Geodesics and gravity

General Relativity (Fall, 2012)

October 15, 2012

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.

  • Parallel transport
  • Tangent vectors
  • Geodesics
  • Spacetime
  • Special relativity
  • Uniform acceleration
  • Uniform gravitational fields