# Integrating the Schwarzschild geodesics equations

I am trying to make a graph similar to this one from the Wikipedia article about Schwarzschild geodesics:

There is an equation like this: $$\varphi = \int\frac{dr}{r^2\sqrt{\frac{1}{b^2}-(1-\frac{r_s}{r})\frac{1}{r^2}}}.$$

I did not get how they manage to draw this picture from that equation. Particular questions:

1. If it is a polar plot, usually it is opposite, like $r(\varphi)$, but not $\varphi(r)$. How can I draw the polar plot if I have $\varphi(r)$ function?

2. What is integration interval here? $\varphi = \int_{?}^{?}...$

• @Countto10 If I will integrate from -Infinity to +Infinity the result will not depend on r. It will be φ(rs, b), but I expect something like φ(rs, b, R). I put capital R to avoid confusion with r inside integral. Commented Feb 24, 2017 at 8:50
• My apologies Zlelik, I read the post too fast, (as usual)
– user146020
Commented Feb 24, 2017 at 11:58

To your first question: Using a parametric plot in $r$ or $\phi$ and plotting $\left\{r\cos(\phi),r\sin(\phi)\right\}$.

To your second question: that is where things become tricky. Depending on what kind of geodesics comes out the radius shrinks or increases and the angle is also kind of problematic for closed orbits. A formulation with proper time as parameter is better suited for implementation.

This Wolfram Demonstration Geodesics in Schwarzschild Space of Niels Walet has a simple implementation of an integration over proper time.

That being said; with a bit of tweaking one can of course implement the differential equation for $\phi$. J. B. Hartle provides a Mathematica Notebook here as supplement to his book Gravity: An Introduction to Einstein's General Relativity.

• Thanks for the really good answer. Mathematica example is very useful. I use Mathematica as well, but I do not have access to Mathematica right now and I will be able to check this example later. What will be integration interval in the most simple case, just to draw the same picture as I attached in the question? I just need a picture, where I can change some parameters like b and rs. The proper time I do not really care. Or it is not possible without proper time? Commented Feb 24, 2017 at 8:48
• Well one can just implement the equation in $\phi$ chose a $\phi(r_0)$ and an $r_0$ and integrate to an $r_1$ for geodesics which fall directly into the black hole that works fine. But for unbound and bound curves one will run into some problems with the integration domain, since $r$ can get smaller first, then bigger and so on. The code of Hartle deals with that.
– N0va
Commented Feb 24, 2017 at 9:59

$\Large \varphi(r_s,b,R_{max}) = \Huge\int_{0}^{R_{max}}\frac{dr}{r^2\sqrt{\frac{1}{b^2}-(1-\frac{r_s}{r})\frac{1}{r^2}}}$