What does gravity look like in a compact space, such as a universe with spatial periodic boundary conditions equivalent to a 3-torus, or a ball with opposite points on the surface of the ball identified? In particular, what is the equivalent to the Schwarzschild vacuum solution to the Einstein field equations? I have only a smattering of General Relativity, and so I am unsure if this is a meaningful question to ask.
Background:
I know that to solve General-Relativity problems, we must solve the Einstein field equations $$G_{\mu \nu} = \kappa T_{\mu \nu}$$ along with the matter/field equations of motion.
The Schwarzschild metric describes a spherically-symmetric, static, vacuum ($T_{\mu \nu}=0$) solution to Einstein's equations.
$$ds^2 = -(1+GM/r)dt^2 + (1+GM/r)^{-1}dr^2 + r^2 (d\theta^2 + \sin^2\theta \ d\varphi^2)$$
According to my understanding, all that we need to do to find this metric is to enforce a spherically-symmetric form for the metric, yielding $$ds^2 = -B(r)dt^2 + A(r)dr^2 + r^2 (d\theta^2 + \sin^2\theta \ d\varphi^2)$$ and to specify $T_{\mu \nu} = 0$.
The partial differential Einstein equations simplify to ordinary differential equations in $A(r)$ and $B(r)$, and solving those yields the form of the Schwarzschild metric. $M$ is a constant of integration that is identified as the mass via the Newtonian far-field limit.
Can I enforce periodic boundary conditions, like $r=r+R$ for some constant $R$? Would I find a different metric? Specifying periodic boundary conditions in space seems to me to overdetermine the ordinary differential equations in $A(r)$ and $B(r)$, so I might be going the wrong way about this.
