Newton’s gravity works brilliantly — but it has a deep problem. It says that if the Sun vanished, Earth would instantly feel the change, even though the Sun is 8 light-minutes away. Nothing should travel faster than light. Einstein fixed this in two steps: special relativity (1905) for motion, then general relativity (1915) for gravity.

Special relativity starts from two postulates:

1. The laws of physics are the same in every inertial frame (no experiment can tell you whether you’re “really” moving or “really” stationary)
2. The speed of light in vacuum is the same for all observers ($c \approx 3 \times 10^8$ m/s), regardless of how fast the source or observer is moving

The first is reasonable — Galileo would agree. The second is shocking. If you’re on a train moving at $0.9c$ and you shine a flashlight forward, you measure the light travelling at $c$. Someone on the platform also measures it at $c$. Not $1.9c$. The same $c$.

Time dilation

If light speed is the same for everyone, then time itself must run differently for different observers. A clock moving relative to you ticks slower:

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

$\Delta t_0$ is the time between ticks as measured by someone travelling with the clock (the proper time). $\gamma$ is the Lorentz factor — it’s 1 at rest and blows up as $v \to c$.

This isn’t a trick of perception. Muons created in the upper atmosphere by cosmic rays have a half-life of 2.2 μs — they should decay before reaching the ground. But they’re moving at ~0.998c, so their clocks run ~15× slower in our frame. They reach the ground. Time dilation is measured routinely.

Length contraction

Space contracts too. An object moving at $v$ is shorter in the direction of motion:

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

A 100-metre spaceship at $0.9c$ measures only 43.6 m to a stationary observer. The spaceship’s occupants notice nothing — to them, your ruler is the one that’s contracted. Both are right. There is no absolute frame.

The speed limit

As $v \to c$, $\gamma \to \infty$. The kinetic energy of a massive object is:

$$K = (\gamma - 1) \, mc^2$$

This diverges as $v \to c$, which means you’d need infinite energy to accelerate a massive object to light speed. The speed of light isn’t just fast — it’s a structural limit woven into the geometry of spacetime.

The invariant: what doesn’t change

Different observers disagree about distances and time intervals. But they all agree on the spacetime interval:

$$\Delta s^2 = -c^2 \Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$

This quantity is the same for every observer. It’s the “distance” in spacetime — but with a crucial minus sign on the time component. That minus sign is the seed of everything that follows: curved spacetime, black holes, the expansion of the universe.