Part III: Relativistic Mechanics
Classical mechanics breaks down at high velocities. This part develops the relativistic mechanics of particles, introducing four-vectors, relativistic momentum and energy, and the famous equation E = mc².
Part Overview
Newtonian mechanics assumes momentum and kinetic energy . These formulas fail at relativistic speeds. We must generalize to and , leading to the energy-momentum relation . This part introduces four-vectors and shows how conservation laws work in special relativity.
Key Topics
- • Four-vectors: position, velocity, momentum, acceleration
- • Relativistic momentum: where is four-velocity
- • Relativistic energy: (total energy including rest mass)
- • Mass-energy equivalence: E = mc² and nuclear reactions
- • Energy-momentum relation:
- • Conservation laws in relativistic collisions
- • Massless particles: photons and the relation E = pc
6 chapters | From particles to E=mc² | Conservation in spacetime
Chapters
Chapter 1: Four-Vectors
Generalizing vectors to spacetime. Four-position , four-velocity , four-acceleration . Index notation, upper/lower indices, the Minkowski metric , and how to raise and lower indices. Lorentz invariance of four-vector dot products.
Chapter 2: Relativistic Momentum
Newtonian momentum fails to conserve in relativistic collisions. The correct form is . As , momentum even for finite mass. Why massive particles can never reach the speed of light. Four-momentum .
Chapter 3: Energy-Momentum Relation
The fundamental relation connecting energy, momentum, and mass: . For particles at rest, this reduces to . For massless particles (photons), . Kinetic energy . The non-relativistic limit.
Chapter 4: Mass-Energy Equivalence
Einstein's most famous equation: E = mc². Mass is a form of energy. Nuclear fission and fusion convert rest mass into kinetic energy. Binding energy in nuclei. Antimatter annihilation. The sun's power from mass loss. Experimental verification: mass defect in nuclear reactions.
Chapter 5: Relativistic Collisions
Analyzing collisions using conservation of four-momentum: . Elastic vs. inelastic collisions. Threshold energies for particle creation. Compton scattering: photon-electron collisions. Pair production and annihilation. Center-of-mass frame vs. lab frame calculations.
Chapter 6: Photons and Massless Particles
Photons have zero rest mass but nonzero energy and momentum: . Photons always move at speed c in all frames. Doppler shift and redshift/blueshift. Radiation pressure. Photon rockets. Why photons can't have rest frames. Massless limit of massive particles.
Course Navigation
Prerequisites:
- • Part I: Spacetime Foundations
- • Part II: Lorentz Transformations
- • Classical mechanics (momentum, energy, collisions)