is easy to verify conceptually that it only makes sense for length contraction to occur along the direction of motion. Otherwise different things could happen in different reference frames. Consider for example a train going through a tunnel at high speed.

Results

These two observations about length contraction and time dilation lead to the Lorentz transform. This transformation gives us a mathematical framework for converting between “events” in different reference frames, given only their relative velocity.