We begin our understanding of the Einstein equations by developing the spherically symmetric vacuum solution, the Schwarzschild metric. This will involve, from a PDE point of view, constants of integration that we must be able to interpret physically. One way to make the physical associations necessary is to position ourselves sufficiently far away from the source that gravitational effects take the form of familiar effective forces, namely the Newtonian force outside of a spherically symmetric distribution of mass.

Weyl's method of varying an action after an ansatz incorporating target symmetries is introduced; this can be a useful tool for finding solutions with specific symmetries, and we will discuss the method initially in the context of spherically symmetric solutions to vacuum electrodynamics. Our first general relativistic job is to actually generate the solution to Einstein's equation in vacuum (i.e. away from source) given a spherically symmetric central body of mass M. We will use Weyl's method to accomplish this. That exploitation of symmetry will leave us with a single (physically relevant) undetermined constant of integration. Using the far-away (linearized) approximation to Einstein's equation, we will see that this constant can naturally be associated with the mass of the central body.