naive relativity

very naively, one might want to write down a relativistic version of Newtonian gravity by enhancing the Poisson equation in this way, \(\partial_i\partial^i\Phi = \Delta\Phi = 4\pi G\rho \quad\rightarrow\quad \partial_\mu\partial^\mu\Phi = (\Delta - \partial^2_{ct})\Phi = 4\pi G\rho\) what phenomena are already contained in this equation and where does it fail?