Editing Circular orbit (section)

=== Derivation ===
For the sake of convenience, the derivation will be written in units in which <math>\scriptstyle c=G=1</math>.

The [[four-velocity]] of a body on a circular orbit is given by:
:<math>u^\mu = (\dot{t}, 0, 0, \dot{\phi})</math>
(<math>\scriptstyle r</math> is constant on a circular orbit, and the coordinates can be chosen so that <math>\scriptstyle \theta=\frac{\pi}{2}</math>). The dot above a variable denotes derivation with respect to proper time <math>\scriptstyle \tau</math>.

For a massive particle, the components of the [[four-velocity]] satisfy the following equation:
:<math>\left(1-\frac{2M}{r}\right) \dot{t}^2 - r^2 \dot{\phi}^2 = 1</math>

We use the geodesic equation:
:<math>\ddot{x}^\mu + \Gamma^\mu_{\nu\sigma}\dot{x}^\nu\dot{x}^\sigma = 0</math>
The only nontrivial equation is the one for <math>\scriptstyle \mu = r</math>. It gives:
:<math>\frac{M}{r^2}\left(1-\frac{2M}{r}\right)\dot{t}^2 - r\left(1-\frac{2M}{r}\right)\dot{\phi}^2 = 0</math>
From this, we get:
:<math>\dot{\phi}^2 = \frac{M}{r^3}\dot{t}^2</math>
Substituting this into the equation for a massive particle gives:
:<math>\left(1-\frac{2M}{r}\right) \dot{t}^2 - \frac{M}{r} \dot{t}^2 = 1</math>
Hence:
:<math>\dot{t}^2 = \frac{r}{r-3M}</math>

Assume we have an observer at radius <math>\scriptstyle r</math>, who is not moving with respect to the central body, that is, their [[four-velocity]] is proportional to the vector <math>\scriptstyle \partial_t</math>. The normalization condition implies that it is equal to:
:<math>v^\mu = \left(\sqrt{\frac{r}{r-2M}},0,0,0\right)</math>
The [[dot product]] of the [[Four-velocity|four-velocities]] of the observer and the orbiting body equals the gamma factor for the orbiting body relative to the observer, hence:
:<math>\gamma = g_{\mu\nu}u^\mu v^\nu = \left(1-\frac{2M}{r}\right) \sqrt{\frac{r}{r-3M}} \sqrt{\frac{r}{r-2M}} = \sqrt{\frac{r-2M}{r-3M}}</math>
This gives the [[Kinetic energy|velocity]]:
:<math>v = \sqrt{\frac{M}{r-2M}}</math>
Or, in SI units:
:<math>v = \sqrt{\frac{GM}{r-r_S}}</math>

[[File:counterintuitive_orbital_mechanics.svg|thumb|250px|At the top of the diagram, a satellite in a clockwise circular orbit (yellow spot) launches objects of negligible mass:<br />(1 - blue) towards Earth,<br />(2 - red) away from Earth,<br />(3 - grey) in the direction of travel, and<br />(4 - black) backwards in the direction of travel.<br /><br />Dashed ellipses are orbits relative to Earth. Solid curves are perturbations relative to the satellite: in one orbit, (1) and (2) return to the satellite having made a clockwise loop on either side of the satellite. Unintuitively, (3) spirals farther and farther behind whereas (4) spirals ahead.]]