# 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.

• 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