Solving the r(λ) Null Schwarzschild Geodesic: Methods and Challenges

[ozone]

Homework Statement

I am looking to solve the r(λ) null Schwarzschild geodesic in terms of the affine parameter λ, but I have not seen this done anywhere and I am not sure that it is even possible to do this somewhat close to analytically. As best I know there is no use-able boundary conditions for this ode, but I will be happy with any method which can give me an answer in terms of a series of polynomials or anything of that sort.

The equation is given as

$(\frac{dr}{dλ})^2 = E^2 - \frac{L^2}{2r^2} + \frac{GML^2}{r^3}$

I tried seperating, solving the radial coordinate and then back-solving for λ, but this was very far from being useful. Mathematica is not much help in solving this ode sadly. I even tried a u=1/r substitution and did not get very far that way.

 

[lippyka]

Homework Equations The equation is given as (\frac{dr}{dλ})^2 = E^2 - \frac{L^2}{2r^2} + \frac{GML^2}{r^3} where E= total energy, L= angular momentum, G= gravitational constant, M= mass of central body, r= distance from center, λ= affine parameter. The Attempt at a Solution I am not sure if there is any way to analytically solve this ode, but I tried to solve it numerically using Mathematica. I used the NDSolve command to compute the solution for different values of the constants. The results were satisfactory, though not perfect.