Leonard Susskind derives the spacetime metric for a gravitational field, and introduces the relativistic mathematics that describe a black hole. In this lecture, Professor Susskind derives the metric for a gravitational field, and introduces the relativistic mathematics that describe a black hole. He begins by reviewing the concept of light cones and space- and time-like intervals from special relativity. He then moves on to review the flat space-time metric and geodesics, and the connection between the mathematics of geodesics and the Lagrangian formulation of classical mechanics. This leads to the mechanics of a particle moving in a gravitational field, and then to the derivation of the metric for a gravitational field, also known as the Schwarzschild metric. These are the fundamental mathematics that show the equivalence of a gravitational field and curved space-time. The metric for a gravitational field has an undefined value at a particular radius from the center of a gravitating body. Where this radius occurs outside of the body, the body is a black hole, and the radius defines the location of the event horizon. The lecture concludes with an introduction to some of the very strange properties of a black hole, including that, to an outside observer, the velocity of light slows and light rays become stuck at the horizon. Topics: - Space-like, time-like, and light-like intervals - Light cone - Black holes - Schwarzschild metric - Event horizon Recorded on October 22, 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.