A.5 Galilean and special relativityIB Physics HL: Revision notes

Section 1

Reference frames and Galilean relativity

A reference frame is a coordinate system (x, y, z, t) from which events are measured. An inertial frame is one that is not accelerating, where Newton's first law holds. Galilean relativity: Newton's laws of motion are the same in all inertial frames, so no mechanics experiment can tell which frame is 'really' at rest.

For frame S′ moving at v along x relative to S (origins coinciding at t = 0):

x′ = x − vt and t′ = t (time is absolute)
velocity addition: u′ = u − v

Key termsreference frameinertial frameGalilean relativity

Section 2

The postulates of special relativity

The laws of physics are the same in all inertial frames.
The speed of light in a vacuum is c in all inertial frames, whatever the motion of the source or observer.

Postulate 2 contradicts Galilean velocity addition, which would predict c + v for light from a moving source. So Galilean transformations are only an approximation valid when v ≪ c.

Key termspostulates of special relativity

Common mistake

Relative velocities can still be up to 2c when calculated by a third observer (two objects each at 0.9c in opposite directions). What cannot exceed c is any object's speed measured in one frame, or its speed relative to another object using relativistic addition.

Section 3

Lorentz transformations and velocity addition

With γ = 1/√(1 − v²/c²):

x′ = γ(x − vt)
t′ = γ(t − vx/c²)

These lead to the relativistic velocity addition equation u′ = (u − v)/(1 − uv/c²), which never gives a speed above c. When v ≪ c, γ ≈ 1 and they reduce to the Galilean equations.

Key termsLorentz factorLorentz transformationrelativistic velocity addition

Section 4

Space-time interval, proper time and proper length

The space-time interval between two events, (Δs)² = (cΔt)² − (Δx)², is invariant: all inertial observers calculate the same value.

Proper time Δt₀: the time between two events measured in the frame where they happen at the same place (the shortest time measured by anyone).
Proper length L₀: the length of an object measured in the frame where it is at rest (the longest length).

Time dilation: Δt = γΔt₀. Length contraction: L = L₀/γ (only along the direction of motion).

Key termsspace-time intervalproper timeproper lengthtime dilationlength contraction

Exam tip

First decide which observer measures the proper quantity: the one for whom the events happen in the same place (time), or the one for whom the object is at rest (length).

Section 5

Relativity of simultaneity and space-time diagrams

Events at different places that are simultaneous in one frame are not simultaneous in another frame moving along the line joining them (from the −vx/c² term). Observers can disagree on the order of events only when the interval is space-like (Δx > cΔt), so no signal could connect them; causality is preserved.

On a space-time diagram (ct upwards, x across), an object's history is its world line. For a particle moving at speed v, the angle θ between its world line and the time axis satisfies tan θ = v/c. Light travels at 45°.

Key termsrelativity of simultaneityworld linespace-time diagram

Section 6

Evidence: muon decay experiments

Muons made in the upper atmosphere have a half-life of about 1.5 μs at rest. Even at 0.995c, without relativity almost all would decay before reaching sea level, yet many are detected.

Earth frame: the muons' clock runs slow; their half-life is dilated by γ ≈ 10.
Muon frame: the half-life is the proper 1.5 μs, but the atmosphere is length-contracted by γ.

Both give the same prediction, which matches the counts: evidence for time dilation and length contraction.

Key termsmuon decaytime dilationlength contraction

That's the notes covered.

Carry on to the next subtopic.