Introduction to General Relativity

Interactive lecture notes based on a 23-lecture course. Equations are rendered with MathJax; proofs and intuition boxes are foldable. The full PDF contains all figures and is the authoritative version.

Interactive Figures

• 🌍 Geodesics on a Gaussian bump interactive
  Parallel geodesics deflected by curvature — toy gravitational lensing
• ⚖ Inertial vs gravitational mass interactive
  Kepler orbits with varying m_g/m_I — universality of free fall
• 🗺️ Two intersecting charts on a manifold interactive
  Drag a point across two overlapping charts — watch it appear and disappear from each coordinate system
• ➡️ Coordinate-basis tangent vectors interactive
  Drag p across a curved surface — watch ∂/∂x¹|_p and ∂/∂x²|_p tilt to follow the local slope
• 🌊 Flows on a manifold interactive
  Click anywhere to set p — watch its orbit Φ_t(p) under the same vector field
• ◇ Lie bracket as parallelogram defect interactive
  Walk ε along v then w vs w then v — the gap reveals [v,w] as ε → 0
• 💧 Fluid flow on a manifold interactive
  200 particles advecting under one field — rotation, saddle, source/sink, vortex pair
• ≡ Covectors as level sets interactive
  Drag v across a stack of parallel lines — ω(v) is the signed number of lines crossed
• ↓ Lowering an index interactive
  Choose a metric (Euclidean / anisotropic / Lorentzian); watch v map to its covector ω = g(·,v)
• ⧓ Metric signature shape-shifter interactive
  Drag the eigenvalues — the unit locus g(v,v)=1 morphs from circle to ellipse to hyperbola
• ⇄ Four-current in two frames interactive
  A cloud of charges flows along a timelike field — boost to S′ and watch (ε̄, J̄) transform as a 2-vector

Lectures

• 1 Prerelativity gravitation live
• 2 The equivalence principle live
• 3 Manifolds live
• 4 Tangent space live
• 5 Flows and tensors live
• 6 Tensors continued live
• 7–13 Connections through Einstein's equations soon