# Einstein field equation,pde and differential geometry

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

• You can. But if you solve it, you only get a local solution. – Si Kucing Aug 15 '19 at 11:36

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.