Metric for a gravitational field

General Relativity (Fall, 2012)

October 23, 2012

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.

  • Space-like, time-like, and light-like intervals
  • Light cone
  • Black holes
  • Schwarzschild metric
  • Event horizon