The Schwarzschild metric is given by
where and .
The geodetic equation is given by
The path for escaping is given by and , so we only need
And we obtain the equations
Now it is clear that . So we obtain the equation
Note that for we obtain – which is the Newton-limit, where we use .
Escaping means that there is some limit and . Thus we obtain
Then we obtain
Let the escape velocity be denoted as then we can write
As we obtain
The only solutions are given by and – thus . Or thus so
The escape velocity for the Schwarzschild metric is given by
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