I Schwarzschild equation of motion: initial conditions

[m4r35n357]

So there are some additional terms to the dot product in your metric, due to the off-diagonal terms in your metric.

OK, thanks for pointing that out, something else to look at!

 

[rrogers]

When I have done calculations similar to this I have taken all vectors to be unity length [-1,0,0,0] (or m*[-1,0,0,0] and then used the metric tensor in sine/sinh form to rotate it to different viewpoints and relative velocities. As a picture/basis I refer almost all situations back to Flamm's paraboloid; which is the space-like slice in a frame where the center is stationary. It provides a basic reference frame that all other items can refer to; to avoid confusion in my simple mind. I did do a little research about a decade ago and came up with a "hyperbolic" path that looped around the black hole once before flying off to infinity :) Getting interesting paths did require a great deal of fine tuning since most paths just shot out or sank into the event horizon. I have tried to extend Flamm's paraboloid to pictorially show different geodesics, time evolution, but have never found a good picture/extrapolation into the third dimension (in this case).

 

[fabsilfab]

I am not sure if anybody has come up with this already, nor if this is the right place to ask, but I'll try anyway.
My aim is initialising a photon in Schwarzschild with a certain initial angle \phi between the radial and polar (or azimuthal) components of the velocity. How would I retrieve v^r, v^{\phi} in this case?

 

[rrogers]

I am not sure if anybody has come up with this already, nor if this is the right place to ask, but I'll try anyway.
My aim is initialising a photon in Schwarzschild with a certain initial angle \phi between the radial and polar (or azimuthal) components of the velocity. How would I retrieve v^r, v^{\phi} in this case?

For the Schwarzschild case, I would just
1)look at Flamm's Paraboloid and then
2) pick/find your starting position,
3) Decide what coordinate system to use (I would use the local Schwarzschild coordinates but there are a lot more)
3) select the spatial direction for r , theta, phi in normal spherical coordinates with spatial length 1
4) Recast it into the selected relativistic coordinates (I even have some code (somewhere) )
5) Compute the time component and make sure the 4-space "length" is zero
6) Promulgate that vector forward by computing the null geodesic with those starting conditions.
Incidentally, Sagemanifolds has the tools to do this faster than you can read the above; although you have to learn their formalism/language, it's quite good. As with any computer implementation/tool you should have a firm idea about the answer before you trust any result. I have been done in so many ways you wouldn't believe it.
Anybody: please clarify or correct any of the above; constructively!

 

[Ibix]

https://www.physicsforums.com/threads/null-geodesics-in-schwarzschild-spacetime.895174/

Do scroll all the way down - my initial solution had a couple of bugs.

 

[fabsilfab]

https://www.physicsforums.com/threads/null-geodesics-in-schwarzschild-spacetime.895174/

Do scroll all the way down - my initial solution had a couple of bugs.

Thanks, this was a very useful suggestion! I read through it, and the conclusion is that I can shoot a photon at a certain angle $\psi$ wrt the $\phi=0$ axis by setting $$ r u^\phi=\sin\psi,$$ $$\frac{u^r}{\sqrt{1-r_S/r}}=\cos\psi.$$
This indeed works. I can get proper Einstein rings knowing the initial shooting angle. Thanks a lot!