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So I am a student and decided (for some bizarre reason) to attempt to tackle general relativity for my final astrophysics and computational physics project this term. I have been doing a lot of reading and in general things make sense conceptually. The tensor math is a bit over my head but once I sit down with a pencil and paper to actually work through them I am hoping things will begin to make more sense in that department. I am however still faced with one question that I can't seem to settle:

How do you move from the geodesic equations derived from, say, the Schwarzschild metric to equations of motion in real 3D Cartesian space that could be used to calculate the orbit of a point mass around a spherically symmetric body?

I'm not sure if I'm missing the point and you don't actually use the geodesic equations to obtain said equations of motion, or I just haven't searched hard enough in my studies to find the connection. Any help would be greatly appreciated, as I need some tangible equations of motion to calculate orbits numerically for my computational physics project.

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If you solve the geodesic equation in Schwarzschild space-time then you will obtain, for the freely falling particle, the coordinates of its worldline $x^{\mu}(\lambda)$ in say the (global) Schwarzschild coordinate system; here $\lambda$ is an affine parameter such as proper time $\tau$ for a massive particle. The $x^i(\lambda)$ will be the spatial coordinates so inverting $t(\lambda)$ to get $\lambda(t)$ we can obtain $x^i(t)$ which corresponds to the spatial trajectory of the freely falling particle in the global coordinates.

Go ahead and try this kind of calculation with the simplest possible case: that of a particle that falls radially from rest at infinity.

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  • $\begingroup$ Awesome, that was really helpful thank you. One more clarifying question, the proper time $\tau$ is the time coordinate relevant to the motion of the point mass we are examining, correct? t is just the time coordinate of an observer infinitely far away. So when handling derivatives with respect to $\tau$, do I have to give them any sort of special treatment? $\endgroup$ – Mr. Frobenius Nov 7 '14 at 15:55
  • $\begingroup$ @Mr.Frobenius: $\tau$ can be any affine parameter that parameterises the curve. For massive particles we normally take it to be the proper time i.e. the time shown on a clock carried by the freely falling observer. However it need not have a physical significance. For example the same equation applies to light, but light has no proper time so in this case $\tau$ has no physical significance. $\endgroup$ – John Rennie Nov 7 '14 at 16:27

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