Chapter 2: Relativistic Momentum
The classical formula p = mv doesn't conserve momentum in relativistic collisions. The correct relativistic momentum includes the Lorentz factor γ, ensuring conservation in all frames.
Relativistic Momentum
\( \vec{p} = \gamma m \vec{v} = \frac{m\vec{v}}{\sqrt{1-v^2/c^2}} \)
Low Speed Limit
When v ≪ c, γ ≈ 1, so p ≈ mv (classical result)
High Speed Limit
As v → c, γ → ∞, so p → ∞. Infinite momentum needed to reach c!
Four-Momentum
\( p^\mu = m u^\mu = (E/c, \vec{p}) = (\gamma mc, \gamma m\vec{v}) \)
The four-momentum unifies energy and momentum into a single four-vector. Its magnitude is invariant:
\( p_\mu p^\mu = E^2/c^2 - p^2 = m^2c^2 \)
Why γm?
The factor γ is required for momentum conservation. In an elastic collision viewed from different frames, only γmv (not mv) gives consistent conservation laws.
Physical interpretation: Sometimes called "relativistic mass" (mrel = γm), but modern physics prefers to keep mass invariant and put γ in the momentum formula.