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Part I: Differential Geometry

General relativity is a geometric theory. To understand how spacetime curves, we need the mathematics of curved spaces: differential geometry. This part develops the mathematical framework of manifolds, tensors, connections, and geodesics.

Part Overview

Differential geometry is the study of smooth manifolds—spaces that locally look like Euclidean space but can have global curvature. We'll learn about tensors on manifolds, the metric tensor that defines distances and angles, the covariant derivative that generalizes differentiation to curved spaces, and geodesics—the "straight lines" of curved spacetime that represent the paths of freely falling particles.

Key Topics

• Manifolds: smooth spaces that generalize Euclidean geometry
• Coordinate charts and transformations on manifolds
• Tensors: coordinate-independent geometric objects
• The metric tensor : measuring distances and angles
• Covariant derivative and the Christoffel symbols
• Parallel transport: moving vectors along curves
• Geodesics: extremal paths and equations of motion

6 chapters | Mathematical foundation of GR | From manifolds to geodesics

Chapters

Chapter 1: Manifolds and Topology

What is a manifold? Topological spaces, coordinate charts, atlases, smooth structures. Tangent spaces and tangent vectors. Examples: spheres, tori, and spacetime. Dimension and topology of manifolds. Why we need manifolds for general relativity.

Topological SpacesCharts & AtlasesTangent Spaces

Chapter 2: Tensors on Manifolds

Tensors as multilinear maps. Contravariant and covariant indices. Tensor transformation laws. The importance of coordinate independence. Tensor operations: contraction, raising/lowering indices. Examples of tensors: vectors, one-forms, the metric.

Tensor AlgebraIndex NotationCoordinate Independence

Chapter 3: The Metric Tensor

The metric defines distances: . Signature of the metric: Riemannian (+,+,+,+) vs. Lorentzian (-,+,+,+). The inverse metric . Raising and lowering indices. Examples: Minkowski, Schwarzschild, FLRW metrics. Proper time and arc length.

Metric TensorLine ElementSignature

Chapter 4: Covariant Derivative

Ordinary derivatives don't respect tensor transformation laws on curved manifolds. The covariant derivative fixes this. Christoffel symbols and the connection. Metric compatibility: . Torsion-free condition. Computing Christoffel symbols from the metric.

Covariant DerivativeChristoffel SymbolsConnection

Chapter 5: Parallel Transport

How do we compare vectors at different points on a curved manifold? Parallel transport keeps a vector "as constant as possible" along a curve. The parallel transport equation: . Path dependence in curved spaces. Holonomy: transporting a vector around a closed loop.

Parallel TransportPath DependenceHolonomy

Chapter 6: Geodesics

Geodesics are the "straightest possible" curves on a manifold. They extremize the proper time (timelike) or arc length (spacelike). The geodesic equation: . Free particles follow geodesics. Variational derivation from the action. Affine parametrization.

General Relativity