Einstein field equation,pde and differential geometry

Question

I'm a math undergraduate student with some interest in mathematical physics with basic knowledge of partial differential equation.

When I was reading a wikipedia article about einstein field equation,it said

when fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic partial differential equations.

my question is if einstein equation is a partial differential equation, why can't you solve it normally,why do you need tensor analysis/riemannian geometry for, and can any partial differential equation be written using the languange of tensor, differential geometry,etc?

I apologize for my minimal understanding of this subject, but I haven't learn any tensor calculus yet

Answer 1

Ten is simply the number of distinct components of a second order symmetric tensor in a space of dimension four. Writing these equations in tensor form enables us to write EFE as a single equation instead of ten, just as $\vec F=m\ddot{\vec x}$ is a single vector equation written in place of three scalar equations.

Answer 2

Since you know Einstein field equations, then you must have heard of metric tensor and curvature tensor and their relationship to the field equations.

In local coordinates (normal coordinates) we can make the first derivative of the metric tensor to vanish at a given point but not the second derivative simultaneously. But curvature tensor is a function of both first and second derivatives of the metric tensor, so local coordinates doesnt always help in finding global solutions.