Geodesic motion in Schwarzschild's spacetime

The geodesic equations are derived from a variational principle; using this approach the equations of motion for massive and massless particles moving in the Schwarzschild spacetime are obtained and discussed. It is shown that the orbits are planar as in Newtonian theory, and that the symmetries of the Schwarzschild spacetime allow for two constants of motion which can be related to the energy and to the angular momentum of the particle. The radial infall of a particle into a black hole is discussed in detail, to clarify the remarkable properties of the event horizon.