# Trajectory of photons in Schwarzschild coordinates 2D + t

I'm french and my english is awfull, excuse me for this. I hope you could however understand my question.

I'm looking for a formula of trajectory of photons in Schwarzschild coordinates. For example in only 1D+t the trajectory is given by :

$$t(r)= \pm \frac{1}{c}\left[r+Rs.ln\mid \frac{r}{Rs}-1\mid \right]$$

(Rs is the Schwarzschild radius, absolute value is for inside and outside and +/- is for ingoing and outgoing nul geodesics)

I'm interrested in a 2D+t version. Maybe $$t(r)$$ and $$\theta(r)$$ with an initial direction ? I dont know… Something that give the trajectory in space-time (not only 2D space curvature).

Remember I'm awfull also in maths and I only use $$y=f(x)$$ formulas... don't waste your time in démonstrations I will not understand.

Thanks for your help

• There is a formula for $r(\theta)$ in terms of Weierstrass elliptic functions. I’m not aware of formulas involving time. I think people tend to just integrate the differential equations numerically. – G. Smith Nov 14 '19 at 3:43
• Are you aware that many differential equations can’t be solved analytically? Having a formula is relatively rare. – G. Smith Nov 14 '19 at 3:59
• This paper is too complicated for me, sorry. Do you know where I can find, if it exists, an example of plot 2D+t solved numerically ? – Mailou75 Nov 14 '19 at 23:08
• If you have Mathematica, maybe this? demonstrations.wolfram.com/GeodesicsInSchwarzschildSpace – G. Smith Nov 15 '19 at 0:55
• Thanks for the link. It’s not really what I expected (2D+t) but it may help me. The problem is that I don’t understand how to use it... Why is the angular momentum value [0;5]? what means the difference between K=+/-1 ? (particle and timelike is the same for me). Maybe I’m too noob... – Mailou75 Nov 18 '19 at 17:15