# Geodesics in 2+1 Schwarzschild metric

Would appreciate if someone could explain the query that, I am trying to understand the code for timelike geodesic in 2+1 Schwarzschild $$(\theta=\frac{\pi}{2})$$ where we have taken $$M=1$$. From this how to get orbit plots? Is this possible to draw orbit without involving effective potential and angular momentum? Here I have taken $$\dot{t}, \dot{r}, \dot{\phi}$$ such way initial condition that satisfy the timelike condition

$$(-\dot{t}^{2}+ \dot{r}^{2}+\dot{\phi}^{2}=-1)$$. We are currently using NDSolve. $$ds^{2}=-(1-\frac{2M}{r})dt^{2}+(1-\frac{2M}{r})^{-1}dr^{2}+r^{2}d \phi^{2}$$

Geodesic equation: $$\ddot{t}+\frac{2 \dot{t} \dot{r}}{(-2+r)r}=0$$ $$\ddot{r}+\frac{(-2+r)\dot{t}^{2}}{r^{3}}+\frac{\dot{r}^{2}}{2r-r^{2}}+(2-r)\dot{\phi}^{2}=0$$ $$\ddot{\phi}+\frac{2 \dot{r} \dot{\phi}}{r}=0$$

Initial Condition: $$t[0]=1, \dot{t}[0]=2$$ $$r[0]=1, \dot{r}[0]=\sqrt{3}$$

$$\phi[0]=2, \dot{\phi}[0]=0$$

Reference: Introduction to Black Hole physics, Valeri P. Frolov and Andrei Zelnikov, Chapter-7, section-7.2.3 (pg.no:192). We are not exactly looking at this picture, similar to this but more concessions on trajectories.

• Welcome to Mathematica.SE. Are you sure you are posting on the right site? There is nothing in your question making it clear that it is concerned with Mathematica software. – m_goldberg Oct 10 '19 at 13:44
• @m_goldberg I believe this related to NDSolve. – Jeevitha T.U. Oct 11 '19 at 10:06

Generally we do NDSolve for geodesics which are 2nd order ODEs. Here is a link of answer given by Alex Trounev Interpreting Mathematica code on black holes