Tensors and Relativity: Chapter 8

Orbits in Schwartzschild spacetime

next up previous index
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

equation1047

Assuming that the orbits remain permanently in the equatorial plane [ as in Newtonian theory ] i.e. tex2html_wrap_inline1303 , the Lagrangian is:

equation1050

The Euler- Lagrange equations are

equation1053

Now

equation1055

so

equation1057

This is just energy conservation. Also

equation1059

so

equation1061

This is conservation of angular momentum.

Remember that tex2html_wrap_inline1305 with tex2html_wrap_inline1307 for timelike orbits and tex2html_wrap_inline1309 for null orbits, so

equation1064

We have

equation1066

Substituting these into the above, we get

equation1068

This is the Newtonian energy equation with a modification to the tex2html_wrap_inline1311 term. Using tex2html_wrap_inline1313 and putting tex2html_wrap_inline1315 we get

equation1070

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:

equation1072

For timelike orbits [ tex2html_wrap_inline1307 ] this is just Newton's equation

equation1074

apart from the last term.