# Light ray paths near schwarzschild blackhole

1. ### nocks

24
hey there, i'm interested in (eventually) simulating light ray paths near black holes, starting with schwarzschild blackholes and working my way to kerr-newman blackholes.
I have a good understanding of the nature of blackholes but have trouble when it comes to the equations.
My background is in computer science and I was wondering if anyone here could put a non-physicist on the right track to plotting the trajectory of a light ray near a black hole. ( in 2d for now ).
Thanks

Last edited: Oct 23, 2009
2. ### nocks

24
I'm currently going over lagrangian mechanics

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5. ### nocks

24
Thanks for the paper, all the others I have read from arxiv have not been very non-physicist friendly.

6. ### George Jones

6,396
Staff Emeritus
Are you interested in what an observer would see with his eyes, or in plotting the trajectory of a photon on a coordinate map, or maybe both? The first thing is somewhat hard, and the second thing is easier.

Let me use an example to illustrate what I mean. Suppose a ship makes a long ocean voyage. The voyage could be watched from a telescope on a satellite in geosynchronous orbit, or the voyage could be plotted as a moving, glowing dot on a map.

7. ### nocks

24
My end goal will be to render what an observer would see , but for now I would like to simply plot the trajectory of several photons on a coordinate map.

I've been studying geodesics (not with much luck) and reading up on equations for the impact parameter and deflection angle in the schwarzschild metric which I have as :
$$\Delta\phi = \int^{r_{observed}}_{r_{omitted}} \stackrel{dr}{r\sqrt{r^{2}/b^{2} - 1 + R_{s}/r}}$$

Last edited: Oct 25, 2009

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9. ### nocks

24
Last edited: Oct 25, 2009
10. ### George Jones

6,396
Staff Emeritus
11. ### nocks

24
Yes something like this is exactly what i'm aiming for at the moment. I noticed you focused on timelike geodesics for the app. Am I right in thinking that I would be using null geodesics for photon trajectories and timelike for the trajectory of an observer.

12. ### George Jones

6,396
Staff Emeritus
Actually, my Exercise 6 is about lightlike (null) geodesics, which is what you want.

13. ### Dmitry67

George, it is wonderful.
Do you know any animations about what observer would see inside the second horizon, near loop singularity?

14. ### nocks

24
Would I still use geodesics for photon paths in polar coordinates?

15. ### nocks

24
Could you give me some more information on plotting the trajectory on a coordinate map? I've been toying with equations for a while now and not making much progress
Thanks

16. ### George Jones

6,396
Staff Emeritus
Yes, exactly. The geodesic equation when $\theta = \pi /2$ then is

$$\begin{equation*} \begin{split} \frac{d \phi}{d \lambda} &= \frac{L}{r^2} \\ \left( \frac{dr}{d \lambda} \right)^2 &= E^2 - L^2 W \left( r \left( \lambda \right) \right), \\ \end{split} \end{equation*}$$

where $W \left( r \right)$ is function that I'll specify later, and $E$ and $L$ are constants of motion.

In its present form, the second equation is a little difficult to implement on a computer since sometimes a positive square root is needed (increasing $r$) and sometimes a negative square root is needed (decreasing $r$) in the same photon orbit. To get aorund this, differentiate the second equation with respect to the affine parameter $\lambda$ taking into account that the $r$ in $W \left( r \right)$ is itself a function of $\lambda$.

What do you get for the second equation after this differentiation?

Last edited: Nov 21, 2009
17. ### nocks

24
Given it's been a while since i've done any differentiation, is it simply:

$$\frac{dr}{d \lambda} \right) &= \frac{ E - L}{W \left( r \right)}$$

18. ### nocks

24
what does W(r) define?

19. ### Nabeshin

2,200
W(r) should be the effective potential of the system.

20. ### nocks

24
Could anyone expand on this please? I would appreciate the help.
I have the the effective potential in the schwarzschild metric as (L being angular momentum)
$$V_{eff} = ( 1 - \frac{r_{s}}{r})(mc^{2} + \frac{L^{2}}{mr^{2}})$$

Would this be enough information to solve for r and $$\Phi$$ so that I could plot the trajectories.

Also could I use the same equation for plotting the course of an observer descending into the black hole?