4-Momentum & Relativistic Collisions

4-Momentum and Collision Invariants

In special relativity, energy and momentum aren't separately conserved—they're components of a single 4-vector called the 4-momentum. When particles collide, the total 4-momentum before equals the total 4-momentum after, but individual energies and momenta can change dramatically depending on your reference frame.

The 4-Momentum Vector

For a particle with rest mass m, velocity v, and Lorentz factor γ = 1/√(1-v²/c²):

• Energy component: p⁰ = E/c = γmc
• Momentum components: pⁱ = γmvⁱ

In natural units (c=1), this simplifies to p^μ = (E, p_x, p_y, p_z). The magnitude of this 4-vector is invariant:

p² = E² - p² = m²

This is the relativistic energy-momentum relation. It holds in every reference frame.

Conservation Law

When two particles collide, the total 4-momentum is conserved:

p₁^μ + p₂^μ = p₃^μ + p₄^μ

This single equation (which is really four equations, one per component) replaces the separate Newtonian conservation laws for energy and momentum. In the lab frame, it means:

• E₁ + E₂ = E₃ + E₄ (energy conservation)
• p₁ + p₂ = p₃ + p₄ (momentum conservation, vector equation)

Invariant Mass

The most important quantity in relativistic collisions is the invariant mass or center-of-momentum energy, denoted √s:

s = (E₁ + E₂)² - (p₁ + p₂)²

This quantity is the same in every reference frame—that's why it's called "invariant." In the center-of-momentum frame (where p₁ + p₂ = 0), it's simply s = (E₁ + E₂)². The square root √s is the total energy available for creating new particles.

Why s is Invariant

The quantity s is the square of the total 4-momentum: s = (p₁ + p₂)². Since 4-momentum transforms as a 4-vector under Lorentz transformations, its squared magnitude is invariant. This is exactly analogous to how the rest mass m² = E² - p² is invariant for a single particle.

Center-of-Momentum Frame

Every collision has a special reference frame called the center-of-momentum (CM) frame where the total momentum is zero. In this frame, the particles approach each other symmetrically, collide, and the products fly apart symmetrically.

Finding the CM frame: The total 3-momentum P = p₁ + p₂ and total energy E = E₁ + E₂ define a 4-momentum (E, P). The velocity of the CM frame is:

β_CM = P / E

Transform to this frame and you'll find the momenta cancel. This frame is ideal for understanding collision dynamics because the physics is symmetric.

Threshold Energy

For inelastic collisions where new particles are created (like p + p → p + p + π⁰), there's a minimum collision energy below which the reaction can't happen. This is the threshold energy.

Example: Pion Production

To create a pion (m_π ≈ 0.14 GeV) in a proton-proton collision where one proton is initially at rest (m_p ≈ 0.94 GeV), you need:

√s ≥ 2m_p + m_π ≈ 1.92 GeV

In the lab frame, the incident proton needs kinetic energy T ≈ 0.28 GeV (about 30% of its rest mass). Below this threshold, conservation laws forbid pion creation.

Energy-Momentum Diagrams

A powerful visualization tool is the energy-momentum diagram, where each particle is a point at (p_x, E). The 4-momentum conservation law becomes a simple geometric statement: the vector sum before equals the vector sum after.

The mass-shell constraint E² - p² = m² means each particle lies on a hyperbola in this diagram. Massless particles (photons) lie on the line E = |p|.

Collider vs Fixed-Target Experiments

Fixed Target

Accelerate particle 1 to energy E₁ and smash it into a stationary target (particle 2 at rest: E₂ = m₂). The invariant mass is:

s = (E₁ + m₂)² - p₁² = m₁² + m₂² + 2E₁m₂

So √s grows only as √E₁ at high energies. Much of the beam energy goes into moving the CM frame, not into the collision itself.

Collider

Accelerate two beams to equal and opposite momenta and collide them head-on. If both particles have energy E, then:

s = (E + E)² - 0² = 4E²

So √s = 2E grows linearly with beam energy. The entire collision happens in the CM frame by construction. This is why modern particle physics uses colliders: you get far more collision energy for the same beam energy.

Real-World Applications

Large Hadron Collider

The LHC collides protons at 6.8 TeV per beam, giving √s = 13.6 TeV. This is enough to produce particles with masses up to ~13.6 TeV. To achieve the same √s with a fixed target would require a beam energy of ~100,000 TeV—impossible with current technology.

Cosmic Ray Collisions

Ultra-high-energy cosmic rays (UHECRs) can have energies exceeding 10²⁰ eV. When they collide with CMB photons (E ≈ 10⁻³ eV), the invariant mass is √s ≈ √(2 × 10²⁰ × 10⁻³) ≈ 10⁹ eV, just enough to produce pions. This is the GZK cutoff: cosmic rays above ~10²⁰ eV lose energy by pion photoproduction on the CMB.

Particle Identification

By measuring energies and momenta of collision products and checking that E² - p² = m², detectors can identify particles. A mismatch indicates measurement error or an unknown particle escaped detection (like a neutrino).

Things to Try

1. Symmetric Collision: Use the preset to set equal masses and opposite velocities. Notice the CM frame has zero momentum. Switch to CM view to see the symmetric collision.
2. Asymmetric Masses: Set m₁ = 2, m₂ = 1 with equal speeds. The CM frame moves because the heavier particle carries more momentum. In the CM frame, the lighter particle moves faster to balance momentum.
3. Glancing Collision: Give the particles perpendicular velocities. Watch how 4-momentum conservation constrains the possible final states. The energy-momentum diagram shows the geometric constraint.
4. Threshold Energy: For inelastic collisions, there's a minimum √s needed. Lower the velocities and watch √s decrease. Below threshold, not enough energy exists to create heavier products.
5. Ultrarelativistic Limit: Set both particles to β ≈ 0.99. Notice E ≈ |p| for each particle (they're nearly massless). The collision becomes similar to photon collisions.
6. Conservation Check: The displayed ΔE and Δp values should be zero (within numerical precision). If not, physics is broken! This is a fundamental symmetry of spacetime.

Beyond Elastic Collisions

This toy shows elastic collisions where particles retain their masses. Real collisions can be inelastic: kinetic energy converts to rest mass (creating new particles) or vice versa. The 4-momentum conservation law still holds, but now there are more than two particles in the final state.

The invariant mass √s determines what's possible. If √s ≥ m₃ + m₄ + ... (sum of product masses), the reaction is kinematically allowed. If √s is too small, the reaction can't happen—this is the threshold condition.

Every collision at the LHC, every cosmic ray interaction, every particle decay obeys 4-momentum conservation. It's one of the most precisely tested laws in physics, verified to extraordinary precision in countless experiments. The 4-momentum formalism elegantly unifies energy and momentum conservation into a single, frame-independent law.