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Part II: Lorentz Transformations

The Lorentz transformations are the mathematical heart of special relativity, describing how space and time coordinates transform between inertial frames. They replace Galilean transformations and reveal the deep structure of spacetime.

Part Overview

The Lorentz transformations are the coordinate transformations that leave the speed of light invariant. They encode time dilation, length contraction, and the relativity of simultaneity in a single mathematical framework. This part develops the transformations from first principles, explores their consequences through spacetime diagrams, and resolves classic relativistic paradoxes.

Key Topics

• Derivation of Lorentz transformations from Einstein's postulates
• The Lorentz factor γ and its physical meaning
• Velocity addition formula: why velocities don't simply add
• Spacetime diagrams and visualizing relativity geometrically
• Classic paradoxes: twin paradox, ladder paradox, train-tunnel paradox
• The invariant spacetime interval: what all observers agree on
• Proper time and worldlines in Minkowski spacetime

6 chapters | Mathematical core of SR | From equations to paradoxes

Chapters

Chapter 1: Lorentz Transformations

Deriving the Lorentz transformations from scratch. How coordinates transform: , . The boost in one spatial direction. Lorentz transformations form a group. Composition of boosts and the Thomas precession.

Coordinate TransformationsLorentz Group

Chapter 2: Velocity Addition Formula

If you move at velocity u in a frame moving at velocity v, your velocity relative to the original frame is NOT u + v. The relativistic formula: . Why nothing exceeds c. Applications to particle accelerators and cosmic rays.

Velocity CompositionSpeed Limit

Chapter 3: Spacetime Diagrams

Visualizing spacetime with Minkowski diagrams. Light cones, worldlines, simultaneity surfaces. How moving observers have tilted time and space axes. Causally connected events vs. spacelike separated events. Geometric interpretation of time dilation and length contraction.

Minkowski DiagramsLight Cones

Chapter 4: Classic Paradoxes

Resolving the famous paradoxes of special relativity. The twin paradox: why the traveling twin ages less. The ladder (barn-pole) paradox: fitting a long ladder into a short barn. The train-tunnel paradox. How relativity of simultaneity resolves apparent contradictions.

Twin ParadoxLadder Paradox

Chapter 5: Invariant Spacetime Interval

The spacetime interval is invariant under Lorentz transformations. All observers agree on even though they disagree on and separately. Timelike, spacelike, and null intervals.

InvarianceInterval Types

Chapter 6: Proper Time and Worldlines

Proper time is the time measured by a clock moving along its worldline. For any worldline, . The principle of maximal aging: free particles follow geodesics that maximize proper time. Connecting to the variational principle and the action.

Special Relativity