Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Coordinate change to remove asymptotic geodesic?

  1. Nov 3, 2011 #1
    Through my mathematical fumblings, I think I have found a metric which gives a solution of the geodesic equation of motion that is asymptotic. It is a diagonal metric, with g00 = (x_1)^(-3) and g11 = 1. I am largely self-taught with SR so I may be miles off, but I think this gives a G.E. of M which has a λ = 1 / (x_1) term in it, so my parameter goes to infinity when x_1 = 0.

    Even if I have the details wrong, I have two questions:

    1) Can you define a metric where geodesics are asymptotic?

    2) Can you define the same metric with different coordinates to remove this behaviour?

    The only thing I can think of is some kind of substitution, but I don't really know what to do and the textbook I am working through is not a lot of help. I even got the massive "Gravitation" book out of my library but if it did contain the solution, I didn't understand it!

    Hopefully someone can give me a clue :)
  2. jcsd
  3. Nov 4, 2011 #2


    User Avatar
    Science Advisor

    In the Schwarzschild metric in the usual coordinates, the inward-going null geodesics run off the coordinate patch, going to t = +∞ as r goes to 2m. Is that what you mean by asymptotic?
  4. Nov 4, 2011 #3


    User Avatar
    Science Advisor
    Gold Member

    If this is what you meant by asymptotic, then whether you can get rid of this behavior via a change of coordinates is dependent on if this behavior is a real singularity (e.g. at the center of a black hole) or a coordinate singularity (e.g. at the event horizon of a black hole). A real singularity would have the curvature tensor also diverging, whereas a coordinate singularity would have a finite curvature. If it's a coordinate singularity then you can find coordinates which remove this asymptotic behavior (for Schwarzschild, you can use Eddington-Finklestein coordinates or Kruskal-Szekeres coordinates).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook