Chapter 1: Galilean Relativity
Before Einstein, the principle of relativity was already well established. Galileo and Newton understood that the laws of mechanics are the same in all uniformly moving reference frames. This "Galilean relativity" worked perfectly for everyday physics—but it would eventually clash with electromagnetism.
The Principle of Relativity
Galileo first articulated the principle of relativity in 1632. Imagine being in the cabin of a smoothly sailing ship:
"Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals... The fish in their water will swim indifferently to all sides; the drops will all fall into the vessel placed underneath; and you, in throwing something to your friend, will need throw it no more strongly in one direction than another."
— Galileo Galilei, Dialogue Concerning the Two Chief World Systems
The key insight: no experiment performed inside the cabin can determine whether the ship is moving or at rest. The laws of mechanics are identical in both cases.
Inertial Reference Frames
An inertial reference frame is one in which Newton's first law holds: objects at rest stay at rest, and objects in motion continue in uniform motion unless acted upon by a force.
✓ Inertial Frames
- • Laboratory on Earth (approximately)
- • Spaceship coasting at constant velocity
- • Train moving at constant speed on straight track
- • Any frame moving at constant velocity relative to an inertial frame
✗ Non-Inertial Frames
- • Accelerating car
- • Rotating merry-go-round
- • Frame fixed to spinning Earth (small effect)
- • Rocket during acceleration
In non-inertial frames, fictitious forces (like centrifugal force) appear. The principle of relativity applies only to inertial frames.
Galilean Transformations
How do we relate measurements in two different inertial frames? Consider frame S (at rest) and frame S′ moving at velocity v in the x-direction.
Galilean Transformation Equations
\( x' = x - vt \)
\( y' = y \)
\( z' = z \)
\( t' = t \)
Note: Time is absolute in Galilean relativity—all observers agree on the time of events.
These transformations have important consequences:
- • Velocity addition: If an object has velocity u in S, its velocity in S′ is simply \( u' = u - v \)
- • Acceleration is invariant: \( a' = a \) (same in all inertial frames)
- • Force is invariant: \( F' = F = ma \)
The Assumption of Absolute Time
Central to Galilean relativity is the assumption that time is absolute. This means:
Universal Simultaneity
Events simultaneous in one frame are simultaneous in all frames.
Time Intervals
The time between events is the same for all observers: \( \Delta t' = \Delta t \)
Synchronized Clocks
Clocks can be synchronized throughout space, independent of motion.
This assumption will be abandoned in special relativity! Einstein showed that time is not absolute—it depends on the observer's motion. This is where Galilean relativity breaks down.
The Clash with Electromagnetism
Galilean relativity works perfectly for mechanics, but Maxwell's equations of electromagnetism create a problem. According to Maxwell, electromagnetic waves travel at speed:
\( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s} \)
But relative to what? If you apply the Galilean velocity addition formula:
- • If you chase a light beam at speed v, shouldn't you see it traveling at c - v?
- • If you run toward an oncoming light beam, shouldn't you measure c + v?
Scientists proposed the luminiferous aether—a medium that light waves propagate through, just as sound waves propagate through air. The speed c would then be relative to this aether. The famous Michelson-Morley experiment set out to detect Earth's motion through this aether...