Special relativity.

Step 1 of 6

Two light clocks

In 1905 Einstein built special relativity on two principles. The first: the laws of physics are the same for everyone moving at a constant velocity, and no experiment can tell who is “really” standing still. The second: the speed of light in a vacuum, $c \approx 3 \times 10^8\ \mathrm{m/s}$$c \approx$$3 \times 10^8\ \mathrm{m/s}$⁠, is the same for every observer, however fast they move.

The Scene has two identical light clocks: a flash of light bouncing between two mirrors, ticking each time it comes back down. One is on the platform and one is on the train. Train speed is 0, the train stands still, and the board counts the same number of ticks for both clocks. Move Train speed and watch the two counts.

Step 2 of 6

The slanted path

Seen from the platform, the train moves on while its flash goes up, so the flash travels on a slant. At Light paths, Show the Lorentz factor draws the triangle of half a tick: the gap between the mirrors $L$⁠, the distance the train has gone $vt$⁠, and the slant the light has covered $ct$⁠. By Pythagoras, $(ct)^2 = L^2 + (vt)^2$$(ct)^2 =$$L^2 + (vt)^2$⁠. On the platform the same trip takes $t_0 = L/c$⁠, and so

$t = \gamma\, t_0, \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$t =$$\gamma\, t_0, \qquad \gamma =$$\frac{1}{\sqrt{1 - v^2/c^2}}$⁠

$\gamma$⁠ is the Lorentz factor. Train speed is locked at 0.6 c.

Step 3 of 6

Time dilation

Each tick of the train’s clock, as measured from the platform, lasts $\gamma$⁠ ticks of the platform clock. So while the platform clock counts $n$⁠ ticks, the train clock counts only $\dfrac{n}{\gamma}$⁠. This is time dilation. There is nothing special about light clocks: every clock on the train, the passengers’ hearts included, ticks at the same rate, or the passengers could find out that they are moving. Here Train speed is locked at 0.8 c.

Step 4 of 6

Length contraction

The same $\gamma$⁠ shortens the train. Measured from the platform, a moving body is $\gamma$⁠ times shorter along its direction of motion, and its height and width do not change. This is length contraction. The carriage is 20 m long at rest, so from the platform

$L = \frac{20\ \mathrm{m}}{\gamma}$$L =$$\frac{20\ \mathrm{m}}{\gamma}$⁠

Step 5 of 6

Whose clock is slow?

By the first principle, a passenger on the train may say that she is standing still and the platform is moving. From where she sits, her flash goes straight up and down, and her clock ticks as usual. What does she see when she looks at the platform clock?

Step 6 of 6

Muons

Muons are particles made about $15$⁠ kilometres up, when cosmic rays strike the atmosphere. At rest a muon lives $2.2$⁠ millionths of a second on average, and in that time, even at the speed of light, it covers only about 660 m. Yet many of them reach the ground. They move at about 0.995 c, which is the Train speed now. The chart beside the Scene shows $\gamma$⁠ at this speed: about $10$⁠.

As the speed approaches $c$⁠, $\gamma$⁠ grows without limit. That is why nothing with mass reaches the speed of light.