I've been wondering:
Are there, still, some advantages, for current research, to study Newtonian gravity? I mean, not experimentally, where Newton gravity is a very good approximation to everyday phenomenon or, even, to the solar system (aside some perihelion problems, gravitational lensing, etc...) but to some study of General Relativity or some approximation to the Einstein field equations where knowing the Newton potential is needed.
In the weak-field case, $$\mathrm{d}s^2 = -\left(1+2\frac{\Phi}{c^2}\right)c^2\mathrm{d}t^2 - \frac{4}{c}A_i\mathrm{d}t\mathrm{d}x^i + \left(1-2\frac{\Phi}{c^2}\right)\mathrm{d}S^2\text{,}$$ where $\Phi$ is the Newtonian potential and $\mathrm{d}S^2 = \mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2$ is the Euclidean metric. In the static case, $A_i = 0$, which is the form used for GPS calculations, but in general it is more interesting as being a direct analogue to classical electromagnetism, first formulated for gravity by Heaviside in 1893: $$\begin{eqnarray*}\mathbf{E}_\text{g} = -\nabla\Phi - \frac{1}{2c}\frac{\partial\mathbf{A}}{\partial t} & \quad\quad & \mathbf{B}_\text{g} = \nabla\times\mathbf{A}\end{eqnarray*}$$ $$\begin{eqnarray*} \nabla\cdot\mathbf{E}_\text{g} = -4 \pi G \rho_\text{g} & \quad\quad & \nabla \times \mathbf{E}_\text{g} = -\frac{1}{2c}\frac{\partial\mathbf{B}_\text{g}}{\partial t} \\ \nabla\cdot\mathbf{B}_\text{g} = 0 & \quad\quad & \nabla\times\frac{1}{2}\mathbf{B}_\text{g} = -\frac{4\pi G}{c}\mathbf{J}_\text{g} + \frac{1}{c}\frac{\partial\mathbf{E}_\text{g}} {\partial t} \end{eqnarray*}$$ This particular version was taken from Einstein's general theory of relativity by Grøn Øyvind and Sigbjørn Hervik; a few variations in defining these fields exist in the literature.
But probably more importantly, the post-Newtonian formalism gives a more general approximation scheme, the first few terms of which are: $$\mathrm{d}s^2 = -(1+2\Phi+2\beta\Phi^2+\ldots)\mathrm{d}t^2 + (1-2\gamma\Phi+\ldots)\mathrm{d}S^2 + (\ldots)\mathrm{d}t\mathrm{d}x^i\text{,}$$ with many other potentials that I'm omitting here. This is very useful for understanding the general predictions of GTR and comparing them to alternative theories of gravity (e.g., GTR predicts $\beta = \gamma = 1$, other theories might not).
As I remember there is a field called MOND or modified Newton's dynamic, where they are hypothesizing that there is a subtle change to the effective force law at extremely low accelerations $\mathrm{10^{-10}\ m/s^2}$ (this is a new empirical constant they've came too in a lab). I haven't had that much time in last few months to catch up, maybe try out to find out more on this theme.
