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I was studying the lecture notes about the schwarzschild solution for general relativity. In a particular example they calculate the equations of motion of a particle falling straight into a black hole. But there are some things about the calculation I really don't get.

First of all they say that because the schwarzschild metric is time independent, so dg/dt = 0 => u_0 = const. Then they show that by using the initial conditions r = R and tau = 0 and t = 0 coincide, that u_0 equals to sqrt( g_00(R) ) = sqrt(1 - 2M/R).

I've tried to prove the constancy of u_0 from the geodesic equation, but all I got were some nasty coupled differential equations.

Somehow it is true that constancy of metric in one component direction implies that the four-velocity in that component is constant. But I don't see why that is true. And how can you show from this, that this means that u_0 = sqrt(1-2M/r) ?

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# Geodesics in schwarzschild solution

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