In Galilean relativity, velocities simply add: a ball thrown forward at 20 m/s from a train moving at 30 m/s goes 50 m/s relative to the ground. And no experiment with balls and springs inside a smoothly moving train can tell you how fast the train is going. Then Maxwell's equations predicted that light moves at one definite speed, $c = 1/\sqrt{\mu_0\varepsilon_0}$. Relative to what? Physicists assumed an invisible medium, the “ether”. In 1887 Michelson and Morley's interferometer looked for Earth's motion through it and found none.
🔑 Key idea
Einstein's two postulates (1905):
The laws of physics are the same in every inertial (non-accelerating) frame.
The speed of light in a vacuum is $c = 3.00 \times 10^8$ m/s in every inertial frame, whatever the motion of the source or the observer.
Since $c$ must come out the same for everyone, our everyday rules for adding velocities, and for time and distance themselves, have to give way at high speeds.
💡 Why it works
The second postulate follows from the first once you trust Maxwell's equations. They are laws of physics, so by postulate 1 they hold in every inertial frame. And they give the speed of light as $1/\sqrt{\mu_0\varepsilon_0}$, built from two constants with no mention of anyone's velocity. So every inertial observer must measure the same $c$. The price is high: if two observers moving relative to each other time the same flash and both get $c$, they cannot also agree about distances and times in the everyday way. The rest of this unit works out what they do agree on.
A light clock bounces a pulse between two mirrors. Seen from the ground, the moving clock's light must travel a longer, slanted path at the same speed $c$, so each tick takes longer. Constant $c$ already forces time to stretch.
Worked examples
✏️ Worked example: A headlight on a fast ship
A spaceship moving at $0.50c$ toward Earth turns on a forward headlight. How fast does the light move according to the crew, and according to observers on Earth?
By postulate 2, the crew measures $c$, since the light moves through vacuum.
By postulate 2 again, Earth observers also measure $c$, not $1.5c$. The source's motion does not add to light's speed.
Answer: $3.00 \times 10^8$ m/s for both
✏️ Worked example: Timing light in a moving cabin
A ship moves past Earth at $0.80c$. A light pulse crosses its 30 m cabin, front to back. How long does the crossing take, as measured by the crew?
In the crew's own frame the cabin is at rest, and by postulate 1 their physics is ordinary.
By postulate 2 the pulse moves at $c$: $t = \frac{30\ \text{m}}{3.00 \times 10^8\ \text{m/s}} = 1.0 \times 10^{-7}$ s.
Answer: $100$ ns, exactly as if the ship were parked
The evidence is strong. In 1964 an experiment at CERN timed gamma rays given off by neutral pions moving at $0.99975c$. Galilean addition would predict nearly $2c$ for photons sent forward; the measured speed was $c$, to within about one part in ten thousand.
⚠️ Common mistake
Thinking light from a moving source travels faster than $c$, the way a ball thrown from a train goes faster than the throw. That rule works for balls at everyday speeds, but light's speed is the same for every inertial observer whatever the source does. A check: binary stars orbit fast toward and away from us, yet their light arrives at the same speed; if it did not, their images would scramble in ways never seen.
⚠️ Common mistake
Reading postulate 1 as “everything is relative”. It says the laws are the same in every inertial frame. Some quantities, like $c$ and (as you will see) the spacetime interval, are the same for everyone.
Practice
Pass with 3 right answers, the last 2 in a row. Each right answer earns 1 XP.