   Riemannian Geometry: Flatness and Curvature
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Video Lecture 3 of 10
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### Lecture Description

In this lecture, Professor Susskind presents the mathematics required to determine whether a spatial geometry is flat or curved. The method presented is to find a diagnostic quantity which, if zero everywhere, indicates that the space is flat. This method is simpler than evaluating all possible metric tensors to determine whether the space is flat. The diagnostic that we are looking for is the curvature tensor. The curvature tensor is computed using covariant derivatives which require the computation of the Christoffel symbols. The Christoffel symbols are computed using the equation for covariant derivative of the metric tensor for Gaussian normal coordinates. We take the second covariant derivative of a vector using two different orders for the indices, and subtract these two derivatives to get the curvature tensor. If the curvature tensor is equal to zero everywhere, the space is flat. Professor Susskind demonstrates the intuitive picture of this computation using a cone, which is a flat two-dimensional space everywhere except at the tip. Topics: - Riemannian geometry - Metric tensor - Gaussian normal coordinates - Covariant derivatives - Christoffel symbols - Curvature tensor - Cones Recorded on October 8, 2012.

### Course Description

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.