Schwarzschild Geodesics: An Overview

[Goldman clarck]

TL;DR Summary

Hi, I started reading this  dealing with the simulation of a Schwarzschild black hole in python, what I didn't understand is how did the author of the paper derive $v_1 (0) = \frac{1}{D \cdot \tan \alpha}$ from $v_0 (0) = \frac{1}{D}$?
Can anyone here kindly provide an answer.
(Below are screenshots from the paper)
[Mentors' note: edited to fix the Latex]

 

[PeterDonis]

what I didn't understand is how did the author of the paper derive v1(0)=1D⋅tanα from v0(0)=1D?

I don't know, and I'm not even sure why those equations make sense. It looks like those equations are taken from reference [5], not derived in the paper itself.

What I don't understand is where the distance $D$ is coming from, since the radial coordinate $r$ in Schwarzschild coordinates does not equal radial distance, and since there is no spatial "center" to the black hole anyway ($r = 0$ is not a point at the center, it's a spacelike like to the future of every event inside the horizon). So I'm not sure what the author of the paper thinks he is doing.

 

[Goldman clarck]

I watched the video he referred to, the video is in french by the way and doesn't provide any explanation to how the second equation has been derived, except mentioning that $u' = \frac{1}{D \cdot tan \alpha}$ is the derivative of $u = \frac{1}{D}$ when $\phi = 0$, where $D$ is the distance between the black hole and the observer and $\alpha$ defining the initial direction of the photon.
I've been trying the whole day but with no success, now I'm confused.

 

[PeterDonis]

xcept mentioning that $u' = \frac{1}{D \cdot \alpha}$ is the derivative of $u = \frac{1}{D}$ when $\phi = 0$, where $D$ is the distance between the black hole and the observer and $\alpha$ defining the initial direction of the photon.

Hm, ok, if he's using the standard method of rewriting the orbital equations, where $u = 1 / r$, then I can see why he would say that the value of $u$ at time $0$ (which is what I think he means by $v_0$, although his notation is very confusing) would be $1 / D$--but that's still only correct in the approximation where we are very far from the hole, so "distance from the hole" can be taken as a good approximation of the radial coordinate $r$. But if he means to include trajectories of photons close to the hole's horizon, then this approximation no longer works.

In general, I don't think this is a good source from which to try to learn how to do numerical simulations in GR.

 

[Goldman clarck]

If $u = \frac{1}{D}$ is correct as an approximation (the source he stated uses the Euler method to solve the light path around a the Schwarzschild spacetime) then how $u' = \frac{1}{D \cdot tan \alpha}$ (which is confusingly denoted as $v_1(0)$ in the paper) was derived from $u = \frac{1}{D}$ when $\phi = 0$, do you have any idea please? and what books, articles or any other source would you suggest for me to start with numerical relativity if you don't mind me asking?

 

[PeterDonis]

If $u = \frac{1}{D}$ is correct as an approximation (the source he stated uses the Euler method to solve the light path around a the Schwarzschild spacetime) then how $u' = \frac{1}{D \cdot tan \alpha}$ (which is confusingly denoted as $v_1(0)$ in the paper) was derived from $u = \frac{1}{D}$ when $\phi = 0$,

The $\phi = 0$ part is just because he has to choose some particular value of $\phi$ as the "initial" value (the word "initial" is a bit misleading since the trajectory can be extended into both the future and the past from this point).

I'm not sure how he's deriving $u' = \frac{1}{D \cdot tan \alpha}$ since the initial value of $u$ should be a position, not a velocity. Again, I don't think this is a good source to learn from.

what books, articles or any other source would you suggest for me to start with numerical relativity

Unfortunately I don't have any good references to suggest, since I have not studied numerical relativity. Possibly some other posters here might.

 

[Goldman clarck]

Thank you @PeterDonis for the valuable time you spent discussing the thread, really appreciate it.

 

[PeterDonis]

Thank you

You're welcome!