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IB DP Physics · Theme A Space, time and motion · A.5 (HL only) · ~8 min read

A.5 Galilean and special relativity

Reference frames and Galilean transformations, Einstein's two postulates, the Lorentz transformations, time dilation, length contraction, simultaneity, the spacetime interval, spacetime diagrams and the muon experiment.

Reference framesGalilean transformationPostulatesLorentz transformationTime dilationLength contractionSpacetime diagramsMuon decay
RSby Riyaz Sir
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No tag = SL and HLHL ONLY HL students onlyDATA BOOKLET given in the examM1A1 how marks are awardedhighlight = comes up again and againTest yourself at the end

In 30 seconds

  • A.5 is HL only. A reference frame is a set of rulers and clocks. Newton's laws hold in every inertial (non-accelerating) frame.
  • Galilean relativity adds velocities simply, u′ = u − v. It fails near the speed of light, because light has the same speed c in every inertial frame (Einstein's second postulate).
  • The Lorentz transformations replace the Galilean ones. They lead to time dilation Δt = γΔt0, length contraction L = L0/γ and the relativity of simultaneity.
  • Observers disagree about times and lengths, but they all agree on the spacetime interval (Δs)2 = (cΔt)2 − (Δx)2.

By the end you can

  • Apply the Galilean transformations and explain why they fail at high speed. HL
  • State the two postulates of special relativity. HL
  • Use the Lorentz transformations, γ and relativistic velocity addition. HL
  • Identify proper time and proper length, and calculate time dilation and length contraction. HL
  • Use the invariant spacetime interval and draw and read spacetime diagrams. HL
  • Explain the muon experiment from both frames. HL

1Reference frames and Galilean relativity HL

xyframe S (ground)x′y′frame S′ (train)veventGalilean: x′ = x − vt, t′ = t, u′ = u − v. Fine at low speed, wrong near c.
S′ moves at speed v along the x-axis of S. At t = 0 the origins coincide.

Inertial frame — a frame moving at constant velocity, where an object with no resultant force keeps a constant velocity.

x′ = x − vt, t′ = t, u′ = u − v DATA BOOKLET
Galilean transformation and velocity addition
Worked example

A train moves at 30 m s−1. A passenger walks forwards at 2.0 m s−1 relative to the train. Find her speed relative to the ground and the speed of a light beam from the train's headlight, according to Galileo.

  1. u = u′ + v = 2.0 + 30 = 32 m s−1 ✓ A1
  2. light: c + 30 m s−1, which contradicts experiment and Maxwell's equations R1

2Einstein's postulates HL

  • Postulate 1: the laws of physics are the same in all inertial frames.
  • Postulate 2: the speed of light in a vacuum is the same, c, for all inertial observers, whatever the motion of the source.
γ = 1√(1 − v2/c2) DATA BOOKLET
the Lorentz factor
v/cγ0.20.40.60.81.01234567v = 0.8c: γ = 1.670.6c: 1.25γ = 1/√(1 − v²/c²)γ ≥ 1, and γ → ∞ as v → c
γ is almost exactly 1 at everyday speeds and shoots up near c.

3Lorentz transformations HL

x′ = γ(x − vt), t′ = γ(t − vxc2) DATA BOOKLET
coordinates of an event in S′
u′ = u − v1 − uv/c2 DATA BOOKLET
relativistic velocity addition; never gives more than c
Worked example

In S, an event happens at x = 100 m, t = 0.50 μs. Find its coordinates in S′ moving at 0.60c.

  1. γ = 1/√(1 − 0.36) = 1.25 A1
  2. x′ = 1.25(100 − 0.60 × 3.0 × 108 × 0.50 × 10−6) = 1.25(100 − 90) = 12.5 m M1A1
  3. t′ = 1.25(0.50 × 10−6 − 0.60 × 100/(3.0 × 108)) = 0.375 μs A1
Worked example

Two spaceships approach the Earth from opposite sides, each at 0.60c. Find the speed of one relative to the other.

  1. Earth frame S: ship B has u = −0.60c; ship A is frame S′, moving at v = +0.60c M1
  2. u′ = (u − v)/(1 − uv/c2) = (−0.60c − 0.60c)/(1 + 0.36) = −0.88c A1
  3. speed 0.88c, not 1.2c A1

4Time dilation and length contraction HL

Proper time Δt0 — the time between two events measured by a clock present at both, so the events happen at the same place in its frame. It is the shortest time any observer measures.

Proper length L0 — the length of an object measured in its own rest frame. It is the longest length any observer measures.

