Special Relativity
Lorentz transformations, time dilation, and the geometry of spacetime.
Einstein's two postulates — the laws of physics are the same in all inertial frames, and the speed of light $c$ is invariant — force the coordinate change between frames to be the Lorentz transformation:
$$t' = \gamma(t - vx/c^2), \quad x' = \gamma(x - vt), \qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}.$$Consequences include time dilation $\Delta t = \gamma \Delta\tau$, length contraction $L = L_0/\gamma$, and a unified spacetime interval
$$ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$$that is invariant under boosts. Energy and momentum unify into the four-momentum $p^\mu = (E/c, \mathbf p)$ with $E^2 = (pc)^2 + (mc^2)^2$.
Interactive: time dilation via light clock
A photon bouncing between two mirrors traces a longer path in the moving frame — so the moving clock ticks slower by factor $\gamma$.