Hello,

I'm still trying to get my head around mathematics, and have a question regarding the paths in the Schwarzschild solution.

I am calculating the gravitational time dilation around the gravitating body by using this equation:

f = \sqrt{1 - \frac{2GM}{c^2r}}

I found the derivative of this to be:

\frac{\partial f}{\partial r} = \frac{GM}{c^2r^2f}

Taking a hint from Newtonian mechanics, I am currently determining the orbit path through an acceleration vector:

\vec{A} = \frac{\partial f}{\partial r}\ \vec{B}\ c^2

Here B is the normalized vector pointing from the orbiting body to the gravitating body.

This produces what appears to be an orbit very close to Newtonian for the planet Mercury (ex: the planet's orbit path appears to be stable).

However, I am concerned that this method is not entirely correct "as is", because it does not take into consideration the momentum of the orbiting body, or whether it's massive or massless.

Is there any way that I can build upon this method to calculate a realistic orbit path for Mercury?

One particular question I have is:

At r = 2GMc^2, does the massive body behave exactly like a photon, in so much that both get "stuck" at the event horizon? I am wondering this because at this radius, the massive body's internal time rate drops to 0.

Any advice is much appreciated. I apologize in advance for any misuse of English or mathematical notation!