# Orbits in Schwartzschild spacetime

**Next:** Solution for timelike orbits **Up:** Title page ** Previous:** Length contraction in a

To find the motion of planets and light rays in a Schwartzschild spacetime we must first find the geodesic equations. This is best done by working from the Lagrangian

Assuming that the orbits remain permanently in the equatorial plane [ as in Newtonian theory ] i.e. , the Lagrangian is:

The Euler- Lagrange equations are

Now

so

This is just energy conservation. Also

so

This is conservation of angular momentum.

Remember that with for timelike orbits and for null orbits, so

We have

Substituting these into the above, we get

This is the Newtonian energy equation with a modification to the term. Using and putting we get

This equation can in fact be integrated immediately, but it leads to elliptical integrals, which are awkward to handle. We therefor differentiate to obtain the equation:

For timelike orbits [ ] this is just Newton's equation

apart from the last term.