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2D Null Geodesic Solution

  1. May 26, 2009 #1
    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.
  2. jcsd
  3. May 26, 2009 #2


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    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/ [Broken].
    Last edited by a moderator: May 4, 2017
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