Non-radial null geodesics in Eddington-Finkelstein coordinates

[Poirot]

Homework Statement

My end goal is to plot null geodesics around a black hole with realistic representations within the horizon (r<2GM, with c=1) using Mathematica. I've done this for outside the horizon using normal Schwarzschild coordinates and gained equation (1) below, and then used this with equation (3) and converted to u=1/r to find equation (4) in which I can plot u as a function of Φ. I then used NDSolve to solve this with varying impact parameters (D) but I know once it crosses the horizon I cannot justify the plot as being meaningful. I'm therefore converting to Eddington-Finkelstein coordinates using equation (5) to go from t -> v so the geodesic can cross the horizon fine. This gained the new metric equation (6). I can't find any help on how to do this properly for any geodesics that aren't radial so I've tried doing it myself a few different ways but all along the same logic as I did before. Every way I've tried it I get the same equation as (1) but with a different constant (Which I set as I plot so is meaningless to changing how the geodesic behaves at the horizon).

Homework Equations

\begin{array}
.Basic \ Schwarzschild \ metric \ in \ Lagrange \ form: \\
L = -\dot{t}^2 + \dot{r}^2 + r^2(\dot{\theta}^2+Sin^2{\theta}\dot{\phi}^2)\\
Relavent \ Equations:\\
\dot{r}^2 = E^2 - \frac{l^2}{r^2}(1-\frac{2GM}{r}) \ (1)\\
where,\ E = \dot{t}(1-\frac{2GM}{r}) \ (2)\\ and\ l = \dot{\phi}r^2 \ (3)\\
(\frac{du}{d\phi})^2 = 2Mu^3 - u^2 + 1/D^2\ (4)\\
where, \ D=\frac{l}{E} \\
t = v -r -2GMln|\frac{r}{2GM}-1| \ (5)\\
L = -(1-\frac{2GM}{r})\dot{v}^2 + 2\dot{v}\dot{r} + r^2\dot{\phi}^2\ (6) \\
Note:\ \theta = \frac{\pi}{2} \ so \ \dot{\theta}=0 \ and \ Sin^2{\theta}=1
\end{array}

The Attempt at a Solution

Attempted method 1:
I tried subbing back in for v and v(dot) so I could use equation (2) so I can use the same conserved quantities but everything just cancels back down to equation (1) like I had before.
Attempted method 2:
Finding a conserved quantity for v(dot) and calling the relevant constant say E' which once running all the maths through gains the same equation as (1) but with E' where E would be.

 

[PeroK]

The Attempt at a Solution

Attempted method 1:
I tried subbing back in for v and v(dot) so I could use equation (2) so I can use the same conserved quantities but everything just cancels back down to equation (1) like I had before.
Attempted method 2:
Finding a conserved quantity for v(dot) and calling the relevant constant say E' which once running all the maths through gains the same equation as (1) but with E' where E would be.

Using Eddington-Finkelstein coordinates, you still have a coordinate singularity at $r = 2GM$. Using the Killing vectors for $\nu$ and $\phi$ and the condition $\mathbf{u} \cdot \mathbf{u} = 0$ you should get the same geodesic equation for light rays as in normal Schwarzschild coordinates:

$E^2 = (\frac{dr}{d\lambda})^2 + \frac{L^2}{r^2}(1 - \frac{2GM}{r})$

You could use the relationship for $E$ to transform this into an equation for $\frac{d\nu}{d\lambda}$, but you still have the singular term in the potential. So, in terms of general light geodescics, you may be no further forward.

To get rid of the remaining coordinate singularity, you could try Kruskal-Szekeres coordinates. Although, I'm not familiar with those myself.