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So im having trouble reparameterizing the geodesic equation in terms of coordinate time.

Normally you have:

[tex] \frac{d^2 x^{\alpha}}{d \tau^2} + \Gamma_{nm}^{\alpha} \frac{d x^{n}}{d \tau}\frac{d x^{m}}{d \tau} = 0 [/tex]

Where [itex] \tau [/itex] is the proper time. I class we were told to express the above in terms of coordinate time and so i reasoned that one would use the chain rule:

[tex] \frac{d }{d \tau} = \frac{d t}{d \tau} \frac{d }{d t} [/tex]

When i do so i get the following:

[tex] \frac{d^2 x^{\alpha}}{d t^2} + \Gamma_{nm}^{\alpha} \frac{d x^{n}}{d t}\frac{d x^{m}}{d t} = - \frac{d^2t/d \tau^2}{dt/d\tau} \frac{d x^{\alpha}}{dt} [/tex]

Which - i guess - has the form that one would expect because t is non-affine.

However i dont know how to proceed from here. I´ve tried to use [itex] d\tau ^2 = g_{nm}dx^ndx^m [/itex] to find [itex]dt/d \tau [/itex], but i can´t seem to get anything meaningful.

Thanks for your time!

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# Geodesic quation coordinate time

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