PU-105 · Special Relativity
Starting from Einstein's two postulates and the failure of Galilean invariance to preserve Maxwell's equations, the unit builds the Lorentz transformation as the symmetry group of flat spacetime and extracts its kinematic consequences—time dilation, length contraction, and the relativity of simultaneity—as geometry rather than paradox. It then reconstructs mechanics on this foundation, culminating in relativistic energy-momentum, four-vector dynamics, and the covariant formulation of electromagnetism that reveals electric and magnetic fields as one geometric object.
Lectures
| L01 | The Crisis in Classical Physics: Ether and Maxwell — |
| L02 | Galilean Relativity and Why It Fails for Light |
| L03 | The Two Postulates of Special Relativity — |
| L04 | Deriving the Lorentz Transformation |
| L05 | Spacetime Diagrams and the Invariant Interval |
| L06 | The Relativity of Simultaneity |
| L07 | Time Dilation and the Light Clock |
| L08 | Length Contraction |
| L09 | Resolving the Paradoxes: Twins, Pole and Barn |
| L10 | Adding Velocities the Relativistic Way |
| L11 | Rapidity: Boosts as Hyperbolic Rotations |
| L12 | Four-Vectors and the Geometry of Spacetime |
| L13 | Proper Time and the Four-Velocity |
| L14 | Why Momentum Must Be Redefined |
| L15 | Mass-Energy Equivalence |
| L16 | The Energy-Momentum Four-Vector and E^2 = (pc)^2 + (mc^2)^2 |
| L17 | Applications: Particle Decays and Collisions |
| L18 | Massless Particles and the Four-Wavevector of Light |
| L19 | The Relativistic Doppler Effect and Aberration |
| L20 | Relativistic Dynamics: Force and Acceleration |
| L21 | Constant Acceleration and the Uniformly Accelerated Observer |
| L22 | Electromagnetism Meets Relativity: The Charge and Current Four-Vector |
| L23 | The Electromagnetic Field Tensor |
| L24 | How E and B Fields Transform Between Frames |
| L25 | The Magnetic Force as a Relativistic Effect |
| L26 | Maxwell's Equations in Covariant Form |
| L27 | Synthesis: Spacetime as the Arena of Physics |
Derivations homed in this unit
Lorentz Transformation from the Two Postulates
Derives the boost equations relating inertial frames from the invariance of the speed of light and the principle of relativity, showing linearity and the form of the gamma factor.
Invariance of the Spacetime Interval
Shows that the quantity s^2 = -c^2 t^2 + x^2 + y^2 + z^2 is preserved under all Lorentz transformations and defines the causal structure of spacetime.
Relativity of Simultaneity
Derives that events simultaneous in one inertial frame are generally not simultaneous in another, quantified by the leading-clocks-lag relation.
Time Dilation
Derives that a moving clock runs slow by the factor gamma relative to the frame in which it moves, both from the Lorentz transformation and from the light-clock argument.
Length Contraction
Derives that the length of a moving object measured along its direction of motion is reduced by the factor gamma.
Relativistic Velocity Addition
Derives the composition law for velocities that keeps the speed of light invariant and forbids superluminal composition of subluminal speeds.
Rapidity and the Additivity of Boosts
Recasts the Lorentz boost as a hyperbolic rotation parameterized by rapidity, which adds linearly and reproduces velocity addition via tanh.
Proper Time and the Four-Velocity
Defines proper time as the invariant arc length along a worldline and constructs the four-velocity as its derivative, a timelike four-vector of constant norm.
Relativistic Momentum from Conservation
Shows that requiring momentum conservation in all frames forces the definition p = gamma m v, recovering the Newtonian form at low speed.
Mass-Energy Equivalence E = mc^2
Derives that the rest energy of a body equals mc^2 and that total energy is gamma m c^2 from consistency of energy-momentum conservation across frames.
The Energy-Momentum Four-Vector
Assembles energy and momentum into a four-vector whose invariant norm yields the relation E^2 = (pc)^2 + (mc^2)^2.
Relativistic Doppler Effect and Aberration
Derives the frequency shift of light between relatively moving frames, including the transverse Doppler effect, and the aberration of light directions.
The Four-Wavevector and Phase Invariance
Shows the phase of a plane wave is a Lorentz invariant, so frequency and wavevector combine into a null four-vector for light.
Relativistic Force and Four-Acceleration
Derives the four-force as the proper-time derivative of four-momentum and shows how three-force relates to acceleration parallel and perpendicular to velocity.
The Electromagnetic Field Tensor
Assembles the electric and magnetic fields into the antisymmetric rank-2 field tensor whose transformation unifies them as one geometric object.
Transformation of Electric and Magnetic Fields
Derives how E and B mix under a boost, showing a pure electric field in one frame acquires a magnetic component in another.
Covariant Form of Maxwell's Equations
Rewrites Maxwell's equations as two tensor equations manifestly invariant under Lorentz transformations, with the four-current as source.
The Four-Current and Charge Conservation
Combines charge density and current density into a four-vector whose vanishing four-divergence expresses charge conservation covariantly.