Solution help for the Geodesic Equation

[Jack3145]

Let’s say an asteroid is about to enter earth’s atmosphere(it will burn up of course). The initially sitting is at:

$$point = [x_{0},y_{0},z_{0},t_{0}] = [r_{0},\\theta_{0},\\phi_{0},t_{0}]$$

With a 4-velocity:

$$V = [v_{1},v_{2},v_{3},v_{4}]$$

The momentum at 0+

$$p_{0+} = mass*V = mass*[v_{1},v_{2},v_{3},v_{4}]$$

The Riemann Geometry is Riemann Space with Torsion = 0. The metric is the Schwarzschild metric. Affinity is the Christoffel Symbol. Curvature is non-zero. And, the Ricci tensor is null.

My question is how do you formulate $$x^{i}$$ to attempt solve for the path, "s" in the second-order geodesic differential equation from the given initial 4-velocity vector and starting point?

 

[Jack3145]

I am looking for how to solve the geodesic path, I. Something like this except taking into account initial conditions:

$$I = \int[(1-2m/r)^{-1} + r^{2}(d\theta/dr)^{2} + r^{2}(sin\theta)^{2}(d\phi/dr)^{2} - (1-2m/r)(dt/dr)^{2}]^{1/2} dr$$

 

[Philosophaie]

The geodesic is a graph of many curves. You have to go piece-meal or one part at a time. Use the Geodesic equation to lesson the amount of unknowns: