Special Relativity Toolkit (IPhO)

This lesson is a timed-exam toolbox: it prioritises what to try first (choose a frame, find an invariant, then compute the quantity asked). The fastest solutions usually come from writing the problem in terms of events and invariants rather than memorising separate “time dilation / length contraction” stories.

Mind-stretcher feedback

• Two photons can have nonzero total invariant mass when their total four-momentum is not null; for opposite momenta, the spatial momenta cancel while their energies add.
• Between fixed timelike-separated events, an inertial straight worldline maximises proper time. Acceleration changes velocity along the route and reduces the accumulated ∫ dt/γ.

1. Definitions (Must Know)

A. Basic notation

• Relative speed: v (take motion along the x axis).
• Dimensionless speed: β = v/c.
• Lorentz factor: γ = 1/(square root of (1-β²))

B. Events and the spacetime interval (the core invariant)

An event is something with coordinates (t,x,y,z) in a chosen frame.

Between two events, define Δ t, Δ x, Δ y, Δ z. The spacetime interval is: s² = c²(Δ t)² - (Δ x)² - (Δ y)² - (Δ z)²

Classification (useful for deciding what “can be made zero” by a frame change):

• Timelike: c²(Δ t)² > (Δ r)² where (Δ r)² = (Δ x)² + (Δ y)² + (Δ z)².
• Lightlike: c²(Δ t)² = (Δ r)².
• Spacelike: c²(Δ t)² < (Δ r)².

C. Proper time and proper length

• Proper time Δ τ is the time measured by a single clock that experiences both events (timelike separation). c²(Δ τ)² = c²(Δ t)² - (Δ r)²

• Proper length L₀ is the length of an object measured in its rest frame. In any other frame, the contracted length is L = L₀/γ when measured with endpoints simultaneous in that frame.

D. Lorentz transformations (standard configuration)

For frames S and S' with S' moving at + v along x relative to S:

E. Relativistic momentum and energy (and the key invariant)

For a particle of rest mass m: p = γ mv, E = γ mc²

Energy-momentum invariant: E² = (pc)² + (mc²)²

For any system (even many particles), define the invariant mass M by: M²c⁴ = Eₜₒₜ² - (pₜₒₜc)²

F. Velocity addition (1D)

If a particle moves at u' in S', then its speed in S is: u = (u' + v)/(1 + u'v/c²)

2. Key Ideas (What Earns Marks)

• Frame choice is a mark-winning move. If the question mentions a particle’s rest properties (rest lifetime, rest length, rest energy), start in its rest frame. If it’s a collision, consider the centre-of-momentum frame (total momentum = 0).
• Translate words into event constraints.
  • “Same place in S” means Δ x = 0 (in S).
  • “Simultaneous in S” means Δ t = 0 (in S).
  • For a rod length measurement in a given frame, endpoints must be measured at the same time in that frame.
• Invariants beat memorisation. If you can write s² or M²c⁴ = E²-(pc)², you can usually avoid multiple Lorentz transforms.
• One-line checks catch most algebra slips.
  • β → 0 gives Newtonian answers.
  • Dimensions: is it time, length, energy?
  • If you add velocities, your result must satisfy |u| < c.

3. Detailed Explanations

A. The “event method” for time dilation and length contraction

Instead of reciting rules, define the two events and use either the Lorentz transform or the invariant interval.

Time dilation (cleanest using Δ x' = 0):

If two events happen at the same place in S' (a single moving clock), then Δ x' = 0.

From the Lorentz transform: 0 = Δ x' = γ(Δ x - vΔ t) ⇒ Δ x = vΔ t Insert into the interval: c²(Δ τ)² = c²(Δ t)² - (vΔ t)² = c²(Δ t)²(1-β²) So: Δ t = γ Δ τ

Interpretation: the lab measures a longer time between ticks than the moving clock’s proper time.

Length contraction (cleanest using Δ t = 0 in the measuring frame):

To measure a moving rod’s length in frame S, you must record endpoints at the same time in S, so Δ t = 0.

Then the Lorentz transform gives: Δ x' = γΔ x Here, Δ x' is the rest-frame endpoint separation (the proper length L₀), while Δ x is the measured length L in S: L₀ = γ L ⇒ L = L₀/γ

B. Using invariant mass for collisions and decays

For a single particle, m is invariant. For a system, the invariant is the system mass M: M²c⁴ = Eₜₒₜ² - (pₜₒₜc)²

Why this is powerful:

• In the centre-of-momentum (COM) frame, pₜₒₜ = 0, so M c² = Eₜₒₜ immediately.
• You can compute M in any convenient frame and it will be the same in all frames.

Common IPhO use cases:

• minimum energy (threshold) for producing new massive particles,
• determining whether a decay is kinematically allowed,
• finding COM energy without doing multiple Lorentz transforms.

C. Velocity addition without panic

In 1D, do not subtract speeds directly. Use: u = (u' + v)/(1 + u'v/c²)

Two quick sanity checks:

• if u' ≪ c and v ≪ c, then u ≈ u' + v,
• if u' = c, then u = c (light speed stays c).

D. Small-β expansions (fast approximations)

When β² ≪ 1: γ = 1/(square root of (1-β²)) ≈ 1 + β²/2

So the fractional time dilation is about: (Δ t-Δ τ)/(Δ τ) ≈ β²/2

This is often enough for quick “estimate the effect size” marks.

4. Common Mistakes

• Using length contraction when the endpoints are not measured simultaneously in the measuring frame.
• Calling any measured time “proper time”. Proper time is measured by a single clock that experiences both events.
• Mixing which frame each symbol belongs to (prime everything in S' consistently).
• Treating relativistic velocity addition as u = u'± v.
• Forgetting that total momentum is a vector when using M²c⁴ = E²-(pc)².
• Dropping c² factors inconsistently (especially when switching between energy and mass).

5. Exam Tips

• Start by writing: “Let S be the lab frame, S' be the object rest frame.” Then define v (direction matters).
• If the question is about a lifetime or decay, use proper time: Δ t = γΔ τ.
• If it’s about a moving object’s length, state explicitly: “Measure endpoints simultaneously in S so Δ t = 0.”
• For collisions/decays, try invariant mass first: M²c⁴ = Eₜₒₜ² - (pₜₒₜc)².
• Do a one-line limit check at the end: β → 0 should reduce to classical physics.

6. Worked Examples

A. Muon survival estimate (time dilation)

Muons are created at altitude L = 10 km and move at v = 0.98c. Their proper lifetime is τ₀ = 2.2 μs.

Estimate whether a significant fraction can reach the ground.

B. Relativistic velocity addition (numbers)

A spaceship moves at v = 0.80c relative to Earth. In the ship frame it launches a probe forward at u' = 0.80c.

Find the probe speed u in the Earth frame.

C. Invariant mass of two photons (system invariant)

Two photons of equal energy E travel head-on (opposite directions). Find the invariant mass M of the two-photon system.

7. Mind Stretchers

• Two events are simultaneous in S ( Δ t = 0) but separated by distance Δ x ≠ 0. Are they simultaneous in every inertial frame? Use the Lorentz transform for Δ t'.
• A particle’s proper time is the “length” of its worldline in spacetime. Why does the straight worldline between two fixed events maximise proper time?
• A single photon has zero invariant mass. How can two photons together have nonzero invariant mass?

8. Practice