  We prove local and global well-posedness for semi-relativistic, nonlinear
Schr\"odinger equations $i \partial_t u = \sqrt{-\Delta + m^2} u + F(u)$ with
initial data in $H^s(\mathbb{R}^3)$, $s \geq 1/2$. Here $F(u)$ is a critical
Hartree nonlinearity that corresponds to Coulomb or Yukawa type
self-interactions. For focusing $F(u)$, which arise in the quantum theory of
boson stars, we derive a sufficient condition for global-in-time existence in
terms of a solitary wave ground state. Our proof of well-posedness does not
rely on Strichartz type estimates, and it enables us to add external potentials
of a general class.
