# Approximating curved spacetime with a grid of cartesian metric tensors?

Let's assume a universe with only some ($$n$$) single point masses $$m_i$$ in it. The point masses have initial positions in space-time, $$x_{i0}$$. The spacetime between them is curved due to general relativity. (n-body-problem)

The curvature of the spacetime between those single point masses must be a solution of Einstein's field equations. However, they are difficult to solve and there aren't solutions for the n-body-problem yet (to the best of my knowledge).

The curvature of spacetime is fully described by the metric tensor. (Or did I get something wrong here?) To derive the metric tensor, Einstein's field equations would have to be solved.

I'm wondering whether it is possible to approximate the true, curved, metric tensor at every point using different cartesian metric tensors. That would be the same principle as doing infinitesimal small steps when calculating the derivative of a function f(x). It's like using the flat tangent space-time at every point to describe the curved space-time.

Does this work? Has it been done somewhere in the literature? Are there circumstances for which this approximation doesn't work anymore?