Walk at 2 m/s along a 30 m/s train and the ground says 32 m/s. Not light: chase it at 0.9$c$ and it still passes at $c$.

A clock made of light#

Bolt two mirrors a distance $d$ apart and bounce a pulse between them. Each round trip is one tick:

$$T_0 = \frac{2d}{c}$$

The simulation runs this clock on a ship, drawn twice: in the ship's own frame on top, and in the ground's frame below.

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Where γ comes from#

The triangle's height is $d$, its base is how far the ship moves in half a tick, and its slanted side is what the light covers, all at $c$. Pythagoras then gives the stretch factor, $\gamma$:

$$T = \gamma T_0, \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$

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Whose clock is slow?#

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Lengths, and only along the motion#

The ground frame measures a moving ship shorter by $\gamma$:

$$\pos{L} = \frac{\pos{L_0}}{\gamma}$$

The clocks explain why. The ground times the ship across a baseline and calls the distance $vt$. The ship's clock times the same crossing as $t/\gamma$, so the ship calls the distance $vt/\gamma$.

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The piece everyone leaves out#

If each frame finds the other's clock slow, how can both be right? Because the two panels are drawn at different instants. Each panel is one frame's "now", and the two frames slice spacetime differently.

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The ship carries a lamp at each end, and both fire at one instant of ship time.

What comes next#

One postulate, three consequences, standing or falling together. Insist that light moves at $c$ in every inertial frame, and moving clocks run slow by $\gamma$, moving lengths shrink along the motion by $\gamma$, and simultaneity depends on the frame. Next comes the other constant of twentieth-century physics: energy that arrives in lumps whose size is set by frequency, and light that arrives one photon at a time.