Special Relativity

Notes on the basics of special relativity — the kinematic framework underlying all of relativistic physics, including quantum field theory.

We work in inertial frames, use natural units $c=1$ where convenient, and adopt the mostly-minus metric ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

Contents

Core kinematics

1. Postulates of Special Relativity — the ether crisis, the two postulates, operational clocks/rods, and relativity of simultaneity.
2. Spacetime and the Invariant Interval — events, worldlines, the interval, causal structure, light cones, proper time.
3. Lorentz Transformations — boosts, rapidity, composition and Thomas–Wigner rotation, the Poincaré group.
4. Kinematic Consequences — time dilation, length contraction, velocity addition, relativistic Doppler and aberration.
5. Relativistic Dynamics — four-momentum, $E=m{c}^2$, the dispersion relation, collisions, four-force.

Formalism and structure

6. Four-Vectors and Tensors — index notation, the metric, covariance.
7. Index Notation and Invariants — building Lorentz scalars; common four-vectors.
8. The Lorentz and Poincaré Groups — components, algebra, $SL(2,\mathbb{C})$, Wigner classification.

Applications and bridges

9. Paradoxes Resolved — twin, ladder–barn, Bell's spaceships.
10. Covariant Electromagnetism — the field tensor and Maxwell's equations.
11. Bridge to Relativistic Quantum Theory — on-shell condition, Wigner, why fields.
12. Tests and Applications — experimental confirmation and GPS.

Relation to other sections

Special relativity supplies the spacetime symmetry (the Lorentz/Poincaré group) that the Quantum Field Theory section takes as a postulate. The relativistic dispersion relation derived here reappears as the on-shell condition for free fields in QFT. The relevant group theory is collected in Group theory.