How can you create a geodesic path using the metric and initial velocity?

[Jack3145]

Let's say there is a small object heading towards Earth (it will burn up). It is first observed at:
$$x^{\\mu}=[x^{1},x^{2},x^{2},x^{4}]=[x_{0},y_{0},z_{0},t_{0}]$$
with a velocity:
$$V_{v}=[v_{1},v_{2},v_{3},v_{4}]$$

The metric is:
$$ds^{2} = dx^{2} + dy^{2} + dz^{2} -c^{2}*dt^{2}$$
$$g_{\\mu\\v} = \\left(\\begin{array}{cccc}<BR>1 & 0 & 0 & 0\\\\<BR>0 & 1 & 0 & 0\\\\<Br>0 & 0 & 1 & 0\\\\<BR>\\\\<BR>0 & 0 & 0 & 1<BR>\\end{array})\\right$$

Affinity is:
$$\\Gamma^{\\rho}{\\mu\\v} = 0$$

Riemann Curvature tensor is:
$$R^{\\rho}{\\mu\\v\\sigma} = 0$$

Ricci Tensor is:
$$R{\\mu\\sigma} = 0$$

My Question is how do you make a geodesic path from the metric and initial velocity?

$$V_{v} = x^{\\mu}*g_{\\mu\\v}$$ and make incremental steps?

 

[rrogers]

http://en.wikipedia.org/wiki/Solving_the_geodesic_equations

 

[Jack3145]

Sorry for the abomination. I know that Minkowski Space has no external gravitational forces acting on it.

 

[Fredrik]

You can still edit post #1. Looks like you need to remove a lot of \ symbols.

 

[rrogers]

You indicated an interest in a numerical solution. I did a Scilab solution for the Schwarzschild metric. Unfortunately I compressed the file using bz2 on Linux and my present (vista) decoder has a problem. If you are interested I will have my Linux machine back up in two weeks or perhaps somebody else will separate out the parts; the mash-up is probably my fault.
I never checked the compression or enhanced the program since nobody ever responded to my scilab post.
Some orbits are shown in the eps file at:
http://www.plaidheron.com/ray/temp/
drdth_f-example.eps
Ray