Flatness and curvature

General Relativity (Fall, 2012)

October 8, 2012

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.

  • Riemannian geometry
  • Metric tensor
  • Gaussian normal coordinates
  • Covariant derivatives
  • Christoffel symbols
  • Curvature tensor
  • Cones