Professor Susskind begins the lecture with a review of covariant and contravariant vectors and derivatives, and the method for determining whether a space is flat. He then introduces the concept of geodesics, which are the straightest paths between two points in a given space. A geodesic is a path that is locally as straight as possible, which means that the derivative of the tangent vector is equal to zero at every point. Professor Susskind then moves on to relate the mathematics of Riemannian geometry (which we have been studying so far) to spacetime. Spacetime is represented by Minkowski space, which has a different metric from that of flat Riemannian space in that the coefficient of the time dimension is negative. Minkowski space is the geometry of special relativity. The rest of the lecture presents uniformly accelerated reference frames and how they transform under special relativity. Professor Susskind shows how uniformly accelerated reference frames produce the same equations of motion as those for a uniform gravitational field, thereby beginning to establish the basis for the equivalence principle which is at the heart of general relativity. Topics: - Parallel transport - Tangent vectors - Geodesics - Spacetime - Special relativity - Uniform acceleration - Uniform gravitational fields Recorded on October 15, 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.