Relativity · 27.1
Frames & Galilean Relativity
Pour coffee on a smoothly cruising train and it falls straight into the cup, exactly as it would at home. Galileo noticed this on ships four centuries ago: below deck, with the sea calm, no experiment can tell you whether the ship is moving. That innocent observation is the first relativity principle.
Build the model
Connect the measurement to the mechanism.
A reference frame is a coordinate grid plus a clock; an inertial frame is one in which Newton’s first law holds — no fictitious pushes appear. The Galilean principle of relativity says the laws of mechanics are identical in every inertial frame: uniform velocity is undetectable from inside. Translating between frames is bookkeeping — the Galilean transformations x′ = x − vt, t′ = t — and differentiating gives velocity addition u′ = u − v, then a′ = a: both frames agree on accelerations, hence on forces, hence on all of mechanics.
The picture cracked when Maxwell’s equations predicted a definite speed for light, c = 3.00 × 10⁸ m s⁻¹, with no frame attached. Galileo insists a moving observer must measure c − v; Maxwell’s equations contain no v. One of them had to give.
- Simple definition
- An inertial frame is one moving at constant velocity in which Newton’s laws hold; Galilean relativity says mechanics is identical in all of them, with frames related by x′ = x − vt, t′ = t and velocities by u′ = u − v.
- Example
- A flight attendant walks forward at 1 m/s in a plane cruising at 250 m/s: the ground measures 251 m/s, the plane measures 1 m/s — and the coffee pours identically for both observers.
The moving observer’s origin has slid a distance vt away, so subtract it. The second equation is the hidden assumption of all pre-1905 physics: one universal clock for everybody.
v = relative speed of the frames along x
Speeds simply add and subtract: a ball thrown at 8 m/s inside a 30 m/s train does 38 m/s past the platform. Utterly reliable at everyday speeds — and, it turns out, only approximately true.
Differentiate x′ = x − vt with t′ = t
Differentiating u′ = u − v kills the constant v. Both frames agree on every acceleration and every force — which is exactly why no mechanics experiment can reveal uniform motion.
v is constant, so it dies under d/dt
Maxwell’s equations fix the speed of light from constants of electricity and magnetism, without asking who is measuring. Galilean addition says different observers must disagree. Both cannot be right.
c = 1/√(μ₀ε₀) — no frame appears
What counts as inertial
A frame gliding at constant velocity is inertial: a puck released at rest stays put. A braking car or a rotating carousel is not — loose objects accelerate with no force in sight, and you must invent fictitious forces to patch Newton up. The Earth’s surface spins and orbits, but the accelerations are so small that it passes as inertial for laboratory physics.
Relativity is about symmetry, not motion
The deep claim is not that things move — it is that the laws cannot tell inertial frames apart. Drop a ball on the train and it lands at your feet, because it shares the train’s velocity and gains none horizontally. Every mechanics experiment comes out identical, so the question ‘are we really moving?’ has no mechanical answer. Only relative velocity between frames is measurable.
The crack Maxwell opened
By 1865 light was an electromagnetic wave with speed c set by μ₀ and ε₀. Waves usually have a medium — sound needs air — so physicists invented the ‘aether’ and expected light’s measured speed to vary with the observer’s motion through it, by u′ = c − v. The Earth orbits at 30 km/s, so the effect had to be there at 1 part in 10⁴. The next lesson is the experiment that went looking — and found nothing.
Change one variable at a time
Make the relationship visible.
Galileo’s whole law is a downward shift of the line u′ = u by v. It works flawlessly for balls and passengers — then try it on light: the train observer should measure c − v, a different light speed in every frame. Maxwell’s equations, and lesson 27.2’s experiment, say that never happens.
Train-frame speed u′ = u − v8 m/s
Ground-frame speed u20 m/s
Same rule applied to lightc − v = 299,792,446 m/s
What experiment finds for lightc in both frames — Galileo fails here
Catch the common trap
Explain before calculating.
You are sealed in a windowless cabin moving at constant velocity. Which experiment can reveal your speed?
Choose an answer to test the model.
Worked examples
State the rule, substitute, then check units.
EasyA passenger walks forward at 2.0 m/s along a train moving at 30.0 m/s. Find her velocity in the ground frame, and her velocity if she walks toward the rear instead.
- Ground frame: u = u′ + v = 2.0 + 30.0 = 32.0 m/s.
- Walking rearward, u′ = −2.0 m/s, so u = −2.0 + 30.0 = 28.0 m/s.
- Both answers come from the same rule u′ = u − v, rearranged — velocities add like ordinary vectors in Galilean physics.
Answer32.0 m/s forward; 28.0 m/s walking rearward
MediumA ball is thrown at 8.0 m/s (train frame) inside a carriage doing 30.0 m/s. Where is the ball after 3.0 s in each frame? Verify the Galilean transformation connects the answers.
- Train frame: x′ = u′t = 8.0 × 3.0 = 24.0 m from the thrower.
- Ground frame: u = 38.0 m/s, so x = 38.0 × 3.0 = 114.0 m.
- Check: x′ = x − vt = 114.0 − 30.0 × 3.0 = 24.0 m ✓ — same event, two coordinate grids.
Answer24.0 m (train frame), 114.0 m (ground frame), linked by x′ = x − vt
HardMaxwell’s equations give c = 1/√(μ₀ε₀) = 3.00 × 10⁸ m/s with no reference frame specified. The Earth orbits the Sun at 30 km/s. What fractional change in the measured speed of light did Galilean physics predict across the year, and why was that a crisis?
- Galilean addition: measured speed should be c ± v depending on the direction of travel through the supposed aether.
- Fractional shift: v/c = 3.0 × 10⁴ ÷ 3.0 × 10⁸ = 1.0 × 10⁻⁴ — one part in ten thousand, reversing over six months.
- But Maxwell’s c is built only from μ₀ and ε₀, constants with no v in them. Either Maxwell’s equations were wrong in moving frames, or the Galilean transformation was. Interferometers were precise enough to referee — see 27.2.
AnswerA ±1 × 10⁻⁴ seasonal shift was predicted; Maxwell’s frame-free c said there should be none
ChallengingStarting from u′ = u − v with v constant, show that both frames agree on accelerations and hence on Newton’s second law — and identify the single assumption that special relativity will later abandon.
- Differentiate with respect to time: du′/dt = du/dt − 0, so a′ = a — the constant v vanishes.
- With identical masses and forces, F = ma holds in both frames: no mechanics experiment can distinguish them. Galilean relativity is proved, not assumed, for mechanics.
- The derivation silently used t′ = t — the same clock for all observers, which let us differentiate both sides ‘with respect to time’ as if time were shared. That is the assumption Einstein drops: once simultaneity is frame-dependent, u′ = u − v must be rebuilt (27.4).
Answera′ = a follows, so mechanics is frame-blind; the buried assumption is universal time t′ = t