Δt = γΔt0 DATA BOOKLET
moving clocks run slow
L = L0γ DATA BOOKLET
moving objects are shorter along their direction of motion only
Worked example

A spaceship travels at 0.80c to a star 12 light-years away (Earth frame). Find the journey time in the Earth frame and for the crew.

  1. Earth: t = 12 ly ÷ 0.80c = 15 years A1
  2. γ = 1/0.60 = 1.67
  3. crew clock is present at departure and arrival, so it shows proper time: 15/1.67 = 9.0 years M1A1
  4. for the crew the distance is contracted to 12/1.67 = 7.2 ly, and 7.2/0.80 = 9.0 years ✓ R1
Common mistakeSwapping proper and dilated time. Ask: which observer has ONE clock present at both events? That clock measures the proper time, and it is the smaller value.

The muon experiment

no relativity:most decay in 0.66 kmEarth frame: timedilated, lifetime ×10muon frame: 10 kmcontracts to 1.0 kmMuons made 10 km up at 0.995c (γ ≈ 10) still reach the ground
Two frames, one result: the muons reach the ground.
  • Muons have a mean lifetime of about 2.2 μs in their own frame. At 0.995c (γ ≈ 10) they would travel only 0.66 km on average without relativity, yet many reach the ground from 10 km up.
  • Earth frame: their lifetime is dilated to about 22 μs, enough to travel about 6.6 km on average, so a measurable fraction survive 10 km. Muon frame: the lifetime is normal, but the atmosphere is contracted to about 1 km.

5Simultaneity and the spacetime interval HL

  • Relativity of simultaneity: events that are simultaneous in one frame are not simultaneous in a frame moving relative to it (unless they happen at the same place). This is the vx/c2 term in the Lorentz transformation for t′.
(Δs)2 = (cΔt)2 − (Δx)2 DATA BOOKLET
invariant: every inertial observer gets the same value
Worked example

Two events are 600 m apart and 3.0 μs apart in the lab. Find the proper time between them.

  1. cΔt = 3.0 × 108 × 3.0 × 10−6 = 900 m M1
  2. (Δs)2 = 9002 − 6002 = 4.5 × 105 m2 M1
  3. in the frame where Δx = 0: cΔt0 = 671 m, so Δt0 = 2.2 μs A1

6Spacetime diagrams HL

xctlight (45°)ct′ (world line of S′)x′θθevent Et′ of Etan θ = v/chere v = 0.5cboth S′ axes tiltby θ towardsthe light line
The S′ axes are tilted towards the light line. A faster frame means a bigger tilt.
tan θ = vc DATA BOOKLET
θ is the angle between the world line (ct′ axis) and the ct axis
  • Reading coordinates in S′: draw lines from the event parallel to the S′ axes. Lines parallel to the x′ axis are lines of equal t′, which is how the diagram shows simultaneity.
  • World lines of objects slower than light are steeper than 45°; light travels along 45° lines.
Exam tipIn "explain using a spacetime diagram" questions, draw lines of simultaneity (parallel to x′) through the two events and show that they meet the ct′ axis at different points.
Linking questionThe muon experiment ties A.5 to E.3 (decay and lifetimes). Particle accelerators are designed with these same equations.

Test yourself — cover the answers first

Quick check

  1. State Einstein's two postulates.
  2. Calculate γ at 0.60c.
  3. Which observer measures proper time?
  4. A rod moves along its length at 0.80c. Its proper length is 3.0 m. What length is measured?
  5. Is the spacetime interval between two events the same for all inertial observers?
  6. On a spacetime diagram, what angle does a light ray make with the axes?

Exam-style

  1. A particle with proper lifetime 26 ns moves at 0.90c in the lab. Find its lifetime and the distance it travels in the lab.
  2. A rocket moving at 0.50c fires a probe forwards at 0.80c relative to the rocket. Find the probe's speed relative to Earth.
Show answers
  1. Physics is the same in all inertial frames; c is the same for all inertial observers.
  2. 1.25
  3. the one for whom both events happen at the same place
  4. 3.0/1.67 = 1.8 m
  5. yes, it is invariant
  6. 45°
  7. γ = 1/√(1 − 0.81) = 2.29
    lab lifetime = 2.29 × 26 = 60 ns
    distance = 0.90 × 3.0 × 108 × 60 × 10−9 = 16 m
  8. u = (u′ + v)/(1 + u′v/c2)
    = 1.30c/(1 + 0.40)
    = 0.93c
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Independent notes written by Riyaz Sir. Not produced, endorsed or approved by the International Baccalaureate Organization. Syllabus codes refer to the IB DP subject guides.