Relativistic Collisions & Invariants (IPhO SR)

Relativistic collision problems feel messy if you chase angles and Lorentz transforms. They become fast if you treat the whole system with 4-momentum conservation and compute an invariant once (usually the system invariant mass). The centre-of-momentum (COM) frame is your best friend.

1. Definitions (Must Know)

A. Particle 4-momentum

For a particle with rest mass m, energy E, and 3-momentum vector p: p^μ = (E/c, pₓ, p_y, p_z)

Invariant (same in all inertial frames): p^μp_μ = (E/c)² - | vector p|² = (mc)²

Equivalent energy-momentum relation: E² = (pc)² + (mc²)²

B. Total 4-momentum and invariant mass of a system

For many particles, define total 4-momentum: P^μ = ∑ᵢ pᵢ^μ

Define the system invariant mass M by: M²c⁴ = Eₜₒₜ² - (pₜₒₜc)² where Eₜₒₜ = ∑ Eᵢ and vector pₜₒₜ = ∑ vector pᵢ.

Key meaning: Mc² is the total energy of the system in the COM frame. In general, M is not the sum of the particles’ rest masses.

C. Centre-of-momentum (COM) frame

The COM frame is defined by: vector pₜₒₜ = 0

In the COM frame: E_COM = Mc²

In 1D (all motion along x), the COM frame speed relative to the lab is: v_COM = pₜₒₜc²/Eₜₒₜ

D. Threshold idea

At threshold for producing final particles of masses m_(f,1), m_(f,2),…, you can take the final state to have no kinetic energy in COM: M_threshold = ∑ⱼ m_(f,j)

2. Key Ideas (What Earns Marks)

• Conserve 4-momentum, not just energy. Energy conservation alone is not enough relativistically.
• Compute an invariant once. The system invariant mass M (or equivalently E_COM) is frame-independent.
• Use COM for thresholds and symmetry. At threshold in COM, final momenta can be taken as zero.
• Fixed-target collisions waste COM energy. Much lab energy can be “locked” into motion of the COM frame.
• Fast checks catch algebra slips. In the low-speed limit, recover Newtonian results.

3. Detailed Explanations

A. The fixed-target invariant (the lab shortcut)

Projectile: mass mₐ, total energy Eₐ in the lab. Target: mass m_b at rest in the lab.

Compute M from initial state only: M²c⁴ = (Eₐ + m_bc²)² - (pₐc)²

Use Eₐ²-(pₐc)² = mₐ²c⁴ to simplify: M²c⁴ = mₐ²c⁴ + m_b²c⁴ + 2m_bc²Eₐ

This single line is the core of many IPhO threshold problems.

B. Why “sticking” increases rest mass

If two objects collide and stick, the final object contains internal energy (heat, deformation, excitations). Relativistically, that internal energy contributes to rest mass: M = (1/c²) square root of (Eₜₒₜ² - (pₜₒₜc)²)

So an inelastic collision can literally create rest mass from kinetic energy.

C. Collider vs fixed target (why COM energy can be small)

For two identical masses m:

• Collider (head-on, symmetric): if each beam has energy E, then E_COM ≈ 2E.
• Fixed target: if one particle has energy E and the other is at rest, then for E ≫ mc²: E_COM ≈ square root of 2mc²E

So fixed-target COM energy grows only like a square root with lab energy.

4. Common Mistakes

• Using energy conservation without momentum conservation (especially in 1D “stick together” questions).
• Forgetting that vector pₜₒₜ is a vector: directions matter.
• Confusing rest mass of a product with the sum of rest masses of constituents (inelastic collisions change M).
• At threshold, accidentally allowing final kinetic energy in COM (threshold means “all at rest in COM”).
• Mixing “kinetic energy” and “total energy” ( E = K + mc²).

5. Exam Tips

• In the first 10 seconds, decide: fixed target or symmetric? If fixed target, write M²c⁴ = mₐ²c⁴ + m_b²c⁴ + 2m_bc²Eₐ.
• If the question says “minimum energy”, you are almost certainly doing a threshold calculation.
• If asked what mass can be created, the maximal rest mass you can produce is M (all COM energy converted to rest mass).
• Do not guess the final speed in inelastic collisions. Use v_COM = pₜₒₜc²/Eₜₒₜ.

6. Worked Examples

A. Perfectly inelastic hit (rest mass creation)

A particle of mass m moves at speed v and hits an identical particle at rest. They stick together. Find:

1. the speed V of the composite in the lab,
2. the rest mass M of the composite.

B. Threshold energy for producing an extra particle (fixed target)

In the lab, a particle of mass m with total energy E hits an identical particle at rest. What is the minimum kinetic energy K required to produce three identical particles of mass m (and nothing else)?

7. Mind Stretchers

A. Why fixed-target accelerators are inefficient (COM energy scaling)

Show that for two identical masses m (one at rest), the COM energy satisfies E_COM ≈ square root of 2mc²E when E ≫ mc².

• A composite system can have M > ∑ mᵢ even if no new particles were created. What physical “stuff” is inside M?
• Two photons can have nonzero system invariant mass even though each photon has zero rest mass. What sets the scale?

8. Practice