text: Orbital velocity in general relativity
  In Schwarzschild metric, the orbital velocity for a circular orbit with radius  is given by the following formula:
  where  is the Schwarzschild radius of the central body.
  Derivation
  For the sake of convenience, the derivation will be written in units in which .
  The four-velocity of a body on a circular orbit is given by:
  ( is constant on a circular orbit, and the coordinates can be chosen so that ). The dot above a variable denotes derivation with respect to proper time .
  For a massive particle, the components of the four-velocity satisfy the following equation:
  We use the geodesic equation:
  The only nontrivial equation is the one for . It gives:
  From this, we get:
  Substituting this into the equation for a massive particle gives:
  Hence:
  Assume we have an observer at radius , who is not moving with respect to the central body, that is, their four-velocity is proportional to the vector . The normalization condition implies that it is equal to:
  The dot product of the four-velocities of the observer and the orbiting body equals the gamma factor for the orbiting body relative to the observer, hence:
  This gives the velocity: