Galilean and special relativity

📚The physics

Galilean relativity is the common-sense version and it works perfectly well at everyday speeds. Velocities simply add: walk forward at 2 m s\(^{-1}\) on a train doing 30 m s\(^{-1}\) and the ground sees 32 m s\(^{-1}\). The transformations are \( x' = x - vt \) and \( t' = t \) — note that second equation, because it quietly assumes everyone shares one universal clock. That assumption is the one Einstein throws away.

The problem is that Maxwell’s equations predict a specific speed for light, with no mention of who is measuring it. Under Galilean addition, chasing a light beam at half its speed should make it recede at half speed. Experiment says otherwise.

The two postulates. First, the laws of physics are identical in all inertial frames. Second, the speed of light in a vacuum is the same for all inertial observers, regardless of the motion of the source or the observer. Everything else in this topic is a consequence — and each consequence is strange precisely because the second postulate is being taken seriously.

Time dilation. A moving clock runs slow:

and \( \Delta t_0 \) is the proper time — the interval measured in the frame where the two events happen at the same place. Getting proper time right is most of the battle. Ask yourself which observer sees both events at one location; that observer measures the proper time, and every other observer measures something longer.

Length contraction. A moving object is shortened along its direction of motion: \( L = L_0/\gamma \), where proper length \(L_0\) is measured in the frame where the object is at rest. Note the asymmetry with time: proper time is the shortest interval, proper length is the longest length. Contraction happens only along the direction of motion; transverse dimensions are unaffected.

Relativity of simultaneity. Two events that are simultaneous in one frame are generally not simultaneous in another. This is not a measurement error or a signal-delay effect — it is a statement about time itself. It is also the resolution of most apparent paradoxes in this topic: when two observers disagree about lengths or times, they are usually disagreeing about which events counted as happening “at once”.

The spacetime interval is what everyone does agree on. While \(\Delta x\) and \(\Delta t\) differ between frames, the quantity \( (c\Delta t)^{2} - (\Delta x)^{2} \) is invariant. Relativity is less a theory about things being subjective than a theory about which quantity is the real one.

Muons are the standard evidence. Created high in the atmosphere, they have a half-life so short that classically almost none should reach the ground. Far more arrive than that. From our frame, their clocks run slow so they live long enough; from the muon’s frame, its lifetime is perfectly ordinary but the atmosphere is length-contracted to a fraction of its thickness. Both frames predict the same number of muons at the detector, which is the point — the physics agrees even when the descriptions do not.