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
Can't help much but this animation of gravitational lensing might be of interest... http://en.wikipedia.org/wiki/Black_hole#History
Thanks for the paper, all the others I have read from arxiv have not been very non-physicist friendly.
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.
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 : [tex]\Delta\phi = \int^{r_{observed}}_{r_{omitted}} \stackrel{dr}{r\sqrt{r^{2}/b^{2} - 1 + R_{s}/r}} [/tex]
I stumbled upon this last week, the paper on 'light deflection near neutron stars' by ute kraus has been quite usefull. Appreciate the help though :)
You mean like Exercise 6 in the (imperfect) animation attached to https://www.physicsforums.com/showthread.php?p=1091901#post1091901? I won't have access to my notes and books until tomorrow.
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.
George, it is wonderful. Do you know any animations about what observer would see inside the second horizon, near loop singularity?
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
Yes, exactly. The geodesic equation when [itex]\theta = \pi /2[/itex] then is [tex]\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*} [/tex] where [itex]W \left( r \right)[/itex] is function that I'll specify later, and [itex]E[/itex] and [itex]L[/itex] 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 [itex]r[/itex]) and sometimes a negative square root is needed (decreasing [itex]r[/itex]) in the same photon orbit. To get aorund this, differentiate the second equation with respect to the affine parameter [itex]\lambda[/itex] taking into account that the [itex]r[/itex] in [itex]W \left( r \right)[/itex] is itself a function of [itex]\lambda[/itex]. What do you get for the second equation after this differentiation?
Given it's been a while since i've done any differentiation, is it simply: [tex]\frac{dr}{d \lambda} \right) &= \frac{ E - L}{W \left( r \right)}[/tex]
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) [tex]V_{eff} = ( 1 - \frac{r_{s}}{r})(mc^{2} + \frac{L^{2}}{mr^{2}})[/tex] Would this be enough information to solve for r and [tex]\Phi[/tex] 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?