*Spacetime Geometry *

**Spacetime **

Line Elements

Circle Trig

Hyperbola Trig

The Geometry of Special Relativity

**Symmetries **

Position and Velocity

Geodesics

Symmetries

Example: Polar Coordinates

Example: The Sphere

**Schwarzschild Geometry **

The Schwarzschild Metric

Properties of the Schwarzschild Geometry

Schwarzschild Geodesics

Newtonian Motion

Orbits

Circular Orbits

Null Orbits

Radial Geodesics

Rain Coordinates

Schwarzschild Observers

**Rindler Geometry **

The Rindler Metric

Properties of Rindler Geometry

Rindler Geodesics

Extending Rindler Geometry

**Black Holes **

Extending Schwarzschild Geometry

Kruskal Geometry

Penrose Diagrams

Charged Black Holes

Rotating Black Holes

Problems

*General Relativity*

Warmup

Differential Forms in a Nutshell

Tensors

The Physics of General Relativity

Problems

**Geodesic Deviation **

Rain Coordinates II

Tidal Forces

Geodesic Deviation

Schwarzschild Connection

Tidal Forces Revisited

**Einstein's Equation **

Matter

Dust

First Guess at Einstein's Equation

Conservation Laws

The Einstein Tensor

Einstein's Equation

The Cosmological Constant

Problems

**Cosmological Models **

Cosmology

The Cosmological Principle

Constant Curvature

Robertson-Walker Metrics

The Big Bang

Friedmann Models

Friedmann Vacuum Cosmologies

Missing Matter

The Standard Models

Cosmological Redshift

Problems

**Solar System Applications **

Bending of Light

Perihelion Shift of Mercury

Global Positioning

*Differential Forms *

Calculus Revisited

Differentials

Integrands

Change of Variables

Multiplying Differentials

**Vector Calculus Revisited **

A Review of Vector Calculus

Differential Forms in Three Dimensions

Multiplication of Differential Forms

Relationships between Differential Forms

Differentiation of Differential Forms

**The Algebra of Differential Forms **

Differential Forms

Higher Rank Forms

Polar Coordinates

Linear Maps and Determinants

The Cross Product

The Dot Product

Products of Differential Forms

Pictures of Differential Forms

Tensors

Inner Products

Polar Coordinates II

**Hodge Duality **

Bases for Differential Forms

The Metric Tensor

Signature

Inner Products of Higher Rank Forms

The Schwarz Inequality

Orientation

The Hodge Dual

Hodge Dual in Minkowski 2-space

Hodge Dual in Euclidean 2-space

Hodge Dual in Polar Coordinates

Dot and Cross Product Revisited

Pseudovectors and Pseudoscalars

The General Case

Technical Note on the Hodge Dual

Application: Decomposable Forms

Problems

**Differentiation of Differential Forms **

Gradient

Exterior Differentiation

Divergence and Curl

Laplacian in Polar Coordinates

Properties of Exterior Differentiation

Product Rules

Maxwell's Equations I

Maxwell's Equations II

Maxwell's Equations III

Orthogonal Coordinates

Div, Grad, Curl in Orthogonal Coordinates

Uniqueness of Exterior Differentiation

Problems

**Integration of Differential Forms**

Vectors and Differential Forms

Line and Surface Integrals

Integrands Revisited

Stokes' Theorem

Calculus Theorems

Integration by Parts

Corollaries of Stokes' Theorem

Problems

**Connections **

Polar Coordinates II

Differential Forms which are also Vector Fields

Exterior Derivatives of Vector Fields

Properties of Differentiation

Connections

The Levi-Civita Connection

Polar Coordinates III

Uniqueness of the Levi-Civita Connection

Tensor Algebra

Commutators

Problems

**Curvature **

Curves

Surfaces

Examples in Three Dimensions

Curvature

Curvature in Three Dimensions

Components

Bianchi Identities

Geodesic Curvature

Geodesic Triangles

The Gauss-Bonnet Theorem

The Torus

Problems

**Geodesics **

Geodesics

Geodesics in Three Dimensions

Examples of Geodesics

Solving the Geodesic Equation

Geodesics in Polar Coordinates

Geodesics on the Sphere

**Applications **

The Equivalence Problem

Lagrangians

Spinors

Topology

Integration on the Sphere

**Appendix A: Detailed Calculations**

Appendix B: Index Gymnastics

**Annotated Bibliography **

**References**