How Do You Calculate a 2D Null Geodesic in the Presence of a Gravitational Mass?

[haytil]

I am interested in solving the null geodesic between two points in the presence of a gravitational mass, assuming that everything takes place in 2 dimensions (i.e., no Z coordinate). The following is known:

-x and y coordinates of first point
-x and y coordinates of second point
-x and y coordinates of gravitational mass
-mass of gravitational body

I need an equation describing the curve of the null geodesic. The purpose of this is for use in a basic computer simulation I'm toying around with, so a basic function with the above variables for input would be very helpful - integrals and derivatives, not so much.

So I am hoping for a solution relating x and y in a 2D flat plane (since that's easiest to represent on a computer screen)

Is there a simple solution that can work in the general case, given the above inputs (or at least an approximation, up to only a few orders in x or y)? If the solution is not so simple (i.e., integrals and derivatives), is there at least just one solution that I could solve with not too much effort?

I hope I've provided enough to describe my problem - feel free to ask more if I haven't.

 

[atyy]

You need a time coordinate, because gravity curves spacetime. After you have time, and the metric (line element), set the line element to zero for a null geodesic. Maybe you need some other steps too, but it's roughly like this.

The null geodesics (photon orbits) of the Schwarzshild solution can be found in Woodhouse's notes: http://people.maths.ox.ac.uk/~nwoodh/gr/ .