I'm a new member on StackExchange. I'm french and my english is awfull, so I beg you to excuse me and I hope you can understand my question anyway…
I'm looking for a very precise equation for free fall in Schwarzschild coordinates (r;t) but with an initial speed superior at release speed.
I already have the formula of the speed with a parameter K
Where Rs is de Schwarzschild radius. This is not a local speed but the "slope" of the path I'm looking for.
For exemple, if K=0 it gives the speed at r for free fall from infinity, if K=-∞ it gives the speed of light (Shapiro's one). I'm interrested in the case K belongs to ]-∞;0[ wich means that a particle could follow a geodesic from an initial r=Ro with an initial speed superior at release speed.
The subject :
This formula can be written v(r). If I had v(t) I could intégrate it and find the equation of the path r(t), but this is not the case. Is anyone here able to write the formula t(r) or r(t) I need to draw the path of this kind of particles in Schwarzschild coordinates (r;t) ?
Either you can find a way to integrate this, either you already have a "ready made" formula without the K parameter. I just want the result, not the whole explanation… waste of time because I can't understand it. I'm not a good mathematician but I'm interrested in black holes and I like to "draw formulas" because it's the only way for me to understand somethis about relativistic équations.
Thank you for your help and sorry for my english…