This lecture focuses on the mathematics of tensors, which represent the core concepts of general relativity. Professor Susskind opens the lecture with a brief review the geometries of flat and curved spaces. He then develops the mathematics of covariant and contravariant vectors, their coordinate transformations, and their relationship to tensors. In the second half of the lecture, Professor Susskind defines tensor operations including addition, multiplication, and contraction, and discusses the properties of the metric tensor. Topics: - Flat space - Metric tensor - Scalar and tensor fields - Tensor analysis - Tensor mathematics: addition, multiplication, contraction Recorded on October 1, 2012.

1. The Equivalence Principle and Tensor Analysis
2. Tensor Mathematics
3. Riemannian Geometry: Flatness and Curvature
4. Geodesics, Gravitational Fields, & Special Relativity
5. The Metric for a Gravitational Field
6. Schwarzschild Radius & Black Hole Singularity
7. Falling into a Black Hole: The Event Horizon
8. Black Hole Formation, Penrose Diagrams & Wormholes
9. Einstein Field Equations of General Relativity
10. Gravity Waves: Gravitational Radiation & Einstein-Hilbert Action

General relativity is the geometric theory of gravitation published by Albert Einstein in 1916 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. This course uses the physics of black holes extensively to develop and illustrate the concepts of general relativity and curved spacetime. This series is the fourth installment of a six-quarter series that explore the foundations of modern physics. In this quarter, Leonard Susskind focuses on Einstein's General Theory of Relativity.