A Metric Ansatz
The line element given above, with f,g, regarded as undetermined functions of the isotropic coordinate r, is often used as a metric Ansatz in deriving static spherically symmetric solutions in general relativity (or other metric theories of gravitation).
As an illustration, we will sketch how to compute the connection and curvature using Cartan's exterior calculus method. First, we read off the line element a coframe field,
where we regard f,g as undetermined smooth functions of r. (The fact that our spacetime admits a frame having this particular trigonometric form is yet another equivalent expression of the notion of an isotropic chart in a static, spherically symmetric Lorentzian manifold). Taking the exterior derivatives and using the first Cartan structural equation, we find the nonvanishing connection one-forms
Taking exterior derivatives again and plugging into the second Cartan structural equation, we find the curvature two-forms.
Read more about this topic: Isotropic Coordinates