So the Schwarzschild metric is given by(adsbygoogle = window.adsbygoogle || []).push({});

ds^{2}= -(1-2M/r)dt^{2}+ (1-2M/r)^{-1}dr^{2}+r^{2}dθ^{2}+r^{2}sin^{2}θ dφ^{2}

and the Lagragian is ##{\frac{d}{dσ}}[{\frac{1}{L}}{\frac{dx^α}{dσ}}] + {\frac{∂L}{∂x^α}}=0##

with L = dτ/dσ. So for each α=0,1,2,3 we have

##{\frac{d^2 x^1}{dτ^2}}=0## for Minkowski spacetime

also the geadesic equation is ##{\frac{d^2 x^a}{dτ^2}}+Γ^a_{bc}{\frac{dx^b}{dτ}}{\frac{dx^c}{dτ}}=0##

So what happens in the Schwarschild metric and how can I find the Christoffel symbols from the Schwarschild metric and the geodesic equations?

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# I Christoffel symbols of Schwarzschild metric with Lagrangian

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