It's a known calculation how to derive the escape velocity of a system following the Schwarschild Metric. It comes out to be, if I am not wrong $$v = c\sqrt{2GM/Rc^2 - (2GM/Rc^2)^2}$$
So, is there a general method of finding out the escape velocity for a given general metric, without assuming any symmetry considerations?
A mathematical help would be highly appreciated
This is certainly not possible in general. The topology of the spacetime is not determined by the Einstein equations, so you could have solutions in situations for which there is no "infinity" to which you could escape. Think something like $R \times T^3$, where $R$ is an unbounded, time-like direction and $T^3$ is a 3-torus. You could find solutions to the Einstein equations for that, but what would it mean to "escape to infinity"?
Someone else might try to make that precise, but the basic idea should be right.
