I need to verify a solution to Einsteins vacuum field equations. I have the solution as follows $$ds^2=a\,dt^2+b\,dr^2+\cdots$$
Is the following the right approach? Einsteins equation reduces to $R_{ab}=0$, which can be expressed in terms of Christoffel symbols which can be defined by the metric.
How does one obtain the metric from the line-element?
Once I have the metric, do I just plug it into $R_{ab}=0$ and check? This seems very tedious, am I on the right track?
Added:
If I change the coordinates, and show that this new metric satisfies Einsteins vacuum Equation, would it follow that the original coordiantes and metric does so as well?